Document [original]

Development of intelligent, robust, and non-linear

Models in Dynamic Equivalencing for

Interconnected Power Systems

Zur Erlangung des akademischen Grades

DOKTORINGENIEUR (Dr.-Ing.)

der Fakultät der Elektrotechnik, Informatik und Mathematik

der Universität Paderborn

genehmigte Dissertation

von

Dipl.-Ing. Oscar Clovis YUCRA LINO

aus Oruro, Bolivien

Referent: PD. Dr.-Ing. habil. Michael Fette

Korreferent: Prof. Dr.-Ing. Jürgen Voß

Tag der mündlichen Prüfung: 04. Juli 2006

Paderborn 2006

Acknowledgments

This research work in context of a doctoral dissertation was realized in the Department of

Electrical Engineering of CINVESTAV Mexico (Centre of Research and Advances Studies),

Consulting System & Dynamic, and the University of Paderborn since Juni 2002.

First of all, I would like to express my sincere gratitude and appreciation to my supervisor

PD. Dr.-Ing. habil. Michael Fette (Consulting System & Dynamic, Germany), Prof. Dr. Zhao

Dong (Department of Electrical Engineering, Queensland University, Australia), Prof. Dr.

Juán Manuel Ramírez (CINVESTAV, Unity Guadalajara-Mexico) for discussions on power

systems dynamics, the skilled guidance, valuable comments, constructive suggestions,

collaboration, and encouragement that they give me over the years of research work on this

doctoral project as well as to my co-advisor Prof. Dr. Jürgen Voß and Prof. Dr. Felix Gausch

for the critical reading and discussion and also the useful suggestions.

Finally, I wish to express my utmost gratitude to my dear soul sister, Melvy for her care and

support, love and friendship. She has stimulated me to begin the doctorate study and she

has contributed to its completion through her encouragement. You have harmonized and

enriched my life in the way I could never dream of; yet.

I will always be grateful heartily to God, the most Beneficent for standing by my side

throughout all the sorrow and happiness, storm, tide, and sunshine days.

To my mother Desideria and father Santiago in eternal gratitude dedicated.

Essen, July 2006

Oscar Clovis Yucra Lino

CONTENTS i

Contents

Contents .......................................................................................................................i

List of Figures .............................................................................................................v

List of Tables..............................................................................................................ix

Nomenclature.............................................................................................................xi

Acronyms .................................................................................................................xvi

Chapter 1 Introduction ..............................................................................................1

1.1 Motivation............................................................................................................1

1.2 Objectives ...........................................................................................................3

1.3 Outlines of the dissertation ..................................................................................5

Chapter 2 Background ..............................................................................................6

2.1 Dynamic equivalencing........................................................................................6

2.1.1 Coherency identification and grouping of generators .....................................8

2.1.2 Network aggregation......................................................................................9

2.1.3 Static network reduction...............................................................................10

2.1.4 Aggregation of control devices.....................................................................11

2.2 Existing approaches ..........................................................................................12

2.3 Power system simulation program.....................................................................13

2.4 Summary...........................................................................................................14

Chapter 3 Electromechanical-based Identity Recognition in Dynamic Equivalencing

.........................................................................................................15

3.1 Introduction........................................................................................................16

CONTENTS

3.2 Coherency-based dynamic equivalencing..........................................................16

3.3 Identity recognition approach.............................................................................20

3.3.1 Conditions....................................................................................................20

3.3.2 Procedure....................................................................................................26

3.3.3 Identity recognition algorithms .....................................................................28

3.3.4 Comparative application ..............................................................................29

3.4 Electromechanical-based identity recognition ....................................................31

3.4.1 Inertia coefficient and electrical power .........................................................31

3.4.2 Electromechanical distance .........................................................................37

3.5 Case studies......................................................................................................38

3.5.1 16 Multi-machine system .............................................................................39

3.5.2 Simulation results and discussion ................................................................40

3.5.3 Interconnected European Network UCTE/CENTREL...................................47

3.5.4 Simulation results and discussion ................................................................49

3.6 Summary...........................................................................................................56

Chapter 4 Splitting Aggregation-based Dynamic Equivalencing.........................58

4.1 Introduction........................................................................................................59

4.2 Conventional aggregation in dynamic equivalencing .........................................59

4.2.1 Inertial Aggregation......................................................................................60

4.2.2 Slow coherency aggregation........................................................................64

4.2.3 Power invariance aggregation......................................................................67

4.2.4 Berg and Ghafurian’s aggregation ...............................................................69

4.3 Splitting-based aggregation approach ...............................................................70

4.3.1 Conditions....................................................................................................70

4.3.2 Aggregated electrical parameters ................................................................72

4.3.3 Splitting factors of generators ......................................................................76

4.4 Case study ........................................................................................................80

4.5 Simulation results and discussion......................................................................81

CONTENTS iii

4.6 Summary...........................................................................................................87

Chapter 5 Dynamic Artificial Neural Network-based Dynamic Equivalencing....89

5.1 Introduction........................................................................................................90

5.2 Conventional dynamic equivalencing.................................................................91

5.3 Dynamic ANN-based dynamic equivalencing ....................................................92

5.3.1 Artificial neural networks (ANN) for modeling...............................................93

5.3.2 Modeling of dynamic system........................................................................96

5.3.3 Mathematical description .............................................................................97

5.3.4 Power system model .................................................................................100

5.3.5 Dynamic ANN model as external area .......................................................104

5.3.6 Robustness................................................................................................113

5.4 Case studies....................................................................................................118

5.4.1 16 Multi-machine system with 2 boundary nodes.......................................118

5.4.2 12 Multi-machine system with 3 to 8 boundary nodes................................119

5.5 Simulation results and discussion....................................................................120

5.5.1 16 Multi-machine system with 2 boundary nodes.......................................120

5.5.2 12 Multi-machine system with 3 to 8 boundary nodes................................130

5.6 Summary.........................................................................................................137

Chapter 6 Closure..................................................................................................139

6.1 Conclusions.....................................................................................................139

6.2 Selection criteria..............................................................................................143

6.3 Suggestions for future work.............................................................................145

6.4 List of publications...........................................................................................146

Bibliography............................................................................................................147

APPENDIX A Classical dynamic equivalencing approaches..............................156

A.1. Ward equivalent ..............................................................................................156

A.2. Modal-based equivalencing.............................................................................159

CONTENTS

A.3. Coherency-based equivalencing......................................................................160

A.4. Hybrid modal procedures.................................................................................162

A.5. Linear model reduction ....................................................................................163

A.6. Model identification methods ...........................................................................165

A.7. Flow chart of the power system simulation tool PSD........................................166

A.8. Flow chart of the coupling of the machine model to the analysis algorithm PSD167

APPENDIX B Identity recognition algorithms.....................................................168

B.1. Hierarchical clustering .....................................................................................168

B.2. Partitioning clustering: K-means ......................................................................169

B.3. Fuzzy clustering...............................................................................................170

B.4. Relationship between K-means (hard clustering) and Fuzzy clustering ...........174

B.5. Self organizing features maps (SOFM)............................................................175

B.6. Clustering quality.............................................................................................176

B.7. Practical comparison of identity recognition algorithms....................................177

APPENDIX C DANN-based dynamic equivalencing ...........................................178

C.1. ANN preliminaries............................................................................................178

C.2. Learning strategies ..........................................................................................179

C.3. Modeling..........................................................................................................179

C.4. Non-linear models ...........................................................................................180

C.5. Power system model .......................................................................................180

APPENDIX D Data sets of 16 Multi-machine system..........................................183

APPENDIX E Data sets of 12 Multi-machine system..........................................188

APPENDIX F Data sets of the interconnected European power system UCTE /

CENTREL ..............................................................................196

LIST OF FIGURES v

List of Figures

Fig. 2.1.- Classical dynamic equivalencing in power systems ............................................................ 7

Fig. 3.1.- Swing oscillating curve of the rotor angular speed representing the conditions for the

identity recognition instead of the classical coherency.............................................................. 21

Fig. 3.2.- Classical Aggregation ........................................................................................................ 22

Fig. 3.3.- Schematic representation of iterative assignment and construction of centroids or

reference generators within a clustering procedure .................................................................. 27

Fig. 3.4.- Schematic representation of similar generators belonging to a cluster group taking into

account its identical properties in phase and amplitude. ........................................................... 27

Fig. 3.5.- Flow chart of applied identity recognition algorithms.......................................................... 29

Fig. 3.6.- One-machine power system for sensitivity analysis ........................................................... 32

Fig. 3.7.- Graphical comparison of the normalized square mean values of the sensitivities of rotor

angle δL with reference to the synchronous machine parameters............................................. 32

Fig. 3.8.- Graphical comparison of the normalized square mean values of the sensitivities of rotor

angle, rotor current and angle with reference to the parameters of the synchronous machines

................................................................................................................................................... 33

Fig. 3.9.- 16 Multi-machine system .................................................................................................... 39

Fig. 3.10.- Schematic representation of the grouped 16-machine System following disturbances at

the boundary nodes C1 on 380kV and C8 on 220kV at the same time. ................................... 41

Fig. 3.11.- Assignment and grouping of identical external machines according to the identity

recognition algorithms in conjunction to Fig. 3.10. .................................................................... 42

Fig. 3.12.a, b.- Comparison of time responses of C12 internal machine calculated with 3 cluster

groups according to the electromechanical Fuzzy and K-means.............................................. 43

Fig. 3.13.- Comparison of time responses of the internal machine C14 depending on the reduction

degree with 1, 3, 6 electromechanical Fuzzy algorithm-based equivalents. ............................. 44

Fig. 3.14.- Time responses of the C2 internal machine calculated with 3 equivalent machines by

different identity recognition-based algorithms. ......................................................................... 45

Fig. 3.15.- Time responses of the C2 internal machine following the disturbance (electrically and

geographically closest) applied on C3 node of internal area..................................................... 45

Fig. 3.16.- Time responses of the C2 internal machine following the disturbance (electrically and

geographically far away) applied on C9 node of internal area. ................................................. 46

Fig. 3.17.-Interconnected European power system UCTE/CENTREL [142]. .................................... 48

LIST OF FIGURES vi

Fig. 3.18.a, b, c.- Comparison of time responses of the KIEL1 German machine calculated with 90

equivalent machines by different identity recognition algorithms and considering their

electromechanical weighting (weighted K-means, weighted Fuzzy and Kohonen-SOFM). ......50

Fig. 3.19.a, b.- Comparison of time responses of the STDE1 German machine calculated with 65

equivalent machines by different identity algorithms with electromechanical distances

(weighted K-means, weighted Fuzzy)........................................................................................52

Fig. 3.20.- Behavior of the BWKUESS German machine using 65 equivalents considering their

electromechanical-based algorithms and following a disturbance at VEHANNSA (VEW)........53

Fig. 3.21.- Behavior of the BWKUESS German machine using 65 equivalents considering their

electromechanical-based algorithms and following a disturbance at EVDELSSAD (EnBW)...53

Fig. 3.22.- Comparison of identity recognition algorithms considering electromechanical weighted

distances by the mean value of J of the 67 German intern machines for a fault located at the

boundary node VEGROUSB(VEW) with different number of external equivalents...................55

Fig. 4.1.- Synchronous machine models: L1-L2-L3 transformed in dq-system.................................61

Fig.4.2.- Slow coherency aggregation...............................................................................................65

Fig. 4.3.- Power invariance aggregation............................................................................................68

Fig. 4.4.a.- Classical aggregation.......................................................................................................71

Fig. 4.4.b.- Proposed splitting based-aggregation .............................................................................71

Fig. 4.5.- Machine-splitting with reference to their nominal power according to the derived splitting

factors a1 and a2........................................................................................................................72

Fig. 4.6.- Splitting-based aggregation ................................................................................................73

Fig. 4.7.- Interconnected 16 Multi-machine System..........................................................................80

Fig. 4.8.- Comparison of time responses of an internal machine calculated with the original external

system and with 3 equivalent machines using the classical aggregation..................................82

Fig. 4.9.- Comparison of time responses of an internal machine calculated with the original external

system and with 3 equivalent machines using the splitting based-aggregation. .......................82

Fig. 4.10.- Comparison of time domain behavior of an internal machine calculated with the original

external system and with 3 equivalent machines using the splitting based-aggregation. .........83

Fig. 4.11.- Behavior of any internal machine following a sequence of disturbances applied on

different nodes electrically and geographically distinct from the equivalent disturbance. It is

simulated with the original external area and with 3 splitting equivalents with Fuzzy factors....84

Fig. 4.12.- Comparison of aggregation algorithms considering the classical inertial aggregation and

the proposed splitting-based aggregation by the mean value of J of the intern machines for a

fault located at the internal node with different number of external equivalents........................86

Fig. 5.1.- Division of complex power networks in areas ....................................................................92

Fig. 5.2.- Neural network structure ....................................................................................................93

Fig. 5.3.- Basic structure of generalized neuron model.....................................................................94

Fig. 5.4.- ANNs applied to the modeling of non-linear systems. Internal and external recurrent ANN

can be used to develop the DANN-based dynamic equivalencing ............................................95

Fig. 5.5.- Modeling structures. The system modeling used in this approach is emphasized as a

hybrid procedure in black-box form with assumptions about the system ..................................96

LIST OF FIGURES vii

Fig. 5.6.- Generator equivalent circuit as voltage and current source ............................................ 103

Fig. 5.7.- Internal and external area of a interconnected power system......................................... 104

Fig. 5.8.- Equivalent external area of a power system.................................................................... 104

Fig. 5.9.- External area as dynamic equivalent............................................................................... 106

Fig. 5.10.- Network structure for approximation of non-linear systems .......................................... 108

Fig. 5.11.- Series-parallel configuration coupling back the observed system output...................... 109

Fig. 5.12.- Parallel configuration with an internal recurrent link to the networks.............................. 110

Fig. 5.13.- System modeling for dynamic equivalencing................................................................. 111

Fig. 5.14.- DANN representing the dynamic equivalent for disturbances applied in the internal area

................................................................................................................................................. 114

Fig. 5.15.- Global dynamic ANN forming the dynamic of the external area.................................... 116

Fig. 5.16 .- Dynamic ANN forming the external dynamic with distributed operating points according

to heterogeneous power levels of the internal area................................................................. 117

Fig.5.17.- Procedure to developing ANN models as dynamic equivalents..................................... 117

Fig. 5.18.- 16 Multi-machine system with 2 interfaces and three areas........................................... 118

Fig. 5.19.- 12 Multi-machine system with multiple number of boundary nodes.............................. 119

Fig. 5.20.i .- Real part of the injected current at the second boundary node following a non-trained

disturbance in area A............................................................................................................... 121

Fig. 5.20.ii .- Imaginary part of the injected current at the second boundary node following a non-

trained disturbance in area A................................................................................................... 121

Fig. 5.21.i .- Real part of the injected current at the first boundary node following a disturbance (100

ms) at node 7 in area A under changed operating point ......................................................... 122

Fig. 5.21.ii .- Imaginary part of the injected current at the first boundary node following a

disturbance (100 ms) at node 7 in area A under changed operating point ............................. 122

Fig. 5.22.i .- Real part of the injected current at the first boundary node following a disturbance (100

ms) on node 6 in area B........................................................................................................... 123

Fig. 5.22.ii .- Imaginary part of the injected current at the first boundary node following a

disturbance (100 ms) on node 6 in area B............................................................................... 123

Fig. 5.23.i .- Active power flow interconnection between nodes 5 in B and 8 in C boundary line of

area B and C according to the currents of Fig. 5.21................................................................ 124

Fig. 5.23.ii .- Reactive power flow interconnection between nodes 5 in B and 8 in C boundary line of

area B and C according to the currents of Fig. 5.21................................................................ 124

Fig. 5.24.i .- Real part of the injected current at the second boundary node following a disturbance

on node 15 in 220 kV of area C ............................................................................................... 125

Fig. 5.24.ii .- Imaginary part of the injected current at the second boundary node following a

disturbance on node 15 in 220 kV of area C ........................................................................... 125

Fig. 5.25.i .- Active power flow interconnection between area C and B following a fault on node 5 in

110 kV of C by changed operation point.................................................................................. 126

Fig. 5.25.ii .- Reactive power flow interconnection between area C and B following a fault on node 5

in 110 kV of C by changed operation point.............................................................................. 126

LIST OF FIGURES viii

Fig. 5.26.i, ii.- Power flow interconnection between area A and C following a sequence of nontrained

faults on C by changed operation point and network topology................................................127

Fig. 5.27.- Evaluation of the prediction capability of ANN considering different non-trained

disturbances and non-trained operating conditions .................................................................129

Fig. 5.28.i .- Active power flow interconnection at the 8th boundary node or between node 39 and 40

following a fault on the node 20 within the internal area..........................................................131

Fig. 5.28.ii .- Reactive power flow interconnection at the 8th boundary node or between node 39 and

40 following a fault on the node 20 within the internal area....................................................131

Fig. 5.29.i .- Real part of the injected current at the 8th boundary node following a disturbance on

the node 20 in internal area under non-trained operating point of case 4 in table 5.2 ............132

Fig. 5.29.ii .- Imaginary part of the injected current at the 8th boundary node following a disturbance

on the node 20 in internal area under non-trained operating point of case 4 in table 5.2 .......132

Fig. 5.30.- Evaluation of the standardized prediction error S

Eof recurrent ANN considering

different disturbance duration (tmin=100 ms, tmax=150 ms) and two sequential disturbances

(t1=100ms after 1s, t2=120 ms after 2s) ..................................................................................133

Fig. 5.31.- S

Eevaluation of the robustness of the recurrent ANN depending on the cases of table

5.2 and in the 12 multi-machine system with different boundary nodes .................................135

Fig. A.1.- Ward static equivalent by eliminating {L} load and {G} generators nodes. .....................156

Fig. A.2.- Reduction of the Ward-PV equivalent with n number of equivalent generator nodes.....157

Fig.A.4.- Recognition of coherent generators based on generators electromechanical eigenvector

..................................................................................................................................................161

Fig. A.3.- Frequency response for input m......................................................................................162

Fig. A.5.- Input Signals u, Output Signals y, and Disturbances e...................................................165

Fig. B.1.- Flow chart of the concept of K-means for a set of clusters ..............................................169

Fig. B.2.- Flow chart of the concept of fuzzy c-means clustering.....................................................174

Fig. B.3.- Schematic representation of SOFM .................................................................................175

Fig. B.4.- Schematic representation of the clustering quality...........................................................176

Fig. C.1.- Interconnections among subsystems ..............................................................................182

Fig. E.1.- 12-machine system with three boundary nodes ..............................................................193

Fig. E.2.- 12-machine system with four boundary nodes ................................................................193

Fig. E.3.- 12 multi-machine system with five boundary nodes ........................................................194

Fig. E.4.- 12 multi-machine system with six boundary nodes .........................................................194

Fig. E.5.- 12 multi-machine system with eight boundary nodes......................................................195

Fig. F.1.- As consequence of the fusion of traditional network operators : BAG, BEWAG, EnBW, PE,

RWE, VEAG and VEW the number of the current operators has been considerably reduced.

..................................................................................................................................................200

LIST OF TABLES ix

List of Tables

Table 3.1.- Disturbance specifications for the 16 multi-machine system........................................... 40

Table 3.2.- Subsystems in the European Interconnected Power System UCTE/CENTREL............. 47

Table 3.3.- Disturbance specifications for the European power system UCTE/CENTREL (the

boundary nodes correspond to NL=Netherlands, CH=Swissland, OEVG=Austria) .................. 49

Table 5.1.- Studied cases in the 16 multi-machine system.............................................................. 120

Table 5.2.- Scenarios to power-flow changes considering Fig.5.19 ................................................ 130

Table 6.1.- Comparison of the proposed approaches to applicability considering power system

relevant aspects....................................................................................................................... 143

Table B.1.- Computation of distances [101]..................................................................................... 168

Table D.1.- Data sets of investigated multi-machine systems ......................................................... 183

Table D.2.- Generator data set of 16 multi-machine system ........................................................... 183

Table D.3.- Transmission line data set of 16- machine system ....................................................... 184

Table D.4.- Comparison of aggregation algorithms considering the classical inertial aggregation and

the proposed splitting-based aggregation by the mean value of J of the intern machines for a

fault located at the internal node with different number of external equivalents...................... 185

Table D.5.- Evaluation of the prediction capability of ANN considering different operating conditions

and points calculating )( jE S............................................................................................... 185

Table D.6.- Sum squared distance error and average error of the predicted boundary behavior

following disturbances at all non-trained nodes of internal area A considering different

operating points ....................................................................................................................... 186

Table D.7.- Sum squared distance error and average error of the predicted boundary behaviour

following disturbances at all non-trained nodes of internal area B considering different

operating points ....................................................................................................................... 186

Table D.8.- Sum squared distance error and average error of the predicted boundary behaviour

following disturbances at all non-trained nodes of internal area C considering different

operating points ....................................................................................................................... 187

Table E.1.- Generator data set of 12-machine system .................................................................... 188

Table E.2.- Transmission line data set of the 12-machine system .................................................. 188

Table E.3.- Evaluation of the standarized prediction error of recurrent ANN considering different

disturbance duration (tmin=100 ms, tmax=150 ms) and two sequential disturbances (t1=100ms

after 1s, t2=120 ms after 2s).................................................................................................... 190

LIST OF TABLES x

Table E.4.- Evaluation of the robustness of the recurrent ANN depending on the cases explained in

table 5.2 and networks based on the 12-multi-machine system with different boundary nodes

..................................................................................................................................................190

Table E.5.- Scenarios to power-flow changes and losses considering 12-machine system with 3

boundary nodes after Fig.B.1...................................................................................................190

Table E.6.- Scenarios to power-flow changes and losses considering 12-machine system with 4

boundary nodes after Fig.B.2...................................................................................................191

Table E.7.- Scenarios to power-flow changes and losses considering 12-machine system with 6

boundary nodes after Fig.E.4...................................................................................................191

Table E.8.- Scenarios to power-flow changes and losses considering 12-machine system with 7

boundary nodes after Fig. E.5..................................................................................................192

Table E.9.- Scenarios to power-flow changes and losses considering 12-machine system with 8

boundary nodes after Fig. 5.18 ................................................................................................192

Table F.1.- Part of the generator data set of the European Interconnected Power System [142] ...196

Table F.2.- Part of the transmission line data set of the European Interconnected Power System.198

Table F.3.- Comparison of identity recognition algorithms considering electromechanical weighted

distances by the mean value of J of the 67 German intern machines for a fault located at the

boundary node VEGROUSB(VEW) with different number of external equivalents.................200

NOMENCLATURE AND ACRONYMS xi

Nomenclature

Latin symbols

A, B, C, D State, input and outputs matrices of a system.

AR, BR1, BR2 Rotor electrical matrices corresponding to the electrical state system.

AM, BM Mechanical matrices corresponding to the mechanical state system.

AE, BE Excitation matrices corresponding to the excitation state system.

AT, BT1 , BT2 Turbine-governor state matrices.

B1, B2 The bias of the hidden and output layer of the ANN, respectively.

ai,j Participation share or splitting factor of external generator j according to the

number of generated equivalents i.

Cx Covariance matrix denoted by cij, representing the covariances between

components xi and xj.

Di Damping coefficient of the ith generator.

dij Minkowski distance between two M-dimensional objects (data points) i and j.

dij Distance between a generator and the reference generator.

E Voltage at generator internal buses.

Ea Complex internal voltage of generator A.

Eb Complex internal voltage of generator B.

Ed’Eq‘ Transient EMF’s of the generator in the d- and q-axes.

EE Internal voltage of the equivalent generator E.

(0)E j Initial driving voltage of external fictive generator j.

Ep Terminal common voltage calculated by the splitted internal voltages.

)( jE N Error function. This measure is realized over the sampling points N of the

whole behavior and for the disturbance j.

E Mean standardized error function.

e Eigenvector defining the axes of the components of the generator data matrix.

NOMENCLATURE AND ACRONYMS xii

)(tey

pi Sensitivity of a determined machine’s behavior y(t) with reference to generator

parameters P.

f, g The state and output function, respectively.

f, g, h Non-linear vector functions, which can be modeled.

hN Functional relationship approximates the real properties of the external area

with a dynamic ANN.

h Non-linear function, which must be modeled.

IG Unit vector with cardinality of {G}.

I Identity matrix.

i subscript for ith generator.

i Index for generator being grouped.

id, iq Stator currents in d- and q-axes, respectively.

ie, ib, ii Injected currents of the external, the boundary and the internal buses,

respectively.

i(t) Injected current is the output vector, and it is a function of all state variables.

I(t) Real-time complex current value.

Io Initial complex injected current value at a static operating point.

ij(t) Injected current following disturbance j considering the original external area.

i’j(t) Injected current following disturbance j predicted by the ANN.

Jm Objective function according to the m fuzziness index.

M Inertia constant in p.u.

Mi Inertia coefficient of the ith generator.

M Number of cluster groups representing the dynamic equivalents.

m Fuzziness index and influences the “fuzziness” of the obtained grouping.

mm Turbine mechanical torque.

N Number of generators or features.

N Number of non-trained disturbances with the same duration applied to the

nodes, which are not considered in the training database.

Ns Number of generators in the internal area.

Np Number of sampling points of the time behavior.

NG Number of coherent or identical machines in a corresponding group.

ny , nu Maximum integer lags in the output and input signal of a system.

P Number of variables or patterns of the time data series.

PG, QG Real and reactive power injections at internal generator buses in p.u.

PL, QL Real and reactive power residual at load buses in p.u.

Pel Generator electric power obtained from a load flow solution.

Pm Mechanical power.

NOMENCLATURE AND ACRONYMS xiii

ref

P Reference power set by LFC (Load Frequency Control).

R Index for the reference generator for the group under consideration.

rE , xE Resistances and inductances of the equivalent generator in p.u.

)(tS Search direction matrix, which contains the gradient search for the minimum of

the performance function.

Sj, Nominal power of the jth generator.

Sri Rated power of the ith single generator.

SrE Resulting nominal power of the equivalent generator.

SE, PE , QE Nominal power, active and reactive power, respectively.

T NxN matrix including the eigenvectors of the covariance matrix C.

Tdo, Tqo Transient time constants of the open circuit and damper winding in the q-axis.

Tm Mechanical torque applied to the shaft of the generator.

TmE Mechanical torque (inertial constant) of the equivalent generator.

Tq Nxq matrix including q significant eigenvectors of C corresponding to the q

largest eigenvalues of C.

U(t) Real-time complex voltage.

Uo Initial complex voltage at a static operating point of the studied power system.

ud, uq, ufd Stator voltage in q and d-axes and the field voltage, respectively.

uS, iS Stator (terminal) voltage and current, respectively.

upss Voltage input signal to the PSS (Power System Stabilizer).

u(t) Input vector, i.e. voltage signal, that consists of state and field voltages, and

mechanical power input of each generator in the system at time t.

ue, ub, ui Voltages of the external, the boundary and the internal buses, respectively.

U Voltage at load buses.

Ua, Ub Load bus voltage magnitude to bus a and b.

u” Driving voltage.

v(t) Random noise signal.

Wj, Weighting factor for the jth generator.

W1 Neural network weights of the first hidden layer into the matrix.

W2 Neural network weights of the output layer into the matrix.

w Weighting electromechanical factor to form the electromechanical distance.

X (N,M) Time response matrix of oscillating swing curves, exhibiting N the number of

generators and M the number of sampling points of the time data set.

X ist the mean of that population of X.

X’dEi Transient reactance connecting the equivalent generator to the ith bus.

x’d Transient reactance.

NOMENCLATURE AND ACRONYMS xiv

s Rotor reactance.

xhd, xhq, xσDq, xσDd, xσfd, xσfDd, x’’d, x’’q Synchronous reactance and subtransient synchronous

reactances in d- and q-axes system.

xi This represents resistances, main-field and linkage inductance of all circuits.

x State variable vector.

xqxdx’d x’q Synchronous reactance and transient synchronous reactance of the machine.

xR Rotor electrical state vector containing the state variables: Ed’Eq‘.

xM Generator mechanical state vector containing the state variables:

Sm.

xE Excitation state vector.

xT Turbine-governor state vector.

x(t) Vector of system states variables, it may contain variables associated with

synchronous generators and their controllers and possible network dynamics.

Y Network admittances in matrix form.

y(t) Machine’s behavior.

Output and input vector of the neural network, respectively.

y(t) Output vector of the system.

Z Matrix, which contains the system output and the regressor.

z Algebraic variable vector.

Greek symbols

a Relationship factor between EE and Ea.

b Relationship factor between EE and Eb.

)(t

It determines the length of the step in the search direction.

Complex constant.

δ Angle at generator internal buses (rotor shaft angle of the generator).

ε User specified tolerance degree.

(t) Time dependent regressor vector.

Neural network activation function.

Eigenvalues, which are equal to the variance of each generator behavior.

ij Fuzzy membership degree of generator xj to cluster ci.

∆ Deviation from a specified steady state operating point.

∆

j(t) Change of the injected current following disturbance j.

NOMENCLATURE AND ACRONYMS xv

∆

i’j(t) Change of the injected current following disturbance j predicted by the ANN.

∆ Current injection variable as phasor form.

∆Ι

b Incrementals for the currents:

a and

∆Pmi Change in mechanical input power in p.u.

∆Pgi Change in electrical output power in p.u.

∆P(i)Original Time domain behavior of the ith internal generator calculated with the original

external area.

∆P(i)Dyn. Equi. Time domain behavior of the ith internal generator calculated with the the

equivalent external area.

∆δ

b Incremental variables for

a and

∆θi Incremental rotor phase angle of the ith generator.

∆θR Incremental rotor phase angle of the reference generator.

∆pi(t) Active power variation.

∆δi(t) Generator angle deviation.

∆ωi(t) Generator angular velocity deviation.

∆ωI Speed deviation in p.u.

∆δi Rotor angle deviation in radians.

θ Angle at load buses.

denotes the collection of involved neural network parameters: W1, W2, B1, B2.

b Bus voltage angle a and b.

Bus angle at the generator terminals obtained from the network solution.

ωo Network frequency referenced to p.u.

ωL, δL Rotor angular velocity and rotor angle, respectively.

ωωo Rotor angular velocity of the generator (synchronous speed of the system).

ψL, ψS Rotor and stator flux linkage, respectively.

))(( tE

∇ Gradient of the error matrix with respect to the ANN parameters.

Σ Diagonal variance matrix.

NOMENCLATURE AND ACRONYMS xvi

Acronyms

AGC Automatic Generation Control

AMS Associative Memory Systems

ANN Artificial Neural Network

BAG Bayernwerk AG

BEWAG Berliner Staedtische Elektrizitaetswerke AG

CENTREL Central European Power System

COI Center Of Inertia

DANN Dynamic Artificial Neural Network

EnBW Energie Baden Wuerttemberg

FACTS Flexible AC Transmission Systems

FC Fuzzy Clustering

HC Hierarchical Clustering

LFC Load Frequency Control

MIMO Multi Input Multi Output

NARX Non-linear AutoRegressive models with eXogenous inputs

NARMAX Non-linear AutoRegressive Moving Average terms with eXogenous inputs

NH Non-Hierarchical Clustering

NOE Non-linear Output Error

OP Operation Point

PCA Principal Component Analysis

PE Preussen Elektra

PSD Power System Dynamic

PSS Power System Stabilizer

RWE Rheinisch-Westfaelisches Elektriizitaetswerk

SOFM Self-Organizing Feature Maps

UCTE Union for the Coordination of Transmission of Electricity

VEAG Vereinigte Energiewerke AG

VEW Vereinigte Elektrizitaetswerke Westfalen

INTRODUCTION 1

“The idea is to try to give all the information to help others to judge the

value of your contribution; not just the information that leads to judgment

in one particular direction or another.” -Richard P. Feyman-

Chapter 1

Introduction

1.1 Motivation

The old electricity supply services were characterized by monopolistic and local market

structures of the electricity industry. In the last two decades, this principle has been

undergoing a radical reform by taking into consideration technical, economical, and political

reasons and replacing those vertical structures with a deregulated electricity market open to

the competition.

Due to the separation of energy producer and network operator and principally through the

competition between the energy producers, a significant increase of energy transport costs in

a complex interconnected power system operation has to be taken into account. This effect

plays an important role considering the fact that plant locations are selected in the future no

longer primarily in the neighborhood of consumer centers, but rather in priority after cost

factors of the production.

With the increased market liberalization, the network operators must perform the

minimization of the generation costs and transmission costs. Thus, significant changes of the

network structures to the increased transmission lines are not plausible. But, it is more

appropriate, that the network operators will operate their transmission lines more closely to

the allowable transmission capability, especially to the stability limit.

In order to guarantee the reliability, the accuracy of stability analysis must be satisfied.

Therefore, the energy providers should be able to estimate the dynamic behavior of their own

power system exactly. In large power systems, it is difficult to perform the dynamic stability

INTRODUCTION 2

analysis accurately because of the large number of transmission lines, system components,

and the network boundaries. Also, the significant influences of the neighborhood networks to

their own area have to be taken into consideration.

Nowadays, a detailed calculation of neighborhood networks is always not possible due to

the complexity of interconnected power systems. Moreover, the hard competition between

network operators due to the deregulated and liberalized electricity market leads to limited

cooperation and restricted internal data availability between operators.

These aspects are clearly presented in the west European power system, which is

coordinated by the UCTE (Union for the Coordination of transmission of Electricity). The

energy market liberalization and the east expansion of the UCTE can be considered as the

most essential change in the European energy economic of the last years. Additional to

UCTE, the central European power system (CENTREL) includes the eastern European

countries.

At present, the expansion of the UCTE network to east is in full swing. In planning

consolidation and extension of the European interconnected power system, new problems

concerning to the power flow and the stability aspects have to be solved.

The calculation of the complete European network would lead to an enormous technical

expense due to its complexity. At the same time, it is not necessary for a national internal

network operator to analyze other networks.

In spite of all, it is impossible to obtain all required data of the complete European network

in detail. Moreover, the global data availability is limited by other local operators, since they

do not want to reveal their own network specific data sets, i.e. the capability of their own

power plants or the load performance, etc. Considering these reasons, it is suitable to

represent the neighbor networks as equivalent networks, which are connected to the local

internal area.

In order to determine such equivalent networks, various conventional procedures in frame

of dynamic equivalencing were developed, which reduce a complex power system to a small

and simplified one. Thus, these simplified power supply models that can be utilized in

network reliability, management and planning to overcome blackouts situations and to affront

new technical circumstances of the deregulated electricity market.

INTRODUCTION 3

The classical dynamic equivalencing consists principally of the following main steps:

- Coherency Identification for grouping of coherent machines

- Aggregation of these machines

- Static network reduction

- Aggregation of control devices

Such equivalent systems are useful both for the planning and operation of interconnected

large power systems. This simplifies the load flow calculations, transient stability calculations,

and general investigations concerning protection and safety aspects. However, the classical

well known procedures to form equivalent systems are mainly based upon linearized system

models around a specific operating point with theoretical constrains validating. Consequently,

they are restricted in its validity on practical non-linear, offline- and online-applications.

1.2 Objectives

Non-linear-based and innovative approaches for the equivalencing of complex large power

systems have been developed in this dissertation. These new techniques satisfy the needs

and requirements of all involved electricity market participants, especially for the network

operators. In this context, the availability of a robust and consistent dynamic equivalencing

using artificial intelligent systems plays an important role.

The objectives are to introduce new dynamic equivalencing concepts, where the linear and

non-linear characteristics and behavior of the power system are essentially considered.

Principally, the following approaches are proposed within the scope of this dissertation:

i. Identity recognition instead of the classical coherency identification

This innovative approach replaces the classical coherency identification in dynamic

equivalencing. It is based on the recognition of real identical machines using their linear and

non-linear properties. The identical machines that swing together in the sense of new defined

criteria are classified into cluster groups. Applying standard pattern recognition methods to

dynamic equivalencing can satisfy the criteria.

INTRODUCTION 4

ii. Electromechanical-based identity recognition

The electromechanical influence of the generator is taken into consideration in the identity

recognition procedure by means of the new proposed electromechanical distance to obtain

‘electrically real identical generators in cluster groups’.

The new defined electromechanical distance is based upon suitable machine model

parameters. In this way, the electromechanical relationship between generators in the

identity process will be considered. This method improves essentially the assignment of

identical generators and the accuracy of dynamic equivalents.

The applicability and consistency of this new approach and its ability to grouping of real

identical machines will be tested both in the 16-generators system and in the large-scale

model of the interconnected European power system with 464 generators.

iii. Splitting-based aggregation instead of the classical aggregation

This research focuses on the splitting of generators, which belong to different groups at the

same time but with different membership degrees. Thus, the generators will be divided into

representative parts, which will be aggregated to virtual equivalent generators.

This innovative approach forms "virtual" generator models. It is based on the application of

principal component analysis (PCA) and the Fuzzy theory for generating a dynamic splitting

of generators. These new aggregated equivalent generators are modeled upon the basis of

the splitted parameters of all original external generators.

iv. DANN-based dynamic equivalencing

By means of the identification of the non-linear behaviors of a power system, a knowledge-

and signal-based robust dynamic artificial neural network (DANN) is developed. The artificial

neural network can be considered as global external non-parametric dynamic equivalent to

represent a non-linear Multi Input Multi Output (MIMO) system model, which replaces and

identifies the static and dynamic behaviors of all elements of an interconnected power

system, i.e. generators, transmission lines, converters, voltage and turbine controllers,

amongst others, as global system.

INTRODUCTION 5

This approach replaces successfully the stages of the classical dynamic equivalencing,

such as coherency identification, aggregation, and network static reduction. It is proved to be

an effective method for the online dynamic equivalencing of interconnected large power

systems.

This approach is applied in a 12- and 16-machine system with many boundary nodes. The

robustness considering different power flow conditions with the ANN-based stability analysis

are examined.

1.3 Outlines of the dissertation

This dissertation is organized consisting of the following chapters:

In CHAPTER 1, an introduction about generic topics of this dissertation is presented.

Following in CHAPTER 2, a general overview of the background and actual state of

development of dynamic equivalencing is given.

The main focus of CHAPTER 3 describes the identity recognition concept as alternative to

the classical coherency identification. The electromechanical distance as an improvement

factor in the procedure will be proposed generating electromechanical-based equivalents.

As an important contribution to aggregation of generators, CHAPTER 4 describes the

splitting-based aggregation instead of the classical aggregation forming virtual equivalent

generators. Their theoretical background and the corresponding algorithm are explained.

An investigation about the system modeling of non-linear MIMO systems that comprise

complex interconnected power systems is introduced in CHAPTER 5 describing the

replacement of external power systems with a dynamic artificial neural network (DANN).

Finally, CHAPTER 6 includes a comparison, conclusion, and summary about the proposed

innovative approaches, their applicability in practical situations of dynamic equivalencing.

Suggestions on future research directions on this topic area are summarized.

BACKGROUND 6

“Analysis of stability, equivalencing of network,…,.are greatly facilitated

by classification of them into appropriate categories. Classification

therefore is essential for meaningful practical analysis and resolution of

a complex system problem”-A quotation from [3]-

Chapter 2

Background

Objective— This chapter briefly presents the key definitions and traditional concepts in

power system dynamic equivalencing, which will be used as basis to develop innovative

approaches throughout this thesis.

Index Terms— Clustering, Coherency Identification, Dynamic Equivalencing,

Electromechanical Distance, Network Reduction, Modal Analysis, System Identification,

Linear Model Reduction, Slow and Inertial Aggregation, Stability Analysis, Ward Reduction.

Organization— Section 2.1 describes the classical dynamic equivalencing. In section 2.2,

the existing approaches are treated in detail and finally, in section 2.3, a brief summary is

presented.

2.1 Dynamic equivalencing

The dynamic equivalencing consists of forming equivalent machines, which represent the

electrical and mechanical characteristics of the original machines. To this end, a complex

large power system network may be divided into the following areas:

- The internal area has to be retained intact and unreduced in detail containing the

internal machines for stability studies. The dynamic behavior of this area is simulated

using dynamic equivalents of the neighborhood area.

- The external area, containing the external generators, transformers, transmission lines,

additional devices, amongst others, will be simplified to a reduced system. The

generators will be classified, grouped, and aggregated to a dynamic equivalent.

BACKGROUND 7

In order to realize the transient stability analysis, it is important to consider the impact of the

external area to the internal area during the disturbance period. Following aspects between

internal and external area should be considered:

• The detailed description of the external area is not important for stability studies.

Therefore, the detailed equivalencing of the external area is not necessary [1-5].

• The external area is not of direct interest in stability studies and is of consequence only

in so far as it influences the response of the internal area to disturbances within it.

• In general, the grouping of generators in the external area is mainly affected by

disturbances coming from the internal area through the boundary nodes and lines.

• The impact of the external generators on the internal area depends generally on the

electrical and geographical distance between the two areas and their boundary nodes.

• In dynamic equivalencing, the influence of particular disturbances in particular locations

of the internal area should be considered.

• This equivalent external area coupled to the detailed model of the internal area must

consistently form the same amplitude and frequency behavior of oscillations in the

boundary buses. Consequently, the reduced system has the same dynamic behavior as

the original interconnected power system.

The following diagram shows schematically the dynamic equivalencing.

Fig. 2.1.- Classical dynamic equivalencing in power systems

External Area

Internal Area

c E

 Aggregation of regulators and governors

 Aggregation of generators

 Coherency identification

 Static network reduction

BACKGROUND 8

The classic dynamic equivalencing consists of the basic steps [6, 7, and 8] mentioned in Fig.

