P oyec o Fin de Ca e a
Ingenie ía de Telecomunicación
Fo ma o de Publicación de la Escuela Técnica
Supe io de Ingenie ía
Au o : F. Ja ie Payán Some
Tu o : Juan José Mu illo Fuen es
Dep. Teo ía de la Señal y Comunicaciones
Escuela Técnica Supe io de Ingenie ía
Uni e sidad de Se illa
Se illa, 2013
Doc o al Disse a ion
Ingenie ía Au oma ica, Elec ónica y de
Telecomunicaciones
Con ibu ions o Con ol Law Designs
and S abili y Analysis in AC Mic og ids
Au ho : Ma ía Camila Me chán-Ri e os
Di ec o s: Ca olina Albea and F ancisco Salas
Ingenie ía de Sis emas y Au omá ica
Escuela Técnica Supe io de Ingenie ía
Uni e sidad de Se illa
Se illa, 2024
Doc o al Disse a ion
Ingenie ía Au oma ica, Elec ónica y de Telecomunicaciones
Con ibu ions o Con ol Law Designs and S abili y Analysis
in AC Mic og ids
Au ho :
Ma ía Camila Me chán-Ri e os
Di ec o s:
Ca olina Albea and F ancisco Salas
P o esso s
Ingenie ´ıa de Sis emas y Au om´a ica
Escuela T´ecnica Supe io de Ingenie ´ıa
Uni e sidad de Se illa
2024
Doc o al Disse a ion:
Con ibu ions o Con ol Law Designs and S abili y Analysis in AC
Mic og ids
Au ho : Ma ía Camila Me chán-Ri e os
Di ec o s: Ca olina Albea and F ancisco Salas
El ibunal nomb ado pa a juzga la Tesis a iba indicada, compues o po los siguien es
doc o es:
P esiden e:
Vocales:
Sec e a io:
acue dan o o ga le la cali icación de:
El Sec e a io del T ibunal
Fecha:
A mi amilia y a Jose
Aknowledgemen s
T
he pe iod o my hesis de elopmen has been a u ning poin in my li e. Since I mo ed
o Spain and le p ac ically all my li e and lo ed ones in Colombia, I ha e lea ned
and g own a lo bo h p o essionally and pe sonally, and he suppo and help o many
people a ound me in di e en a eas o li e ha e been key o do his hesis.
Fi s o all, I would like o exp ess my g a i ude o my supe iso s, Ca olina Albea
and Paco Salas, o hei suppo , guidance and eaching, as well as o hei ideas and
commen s con ibu ed o he de elopmen o he hesis and hei e ision and co ec ion
o his documen . Ca olina, I would also like o hank you no only o you i eless wo k
bu also o you ad ice and emo ional suppo , which was e y impo an o me.
I would like o hank P o esso Paco Go dillo, he i s pe son who welcomed me when
I a i ed in Se ille, o he oppo uni y o joining he esea ch g oup, and o his us and
his encou aging guidance. Also I would like o hank P o esso Alexand e Seu e o his
ideas and con ibu ions o his hesis.
I would like o hank P o esso Luca Zacca ian o accep ing o e iew my wo k as well
as o pa icipa ing in my ju y commi ee. I am also g a e ul o P o esso Gio anni Ga a a
o accep ing o e iew my wo k and helping wi h he expe imen al es s in Pale mo. I
deeply app ecia e hei cons uc i e sugges ions on he hesis, and I am g a e ul hey ake
he ime o unde ake hese asks despi e hei busy schedules.
Many hanks o P o esso Suzanne Lesecq, P o esso Fabio Gómez-Es e n, P o esso
Ca los Bo dons, and P o esso Ascensión Za a, o hei commi men o ake pa in my
ju y commi ee.
I would like o hank P o esso An onino S e lazza o in i ing me o Pale mo o doing
my in e na ional pe iod and o his guidance he e.
III
XCon en s
3.1 In oduc ion 25
3.1.1 P oblem o mula ion 26
3.2 P oposed con ol scheme 29
3.3 Con ol objec i es 30
3.4 Dynamical models in he p ima y and seconda y con ol loops 30
3.4.1 Consensus algo i hm 30
3.4.2 D oop con ol 31
3.4.3 Con inuous- ime con ol law o he in e e s 32
3.5 Comple e con ol dynamic 34
3.5.1 Closed-loop dynamics o a BESSi35
3.5.2 Comple e con ol dynamic 35
3.5.3 Th ee ime-scale sepa a ion 37
3.6 S abili y analysis o he comple e con inuous- ime sys em 37
3.6.1 Singula pe u ba ion o m 37
3.6.2 The quasi-s eady-s a e equilib ium ’mani old’ 38
3.6.3 S abili y o he bounda y laye sys em 38
3.6.4 S abili y o he educed sys em 39
3.7 Expe imen al alida ion 39
3.7.1 Scena io 1 43
3.7.2 Scena io 2 43
3.7.3 Compa ison wi h [1] 46
3.8 Conclusion 49
4 Hyb id dynamical con ol scheme o discha ging a e consensus 51
4.1 Con ol objec i es 52
4.2 In e e model and con ol 52
4.2.1 Vol age ou pu egula ion o in e e s 53
4.3 Hyb id con ol s uc u e 54
4.4 Global hyb id s uc u e 56
4.4.1 Th ee ime-scale sepa a ion 57
4.5 S abili y analysis o he comple e hyb id sys em 58
4.5.1 Singula pe u ba ion o m 58
4.5.2 Regula i y o Sys em’s Da a 59
4.5.3 Regula i y o he ‘mani old’ 59
4.5.4 S abili y o he bounda y laye sys em 60
4.5.5 S abili y o he educed sys em 61
4.6 Simula ion esul s 62
4.6.1 Compa ison wi h [1] 65
4.7 Conclusions 67
5 Hyb id dynamical con ol scheme o eac i e powe sha ing 69
5.1 In oduc ion 69
5.2 P oposed con ol scheme 71
Con en s XI
5.3 Con ol objec i es 72
5.4 Dynamical models o eac i e powe sha ing 72
5.4.1 Consensus algo i hm 72
5.4.2 D oop con ol 72
5.5 Hyb id con ol s uc u e 73
5.6 Global hyb id s uc u e 74
5.6.1 Th ee ime-scale sepa a ion 76
5.7 S abili y analysis o he comple e hyb id sys em 76
5.7.1 Regula i y o Sys em’s Da a 77
5.7.2 Regula i y o he ’mani old’ 78
5.7.3 S abili y o he ‘bounda y laye ’ 78
5.7.4 S abili y o he educed sys em 79
5.8 Simula ion esul s 80
5.8.1 Scena io 1 82
5.8.2 Scena io 2 85
5.9 Conclusion 88
6 Bi u ca ion analysis o islanded AC mic og ids wi h Cons an Powe Loads 91
6.1 In oduc ion 91
6.2 P elimina y concep s 92
6.2.1 Some no ions o bi u ca ion heo y 92
Poinca é-And ono -Hop bi u ca ion 92
6.3 AC mic og id model 96
6.3.1 Modelling o a h ee phase in e e 96
6.3.2 Model o a CPL 99
Selec ed CPL model 101
6.3.3 Model o he mic og id wi h a CPL 102
6.4 Nume ical s abili y analysis 104
6.5 Simula ion esul s 106
6.6 Expe imen al esul s 109
6.6.1 CPL ime cons an es 109
6.6.2 Expe imen al esul s o Bi u ca ion analysis 110
6.7 Conclusions 116
7 Conclusions and u u e wo k 117
7.1 Conclusions and con ibu ion summa y 117
7.2 Fu u e wo ks 118
Appendix A Appendix o Chap e 1 121
Lis o Figu es 127
Lis o Tables 131
Bibliog aphy 133
No a ion
Lis o Symbols
RThe se o eal numbe s
R≥0The se o nonnega i e eal numbe s
RnThe n-dimensional euclidean space
Rn×mThe eal n×mma ices space
ZThe se o in ege numbe s
NThe se o posi i e in ege numbe s
A⊂B The se Ais subse o B
A∪B The union o se s Aand B
A∩B The in e sec ion o se s Aand B
A B The di e ence be ween se s Aand B
A×B The ca esian p oduc be ween se s Aand B
AThe closu e o he se A
ATThe anspose o ma ix A
A=Si∈N AiThe union o se s Ai o i∈N
A=Ti∈N AiThe in e sec ion o se s Ai o i∈N
IThe iden i y ma ix
∅The emp y se
1and 0The ec o s o ones and ze os, esp. o sui ed dimension
eig(M)The eigen alues o ma ix M
Re(a)The eal pa o he complex numbe a
M≻0(M≺0) The eigen alues o Ma e s ic ly posi i e (nega i e)
diag{a1,a2,...,aN}The diagonal ma ix whose elemen s a e a1,a2,...,aN
˙xThe s a e ime de i a i e
x+The s a e alue a e an ins an aneous change (jump)
|x|The absolu e alue o x
∥x∥The euclidean no m o ec o x
XIII
XIV Chap e 0. No a ion
min(A)The minimum elemen o se A
:Rm→Rn
This no a ion indica es ha
is a unc ion om a space
Rm
o a space
Rn
F:Rm⇒Rn
This no a ion indica es ha
F
is a se - alued mapping wi h
F(x)⊂Rn
o each x∈Rm
a gmin( )s and o he a gumen ha minimize he unc ion
ρiBiThe closed ball o adius ρi>0
ν→0+A posi i e pa ame e νis small enough.
Ac onyms
AC Al e na ing Cu en
BESS Ba e y Ene gy S o age Sys em
CPL Cons an Powe Load
CSI Cu en -Sou ce In e e s
DC Di ec Cu en
DDE Delay-Di e en ial Equa ion
DG Dis ibu ed Gene a ion
ESS Ene gy S o age Sys ems
HB Hal -B idge
HDS Hyb id Dynamical Sys em
LQR Linea -Quad a ic Regula o
MAS Mul i-Agen Sys ems
MG Mic og id
MPPT Maximum Powe Poin T acking
MOSFET Me al-oxide semiconduc o ield-e ec ansis o
OL Ou e Loop
PCC Poin o Common Coupling
PI P opo ional In eg al (con olle )
PLL Phase-Locked Loop
PR P opo ional Resonan (con olle )
PV Pho o-Vol aic
PWM Pulse-Wid h Modula ion
RES Renewable Ene gy Sou ces
RMS Roo -Mean Squa e
SOC S a e o Cha ge
SPAS Semi-Global P ac ical S abili y
VSI Vol age-Sou ce In e e s
1 In oduc ion
The e is a d i ing o ce mo e powe ul han s eam, elec ici y
and nuclea powe : he will.
a ibu ed o Albe Eins ein
T
his chap e p esen s he backg ound and a b ie summa y o he con ex o his hesis’
con ibu ions, wi h he objec i e o o ien a ing he eade wi hin he b oade scope
o he wo k’s ield o s udy. Fi s ly, a gene al o e iew o AC mic og ids is p o ided.
Secondly, he main con ibu ions a e p esen ed, ollowed by a desc ip ion o he s uc u e
o he manusc ip . Finally, a lis o publica ions and con ibu ions is p o ided.
1.1 In oduc ion o AC Mic og ids
In ou apidly changing wo ld, ene gy e iciency and sus ainabili y a e c ucial, especially
o u u e gene a ions’ well-being. Al e na i e ene gies a e now i al, mee ing he g owing
global demand o cleane , sus ainable powe sou ces. As a esul , he gene a ion o g een
and clean ene gy h ough Renewable Ene gy Sou ces (RESs), such as pho o ol aic (PV)
and wind ene gy, has eme ged as a p ominen opic in ecen decades.
The concep o Mic og id has been widely accep ed as a p omising solu ion o in eg a e
RESs in o he con en ional elec ici y g id in a lexible, eliable, and sus ainable way, as is
in oduced by USA’s CERTS (Conso ium o Elec ic Reliabili y Technology Solu ions)
in 2002 [2].
Mic og ids (MGs) a e de ined as powe dis ibu ion sys ems which consis o Dis-
ibu ed Gene a ion (DG), Ene gy S o age Sys ems (ESSs), and loads ope a ing as a single
con olled subsys em. Gene ally, hey can ope a e in pa allel wi h he b oade u ili y
g id (g id-connec ed mode) wi h a g id- ollowing (cu en dependen ) con ol [3] o as
au onomous powe sys em (islanded mode) wi h a g id- o ming ( ol age dependen ) con-
ol [4]. To ope a e in islanded mode allows o ensu e ene gy supply in emo e loca ions o
in he e en o elec ical ou ages. In g id-connec ed mode, he mic og id is in eg a ed in o
1
2Chap e 1. In oduc ion
Figu e 1.1 Block diag am o an AC mic og id.
he u ili y g id h ough a single poin o connec ion called he Poin o Common Coupling
(PCC), ypically wi h an elec omechanical ci cui b eake .
The main ad an ages o mic og ids a e: imp o ed eliabili y h ough dis ibu ed gen-
e a ion, highe e iciency ia educed ansmission, and in eg a ion o al e na i e ene gy
sou ces. To maximize hese ad an ages, mic og ids mus ensu e minimal dis ibu ion
losses, high eliabili y, ene gy e iciency, blackou esilience, and scalabili y.
In e ms o he ype o powe opology, mic og ids can be di ided in o h ee main g oups:
AC, DC, and hyb id. Because o hei abili y o p o ide a di ec way o in eg a e DGs in o
he main g id, hei simple s uc u e, and cos -e ec i eness, AC mic og ids a e now mo e
in demand [5]. A ypical diag am o an AC mic og id is shown in Fig. 1.1 [6]. The low
o ene gy om he DGs mus be con olled acco ding o he load demand. The con ol
scheme, whe he cen alized o decen alized, mus main ain he quali y and eliabili y o
he supply. DGs can be classi ied in o wo ypes depending on whe he powe ou pu can
be dispa ched on demand o no [6]. A dispa chable uni , like a diesel gene a o , can be
ully con olled. In con as , nondispa chable uni s, such as DGs based on PV and wind
ene gy, a e gene ally in e mi en and di icul o con ol. Consequen ly, mic og ids mus
necessa ily include powe elec onics con e e s o ac as he in e ace medium be ween
nondispa chable DGs and he mic og id. Fu he mo e, ESSs mus be in eg a ed no only o
s o e he powe p oduced by dis ibu ed gene a o s (DGs), bu also o enhance he o e all
1.1 In oduc ion o AC Mic og ids 3
pe o mance and s abili y o he mic og id. This is achie ed by allowing he DGs o un a
a cons an ou pu while hey compensa e o luc ua ions in load powe .
All hese componen s a e in eg a ed in o a complex sys em wi h se e al con ol le els,
whose con ol laws mus be designed o exploi he ad an ages o mic og ids, gua an ee-
ing minimal dis ibu ion losses, ele a ed eliabili y, ene gy e iciency, esilience agains
blackou s, and scalabili y.
1.1.1 Con ol echniques in AC mic og ids
As men ioned be o e, con ol s uc u es play an impo an ole in ensu ing synch oniza ion,
ol age egula ion, powe balance, and load sha ing ac oss all buses in he ne wo k,
con ibu ing o a s able and obus ope a ing poin . The main aim o any con ol echnique
used in AC mic og ids is o main ain a cons an ol age and equency, he e o e some o
he main con ol a ge s a e lis ed below [7]:
•To egula e ol age and equency unde any ope a ing condi ion.
•To ob ain an op imal powe sha ing o ac i e and eac i e powe .
•
To ob ain a consensus on he cha ging/discha ging s a e o ba e ies in islanded mode.
•
To achie e a smoo h ansi ion om g id-connec ed o islanded mode and ice- e sa
•To op imize ope a ing cos o p oduc ion and powe exchanges wi h he u ili y g id.
The con ol s a egy employed depends on he ope a ing mode: g id-connec ed, o islanded.
Du ing g id-connec ed mode, he ol age and equency magni udes a e egula ed by he
u ili y g id. In islanded mode, a g id o ming/suppo ing con e e de e mines ol age and
equency, using ei he a cen alized o decen alized con ol app oach [7]. In his hesis,
we will ocus on con ol app oaches applied o islanded mic og ids.
To accomplish hese con ol equi emen s, di e en hie a chical laye s a e es ablished [6].
The Union o he Coo dina ion o T ansmission o Elec ici y (UCTE, Con inen al Eu ope)
has p esen ed a hie a chical con ol scheme o la ge powe sys ems, whe e h ee main
con ols a e de ined: p ima y, seconda y, and e ia y, which oge he a e used o p o ide
coo dina ion among all DGs, ESSs, and loads h ough managemen s a egies wi h and
wi hou communica ion ne wo ks [8]. The p ima y con ol laye is esponsible o local
ol age egula ion and o ensu ing powe sha ing be ween DGs. The seconda y con ol
ocuses on mi iga ing ol age and equency de ia ions caused by p ima y con ol, and he
e ia y con ol is esponsible o con olling he low o ene gy be ween he u ili y g id
and he mic og id a he PCC, and akes in o accoun economic issues.
1.1.2 P ima y con ol
This is he i s le el o he hie a chical con ol scheme, which is an independen and
locally execu ed s a egy ha allows an au onomous ope a ion o each DG [9], i.e., he
p ima y con ol does no need o communica e and only uses measu ed signals. This
con ol laye aims o achie e he DG ope a ing poin , ypically in ol ing DC/AC powe
con e e s (in e e s) con olled by he inne con ol. A he same ime, app op ia e powe
sha ing is gene ally ensu ed by a d oop con ol, which applies he well-known
P/ω
(ac i e
4Chap e 1. In oduc ion
Figu e 1.2 P ima y con ol loop.
powe / equency ) and
Q/V
( eac i e powe / ol age) cu es, which de e mine he ol age
and equency e e ences o he inne con ol o enhance he pe o mance o he sys em. A
gene al scheme o a DG wi h p ima y con ol is shown in Fig. 1.2.
Inne con ol (in e e ou pu con ol) and powe sha ing (d oop con ol) s a egies a e
discussed below:
Inne con ol
The con ol objec i e o he inne con ol loop is o achie e he ope a ing poin o a powe
con e e ha wo ks as an in e ace o a DG o ESS wi h he mic og id. Gene ally, hese
powe con e e s a e in e e s ha can be con olled as Cu en -Sou ce In e e s (CSIs) o
Vol age-Sou ce In e e s (VSIs) [5]. The con ol o CSIs consis s o an in e nal cu en
con ol loop and a Phase-Locked Loop (PLL) o s ay synch onized wi h he g id, while
VSIs use an ex e nal ol age con ol loop and an in e nal cu en con ol loop.
CSIs a e used mainly in g id-connec ed ope a ion as hey wo k as a g id- eeding con-
e e capable o injec ing ac i e and eac i e powe in o he PCC, while he ol age a he
PCC is go e ned by he u ili y g id o a g id- o ming con e e [7]. These con e e s a e
o en connec ed o PV o wind u bines, which a e con olled by a maximum powe poin
acking algo i hm. In con as , VSIs wo k as a g id- o ming con e e , which se s ol age
and equency e e ences o he mic og id in islanded mode ope a ion. These con e e s
a e o en connec ed o ESS uni s as hey se he ol age and equency a he PCC. Bo h
VSIs and CSIs can wo k oge he in a mic og id, so se e al VSIs and CSIs, o only VSIs,
can be connec ed in pa allel o o m a mic og id.
The use o VSIs is mo e lexible because hey do no need an ex e nal e e ence o
s ay synch onized and can p o ide ide- h ough capabili y ( esilience o sho -du a ion
aul s o dis u bances) and imp o e powe quali y [8]. In addi ion, i is he main ype o
con e e ha is used in islanded mic og ids. The in e nal and ex e nal con ol loops o
1.1 In oduc ion o AC Mic og ids 5
Figu e 1.3 Inne con ol loop.
VSIs a e desc ibed in Fig. 1.3 [10]. The ex e nal ol age egula o mus minimise he e o
be ween he e e ence and he measu ed ou pu ol age and gene a e he cu en e e ence
o he in e nal cu en loop which gene a es he desi ed ou pu ol age o be modula ed.
Then, a modula o is used o con e he con inuous con ol signal
u
in o a disc e e one o
he swi ching o he con e e , whe e he mos common is he Pulse-Wid h Modula ion
(PWM), which changes he du y cycle o a ixed equency squa e signal in esponse o
he modula ed signal (con ol signal). Howe e , he e a e a numbe o con ol s a egies
ha do no use he common scheme o a con inuous con olle plus a modula o . Ins ead,
hey simply calcula e he disc e e signals, aking in o accoun all possible combina ions o
ou pu s.
The con olle s o bo h loops can be linea , such as P opo ional-In eg al (PI), P opo ional-
Resonan (PR), p edic i e con olle s, o nonlinea , such as hys e esis, sliding mode, uzzy
logic o hyb id dynamic con olle s. A pa icula in e es has eme ged o apply hyb id
con ol echniques o powe con e e s wi h he aim o conside ing he sub le y o he
hyb id na u e o hese sys ems [11
–
13]. Indeed, hey a e composed o con inuous- ime
dynamics ( ol ages and cu en s) and disc e e- ime dynamics (swi ching). Some o he
main con ibu ions o his wo k a e based on Hyb id Dynamical Sys ems (HDS) hence in
he nex chap e his heo y is explained in de ail.
The ollowing pa desc ibes he mos commonly used linea ype con olle s: The PI
con olle and he PR con olle :
PI con olle :
The PI con olle is used ypically in enginee ing due o i s simple s uc u e, easy imple-
men a ion, and good pe o mance. I s classical ans e unc ion is depic ed below:
CPI(s) = Kp+Ki
s,(1.1)
whe e
Kp
and
Ki
a e he p opo ional and in eg al gain espec i ely. This con olle is
sui able o DC signals due o i s high gain a low equencies. Fo AC magni udes, i has
a limi ed esponse, hus i equi es ansla ion o he con olle signals om sinusoidal o
cons an alues, ha is, he
dq
e e ence ame (di ec -quad a u e). This e e ence ame
o a es a he same equency, as he magni ude conside ed in AC, and hus he magni ude
exp essed in hese axes appea s as a DC componen [14].
12 Chap e 2. P elimina y concep s
dom M={x∈Rm:M(x)=∅}.
Rema k 1:
No e ha a se - alued mapping
M:Rm⇒Rn
associa es wi h e e y poin
x∈Rm
a subse o
Rn
. The double a ow no a ion
M:Rm⇒Rn
dis inguishes a se -
alued mapping
M
om a unc ion. Also, we can w i e
M:Rm⇒S
, o
S⊂Rn
, which
indica es ha M:Rm⇒Rnis a se - alued mapping wi h M(x)⊂S o all x∈Rm.
