Resonance Analysis in Parallel Voltage-Controlled Distributed Generation Inverters

Transcription

1 Reonance Analy n Parallel Voltage-Controlled Dtrbuted Generaton Inverter Xongfe Wang Frede Blaabjerg and Zhe Chen Department of Energy Technology Aalborg Unverty Pontoppdantraede Aalborg Denmark Emal: {xwa fbl zch}@et.aau.dk Wemn Wu Department of Electrcal Engneerng Shangha Martme Unverty Shangha Chna wmwu@cle.hmtu.edu.cn Abtract Thank to the fat repone of the nner voltage and current control loop the dynamc behavor of parallel voltagecontrolled Dtrbuted Generaton (DG) nverter not only rele on the tablty of load harng among them but ubject to the nteracton between the voltage control loop of the nverter and the remanng ytem dynamc. Th paper addree the later nteracton and the conequent reonance through the frequency-doman analy of the nverter output mpedance and the remanng equvalent network mpedance. Furthermore mpact of the vrtual output mpedance loop and the voltage feedforward loop n the current controller are evaluated baed on uch an mpedance nteracton analy. Smulaton reult are preented to confrm the valdty of the theoretcal analy. I. INTRODUCTION Drven by the emergng actve dtrbuton network n the development of mart grd the nverter-nterfaced Dtrbuted Generaton (DG) ytem have been undergong a fat growth n the power grd [1] [2]. Thee DG nverter are expected to operate n both current- and voltage-controlled mode thereby provdng more relable and effcent electrcty ervce to the cutomer [3]. Conequently the landed network that are domnated by multple DG nverter are becomng mportant component of the mart dtrbuton grd [4]. The ncreang concern over the control and tablty of uch nverter-baed landed ytem have been raed [5]. The actve power-frequency (P-ω) droop that mmc the peed-governng mechanm of ynchronou generator and the reactve power-voltage (Q-V) droop are uually ued for the load harng n the parallel DG nverter [6]. Furthermore to overcome the power couplng caued by a hgh R/X rato of the low-voltage dtrbuton lne everal mproved droopbaed power control cheme have been reported [7]. Among them the vrtual output mpedance loop that mlar to the load compenator n the exctaton ytem of the ynchronou generator how uperor performance [8]. In order to properly degn the droop coeffcent and the vrtual output mpedance mall-gnal tablty analy of the parallel voltage-controlled nverter have been well dcued n [9] [1]. However notce that the dynamc behavor of the nner current and voltage control loop are neglected n thoe analye due to the well eparaton between the bandwdth of the outer power control loop and the nner control loop. A a reult the nteracton between the output voltage control loop of the DG nverter and the remanng network dynamc are often overlooked [11]. Unlke tradtonal power ytem where ynchronou generator uually have large tme contant the much fater repone of the voltage and current control loop n the DG nverter tend to reult n the addtonal reonance and mall-gnal ocllaton at the frequence hgher than the ytem fundamental frequence. Therefore there an urgent need to explore the dynamc nteracton between the nner control loop of the parallel DG nverter. In th paper the dynamc nteracton between the output voltage control loop of the parallel voltage-controlled DG nverter are addreed. The potental voltage reonance and untable ocllaton caued by uch nteracton are dentfed through the mpedance-baed tablty analy. The nfluence of the vrtual output mpedance loop and the output voltage feedforward term n the nner current control loop are aeed n the frequency-doman. Fnally mulaton are performed to valdate the theoretcal analy reult. II. MODELING OF PARALLEL DG INVERTERS Fg. 1 repreent an landed three-phae network where two parallel nverter-nterfaced DG unt are connected va the dtrbuton feeder to the common load bu repectvely. Contant DC-lnk voltage of the DG nverter are aumed. Fg. 2 llutrate the control block dagram for the -th DG nverter (=1 2). The multloop control cheme employed ncludng 1) the nner voltage and current control loop 2) the ntermedate vrtual output mpedance loop and 3) the outer droop-baed power controller. Snce the nteracton between the nner voltage and current control loop of DG nverter are of the man concern the low-frequency power ocllaton caued by the dynamc of the droop control are dregarded. In the nner control loop the proportonal current controller adopted for the over-current protecton and a better LC-flter reonance dampng [12] and the Proportonal Reonant (PR) voltage controller ued for the zero teady-tate error.

