Analysis of Droop Controlled Parallel Inverters in Islanded Microgrids

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1 Downloae from vn.aau. on: januar 6 19 Aalorg Unverstet Analyss of Droop Controlle Parallel Inverters n Islane Mrogrs Maran Valero; Vasa Franeso; Zapata Josep Mara Guerrero Pulshe n: Proeengs of the 14 IEEE Internatonal Energy Conferene ENERGYCON) DOI ln to pulaton from Pulsher): 1.119/ENERGYCON Pulaton ate: 14 Doument Verson Early verson also nown as pre-prnt Ln to pulaton from Aalorg Unversty Ctaton for pulshe verson APA): Maran V. Vasa F. & Guerrero J. M. 14). Analyss of Droop Controlle Parallel Inverters n Islane Mrogrs. In Proeengs of the 14 IEEE Internatonal Energy Conferene ENERGYCON) pp ). IEEE Press. I E E E Internatonal Energy Conferene. ENERGYCON proeengs General rghts Copyrght an moral rghts for the pulatons mae aessle n the pul portal are retane y the authors an/or other opyrght owners an t s a onton of aessng pulatons that users reognse an ae y the legal requrements assoate wth these rghts.? Users may ownloa an prnt one opy of any pulaton from the pul portal for the purpose of prvate stuy or researh.? You may not further strute the materal or use t for any proft-mang atvty or ommeral gan? You may freely strute the URL entfyng the pulaton n the pul portal? Tae own poly If you eleve that ths oument reahes opyrght please ontat us at vn@au.aau. provng etals an we wll remove aess to the wor mmeately an nvestgate your lam.

2 Analyss of Droop Controlle Parallel Inverters n Islane Mrogrs V. Maran #1 F. Vasa # J. M. Guerrero 1 # Department of Engneerng Unversty of Sanno Pazza Roma 1 81 Benevento Italy 1 valero.maran@unsanno.t franeso.vasa@unsanno.t Department of Energy Tehnology Aalorg Unversty Pontoppanstræe Aalorg Denmar 1 joz@et.aau. Astrat Three-phase roop ontrolle nverters are wely use n slane mrogrs to nterfae strute energy resoures an to prove for the loas atve an reatve powers eman. The assessment of mrogrs stalty affete y the ontrol an lne parameters s a strngent ssue. Ths paper shows a systemat approah to erve a lose loop moel of the mrogr an then to perform an egenvalues analyss that hghlghts how the system s parameters affet the stalty of the networ. It s also shown that y means of a sngular perturaton approah the resultng reue orer system well approxmate the orgnal full orer system. Inex Terms Power Systems Islane Mrogrs Small- Sgnal Stalty Analyss Sngularly Perture Systems. Fg. 1. loas. u rst a rst oa rst a Z a Z rst rst Z rst o u rst AC mrogr n slane-moe wth two nverters an two loal I. INTRODUCTION Mrogrs onsst of an nteronneton of strute energy resoures storages an loas arrange va varous topologes. Tpally eah of the aforementone elements uses nverters to nterfae to the networ n orer to ensure the proper ehavor of the mrogr n the west range of the worng ontons. A wely use ontrol tehnque for the strute energy resoures nverters s the roop ontrol eause t allows to properly support for the atve an reatve power loas eman [1] [] [3]. Beause of the nonlnearty of the ontrol tehnque even n presene of mrogrs wth few nverters the lose loop moel of suh systems may eome very omplex. Neverthless a prese an rgorous assessment of the stalty of the mrogrs s very mportant. Many papers show through a small-sgnal approah an expermental valatons that the frequeny roop ontrol parameter whh s tpally esgne n orer to ensure the proper loa sharng strongly affets the stalty of the system [4] [5] [6] [7]. More theoretal analyss of the nfluene of the ontrol parameters on the stalty of strute generaton systems wth roop ontrolle nverters an e foun n [8] [9] [1]. In ths paper a systemat approah to the moelng of three-phase AC mrogrs wth nverters uner voltage an frequeny roop ontrols s presente. In Se. II a moel of a three-phase AC mrogr onsstng of two