Wi h he p e ious de ini ion, he in eg a ion o con inuous and disc e e beha io in
(2.1)
o a hyb id sys em in Rnis ep esen ed by ou objec s desc ibed below:
•
A se
C ⊂ Rn
, called he low se , which de ines he egion whe e he sys ems
p esen s a con inuous- ime e olu ion.
•
A se - alued mapping
F(x) : Rn⇒Rn
(o unc ion
) wi h
C ⊂dom F
, called he
low map. I indica es how he sys em e ol es o lows.
•
A se
D ⊂ Rn
, called he jump se , which de ines he egion whe e he sys ems
p esen s a disc e e- ime beha iou .
•
A se - alued mapping
G(x) : Rn⇒Rn
(o unc ion
g
) wi h
D⊂dom G
, called he
jump map. I indica es how he sys em ins an aneously changes o jumps.
The e o e, a hyb id sys em wi h he da a as abo e can be ep esen ed by he no a ion
H(C,F,D,G), o b ie ly by H.
2.1.2 Hyb id ime domains and hyb id a cs
A solu ion o a hyb id sys em is pa ame e ized by a hyb id ime domain which is com-
posed o con inuous- ime and disc e e- ime a iables. The elapsed con inuous ime is
pa ame e ized by
∈R≥0
, and he numbe o jumps ha ha e occu ed is pa ame e ized
by j∈N.
De ini ion 2:
(Hyb id ime domain) [18, De ini ion 2.3] A subse
E⊂R≥0×N
is a hyb id
domain i
E=
J−1
[
j=0
([ j, j+1],j)
whe e jcan be ini e o in ini e. I is a compac hyb id domain i jis ini e.
In o he wo ds,
E
is a hyb id ime domain i i is a union o a ini e o in ini e sequence
o in e als
[ j, j+1]×{j}
, while
E
is a compac hyb id ime domain i his sequence is
ini e.
De ini ion 3: (Hyb id a c) [18, De ini ion 2.4] A unc ion ϕ:E→Rnis a hyb id a c i :
•Eis a hyb id ime domain and,
•
i o each
j⊂N
, he unc ion
→ϕ( ,j)
is locally absolu ely con inuous on he
in e al Ij={ : ( ,j)∈E}
2.1 Hyb id dynamical sys ems 13
Figu e 2.1 Hyb id a c.
Fig. 2.1 shows a g aphical ep esen a ion o a hyb id a c
ϕ
wi h hyb id ime domain
dom ϕ
. Fu he mo e, a hyb id a c can be classi ied based on he s uc u e o hei domains.
To ci e some o hem, a hyb id a c is:
•non i ial i dom ϕhas a leas wo poin s;
•con inuous i i is non i ial and dom ϕ⊂R≥0×{0};
•disc e e i i i is non i ial and dom ϕ⊂{0}×N, and
•comple e i dom is unbounded i.e., i leng h(E) = ∞.
The solu ions o a hyb id sys em
H
a e gi en by a hyb id a c ha sa is ies ce ain condi ions
de e mina ed by he hyb id ime domain and he da a o H.
De ini ion 4:
(Solu ion o a hyb id sys em) [18, De ini ion 2.6] A unc ion
ϕ
is a solu ion
o he hyb id sys em H(C,F,D,G), i :
1. The ini ial condi ion ϕ(0,0) ∈C∪D, whe e Cdeno es he closu e o he se C.
2. Fo all j∈Nsuch ha Ij={ : ( ,j)∈E}has nonemp y in e io
ϕ( ,j)∈C o all ∈in Ij,
˙
ϕ( ,j)∈F(ϕ( ,j)) o almos all ∈Ij;
3. Fo all ( ,j)∈dom ϕsuch ha ( ,j +1) ∈dom ϕ,
ϕ( ,j)∈D,
ϕ( ,j +1) ∈G(ϕ( ,j)).
Fig. 2.2 illus a es a solu ion admi ed by he p e ious de ini ion. He e a e ep esen ed
he lows and jumps o he solu ion
ϕ
, whe e he lows a e allowed only on he low
se Cand he jumps mus o igina e om he jump se D.
14 Chap e 2. P elimina y concep s
Figu e 2.2 E olu ion o a solu ion o a hyb id sys em.
2.1.3 Basic assump ions
In his sec ion, we p esen he basic assump ions on a hyb id sys em o ensu e ha i is
well-posed, which is an impo an p ope y equi ed o he applicabili y o many esul s
in hyb id sys ems heo y, as shown in [18].
Assump ion 1: (Hyb id basic condi ions) [18, Sec ion 6.2]
1. Cand Da e closed subse s o Rn;
2. F:Rn⇒Rn
is ou e semicon inuos and locally bounded ela i e o
C
,
C ⊂dom F
,
and F(x)is con ex o e e y x∈C;
3. G:Rn⇒Rn
is ou e semicon inuos and locally bounded ela i e o
D
, and
D⊂
dom G.
Theo em 1:
(Basic condi ions and well-posedness) [18, Theo em 6.30] I a hyb id sys em
H(C,F,D,G)sa is ies Assump ion 1 hen i is well-posed. □
The p oo o he well-posedness heo em is depic ed in [18, Chap e 6].
I
F
and
G
a e single alue maps, ha is, unc ions, he equi emen is ha
(x)
and
g(x)
mus be con inuous (ou e semicon inuous as a se - alued mapping), hence he di e en ial
equa ion
˙z= (x)
and he di e ence equa ion
x+=g(x)
can be iden i ied wi h a hyb id
sys em ha sa is ies he hyb id basic condi ions.
De ini ion 5:
(De ini ion 5.9 [18]) A se - alued mapping
F:Rn⇒Rn
is said o be ou e
semicon inuous (osc) i each sequence
(xi,yi)∈Rn×Rn
ha sa is ies
yi∈F(xi)
o each
i
and con e ges o a poin
(x,y)∈Rn×Rn
, has he p ope y ha
y∈F(x)
. Mo eo e ,
ano he way o see his p ope y is ha a se - alued mapping
F
is ou e semicon inuous i
and only i he g aph o Fis closed (Fig. 2.3).
De ini ion 6:
(De ini ion 5.14 [18]) A se - alued mapping
F:Rm⇒Rn
is locally bounded
a
x∈Rm
i he e exis s a neighbo hood
Ux
o
x
such ha
F(Ux)⊂Rn
is bounded. Gi en
a se
S⊂Rm
, he mapping
F
is locally bounded ela i e o
S
i he se - alued mapping
om
Rm
o
Rn
de ined by
F(x)
o
x∈S
and
∅
o
x /∈S
is locally bounded a each
x∈S.
2.1 Hyb id dynamical sys ems 15
Figu e 2.3 Mappings:(a) no ou e semicon inuous and (b) ou e semicon inuous.
2.1.4 Lyapuno condi ions o hyb id sys ems
Based on sec ion [18, Sec ion 3.2], we p esen he ollowing heo em add essing he
Lyapuno unc ion as a su icien condi ion o s abili y p ope ies o a compac se
A
ela ed o a hyb id sys em. This compac se is de ined as he a ac o o he sys em.
Mo eo e , he da a o he hyb id sys em mus sa is y he basic hyb id condi ions depic ed
in he p e ious sec ion.
Rema k 2:
An a ac o is de ined as he se o all s a es owa d which a sys em ends o
e ol e, ega dless o he s a ing condi ions o he sys em.
Theo em 2:
Gi en a hyb id sys em
H
wi h s a e
x
, and a compac se
A⊂Rn
, i he e
exis a Lyapuno unc ion candida e V o Hsuch ha
1. (Lyapuno unc ion candida e)
V(x)=0,∀x∈A,
V(x)>0,∀x∈C∪D∪G(D) A,
lim
|x|→∞V(x) = ∞,∀x∈C∪D∪G(D)
2. (Flow)
˙
V=⟨∇V(x), (x)⟩<0,∀x∈C A, ∈F(x)
3. (Jump)
∆V=V(g(x))−V(x)<0,∀x∈D A, g ∈G(x)
G(A∩D)⊂A
hen Ais Uni o mly Globally Asymp o ically S able (UGAS) o H.□
Rema k 3: A se Ais a compac se i and only i Ais closed and bounded.
In his heo em, he i s poin s a es he condi ion o a unc ion
V
o be a Lyapuno
unc ion candida e wi h espec o
A
o
H
. Poin s 2 and 3 a e called he Lyapuno
condi ions o low and jump, espec i ely. A Lyapuno unc ion candida e
V
mus be
16 Chap e 2. P elimina y concep s
con inuous and posi i e de ini e wi h espec o
A
, con inuously di e en iable in an open
se con aining
C
, and adially unbounded. Fu he mo e, he Lyapuno condi ions s a e ha
o low: he e a e s ic dec emen s whene e he sys em lows away om he se
A
; and
o jump: he di e ence be ween he alue a e he jump and he alue be o e he jump
mus be nega i e.
A sui able Lyapuno unc ion candida e om Theo em 2 is illus a ed in Fig. 2.4.
Figu e 2.4 E olu ion o a lyapuno unc ion candida e V(x)wi h x∈A o H.
2.2 Singula pe u ba ion heo y
In his sec ion, we p esen he gene al heo y o singula pe u ba ions applied o s abili y
analysis o bo h con inuous- ime and hyb id sys ems. Fi s , some basic concep s o
singula pe u ba ion heo y a e gi en, and hen s abili y app oaches a e explained.
2.2.1 S anda d singula pe u ba ion model
To simpli y complex sys ems, singula pe u ba ion me hods a e used o spli he dynamics
o he sys em in o wo pa s: slow and as dynamics. Singula pe u ba ions in oduce
mul i ime-scale beha io in he sys em’s esponse o ex e nal s imulus. By in oducing an
ex e nal pa ame e and hen se ing i o ze o, a s ep owa ds " educed-o de modeling"
is achie ed. This o de educ ion is ans o med in o a pa ame e pe u ba ion called
"singula ".
As is p esen ed in [19, Chap e 11] and [20, Chap e 1], he singula pe u ba ion model
o a dynamical sys em is ob ained by mul iplying he de i a i es o some o he s a es by a
small posi i e pa ame e ε, ha is,
˙x= ( ,x,z,ε)(2.2)
ε˙z=g( ,x,z,ε)(2.3)
whe e
x∈Rn
ep esen s he slow a iables whe eas
z∈Rm
he as ones, wi h
and
g
being con inuously di e en iable unc ions in hei a gumen s
( ,x,z,e)
. When we se
ε= 0
in
(2.2)
and
(2.3)
he dimension o he s a e space equa ion educes om
n+m
o
2.2 Singula pe u ba ion heo y 17
nbecause he di e en ial equa ion (2.3) u ned in o he algeb aic equa ion:
0 = g( ,x,z,0).(2.4)
I (2.4) has k≥1dis inc ("isola ed") eal oo s:
z=hi( ,x)i= 1,2,...,k, (2.5)
he model
(2.2)
–
(2.3)
is in s anda d o m. Equa ion
(2.5)
desc ibes an
n
-dimensional
mani old o he s a e space. Fo each oo o
(2.4)
a well-de ined educed model is
ob ained, subs i u ing he oo (2.5) in o he slow dynamics equa ion (2.2), a ε= 0, so,
˙x= ( ,x,h( ,x),0).(2.6)
This model is known as he slow model, also e e ed o as he quasi-s eady-s a e model,
because he as dynamical a iable
z
has been eplaced by i s equilib ium, o quasi-s eady-
s a e,h( ,x)(2.4).
Howe e , we ha e no in o ma ion abou how
z
e ol es owa d i s equilib ium. We know
ha he eloci y o
z
can be la ge, since
˙z=g/ε
. And in ac , by se ing
ε= 0
in
(2.3)
,
he ansien beha io o
z
becomes ins an aneous whene e
g= 0
. The e o e, we need o
analyze whe he his ansien zescapes o in ini y o con e ges o i s equilib ium.
To analyze
(2.3)
, we i s no e ha
ε˙z
can emain ini e e en i
ε
ends o ze o and
˙z
ends o in ini y.
To check his, we s e ch o scale he ime by a ac o o 1/ε. We se
εdz
d =dz
dτ ,hence, dτ
d =1
ε.
On he τ imescale, (2.3) is ew i en as:
dz
dτ =g( ,x,y +h( ,x),0),(2.7)
whe e
y=z−h( ,x)
. This shi s he equilib ium o
z
o he o igin. By se ing
ε= 0
,
( ,x)
a e ozen a hei ini ial alues and ea ed as ixed pa ame e s (cons an s). This
au onomous sys em (2.7) is known as he bounda y-laye sys em.
2.2.2 Con inuous- ime s abili y analysis
Singula pe u ba ion heo y can be applied o nonlinea sys ems cha ac e ized by bo h
slow and as dynamics o analyze hei s abili y p ope ies [19, Chap e 11.5]. To apply
singula pe u ba ion analysis, we i s need o w i e he sys em in he s anda d singula
pe u ba ion o m (2.2)–(2.3), as desc ibed in he p e ious sec ion.
Then, he nex h ee s eps in singula pe u ba ion analysis should be ollowed and some
basic assump ions a e conside ed:
a) Mani old:
18 Chap e 2. P elimina y concep s
Fi s , we need o ind he quasi-s eady-s a e equilib ium mani old o he as sub-
sys em
(2.3)
, which co esponds o he "isola ed oo ",
z=h( ,x)
, de i ed om
(2.4) wi h ε= 0.
Assump ion 2:
The equa ion
(2.4)
has an isola ed oo
z=h( ,x)
, which is he
quasi-s eady-s a e mani old and he s a iona y solu ion o he as subsys em.
b) Bounda y-laye sys em ( as subsys em):
In he second s ep, we conside he bounda y laye sys em
(2.7)
, which ep esen s
he apid ansien o he as subsys em in he ini ial ime in e al be o e i con e ges
o i s quasi-s eady s a e. Since hese dynamics a e conside ed ins an aneous as
ε
ends o
0
, we ha e o scale he ime
by
1/ε
in
(2.3)
. He e, he slow dynamics a e
ea ed as cons an s ( ˙x= 0).
Assump ion 3:
The o igin o he bounda y laye sys em de ined in
(2.7)
is asymp-
o ically s able.
c) Reduced sys em (slow subsys em):
In he las s ep, we conside he educed sys em ob ained by eplacing he isola ed
oo in he slow subsys em,
(2.2)
. He e, he as dynamics a e in s eady s a e and
only he slow a iables a e conside ed.
Assump ion 4:
The o igin o he educed sys em de ined in
(2.6)
is asymp o ically
s able.
Now, we p esen Theo em 3, based on Assump ions 2– 4, o s a e ha he o igin o
a singula pe u bed sys em in he s anda d o m
(2.2)
–
(2.3)
is asymp o ically s able.
The ollowing assump ions s a e asymp o ic s abili y equi emen s on he educed and
bounda y-laye sys ems, exp essed by he exis ence o Lyapuno unc ions o each sys em.
Theo em 3:
[19, Theo em 11.4] Conside he singula pe u bed sys em
(2.2)
–
(2.3)
, and
Assump ions 2– 4 a e sa is ied. Then he e exis s
ε∗>0
such ha o all
ε<ε∗
, he o igin
o (2.2)–(2.3) is asymp o ically s able. □
The p oo o Theo em 3 is de ailed in [19].
2.2.3 S abili y analysis o hyb id dynamical sys ems
The singula pe u ba ion s abili y analysis could be adap ed and applied o hyb id dynam-
ical sys ems wi h slow and as dynamics, as de ailed in [21].
Conside a hyb id sys em
(2.1)
, whose s a e
x∈Rn
, whe e
n=n1+n2
, is composed
o slow a iables
x1∈Rn1
and as a iables
x2∈Rn2
. Then, we can w i e i in singula
pe u ba ion o m as ollows:
2.2 Singula pe u ba ion heo y 19
(diag(In1,εIn2) ˙x∈F(x), x ∈C1×C2
x+∈G(x), x ∈D1×D2
(2.8)
whe e
ε > 0
is a small scala ,
Ini
deno es he
ni×ni
iden i y ma ix,
C1,D1⊂Rn1
and
C2,D2⊂Rn2.
In [21] he condi ions o semi-global p ac ical asymp o ic s abili y o a compac se
A=A ×C2
as
ε→0+
associa ed wi h he hyb id sys em a e p esen ed. As in Sec-
ion 2.2.2, hese condi ions a e exp essed conside ing he h ee key s eps in singula
pe u ba ion analysis: he quasi-s eady-s a e mani old, he bounda y laye sys em and he
educed sys em. Fu he mo e, he well-posedness o he hyb id sys em is equi ed in his
case.
The ollowing s eps and basic assump ions a e conside ed:
a) Well-posed hyb id sys em:
In he i s s ep o his s abili y analysis, he well-posedness o he hyb id sys em is
conside ed. The e o e, he egula i y o he da a o he sys em mus be gua an eed
by sa is ying he h ee poin s o he Assump ion 1.
b) Mani old:
The quasi-s eady-s a e mani old appea s in he case o he hyb id app oach as a se -
alued mapping
Ξ : Rn1⇒Rn2
and ep esen s, as in classical singula pe u ba ion
heo y, he s a iona y solu ion o he as subsys em.
Assump ion 5:
(Regula i y o "Mani old"). The se - alued mapping
Ξ : Rn1⇒Rn2
is ou e semicon inuous and locally bounded, and o each
x1∈C1
hen
Ξ(x1)
is
nonemp y subse o C2.
c) Bounda y-laye sys em ( as subsys em):
The amily o bounda y laye sys ems is gi en by
˙x∈diag(0,In2)F(x), x ∈(C1∪ρB)×C2(2.9)
wi h
ρ > 0
, which makes he low se compac . The bounda y laye is ob ained by
scaling he ime
by
1/ε
in
(2.8)
. Mo eo e , he bounda y laye sys ems igno e
jumps and slow a iables emain cons an s, i.e. ˙x1= 0 du ing lows.
Assump ion 6:
(S abili y o bounda y laye ). Fo each
ρ > 0
he e is a closed ball
ρB, such ha he compac se
M:= {(x1,x2) : x1∈(C1∩ρB),x2∈Ξ(x1)}
associa ed wi h he bounda y laye sys em
(2.9)
, is Globally Asymp o ically S able
(GAS).
20 Chap e 2. P elimina y concep s
d) Reduced sys em (slow subsys em):
The educed sys em is ob ained conside ing ha he as dynamics a e in s eady
s a e gi en by
Ξ(x1)
, he e o e only slow a iables a e conside ed. The educed
sys em is gi en by:
(˙x∈F (x1), x ∈C1
x+∈G (x1), x ∈D1,(2.10)
whe e
F (x1) := diag(In1,0)F(x1,x2), x2∈Ξ(x1)
G (x1) := diag(In1,0)G(x1,x2)
The jump map o he educed sys em is no exp essed in e ms o
x2∈Ξ(x1)
since
he bounda y laye sys em igno es jumps. In he mos s aigh o wa d case,
G
does
no depend on
x2
, and hus he educed sys em igno es
x2
when de e mining jumps.
Ou ocus is on he s abili y o he compac se
A1:= {x1∈C :x1=x∗
1}
ep esen s
he equilib ium s a e o x1in he educed model.
Assump ion 7:
(S abili y o educed sys em). Fo he educed sys em
(2.10)
, he
compac se A1⊂Rn1is globally asymp o ically s able.
Theo em 4:
[21, Theo em 1] Unde Assump ion 1 and Assump ions 5– 7, he compac
se A1×C2is Semi-globally P ac ically Asymp o ically S able (SPAS) as ε→0+.□
The p oo o Theo em 4 is de ailed in [21].
2.3 Mul i-Agen Sys em heo y
A mul i-agen sys em consis s o a eam o elemen s, called agen s, ha exchange in o ma-
ion o e a communica ion ne wo k. In dis ibu ed con ol s a egies, hese agen s mus
espond o unexpec ed si ua ions o changes in he sys em. To achie e e ec i e con ol
coo dina ion, he eam o agen s mus each a consensus on he coo dina ion da a h ough
aconsensus algo i hm.
Consensus algo i hm:
A consensus algo i hm is an in e ac ion ule ha es ablishes how agen s sha e in o ma-
ion wi h hei neighbo ing agen s in he ne wo k. Th ough hese algo i hms, he en i e
eam can each ag eemen on ele an quan i ies ha depend on he s a es o all agen s [22].
An example o a consensus algo i hm is p esen ed below:
2.3 Mul i-Agen Sys em heo y 21
Conside a mul i-agen sys em o med by
i={1,...,N}
agen s. The objec i e is o design
a dis ibu ed con ol law which ensu es ha he ou pu ec o o each agen , ep esen ed
by
yi
, achie es ag eemen . The e o e, he consensus algo i hm can be w i en as ollows:
ui=−KX
j∈Vi
(yi−yj),(2.11)
whe e
K > 0
is a con ol ma ix, and
Vi
he neighbo hood o agen
i
. This model was
used in [23].
G aph heo y:
MAS uses g aph heo y o ep esen he ne wo k communica ion be ween agen s. G aphs
a e pic u es consis ing o a se o poin s, some o which a e connec ed by lines. The poin s,
called e ices o nodes, ep esen he agen s, while he lines be ween poin s, called edges,
ep esen he communica ion links. G aphs in which edges ha e no di ec ion, allowing
bidi ec ional communica ion be ween e ices, a e called undi ec ed g aphs. In con as ,
di ec ed g aphs ha e edges wi h ixed di ec ions [24]. Hence, he ollowing de ini ions a e
gi en:
De ini ion 7:
[22] A di ec g aph is de ined as
G(N,E)
consis ing o a se o
N
ele-
men s called e ices,
N={1,2,...,N}
and a se o o de ed pai s o e ices called edges,
ep esen ed by E ⊆N ×N. The pai {i,j}deno es an edge om he elemen i o j.
De ini ion 8:
[22] An undi ec ed g aph consis s o a se o e ices
N
and a se o edges
Esuch ha , o all pai s o elemen s i,j ∈N,{i,j}∈Eand {i,j}∈E.
Fig. 2.5 ep esen s an example o a g aph wi h ou e ices (1,2,3,4) and i e edges
((1,2), (1,3), (2,3), (2,4) and (3,4)). No e ha his g aph is undi ec ed; hence he edge (1,2)
is he same as he edge (2,1).
