Aalborg Universitet. Published in: Electric Power Systems Research. DOI (link to publication from Publisher): /j.epsr

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1 Aalbor Univeritet A comprehenive control ytem or multi-parallel rid-connected inverter with LCL ilter in weak rid condition Akhavan, Ali; Mohammadi, Hamid Reza; Guerrero, Joep M. Publihed in: Electric Power Sytem Reearch DOI (link to publication rom Publiher):.6/j.epr Publication date: 8 Document Verion Early verion, alo known a pre-print Link to publication rom Aalbor Univerity Citation or publihed verion (APA): Akhavan, A., Mohammadi, H. R., & Guerrero, J. M. (8). A comprehenive control ytem or multi-parallel rid-connected inverter with LCL ilter in weak rid condition. Electric Power Sytem Reearch, 63, DOI:.6/j.epr General riht Copyriht and moral riht or the publication made acceible in the public portal are retained by the author and/or other copyriht owner and it i a condition o accein publication that uer reconie and abide by the leal requirement aociated with thee riht.? Uer may download and print one copy o any publication rom the public portal or the purpoe o private tudy or reearch.? You may not urther ditribute the material or ue it or any proit-makin activity or commercial ain? You may reely ditribute the URL identiyin the publication in the public portal? Take down policy I you believe that thi document breache copyriht pleae contact u at vbn@aub.aau.dk providin detail, and we will remove acce to the work immediately and invetiate your claim. Downloaded rom vbn.aau.dk on: oktober 7, 8

2 A Comprehenive Control Sytem or Multi-Parallel Grid-Connected Inverter with LCL Filter in Weak Grid Condition Ali Akhavan, Hamid Reza Mohammadi, and Joep M. Guerrero Abtract: Active dampin method are ued or reonance dampin in rid-connected inverter with LCL ilter. In microrid, parallel rid-connected inverter are coupled due to rid impedance introducin multiple reonance. In eneral, uch couplin eect i not taken into account or modelin and controller dein. For inle rid-connected inverter, depite ood perormance, the ytem tend to become intable with parallel connection o other inverter. Moreover, the rid injected current can be ditorted by the rid voltae harmonic. In traditional control ytem, rid voltae i ued a a eedorward inal to achieve harmonic rejection capability by bootin the inverter output impedance. However, thi method introduce neative phae anle which could lead to control ytem intability. In thi paper, the control ytem dein or multi-parallel rid-connected inverter uin active dampin i clariied. Inverter with dierent characteritic are alo modeled in a weak rid a a multivariable ytem while couplin eect with a wide variation o rid impedance i taken into account. An improved rid voltae eedorward method i propoed to eliminate neative apect o the traditional method. The imulation reult in MATLAB/SIMULINK otware demontrate the eectivene o the propoed control ytem. Keyword: Multi-parallel inverter, Active dampin, Couplin eect, Grid voltae eedorward method, LCL ilter. Introduction Grid-connected inverter are eential element in convertin nearly all kind o enerated power in ditributed eneration plant into a hih quality AC power to be injected reliably into the rid []. The quality o rid injected current in rid-connected ytem i a matter o concern []. Thu, a low-pa ilter i ued to ilter out the witchin requency harmonic o the inverter output current. The LCL ilter i preerred in comparion with other lowpa ilter uch a L and LC becaue o better witchin harmonic attenuation and reduced ilter ize at the ame time [3], [4]. Neverthele, due to reonance o the LCL ilter, a dampin method i needed to tabilize the ytem [5]. Reonance dampin method or LCL ilter, includin paive and active dampin method have been extenively dicued in literature [6-9]. Power lo in paive dampin method i the biet drawback. Hence, active dampin i preerred over paive mainly dampin due to it hih eiciency and lexibility. Dual-loop active dampin method are widely ued or reonance dampin. Thee method are baed on eedback o the tate variable o LCL ilter uch a capacitor current [], [-3], capacitor voltae [4], rid-ide inductor current [5], [6] and inverter-ide inductor current [7]. A main challene i encountered when multi-parallel rid-connected inverter are coupled throuh rid impedance Z. In Fi., the voltae o Point o Common Couplin (PCC), V pcc, i hared by all inverter and can be modiied by their injected current [8]. Thereore, all inverter inluence each other due to rid impedance exitence. Dependin on the number o parallel inverter and rid impedance value, the inverter intalled in a microrid may not behave a expected. It i worth nothin that, multi-parallel rid-connected inverter alo introduce multiple reonance whoe requency and peak can be varied with variation o the rid impedance and number o inverter [9]. Furthermore, rid voltae in weak rid uually contain harmonic component which are created by nonlinear load connection at other bue. The rid voltae harmonic can thereore reatly ditort the rid-injected current. Hence, the inverter control ytem hould be deined with harmonic rejection capability which i cloely correlated with the inverter output impedance []. Recently, a popular approach attractin attention i employin the rid voltae eedorward method in the control ytem to enhance the inverter output impedance [5]. V dc_ V dc_ V dc_n VSC VSC VSC n i Z Z i V inv_ Z 3 i Z Z i V inv_ i n V inv_n Z n i 3 i 3 i 3n Z 3 i n Z 3n Z n V pcc Fi.. Typical multi-parallel rid-connected inverter. i Z V