nverters onnete va a resstve-nutve lne an of two loal resstve-nutve loas s otane. Then n Se. III the lose-loop moel an a orresponng reue orer moel etermne va a sngular perturaton approah are presente. Numeral analyss of the full orer moel showng the tme evolutons of the urrents the powers the angle elay etween the two nverters the nverter frequenes s presente n Se. IV. The moels egenlous for nreasng frequeny ontrol parameter an power flters tme onstants are nvestgate. By omparson t s hghlghte that the reue orer moel s ale to pret the nstalty for the same value of the frequeny ontrol parameter for whh the full orer moel s unstale. Se. V onlues the paper. II. THREE-PHASE SYSTEM MODELING An equvalent rut of the three-phase AC mrogr uner nvestgaton s epte n Fg. 1. Two nverters are onnete y means of a alane resstve-nutve lne wth phase mpeanes Z = R jωl. The two nverters have orresponng alane resstve-nutve loal loas wth phase mpeanes Z a = R a jωl a an Z = R jωl respetvely. We assume that the -th nverter represente wth the orresponng loal loa y the rut epte Fg. proves the three-phase voltage sequene u rst = u r u s u t gven y u r u s = U os θ U osθ /3π) 1) U osθ /3π) u t wth = a an where U θ are the tme-varyng voltage ampltue an the tme-varyng angle of the -th nverter

3 etermne wth respet the -th loal referene frame. Sne we o not assume ommunaton etween the two nverters the orresponng ontrols are not assume to e synhronze n phase.e. θ a ) an θ ) are nepenent. The three-phase -th nverter output urrent sequene rst o = r o s o t o an loal loa urrent sequene rst = [ r s ] t are efne as r o s o = I o osθ ϕ ) I o osθ ϕ /3π) ) t o I o osθ ϕ /3π) an r s = I osθ φ ) I osθ φ /3π) 3) t I osθ φ /3π) wth = a an where I o I ϕ an φ are the tmevaryng output urrent ampltue the tme-varyng loa urrent ampltue the tme-varyng angle elay of the -th output nverter urrent an the tme-varyng angle elay of the -th loa urrent wth respet to the orresponng voltages nstantaneous phases θ. Furthermore the three-phase lne urrent rst = [ r s ] t an e wrtten as r s = I os θ I osθ /3π) t I osθ /3π) = I osθ a x a ) I osθ a x a /3π) 4a) I osθ a x a /3π) where θ s the tme-varyng angle etermne wth respet to the loal lne referene frame an x a = θ θ a s the angle elay etween the angle of u a an the angle of the lne urrent. Eah -th three-phase voltage an urrent s represente n a loal β referene frame y means of the well nown Clare transformaton [11] T = 1 1/ 1/ 3/ 3/ = [ ] T 3to 3 3 1/ 1/ 1/ 1/ 1/ 1/ 5) wth T 3to R 3 eng the matrx rawn from T R 3 3 tang the frst an the seon row. For alane three-wre systems the thr row an e neglete. The nverse Clare transformaton s efne as T 1 = 3 /3 /3 1/3 3/3 /3 1/3 = 3 /3 T to3 /3 3/3 /3 /3 6) wth T to3 R 3 eng the matrx rawn from T 1 R 3 3 tang the frst an the seon olumn. By applyng the Krhhoff voltage laws one otans L t rst = R rst u rst 7a) L t rst = R rst u rst a u rst 7) where = a. By representng 7) n the orresponng -th β loal referene frame that s y susttung to eah voltage an urrent n 7) the orresponng representaton n terms of β omponents gven y y rst j = T to3 y β j 8) where j {a } an y {u o } t follows an L t β = R β u β 9a) L t β = R β u β a u β 9) u β u = u β = U os θ 1a) U sn θ β o = o β = Io osθ ϕ ) 1) I o o snθ ϕ ) β = β = I osθ φ ) 1) I snθ φ ) β = = I osθ a x a ). 1) I snθ a x a ) β For eah -th voltage an urrent expresse wth respet to the orresponng -th referene frame efne then the -th rotaton matrx os θj sn θ Γθ j ) = j 11) sn θ j os θ j where j {a }. Let us apply 11) to eah -th voltage an urrent n 9) n orer to eal wth emoulate voltages an urrents that s efne y β j = Γθ j ) y q j 1) wth y {u o }. Therefore y susttutng 1) n 9) t follows L t q = R q L Ω t θ q uq 13a) L t q = R q L Ω t θ a q u q a Γδ) 1 u q 13) where = a δ = θ a θ an we use Γθ ) 1 1 t Γθ ) = 1 t θ = Ω t θ 14) an u q q q = u = U u q = 15a) = I os φ q = 15) I sn φ I os x = a. 15) I sn x a q The -th nverter output urrent rst o efne n ) an e wrtten n the rotatng q referene frame y susttutng ) n 1) wth 8) gettng q o = Γθ ) T 3to rst o 16)