Figu e 2.5 Example o a g aph [24].
I a g aph has an edge be ween e e y pai o e ices, i is said o be a comple e g aph.
The numbe o edges inciden on a e ex is he deg ee o a e ex ep esen ed by
δi
, in
o he wo ds he numbe o i s neighbo s.
28 Chap e 3. Con inuous con ol scheme o discha ging a e consensus
Figu e 3.1 Islanded AC mic og id wi h BESS uni s.
Figu e 3.2 S uc u e o a BESSiin an AC islanded mic og id .
As men ioned abo e, he main con ol objec i e o manage he se o
N
-BESSs in
discha ging mode is SOC balancing o consensus. Taking in o accoun his s a emen and
he p e ious li e a u e o e iew, he con ibu ions o his chap e ocus on he design o a
dis ibu ed con ol scheme ela ed o he
N
-BESSs in discha ging mode, using a con ol
app oach ha achie es he ollowing objec i es:
3.2 P oposed con ol scheme 29
Figu e 3.3
P oposed dis ibu ed con ol scheme o discha ging a e consensus in an is-
landed AC mic og id.
1) To ensu e ha he es ima ed SOCs o he se o BESS in discha ging mode in an
islanded AC mic og id con e ge o a consensus in o de o inc ease he ba e y li espan.
2) To design a dis ibu ed comple e con ol scheme ha conside s bo h seconda y and
p ima y con ol loops.
3) To p o ide la ge-signal s abili y analysis o he comple e con ol sys em.
3.2 P oposed con ol scheme
As mo i a ed abo e and shown in Fig. 3.3, we p opose a dis ibu ed con ol scheme wi h:
•Seconda y con ol loop:
–
Consensus algo i hm: a con olle ha eaches a balancing o SOCs conside -
ing any communica ion ailu e.
•P ima y con ol loop:
–
D oop con ol: a mechanism ha ensu es he con e gence o he SOC and ac-
i e powe s o hei e e ences, and gene a es ol age and equency e e ences
o he inne con ol loop.
–Inne con ol: a con ol law o hal -b idge in e e s.
The objec i e o he seconda y con ol loop is o achie e a consensus among he es-
ima ed SOCs,
Ψi
, o he in e connec ed BESSs. This is eached indi ec ly h ough a
consensus algo i hm based on MAS applied o he e e ence a iables o he es ima ed
SOCs,
Ψ ,i
. The neighbo ing BESSs a e conside ed as agen s ha sha e he alue o hei
SOC e e ences and independen ly each calcula es he consensus alue
Ψ∗
. The consensus
e e ence eeds he p ima y con ol composed o a d oop con ol and an inne con ol loop.
The d oop con ol based on con en ional
P/w
and
Q/V
cu es is designed o ensu e ha
he es ima ed SOCs and ac i e powe s con e ge o hei e e ences, espec i ely. This
30 Chap e 3. Con inuous con ol scheme o discha ging a e consensus
d oop con ol gene a es he equency,
w ,i
, and ol age,
V ,i
, e e ences o he inne
con ol loop o he in e e . These h ee con olle s, namely, consensus algo i hm, d oop
con ol, and in e e con ol, exhibi a h ee- ime-scale sepa a ion h ough an app op ia e
choice o con ol pa ame e s. The aim is o ob ain a la ge signal s abili y analysis using
singula pe u ba ion heo y.
3.3 Con ol objec i es
The con ol objec i es conside ed he e a e o mula ed as ollows:
Design a comple e con inuous- ime dynamic model o he p ima y and seconda y
con ol loops in an islanded AC mic og id wi h a se o
N
-BESSs in discha ging mode,
such ha i accomplishes he ollowing con ol objec i es o each
i∈N := {1,2,...,N}
agen :
1.
Con e gence o he in e e s a e
xi:= [iL,i, C,i]⊤
, wi h
iL,i
he induc ance cu en
and
C,i
he capaci o ol age, o a e e ence
x ,i
h ough a con inuous- ime con ol
law ha gua an ees GAS o a small neighbo hood o xi=x ,i(V ,i,ω ,i).
2.
Con e gence o he SOC e e ences,
Ψ ,i
, o he in e connec ed BESSs o a consensus
alue Ψ∗.
3. Con e gence o Ψiand Pi, o hei e e ences Ψ ,i and P ,i, espec i ely.
4.
SPAS p ope y o he comple e closed-loop con ol sys em, conside ing a la ge-
signal analysis, ha includes powe con e e con ol, d oop con ol, and e e ence
consensus, by using a h ee ime-scale sepa a ion model and applying singula pe -
u ba ion heo y.
3.4 Dynamical models in he p ima y and seconda y con ol loops
This sec ion desc ibes he h ee p oposed closed-loop dynamic models ha achie e he
con ol objec i es men ioned abo e.
3.4.1 Consensus algo i hm
The SOC e e ences consensus algo i hm o he BESSs is designed using MAS heo y,
whe e each BESS
i
is conside ed as an agen and he ene gy exchange be ween wo agen s
a e conside ed as edges. The ene gy exchange be ween di e en agen s is co e ed using
g aph heo y and will be p esen ed la e in Sec ion 3.5.2.
3.4 Dynamical models in he p ima y and seconda y con ol loops 31
The dynamic o he consensus algo i hm o SOC e e ences (
Ψ ,i
) is p oposed he e
∀i∈N acco ding o
˙
Ψ ,i =−αiKc
N
X
j=1
αj(Ψ ,i −Ψ ,j)
−Ke(Ψ ,i −Ψi(xi)) (3.3)
wi h
Kc
and
Ke
posi i e pa ame e s. The in ui i e idea o his exp ession is he ollowing.
The i s e m o
(3.3)
looks o achie ing a consensus be ween all
Ψ ,i
, while he second
e m collec s he de ia ion be ween he es ima ed SOC o BESS
i
,
Ψi
, and i s e e ence
Ψ ,i ∈[0,1]
.
αi= 1
ep esen s i he BESS
i
is communica ing and
αi= 0
a communica ion
ailu e o BESSi.
The es ima ion me hod o he SOC is adop ed om [1] based on he Coulomb coun ing
es ima ion echnique desc ibed in Sec ion 3.1.1. I we ake in o accoun ha he powe
loss in he con e sion is omi ed and i is assumed ha he ba e ies ha e he same ou pu
ol age,
Vin,i
, he ou pu cu en could be ew i en as
ib,i =Pi
Vin,i
in
(3.2)
. Thus, he SOC
o he BESSidepends on he ac i e powe as ollows,
dΨi
d =−δiPi∀i∈N (3.4)
wi h
δi:= k
Cba ,iVin,i
, whe e
k
is a ime scale a io, and
Cba ,i
is exp essed in ampe e
hou s (Ah).
The nex d oop con ol is key o gua an ee ha he es ima ed SOC con e ges o i s
e e ence, ending his de ia ion o ze o in a ini e ime, as will be p o en la e .
3.4.2 D oop con ol
A d oop con ol is used he e o gua an ee ha he es ima ed SOCs and ac i e powe s
con e ge o hei e e ences, which a e gi en by he consensus algo i hm
(3.3)
. We
conside he
P/ω
d oop ela ion o be c ucial o he con ol objec i es, as i has a di ec
in luence on he ac i e powe and hus he SOC o he BESSi.
Based on he well-known
P/ω
and
Q/V
cu es, and adding a p opo ional-in eg al (PI)
adjus men e m abou he SOC e e ence alue, he d oop con ol modi ies he equency
and ampli ude o he in e e ol age e e ence ∀i∈N, as ollows:
ω ,i =ωn−Kd,1(Ψi−Ψ ,i)−Kd,2Z(Ψi−Ψ ,i)d
−Kd,3(Pi−P ,i)(3.5)
V ,i =Vn−KV,iQi,(3.6)
whe e
ωn
and
Vn
a e he nominal equency and ol age, espec i ely, and
Pi
and
Qi
a e
he ac i e and eac i e powe s. We will p o e la e , ha
Ψi
and
Pi
con e ge asimp o ically
32 Chap e 3. Con inuous con ol scheme o discha ging a e consensus
o hei e e ence alues Ψ ,i and P ,i i he d oop con ol pa ame e s a e de ined as
Kd,1=Kω
Kc1,
Kd,2=Kω
Kc1KΨ+K2
c1
Kω+K2
0
KdP
Kd,3=KωK0
Kc1KdP ,
(3.7)
whe e
Kω,KΨ,KdP ,K0,
and
Kc1
a e posi i e pa ame e s, which will be used la e in he
hyb id scheme.
The e o dynamics ob ained om (3.5), ∀i∈N a e:
˙
˜ωi=−Kω˜ωi−Kc1˜
Ψi
˙
˜
Ψi=−KΨ˜
Ψi+Kc1˜ωi+K0˜
Pi
˙
˜
Pi=−KdP ˜
Pi−K0˜
Ψi,
(3.8)
wi h
˜
Ψi= Ψi−Ψ ,i
,
˜
Pi=Pi−P ,i
, and
˜ωi=ω ,i −ωn,p,i
, whe e
ωn,p,i
is he nominal
equency associa ed wi h he desi ed e e ence
x ,i
, such ha
˜ωn,i =ωn,p,i −ωn
[27].
No e ha
(3.5)
is ob ained om simple ma hema ical manipula ions o
(3.8)
, which a e
based on he c oss-coupling o e o s among
˜
Ψi
,
˜
Pi
, and
˜ωi
, designed o be asymp o ically
s able using a Lyapuno unc ion candida e, which will be p o ed in Sec ion 3.6.
3.4.3 Con inuous- ime con ol law o he in e e s
We conside a hal -b idge con e e as he in e e o he
N
-BESSs in discha ging mode.
As shown in Fig. 3.4, his in e e is ed by a DC sou ce and, by he commu a ion o wo
swi ches (
U1,i,U2,i
), gene a es a single-phase AC ou pu ollowed by an LC il e . The
dynamical model o his in e e can be ew i en acco ding o [38] as ollows:
˙xi=Aixi+Biui,∀i∈N (3.9)
whe e
Ai="−RLS,i
Li−1
Li
1
Ci−1
RiCi#
,
Bi=Vin,i
Li
0.
wi h
Ci
and
Li
, he capaci ance and induc-
ance espec i ely,
Ri
and
RLS,i
, he nominal con e e load and he pa asi ic esis ance
espec i ely, and
Vin,i
he inpu ol age.
xi= [iL,i, C,i]⊤∈R2
ep esen s he con inuous-
ime s a e ec o , such ha
iL,i
is he induc ance cu en and
C,i
is he capaci o ol age,
while
ui=U1,i −U2,i ∈{−1,1}
is he he disc e e- ime con ol inpu , ep esen ing wo
ope a ing modes:
Mode 1: ui=−1,i U1,i =OFF and U2,i =ON → C,i =−Vin,i
Mode 2: ui= 1,i U1,i =ON and U2,i =OFF → C,i = +Vin,i
No e ha capaci o s C1,i and C2,i a e la ge enough such ha he ipple is negligible.
The desi ed ou pu ol age e e ence is de ined by
C ,i =V ,i sin(ω ,i )
3.4 Dynamical models in he p ima y and seconda y con ol loops 33
Figu e 3.4 Hal -b idge in e e .
whe e
ω ,i
and
V ,i
ep esen he desi ed equency and ol age, espec i ely. F om
C ,i
, we can de i e he induc ance cu en e e ence,
iL ,i =Ciω ,iV ,i cos(ω ,i ) +
V ,i
Risin(ω ,i )
. Then, he e e ence signal o he comple e s a e
x ,i
can be gene a ed
om a gene ic oscilla o :
˙zi= Θizi(3.10)
wi h zi(0) = h0
V ,i iand Θi=h0−ω ,i
ω ,i 0i, such ha z2,i = C ,i . Hence,
zi∈Φi,Φi={z1,i,z2,i ∈R:z2
1,i +z2
2,i =V2
,i},
he e o e he comple e s a e e e ence x ,i = [iL ,i , C ,i ]⊤is de ined as ollows
x ,i = Πizi,(3.11)
wi h Πi=hω ,iCi1
Ri
0 1 i.
The con ol law used o hese in e e s a e wo PIs con olle s, as shown in Fig. 3.5,
which a e desc ibed in he ime domain as ollows:
˙ei=K1˙
˜x2,i +K2˜x2,i
˙ i=KP˙
˜x1,i +KI˜x1,i
ui= i+Γizi,
(3.12)
whe e
K1,K2,KP,KI
a e pa ame e s,
˜x2,i = C,i − C ,i
and
˜x1,i =iL,i −iL ,i +ei
,
while
ui∈R
is he con ol inpu . Now, le us de ine he acking e o as
˜xi=xi−Πizi+
[1
0]eiwi h ˜xi= [˜xi,1,˜xi,2]⊤. Then, conside ing AiΠi+BiΓi= ΠiΘi, wi h
Γi=hω ,iLi
RiVin,i +ω ,iRLS,iCi
Vin,i 1
Li−Ciω2
,i+RLS,i
LiRiLi
Vin,i i,
he acking e o dynamic is go e ned by
˙
˜xi= ˙xi−Πi˙zi+[1
0]˙ei=Aixi+Biui−ΠiΘizi+[1
0]˙ei
=Ai(˜xi+Πizi−[1
0]ei)+Biui−(AiΠi+BiΓi)zi+[1
0] ˙ei
=Ai(˜xi−[1
0]ei)+Bi i+[1
0]˙ei.
(3.13)
34 Chap e 3. Con inuous con ol scheme o discha ging a e consensus
Figu e 3.5 Block diag am o he con olled in e e .
Finally, he closed loop o he acking e o dynamic
(3.13)
and he con ol law
(3.12)
can be w i en as ollows:
˙χi=Miχi,(3.14)
wi h χi=˙
˜xi
˜xi,Mi=¯
Ci−1¯
Ai¯
Ci−1¯
Bi
I 0 ,
¯
Ai=AiI−0K1
0 0 +BiKP0+0K2
0 0 ,
¯
Bi=Ai0K2
0 0 +BiKI0and ¯
Ci=I−0K1
0 0 .
Then, nex lemma p o ides a s abili y p ope y o he closed-loop sys em (3.12).
Lemma 1:
Conside he closed-loop dynamic
(3.14)
o an in e e
i∈N
. Fo a gi en
K1,K2,KP,KI∈R, i he e exis s any PL,i,QL,i ≻0∈R4×4, such ha
M⊤
iPi+PiMi≺−2QL,i.(3.15)
Then, χi=0is GAS.
P oo : The p oo is di ec , selec ing V(χi) = χ⊤
iPiχias Lyapuno unc ion.
■
No e ha his lemma es ablishes, unde some assump ions, ha he ou pu ol age
eaches he desi ed e e ence, ensu ing ou pu ol age egula ion o he in e e , while he
induc ance cu en con e ges o ano he equilib ium modi ied by ei.
3.5 Comple e con ol dynamic
In his sec ion, we de ine a comple e con ol sys em ha collec s all dynamics o
i∈N
o a ixed communica ion ne wo k. This comple e con ol dynamic s uc u e wi h h ee
ime-scale sepa a ion allows us o p o ide a la ge-signal s abili y analysis o he comple e
sys em applying singula pe u ba ion me hod [31], which will be p esen ed in he ollowing
sec ion. Fi s , we de ine he comple e closed-loop dynamics o a BESS
i
. Then, comple e
3.5 Comple e con ol dynamic 35
con ol dynamics a e p esen ed, ollowed by h ee ime-scale sepa a ion gua an ees be ween
he h ee con ol dynamics.
3.5.1 Closed-loop dynamics o a BESSi
Fi s , we de ine he comple e closed-loop dynamics o a BESS
i
and a ixed communica ion
ne wo k, which includes he d oop con ol, he p oposed con inuous- ime con ol law o
he in e e , and he SOC e e ence consensus algo i hm.
Le us de ine he dynamic a iables o each con ol loop o a BESSias ollows:
•D oop con ol: ξ1,i = [˜ωi,˜
Ψi,˜
Pi]⊤.
•In e e con ol: ξ2,i = [˜x⊤
i,ei,ui,zi]⊤.
•Consensus algo i hm: Ψ ,i.
Then, he comple e closed-loop sys em o agen iis
˙
ξ1,i
˙
ξ2,i
˙
Ψ ,i
= i(ξi),(3.16)
whe e ξi= [ξ⊤
1,i,ξ⊤
2,i,Ψ ,i]⊤and
i(ξi)=
−Kω˜ωi−Kc1˜
Ψi
−KΨ˜
Ψi+Kc1˜ωi+K0˜
Pi
−KdP ˜
Pi−K0˜
Ψi
Ai(˜xi−[1
0]ei)+Bi i+[1
0] ˙ei
K1˙
˜x2,i +K2˜x2,i
KP˙
˜x1,i +KI˜x1,i +Γi(ω ,i,V ,i)Θ(ω ,i)zi
Θi(ω ,i)zi
−αiKc N
P
j=1
αj(Ψ ,i−Ψ ,j)
!−Ke(Ψ ,i −Ψi(xi))
.
The dynamics o
ξ1,i
(d oop con ol) a e de ined in
(3.8)
and he dynamics o
Ψ ,i
(consensus) in
(3.3)
. Las ly, he dynamics o
ξ2,i
(in e e ) a e gi en in
(3.12)
. Mo eo e ,
le us ecall ha αi,αj∈{0,1} ep esen s he communica ion be ween BESSs.
3.5.2 Comple e con ol dynamic
Now, we de ine a comple e con ol sys em ha collec s all dynamics o
i∈ N
. We
i s p o ide a Laplacian ma ix de ini ion, which is used o ep esen he undi ec ed
communica ion g aph among BESSs acco ding o MAS heo y and we will conside
ixed. This g aph is o mally desc ibed h ough algeb aic g aph heo y and ma hema ically
ep esen ed by a Laplacian ma ix.
De ini ion 1:
Conside a communica ion ne wo k de ined by an undi ec ed g aph
G(N,E)
,
whe e
N
is he se o BESSs in discha ging mode, and
E ⊆N ×N
he edges, whe e each
36 Chap e 3. Con inuous con ol scheme o discha ging a e consensus
edge
{i,j}∈E
ep esen s a bidi ec ional communica ion link be ween wo di e en agen s.
The Laplacian ma ix ep esen s he communica ion be ween neighbou s and depends on
αij =αiαj∈{0,1}
and
α={αij ∈{0,1},i,j ∈N}
.
αij = 0
de ines an unin en ional
disconnec ion be ween agen
i
and
j
, due o a connec ion ailu e o agen
i
o he PCC o
communica ion ailu e, o ins ance, and
αij = 1
he connec ion be ween hem. No e ha
αij
is an exogenous a iable. Then, he Laplacian ma ix is
L(α) = ∆(α)−Ad(α)
, wi h
∆(α) he deg ee ma ix and Ad(α) he adjacency(o connec i i y) ma ix, whe e
∆(α):= diag([δ1(α),δ2(α),...,δN(α)]),wi h δi(α):=
N
X
j=1
aij(α)
and,
Ad(α):= [aij(α)],wi h aij(α) : αij i i=jand ∀(i,j)∈E
0i i=jo ∀(i,j)/∈E.
The Laplacian ma ix L(α)is posi i e semi-de ini e [39].
F om his de ini ion, o a ixed α, he ollowing comple e con ol sys em is p esen ed
˙
ξ1
˙
ξ2
˙
Ψ
=
−Kω˜ω−Kc1˜
Ψ
−KΨ˜
Ψ+Kc1˜ω+K0˜
P
−KdP ˜
P−K0˜
Ψ
A(˜x−[1
0]e)+B +[1
0]˙e
K1˙
˜x2+K2˜x2
KP˙
˜x1+KI˜x1+Γ(ω ,V )Θ(ω )z
Θ(ω )z
−KcL(α)Ψ +Ke˜
Ψ
(3.17)
wi h
ξ1:= [˜ω,˜
Ψ,˜
P]⊤, ξ2:= [˜x,z, ,τ]⊤
Ψ := [Ψ ,1,Ψ ,2,..,Ψ ,N ]⊤ξ:= [ξ⊤
1,ξ⊤
2,Ψ ]⊤
˜ω:= [˜ω1,˜ω2,..,˜ωN]⊤˜
Ψ := [˜
Ψ1,˜
Ψ2,..,˜
ΨN]⊤
˜
P:= [ ˜
P1,˜
P2,.., ˜
PN]⊤,˜x:= [˜x1,˜x2,..,˜xN]⊤
e:= [e1,e2,..,eN], := [ 1, 2,.., N]
z:= [z1,z2,..,zN]⊤, A := diag{A1,A2,..,AN}
B:= [B1,B2,..,BN]⊤,Θ := diag{Θ1,Θ2,..,ΘN}
Γ := diag{Γ1,Γ2,..,ΓN}
The desi ed equilib ium associa ed wi h sys em (3.17) is
{ξe
1,ξe
2,Ψe
}∈Se
1×Se
2
Se
1={ξ1= 0
0
0,˜x= 0
0
0,e, ∈RN,zi∈Φi∀i∈N}
Se
2={L(α)Ψ = 0
0
0}.(3.18)
3.6 S abili y analysis o he comple e con inuous- ime sys em 37
3.5.3 Th ee ime-scale sepa a ion
As men ioned be o e, we can iden i y h ee con ol objec i es: powe con e e con ol,
d oop con ol, and e e ence consensus, which a e equi ed o wo k in di e en ime-scales,
as is an in insic p ope y in mic og ids.
Gene ally, he d oop con ol e ol es slowly enough so ha ol age and equency
e e ences can be kep app oxima ely ’cons an ’ by he con e e s. I is desi able o he
consensus e e ence loop in he seconda y con ol o be as e han he con olled powe
con e e s and he d oop con ol. In his con ex , whe e a h ee- imescale sepa a ion can
be es ablished o comple e sys em
(3.17)
, singula pe u ba ion heo y [19] is appealed o
p o ide nonlinea s abili y gua an ees.
Assump ion 8:
Conside
(3.17)
. Then, he e exis some pa ame e s
Kc>> Ke>0
,
K0
,
Kc1
,
Kω
,
KΨ
,
KdP >0
,
Kdc = min
i∈N(min|eig(K)|)
wi h
K=−Kω−Kc10
Kc1−KΨK0
0−K0−KdP
,
K1,K2,KP,KI>0and Kin = min
i∈N(min|eig(Mi)|)such ha condi ion (3.15) and
Kc>> Kin >> Kdc
a e sa is ied.