3 Althouh thi method can boot coniderably the inverter output impedance, it neverthele introduce neative phae anle to the control ytem which could lead to intability in weak rid []. Many literature reardin active dampin trateie are publihed or inle rid-connected inverter ytem [- 7], [-3]. However, the couplin eect amon inverter due to rid impedance i not conidered. In [4], an optimal virtual lux predictive direct power control (VF-PDPC) i propoed or a three-phae rid-connected inverter which operatin under unbalanced and ditorted rid voltae. Depite ood perormance o the control ytem, only one inverter i conidered and couplin eect amon inverter in a microrid containin everal inverter i not addreed. In [8], a PV power plant contain N parallel rid-connected inverter ha been modeled a a multivariable ytem. However, all inverter are aumed to be the ame. In [5], the inverter with dierent characteritic in a microrid are modeled a a multivariable ytem and controller dein proce i introduced praieworthy, but the eect o rid voltae harmonic on the rid injected current i not conidered. Multiple reonance in a microrid are invetiated in [9]. However, the control ytem dein i done without coniderin the couplin eect amon inverter. In thi paper, modelin and control o three parallel inle-phae rid-connected inverter in a weak rid condition i decribed. In the propoed cheme, the wide variation o rid impedance and alo rid voltae harmonic are conidered. Unlike [8], all inverter may have dierent characteritic. Alo, dual-loop active dampin control uin capacitor current eedback i choen or it imple and eective implementation. A imple but eective improved rid voltae eedorward method i propoed which uppre the eect o rid voltae harmonic by bootin the inverter output impedance, while eliminate the neative apect o traditional method uch a introducin neative phae anle to the control ytem in weak rid. Then, the ytem tability i invetiated baed on impedance-baed tability criterion [6]. The deiciencie o traditional rid voltae eedorward method are:. Introducin a neative phae anle to cloe loop control ytem. Conequently, the phae marin i reduced and it may be caue the ytem intability.. Inorin the couplin eect between inverter. Conequently, the overall ytem maybe untable wherea the connection o each individual inverter i table. The uetion in the propoed comprehenive control ytem are:. Uin a proportional ain in an improved rid voltae eedorward method which mitiate the irt deiciency.. Precie modelin and controller dein or multiparallel rid-connected inverter (with dierent parameter) coniderin the couplin eect between inverter. In thi way, the econd deiciency i mitiated. Thi paper i oranized a ollow. In Section, modelin and control o a inle rid-connected inverter i decribed. Norton equivalent circuit and the propoed rid voltae eedorward method are preented in thi ection. In Section 3, modelin and control o three parallel ridconnected inverter with dierent characteritic are decribed. In Section 4, controller dein proce or the multivariable control ytem that i modeled in ection 3 i decribed. In Section 5, the theoretical tudy i validated throuh numerou imulation in MATLAB/Simulink otware. Finally, Section 6 conclude the paper.. Control o a Sinle Grid-Connected Inverter in a Weak Grid In thi ection, the modelin and control o a inle rid-connected inverter i decribed. Then, a imple but eective improved rid voltae eedorward method i propoed... Modelin o a Sinle Grid-Connected Inverter Fi. how the eneral tructure o an LCL-iltered rid-connected inverter. The LCL ilter conit o an inverter-ide inductor L, a rid-ide inductor L, and a ilter capacitor C. Z L., Z L., Z 3, Z L. C3. () In thi iure, V dc i the input DC voltae, V inv i the output voltae o the inverter bride, i, i, and i C are inverter-ide current, rid-ide current and capacitor current, repectively. Alo, G i() i the current reulator and i C i ed back to damp the LCL ilter reonance. At the PCC, the rid i modeled by it Thevenin equivalent circuit or implicity, conitin o a voltae ource V in erie with rid impedance Z. G d() i the traner unction which combine the computational and PWM delay [8]..5.. T Gd () () (.5.. T ) where, T reer to amplin period. With the aorementioned model, the linearized model o a inle rid-connected inverter with LCL ilter in - domain can be derived a hown in Fi. 3. In thi iure, K P_inner i the proportional controller in the inner loop. Coniderin Fi. 3, adequate controller dein i quite complicated due to interactin loop. In order to impliy the dein procedure, an equivalent model with decoupled reulatin loop would be deirable. The model in Fi. 3 can be impliied by addin capacitor voltae (v C) to output inal o the traner unction K P_inner, and by replacin eedback inal i C with i -i a hown in Fi. 3. Uin the equivalent tranormation preented in [] and [], the block diaram o Fi. 3 can be tranormed into Fi. 4. G x() and G x() traner unction are iven by V dc VSC PWM K P _ inner i V inv Z Z i c Z 3 Gd () G () Fi.. Coniuration o a inle rid-connected inverter with LCL ilter. i c i i V pcc Z i ire V

