Load flow analysis in power system network incorporating statcom: A comparison of the direct and indirect algorithm of the newton- raphson method

POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 17 |NUMBER: 1 |2019 |MARCH Load Flow Analysis in Power System Network Incorporating STATCOM: A Comparison of the Direct and Indirect Algorithm of the Newton-Raphson Method Dung VO TIEN1, Radomir GONO2, Zbigniew LEONOWICZ3, Petr KREJCI2 1Department of Electrical Power Engineering, Faculty of Electric Power Systems, Vinh University of Technology Education, Hung Dung street, Vinh, Vietnam 2Department of Electrical Power Engineering, Faculty of Electrical Engineering and Computer Science, VSB–Technical University of Ostrava, 17. listopadu 15/2172, 708 00 Ostrava, Czech Republic 3Department of Electrical Engineering Fundamentals, Faculty of Electrical Engineering, Wroclaw University of Science and Technology, Wybrzeze Stanislawa Wyspianskiego 27, 50-370 Wroclaw, Poland tdungtm[email protected], [email protected], [email protected], p[email protected] DOI: 10.15598/aeee.v17i1.3054 Abstract. This paper presents load flow analysis and the mathematical steady-state modeling of Static Synchronous Compensator (STATCOM) to study its effect on the power system network. More precisely, we propose a new approach method so-called direct algorithm and then compare it with the indirect algorithm in testing cases: IEEE 5-bus, IEEE 14-bus, and IEEE 30-bus systems. We compare the accuracy, the number of iterations and the computational time. The simulation results show that the direct algorithm is effective in terms of accuracy, speed of computation and various practical applications. Keywords FACTS, iteration, load flow, Newton-Raphson, power flow, power system analysis, STATCOM. 1. Nomenclature PLi: Active power of load at ith bus in pu (per unit system). QLi: Reactive power of load at ith bus in pu. PSi: Active power obtained from STATCOM at ith bus in pu. QSi: Reactive power obtained from STATCOM at ith bus in pu. Vi: System bus voltage magnitude at ith bus in pu. δi: Phase angle of bus voltage at ith bus. VSi: STATCOM output voltage at ith bus in pu. δSi: Phase angle of STATCOM output voltage at ith bus. Vcon i: Bus voltage control reference in pu. ISi: STATCOM current in pu. αSi: Phase angle of STATCOM current. Rij, Xij: Resistance and reactance of branch ij in pu. Admittance of branch ij in pu: 1/Zij = 1/(Rij +jXij ) = |Yij|∠θij =Gij +jBij. RSi, XSi: Resistance and reactance of STATCOM in pu. Admittance of STATCOM in pu: 1/ZSi = 1/(RSi +jXSi) = gSi +jbSi. VSi min: The minimum of STATCOM output voltage in pu. VSi max: The maximum of STATCOM output voltage in pu. QSi min: The minimum value of reactive power of STATCOM in pu. QSi max: The maximum value of reactive power of STATCOM in pu. c 2019 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 13 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 17 |NUMBER: 1 |2019 |MARCH 2. Introduction Together with the development of power electronics, the Flexible AC Transmission system (FACTs) devices have been proposed and used in the electrical power system to improve power quality. Various FACTs (e.g., Static Synchronous Compensator (STATCOM), Static VAR Compensator (VSC), Thyristor Controlled Series Compensator (TCSC), Static Synchronous Series Compensator (SSSC) and Unified Power Flow Controller (UPFC)) are used to control the bus voltage magnitude and the power flow along the transmission lines. Among these devices, STATCOM is one of the most useful ones in power systems because it can regulate the voltage very fast, improve transient stability and compensate variable reactive power [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] and [11]. Load flow (or power flow) is a solution for the steady state of the power