2.1, which will be described in the following sections:

2.1.1 Coherency identification and grouping of generators

In this step, coherent generators are identified and grouped together. These aspects are

evaluated in appendix A.3. This identification procedure can be significantly simplified by

using the following assumptions:

- Non-generator dynamics may be ignored or simplified.

- Classical generator models can be used.

- The linearized system model preserves well the properties of coherency.

The following methods are available for determination of the coherent groups of generators:

• Weak links coupling [9].- In this method, the coherency is determined by analyzing the

coupling of generators in the state matrix. A group of generators are identified as

coherent if the coupling coefficients among them are high.

• Two-time scale [10].- This method is based on the concept that a slow oscillation is

caused by two groups of strongly coherent generators interconnected through weak ties.

In this case, the two oscillating generator groups can be easily identified by means of the

eigenvector associated with the mode of oscillation. With this method, the system can be

partitioned into an arbitrary number of coherent external generator groups by analyzing

the same number of the slowest modes of oscillations. This method is extended not only

for generators [11, 12] but also for generators and weak tie lines [13].

• Slow coherency [14] and tolerance-based slow coherency [15].- These contributions

describe an algorithm for fault independent area grouping. However, they don’t indicate

how more complicated generating unit models can be handled. The slow coherent-

decomposition is defined with respect to a select subset of modes of a linearized model,

in which only the lowest electromechanical modes of the system are selected. The

tolerance-based coherency procedure relaxes slow-coherency in the direction of slow-

synchrony, although the dynamical implications are not exactly explored.

• Linear time simulation [17, 18].- This is the classical method to identify coherent

generators. The time domain response of the system is solved for a specified fault in the

power system, and the rotor angles of generators are compared. Those generators with

BACKGROUND 9

rotor angles swinging together with respect to the phase angle are identified as coherent

(Podmore condition) 1. This will be in detail explained in chapter 3.

Remarks:

• The ‘tolerance-based slow coherency’ is similar to the previous ‘two-time scale for

generators only’ method, but includes additional constraints to ensure that widely-

separated generators are not aggregated.

• The ‘two-time scale’ and ‘slow coherency’ methods require the calculation of selected

eigenvalues and eigenvectors of the full system and complicate its practical application.

The modified Arnoldi eigenvalue solver used in [16] can be utilized to simulate large

interconnected power systems.

• There are different possibilities to group coherent generators, which can be realized

through the analytical method, such as ‘the two-time scale method’ and ‘the weak-link

method’. But the analytical method often produces incorrect results. To overcome the

problem, heuristic methods are carried out in parallel with the analytical method [25].

• The method of Lee and Schweppe in [20] offers several methods [19, 21-24, 27-29] with

reference to heuristic approaches based on the Podmore condition [17-18].

• An important technique in this step is the building of standard equivalents according to ‘a

pre-reduction of dynamic states identifying similar dynamic models’ or control devices to

the corresponding generators at the same bus.

• To this aim, weighted average and least squares frequency domain algorithms to

calculate the parameters of the aggregated generators and their control models are

used. After this pre-procedure, the classical coherency methods are applied. However,

the results are not promising [26].

2.1.2 Network aggregation

In this step, an aggregated network is constructed on the basis of the equivalent generator

parameters for each coherent group of generators.

This obtained dynamic equivalent is a single unit that exhibits the same voltage, speed,

and total mechanical and electrical power as grouping generators during any disturbances,

where those generators show coherent properties.

1 This method is developed on the basis of a simplified and linearized power system model representing the mechanical

equations for the motion of synchronous generators. The swing curve simulation has been realized faster by using a

linear model and solving it in many cases by means of trapezoidal integration algorithm.

BACKGROUND 10

Retaining the original steady state, power flows, and voltages in the network, only the

network equations are modified, which is replace by several coherent generators.

In the literature, the following classical aggregations which are given in detail in chapter 4,

are presented:

• Inertial and slow aggregation [15, 30-33].- The generators in a coherent group are

represented by an equivalent classical generator model. In its simplest form, the

equivalent inertia is the sum of the inertia of all coherent generators in the group, and the

equivalent transient reactance is obtained by paralleling the transient reactance of all

generators in the group.

• Detailed aggregation [34].- In this method, if some or all generators in a coherent group

have similar control systems, they can be aggregated to a detailed generator model with

an equivalent exciter, stabilizer, and governor. The parameters of the equivalent models

are obtained using a combination of two approaches: a least square fit of the frequencies

responses to determine the linear characteristics, and an evaluation of the time domain

constraints to set the non-linear characteristics.

• Power invariance principle [35].- This method summarizes the equivalent generator

representation and the network reduction retaining the terminal buses of each of the

coherent generators. Hence, it preserves the basic physical structure of the original

system. For each coherent group, a fictitious point is constructed such it connects all the

internal voltage sources of the generators with one end of the transient reactance.

• Berg and Ghafurian method [36, 37].- In this method, the coherent group of generators

can be replaced by an equivalent generator modifying the network by mathematical

formulations according to complex ratios as weighting factors.

2.1.3 Static network reduction

Once equivalent generators are determined for the generators groups, a network reduction

is performed 2. This reduction is generally achieved in two steps:

• The equivalent generators are inserted into the system and the generators in the

associated coherent groups are removed. The network is modified to maintain the

balanced steady state power flow conditions.

2 After the static network reduction procedure, the reduced network must have the exact electrical behavior as the

original one.

BACKGROUND 11

• The network is reduced to restricted number of nodes. In this way, nodes are eliminated,

and new transmission lines can be created or by using an adaptive reduction technique,

similar to the one introduced in [40, 41]. The criterion for the nodes elimination is the

network sparsity.

The network nodes can be eliminated mathematically. This is a simple network

transfiguration. Equivalent lines and shunts as result can be obtained. Only if non-linear load

are connected to the nodes, it is necessary to involve linearized load models into the

transfiguration. The intention of transfiguration is the encoding of the real load flow properties

rather than the reduction of computational efforts 3.

2.1.4 Aggregation of control devices

Once equivalent generators are determined, and network reduction is performed, voltage

regulators and governors will be aggregated in the dynamic behavior of power plants 4.

In order to obtain accurate results, it is necessary to model these devices on the basis of

their real structure and operating mode. This aspect leads to different types of control

models. To meet the requirements in the practice, different modeling techniques are

implemented into the stability analysis. However, this generates a large number of different

controller models enclosed in data sets of real electric power systems. However, techniques

for controller aggregation are not explored satisfactory yet.

In the corresponding literature, the following methods are presented:

• Splitting of coherent generator group in subgroups with similar controllers [26, 37].

• A controller with the best similarities to all controllers in the coherent group is chosen.

The parameters fitting is realized by simulating disturbances in the frequency range.

Usually, it is carried out in a network, where group members and the equivalent one are

connected to the same bus [34, 38, and 39].

• A suitable fact is to choose the optimal equivalent controller using the controller of the

largest subgroup. This optimal equivalent controller is used for the equivalent of the whole

group [39].

3 Actually, if too much nodes are eliminated, due to the large number of created equivalent lines, the effort can be

increased significantly. It should emphasize, that for the simulation it is not necessary to reduce the network.

4 If within a coherent group the controllers don’t have the same structure and parameter settings, it is suitable to create

an equivalent controller, which will be assigned to the equivalent generator.

BACKGROUND 12

• Another alternative is to select the controller of the greatest generator of the coherent

group to build a standard controller.

The influence of controllers on the dynamic behavior of power systems is commonly

significant. Consequently, it neither in the coherency identification nor in the aggregation of

power plants can be neglected.

2.2 Existing approaches

In the past, the equivalencing procedure was realized on the basis of performing a static

reduction of the equivalent area by essentially ‘Gaussian elimination techniques’ [42].

According to this method small generators less than 50 MW were simply netted as negative

load. Larger generators were retained and equivalenced by classical approaches [25].

Various approaches are proposed for dynamic equivalencing, in particular by exploiting

modal and coherency properties of the machines. A detailed explanation of these

approaches will be explained in appendix A. Subsequent developments led to the following

classical types of dynamic equivalents:

• Ward Types.- In a preliminary way, the dynamic equivalencing was developed on the

basis of Ward type equivalents, which are based on distribution factors used in power

flow studies [48]. An interesting approach was proposed in [56] as dynamic Ward

equivalent. Here, a transient energy function for a reduced system is built after the

elimination of load buses provided with constant current and constant power loads 5.

• Modal equivalents.- It involves two steps:

- Construction of matrices, which represent equivalents of the external system.

- Interfacing these matrices with the transient stability simulation of the internal area to

simulate the complete system.

State changes in the external area are captured by a linearized model. The dynamic

characteristics of this area are then expressed by using voltages and injected currents at

the interconnected nodes as inputs and outputs, and they are linearized at the operating

point as base case specified by a power flow 6.

5 This reduction employs a Ward equivalencing method in which the equivalent current injections are updated at each

integration step of the path-dependent term of the energy function. Each step involves a single iteration of the Newton-

Raphson procedure on the unreduced system.

6 In the modal equation of the state space equation of the selected group of the generators, the exact coherency is

equivalent to the situation where only one mode is excited by any disturbance and all other modes are equal to zero.

The excited mode represents oscillations between the given group of generators and the rest of the system.

BACKGROUND 13

• Coherency-based equivalents.- An alternative approach to modal analysis is the

coherency identification. Some previous attempts at the problem of identifying coherency

have been heuristically-based and have utilized the concept of electrical distance [20-24].

A common limitation of the heuristic methods is the lack of accuracy and consistency

demanded for using in routine planning applications.

• Model reduction and identification methods.- The need to use low-order dynamic

models of a complex power system, especially for its stability analysis considering

damping inter-area and local oscillations, is an important reason for a model reduction.

Identification methods can be used in dynamic equivalencing. System identification deals

with building dynamic models in form of a state-space system structure [84-86]. Identified

models describe linear difference relationships between input and output signals.

2.3 Power system simulation program

In this research, the power system simulation program PSD (Power System Dynamic) is

employed for dynamic studies. The appendix part A.7 shows the implementation of the PSD

flow chart. The PSD is a program for modeling and simulating power systems including

steady state analysis and online transient simulation. The applications and results of the

program are reported in [87, 88]. PSD is based upon the principles of electromechanical-

transient computations in electrical networks. The program has a strong library providing

accurate dynamic models for most known elements in power systems, such as synchronous

machine, two and three winding transformers, and transmission lines, among others. Thus,

the user can chose a suitable model for the synchronous machine among the second-, fifth-

and sixth-order models and define the parameters. It is also possible to build special units

like FACTS, fuel cell devices, among others in a so-called “regulator files” depending on their

block diagram models. These models are integrated into the network through connecting

nodes, which helps the operator to build his own models for the regularly variant components

like the voltage and speed governor regulators. The interaction between the built units and

the network is accomplished through selected variables, which are exchangeable during the

simulation process. The model structures of the regulator devices are implemented using a

special standard code in the PSD simulation package. The interface of these units with the

network is accomplished through the output active and reactive power at each time interval.

The program was carried out using FORTRAN and contains the coherency, aggregation

and static network module as well the proposed approaches presented in chapter 3, 4 and 5,

which have been implemented in frame of this research on the PSD platform [141].

Each power system studied is first being simulated by PSD in its original state using the

standard models found in the library of the PSD validated in [87]. Following that, the new-

BACKGROUND 14

implemented modules in PSD will perform the dynamic simulation of the power system using

the identity recognition, the splitting aggregation and the ANN-based equivalencing in

interaction with additional special programs developed in some cases in MATLAB and

FORTRAN [141].

Hence, after simulating the whole power system in the PSD simulation package, the

dynamic performance of the one is studied taking into consideration the dynamic

equivalencing. Firstly, a power flow calculation is carried out to define the initial operating

condition of the power system. Different disturbances are then simulated in different areas of

the system. The results of the simulations will show how close enough the proposed model’s

responses are to the original system’s responses in the event when disturbance is applied.

2.4 Summary

• In above, the basic steps of the classical dynamic equivalencing, such as: (i) coherency

identification, (ii) generator aggregation, (iii) network reduction and (iv) control

aggregation, and their corresponding methods are summarized.

• The definition of coherency identification can be defined as: closely those of the similarity

rotor angle behavior of machines. All coherency-based approaches are based on this

condition applied to linearized models of the power system.

• The modal-based approaches are based on the inspection of the eigenvalues of the

linearized state matrix. However, it provides limited information about the mode behavior

of generators for a given operating point.

• The determination of parametric properties of the multi-machine power system requires

the use of linear model reduction approaches and linear parametric identification

methods.

The next chapters will present innovative non-linear dynamic equivalencing approaches

using modern and non-conventional techniques.

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 15

“The problem becomes highly complex when dealing with real

disturbances, since linearization is totally ruled out. For these cases

solution techniques using non-conventional methods should be

discussed beginning with the statement of the problem under physical

insight aspects” - M. A. Pai [135] -

Chapter 3

Electromechanical-based

Identity Recognition in

Dynamic Equivalencing

Objective— Aim of this chapter is to present a new electromechanical-based approach in

transient stability of power systems for recognition of identical behaviors of machines. The

approach reformulates the classical coherency condition on the basis of the identity

recognition. Those conditions are used as a basis for generating dynamic equivalents

incorporating physical and model system parameters of the external generators with high

accuracy in the results. Hereby, it consists of the introduction of an electromechanical

distance additional to the geometrical distance that it significantly improves the accuracy and

efficiency of identity-based dynamic equivalents.

Test of this approach have been performed and evaluated on a 16 multi-machine system and

on large-scale model of the interconnected European power system (UCTE/CENTREL).

Index Terms— Clustering, Coherency Identification, Dynamic Equivalent, Electromechanical

Distance, Electromechanical Parameters, Identity Recognition, Network Reduction, Stability

Analysis in Power System.

Organization— Section 3.1 and 3.2 of this chapter describe the introduction and the

classical coherency identification, respectively. In section 3.3 the proposed identity

recognition is presented, and mainly, the electromechanical-based identity recognition

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 16

approach is treated in section 3.4, followed in section 3.5 by the application in interconnected

power systems. In section 3.6 the simulation results are evaluated and the summary in

section 3.7.

3.1 Introduction

This chapter addresses a new alternative to coherency identification. It is the identity

recognition as an efficient approach for recognizing the identical behavior of external

generators forming cluster groups of identical generators and to show that once the identical

behavior was known, significantly reduced dynamic equivalents could be obtained.

The approach presented is based upon necessary conditions to process the swing

oscillating curves in time domain of the external machines. Those reformulated conditions

may be realized using standard pattern recognition algorithms (clustering algorithms). In

order to generate more accurate dynamic equivalents, the definition of an electromechanical

distance considering the physical characteristics and model properties of the external

generators in the identity recognition procedure will be proposed.

3.2 Coherency-based dynamic equivalencing

A coherent group of generators is defined as a group of generators oscillating with the

same rotor angular speed. For this purpose, two generators buses are defined as coherent if

their angular difference is constant within a certain tolerance over a certain time interval [17].

The classical coherency identification of Podmore [17] is based upon the determining the

difference of the voltage angles of the terminal generator nodes, which is extended to the

rotor phase angle behavior. The coherency of both generator internal and terminal buses is

of interest.

Therefore, the generator time responses are evaluated only regarding the rotor angular

phase. This condition is necessary to form dynamic equivalents [17, 18]. This aspect

represents a significant limitation of the equivalencing, because only linear processes and

behaviors of the power system are considered without taking into consideration important

factors of the real generator system, such as no-linear characteristics of governor-, turbine

devices, real modeling parameters, among others.

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 17

Disadvantages:

The formation of a simplified linear model of the power system is realized and solved using

a fast trapezoidal integration algorithm. Considering this linearized simplified system model,

the following disadvantages should be taken into account:

• Coherent groups forming the corresponding equivalents are not exact enough, since the

performance of an equivalent system tends to depend on the applied disturbance.

• The coherent groups are independent of the amount of detail in the machine model, i.e.

in the coherency procedure the real parameters and properties of the machines and

governors in form of non-linear behavior are not considered in spite of all enhancements

in the coherency strategy [19-25].

• A classical synchronous generator model is considered and the excitation and turbine-

governor systems are ignored. This aspect is based upon the observation that the

amount of detail in the generator unit models has a significant effect upon the swing

oscillation curves, particularly the damping [17].

Linearized system

The dynamic equation of the ith generator in a power system with the damping coefficient

included can be formulated as linearized in the following form7:

iiGm

iDPPM ii

ωδ

∆−∆−∆=∆ •• i=1,..,n (3.1)

where

• i is a subscript for ith generator.

• ∆ indicates that this variable represents a deviation from a specified steady state

operating point.

• Mi is the inertia constant in p.u.

• ∆

I is the speed deviation in p.u.

• ∆

i is the rotor angle deviation in radians.

• Di is the damping constant in p.u.

• ∆Pmi is the change in mechanical input power in p.u.

• ∆PGi is the change in electrical output power in p.u.8

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 18

The changes in the complex voltages and power injections at the generator and load buses

may be expressed using the Jacobian matrix in (3.2) and in simplified form in (3.3):













∆













∂













∆

LLLL

GGGG

LLLL

GGGG

θδ

(3.2)

The variables used in the Jacobian matrix are defined as [17]:

• PG, QG are real and reactive power injections at internal generator buses in p.u.

• PL, QL are real and reactive power residual at load buses in p.u.

• E,

are voltages and angles at generator internal buses.

• U,

are voltages and angles at load buses.

The voltage dependence of the load powers is included in the

∂

∂ and

∂

∂ terms and the

changes in the power residuals L

P∆ and QL

∆ are normally zero but may be assigned certain

values in order to model a disturbance such as bus load shedding. Equation (3.2) can be

simplified by accounting for the decoupling, which exists between the real and reactive power

flows for a transmission system with high impedance ratios. The real power flows are largely

dependent upon the voltage angles and as a first order approximation; the effect of variations

in load bus voltage magnitude may be neglected by setting the terms

∂

∂ and

∂

∂ to zero.

The voltage behind the generator transient reactance is constant thus, 0 E =∆ . According to

these assumptions the incremental decoupled active power flow equation may be derived to:













∆













∂













∆

θδ

G (3.3)

7 Equation (3.1) merely states that the accelerating power for each machine is balanced by the increase in kinetic energy

of the rotor and the power absorbed by the damping forces with respect to a synchronous rotating reference frame.

8 ∆PG in (3.1), in general is a very complicated expression calculated from the non-linear differential equations of the

electrical part of the machine and the algebraic equations of the transmission network and the synchronous machine.

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 19

This equation may be arranged for notation convenience to the following form:













∆

























∆

H H

LLLG

GLGG

G (3.4)

Hereby, the partial derivatives in (3.3) are most precisely calculated using the voltages and

angles at the pre-fault steady state operating point.

The electrical power output of the generating units during a fault is calculated by solving the

faulted network equations with the generator transient voltages fixed at the pre-fault values.

In (3.4) the bus load-dropping disturbance can be modeled by introducing step changes in

the L

P∆ and L

Q∆ variables for the selected bus at the appropriate time.

Coherency identification condition

The coherency algorithm minimizes the number of data curve comparisons by recognizing.

Thus, the coherency of generators is a transitive process. A reference generator is defined in

each group and other generators are always compared against this reference in order to

determine whether they should fall in the same group.

The remainder of the generating units are evaluated in turn with two alternative

consequences, either the unit is combined with an existing group or the unit does not

combine with any existing group and a new group is generated.

The Podmore’s coherency criterion, which is based on the examination of the phase

behavior of the rotor angle oscillation and indirect of the generator voltage angle, is used for

determining whether a generator should be added according to its behavior to an existing

group as follows:

εθθ

tt Ri <∆−∆ )()( (3.5)

For all the samples of time, where:

•

is a specified tolerance degree.

• i is the index for generator being grouped.

• R is the index for the reference generator for the group under consideration.

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 20

3.3 Identity recognition approach

The identity recognition leads to determining of grouping of identical generators, which are

replaced by accurate equivalents one without changing the power flow relationships9. These

generators, which are grouped together, have practically a strong coupling with reference to

their

- physical, mechanical and electrical properties and

- linear and non-linear characteristics.

In order to consider these aspects, the classical coherency-based condition has to be

reformulated. This fact implies the consideration of an additional and important condition for

the grouping criterion, whose performance can identify identical linear and non-linear

properties of external machines satisfactorily.

The main advantage of this approach is the possibility to consider the model parameters of

machines within the identity recognition process.

Remarks:

• Usually, the analysis of identity recognition may be realized based on the time responses

of generators to selected faults.

• The faults influence properly on the dynamic equivalencing. Location, kind, and duration

of the faults influences on the form and quality of the dynamic equivalents.

• Time responses of machines to each fault can correspond to the active power variation

()

t∆pi. Further quantities and behaviors, e.g. the angular velocity

()

t∆ωi or the angle

deviation

()

t∆δi to the center of inertia (COI), can be used properly in the identity

recognition procedure too.

• One of the advantages of using active power as time responses to be identical

recognized, is its independence from reference frames.

3.3.1 Conditions

The following figure illustrates the difference between the classical coherency identification

and the proposed identity recognition:

9 The resulted dynamic equivalents are single generators that exhibit the same voltage, speed, total mechanical and

electrical power as grouping generators during the studied disturbance, where those generators show identical

properties.

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 21

Fig. 3.1.- Swing oscillating curve of the rotor angular speed representing the conditions for the identity

recognition instead of the classical coherency.

The first illustration shows the necessary condition for forming the coherency identification.

This consists principally in the evaluation of the phase identity of the rotor angle to identify

similar machines in terms of coherency. As it can be seen in the second illustration, an

additional and imperative condition is the evaluation of the amplitude and phase identity of

the rotor angle of together oscillating machines.

Therefore, the proposed identity recognition is based upon:

• both the identity of amplitude size and

• the identity of phase angle or frequency of the rotor angle behavior.

For the recognition and grouping of identical oscillating generators, an analysis of the

behavior of the generators has to be realized regarding these two important identities. Both

indispensable conditions are criteria for the grouping of identical oscillating generators.

∆

)()( 21 t t

∆≈∆

)()( 21 tt

δδ θθ

∆∆ ≈

COHERENCY IDENTIFICATION

condition according to:

∆

IDENTITY RECOGNITION

condition according to:

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 22

Verification of the identity recognition condition for two machines

These identity recognition conditions can be derived on the basis of the inertial aggregation

[30, 34] considering that equivalent generator should represent the individual behavior of the

single generators both in the initial state and in the whole time period. To this aim, it is more

appropriate to perform aggregation at the machines internal nodes and not at the machine

terminal buses, because the machine rotor angle is the phase angle of the internal node

voltage phasor.

Fig. 3.2.- Classical Aggregation

The machine internal node voltages are tied to a common bus with appropriate transformers

and phase shifters to preserve the power flow.

Defining the voltage transformation ratios from the internal voltages Ea and Eb to the

common bus voltage EE and the corresponding reference frame ‘dq’ to ‘N’ from the steady

state by :

)0(

ieE

eE E

= i=a,b (3.6)

and considering for the whole time period that the changes in the voltages are derived from

the rotor flux linkages and caused by the rotor torsion ∆δ(t):

()

)()(

)(

)( ttj

etE

δδ

∆−∆

== i=a,b (3.7)

ggregated model of Gen. a and Gen. b

() ()

()()

t∆δδj0

ett bb

NEE +

Generators a and b

() ()

()()

t∆δδj0

ett EE

NEE +

() ()

()()

t∆δδj0

ett aa

NEE +

Xda“

Xdb“

XE“

-j XE“

Xda“

Xdb“

GbGa

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 23

the following relationship may be supposed:

etE

=≈ 0

)0(

)(

)( i=a,b (3.8)

Considering this relationship, the expression (3.7) can be satisfied by:

0)()( =∆−∆ tt iE

δδ

i=a,b (3.9)

)()( tt ba

δδ

∆=∆⇒ (3.10)

This condition implies, that two generators are defined as identical, if both amplitude and

phase difference of their rotor angles are constants and minimal within a specified tolerance

over a certain time interval.

Verification of the identity recognition condition for n-machines

The verification of the identity condition extending to a n-machine power system is based

upon the coupled active power flow equation of the external power system. It is expressed on

the equation (3.4) in the decomposed form as:

LGLGGGG θHδHP ∆+∆=∆ (3.11)

LLLGLGL θHδHP ∆+∆=∆ (3.12)

where index ‘G’ denotes the set of selected internal generator nodes assumed to be identical

for any changes in form of disturbances on the internal area and index ‘L’ is the set of the

remaining nodes in form of load buses of the external area.

Taking into account the square matrix HLL is regular and hence, it is invertible the following

expression may be obtained from (3.12):

GLG

LLL

LLL δHHPHθ∆−∆=∆ 11 (3.13)

Replacing the voltage angles at the load buses in (3.11) the following transformed equation

may be derived:

LGGGLLGLGLGLLGLGGG - PRδHPHHδHHHHP L∆+∆=∆+∆=∆ −− 11 ][ (3.14)

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 24

where

LGLLGLGGG - HHHHH 1−

= (3.15)

1−

=LLGLG HHR (3.16)

The above derivation for G

P∆ can be considered in the generator rotor movement. It is

expressed as follows:

iGm

ii PPDM ∆−∆=∆+∆ •••

δδ

i=1,..,n (3.17)

In this case, the mechanical power is assumed as constant for a short duration of the fault

period:

0 Pm=∆ (3.18)

According to G

P∆ (3.14) in (3.17) the following matrix notation can be obtained10:

LMGM

G PRδHδDδ∆+∆=∆+∆ ••• (3.19)

with

)(,

GM H diag h

, h ===−= GG H IHMH i=1,..,n (3.20)

)(,

GM R diag r

, r ===−= GG R IRMR i=1,..,n (3.21)

)(,

GM D diag d

, d ==== GG D IDMD i=1,..,n (3.22)

)( iG M diag =M i=1,..,n (3.23)

where i

M is the inertia coefficient of the ith generator. In this case, equation (3.19) is the

second order state-space equation describing a selected identical group of generators, which

can be solved in a similar form for the non-uniform damping case, for the uniform damping

case and the zero damping case.

10 Increments L

P∆ of the active power of the remaining nodes modeled as bus load-dropping changes can be treated as

the effect disturbance at a certain internal node on the external area.

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 25

The generators of group {G} are said to be exactly identical if increments )(t

∆ of all grouping

generators resulting from equation (3.19) are identical for all

i {G}.

This may be expressed in matrix notation in the following way:

GG t t Iδ)()(

∆=∆ (3.24)

where )(t

∆ is the scalar value and IG the unit vector with cardinality of {G}, i.e. of all identical

components of the group. In this expression should be denoted that the magnitudes and

phase angles of the rotor movement of the identical generators should be identical in sense

of the identity recognition.

Relevant in this case is that the solution of the second order state-space equation system

(3.19) must satisfy and fulfill above equation (3.24) for any disturbances expressed as step

changes in L

P∆ if all rows of certain sub matrix, describing the external area, are identical

considering phase and amplitude.

Therefore, this identity criterion is specified by the property that following any disturbance

the difference of the rotor angle behaviors of identical generators with reference to their

phase and amplitude remains time-dependent and significant small. Thus, the following

relationship 11 in terms of )(t

δ∆ can be derived for two identical generators i and j as:

t tt ji

εδδδ

≤∆=∆−∆ )()()(

)()( tt ji

δδ

∆≈∆ (3.25)

where )(t

∆ is the behavior difference during and after the fault and ε the specific tolerance.

Taking into consideration this identity, the following additional identities may be obtained by

differentiating:

)()( tt ji

•• ∆≈∆

δδ

(3.26)

)()( tt ji

•••• ∆≈∆

δδ

(3.27)

11 If (3.25) in (3.19) is fulfilled approximately for the linearized state space model then the external grouped generators

are approximately in the same manner identical in the non-linearized model.

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 26

It means, if two generators are identical, they will have similar velocity and acceleration

behaviors considering amplitude and phase.

3.3.2 Procedure

In a time period, the behavior of the machines as a time series matrix X is established. This

time data set can include voltage, current, power or rotor angle behavior of the generators

considering the non-linear characteristics of governor and excite control devices.

The procedure to detect the identity consists of the following aspects:

(i) Generate a time domain matrix that reveals the behavior of the machines following

simulated disturbances on a particular node of the system, like a three-phase fault at a

particular node of the internal area.

The generator matrix may be characterized by following attribute vectors:

[]

NMNNN

x..x x x

xx x x

.......,,

............,,

.............,,

321

2232221

1131211

=→

(3.28)

Where the time data set matrix X (N,M) includes the time responses in form of oscillating

swing curves, exhibiting N the number of generators or features and M the number of

variables or patterns or sampling points of the time data set.

(ii) Initial cluster centers, this procedure starts with a predefined number of cluster groups

and corresponding cluster centers that can be considered as initial reference generators.

(iii) Identity recognition according to clustering algorithm. All generators are compared

against the reference generators as cluster centers to determine whether they should fall in

the cluster groups. All clustering algorithms are based on the minimizing of the within cluster

squared distances by means of an iterative way or an optimization procedure.

This aspect can be illustrated iteratively in the following figure:

Generator 1

Generator 2

Generator N

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 27

Fig. 3.3.- Schematic representation of iterative assignment and construction of centroids or

reference generators within a clustering procedure

The assignment of generators to the cluster groups regarding their time behavior

characteristics can be realized by means of the evaluation of a distance criterion between the

grouping generator and the reference generator. In this case the following illustration can be

sketched:

Fig. 3.4.- Schematic representation of similar generators belonging to a cluster group taking into

account its identical properties in phase and amplitude.

Calculation of the

centroids

Assignment to the next centroid

Calculation of the new

centroids

∆

∆P[p.u.]

t [s]

ssignment of identical generators

X1, X2 and X3 to a cluster

___Reference Generator C

___Generator X1

___Generator X2

___Generator X3

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 28

3.3.3 Identity recognition algorithms

The identity recognition algorithms as clustering process consist of procedures, which

divide the multidimensional data space of the features into a number of separated groups

called clusters, whose features have identical patterns in a multidimensional space [91-98]

and the profiles of objects in the same groups are relatively homogenous whereas the

profiles of objects in different groups are relatively heterogeneous.

This assignment in groups is based on the minimizing of the within cluster squared

distances directly in an iterative way (partitioning), in a transitive process (hierarchical) or

indirectly in an optimization procedure (Fuzzy and SOFM) and thus, the machines can be

assigned to the corresponding most nearby clusters.

In detail, the distance criterion can be described as follows:

• The identity between objects can be recognized by means of distance measurement

criteria between the data vectors in the multidimensional space in different ways.

• For an extensive review of measures can be implemented the LP-Metric or Minkowski

distance. The Minkowski distance dij between two M-dimensional objects (data points) i

and j is defined by the following expression:

()

jlil yxyxd

,











−= ∑

(3.29)

where xil and yjl are objects with M sampling points, whose distance has to be

calculated. M corresponds to the number of variables.

• By these distance definition may be diverted other distance functions by means of new

definition of p. An important distance measure is the Euclidean distance metric, which is

defined by p=2.

The wide variety of existing clustering can be divided into four main groups, such as:

- Hierarchical [92-95],

- Partition [94-101] in form of K-means,

- Fuzzy C-means [102-105] and

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 29

- Unsupervised neural networks [106-109] in form of self-organizing feature maps

(SOFM) or competitive networks, although other techniques are possible.

Facts of these standard algorithms are explained in detail in appendix B.

3.3.4 Comparative application

To compare the standard identity recognition algorithms, their properties and advantages

should be discussed. In fact, each method works in a different way and very often yields

results different from the others.

The identity recognition algorithms may be showed schematically in the following flowchart.

Bold lines indicate the more successful methods with significant accuracy.

Fig. 3.5.- Flow chart of applied identity recognition algorithms.

Hierarchical clustering (HC)

• It is the most widely used algorithm due to the computational simplicity, sometimes it is

stated that this technique is no longer valid if applied to large data sets [92].

• Its main purpose is to provide the user with suitable initial cluster centers, and thus it is

able to identify the initial seed points.

• The main disadvantage lies in the fact that, according to the logic of hierarchical

clustering, a kind of hierarchical structure is imposed into the data, even if the data do not

possess such structure. This can lead to a misclassification of the data structure [93].

K-means SOFM-ANN

Features of Generators

Hierarchical Fuzzy C-means

Groups consisting of identical oscillating machines

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 30

Partitioning or non-hierarchical clustering (NH) as K-means

• It is free from the hierarchical disadvantage.

• Its main properties are its computational simplicity, the re-determination, and adjustment

of the composition and population of clusters with non-hierarchical clustering.

• This clustering technique is referred to as ‘hard clustering’ schemes, where each object is

assigned to one and only one cluster.

• In the case of touching or overlapped clusters this assumption is apparently invalid,

leading consequently to misclassification.

Fuzzy clustering (FC)

• The procedure of dividing the N objects into k clusters is replaced by a procedure of

determining the membership degree of each object belonging to each cluster group.

From this point of view, ‘hard clustering’ can be considered as a particular case of Fuzzy

clustering, i.e. when all membership coefficients, except one, of a single machine are

equal to zero.

• Although this is yet another mathematical model that does not necessarily describe the

real data structure, it seems that such an approach should lead to appropriate results.

• Fuzzy provides a clustering classification by using of membership degree coefficients

between 0 and 1, which can contribute to divide an object in shares. The shares number

corresponds to the number of defined cluster groups.

• The internal structure of clusters and their interrelationships can be determined with

Fuzzy-c means. Moreover, the ‘dispersion level’ or ‘splitting’ of the grouped objects can

be defined with help of the membership degrees.

Kohonen self organizing features maps (SOFM)

• Their application is restricted due to the large computation time with large number of

objects, by which the network can be trained.

• A limitation of this method is that the number of groups not always corresponds to the

defined number of output neurons, and thus the Kohonen network can fail.

• The procedure of Fuzzy c-means and K-means clustering has not this difficulty, because

the cluster center forming is forced by the input of the number of objects. Nevertheless

Kohonen maps can be considered as a complementary method to verify the results of

previous clustering methods.

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 31

3.4 Electromechanical-based identity recognition

In spite of all identity recognition technique’s properties, an important improvement in

identity recognition can be reached by using electromechanical weighted distances to obtain

‘electrically real identical generators grouped in clusters’. In this context, following aspects

should be considered:

• Factors, such as the model parameter properties, physical characteristics and the

particular influences of single generators on the power system, must be considered to

develop a real electrical assignment preserving the singularity of each machine.

• Thus, it is indispensable to introduce characteristic sizes of generators, which

characterize the particular behavior and the proper impact of the machines on the

external power system.

• The electromechanical distance is introduced in the identity recognition process

additionally to the geometrical distance.

• Thus, it is not appropriate to apply the same distance criteria to generators located far

from the area of interest as the nearest one.

• Electromechanical small machines with small characteristics have less influence on the

whole system dynamic than larger machines.

3.4.1 Inertia coefficient and electrical power

According to the following sensitivity analysis, model parameters, as pre- and during-fault

relevant electromechanical factors may be selected, to improve the assignment to groups of

machines according to their physical and model properties.

Sensitivity analysis

The sensitivity of a determined machine’s behavior y(t) with reference to the parameters P

may be defined mathematically as:

PioPi

pi P

Pte =

∂

=)(

)( (3.30)

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 32

The sensitivity is a time function, which expresses the influence of parameter Pi on the

investigated behavior y(t), such as rotor speed, electrical power, rotor current, rotor voltage,

etc. and a certain time during this behavior.

The rotor speed behavior is an important electromechanical behavior to be evaluated

according to the sensitivity. For the sensitivity analysis the following one-machine system

was evaluated:

Fig. 3.6.- One-machine power system for sensitivity analysis

On the basis of this evaluation, the influence of the parameters on the electromechanical

behavior of the machines may be quantitatively detected. In order to derive a quantity

evaluation for sensitivity the square mean value is formed and afterwards normalization to

unit is defined.

The following illustration shows a summarized comparison of the normalized square mean

values of the sensitivities of rotor angle δL with reference to important synchronous machine

parameters.

Fig. 3.7.- Graphical comparison of the normalized square mean values of the sensitivities of rotor

angle δL with reference to the synchronous machine parameters

Normalized Squared

Mean Value of

Sensitivity δ

δδ

δL(%)

(%)(%)

(%)

x‘d x‘‘d xd T‘‘ x

s rs T‘d T‘‘

x‘‘

MachineParameters

Before max. amplitude

After max. amplitude

Before the max. amplitude of the

L behavior

After the max. amplitude of the δL behavior

x=0.25 p.u.

P=1.0 p.u.

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 33

This illustration shows the sensitivity of the rotor angle with reference to different machine

parameters considering the time setting behavior before and after the maximum amplitude,

which is generated following a three-phase short circuit on the one-machine system. On the

basis of this illustration following aspects can be detected:

• The large influence of the inertial coefficient parameter Tm as well the small effect of

other parameters on the behavior of

L can be depicted.

• However, Tm decreases its influence after the maximum amplitude. On the other hand,

the sensitivity influence of the other parameters, such as the transient reactance x’d is

increased in this case, but the sensitivity of Tm cannot be reached.

• The sensitivity has been evaluated taking into account the parameters defined in Fig.

3.6. With this wide range of model parameters the synchronous machine may be

described completely.

Instead of the

L other machine behaviors are considered, which can be showed in the

following figure. It shows the squared mean values of the sensitivities.

Fig. 3.8.- Graphical comparison of the normalized square mean values of the sensitivities of rotor

angle, rotor current and angle with reference to the parameters of the synchronous machines

In this illustration, the sensitivity of different behaviors, i.e. electrical power, rotor current

and voltage is described. In this case, a similar tendency as in the previous figure can be

detected.

In summary, Tm and x’d are relevant machine model parameters with significant influence on

the behavior of the machine.

Electrical Power

Rotor Current

Rotor Voltage

Normalized Squared

Mean Value of

Sensitivity δ

δδ

δ L

(%)

(%)(%)

(%)

‘d x‘‘d

‘‘

σs rs

‘d T‘‘

‘‘

MachineParameters

x‘d x‘‘d xd T‘‘ xσs rs T‘d T‘‘q x‘‘q xq Tm

Electrical Power

Rotor Current

Rotor Voltage

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 34

According to the sensitivity analysis, following main conclusions can be summarized:

• The inertial constant Tm determines significantly the behavior of the generator during

short circuit period. Under all machine model parameters, this constant has the largest

influence on the system dynamics during the total stability analysis.

• Additional to this factor, the transient reactance x’d plays an important role, but in a

substantially lower grade than the inertial constant Tm.

• The sensitivity of the rotor angle speed, rotor current, and voltage with reference to the

rotor reactance x

s is insignificantly low.

Analytical verification to determine the relevance of the inertial coefficient Tm

In the following, an analytical verification will be realized to determine the influence of the

inertial coefficient on the identity recognition. By differentiating the equation of identity (3.26),

two machines will have equal velocity and acceleration according to (3.27) and (3.28). Since

the proposed identity of generators does not depend on the disturbance, the linearized

equation (3.17) can be simplified according to the following assumptions:

Assumptions:

• The electric power using the classical generator model may be found taking into account

the admittance matrix denoted by Yij 12.

]eEYE[ Re]EY [E Re]I[E Re P )-j(-

jij

iji

iiG

ijij

δδθ

∑∑ ===

(3.31)

Let

ii eEE

= ij

ijijijij eYBjGY

=+= (3.32)

Then

[]

))(cos(

jiijijj

iiiiG YEE GE P

δδθ

−−+= ∑

≠

(3.33)

• The mechanical power is considered as constant value for the fault period.

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 35

0 Pm=∆ (3.34)

• Internal voltages of the generators ji E E , are nearly equal to 1.0 p.u.

• The network is considered as highly reactive.

• The angular differences (

j) are within 30°.

In consequence, the change in electrical power in (3.33) can be expressed as:

()

∑

∆−∆=∆

jiijG Y P

δδ

(3.35)

Substituting expressions (3.34) and (3.35) in (3.17), it becomes:

()

ijiji DY M i

•

•• ∆−∆−∆=∆ ∑

δδδδ

(3.36)

The linearized power variation-based swing equation of the generators for the ith and jth

generator can be expressed according to (3.36) as follows:

()

()(){}

[]

iiiininiiijijii DMYYYM

ωδδδδδδδ

∆−∆−∆++∆−∆+∆−∆=∆ −−

•• 1

1.. (3.37)

()()(){}

[]

jjjjnjnjjjijijj DMYYYM

ωδδδδδδδ

∆−∆−∆++∆−∆+∆−∆=∆ −−

•• 1

1.. (3.38)

The sign of the first term of the above two expressions are opposite to each other.

Therefore, for any disturbance in power system an increase in acceleration of the ith

generator causes an increase in

∆δ

i and

∆ω

i. The increase in

∆δ

i and

∆ω

i will tend to cause a

decrease in the acceleration of the ith generator based on (3.37) and at the same time to

increase the acceleration of the jth generator in (3.38). As the acceleration of the jth

generator starts increasing, it causes to increase

∆δ

j and

∆ω

12 On the basis of the linearized model of the dynamic equation of the ith generator expressed in (3.1), Pe is the

generator electric power obtained from a load flow solution.

Terms relating other generators as the jth with ith generator

Terms relating other generators as the ith with jth generator

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 36

Thus, the generators will try to swing together in terms of the synchronism through the

mutual change effect defined by the first term. Therefore, the coefficient terms of expressions

(3.37) and (3.38):

ij −−

together give an estimate of the relative variation between the acceleration i

••

∆

and j

••

∆

13.