4 r o s o where = a Ω s efne n 14) an u q an q o are the -th q voltage an output urrent gven y 15). Then y susttutng 18) n 1) an then n ) t follows that the -th atve power P an reatve power Q are gven y u r u s u t t o Fg.. Equvalent rut of the -th three-phase nverter an the orresponng loal loa. where = a. Let Γ δ) = t L R { I = a Γδ) = s L R r L R 17) where I s the entty matrx of sutale mensons. Then the Krhhoff urrent laws allow to wrte q o = q Γ δ) 1 q 18) wth = a. The ynam moel 13) together wth the frequeny an voltage roop ontroller equatons wll e use n next Setons for the lose loop stalty analyss. III. CLOSED LOOP MODEL The lose loop moel of 13) s otane y etermnng the ampltue U an the frequeny /t)θ of eah -th voltage wth = a aorng to the frequeny an voltage roop laws t θ =ω m P P ) U =Ū n Q Q ) 19a) 19) where ω s the referene frequeny of eah nverter Ū s the -th voltage referene m an n are the -th frequeny an voltage roop oeffents P an Q are the -th atve an reatve power referenes P an Q are the -th atve an reatve powers prove y the -th nverter ownstream the -th measurng flter an = a. Eah -th P an Q s gven y τ P τ Q t P = P p t Q = Q q a) ) where τ P an τ Q are the tme-onstants of the -th frst orer measurng flter an p an q are the -th nstantaneous atve an reatve powers efne as p =u q ) q o q =u q ) Ω q o 1a) 1) τ P τ Q t P = P u q ) q Γ δ) 1 q ) a) t Q = Q u q ) Ω q Γ δ) 1 q ). ) Therefore the lose loop moel of the mrogr s otane y susttutng 19) n 15a) an then n 13) an ) 19a) n 13) an reallng that δ = θ a θ : L R t q = I ω L ) Ω q R L L R t q = τ P τ Q Ωm R P P ) q 1 [Ū n R Q 1 Q )] 3a) I ω L ) Ω q L Ωm a R R P a P a ) q 1 R =a t P = P [ 1 Γ δ) 1 [Ū n Q Q )] ] q [Ū n Q Q )] t Q = Q [ 1 ] Ω q Γ δ) 1 q ) [Ū n Q Q )] t δ =m a P a P a ) m P P ) Γ δ) 1 q ) [ 1 ] 3) 3) 3) 3e) where = a an Γ δ) an Ω are efne n 17) an 14) respetvely. Typal values of the loas nutanes L an resstanes R where = a allow to assume L L R R L τ P R L τ Q R. 4a) 4) 4) Therefore y applyng the sngular perturaton tehnque [1] to 3) from 3a) one otans = I ω L ) Ω z q R 1 [Ū n R Q 1 Q z )] 5) where = a z an Q z nate the urrents an the reatve power of the reue orer system respetvely. Note that the assumpton of small ratos L /R oes not mply that ωl /R ) s small too. The reue orer moel s otane y susttutng the soultons of 5) nto 3)

5 3e): L R t zq = τ P τ Q I ω L ) Ω z q L R 1 R =a t P z = P z [ 1 Ωm a R P a P za ) z q [ 1 ] Γ δ) 1 [Ū n Q Q z )] ] z q [Ū n Q Q z )] t Q z = Q z [ 1 ] Ω z q Γ δ z ) 1 z q ) [Ū n Q Q z )] t δ z =m a P a P za ) m P P z ) Γ δ z ) 1 z q ) 6a) 6) 6) 6) where = a an where δ z an P z are the angle an the atve power of the reue orer moel. In next Seton t s shown that the 7-th orer moel 6) s ale to apture the stalty property of the 11-th orer lose loop moel 3). IV. SIMULATION RESULTS The smulatons have een arre out y onserng the followng realst values of the ontrol loas an lne parameters [13] for the mrogr epte n Fg. 1: m a = ra/sw n a = V/VAr τa P =.17 s τa Q =.17 s Pa = 86 W Qa = 384 VAr Ū a = 17 V R a = 13 Ω L a = 16 mh m = ra/sw n = V/VAr τ P =.17 s τ Q =.17 s P = 75 W Q = 375 VAr Ū = 13 V R = 5 Ω L = 35 mh R =.5 Ω L = 8 mh ω = π 6 ra/s. An equlrum pont of the lose loop system 3) s gven y eq a = 8.1 A qeq a = 3.7 A Pa eq = 86.3 W Q eq a = VAr eq = 4.1 A qeq =.1 A P eq = 75.3 W Q eq = VAr eq = 1.7 A qeq =.7 A δ eq =.368 ra. The egenvalues of the system s Jaoan ompute aroun suh equlrum pont are λ 1 = ± j λ 34 = ± j λ 56 = 6.7 ± j λ 78 =.98 ± j4.8 λ 9 = 6.8 λ 1 = 6. λ 11 = a) 7) 7) 7) 7e) 7f) 7g) whh show that the system s loally stale for the