Frequency Adaptive Repetitive Control of Grid-Tied Single-Phase PV Inverters Zhou, Keliang; Yang, Yongheng; Blaabjerg, Frede

Transcription

1 Aalborg Uiversitet Frequecy Adaptive Repetitive Cotrol of Grid-Tied Sigle-Phase PV Iverters Zhou, Keliag; Yag, Yogheg; Blaabjerg, Frede Published i: Proceedigs of the 205 IEEE Eergy Coversio Cogress ad Expositio (ECCE) DOI (lik to publicatio from Publisher): 0.09/ECCE Publicatio date: 205 Documet Versio Early versio, also kow as pre-prit Lik to publicatio from Aalborg Uiversity Citatio for published versio (APA): Zhou, K., Yag, Y., & Blaabjerg, F. (205). Frequecy Adaptive Repetitive Cotrol of Grid-Tied Sigle-Phase PV Iverters. I Proceedigs of the 205 IEEE Eergy Coversio Cogress ad Expositio (ECCE) (pp ). IEEE Press. DOI: 0.09/ECCE Geeral rights Copyright ad moral rights for the publicatios made accessible i the public portal are retaied by the authors ad/or other copyright owers ad it is a coditio of accessig publicatios that users recogise ad abide by the legal requiremets associated with these rights.? Users may dowload ad prit oe copy of ay publicatio from the public portal for the purpose of private study or research.? You may ot further distribute the material or use it for ay profit-makig activity or commercial gai? You may freely distribute the URL idetifyig the publicatio i the public portal? Take dow policy If you believe that this documet breaches copyright please cotact us at vb@aub.aau.dk providig details, ad we will remove access to the work immediately ad ivestigate your claim. Dowloaded from vb.aau.dk o: maj 2, 208

2 Frequecy Adaptive Repetitive Cotrol of Grid- Tied Sigle-Phase PV Iverters Keliag Zhou, Seior Member, IEEE, Yogheg Yag 2, Member, IEEE, ad Frede Blaabjerg 3, Fellow, IEEE School of Egieerig Uiversity of Glasgow Glasgow, UK Departmet of Eergy Techology Aalborg Uiversity Aalborg, Demark 2 yoy@et.au.dk; 3 fbl@et.au.dk Abstract The iteral model priciple based Repetitive Cotrol (RC) offers a accurate cotrol strategy for grid-tied power coverters to feed siusoidal curret ito the grid. However, i the presece of grid frequecy variatios, the covetioal RC fails to produce high quality feedig curret. This paper thus explores a frequecy adaptive repetitive cotrol strategy for grid coverters, which employs fractioal delay filters i order to adapt to the chage of the grid frequecy. Case studies with experimetal results of a sigle-phase grid-coected PV iverter system are provided to verify the proposed cotroller. Keywords frequecy adaptive; power coverter; repetitive cotrol; fractioal delay filter; PV iverters I. INTRODUCTION Accordig to the iteral model priciple, a Repetitive Cotroller (RC) []-[8] ca achieve ero steady-state error trackig of ay periodic sigal with kow period due to the itroductio of high gais at the iterested harmoic frequecies. It offers a very simple but effective ad accurate cotrol solutio for power coverters to produce high quality siusoidal voltages/currets. The covetioal RC cotroller i its digital form of N /( N ) ca track ay periodic referece sigal with a iteger period of N =f s /f, where f is the fudametal frequecy of referece sigal ad f s is the samplig rate. However, i grid-coected applicatios, the grid frequecy is usually variable i a certai rage (e.g., 