2 Fg. 1. An landed three-phae network wth two nverter-nterfaced DG unt and a common load. Energy ource Inner voltage and current control loop Gv IL Kpc PWM DG V dc IL Vo (feedforward term) IL Vo Vo Vof Ref. Generator Voc V V Droop controller V V P Q P Q P & Q Cal. Io Vo Io L C Z l Fg. 3. Smplfed nner current control loop for the -th DG nverter. ZV () Vrtual output mpedance loop Common load bu Fg. 2. Control block dagram for the -th DG nverter (=1 2). It known that the termnal behavoral model of the DG nverter eental to perform an mpedance-baed tablty analy [13]. Snce the DC voltage aumed to be contant the DC nverter can be een a a lnear ytem for mall-gnal tablty analy [14]. From Fg. 2 t oberved that the output LC-flter can be modeled by a two-nput two-output ytem a follow Fg. 4. Smplfed nner voltage and current control loop for the -th DG nverter. where G cl () the cloed-loop gan of the current control loop and G Ioc () denote the cloed-loop effect of the load current whch are gven by T G G G 1 T ( ) 1 T ( ) c Io cl Ioc c c (5) IL G () () o () Io Y I L V Z G V o o Vo PWM Z 1 G Y Z Z Z Z C Io L L C L C Z Z Z Z G Z Z Z Z L C C o Vo L C L C (1) (2) (3) T K G Y (6) c pc d L 2 1.5T 1.75T T d () T 1.5T 12 G e (7) where T c () the open-loop gan of the current control loop. G d () the approxmated 1.5 amplng perod (T ) delay. By clong the output voltage control loop the nner voltage and current control loop are mply repreented n Fg. 4. Thu the output voltage control loop can be modeled a where V PWM () the output voltage of the -th nverter Z C () and Z L () are the mpedance for the output capactor and nductor repectvely. Thu the nner current control loop can be mplfed a hown n Fg. 3. Suppong that wthout the voltage feedforward term frt the dynamc behavor of the current control loop can be gven a I G I G I (4) L cl L Ioc o V G V Z I (8) o clv o ov o T Z G Z 1 G v C clv ov Ioc 1 Tv ( ) 1 Tv ( ) v v cl C (9) T G G Z (1)

3 where G clv () the cloed-loop gan of the voltage control loop Z ov () the cloed-loop output mpedance and T v () the open-loop gan of the voltage control loop. G v () the PR voltage controller whch gven by [15] G K K rv c v pv 2 2 c (11) Conderng the vrtual output mpedance loop Z V () the voltage control ytem can be derved by Fg. 5. Cloed-loop model of the landed three-phae network. V G V Z G Z I (12) o clv o ov clv V o Z Z G Z (13) tov ov clv V where Z tov () the total cloed-loop output mpedance. It can be een that the actual effect of the vrtual output mpedance affected by the bandwdth of the nner control loop and the vrtual mpedance loop ha no effect on the cloed-loop gan of the nner control loop. On the other hand takng the output voltage feedforward term nto account the dynamc behavor of the current control loop modfed a follow T T c c (14) 1 GVo ( ) Gd ( ) G cl T c 1 T ( ) GIo Zo Gd YL G Ioc 1 T ( ) 1 T 1 G G c c c d Vo (15) (16) Subttutng (14) (15) and (16) nto (9) and (1) the termnal behavoral model of the -th nverter modfed a V G V Z I (17) o clv o ov o T Z G Z 1 G v C clv ov Ioc 1 T v ( ) 1 T v ( ) v v cl C (18) T G G Z (19) It worthy to note that the output voltage feedforward term affect both the cloed-loop gan of the nner control loop and the cloed-loop output mpedance whch markedly dfferent from the vrtual output mpedance loop. Followng the dervaton of the termnal behavoral model of the -th DG nverter the cloed-loop model for the landed three-phae network hown n Fg. 1 thu bult a hown n Fg. 5. The equvalent load mpedance Z load () can be ued to repreent the remanng network dynamc whch nclude the cloed-loop output mpedance of the other DG nverter and the Fg. 6. Mnor feedback loop for the -th DG nverter. dtrbuton lne a well a the common load. Conequently the nteracton between the nner control loop of the -th DG nverter and the remanng network dynamc can be modeled by a mnor feedback loop a hown n Fg. 6. The loop gan of uch a mnor feedback loop T m () can be gven by T m Z tov Zload (2) Thu the voltage reonance caued by uch nteracton can be