systems network. The studies of load flow provide methods for calculating various bus voltage magnitudes, phase angles, active and reactive power flowing through the line of the power system. Load flow analysis of power systems incorporating STATCOM is an important tool to further determine the inject power of STATCOM to regulate voltage under steady state conditions. The power system embedded with STATCOM in load flow requires an accurate method for computation and controlling of the bus voltage magnitudes to determine the steady states and power planning purposes of the power system. Many different researches have been proposed for the load flow incorporating STATCOM [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] [11], [14], [15], [16] and [17]. In general, there are two main categories of load flow techniques: Current Injection model (CI) and Power Injection model (PI) or Voltage Source model. In the CI model, the STATCOM is represented as the current source connected in shunt at the bus for controlling of voltage magnitudes. In the PI model, it is represented as shunt voltage source behind an equivalent impedance, and controls bus voltage magnitudes by adjusting its voltage magnitude and phase angle. Techniques in both categories have advantages and disadvantages. However, the PI proved its effectiveness in term of computation speed and accuracy [2] and [17]. In the PI model, the buses with STATCOM in power system network can be solved by Eq. (1) a STATCOM represented as a PV bus and Eq. (2) a STATCOM represented as an independent variable. The parameters of Eq. (1) are calculated by the voltage at the bus where the STATCOM is placed, and the main equations of load flow and the STATCOM are solved separately (indirect algorithm) [6], [7], [8] and [9]. In Eq. (2), the Jacobian matrix in the main equation of load flow is modified (direct algorithm) [14], [15], [16] and [17]. In this paper, a MATLAB program is developed for the load flow analysis of power system network incorporating STATCOM. We provide two algorithms for this issue in Sec. 5. and Sec. 6. The load flow study is then performed in IEEE 5-bus, IEEE 14-bus and IEEE 30-bus systems in Sec. 7. 3. Newton-Raphson Load Flow in Power System without STATCOM The Newton-Raphson method is robust load flow method used in power system. From the node-voltage equation: Ibus =Ybus ·Vbus,(1) where Ibus is the vector of the injected bus currents, Vbus is the vector of bus voltages and Ybus is known as the bus admittance matrix. This equation can be rewritten in form for an n-bus system: Ii= n X j=1 Yij ·Vj= n X j=1 |Yij||Vj|∠θij +δj.(2) The active and reactive power at bus iis: Pi−jQi=V∗ i·Ii=Vi∠−δi·( n X j=1 |Yij||Vj|∠θij+δj).(3) Therefore: Pi=Re{V∗ i·Ii}= = n P j=1 |Vi||Vj||Yij|cos(θij −δi+δj),(4) Qi=−Im{V∗ i·Ii}= = n P j=1 |Vi||Vj||Yij|sin(θij −δi+δj).(5) The main equation of Newton-Raphson power flow can be expressed as follow: "∆P ∆Q#="J11 J21 J21 J22#."∆δ ∆|V|#.(6) The elements of the Jacobian matrix are: J11 =∂Pi ∂δj ,J12 =∂Pi ∂|Vj|,J21 =∂Qi ∂δj ,J22 =∂Qi ∂|Vj|.(7) The term ∆Pand ∆Qare the difference between the specified and calculate values, given by: ∆P(k) i=PSpec i−P(k) i,(8) ∆Q(k) i=QSpec i−Q(k) i.(9) c 2019 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 14 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 17 |NUMBER: 1 |2019 |MARCH The new estimates for bus voltage are: δ(k+1) i=δ(k) i+ ∆δ(k) i,(10) |Vi|(k+1) =|Vi|(k)+ ∆|Vi|(k).