49 H ~ 5 H) i practice ad specified i the grid codes. Thus N would ofte be fractioal i the case of a fixed samplig rate f s. Sice oly N with a iteger N ca be implemeted i practice, the covetioal RC is sesitive to grid frequecy variatios, ad it thus caot exactly compesate periodic voltages/currets of variable frequecy. Esurig the iteger period of N is always the same i the presece of grid frequecy variatios, the variable samplig rate approach eables the RC to reject harmoics completely [9]-[5]. However, a variable samplig rate will sigificatly icrease the real-time implemetatio complexity of the cotrol systems, such as olie cotroller redesig [6], [7]. I order to address this issue, a frequecy adaptive RC strategy at fixed samplig rate is proposed to for grid-tied coverters to feed siusoidal curret ito grid i the presece of a variable grid frequecy. The fractioal delay N with a fractioal umber N will be replaced with a approximated Fiite Impulse Respose (FIR) factioal delay filter. The proposed FIR fractioal delay filter oly cosumes a small umber of multiplicatios ad additios to update its coefficiets, ad it is well suited to fast olie tuig of the fractioal delay. The proposed frequecy adaptive RC will eable grid-tied coverters always to produce siusoidal feedig currets uder variable grid frequecy. The aalysis ad sythesis of such frequecy adaptive RC systems are addressed. Case studies of grid-tied coverters are provided to evaluate the proposed frequecy adaptive RC. II. FREQUENCY ADAPTIVE REPETITIVE CONTROL Fig. shows the typical closed-loop cotrol system with a plug-i Covetioal Repetitive Cotroller (CRC), where R() is the referece iput, Y() is the output, E() = R() Y() is the trackig error, D() is the disturbace, G p () is the plat, G c () is the covetioal feedback cotroller, G r () is a feedforward plug-i CRC, k r is the RC gai, U r () is the output of the CRC, G f () is a phase lead compesatio filter to stabilie the overall closed-loop system [8]-[22], ad Q() = a + a 0 + a with 2a + a 0 = is a low pass filter to ehace the etire cotrol system robustess [2]. Fig.. Plug-i repetitive cotrol system i a geeral coverter structure. The trasfer fuctio G r () of the plug-i CRC show i Fig. ca be writte as N Ur () Q() Gr() kr Gf () N () E () Q( ) where N =f s /f with f beig the fudametal frequecy of the referece sigal R() ad/or disturbace D(), ad f s beig the samplig rate, N is the order of the RC; the poles of G r () are located aroud 2m f, with m =0,, 2,, M (M = N/2 for a eve N ad M = (N )/2 for a odd N). It is clearly see /5/$ IEEE

3 that the amplitudes of G r () at frequecies 2m f approach ifiity if Q()=. Cosequetly, the CRC provides ero steady-state error trackig of all harmoic compoets below the Nyquist frequecy if Q()= ad its order N is a iteger [6]-[8]. Moreover, N with a iteger N ca be easily implemeted i practice. However, i the case of a timevaryig frequecy f, N =f s /f would ofte be fractioal with a fixed samplig rate f s. As a result, high cotrol gais will be shifted away from the iterested harmoic frequecies. Thus, the CRC is sesitive to the chage of the grid frequecy f. I order to adapt to a variable frequecy f, the factioal delay term of -N ca be approximated by Fractioal Delay (FD) filters [23]-[28]. Assumig that N = Ni F