Due to that M is considered as the per-unit value and proportional to Tm, those value can be

found in the machine data set the following expression conveniently may be obtained:

ωπ

== (3.39)

T M m

~⇒ (3.40)

From this viewpoint and the previous sensitivity analysis, the relevant role of the inertia

constant Tm is determined with respect to identical recognition of together oscillating

machines, which give an estimate of the electromechanical effect of the generators.

Following aspects are relevant to select the weighting factors:

• The inertial constant Tm determines significantly under all machine model parameters

the behavior of the generator during short circuit period, which has been demonstrated

by the sensitivity analysis and in expressions (3.37-3.39).

• This widely used parameter informs about the machine’s ability to ‘absorb’ and is used to

quantify abrupt changes of their mechanical torque. In consequence, this constant

characterizes the disturbed rotor angle trajectory.

• A significantly larger impact of this constant can be observed in nominal power, current,

voltage or rotor angle behaviors of the machines according to Fig. 3.7 and Fig. 3.8.

• Because of this influencing factor, it is possible, to treat differently in electromechanical

terms all the generators in the identity recognition procedure.

13 Since Yij=Yji, the difference in the values of the coefficients of

∆ in terms to find an equilibrium point, synchronism

effect between together oscillating machines and minimal difference of the rotor angle behavior, is only due to the

difference in Mi and Mj.

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 37

• Another characteristic size of machines is the nominal power Sj, which characterizes

the power balance experienced by each generator and its participation in the total power

balance of the power system. Thus, it is a proper machine characteristic behavior.

• The nominal power Sj are particular size and characteristic of each machine that inform

about the ‘degree of participation and effect’ of each machine in the during- and post-

fault periods.

• Therefore, large generators with large nominal power and large inertial constants have

more electromechanical influence on the system dynamic than smaller generators.

In consequence, according to the sensitivity analysis and the above verification, the following

weighting factor may be defined:

Nj1 ST W jmjj <<= ,* (3.41)

where Tmj is the inertial constant an Sj the nominal power of the corresponding generator.

3.4.2 Electromechanical distance

The specific physical effect and electromechanical influence of the generators with regard

to their properties and structure derived previously have to be considered into the clustering

of the machines. To this end, it is indispensable to include the weighting factors in the

calculations of minimizing the cluster square distances to a centroid within each cluster

resulting the so-called “electromechanical distance”.

This situation contributes to a more accurate equivalencing that forms clusters to represent

groups of entities with real electrical and physical identical properties.

Hence, the grouping procedure can be adjusted by a weighted distance. These weights

can be implemented in hierarchical, K-means and Fuzzy algorithms as distance criterion

resulting the electromechanical distance. As can be obtained in the following way:

mincxSTcxWd

líljjmj

líljj

hanicalelectromec ∑∑∑∑∑∑

======

→−=−=

(3.42)

where Wj, is the weighting expression consisting of the inertia constant and the nominal

power according to the expression (3.41).

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 38

The expression in (3.42) has to be introduced into the algorithms, such as K-means and

hierarchical to be minimized.

In Fuzzy clustering the electromechanical distance has to be integrated both in the distance

calculation and in the objective function Jm (B.6), which is optimized iteratively according to

the expression:

),,;( jij

mdw CUWXJ ∑∑

(3.43)

where

• dij is the distance between the generator and cluster centers,

• j

w is the weighting electromechanical factor to form the electromechanical distance,

• the membership degree of generator xj to cluster ci is denoted by

ij and

• the parameter m>1 is called fuzziness index and influences the “fuzziness” of the

obtained grouping.

Accurate model parameter-based dynamic equivalents may be obtained by the

electromechanical neighborhood relationships using the electromechanical distance between

generators into the identity recognition.

3.5 Case studies

In order to verify the effectiveness, accuracy, and applicability of the electromechanical-

based identity recognition using K-means, hierarchical, Fuzzy and Kohonen-SOFM, a small-

scaled and large sized interconnected power system are examined.

The investigated systems are:

1. 16 Multi-machine system consisting of 16 generators.

2. Interconnected European power system, known as UCTE/CENTREL consisting of the

western European Union for the Coordination of transmission of Electricity (UCTE)

and the central European power system (CENTREL) [142].

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 39

3.5.1 16 Multi-machine system

This system consists of three strongly meshed areas with different voltage levels (380kV,

220kV and 110kV) characterized by the area A, B and C. Each area has 5 or 6 generators.

This system comprises hydro, nuclear, and thermal generators with ratings of 220MW,

247MW, and 259MW respectively. It can be seen in Fig. 3.9.

Fig. 3.9.- 16 Multi-machine system

Following system characteristics are important:

• In general, the system comprises 16 generators with their corresponding excitation and

governor systems, 66 nodes, 16 two-winding transformation units, 12 three-winding

transformation units and 54 transmission lines.

• The machines are described by 5th, exciters by 2nd and in some cases by 3rd order

models. IEEE standard controller parameters are used for the governors and the

excitation systems. Thus, a state vector of large dimension characterizes the models.

• Area A contains mostly hydro power plants and it is structured to be a power exporting

area and B and C as demanding distribution systems.

• Areas A, B and C are considered as internal areas separately and retained individually in

detail. The rest machines located outside of this corresponding area, assumed as

AREA C

AREA A

AREA B

1a A1b

A2a A2b

B2a B2b

B10

C10

C12

C14

380 kV

220 kV

110 kV

15.75 kV

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 40

external area, have to be replaced by dynamic equivalents involving the

electromechanical-based identity recognition and the classical inertial aggregation

procedure.

• Using the equivalent models of the external area, the dynamic behavior of the time

response of an internal machine following a certain disturbance is simulated.

• The behavior of the internal area machines calculated with help of the dynamic

equivalents is compared with them calculated on the basis of the original external area.

• In order to realize an adequate testing of the proposed electromechanical identity

recognition, various stability scenarios was investigated taking into consideration as

study areas, A, B, and C respectively. Different disturbances as three-phase short circuit

with duration between 80 to 200 ms were selected on the specifications, as described in

the following table.

Table 3.1.- Disturbance specifications for the 16 multi-machine system

Internal Area Fault duration (ms) Fault location (node)

100 C5 on 380kV

100 C1 on 380kV

Area C

100 C8 on 220kV and C1 on

380kV

150 B1 on 380kV

150 B10 on 220kV

Area B 150 B2 on 220kV

200 A5a on 380kV

200 A5b on 380kV

Area A

200 A2 on 380kV

These disturbances were applied to the boundary nodes between the areas. The boundary

node location is a suitable choice for the application of a disturbance because electrically and

geographically the disturbance is strongly coupled to all areas.

3.5.2 Simulation results and discussion

All results, considering the disturbances of table 3.1, are accurate enough. However, a

representative scenario may be described. In this case, the internal area consisting of 5

internal machines of area C, which dynamics are of interest, is retained in detail, and the

external area, consisting of 11 machines of areas A and B, is performed on basis of 3

equivalent machines applying the proposed identity recognition algorithms.

The disturbance is a three-phase short circuit during 100ms at the nodes C01 on 380kV

and C08 on 220kV applied at the same time. This fault creates a major system-wide

disturbance and is applied after 1 sec., the transient stability simulation duration is 10 sec.

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 41

Considering this disturbance, following aspects has been examined:

i. Clustering or grouping of external machines according to the identity recognition.

ii. Internal machine behavior using the identity recognition-based dynamic equivalents.

iii. Internal machine behavior depending on different number of dynamic equivalents.

iv. Identity recognition accuracy depending geographically on the disturbance location.

i. Clustering

According to the above scenario, the following external machine grouping assignment can

be obtained independent on the used identity recognition algorithm.

Fig. 3.10.- Schematic representation of the grouped 16-machine System following disturbances at

the boundary nodes C1 on 380kV and C8 on 220kV at the same time.

This grouping, showed in Fig. 3.10, corresponds to the grouping of time responses

represented in Fig. 3.11. These generator responses are simulated with PSD. Same

machine assignment to the cluster groups and consequently similar dynamic equivalents are

obtained using K-means, Hierarchical, Fuzzy, and SOFM.

1a A1b

2a A2b

B2a B2b

B10

C10

C12

C14

1. Group

2. Group 3. Group

C1 C8

INTERNAL AREA

Disturbance

AREA A

AREAB

380 kV

220 kV

110 kV

15.75 kV

EXTERNAL

REA

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 42

Fig. 3.11.- Assignment and grouping of identical external machines according to the identity

recognition algorithms in conjunction to Fig. 3.10.

However, another cluster assignment can be obtained using the electromechanical-based

recognition algorithms.

To compare the algorithms, the internal machines will be simulated using the equivalents and

their behaviors evaluated tacking into account their phase and amplitude.

-0,06

-0,02

0,02

0,06

0,1

012345678910

B03

B10

-0,03

0,01

0,05

0,09

012345678910

B02a

B02b

B08

-0,05

0,05

0,1

0,15

0,2

012345678910

A01a

A01b

A02a5

A02b

A03

A06

Identical external machines belonging to 1. Group

Identical external machines belonging to 3. Group

Identical external machines belonging to 2. Group

A1a

A1b

A2a

A2b

B2a

B2b

B10

t [s]

∆

∆P

.u.

]

t [s]

∆

∆P

.u.

]

t [s]

∆

∆P

.u.

]

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 43

ii. Internal machine behavior with identity-based dynamic equivalents

The internal machine C12 will be simulated in the stability analysis one hand with help of

the dynamic equivalent of the external area on the basis of the identity recognition methods,

and another hand with help of the original external area using PSD. For more details on the

validity of PSD, the reader can refer to section 2.3.

All internal machine behaviors show the same accuracy independent of the used identity

recognition algorithm, as can be seen in the following Fig. 3.12; the behavior of the internal

generator C12 is represented using the electromechanical-based Fuzzy and K-means.

Fig. 3.12.a, b.- Comparison of time responses of C12 internal machine calculated with 3 cluster

groups according to the electromechanical Fuzzy and K-means

As it can be seen in Fig. 3.12.b, an improvement in accuracy and agreement can be

obtained using the electromechanical-based equivalents. Although, Fuzzy, hierarchical, K-

-0,09

-0,05

-0,01

0,03

0,07

0,11

0,15

0,19

012345678910

Original

Electromechanical Fuzzy

Electromechanical K-means

∆

∆∆

∆p

/p.u.

3 Equivalent generators reduced from 11

C12 Internal machine behavior calculated with:

-0,09

-0,03

0,03

0,09

0,15

012345678910

Original

Electromechanical K-means

K-means

∆

∆∆

∆p

/p.u.

3 Equivalent generators reduced from 11

C12 Internal machine behavior calculated with:

Original (PSD)

Electromechanical Fuzzy

Electromechanical K-means

Original (PSD)

Electromechanical K-means

K-means

Time [s]

3 equivalent generators reduced from 11 Gen.

∆

∆P [p.u.]

Time [s]

3 equivalent generators reduced from 11 Gen.

∆

∆P [p.u.]

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 44

means, and SOFM generate similar grouping assignments, the electromechanical-based

algorithms of Fuzzy, K-means and hierarchical generate other groups.

iii. Internal machine behaviors with different number of dynamic equivalents

The following figure examines the internal machine C14 using different numbers of identity

recognition-based equivalents reduced from 11 machines.

Fig. 3.13.- Comparison of time responses of the internal machine C14 depending on the reduction

degree with 1, 3, 6 electromechanical Fuzzy algorithm-based equivalents.

It can be detected that the oscillating swing curve for C14 calculated with different number

of equivalents is very close to that of the original behavior. However, a degradation in

accuracy with one equivalent can be determined.

iv. Internal machine behaviors depending on the disturbance location

A detectable degradation of agreement and accuracy can be discerned in the internal

machines depending on the disturbance location.

- The internal machines, i.e. the C10, C12 and C14, located far away from the

disturbance, with which the identity recognition was realized, are less accurate (see Fig.

3.12 and Fig. 3.13).

- In contrast with this, the C2 and C7 machine behaviors are high accurate, as can be

seen in the Fig. 3.14. These machines are closest to the disturbances of the equivalents

derived (disturbances on node C1 and C8 of area C, see Fig. 3.10).

-0,07

-0,01

0,05

0,11

0,17

0,23

012345678910

Original

Elec. Fuzzy with 1 cluster

Elec. Fuzzy with 3 clusters

Elec. Fuzzy with 6 clusters

C14 Internal machine behavior calculated with different

number of groups or clusters:

Original (PSD)

Elec. Fuzzy with 1 equivalent

Elec. Fuzzy with 3 equivalents

Elec. Fuzz

with 6 e

uivalents

Time

[

]

Equivalent generators reduced from 11 Gen.

∆

∆P [p.u.]

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 45

Fig. 3.14.- Time responses of the C2 internal machine calculated with 3 equivalent machines by

different identity recognition-based algorithms.

The same identity recognition-based equivalents can be used to simulate other

disturbances, which have to be coupled electrically and geographically to the one. The

following cases can be simulated:

- In Fig. 3.15, the behavior of the C2 generator following the disturbance on the node C3 in

380kV area is represented. This disturbance is electrically and geographically in close to

the disturbance of the equivalents derived.

- In Fig. 3.16, the behavior of the C2 generator following the disturbance on the node C9 in

220kV area is represented. This disturbance is electrically and geographically far away

from the disturbance of the equivalents derived.

Fig. 3.15.- Time responses of the C2 internal machine following the disturbance (electrically and

geographically closest) applied on C3 node of internal area.

-0,2

-0,1

0,1

0,2

0,3

0,4

0,5

0,6

012345678910

Original

Electromechanical K-means

Electromechanical Fuzzy

C2 Internal machine behavior calculated with:

Original (PSD)

Electromechanical K-means

Electromechanical Fuzzy

Time

[

]

3 Equivalent generators reduced from 11 Gen.

∆

∆P [p.u.]

-0,3

-0,1

0,1

0,3

0,5

0,7

0,9

012345678910

Original

Elec. Fuzzy equivalent in 380 kV area

C2 Internal machine behavior following the closest disturbance on

C3 node calculated with:

Original (PSD)

Electromechanical Fuzzy in 380 kV

Time

[

]

3 Equivalent generators reduced from 11 Gen.

∆

∆P [p.u.]

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 46

Fig. 3.16.- Time responses of the C2 internal machine following the disturbance (electrically and

geographically far away) applied on C9 node of internal area.

The oscillation behavior of Fig. 3.15 shows a higher degree of agreement than the one of

Fig. 3.16. Thus, the electromechanically derived equivalents are also valid for other

disturbances limitedly, that are electrically and geographically in close distance to the fault of

the equivalents derived, i.e. the closer these disturbances, higher the accuracy of the

equivalents.

Following aspects can be briefly summarized:

• All identity recognition algorithms provide the same machine assignment and grouping of

external machines for this 16 multi-machine system. Consequently, they generate

identical dynamic equivalents.

• Improved dynamic equivalents are obtained using electromechanical-based identity

recognition algorithms.

• The accuracy of the identity recognition depends upon important aspects, such as:

- the identity recognition capability of the algorithms.

- number of dynamic equivalents.

- the geographical and electrical distance between internal machines and

disturbances.

A discernible accuracy improvement by means of the electromechanical distances can be

detected at the stability analysis in a large interconnected power system, as it can be

presented in the following case study.

-0,05

0,05

0,1

0,15

012345678910

Original

Elec. Fuzzy equivalent in 220kV area

Original (PSD)

Electromechanical Fuzzy in 220 kV

Time

[

]

3 Equivalent generators reduced from 11 Gen.

∆

∆P [p.u.] C2 Internal machine behavior following the far away disturbance

on C9 node calculated with:

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 47

3.5.3 Interconnected European Network UCTE/CENTREL

The electromechanical identity recognition was applied to the Interconnected European

power system UCTE/CENTREL, consisting of 464 machines, 2016 nodes and 2098

transmission lines with their excitation and governor control models [142].

Additional to the western European Union for the Coordination of Transmission of

Electricity (UCTE), the central European power system (CENTREL) includes the eastern

European countries. The following table shows different European network subsystems.

Table 3.2.- Subsystems in the European Interconnected Power System UCTE/CENTREL

Power Subsystem Country Power Subsystem Country

ALB Albania MAZ Macedonia

B Belgium NL The Netherlands

BAG Germany OEVG Austria

BEWAG Germany PE Germany

BG Bulgaria PL Poland

BiH Bosnia/Herzegovina Po Portugal

CEZ Czech Republic ROM Romania

CH Switzerland RWE Germany

ELSAM Denmark SEP Slovakia

EnBW Germany SL_HR Slovenia/Croatia

FR France Sp Spain

GR Greece VEAG Germany

HU Hungary VEW Germany

IT Italy YU Yugoslavia

LVOV Ukraine

The integration of the systems UCTE and CENTREL leads to a complex stability behavior

and reciprocal dynamic impact between strongly mashed subsystems. Taking into

consideration the liberalization of electric market in Europe, the operators are forced to

operate the subsystems near to the stability limits and outside their local area. Consequently,

the operators due to safety reasons have to realize different stability assessment and

analysis actions within their internal subsystems. To this end, information, and knowledge

over the network structure and the actual operation situation of the external subsystems, i.e.

outside of the internal subsystem, should be available limitedly involving dynamic

equivalents.

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 48

In this study, the available data for the UCTE/CENTREL system include transformers,

machines, loads, transmission lines, amongst others. The machine model is either described

by 5th or 6th order models including governors and exciters by 2nd and 3rd order models.

Fig. 3.17.-Interconnected European power system UCTE/CENTREL [142].

For the simulation following aspects are relevant:

• The power system is divided into the German system consisting of the old operators:

BAG, BEWAG, EnBW, HEW, PE, RWE, VEAG and VEW forming the internal area. It has

to be preserved in the original form, consisting of 67 machines, which are available

physically in the actuality in spite of all liberalized market operations (For details see

appendix Fig. F.1). All these machines are retained individually.

• The rest 397 machines located outside the German area, considered as external area,

have to be replaced by dynamic equivalents involving the electromechanical identity

recognition and the classical inertial aggregation procedure.

• The electromechanical identity recognition performance will be evaluated by comparing

the oscillating swing curves of the internal area machines. They will be simulated using

PSD in one hand with help of the full external area and another hand by the dynamic

equivalent of the external area. (For more details on PSD see section 2.3)

• To this end, several disturbances as three-phase short circuit with a duration between 80

to 200 ms were selected, which are applied at the boundary between the German and

external area. These disturbances are described in table 3.3.

External Area

Internal

Area

pplied disturbances

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 49

Table 3.3.- Disturbance specifications for the European power system UCTE/CENTREL (the boundary

nodes correspond to NL=Netherlands, CH=Swissland, OEVG=Austria)

Cases

Fault location

(German network

operator)

Neighborhood nodes

1 VEGROUSB (VEW) NLHENGL4 (NL)

2 BWKUESSA (EnBW)

CHLAC SA (CH)

CHLAUBSB (CH)

CHLAVOG4 (CH)

CHGOEESA (CH)

CHBASUSA (CH)

3 RWHERBG2 (RWE ) OBUERS2 (OEVG)

CHWINKL2 (CH)

4 EVPULDSB (EnBW) CHLAUBSB (CH)

5 RWLEULSA (RWE) OWESTT1 (OEVG)

Considering these faults, all simulations are similar and show a high accuracy and

agreement. A representative scenario is the disturbance, which is based on the case 1

during 100 ms at the boundary node VEGROUSB (VEW). This node is located in the near to

the Dutch node NLHENGL4 (NL). The three-phase short circuit is applied to a 500kV bus,

which connects the Dutch and the German system.

This fault creates a major system-wide disturbance and is applied after 2 sec., the transient

stability simulation duration is 10 sec.

3.5.4 Simulation results and discussion

This disturbance is simulated using the original external area and the reduced external

models, which are based on different number of dynamic equivalents from 180 until 20. The

proposed identical recognition algorithms will be applied. Hence, the following representative

cases are presented:

- Case 1 — 90 European dynamic equivalents reduced from 397 machines

- Case 2 — 65 European dynamic equivalents reduced from 397 machines

- Case 3 — Electrical and geographical coupled 65 European dynamic equivalents

Case 1 — 90 European dynamic equivalents reduced from 397 machines

In this case, the time behavior of the German machine KIEL1 following the disturbance at

the boundary node VEGROUSB (VEW) is calculated, in one hand, with the unreduced 397-

machine external area and in the other hand with 90 equivalents. K-means, Fuzzy and

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 50

SOFM and their corresponding electromechanical algorithms have been applied, as can be

shown in the following figures:

Fig. 3.18.a, b, c.- Comparison of time responses of the KIEL1 German machine calculated with 90

equivalent machines by different identity recognition algorithms and considering their

electromechanical weighting (weighted K-means, weighted Fuzzy and Kohonen-SOFM).

Upon the basis of Fig. 3.18 following aspects are detected:

-0,04

-0,02

0,02

0,04

0,06

0,08

0,1

012345678910

Or SOFM E F

-0,04

-0,02

0,02

0,04

0,06

0,08

0,1

012345678910

O Fu E

-0,04

-0,02

0,02

0,04

0,06

0,08

0,1

012345678910

Or K K

Time [s]

90 equivalent generators reduced from 397 Gen.

K-means

∆

∆P [p.u.]

Original (PSD) Electromechanical

K-means

Time [s]

90 equivalent generators reduced from 397 Gen.

Fuzzy

∆

∆P [p.u.]

Original (PSD) Electromechanical

Fuzzy

Internal German machine KIEL1 calculated with:

Time [s]

90 equivalent generators reduced from 397 Gen.

NN-SOFM

∆

∆P [p.u.]

Original (PSD) Electromechanical

Fuzzy

Internal German machine KIEL1 calculated with:

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 51

• The time responses of the internal machines with the dynamic equivalents considering

the electromechanical distances in all identity recognition algorithms (as K-means in Fig.

3.18.a, and Fuzzy in Fig. 3.18.b) is very close to that of the unreduced system for the

whole time simulation.

• Considering only the algorithms without electromechanical distance, a notable accuracy

of K-means in Fig. 3.18.a can be detected in comparison to Fuzzy in Fig. 3.18.b and

SOFM in Fig. 3.18.c, which show a great deviation over 3 sec. to 6 sec. concerning

phase and amplitude.

• However, it should be mentioned that same accuracy by Fuzzy and SOFM is achieved

directly after the disturbance, i.e. during the first 3 seconds and over 6 seconds of the

simulation.

• Therefore, depending on the applied algorithm a detectable degradation in accuracy may

be discerned in large-scaled power systems.

• K-means and relatively Fuzzy recognizes better identical groups of generators than

SOFM, but through the electromechanical distance all algorithms recognizes

electromechanically close identical groups taking into consideration the physical

parameters of the grouped generators and consequently they form more accurate

dynamic equivalents.

Case 2 — 65 European dynamic equivalents reduced from 397 machines

In this case, the Figs. 3.19.a and 3.19.b show the oscillation time responses of the STDE1

German machine. They are simulated using the dynamic equivalent consisting of 65

equivalent machines and following the disturbance at node VEGROUSB (VEW).

Remarks to Fig. 3.19.a and Fig. 3.19.b:

• The same degradation of accuracy in K-means and Fuzzy of oscillation of the STDE1

German machine following the same disturbance in case 1 of table 3.3 can be observed

in Fig. 3.19.a and Fig. 3.19.b, respectively. In this context, a considerable deviation of the

time oscillation both in K-means and Fuzzy in amplitude and phase over 4 seconds until

to 6 seconds during the simulation can be detected.

• However, a significant enhancement of the agreement can be discerned considering the

electromechanical parameters as weighting distances in K-means (Fig. 3.19.a) and

Fuzzy (Fig. 3.19.b), as well in the previous case.

• Both above aspects, such as the degradation of accuracy and enhancement through the

electromechanical distance are observed in dynamic equivalencing procedures on the

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 52

European power system independent of the reduction degree of the external area, i.e.

equivalents of 90, 65, and lower number of equivalent machines reduced from 397

external machines outsides of the German area.

Fig. 3.19.a, b.- Comparison of time responses of the STDE1 German machine calculated with 65

equivalent machines by different identity algorithms with electromechanical distances

(weighted K-means, weighted Fuzzy).

The high accuracy through the electromechanical distance was determined for all

machines in the German system.

Case 3 — Electrical and geographical coupled 65 European dynamic equivalents

The derived equivalents calculated by the electromechanically identical groups following

the disturbance at VEGROUSB (VEW) (equivalent disturbance) were valid in the same

manner for disturbances too, whose locations are electrically and geographically in the near

or close to this equivalent disturbance. The following cases can be simulated:

-0,15

-0,1

-0,05

0,05

0,1

0,15

0,2

012345678910

Ori K W K

Time [s]

65 equivalent generators reduced from 397 Gen.

K-means

∆

∆P [p.u.]

Original (PSD) Electromechanical

K-means

Internal German machine STDE1 calculated with:

-0,15

-0,1

-0,05

0,05

0,1

0,15

0,2

012345678910

Ori F We Fu

Time [s]

65 equivalent generators reduced from 397 Gen.

Fuzzy

∆

∆P [p.u.]

Original (PSD) Electromechanical

Fuzz

Internal German machine STDE1 calculated with:

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 53

- In Fig. 3.20, the behavior of the BWKUESS German machine following the disturbance

on the node VEHANNSA (VEW) is represented. This disturbance is electrically and

geographically in close to the disturbance of the equivalents derived.

- In Fig. 3.21, the behavior of the same German machine following the disturbance on the

node EVDELSSAD (EnBW) is represented. This disturbance is electrically and

geographically far away from the disturbance of the equivalents derived.

Fig. 3.20.- Behavior of the BWKUESS German machine using 65 equivalents considering their

electromechanical-based algorithms and following a disturbance at VEHANNSA (VEW).

Fig. 3.21.- Behavior of the BWKUESS German machine using 65 equivalents considering their

electromechanical-based algorithms and following a disturbance at EVDELSSAD (EnBW).

In Fig. 3.20, it can be seen a quite agreement of the time responses in the first 5 sec.

following a degradation in accuracy over 7 sec. Thus, the electromechanical weighted

algorithms show a detectable improvement.

-0,03

-0,02

-0,01

0,01

0,02

0,03

0,04

012345678910

Original E K El F

German machine BWKUESS following the closest disturbance on VEHANNSA

node calculated with:

Time [s]

65 equivalent generators reduced from 397 Gen.

Fuzzy

∆

∆P [p.u.]

Original (PSD) Electr.

Fuzzy

Electr.

K-means

-0,06

-0,01

0,04

0,09

0,14

012345678910

Original Electr. K-means Electr. Fuzzy

German machine BWKUESS following the far away disturbance on

EVDELSSAD node calculated with:

Time [s]

65 equivalent generators reduced from 397 Gen.

∆

∆P [p.u.]

Original

(PSD)

Electr.

Fuzzy

Electr.

K-means

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 54

Fig. 3.20 and Fig. 3.21 show that electromechanical identity recognition-based dynamic

equivalents are valid for other disturbances, whose location have to be geographically and

electrically coupled to the disturbance, for which the equivalent was developed.

In both cases, the oscillation behavior shows a high degree of agreement. Although, the

disturbance on EnBW subsystem (in Fig. 3.21) is far away geographically from the

disturbance on VEW subsystem, a significant accuracy of the derived equivalents can be

obtained. However, considering all German machines, the electromechanically derived

equivalents are also valid for other disturbances that are independent electrically and

geographically of close distance to the disturbance of the equivalents derived. Therefore,

electromechanical-based identity recognition makes it possible that disturbance-independent

dynamic equivalents can be generated.

Quality measurement of the identity recognition approach

In this section, the behavior of all German machines is investigated using the following

proposed algorithms regarding their suitability and accuracy:

- Identity recognition algorithms, such as Fuzzy, K-means, Hierarchical, and SOFM.

- Electromechanical-based algorithms, such as electromechanical Fuzzy, K-means and

Hierarchical.

The measure for evaluating the three methods is defined as follows:

()

EquiDyn

Original

iPiP

,..,1

)()(

1)( 1

=∴

∆−∆

−=

∑

= (3.44)

where

• ∆P(i)Original and ∆P(i)Dyn. Equi. are the time domain behavior of the ith generator in the

German system, which are calculated with the original European external area and

the equivalented one. (∆P(i)Original-∆P(i)Dyn. Equi.)2 is defined as squared distance error.

• Np is the number of sampling points and

• Ns is the total number of generators in the internal area.

The “best” identity recognition algorithm is the one that gives the minimum squared

distance error or maximizes J(i). Taking into consideration all German machines, the

following mean value J may be defined as:

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 55

∑

)(

(3.45)

This J may be called the identity coefficient corresponding to a used algorithm.

Of course, by means of this value the quality, accuracy and grouping capability of a identity

recognition algorithm can be characterized and compared depending on the reduction

degree of 397 external machines from 20 to 180 dynamic equivalents, as it can be seen in

the following figure:

Fig. 3.22.- Comparison of identity recognition algorithms considering electromechanical weighted

distances by the mean value of J of the 67 German intern machines for a fault located at the

boundary node VEGROUSB(VEW) with different number of external equivalents.

K-means Weigh.

K-means Fuzzy Weigh.

Fuzzy SOFM

20 65 90145

180

0,984

0,988

0,992

0,996

1,000

Identity

Coefficient

Identity Recognition Algorithms

Number of

Equivalent

Generators

Best Identity Recognition Algorithms regarding

accuracy and agreement

1.00

0.996

0.992

0.988

0.984

20 65 90 145

180

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 56

As it can be seen in Fig. 3.22, the proposed electromechanical-based identity recognition

with Fuzzy and K-means, i.e. weighted Fuzzy and weighted K-means, gives the best result

and accuracy, as accentuated and highlighted beams illustrated, independent of the number

of equivalents. The detailed values are given in the appendix table F.3.

Remarks to Fig. 3.22:

• Both SOFM and Fuzzy algorithms show a considerable degradation in agreement and

accuracy in contrast to K-means.

• The identity recognition capability of the SOFM is limited due to the slow learning

process and the unsupervised nature of the ANN learning.

• In Fig. 3.22, it is depicted that lower the number of equivalents, the less accurate are the

algorithms, i.e. J for 20 dynamic equivalents are lower J for 180 dynamic equivalents. It

is due to the nature of the clustering procedure.

• The best identity algorithms and consequently the small distance errors are the

electromechanical-based algorithms independent of the number of equivalents.

• This aspect leads to electrically efficient and more accurate dynamic equivalents.

Consequently, in the grouping process is taken into account the electromechanical

properties of the interconnection among the external machines.

It should be noted that the electromechanical-based identity recognition is applicable

without restriction to a power system independent of its structure, size and complexity. This

is fairly demonstrated both on the 16 multi-machine system and the interconnected European

system UCTE/CENTREL obtaining strongly accurate equivalents.

3.6 Summary

• A new approach in dynamic equivalencing of power systems, called electromechanical-

based identity recognition, as alternative to the classical coherency identification is

proposed to obtain identity-based equivalent generators of external large power systems.

• Through this approach, the grouping of generators is considered as an identity analysis

task. Together oscillating machines have to be determined by evaluating the amplitude

and phase identity of the rotor angle behavior of the generators, and not only the phase

of the rotor angle (coherency identification).

• The phase and amplitude of the behavior have to be examined applying standard pattern

recognition algorithms, such as hierarchical, K-means, Fuzzy and SOFM.

ELECTROMECHANICAL-BASED IDENTITY RECOGNITION 57

• The proposed identity recognition can incorporate machine system model parameters in

dynamic equivalencing process.

• Thus, specific physical effects and electromechanical influences of the generators with

regard to their properties, modeling and structure can be considered to cluster the

machines according to the proposed electromechanical distance. These parameters are

defined using the sensitivity analysis. Thus, large generators with large nominal power

and large inertial constants have more electromechanical influence on the system

dynamic than smaller generators.

• Therefore, the obtained electromechanical-based identity recognition forms improved

and high accurate dynamic equivalents.

• This approach is verified both on the 16 multi-machine system and the interconnected

European power system UCTE/CENTRAL without restriction. It is applicable to all forms

of power systems independent of their structure, size, and complexity.

• In small power systems, all electromechanical identity recognition algorithms generate

similar accurate dynamic equivalents with high agreement. But, in the Interconnected

European power system UCTE/CENTREL, in which the German network as internal area

is simulated, the results are significantly accurate.

• Best results with a high degree of accuracy are achieved with help of the

electromechanical distances forming weighted K-means and weighted Fuzzy algorithms.

SOFM are not appropriate due to the ineffectiveness of its learning process and neural

network topology for complex systems.

• Further, it has been determined, that grouping process is partially independent of the

fault location. However, more accurate dynamic equivalents can be achieved by

disturbances, which are electrically and geographically strong connected to the one.

• But, the derived equivalents calculated by the electromechanical identity recognition

following a certain disturbance are valid in the same manner for other disturbances too,

whose locations are electrically and geographically in close to the equivalent disturbance

or far away from the one.

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 58

“He was under the impression that a system description (by the system

decomposition) is exactly the same as a system in a concrete physical

sense.”-R. Kalman [138]-

Chapter 4

Splitting Aggregation-based

Dynamic Equivalencing

Objective— The objective of this chapter is to develop an innovative aggregation approach

to construct representative machines with improved accuracy and to consider essentially

characteristics of the generators regarding their electromechanical parameters.

This chapter mainly describes the concept and strategy of an aggregation approach on the

basis of the fictive splitting of external machines. This splitting strategy is used as a basis for

implementing mathematical reduction techniques in dynamic equivalencing and thus

obtaining a reduced electromechanical system of the area being equivalenced, which can be

applied in transient stability studies.

The splitting factors may be derived by mathematical reduction techniques, such as the

Fuzzy theory or principal component analysis. The relevant property of this approach

consists of incorporating of new defined electromechanical splitting-based machine

parameters that it significantly improves the accuracy and efficiency of dynamic equivalents

and thereby enhances their effectiveness and application.

Simulations of this approach have been performed and evaluated in an interconnected 16

multi-machine power system.

Index Terms— Aggregation, Coherency Identification, Dynamic Equivalent,

Electromechanical Parameters, Identity Recognition, Splitting-based Parameters.

Organization— Section 4.1 describes the introduction and section 4.2 of this chapter the

classical methodologies concerning generator aggregation. The proposed splitting-based

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 59

aggregation approach is discussed in section 4.3, followed in section 4.4 by the application.

In section 4.5 the simulation results are evaluated and in section 4.6 the summary is given.

4.1 Introduction

Generally, a complex power system can be divided into two areas, the internal area that

has to be retained for analysis, and the external area that is to be reduced to a simplified

aggregated model of the external machines. The conventional aggregation in dynamic

equivalencing generates aggregated equivalents on the external area on the basis of inertial

and slow aggregation [30-32], which can be replaced by an innovative approach, i.e. the

splitting-based aggregation. Using this concept, the classification and grouping of together

oscillating machines are replaced by a virtual splitting of machines.

The developed approach is based upon the splitting strategy to process the swing

oscillating curves in time domain of the external machines. This strategy generates splitting-

based electromechanical parameters considering the physical characteristics of the whole

number of external machines.

4.2 Conventional aggregation in dynamic equivalencing

The classical dynamic equivalencing involves mainly a three-stage procedure of:

- Identification of coherent generators in the external area forming groups [17, 18].

- Conventional Aggregation of the grouped external generators without changing the

power flow relationships where these generators show coherent [17] properties.

- Static network reduction

A coherent group of generating units is defined as a group of generators oscillating with the

same rotor angular speed. The representative machine parameters of this group can be

calculated according to the classical aggregation approaches, such as the inertial and slow

aggregation, which are zero and first order approximations of singularly perturbed two-time-

scale power system models, respectively [30, 31, and 35].

The inertial aggregation does not involve linearization aspects, because its procedure

operates only on the generator terminal buses of the grouped generators. However, these

approaches require the determination and classification of coherent machines, which is not

necessary according to the proposed splitting based aggregation. It represents a significant

change of the classical dynamic equivalencing procedure because non-linear processes and

behaviors of the power system can be taken into account.

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 60

Disadvantages:

Considering the grouping-based simplified models, which are required by the classical

aggregation, the following disadvantages may be determined:

• The inertial and slow aggregation are performed only and limitedly on a per coherent

area of external subsystem.

• The aggregation of the network in one coherent or identical area can affect electrically

the network aggregation in other coherent areas. Consequently, the error in the

aggregation approximation may result in a decrease of accuracy.

• On the coherent-based models of power systems, the aggregated area that includes

generator and non-generator buses (PV and PQ nodes) is aggregated to the machine

terminal node and non-generator buses are not considered in the aggregation procedure.

As a result, the accuracy of aggregated network is reduced.

• The aggregated groups are independent upon the detailed machine model, i.e. the

physical parameters of the machines. Moreover, although many improvements in the

classical aggregation strategies [30] and coherency [19] were made, the related non-

linear behavior cannot be considered accurate enough for forming equivalent

parameters.

• Due to the grouping procedure, the corresponding equivalents are not exact enough,

since the whole aggregation structure of the external area can be lost.

4.2.1 Inertial Aggregation

Modeling of synchronous machines

There are various types of models of synchronous machines for power system transient

stability studies. In aggregation simulation using PSD [87, 141] (for details see section 2.3),

the machine model can be coupled into the analysis program PSD according to the flow

chart in appendix A.8. These machines are expressed by the direct- and quadrature axis

having one damper circuit both in d- and q-axis, which can be transformed from the L1, L2

and L3 stator to dq0 system, as it can be graphically represented as follows:

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 61

Fig. 4.1.- Synchronous machine models: L1-L2-L3 transformed in dq-system

In this model following aspects are important:

• The quasi-state synchronous machine model is based upon the voltage:

00000

0000ω

000ω0

r0000

0r000

00r00

000r0

0000r



































−





































(4.1a)

flux linkage:

xx00x0

0xxxxx0x

x00xx0

0xx0xx

σDqhqhq

σfDdσDdhdσfDdhdhd

σfDdhdσfDdσfdhdhd

hqσShq

hdhdσShd

























+++

−=













(4.1b)

and mechanical equations:

)( oL

ωω

−= (4.1c)

[]

mdqqd

Lmii

Tdt

d+−= )(

ψψ

(4.1d)

• All state variables will be transformed into the rotor coordinate system with the aim that

the stator flux linkage is not a state variable and thus, it follows changes of the stator

voltage and rotor flux linkage. Hence, the stator flux linkage in the rest state space

equations for rotor flux linkage and mechanical behavior may be eliminated [87]. For

transient stability analysis a quasi-state model is sufficient.

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 62

• Thus, 0=

D and 0=

D lead to the following simplified state space form of

synchronous machines:













+−

























−

−−−

−=













−

−−

mqd

qDqDq

dfdDd

fddfdfd

dDq

q Dd

qfd

fDd

Dfd

mi i )x(x

i k r

ui k r

01000

i k

00T00

000TT

(4.2)

where the time constants are:

)( fDd Dd hdfd

fd xxx r

Det

σσ

= )( fDd hdfd

fDd xx r

Det

−

Dq hd

Dq r

T)(

)( fDd hdDd

Dfd xx r

Det

−

= )( fDd fd hdDd

Dd xxx r

Det

σσ

= (4.3)

Hereby are:

Dd fd Dd fd fDd hd x xxxxx Det

σσσσσ

+++= ))((













fdDd

khdfD

= De

khdDd

hqDq

fd xx

(4.4)

where

• xhd, xhq, xσDq, xσDd, xσfd, xσfDd, x’’d, x’’q are the synchronous reactance and

subtransient synchronous reactance, respectively.

• ud, uq, ufd are the stator voltage in q and d-axes and the exciter voltage

respectively.

• uS, iS are the stator (terminal) voltage and current, respectively.

• mm is the turbine mechanical torque.

• id, iq, are the stator currents in q and d-axes, respectively.

• Tm is the shaft inertia constant.

• ωo is the network frequency referenced to p.u.

• ψL, ψS are the rotor and stator flux linkage, respectively.

• ωL, δL are the rotor angular velocity and rotor angle, respectively.

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 63

• The state space equation of the machine may be coupled to the power system through

the following complex algebraic equation and according to the appendix A.8:

sui )x j(r u ++= (4.5)

where the driving voltage ''

u is a function of the state variables flux linkages, rotor

position angle and partly of the stator current, as follows:

LL jδ

jδ

'''' e)ixj(xeuj uuu u −++=∆+= )( (4.6)

()

{}

jδ

qDdDdfdfdDqDq0

'' e )ixj(x)ψkψj(kψk-ω u −+++= (4.7)

• By means of L

, a transformation from the rotor to the network coordinate system is

realized. The quasi-state space equation (4.2) is solved by numerical integration in PSD,

when id, iq are known [87]. This model is used within classical aggregation [15, 30-33].

Following aspects are relevant to the machine equivalent:

• This model of the classical aggregation creates a new terminal bus and connects it to the

internal busses of each individual generator via pseudo transformer. The transformation

ratios are chosen so that the driving voltages of generators will be transformed to the

aggregated driving voltage of the equivalent generator to a uniform one, according to the

relationships (3.6) and (3.7).

• The circuit must be completed by a branch with the negative transient reactance of the

equivalent generator to get the new terminal bus according to Fig. 3.2. The transient

reactances combined with the transformers, allow the elimination of the internal nodes of

original generators.

Aspects of the machine parameters are summarized as follows:

• The machine parameters are calculated from the weighted mean values of generators

inductances and resistances as reciprocal values in p.u., as follows:

∑

=NG

∑

=NG

x1 (4.8)

where

• xi represents resistances, main-field and linkage inductance of all circuits.

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 64

• NG is the number of coherent or identical machines in the corresponding group.

• Equation (4.8) means that the parameters will be connected parallel and it is one of the

possible solutions. Another way would be the calculation of aggregated parameters for

the transient, subtransient reactance, time constants system parameters directly.