(11) 4. Modeling of STATCOM Voltage Source Converter VDC - + Transformer Transmission Line Transmission Line i PLi+jQLi Fig. 1: Schematic diagram of STATCOM. i PLi+jQLi PSi+jQSi ZSi 𝑉𝑖∠𝛿𝑖 ~ 𝑉 𝑆𝑖∠𝛿𝑆𝑖 Fig. 2: Equivalent circuit of STATCOM. Generally, a STATCOM including a coupling transformer, Voltage Source Converter (VSC) and a DC energy storage device (Fig. 1). In its simplest form, a DC capacitor is used to replace the energy storage device, thus the STATCOM is capable of only exchange reactive power with the power systems. In the 90s of previous century, the internal loss power of the transformer and the STATCOM are neglected so that the result is not accurate [6]. Then the STATCOM can be represented by an equivalent circuit as shown in Fig. 2. It is able to regulate the bus voltage magnitude by injecting or absorbing reactive power to or from the bus where it is connected. The equation for the ith bus with STATCOM can be written as follows: Si= (PLi +jQLi)+(PSi +jQSi),(12) Pi=PLi +PSi, Qi=QLi +QSi,(13) ISi = (PSi −jQSi)/V ∗ i,(14) VSi =Vi−ISiZSi.(15) According to the equivalent circuit of the STATCOM shown in Fig. 2, the power constrains of STATCOM are: PSi =|Vi|2·gSi − |Vi||VSi|· ·[gSi ·cos(δi−δSi) + bSi ·sin(δi−δSi)],(16) QSi =−|Vi|2·bSi +|Vi||VSi|· ·[−gSi ·sin(δi−δSi) + bSi ·cos(δi−δSi)],(17) that are the active and reactive power equations obtained from STATCOM at bus i, respectively. The voltage injection and capacity of STATCOM are bounded as follows: VSi min ≤VSi ≤VSi max, QSi min ≤QSi ≤QSi max.(18) The active and reactive power exchange via the DClink are described by: PST AT,i −jQST AT,i =V∗ Si ·ISi.(19) Therefore, we have: PST AT,i =|VSi|2·gSi − |Vi||VSi| ·[gSi ·cos(δi−δSi)−bSi ·sin(δi−δSi)],(20) QST AT,i =−|VSi|2·bSi +|Vi||VSi| ·[gSi ·sin(δi−δSi) + bSi ·cos(δi−δSi)].(21) 5. Indirect Algorithm This algorithm was researched and published, for example in [6], [7], [8] and [9]. According to this algorithm, the system buses, where the STATCOM are installed, are made PV buses. In [6] and [7] the internal losses in the STATCOM (the losses in switching and transformer) were neglected and therefore, it is not accurate. In [8] and [9] the algorithm was improved, such losses are taken into consideration (they have been represented by resistance and reactance of STATCOM in the equivalent circuit). The procedure of this algorithm can be summarized as follows: c 2019 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 15 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 17 |NUMBER: 1 |2019 |MARCH •Step 1. Input the system data and the information of buses (voltage magnitude and phase angle at a slack bus, voltage magnitude and active power at PV buses, active and reactive power at PQ buses). The system buses, where the STATCOMs are placed, are made PV buses. On the STATCOM buses, the limits of the voltage and reactive power of STATCOM should be included. •Step 2. Set the initial value of reactive power exchange via the DC-link, for example P(0) ST AT,i = 0.01. •Step 3. Run normal load flow to calculate bus voltages, phase angle and injected reactive powers. •Step 4. Calculate new value of PST AT,i using the following procedure: – Calculate QSi from the Eq. (13). – Calculate ISi from the Eq. (14). – Calculate VSi from the Eq. (15). – Calculate PST AT,i from the Eq. (20). – The errors of PST AT,i can be calculated as follows: ∆PST AT,i = 0 −PST AT,i.(22) •Step 5. Update the active power of STATCOM and return to step 2. •Step 6. The process is continued until the residuals ∆PST AT,i are less than the specified accuracy. 6. Direct Algorithm 6.1. Newton-Raphson Load Flow Formulation with STATCOM The direct algorithm was presented, for example, in [10], [14], [15], [16] and [17]. In [10], a STATCOM was presented as a new bus and new branch. In other publications [14], [15], [16] and [17], a STATCOM is represented as independent variables. In this method, the Newton-Raphson load flow algorithm for electrical power system incorporating STATCOM requires some modifications: •The Jacobian matrix needs to be extend. The new sub-blocks related to the STATCOM device should be included. •The mismatch vector