clarfed va the frequency repone of the loop gan T m (). III. IMPEDANCE-BASED STABILITY ANALYSIS On the ba of the derved cloed-loop ytem model the mpedance-baed tablty analy performed n frequencydoman. For the ake of mplcty only the balanced network condered here and the two DG nverter are aumed to have the ame flter contant and controller parameter. Table I ummarze the ytem electrcal contant and Table II lt the man controller parameter of the DG nverter. Frt uppong that nether the vrtual output mpedance loop nor the output voltage feedforward term appled the frequency repone of the open-loop gan for the nner voltage control loop T v () hown n Fg. 7. It een that a table cloed-loop termnal behavor of the -th DG nverter acro the output capactor obtaned. Then conderng the loadng effect of the equvalent load mpedance Z load () the nteracton between the -th nverter and the remanng network dynamc aeed. Fg. 8 how the frequency repone of Z ov () and Z load (). The phae dfference at the nterecton pont of two mpedance mply the phae margn of the mnor feedback loop. The nterecton pont where the phae dfference are larger than 18 denote the reonance frequence. From Fg. 8 notce that Z o () behave a a capactance at hgh frequence wherea Z load () become an nductance due to the feeder mpedance. A a conequence the ere RLC crcut formed n the mnor feedback loop. Furthermore t clear that the phae dfference at the nterecton pont 177 Hz larger than 18 degree whch repreent the reonance

4 4 4 Magntude (db) Magntude (db) Zov Zload (.45 mh R/X=3) Zload (.9 mh R/X=3) Zload (1.8 mh R/X=3) Fg. 7. Frequency repone of the open-loop gan of the nner voltage control loop for the -th DG nverter Fg. 9. Frequency repone of Z load () wth the dfferent dtrbuton lne mpedance and Z ov (). Magntude (db) Zov Zload Magntude (db) Zov Ztov (Rv=.6 ohm) Ztov (Rv=1.2 ohm) Ztov (Rv=2.4 ohm) Fg. 8. Frequency repone of Z ov () and the equvalent load mpedance Z load (). frequency. Hence even though a table termnal behavor of ngle nverter obtaned the nteracton between the nner control loop of parallel nverter can tll lead to mall-gnal ntablty problem. Fg. 9 compare the frequency repone of Z load () wth the dfferent dtrbuton lne mpedance and Z ov (). It can be oberved that the phae margn of the mnor feedback loop ncreaed wth the ncreae of the lne mpedance. Th fact mple that uch an undered reonance can be avoded by rehapng the nterconnectng mpedance between the parallel DG nverter. Under the gven common load and dtrbuton lne th can be acheved by changng the cloed-loop output mpedance of nverter. From (9) (13) and (18) t can be noted that there are three way to change the cloed-loop output mpedance of the -th nverter whch nclude 1) adjutng the open-loop gan of the voltage control loop 2) applyng the vrtual output mpedance loop and 3) ung the output voltage feedforward term n the current control loop. In [11] the dampng of reonance va reducng the open-loop gan of the voltage control ytem ha been dcued. Thu the dampng effect of the latter two control method are nvetgated n th work. Fg. 1 how the frequency behavor of the total cloedloop output mpedance Z tov () wth the dfferent vrtual output Fg. 1. Frequency repone of Z ov () and the equvalent load mpedance Z load (). mpedance and the orgnal output mpedance Z ov (). Notce that only the vrtual retance condered. It oberved that the total cloed-loop output mpedance ha very few change at the reonance frequency whch mple the vrtual output retance loop ha no effect on dampng of uch a reonance. In order to ee the effect of the output voltage feedforward term n the current control loop the frequency repone of the open-loop gan of the current control loop and the cloed-loop effect of load current are plotted n Fg. 11. Fg. 11 (a) how a comparon of the open-loop gan before and after ung the output voltage feedforward term. It clear that the reonance peak caued by the output LC-flter damped by the voltage feedforward term. Fg. 11 (b) compare the cloed-loop effect of the load current derved n (5) and (16). It can be een that the repone magntude reduced. Fg. 12 (a) compare the frequency repone of the openloop gan of the voltage control loop derved n (1) and (19). It een that the loop gan at the low frequence ncreaed wth the output voltage feedforward term. Fg. 12 (b) how the frequency behavor of the cloed-loop output mpedance n (9) and (18). It obvou that the reonance peak n Z ov () damped by the output voltage feedforward term. Fg. 13 depct the frequency repone of the cloed-loop output mpedance and the equvalent load mpedance wth the