4 G K G () G ( ) Gi( ) L C CK G ( ) P _ inner d x P _ inner d L C CKP _ innergd ( ) x () 3 L LC L CKP _ innergd L L ( ) ( ) Hence, accordin to Fi. 4, the current loop ain T() can be derived a T( ) G ( ) G ( ) x x K G ( ) G ( ) L L C L CK G ( ) ( L L ) P _ inner d i 3 P _ inner d A hown in Fi. 5, a rid-connected inverter can be modeled by it Norton equivalent circuit, conitin o an ideal current ource i () in parallel with an admittance Y o(). In Fi. 5, the rid i modeled by it Thevenin equivalent circuit conitin o an ideal voltae ource V () in erie with the rid impedance Z (). Accordin to thi iure, the injected rid current i () can be obtained a vpcc() i( ) i( ) Zo() ; Yo () Zo() (6) The Norton equivalent current ource (i ()) which i equal to hort circuit current at PCC (V pcc=), can be obtained uin Fi. 4 a () i ( ) T ire ( ) Gcl ire ( ) (7) T ( ) where G cl i cloed-loop traner unction o control ytem. Alo, Z o() i equal to inverter output impedance, which can be obtained uin Fi.4 a vpcc T ( ) Zo () i ( ) G ( ) i ( ) Uin uperpoition theorem in Fi. 5, the injected rid current i () can be obtained a (9). Zo () i ( ) i ( ) V ( ) (9) Z ( ) Z ( ) Z ( ) Z ( ) x o o (3) (4) (5) (8) Reerrin to (9), in order to uppre the rid current ditortion caued by V (), the manitude o Z o() + Z () hould be a hih a poible... Grid Voltae Feedorward Method A demontrated in previou ubection, the rid voltae harmonic rejection capability o a rid-connected inverter can be achieved by increain the manitude o Z o() + Z (). Since Z () i determined by power rid, only Z o() can be haped to achieve the hih manitude o Z o() + Z (). Uin rid voltae eedorward method can boot the inverter output impedance, praieworthy. However, thi method introduce neative phae anle to the control ytem which could lead to intability in weak rid []. The main idea i introducin a virtual admittance uch a Y op() in Fi. 6 in order to boot the equivalent output impedance. Z Zo ( ) Zop ( ) Z o () Z ( ) Z ( ) o op ; Z op () Y () () From (), it can be een that could be increaed ininitely i the parallel impedance Z op() i et to Z o(). Vpcc () G () i () x G () re x i () op Z Fi. 4. The equivalent tranormation o control block diaram o the rid-connected inverter. i () i Yo () Z () o V pcc Z V () Fi. 5. Equivalent circuit o rid-connected inverter. vpcc () ire () i Gi () Gd () K () C P _ inner L. C. L. v () C i () vpcc () ire () Gi () Gd () K i () () ic P _ inner L. C. L. i () Fi. 3. Block diaram o the dual-loop control tratey baed on capacitor current eedback. Simpliied block diaram o the aorementioned dualloop control tratey. 3

5 Complete analyi about virtual admittance i done in []. The control block diaram o rid-connected inverter with virtual admittance i hown in Fi. 7. In thi iure, the implementation unction o the parallel admittance G z() can be expreed a () []. L C CKP _ innergd ( ) Gz () () K P _ inner.3. Propoed Grid Voltae Feedorward Method A hown in (), G z() ha a econd-order derivative element which increae the noie enitivity. Hence, the implementation unction can be cloely approximated a CKP _ innergd ( ) Gz( ) G z( ) () K P _ inner Uin (), Fi. 7 and inal low raph (SFG), the haped output impedance Z () can be obtained a vpcc Z o() i () o T( ) [ ] Gx() ZCGd ( ) KP _ innergd ( ) Z Z K G () L C P _ inner d Zo( ) [ ] ZCGd ( ) KP _ innergd ( ) Z Z K G () L C P _ inner d (3) The main challene in implementation o rid voltae eedorward method i that thi method introduce neative phae anle to the control ytem which could lead to intability in weak rid. Typical requency repone o the inverter output impedance Z o() and the haped output impedance are plotted in Fi. 8. The parameter which are ued or thee plot are related to inverter parameter in Table and Set II parameter in Table. A hown in Fi. 8, the haped output impedance ha hiher manitude than inverter output impedance Z o() which validate the harmonic rejection capability o traditional rid voltae eedorward method. However, a deep phae la i introduced a hown in phae plot o. The ytem will be table i the phae marin (PM) in the interection point o Z () and ( i ) to be a poitive value, i.e., PM > [], [6]. The PM i expreed a PM 8 [ Z ( ) Z( )] (4) i () Z Z Yo () Z () Z Z i o i o op i Yop () Z () V pcc Fi. 6. Equivalent circuit o rid-connected inverter includin virtual admittance Y op(). G() Vpcc () G i () Gi () G () re () x x i () Gi () Fi. 7. The equivalent tranormation o control block diaram o the rid-connected inverter with rid voltae eedorward method. z Manitude (db) Phae (de) Frequency (Hz) Fi. 8. Frequency repone o inverter output impedance Z o, haped output impedance Z' o and propoed output impedance Z o_ pro. Thereore, to improve tability-robutne, it i neceary to boot the phae anle o Z o() to acquire uicient PM. In thi paper, a imple and improved rid voltae eedorward method i propoed which olve the problem o traditional method. Accordin to (3) i compoed o two part, i.e. Z o() and a ractional part. Z o() i determined by control ytem and LCL ilter. Thereore, i Z o() i modiied in order to enhance phae anle o, then the loop ain will chane which i not deirable. Beide, increain the phae anle o i limited in thi method due to tability contraint. However, the econd part o (3) can be ued or thi purpoe becaue it i introduced due to eedorward path and ha not iniicant eect on main loop ain. Reerrin (3), the econd part o i written a Z Z Z Z Z ( ) [ ] Z G ( ) K G ( ) Z Z K G () p C d P _ inner d L C P _ inner d (5) The requency repone o Zp i plotted in Fi. 9. A hown in thi iure, it introduce a deep phae la about -9 in lower requencie. The main reaon or thi phae la i related to ractional part in denominator o (5). The ractional part i iven by (6). ZCGd ( ) KP _ innergd ( ) Zp_ () (6) Z Z K G () L C P _ inner d The requency repone o Zp_ i plotted in Fi.. A hown in thi iure, the manitude o Z p_ i very cloe to unity in lower requencie. In act, it i a bit reater than, which i hown in zoomed area. In lower requencie, G d can be nelected (e.. G d =) and thereore, Zp_ can be written a (7). KP _ inner ZC Zp_ () (7) K Z Z In requency domain, Zp_ P _ inner L C Bode Diaram can be written a Zo Z ' o Zo_pro 4