also needs to be extended. The residuals of power contributed by STATCOM at the connected buses should be included. With these modifications, the Jacobian matrix is increased according to the number of STATCOMs. Acha et al. in [16] proposed the phase angles and voltage magnitudes of STATCOM are the independent variables with the mismatch vectors ∆PSand ∆QS(the difference between specified and calculate values of active and reactive power of STATCOM, respectively). However, the disadvantage of this method is the unspecified reactive power of STATCOM. We introduce a new approach to solve this problem. Below, the system of linearized load flow equations of the power systems with STATCOM is shown:          ∆P ∆Q ∆PE ∆F          =              ∂P ∂δ ∂P ∂|V| ∂P ∂δS ∂P ∂|VS| ∂Q ∂δ ∂Q ∂|V| ∂Q ∂δS ∂Q ∂|VS| ∂PE ∂δ ∂PE ∂|V| ∂PE ∂δS ∂PE ∂|VS| ∂F ∂δ ∂F ∂|V| ∂F ∂δS ∂F ∂|VS|              .          ∆δ ∆|V| ∆δS ∆|VS|          . (23) According to Eq. (23): •PE is the active power exchange via the DC-link of STATCOM, it is given in Eq. (20). •Fis the voltage magnitude of the bus where the STATCOM is installed. •∆Pand ∆Qare given in Eq. (8) and Eq. (9). •For a trial set of variables ∆δS,∆|VS|, the mismatch vectors are added, represented by: ∆PE(k) i= 0 −P(k) ST AT,i,(24) ∆F(k) i=|VCon i|−|V(k) i|.(25) Assume that the power system has n buses including: 1 slack bus, mPV buses, (n−1−m)PQ buses and p STATCOMs. Accordingly, the Jacobian matrix of this algorithm is in the order (2n−2−m+2p)×(2n−2−m+ 2p)while it is in the order (2n−2−m)×(2n−2−m) in indirect algorithm. The elements of the Jacobian matrix in Eq. (23) are given in App. B. 6.2. The Newton-Raphson Load Flow Algorithm with STATCOM The procedure for the load flow solution with the power system incorporating STATCOM of this algorithm is summarized as follows: c 2019 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 16 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 17 |NUMBER: 1 |2019 |MARCH Tab. 1: Comparison of the complex bus voltage of the IEEE 5-bus system with STATCOM between two algorithms. Bus Indirect algorithm Direct algorithm Differences No. |V|(p.u)δ(degree)|V|(p.u)δ(degree)|V|(p.u)δ(degree) 1 1.0600 0.00 1.0600 0.00 0.0000 0.00 2 1.0000 -2.05 1.0000 -2.05 0.0000 0.00 3 1.0000 -4.84 1.0000 -4.84 0.0000 0.00 4 0.9942 -5.11 0.9944 -5.11 0.0002 0.00 5 0.9751 -5.80 0.9752 -5.80 0.0001 0.00 •Step 1. Input the system data and the information of buses (voltage magnitude and phase angle at a slack bus, voltage magnitude and active power at PV buses, active and reactive power at PQ buses) and STATCOM (location, impedance, the limits of voltage and capacity). •Step 2. Calculate ∆Pi,∆Qi,∆PEi,∆Fifrom Eq. (8), Eq. (9), Eq. (24) and Eq. (25). •Step 3. The elements of the Jacobian matrix are calculated from Eq. (26) to Eq. (58) (see App. B). •Step 4. Solve Eq. (23) for corrections of voltage magnitudes and phase angles at buses and STATCOMs. The limits of voltage and capacity of STATCOM should be included. •Step 5. Update the voltage magnitude and phase angle by adding to the previous values and return to step 2. •Step 6. The process is continued until the residuals ∆Pi,∆Qi,∆PEi,∆Fiare less than the specified accuracy. 7. Case Studies and Results MATLAB programs based on these algorithms were developed for the load flow analysis of electrical power systems incorporating STATCOM. In order to investigate the performance of the indirect and direct algorithm, these flow programs were tested on IEEE 5-bus, modified IEEE 14-bus and IEEE 30-bus systems (see App. A). 