with N i = N beig the iteger part of N ad F = N N i (0 F < ) beig the fractioal part of N, the fractioal delay F ca thus be approximated by a Lagrage iterpolatio polyomial FIR filter as give i the followig [23]-[26] F k Ak (2) k 0 where k = 0,,,, ad A k are the iterpolatio polyomial coefficiets that ca be give as F i Ak k, i = 0,,, (3) k i i 0 i k It should be oted that, if = i (2), a liear iterpolatio polyomial -F (-F) + F - will be attaied. Substitutig (2) ad (3) ito (), a Frequecy Adaptive RC (FARC) will be obtaied as Ni k Ak Q() Ur () k 0 fr () r f () E () Ni k Ak Q( ) k 0 G k G which will become the CRC of () whe F = 0. The FARC of (4) provides a geeral approach to track or elimiate ay periodic sigal with a arbitrary fudametal frequecy. It should be poited out that both N i ad F should chage slowly i practical applicatios. Fig. 2 shows the magitude resposes of the Lagrage iterpolatio based FD filter of (2) with the order = ad = 3 for various fractioal F from 0 to 0.9. It is see that the FD filter of (2) with order = 3 gives a excellet approximatio of the fractioal delay F at low frequecies withi the badwidth of 75 % of the Nyquist frequecy. I cotrast, the badwidth of 50 % of the Nyquist frequecy is observed for the FD filter of order =. Notably, the Lagrage iterpolatio is oe of the easiest ways to desig a FD filter to approximate a give fractioal delay. Moreover, the coefficiet of (2) for the FD filter oly cosumes a small umber of additios ad multiplicatios for a fast olie update of the coefficiets. Such a FIR FD filterbased FARC offers a attractive method for the real-time cotrol of high switchig-frequecy grid-tied coverters. (4) Magitude Magitude dB F =0 F =0.,0.9 F =0.2,0.8 F =0.3,0.7 F =0.4,0.6 F = Normalied Frequecy -3dB F =0.9 F =0 F = 0.5 F = 0.2 F = 0.4 F =0.3 F = Normalied Frequecy Fig. 2. Frequecy resposes of the Lagrage iterpolatio based fractioal delay filters with differet F, where the frequecy is ormalied takig the Nyquist frequecy as the base value: = ad = 3. III. CASE STUDY: GRID-TIED SINGLE-PHASE PV INVERTER Fig. 3 shows a grid-coected sigle-phase iverter for PV applicatios with a LCL-filter, which is used to feed currets ito the grid. The ier curret cotrol loop comprises a deadbeat feedback ad the proposed plug-i FARC cotroller. The outer cotrol loop is resposible for geeratig accurate curret refereces for the ier cotrol loop []. A. Modelig ad Cotrol PV V dc Iverter PWM u* v* iv v iv L R L 2 R 2 C f Deadbeat Cotroller DB CRC/ FARC CRC/FARC Cotroller * I g * si Isolatio Trasformer Grid Fig. 3. Schematic ad overall cotrol structure of a sigle-phase sigle-stage grid-coected PV iverter system with a LCL filter (PLL Phase Locked Loop, PWM Pulse Width Modulatio). N PLL ωt φ* Cotrol System

4 As it is show i Fig. 3, the capacitor C f is used to elimiate high-order harmoic currets of switchig frequecies, ad together with the grid-side iductor L 2, it is referred to as a ideal load. Hece, the dyamics of the PV iverter ca simply be described as dig L Ri g vi v vg (5) dt where is the grid voltages, is the grid currets, L ad R are the omial values of ac-side iductor (L ) ad resistor (R ) of the LCL filter, respectively. Oe cotrol objective of the iverter is typically to achieve a uity power factor ad thus a Secod-Order Geeralied