6 Manitude (db) Phae (de) Frequency (Hz) Fi. 9. Frequency repone o econd part o the inverter output impedance ( ): traditional method and propoed method. Z Z p p_ ( j) K K jc jl jc P _ inner P _ inner j KP _ inner C (8) KP _ inner j( L ) C I C L, then the real part o Z p_ ( j) will be reater than, ( realz p j _ ( ) ). Aorementioned contraint i conirmed at lower requencie in nearly all cae becaue the ilter capacitor C, ha a very mall value in comparion with inverter-ide inductor L, a in our cae tudy. When real part o i reater than in lower requencie, Z p Bode Diaram Z p_ ha a neative real part which caue phae la. A proportional ain can be ued to prevent thi phae la. I a proportional ain that i maller than inerted in the eedorward path, denominator o (5) will have a poitive real part and phae la will be mitiated. The proportional ain can be elected a Gm (9) Z ( j) p_ Z ' traditional method p Z ' propoed method p It hould be noted that the proportional ain G m hould not be elected much maller than ince Z ( j) p_ it reduce the manitude o inverter output impedance in lower requencie. However, i in a peciic cae, Z p_ ( j) to be maller than in lower requencie, then there i no need to add any proportional ain. The proportional ain or inverter (G m) can be calculated uin (9). Uin parameter in Table and Set II parameter in Table the ain are calculated a G m.988, G m.956 and G m3.979 or irt, econd and third inverter, repectively. Reardin calculated value, the proportional ain i elected G m=.95 or all three inverter. The requency repone o propoed Zp or inverter i alo plotted in Fi. 9. A hown in thi iure, the phae la i mitiated in lower requencie. Accordin to aorementioned analyi, the propoed output impedance Z o_ pro can be repreented a Z () o _ pro Zo( ) [ ] ZCGd ( ) KP _ innergd ( ) Gm Z Z K G () L C P _ inner d () The requency repone o the propoed inverter output impedance Z o_ pro i alo hown in Fi. 8. It i clearly hown that the phae anle o Z o_ pro i booted in comparion with, praieworthy. Indeed, by bootin the phae anle o inverter output impedance, the ytem tability i uaranteed accordin to (4). It i the main advantae o the propoed rid voltae eedorward method with repect to traditional one. Accordin to manitude plot o Fi. 8, the manitude o Z o_ pro i maller than Z o() in requencie below 5 Hz, epecially in requencie below 5 Hz. It i not a bi challene becaue enerally harmonic component in the rid voltae have requencie hiher than 5 Hz (third harmonic). In act, the propoed method achieve a trade-o between the tability and harmonic rejection capability o rid-connected inverter. With the propoed method, the rid-injected current harmonic caued by rid voltae harmonic can be eectively uppreed while the tability i alo uaranteed under weak rid condition. The equivalent block diaram model o the ytem with propoed rid voltae eedorward method i hown in Fi.. Manitude (db) Manitude (db) Phae (de) Z Bode Diaram Frequency (Hz) Bode Diaram Frequency (Hz) Fi.. Frequency repone o Z. p_ 5

7 ire () G m G() Gi () Gd () K i () () ic P _ inner L. C. L. z vpcc () i () Fi.. Block diaram o dual-loop control tratey baed on capacitor current eedback with the propoed rid voltae eedorward method. 3. Modelin and Control o Multi-Parallel Grid- Connected Inverter 3.. Sytem Decription A et o N-parallel LCL-iltered rid-connected inverter i hown in Fi.. The dynamic o thee inverter are coupled due to the rid impedance. The equivalent circuit o the N-parallel inverter o Fi. i hown in Fi., where Z i, Z i and Z 3i (i = N) are inverter-ide inductor impedance, rid-ide inductor impedance and capacitor impedance, repectively, all or i-th inverter. Moreover, i i, i i and i 3i are inverter-ide current, rid-ide current and capacitor current, repectively. The v inv-i i inverter output voltae and i i rid-injected current. 3.. Modelin Multivariable control loop correpondin to three parallel rid-connected inverter with LCL ilter that coupled due to the rid impedance in a microrid are hown in Fi. 3. Thi i the Multiple Input Multiple Output (MIMO) verion o the Sinle Input Sinle Output (SISO) control loop o Fi.. In thi iure, i the matrix traner unction that contain the controller G i(); Gd () i the diaonal matrix traner unction that contain the delay traner unction G d(); K i the matrix traner P _ inner Gi () unction o the inner loop reulator; i the matrix traner unction repreentin the relation between inverteride current (i i ; i=,,3) and inverter output voltae (v inv-i; i=,,3). Due to couplin eect, thi matrix traner unction contain diaonal and non-diaonal element which will be obtained in the next ubection. V inv_ V inv_ V inv_n i Z Z i Z 3 i Z Z i i n Z n i 3 i 3 i 3n Z 3 i n Z 3n Z n i V pcc G () Fi.. The equivalent circuit o the N-parallel inverter. Z V 3.3. Calculation o the Matrix Traner Function Matrix traner unction i repreented in (). Thi matrix ha non-diaonal element ince each inverter output voltae v inv_i inluence the output current o other inverter. The element o the matrix traner unction are calculated uin uperpoition and Thevenin equivalent circuit theorem. i G() v n inv G () i G G... G n vinv i G G... G v n inv i G G... G v n n n nn inv _ n G () () The diaonal element are rearded a the traner unction between the inverter-ide current and it own output voltae. Accordinly, or example G can be calculated i rid voltae and all the inverter output voltae v inv_i are uppoed to be zero except v inv_. For thi purpoe, the auxiliary circuit o Fi. 4 i derived rom Fi.. In thi circuit, the output current i i and the only voltae ource i v inv_. Thereore, the diaonal element G, i directly obtained a ollow: i G () v inv Z Z Z3 Z3 Z33 Z Z Z3 Z Z3 Similarly, other element o the matrix traner unction are calculated a preented in Appendix. G () 4. Control Sytem Dein In order to determine the interaction between loop o a MIMO ytem, relative ain array (RGA) method can be ued [7]. The RGA o a non-inular quare matrix G () i deined a G(). G() T (3) Where and G() i the matrix G () in the teady-tate condition (ω=). The RGA i a quare matrix which ha ome unique propertie, i.e. the um o it row a it column are equal to. I diaonal element o the RGA matrix be cloe to unity, the ytem i diaonally dominant. In other word, interaction o loop in the ytem i relatively low. Since in decribed control loop, only matrix traner unction G () ha non-diaonal element, thereore, the RGA o thi matrix hould be calculated. By uin parameter iven in. denote element-by-element multiplication 6