• The sum of active and reactive power at the internal nodes behind transient reactances

must be supplied by equivalent generator, too. The resulting nominal power of the

equivalent machine is the sum of all generator powers given by:

∑

rirE SS (4.9)

• The nominal voltage of the machine is not a relevant variable, but it has to be considered

in the tape ratio of pseudo transformers.

• The resulting inertial constant of the equivalent machine is given by:

∑

mTT (4.10)

• When all parameters are given, the driving voltage of the equivalent behind of the

resulting transient reactance can be calculated.

• After then the equalizer transformers are included into the network. These transformers

don’t have inner impedance. Therefore the individual generator reactance can be

interpreted as transformer impedance, so that the transformers are placed between the

individual terminal buses and the joined aggregated point.

The advantage is its simplicity. In dynamic behavior, it shows a sufficient accuracy.

4.2.2 Slow coherency aggregation

The classical slow coherency aggregation [15, 30-33] is based upon an impedance

modification to the inertial aggregation.

This slow aggregation starts with a linearization at the generator terminal buses. Then, the

fast inter-machine variables, defined by the singular perturbations theory, are eliminated, and

a power network is reconstructed from the reduced linearized model.

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 65

A schematic illustration of the slow coherency aggregation is shown in the following figure.

(i) (ii) (iii)

Fig.4.2.- Slow coherency aggregation

Linearized model of the slow and fast subsystem

The slow coherency aggregation is characterized by the linearizing about the generator

buses and internal nodes. These linearized swing equations for the generators at the

operating power flow equilibrium:

UU EE ioiioiioiioi

==== ,,, bai ,= (4.11)

are

ioioioio

ioioio

ioioioio

∆

−

+∆

−

−∆

−

−=∆ '''

.. )cos()sin()cos(

2 (4.12)

θδ

∆+∆+∆−=∆ ''' bai ,= (4.13)

where

•

∆δ

a and

∆δ

b are the incremental variables for

a and

•

∆Ι

a and

∆Ι

b the incremental for

a and

• Ua, Ub are the bus voltage magnitude to bus a and b.

•

a and

b are the bus voltage angle a and b.

• Ea, Eb are the individual generator internal voltage of generator a and b.

• Mi is the inertia constant of the ith generator.

jx"db

jx"

jx"−

jx"db

jx"

jx"db

jx"

()

a b

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 66

Defining the state variable vector x, the algebraic variable vector z and the current injection

variable

∆ as phasor form with:













∆













∆

z 













∆

I∆ (4.14)

the matrix form of the state system can be obtained as:

zxx 21 KK +=

• (4.15a)

zxI∆43 KK += (4.15b)

Another important step in slow aggregation is the transforming to slow and fast variables. In

this case, when the machines form a slow coherent group, their center of angle is considered

as the slow variable and the inter-machine oscillations as the fast variables.

In order to perform the slow aggregation, the original machine angles have to be

transformed to these new slow and fast variables. These transformation and aggregation can

be realized on a per coherent area, sequentially in any order of the coherent areas.

The slow aggregate variable δs and the fast local variable δf are defined as:

sTT

(4.16)

abf

δδδ

−= (4.17)

Applying the transformation (4.16) and (4.17) to the linearized model (4.15), in matrix form

the linearized system in two time-scales can be obtained:

s























∆

























∆

••

1413

1211

(4.18)

[]

zI∆ K KK

43231 +













∆

(4.19)

Assuming that the fast dynamics in

∆δ

f have decayed, i.e. 0=∆ ••

, the quasi-steady of

∆δ

f is:

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 67

()

z K K K sf 2213

14 +∆−=∆ −

δδ

(4.20)

Eliminating

∆δ

f from other variables, the following expressions can be obtained:

sss

sK K 21 +∆=∆ ••

δδ

(4.21)

zI∆sss K K 43 +∆=

(4.22)

Hereby

1412111 KKK KK s

−

−= 22

1412212 KKK KK s

−

−= (4.23)

1432313 KKK KK s

−

−= 22

143244 KKK KK s

−

−= (4.24)

System (4.21) represents the linearized model of the slow subsystem. A power network must

be reconstructed whose linearization would yield (4.21).

The machine equivalent parameters are defined as follows:

• The terms K1s, K2s and K3s are needed to construct lines connecting bus ‘p’ to the original

generator terminal buses ‘a’ and ‘b’, and the term K4s is needed for the lines

interconnecting buses ‘a’ and ‘b’ according to Fig. 4.2 ii.

• In addition, the reconstruction from K4s will not satisfy the network flow condition.

Therefore, after the line reconstruction for buses ‘a’ and ‘b’ is completed, the balance of

the power flow by adding loads to these buses has to be realized.

• The creation of bus ‘q’ and adjusting generation on buses ‘a’, ‘b’ and ‘q’ are similar to the

procedure on inertial aggregation, as can be seen in Fig. 4.2 iii.

This analytical aggregation approach is suitable to be used, when linearized modes only are

considered.

4.2.3 Power invariance aggregation

This method is based on the concept of the power invariance [35] at the generator internal

buses and at the terminal buses, in which the generators of a coherent group are connected.

Since a generator is represented by a voltage source behind transient reactance in series

with its transient reactance and connected to the terminal bus, a fictitious point can be

assumed in between the internal voltage source and the transient reactance (see Fig. 4.3).

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 68

Following aspects to the machine equivalent are important:

• For a group containing NG generators (see Fig. 4.3 i), fictitious points are connected

together such that it connects all the internal voltage sources of that group and one end

of the transient reactances in a common point, as it can be shown in Fig. 4.3 ii.

• The paralleled internal voltage sources of the group are now replaced by an equivalent

voltage source considering in terms of the generator current injection into corresponding

buses, which may be observed in Fig. 4.3 iii:

∗













∑

1 NGai ,.,= (4.25)

• It will be capable of supplying the active and reactive power equal to the sum of the

active and reactive powers delivered by all the generators of the group.

• This changing structure can be showed schematically as follows:

(i) (ii) (iii)

Fig. 4.3.- Power invariance aggregation

Aspects to the machine parameters are summarized as follows:

• The transient reactance of the individual machine of coherent group is modified to retain

the original power division from the equivalent voltage source to the points of the

connection in the network as:

dNG

jx'

dEa

jx'dEi

jx'dENG

jx'

jx'di

jx'

dNG

jx'

...

()

()()

NG ..,ai , EE ii

i,et t∆δδj0== +

()

()()

t∆δδjE0E

et +

EEE

Generators. a,.. ; m Fictitious structure A

ated model

NGi

... ... ...

jx'

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 69

dEi X

X '' 













−

= NGai ,.,= (4.26)

where, X’dEi is the transient reactance connecting the equivalent generator to the ith bus.

• After modifying the reactances of all the coherent generators, all the load buses and the

generator terminal buses are eliminated retaining only the generator internal buses.

• The loads are converted to constant admittance to ground.

• Thus, the reduced order system admittance matrix obtained from the bus admittance is

used in the dynamic equation for transient stability analysis.

• The inertia constant, damping coefficient, electrical and mechanical power of the

equivalent machine are obtained respectively as:

∑

iE MM

; ∑

iE DD

; ∑

mimE PP

(4.27)

• This method is suitable in case of equivalent generator representation and in network

reduction retaining the terminal bus of each of the coherent generators. Hence, it

preserves the basic structure of the original system.

4.2.4 Berg and Ghafurian’s aggregation

In this method an equivalent generator can replace the coherent group of generators and the

following mathematical steps modify the network [36, 37]:











+











+= ∗

−

∗

−

NGNG

dNG

EEE

ααα

' (4.28)

−

∗

−











+=

αα

dNG

dNGdE X

X X (4.29)

where NGth and (NG-1)th generators are coherent,

NG =

−1

α is a complex constant. (4.30)

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 70

A coherent group of NG generators can be reduced to an equivalent generator by a

sequential procedure of elimination of one generator at a time. The inertia, damping and

mechanical power of the equivalent are obtained in terms of the power invariance equations.

4.3 Splitting-based aggregation approach

According to the classical aggregation, external generators [17, 18] can be replaced by

means of an equivalent generator using the concept of ‘direct assignment’ (see Fig. 4.4.a).

The proposed approach presents an innovative concept based on the splitting of the

complete number of external generators into fictive shares in terms of virtual generators.

The impact of power generation of the generators on the dynamic behavior of power

systems is splitted into significant parts, which represent the influences of the original

generators on the dynamic equivalents according to a ‘share assignment’ or effect degree,

as shown in Fig. 4.4.b. In this context, following aspects are important:

• Completely, all external generators will be considered in the calculation of the

aggregated electrical splitting based-parameters shown in Fig. 4.4.b.

• Instead of the classical coherency identification and grouping, by means of this share

assignment, new methods of consistent dynamic equivalencing can be explored

satisfactory according to mathematical approaches and reduction techniques.

• Of course, this splitting based-aggregation provides a new viewpoint and theoretical

founding in dynamic equivalencing.

This procedure implies the reformulation of the classical aggregation criterion.

4.3.1 Conditions

Fig. 4.4 illustrates schematically important aspects and the differences between the

classical aggregation and the proposed splitting technique:

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 71

Fig. 4.4.a.- Classical aggregation

Fig. 4.4.b.- Proposed splitting based-aggregation

Remarks to Fig. 4.4:

• Fig. 4.4.a shows schematically the necessary condition for forming classical aggregated

dynamic equivalents. This consists of forming of similar generator grouping of the

external area to generate equivalents in terms of a direct assignment.

• As can be seen in Fig. 4.4.b, all external generators have a participation assignment to

form equivalent aggregated generators. The participation assignment is evaluated by

means of a splitting power factor.

This novel splitting method is simulated in the following two-machine power system, as given

in Fig. 4.5. Here an external machine in Fig 4.5 i, can be divided fictively considering the

participation assignment to form virtual generators, as shown in Fig.4.5 ii. On the basis of

these virtual generators, the equivalents can be calculated, as can be seen in Fig.4.5 iii:

CLASSICAL AGGREGATION

Coherent or

Identical external

generators

Classical equivalent

generators

~ ~ ~

Direct Assignment

SPLITTING-BASED AGGREGATION

External generators

divided into shares

Splitting-based

equivalent generators

~ ~ ~

E2 E1

Share Assignment

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 72

(i) (ii) (iii)

Fig. 4.5.- Machine-splitting with reference to their nominal power according to the derived splitting

factors a1 and a2.

Following aspects can be detected on basis of Fig. 4.5:

• The generator ‘a’ is splitted into 70% and 30% of its rated power and generator ‘b’ into

80% and 20%. Under following conditions, the splitting can be defined.

• A necessary condition is the invariance of the dynamic behavior: i.e. after a disturbance,

the dynamic generator output in node N1 must be similar to the behavior in node N2

considering the splitting factors.

• After the splitting of the external generators, the corresponding shares, for example parts

1 of generators ‘a’ and ‘b’, have to be aggregated to generate the equivalent generator

‘E1’ and the 2 shares of ‘a’ and ‘b’ to ‘E2’ as well.

• Another essential condition is that the dynamic behavior at the original node N3 must be

relatively equal to the behavior at aggregated node N4.

• Considering the above aspects, the splitting-based electromechanical parameters of the

equivalent generator can be determined.

4.3.2 Aggregated electrical parameters

These generator parameters can be derived on the basis of the proposed splitting

approach that the aggregated equivalent should have the same behavior of all the external

generators in the whole time period. This aggregation is more appropriate to be performed at

the internal nodes and not at the generator terminal buses. The proposed splitting approach

is shown schematically in Fig.4.6 as follows:

N3 N4

Generators

aandb

Splitting of generators Splitting-based dynamic

equivalents E1 and E2

SE2=2360 MV

SE1=1840 MV

Sb1=440 MV

Sb2=1760 MV

Sa1=1400 MV

Sa2=600 MV

Sb=2200 MV

Sa=2000 MV

a1 a2 b1 b2 E1

~ E2

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 73

(i) (ii) (ii)

Fig. 4.6.- Splitting-based aggregation

The machine equivalent can be generated as follows:

• From the splitted bus injection jiji, jQP ,

+ (i=number of external machines, j=number of

splitting factors or equivalents) and the generator current injection, the corresponding

internal voltage ji,

E of the splitted generators can be computed, as shown in Fig. 4.6 i.

• The internal nodes of the individual non-coherent external virtual generators in Fig. 4.5 ii,

i.e. generators ‘a1’ and ‘b1’ are connected to a common bus with appropriate pseudo

transformers and phase shifters to preserve the power flow relationships (see Fig. 4.6 ii).

• Defining the ratio of the pseudo transformers from the internal voltages a1

E, b1

E ( ji,

of the virtual splitted generators and the common bus voltage E1

E (see Fig. 4.6 iii). E1

is defined as the aggregated driving voltage of the equivalent generator ‘E1’. The circuit

must be implemented with the negative subtransient reactance of the equivalent

generator to create a single generator internal node and new terminal bus q

• The pseudo transformers are needed for transforming the different driving voltages of the

virtual generators to an uniform one.

• The subtransient reactances combined with the transformers, allow the elimination of the

internal nodes of the original fictive generator in terms of the static network reduction.

• The nominal voltage of the internal node and of the connected equivalent generator must

be defined properly. However, it can affect the transformer ratio.

• Equivalent generators must supply the sum of the splitted active and reactive power.

Non-coherent generators a

and b splitted into the first

egated equivalent model of

enerators a and b

() ()

))((

tt tj

bob

eEE

δδ

∆+

() ()

))((

tt tj

aoa

eEE

δδ

∆+

() ()

))((

tt tj

EoE

eEE

δδ

∆+

b1 a1

"da

jx 1

"db

jx−

"da

jx 1

"db

"da

jx 1

"db

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 74

• The subtransient reactance d

jx '' is used instead of the transient reactance d

jx' because

it is the small possible reactance that the machine can take within a disturbance.

Taking into consideration the splitting factors in this procedure, the following matrix can be

represented. It can be used to calculate the machine parameters.













M,NM,2M

2,N2,2

1,N1,2

aaa

aa a

aaa

...

1,2

1,1

A (4.31)

Machine parameters can be determined as follows:

• The voltage of the common bus ‘p’ is calculated using a splitting-based nominal power as

weighted average of the individual virtual generator internal voltage.

• The common driving voltage can be calculated using the splitting internal voltages:

∑

jij

EaS

,)0(

)0( i=1…M (4.32)

and the nominal power, active and reactive power by

∑

jijE aSS i, ∑

jijE aPP i, ∑

jijE aQQ i, i=1…M (4.33)

where

• (0)E j is initial driving voltage of external fictive generator j in p.u.,

• j

S is the rated unit power and

• ai,j the participation share or splitting factor of external generator j according to the

number of generated equivalents i.

• Pseudo transformation ratios from the internal voltages of the virtual generators to the

common bus voltage i

E are defined by:

Number of dynamic

equivalents i=1…M

Number of external generators j=1...N

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 75

()

)()(

)(

)( ttj

ii e

etE

δδ

∆−∆

== i=1..M; j=1..N (4.34)

• Though there are pseudo transformers, pseudo transmission lines and transient

reactances, the power flow relationships result of Fig.4.6 iii remains the same as which in

Fig.4.6 i.

• Considering the splitting factors of (4.31), the parameters of the equivalent generator can

be calculated from the mean values of the splitted generator inductances and

resistances as reciprocal values:

∑

jij

∑

jij

i=1..M (4.35)

Where the relationship in (4.35) means that the splitting-based parameters are

connected in parallel. The time constants can be calculated based on these parameters.

• Thus, the resulting inertial constant of the equivalent generator is given by:

∑

jij

aST

, i=1..M (4.36)

• In order to create an internal node ‘E’ for the equivalent generator, bus ‘p’ should be

extended to an additional bus ‘q’ with the impedance i

jx ''

−, and then to the bus ‘E’

with impedance i

jx '' . Therefore, the bus ‘E’ has the same voltage as bus ‘p’ according

to Fig. 4.6 iii.

• Bus ‘q’ can be considered as the terminal bus and bus ‘E’ as the internal bus of the

equivalent.

• Consequently, the voltage at bus ‘q’ is set by (t)E i

p and it is connected to the individual

buses a and b. Therefore, the voltage of the equivalent can be calculated starting from

the internal voltage, which is given in (4.32) as:

))((

)()( tj

ii e tE tE

δδ

∆+

= i=1…M (4.37)

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 76

• Because the individual buses a and b are no longer generator terminal buses and the

system size is already reduced, it is possible to eliminate these buses, nodes, and

transmission lines by the classical static network reduction. It is suitable to couple this

step with the factorization of the admittance matrix.

4.3.3 Splitting factors of generators

The aggregated parameters in equations (4.32-4.36) imply that non-linear reduction

techniques may be applied to generate accurate and representative equivalent generators.

This aspect can be realized by mathematical approaches, such as Fuzzy theory or the

principal component analysis.

i. Fuzzy clustering

The splitting or virtual division of machines using the splitting factors can be described by

means of their Fuzzy membership degrees. Hereby, the similarity of one generator to the

other generators can be expressed quantitatively by partitioning the generator using the

membership degree.

Remarks:

• This algorithm extends the identity analysis to an optimization problem [102-105]. With

Fuzzy clustering, each generator belongs to all classification cluster groups

simultaneously. However, it has different degrees according to its identity with other

generators.

• Fuzzy logic is a generalization of yes and-no Boolean logic. Assigning 0 to false values

and 1 to true ones. Fuzzy logic also allows in-between values. Assuming that

is asset

of values of member degrees, Fuzzy logic defines a mapping from

to the unit interval

trough a membership function.

• This Fuzzy similarity should be understood as mathematical similarity, measured in some

well-defined sense, for example by using a distance norm.

• The Fuzzy clustering is based on the within groups sum of squared errors objective

function Jm,. The set of solutions that satisfy the minimum Jm is simplified by the

weighting factor

ij, as can be seen in the following equation:

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 77

mincxdCUXJ

m→−== ∑∑∑∑

==== 1

),,(

µµ

(4.38)

where

• N corresponds to the number of external generators,

• M the number of cluster groups representing the dynamic equivalents and

• di,j is the distance between the machine and the reference generators.

•

i,j denotes the membership degree of generator xj to cluster group ci.

• m>1 is the fuzziness index and influences the “fuzziness” of the obtained partition.

• This Fuzzy clustering is subjected by the following constraints [105]:

0:1

, Mi

ji >≤<∀ ∑

(4.39)

1:1

,=≤<∀ ∑

(4.40)

• Because of the optimal nature of the problem (4.38), methods of calculus of variations

are used to derive the necessary conditions, such as the membership degree condition:

)1(

,)(

)(

−













== ∑

∑

ji d

i=1..M; j=1..N (4.41)

• For the initial values of the membership degree can be derived the following formulation:

∑

−

(4.42)

• According with the solution for local extreme, the cluster centers as reference generators

can be reached as other important condition:

∑

Mi ≤≤1 (4.43)

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 78

• Conditions (4.41) and (4.43) are first-order necessary conditions for local extreme of Jm.

All procedures used to solve (4.38) should satisfy both (4.41) and (4.43). These

conditions are derived in appendix B.3.

The relevant result of Fuzzy clustering are the membership coefficients expressing the

identity degree that a machine is similar to other generators with identical properties

belonging to a cluster, i.e. the membership factor expresses the weakness or strength of the

assignment of the generators to all groups with identical properties in time domain. This

aspect will be applied to the virtual splitting of generators, which belong to all clusters

simultaneously but with different ‘weakness or strength degree’.

ii. Principal component analysis

The principal component analysis, known as eigenvalue analysis, is a mathematical way of

determining that linear transformation of a sample of time behaviors in N dimensional space

along the coordinate axes, whose sample variances are extremes and uncorrelated.

According to the proposed generator aggregation, no significant and redundant generator

behaviors can be neglected. The property of transforming the coordinates along the principal

axes into physical meaningful parameters can be reached by this approach.

The PCA is applied to the time behavior matrix of external generators by projection onto a

smaller number of orthonormal axes. This leads to a coordinate system with the axes of

largest spread. The property of the PCA is that the original features of the generators

described by the time behavior will be transformed into new meaningful ones. These

significant generator features represent the whole original system, which is reduced

neglecting its redundancy.

Remarks:

• According to the machine matrix (3.28), the covariance matrix of the same data set is 14:

}))({( T

xXXE −−= XXC (4.44)

14 The mean of that population is denoted by }{XEX =

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 79

• The covariance matrix of Cx, denoted by cij, represents the covariances between

components xi and xj. These generator behaviors should span the same subspace as the

original vectors of original generators; however they are now characterized by a set of

eigenvalues and eigenvectors.

• Consequently, from the symmetric matrix such as the covariance matrix, an orthogonal

basis by findings its eigenvalues i

and eigenvector i

e can be calculated by:

ii eeC ix

= i=1…N (4.45)

• These values can be found by finding the solutions of the characteristic equation

0=− IC

x (4.46)

• If T is a NxN matrix including the eigenvectors of the covariance matrix C, the diagonal

variance matrix 2

Σ is given by:

TCTΣT⋅⋅=

2 (4.47)

• The eigenvalues i

of the covariance matrix C are equal to the elements of the variance

matrix 2

Σ, which includes the variances i

• By ordering the eigenvectors in the order of descending eigenvalues N

λλλ

≥≥≥ ...

21 ,

one can create an orthogonal basis with the first eigenvector having the direction of

largest variance of the data set of generators. In this way, directions can be found, in

which the data set has the most significant amounts of energy.

• The time behavior of generators represented as multi space matrix could be well reduced

by approximation with a reduced dimensional representation concentrated along

particular and significant eigenvectors.

• In comparison to T as N dimensional matrix, Tq is a Nxq matrix including q significant

eigenvectors of C corresponding to the q largest eigenvalues of C. The value of q

determines the size of the new dimension and is smaller than N.

• Let Tq be a matrix consisting of eigenvectors of the covariance matrix as the row vectors.

By transforming a data vector X, the orthogonal space representation as reduced

generator time behavior can be obtained as:

)( XXTY −= qq (4.48)

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 80

which is a reduced generator data matrix in the orthogonal coordinate system defined

by the eigenvectors 15.

Splitting of the oscillating curves of the external generators in orthogonal part oscillating

can be obtained according to the eigenvectors as splitting factors, which provide the share of

the principal components in the complete oscillating.

4.4 Case study

The accuracy of the proposed splitting-based dynamic equivalencing are evaluated in a 16

multi-machine system. This power system is described in chapter section 3.5.1. Thus, the

order of the dynamic model of the test system is relatively high. The following illustration

shows the topology of the 16 multi-machine system.

Fig. 4.7.- Interconnected 16 Multi-machine System.

15 Components of Yq can be seen as coordinates in the orthogonal base and vectors of the reduced behavior matrix of

the external generators.

INTERNAL

REA

Disturbance

EA A

AREAB

380 kV

220 kV

110 kV

15.75 kV

1a A1b A2a A2b

B2a B2b

B10

C10

C12

C14

EXTERNAL

REA

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 81

This power system was divided into two areas, where the area C as internal area, will be

preserved in the original form. This area consists of 5 machines. The rest 11 machines

located outside this area, as external area, will be replaced by dynamic equivalents. For the

simulation following aspects are relevant:

• Following disturbances, the dynamic performance of the internal area will be simulated in

two steps: First, the original whole external area is simulated. Then, based on the

aggregated equivalent external area, new simulation will also be preceded.

• This external area will be aggregated based on both the splitting based aggregation and

classic aggregation approach. Thus, the comparison between these two approaches can

be realized.

4.5 Simulation results and discussion

After simulating the whole power system and aggregating the external area with the

abovementioned configuration, the dynamic performance of the internal area is studied.

Firstly, a power flow calculation is carried out to define the initial operating condition of the

power system.

The accuracy of the splitting-aggregated equivalents can be evaluated by comparing the

oscillating swing curves of the internal area machines with the original power system. The

simulation is realized using PSD. For more details on the validity of PSD see section 2.3.

A representative scenario was simulated, where the fault is located at the node 5 of internal

area C with duration of 100 ms as shown in Fig. 4.7. This disturbance begins at 1.0 second

and it has great impact on the whole system. It is simulated for 10 seconds.

In Fig. 4.8 and Fig. 4.9, the time domain behavior of an internal machine in area C is

illustrated on the abovementioned conditions. In Fig. 4.8, 3 dynamic equivalent generators of

the external area are employed using the classical inertial aggregation and the enhanced

electromechanical identity recognition with K-means as grouping technique. The oscillating

swing curves of the internal machines are then compared with the original power system

behavior.

It can be seen in the following figure, that the phase and amplitude for all generators in the

internal area using the classical inertial aggregation are relatively accurate.

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 82

Fig. 4.8.- Comparison of time responses of an internal machine calculated with the original external

system and with 3 equivalent machines using the classical aggregation.

In comparison to the behavior in Fig. 4.8, the time domain behavior of the same internal

machine using the proposed splitting aggregation approach in the external area is presented

in the following figure:

Fig. 4.9.- Comparison of time responses of an internal machine calculated with the original external

system and with 3 equivalent machines using the splitting based-aggregation.

-0,2

-0,1

0,1

0,2

0,3

0,4

0,5

012345678910

ORIGINAL CLASSICAL Aggregation

3 Equivalent generators reduced of 11

3 Equivalent generators reduced from 11 Gen.

External system aggregated by

classical Aggregation

Original unreduced full

external system

(PSD)

∆

∆∆

∆P [p.u.]

Time [s]

Internal machine behavior calculated with:

-0,2

-0,1

0,1

0,2

0,3

0,4

0,5

012345678910

ORIGINAL SPLITTING Aggregation

3 Equivalent generators reduced from 11 Gen.

Original unreduced

full external system

(PSD)

∆

∆∆

∆P [p.u.]

Time [s]

External system aggregated by splitting-

based Aggregation

Internal machine behavior calculated with:

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 83

The oscillating swing curves of an internal machine are calculated with 3 dynamic equivalent

machines, whose aggregation is based upon the splitting approach with Fuzzy membership

degrees according to expression (4.41) as splitting factors. Following aspects in Fig. 4.9 can

be detected:

• Hereby, the performance of the internal machines is accurate during the whole

simulation, i.e. before and after the disturbance. These simulation results are much

better than the results calculated by the classical inertial aggregation (see Fig. 4.8).

Although only the simulation result of one internal machine is presented, the results of

other internal machines in same manner are also accurate.

• Moreover, the behavior of all internal machines shows a notable accuracy and

agreement with different number of dynamic equivalents aggregated using the

splitting-based aggregation.

The time domain simulation results during the first 3 seconds of Fig. 4.8 and Fig. 4.9 are

given in detail in Fig. 4.10.

Fig. 4.10.- Comparison of time domain behavior of an internal machine calculated with the original

external system and with 3 equivalent machines using the splitting based-aggregation.

In Fig. 4.10, it can be detected an enhanced agreement of the power oscillating curve

applying the splitting aggregation both within the damping zone and during the disturbance.

-0,2

-0,1

0,1

0,2

0,3

0,4

0,5

1 2

ORIGINAL

CLASSICAL Aggregation

SPLITTING Aggregation

t/s

3 Equivalent generators reduced from 11 Gen.

∆

∆∆

∆P [p.u.]

Time [s]

Classical Aggregation

Original (PSD)

Splitting Aggregation

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 84

The splitting equivalents are valid for disturbances too, which are electrically and

geographically independent of the equivalent disturbance and too of the extent of severity

(fault duration) and of the load generation balance prior the occurrence of the faults.

This equivalencing robustness can be demonstrated in the following figure, where the

behavior of any internal machine using the splitting aggregation following a sequence of

faults (The first fault after 1 sec. and the second fault after 2 sec. with a duration of 150 ms.

and 200 ms., respectively) is simulated.

These faults are applied to node 3 (380kv area) and node 9 (220kv area) in area C,

respectively, i.e., these disturbances are electrically and geographically closest and far away

from the equivalent disturbance, with which the equivalents are derived.

Fig. 4.11.- Behavior of any internal machine following a sequence of disturbances applied on

different nodes electrically and geographically distinct from the equivalent disturbance. It is

simulated with the original external area and with 3 splitting equivalents with Fuzzy factors.

Hereby a quite agreement of the internal machine responses over the whole time period

can be detected. This notable accuracy in robustness can be determined for all generators in

the internal area C using the splitting-based aggregation.

In contrast to this aggregation the classical inertial aggregation with coherency grouping is

not enable to obtain this robustness and approximation capability.

Quality measurement by standardized sum distance error

In this section, the behavior of all internal machines is investigated using the following three

methods regarding their suitability and accuracy:

-0,2

-0,1

0,1

0,2

0,3

0,4

012345678910

Original

SPLITTING Aggregation

3 Equivalent generators reduced from 11 Gen.

∆

∆∆

∆P [p.u.]

Time [s]

Original (PSD)

Splitting Aggregation

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 85

- Splitting based-aggregation method with the Fuzzy membership degrees as splitting

factors

- The classical aggregation method with Fuzzy clustering

- The classical aggregation method with K-Means

The measure for evaluating the three methods is defined as follows:

()

EquiDyn

Original

iPiP

,..,1,

)()(

1)( 1

=∴

∆−∆

−=

∑

= (4.49)

where

• ∆P(i)Original and ∆P(i)Dyn. Equi. are the time domain behavior of the ith generator in the

internal system, which are calculated with the full external system and the aggregated

external system respectively. In this research, (∆P(i)Original-∆P(i)Dyn. Equi.)2 is defined as

squared distance error.

• Np is the number of sampling points and

• Ns is the total number of generators in the internal area.

The “best” aggregation method is the one that gives the minimum squared distance error or

maximizes J(i). Taking into consideration all internal machines, the following mean value J

may be defined:

∑

)(

(4.50)

Of course, by means of this value the quality, accuracy and reduction capability of the

splitting aggregation approach in comparison to the classical inertial aggregation can be

characterized and evaluated depending on the reduction degree of 11 external machines to 3

and 6 dynamic equivalents, as it can be seen in the following figure.

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 86

Fig. 4.12.- Comparison of aggregation algorithms considering the classical inertial aggregation and

the proposed splitting-based aggregation by the mean value of J of the intern machines for a

fault located at the internal node with different number of external equivalents.

Fig. 4.12 can be interpreted as follows:

• The best results and accuracy, as shaded beams illustrated, are to be found in the cases

for which the proposed splitting-based aggregation is used in dynamic equivalencing with

Fuzzy membership degrees independent on the number of equivalents. The detailed

values in Fig. 4.12 are given in the appendix table D.4.

• Comparable agreement and similar accuracy between the splitting-based equivalents

and equivalents, which are calculated by the classical aggregation and

electromechanical identity recognition (K-means algorithm), are provided.

• The splitting technique shows a significant enhancement in accuracy in comparison with

the equivalents calculated by the classical aggregation wizh Fuzzy clustering as identity

recognition.

• In Fig. 4.12, the accuracy is independent of the number of equivalent machines.

• The best aggregation and consequently the small distance errors defined in (4.49) are

given by the splitting technique.

Coefficient

Splitting with

Fuzzy Classical

Aggregation

with Fuzzy

Classical

Aggregation

with K-means

0,986

0,988

0,990

0,992

0,994

ggregation Algorithms

Number of

Equivalent

Generators

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 87

This approach has been applied to the interconnected European power system too, but the

result is not satisfactory. Some restrictions arise when applying this approach to large-scaled

power systems and also the number of external machines can cause some limitations in the

splitting procedure. Thus, the splitting factors, which determine the splitting electrical

parameters of the equivalent generator, can be dispersed. For instance, each of the 397

external machines of the European power system outside of the German system can be

splitted into 40 share factors. Some of them can tend to an insignificant factor, and so its

influence to build the equivalent, can disappear. This leads to an accumulation of error

yielding inaccuracy in the results. Hence, all splitting parameters of the external machines

are required to calculate the electrical parameters of equivalent according to equations (4.32)

to (4.37). This new proposed aggregation approach leads, however, to electrically efficient, in

terms of splitting-based machine parameters and more accurate dynamic equivalents.

4.6 Summary

• A new aggregation concept and strategy for the dynamic equivalencing, which is called

splitting-based aggregation, as alternative to the classic inertial and slow aggregation, is

proposed to obtain accurate, non-linear, splitting-based aggregated equivalent machines

of external large power systems.

• The fictional splitting of generators in virtual generators is based on the share factors,

such as the Fuzzy membership degrees and eigenvectors. They are considered in

forming accurate dynamic equivalents on the basis of the time behaviors of the external

machines. The share factors can be derived from mathematical reduction techniques

according to the Fuzzy theory or principal components.

• The significant benefit of splitting-based aggregation is principally that the resulting

aggregated dynamic equivalent is composed of splitting electrical system parameters.

• The main advantages of this approach is that the splitting factors or participation shares

of the machines are considered to define splitting-based electromechanical equivalent

parameters incorporating significantly mathematical reduction techniques, which

generate highly accurate aggregated equivalents in terms of:

- Linear independent dynamic equivalents with orthogonal oscillating swing curves by

principal components or

- Representative non-linear equivalents with oscillating swing curves by Fuzzy

membership degrees.

• The splitting equivalents are valid for disturbances too, which are electrically and

geographically independent of the equivalent disturbance and too of the extent of

severity (fault duration).

SPLITTING AGGREGATION-BASED DYNAMIC EQUIVALENCING 88

• Notable accuracy using splitting-aggregated dynamic equivalents can be reached in

comparison to the equivalent machines calculated with the classical inertial aggregation

in the 16 multi-machine power system.

• This splitting-based aggregation approach is applied to a 16 machine interconnected

power system, where the internal area is simulated regarding the transient stability. The

results are accurate independent of the number of dynamic equivalents. Best results with

a high degree of accuracy are achieved using splitting by Fuzzy membership factors.

• In comparison with the classical aggregations which are performed only and limitedly on

a per coherent area basis, the proposed splitting-based aggregation is extended to the

complete external area.

• In consequence, the aggregation approximation of the network in the complete external

area may result in an increase in accuracy.

• In comparison with the electromechanical-based identity recognition, this approach omits

the first step of the classical dynamic equivalencing, i.e. the grouping or

electromechanical clustering of similar generators on a coherent or identical area basis.

• This approach can be applied in small-scale power systems effectively obtaining

significant accurate dynamic equivalents. A similar accuracy is obtained by using the

electromechanical-based identity recognition applied to small-scale power systems

independent of the number of equivalents, too.

• However, its application in large-scale power systems, such as the European power

system, is limited because of the influence of other electromechanical factors of the

power system on the splitting process of the external machines. Therefore, in

comparison with the electromechanical-based identity recognition, the results are not

enough accurate. Thus, a drawback of this method is the accumulation of error when it is

applied to large number of external machines.

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 89

“All these constructions and the laws connecting them can be arrived at

by the principle of looking for the mathematically simplest concepts and

the link between them”-A. Einstein- “A basic rule in estimation is not

estimate what you already know”-A quotation from [129]-

Chapter 5

Dynamic Artificial Neural

Network-based Dynamic

Equivalencing

Objective— The aim of this chapter is to present a novel approach to construct an intelligent

system as interconnected external area. It considers and captures essentially non-linear

characteristics and behavior of the power system components.

This intelligent system is developed using dynamic artificial neural networks (DANN) as

dynamic models, which is proposed as alternative to the conventional dynamic

equivalencing. The conventional steps to generate dynamic equivalents are replaced by the

properly chosen recurrent artificial neural network taking into consideration a suitable off-line

training process, in which the effect of the disturbance influence of the internal area on the

external area has to be considered globally.

Thus, the proposed approach is based upon the modeling of non-linear systems using

dynamic ANNs in form of dynamic equivalents, which can be applied to dynamic stability

studies.

Simulation results demonstrate the effectiveness, high accuracy, and robustness of this

approach in different large multi-machine power systems with 2 to 8 boundary nodes.

Index Terms— Dynamic Equivalents, Model Reduction, Recurrent Artificial Neural Network,

Stability in Power Systems, System Modeling.

Organization— Section 5.1 describes the introduction and section 5.2 of this chapter the

classical methods. The proposed recurrent ANN-based approach with mathematical

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 90

preliminaries is treated in section 5.3, followed in section 5.4 by the application. In section

5.5, the simulation results are discussed and the conclusion in section 5.6.

5.1 Introduction

Interconnected power system can be simulated difficultly for stability analysis due to their

large size, non-linear behavior of generators and components, such as voltage and turbine

governors, exciters, loads, electronic converters, among others. In order to utilize limited

technical resources and limited data exchange between energy utilities the dynamic

equivalencing of power system is indispensable.

The conventional dynamic equivalencing [6, 7, 15, 17, and 30] consists mainly of the

following steps:

- Coherency identification

- Aggregation of generators

- Static network reduction

- Aggregation of control devices

The nature of power system is essentially non-linear and consequently, the nature of

equivalents should be non-linear, too. Mathematically speaking, non-linear systems are

known to be very hard to manage. To overcome this problem, when studying the behavior of

a power system in a neighborhood of an equilibrium point, it is a common assumption that

the power system is a linear, time-invariant system [3, 133]. Thus, the initial non-linear

system is approximated by linear one. In many cases of practical importance, this

assumption works quite well yielding numerous advantages. However, when transient

stability of the system is investigated, the use of a linear model cannot be justified. There are

several reasons for questioning the validity of the linear model. The main reason is the

dependence of the qualitative behavior of the power system model on the non-linear nature

of its components. Therefore, it is important to find a dynamic non-linear model of an

interconnected power system.

In this chapter, an innovative approach for forming dynamic equivalents of the external

area on the basis of the dynamic ANN is proposed. The basic concept underlies the

replacement of the non-linear external area by a robustly trained recurrent ANN, which is

connected to the internal area through tie lines and busses. Using this ANN-based approach

the abovementioned classical steps of dynamic equivalencing will be omitted.

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 91

5.2 Conventional dynamic equivalencing

Various dynamic equivalencing methods are proposed, in particular by exploiting modal,

coherency, linear model reduction and model identification properties.

Linearization

The most important dynamic equivalencing methods, which are described in appendix A,

are based upon linearization of the mathematical description of the system. They have had a

limited success, and their use is justified by the following facts:

- Linearization around the equilibrium yield mathematically tractable linear models.

- The output of system can be computed for any arbitrary input.

- Most non-linear systems could be approximated satisfactorily in their normal ranges

of operation.

Remarks to linearization:

• However, in many cases, systems are required to operate in regions in the state space

where linear models do not give satisfactory results. In order to cope with this, research

on developing input-output models (empirical approach), i.e., models that rely completely

on the inputs and the outputs of the system, has increased [116-120].

• Several model structures are available for developing input-output models of complex

systems. These include models based on spline functions, polynomial models, and

threshold models. Their limitation is that they can only be used for interpolation.

• If the system behavior is understood, but not so well that the adequate mathematical

model based on the fundamental laws could be developed, it can be reasonable to

construct a model based on unconventional methods.

• An unified framework for developing a non-linear model is not available. The true

modeling capability of any given system model depends on its structure and dynamic.

Thus, search of an appropriate structure and global description using non-conventional

or empirical methods can be incorporated in the modeling phenomenon.

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 92

5.3 Dynamic ANN-based dynamic equivalencing

The proposed approach is based on a robustly trained recurrent (dynamic) ANN that

models the external area including the non-linear properties of all power system components.

The behavior of the external area will be described through the dynamic of a ANN in contrast

to the methods described in [116, 143], which are disturbance-dependent and non-robust.

The power system can be divided in the following form:

Fig. 5.1.- Division of complex power networks in areas

Essential aspects of this approach can be briefly summarized as:

• An external area is replaced by a dynamic ANN considering important aspects, such as:

- The dynamic structure of the static ANN system is preliminary determined on the

estimated order of the external area after a linear modeling (knowledge-based).

Through a tuning process, it can be adjusted gradually to the electromechanical

order of the external area.

- The DANN-based model requires simulation results or measurements only at the

boundary buses obtaining Multi Input Multi Output (MIMO) magnitudes that are

defined using the Norton model of machines (signal-based).

• In consequence, this approach may be defined as a knowledge- and signal-based

system modeling. The key issues of the ANN based system modeling are both the

parameter determination, ANN-structure selection and the quality of selected input,

output signals.

• In the training procedure of the ANN, in which the parameters are determined, the

external area has to be excited through efficiently generated disturbance sets located in

the internal area.

Internal Area

External Area

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 93

• The well-trained dynamic ANN must be able to approximate and describe globally the

non-linear behavior of the external area following disturbance sets in the internal area.

To obtain a robust, operating point-independent ANN, a normalized MIMO with reference

to a certain operating point will be used.

In the following, key approach steps will be explained in detail.

5.3.1 Artificial neural networks (ANN) for modeling

To predict the behavior of an unknown system, ANN can learn and identify the system from

experience. Its highly parallel distributed architecture and the ability to learn based on limited

data makes ANN a powerful computing resource. Following ANN-aspects are important:

• A multi-layer neural network, consisting of one input layer, one output layer and an

appropriate number of hidden layers, can be used either as a static or dynamic

approximator, where information flows in one direction, as shown in Fig. 5.2.

• A desired accuracy in non-linear problems is achieved by suitable number of hidden

layers and neurons. The multi-layer neural network of hyperbolic tangent units and the

output layer of linear units are capable of approximating the non-linear dynamic of a

complex system. A one multi-layer feedforward ANN can be represented, as follows:

Fig. 5.2.- Neural network structure

• Its mathematical description is given by:

)(

),h( 112 κ

Wθ

BBy WW

)

(

++==

ϕϕ

ϕθ

θθ

θϕ

ϕϕ

(5.1)

Input layer Hidden layer Output layer

θθ

∑

κ→

∑

κ→

∑

κ→

∑

θθ

∑

κ→

∑

κ→

∑

κ→

∑

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 94

where ϕ

ϕϕ

ϕ is the input vector, κ the activation function, θ

θθ

parameter set, W the weight

matrix and B denotes the bias that is considered, for simplicity, a weight associated with

an unitary input.