Itegrator based Phase Locked loop (PLL) system [29] is adopted. The secod objective is to maitai a low harmoic distortio siusoidal curret usig advaced cotrol schemes. The sampled-data model of (5) ca be writte as b b ( ) 2 uk ( ) vg k ig( k ) ig( k) vdc( k) (6) b b b where b =L /T s, b 2 =R, u is the modulatio sigal with v iv (t)=u(t)v dc (t), ad T s is the samplig period. For the plat i (6), a Dead-Beat (DB) curret cotroller is adopted as uk ( ) vg( k) bi gref ( k) b b2 ig( k) vdc ( k) (7) which makes (k+)=ref (k). As it is show i Fig. 3, the CRC G r () ad the proposed FARC G fr () of (4) are plugged ito the curret cotrol loop to esure high accuracy curret trackig. For the FARC of (4), = 3 is chose to be the Lagrage polyomial degree. Hece, the correspodig fractioal delay will be N Ni Ni Ni 2 Ni 3 A0 A A2 A 3 (8) B. Experimetal Setup A test rig is built-up, where a sigle-phase commercial power coverter is coected to the grid through a LCLfilter, ad the cotrol system was implemeted i a dspace DS 03 rapid prototypig kit. Parameters of the test setup are listed i Table I. To achieve approximately ero phase compesatio, a filter G f () = p is used to compesate samplig delays, model mismatches, ad u-modeled delay, where the lead step p = 3 is determied by experimets. TABLE I. SYSTEM PARAMETERS OF A SINGLE-PHASE GRID-TIED INVERTER SYSTEM. LCL-filter L = L 2 = 3.6 mh, C f = 2.35 μf, Trasformer leakage iductace L g= 2 mh Switchig ad samplig frequecy f s = f sw = 0 kh DC voltage V dc = 400 V Power ratig P = kw Nomial grid voltage 50 H, 325 V (peak) * Grid curret referece I g 5 A (peak) at uity power factor Repetitive cotrol gai k rc =.8 Low pass filter Q() C. Experimetal Results Fig. 4 gives the steady-state resposes of the DB plus CRC cotrolled sigle-phase iverter. It ca be see that the CRC is sesitive to the chage of the grid frequecy f whe f drops from omial 50 H to 49 H, the Total Harmoic Distortio (THD) of the feedig curret icreases from.4% to 6.25%; whe f rises from omial 50 H to 5 H, the THD of feedig curret icreases from.4% to 6.5%. The, the FARC cotroller is added to improve the curret cotrol. Fig. 5 gives the steady-state resposes of the DB plus FARC cotrolled sigle-phase iverter. It ca be see that the CRC is much less sesitive to the chage of the grid frequecy f. Specifically, whe f drops from omial 50 H to 49 H, the THD of feedig curret icreases from.4% to 3.0%; whe f rises from omial 50 H to 5 H, the THD of feedig curret icreases from.4% to 3.6%. f = 50 H, THD i =.4% ig f = 49 H, THD i =6.25% f = 5 H, THD i = 6.5 % (c) Fig. 4. Steady-state resposes of the DB plus CRC cotrolled sigle-phase iverter system (grid voltage [00 V/div]; grid curret [5 A/div]; time [4 ms/div]): f = 50 H, N = 200, THD of =.4%, f = 49 H, N = 200, THD of = 6.25%, ad (c) f = 5 H, N = 200, THD of = 6.5%.