8 v pcc G m G () z G m G () z G m G () z3 i re i re i re 3 Gi () Gd () K P _ inner G () C. 3 C. 3 C. 33 L. L. L. 3 i i i 3 Fi. 3. Multivariable control loop or three parallel rid-connected inverter with LCL ilter. Vinv_ i Z Z3 Z Z Z Fi. 4. Auxiliary circuit o the three parallel inverter provided that rid voltae and all inverter voltae v inv-i are zero except v inv-. Table, the matrix traner unction a: G() (4) Thereore, the RGA matrix can be eaily calculated a: G(). G() T Z3 Z3 Z3 Z33 Z G() can be calculated (5) Accordin to calculated RGA matrix, diaonal element are cloe to unity and non-diaonal element are cloe to zero. Hence, i cut-o requency o each main loop deined a hih a poible, the interaction o loop can be nelected. It hould be noted that with thi aumption, the couplin due to the rid impedance remain becaue any diaonal element o matrix G (), or example G, include the element o other inverter (Z, Z, Z 3, Z 3 and etc.). However, i in another cae tudy, the diaonal element o RGA matrix be ar rom unity, the pre-compenator matrix can be ued to reduce the interaction [7]. In order to dein the inner loop and outer loop controller, the control ytem block diaram o three parallel rid-connected inverter are hown in Fi. 5-. In thee iure G, G and G 33 are diaonal element o G (). Alo, the PR controller a (6) i ued due to it hih ain at undamental requency. Table Parameter o the inverter and rid Input DC voltae, Vdc_ Inverter-ide impedance Z (R, L) Parameter o the inverter Grid-ide impedance Z (R, L) Impedance o ilter capacitor Z3 (R3, L3) Samplin requency Input DC voltae,vdc_ Parameter o the inverter Inverter-ide impedance Z (R, L) Grid-ide impedance Z (R, L) Impedance o ilter capacitor Z3 (R3, L3) Samplin requency Input DC voltae,vdc_3 Parameter o the inverter3 Inverter-ide impedance Z3 (R3, L3) Grid-ide impedance Z3 (R3, L3) Impedance o ilter capacitor Z33 (R33, L33) Samplin requency Grid Voltae, V (RMS) Fundamental requency Grid impedance Z (R, L) Parameter o the rid 36 V L=33 µh R=. Ω L=33 µh R=.3 Ω C3= µf R3=. Ω 3 khz 36 V L= mh R=. Ω L= mh R=. Ω C3=3 µf R3=.3 Ω 3 khz 36 V L3=6 µh R3=.3 Ω L3= µh R3=. Ω C33= µf R33=. Ω 3 khz V 5 Hz L=.3 mh R= Ω 7

9 G m G i () re G () i Gd () KP _ inner 3 C. L. 3 i () () z v pcc G m G i () re G () i Gd () KP _ inner 3 C. L. 3 i () () z v pcc G m G i () re 3 G () i3 Gd () KP _ inner C. L i () 3 () z3 v pcc k r Gi() k p where, and Fi. 5. Control ytem block diaram Inverter. Inverter. Inverter3. (6) i the undamental requency. 5. Simulation Reult In thi ection, a inle phae microrid with three parallel rid-connected inverter with LCL ilter i imulated uin MATLAB/Simulink otware. The imulation reult are analyzed to validate the theoretical tudy in previou ection and dierent apect o the propoed method are invetiated. The key parameter o the inverter and rid are iven in Table. 5.. Sinuoidal Grid Voltae Condition In the irt tep, to how the neceity o coniderin the couplin eect in multi-parallel rid-connected inverter, two imulation with and without coniderin the couplin eect are perormed in inuoidal rid voltae condition. The control parameter o Set I in Table are deined without coniderin the couplin eect o three inverter. In other word, the block diaram o Fi. i ued or dein o controller o each inverter, individually. The manitude o reerence injected current (i re) o thee inverter are, 3 and 4A, repectively and the correpondin phae anle are. The bode diaram o the total equivalent output impedance ( Zeq Zo Zo Zo3 ) and the rid impedance are hown in Fi. 6. Note that, the rid voltae eedorward method i not applied in thi tep. Accordin to (4), i Z and Z eq interect at i, PM mut be a poitive value to how the ytem tability [], i.e. PM 8 [ Z ( ) Z ( )] (7) i eq i A hown in Fi. 6, the PM ha a neative value at interection point (86 Hz) which how the connection o three parallel inverter i untable. In Fi. 7-, the imulation reult or inle rid-connected inverter are hown or each inverter. A hown in thee iure, Table Parameter o the controller SET I Parameter o the controller o inverter Parameter o the controller o inverter Parameter o the controller o inverter3 Kp_inner 7.35 Kp_inner 37. Kp_inner 5. Kp.7 Kp.9 Kp.65 Kr 35 Kr 33 Kr 8 SET II Parameter o the controller o inverter Parameter o the controller o inverter Parameter o the controller o inverter3 Kp_inner 5.37 Kp_inner.6 Kp_inner 6.4 Kp.66 Kp.34 Kp.6 Kr 38 Kr 66.7 Kr 67 8