5 TABLE I. SYSTEM ELECTRICAL CONSTANTS (SEE FIG.1) TABLE II. MAIN CONTROLLER PARAMETERS OF DG INVERTERS Electrcal Contant DC voltage (V dc1 = V dc2 ) Value 75 V Controller Parameter Value Samplng perod T 1 = T DG nverter (DG1 and DG2) Flter nductor (L 1 = L 2 ) Flter capactor (C 1 = C 2) 1.5 mh 25 μf Current controller K pc1 = K pc2 5 K pv1 = K pv2.6 Dtrbuton feeder (Z l1 = Z l2 ) Swtchng frequency (f w ) 1 khz Lne nductance (L l1 = L l2).45 mh R/X rato 3 PR voltage controller K rv1 = K pv2 1 ω c1= ω c2 8 rad/ ω 1π rad/ Common load (Z L ) Retance load (R L ) Inductance load (L L) 8 Ω 166 mh Actve power droop controller n 1 = n Reactve power droop controller m 1 = m Magntude (db) 5-5 Tc (no feedforward term) Tc' (wth feedforward term) Magntude (db) Tv (no feedforward term) Tv' (wth feedforward term) (a) (a) Magntude (db) 2-2 GIoc (no feedforward term) GIoc' (wth feedforward term) Magntude (db) Zov (no feedforward term) Zov' (wth feedforward term) (b) Fg. 11. Frequency repone of the current control loop wth and wthout the output voltage feedforward term. (a) Open-loop gan. (b) Cloed-loop effect of the load current. output voltage feedforward term n the current control loop. Compared to Fg. 8 t evdent that the phae dfference at the nterecton pont are maller than 18 whch ndcate that the reonance reultng from the nteracton between the nner control loop of the parallel DG nverter are effectvely damped by ung the voltage feedforward term. IV. SIMULATION RESULTS To valdate the above frequency-doman tablty analy the tme-doman mulaton of the landed network hown n (b) Fg. 12. Frequency repone of the voltage control loop wth and wthout the output voltage feedforward term. (a) Open-loop gan. (b) Cloed-loop output mpedance. Fg. 1 performed n MATLAB/Smulnk and the electrcal crcut bult wth the SmPowerSytem toolbox. The model parameter are referred to Table I and Table II. Fg. 14 how the mulated output voltage of the parallel nverter wthout ung the vrtual output mpedance loop and the output voltage feedforward term. The two DG nverter are connected n parallel at the ntant of.2. It evdent that the voltage reonance are when the nverter are connected n parallel. Fg. 15 gve the harmonc pectra for the reonant voltage where a good match wth the mpedance nteracton

6 4 Magntude (db) Zov' Zload' Fg. 13. Frequency repone of the cloed-loop output mpedance and the equvalent load mpedance wth the voltage feedforward term. 5 DG nverter 1 5 (a) DG nverter DG nverter 2 5 DG nverter Tme () Fg. 14. Smulated output voltage of the parallel nverter wthout ung the vrtual output mpedance loop and the output voltage feedforward term Tme () (b) Fg. 16. Smulated output voltage of the parallel nverter wth the ncreae of the dtrbuton lne mpedance. (a) L l =.9 mh. (b) L l =1.8 mh. 1. DG nverter DG nverter Fg. 15. Harmonc pectra for the reonant voltage hown n Fg. 14. analy n Fg. 8 can be oberved. Fg. 16 how the output voltage of the parallel nverter wth the ncreae of the dtrbuton lne mpedance. It een that once connectng the two nverter n parallel at the ntant of.2 the reonance tll are when the lne nductance are.9 mh but dappear for the cae that the lne nductance Fg. 17. Smulated output voltage of the parallel nverter when the vrtual retance 2.4 Ω appled to the control ytem. are 1.8 mh. Th phenomenon valdate the frequency-doman analy of the mnor-loop gan n Fg. 9. Fg. 17 how the output voltage of the parallel nverter when the vrtual retance 2.4 Ω appled. It obvou that the voltage reonance tll are after.2 whch mple that