7.1. Case I: IEEE 5-bus From the results obtained from load flow for the 5-bus system without STATCOM (see Tab. 2), one can observe that the voltage magnitudes at bus 3, 4 and 5 are lower than 1.0 pu. Therefore, one STATCOM was installed at bus 3 to regulate the voltage magnitude at 1.0 pu. The STATCOM data is given in Tab. 3. Table 1 and Tab. 4 provide the details of simulation results of two algorithms including a comparison of voltage magnitudes and phase angles at all buses, and the injected power of STATCOM. Tab. 2: The complex bus voltage of the IEEE 5-bus system without STATCOM. Bus No. 1 2 3 4 5 (Slack) (PV) (PQ) (PQ) (PQ) |V|(pu)1.0600 1.000 0.9872 0.9841 0.9717 δ(degree)0.00 -2.06 -4.63 -4.96 -5.77 Tab. 3: STATCOM parameters. RS(pu)XS(pu)QSmin(pu)QSmax(pu) 0.01 0.10 -0.5 0.5 Tab. 4: Comparison of the voltage and reactive power of the STATCOM between two algorithms in the IEEE 5-bus system. |VS|(pu)δS(degree)QS(pu) Indirect algorithm 1.0198 -4.96 -0.2047 Direct algorithm 1.0205 -4.96 -0.2049 Differences 0.0007 0.00 0.0002 7.2. Case II: IEEE 14-bus The second case study in this paper used the IEEE-14 bus system with a small modification, PV buses at bus 6 and 8 are 1.07 pu and 1.09 pu, respectively, which are changed to 1.05 pu. From the results obtained by a program for this system without STATCOM (see Tab. 5), we can observe that all the voltage magnitudes are greater than 1.0 pu. Assume that a STATCOM is installed at bus 11 to ensure the voltage at bus 11 is 1.0 pu. The simulation results of two algorithms are illustrated in Tab. 6 and Tab. 7, including the comparison of voltage magnitudes, phase angles at all buses, and the injected power of STATCOM. In this case, the STATCOM absorbed the reactive power, therefore, its voltage magnitude is smaller than at the bus which the STATCOM was installed. 7.3. Case III: IEEE 30-bus Similar results was also obtained from the modified IEEE 30-bus system and hence these are not repeated here. Two STATCOMs were installed at bus 26 and 30 in order to regulate the voltage to 1.0 pu. The voltage magnitudes of all buses with and without STATc 2019 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 17 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 17 |NUMBER: 1 |2019 |MARCH Tab. 5: The complex bus voltage of the IEEE 14-bus system without STATCOM. Bus No. 1 (slack) 2 (PV) 3 (PV) 4 (PQ) 5 (PQ) 6 (PV) 7 (PQ) |V|(pu)1.0600 1.0450 1.0100 1.0067 1.0104 1.0500 1.0270 δ(degree)0.00 -5.01 -12.81 -10.18 -8.69 -14.49 -13.26 Bus No. 8 (PV) 9 (PQ) 10 (PQ) 11 (PQ) 12 (PQ) 13 (PQ) 14 (PQ) |V|(pu)1.0500 1.0127 1.0116 1.0269 1.0331 1.0266 1.0001 δ(degree)-13.26 -14.91 -15.13 -14.93 -15.37 -15.41 -16.20 Tab. 6: Comparison of the complex bus voltage of the IEEE 14-bus system with STATCOM between two algorithms. Bus Indirect algorithm Direct algorithm Differences No. |V|(pu)δ(degree)|V|(pu)δ(degree)|V|(pu)δ(degree) 1 1.0600 0.00 1.0600 0.00 0.0000 0.00 2 1.0450 -5.01 1.0450 -5.02 0.0000 0.01 3 1.0100 -12.83 1.0100 -12.84 0.0000 0.01 4 1.0052 -10.17 1.0051 -10.17 0.0001 0.00 5 1.0093 -8.70 1.0092 -8.70 0.0001 0.00 6 1.0500 -14.63 1.0500 -14.62 0.0000 -0.01 7 1.0224 -13.24 1.0222 -13.24 0.0002 0.00 8 1.0500 -13.24 1.0500 -13.24 0.0000 0.00 9 1.0035 -14.89 1.0033 -14.90 0.0002 0.01 10 0.9969 -14.98 0.9967 -14.99 0.0002 0.01 11 1.0000 -14.44 1.0000 -14.45 0.0000 0.01 12 1.0324 -15.51 1.0315 -15.50 0.0009 -0.01 13 1.0252 -15.53 1.0243 -15.53 0.0009 0.00 14 0.9942 -16.25 0.9937 -16.25 0.0005 0.00 Tab. 7: Comparison of the complex voltage and reactive power of the STATCOM between two algorithms in the IEEE 14-bus system. |VS|(pu)δS(degree)QS(pu) Indirect algorithm 0.9806 -14.33 0.1920 Direct algorithm 0.9808 -14.34 0.1916 Differences -0.0002 0.01 0.0004 COM corresponding to two algorithms are shown in Fig. 3. Table 8 shows the simulation results of two algorithms, including the comparison of voltage magnitudes, phase angles and the injected powers of STATCOMs. The maximum difference of voltage magnitudes and phase angles between two algorithms are 0.0011 pu and 0.02 degrees, respectively. In