partular hoe of the lne ontrol an loa parameteres. Fg. 3 an Fg. 4 show the tme oman evoluton of the loal loas ret an quarature urrents an the tme oman evoluton of the nverters an lne ret an quarature urrents respetvely. Fg. 4 partularly shows the presene of fast an slow moes n the nverters an lne urrents ynams. By omparng Fg. 3 an Fg. 4 one an note how the loas urrents evolve on the same small temporal sale of the fast moes whh affet the nverters an lne urrents. In Fg. 5 are reporte the tme evolutons of the atve powers P a P an reatve powers Q a Q of eah nverter. Fg. 6 shows the tme oman evoluton of δ = θ a θ. The nverters a q a q [A] Fg. 3. Tme oman evoluton of the ret omponents a sol lne) an ashe lne) of the loa a an urrents an quarature omponents q a sol th lne) an q ashe th lne) of the loa a an urrents for zero ntal ontons for m a = start wth the same angle that s δ) = an n orer to prove for the requre atve an reatve powers aorng to the orresponng referene values at steay-state a nonzero angle fferene s neee. Clearly ths epens also y the loal loas. In Fg. 7 t s shown the tme evoluton of the nstantaneous frequenes /t)θ a sol lne) an /t)θ ashe lne) etermne y the frequeny roop laws. Eah /t)θ starts from the orresponng ntal value gven y ω m P where = a. Then oth /t)θ a an /t)θ onverge to the same steay-state value gven y the frequeny referene ω = π 6 ra/s sne also P a an P onverge to P a an P respetvely. The stalty of the lose loop system s nfluene y the ontrol parameters. Fg. 8 shows the egenlous for m a [ ]. For eah m a the new equlrum pont gven y the soluton of 3) otane settng all the ervatve terms to zero has een ompute an then use to etermne the egenvalues of the system s Jaoan. As t s shown y nreasng the frequeny ontrol parameter m a some of the egenvalues eome postve real an for m a the system s unstale. Fg. 9 shows the egenlous for τa P [.17.17] s. Partularly the fnal value of the atve power flter tme-onstant τa P of the nverter a s τa P =.17 s that s the ut-off frequeny of suh atve power flter s approxmately an orer of magntue lower than the nomnal frequeny ω. The orresponng egenlous s rawn for m a = 5 1 an for τa P =.36 s some egenvalues eome postve real. By omparng ths result wth those n [14] where t s shown that f no flters at all are use the system s unstale for m a =.4 1 t s lear that y means of the atve an reatve power flters the system loal stalty s mprove for hgher value of m a. On the q a a q

6 oa o q oa q o q [A] 5 oa o q q o δ [ra].368 q oa Fg. 4. Tme oman evoluton of the ret omponents oa sol lne) o ashe lne) otte lne) an quarature omponent q oa sol th lne) q o ashe th lne) q otte th lne) of the nverters an lne ret an quarature urrents for zero ntal ontons for m a = Fg. 6. Tme oman evoluton of the angle elay δ exstng etween the voltage u rst a an the voltage urst for zero ntal ontons for m a = P a Pa P [W]; Qa Q [VAr] 6 4 P Q a Q /t)θa /t)θ [ra/s] /t)θ /t)θ a Fg. 5. Tme oman evoluton of the atve powers P a sol lne) P ashe lne) an reatve powers Q a sol th lne) Q ashe th lne) prove y the nverters for m a = Fg. 7. Tme oman evoluton of the frequeny /t)θ a sol lne) an /t)θ ashe lne) of the voltage u a an the voltage u prove y the mrogr nverters for m a = other han omparng the egenvalues aroun the equlrum pont wth those showe n [14] n ase the flters are use the system response s slower for the same value of m a. Fg. 8 an 7) show that for m a = the lnearze ynam s approxmately gven y a seventh orer system otane y negletng the fast egenvalues λ 1 an λ 34. Ths s also justfe y the sngular perturaton approah use n Se. III. As Fg. 1 shows the egenlous of the lnearze reue orer moel system prets the nstaltes for the same value of m a for whh the full orer system s unstale. Im Re ma = Fg. 8. Egenlous of the lnearze system aroun the equlrum pont for m a [ ].