No e ha
1
Kc,1
Kin ,1
Kdc
ep esen he es ima ions o he con e gence speed o each
con ol loop. Then, his assump ion implies ha
Ψ
is as e han
ξ2
, because
1
Kc<< 1
Kin
and, ha
ξ2
is as e han
ξ1
, because
1
Kin << 1
Kdc
. Mo eo e , no e ha
Kin
esul s
om he selec ed pa ame e s K1,K2,KP,KI>0and he in e e pa ame e s.
3.6 S abili y analysis o he comple e con inuous- ime sys em
Inspi ed by [40], his sec ion is de o ed o he s abili y analysis o he comple e con inuous-
ime sys em gi en by
(3.17)
. Following Sec ion 2.2.2, we p o ide he e a la ge-signal
s abili y analysis o he dis ibu ed con ol scheme p oposed o an islanded AC mic og id,
using h ee- ime-scale sepa a ion and singula pe u ba ion heo y. Singula pe u ba ions
app oaches a e used o simpli y complex sys ems by sepa a ing he dynamics o he whole
sys em in o slow and as dynamics i Assump ion 8 is sa is ied, we ind ha
ξ1
is slowe
han
ξ2
, which in u n is slowe han
Ψ
. This esul s in a h ee- ime-scale sepa a ion,
making he sys em sui able o applying he singula pe u ba ion me hod.
3.6.1 Singula pe u ba ion o m
Then, in oducing he pa ame e s
ν1
and
ν2
, he sys em
(3.17)
is ew i en in he ollowing
singula pe u ba ion o m:
44 Chap e 3. Con inuous con ol scheme o discha ging a e consensus
Figu e 3.10
Scena io 1: In he op, e olu ion o he SOC e e ences,
Ψ ,i
, he SOCs,
Ψi
,
and he ac i e powe s,
Pi
o
i={1,2,3}
, when a load o 22
Ω
is connec ed a
T2
and hen i is disconnec ed back a
T3
. In
T3
,
α1,2= 0
. In he bo om a
zoom o SOC e e ences a T1.
Fig. 3.13. A ime
T1
, he SOCs, whose ini ial condi ions a e di e en , s a o con e ge o a
consensus. Du ing he ime pe iod
T3−T2
, he SOC and ac i e powe o he disconnec ed
BESS
1
emain cons an , and he o he BESSs emp y ou as e because hey ha e o abso b
he capaci y o he disconnec ed BESS. F om
T3
BESS
1
eco e s i s ac i i y, compensa ing
o i s s o age ene gy wi h he o he BESSs. Mo eo e , he e olu ion o ol age e o s,
capaci ance ol ages, and induc ance cu en s a e shown in Fig. 3.14, alida ing he
3.7 Expe imen al alida ion 45
0 20 40 60
-100
0
100
0 20 40 60
-100
0
100
0 20 40 60
-100
0
100
0 20 40 60
-100
0
100
0 20 40 60
-100
0
100
0 20 40 60
-100
0
100
0 20 40
-5
0
5
0 20 40 60
-5
0
5
0 20 40 60
-5
0
5
T2 T3
T1 T1 T2 T3 T3
T2
T1
Figu e 3.11
Scena io 1: E olu ion o ol age e o s,
˜x2,i
, capaci ance ol ages,
C,i
, and
induc ance cu en s, iL,i o i={1,2,3}.
Figu e 3.12
Scena io 1: Zoom a T1 o SOC e e ences
Ψ ,i
, and ol age e o s
˜x2,i
, o
i={1,2,3}.
obus ness o he powe con e e con ol loop a e any connec ion/disconnec e en .
Once again, no e he h ee- ime-scale sepa a ion. The SOC e e ences con e ge o a
consensus wi h ansien imes o less han 0.2 ms. This e olu ion is as e han he SOCs
and he ac i e powe s con e gence o hei e e ences. Mo eo e , as shown in Fig. 3.15, he
ime esponse o he DC/AC con e e s (app oxima ely 2ms) is longe han he con e gence
speed o he SOC e e ences, bu sho e han con e gence speed o he SOCs and he
46 Chap e 3. Con inuous con ol scheme o discha ging a e consensus
Figu e 3.13
Scena io 2: In he op, e olu ion o he SOC e e ences,
Ψ ,i
, he SOCs,
Ψi
,
and he ac i e powe s,
Pi
, o
i={1,2,3}
, when BESS
1
is disconnec ed a
T2
and hen i is connec ed back a
T3
. In he bo om, a zoom o SOC e e ences
a : a) T1and b) T2.
ac i e powe s con e gence speed. F om hese scena ios, we can alida e he s a emen o
Theo em 5.
3.7.3 Compa ison wi h [1]
This sec ion ocuses on he expe imen al compa ison be ween ou con ol app oach and he
one p oposed in [1], which in oduces a disc e e- ime a e age MAS consensus algo i hm.
3.7 Expe imen al alida ion 47
0 50
-200
0
200
0 50
-200
0
200
0 50
-200
0
200
0 50
-200
0
200
0 50
-200
0
200
0 50
-200
0
200
0 50
-5
0
5
0 50
-5
0
5
0 50
-5
0
5
T1 T2 T3 T1 T2 T3 T1 T2 T3
Figu e 3.14
Scena io 2: E olu ion o ol age e o s,
˜x2,i
, capaci ance ol ages,
C,i
and
induc ance cu en s, iL,i o i={1,2,3}.
Figu e 3.15
Scena io 2: Zoom a T2 o SOC e e ences,
Ψ ,i
, and ol age e o s,
˜x2,i
, o
i={1,2,3}.
As p e iously desc ibed in Sec ion 4.6.1, his algo i hm calcula ed he a e age alue o
i s neighbo s,
SOCmean,i
, o he SOC
i
o he se o
Ni
BESSs uni s. Fu he mo e, he
48 Chap e 3. Con inuous con ol scheme o discha ging a e consensus
Figu e 3.16
Compa ison o he SOC and ac i e powe e olu ion be ween a) ou con ol
and b) he con olle p oposed in [1].
d oop con ol modi ies he e e ence o he equency wi h espec o he de ia ion be ween
he SOCiand he a e age alue o i s neighbo s.
We implemen he con ol algo i hm gi en in [1], selec ing
KP=Kd,3
, using he
same con ol pa ame e s gi en in Table 3.2, and se ing
σ= 0.3333
and
KSOC = 0.05
acco ding o [1]. The compa ison is pe o med unde he condi ions o Scena io 1, which
mimics a scena io simila o he one in [1]. A ins an ime
T1
, bo h consensus algo i hms
a e ac i a ed. Then, a ins an ime
T2
a second load o 22
Ω
is connec ed, and hen, in
T3
,
i is disconnec ed.
Figu e 3.16 shows a compa ison o he SOCs and he ac i e powe e olu ion be ween ou
con ol p oposal and he con olle p oposed in [1]. Each
SOCmean,i
and
Ψ ,i
con e ge
o a consensus wi h hei neighbo s, espec i ely. Likewise, each SOC con e ges o
SOCmean,i
and
Ψ ,i
, espec i ely. No e ha he con e gence speed o he SOCs and
he ac i e powe s is as e wi h ou p oposed con olle . Mo eo e , ou con ol algo i hm
ensu es a la ge-signal s abili y o he comple e nonlinea con inuous- ime sys em, while [1]
p o ides a small-signal s abili y analysis o a disc e e sys em.
3.8 Conclusion 49
3.8 Conclusion
This chap e p o ides a dis ibu ed con ol scheme o h ee- ime scales, composed o a
p ima y and seconda y con ol loops o AC-bus islanded mic og ids. The seconda y loop
consis s o a consensus algo i hm based on MAS heo y connec ed o a d oop con ol,
ensu ing he SOC con e gence o a consensus, he eby inc easing ba e y li espan and
p o iding obus ness wi h espec o any communica ion ailu e. Mo eo e , he p ima y
loop consis s o a d oop con ol and an in e e con ol, which includes ol age and
cu en con inuous- ime con ol loops o enable he use o PWM, commonly equi ed
in he indus y. Unde he sa is ac ion o Assump ion 10, a h ee- ime scale sepa a ion
is ensu ed be ween each con ol goal (consensus algo i hm, d oop con ol, and powe
con e e con ol). This p ope y allows us o apply singula pe u ba ion heo y o ensu e
la ge-signal s abili y o he comple e nonlinea sys em. The p oposed dis ibu ed con ol
is expe imen ally alida ed h ough an Impe ix powe es bench unde a ious scena ios.
In addi ion, a compa ison wi h he con olle gi en in [1] is p o ided.
Fu he mo e, u u e wo k en isions ex ending hese esul s by adding an ex a con ol
loop o manage he ope a ion modes o ba e ies, loads, and sou ces (cha ging mode o
discha ging mode) o imp o e ba e y deg ada ion. Fu he mo e, ex ending he esul s o
conside ing disc e e- ime dynamics is also p oposed.
In he nex chap e , a hyb id model o he p oposed con ol scheme will be discussed.
4 Hyb id dynamical con ol scheme
o discha ging a e consensus
T
he p e ious chap e exhibi ed he con ibu ion o discha ging a e consensus h ough
a con inuous- ime con ol scheme o he p ima y and seconda y con ol loops o
he se o
N
-BESSs in an AC islanded mic og id. In con as wi h he p eceding chap e ,
his chap e p esen s a hyb id dynamical con ol app oach o he p oposed dis ibu ed
con ol scheme ha add esses he p oblem o mula ion men ioned in Sec ion 3.1.1. The
objec i e o his dis ibu ed con ol scheme is o achie e SOC balancing using he a o e-
men ioned p oposed seconda y con ol loop and d oop con ol. Howe e , in he p ima y
loop, we conside a hyb id dynamical powe con e e con ol loop, aking in o accoun
he nonlinea i ies o he powe con e e models (swi ches and a ine e ms) and o he
signals (dwell ime cons ain s).
A pa icula in e es has eme ged o apply hyb id con ol echniques o powe con e e s
wi h he aim o conside ing he sub le y o he hyb id na u e o hese sys ems. Indeed,
hey a e composed o con inuous- ime dynamics ( ol ages and cu en s) and disc e e- ime
dynamics (swi ching). P e ious wo ks on swi ched a ine sys ems ha e conside ed a hyb id
amewo k [38], [39], [40], [41] based on HDS heo y gi en in [42]. Some o hem ha e
also aken in o accoun p ac ical issues, such as minimum dwell ime, obus ness wi h
espec o pa ame e a ia ion, absence o senso signal o e en PWM implemen a ion.
In his chap e , he con ol objec i es a e i s p esen ed, and a e wa ds he hyb id
con ol loop o he powe con e e wi h ol age ou pu egula ion is p esen ed. Then,
a comple e hyb id dynamic model o he seconda y and p ima y con ol loops using
HDS heo y is discussed, ollowed by he s abili y analysis using singula pe u ba ion
heo y applied o HDS. Finally, he chap e concludes wi h simula ion alida ion esul s,
p o iding a compa ison be ween he p oposed con ol scheme and he one p esen ed in [1].
51
52 Chap e 4. Hyb id dynamical con ol scheme o discha ging a e consensus
4.1 Con ol objec i es
The goal he e is o design a comple e hyb id scheme o he p ima y and seconda y con ol
loops in an islanded AC mic og id wi h a se o
N
-BESSs in discha ging mode, such ha
i accomplishes he ollowing con ol objec i es o each i∈N := {1,2,...,N}agen :
1.
Con e gence o he hal -b idge in e e s a e
xi:= [iL,i, C,i]⊤
, wi h
iL,i
he induc-
ance cu en and
C,i
he capaci o ol age, o a e e ence
x ,i
, h ough a hyb id
con ol ha conside s a swi ched a ine model and a minimum dwell ime. Mo eo e ,
UGAS o a small neighbo hood o
xi=x ,i(V ,i,ω ,i)
, as well as obus ness ou pu
ol age egula ion should be ensu ed.
2.
Con e gence o he SOC e e ences,
Ψ ,i
, o he in e connec ed BESSs o a consensus
alue Ψ∗.
3. Con e gence o Ψiand Pi, o hei e e ences Ψ ,i and P ,i, espec i ely.
4.
SPAS o he comple e hyb id dynamical sys em, h ough a la ge-signal analysis, ha
includes in e e con ol, d oop con ol, and consensus algo i hm, by using a h ee
ime-scale sepa a ion model and applying singula pe u ba ion heo y in HDS.
4.2 In e e model and con ol
In his sec ion, we p esen a hyb id con ol law o he in e e s o he
N
-BESSs in
discha ging mode in AC islanded mic og id. As in Sec ion 3.4.3 we conside hal -b idge
con e e s as he in e e s o he BESSs as well as he dynamical model p esen ed in [38].
Le us o ecall om Chap e 3, ha
zi
is he s a e o a gene ic oscilla o used by
he e e ence gene a o block wi h ma ices
Θi(ω ,i) = h0−ω ,i
ω ,i 0i
and
Πi(ω ,i) :=
hω ,iCi1
Ri
0 1 i
o gene a e he comple e s a e e e ence
x ,i = [iL ,i , C ,i ]⊤
, p e iously
de ined in (3.11). Mo eo e ,
Ai="−RLS,i
Li−1
Li
1
Ci−1
RiCi#and Bi=Vin,i
Li
0.
The dynamics o he o e all sys em is desc ibed by
˙xi=Aixi+Biui
˙zi= Θi(ω ,i)zi
˜xi=xi−Πi(ω ,i)zi,
(4.1)
whe e ˜xiis he acking e o wi h ˜xi= [˜xi,1,˜xi,2]⊤.
4.2 In e e model and con ol 53
Then, conside he ollowing algeb aic equa ion:
AiΠi+BiΓi= ΠiΘi.(4.2)
F om simple calcula ions, we yield
Γi=hω ,iLi
RiVin,i +ω ,iRLS,iCi
Vin,i 1
Li−Ciω2
,i+RLS,i
LiRiLi
Vin,i i,
om (4.1) and ollowing (4.2) we can de ine he acking e o dynamic as ollows:
˙
˜xi= ˙xi−Πi˙zi=Aixi+Biui−ΠiΘizi
=Ai(˜xi+Πizi)+Biui−(AiΠi+BiΓi)zi
=Ai˜xi+Bi i,
(4.3)
wi h
i=ui−Γizi∈R.(4.4)
The new con ol inpu o be designed,
i
, is composed o a con inuous- ime signal
Γizi
and a swi ching signal o wo logical modes
ui∈K={−1,1}
. Ne e heless, we need o
keep in mind ha he con ol applied o he hal -b idge in e e is
ui
. This con ol law o
he swi ching signal
ui
mus ensu e sui able con e gence p ope ies o he in e e e o
a iable ˜xi o 0. To do so, we in oduce nex assump ion.
Assump ion 9:
[38] Conside model
(4.1)
. Fo a gi en ma ix
QL,i ≻0∈R2×2
, he e
exis s a ma ix PL,i ≻0∈R2×2such ha
1. ma ix Ai e i ies
AT
iPL,i +PL,iAi+2QL,i ≺0(4.5)
2.
and, o any pe iodic signal
zi( )∈Φi
o pe iod
Tp
, he e exis s a
λj,i( )
o each
j∈Ksuch ha λ−1,i( )+λ1,i( )=1such ha
λ1,i( )−λ−1,i −Γizi( )=0.(4.6)
The Hu wi z equi emen o ma ices
Ai
is a common p ope y in con e e models
[38,41].
The con ol law used he e o i(o equi alen ly ui) is [38]
i= (a gmin
ui∈K˜x⊤
iPL,i(Ai˜xi+Bi(ui−Γizi)))−Γizi.(4.7)
The “a gmin” ope a o inds he alue o
ui
which minimizes he Lyapuno unc ion,
V(˜xi) = ˜x⊤
iPi˜xi. The UGAS p ope y is gi en in [38].
4.2.1 Vol age ou pu egula ion o in e e s
I is well known ha powe con e e s su e a ia ions in he load, he ol age inpu , o
o he componen s. The e o e, i is necessa y o gua an ee obus ness in he ou pu ol age
60 Chap e 4. Hyb id dynamical con ol scheme o discha ging a e consensus
P oposi ion 4:
Unde Assump ion 10 and o a gi en
α
, he quasi-s eady-s a e equilib ium
‘mani old’ o Hsp is egula .
P oo :
No e ha he dynamic o a iable
Ψ
in
(4.16)
–
(4.17)
is pu ely con iniuous.
The quasi-s eady-s a e equilib ium ’mani old’ o
Hsp
is compu ed when
ν1= 0
and
−L(α)Ψ +ν1Ke˜
Ψ = 0. Then, i is:
Ξ(ξ1,ξ2) = (Ψ∗diag(αi)1(ξ1,ξ2)∈C
∅(ξ1,ξ2)/∈C
wi h
Ψ∗
a pa ame e . Then,
Ξ
is a se - alued mapping, which is emp y ou side o se
C
.
■
4.5.4 S abili y o he bounda y laye sys em
The nex p oposi ion p o ides a s abili y analysis o he ‘bounda y laye ’ be ween
(ξ1,ξ2)
and
Ψ
, which is ob ained by scaling o dina y ime by
1/ν1
o
τ= /ν1
in he o iginal
sys em (4.20)–(4.21) and hen se ing ν1= 0.
P oposi ion 5:
Unde Assump ion 10 and o a gi en
α
, he e exis s a closed ball
ρiBi
o
adius ρi>0such ha he compac se
M={i∈N : (ξ1,i,ξ2,i)∈(Ci∩ρiBi),Ψ ∈Ξ(ξ1,ξ2)}
associa ed wi h he bounda y laye
˙
ξ1
˙
ξ2
˙
Ψ
=
0
0
−L(α)Ψ
(ξ,Ψ )∈
i∈N
(Ci∩ρiBi)×RN,(4.22)
is GAS.
P oo :
Fi s no e ha i Assump ion 10 is sa is ied he e is a ime-scale sepa a ion be ween
be ween
(ξ1,ξ2)
and
Ψ
ha allows ge ing he bounda y laye
(4.22)
as ollows, because
ν1<< 1/Ke
. In o he wo ds, we zoom in on he as e a iable
Ψ
, causing he slowe
a iables o become ozen, emaining
ξ1
and
ξ2
cons an du ing lows. Mo eo e , we
s ess ha he bounda y laye sys em igno es jumps. The as subsys em dynamics om
(4.20) can be w i en as:
ν1
dΨ
d =−L(α)Ψ +ν1Ke˜
Ψ
eplacing τ= /ν1, and se ing ν1= 0
ν1
dΨ
d(ν1τ)=dΨ
dτ =−L(α)Ψ +ν1Ke˜
Ψ
=−L(α)Ψ
4.5 S abili y analysis o he comple e hyb id sys em 61
Now, conside he Lyapuno unc ion candida e
Vc:= 1
2Ψ ⊤L(α)Ψ .
Then,
⟨▽Vc(Ψ ), bl⟩=−Ψ ⊤L(α)L(α)Ψ ≤0,
wi h
bl := [0⊤0⊤−Ψ⊤
L(α)]⊤
No e
ha
⟨▽Vc(Ψ ), bl⟩
is nega i e semide ini e. Then by LaSalle’s p inciple, we can conclude
he p oo s a ing ha all agen s in e connec ed con e ge o a consensus. Consequen ly,
(4.22)
is GAS. The dynamic de ined in
(4.22)
gua an ees ha each
Ψ ,i
wi h associa ed
αi= 1
con e ges o a neighbo hood o
Ψ∗
which de ines a consensus be ween he in e -
connec ed agen s. ■
4.5.5 S abili y o he educed sys em
Now, we es ablish he nex esul o he educed sys em associa ed wi h
Hsp
. No e ha
he educed sys em’s jump map is no exp essed in e ms o
Ψ ∈Ξ(ξ)
since he bounda y
laye sys em igno es jumps.
P oposi ion 6:
Unde Assump ion 10 and o a gi en
α
, he a ac o
A := S
i∈N A ,i
wi h
A ,i := {(ξ1,i,ξ2,i)∈Hi:∥˜xi∥< Xi,ξ1,i =0}associa ed o he educed sys em:
H (ξ):
˙
ξ1
ν2˙
ξ2
=
−Kω˜ω−Kc1˜
Ψ
−KΨ˜
Ψ+Kc1˜ω+K0˜
P
−KdP ˜
P−K0˜
Ψ
ν2A˜x+ν2B
ν2Θi(ω ,V )z
−ν2Γ(ω ,V )Θ(ω )z
ν21
ξ∈C
ξ1
+
ν2ξ+
2
∈
ξ1
ν2˜x
ν2z
ν2h(ξ)
0
ξ∈D,
(4.23)
is SPAS as ν2→0+.
P oo : Conside he ollowing Lyapuno unc ion o he educed sys em H ,
V(ξ1,˜x) := X
i∈N
Vi(ξ1,i,˜xi)
Vi(ξ1,i,˜xi) = 1
2(ξ⊤
1,iξ1,i + ˜x⊤
iPL,i ˜xi)∀(ξ1,i,ξ2,i)∈Hi A ,i.
62 Chap e 4. Hyb id dynamical con ol scheme o discha ging a e consensus
Applying [38, Theo em 1] o hyb id sys em H , i is go
⟨∇Vi(ξ1,i,˜xi)⟩=−ξ⊤
1,iSξ1,i +αi˜xiPL,i(Ai˜xi+Bi i)
≤−ξ⊤
1,iSξ1,i −αiηi˜x⊤
iQL,i ˜xi<0∀(ξ1,i,ξ2,i)∈Ci A ,i
(4.24)
Vi(ξ+
1,i,˜x+
i)−Vi(ξ1,i,˜xi)=0 ∀(ξ1,i,ξ2,i)∈Di A ,i
(4.25)
wi h S:= diag{Kω,KΨ,KP}.
No e ha he dynamic o ˙xi1is a con inuous- ime au onomous sys em.
The p oo is concluded by applying [38, Theo em 3].
■
Theo em 6:
Unde Assump ion 9, 10 and o a gi en
α
, he se
A=A ×{L(α)Ψ =0}
associa ed wi h he hyb id sys em H, (4.16)–(4.19), is SPAS as ν1,ν2→0+.□
P oo :
No e ha i Assump ions 9 and 10 a e sa is ied, hen P oposi ions 3–6 a e hold.