10 all three inverter are table when they are connected to the rid, individually. Total harmonic ditortion (THD) o each inverter injected current i iven in Table 3 (cae ). The imulation reult or parallel connection o all three inverter to the rid, with previou control parameter, are hown in Fi. 8-. A hown in thee iure, althouh the individual connection o each inverter to the rid i table, parallel connection o inverter to the rid will be untable which validate the theoritical reult o Fi. 6. The THD o each inverter injected current when all three inverter are paralleled i alo iven in Table 3 (cae ). The imulation reult how that conideration o couplin eect in multi-parallel rid-connected inverter i neceary. In the next tep, Set II control parameter lited in Table i ued or imulation. The controller parameter are deined with conideration o couplin eect a hown in Fi. 5-. The bode diaram o the total equivalent output impedance and the rid impedance are hown in Fi. 9. A hown in thi iure, PM ha a poitive value at the interection point (7 Hz) which how that the connection o three parallel inverter i table. The injected current o three inverter are hown in Fi. -. It can be een that, depite dierence in inverter and their control parameter, injected current track their reerence value. The THD o each inverter injected current when all three inverter are paralleled and with conideration o the couplin eect i alo iven in Table 3 (cae 3). Fi. (d) how the total rid-injected current which i exactly in phae with the PCC voltae and track the um o reerence current thank to PR controller. The imulation reult how that conideration o couplin eect in multi-parallel inverter i neceary and alo validate the propoed control ytem in Fi. 5. Alo, the analytical reult obtained by impedance-baed tability criterion in Fi. 6 and Fi. 9 are validated by imulation reult. The neative PM at interection point lead to ytem intability a hown in Fi. 8 and on the other hand, poitive PM lead to ytem tablity a hown in Fi.. Fi. 6. Bode diaram o the total equivalent output impedance Z eq and the rid impedance without coniderin couplin eect. I(A) I(A) Time() Time() Time() Fi. 7. Simulated waveorm or inle rid-connected inverter without coniderin the couplin eect amon inverter in dein o controller parameter. Injected current by inverter Injected current by inverter Injected current by inverter3. I3(A) Time() Time() Time() Fi. 8. Simulated waveorm or three parallel rid-connected inverter without conideration o couplin eect amon inverter in dein o controller parameter. Injected current by inverter Injected current by inverter Injected current by inverter3. I(A) I(A) I3(A) 9

11 Table 3 The THD o each inverter injected current (inuoidal rid voltae condition) Cae tudy I I I3 Cae- Sinle rid-connected inverter without coniderin the couplin eect in controller dein proce Cae- Three rid-connected inverter without coniderin the couplin eect in controller dein proce Cae3- Three rid-connected inverter with coniderin the couplin eect in controller dein proce 3.83%.65%.48% 47.3% 34.6% 4.55% 3.46%.8%.9% 5.. Non-Sinuoidal Grid Voltae Condition In order to validate the harmonic rejection capability o the propoed rid voltae eedorward method, two cae tudie are imulated in non-inuoidal rid voltae condition: ) without application o rid voltae eedorward method; ) with application o propoed rid voltae eedorward method. In thi cae tudy, the rid voltae i ditorted by third, ith, eventh, ninth, th and 3th harmonic. The manitude o harmonic component with repect to the rid undamental voltae ( V) are 4%, 3%, %,.5%, % and.5%, repectively, and the correpondin phae anle are, 9,, 7, 9, and 5. Fi. -(d) how the imulation reult when the rid voltae eedorward method i not ued. The THD o each inverter injected current and alo, total rid-injected current are iven in Table 4 (Cae ). It can be een that the rid-injected current o each inverter and alo, the total ridinjected current are ditorted and power quality i not acceptable at all. The imulation reult or propoed rid voltae eedorward method are hown in Fi. -(d). It can be een that the power quality o injected current are improved eectively thank to propoed rid voltae eedorward method. The THD o each inverter injected current and total rid-injected current when the propoed rid voltae eedorward method i ued are alo iven in Table 4 (cae ). Fi. 9. Bode diaram o the total equivalent output impedance Z eq and the rid impedance with conideration o couplin eect. I(A) I3(A) I(A) I(A) I3(A) Time() Time() Time() i V pcc Time() (d) Fi.. Simulated waveorm or three parallel rid-connected inverter 4 with conideration o couplin eect amon inverter in dein o controller parameter. Injected current by inverter. Injected current by inverter. Injected current by inverter3. (d) Total rid-injected current (i ) and PCC voltae (V - pcc). -4 A expected, bootin o inverter output impedance uin.5 propoed. method ha.5 a reat impact. on quality.5 improvement o rid-injected Time() current and rid voltae harmonic rejection capability. Simulation reult validate the capability 4 o the propoed rid voltae eedorward method in non-inuoidal rid voltae condition which i compatible with analytical reult o Fi Non-Sinuoidal Grid Voltae in Weak Grid Condition.5 In the next. tep, a comparative.5 tudy.i done, which.5 validate uperiority o Time() the propoed rid voltae eedorward method with repect to traditional one in weak rid condition with L =6mH. A aid earlier, traditional rid voltae eedorward method introduce a deep phae la which could lead to intability when rid impedance increae. Bode diaram o the total equivalent output impedance with traditional and propoed method are hown in Fi. 3 with rid inductance equal to L =6mH. A hown in Fi. 3, when the traditional method i ued, PM ha a neative value at interection point (38 Hz) which how that the ytem i untable. In contrat, when the propoed rid voltae eedorward method i ued, the ytem ha a poitive PM. The imulation reult or traditional rid