• Neuron models have in common the structure according to three criteria: Input operator,

activity function, and learning rule (These aspects are explained in appendix C.1 and

C.2.) shown in the following diagram:

Fig. 5.3.- Basic structure of generalized neuron model

ANN advantages:

In neural networks, the relations are not explicitly given, but are coded in a network and its

weights. Its main advantages are [120-132]:

• Non-linearity: The mathematical interconnection of the ANN structure provides non-

linear characteristics to the complex system.

• Input/Output Mapping: The identification paradigm used is based on nonparametric

statistical inference, where no assumptions have to be made about the model under

study. Thus, the network learns from experience.

• Adaptability: An ANN adapts its synaptic weights to changes in the environment. An

initial training is usually made to model the stationary state of the system providing the

initial estimates for the ANN parameters. Real time adaptivity is required to model non-

stationary environments where some characteristics are dynamic and time variant.

ANN systems:

ANNs can be employed in a wide spectrum of problems in technical aspects. Thus, four

categories of ANN may be distinguished according to its applicability. From the viewpoint of

non-linear modeling three dynamic structures of ANNs subdivided into structures, such as

Learning

Rule Input Operator

B)W ( +

ϕϕ

ctivity Function

κ(x)

ϕϕ

Neuron Transfer Function

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 95

the internal recurrent networks, external recurrent with lamped dynamics and distributed

dynamics [119], can be determined, as it can be shown in the following diagram.

Fig. 5.4.- ANNs applied to the modeling of non-linear systems. Internal and external recurrent ANN

can be used to develop the DANN-based dynamic equivalencing

Remarks to Fig. 5.4:

• Mapping ANN represents non-linear parametric models without requiring a specified a

priori accurate assumption about the system structure. Thus, the learning procedure

estimates the unknown network’s weights, which represent the model’s parameters.

• ANN, which belongs to the class of mapping neural networks, performs mathematically a

mapping action from a domain of its input space to the output space [117-122].

• The mapping task is labeled in spatial if there is no time-dependency. In general, these

ANNs can be applied to identify static non-liner systems.

• Another mapping task is the spatiotemporal mapping. In this case, the modeling and

identification of dynamic non-linearities involved in subjects, such as behavior, response,

and non-parametric modeling of complex systems, can be considered as an

approximation of spatiotemporal rules.

In power systems, the dynamic external area can be approximated by means of dynamic

ANNs. The dynamics are realized either using static ANN combined with an external

-Recurrent Backpropagation

-Backpropagation through time

-Fully Recurrent

-Diagonal Recurrent

-Partially Recurrent

Internal Recurrent External Recurrent-

Lumped Dynamics

Stochastic NN Hierarchical NN Associative NN

Spatiotemporal

(dynamic)

Spatial

(static)

Distributed Dynamics

-Time Delay NN

-Dynamic Learning

-Associative Memory

Systems (AMS)

-Self-Organized AMS

-Cerebellum Model

Articulation Controller

- Dynamic Perceptron

-Dynamic Radial Unit

-Memory Neuron

Network

rtificial Neural Network

Mapping NN

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 96

feedback connection (recurrent ANN) or using internal recurrent dynamic ANN, where

feedback is introduced internally to the inputs of antecedent neurons.

In this study, both options are examined and assessed, and the external recurrent ANN

was found as the suitable resource to describe accurately the whole behavior of a complex

power system. These options are emphasized in the Fig. 5.4.

5.3.2 Modeling of dynamic system

Its main purpose is to identify a model of an unknown complex system in order to predict

and gain insight into the behavior of the system [116]. The key problem in system modeling

is to find a suitable model structure. To this end, prior knowledge and physical insight about

the system should be utilized.

Types of models:

The system modeling based on physical and experimental aspects of non-linear systems can

be summarized in the following diagram:

Fig. 5.5.- Modeling structures. The system modeling used in this approach is emphasized as a

hybrid procedure in black-box form with assumptions about the system

Following aspects of Fig. 5.5 can be emphasized:

• In white-box models, some prior physical information is available (e.g. physical

knowledge, linguistic rules, vague state-space model).

• But, in comparison in grey-box models this information is not complete. In order to get

better model some parameters must be determined from the data.

Physical (White-Box)

Experimental

Parametric

Non-Parametric

Physically motivated (Grey-Box)

With assumptions about system

structure

Without assumptions about system

structure (Black-box)

Modeling

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 97

• A grey-box approach is an approach to hybrid modeling. Another important alternative is

an approach where an a priori mathematical model is used as a starting point and a

radial basis function expansion compensates the mismatch between the mathematical

model and the process data.

• The system modeling on experimental variables involves parametric and non-parametric

models.

• Modeling of parametric models entails the estimation of unknown parameters of an

appropriate model structure, which is chosen a priori (black-box) or partially motivated

by physical analysis (grey-box).

• In black-box structure for a dynamical system, no prior information about the system is

available. The modeling must be done relying on the observed data and describing the

system in an input/output sense. The non-linear black-box is difficult, because nothing

system behavior should be excluded and consequently a very rich spectrum of possible

models descriptions must be taken into consideration.

• The main futures of parametric models are the explicit small set of used parameters and

the model structure. In contrast, non-parametric models don’t posses a given structure

and, furthermore, use a large set of parameters.

• The system modeling used in this approach is realized as a hybrid procedure in black-

box form with assumptions about the estimated linear system.

• There are different non-linear models, such as the non-linear Output Error (NOE),

non-linear AutoRegressive models with eXogenous inputs (NARX), non-linear

AutoRegressive Moving Average terms with eXogenous inputs (NARMAX), etc.

These models area treated in detail in appendix C.4.

5.3.3 Mathematical description

The modeling can be divided into two basic functions (explained in Appendix C.3 and C.4):

• Mapping from past observed data to a regression vector (by preprocessing of data).

• Non-linear mapping from regressor space to the output space, which is typically

formed as a basis function expansion.

Dynamic systems can be described using the regressor vector. It starts from the state form:

))(,()(

))(),(,()(

tt t

ttt t

xgy

uxfx

•

(5.2a)

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 98

Discretizing (5.2a) yields to:

))(,( )(

))(),(,( )1(

ttt

tttt

xgy

uxfx

=+ (5.2b)

where x(t) is the vector of system states at time t, u(t) is an input, i.e. control signal, and y(t)

is the output of the system. Linear models can be formed by linearizing (5.2):

)()(

)()()1(

ttt

Cxy

BuAxx

+=+ (5.3)

where

000

A∂

∂

, , (5.4)

This non-linear time-discrete system (5.2b) can be described by the following equation:

))(),1(),1(( )( tttt uuxg

−−= (5.5)

Usually, (5.2b) can be reformulated too as a general discrete time dynamic system looking

for a relationship between past observations and future outputs in terms of the regression

vector as:

))(),1(()( tt t u

−= (5.6)

Hereby, the goal is to model a dynamic system in discrete time with input and output using

observations. The fact that the next output y(t) will not be an exact function of past data is

described by the additive term v(t) which usually is described as a random noise signal. The

goal is to find a model of ‘h’ which can be used to predict future y(t) with a system mapping

and a regressor vector as:

[]

ntututut

ntytytyt

)](),..,1(),([)(

)(),...,1(),()(

−−=

y (5.7)

where integers nu , ny are the maximum lags in the input and output and they determine the

system order. If their values are too large, the model is overparametrized and thus the

generalization property of the model is affected. As a result of choosing too small values the

model cannot model all the important dynamics of the system.

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 99

Thus, the general description in (5.6) is normally divided into two mappings. The first

mapping gives a regressor from the data as:

(

)

)..(),...,2(),1(),(),..,1(),()( yu ntytytyntututu t −−−−−=

ϕϕ

(5.8)

where

(t) is the parameter vector of fixed dimension. The second transformation maps

(t)

to the space of the outputs in the following form:

()()

θh θhy

t tytu t ),(),..,1(),..,()(

=−= (5.9)

where

y(t) is the model estimate of y(t) and

θθ

contains the parameter matrix of the model.

The regressor vector (5.8) can be parametrized as:

(

)

ηη

ηϕ

ϕϕ

),..,(),...,2(),1(),(),..,1(),()( yu ntytytyntututu t −−−−−= (5.10)

which may be denoted as

(t,

). Sometimes

η=

, i.e. the regression vector depends on all

the model parameters.

Following aspects of modeling are relevant:

• The data observation set y(n) and u(n) is a projection of the multivariate state space of

the system onto the reduced dimensional space.

• In order to realize the prediction, it is needed to reconstruct as well as possible the state

space of the system using the input output data information of the observation set.

• Recurrent ANN can also be described with the recurrence in the regressor combined

with a static mapping ‘h’.

• There are several advantages with describing the model as a concatenation of two

mappings instead of using one single function to capture the entire model as is common

in the neural network.

• First the mapping ‘h’ is static and all the dynamics are described by

(t).

• Second, as mentioned above, it is possible to introduce non-linear black-box models as

generalizations of linear black-box models by keeping the same

(t) but changing ‘h’

from linear to non-linear.

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 100

5.3.4 Power system model

The major component of the dynamics within a power system, which can be described by

the recurrent ANN is generated by the generator and its controllers. Therefore, appreciating

the importance of the generators, its description will be treated in the following section.

Modeling of synchronous machines

Depending on the nature of a study, several models of a synchronous generator, having

different levels of complexity, can be utilized [3, 134, 135, and 136]. In the simplest case, a

synchronous generator is represented by a second-order differential equation, while studying

fast transients in the generator’s windings would require the use of a more detailed model,

e.g., 8th order or Krause model [135].

The dynamical characteristics of a generator, whose structure modeling are based upon a

field coil on the d-axis and a damper coil on the q-axis, can be accurately represented by the

following differential equations:

[]

fddddq

qEixxE

dE +−+−= )(

1''

(5.11)

[]

qqqd

dixxE

dE )(

1''

−−−= (5.12)

ωω

−= (5.13)

[]

Gom P DP

d−−−= )(

ωω

(5.14)

qdqdddqqG ii xxiEiEP )( '''' −++= (5.15)

In the equations above, the following symbols are used to denote:

• Eq’, Ed’ denominate the transient EMF’s of the machine in the q- and d-axes.

• δ is the rotor shaft angle of the generator.

• ω is the rotor angular velocity of the generator. The ωo is the synchronous speed of

the system.

• M is the shaft inertia constant of the generator.

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 101

• Pm is the mechanical torque applied to the shaft of the generator.

• PG is the output electrical power of the generator.

• iqid are the equivalent currents of the synchronous machine in the q- and d-axes.

• D is the damping coefficient of the generator.

• T’doT’qo are transient time constants of the open circuit and a damper winding in the

q-axis.

• xq xd x’d x’q stand for the synchronous reactance and transient synchronous

reactance of the machine.

According to the equivalent circuit form of the two axes model [3] the voltage equations in d-

and q-coordinate system may be simplified neglecting stator transients and as well the flux

linkage equations may be simplified ignoring the damper winding. Defining the stator flux

transient flux linkages and the corresponding speed voltages, the stator equations are:

qqaddq u iR i x E =−+ '' (5.16)

ddaqqd u iR i x E =−− '' (5.17)

From the above equations, id and iq are solved as:

























−













adq

u- E

R x

x R

R x x

2''

1 (5.18)

These can be substituted in equations (5.11) and (5.12) and the machine rotor electrical

equations can be expressed as state space representation:













++=

•

RRRR 21 BBA xx , T

R











=''

x (5.19)

The mechanical equations (5.13) to (5.15) may be expressed as:

MMMM BBA xx ++=

•,

[]

ωδ

=x (5.20)

),,( dqGG v v PP R

x= (5.21)

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 102

When equation (5.18) is substituted in (5.15) PG becomes a non-linear function f2 of Eq’, Ed’

and vd, vq, which is depicted in appendix Fig. C.1. In the same manner, the modeling of the

excitation system, the turbine and governor system are explained in appendix C.5.

Equivalent generator

In order to realize a steady state analysis, the generator stator can be incorporated in the

network. Thus, the network may be represented on a single-phase basis using phasor

quantities for slowly varying sinusoidal voltages and currents in the network. Therefore, the

generator stator can also be represented on a single-phase basis.

Equation (5.18) can be expressed as a single equation in phasor quantities if transients

saliency is neglected, that is, ''' xx x qd == obtaining:

[]

)(

1''

'dqdq

dq juu(- )jEE

jxR

jii ++

=+ (5.22)

Hereby,

−

is the phasor voltage behind the transient impedance of the machine known as

the transient internal voltage and has two components

−

E and

−

E. According to this basis,

the current phasor

−

I can also be represented on a synchronously rotating reference frame

as:













=+=+= −−−

dqDQ

aU- 'E

jxR

ejii( jiiI '

)

(5.23)

where

θδ

dqDQ

eUejuu( juuU

ejEE( 'jEEE

=+=+=

+=+=

−

)

)'''' (5.24)

On the basis of equation (5.23), an equivalent circuit of the generator using the Norton model

can be represented as follows:

jxR

= (5.25)

jxR

e )jEE

Y 'E I

GG '

(

== −−

(5.26)

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 103

Therefore, I G

−

is a function of all state variables ( dq ' E ,' E and

). Hence, it does not

change suddenly whenever there is a network switching. The equivalent circuit can be

merged with the power network external to the generator.

This equivalent circuit can be represented schematically according to expressions (5.25-

5.26) as follows:

Fig. 5.6.- Generator equivalent circuit as voltage and current source

Considering this representation of generators using the Norton model in the network

equation for a interconnected power system, the boundary voltages between the internal and

external area may be used as the inputs to the model (independent variables), while the

injected current as the outputs (dependent variables), described as follows:

I−= (5.27)

Thus, this injected current is a function of all state variables of the generator, including its

control devices, such as voltage regulator and governor.

Thus, extending this model to complex systems the injected generator currents of the

external network presented in Fig. 5.7 as currents iG1, iG2, iG3 and iG4 are functions of the state

variables of the generators 1, 2, 3 and 4, respectively. As it may be seen in the following

figure 16:

16 Extending the Norton model to a complex power system divided in external and internal area, which are connected

over the boundary busses, the injected currents within the boundary busses (see Fig. 5.7) are in same manner functions

of all generator state variables of the whole external area including all their components.

−

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 104

Fig. 5.7.- Internal and external area of a interconnected power system

As a result injected currents within the boundary busses appear according to Fig. 5.8.

These currents (iE1 and iE2) are functions of all generator state variables of the external

network on the basis of injected currents iG1, iG2, iG3 and iG4. At the same time, the equivalent

admittances at the boundary busses are a result of the passive network reduction as follows:

Fig. 5.8.- Equivalent external area of a power system

5.3.5 Dynamic ANN model as external area

The Norton model replaces a system with dependent and independent sources. It requires

the voltage as independent input variable and the injected current as output of the system.

Thus, the power system which may be replaced, can be described as follows:





































(5.28)

Generator 2

Boundary bussesGenerator 1

Internal Area

iG4

iG2

iG1

Generator 3

iG3

External Area

Generator 4

Equivalent

Admittances

Boundary busses

External Area

iE1

iE2

Internal Area

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 105

where

• ie, ib, ii are injected currents of the external, the boundary and the internal buses,

respectively.

• ue, ub, ui are voltages of the external, the boundary and the internal buses,

respectively.

• Y is the network admittance including external, internal and boundary areas.

In mathematical terms, this equation can be solved using step by step triangular

factorization of the admittance matrix and transfiguration of the node currents. Given the

partition of the network equations according to equation (5.28), the external nodes will be

transformed first.

More specifically, the dynamic ANN model can replace an external area taking into

consideration following fields:

i. The discrete-time input-output representation

The dynamic behavior of the external area in terms of a general continuous state space

model is described as:

DuCxy

BuAxx

•

(5.29)

It may be modeled identifying the corresponding matrices A, B, C, and D in parametric

form. But, a non-linear model of a power system in system modeling is suitable to describe

the real properties of a power system. It can be obtained using input and output information,

which are defined on the injected current and voltage measurements on the boundary nodes,

i.e. considering the Norton model. This system modeling allows the determination of the

model structure as well as the description of the behavior of system in non-parametric form.

Thus, now the methods of transient stability analysis can be applied in conjunction with the

system equivalent as non-linear model (see the Fig. 5.9). It can be illustratively shown as

follows:

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 106

Fig. 5.9.- External area as dynamic equivalent

Thus, considering the input-output representation of a time-discrete approximation of a

continuous time dynamic system with a recurrent ANN during a simulation transient period,

the external area (according to Fig. 5.9) can be described by a set of differential and

algebraic equations in discretized non-linear form as:

))(),(,()1( tttt uxfx =+ (5.30)

))(,()( ttt xgi = (5.31)

In that case, the actual output vector i(t) depends on the actual time dependency and also

the actual value of the state variables x(t). Whereas the next value of the state vector x(t+1)

depends on the actual input u(t) and the actual state value x(t).

In equations (5.30-5.31) above, the following symbols are used to denote:

• u(t) is an input vector, i.e. voltage signal, that generally consists of state and field

voltages, and mechanical power input of each generator in the system at time t,

• x(t) is the vector of system states variables, it may contain variables associated with

synchronous generators and their excitation systems including turbines and

governing systems, other controllers, and possible network dynamics.

• The injected current i(t) is the output vector, and it is a function of all state variables.

• f, g are non-linear vector functions.

Boundary busses

Internal Area

Outputs:

Injected

currents

Inputs:

Voltages

Equivalent

Admittances

Dynamic

System

))(,()(

))(),(,()1(

ttt

tttt

xgi

uxfx

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 107

This way of representing a dynamic system will be called the discrete-time state-space

representation 17. Substituting (5.30) in (5.31) the output can be expressed as:

()

)1()),(),(,(,1)1( ++=+ tttttt uuxfgi (5.32)

This equation can be expressed according to a new vector function ‘h’ as:

()

)1(),(),(,,1)1( ++=+ tttttt uuxhi (5.33)

Usually, this expression can be reformulated as a general discrete time dynamic system as:

()

)(),1()( ttt uihi −= (5.34)

This type of representation will be called the discrete-time input-output representation.

This functional relationship ‘h’ of discrete-time dynamic systems can be identified with a

system based upon the optimal mapping and a suitable regressor vector as:

[]

)(),...,2(),1(

)( i

ntititi T

t−−−

=i (5.35)

[]

)(),...,2(),1(

)( u

ntututu T

t−−−

=u (5.36)

In the input output representation following aspects can be emphasized:

• The state space system (5.30-5.31) in input-output form consists of a collection of finite

sequences of input samples i(t) and corresponding sequences of output samples u(t) as

follows:

),()( )(),...,1(),()(),...,1( ntututuntiti ui

t −−−−=hi (5.37)

• i(t) and u(t) are past output and input values, respectively. Integers ni, nu (maximum lags)

reflex relatively the order of the system 18. These values can be estimated initially based

on the linear system modeling of the non-linear complex system.

• The suitable number of past inputs and outputs are collected into the regressor vector:

17 Thus, in mathematical terms, the objective is to study the stability of the dynamic system by solving the system (5.30-

5.31) with steady-state operating conditions or by describing the dynamic system using dynamic ANNs.

18 ni, nu must be chosen properly. If their values are too large, the model is overparametrized and thus the generalization

property of the model is affected. As a result of choosing too small values, the model cannot describe the whole system.

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 108

[]

)(),...,1(),(),(),...,1( u

ntututu

ntiti T

(t) −−−−

ϕϕ

(5.38)

• Then the problem is to map

ϕϕ

(t) to the next output i(t) with a non-linear function ‘h’:

))(()( tt

ϕϕ

hi → (5.39)

• This mapping can be realized by the NARX model (Nonlinear Auto Regressive model

with Exogenous inputs), which provides an unified representation for a wide class of

discrete-time non-linear systems. In a NARX description, the system is modeled in terms

of a non-linear functional expansion of past inputs and outputs:

))(()|(' ,θθ N

hi tt

ϕϕ

= (5.40)

• The functional relationship ‘hN‘ approximating the real-properties of the external area can

be identified with a DANN as external recurrent artificial neural network (time delay

ANN) with proper weighting parameters

θθ

and a suitable regressor vector

ϕϕ

(t).

ii. ANN Model structure

This dynamic ANN structure has to capture fitting function ‘hN’ representing a global

approximation of the dynamic non-linear function of the external area. A model based on

estimated function ‘hN‘ in (5.40) can be constructed according to the following time-delay

neural network structure:

Fig. 5.10.- Network structure for approximation of non-linear systems

ANN

z-1 z-1 z-1

z-1 z-1

z-1

W( +

∑

ϕκ

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 109

The ANN dynamics are realized either using static ANN combined with an external

feedback connection (recurrent ANN) or using internal recurrent dynamic ANN, where

feedback is introduced internally to the inputs of antecedent neurons. In this study both

options are examined and evaluated. Thus, the external recurrent ANN is more enable to

provide better and accurate results (For details see Fig. 5.4).

This static ANN depicted in Fig. 5.10 is provided with current and delayed values of the

external area inputs and outputs magnitudes. According to equation (5.34) in Fig. 5.10, the

implicit time dependence of the system mapping function is thus transformed to an explicit

spatial representation by additional inputs to the network, i.e. the neural network learns to

associate an output value i(t) depending on the trajectory determined by the input vector

elements i(t-1),..,i(t-ni), u(t),…,u(t-nu) at the neural network’s input.

According to the discrete-time input-output dynamic system representation in (5.37), there

are principally two possibilities to implement the ANN structure of Fig. 5.10 in the procedure

of the modeling of the complex external area. It can be realized either by means of a

configuration presented in the following figure and termed as series-parallel model or by

the parallel configuration represented in Fig. 5.12.

Fig. 5.11.- Series-parallel configuration coupling back the observed system output

Wσ(+

∑

External Area

System Model

I’

U I

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 110

Fig. 5.12.- Parallel configuration with an internal recurrent link to the networks

Remarks to Fig. 5.11 and Fig. 5.12:

• The series-parallel model in Fig. 5.11 is connected to the external area in a parallel

manner with respect to its input and in a serial manner to its output (observer).

• This configuration is advantageous for training purposes, where the desired external

area output can be provided to the neural network. Once the ANN has been trained, the

identified model can be used independent of the complexity of external area by feeding

back the ANN output instead of the real external area output.

• In contrast to this configuration, the parallel model in Fig. 5.12 requires an input that is

provided and estimated from the ANN self and not from the external area.

• Through this input sequence the error training and cost function evaluated over time

intervals will increment significantly.

• However, the primary benefit of this parallel model refers to the very simple synthesis

procedure of the ANN whose intern dynamic is implemented requiring the development

of time-dependent estimation schemes.

• According to these evaluated factors, the series parallel model has the capability to

identify the complex non-linear external area based on the linear system order, which

can be implemented according to the number of delays of the external recurrent links of

the ANN. The delays approximate the dynamic of the external area.

Wσ(+

∑

External Area

System Model

I’

U I

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 111

iii. Recurrent ANN model as external area

The recurrent ANN is able to capture the dynamic of the external area in the following

configuration and on this way it will be interfaced to the internal area for stability studies 19.

Fig. 5.13.- System modeling for dynamic equivalencing

In this figure the dynamic behavior of the external area can be modeled using voltage at

the boundary buses as input magnitudes and the injected currents into the boundary lines as

output (Norton model). All variables are complex quantities and thus they are treated

separately. Therefore, the selected parameter and structure of the recurrent ANN must be

able to describe a MIMO-system.

The multi-layer network, consisting of an input layer, one output layer, and an appropriate

number of hidden layers (containing two to four hidden layers with 10 to 24 neurons)

depending of the complexity of the system, may be used as a dynamic approximator. The

multi-layer neural network of hyperbolic tangent units and the output layer of linear units are

capable of approximating any non-linear dynamic of the external area and to obtain optimal

non-linear mappings.

The number of time delays in terms of ni, nu in (5.35-5.36) must be chosen properly to

generate an optimal regressor vector and estimated initially based on the linear system

Wσ(+ϕ

∑

I’

U I

External Area

Internal Area

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 112

modeling of the non-linear complex system, i.e. according to the linear order model of the

system and then, it may be adjusted gradually to reach the non-linear description.

It should be noted that the data set as inputs and outputs of the recurrent ANN is chosen

taking into account the variance of the sampling points of the observed data set. In this case,

the littler the distance between sampling points, more accurate will be the modeling of the

state space of the complex system, but the mapping function may be too complicated to

approximate the external area from the learning and estimation viewpoint and in

consequence, the integration time interval between sampling points of the model must be

suitably chosen.

iv. Estimation of the ANN-model parameters

The structure model of Fig. 5.13 represents the predictor. Its error is evaluated to select the

best model using the equation (5.42). Thus, the predicted output i(t)’ of the model is:

[]

tt Et )(|)()(

ii' = (5.41)

The prediction errors e(t) are a white noise sequence whose variance is minimized. The

variance of the prediction error is minimized with respect to the some performance function

(5.43) using the minimization of a fit criterion, which is the sum of square errors:

∑

−=

N tE t

)|(')(

),( θiiZ (5.42)

∑∑

=−=

NN te

h t

θE t

2),(

)(

),( )),(( θiZ θ

(5.43)

where

• N is the number of samples or the length of data set and

• Z is a matrix, which contains the system output, and the regressor matrix [128].

Minimization of (5.43) is realized by a parameter estimation algorithm, which iteratively

adjusts model parameters

until they achieve optimal values *

defined as:

),(minarg*ZθθE = (5.44)

19 Thus, this recurrent neural network structure represented as a partial Finite-Impulse-Response system (causal filter),

can replace the external area in terms of a series-parallel configuration as it may be seen in Fig. 5.13.

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 113

As can be seen in Fig. 5.13, when the recurrent ANN is used as a parametrized function ‘hN’,

its weights are the model parameters that have to be adjusted as:

[]

θθ

,...,1

=θ (5.45)

Parameter estimation algorithms are those based on gradient numerical optimization, which

adjust model parameters according to the following iterative expression:

))(()()()()()()1( tE t ttttt

θθ

αα

∇+=∆+=+ Sθθθθ (5.46)

where:

• )(tS is the search direction matrix, which contains the gradient search for the

minimum of the performance function. Alternative search directions are the gradient

direction, Gauss Newton direction and the Lavenberg Marquardt direction.

• )(t

determines the length of the step in the search direction.

• ))(( tE

∇ is the gradient of (5.43) with respect to the ANN parameters.

5.3.6 Robustness

The approach robustness is characterized by the training, learning procedure and the local

distributed dynamics of the recurrent ANN, which will be described as follows:

(i) Training procedure

The objective in the training phase is to take into account extreme and representative

disturbance situations obtaining global MIMO data sets at the boundary nodes. To generate

a suitable set of disturbance, its form and magnitude in the internal area, extended to all

disturbance variations, have to be considered. Following training aspects are important:

• In order to generate global training data sets, relevant disturbance scenarios in form of

three phase short circuits must be carried out involving the whole external area

according to:

- Faults with defined minimal and maximal duration till to reach the critical time. A wide

range of the fault is suitable.

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 114

- Faults applied on different nodes of the internal area, which are located principally in

different areas of the internal area.

• These aspects will avoid a redundancy and concentration of training data sets in any

small sub area in the internal area.

• The factors that influence the transient stability, such as:

- nature, extent of severity, and location of faults,

- load generation balance prior the occurrence of faults and

- network configuration

must be taken into account in the training range reflecting the global performance of the

replaced external area under various operating conditions.

• The dynamic ANN-model can predict injected currents following a certain disturbance in

the internal area, where these faults are located in the trained area, i.e. for faults located

in the training time range and inside the training location area of internal area.

• The simulation of each fault is carried out for 10 s, which is enough to restore the

dynamic behavior conditions after the disturbance.

• A 10 ms integration time step is used in the simulation generating the pattern variables.

• Complex injected currents and voltages at the boundary buses are stored during the

stability simulation and subsequently used to prepare suitable patterns for training the

ANN in off-line form.

(i) Learning procedure

To generate a robust ANN, which is valid for different power flow conditions in the internal

area, normalized deviations of the corresponding boundary currents and voltages are used to

ANN-learning, as it can be seen in Fig. 5.14.

Fig. 5.14.- DANN representing the dynamic equivalent for disturbances applied in the internal area

I(t) Dynamic Neural

Network

∆

i(t)·I0

Internal

Area

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 115

These referenced boundary complex values can be expressed as follows:

UtU

u−

∆

=∆ )(

)( (5.47)

ItI

i−

∆

=∆ )(

)( (5.48)

where

• U(t), I(t) correspond to the real-time complex voltage and current value, respectively

• Uo, Io corresponds to the initial complex voltage and injected current value at a static

operating point of the external area of power system.

Following aspects are important in the learning strategy:

• The use of normalized deviations according to (5.47) and (5.48) allows that the ANN

learns during the training process only the changes of the boundary input and output

magnitudes at a certain operating point with reference to the static operating point or

base case specified by loads flow of the power system, characterizing the robustness of

the dynamic ANN.

• In this case, this ANN represents a normalized “per unit” model scaled on initial

conditions at the boundary buses. Thus, the robust ANN can be used with trained and

changed power flow conditions outside the ANN.

• The data pre- and post-processing, i.e. normalization and back normalization,

respectively have to be realized outside the ANN.

• The ANN itself acts as a Norton model, where the normalized deviations of voltages are

used as main inputs and the normalized deviations of currents represent the outputs.

• Other magnitudes, such as the active and reactive power, or the absolute value and

phase of the voltage and current as input and output for the ANN, respectively can be

used. However, these measured data signals don’t provide the physical and non-linear

combination between input and output sets.

• Thus, the use of voltage and current gives better convergence in the training process in

comparison with the use of power magnitudes due to the complete decoupling and

independency between inputs and outputs of the ANN.

• It should be mentioned, that the results with ANN trained by power variables as input and

output variables in one-boundary power systems are high accurate as well as with the

ANN trained with injected currents and voltages. But, this aspect can be detected only in

power systems with one boundary node between internal and external area.

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 116

(iii) Dynamic ANN with locally distributed dynamic areas

In order to achieve a more accurate ANN-based system model, whose training procedure

is less time consuming and offers limited computational requirements, the learning operating

point and dynamic of this ANN-based system model can be distributed according to the

power levels of the retained or internal area.

In comparison to the ANN with distributed dynamics, in the following figure, the global ANN

captures the dynamic of the external area excited by internal area with heterogeneous power

levels.

Fig. 5.15.- Global dynamic ANN forming the dynamic of the external area

In order to predict accurate injected currents with less computational requirements, ANNs

with locally distributed dynamics are proposed. This ANN structure with distributed dynamics

can interact with an internal area, which consists of heterogeneous power levels, according

to its training operating point, i.e. for each power level of the internal area a corresponding

ANN will be trained. Thus, following aspects are relevant to the ANN-dynamic distribution:

• The necessary input variables for the training procedure can be obtained by disturbance

scenarios in the area of the corresponding power level to form an adequate ANN locally

trained.

• The offline locally trained ANN structure will replace the external area dependent on the

power level-located disturbance.

The ANN structure with distributed dynamics can be illustrated schematically in the following

diagram:

Global dynamic artificial neural

network

I(t)

∆i(t)·I0

Internal Area

220 KV

380 K

110 KV

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 117

Fig. 5.16 .- Dynamic ANN forming the external dynamic with distributed operating points according

to heterogeneous power levels of the internal area

Remarks:

Briefly, recurrent ANN-based dynamic equivalencing involves following aspects, which are

discussed in a previous way extensively:

Fig.5.17.- Procedure to developing ANN models as dynamic equivalents

However, in order to develop accurate results considering computational aspects, ANN with

locally distributed dynamics can be realized according to the previous procedure.

NN structure with distributed dynamics

Dynamic ANN that captures

the behavior of

220 kV area

Dynamic ANN that captures

the behavior of

110 kV area

Dynamic ANN that captures

the behavior of

380 kV area

I(t)

∆

i(t)·I0

Internal Area

220 K

380 K

110 K

Accepted

Not

accepted

Estimation by ANN parameter optimization algorithms and

Validation of the ANN model

Generating global data set disturbance as MIMO magnitudes

according to relevant disturbance scenarios

Generating robustness by means of preprocessing of data sets for

training procedure

Suitable selection of the ANN parameters and its structure

Generating optimal regressor vectors

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 118

5.4 Case studies

The accuracy, effectiveness, and robustness of the recurrent-ANN as dynamic equivalent

are evaluated on basis of two multi-machine systems. The following power systems are

studied:

- 16 Multi-machine system with 2 boundary nodes and

- 12 Multi-machine system topologically adapted from 3 to 8 boundary nodes.

In both power systems, the synchronous machines are described by 5th order models,

exciters by 2nd and in some cases by 3rd order models. A state vector of large dimension

characterizes the models of the external areas in both cases.

5.4.1 16 Multi-machine system with 2 boundary nodes

The 16 multi-machine system shown in Fig. 5.18 comprises 16 hydro, nuclear and thermal

generators with their corresponding excitation systems.

Fig. 5.18.- 16 Multi-machine system with 2 interfaces and three areas

AREA A

REA B

380 kV

220 kV

110 kV

15.75 kV

REA C

1a A1b A2a A2b

B2a B2b

B10

C10

C12

C14

Boundary Nodes

Training

disturbances

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 119

This system consists of three strongly meshed areas A, B and C with different voltage

levels, which are connected by boundary buses. Each area has 5 or 6 generators and it has

been considered as internal area. Area A is structured to be a power exporting area.

5.4.2 12 Multi-machine system with 3 to 8 boundary nodes

This power system contains 12 generators as hydro power plants with their corresponding

voltage regulators and governors. On the basis of this network topology, different power

systems with different boundary nodes between 3 to 8 nodes may be derived, which may be

seen in appendix in Fig. B.1 to Fig. B.5. All systems consist of a 380 kV network with the

corresponding internal and external area.

Fig. 5.19.- 12 Multi-machine system with multiple number of boundary nodes

The internal area contains 5 generators and it is a load demanding area. This system is

designed in such a way that successively any desired number of boundary interconnections

between 3 to 8 can be implemented (MIMO system). The accuracy and principally the

robustness of this ANN-based equivalencing approach will be verified in this power system.

23 24 25

27 28 29 47

42 43 44 45

36 35

G11

G10

G12

EXTERNAL AREA

L1 L2

INTERNAL AREA

G9 G12

Training

disturbances

380 kV

15.75 kV

Boundary Nodes

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 120

5.5 Simulation results and discussion

To verify the performance of the recurrent ANN as dynamic equivalent, the ANNs are

trained with data sets generated by appropriate disturbance scenarios on various nodes in

the internal areas. The location of the disturbances is shown in Fig. 5.18 and Fig. 5.19 and

with duration from 100 ms to 150 ms. In case of using the ANN-based equivalent as external

area, these disturbances (as three-phase short circuits) and nodes will not be considered

within the transient stability simulation. This disturbance begins at 1.0 seconds. The whole

system is simulated for 10 second.

The accuracy of the ANN-based dynamic equivalents can be evaluated by comparing the

oscillating swing curves of the boundary injected currents and the boundary power flow

interconnection with the original power system. The injected currents are calculated with

respect to a synchronously rotating reference frame. The original boundary behavior is

simulated with the unreduced external area using PSD. For details on PSD see section 2.3.

These simulations are realized in both power systems: i) 16 multi-machine system and ii) 12

multi-machine system.

5.5.1 16 Multi-machine system with 2 boundary nodes

This power system is investigated considering following internal and external areas:

Table 5.1.- Studied cases in the 16 multi-machine system

Cases Internal area External area

1 380 kV area A B and C

2 380 kV and 220 kV area B A and C

3 110 kV, 220 kV and

380 kV area C A and B

Considering that areas A and B have an uniform voltage level and C incorporates varying

voltage levels (see Fig. 5.19), the ANNs are capable to replace heterogeneous external

areas from Table 5.1. Areas A, B and C are connected by two boundary lines.

Interfacing the robust trained ANN with the transient stability simulation of the internal area,

following cases from Table 5.1 can be evaluated.

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 121

Case 1 — Area A is the internal area, external areas B and C are replaced by the ANN.

The following figures demonstrate the boundary-injected currents following a non-trained

disturbance with duration of 100 ms on the node 4 in area A.

Fig. 5.20.i .- Real part of the injected current at the second boundary node following a non-trained

disturbance in area A

Fig. 5.20.ii .- Imaginary part of the injected current at the second boundary node following a non-

trained disturbance in area A

Fig. 5.20.i and Fig. 5.20.ii show a high degree of accuracy of the predicted current from the

ANN with respect to them simulated with the original external area. Relatively, the same

prediction quality is obtained at boundary nodes following other non-trained faults on the

nodes 2, 6, 7 and 8 (see Fig. 5.18) in the internal area A.

0.5

0.25

-0.25

-0.5

∆

∆i ire (p.u.)

0 2 4 6 8 10

Time (s)

Original unreduced external(PSD)

Recurrent ANN-based Equivalent

Time (s)

∆

∆i im (p.u.)

1.25

0.5

-0.25

0 2 4 6 8 10

Original unreduced external (PSD)

Recurrent ANN-based Equivalent

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 122

Successfully, the robustly trained ANN captures the external area dynamics considering

other load flow condition, i.e. the ANN can be interconnected to a changed internal area with

other operating point, where the load in the internal area A was reduced to half of its previous

value of the training. The simulation results are shown in Fig. 5.21:

Fig. 5.21.i .- Real part of the injected current at the first boundary node following a disturbance (100

ms) at node 7 in area A under changed operating point

Fig. 5.21.ii .- Imaginary part of the injected current at the first boundary node following a

disturbance (100 ms) at node 7 in area A under changed operating point

It can be seen in Fig. 5.20 and Fig. 5.21 that the phase and amplitude of all predicted

boundary injected currents shows a notable accuracy and agreement with respect to them

simulated with the original area. The dynamic behavior of these currents under different

operating points (trained and non-trained) is quite identical over the whole time period.

Time (s)

∆

∆i ire (p.u.)

0.5

-0.5

-1

0 2 4 6 8 10

Original unreduced external (PSD)

Recurrent ANN-based Equivalent

∆

∆i im (p.u.)

Time (s)

1.5

0.5

-0.25 0 2 4 6 8 10

Original unreduced external (PSD)

Recurrent ANN-based Equivalent

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 123

The sum squared distance error and average error of the predicted boundary behaviors

following disturbances at all non-trained nodes of internal area A and considering different

operating points (trained and non-trained) are summarized in appendix in table D.5 and D.6.

Case 2 — Area B constitutes the internal area, areas A and C are replaced by ANN

In same manner, Fig. 5.22 shows a high degree of accuracy of the ANN-predicted injected

current in comparison to the real time currents using the original external area. This stability

simulation is realized following a non-trained disturbance of 100 ms at the node 6 in area B.

Fig. 5.22.i .- Real part of the injected current at the first boundary node following a disturbance

(100 ms) on node 6 in area B

Fig. 5.22.ii .- Imaginary part of the injected current at the first boundary node following a

disturbance (100 ms) on node 6 in area B

Time (s)

0.2

0.1

-0.1

-0.2

∆

∆i ire (p.u.)

0 2 4 6 8 10

Original unreduced external (PSD)

Recurrent ANN-based Equivalent

Time (s)

0.5

-0.5

-1

0 2 4 6 8 10

∆

∆i im (p.u.)

Original unreduced external (PSD)

Recurrent ANN-based Equivalent

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 124

In Fig. 23.i and Fig. 23.ii, active and reactive power flows in interconnection between nodes

5 in B and 8 in C (boundary line of area B and C) are shown, together with waveforms

obtained in the original system.

Fig. 5.23.i .- Active power flow interconnection between nodes 5 in B and 8 in C boundary line of

area B and C according to the currents of Fig. 5.21

Fig. 5.23.ii .- Reactive power flow interconnection between nodes 5 in B and 8 in C boundary line of

area B and C according to the currents of Fig. 5.21

This power transmission between area B and C suggest that the ANN-based equivalent

successfully capture the external area dynamic with a high degree of accuracy after and

before the disturbance and the overall response quality is satisfactory.

200

160

120

Active power (MW)

0 2 4 6 8 10

Time (s)

Original unreduced external (PSD)

RecurrentANN

basedEquivalent

Time (s)

Reactive power (Mvar)

200

100

-100

-200

0 2 4 6 8 10

Original unreduced external (PSD)

Recurrent ANN-based Equivalent

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 125

The sum squared distance error and average error of the ANN-predicted boundary

behaviors following disturbances at all non-trained nodes of internal area B and considering

different operating points are summarized in appendix part in table D.5 and D.7.

Case 3 — C is the internal area, areas A and B are replaced by the ANN

The ANN-predicted current is quite accurate, compared with the current simulated with help

of the original external area. This simulation is realized following a non-trained disturbance of

120 ms at node 15 in 220 kV of C, as shown in Fig. 5.24.

Fig. 5.24.i .- Real part of the injected current at the second boundary node following a disturbance

on node 15 in 220 kV of area C

Fig. 5.24.ii .- Imaginary part of the injected current at the second boundary node following a

disturbance on node 15 in 220 kV of area C

∆

∆i ire (p.u.)

-0.5

-1

-1.5

-2

0 2 4 6 8 10

Time (s)

Original unreduced external (PSD)

Recurrent ANN-based Equivalent

∆

∆i im (p.u.)