Consequen ly, he p oo is di ec om [21, Theo em 1]. ■
4.6 Simula ion esul s
In his sec ion, he hyb id con ol p oposed abo e is alida ed in Ma lab/Simulink using he
Elec ical Toolbox o an AC islanded mic og id composed o h ee BESSs in discha ging
mode, as shown in Fig. 4.1. A simila scena io was gi en in [1].
Each hal -b idge is emula ed as an AC con olled ol age sou ce whose inpu is he
equi alen ol age modula ed by he wo swi ches, gi en by
Vm,i =uiVin,i
, ollowed by
i s co esponding LC il e . The ou pu cu en ,
io,i
is also measu ed o calcula e he ou pu
ac i e powe and he line impedances a e included.
The bus line is
Vbus = 120√2sin(2π60 )V
, hen
Vn= 120√2
V and
ωn= 2π60 ad/s
.
Tables 4.1 and 4.2 p o ide he pa ame e s o he seconda y and p ima y con ol loops
espec i ely, i.e., he pa ame e s o he hyb id model
(4.11)
–
(4.14)
. F om hese pa am-
e e s, we ha e
Ke/Kc= 0.01
,
Kin = 34.7
and
Kdc = 0.51
, hen i is easy o see ha
Assump ion 10 is sa is ied.
Fu he mo e, ma ices
PL,i
and
QL,i
a e selec ed p o iding some LQR (Linea -Quad a ic
Regula o ) pe o mance gua an ee in o de o educe, o ins ance, he ene gy cos , cu en
peaks, and esponse ime o he ol age o cu en . To his end, we apply [45, Theo em 2].
The e o e, sa is ying Linea Ma ix Inequali y (LMI) ools wi h he in e e pa ame e s
gi en in Table 4.2, we ob ain:
PL,i =28.20 0.12
0.12 0.08and QL,i =1.50 0
0 5.55.
4.6 Simula ion esul s 63
Figu e 4.1 Mic og id con igu a ion scheme.
Table 4.1 Consensus algo i hm and d oop con ol pa ame e s.
Pa ame e
Value
Pa ame e
Value
Kc10000 KdP 50
Ke100 Kc10.5
KΨ0.45 K00.05
Kω5
Table 4.2 In e e pa ame e s.
Pa ame e ∀iValue Pa ame e ∀iValue
Vin,i 48 V ωn2π60 ad/s
Li50 mHVn120√2V
Ci140.72 µFZline,i
R=0.1
Ω
L=1.6
m
H
RLS,i 1.5 ΩT0.1 ms
Ri180 ΩCba ,i 0.1615 Wh
Xi0.041 k1/3600
K1
The Laplacian ma ix ha ep esen s he communica ion in e connec ions among he
h ee BESSs is he same as p e iously gi en in (3.22), as shown in Fig. 3.9.
To alida e he e ec i eness o ou p oposed s a egy, we pe o med he SOC balancing
e i ica ion h ough a simula ion scena io. This scena io will es he pe o mance o
he sys em agains communica ion ailu es and he capabili y o ide ou plug-and-play
e en s (i.e. in en ional o unin en ional shu down o a BESS). A compa ison wi h a simila
consensus p oposal in [1] is also p esen ed below.
64 Chap e 4. Hyb id dynamical con ol scheme o discha ging a e consensus
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
0.5
1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
50
100
150
200
250
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
0.5
1
012
10-4
0.8
0.9
1
1.7 1.7001 1.7002
0.6
0.65
Figu e 4.2
E olu ion o
Ψ ,i
,
Ψi
, and he ac i e powe s
Pi
, o
i={1,2,3}
, when he
ba e y 1 is disconnec ed a
T2
and hen connec ed back a
T3
. In
T1
,
α1,3
u ns
o 0.
We conside ha he ne powe (load powe minus DG powe ) is
Pn= 300
W, which is
ep esen ed by he load connec ed in he con igu a ion scheme in Fig. 4.1. The ini ial con-
di ions o he SOCs a e di e en among hem, wi h alues o
Ψ1(0) = 1
,
Ψ2(0) = 0.9
, and
Ψ3(0) = 0.8
, espec i ely. A he ins an ime
T2= 1.5
s he BESS
1
is disconnec ed (i.e.,
α1= 0
), and hen i is connec ed back in
T3= 1.7
s. Also, a
T1= 0.5
s, a communica ion
ailu e occu s be ween BESS1and BESS3, i.e., α1,3=α3,1= 0.
Fig. 4.2 shows he con e gence o he SOC e e ences, (o
Ψ ,i
), o a consensus om
he ini ial condi ions and a e he BESS plug-and-play e en . No e ha , a he s a o he
simula ion, he SOC e e ences con e ge o a consensus wi h ansien imes o less han
0.1ms app oxima ely. Howe e , he ac i e powe s a e pe u bed on his ime scale. In ac ,
hey ini ially di e ge because he BESS wi h a highe SOC (
Ψ1
) sha es mo e powe , while
he uni wi h lowe SOC (
Ψ3
) sha es less powe . Hence, on a la ge ime scale, ac i e
powe s and es ima ed SOCs con e ge o a consensus. Indeed, he SOCs and he ac i e
powe s con e ge o hei e e ences be o e o 2.4s app oxima ely (no e ha
1/Kdc = 2.4
)
and
Ψ
in 0.1ms app oxima ely (no e ha
1/Kc= 10−4
). Mo eo e , no e ha he signals
e ol e obus ly a e he communica ion ailu e.
4.6 Simula ion esul s 65
1.5 1.6 1.7 1.8
-200
0
200
1.5 1.6 1.7 1.8
-10
0
10
1.5 1.6 1.7 1.8
-1
0
1
1.5 1.6 1.7 1.8
-200
0
200
1.5 1.6 1.7 1.8
-10
0
10
1.5 1.6 1.7 1.8
-1
0
1
1.5 1.6 1.7 1.8
-200
0
200
1.5 1.6 1.7 1.8
-10
0
10
1.5 1.6 1.7 1.8
-1
0
1
Figu e 4.3
E olu ion o he ol ages, cu en s and du y cycles o in e e s o
i={1,2,3}
be ween 1.45s o 1.8s.
024
-200
0
200
0 2 4
-200
0
200
024
-200
0
200
Figu e 4.4 E olu ion o he ol age e o s.
Fo he in e al
T3−T2
, he emaining BESSs ha e o discha ge as e as hey ha e
o abso b he capaci y o he disconnec ed BESS
1
. A
T3
BESS
1
esumes i s ope a ion,
balancing i s s o ed ene gy wi h he o he BESSs. In addi ion, Fig 4.3 shows he e olu ion o
he capaci ance ol ages, induc ance cu en s, and con ol inpu s, alida ing he obus ness
o he powe con e e con ol loop a e any connec ion/disconnec e en . Fu he mo e, in
Fig. 4.4 he ol age e o s a e shown. No e ha he ol age e o s con e ge o ze o, ensu ing
ha he ol age ou pu s ollow hei e e ences, alida ing ol age ou pu egula ion. No e
ha hese e o s p esen a ansien ime less han 30ms (no e ha
1/Kin = 34.48
).
The e o e, in e e s e ol e slowe han he SOC e e ences o a consensus and, as e han
he d oop con ol, alida ing he ime-scale sepa a ion assump ion, hence he s a emen o
Theo em 6.
4.6.1 Compa ison wi h [1]
This sec ion ocuses on a compa ison be ween ou con ol app oach and he one p oposed
in [1], which in oduces a disc e e- ime a e age MAS consensus algo i hm. In [1], he
d oop con ol modi ies he e e ence o he equency wi h espec o he de ia ion be ween
66 Chap e 4. Hyb id dynamical con ol scheme o discha ging a e consensus
01234
0
0.2
0.4
0.6
0.8
1
01234
0
0.2
0.4
0.6
0.8
1
01234
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4
0
0.2
0.4
0.6
0.8
1
(b)(a)
(a) (b)
Figu e 4.5
Compa ison o simula ion esul s o he SOC e olu ions be ween he con olle
p oposed in [1] (a) and ou hyb id con ol p oposal (b).
he SOCiand he a e age alue o i s neighbou s, SOCmean,i, as ollows:
ω ,i =ωn+KSOC(Ψi−SoCmean,i)−KPPi,(4.26)
SOCmean,i =SOCi+σZ
0X
j∈Ni
(SOCmean,j(τ)−SOCmean,i(τ))dτ, (4.27)
whe e
Ni
is he se o neighbou s o agen
i
,
KSOC
and
KP
a e he d oop gains, and
σ
is a scala uning pa ame e which depends on
N
. No e ha
(4.26)
is simila o he one
gi en in
(3.5)
. Bo h equa ions ha e he p opo ional e m ela ed o he e o be ween he
es ima ed SOC wi h espec o
SOCmean,i
and
Ψ ,i
, espec i ely. Howe e , ou p oposal
adds an in eg al e m o he e o be ween he es ima ed SOC and a e m wi h he di e ence
be ween he ac i e powe and i s e e ence. Addi ionally, no e ha
(4.27)
lacks he e m on
he le side in
(3.3)
, which ga he s he de ia ion be ween he es ima ed SOC,
Ψi
, and i s
e e ence
Ψ ,i
. To simula e his algo i hm,
σ= 0.3333
and
KSoC = 0.0015
a e op imally
selec ed acco ding o [1] and he ba e ies pa ame e s a e gi en in Table 4.2. Mo eo e , in
T1= 2s
a load o 150W is connec ed and hen, in
T2= 3s
, is disconnec ed o mimic he
scena io gi en in [1].
Figu e 4.5 shows a compa ison o he SOC e olu ions be ween he con olle p oposed
in [1] (a) and ou con ol p oposal (b). No e ha each
SOCmean,i
and each
Ψ ,i
con e ges
4.7 Conclusions 67
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
50
100
150
200
250
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
0
50
100
150
200
250
(a)
(b)
Figu e 4.6
Compa ison o simula ion esul s o he ac i e powe e olu ions be ween he
con olle p oposed in [1] (a) and ou hyb id con ol p oposal (b).
o a consensus, espec i ely. Mo eo e , he SOCs con e ge o
SOCmean
and o hei
Ψ ,i
when using he con olle gi en in [1] and ou con ol solu ion, espec i ely. The
con e gence speed o he SOCs is as e in ou p oposi ion. Fu he mo e, in Fig. 4.6 he e
is a compa ison o he ac i e powe e olu ions. He e, we again no e he imp o emen
ob ained wi h ou con ol algo i hm wi h espec o he esponse ime. The oscilla ions
gi en in he s eady s a e a e due o he e en - igge ed con ol used in ou p oposi ion.
Indeed, di e en o [1], we do no use PWM in he in e e con ol signal, bu a hyb id
con ol which may no be sui able o indus ial applica ions due o high sampling equency
equi emen s.
No e ha ou p oposi ion also p o ides a la ge-signal s abili y analysis o he comple e
hyb id sys em, i.e., conside ing he con inuous- ime and disc e e- ime dynamics. Indeed,
in [1], a small-signal s abili y analysis is gi en o a disc e e sys em.
4.7 Conclusions
This chap e p o ides a comple e hyb id scheme o he p ima y and seconda y con ol
loops in an islanded AC mic og id o achie e SOC consensus among he BESSs in dis-
cha ging mode. The scheme conside s nonlinea i ies in he in e e model (swi ching
and a ine e ms) and in he signals (minimum-dwell ime cons ain ). Mo eo e , he
seconda y loop is composed o a consensus algo i hm o inc ease he ba e y li espan,
68 Chap e 4. Hyb id dynamical con ol scheme o discha ging a e consensus
as well as p o ide scalabili y o he mic og id wi h espec o plug-and-play e en s and
communica ion ailu es. Fu he mo e, a la ge-signal s abili y analysis o he h ee con ol
loops in a whole: in e e con ol, d oop con ol, and consensus algo i hm is ob ained by
applying singula pe u ba ion heo y o a ime-scale sepa a ion hyb id model ob ained by
selec ing he con olle gains app op ia ely. Then, i is concluded ha he a ac o
A
is
SPAS gi en a gi en in e connec ed g aph L(α).
5 Hyb id dynamical con ol scheme
o eac i e powe sha ing
T
his chap e p esen s an ex ended e sion o he hyb id con ol scheme in oduced in
Chap e 4, inco po a ing eac i e powe sha ing in o he con ol loops o a se o
N
-BESSs ope a ing in discha ging mode wi hin an AC islanded mic og id. The p ima y
objec i es o his enhanced hyb id con ol scheme a e o achie e SOC balancing and
op imal eac i e powe sha ing, u ilizing a modi ied e sion o he p e iously p oposed
seconda y con ol loop and d oop con ol.
This chap e begins by in oducing a li e a u e e iew and desc ibing he eac i e
powe consensus s a egy. Then, a comple e hyb id dynamical model o he p ima y and
seconda y con ol loops is p o ided. This model exhibi s a h ee- ime-scale sepa a ion,
enabling he applica ion o he singula pe u ba ion me hod o gi e s abili y gua an ees o
he comple e sys em, which will be de ailed below. The chap e concludes wi h simula ion
esul s ha illus a e he model’s pe o mance ac oss a ious applica ion scena ios.
5.1 In oduc ion
Op imal eac i e powe sha ing is one o he main objec i es o pa allel in e connec ed
DG uni s in a mic og id, add essing c ucial issues such as diminishing ol age de ia ions
and unce ain ies in he line impedances. In high- ol age ne wo ks, eac i e powe sha ing
be ween gene a o s is usually no a majo issue due o capaci i e compensa ion o bo h he
loads and he ansmission lines. The e o e, gene a o s ol ages a e con olled o ixed
alues. Howe e , he si ua ion is di e en in small-scale mic og id applica ions whe e
islanded AC mic og ids a e in ol ed. In hese cases, he low capaci ies o he DGs, he
sho elec ical dis ances be ween he uni s, and he lack o s a ic compensa ion equi e a
p ecise dis ibu ion o he eac i e powe demand be ween he DGs. This p ecise sha ing
is necessa y o a oid o e loading o he indi idual uni s [47].
D oop con ol me hods a e ex ensi ely used o ob ain an accep able powe sha ing
[48,49]. These ne wo k con ol me hods a e o pa icula in e es due o hei scalabili y,
69
76 Chap e 5. Hyb id dynamical con ol scheme o eac i e powe sha ing
5.6.1 Th ee ime-scale sepa a ion
We ew i e Assump ion 10 o include he eac i e powe consensus pa ame e s, es ablishing
he condi ions necessa y o achie ing a h ee- ime-scale sepa a ion among he h ee con ol
dynamics.
The in e e and d oop con ol esponse speed a e es ima ed om
Kin := min
i∈N(min|Re(eig(Ai))|)and
Kdc := min
i∈N(min|Re(eig(K))|),
such ha K:
−Kω−Kc10 0 0
Kc1−KΨ0 0 K0
0 0 −KV−K20
0 0 K2−KQ0
0−K00 0 −KP
,
espec i ely. The e o e, he h ee- ime scales among ξ1,ξ2and (Ψ ,Q )a e gi en i he
nex assump ion is sa is ied.
Assump ion 11:
Conside
(5.8)
–
(5.11)
. Then, selec
T
,
Kc>> Ke>0
and ma ix
K
,
such ha
(i) Kc>1
T>> Kin
(ii) Kc>> Kin >> Kdc
a e sa is ied.
No e ha
1
Kc
,
1
Kin
and
1
Kdc
ep esen he es ima ions o he con e gence speeds o
(Ψ ,Q ),ξ2and ξ1, espec i ely.
The nex pa ame e design guideline is gi en o p o ide a me hod o uning he pa ame-
e s in o de o sa is y Assump ion 11.
Algo i hm 2 Pa ame e design guideline 2
1: Es ima e: Kin .
2: Selec : T,Kcand Ke, s. . Ass. 11, i em (i) is sa is ied.
3: Selec : Kω,KΨ,KV,K0and K1s. . Kd,1>Kd,2>Kd,3.
4: Selec : KQ,KPand K2s. . Kd,4<Kd,5.
5: Tes : Kin >> Kdc s. . Ass. 11, i em (ii) is sa is ied.
5.7 S abili y analysis o he comple e hyb id sys em
This sec ion ocuses on analyzing he s abili y o
H
. This analysis is g ounded in he
heo y o singula pe u ba ion applied o hyb id con ol sys ems, as desc ibed in [21]
simila o he de elopmen p o ided in he Sec ion 4.5. Le us conside ha he uned
pa ame e s a e selec ed sa is ying Assump ion 11 such ha sys em
H
is composed by
5.7 S abili y analysis o he comple e hyb id sys em 77
h ee ime-scale sepa a ed dynamic loops whe e
ξ1
is slowe han
ξ2
and hese las ones
slowe han (Ψ ,Q ).
Then, His ew i en in he ollowing singula pe u ba ion o m:
Hsp(ξ) :
˙
ξ1
ν2˙
ξ2
ν1˙
Ψ
ν1˙
Q
= sp(ξ,Ψ ,Q ),(ξ,Ψ ,Q )∈C×R2N,
ξ1
+
ν2ξ+
2
Ψ+
Q+
∈Gsp(ξ,Ψ ,Q ),(ξ,Ψ ,Q )∈D×R2N,
(5.12)
such ha
sp(ξ) :=
−Kω˜ω−Kc1˜
Ψ
−KΨ˜
Ψ+Kc1˜ω+K0˜
P
−KV˜
V−K2˜
Q
−KQ˜
Q+K2˜
V
−KdP ˜
P−K0˜
Ψ
ν2A˜x+ν2B
ν2Θ(ω )z
−ν2Γ(ω ,V )Θ(ω )z
ν21
−L(α)Ψ +ν1Ke˜
Ψ
−L(α)Q +ν1KeQ
, Gsp(ξ) =
ξ1
ν2˜x
ν2z
ν2h(ξ)
0
Ψ
Q
(5.13)
wi h
ν1:= 1/Kc
and
ν2= 1/Kin
. Highligh ing ha he jump se
G
does no change o
dynamics ξ1and (Ψ ,Q )in a singula pe u ba ion o m.
5.7.1 Regula i y o Sys em’s Da a
Following [21], we impose basic assump ions on (5.12).
P oposi ion 7: (Assump ion 1) Fo a gi en αsys em Hsp(C, sp,D,Gsp)is well-posed.
P oo :Hsp(C, sp,D,Gsp) e i ies he ollowing p ope ies
•C×R2Nand D×R2Ngi en in (5.10) and (5.11) espec i ely, a e closed se s;
• sp
is a con inuous unc ion, hus locally bounded and ou e semicon inuous. Fu -
he mo e, i is con ex and non-emp y o each ξ∈C×R2N;
•Gsp
is ou e semicon inuous and locally bounded. Mo eo e , o each
ξ∈D×R2N
,
i is non-emp y;
Then,
Hsp
sa is ies he hyb id basic condi ions and he e o e i is well posed (Theo em 1).
■
78 Chap e 5. Hyb id dynamical con ol scheme o eac i e powe sha ing
5.7.2 Regula i y o he ’mani old’
The nex esul es ablishes a p ope y o he quasi s eady s a e mani old. I is a se - alued
map om he slow a iables o he as a iables, as Ξ : H⇒R2N.
P oposi ion 8:
Fo a gi en
α
and unde Assump ion 11, he quasi-s eady-s a e equilib ium
’mani old’ o Hsp is egula .
P oo :
Obse e ha he a iables
Ψ
and
Q
in equa ions
(5.8)
–
(5.9)
exhibi pu ely con-
inuous dynamics. Hence, we calcula ed he he quasi-s eady-s a e equilib ium ’mani old’
doing ν1= 0 and,
−L(α)Ψ +ν1Ke˜
Ψ = 0
−L(α)Q +ν1Ke˜
Q=0
The e o e,
Ξ(ξ1,ξ2) =
Ψ∗diag(αi)1
Q∗diag(αi)1(ξ1,ξ2)∈C
∅(ξ1,ξ2)/∈C
whe e
Ψ∗
and
Q∗
a e pa ame e s. Then,
Ξ
is a se - alued mapping, which is emp y ou side
o se C.
■
5.7.3 S abili y o he ‘bounda y laye ’
The ollowing p oposi ion p esen s a s abili y analysis o he ‘bounda y laye ’ be ween
(ξ1,ξ2)
and
(Ψ ,Q )
, which is ob ained by scaling he o dina y ime by
1/ν1
in he
o iginal sys em (5.12)–(5.13) and hen se ing ν1= 0.
P oposi ion 9:
Conside ha Assump ion 11 is sa is ied hen, he e exis s a closed ball
ρiBio adius ρi>0such ha he compac se
M={i∈N : (ξ1,i,ξ2,i)∈(Ci∩ρiBi),(Ψ ,Q )∈Ξ(ξ1,ξ2)}
associa ed wi h he bounda y laye be ween (ξ1,ξ2)and (Ψ ,Q )
˙
ξ1
˙
ξ2
˙
Ψ
˙
Q
=
0
0
−L(α)Ψ
−L(α)Q
(ξ,Ψ ,Q )∈
i∈N
(Ci∩ρiBi)×R2N,(5.14)
is GAS.
P oo :
No e ha i Assump ion 11 is sa is ied, he e is a ime-scale sepa a ion be ween
be ween
(ξ1,ξ2)
and
(Ψ ,Q )
ha allows ge ing he bounda y laye
(5.14)
, which is
ob ained e-scaling he o dina y ime,
, by
1/ν1
, in
(5.12)
, and doing
ν1= 0
. Tha is, we
zoom in o he as e a iables
(Ψ ,Q )
, eezing he slowe a iables so ha
ξ1
and
ξ2
emain cons an du ing lows. Likewise, no e ha he e is no jump in he bounda y laye .
5.7 S abili y analysis o he comple e hyb id sys em 79
The as subsys em dynamics om (5.12) can be w i en as:
ν1
dΨ
d =−L(α)Ψ +ν1Ke˜
Ψ
ν1
dQ
d =−L(α)Q +ν1Ke˜
Q
eplacing τ= /ν1, and se ing ν1= 0
ν1
dΨ
d(ν1τ)=dΨ
dτ =−L(α)Ψ +ν1Ke˜
Ψ
=−L(α)Ψ
ν1
dQ
d(ν1τ)=dQ
dτ =−L(α)Q +ν1Ke˜
Q
=−L(α)Q
Now, le us conside he ollowing Lyapuno unc ion candida e
Vc:= 1
2Ψ ⊤L(α)Ψ +1
2Q ⊤L(α)Q ≥0.