12 (d) Fi.. Simulated waveorm or three parallel rid-connected inverter without rid voltae eedorward. Injected current by inverter. Injected current by inverter. Injected current by inverter3. (d) Total rid-injected current (i ). voltae eedorward method are hown in Fi. 4-(d). The harmonic component o the rid voltae are ame a Section 5.. i.e. the manitude o harmonic component with repect to the rid undamental voltae ( V) are 4%, 3%, %,.5%, % and.5%, repectively, and the correpondin phae anle are, 9,, 7, 9, and 5. Table 4 The THD o each inverter injected current (non-inuoidal rid voltae condition) Cae tudy I I I3 I Cae- Three ridconnected inverter without uin rid voltae eedorward method Cae- Three ridconnected inverter with uin propoed rid voltae eedorward method 9.6%.93% 9.44%.57% 4.97%.85%.6%.86% (d) Fi.. Simulated waveorm or three parallel rid-connected inverter with rid voltae eedorward. Injected current by inverter. Injected current by inverter. Injected current by inverter3. (d) Total rid-injected current (i ). It can be een that the injected current are ditorted eriouly, which i compatible with Fi. 3. A tated in Section, the traditional rid voltae eedorward method introduce neative phae anle to the control ytem which could lead to intability in weak rid. The introduced neative phae anle could aect the PM and in cae o hih rid impedance caue neative PM a hown in Fi. 3. The THD o each inverter injected current and total injected current are hown in Table 5 (Cae ). Table 5 The THD o each inverter injected current (non-inuoidal rid voltae in weak rid condition) Cae tudy I I I3 I Cae- Three ridconnected inverter with uin traditional rid voltae eedorward method Cae- Three ridconnected inverter with uin propoed rid voltae eedorward method 4.8% 6.6% 7.5% 8.39% 5.59% 3.77% 4.% 4.%

13 Alo, the imulation reult or propoed rid voltae eedorward method are hown in Fi. 5-(d). The THD o each inverter injected current and total injected current when the propoed rid voltae eedorward method i ued are alo iven in Table 5 (cae ). It hould be noted that the quality o injected current are improved, praieworthy. A hown in Fi. 3, the phae anle increment uin propoed method, caue robut tability o the rid connected ytem aaint rid impedance variation. The imulation reult in lat two cae tudie how the eectivene o the propoed rid voltae eedorward method in comparion with traditional one Preence o Non-linear Local Load The quality o rid-injected current i examined in cae o non-linear load connection. In addition to the rid voltae harmonic ditortion, another ource o current harmonic in a microrid are nonlinear load. Hence, to evaluate the eect o non-linear load on the quality o ridinjected current, a thyritor bride rectiier i connected at the PCC. In thi cae, the rid voltae i conidered inuoidal and the rid inductance i L =.3mH. The load o the thyritor rectiier i an RL load (R=5Ω, L= mh). Manitude (db) Phae (de) Sytem: Zeq propoed Frequency (Hz): 38 Phae (de): Bode Diaram Sytem: Zeq propoed Frequency (Hz): 38 Manitude (db): Sytem: Zeq traditional -8 Frequency (Hz): 38 Phae (de): Frequency (Hz) Zeq traditional Zeq propoed rid impedance Fi. 3. Bode diaram o the total equivalent output impedance Z eq with traditional and propoed method and alo the rid impedance. (d) Fi. 4. Simulated waveorm or three parallel rid-connected inverter with traditional rid voltae eedorward with L =6mH. Injected current by inverter. Injected current by inverter. Injected current by inverter3. (d) Total injected current (i ). (d) Fi. 5. Simulated waveorm or three parallel rid-connected inverter with propoed rid voltae eedorward with L =6mH. Injected current by inverter. Injected current by inverter. Injected current by inverter3. (d) Total injected current (i ).