7 Fg. 18. Smulated output voltage of the parallel nverter when the output voltage feedforward term ued n the current control loop. very lttle dampng effect can be obtaned va ung the vrtual output mpedance loop. Th confrm the frequency behavor of the total output mpedance hown n Fg. 1. In contrat Fg. 18 depct the output voltage of parallel nverter for the cae that the output voltage feedforward term appled n the current control loop. It oberved that the voltage waveform keep nuodal after connectng the DG nverter n parallel whch valdate the frequency-doman analy hown n Fg. 13. Hence t can be concluded that the ue of the output voltage feedforward term n the nner current control loop can help to tablze the nteracton of parallel voltage-controlled DG nverter. V. CONCLUSIONS Th paper ha dcued the potental voltage reonance caued by the control loop nteracton of the parallel voltagecontrolled DG nverter. In order to clarfy the nature of uch undered reonance the mpedance-baed analy ha been performed on an landed network wth two parallel voltagecontrolled nverter. It ha hown that a ere LC reonance may be raed dependng on the phae dfference between the nverter output mpedance and the equvalent load mpedance of the remanng network. The dampng effect of the vrtual output mpedance loop and the output voltage feedforward term n the nner current control loop have been evaluated n the frequency doman. It ha found that the vrtual mpedance ha no effect on tablzng the controller nteracton wherea an output voltage feedforward term can help to dampen out the potental reonance. Smulaton reult are preented to valdate the frequency-doman analy. [3] C. L. Chen Y. Wang J. S. La Y. S. Lee and D. Martn Degn of parallel nverter for mooth-mode tranfer mcrogrd applcaton IEEE Tran. Power Electron. vol. 25 pp Jan. 21. [4] R. Laeter Smart dtrbuton: Coupled mcrogrd IEEE Proc. vol. 99 no. 6 pp Jun [5] A. Radwan and Y. A.-R. I. Mohamed Modelng analy and tablzaton of converter-fed AC mcro-grd wth hgh penetraton of converter-nteraced load IEEE Tran. Smart Grd vol. 3 no. 3 pp Sep [6] Y. W. L D. M. Vlathgamuwa and P. C. Loh Degn analy and real-tme tetng of a controller for multbu mcrogrd ytem IEEE Tran. Power Electron. vol. 19 pp Sep. 24. [7] J. Rocabert A. Luna F. Blaabjerg and P. Rodrguez Control of power converter n AC mcrogrd IEEE Tran. Power Electron. vol. 27 no. 11 pp Nov [8] S. J. Chang and J. M. Chang Parallel control of the UPS nverter wth frequency-dependent droop cheme n Proc. IEEE PESC 21 pp [9] E. A. A. Coelho P. C. Cortzo and P. F. Garca Small-gnal tablty for parallel-connected nverter n tand-alone AC upply ytem IEEE Tran. Ind. Appl. vol. 38 pp Mar./Apr. 22. [1] J. He and Y. W. L Analy degn and mplementaton of vrtual mpedance for power electronc nterfaced dtrbuted generaton IEEE Tran. Ind. Appl. vol. 47 pp Nov./Dec [11] L. Corradn P. Mattavell M. Corradn and F. Polo Analy of parallel operaton of unnterruptble power upple loaded through long wrng cable IEEE Tran. Power Electron. vol. 25 pp Apr. 21. [12] Y. W. L D. M. Vlathgamuwa and P. C. Loh Robut control cheme for a mcrogrd wth PFC capactor connected IEEE Tran. Ind. Appl. vol. 43 pp Sep./Oct. 27. [13] J. Sun Small-gnal method for AC dtrbuted power ytem a revew IEEE Tran. Power Electron. vol. 24 no. 11 pp Nov. 29. [14] S. Ht D. Boroyevch and C. Cuadro Small-gnal modelng and control of three-phae PWM converter n Proc. IEEE IAS 1994 pp [15] X. Wang F. Blaabjerg and Z. Chen Synthe of varable harmonc mpedance n nverter-nterfaced dtrbuted generaton unt for harmonc dampng throughout a dtrbuton network IEEE Tran. Ind. Appl. vol. 48 pp Jul./Aug REFERENCES [1] F. Blaabjerg Z. Chen and S. B. Kjaer Power electronc a effcent nterface n dpered power generaton ytem IEEE Tran. Power Electron. vol. 19 pp Sep. 24. [2] Y. Xue L. Chang S. B. Kjaer J. Bordoau and T. Shmzu Topologe of ngle-phae nverter for mall dtrbuted power generator: an overvew IEEE Tran. Power Electron. vol. 19 pp Sep. 24.