Then,
⟨▽Vc(Ψ ,Q ), bl⟩=−Ψ ⊤L(α)L(α)Ψ −Q ⊤L(α)L(α)Q ≤0,
being
bl := [0 0 −L(α)Ψ −L(α)Q ]⊤
. No ice ha
⟨▽Vc(Ψ ,Q ), bl⟩
is nega i e
semide ini e. Consequen ly, om he s uc u e o
L(α)
and s anda d consensus esul s [39],
(5.14) is GAS.
The dynamic de ined in
(5.14)
gua an ees ha each
Ψ ,i
and
Q ,i
, o agen s wi h
associa ed
αi= 1
, con e ges o a neighbo hood o
Ψ∗
and
Q∗
, espec i ely, which de ines
a consensus be ween he in e connec ed agen s. ■
5.7.4 S abili y o he educed sys em
Fo he educed sys em associa ed wi h Hsp, we now ob ain he nex esul .
P oposi ion 10:
Unde Assump ion 11 and o a gi en
α
, he a ac o
A := S
i∈N A ,i
wi h A ,i := {(ξ1,i,ξ2,i)∈Hi:∥˜xi∥< Xi,ξ1,i =0}associa ed o he educed sys em:
80 Chap e 5. Hyb id dynamical con ol scheme o eac i e powe sha ing
H (ξ1,ξ2):
˙
ξ1
ν2˙
ξ2
=
−Kω˜ω−Kc1˜
Ψ
−KΨ˜
Ψ+Kc1˜ω+K0˜
P
−KV˜
V−K2˜
Q
−KQ˜
Q+K2˜
V
−KdP ˜
P−K0˜
Ψ
ν2A˜x+ν2B
ν2Θi(ω ,V )z
−ν2Γ(ω ,V )Θ(ω )z
ν21
(ξ1,ξ2)∈C
ξ1
+
ν2ξ+
2
∈
ξ1
ν2˜x
ν2z
ν2h(ξ)
0
(ξ1,ξ2)∈D,
(5.15)
is SPAS as ν2→0+.
P oo :
The p oo o P oposi ion 10 ollows he same app oach p esen ed in Sec ion 4.5.5,
wi h S:= diag{Kω,KΨ,KV,KQ,KP}.■
Theo em 7:
Unde Assump ion 9, 11 and o a gi en
α
, he se
A=A ×{L(α)Ψ =
0,L(α)Q =0}
associa ed wi h he hyb id sys em
H
,
(5.8)
–
(5.11)
, is SPAS as
ν1,ν2→0+
.
□
P oo :
No e ha
A ∈A
and ha i Assump ions 9 and 11 a e sa is ied, hen P oposi ions 7–
10 a e hold. As a esul , he p oo is di ec ollowing [21, Theo em 1]. ■
5.8 Simula ion esul s
The hyb id con ol scheme p oposed he e is alida ed o an islanded AC-bus mic og id
composed o 6 BESSs simula ed in Ma lab/Simulink using he Elec ical Toolbox, as
shown in Fig. 5.2. Each hal -b idge is emula ed as an AC con olled ol age sou ce whose
inpu is he equi alen modula ed ol age, ollowed by i s co esponding LC il e and line
impedance. The line impedances a e he e ogeneous and unknown o he con olle ; hence,
he ou pu cu en ,
io,i
, is also measu ed o calcula e he ou pu ac i e and eac i e powe s.
The nominal ol age o he bus line is
Vbus = 120√2sin(2π60 )V
, hen
Vn= 120√2
V
and
ωn= 2π60 ad/s
. Table 5.1 and 5.2 p o ide he pa ame e s o he seconda y and
p ima y con ol loops espec i ely, i.e., he pa ame e s o he hyb id model
(5.6)
. F om
hese pa ame e s and
PL,i =28.20 0.12
0.12 0.08
,
QL,i =1.50 0
0 5.55
, i is easy o see
ha Assump ion 9 is sa is ied. Mo eo e , no e ha
Kin = 34.48
and
Kdc = 0.42
, hen
Assump ion 11 is also sa is ied. Finally, om Table 5.1, he d oop con ol
(3.5)
and
(5.2)
pa ame e s a e: Kd,1:= 10,Kd,2:= 5,Kd,3= 0.01,Kd,4= 10−4and Kd,5= 0.1.
5.8 Simula ion esul s 81
Figu e 5.2 Mic og id con igu a ion scheme.
Table 5.1 Mic og id and d oop con ol pa ame e s.
Pa ame e Value Pa ame e Value
Kc10000 KV0.0049
Ke100 KdP 50
KΨ0.45 Kc10.5
Kω5K20.49
KQ1K00.05
82 Chap e 5. Hyb id dynamical con ol scheme o eac i e powe sha ing
BESS1
BESS2
BESS3
BESS4
α1,2=α2,1
α6,1=α1,6
α2,3=α3,2
BESS6
α3,4=α4,3
α4,5=α5,4
α5,6=α6,5
BESS5
Figu e 5.3 Communica ion g aph among he BESSs.
Table 5.2 In e e pa ame e s.
Pa ame e ∀iValue Pa ame e ∀iValue
Vin,i 48 V Zline1R=0.1ΩL=1.6mH
Li50 mHZline2R=0.4ΩL=1.8mH
Ci140.72 µFZline3R=0.7ΩL=1.9mH
RLS,i 1.5 ΩZline4R=0.6ΩL=1.7mH
Ri180 ΩZline5R=0.9ΩL=2mH
ωn2π60
ad/s
Zline6R=1ΩL=2.2mH
Vn120√2VK1
Xi0.041 k1/3600
T0.1 ms Cba ,i 0.1615 Wh
The Laplacian ma ix ha ep esen s he communica ion in e connec ions among he 6
BESSs as shown in Fig. 5.3 is
L=
α1,2+α1,6−α1,20 0 0 −α1,6
−α2,1α2,1+α2,3−α2,30 0 0
0−α3,2α3,2+α3,4−α3,40 0
0 0 −α4,3α4,3+α4,5−α4,10
0 0 0 −α5,4α5,4+α5,6−α5,6
−α6,10 0 0 −α6,5α6,5+α6,1
.
To alida e he e ec i eness o ou p oposed s a egy, we e i y he SOC balancing
and eac i e powe consensus h ough wo simula ion scena ios. The i s scena io es s
pe o mance unde load changes and communica ion ailu es, while he second e alua es
he pe o mance agains plug-and-play e en s.
5.8.1 Scena io 1
In his scena io, he p oposed con ol hyb id scheme is alida ed conside ing ha a load
change and a connec ion ailu e occu . Ini ially, he o al load is 810W and 150VA . Then,
5.8 Simula ion esul s 83
one load o 300W and 60 VA is disconnec ed a
T1= 1.5
s and connec ed back a
T2= 2.5
s.
Mo eo e , a
T3= 3s
, a connec ion ailu e occu s be ween he BESS
1
and he BESS
2
, i.e.,
α1,2=α2,1= 0.
Fig. 5.4 shows he SOC, hei e e ences, and he ac i e powe s o each BESS in
discha ging mode. No e ha hey con e ge o a consensus, e en when he be o e-men ioned
pe u ba ions occu . No e ha in his igu e, he SOC e e ences (o
Ψ ,i
) con e ge as e
o a consensus han he SOCs and he ac i e powe s o hei e e ences. Indeed, he SOCs
and he ac i e powe s con e ge o hei e e ences be o e o 2.4s app oxima ely (no e ha
1/Kdc = 2.4) and Ψ in 0.1ms app oxima ely (no e ha 1/Kc= 10−4).
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3 3.5 4
0
100
200
300
0 0.5 1 1.5 2 2.5 3 3.5 4
0
0.2
0.4
0.6
0.8
1
0 1 2
10-4
0.8
0.9
1
Figu e 5.4 E olu ion o he SOC e e ences Ψ ,i, he SOCs Ψiand he ac i e powe s Pi,
o
i={1,2,3,4,5,6}
, when a load o 300W and 60VA is disconnec ed a
T1
and hen i is connec ed back a T2. In T3α1,2= 0.
Fig. 5.5 shows he eac i e powe s con e ging o hei e e ences and hei e e ences
con e ging o a consensus, again no ing he di e en ime scales be ween bo h dynamics.
Finally, Fig. 5.6 shows he e o a iables
˜
Ψi
,
˜
Pi
,
˜
Qi
, no ing ha hey con e ge o ze o. I
can be seen ha he connec ed and disconnec ed load modi ies he slope o
Ψ ,i
and he
cons an consensus alue o
Q∗
. Howe e , he disconnec ion ailu e does no change any
e olu ion.
Now, ocusing on he in e e s, Fig. 5.7 shows a zoom o he a iables o each DC-AC
con e e . No e ha he ol age e o s con e ge o ze o, ensu ing ha he ol age ou pu s
ollow hei e e ences. Mo eo e , we highligh ha hese dynamics (
ξ2,i
) a e slowe han
Ψ ,i
and
Q ,i
and, as e han SOCs, eac i e and ac i e powe s (
ξ1,i
). Indeed, he ol age
84 Chap e 5. Hyb id dynamical con ol scheme o eac i e powe sha ing
esponse speed is 30ms app oxima ely (no e ha
1/Kin = 0.029
). Then, we can see as
he h ee ime-scale sepa a ion model is alida ed.
0 0.5 1 1.5 2 2.5 3 3.5
-20
0
20
40
60
80
0 0.5 1 1.5 2 2.5 3 3.5 4
-50
0
50
100
150
Figu e 5.5
E olu ion o he eac i e powe e e ences
Q ,i
and he eac i e powe s
Qi
o
i={1,2,3,4,5,6}
, when a load o 300W and 60VA is disconnec ed a
T1
and
hen i is connec ed back a T2. In T3α1,2= 0.
0 0.5 1 1.5 2 2.5 3 3.5 4
-0.2
-0.1
0
0.1
0.2
0.3
0 0.5 1 1.5 2 2.5 3 3.5 4
-100
0
100
200
0 0.5 1 1.5 2 2.5 3 3.5 4
-100
-50
0
50
100
150
Figu e 5.6
E olu ion
˜
Ψi
,
˜
Pi
,
˜
Qi
in Scena io 1 o
i={1,2,3,4,5,6}
, when a load o 300W
and 60VA is disconnec ed a
T1
and hen i is connec ed back a
T2
. In
T3
α1,2= 0.
5.8 Simula ion esul s 85
Figu e 5.7
E olu ion o ol age e o s, ol ages, cu en s and du y cycles o con e e s o
he i s 0.2s in Scena io 1.
5.8.2 Scena io 2
In his scena io, he mic og id su e s he connec ion/disconnec ion o a BESS
1
, which
can be ela ed o a plug-and-play e en o a ailu e o his BESS. We conside a load o
400W and 80 VA . The disconnec ion occu s a he ins an ime
T1= 1.5
s and hen, i is
connec ed back a T2= 1.7s.
Fig. 5.8 shows he con e gence o he SOC e e ences o a consensus om ini ial
condi ions and a e connec ion/disconnec ion o he BESS
1
, wi h ansien imes less han
0.1
ms. These e olu ions a e as e han he SOCs and he ac i e powe s con e gence o a
consensus, as seen in he Scena io 1. Fig. 5.9 shows he e olu ion o he eac i e powe
e e ences,
Q ,i
, and he eac i e powe s,
Qi
du ing he BESS
1
connec ion/disconnec ion.
Mo eo e , in Fig. 5.10, he e olu ion o ze o o he e o a iables
˜
Ψi
,
˜
Pi
,
˜
Qi
is shown,
e en a e he pe u ba ions. No e ha he esponse speed o all
Q ,i
o a consensus is
as e han he e olu ion o each ˜
Qi o ze o.
Finally, in Fig. 5.11, a zoom in on he ol age e o s is shown, no ing he obus ness o
he ou pu ol age egula ion a e any connec ion/disconnec ion e en . I can be seen ha
he cu en equilib ium is changing wi h espec o he nominal e e ence o abso b he
pa ame e a ia ion, gua an eeing his ol age egula ion. In addi ion, his igu e shows
he e olu ion o he ol age, cu en , and du y cycle. Again, we can app ecia e he h ee
ime-scale sepa a ion. Indeed, he ime esponse o he in e e (less han 30ms) is la ge
han he SOC and he eac i e powe e e ences (less han
0.1m
s), bu smalle han he
esponse speed o ˜
Ψi,˜
Pi,˜
Qi(2.4s app oxima ely).
F om hese scena ios, we can alida e he s a emen o Theo em 7 and, he e o e, all he
objec i es o his chap e .
92 Chap e 6. Bi u ca ion analysis o islanded AC mic og ids wi h Cons an Powe Loads
mic og ids, is o apply bi u ca ion heo y o hei nonlinea models [63]. By applying bi u -
ca ion analysis echniques, i is possible o iden i y a sa e ope a ing egion o mic og ids,
and his analysis can hen be used o design app op ia e p o ec ion mechanisms, imp o ing
he o e all eliabili y and s abili y o he sys em. One o he mos c i ical elemen s ha a -
ec s abili y is CPLs, so in his chap e , we will ocus on a mic og id wi h a load o his ype.
In he li e a u e, a ious app oaches o la ge signal s abili y analysis applied o powe
sys ems and mic og ids wi h CPL ha e eme ged, mos o hem using bi u ca ion echniques
[61,64
–
67]. In [68], he s abili y analysis o a h ee-phase AC mic og id wi h a CPL is
p esen ed. Assuming ha he e is no eac i e powe consump ion, he bi u ca ion analysis
o he h ee-phase CPL is educed o he analysis o he model o he DC mic og id
sys em. Also, he bi u ca ions a e compu ed wi h a di ec nume ical me hod using he
New on-Raphson p ocedu e. The au ho s in [64] p oposed a s abili y s udy o a Dual
Ac i e B idge con e e using bi u ca ion analysis wi h Poinca é map heo y and Jacobian
ma ix, conside ing he e ec o inpu il e and a CPL using a disc e e- ime model. In [65]
and [66], CPL models conside ing delays a e p esen ed. Speci ically, [65] p esen ed
he CPL model wi h Delay-Di e en ial Equa ions (DDEs); he model is p esen ed o a
single-phase AC mic og id bu could be applied o balanced h ee-phase sys ems.
These s udies ha e examined di e en mic og id s uc u es and mul iple ope a ing
modes, and mos o hem p oposed di e en CPL models o hei s udies. Howe e ,
despi e achie ing g ea e model p ecision, hese me hods esul in an excessi e numbe o
equa ions e en o basic mic og id sys ems, complica ing he analysis p ocess, and almos
none o hem p esen s expe imen al alida ions. In [61], a nume ical bi u ca ion analysis
using XPP-AUTO is p esen ed o an educed model o an h ee-phase AC mic og id wi h
a CPL, whe e is p oposed a educed model o AC-DC con e e s as loads.
Fu he mo e, la ge signal s abili y analysis o AC mic og ids wi h CPL also conside s
o he s udy echniques. In [67], a la ge signal s abili y analysis o AC mic og ids based on
mixed po en ial heo y is p esen ed, also conside ing he in luences o he s o age sys em.
6.2 P elimina y concep s
6.2.1 Some no ions o bi u ca ion heo y
As men ioned abo e, bi u ca ions heo y is he s udy o changes in he quali a i e s uc u e
o he low o a nonlinea dynamical as pa ame e s a e a ied. The pa ame e alues a
which hese bi u ca ions occu a e called he bi u ca ion alues o poin s [69]. Some
bi u ca ions, known as local bi u ca ions, can be iden i ied simply by analyzing he equi-
lib ium equa ions
˙x= (x) = 0
and he Jacobian ma ix
∂ (x)/∂x
. Howe e , o he mo e
complex bi u ca ions equi e nume ical me hods o de ec hem.
In pa icula , one o he mos impo an bi u ca ions in powe elec onic sys ems is he
Hop bi u ca ion, as i o en leads o he ins abili y o he sys em.
Poinca é-And ono -Hop bi u ca ion
This bi u ca ion occu s when an equilib ium poin swi ches i s s abili y as a pa ame e
a ies, accompanied by he occu ence o a pe iodic solu ion (limi cycle) a ound his poin
6.2 P elimina y concep s 93
[70]. A ypical example occu s when inc easing he load powe (a sys em pa ame e ) leads
o ins abili y. F om a local pe spec i e, his bi u ca ion is iden i ied by wo eigen alues
c ossing he imagina y axis. Hop bi u ca ions can be classi ied in o wo ca ego ies:
supe c i ical and subc i ical.
Supe c i ical Hop bi u ca ion: To s udy his bi u ca ion, we conside he ollowing
R2sys em [71] :
˙x1=x2+x1λ−x2
1−x2
2
˙x2=−x1+x2λ−x2
1−x2
2.
whe e
λ
is a pa ame e . When
λ≤0
, he sys em solu ions a e a clockwise spi al con e ging
on he o igin, which is a s able ocus, as inc eases. Howe e , when
λ > 0
, he o igin
becomes an uns able ocus and a pe iodic o bi o limi cycle appea s in such a way ha
all solu ions con e ge o i as inc eases. Fig.6.1 and Fig.6.2 show he phase po ai (o
plane) and he e olu ion o he s a e a iables o λ≤0and λ > 0, espec i ely.
Figu e 6.1 Phase po ai o he supe c i ical hop bi u ca ion o pa ame e λ.
Figu e 6.2
E olu ion o he sys em in he s able egion when
λ≤0
and in he uns able
egion when λ > 0.
94 Chap e 6. Bi u ca ion analysis o islanded AC mic og ids wi h Cons an Powe Loads
Figu e 6.3 Bi u ca ion diag am o a supe c i ical Hop bi u ca ion.
When he pa ame e
λ
changes om nega i e o posi i e alues, he eigen alues o
he Jacobian ma ix, which a e complex conjuga es, mo e om he le hal -plane o he
igh hal -plane o he complex plane. Fo
λ= 0
he eal pa o he eigen alues is ze o
(imagina y eigen alues), and a his poin he Hop bi u ca ion appea s. Fo nega i e
alues, he sys em is locally s able, and o posi i e alues i is locally uns able. The
supe c i ical Hop bi u ca ion diag am o he plane {x1,λ}is shown in Fig. 6.3.
No ice ha he pa ame e is assigned o he ho izon al axis; he s able equilib ium poin
is d awn in solid lines, and he uns able equilib ium poin is d awn in dashed lines, as well
as o limi cycles. We will ollow hese con en ions in bi u ca ion diag ams.
Subc i ical Hop bi u ca ion: To s udy his bi u ca ion, we conside he ollowing
sys em [63]:
˙x1=λx1−x2+x1x2
1+x2
2
˙x2=x1+λx2+x2x2
1+x2
2
whe e λis a pa ame e .
when
λ > 0
, he sys em has an uns able equilib ium; while, o alues o
λ < 0
, an
uns able limi cycle appea s, which su ounds a s able equilib ium. Thus, in his case,
he sys em e ol es ollowing spi als away om he uns able limi cycle as inc eases,
ending owa ds he equilib ium a he o igin i
x(0)
is in he in e io o he uns able o bi
o owa ds in ini y i i is in he ex e io . Fig.6.4 and Fig.6.5 show he phase po ai and
he e olu ion o he s a e a iables o λ < 0and λ > 0, espec i ely.
6.2 P elimina y concep s 95
Figu e 6.4 Phase po ai o he subc i ical hop bi u ca ion o pa ame e λ.
Figu e 6.5
E olu ion o he sys em in he s able egion when
λ≤0
and in he uns able
egion when λ > 0.
Figu e 6.6 Bi u ca ion diag am o a subc i ical Hop bi u ca ion.
The subc i ical Hop bi u ca ion diag am o he plane
{x1,λ}
is shown in Fig. 6.6. The
subc i ical Hop bi u ca ion indica es ha a ound he s able equilib ium poin he e is an
96 Chap e 6. Bi u ca ion analysis o islanded AC mic og ids wi h Cons an Powe Loads
uns able limi cycle. This uns able limi cycle delimi s he egion (o basin) o a ac ion
o he s able equilib ium, and i he sys em c osses his egion, i becomes uns able, e en
hough he Jacobian ma ix has all s able eigen alues. In his si ua ion he non-linea e ms
o he model a e making he sys em uns able, bu no e ha close o equilib ium he sys em
emains s able.
The supe c i ical Hop bi u ca ion indica es ha a ound he uns able equilib ium poin
he e is a s able limi cycle, and he sys em oscilla es in his egion. The s able limi cycle
p o ides he maximum and minimum alues o he oscilla ions o which he sys em is
subjec when he equilib ium becomes uns able. Al hough he Jacobian ma ix does no
explain he o igin o he ipples, in o de o gua an ee s abili y i is necessa y o know only
he linea pa o he model o his ype o Hop bi u ca ion. No e ha he subc i ical ype
is wo se since local s abili y is no su icien when he sys em is a om equilib ium [63].
Typically, CPLs ins abilize an elec ical sys em h ough a subc i ical Hop , as will be
discussed in de ail in he nex sec ions.
6.3 AC mic og id model
In his sec ion, we p esen he ma hema ical model o he selec ed AC mic og id opol-
ogy. Fi s , we p esen he model o a h ee-phase con e e based on he g aphical
dq
-
ans o ma ion o powe swi ching con e e s depic ed in [72]. We ob ained he ma he-
ma ical model o a h ee-phase con e e wi h an LC il e and esis i e load. Then, he
comple e model wi h wo h ee-phase con e e s wi h line impedances and he addi ion o
a CPL is p esen ed.
6.3.1 Modelling o a h ee phase in e e
Based on he g aphical
dq
ans o ma ion me hod o powe swi ching con e e s [72],
we model an islanded h ee-phase con e e wi h an LC il e and esis i e load. The
h ee-phase con e e is ep esen ed in a educed o m as a con olled h ee-phase ol age
sou ce equi alen o he modula ed ol age by swi ching, as shown in Fig. 6.7.
The g aphical modeling me hod educes he e o o ob ain he model. The h ee-phase
in e e is modeled as a linea ime-in a ian ci cui ob ained by changing balanced AC
eac o s in o equi alen DC eac o s combined by gy a o s [72]. The ci cui is decomposed
in o ou subci cui s: h ee-phase ol age sou ce, induc o , capaci o , and esis o se s.
Bo owing om [72], he g aphical
dq
ans o ma ion o hese subci cui s is p esen ed,
assuming iden ical pa ame e s o each phase.