this case, the STATCOMs compensate reactive power, therefore their voltage magnitudes are larger than at the buses which the STATCOMs were installed at. 7.4. Discussion In three case studies, it is easy to see that the accuracy of the two algorithms is nearly the same. To determine the effectiveness of the different algorithms, the number of iterations, the total time took off the execution to complete the solution on three systems have been studied and compared. If the desired accuracy decreases, the accuracy of the solution and the number of iterations increase. In this paper, the desired accuracy is 0.0001. Table 9 provides detailed information of the comparison between two algorithms for IEEE 5bus, 14-bus and 30-bus systems. From the information, we can see that: •The results of two algorithms are slightly different. This demonstrates that the direct algorithm is quite good in comparison with the well-known indirect algorithm. •The convergence rate for Newton-Raphson method is fast and the number of iterations are less dependent on the number of buses in the system [12], [13], [14], [15], [16], [17], [18], [19] and [20]. In three case studies, the number of iterations of indirect algorithm is less than or equal to the direct one. •However, the indirect algorithm is slower than the direct algorithm. Indeed, the loop of indirect algorithm includes the computing time of solving the n-bus system that does not appear in the direct algorithm. As a result, it is profitable when applying the latter to large scale systems. 8. Conclusion In this paper, two algorithms of Newton-Raphson method for load flow analysis in power system incorporating STATCOM were presented. MATLAB programs based on these algorithms were developed and implemented for the IEEE 5-bus, IEEE 14-bus and IEEE 30-bus systems. The simulation results show that the STATCOM is able to inject or absorb reactive c 2019 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 18 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 17 |NUMBER: 1 |2019 |MARCH 0.8 0.85 0.9 0.95 1 1.05 1.1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Bus Voltage Magnitude [p.u] Without STATCOM Direct algorithm Indirect algorithm Fig. 3: The voltage of all buses of the IEEE 30-bus system without and with STATCOMs calculated by two algorithms. Tab. 8: Comparison of the complex voltage and reactive power of the STATCOM between two algorithms in the IEEE 30-bus system. Bus Indirect algorithm Direct algorithm Differences No. |VS|(pu)δS(degree)QS(pu)|VS|(p.u)δS(degree)QS(pu)|VS|(p.u)δS(degree)QS(p.u) 26 1.0041 -20.29 -0.0436 1.0043 -20.32 -0.0432 -0.0002 0.03 -0.0004 30 1.0170 -23.69 -0.1735 1.0173 -23.71 -0.1730 -0.0003 0.02 -0.0005 Tab. 9: The computing time of two algorithms. System Indirect algorithm Direct algorithm Number of Time per Computing Number of Time per Computing Iterations Iteration (ms) time (ms) Iterations Iteration (ms) time (ms) 5-bus 3 1.909 5.727 4 0.748 2.993 14-bus 4 13.277 53.109 5 3.243 16.217 30-bus 4 48.413 193.652 5 12.528 62.643 power to regulate the voltage magnitude of the buses where it is connected. From the obtained results, the accuracy of two algorithms was confirmed. However, the direct algorithm is more effective and reliable in terms of computing speed. 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[16] ACHA, E., C. FUERTE-ESQUIVEL, H. AMBRIZ-PEREZ and C. ANGELES CAMACHO. FACTS: modelling and simulation in power networks. Hoboken: John Wiley &son, 2004. ISBN 0-470-85271-2. [17] SHAHNIA, F., S. RAJAKARUNA and A. GHOSH. Static Compensators (STATCOMs) in Power Systems. Singapore: Springer, 2015. ISBN 978-981-287-281-4. [18] SAATDAT, H. Power system analysis. New York: McGraw Hill, 1999. ISBN 0-07-561634-3. [19] MILANO, F. Power System Modelling and Scripting. London: Springer, 2010. ISBN 978-3-64213668-9. [20] MUSTI, K. S. S. and R. B. RAMKHELAWAN. Power System Load Flow Analysis using Microsoft Excel. Spreadsheets in Education (eJSiE). 