5 f = 49 H, THD i =3.% ig 49.5 H f pll 50.5 H f = 5 H, THD i = 3.6 % ig 50.5 H f pll 49.5 H Fig. 5. Steady-state resposes of the DB plus FARC cotrolled sigle-phase iverter system (grid voltage [00 V/div]; grid curret [5 A/div]; time [4 ms/div]): f = 49 H, THD of = 3.% ad f = 5 H, THD of = 3.6%. Furthermore, the THD of the feedig curret with these two repetitive cotrol schemes uder various grid frequecies is show i Fig. 6. It ca be observed that the FARC ca esure a satisfactory feedig curret quality with THD < 5% i the presece of time-varyig grid frequecy, but the CRC caot maitai a lower THD i the case of grid frequecy variatios. Addtioally, Fig. 7 shows that the FARC cotrolled iverter keeps feedig almost costat good quality curret ito the grid regardless of the step-chages of the grid frequecy betwee 49.5 H to 50.5 H. This further cofirms the effectiveess of the FARC i terms of dyamics. Grid curret THD (%) DB + CRC DB + FARC Grid frequecy (H) Fig. 6. THD of the feedig curret of the CRC ad FARC cotrolled siglephase grid-tied system uder various grid frequecies. Fig. 7. Dyamic performace of the DB plus FARC cotrolled sigle-phase iverter system (grid voltage [250 V/div]; grid curret [5 A/div]; PLL estimated grid frequecy f pll [ H/div]; time [20 ms/div]): the grid frequecy chaged from 49.5 H to 50.5 H ad the grid frequecy chaged from 50.5 H to 49.5 H. IV. CONCLUSIONS A frequecy adaptive repetitive cotrol method has bee proposed for grid-tied coverters to feed siusoidal currets ito electricity etwork i the presece of a time-varyig grid frequecy. The proposed frequecy adaptive repetitive cotrol scheme offers a fast o-lie tuig of the fractioal delay ad a fast update of the coefficiets. It provides to grid-tied coverters with a simple but very accurate cotrol solutio uder grid frequecy variatios. A applicatio example of grid-tied sigle-phase PV iverters has preseted to demostrate the effectiveess ad advatages of the proposed frequecy adaptive repetitive cotrol solutio. REFERENCES [] F. Blaabjerg, M. Liserre, R. Teodorescu, A.V. Timbus Overview of Cotrol ad Grid Sychroiatio for Distributed Power Geeratio Systems. IEEE Tras. Id. Electro., vol. 53, o. 5, pp , [2] P. C. Loh, Y. Tag, F. Blaabjerg, ad P. Wag, Mixed-frame ad statioary-frame repetitive cotrol schemes for compesatig typical load ad grid harmoics, IET Power Electro., vol. 4, o. 2, pp , 20. [3] K. Zhou, Z. Qiu, N.R. Watso, ad Y. Liu, Mechaism ad elimiatio of harmoic curret ijectio from sigle-phase grid-coected PWM coverters, IET Power Electroics, vol. 6, o., pp , 203. [4] L. He, K. Zhag, J. Xiog, ad S. Fa, "A repetitive cotrol scheme for harmoic suppressio of circulatig curret i modular multilevel coverters," IEEE Tras. Power Electro., vol. 30, o., pp , Ja. 205.

6 [5] S. Jiag, D. Cao, Y. Li, J. Liu, ad F.Z. Peg, "Low-THD, fast-trasiet, ad cost-effective sychroous-frame repetitive cotroller for threephase UPS iverters," IEEE Tras. Power Electro., vol. 27, o. 6, pp , Ju [6] K. Zhou ad D. Wag, Zero trackig error cotroller for three-phase CVCF PWM iverter, Electroics Letters, vol. 36, o. 0, pp , [7] Q. C. Zhog ad T. Horik, Cascaded curret voltage cotrol to improve the power quality for a grid-coected iverter with a local load, IEEE Tras. Id. Electro., vol. 60, o. 4, pp , 203. [8] K. Zhag, Y. Kag, J. Xiog, ad J. Che, Direct repetitive cotrol of SPWM iverter for UPS purpose, IEEE Tras. Power Electro., vol. 8, o. 3, pp , [9] B. Zhag, K. Zhou, ad D. Wag, Multirate repetitive cotrol for PWM DC/AC coverters, IEEE Tras. Id. Electro., vol. 6, o. 6, pp , 204. [0] W. Lu, K. Zhou, D. Wag, ad M. Cheg, A geeral parallel structure repetitive cotrol scheme for multi-phase DC-AC PWM coverters, IEEE Tras. Power Electro., vol. 28, o. 8, pp , 203. [] W. Lu, K. Zhou, D. Wag, ad M. Cheg, A geeric digital k±m order harmoic repetitive cotrol scheme for PWM coverters, IEEE Tras. Id. Electro., vol. 6, o. 3, pp , 204. [2] W. Zhu, K. Zhou, ad M. Cheg, A bidirectioal high-frequecy-lik sigle-phase iverter: modulatio, modelig, ad cotrol, IEEE Tras. Power Electro., vol. 29, o. 8, pp , 204. [3] Y.