6.3 AC mic og id model 97
Figu e 6.7 Elec ical ci cui o a h ee-phase in e e wi h an LC il e .
a)
Th ee-phase ol age sou ce: The
dq
ans o med ci cui o a h ee-phase ol age
sou ce se is shown in Fig. 6.8. F om Fig. 6.8 (b) we ha e:
dq
e =Kabc→dq abc
e ,
whe e abc
e = [ a
e , b
e , c
e ]Tand dq
e = [ d
e , q
e ]T, and
Kabc→dq = 2
3
cos(θ) cosθ−2π
3cosθ+2π
3
−sin(θ)−sinθ−2π
3−sinθ+2π
3
√2
2
√2
2
√2
2
Figu e 6.8 dq
ans o ma ion o a ol age sou ce se . (a) O iginal ci cui . (b)
dq
ans-
o med ci cui .
b)
Th ee-phase induc o se : The
dq
ans o med ci cui o a h ee-phase induc o
se is shown in Fig. 6.9. The equi alen dynamics om Fig. 6.9 (b) a e:
L˙
iq
L=−wLid
L+ q
L˙
id
L=wLiq
L+ d
L˙
i0
L= 0
L= 0; balanced condi ion.
(6.1)
98 Chap e 6. Bi u ca ion analysis o islanded AC mic og ids wi h Cons an Powe Loads
The p oo is p esen ed in [72].
Figu e 6.9 dq
ans o ma ion o a induc o se . (a) O iginal ci cui . (b)
dq
ans o med
ci cui .
As he sou ces a e balanced, he 0-axis a iable can be excluded om ou discussion.
c)
Th ee-phase capaci o se : The
dq
ans o med ci cui o a h ee-phase capaci o
se is shown in Fig. 6.10. The equi alen dynamics om Fig. 6.10 (b) a e:
C˙ q
C=−wC d
C+iq
C˙ d
C=wC q
C+id
C˙ 0
C=i0
C= 0; balanced condi ion.
(6.2)
The p oo is p esen ed in [72].
Figu e 6.10 dq
ans o ma ion o a capaci o se . (a) O iginal ci cui . (b)
dq
ans o med
ci cui .
d)
Th ee-phase esis o se : The
dq
ans o med ci cui o a h ee-phase esis o se
is shown in Fig. 6.11. The equi alen dynamics om Fig. 6.11 (b) a e:
dq =Kabc→dq abc =Kabc→dqRiabc =Ridq (6.3)
The 0 axis a iables a e excluded o bo h he balanced sou ces and he balanced
ini ial condi ions.
6.3 AC mic og id model 99
Figu e 6.11 dq
ans o ma ion o a esis o se . (a) O iginal ci cui . (b)
dq
ans o med
ci cui .
Now, om he equi alen equa ions ob ained om he g aphical
dq
ans o ma ion o each
subci cui we can ob ained he comple e model o he h ee-phase in e e o Fig. 6.7 o a
dq equi alen ci cui gi en by
Ldid
L
d = d
e −RLid
L+wLiq
L− d
c
Ldiq
L
d = q
e −RLiq
L−wLid
L− q
c
Cd d
c
d =wC q
c+id
L− d
c
R
Cd q
c
d =−wC d
c+iq
L− q
c
R
(6.4)
In he nex sec ion, we in oduced he CPL selec ed model o be used in he mic og id
model.
6.3.2 Model o a CPL
In AC mic og ids, mos componen s connec ed o he bus a e ol age-con olled ec i ie s
supplying a load, such as a ba e y in cha ging mode, d awing cons an powe om he
bus. In balanced h ee-phase AC ne wo ks, as well as in DC mic og ids, his combina ion
o a ol age-con olled con e e and a load can be modeled as a CPL.
Cons an powe loads usually p esen a nega i e impedance cha ac e is ic, which can
des abilize he powe sys em [66]. As explained in [63,73], he basic diag am o a cons an
powe load, such as a DC/DC con e e connec ed o a linea load, is shown in Fig. 6.12.
Figu e 6.12 CPL block diag am. (a) Con e e + load. (b) Equi alen ci cui .
100 Chap e 6. Bi u ca ion analysis o islanded AC mic og ids wi h Cons an Powe Loads
As his con e e egula es i s ou pu ol age, any dis u bance in he bus ol age will
be compensa ed by he con ol loop, ensu ing ou pu ol age egula ion. This adjus men
equi es he cu en o compensa e o changes in he bus ol age o main ain cons an
powe .
Figu e 6.13 Nega i e impedance beha io o CPLs.
The s a ic ol age-cu en cha ac e is ic o a CPL load is depic ed in Fig. 6.13. Since
Pload
is cons an , he plo o ms a hype bolic cu e. The inpu esis ance,
Rload
, is
gi en by he a io o small-signal ol age changes o e small-signal cu en changes,
∆
∆i
,
and his alue depends on he con e e ’s ope a ing poin . As shown in he g aph, he
slope is nega i e, indica ing ha he inpu esis ance o he con e e exhibi s a nega i e
cha ac e is ic. This means ha any dec ease in ol age will cause an inc ease in cu en ,
and ice e sa. Assuming ha he con e e has no losses, he nega i e inpu esis ance
can be calcula ed by di e en ia ing he ol age wi h espec o he cu en in
=Pload
i
, as
ollows.
d
di =−i−2Pload =−| 2
Pload |=−Rload (6.5)
This nega i e esis ance is also nonlinea depending upon he cu en and ol age, and
would causes ins abili y in he sys em as depic ed in [73].
In he li e a u e, mos CPL models o DC mic og ids a e based on hei nega i e
impedance cha ac e is ics. The wo ks in [74,75] analyze he cu en - ol age cha ac e is ics
o DC/DC con e e s as CPLs, conside ing limi s in du y cycle and maximum egula ed
ol age. In ac , he con e e beha es as a CPL when ope a ing unde closed-loop con ol,
and as a posi i e esis ance in egions whe e i is in open-loop ope a ion o he con ol
loop is sa u a ed, as shown in Fig. 6.14.
Figu e 6.14 Cu en - ol age cha ac e is ic o a load con e e as a CPL.
6.3 AC mic og id model 101
Following Fig. 6.14, he CPL is gi en by:
i=
Pload
,i ≥V h
R h ,i < V h
whe e R h =V2
h
Pload is he esis ance o he linea egion.
Fo AC mic og ids, models exis o single-phase and h ee-phase sys ems. The wo ks
in [65,66] p opose CPL models o single-phase sys ems based on he concep o a e age
powe , which in ol es in eg a ing he p oduc o ol age and cu en signals o e a ini e
mo ing window. In [65], a model based on DDEs is also p oposed.
Fo balanced h ee-phase sys ems, he same model based on nega i e esis ance cha ac-
e is ics is applicable since, a e o a ing he coo dina es o he synch onous ame, he
sys em can be ea ed as DC [65]. In [63], a h ee-phase AC/DC ec i ie con olled wi h
powe ac o co ec ion (PFC) is modeled as a cons an h ee-phase ac i e powe load.
This modeling is based on PQ heo y, which assumes ze o ins an aneous eac i e powe .
Howe e , modeling CPLs o single-phase (o unbalanced mul i-phase) AC sys ems
emains la gely unexplo ed due o he complexi y o de ining ac i e powe P( ) in hese
sys ems [65].
Selec ed CPL model
In his wo k, we a e going o conside he model p oposed in [61]. He e, a CPL model
is p esen ed as an al e na i e ep esen a ion o mul iple iden ical h ee-phase con olled
ec i ie s, modeled as a single ime-in a ian nonlinea dynamic load. This p oposed model
is exp essed in unc ion o he o al demanding load cu en gi en by
did
load
d =−λid
load +λPo
d
bus
(6.6)
diq
load
d =−λiq
load (6.7)
whe e
λ
is he in e se o he load ime-cons an ,
Po
he ex ac ed cons an ac i e powe
om he common bus, and
d
bus
he di ec componen o he bus ol age. Equa ions
(6.6)
and
(6.7)
ep esen he idea o using a low-pass il e o model a powe con e e a he
equilib ium poin . I is assumed ha he e is ideal powe ac o co ec ion (PFC) on he
common bus, so he eac i e powe is ze o. Fu he mo e, his model conside s sa u a ion
o he cu en o a maximum d awn cu en
Imax
, so he equi alen load dynamic changes
o
did
Load
d =−λid
Load +λImax (6.8)
diq
Load
d =−λiq
Load (6.9)
108 Chap e 6. Bi u ca ion analysis o islanded AC mic og ids wi h Cons an Powe Loads
powe e e ence o
Po≈2940
W, whe e
id
load
is sa u a ed o
Imax = 40
A, leading he
sys em back o a s able ope a ing poin . The e ec o he supe c i ical Hop bi u ca ion
can also be obse ed on bus ol age
bus
and in he cu en
iL1
, as depic ed in Fig. 6.22
and Fig. 6.21, espec i ely. He e, we obse e ha
id
L1
is sa u a ed o
≈20.8
A because he
wo in e e s a e also d awing powe o he esis o s connec ed o he common bus.
0.11 0.12 0.13 0.14 0.15 0.16 0.17
0
500
1000
1500
2000
2500
3000
3500
4000
Figu e 6.20
CPL powe e olu ions o powe e e ence s eps a ound bi u ca ion poin
H2
.
0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17
-30
-20
-10
0
10
20
30
Figu e 6.21
Cu en
iL1
e olu ions in
abc
and
dq
coo dina es a ound bi u ca ion poin
H2.
6.6 Expe imen al esul s 109
0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17
-200
-150
-100
-50
0
50
100
150
200
250
Figu e 6.22
Bus ol age e olu ions in
abc
and
dq
coo dina es a ound bi u ca ion poin
H2
.
6.6 Expe imen al esul s
This sec ion p o ides expe imen al esul s o alida e he bi u ca ion analysis desc ibed
abo e. Fi s , we p esen expe imen al es s o ob ain he ime cons an o an elec onic CPL.
Then, we desc ibe he expe imen al se up o he AC mic og id in Fig. 6.15 and p esen
expe imen al esul s o selec ed ope a ing poin s ha alida e he bi u ca ion analysis.
6.6.1 CPL ime cons an es
Fi s , we ca ied ou es s o de e mine he ime cons an o he elec onic CPL used in he
expe imen al alida ion. We used he Cine gia EL+ AC AC elec onic load, con igu ed in
p og ammable powe mode, which can be con olled by powe e e ence s eps. We ob ained
he ime cons an measu emen h ough es s in CPL mode by connec ing i di ec ly o a
g id gene a o (NHR 9410-12). We hen applied g id ol age s eps o obse e he cu en
a ia ions, as shown in Fig.6.23(c). He e, he in e ace o he CPL con ol so wa e is
shown, whe e he s eps in g id ol age, hei e ec on CPL cu en , and he cons an
powe alue in he load can be obse ed. Mo eo e , Fig.6.23(a) shows he schema ic
connec ion o he es , and Fig. 6.23(b) shows he Impe ix so wa e in e ace o cu en
measu emen s.
Fig. 6.24 shows he CPL cu en
iLoad
measu emen s ob ained using Impe ix ex e nal
ModuLink ±50 A isola ed cu en senso s o a ol age s ep esul ing in a cu en s ep
( om
i e ,1
o
i e ,2
). Mo eo e , a zoom in on he
id
Load
ansien is highligh ed, showing
a ime cons an
∆ ≈0.0625
ms, which is he ime in e al o
id
Load
, om
i e ,1
, o each
app oxima ely 63.2% o i e ,2, leading λ≈1/0.0625 ≈1600.
110 Chap e 6. Bi u ca ion analysis o islanded AC mic og ids wi h Cons an Powe Loads
Figu e 6.23
CPL ime cons an expe imen al es . (a) Connec ion scheme. (b) Impe ix
so wa e in e ace. (c) In e ace o he CPL con ol so wa e..
0.88 0.9 0.92 0.94 0.96 0.98 1
-6
-4
-2
0
2
4
6
0.9 0.91 0.92 0.93 0.94 0.95 0.96
-6
-4
-2
0
2
4
X 0.937015
Y 2.91008
X 0.936389
Y 2.15
Figu e 6.24 CPL cu en measu emen s o ime cons an calcula ion.
6.6.2 Expe imen al esul s o Bi u ca ion analysis
We alida e he egions a ound he bi u ca ion poin
H1
whe e a subc i ical Hop bi u -
ca ion occu s, howe e , he supe c i ical Hop bi u ca ion canno be alida ed, as i is a
consequence o cu en sa u a ion in he CPL model, and we use a ixed elec onic CPL.
6.6 Expe imen al esul s 111
The expe imen al se up shown in Fig. 6.25 ep esen s an islanded AC mic og id com-
posed o 2 h ee-phase in e e s wi h LC ou pu il e s implemen ed wi h Impe ix powe
es bench, 2 DC sou ces, 2 h ee-phase induc o s, 3 esis o s and an elec onic CPL, as
shown in Fig. 3.7. The mic og id pa ame e s a e he same as hose p esen ed in Table 6.1.
Each DC/AC con e e is con igu ed om 3 PEB8038 hal -b idge modules, ollowed by
an LC ou pu il e , which is composed o 3 powe induc o s
Li
, and h ee ilm capaci o s
Ci
, embedded in a passi e il e box, as well as line impedances, which a e 3 powe
induc o s
L ,i
. Fo DC sou ces, wo PSB 9360-40 uni s a e used. The induc ance cu en s,
iL,i
and he capaci o ol ages,
C,i
, a e measu ed using gal anically isola ed senso s
onboa d he PEB8038 module, speci ically LEM CKSR 50-P and A ago ACP-C87B,
espec i ely. In addi ion, ex e nal ModuLink ±50A isola ed cu en senso s a e used o
measu e he ou pu cu en iLoad o he CPL.
Figu e 6.25 Expe imen al se up o an islanded AC mic og id wi h a CPL.
Fu he mo e, each DC/AC con e e ope a es as a ol age sou ce in e e (VSI), con-
olled by a dual con ol loop in
αβ
coo dina es, simila o Fig. 1.3. The ol age loop uses
a PR con olle , while he cu en loop uses a p opo ional-in eg al PI con olle . These
con ol algo i hms a e p og ammed using he Simulink blockse o he Impe ix ACG SDK
and hen assembled in he B-BOX RCP. Recall ha he in e nal dynamics o he con ol
loop a e neglec ed in he mic og id model as hey a e assumed o be as e han he es
o he sys em dynamics and in a s able s a e. The e o e, o ensu e he s abili y o he wo
in e e s, some es s a e ca ied ou wi hou he CPL, using only esis i e loads a di e en
ope a ing poin s. Fig. 6.26 shows he AC mic og id ol ages (
abc
C1
and
abc
C2
), cu en s
(
iabc
L1
and
iabc
L2
) and powe measu emen s o bo h in e e s unde di e en powe s eps
gene a ed by he elec onic load ope a ing as a cons an esis o .
To alida e he bi u ca ion analysis ob ained abo e, we implemen some powe s eps in
he elec onic load ope a ing in CPL mode o each he i s bi u ca ion poin
H1
, as shown
112 Chap e 6. Bi u ca ion analysis o islanded AC mic og ids wi h Cons an Powe Loads
Figu e 6.26
E olu ion o mic og id ol ages (
abc
C1
and
abc
C2
), cu en s (
iabc
L1
and
iabc
L2
) and
powe s P1and P2unde s eps o cons an esis o s.
Figu e 6.27
E olu ion o mic og id a iables a ound bi u ca ion poin
H1
. (a) CPL powe
and cu en id
L1. (b) ol ages abc
C1and abc
C2, cu en s iabc
L1,iabc
L2and iabc
Load.
in Fig.6.27. A ime
T1
,
Po≈1310
W, and he sys em s a s o oscilla e un il
T2
, whe e
poo g id quali y causes he CPL o disconnec , lea ing only he powe consumed by he
esis o s (some imes Impe ix p o ec ions a e igge ed as well). The e o e, we con i m
ha he expe imen al ope a ing poin is e y close o he bi u ca ion poin
H1
ob ained
6.6 Expe imen al esul s 113
in he nume ical and simula ion analyses, whe e a subc i ical Hop bi u ca ion occu s.
Fu he mo e, Fig. 6.27(b) shows he capaci o ol ages and induc o cu en s o bo h
in e e s, as well as he CPL cu en a ound he bi u ca ion H1.
Addi ionally, we ha e ca ied ou es s o ob ain expe imen al alida ion o he e ec
on he basin o a ac ion o he s able equilib ium in egion 2 in Fig.6.16(b). To his end,
when he sys em is a he equilib ium poin , we in oduce a pe u ba ion o mo e away
om i . The pe u ba ion consis s o a ying he e e ence alue o he in e e ou pu
ol age (
Vn
) and, a e a ew momen s, e u ning o he ini ial alue. We ha e done his
o di e en alues o
Po
as depic ed in Fig. 6.28, which shows he e olu ion o he CPL
powe (
Po
, op window) and he
d
componen o he il e induc o cu en o in e e 1
(
id
L1
, bo om window). A he h ee poin s, he alue o
Vn
changes om
80
V o
65
V a
ime T1and hen e u ns o 80V a T2.
A poin 1, which co esponds o
Po≈900
W, he sys em emains s able a e e u ning
om he change. A he second poin , co esponding o
Po≈1100
W, i s a s o become
uns able when he change occu s, bu s abilizes when i e u ns. Howe e , a poin 3, a
Po≈1250
W, a powe close o he Hop bi u ca ion, when i e u ns, i s a s o become
uns able and hen he sys em shu s down (Impe ix p o ec ions a e igge ed).
Figu e 6.28
E olu ion o
Po
( op) and
id
L1
(bo om) o 3 di e en ope a ion poin s unde
Vnchanges.
We compa ed hese esul s wi h he MATCONT bi u ca ion diag am as shown in
Fig. 6.29. Bo h poin s 1 and 3 coincide well wi h he beha io o he limi cycle a ound
he subc i ical Hop bi u ca ion (
H1
). A he i s poin wi h a powe o
Po≈900
W, he
cu en
id
L1
goes om
6.25
A o
7.25
A, which is wi hin he uns able limi cycle. Howe e ,
a poin 3, a
Po≈1250
W, he cu en goes om
9
A o
10.5
A, which is al eady on he
114 Chap e 6. Bi u ca ion analysis o islanded AC mic og ids wi h Cons an Powe Loads
edge o he uns able limi cycle and makes he ajec o y no e u n o equilib ium. This
e eals ha he domain o a ac ion o each equilib ium becomes e y hin as we app oach
he Hop bi u ca ion.
Figu e 6.29
Compa ison o
Po
and cu en
id
L1
alues o 2 ope a ion poin s be ween
expe imen al esul s and bi u ca ion diag am o Fig 6.16(b).
Fu he mo e, Fig.6.30 and Fig. 6.31 show a zoom in on he capaci o ol ages and
induc o cu en s o bo h in e e s, as well as he CPL cu en a ound o ope a ing poin s
1 and 3, espec i ely.
6.6 Expe imen al esul s 115
Figu e 6.30
Zoom in on he capaci o ol ages, induc o cu en s and CPL cu en a ound
o ope a ion poin 1.
Figu e 6.31
Zoom in on he capaci o ol ages, induc o cu en s and CPL cu en a ound
o ope a ion poin 3.
116 Chap e 6. Bi u ca ion analysis o islanded AC mic og ids wi h Cons an Powe Loads
6.7 Conclusions
This chap e p o ides he bi u ca ion analysis and expe imen al alida ion o a h ee-phase
AC mic og id in islanded mode wi h a CPL conside ing line impedances and esis i e loads
connec ed o he bus. Al hough his de elopmen was inspi ed by [61], he bi u ca ion
diag ams a e qui e di e en , being alida ed by simula ions and se e al expe imen al
esul s h ough an Impe ix powe es bench unde a ious scena ios. The expe imen al
esul s ocused on he subc i ical Hop bi u ca ion, as i is he mos impo an p ac ical
ope a ing poin o he sys em.
The esul s ob ained p o ide in o ma ion on he sa e ope a ing poin o he mic og id
and whe e he ins abili y egion is, by es ima ing he egion o a ac ion o he s able
ope a ing poin in egion 2 o he bi u ca ion diag am (Fig. 6.16). Based on his analysis,
we can es ablish sa e y limi s o gene al ope a ion and allowable dis u bance anges.
Mo eo e , eac i e powe analysis and a ia ions in d oop con ol pa ame e s a e expec ed
o be s udied in ex ended u u e e sions.
7 Conclusions and u u e wo k
7.1 Conclusions and con ibu ion summa y
This PhD manusc ip con ibu es o he design o con ol laws and o la ge-signal s abili y
analysis o AC islanded mic og ids. Speci ically, i add esses he design o dis ibu ed
con ol loops o s a e-o -cha ge consensus and eac i e powe sha ing among a se o
ba e ies in discha ging mode. The e o e, con inuous- ime and hyb id dynamic h ee- ime
scale con ol schemes o seconda y and p ima y con ol loops a e p oposed wi h la ge
signal s abili y p ope ies using singula pe u ba ion heo y. Fu he mo e, a quali a i e
la ge-signal s abili y analysis using bi u ca ion heo y o AC islanded mic og ids wi h con-
s an powe loads was p esen ed and alida ed h ough simula ion and expe imen al esul s.
Due o he main esul s no being di ec ly co ela ed, his hesis was di ided in o wo
pa s: The i s pa ocuses on he design o con ol laws o consensus algo i hms and
op imal ac i e and eac i e powe sha ing. And he second pa is de o ed o bi u ca ion
analyses o AC mic og ids wi h nonlinea loads.
Nex , we highligh he main impo an con ibu ions o his hesis.
Fo he i s pa , he main con ibu ions a e lis ed below:
•
A dis ibu ed con ol scheme wi h seconda y and p ima y con ol loops has been de-
eloped o achie e SOC consensus o a se o BESS in discha ging mode. BESS a e
c ucial elemen s o eliabili y and g id- o ming gene a ion in islanded mic og ids,
and SOC is an impo an ac o o ba e ies’ li espan. The seconda y con ol loop
co esponds o a SOC e e ence consensus algo i hm designed acco ding o MAS
heo y, p o iding obus ness agains communica ion ailu es and plug-and-play
capabili y. In addi ion, a d oop con ol me hod has been designed in he p ima y
con ol, which uses c oss-coupling e o s o SOC, ac i e powe and equency o
achie e asymp o ic s abili y in he Lyapuno sense. This p ima y con ol is essen ial
117
4238 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 71, NO. 9, SEPTEMBER 2024
poin since powe con e e s can ope a e in a limi ed se o
ope a ing poin s. Hence, we include he ollowing assump ion.