2012, vol. 6, iss. 1, pp. 1–5. ISSN 1448-6156. About Authors Dung VO TIEN received his M.Sc., Ph.D. degrees, all in Electrical Power Engineering, in 2007, and 2018, respectively. He has been with the Department of Electrical Engineering, Vinh University of Technology Education, Vietnam. His current research interests are in the areas of power system analysis, reliability of electric power systems and power quality. Radomir GONO (IEEE M’12-SM’16) received his M.Sc., Ph.D. and Habilitate Doctorate degrees, all in Electrical Power Engineering, in 1995, 2000, and 2008, respectively. He has been with the Department of Electrical Power Engineering, VSB–Technical University of Ostrava, Czech Republic, since 1999 where he currently holds the position of Associate Professor and Vice-head of the department. He works also as a Senior researcher of the research centre - c 2019 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 20 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 17 |NUMBER: 1 |2019 |MARCH Energy Units for Utilization of non Traditional Energy Sources at the same university. His current research interests are in the areas of electric power systems reliability, optimization of maintenance and renewable energy sources. Zbigniew LEONOWICZ (IEEE M’03–SM’12) received his M.Sc., Ph.D. and Habilitate Doctorate (Dr Sc.) degrees, all in Electrical Engineering, in 1997, 2001, and 2012, respectively. He has been with the Department of Electrical Engineering, Wroclaw University of Science and Technology, Poland, since 1997 where he currently holds the position of Associate Professor. His current research interests are in the areas of power quality, control and protection of power systems, renewables, industrial ecology and applications of advanced signal processing methods in power systems. Petr KREJCI received his M.Sc., Ph.D. and Habilitate Doctorate degrees, all in Electrical Power Engineering, in 1998, 2001, and 2009, respectively. He has been with the Department of Electrical Power Engineering, VSB–Technical University of Ostrava, Czech Republic, since 2001 where he currently holds the position of Associate Professor. He holds the post of Vice-dean of Faculty of Electrical Engineering and Computer Science. His current research interests are in the areas of power quality and distribution networks. Appendix A The data of modified IEEE system Tab. 10: Modified IEEE-14 bus system. Bus Initial IEEE-14 bus Modified No. |V|(pu)|V|(pu) 6 (PV) 1.07 1.05 8 (PV) 1.09 1.05 Tab. 11: Modified IEEE-30 bus system. Bus Initial IEEE-30 bus Modified No. P (MW) W (Mvar) P (MW) W (Mvar) 26 3.5 2.3 7.0 5.6 30 10.6 1.9 21.2 13.1 Appendix B Jacobian matrix •The sub-matrix J11 is of order (n−1) ×(n−1) and the elements of J11 are: ∂Pi ∂δi = n P j6=i |Vi||Vj|[−Gij sin(δi−δj) +Bij cos(δi−δj)], (26) ∂Pi ∂δj =|Vi||Vj|[Gij sin(δi−δj) −Bij cos(δi−δj)], j6=i. (27) If bus iconnect with STATCOM, ∂Pi ∂δi = n P j=1 |Vi||Vj|[−Gij sin(δi−δj) +Bij cos(δi−δj)] + |Vi||VSi|[gSi sin(δi−δSi) −bSi cos(δi−δSi)]. (28) •The sub-matrix J12 is of order (n−1)×(n−1−m) and the elements of J12 are: ∂Pi ∂|Vi|= 2|Vi|Gii +P j6=i |Vj|[Gij cos(δi−δj) +Bij sin(δi−δj)], (29) ∂Pi ∂|Vj|=|Vi||[Gij cos(δi−δj) + Bij sin(δi−δj)], j6=i. (30) If bus iconnect with STATCOM, ∂Pi ∂|Vi|= 2|Vi|Gii + 2|Vi|gSi+ P j6=i |Vj|[Gij cos(δi−δj) + Bij sin(δi−δj)] −|VSi|[gSi cos(δi−δSi) + bSi sin(δi−δSi)]. (31) •The sub-matrix J13 is of order (n−1) ×pand the elements of J13 are: If bus iconnect with STATCOM, ∂Pi ∂δSi = −|Vi||VSi|[gSi sin(δi−δSi)−bSi cos(δi−δSi)]. (32) Otherwise, ∂Pi ∂δSj = 0, j 6=i. (33) c 2019 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 21