1.5

0.5

-0.5

0 2 4 6 8 10

Time (s)

Original unreduced external (PSD)

Recurrent ANN-based Equivalent

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 126

The robustness of the ANN equivalent can be evaluated under changed operating points,

caused by load changes, generator disconnection and disconnection of the transmission line

between node 3 and 4 in area C. From the ANN-predicted currents the following power flow

interconnection between area C and B can be derived following a fault of 120 ms at node 5 in

110kV of area C.

Fig. 5.25.i .- Active power flow interconnection between area C and B following a fault on node 5 in

110 kV of C by changed operation point

Fig. 5.25.ii .- Reactive power flow interconnection between area C and B following a fault on node 5

in 110 kV of C by changed operation point

In Fig. 5.25, the active and reactive power flow behaviors show a low loss of accuracy. This

is not surprising, since the disturbance and operating point are not in the training database.

0 2 4 6 8 10

Active power (MW)

Time (s)

Original unreduced external (PSD)

Recurrent ANN-based Equivalent

-40

0 2 4 6 8 10

Reactive power (MVar)

Time (s)

Original unreduced external (PSD)

Recurrent ANN-based Equivalent

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 127

Moreover, this ANN captures the dynamic of the external area independent of the

disturbance and under changed operating point, caused by generator disconnections, line

disconnections and strong load reduction in the internal area, i.e. the ANN can be

interconnected to the changed internal area with regard to the network topology, and power

generating.

Thus, on the basis of the ANN-predicted injected current the power flow transmission

between area A and C can be on-line simulated in conjunction with PSD. This simulation

following a sequence of non-trained faults (after 1 sec. and 3 sec. with 150 ms and 200 ms

duration, respectively) applied on non-trained nodes is shown in the following figures:

Fig. 5.26.i, ii.- Power flow interconnection between area A and C following a sequence of nontrained

faults on C by changed operation point and network topology

This predicted power flow transmission in Fig 5.26 suggests that the ANN-based equivalent

successfully capture the system dynamic of the interconnected external area with a high

degree of accuracy considering the changed operating point condition. Through this aspect

the robustness of this ANN-based approach is being extensively demonstrated.

-350

-300

-250

-200

-150

-100 012345678910

Original DANN-based Equivalent

Time (s)

Active power (MW)

DANN-based E

uivalent Ori

inal

(

PSD

)

-300

-200

-100

100

012345678910

Original DANN-based Equivalent

Time (s)

Reactive power (Mvar.)

DANN-based Equivalent

Original (PSD)

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 128

The sum squared distance error and average error of the predicted boundary behaviors

following disturbances at all non-trained nodes of internal area C and considering different

non-trained operating points are summarized in appendix part in table D.5 and D.8.

Quality measurement of the ANN-based dynamic equivalent

For a more precise evaluation of the approximation between the injected currents

calculated with the original external area and the injected currents predicted by the ANN, the

following sum of square error can be defined:

∑

∆−∆=

NtE iti

]')([

)( )( (5.49)

where

•

∆

j(t) the change of the injected current following disturbance j considering the original

model of the external area,

•

∆

i’j(t) the change of the injected current following disturbance j predicted by the ANN

as dynamic equivalent,

• )( jE p

Nis the error function. This measure is realized over the sampling points Np of

the whole behavior and for the disturbance j.

The standardized form of this error function can be reformulated as:

)(1)( jEj p

E−= (5.50)

∑

SjE

)(

1 (5.51)

The standardized error function of the injected current of all non-trained disturbances is

summarized and averaged to form their mean value as expression (5.51). Where ND is the

number of non-trained disturbances with the same duration applied to the nodes, which are

not considered in the training database.

By means of the standardized error function S

E, the ANN-based dynamic equivalencing will

be evaluated regarding the quality, accuracy, and modeling capability. The disturbances and

the nodes (on which the disturbances are applied) are not trained.

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 129

The result simulations of the 16 multi-machine system which was investigated according to

cases 1, 2, and 3 of table 5.1, are compared with reference to their accuracy considering

separately area A, B and C as internal areas.

Considering the non-trained disturbances in the internal area A, B and C, and the non-

trained operating points, such as load reduction on the corresponding internal area, the

prediction capability and robustness of the ANN are evaluated using the beam representation

of the S

E value in the following illustration 20:

Fig. 5.27.- Evaluation of the prediction capability of ANN considering different non-trained

disturbances and non-trained operating conditions

Following aspects in Fig. 5.27 can be detected:

• The ANN replacing the areas A and B as external area, i.e. C as internal area, shows

results, which are less satisfactory considering both operating points. This behavior is

because of the heterogeneous voltage levels of the internal area C and in consequence

due to the different operating points of the dynamic ANN to capture the dynamic of the

external area. This degradation in accuracy may be observed representatively in Fig.

5.24 and Fig. 5.25.

• In this case, an improvement in accuracy can be obtained using the ANN structure with

locally distributed dynamics.

20 The values are summarized in appendix part in table D.5, D.6, D.7 and D.8, in which the sum squared distance error

and average error of the predicted boundary behavior are presented.

0,94

0,96

0,98

ABC

Changed operating point Training operating point

0.98

0.96

0.94

A B C

Internal Areas

Changed operating point Training operating point

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 130

• Moreover, in case of considering area A and B as internal areas, a low mean error

function or significant accuracy can be obtained. However, in the three cases, area A, B

and C as internal area, best results are achieved under the training operating points.

• The mean error function under changed operating points (illustrated by dark bar) is low

enough considering A, B and C as internal areas, which may be observed in Fig. 5.21

and Fig. 5.25 as extreme cases. With this evaluation the robustness of this approach is

successfully verified.

• Its reduced accuracy with reference to the training operating point in the three cases can

be accepted. As can be seen in Fig. 5.21 and Fig. 5.25, the injected currents and power

flow interconnections are satisfactorily accurate.

5.5.2 12 Multi-machine system with 3 to 8 boundary nodes

In order to evaluate the robustness of the ANN-based equivalencing, the 12 multi-machine

system, which is topologically adapted from 3 to 8 boundary nodes, i.e. to 6 different power

systems with 12 machines, is evaluated according to different operating conditions 21. These

conditions are summarized in the following Table:

Table 5.2.- Scenarios to power-flow changes considering Fig.5.19

Cases Load Conditions Location in network

1 Initial Loading Condition

(Training operating point)

2 Generator Disconnection G1

3 Transmission Line

Disconnection L1, L2

Generator, Line

Disconnection and Load

Reduction

G1, L1, L2 and load

reduction on all nodes of

internal area

5 Load Reduction to half On almost all load nodes in

internal area

In case 1, the ANN replaces the original external area under the same operating point, under

which the training of the ANN was realized. Consequently, this case may be considered as

an initial or reference operating point. In the subsequent cases, the global trained ANN will

replace the external area under new non-trained operating conditions.

21 The corresponding power flow changes and losses of the 12-machine system with different boundary nodes are

summarized in appendix part in tables E.5 to E.9.

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 131

All results of the power systems with 3 to 8 boundary nodes are similar accurate. However,

a representative scenario as worst case will be presented in Fig. 5.28. It is based on the 12

multi-machine system with 8 boundary nodes, where the non-trained fault of 120 ms is

applied on the node 20 in the internal area under the trained operating point according to

case 1 in table 5.2.

In Fig. 5.28, the power flow interconnection between node 39 and 40 on the 8th boundary

line is demonstrated (see Fig. 5.19). Hereby, it may be detected that the performance of the

ANN-predicted active and reactive power shows a quite agreement and accuracy during the

whole simulation, i.e. before and after the disturbance.

Fig. 5.28.i .- Active power flow interconnection at the 8th boundary node or between node 39 and 40

following a fault on the node 20 within the internal area

Fig. 5.28.ii .- Reactive power flow interconnection at the 8th boundary node or between node 39 and

40 following a fault on the node 20 within the internal area

Active power (MW)

0 2 4 6 8 10

Time (s)

Original unreduced external (PSD)

Recurrent ANN-based Equivalent

Reactive power (MVar)

-5

-10

-15

0 2 4 6 8 10

Time (s)

Original unreduced external (PSD)

Recurrent ANN-based Equivalent

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 132

This high accuracy has been determined too in the 12 machine systems whose internal

and external area are connected by 3 until 8 boundaries and considering extremely changed

operating points based upon cases 2 to 4 from table 5.2.

In Fig. 5.29, a representative scenario as the worse case is presented, where a non-trained

disturbance of 100 ms is located on node 26 in internal area in the 12-machine power system

with 8 boundary nodes under extremely changed operating conditions (case 4 from Table

5.2), i.e. considering at the same time generator disconnection G1, lines disconnections L1

and L2 and a considerable load reduction in the internal area.

Fig. 5.29.i .- Real part of the injected current at the 8th boundary node following a disturbance on

the node 20 in internal area under non-trained operating point of case 4 in table 5.2

Fig. 5.29.ii .- Imaginary part of the injected current at the 8th boundary node following a disturbance

on the node 20 in internal area under non-trained operating point of case 4 in table 5.2

∆

∆i ire (p.u.)

0.05

-0.05

-0.1

-0.15

0 2 4 6 8 10

Time (s)

Original unreduced external (PSD)

Recurrent ANN-based Equivalent

∆

∆i im (p.u.)

0.1

0.05

-0.05

-0.1

0 2 4 6 8 10

Time (s)

Original unreduced external (PSD)

Recurrent ANN-based Equivalent

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 133

The Fig. 5.29.i and Fig. 5.29.ii show an acceptable accuracy. However, this is not surprising,

since both the disturbance and the operating point are not in the training database.

Both Fig. 5.28 and Fig. 5.29 as worse cases show a good agreement of the ANN-predicted

active, reactive power and complex injected currents in its dynamic performance.

However, degradation in accuracy may be detected after the number of boundaries. The

behavior waveforms are more accurate and agreement, the smaller the number of

boundaries in the power system, as it can be showed illustratively in Fig. 5.31.

Evaluation of the ANN-based equivalencing in the 12 multi-machine system

The mean square error function S

E of the ANN-predicted injected currents according to

(5.51) in power systems with 3 to 8 boundary nodes may be calculated considering:

a) Different durations of non-trained disturbances and

b) Different non-trained operating conditions according to table 5.2

a) Different duration of non-trained disturbances

In the Fig. 5.30, the abovementioned evaluation can be realized using the beam repre-

sentation of the S

E value for the 12-machine system with 3 to 8 boundary nodes.

Fig. 5.30.- Evaluation of the standardized prediction error S

Eof recurrent ANN considering

different disturbance duration (tmin=100 ms, tmax=150 ms) and two sequential disturbances

(t1=100ms after 1s, t2=120 ms after 2s)

0,97

0,98

0,99

1,00

35678

tmin tmax t1 - t2

Number of Boundary Nodes

0.99

0.98

0.97

3 5 6 7 8

Es t min t max t 1 - t 2

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 134

The disturbance durations 22 correspond to:

- tmin=100ms

- tmax=150ms and

- the sequence of 2 different disturbances with a duration of t1=100ms applied after 1sec.,

t2=120ms after 2 sec. on different nodes.

Fig. 5.30 may be interpreted as follows:

• It shows a low mean prediction error in the 12 machine system with 3 and 5 boundary

nodes for all disturbance durations and the sequence of disturbances.

• However, with the increase of the number of boundary nodes, this mean error function is

also increased, but a satisfactory accuracy in the 12 machine system with 3 to 8

boundary nodes can be obtained, as can be observed in the presented worst cases in

Fig. 5.28 and Fig. 5.29.

• From the Fig. 5.30 is depicted that following a sequence of disturbances, not enough

accurate results are determined in the 12 machine system with 7 and 8 boundary nodes.

• However, these results for stability studies can be accepted, because a good agreement

between original and ANN-predicted injected currents is obtained.

b) Different operating points

In order to evaluate the robustness of the ANN-based dynamic equivalent under different

operating power flow conditions 23 expressed in cases 2 to 5 (i.e. non-trained operating

points) in table 5.2, the mean standardized square error function S

E of all injected currents

following the non-trained faults applied on non-trained nodes in the 12 machine system with

3 to 8 boundary nodes are calculated. The power flow relationships of these power systems

are summarized in the appendix table E.5 to table E.9.

Illustratively, these evaluation results are presented using the beam representation of the

E value in the following figure. Numerically, the results are summarized in the appendix in

table E.4.

22 These disturbance durations must be within the range, in which the training procedure was realized.

23 The offline trained ANN can be used in a transient stability analysis under different operating points. Transmission line

disconnection, loss of a large generator and load modulation are impacts, which create an imbalance between

generation and load with respect to the initial training operating point.

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 135

Fig. 5.31.- S

Eevaluation of the robustness of the recurrent ANN depending on the cases of table

5.2 and in the 12 multi-machine system with different boundary nodes 24

Fig. 5.31 shows following characteristics:

• Case 1 shows the lowest mean standardized square errors S

E considering the

operating point, in which the ANN was off-line trained for all adapted networks of the 12

machine system with 3, 4, 6 and 8 boundary nodes. Therefore, in this case, the best

agreement and accuracy of the ANN-predicted currents are obtained.

• In case 2 (in spite of a large generator disconnection, i.e. changed operating point), the

mean error function is similar to them of case 1 (trained operating point) in all 12

machine systems with different number of boundary nodes.

• In cases 3 to 5 (extremely changed power flow conditions), the light bars (corresponding

to systems with 6 and 8 boundary nodes) show a detectable increasing of the mean error

function depending on the strong of the operating point change.

• In all cases, the mean errors corresponding to the 12 machine systems with 3 and 4

boundary nodes (1.and 2. hatched bars) are lower than one of systems with 6 and 8

boundary nodes and relatively constant.

• Briefly, considering all cases, i.e. all examined changed operating points, with the

increase of the number of boundary nodes, the mean error of the predicted currents will

be increased or S

E is decreased. This aspect is due to the limited MIMO modeling

0,97

0,98

0,99

1(Init.) 2(Gen.) 3(Lines) 4(Gen.,

Line,Load)

5(Load)

3 4 6 8

Cases

1(Trained 2(Gen. 3(Line) 4(Gen.,Line, 5(Load)

O.P.) Loss) Load)

0.99

0.98

0.97

Number of Boundary Nodes:

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 136

capability of the recurrent ANN. Moreover, because of the limited computational

resources.

• However, an accurate modeling capability of the ANN for power systems with small

number of boundary nodes can be detected independent of the change of the operating

point.

• In spite of the lowest mean standardized error of worst cases 4 and 5 (extremely

changed power flow conditions) in the 12 multi-machine system with 8 boundary nodes,

the ANN-based equivalent predicts accurate injected currents (whose response quality

are satisfactory), which can be observed in Figure 5.29.

This approach is very useful and applicable as dynamic equivalent of a power system

independent of its size, complexity and number of boundary nodes. Because it does not

require the full system parameters and the state variables of the external area. This aspect

was fairly demonstrated in the 16 multi-machine system and 12 multi-machine system, with 2

and 8 boundary nodes, respectively.

In practical application following aspects can be determined:

• Some restrictions arise when applying this approach to large number of external

machines, such as in the interconnected European power system and also the non-linear

system dimension can cause some limitations in the computational procedures.

• However, this approach can be suitably applied to an analogous strongly meshed power

system, such as the Western North American System (WSCC) [144]. This system

comprises 46 nodes, 19 generators; the full model is characterized with a 123-

dimensional state vector, which is analogue to the 16 multi-machine system.

• The problem of the interconnected European power system is the complex order of the

state system of the external area, because higher is the dimension of the state vector,

higher is the number of hidden units, too.

• Thus, the increase of hidden units influences the computational complexity of the DANN-

based dynamic equivalencing. However, a suitable alternative can be to generate

homogeneous local sub areas from the whole external area.

• For instance, in the European external area involving the 397 machines can be

composed on the basis of the power subsystems given in table 3.2 and using the

proposed DANN with locally distributed dynamics.

24 In case 1, 2 3, 4 and 5, the 4 bars represent the mean error in the adapted 12 machine systems with 3, 4, 6, and 8

boundary nodes.

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 137

5.6 Summary

• With the limited data exchange and sporadic cooperation between energy utilities due to

the economic interests in the increasingly liberalized and deregulated energy markets,

this proposed novel approach is suitable to be used in transient analysis, because it

needs only reduced boundary data sets between internal and external area operators.

• The proposed approach replaces the classical steps of the dynamic equivalencing, such

as grouping, generator aggregation, control aggregation, and static network reduction by

means of a robust recurrent ANN as dynamic equivalent.

• Its main advantage is that it describes the non-linear dynamic behavior of the external

area considering all power system components, i.e. transmission lines, converters,

generators with their additional devices such as governor, excitation systems, etc.

• This global trained recurrent ANN-based dynamic equivalent is subject to a wide range of

disturbances applied geographically and electrically in an extensive way in the internal

area. The so generated MIMO disturbance data sets are provided to the ANN.

• Through the wide range and the normalization of the boundary MIMO magnitudes with

respect to an initial static operating point, a robust ANN equivalent is obtained.

• The ANN structure can be realized either as global recurrent ANN or as ANN with locally

distributed dynamics according to its operating points in conjunction to a heterogeneous

voltage level of the internal area.

• This novel approach is tested in various power systems with 2 to 8 boundary nodes

under different power flow conditions.

- Tests in the 16 multi-machine system have demonstrated that this ANN-based

equivalent is applicable and enough accurate in power systems with heterogeneous

voltage levels subject to non-trained disturbances of different durations.

- Tests in the 12 multi-machine system have verified that the ANN-based equivalent is

extremely robust for stability analysis of internal areas with changed power flow

conditions caused by disconnections of transmission lines, loss of generators and

changes in the load-generation balance.

• The stability analysis using the ANN-based dynamic equivalent is highly efficient,

because different non-trained disturbances on the internal area can be applied under

non-trained operating conditions and the ANN captures adequately the external area

dynamic. Although, the disturbance and the operating point differs extremely from the

ANN training case.

DYNAMIC ANN-BASED DYNAMIC EQUIVALENCING 138

• In comparison with the conventional proposed approaches, such as the

electromechanical-based identity recognition and splitting-based dynamic equivalencing,

the ANN-based dynamic equivalents can be applied to complex interconnected power

systems independent of the operating point and independent of the disturbance in terms

of time duration and location applied on the corresponding internal area.

• This approach is based on an intelligent robust model of the external area allowing its

complete replacement in comparison to the classical development of dynamic

equivalents by identity recognition and splitting-based aggregation.

• The robust ANN-based dynamic equivalent can be integrated in transient stability

analysis for all forms, extent, and locations of disturbances on the internal area. The

dynamic equivalents by identity recognition and splitting-based aggregation are validated

only for a specific disturbance and for other disturbances geographically and electrically

in the near. Moreover, these dynamic equivalents may be applied only for a specific

operating point of the internal area.

• This approach can be applied to the interconnected European system difficultly due to

the increase of hidden units, which influences the computational complexity of the

DANN-based dynamic equivalencing. However, a suitable alternative can be to generate

homogeneous local sub areas derived from the whole external area.

• With reference to the accuracy and agreement, this ANN-based approach is more

accurate than the dynamic equivalencing using the electromechanical-based identity

recognition and splitting-based aggregation in a small-scale power system, such as the

16-machine-system. However, it can work in conjunction with these classical equivalents

to reduce the effects of uncertainties.

CLOSURE 139

“In all human affairs there are efforts, and there are results, and the

strength of the effort is the measure of the result.”-J. Allen [140]-

Chapter 6

Closure

6.1 Conclusions

The major focus of this work has been the development of models in dynamic equivalen-

cing using intelligent systems generating innovative approaches as alternatives to the

classical dynamic equivalencing in power systems. The growing importance of considering

the properties of the components of power systems encourages the investigation of this

research area.

The proposed approaches can be roughly categorized into three main groups:

- Grouping of generators,

- Aggregation procedure of generators, and

- Construction of robust and intelligent dynamic equivalents.

The main conclusions obtained in this research can be summarized as follows:

1 Electromechanical-based identity recognition in dynamic

equivalencing

• In a preliminary way, an innovative approach in dynamic equivalencing, called identity

recognition, as alternative to the classical coherency identification is proposed to obtain

identity-based equivalents.

• Through this approach, the grouping of generators is considered as an identity analysis

task according to the introduced conditions for together oscillating machines, i.e.

evaluating the identical rotor angle behavior of the machines. The condition of the identity

recognition is reformulated in a way, in which the identity of the phase and amplitude of

the behavior of the machines has to be considered. By this rigorous identity evaluation,

CLOSURE 140

together oscillating machines can be grouped. These conditions are satisfactorily verified

on a theoretical way.

• The evaluation can be practically realized using standard pattern recognition algorithms,

such as hierarchical, K-means, Fuzzy and static ANN-based SOFM.

• Due to its nature, the proposed identity recognition can incorporate machine model

parameters in dynamic equivalencing establishing the proposed electromechanical-based

identity recognition.

• Thus, specific physical effects and electromechanical influences of the generators can be

considered to group identical generators.

• These effects are selected by realizing a sensitivity analysis on a one-machine system,

suitable machine parameters, such as the inertial constant and nominal power, are

determined. In conjunction with these parameters, the geometrical distance of the pattern

recognition algorithms is reformulated in an electromechanical distance.

• By means of the obtained electromechanical-based identity recognition, improved and

accurate dynamic equivalents are generated.

• In order to verify the effectiveness of the proposed approach, it was tested both in the16

multi-machine system and the interconnected European power system UCTE/ CENTRAL.

In the small-scaled power system, all algorithms generate similar grouping compositions

and in consequence, same accurate dynamic equivalents with high agreement. But, in

the interconnected European power system, in which the German network was simulated

as internal area, dynamic equivalents with different degrees of accuracy can be obtained

depending from the applied algorithm.

• Thus, the results with a high degree of accuracy are obtained using the electromecha-

nical weighted K-means and electromechanical weighted Fuzzy algorithms. SOFM are

not appropriate to this identity task in complex power systems due to the ineffectiveness

of its learning process.

• Further, the electromechanically derived equivalents following a disturbance are valid in

the same manner for other disturbances too, whose locations are electrically and

geographically in close to the one. Thus, this approach makes it possible that

disturbance-independent dynamic equivalents can be generated in limited terms.

• The accuracy of this approach depends upon important aspects, namely upon:

- The identity recognition capability of the algorithms.

- Number of dynamic equivalents.

- The geographical and electrical distance between internal machines and

disturbances.

CLOSURE 141

2 Splitting aggregation-based dynamic equivalencing

• An innovative aggregation approach in dynamic equivalencing, called splitting-based

aggregation, as an alternative to the classical aggregation, is proposed to obtain accurate

splitting-based aggregated equivalent machines.

• The fictional splitting of generators in virtual generators is based upon the share factors.

They can be derived from mathematical reduction techniques.

• The main advantage of this approach is that it can incorporate mathematical reduction

techniques in dynamic equivalencing, which can generate highly accurate aggregated

equivalents in terms of:

- Linear independent dynamic equivalents with orthogonal oscillating swing curves by

principal components. The dynamic behavior of the external machines is splitted into

orthogonal part oscillating swings involving principal components (eigenvectors).

- Representative non-linear equivalents by Fuzzy membership degrees. The identity

assignment of the external machines is realized by fuzziness membership degrees.

• The proposed splitting-based aggregation is extended to the complete external area.

This is in contrast to the classical aggregations, which are performed only and

restrictively on a per coherent area basis.

• In comparison to the electromechanical-based identity recognition, this approach omits

the first step of the dynamic equivalencing, i.e. the electromechanical grouping of

identical generators on the external area.

• This splitting-based approach was tested in the 16 multi-machine system. Notable

accuracy of splitting-aggregated dynamic equivalents can be obtained independent of

disturbance in terms of duration, location, and sequence faults, in comparison to the

equivalent machines calculated with the classical inertial aggregation. Further, the

accuracy is independent of the number of dynamic equivalents. Best results with a high

degree of accuracy are achieved using splitting by Fuzzy membership factors.

• Independent of the number of equivalents, a similar high accuracy is obtained using the

electromechanical-based identity recognition applied to small-scale power systems, e.g.

the 16 multi-machine system, too.

• However, its application in large-scale power systems, such as the interconnected

European power system UCTE/CENTRAL, is restricted. This is due to the influence of

other electromechanical factors of the power system on the splitting process of the

external machines. Thus, the splitting factors, which determine the splitting electrical

parameters of the equivalent generator, can be dispersed. Therefore, in contrast to the

electromechanical-based identity recognition, the results are not sufficiently accurate.

CLOSURE 142

3 DANN-based dynamic equivalencing

• This proposed approach omits the classical steps of the dynamic equivalencing, such as

grouping, generator aggregation, control aggregation, and static network reduction by

means of a robust recurrent ANN as dynamic equivalent. This intelligent robust model

replaces completely the external area.

• The main advantage is that it describes and captures properly the non-linear behavior of

an external area considering all power system components, i.e. transmission lines,

converters, generators with their governor, excitation systems, and amongst others.

• This recurrent ANN-based dynamic equivalent is globally trained with a wide range of

disturbances. There are applied geographically and electrically in an extensive way in the

internal area. The generated MIMO disturbance data sets are provided to the ANN.

• A robust ANN equivalent is obtained through this wide range and the normalization of the

boundary MIMO magnitudes with respect to an initial static operating point.

• The ANN structure can be realized either as global recurrent ANN or as ANN with locally

distributed dynamics according to the ANN operating points in conjunction with the

heterogeneous voltage levels of the internal area.

• This innovative ANN-based approach was tested in various power systems with 2 to 8

boundary nodes under different power flow conditions.

- Tests in the 16 multi-machine system have demonstrated that this ANN-based

equivalent is applicable and enough accurate in power systems with heterogeneous

voltage levels subject to non-trained disturbances of different durations.

- Tests in the 12 multi-machine system have verified that the ANN-based equivalent is

extremely robust for stability analysis of internal areas with changed power flow

conditions caused by disconnections of transmission lines, loss of generators and

changes in the load-generation balance.

• The ANN-based dynamic equivalents can be applied to strongly meshed interconnected

power systems with high accuracy independent of the operating point and the

disturbance in terms of time duration, location, and fault sequence. However, this

approach can be difficultly applied to the interconnected European system due to the

increase of hidden units, which influences the computational complexity of the ANN-

based dynamic equiva-lencing. A suitable alternative can be to generate ANN models for

small homogeneous local sub areas, which are splitted from the whole external area.

• The robust ANN-based dynamic equivalent is valid for all forms, extent, and locations of

disturbances and operating points on the internal area. This is in contrast to the dynamic

CLOSURE 143

equivalent using identity recognition and splitting-based aggregation, which are

generated mainly for a specific operating point of the power system.

6.2 Selection criteria

In the following table, a summarized selection criteria schema is presented.

Table 6.1.- Comparison of the proposed approaches to applicability considering power system

relevant aspects

Selection aspects

Electromechanical

identity

recognition-based

equivalencing

Splitting-

based

equivalencing

Recurrent ANN-

based

equivalents

Small-scale power system +++ +++ +++

Large-scale power system +++ - -

Disturbance

independent ++ +++ +++

Operating point

independent --- --- +++

Number of equivalents + ++

Data availability of external

area - - +++

Physical structure ++ +++ --

Boundary magnitudes - - ++

Resources Electromechanical

distance

Mathematical

reduction

techniques

Intelligent model

system

In the previous table according to the comparison technical aspects, the applicability of the

approaches has been summarily evaluated involving different usages and practical factors.

According to the relevant aspects of the dynamic equivalencing, the most appropriate

approaches are assigned by the symbols ‘+++’, and the less appropriate by ‘---‘. These

symbols can be chosen as an indicator to assess and use suitably the approaches for

forming dynamic equivalents in transient and dynamic stability studies according to the

corresponding practical aspect.

As small-scale power system, the 16 multi-machine and 12 multi-machine systems are

considered, and as large-scale power system the interconnected European power system,

as well. However, these proposed approaches can work in conjunction to reduce the effects

of uncertainties in dynamic equivalencing.

Under the following criteria aspects an appropriate approach could be selected in detail:

CLOSURE 144

Electromechanical-based identity recognition in dynamic equiva-

lencing

• This electromechanical-based approach is suitable to develop dynamic equivalents both

in small-scaled power systems and in the interconnected European power system

UCTE/CENTRAL without restriction independent of its size and complexity.

• The accuracy of the dynamic equivalents depends upon the used algorithm, the number

of equivalents and the electrical distance of internal machines to the disturbance.

• Further, the electromechanical derived equivalents are also valid with a degradation of

accuracy for other disturbances that are electrically and geographically in close distance

to the fault of the equivalents derived.

• These dynamic equivalents are operating point dependent. Moreover, this approach

needs considerable data sets of the external area to generate the dynamic equivalents.

Splitting-aggregation based dynamic equivalencing

• This approach is suitable for small-scale power systems obtaining strongly accurate

equivalents. Further, its high accuracy is mainly independent of the number of

equivalents and disturbance severity, form, location and duration.

• A drawback of this method is the accumulation of error when it is applied to large number

of external machines, such as in the interconnected European power system.

• From practical viewpoint, this approach is appropriate, when

- Electromechanical external machine parameters are available.

- The fictional splitting of external machines enables the implementation of

mathematical reduction techniques in dynamic equivalencing.

• However, the splitting-based dynamic equivalents are operating point dependent.

Dynamic ANN-based dynamic equivalencing

• The ANN-based dynamic equivalents can be applied without restriction to strongly

meshed, small-scaled power systems independent of the operating point and the

disturbance in terms of time duration, location, and fault sequence.

• Some restrictions arise when applying this approach to large number of external

machines, such as in the interconnected European power system and also the non-linear

system dimension can cause some limitations in the computational resources.

CLOSURE 145

• In comparison to others approaches, the ANN-based equivalent is valid for diverse

operating points far away (caused by disconnections of transmission lines, loss of

generators and changes in the load-generation balance) from the derived one.

• This intelligent robust model of the external area captures the complete dynamic of an

external area of a power system on the basis of an extremely reduced MIMO data set.

• From practical viewpoint, the network operator needs only limited boundary magnitudes

of the neighborhood areas to generate the ANN-based dynamic equivalent.

• This approach is more accurate as the results obtained by electromechanical-based

identity recognition and splitting-based aggregation in a small-scaled power system, such

as the 16 multi-machine system.

6.3 Suggestions for future work

This research represents a new way for implementing intelligent systems, such as pattern

recognition algorithms, Fuzzy theory, and ANN in dynamic equivalencing for power systems.

In the future, this study can be extended to the following aspects:

Electromechanical-based identity recognition

• Using the reformulated conditions of the proposed electromechanical-based identity

recognition, it would be interesting to generate dynamic equivalents for small stability

studies in terms of damping and modal analysis.

• Future prospects should focus on the implementation of this approach in damping

analysis of the interconnected European power system UCTE/CENTREL.

Splitting-based aggregation

• Different mathematical reduction techniques (based upon system decomposition) can be

implemented in this approach to form the splitting of the external area machines.

• The impact that splitting of generators in virtual machines would have on the dynamic

performance of the reduced power system, should be studied to determine the inaccuracy

in large-scale power systems. From such a study, more insights about the role of virtual

generators in providing additional electromechanical properties could be gained.

CLOSURE 146

Dynamic ANN-based dynamic equivalencing

With the increasing concurrence between energy utilities, dynamic ANN as non-linear

dynamic equivalents could be employed widely in online transient stability studies of real

complex power systems. Thus, this approach as proposed ANN-based equivalent with locally

distributed dynamics, could be applied on the homogenous sub areas derived from the

interconnected European power system UCTE/CENTREL, taking into account sufficient

computational resources.

6.4 List of publications

The work in this doctoral project resulted in a number of publications, which are listed below:

[1] O. Yucra Lino, “Recurrent Neural Network-based Dynamic Equivalencing in Power

Systems”, IEEE Trans., PSCE Power Systems Conference & Exposition 2004, New York,

10-13 Oct. 2004.

[2] O. Yucra Lino, Michael Fette, “Electromechanical Identity Recognition in Dynamic

Equivalencing”, 39th Universities Power Engineering Conference 2004, pp. 1078-1085,

Vol.3, Bristol-England, 6-8 Sep. 2004.

[3] O. Yucra Lino, Michael Fette, Zhao Dong, “Splitting-based Aggregation in Dynamic

Equivalencing”, PES Power Tech 2005, St. Petersburg-Russia, 27-30 Jun. 2005.

[4] O. Yucra Lino, Michael Fette, Juan Manuel Ramírez, “Electromechanical Distance and

Identity Recognition in Dynamic Equivalencing”, PES Power Tech 2005, St. Petersburg-

Russia, 27-30 Jun. 2005.

[5] O. Yucra Lino, Michael Fette, Zhao Dong, Juan Manuel Ramírez, “Non-linear

Approaches for Dynamic Equivalencing in Power Systems”, accepted for Publication in

the IEEE Trans., Power Systems Conference & Exposition 2006, Atlanta-Georgia, Oct.

29-Nov. 1, 2006.

BIBLIOGRAPHY 147

Bibliography

[1] H.E. Brown, R.B. Shipley, D. Coleman and R. Neid Jr., “A Study of Stability Equivalents”,

IEEE Trans. Vol. PAS-88, Nr. 3, pp. 200-207, 1969.

[2] J.M. Undrill, J. Casazza, L.K. Kirchmayer, “Electromechanical Equivalents for Use in Power

System Studies”, IEEE Trans. Vol. PAS-90, pp. 2060-2071, 1971.

[3] P. Kundur, Power System Stability and Control, McGraw-Hill, New York, 1994.

[4] P. Kundur; L. Wang; S. Yirga, “Dynamic Equivalents for Power System Stability Studies”,

Fifth Symposium of Spezialists in Electric Operational and Expansion Planning, Recife,

Brazil, May 1996.

[5] T. M. McCauley, “Disturbance dependent electromechanical equivalents for transient

stability studies”, IEEE Winter Power Meeting, New York, January, 1975.

[6] L. Wang, M. Klein, S. Yirga and P. Kundur, “Dynamic reduction of large power systems for

stability studies”, Power Systems, IEEE Transactions on Power Systems, Vol. 12, No. 2, pp.

889-895, May 1997.

[7] W. Price, J. Choe, A. Hargrave, B. Hurysz, P. Hirsch, “Large-scale system testing of a power

system dynamic equivalencing program”, IEEE Transactions on Power Systems, Vol. 13,

No. 3, pp. 768- 773, August 1998.

[8] W. W. Price, B. A. Roth, “Large-scale implementation for dynamic equivalents”, IEEE Trans.,

Vol. PAS-100, pp. 3811-3817, 1981.

[9] R. Nath, S. S. Lamba, K. S. Prakasa, “Coherency based system decomposition into study

and external areas using weak coupling”, IEEE Trans., Vol. PAS-104, pp. 1443-1449, June,

1985.

[10] J. R. Winkelman, J. H. Chow, B. C. Bowler, B. Avramovic, P. V. Kotokovic, “An Analysis of

interarea dynamics of multi-machine systems”, IEEE Trans., Vol. PAS-100, pp. 754-763,

February, 1981.

[11] J. H. Chow, ”A toolbox for power system dynamics and control engineering education and

research”, IEEE Transaction on Power Systems, Vol. 7, No. 4, pp. 1559-1564, November,

1992.

[12] J. H. Chow, “Time-scale modeling of dynamic networks with application to power systems”,

Lecture Notes in Control and Information Sciences, Vol. 46, Springer-Verlag, 1982.

BIBLIOGRAPHY 148

[13] S. B. Yusof, G. J. Rogers, “Slow coherency based network partitioning including load buses”,

IEEE Trans. On Power Systems, Vol. 8, pp. 1375-1382, 1993.

[14] J. H. Chow, J. Cullum, R. A. Willoughby, “A sparsity-based technique for identifying slow-

coherent areas in large power systems”, IEEE Trans. PAS-103, No. 3, pp. 463-473, march

1984.

[15] J. H. Chow “New algorithms for slow coherency aggregation of large power systems”,

Systems and Control Theory for Power Systems, IMA Volumens in Mathematics and its

Applications, Vol. 64, pp. 95-115, Springer-Verlag, 1995.

[16] L. Wang, A. Semlyem, “Application of sparse eigenvalue techniques to the small signal

analysis of large power systems“, IEEE Trans. On Power Systems, Vol. 5, pp. 635-642,

1990.

[17] R. Podmore, “Identification of coherent generators of dynamic equivalents”, IEEE Trans..,

Vol. PAS-97, pp. 1344-1354, 1978.

[18] U. DiCaprio, “Condition for theoretical coherency in multimachine power system”,

Automatica, Vol. 17, pp. 687-701, 1981.

[19] Sung-Kwan Joo, Ch. Liu, J. Choe, “Enhancement of Coherency Identification Techniques for

Power System Dynamic Equivalents”, Power Engineering Society Summer Meeting 2001,

IEEE, Vancouver, Canada, Vol. 3 ,15-19, pp. 1811-1816, July 2001.

[20] S.T.Lee, F.C. Schweepe, “Distance measures and Coherency Recognition”, IEEE Trans.,

Vol. PAS-82, pp. 1550-1557, Sept/Oct. 1973.

[21] B. Avramovic, P. V. Kokotovic, J.R. Winkelman, J. H. Chow, “Area decomposition for

electromechanical models of power systems”, Automatica, Vol. 16, pp. 637-648, 1980.

[22] P. V. Kokotovic, B. Avramovic, J. H. Chow, J. R. Winkelman, “Coherency based

decomposition and aggregation”, Automatica, Vol. 18, pp. 47-56, 1982.

[23] R. Baiasubramanian, S. C. Tripathy and Shivanna, “A Fast Algorithm for Coherency

Identification and Dynamic Equivalencing”, ACE’90 Proceedings of IEEE India, pp.71-75,

Jan. 22-25, 1990.

[24] A. M. Miah, “Simple dynamic equivalent for fast online transient stability assessment”, IEE

Proceeding, Generation, Transmission and Distribution, Vol. 145, No. 1, pp. 49-55, January,

1998.

[25] Byung Chang, Jin Choo, Sae Kwon, “A reduced order equivalent model of large power

systems for the stability analysis”, IEEE, 2000.

[26] G. Jang, B. Lee, S. Kwon, H. Kim, Y. Yoon, J. Choo, “Development of KEPCO equivalent

systems for the KEPCO enhanced power system simulator”, Electrical Power and Energy

Systems Vol. 23, pp. 577-583, 2001.

[27] S. K. Joo, C. C. Liu, L. Jones and J. W. Choe, “Coherency techniques for dynamic

equivalents incorporating rotor and voltage dynamics” Bulk Power Systems Dynamics and

Control–V, Security and Reliability in a Changing Environment, Hiroshima, Japan, Aug.

2001.

BIBLIOGRAPHY 149

[28] T. Krishnaparandhama, S. Elangovan, A. Kuppurajulu, “Method for identifying coherent

generators”, Electrical Power and Energy Systems, Vol. 3, No. 2, pp. 85-90, 1981.

[29] V. Sankaranarayanan, M. Venugopal, S. Elangovan, N. Dahrma, “Coherency identification

and equivalents for transient stability studies”, Electric Power Systems Research, Vol. 6, pp.

51-60, 1983.

[30] J. H. Chow, R. Galarza, P. Accaci, W. Price, “Inertial and Slow Coherency Aggregation

Algorithms for Power System Dynamic Model Reduction”, IEEE Trans., Vol. 10, pp. 680-685,

1995.

[31] J. H. Chow, “Singular Perturbation, Coherency and Aggregation of Dynamic Systems”, IEEE

Trans., pp. 6-42, July 19981.

[32] Pierre Accari, Joh H., “Dynamic Aggregation Algorithms for Power System Model

Reduction”, IEEE Trans.., Vol. 10, No. 3, May 1994.

[33] Lj. B. Jocic, M. Ribbens-Pavella and D. Siljak, “Multimachine Power Systems: Stability,

Decomposition and Aggregation”, IEEE Trans., Vol. AC-23, pp. 325-332, 1978.

[34] A. J. Germond, R. Podmore, “Dynamic Aggregation of Generating Unit Models”, IEEE Trans.

Vol. PAS-97, Nr. 4, pp. 1060-1069, 1978.

[35] M. Hussain, V. Rau, “An efficient and simple method of dynamic equivalent construction for

large multi-machine power system”, IEEE International Conference on Advances in Power

System Control, Operation and Management, pp. 90-94, Hong Kong, November 1991.

[36] A. Ghafurian, G. J. Berg, “Coherency-based multi-machine stability study”, Proc. IEE, Vol.

129, Part-C, pp. 153-160, 1982.

[37] G. J. Berg, A. Ghafurian, “Representation of coherency-based equivalents in transient

stability studies”, Electric Power Systems, Research, Vol. 6, pp. 235-241, 1983.

[38] R. J. Galarza, J. H. Chow, W. W. Price, A. W. Hargrave, P. M. Hirsch, “Aggregation of

exciter models for constructing power system dynamic equivalents”, IEEE Transaction on

Power Systems, Vol. 13, No. 3, pp. 782-788, August 1998.

[39] R. A. Date, J. H. Chow, “Aggregation properties of linearized two-time-scale power

networks”, IEEE Trans. On Circuits and Systems, Vol. 38, No.7, pp. 720-730, July, 1991.

[40] J. D. McCalley, J. F. Dorsey, J. F. Luini, R. Peter Mackin, G. H. Molina, ”Subtransmission

reduction for voltage instability analysis”, IEEE Transaction on Applied Superconductivity,

Vol. 3, No. 1, pp. 349-356, March 1993.