-Y. Tou, S.-L. Jug, ad H.-C. Yeh, "Adaptive repetitive cotrol of PWM iverters for very low THD AC-voltage regulatio with ukow loads," IEEE Tras. Power Electro., vol. 4, o. 5, pp , Sept [4] D. Che, J. Zhag, ad Z. Qia, "a improved repetitive cotrol scheme for grid-coected iverter with frequecy-adaptive capability," IEEE Tras. Id. Electro., vol. 60, o. 2, pp , Feb [5] K. Zhou, Y. Yag, F. Blaabjerg, ad D. Wag, Optimal selective harmoic cotrol for power harmoics mitigatio, IEEE Tras. Id. Electro., vol. 62, o. 2, pp , 205. [6] M.A. Herra, J.R. Fischer, S.A. Goale, M.G. Judewic, I. Carugati, ad D.O. Carrica, "Repetitive cotrol with adaptive samplig frequecy for wid power geeratio systems," IEEE J. Emerg. Sel. Topics Power Electro., vol. 2, o., pp , Mar [7] Z. Cao ad G.F. Ledwich, "Adaptive repetitive cotrol to track variable periodic sigals with fixed samplig rate," IEEE/ASME Tras. Mechatro., vol. 7, o. 3, pp , Sept [8] B. Zhag, D. Wag, K. Zhou, ad Y. Wag. Liear phase lead compesatio repetitive cotrol of a CVCF PWM iverter. IEEE Tras. Id. Electro., vol. 55, o. 4, pp , [9] B. Zhag, K. Zhou, Y. Wag, ad D. Wag, Performace improvemet of repetitive cotrolled PWM iverters: A phase-lead compesatio solutio, Iteratioal Joural of Circuit Theory ad Applicatios, vol. 38, o. 5, pp , 200. [20] B. Zhag, K. Zhou, Y. Ye, ad D. Wag, Desig of liear phase lead repetitive cotrol for CVCF PWM DC-AC coverters, i Proc. of ACC, pp , [2] G.A. Ramos ad R. Costa-Castelló, "Power factor correctio ad harmoic compesatio usig secod-order odd-harmoic repetitive cotrol," IET Cotrol Theory & Applicatios, vol. 6, o., pp , Jul [22] Y.H. Yag, K. Zhou, M. Cheg, ad B. Zhag, Phase compesatio multi-resoat cotrol of CVCF PWM coverters, IEEE Tras. Power Electro., vol. 28, o. 8, pp , 203. [23] Y. Wag, D. Wag, B. Zhag, ad K. Zhou, Fractioal delay based repetitive cotrol with applicatio to PWM DC/AC coverters, i Proc. IEEE ICCA, pp , [24] Y. Yag, K. Zhou, H. Wag, F. Blaabjerg, D. Wag, ad B. Zhag, Frequecy adaptive selective harmoic cotrol for grid-coected iverters, IEEE Tras. Power Electro., vol. 30, o. 7, pp , 205. [25] Z. Zou, K. Zhou, Z. Wag, ad M Cheg, Frequecy adaptive fractioal order repetitive cotrol of shut active power filters, IEEE Tras. Id. Electro., vol. 62, o. 3, pp , 205. [26] Z. Zou, K. Zhou, Z. Wag, ad M. Cheg, Fractioal-order repetitive cotrol of programmable AC power sources, IET Power Electro., vol. 7, o. 2, pp , Feb [27] D. Che, J. Zhag, ad Z. Qia, A improved repetitive cotrol scheme for grid-coected iverter with frequecy-adaptive capability, IEEE Tras. Id. Electro., vol. 60, o. 2, pp , Feb [28] M. Rashed, C. Klumper, ad G. Asher, Repetitive ad resoat cotrol for a sigle-phase grid-coected hybrid cascaded multilevel coverter, IEEE Tras. Power Electro., vol. 28, o. 5, pp , May 203. [29] M. Ciobotaru, R. Teodorescu, ad F. Blaabjerg, "A ew sigle-phase PLL structure based o secod order geeralied itegrator," i Proc. of PESC, pp. -6, 8-22 Jue 2006.