Assump ion 2: Fo a gi en desi ed ope a ing ol age Ce,
he e exis s a co esponding s a e xeand weigh s λe=[λ1λ2]
such ha λ1in [0,1] (and λ2=1−λ1), and
2
j=1
λj¯
Bj:=
2
j=1
λj(Aj−I)xe+Bj=0.(4)
This assump ion is based on he compu a ion o he solu-
ions Ajand Bj o he leas squa es op imiza ion p oblem.
The alue o λ1in his assump ion gi es an in o ma ion abou
he du y cycle ha co esponds o he desi ed ol age Ce.
To ansla e his in o ma ion in o a cycle ν, we p opose he
ollowing p ocedu e o build he co esponding cycle. Fo a
gi en Nν∈N, de ine he cycle νgi en by
∀∈[1,Nν],ν
()=1i ≤Round(Nνλ1),
2i >Round(Nνλ1).(5)
In p ac ice, one has o selec Nνsuch ha Round(Nνλ1)is
su icien ly close o Nνλ1.
D. Da a-D i en Con ol Design
We a e now in posi ion o s a e ou main esul .
Theo em 1: Gi en da a expe imen s Xj,X+
j, a desi ed
ope a ing ol age Ce, associa ed wi h co esponding ope a ing
ec o xeand cycle ν(i.e., λ1)such ha Assump ions 1and 2
is sa is ied, and o a gi en a pa ame e μ∈(0,1), conside
{(Wi,ρ
i,η
i)}i∈Dνin S2×R2×(R>0)2as he solu ion o he
ollowing p oblem.
min
{(Wi,ρi,ηi)}i∈Dν
ε
s. . iX+
ν(i),Xν(i),xe0,η
i>0,
εIWi0,∀i∈Dν,(6)
whe e ma ices ia e gi en by
iX+
ν(i),Xν(i),xe:=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
(1−μ)Wi0WiA
ν(i)Wi0
∗μ
(23)
i(ρi−xe)1
∗∗(33)
i00
∗∗∗ηiXν(i)X
ν(i)ηXν(i)1ν(i)
∗∗∗ ∗ ηipν(i)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
,
(7)
whe e (23)
i=ρ
iA
ν(i)+¯
B
ν(i)−ρ
i+1ν,(33)
i=Wi+1ν−
ηiαν(i)ν(i). Then, unde Assump ion 1, hese
Sν:=
i∈Dν
EW−1
i,ρ
i+xe(8)
is obus ly globally exponen ially s able o sys em (2) wi h
˜x=x−xeand he swi ching con ol law
u(x):=ν(θ),θ∈a gmin
i∈Dν
(˜x−ρi)W−1
i(˜x−ρi)⊂K.
(9)
P oo : Following [17,Th.2], he conclusion o he p e ious
heo em holds i he ollowing condi ions
0≺iAν(i),Bν(i),xe:=
⎡
⎢
⎣
(1−μ)Wi0WiA
ν(i)
∗μ(ρi−xe)A
ν(i)+B
ν(i)−ρi+1ν−xe
∗∗ Wi+1ν
⎤
⎥
⎦.
a e ensu ed o all unce ain ma ices Aν(i),Bν(i). Then, eplac-
ing Aν(i)and Bν(i)by Aj−(Aν(i)−Aν(i))and Bν(i)−(Bν(i)−
Bν(i)), espec i ely, he p e ious LMI can be ew i en as
iAν(i),¯
Bν(i)+HeWi0
(ρi−xe)1
003×2A
ν(i)−A
ν(i)
B
ν(i)−B
ν(i)0,
(10)
whe e we ecall ha ¯
Bν(i)=(Aj−I)xe+Bj. Then, Young’s
inequali y ensu es ha he p e ious inequali y is equi alen o
he exis ence o ηi>0 such ha
iAν(i),¯
Bν(i)
−η−1
iWi0
(ρi−xe)1
0Xj
1
jXj
1
j−1Wi0
(ρi−xe)1
0
−ηi⎡
⎣
00
0A
ν(i)−A
ν(i)
B
ν(i)−B
ν(i)Xj
1
jXj
1
jA
ν(i)−A
ν(i)
B
ν(i)−B
ν(i)⎤
⎦0.
The p oo is concluded by using Assump ion 1, s a ing ha
ma ices Ajand Bja e in j, so ha he e m in αjjappea s
na u ally and by applica ion o he Schu Complemen o he
second e m o he p e ious inequali y.
Rema k 1: I ma ices Ajand Bja e cons an , he leas
squa es app oxima ion Ajand ¯
Bja e equal o he co espond-
ing ma ices Ajand Bj. In his si ua ion, [17, P oposi ion 2]
gua an ees ha he solu ion o he LMI p oblem is equi alen
o condi ions A
ν(i)W−1
i+1νAν(i)−W−1
i≺0, which acco ding
o [4], is equi alen o ha ing a Schu s able monod omy
ma ix ν:=ARound(Nνλ1)
1ANν−Round(Nνλ1)
2. In o he wo ds,
he eigen alues o νa e s ic ly inside he uni ci cle. A
consequence o he p e ious s a emen is ha , in he case o
ime- a ying ma ices Ajand Bj, a necessa y condi ion o he
sol abili y o he LMI is ha νis Schu .
Rema k 2: P oblem (6) is con ex i he scala pa ame e
μ∈(0,1)is ixed, and he e o e has a unique global minima.
A g id sea ch on μcan be pe o med o op imize he esul .
E. Summa y o he Da a-D i en Me hod
To ge a be e unde s anding o he p e ious de elopmen s,
we p opose in his sec ion a summa y o he p ocedu e o
he da a-d i en con ol design, o mula ed as Algo i hm 1.
In his algo i hm, we speci y he mechanism o choosing
con olle pa ame e s ρiand Wi,∀i∈Dν, om a selec ing
ou pu ol age, Ce.Fi s ,weha eawhile loop o selec
he app op ia e amoun o da a such ha ank(Xj
1
j)=
3, o iden i y he ma ices om he da a acquisi ion. This
ensu es ha a desi ed ope a ing poin xeexis s, along wi h
i s associa ed λe=λ1, which sa is ies Assump ion 2. Then,
a cycle ha app oxima es λeis selec ed acco ding o (5),
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MERCHÁN-RIVEROS e al.: DATA-DRIVEN CONTROL DESIGN 4239
Algo i hm 1 Con ol Ma ices and Pa ame e
epea
Take: Ex ac Xj,X+
j∈Rn×pj om Xand .
Iden i y: Ajand Bjusing Xj,X+
j,∀j∈{1,2}.
Tes : o a desi ed Ce,∃any xeand λe.
un il Assump ion 2is sa is ied
Selec : ν() ∀∈N, acco ding o (5) and μ∈(0,1),αj.
Compu e: ρi,ηiand Wi,∀i∈Dν, s. . (6) a e sa is ied.
i.e., ensu ing ha Round(Nνλ1)≈1. Finally, o a gi en
μ, he op imal solu ion o (6) p o ides con ol ma ices and
pa ame e s.
III. EXPERIMENTAL VALIDATION
This sec ion is de o ed o he expe imen al alida ion o
he da a-d i en design on a boos con e e , conside ing ha
i can be app oxima ed by a swi ched a ine model (2) wi h
wo unc ioning modes. The powe con e e is implemen ed
in an Impe ix powe es bench, as shown in Fig. 1and
i s gene al a chi ec u e o he expe imen al se up in Fig. 2.
Mo e p ecisely, he boos con e e is con igu ed om a
PEB8038 hal -b idge powe module wi h silicon ca bide (SiC)
MOSFETs a ed o 800V and 38A. Mo eo e , a powe
induc anceLand a ilm capaci o C0a e embedded in a passi e
il e box.
Fu he mo e, a DC sou ce Elek o-Au om ik PSB 9360-
40 is employed o p o ide he con e e inpu ol age, in.
The con ol algo i hm is implemen ed in a B-Box apid
p o o yping con olle (RCP), which sends ga e d i e signals
ia op ical channels and acqui es senso s signals h ough
analog RJ45 connec o s, unning wi h an in e nal sampling
equency o 50 kHz. The induc ance cu en , iL, and capaci o
ol age, C, a e measu ed using gal anically isola ed onboa d
senso s in he PEB8038 module, which a e LEM CKSR 50-
P and A ago ACP-C87B, espec i ely. The B-BOX RCP is
p og ammed using he Simulink blockse o he Impe ix ACG
so wa e de elopmen ki (SDK). Con ol es s, debugging,
da a logging, and signal moni o ing a e acqui ed om he
Impe ix Cockpi eal- ime moni o ing so wa e.
The elec ical block diag am illus a es wo di e en con ol
p og ams con igu ed, namely, da a acquisi ion and closed-loop
con ol. Bo h p og ams use an ADC and a DO-PWM (Di ec
ou pu PMW) modules, whe e i is necessa y o con e σin o
1 and 0 o he on/o s a e o he MOSFET, espec i ely.
A. Da a Gene a ion
Some da a we e collec ed by eeding he sys em by a
andom inpu signal, σc∈{1,2}.
The selec ed da a se is depic ed in Fig. 3. The selec ed
equency o he andom signal σcis 50KHz, and he da a
sample has a equency also o 50KHz.
B. Closed-Loop Con ol Valida ion
The objec i e is o d i e he ou pu ol age o a neighbou -
hood o Ce=66V. We highligh ha he iden i ied sys em
and p oposed con ol law depend on he sampling equency
o 50 KHz. Following he Algo i hm 1, he expe imen al
Fig. 1. Con igu ed Impe ix uni .
Fig. 2. Block diag am o he expe imen al se up.
Fig. 3. Expe imen al da a o iLand Cob ained wi h a andom inpu σ.
da a deli e s he ollowing desi ed ope a ing poin xe=
[1.265.9]wi h associa ed λ1=0.13 and iden i ied ma ices
A1=1.08 0
01.01 ,A2=0.91 0
−0.01 0.97 ,¯
B1=0.38
1.81 ,¯
B2=−0.44
−2.13 ,
which sa is y Assump ion 2. Then, we selec he cycle
ν=[1222222
],
which is consis en wi h he equi ed λ1, i.e., i has been buil
so ha Round(Nνλ1)=0.91 ≈1. The obus limi cycle
associa ed o his cycle, which is solu ion o he op imiza ion
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4240 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 71, NO. 9, SEPTEMBER 2024
Fig. 4. Impe ix Cockpi eal- ime moni o ing so wa e.
p oblem (6) wi h α1=α2=1, is ob ained by he composi ion
o he cen e s ρi ha a e s acked in he ollowing ma ix
ρ=−0.09 0.07 0.03 0.00 −0.03 −0.06 −0.07
0.06 0.94 0.77 0.61 0.45 0.30 0.15 ,
and he ellipsoids ob ained wi h he ollowing symme ic
posi i e de ini e ma ices
W1=210.45 −3.22
−3.22 171.53 ,W2=170.1−4.9
−4.9 158.9,W3=173.58 −4.79
−4.79 161.69 ,
W4=176.53 −4.70
−4.70 163.92 ,W5=178.39 −4.56
−4.56 165.69 ,W6=180.67 −4.37
−4.37 167.37 ,
W7=189.74 −4.1
−4.1 169.29 .
The compu a ional ime o sol e he LMIs by using CVX-
Sedumi in MATLAB on a compu e wi h a 2 GHz In el Co e
i5 p ocesso and 4 co es is 1.19s.
We ha e implemen ed an ex e nal loop h ough a PI
con ol such ha he ou pu ol age is egula ed in Ce.
Fig. 4shows he ol age and cu en e olu ion o he boos
con e e con igu ed in he Impe ix uni con olled wi h he
p esen ed da a-d i en app oach, including he p edic o and
he ol age ou pu egula ion. The load in he boos con e e
has been changed om 69 o 34.5. The igu e also shows
he ol age e o . The expe imen alida es he da a-d i en
app oach.
IV. CONCLUSION
In his b ie , we p esen a heo e ical con ibu ion add essing
he da a-d i en con ol design o powe con e e s modelled
as a swi ched a ine sys em. The dis inc i e ea u e o his
con ol design lies in he unknown dynamics o he sys em.
Howe e , his lack o knowledge is mi iga ed by le e aging
p io expe imen al da a. Building upon se e al non- es ic i e
assump ions, we alida e he heo e ical esul s using an
expe imen al se up, speci ically an Impe ix boos con e e .
Fu he mo e, we inco po a e an ex e nal con ol loop ha
egula es he ou pu ol age, which is alida ed h ough
expe imen s.
This con ibu ion ma ks a signi ican miles one o u u e
esea ch. We ha e al eady iden i ied se e al c ucial issues ha
me i u he in es iga ion. Fo example, he me hod elies on
he sa is ac ion o Assump ion 1. I would be o in e es o
de ise a me hodology o e alua ing he alues o pa ame e s
αjdi ec ly om he da a. Ano he impo an conce n lies in
mode iden i ica ion, accoun ing o addi ional powe con e e
ea u es such as he discon inuous conduc ion ope a ion mode.
Tho ough in es iga ion is needed o ex end his esul o a
b oade class o powe con e e s wi h N unc ioning modes.
Finally, a da a-d i en algo i hm ha adap s he con ol law
and de ec s any alse o co up ed da a om a cybe a ack is
ano he in es iga ion line.
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Lis o Figu es
1.1 Block diag am o an AC mic og id 2
1.2 P ima y con ol loop 4
1.3 Inne con ol loop 5
1.4 D oop con ol cha ac e is ics: (a) P/ω cu e and (b) Q/V cu e 7
2.1 Hyb id a c 13
2.2 E olu ion o a solu ion o a hyb id sys em 14
2.3 Mappings:(a) no ou e semicon inuous and (b) ou e semicon inuous 15
2.4 E olu ion o a lyapuno unc ion candida e V(x)wi h x∈A o H16
2.5 Example o a g aph [24] 21
3.1 Islanded AC mic og id wi h BESS uni s 28
3.2 S uc u e o a BESSiin an AC islanded mic og id 28
3.3 P oposed dis ibu ed con ol scheme o discha ging a e consensus in an is-
landed AC mic og id 29
3.4 Hal -b idge in e e 33
3.5 Block diag am o he con olled in e e 34
3.6 Expe imen al se up 40
3.7 Mic og id con igu a ion scheme 40
3.8 Block s uc u e o a BESSiuni 41
3.9 Communica ion g aph among he BESSs 42
3.10 Scena io 1: In he op, e olu ion o he SOC e e ences, Ψ ,i, he SOCs, Ψi,
and he ac i e powe s, Pi o i={1,2,3}, when a load o 22Ωis connec ed
a T2and hen i is disconnec ed back a T3. In T3,α1,2= 0. In he bo om a
zoom o SOC e e ences a T144
3.11 Scena io 1: E olu ion o ol age e o s, ˜x2,i, capaci ance ol ages, C,i, and
induc ance cu en s, iL,i o i={1,2,3}45
3.12 Scena io 1: Zoom a T1 o SOC e e ences Ψ ,i, and ol age e o s ˜x2,i, o
i={1,2,3}45
127
128 Lis o Figu es
3.13 Scena io 2: In he op, e olu ion o he SOC e e ences, Ψ ,i, he SOCs, Ψi,
and he ac i e powe s, Pi, o i={1,2,3}, when BESS1is disconnec ed a T2
and hen i is connec ed back a T3. In he bo om, a zoom o SOC e e ences
a : a) T1and b) T246
3.14 Scena io 2: E olu ion o ol age e o s, ˜x2,i, capaci ance ol ages, C,i and
induc ance cu en s, iL,i o i={1,2,3}47
3.15 Scena io 2: Zoom a T2 o SOC e e ences, Ψ ,i, and ol age e o s, ˜x2,i, o
i={1,2,3}47
3.16 Compa ison o he SOC and ac i e powe e olu ion be ween a) ou con ol and
b) he con olle p oposed in [1] 48
4.1 Mic og id con igu a ion scheme 63
4.2 E olu ion o Ψ ,i,Ψi, and he ac i e powe s Pi, o i={1,2,3}, when he
ba e y 1 is disconnec ed a T2and hen connec ed back a T3. In T1,α1,3
u ns o 0 64
4.3 E olu ion o he ol ages, cu en s and du y cycles o in e e s o i={1,2,3}
be ween 1.45s o 1.8s 65
4.4 E olu ion o he ol age e o s 65
4.5 Compa ison o simula ion esul s o he SOC e olu ions be ween he con olle
p oposed in [1] (a) and ou hyb id con ol p oposal (b) 66
4.6 Compa ison o simula ion esul s o he ac i e powe e olu ions be ween he
con olle p oposed in [1] (a) and ou hyb id con ol p oposal (b) 67
5.1 Dis ibu ed con ol scheme o discha ging a e and eac i e powe consensus
in an islanded AC mic og id 71
5.2 Mic og id con igu a ion scheme 81
5.3 Communica ion g aph among he BESSs 82
5.4 E olu ion o he SOC e e ences Ψ ,i, he SOCs Ψiand he ac i e powe s Pi,
o i={1,2,3,4,5,6}, when a load o 300W and 60VA is disconnec ed a T1
and hen i is connec ed back a T2. In T3α1,2= 0 83
5.5 E olu ion o he eac i e powe e e ences Q ,i and he eac i e powe s Qi
o i={1,2,3,4,5,6}, when a load o 300W and 60VA is disconnec ed a T1
and hen i is connec ed back a T2. In T3α1,2= 0 84
5.6 E olu ion ˜
Ψi,˜
Pi,˜
Qiin Scena io 1 o i={1,2,3,4,5,6}, when a load o
300W and 60VA is disconnec ed a T1and hen i is connec ed back a T2. In
T3α1,2= 0 84
5.7 E olu ion o ol age e o s, ol ages, cu en s and du y cycles o con e e s o
he i s 0.2s in Scena io 1 85
5.8 E olu ion o he SOC e e ences Ψ ,i, he SOCs Ψiand he ac i e powe s
Pi, o i={1,2,3,4,5,6}, when he BESS1is disconnec ed a T1and hen
connec ed back a T286
5.9 E olu ion o he eac i e powe e e ences Q ,i and he eac i e powe s Qi
o i={1,2,3,4,5,6}, when he BESS1is disconnec ed a T1and hen con-
nec ed back a T286
5.10 E olu ion ˜
Ψi,˜
Pi,˜
Qi o Scena io 2 87
Lis o Figu es 129
5.11 E olu ion o ol age, cu en and du y cycle o con e e s o 1.45s o 1.95s o
Scena io 2 87
6.1 Phase po ai o he supe c i ical hop bi u ca ion o pa ame e λ93
6.2 E olu ion o he sys em in he s able egion when λ≤0and in he uns able
egion when λ > 093
6.3 Bi u ca ion diag am o a supe c i ical Hop bi u ca ion 94
6.4 Phase po ai o he subc i ical hop bi u ca ion o pa ame e λ95
6.5 E olu ion o he sys em in he s able egion when λ≤0and in he uns able
egion when λ > 095
6.6 Bi u ca ion diag am o a subc i ical Hop bi u ca ion 95
6.7 Elec ical ci cui o a h ee-phase in e e wi h an LC il e 97
6.8 dq ans o ma ion o a ol age sou ce se . (a) O iginal ci cui . (b) dq ans-
o med ci cui 97
6.9 dq ans o ma ion o a induc o se . (a) O iginal ci cui . (b) dq ans o med ci cui 98
6.10 dq ans o ma ion o a capaci o se . (a) O iginal ci cui . (b) dq ans o med ci cui 98
6.11 dq ans o ma ion o a esis o se . (a) O iginal ci cui . (b) dq ans o med ci cui 99
6.12 CPL block diag am. (a) Con e e + load. (b) Equi alen ci cui 99
6.13 Nega i e impedance beha io o CPLs 100
6.14 Cu en - ol age cha ac e is ic o a load con e e as a CPL 100
6.15 Elec ical ci cui o a islanded AC mic og id wi h a CPL 102
6.16 Bi u ca ion diag am in he {id
L1,Po}plane. (a) Bi u ca ion poin s and equi-
lib ium s abili y. (b) Uns able limi cycle and egion o a ac ion o he s able
equilib ium 105
6.17 CPL powe e olu ions o powe e e ence s eps a ound bi u ca ion poin H1106
6.18 Cu en iL1e olu ions in abc and dq coo dina es a ound bi u ca ion poin H1107
6.19 Bus ol age e olu ions in abc and dq coo dina es a ound bi u ca ion poin H1107
6.20 CPL powe e olu ions o powe e e ence s eps a ound bi u ca ion poin H2108
6.21 Cu en iL1e olu ions in abc and dq coo dina es a ound bi u ca ion poin H2108
6.22 Bus ol age e olu ions in abc and dq coo dina es a ound bi u ca ion poin H2109
6.23 CPL ime cons an expe imen al es . (a) Connec ion scheme. (b) Impe ix so -
wa e in e ace. (c) In e ace o he CPL con ol so wa e. 110
6.24 CPL cu en measu emen s o ime cons an calcula ion 110
6.25 Expe imen al se up o an islanded AC mic og id wi h a CPL 111
6.26 E olu ion o mic og id ol ages ( abc
C1and abc
C2), cu en s (iabc
L1and iabc
L2) and
powe s P1and P2unde s eps o cons an esis o s 112
6.27 E olu ion o mic og id a iables a ound bi u ca ion poin H1. (a) CPL powe
and cu en id
L1. (b) ol ages abc
C1and abc
C2, cu en s iabc
L1,iabc
L2and iabc
Load 112
6.28 E olu ion o Po( op) and id
L1(bo om) o 3 di e en ope a ion poin s unde Vn
changes 113
6.29 Compa ison o Poand cu en id
L1 alues o 2 ope a ion poin s be ween ex-
pe imen al esul s and bi u ca ion diag am o Fig 6.16(b) 114
6.30 Zoom in on he capaci o ol ages, induc o cu en s and CPL cu en a ound
o ope a ion poin 1 115
130 Lis o Figu es
6.31 Zoom in on he capaci o ol ages, induc o cu en s and CPL cu en a ound
o ope a ion poin 3 115
Lis o Tables
3.1 Mic og id and d oop con ol pa ame e s 42
3.2 In e e pa ame e s 42
4.1 Consensus algo i hm and d oop con ol pa ame e s 63
4.2 In e e pa ame e s 63
5.1 Mic og id and d oop con ol pa ame e s 81
5.2 In e e pa ame e s 82
6.1 Mic og id pa ame e s 104
131
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