[41] W. F. Tinney, J. M. Bright, “Adaptive reductions for power flow equivalents”, IEEE Trans.,

Vol. PWRS-2, pp. 351-360, May, 1987.

[42] R. J. Newell, M. D. Risan, L. Allen, K. S. Rao, D. L. Stuehm, “Utility experience with

coherency-based dynamic equivalents of very large systems”, IEEE Transaction on Power

Apparatus and Systems, Vol. PAS-104, No. 11, pp. 3056-3063, November 1985.

[43] K.Tanaka, M Takemura, “Development of Reduction Program for Bulk Power System

Stability Study,” IEEE trans., pp. 47-54, April 1998.

BIBLIOGRAPHY 150

[44] A. A. Fouad, S. E. Staton, “Transient stability of a multi-machine power system”, IEEE

Trans., 1981, PAS-100, pp. 3408-3424, 1981.

[45] A. A. Fouad, V. Vittal, T. Kyoo, “Critical energy for direct transient stability assessment of a

multi-machine power system”, IEEE Trans., PAS-103, pp. 2199-2206, 1984.

[46] Y. Xue, TH. Cutsem, M. Ribbens-Pavella, “A simple direct method for a fast transient

stability assessment of large power systems”, IEEE Trans., PWSR-3, pp. 400-412, 1988.

[47] Y. Xue, TH. Cutsem, M. Rubbens-Pavella, “Extended equal area criterion: justifications,

generations, applications”, IEEE Trans., PWSR-4, pp. 44-52, 1989.

[48] W. Brown, W. J. Cloues, “Combination load-flow stability equivalents for power system

representation on a.c. analyzers”, AIEE Transactions, Vol. 74, pp. 782-785, 1985.

[49] H. E. Brown, R. B. Shipley, O. Coleman, R. E,. Nied, “A study of stability equivalents”, IEEE

Trans. On PAS, Vol. 88, pp. 200-207, March, 1969.

[50] F. F. Wu, A. Monticelli, “Critical review of external network modeling for online security

analysis”, Electric Power and Energy Systems, Vol. 5, No. 4, pp. 222-235, October, 1983.

[51] J. Machowski, A. Cichy, “External subsystem equivalent model for steady-state and dynamic

security assessment”, IEEE Transactions on Power Systems, Vol. 3, No.4, pp. 1456-1463,

November 1988.

[52] S. Deckmann, A. Pizzolanzte, A. Monticelli, B. Stott, “Numerical testing of power system

load-flow equivalencing”, IEEE Trans. On PAS, PAS-99, pp. 2292-2300, November, 1980.

[53] Thomas Baldwin, Lamine Mili, Arun Phadke, “Dynamic ward equivalent for transient stability

analysis”, IEEE Transactions on Power Systems, Vol. 9, No.1, pp. 59-67, February 1994.

[54] A. Bergen, D. Hill, “A structure preserving model for power system stability analysis”, IEEE

Transactions on Power Apparatus and Systems, Vol. 100, No. 1, pp. 25-35, January, 1981.

[55] M. A. Pai, K. R. Padiyar, C. Radhakrishna, “Transient stability analysis of multimachine

AC/DC power system via energy function method”, IEEE Transactions on Power Apparatus

and Systems, Vol. 100, No.12, pp. 5027-5035, December 1981.

[56] V. Vital, N. Bhatia, A. A. Fouad, G. A. Maria, H. M. Zein El-Din, “Incorporating of non-linear

load models in the transient energy function method”, IEEE Transaction on Power Systems,

Vol. 4, No. 3, pp. 1031-1036, August, 1989.

[57] T. L. Baldwin, L. Mili, A. G. Phadke, “Ward-type equivalents for transient stability analysis”,

Proceeding of the IFAC International Symposium on Control of Power Plants and Power

Systems, March 9-11, Munich, Germany, pp.251-255, 1992.

[58] J.M. Undrill, A. Tuner “Construction of Equivalents by Modal Analysis”, IEEE Trans. Vol.

PAS-90, pp. 2049-2059, 1971.

[59] W. Price, E. Gulashenski, P. Kundur, G. Loehr, B. Roth, R, Silva, “Testing of the modal

dynamic equivalents techniques”, IEEE Tras. on PAS, Vol. 97, pp. 1366-1372, July 1978.

[60] S. D. Spalding, D. B. Goudi, H. Yee, “Use of equivalents and reduction techniques in power

system transient stability computation”, PSCC Proceeding, Vol. 2, 1975

BIBLIOGRAPHY 151

[61] S. E. M. de Oliveira, J. F. de Queiroz, “Modal dynamic equivalent for electric power

systems”, IEEE Transaction on Power Systems, Vol. 3, No. 4, pp. 1723-1730, November

1998.

[62] S. Takeda, S. Nishida, T. Yamaguchi, “Derivation of dynamic equivalents for stability

analysis”, Electrical Power and Energy Systems, Vol. 2, pp. 96-102, April, 1980.

[63] S. Nishida, S. Takeda, “Derivation of equivalents for dynamic security assessment”,

Electrical Power and Energy Systems, Vol. 6, No. 1, January, 1984.

[64] W. Price, “Testing of Modal Dynamic equivalents Technique”, IEEE Trans. Vol. PAS-97, Nr.

4, pp. 1366-1372, 1978.

[65] I. J. Perez-Arriaga, C. G. Verghese, F. C. Schwepe, “Selective modal analysis with

applications to electric power systems”, IEEE Trans. On Power Systems, Vol. 101, No. 9,

September 1982.

[66] S. Geeves, “A modal-coherency technique for deriving dynamic equivalents”, IEEE Trans.

On Power Systems, Vol. 3, No. 1, February 1988.

[67] P. M. van Oirsouw, “A dynamic equivalent using modal coherency and frequency response”,

IEEE Transaction on Power Systems, Vol. 5, No. 1, pp. 289-295, Febraury 1990.

[68] T. Hiyama, “Identification of coherent generators using frequency response”, IEEE

Proceeding, C 128, pp. 262-268, 1981.

[69] G. Ramaswamy, P. Panciatici, L. Rouco, B. Lesieutre, O. Filiatre, D. Peltiers, G. Verghese,

“Synchronic modal equivalencing for structure-preserving dynamic equivalents”, IEEE

Transaction on Power Systems, Vol. 11, No. 1, pp. 19-29, February 1996.

[70] I. J. Perez, G. Verghese, F. L. Pagola, J. Sancha, F. C. Schweppe, “Developments in

selective modal analysis of small signal stability in electric power systems”, Automatica, Vol.

26, No.2, pp. 215-231, 1990.

[71] L. Rouco, I. J. Perez, “Multi-area analysis of small signal stability in large electric power

systems by SMA”, IEEE Transaction on Power Systems, Vol. 8, No.3, pp. 1257-1265,

August, 1993.

[72] A. Chang, M. M. Adibi, “Power system dynamic equivalents”, IEEE Trans. On PAS, Vol. 89,

pp. 173-175, November, 1970.

[73] R. W. de Mello, R. Podmore, K. N. Stanton, “Coherency based dynamic equivalents:

application for stability studies”, PICA Conference Proceedings, pp. 21-23, 1075.

[74] Y. Ohsaya, M. Hayashi, “Coherency recognition for transient stability equivalents using

Liapunov function”, 6 th Power System Computation Conference, Vol.2, pp. 815-818, 1978.

[75] M. A. Pai, C. L. Narayama, “Dynamic equivalents using energy functions”, IEEE PES

Summer Meeting, July 1977.

[76] G. Troullinos, J. Dorsdey, H. Wong, J. Myers, “Reducing the order of very large power

system models”, IEEE Trans. on Power Systems, Vol. 3, No. 1, pp. 127-133, Feb. 1988.

BIBLIOGRAPHY 152

[77] O. Wasynczuk, Y-M. Diao, P.C. Krause, 'Theory and comparison of reduced order models of

induction machines', IEEE Transactions on Power Apparatus and Systems, Vol. 24, No.3,

pp. 598-606, March 1985.

[78] F.D. Rodriguez, O. Wasynczuk, 'A refined method of deriving reduced order models of

induction machines', IEEE Transactions on Energy Conversion, Vol. 2, No.1, pp. 31-37,

March 1987.

[79] T. Noda, M. Takasaki, Okamoto, Fujibayashi, H. Nishigaito, T. Okada, “Low-Order Linear

Model Identification Method of Power System”, IEE Transactions of the Institute of Electrical

Engineers of Japan, Vol. 121-B, No.1, pp. 52-57, January 2001.

[80] A. H. Whitfield, “Transfer Function Synthesis Using Frequency Response Data,” Int. J.

Control, Vol. 43, No. 5, pp. 1413–1426, 1986.

[81] A. Soysal, A. Semlyen, “Practical Transfer Function Estimation and its Application to Wide

Frequency Range Representation of Transformers,” IEEE Trans. on Power Delivery, Vol. 8,

No. 3, pp. 1627–1637, January, 1993.

[82] J. M. Ramírez, R. Garcia, “A technique to reduce power systems electromechanical models”,

Conference Proceeding of the IEEE Transmission and Distribution 2002, Sao Paulo-Brazil,

pp. 51-57, March 2002.

[83] J. M. Ramirez Arredondo, “Obtaining dynamic equivalents through the minimization of a line

flows function”, Electrical Power and Energy Systems, Vol. 21, pp. 365-373, 1999.

[84] P. Ganapati, M. Sanjay Kumar, M. Rasmi Rekha, “Adaptive Identification of the Parameters

of Nonlinear Systems Using Higher Order Statistics”, Conference Proceeding of the

ICSPAT-96, Boston, MA, pp. 200-204, 1996.

[85] C. K. Sanathanan, J.Koerner, “Transfer Function Synthesis as a Ratio of Two Complex

Polynomials,” IEEE Trans. on Automatic Control, Vol. 8, pp. 56–58 , 1963.

[86] M. Burth, G. C. Verghese and M. Vélez-Reyes, “Subset selection for improved parameter

estimation in on-line identification of a synchronous generator”, IEEE Trans. on Power

Systems, vol. 14, pp. 218 225, Feb. 1999.

[87] I. Erlich, H. Pundt, “Digital program for the simulation of electromechanic transients in the

medium”, PSCC Helsinki, pp. 19-24, August 1994.

[88] B. Kulicke, I. Erlich, S. Demmig, U. Bachmann, W. Glaunsinger, U. Zimmermann,

“Netzaequivalentierung für Stabilitaetsberechnungen basierend auf der Analyse kohärenter

Generatorgruppen“, Elektrizitätswirtschaft Jg. 99, Heft 7, 37-43, 2000.

[89] R. Isermann. Identifikation dynamischer Systeme, Bd. I, II., Berlin Heidelberg New York:

Springer 1992.

[90] I. A. Hiskens, “Nonlinear dynamic model evaluation from disturbance measurements”, IEEE

Trans. on Power Systems, vol. 16, pp. 702-710, Nov. 2001.

[91] P.Michaud, “Pattern recognition techniques”, Future generation computer systems, pp.145-

147, 1997.

[92] D. L. Massart and L. Kaufman, Cluster Analysis, Wiley, New York, 1983.

BIBLIOGRAPHY 153

[93] H. J. Mucha, Clusteranalyse mit Microcomputern, Akademie Verlag, Berlin, 1992.

[94] B. Mirkin, Mathematical Classification and Clustering, Kluwer Academic Publishers,

Dortrecht, Boston, London, 1996.

[95] R. Dubes, A.K. Jain, Algorithms for Clustering Data, Prentice Hall, Englewood Cliffs, New

Jersey, 1998.

[96] A. Hartigan, Cluster Analysis, Wiley, New York, 1969.

[97] Y. Lu, “Pattern recognition and Classification”, Proc. Int. Joint Conf. On Neural Networks, Vol

I, pp. 471-476, 1990.

[98] A. Jain, Clustering Algorithms, Prentice Hall, New York, 1991.

[99] H. P. Kriegel, Report Knowledge Discovery in Databases, Ludwig-Maximilian-University,

Munich, 2002.

[100] R. Dave, “Characterization of noise in clustering”, Pattern Recognition Letters, 12, pp.

657-664, 1991.

[101] Visual Numerics: IMSL Fortran 90 MP Library Help, Stat/Library, Volume 2, Chapter 11:

Cluster Analysis.

[102] J. C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum

Press, New York, 1981.

[103] F. Höppner, Obtaining Interpretable Fuzzy Models from Fuzzy Clustering, http://www.et-

inf.fho-emden.de/~dmlab/fc, Emden, 2002.

[104] J. Keller, R. Krishnaparum, “A Possibilistic Approach to Clustering”, IEEE Tran. On Fuzzy

Systems, 1(2), pp. 98-110, 1993.

[105] J. Bezdek, “On Cluster Validity for the Fuzzy Model”, IEEE Trans. On Fuzzy Systems, Vol

3, pp. 3-5, 1995.

[106] T. Kohonen, Self Organizing and Memories, Springer Verlag, Berlin, 1995.

[107] S. Mitra, “Self Organizing Maps as a fuzzy Classifier”, IEEE Transactions on Fuzzy

Systems, 24, pp. 101-123, 1994.

[108] G. Carpenter, S. Grossberg, “ART2: Self-organization of stable category recognition

category of analog input patterns,” Applied Optics 26, 4919-4949, 1987.

[109] D. Niebur, A. J. Germond, “Power system static security assessment using the Kohonen

neural network classifier,” Procs. of the 7th Power Industry Computer Applications

Conference, Baltimore, May 1991, IEEE Transaction on Power Systems, vol. PWRS-7, No.

2, 865-872, May 1992.

[110] D. Niebur, “Neural network applications in power systems,” Int. Journal of Engineering

Intelligent Systems, Vol. 1, No, 3, 133-158, December 1993.

[111] Y. H. Pao, Adaptive Pattern Recognition and Neural Networks, Addison-Wesley, Reading,

MA, 1989.

[112] I. T. Jollife, Principal Component Analysis, Springer Verlag, New York, 1986.

[113] O. Alter, P. O. Brown, “Singular value decomposition for genome-wide expression data

processing and modeling”, Proc. Nat. Acad. Sc. USA, No. 97, pp. 10101-10106, 2000.

BIBLIOGRAPHY 154

[114] M. W. Berry, “Large scale sparse singular value computations”, Int. Journal of

Supercomputer Applications, Vol. 6, pp. 13-49, 1992.

[115] F. Deprettere, “SVD and Signal Processing: Algorithms, Analysis and Application”,

Amsterdam: Elsevier Science Publishers, 1998.

[116] G. Crapanzano, Dynamische modellierung von generatoren und generatorgruppen mit

künstlichen neuronalen netzen (PhD Thesis), Shaker Verlag, Kaiserslautern, 2004.

[117] R. Rivera-Sampayo and M. Vélez-Reyes, “Gray-box modeling of electric drive systems

using neural networks”, Proceedings of the 2001 IEEE International Conference on Control

Applications, pp. 146-151, Sep. 2001.

[118] K. Narendra, K. Parthasarathy, “Identification and Control of dynamical systems using

neural networks”, IEEE Transactions on Neural Networks, Vol.1, pp 4-27, 1990.

[119] Z. Zhao, A. M. Stankovic and G. Tadmor, “A study of power system dissipativity using

artificial neural networks”, in Proceedings of the 1998 North American Power Symposium,

pp. 241-246, Oct. 1998.

[120] R. Rico-Martinez and I. G. Kevrekidis, “Nonlinear system identification using neural

networks: dynamics and instabilities”, Neural Networks for Chemical Engineers, pp. 409-442,

Elsevier, 1995.

[121] M. A. Kramer, “Nonlinear principal component analysis using auto-associative neural

networks”, AIChE Journal, vol. 37, pp. 233-243, Feb. 1991.

[122] M. Kirby and R. Miranda, “Nonlinear reduction of high-dimensional dynamical systems via

neural networks”, Physical Review Letters, vol. 72, pp. 1822-1825, Mar. 1994.

[123] J. S. Anderson, I. G. Kevrekidis and R. Rico-Martinez, “A comparison of recurrent training

algorithms for time series analysis and system identification”, Computers Chem. Eng., vol.

20, pp. S751-S756, 1996.

[124] F. J. Pineda, “Generalization of back-propagation to recurrent neural networks”, Physical

Review Letters, vol. 59, pp. 2229-2232, Nov. 1987.

[125] C. W. Liu and J. S. Thorp, “New methods for computing power system dynamic response

for real-time transient stability prediction”, IEEE Trans. on Circuit and Systems – I:

Fundamental Theory and Applications, vol. 47, pp. 324-337, Mar. 2000.

[126] R. Rico-Martinez, I. G. Kevrekidis, M. C. Kube and J. L. Hudson, “Discrete- vs.

continuous-time nonlinear signal processing: attractors, transitions and parallel

implementation issues”, in Proceedings of the 1993 American Control Conference, vol. 2, pp.

1475-1479, Jun. 1993.

[127] R. Rico-Martinez and I. G. Kevrekidis, “Continuous time modeling of nonlinear systems: a

neural network-based approach”, in Proceedings of the 1993 IEEE International Conference

on Neural Networks, vol. III, pp. 1522-1525, 1993.

[128] Y. J. Wang and C. T. Lin, “Runge-Kutta neural network for identification of dynamical

systems in high accuracy”, IEEE Trans. on Neural Networks, vol. 9, pp. 294-307, Mar. 1998.

BIBLIOGRAPHY 155

[129] A. Juditsky, H. Hjalmarsson, A. Benveniste, B. Delyon, L. Ljung, J. Sjöberg, Q. Zhang,

“Nonlinear black-box modeling in system identification: Mathematical foundations.”

Automatica 31(12), pp. 1691-1742, 1995.

[130] I. J. Leontaritis, S. A.Billings, “Input-output parametric models for nonlinear systems. Part

1: Deterministic nonlinear systems”, Int. J. Control 41, pp. 303-344, 1985.

[131] H. Tolle, E. Ersü, “Neurocontrol-Learning control systems inspired by neuronal

architectures and human problem solving strategies”, Lecture Notes in Control and

Information Science No. 172, Berlin Heidelberg New York: Springer, 1992.

[132] R. J. Williams, D. Zipser, “A learning algorithm for continually running fully recurrent

neural networks.”, Neural Comput. 1, 1989.

[133] G. Rogers. “Power system oscillations”, the Kluwer international series in engineering and

computer science, Kluwer Academic Publishers, 2000.

[134] Yaonan Yu. Electric Power System Dynamics. Academic Press, New York, 1983.

[135] Fette, Michael. Strukturelle Analyse elektrischer Energieversorgungssysteme (struktural

Analysis of electric Power Systems), Reihe21: Elektrotechnik, Nr. 140, VDI Verlag, 1993.

[136] P. W. Sauer and M. A. Pai. Power system dynamics and stability, Prentice Hall, 1998.

[137] T. Van Cutsem and C. Vournas. Voltage Stability of Electric Power Systems. Kluwer

Academic Publishers, 1998.

[138] J. C. Das and J. Casey. Effects of excitation controls on operation of synchronous motors.

Industrial & Commercial Power Systems Technical Conference, 1999.

[139] R. Kalman. Discovery and invention: The newtonian revolution in systems technology.

Journal of Guidance and Control, 26(6), pp.833–837, October 2003.

[140] James Allen. As You Thinketh. Putnam Publishing Group, 1959.

[141] O. Yucra Lino. Software handbook to PSD and development of intelligent, robust and

non-linear models in dynamic equivalencing for interconnected power systems. Report of

department of electrical and information systems, sustainable energy concepts. University of

Paderborn, October 2005.

[142] U. Bachmann, H. Breulmann, W. Glaunsinger, P. Hoiss, M. Loesing, A. Menze, S.

Roemelt, U. Zimmermann, “Verbundstabilitaet nach dem Synchronanschluss der Netze der

Elektrizitaetsunternehmen von Bulgarien und Rumaenien an UCTE/CENTREL-Verbund”,

Munich, ETG Tage 1999, ETG-Fachbericht 79, ISBN 3-8007-2502-9, pp. 251-257, 1999.

[143] Aleksander M. Stankovic, Andrija T. Saric and Mirjana Milosevic, “Identification of non-

parametric dynamic power system equivalents with artificial neural network” Power Sys-

tems, IEEE Transactions on, Vol. 18, Issue: 4, pp. 1478-1486, Nov. 2003.

[144] J. R. Smith, C. S. Woods, F. Fatehi, G. L. Keenan and J. F. Hauer, “A low order power

system model with dynamic characteristics of the Western North American System”, in

Proceedings of the 1994 North American Power Symposium, Part II, pp. 533-539, 1994.

APPENDIX 156

APPENDIX A Classical dynamic

equivalencing approaches

A.1. Ward equivalent

The Ward method consists in eliminating selected nodes of the network [50], which are

denoted by index ‘l’ and generator nodes by ‘g’, as can be seen in Fig. A.1.

Fig. A.1.- Ward static equivalent by eliminating {L} load and {G} generators nodes.

According to Fig.A.1, the bus current is related to the bus voltage through:



































=

lllg

glgg

Y Y

(A.1)

The current-voltage relationships are reduced to:

eqeq

gEYI = (A.2)

where

lllglg

gIYYII 1−

−= (A.3)

…..

INTERNAL

AREA

EXTERNAL

AREA

Ib Ig

…… …….

APPENDIX 157

It denotes the equivalent current injection vector and

1YYYYY −

−= llglgg

gg (A.4)

denotes the equivalent bus admittance matrix [50].

• The alternative way to improve accuracy of the Ward-type is to retain selected generator

nodes yielding ‘the Ward-PV equivalent’ [52-50].

• Replacing, by ‘grouping and aggregation of generator nodes’ [51], each selected group

by one equivalent generator node, ‘the reduced Ward-PV equivalent’ can be obtained.

Fig. A.2 shows schematically the reduced Ward PV-equivalencing.

Fig. A.2.- Reduction of the Ward-PV equivalent with n number of equivalent generator nodes.

Disadvantages:

• This approach is based upon the linearized differential equations of the generator rotor

movement. Hence, such equivalents do not retain the dynamic properties of the system.

• A major drawback of all these approaches is the unreliability of the analysis due to the

oversimplification of the load nodes [53, 57].

In order to overcome this weakness,

Reduced

WARD-PV

Equiv.

Ib .

WARD-PV

Equiv.

{g1}

{g2}

{gn}

APPENDIX 158

• Bergen [54], Podmore and Germond considered non-linear loads.

• Pai [55] initiated a method based on the assumption that the complex ratios of the

voltage phasors at the generator buses to those at the load buses are constant.

Dynamic Ward equivalent

The main property of this method is a correction formula that allows to update the equivalent

current injections at the retained buses as follows:

gΙΙΙ ∆+= 0 (A.5)

Here the subscript ‘0’ refers to the current in the base case operating point. The equivalent

current increments can be considered into the generator electric power formulation by:

ii em

iPPDM −=+ •••

δδ

(A.6)

where Pei is the generator electric power obtained from a load flow solution.

When the system is reduced to the internal generator nodes, the equivalent electric powers

and their increments are found through:

∑

−−=

gojgoi

gogo

eo YEEP ji

)cos(

ϑδδ

(A.7)

)cos( eq

goi

eIEP i

ψδ

∆−∆=∆ (A.8)

eiii PPP ∆+= (A.9)

where

• eq

∠are the entries of the equivalent bus admittance matrix from (A.4).

• eq

∆∠∆ are updated by the sensitivity matrix.

Disadvantages:

• The sensitivity matrix is based upon the linearized differential equations of the generator

rotor movement.

• The correction formula is performed on linear load models and power flow solutions.

APPENDIX 159

A.2. Modal-based equivalencing

The external area dynamic, linearized at the operating point, is expressed by [67]:

UDXCI

UBXAX

∆+∆=∆

∆+∆=∆ •

(A.10)

where

• X is the original state variables

• UT is the node terminal bus voltage at interconnected points

• IT is the node injection current at interconnected points

• A, B, C, D are the coefficient matrices composed of generator equations, coordinate

transformation equations, algebraic equations of transmission network, etc.

• ∆

∆∆

∆ is the small deviation from initial value.

The Laplace transformed frequency response formula can be written as:

()

j UBAIX ∆−=∆ −1

(A.11)

The frequency response can be simplified by diagonalizing the system using eigenvalues

and eigenvectors as follows:

−

YTX = (A.12)

with:

diag

{}

ee ,....,

1 (A.13)

{}

vvT ,...,= (A.14)

Using this transformation, the Laplace transformed linearized system can be diagonalized:

()

-1 j UBTΛIY ∆−=∆ −1

(A.15)

where

APPENDIX 160

• Λ

ΛΛ

Λ, T is the diagonal matrix of eigenvalues and the matrix containing eigenvectors,

respectively.

• Y represents the transformed state variables or state vector in frequency domain on

eigenvector basis.

After the transformation, it is easy to calculate the frequency responses for all input

frequencies. In general, these equations cannot be interpreted as models of physical

devices. Furthermore, model reduction based on modal analysis requires computation of

eigenanalysis. Basic concepts of reduction by modal are the reduction by aggregation of

similar nodes and elimination of modes, which have large eigenvalues and not included in a

similar mode group. This reduction is executed until the number of modes is reduced to a

specific number [58-63].

Disadvantages:

• The modal dynamic equivalent, however, has a limited application. it doesn’t have a

structural and physical identity. It is a purely mathematical representation the external

area and cannot retain its non-linear characteristics.

• The modal technique deals with the modes of the linearized system, in order to eliminate

the less significant ones for the disturbance of concern [64].

A.3. Coherency-based equivalencing

• The coherency was proposed by Podmore [17]. In this context, simplified and linearized

equations were defined to express the accelerating power deviations of each generator.

• The swing curves obtained are processed to determine the coherent groups of

generators. Thus, two generators buses are defined as coherent if their phase angular

difference is constant within a certain tolerance over a certain time interval.

Disadvantages:

• The determined coherent groups are independent of the amount of detail in the machine

model, i.e. the real modeling parameters and physical properties of the generators.

• Coherent groups are dependent of the size and art of the disturbances. Therefore, the

determined equivalents are not exact enough.

APPENDIX 161

Remarks:

• Methods proposed by Di Caprio [18], Avramovic [21], Kokotovic [22] and

Balasubramanian [23] are based on the coherency concept.

• The coherency is determined using different magnitudes of the generator, such as

moment inertia [23], rotor angle [17], relative rotor angle [67], modes of the eigenvalues

[66], and similarity of eigenvectors, amongst others.

• With the coherency-based procedure, which is presented in [23, 24], direct methods,

such as the transient energy function methods [44, 45] and extended equal area criterion

[46, 47] can be made even faster and optimatized.

• The property of coherency of Chang and Abibi [72], Ohsawa and Hayashi [74], Pai and

Narayama [75], De Mello [73], Podmore [17], Germond and Podmore [34] are taken from

stability simulation cases of the original system, or from inspection of the contribution for

the system potential energy associated with the relative motion between each pair of

external area generators.

Coherency in frequency domain

• Coherency of linear system swings depends on the frequency and damping of a

particular swing may be detected. For one selected mode the frequency and damping

are evaluated. Thus, coherency is described by the corresponding right eigenvector.

• Considering only that the generator are characterized by the reduced right eigenvector,

which describes the so-called rotor mode shape, coherent generator groups can be

identified based on the small angles between the elements of this reduced eigenvector.

• It is shown in Figure A.4. The magnitudes of the vectors are not significant.

Fig.A.4.- Recognition of coherent generators based on generators electromechanical eigenvector

Coherent generator

group A

Non-coherent

generators

Non-coherent

generators

Coherent generator

group B

Coherent generator

group C

Non-coherent

generator

Right Eigen-

vector to λ

λλ

λi

APPENDIX 162

A.4. Hybrid modal procedures

Based on ‘modal analysis’ and ‘coherency identification’, following hybrid procedures show

important alternatives to develop dynamic equivalents:

(i) Modal coherency using frequency response and

(ii) Synchronic modal equivalencing.

(i) Modal coherency using frequency response [65-68]

• After the linearization, diagonalization and transformation of (A.10), the frequency

response for all input frequencies is calculated, since the matrix to be inverted for each

frequency now is a complex diagonal matrix, as can be seen in Fig. A.3:

Fig. A.3.- Frequency response for input m.

In Fig. A.3 all inputs ui are zero, except input m. The excitation frequency is ωs. The output

vector element δ k

(ωs, m) represents the relative rotor angle in frequency domain of

generator k for a perturbation with frequency ωs, applied at input m.

According to the frequency analysis, an index Ci,j is calculated and compared between the

external generators i and j for a sine shaped perturbation of frequency ωs, applied at a single

input m as follows.

11 11

,),(),(

),(),(

∑∑

== 













−

sji

sjsi

ji mm

ωδωδ

(A.16)

Two generators i and j are coherent with respect to the perturbation in the internal area if

their index is less than a certain tolerance. Thus, a grouping process according to this index

is realized.

System

Model

u1=0

um=1

uM=0

δ1(ωs,m)

δk(ωs,m)

δK(ωs,m)

APPENDIX 163

(ii) Synchronic modal equivalencing [69]

It generates equivalents using both a linear multiport admittance, replaced by voltage

controlled injected currents at the replaced generator buses and external reference

generators, one for each external synchronic group.

• The decomposition into synchronic areas is defined with respect to a selected subset of

modes

of a linearized model.

• Two generators are synchronic, if their angular variations are exactly or approximately in

constant proportion for any transient, in which only the modes in

are excited.

• The injected current as multiport admittance is a linear function of the voltages of the

buses of the equivalenced generators. This approach grows out of multi-area ‘Selective

Modal Analysis’ [70, 71] and ‘slow-coherency‘[12].

A.5. Linear model reduction

This analytical method reduces the order of a full linear system (A.10) [76-78]. The transfer

function matrix between outputs Y and inputs U in (A.10) is given by:

() ( )

DBAICG +−= −1

ωω

jj (A.17)

One of the main reasons for model reduction bases on the fact, that many poles of the

transfer function are compensated by zeros. The positions of the zeros depend on the

chosen inputs and outputs, whereas the poles are the same for the whole model.

The task of model reduction, in terms of system theory, can be formulated for a

asymptotically stable system, i.e. Re {λi (A)}<0 as follows:

uBxAx RRRR ˆ

ˆ+=



(A.18)

uDxCy RR += ˆ

ˆ (A.19)

with R

ˆas state-vector of the reduced system with nR < n and Re {λi ( R

ˆ)} <0.

For the approximation error can be used:

(i) the H∞−Norm or

(ii) an optimization procedure.

APPENDIX 164

(i) The H∞−Norm of the difference between the frequency response matrices of the non-

reduced and reduced system is:

)}(

)({)(

)(

ωωσωω

jjpusjj RR GGGG −=− ∞ (A.20)

where

•

=)(

is greatest singular value.

• )(

jG is the transfer function matrix of the system.

• )(

Gis the transfer function matrix of the reduced one in frequency domain.

The greater the dimension nR the smaller is the approximation error.

(ii) The dynamic equivalent can be calculated solving the following minimization problem as

optimization procedure using e.g. genetic algorithms,

{

}

∑−

kRk

λλ

)

()(min _

AA (A.21)

where

• K is the set of operating conditions under study;

• λ(Ak) is the set of electromechanical modes with relevant contributions of generators

of the studied system;

• λ(AR_k) is the associated set of electromechanical modes of the reduced system.

A multi-machine power system model linearized around a equilibrium point is represented

by Ak and defining AR_k as the state matrix of the corresponding linearized reduced model in

(A.17) around the kth equilibrium point, when just generators of the internal area and some

fictitious generators representing the external area are retained [79-83].

On this base, least-squares methods [79-81], genetic algorithms in [82] according to balance

realizations and statistical approaches [83] can be used too.

Disadvantages:

• The significant drawback of this method is principally that the resulting equivalent system

is not composed of physical components.

• The estimation of a set of state variable parameters is based upon the linear sate space

system that is assumed to describe only linear parts of the reduced power system.

APPENDIX 165

A.6. Model identification methods

The relationship between inputs, outputs, and unmeasured inputs can be described by the

state-space representation described in (A.10) and illustrated as follows:

Fig. A.5.- Input Signals u, Output Signals y, and Disturbances e

The identification of the model parameters from (A.10), i.e. A, B, C and D should be

realized by several modeling methods in parametric form.

Assuming the discretized signals are related by a linear system, the relationship of general

linear models can be written as.

() ()() ()()

t qHt qGt euy += (A.23)

where

• q is the shift operator.

• G(q) is the transfer function of the system.

• H(q) represents the disturbance filter.

The estimation of the transfer functions G(q) and H(q) of a model can be realized with

different parametric or nonparametric models [89, 90].

The identification of dynamic equivalents can be solved using the parametric ARX model with

() ()

() () ()

q=qG nk- = (A.24)

where nk corresponds to the number of delays from input to output and A and B are

polynomials in the delay operator q-1 described as follows:

()

1nb+

qbqbbqB

qaqa1qA

−−

+++=

(A.25)

The model is usually written as:

()() ()( ) ()

tenktuqBtyqA +−= (A.26)

Linear System

APPENDIX 166

The identification of the ARX-model parameters ai and bi is based on the least squares

estimation method.

Disadvantages:

• Linearization around the equilibrium yields mathematically tractable linear models but

according to the power system components not completely non-linear compatible.

• Non-linear systems could be approximated partly in their normal ranges of operation.

• The disadvantage of this method is principally that the resulting dynamic equivalent

model is not composed of physical components.

A.7. Flow chart of the power system simulation tool PSD

Disturbance

Network equation

Algebraic Solution->Power flow calculation

(Obtaining the initial state)

New generation of the network equation, because some values

of the admittance matrix have been changed. (Yjj,Yij, Yji, Yii).

Under certain conditions new network topology. and eventually

omission of the whole network.

Solving of the new network equation

Generator / Grouping of generators / Aggregation-Splitting

/ Static network reduction

Integration of regulators

Values for the first (to n) time interval (

∆

APPENDIX 167

A.8. Flow chart of the coupling of the machine model to the analysis algorithm PSD

Terminal

voltage

Frequency

u"∆q

u"∆qqo uu "" ∆+

do uu "" ∆+

u"K

Differential

equations of the

control and turbine

system

Quasi-state equation

according to (4.2) of

the rotor flux linkages

and mechanical

system

Transformation into

the network

coordinate system

and conversion in

the driving current

Transformation into

the rotor coordinate

system

Electromechanical

unsymmetry

Z∆

Non-linear static loads

uu q

P























,

Differential

equations of the

exciter system

dmittance

Matrix

∆

Power

Terminal

voltage

Signals

i∆

Synchronous machine model

Iteration

Po, Qo, Uo, foare static values from the

load flow calculation.

pu=qu=0 corresponds to constant power.

pu=qu=1 corresponds to constant current.

pu=qu=2 corresponds to constant

admittance.

APPENDIX 168

APPENDIX B Identity recognition

algorithms

B.1. Hierarchical clustering

Hierarchical clustering starts with the calculation of all distances dij between all the objects

in the multidimensional space. Thus, the number of cluster c initially corresponds to the

number of objects of generatorts N. In total, N(N+1)/2 distances are computed hierarchically.

If at a certain step during the clustering process two clusters P and Q are agglomerated into

a new cluster K, then the distance between cluster K and any other cluster R can be

computed according to the following general hierarchical distance form:

),(),(4),(3),(2),(1),( RQRPQPRQRPRK dddddd −+++=

δδδδ

(B.1)

Where the coefficients are different for different strategies, such as the single-, complete-,

simple average-, average linkage, centroid, median and Ward’s method after table B.1.

Table B.1.- Computation of distances [101]

Name 1

Single linkage 1/2 1/2 0 -1/2

Complete

linkage 1/2 1/2 0 1/2

Simple

Average

linkage

1/2 1/2 0 0

Average

linkage QP

+ 0 0

Centroid

()

n n

+ 0

Median 1/2 1/2 -1/4 0

Ward

QPR

nnn

QPR

nnn

QPR

nnn

− 0

APPENDIX 169

B.2. Partitioning clustering: K-means

This clustering permits objects to change group membership through a cluster formation

process [99]. The reallocation occurs according to the following optimality criterion [101]:

()

mincxKES

lílj

inerrorsquared →−= ∑∑∑

===1

)( (B.2)

where xj is a data vector, ci is a centroid vector of a group, M denotes the number of

variables, N the number of objects and K the number of cluster groups.

The total squared error of the procedure is constant; the minimum of squared error in the

clusters corresponds to a maximization of the squared error between the clusters, as follows:

inerrorsquaredTotalbetweenerrorsquared KESSEKES )()( −= (B.3)

The flow chat of K-means algorithm is described schematically as:

Fig. B.1.- Flow chart of the concept of K-means for a set of clusters

Iter ++

Maximal number of iterations or change the

centres within a

res

ecified accurac

Calculate the cluster criteria for the given c

Select the best clustering based on the examination of the

various cluster quality criteria

For each object xj the distance to all cluster centres cis

computed and assigned to the nearest cluster

Initial cluster centres for c clusters according to a random way

Recompute the cluster centres ci according to (B.4)

c = cmax

Define a set {cmin,..,cmax} . Set the current number of

clusters c to the first value of the set, cmin

Iter ++

APPENDIX 170

The algorithm for this method can be modified to realize this procedure automatically not

only for one specified number of cluster, but rather for a set of clusters between cmin and cmax.

The iterative partitioning of reassignment in the cluster centers is calculated by:

Kix

i,..,1

=∴= ∑

(B.4)

B.3. Fuzzy clustering

The key idea of this clustering is to extend the classical within groups sum of squared error

objective function to a Fuzzy version by minimizing an objective function [102-105]. The

Fuzzy clustering conditions can be derived as the following way.

In classical variational theory, a fundamental continuous-time continuous-state functional

optimisation problem (minimizing) in its simplest form is defined as:

mindttytytFJ

→













=∫⋅

)(),(, (B.5)

If bundle y is a weak relative minimum to (B.5), and if t is any point in [to, t1] where

derivative )(ty

⋅ exists, then the Euler-Lagrange equation in its differentiated form holds:

∂

−

∂

•

F (B.6)

A functional optimisation problem in discrete time can be defined as:

()

mintytytFJ

y→+= ∑

)1(),(, (B.7)

Such a functional is solved by discrete-time Euler-Lagrange equations. Although these

results do not consider constraints, side constraints, including Lagrange constraints and

isoperimetric constraints, can be included. Constraints are typically handled by using

Lagrange multipliers to transform a constrained problem into an unconstrained one, and by

applying the Euler-Lagrange condition on the Lagrangian function to get the Euler Lagrange

equations, as follows:

APPENDIX 171

Thus, a generalized Lagrangian function (constrained functional):

mindttytytLJ

→













=∫⋅∧ 1

),(),(,

(B.8)

subject to 0))(),(,( =

⋅

tytytg (B.9)

with

))(),(,())(),(,()),(),(,( tytytgtytytFtytytL T⋅⋅⋅ +=

λλ

(B.10)

may be solved by the Euler-Lagrange equation, too:

∂

−

∂

•

L (B.11a)

∂

−

∂

•

dL (B.11b)

This concept can be extended to consider a discrete constrained optimisation problem of

Fuzzy clustering. The objective function is:

mincxdCXJ

m→−== ∑∑∑∑

==== 1

),,(

µµµ

(B.12)

where

• N corresponds to the number of objects and K to the number of clusters.

• dij is the distance between the datum vector and cluster centers.

• µij is the membership degree of datum xj to cluster ci.

• m>1 is the fuzziness index and influences the “fuzziness” of the obtained partition.

The objective function is subjected to the following discrete variable constraints of

ij:

0:1

,>≤<∀ ∑

, 1:1

,=≤<∀ ∑

(B.13)

APPENDIX 172

(B.12) can be minimized, while too for all µ the sum ∑

∈Cc

ccxd ),(

will be minimized. A

generalized discrete Lagrangian function of (B.12) is:

()

cxdL

c)1)((,),( 2−+= ∑∑

εε

µλµλµ

(B.14)

This applicability doesn’t pose problems here because it doesn’t require the differentiability.

Applying the Euler-Lagrange equation following equations are obtained:

01)(),( =−=

∂

∂∑

µλµ

(B.15)

0),(**),( 21 =+=

∂

∂−

λµλµ

cxdmL m

(B.16)

Dissolving the membership degree in (B.16) and replacing in (B.15) results:

2),(*

−











−= m

ccxdm

(B.17)

∑∑

−−

−==

CiCi

iixd

εε

)

),(

()(1 (B.18)

The resulting constant:

∑−

−=−

ixd

)

),(

(

)(

together with (B.17) forms the membership degree condition as:

)

),(

)

),(

(

1−

−

∑

Ccxd

ixd

)1(

)

),(

(

−

−













=⇒∑

cixd

cxd

)1(

,)(

−

−













=∑

ji d

i=1..K; j=1..N (B.19)

For the initial values of the membership degree, the following formulation can be used:

APPENDIX 173

∑

−

(B.20)

Another condition for local extreme is based upon the differentiation of Jm to ci using the

residue principle as:

,||||||||0 ij

c−

∂

=−

∂

=∑∑∑

===

µµ

(B.21)

cxtcx ijij

||||||)(||

lim0 −−+−

=→

∑

))()()))(())(((

lim0 0

,ij

ijij

ji cxcxtcxtcx

−−−−−−−= →

∑

ξξµ

∑∑

→

−−=

+−−

ji cx

tcxt

,)(2

)(2

lim0

ξµ

ξξξ

According with the condition for local extreme:

∑

=−⇒=

∂

∂N

cxJ

,0)(0

(B.22)

following condition can be reached:

i≤≤∴=⇒

∑

(B.23)

Conditions (B.19), (B.20) and (B.23) are first-order necessary conditions for local extreme of

Jm. Thus, all Fuzzy algorithms used to solve (B.12) should satisfy (B.19), (B.20) and (B.23).