7 Im Re τa P =.38 s Fg. 9. Egenlous of the lnearze system aroun the equlrum pont for τ P a [.17.17] s an ma = 5 1. Im Re ma = Fg. 1. Egenlous of the reue orer lnearze system aroun the equlrum pont for m a [ ]. V. CONCLUSIONS REFERENCES [1] J. M. Guerrero M. Chanorar T.-L. Lee an P. C. Loh Avane Control Arhtetures for Intellgent Mrogrs-Part I: Deentralze an Herarhal Control IEEE Trans. on Inustral Eletrons vol. 6 no. 4 pp [] Q.-C. Zhong Roust Droop Controller for Aurate Proportonal Loa Sharng Among Inverters Operate n Parallel IEEE Trans. on Inustral Eletrons vol. 6 no. 4 pp [3] Y. Mohame an E. El-Saaany Aaptve Deentralze Droop Controller to Preserve Power Sharng Stalty of Parallele Inverters n Dstrute Generaton Mrogrs IEEE Trans. on Power Eletrons vol. 3 no. 6 pp [4] N. Pogau M. Proanov an T. C. Green Moelng Analyss an Testng of Autonomous Operaton of an Inverter-Base Mrogr IEEE Trans. on Power Eletrons vol. no. pp [5] I. Seran an C. Marnesu Enhane Control Strategy of Three-Phase Battery Energy Storage Systems for Frequeny Support n Mrogrs an Wth Unnterrupte Supply of Loal Loas IEEE Trans. on Power Eletrons no. 99 p [6] L. Wang X. Q. Guo H. R. Gu W. Y. Wu an J. M. Guerrero Prese Moelng Base on Dynam Phasors for Droop-Controlle Parallel- Connete Inverters n Pro. of the 1st Internatonal Symposum on Inustral Eletrons Hangzhou Chna 1 pp [7] S. Taataaee H. R. Karshenas A. Bahsha an P. Jan Investgaton of Droop Charatersts an X/R Rato on Small-Sgnal Stalty of Autonomous Mrogr n Pro. of the n Power Eletrons Drve Systems an Tehnologes Conferene Tehran Iran Feruary 11 pp [8] J. W. S.-Poro F. Dorfler an F. Bullo Synhronzaton an Power Sharng for Droop-Controlle Inverters n Islane Mrogrs Automata vol. 49 no. 9 pp [9] V. Maran an F. Vasa Stalty Analyss of Droop Controlle Inverters va Dynam Phasors an Contraton Theory n Pro. of the 1th European Control Conferene Zurh Swtzerlan July 13 pp [1] Partal Contraton Analyss for Droop Controlle Inverters n Pro. of the 3r Internatonal Conferene on Systems an Control Algers Algera Otoer 13. [11] E. Clare Crut Analyss of AC Power Systems. New Yor: Wley 1943 vol. 1. [1] H. K. Khall Nonlnear Systems 3r e. New Jersey: Prente Hall. [13] E. A. A. Coelho P. C. Cortzo an P. F. D. Gara Small-Sgnal Stalty for Parallel-Connete Inverters n Stan-Alone AC Supply Systems IEEE Trans. on Inustry Applatons vol. 38 no. pp [14] V. Maran F. Vasa an J. M. Guerrero Dynam-Phasor-Base Nonlnear Moellng of AC Islane Mrogrs Uner Droop Control n Pro. of 11th Internatonal Mult-Conferene on Systems Sgnals an Deves Barelona Span Feruary 14. The propose approah to moel two roop ontrolle nverters formng a mrogr allows to ompute the equlrum ponts of the lose loop system an to arry out an egenvalue analyss showng that for some values of the frequeny roop ontrol parameter m a the system eomes unstale. The tehnque use allows also to assess that the nomnal value of m a the large sgnals ynams are well approxmate y those of a reue orer moel erve y negletng the loas ynams that s y applyng a sngular perturaton approah to the orgnal moel. Inee the reue orer moel stll allows to etermne the value of m a whh maes the system unstale. The loal stalty propertes of the system are mprove y the presene of power flters also for qute large value of m a even though the ynam response s slower than that of the same system wthout power flters. Smlar onseratons an e operate wth fferent values of the remanng parameters.