The algorithm realized for Fuzzy clustering is described schematically as follows:

APPENDIX 174

Maximal number of iterations or change the

optimization tolerance?

Fig. B.2.- Flow chart of the concept of fuzzy c-means clustering.

B.4. Relationship between K-means (hard clustering) and Fuzzy clustering

The difference between K-means and Fuzzy clustering can be described mathematically

according to the conditions (B.19) and (B.23) as follows:

• Considering the convergence m→1, following equations can be obtained:

















≠∀<

























∑−

+→+→ sonst

ciixdcxd

ixd

cxd

m;0

),(),(;1

)

),(

(

lim}{lim

(B.24)

ik ≤≤=













∑

+→+→ 1,lim}{lim

(B.25)

Centroids C and fuzzy labels are available using the

constraints (B.13)

Initial cluster centres by means of random way.

Iter ++

Compute the matrix of distances between object

and the centre of

cluster i and the membership coefficient matrix {

µµ

ij} based on (B.20)

Compute the cluster centres using the derived necessary conditions

defined in (B.23)

Recompute the distance of data to centroid

The membership coefficients are updated according to the necessary

condition defined in (B.19)

APPENDIX 175

This convergence corresponds to the K-means partition and represents the classical

within-groups sum of squared errors. The cluster centers are defined as the average sum

of objects assigned to the corresponding clusters.

• Considering the convergence m∞→ , i.e. in this case, the clusters are nearly

undistinguishable. The following equations can be obtained:

ixd

cxd

)

),(

(

lim}{lim

























∑−

∞→∞→

(B.26)

Ki C

m≤≤∴==













∑

∞→∞→ 1

lim}{lim 1

(B.27)

Where uk and ci represent a single large cluster group, in which is included all data

vectors. All those objects show same membership degree [105].

B.5. Self organizing features maps (SOFM)

The unsupervised ANN learn to recognize groups of similar input vectors in such a way that

neurons physically close together in the neuron layer respond to identical objects [110].

The following graphic shows the schematic representation of a SOFM.

Fig. B.3.- Schematic representation of SOFM

‘Winner’ neuron and neighbor neurons

Output

neurons

(i=1..k)

1 j N

Input neurons (j=1…N)

Weight W

APPENDIX 176

Thus, the weights of the winning neuron has to be adjusted so as to move it closer to the

input vector by the following learning rule (Kohonen learning rule):

)]1()()[,()()1()( −−+−= tWtxdtNttWtW gjggg

(B.28)

Hereby, the corresponding vector of the input weight matrix Wg of the neurons has to be

adjusted and updated according to a learning function α(t) and neighborhood function N(t).

The ‘winner’ neuron can be determined by the following Euclidean distance criterion that for

the gth neuron obtains minimum value.

minwxgD

gljl →−= ∑

)()( (B.29)

In this expression xjl is the pattern of the time response of matrix X(M,N) and wgl is the weight

vectors of W matrix of order (k,M) with k corresponding to number of clusters.

B.6. Clustering quality

A problem in clustering is the choice of the correct number of clusters and its quality.

Clustering criteria can help to suggest these aspects. The Fig. B.4 shows these aspects:

Fig. B.4.- Schematic representation of the clustering quality

Worse assignmen

within the clustering

due to the high

degree of dispersion

in clusters.

Best assignmen

within the clustering

based on compac

clusters.

APPENDIX 177

The problem has been discussed extensively in the literature in [93-98]. The question,

"which clustering criterion gives the best results?" remains. It is suitable that a fruitful

approach implies the application of several criteria because each has strong and weak

points. The criteria implemented are the total error sum of squares, Davies-Bouldin index and

the Silhoutten coefficient discussed extensively in [93-99], whose criteria principally are

based on the within-cluster variation.

B.7. Practical comparison of identity recognition algorithms

The strengths and weaknesses of each algorithm implemented in [141] are:

- Accuracy.- The electromechanical K-means and Fuzzy show a similar high accuracy in

the forming of identical groups of generators with a non-significant computational

requirement. In contrast to this, Kohonen SOFM shows the worst accuracy.

- Complexity and processing time.- The resource consumption varies considerably

between the K-means and SOFM. SOFM are impractical for large power system. SOFM

inefficiency, resulting from search of the winner neuron, increases the time cost and it

shows a slower convergence rate. The simplicity of the K-means and its speed of

convergence are obvious advantages for the applicability in large power systems. Fuzzy

has own computational advantages due to its optimization nature, but its processing time

due to the iterative operation of the objective function, is computing resource-based.

- Efficiency.- K-means, hierarchical and Fuzzy efficiency depends largely on the inputs

patterns. The efficiency of SOFM depends on how accurately the learning and

neighborhood function represent the input patterns in the weight vectors.

Electromechanical Fuzzy and K-means are numerically stable as well consistent, and it is

applicable to power systems with larger amounts of generators.

APPENDIX 178

APPENDIX C DANN-based dynamic

equivalencing

C.1. ANN preliminaries

Architecture and topology

A non-linear system can be captured by the following ANN structure:

- The feedforward ANN mathematical description is given by:

2112

),h( BBWW ++== )

ϕϕ

ϕκ

κκ

κθ

θθ

θϕ

ϕϕ

(C.1)

where ϕ

ϕϕ

ϕ are neuron inputs, Wi is the weight matrix and Bi denotes the bias that is

considered, for simplicity, a weight associated with an unitary input.

- The activation function κ(.) representing the non-linearity property of the ANN, may be:

The tangent hyperbolic: x

xx −

−

== )tanh()(

(C.2)

Sigmoidal type: x

)()(

σκ

(C.3)

Gaussian: 2

)(

−

(C.4)

- The ANN weights are updated by the back-propagation according to:

)1(

)(

)( −∆+

∂

−=∆ t

θα

ηθ

(C.5)

APPENDIX 179

where η is the learning rate, which controls the rate at which the ANN learns, )(

)(

∂

∂ is

the derivative of the error with respect to the weight and α is the momentum.

- The output layer is a linear type function.

C.2. Learning strategies

The learning strategy used is to minimize the difference between the desired and the actual

output of the network, using following optimization strategies [124-128]:

• The Back-propagation is based on the propagation of the output errors until it reaches

the first layer of the neural network. The ANN weights in (C.5) can be updated by the

back-propagation according to the sensitivity of the error with respect to the weighting.

• The Levenberg-Marquardt optimization uses this approximation to the Hessian matrix

in the following Newton-like update:

[]

)()()1( 1ttt TT EJIJJ −

+−=+

µµ

µθ

θθ

(C.6)

JJH T

= (C.7)

)(t

TEJg = (C.8)

where xk is a vector of current weights and biases. When the scalar µ is zero, this is just

Newton's method, using the approximate Hessian matrix. When µ is large, this becomes

gradient descent with a small step size. The Hessian matrix and the gradient can be

approximated in (C.7) and (C.8). J is the Jacobian matrix that contains first derivatives of

the network errors with respect to the weights and biases, and E is a vector of errors.

C.3. Modeling

The modeling may be divided into the following basic functions:

• Regression vector.- It may be implemented in form of a delay space embedding of

input and output variables and represents the long-term prediction capability of a

model providing sufficient information to reconstruct the states of the system.

A collection of time lags in a regressor vector space of d dimensions can be:

()

)..)1((),..,2(),(),)1((),..,(),()( TdtyTtyTtyTdtuTtutu t −−−−−−−=

ϕϕ

(C.9)

APPENDIX 180

• Non-linear mapping.- The non-linear mapping is described by the modeling

capability of the ANN. The most difficult part of the system modeling is not the

parameter estimation but the selection of the suitable model structure and the feature

extraction for the nonlinear mapping

C.4. Non-linear models

The following non-linear models may be considered [117, 118]:

- Non-linear Output Error (NOE structure):











+−+−=+ )1(),...,(),1(),...,(h)1(

^^^^

uy ntutuntyty ty (C.10)

- Non-linear AutoRegressive models with eXogenous inputs (NARX structure):

()

)1(),...,(),1(),...,(h)1(

^^ +−+−=+ uy ntutuntyty ty (C.11)

- Non-linear AutoRegressive Moving Average with eXogenous inputs (NARMAX):

)()1(),...,(),1(),...,(h)1(

^^^^

tentutuntyty ty uy +











+−+−=+ (C.12)

where

y is the output of the identification model

h, y of the non-linear system.

C.5. Power system model

Modeling of the excitation system

The excitation system including PSS, whose basic function is to add damping to the

generator rotor oscillations by controlling its excitation, may be represented by the equations:

PSS

EEEE uuxx BBA 21 ++=

• (C.13)

)( E

fdfd EE = (C.14)

APPENDIX 181

where upss is the input signal to the PSS or the exciter input voltage.

If it is derived from the rotor velocity then upss =

. Ut corresponds to the terminal voltage

magnitude. In general, Efd is a linear function of xE except when limits of Efd are to be

considered.

Control of the excitation system of a synchronous machine has a very strong influence on its

performance, voltage regulation, and stability [135, 137, and 138].

Modeling of the turbine and governor

The turbine-governor system can be expressed generally by the equations [136]:

ref

TTTT 21 BBA xx ++=

•

(C.15)

)( T

mm TT = (C.16)

where ref

P is the reference power set by Load Frequency Control (LFC) or Automatic

Generation Control (AGC).

The variables and parameters of equations (C.13–C.16) are given in [136]. They are

important pieces of power system equipment.

The dynamics of the turbine and governor are normally much slower than that of the exciter.

Hence, the dynamics of these devices can be neglected.

Remarks:

Thus, the generator equations comprise following important parts [136]:

- Rotor electrical,

- Mechanical,

- Excitation and

- Turbine-governor equations according to the basic components of power systems.

The interconnections among the various subsystems of the generator are shown

schematically as follows:

APPENDIX 182

Fig. C.1.- Interconnections among subsystems

Hereby f1, f2 and f3 are non-linear functions of the input variables obtained from equations

given above, uq and ud are derived from the knowledge of the phasor t

−

expressed with

reference to a common reference frame and the generator rotor angle δ, as given below

according to f3 depicted in Fig. C.1.

)(

δθδ

−−

−==+ j

tdq eUeUjuu (C.17)

−

U and

−

I are the terminal voltage and terminal current phasors of the generator.

From this expression, the following relationship can be obtained:

Uu tq )cos(

δθ

−=

Uu td )sin(

δθ

−= (C.18)

and t

U based on function f1 in Fig. C.1 can be expressed as:

qdt uuU += (C.19)

where θ is the bus angle at the generator terminals obtained from the network solution.

Pm(xT) xM(δ, ω)

Excitation

System

Turbine

Governor

Rotor

Electrical

Mechanical

AC Network

UPSS

xR(E’q, E’d)

ud, uq

−

ud, uq

Efd(xE)

APPENDIX 183

APPENDIX D Data sets of 16 Multi-

machine system

Table D.1.- Data sets of investigated multi-machine systems

16-machine

system

12-machine

system

European Interconnected

Power System UCTE/CENTRAL

Number of generating

units 16 12 496

Number of transmission

lines 54 60 2098

Number of Nodes 66 58 2016

Number of two winding

transformator units 16 12 1032

Number of three winding

transformator units 12 0 0

Table D.2.- Generator data set of 16 multi-machine system

Name Mod.

Order

[MVA] Tm UN[kV] ra

[p.u.]

xd‘

[p.u.]

xd’’

[p.u.]

A01aG 5 220 1518 15,75 0,001 0,195 0,43 0,225

A01bG 5 220 1518 15,75 0,001 0,195 0,43 0,225

A02aG 5 220 1518 15,75 0,001 0,195 0,43 0,225

A02bG 5 220 1518 15,75 0,001 0,195 0,43 0,225

A03_G 5 220 1518 15,75 0,001 0,195 0,43 0,225

A06_G 5 247 1729 15,75 0,002 0,19 0,36 0,24

B02aG 5 259 2719,5 15,75 0,001 0,156 0,29 0,2

B02bG 5 259 2719,5 15,75 0,001 0,156 0,29 0,2

B03_G 5 259 2719,5 15,75 0,001 0,156 0,29 0,2

B08_G 5 247 1729 15,75 0,002 0,19 0,36 0,24

B10_G 5 259 2719,5 15,75 0,001 0,156 0,29 0,2

C02_G 5 247 1729 15,75 0,002 0,19 0,36 0,24

C07_G 5 247 1729 15,75 0,002 0,19 0,36 0,24

C10_G 5 247 1729 15,75 0,002 0,19 0,36 0,24

C12_G 5 247 1729 15,75 0,002 0,19 0,36 0,24

C14_G 5 247 1729 15,75 0,002 0,19 0,36 0,24

APPENDIX 184

Table D.3.- Transmission line data set of 16- machine system

From bus To bus Long r [p.u.] x [p.u.] b [p.u.]

A01_L380 A04_L380 50 0,0155 0,1358 0,0267

A01_L380 A02_L380 100 0,0155 0,1358 0,0267

A04_L380 A05aL380 100 0,0155 0,1358 0,0267

A02_L380 A05aL380 100 0,0309 0,266 0,0136

A02_L380 A05bL380 100 0,0309 0,266 0,0136

A02_L380 A03_L380 100 0,0309 0,266 0,0136

A05aL380 A07_L380 50 0,0309 0,266 0,0136

A05aL380 C01_L380 200 0,0309 0,266 0,0136

A05aL380 A05bL380 0,1 0,01 0,01 0,01

A05bL380 A07_L380 50 0,0309 0,266 0,0136

A05bL380 B01_L380 220 0,0309 0,266 0,0136

A06_L220 A07_L220 50 0,0792 0,2901 0,0126

B01_L380 B02_L380 100 0,0309 0,266 0,0136

B02_L380 B03_L380 100 0,0309 0,266 0,0136

B01_L220 B04_L220 40 0,0792 0,2901 0,0126

B01_L220 B07_L220 40 0,0792 0,2901 0,0126

B02_L220 B04_L220 40 0,0792 0,2901 0,0126

B02_L220 B05_L220 50 0,0395 0,1474 0,0248

B02_L220 B06_L220 50 0,0792 0,2901 0,0126

B02_L220 C08_L220 180 0,0395 0,1474 0,0248

B03_L220 B06_L220 40 0,0792 0,2901 0,0126

B03_L220 B11_L220 70 0,0395 0,1474 0,0248

B05_L220 B09_L220 40 0,0395 0,1474 0,0248

B07_L220 B08_L220 50 0,0792 0,2901 0,0126

B08_L220 B09_L220 60 0,0792 0,2901 0,0126

B09_L220 B10_L220 50 0,0792 0,2901 0,0126

B10_L220 B11_L220 40 0,0792 0,2901 0,0126

C01_L380 C02_L380 50 0,0309 0,266 0,0136

C01_L380 C07_L380 80 0,0309 0,266 0,0136

C02_L380 C03_L380 50 0,0155 0,1358 0,0267

C03_L380 C04_L380 90 0,0309 0,266 0,0136

C03_L380 C05_L380 70 0,0309 0,266 0,0136

C04_L380 C06_L380 70 0,0155 0,1358 0,0267

C04_L380 C07_L380 70 0,0155 0,1358 0,0267

C05_L380 C06_L380 80 0,0309 0,266 0,0136

C05_L110 C16_L110 30 0,192 0,4 0,0085

APPENDIX 185

C05_L110 C19_L110 20 0,096 0,2 0,017

C06_L220 C08_L220 40 0,0792 0,2901 0,0126

C06_L220 C15_L220 40 0,0792 0,2901 0,0126

C07_L380 C08_L380 80 0,0155 0,1358 0,0267

C08_L220 C09_L220 40 0,0792 0,2901 0,0126

C09_L220 C10_L220 50 0,0792 0,2901 0,0126

C10_L220 C11_L220 30 0,0792 0,2901 0,0126

C11_L220 C12_L220 30 0,0792 0,2901 0,0126

C12_L220 C13_L220 40 0,0395 0,1474 0,0248

C13_L220 C14_L220 40 0,0792 0,2901 0,0126

C14_L220 C15_L220 40 0,0395 0,1474 0,0248

C14_L110 C16_L110 20 0,192 0,4 0,0085

C14_L110 C17_L110 20 0,192 0,4 0,0085

C17_L110 C18_L110 20 0,192 0,4 0,0085

C18_L110 C19_L110 20 0,192 0,4 0,0085

Table D.4.- Comparison of aggregation algorithms considering the classical inertial aggregation and

the proposed splitting-based aggregation by the mean value of J of the intern machines for a fault

located at the internal node with different number of external equivalents.

Number of

Dynamic

equivalents

3 6

Splitting

with Fuzzy 0,9925 0,9935

Classical

Aggregation

with Fuzzy

0,989 0,9926

Classical

Aggregation

with K-

means

0,992 0,9938

Table D.5.- Evaluation of the prediction capability of ANN considering different operating conditions

and points calculating )( jE S

Internal

Area

Changed

operating

point

Training

operating

point

A 0,972 0,991

B 0,9845 0,994

C 0,959 0,968

APPENDIX 186

Table D.6.- Sum squared distance error and average error of the predicted boundary behavior

following disturbances at all non-trained nodes of internal area A considering different operating points

Internal Area A part 380 kV

Nodes A1 A2 A3 A4 A5a A5b A7

Training

operating point 0,034 1,4*10-3 4,2*10-3 3,9*10-3 0,054 1,4*10-3 0,003

Changed

operating point 0,031 5,2*10-3 4,1*10-3 3,9*10-3 0,147 4,4*10-3 3,2*10-3

Internal Area A

part 220 kV

Internal Area A

part 110 kV

A6 A7 A4 A6 Sum squared

error )( jEN

Average

standardized

error function

)( jE S

2,2*10-4 7,1*10-4 1,1*10-4 1,6*10-4 0,099 0,991

2,3*10-4 6,3*10-3 1,2*10-3 0,0101 0,308 0,972

Table D.7.- Sum squared distance error and average error of the predicted boundary behaviour

following disturbances at all non-trained nodes of internal area B considering different operating points

Internal Area B part 220 kV

Nodes B1 B2 B3 B4 B5 B6 B7 B8 B9 B10 B11

Training

operating

point

3,02

*10-4 0,0454 8,52

*10-4

4,65

*10-4

2,29

*10-4

1,57

*10-4

2,67

*10-4

2,69

*10-4

2,37

*10-4

3,68

*10-4

3,65

*10-4

Changed

operating

point

3,87

*10-4 0,0247 0,0016 5,13

*10-4

7,18

*10-4

1,65

*10-4

3,52

*10-4

7,39

*10-4

9,37

*10-4

8,46

*10-4

6,0

*10-4

Internal Area B part 380

B1 B2 B3 Sum squared

error )( jEN

Average

standardized error

function )( jE S

0,026 1,3*10-3 2,1*10-3 0,0784 0,9944

0,072 0,052 0,0276 0,217 0,9845

APPENDIX 187

Table D.8.- Sum squared distance error and average error of the predicted boundary behaviour

following disturbances at all non-trained nodes of internal area C considering different operating points

Internal Area C part 380 kV

Disturbanc

e applied at

nodes

C1 C2 C3 C4 C5 C6 C7 C8

Training

operating

point

0,04 0,0156 2,6

*10-3

4,6

*10-3

6,0*10-

2,7*10-

3 0,16 5,4*10-3

Changed

operating

point

0,2 0,12 0,05 0,09 0,043 0,09 0,3 0,1

Internal Area C part 220 kV

C6 C8 C9 C10 C11 C12 C13 C14 C15

3,1*10-3 0,0185 1,7*10-3 0,40 4,1*10-3 0,027 1,9*10-3 0,06 5,4*10-4

0,067 0,091 0,025 0,237 0,268 0,018 0,07 0,016 0,05

Internal Area C part 110 kV

C5 C14 C16 C17 C18 C19 Sum squared error

)( jEN

verage standardized

error function

)( jE S

3,4

*10-5

1,76

*10-4 4,8

*10-5

1,04

*10-4

8,7

*10-5 6,4

*10-5 0,672 0,968

3,6

*10-3

7,8

*10-3 2,2

*10-3

2,1

*10-3

1,4

*10-5 1,8

*10-3 0,943 0,959

APPENDIX 188

APPENDIX E Data sets of 12 Multi-

machine system

Table E.1.- Generator data set of 12-machine system

Name Mod.

Order

[MVA] Tm UN[kV] ra

[p.u.] xs [p.u.] xd’

[p.u.]

xd’’

[p.u.]

A2_G 5 225 690 15,75 0,001 0,195 0,43 0,25

A3_G 5 275 1518 15,75 0,001 0,195 0,43 0,225

A8_G 5 575 1518 15,75 0,002 0,195 0,43 0,25

A5_G 5 225 1729 15,75 0,002 0,195 0,43 0,225

A12_G 5 225 2719,5 15,75 0 0,195 0,29 0,225

A13_G 5 225 690 15,75 0,002 0,195 0,36 0,225

A14_G 5 225 690 15,75 0,001 0,195 0,29 0,225

A15_G 5 225 690 15,75 0,001 0,19 0,43 0,225

A20_G 5 225 1200 15,75 0 0,156 0,43 0,225

A25_G 5 275 1200 15,75 0 0,156 0,43 0,24

A29_G 5 175 1200 15,75 0 0,195 0,29 0,24

A45_G 5 225 690 15,75 0 0,195 0,43 0,225

Table E.2.- Transmission line data set of the 12-machine system

From bus To bus Long r [p.u.] x [p.u.] b [p.u.]

A16 A4 50 0,0309 0,266 0,0136

A4 A3 50 0,0309 0,266 0,0136

A3 A7 50 0,0309 0,266 0,0136

A31 A6 50 0,0309 0,266 0,0136

A5 A6 50 0,0309 0,266 0,0136

A9 A21 50 0,0792 0,2901 0,0126

A8 A9 50 0,0309 0,266 0,0136

A8 A39 50 0,0309 0,266 0,0136

A39 A12 50 0,0309 0,266 0,0136

A8 A18 50 0,0309 0,266 0,0136

A12 A18 50 0,0792 0,2901 0,0126

A18 A19 50 0,0309 0,266 0,0136

A19 A22 50 0,0309 0,266 0,0136

APPENDIX 189

A12 A13 50 0,0155 0,1358 0,0267

A12 A17 50 0,0309 0,266 0,0136

A13 A38 20 0,0309 0,266 0,0136

A38 A1 20 0,0309 0,266 0,0136

A1 A14 50 0,0309 0,266 0,0136

A17 A37 20 0,0309 0,266 0,0136

A37 A1 20 0,0309 0,266 0,0136

A14 A15 50 0,0309 0,266 0,0136

A15 A36 50 0,0309 0,266 0,0136

A36 A35 50 0,0309 0,266 0,0136

A35 A34 50 0,0155 0,1358 0,0267

A34 A5 20 0,0395 0,1474 0,0248

A13 A10 50 0,0792 0,2901 0,0126

A10 A16 50 0,0395 0,1474 0,0248

A30 A11 50 0,0792 0,2901 0,0126

A14 A11 50 0,0395 0,1474 0,0248

A20 A21 50 0,0309 0,266 0,0136

A21 A23 50 0,0309 0,266 0,0136

A21 A22 50 0,0309 0,266 0,0136

A22 A3 50 0,0792 0,2901 0,0126

A23 A24 50 0,0792 0,2901 0,0126

A24 A25 50 0,0792 0,2901 0,0126

A25 A26 50 0,0309 0,266 0,0136

A26 A3 50 0,0309 0,266 0,0136

A26 A27 50 0,0395 0,1474 0,0248

A27 A28 50 0,0395 0,1474 0,0248

A28 A29 80 0,0309 0,266 0,0136

A29 A47 80 0,0309 0,266 0,0136

A31 A47 80 0,0309 0,266 0,0136

A31 A30 50 0,0155 0,1358 0,0267

A30 A3 50 0,0155 0,1358 0,0267

A28 A32 50 0,0155 0,1358 0,0267

A29 A32 50 0,0155 0,1358 0,0267

A32 A33 50 0,0309 0,266 0,0136

A34 A33 50 0,0309 0,266 0,0136

A25 A40 80 0,0309 0,266 0,0136

A24 A40 80 0,0309 0,266 0,0136

A39 A41 50 0,0309 0,266 0,0136

A41 A40 80 0,0395 0,1474 0,0248

A14 A42 50 0,0395 0,1474 0,0248

APPENDIX 190

Table E.3.- Evaluation of the standarized prediction error of recurrent ANN considering different

disturbance duration (tmin=100 ms, tmax=150 ms) and two sequential disturbances (t1=100ms after

1s, t2=120 ms after 2s)

Number of

Boundary

Nodes

Minimal

disturbance

duration

Tmin

Maximal

disturbance

duration

Tmax

Sequential

disturbances

T1 - T2

3 0,9994 0,9991 0,9981

5 0,9956 0,9952 0,9948

6 0,9874 0,9844 0,9838

7 0,9832 0,981 0,9781

8 0,9816 0,9798 0,978

Table E.4.- Evaluation of the robustness of the recurrent ANN depending on the cases explained in

table 5.2 and networks based on the 12-multi-machine system with different boundary nodes

Number of boundary nodes

Cases of changed operating

points after table 5.2 3 4 6 8

1 (Initial o.p. or trained o.p.) 0,9994 0,9974 0,9874 0,9853

2 (Gen. disconnection) 0,9993 0,9973 0,9865 0,9820

3 (Lines disconnection) 0,9992 0,9975 0,9848 0,9758

4 (Gen.,Line disconnection and

Load reduction) 0,9985 0,9943 0,9830 0,9745

5 (Load reduction to half) 0,9974 0,9931 0,9811 0,9752

Table E.5.- Scenarios to power-flow changes and losses considering 12-machine system with 3

boundary nodes after Fig.B.1

Cases Load Conditions Location in

network

Network power

flow loss (MVA)

1 Initial Loading Condition

(Training operating point) 291,61+j1720,88

2 Generator Disconnection G1 312,93+j1942,50

3 Transmission Line

Disconnection L1, L2 295,42+j1791,61

Generator, Line

Disconnection and Load

Reduction

G1, L1, L2 and

load reduction on

all nodes of

internal area

174,05+j666,69

5 Load Reduction to half

On almost all load

nodes in internal

area

159,72+j488,42

APPENDIX 191

Table E.6.- Scenarios to power-flow changes and losses considering 12-machine system with 4

boundary nodes after Fig.B.2

Cases Load Conditions Location in

network

Network power

flow loss (MVA)

1 Initial Loading Condition

(Training operating point) 218,08+j980,68

2 Generator Disconnection G1 232,93+j1135,52

3 Transmission Line

Disconnection L1, L2 218,93+j1020,40

Generator, Line

Disconnection and Load

Reduction

G1, L1, L2 and

load reduction on

all nodes of

internal area

128,67+j183,49

5 Load Reduction to half

On almost all load

nodes in internal

area

119,32+j56,2

Table E.7.- Scenarios to power-flow changes and losses considering 12-machine system with 6

boundary nodes after Fig.E.4

Cases Load Conditions Location in

network

Network power

flow loss (MVA)

1 Initial Loading Condition

(Training operating point) 244,48+j927,08

2 Generator Disconnection G1 256,43+j1058,50

3 Transmission Line

Disconnection L1, L2 245,6+j975,31

Generator, Line

Disconnection and Load

Reduction

G1, L1, L2 and

load reduction on

all nodes of

internal area

236,82+j895,43

5 Load Reduction to half

On almost all load

nodes in internal

area

225,08+j733,52

APPENDIX 192

Table E.8.- Scenarios to power-flow changes and losses considering 12-machine system with 7

boundary nodes after Fig. E.5

Cases Load Conditions Location in

network

Network power

flow loss (MVA)

1 Initial Loading Condition

(Training operating point) 262,62+j815,51

2 Generator Disconnection G1 275,46+j962,46

3 Transmission Line

Disconnection L1, L2 263,03+j859,40

Generator, Line

Disconnection and Load

Reduction

G1, L1, L2 and

load reduction on

all nodes of

internal area

216,36+j426,08

5 Load Reduction to half

On almost all load

nodes in internal

area

178,39+j14,99

Table E.9.- Scenarios to power-flow changes and losses considering 12-machine system with 8

boundary nodes after Fig. 5.18

Cases Load Conditions Location in

network

Network power

flow loss (MVA)

1 Initial Loading Condition

(Training operating point) 312,65+j1015,60

2 Generator Disconnection G1 325,89+j1164,65

3 Transmission Line

Disconnection L1, L2 316,74+j1094,22

Generator, Line

Disconnection and Load

Reduction

G1, L1, L2 and

load reduction on

all nodes of

internal area

272,31+j680,40

5 Load Reduction to half

On almost all load

nodes in internal

area

239,04+j291,85

APPENDIX 193

12 Multi-machine systems topologically adapted from 3 to 8

boundary nodes and illustrated in Fig. E.1 to Fig. E.5

Fig. E.1.- 12-machine system with three boundary nodes

Fig. E.2.- 12-machine system with four boundary nodes

23 24 25

27 28 29

12 38

G11

G10

G12

EXTERNAL AREA

L1 L2

INTERNAL ARE

G9 G12

Training

disturbances

380 kV

15.75 kV

Boundary Nodes

23 24 25

27 28 29

G11

G10

G12

EXTERNAL AREA

L1 L2

INTERNAL AREA

G9 G12

Training

disturbances

380 kV

15.75 kV

Boundary Nodes

APPENDIX 194

Fig. E.3.- 12 multi-machine system with five boundary nodes

Fig. E.4.- 12 multi-machine system with six boundary nodes

23 24 25

27 28 29

12 38

G11

G10

G12

EXTERNAL AREA

L1 L2

INTERNAL AREA

G9 G12

Training

disturbances

380 kV

15.75 kV

Boundary Nodes

23 24 25

27 28 29

36 35

12 38

G11

G10

G12

EXTERNAL AREA

L1 L2

INTERNAL ARE

G9 G12

~G4

Training

disturbances

380 kV

15.75 kV

Boundary Nodes

APPENDIX 195

Fig. E.5.- 12 multi-machine system with eight boundary nodes

23 24 25

27 28 29

36 35

G11

G10

G12

EXTERNAL AREA

L1 L2

INTERNAL AREA

G9 G12

Training

disturbances

380 kV

15.75 kV

Boundary Nodes

APPENDIX 196

APPENDIX F Data sets of the

interconnected European power system

UCTE / CENTREL

Table F.1.- Part of the generator data set of the European Interconnected Power System [142]

Name Mod.

Order

[MVA] Tm UN[kV] ra

[p.u.]

xd‘

[p.u.]

xd’’

[p.u.]

ALMARAUS G5 3600 5,6 21 0,16 0,326 0,255

ASCO US G5 2100 7 21 0,27 0,45 0,33

FRLEC S- G5 18000 11,2 21 0,005 0,255 0,51 0,34

FRBLA S- G5 7000 12,6 24 0,004 0,22 0,4 0,3

FRTAV S- G5 13300 8 21 0,25 0,45 0,33

FRVIG S- G5 5650 11,2 20 0,005 0,255 0,509 0,344

RWBUETA- G5 1500 9,4 27 0,29 0,5 0,36

RWBUETB- G5 1530 9,2 27 0,295 0,51 0,37

RWGUNLA- G5 1530 6,8 27 0,295 0,51 0,37

RWGUNLB- G5 1530 6,8 27 0,295 0,51 0,37

RWWTHMS- G5 1635 8,55 27 1,759 1,709 0,153 0,00025

BWDAXNS- G5 622 11,6 21 0,2 0,33 0,231

BWKUESS- G5 870 7,1 21 0,2 0,34 0,23

NLDIEMN- G5 2400 8 21 0,25 0,41 0,33

NLDODEA- G5 4050 7 21 0,24 0,42 0,33

ITBENE4- G5 370 8,2 20 0,24 0,4 0,32

ITBRIS4- G5 2300 5,7 20 0,12 0,3 0,15

62-ELSAM G5 4800 15 21 2,41 2,11 0,10129 0,03039

63-SHEG8 G5 825 7,2 21 0,00143 0,233 0,347 0,274

KIEL 1 1 G5 400 6,6 21 2,14 1,84 0,13644 0,04093

BDOR 1 1 G5 1640 8,6 27 1,91 1,81 0,1947 0,05842

BRUH 1 1 G5 1006 9,7 27 1,6 1,5 0,18655 0,05597

KRUH 1 1 G5 1530 9,2 27 1,83 1,73 0,18565 0,05569

68-SIEMS G5 537,5 8,8 21 0,221 0,321 0,284

STDE 1 1 G5 780 12 21 1,89 1,72 0,24577 0,07373

UWES 1 1 G5 1530 9,2 27 1,8462 1,744 0,17427 0,05228

FARG 1 1 G5 400 6,3 21 2,16 1,89 0,09624 0,02887

G4LIP1 G5 1167 8 27 0,001 0,101 0,291 0,196

APPENDIX 197

G4LIP2 G5 1167 8 27 0,001 0,101 0,291 0,196

G4SWPA G5 1000 9,03 27 0,0012 0,235 0,34 0,275

G4SWPB G5 1000 9,03 27 0,0012 0,235 0,34 0,275

P4WIE-G1 G5 470 7 15,75 0,11 0,275 0,191

P4ROG-G3 G5 1278 6,45 22 0,199 0,319 0,235

P4PEL-G1 G5 705 7 15,75 0,11 0,275 0,191

EBOV23 G5 518 9,89 15,75 0,0017 0,166 0,267 0,199

EBOV24 G5 518 9,89 15,75 0,0017 0,166 0,267 0,199

EMO1 G5 518 9,89 15,75 0,0017 0,166 0,267 0,199

EMO2 G5 518 9,89 15,75 0,0017 0,166 0,267 0,199

COFRENUS G5 1200 7 21 0,27 0,38 0,33

GUENESUS G5 1800 8 21 0,27 0,45 0,33

MEQUINUS G5 1800 7,8 21 0,27 0,45 0,33

MONTEAUS G5 1800 8,5 21 0,27 0,4 0,33

PLANA US G5 1800 7 21 0,27 0,45 0,33

ROBLA US G5 750 7,5 21 0,27 0,4 0,33

SANTURUS G5 700 7 21 0,27 0,4 0,33

TERUELUS G5 1250 7 21 0,27 0,45 0,33

TRILLOUS G5 1300 7,41 21 0,127 0,3415 0,2426

VALDECUS G5 3500 8 21 0,27 0,4 0,33

VANDELUS G5 1300 7,8 21 0,27 0,45 0,33

KREMS G5 330 8 21 0,2 0,3 0,27

LAVRI.1 G5 350 8 21 0,2 0,3 0,27

LAVRI.2 G5 200 8 21 0,2 0,3 0,27

PTOLE.1 G5 82 10 15,8 0,00186 0,08 0,21 0,135

PTOLE.4 G5 375 8,9 21 0,00144 0,13 0,3 0,18

MEGAL.1 G5 370 8 21 0,2 0,3 0,27

MEGAL.2 G5 280 8 21 0,2 0,3 0,27

WGACKO G5 353 3,53 22 0,0035 0,16 0,39 0,23

WJABLA G5 113 7 13,8 0,0045 0,22 0,48 0,29

WKAKAN G5 287 5,72 15,75 0,0016 0,15 0,24 0,18

WMOSTA G5 240 10 22 0,0035 0,16 0,39 0,23

WSALAK G5 62 6,33 15 0,0049 0,16 0,23 0,19

WTREBI G5 140 6,11 15 0,0049 0,16 0,23 0,19

APPENDIX 198

Table F.2.- Part of the transmission line data set of the European Interconnected Power System

From bus To bus r [p.u.] x [p.u.] b [p.u.]

ITGORL41 CHROBB41 2,32 31,63 1,542

ITBULC41 CHSOZZ41 2,71 32,21 1,004

ITMUSI41 CHLAVOG4 1,18 19,77 0,749

BGAUB SA FRMOU SA 0,21 2,41 0

BGAVELE4 FRAVE SA 1,32 14,26 0

BGGRAESA FRLON SA 2,8685 34,016 0

BGDOELA4 NLGEERR4 1,5773 18,965 0

BGHERRAB NLMAARSA 1,5467 17,32 0

BGCOO_A4 BGGRAESA 1,4146 14,998 0

BGDOELA4 BGDOELB4 1,3359 13,768 0

BGDOELA4 BGGRAESA 7,0571 75,847 0

BGDOELB4 BGGRAESA 3,986 38,13 0

BGGRAESA BGHERRAB 1,2424 13,901 0

BWLAUFB2 CHLAUBOR 0,01 0,111 0

BWKUESSB CHLAUBSB 0,293 2,307 0

BWKUESSA CHLAUBSA 0,2708 2,2122 0

BWKUESSB CHLAC SA 2,1111 21,212 0

CHLAUBSB EVPULDSA 8,5708 105,76 0

CHLAUBSB EVPULDSB 8,5455 105,97 0

CHAIROSA CHGOEESA 4,26 48 0

EVPULDSB RWVOEGSA 3,4894 34,889 0

EVOMOOE2 RWDELL 2 2,2939 19,394 0

FRCOR S1 FRZME S1 0,01 367,93 0

FRALB SA FRAVE SA 0,01 150,83 0

FRALB SA FRB.T SA 0,0016 40,42 0

FRALB SA FRLON SA 0,01 389,01 0

FRALB SA FRMAM SA 0,01 228,36 0

NLMAARSA RWOBZRSB 1,6871 16,1 0

NLMAARSA RWROK SA 2,358 22,291 0

RWKW_IB2 VEHANEN2 3,899 24,596 0

RWNSTEM2 RWURB GN 53,83 188 0

RWBUETSA RWBUETSB 14,288 140,61 0

RWBUETSA RWUCHFSA 20,438 229,17 0

RWBUETSA RWURB SA 1,7543 16,228 0

BRUH4201 HAMH4201 1,07 13,95 0,86

DOLL4201 HAMS4201 0,91 7,07 0,4145

APPENDIX 199

DOLL4201 HAMS4201 0,91 7,07 0,4145

HAMH4201 HAMO4201 1,04 7,82 0,4265

HAMO4201 HAMS4201 0,99 7,44 0,447

HAMO4201 KRUH4201 0,47 6,26 0,429

KRUE4201 KRUH4201 0,006 0,05 0

JKRALJ2 JUPOZE2 5,23 27,64 0,5398

JPLJEV2 JUPOZE2 7,37 40,14 0,7633

JPANCE2 JZRENJ2 5,87 32,23 0,6006

ABURRE21 AELBAS21 6,59 25,7 0,6525

ABURRE21 AFIERZ21 7,54 38,87 0,9899

AELBAS21 AELBAS22 0,26 1,62 0,0413

ASHKOP52 AULZA 51 2,15 2,7 0,0477

ASLT4 51 ATIRAN51 3,64 5,69 0,0986

ATIRAN51 AUTR4 51 1,89 3,03 0,0509

WGACKO11 WTREBI11 1,92 21,12 0,7171

WGACKO11 WMOSTA11 1,8 20,92 0,6815

WMOSTA11 WSARAJ11 2,76 32 1,042

WSARAJ11 WTUZLA11 2,5 28,98 0,9441

WTUZLA11 WUGLJE11 1,26 13,86 0,5357

WMOSTA2B WTREBI21 6,74 35,72 0,7174

WGRADA21 WTUZLA21 4,22 21,8 0,4548

WJABLA21 WJAJCE21 7,46 39,55 0,7945

WJABLA21 WKAKAN21 4,83 25,62 0,5147

WJABLA21 WMOSTA2B 3,64 19,3 0,3877

WKAKAN21 WPRIJE21 10,61 55,06 2,02

WKAKAN21 WSALAK21 6,94 37,31 0,7496

WKAKAN21 WTUZLA21 5,68 30,11 0,6047

WKAKAN21 WZENIC21 1,81 9,59 0,1925

APPENDIX 200

Table F.3.- Comparison of identity recognition algorithms considering electromechanical weighted

distances by the mean value of J of the 67 German intern machines for a fault located at the boundary

node VEGROUSB(VEW) with different number of external equivalents.

Number of reduced equivalent machines and their reduction degree

(Reduction degree in percentage)

5,04% 8,82% 16,37% 22,67% 36,52% 45,34%

Identity

techniques

(K=K-means,

H=Hierarchical,

P=Preclustering,

E=Electromechanical

distance, F=Fuzzy)

20 35 65 90 145 180

K-means 0,99247 0,99506 0,99594 0,9956 0,99603 0,99752

Electrom. K-means 0,99425 0,99538 0,9976 0,99789 0,9984 0,99826

K-means with

Preclustering 0,9893 0,9951 0,9969 0,99695 0,9967 0,9979

Electrom. K-means

with Preclust. 0,9923 0,99524 0,9969 0,9971 0,99792 0,9978

Hierarchical 0,9935 0,9954 0,9976 0,99783 0,998 0,9978

Electrom.

Hierarchical 0,99296 0,99227 0,9938 0,99694 0,99775 0,9981

Fuzzy 0,9889 0,9927 0,9943 0,99087 0,9932 0,99579

Fuzzy with Preclus. 0,99225 0,99485 0,99745 0,99482 0,9952 0,99647

Electrom. Fuzzy 0,99287 0,9956 0,9976 0,9963 0,9976 0,99798

SOFM 0,99106 0,9943 0,99571 0,993 0,9937 0,992

German power system operators in the present liberalized power markets

Fig. F.1.- As consequence of the fusion of traditional network operators : BAG, BEWAG, EnBW, PE,

RWE, VEAG and VEW the number of the current operators has been considerably reduced.

2000 2002