14 Alo, the irin anle o the rectiier i et to 45. Fi. 6 how the non-linear load current which it harmonic content i extremely hih. The rid-injected current without uin the rid voltae eedorward method i hown in Fi. 6. A hown in thi iure, the harmonic ditortion i created due to non-linear load connection. The THD value o the rid-injected current i equal to 5.5%. Fi. 6 how the rid-injected current with uin the propoed rid voltae eedorward method. A hown in thi iure, the harmonic ditortion o the rid-injected current i mitiated and the THD value i reduced to.4% thank to propoed rid voltae eedorward method. A hown in thi cae tudy, the rid voltae eedorward method can improve the quality o the rid-injected current even in cae o non-linear load connection. Phae hit No Phae Shit 5.5. The Impact o Grid Frequency Variation Finally, the impact o rid requency variation on the rid-injected current with and without uin the propoed rid voltae eedorward method i invetiated in thi ubection. The previou ytem contain three parallel ridconnected inverter i imulated coniderin inuoidal rid voltae and nelectin rid impedance to demontrate the impact o rid requency variation. Manitude o the reerence current (i re) or the invertr are, 3 and 4A, repectively. A tep chane in rid requency i occurred at t=.5 rom 5Hz to 49 Hz. Fi. 7 how the total rid-injected current a well a rid voltae when the rid voltae eedorward method i not ued. A hown in thi iure, the total rid-injected current track the um o reerence current but there i a bit phae hit between the rid voltae and rid-injected current ater requency variation. Thi lead to a bit reactive power exchane. Fi. 6. Simulated waveorm in cae o nonlinear load connection. nonlinear load current rid injected current without uin the propoed method. Grid injected current with uin the propoed method. Fi. 7. Simulated waveorm in cae o rid requency variation. Without uin rid voltae eedorward method. With uin propoed rid voltae eedorward method. Fi. 7 how the total rid-injected current a well a rid voltae uin propoed rid voltae eedorward method. A hown in thi iure, the total rid-injected current i exactly in phae with rid voltae and track the um o reerence current in pite o requency variation. Indeed, by bootin the inverter output impedance the eect o rid voltae i uppreed and the quality o rid-injected current i remained atiactory even in cae o rid requency variation. 6. Concluion In thi paper, an improved rid voltae eedorward method i propoed which enhance the rid voltae harmonic rejection capability in multi-parallel ridconnected inverter. By uin the propoed method, the neative apect o the traditional method, that are introducin neative phae anle to the control ytem and inorin couplin eect amon inverter, are eliminated. The mathematical analyi how that a proportional ain can be ued to prevent phae la in the control ytem and improve the ytem tability in cae o rid impedance variation. Alo, in the propoed method, the couplin eect amon inverter i conidered and the ytem i preciely modeled a a multivariable control ytem. The preented model acilitate the tudy o couplin eect amon inverter with dierent characteritic uch a LCL ilter and rated power. Three parallel rid-connected inverter are conidered a a cae tudy. Then, the control ytem dein uideline are ueted baed on multivariable control theory with coniderin the propoed rid voltae eedorward method and couplin eect amon inverter. Numerou imulation were perormed which validate the eectivene o the propoed rid voltae eedorward method in comparion with the traditional one in cae o rid impedance variation. Theoretical analyi and imulation reult conirm the validity o the model and the neceity to coniderin the couplin eect amon inverter. 3

15 G G G G 3 Appendix i vinv Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z3 Z Z Z Z3 Z3 Z33 Z3 Z Z Z3 i vinv _3 Z3 Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z33 Z3 Z Z Z Z Z3 Z3 Z Z3 Z3 i vinv _ Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z3 Z Z Z Z3 Z3 Z33 Z3 Z Z Z3 i vinv _ Z Z Z3 Z3 Z33 Z Z Z3 Z Z3 G G G i v i v inv _3 3 i v inv _ 3 inv _ Z3 Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z33 Z3 Z Z Z Z Z3 Z3 Z Z3 Z3 Z Z Z Z Z Z Z Z Z Z Z Z3 Z 3 Z3 Z3 Z3 Z Z3 Z Z Z Z3 Z33 Z 3 Z Z33 Z Z Z Z Z Z Z Z Z Z Z Z3 Z 3 Z3 Z3 Z3 Z Z3 Z Z Z Z3 Z33 Z 3 Z Z33 G 33 i 3 vinv _3 Z3 Z Z Z Z 3 Z Z Z 3 Z3 Z33 Reerence [] D. Yan, X. Ruan, and H. Wu, Impedance hapin o the ridconnected inverter with LCL ilter to improve it adaptability to the weak rid condition, IEEE Tran. Power Electron., vol. 9, no., pp , Nov. 4. [] IEEE Standard or Interconnectin Ditributed Reource With Electric Power Sytem, IEEE Std , Jul. 8, 3. [3] S. Jayalath and M. Hani, Generalized LCL-ilter dein alorithm or rid-connected voltae-ource inverter, IEEE Tran. Ind. Electron., vol. 64, no. 3, pp , Mar. 7. [4] E. Kantar and A. M. Hava, LCL-ilter dein or low-voltae hihpower rid-tied voltae-ource converter coniderin variou dampin method, in Proc. IEEE 7th Workhop on Control and Modelin or Power Electronic (COMPEL), 6, pp. -8. [5] W. Li, X. Ruan, D. Pan, and X. Wan, Full-eedorward cheme o rid voltae or a three-phae LCL-type rid-connected inverter, IEEE Tran. Ind. Electron., vol. 6, no. 6, pp. 37 5, Jun. 3. [6] A. K. Balaubramanian and V. John, Analyi and dein o plitcapacitor reitive-inductive paive dampin or LCL ilter in ridconnected inverter, IET Power Electron., vol. 6, no. 9, pp. 8 83, Nov. 3. [7] W. Yao, Y. Yan, X. Zhan, F. Blaabjer, and P. C. Loh, Dein and analyi o robut active dampin or LCL ilter uin diital notch ilter, IEEE Tran. Power Electron., vol. 3, no. 3, pp , Mar. 7. [8] X. Li, X. Wu, Y. Gen, X. Yuan, C. Xia, and X. Zhan, Wide dampin reion or LCL-type rid-connected inverter with an improved capacitor-current-eedback method, IEEE Tran. Power Electron., vol. 3, no. 9, pp , Sep. 5. [9] F. Wu, J. Ye, X. Luo, Z. Zhan, and Y. Li, Reonance characteritic analyi and dampin control or LCL-ilter-baed voltae ource converter durin bu traner, Electr. Pow. Syt. Re., vol. 5, pp. 44 5, Sep. 7. [] X. Wan, X. Ruan, S. Liu, and C. K. Te, Full eedorward o rid voltae or rid-connected Inverter with LCL ilter to uppre current ditortion Due to rid voltae harmonic, IEEE Tran. Power Electron., vol. 5, no., pp , Dec.. [] X. Wan, F. Blaabjer, and P. C. Loh, Virtual RC dampin o LCLiltered voltae ource converter with extended elective harmonic compenation, IEEE Tran. Power Electron., vol. 3, no. 9, pp , Sep. 5. 4

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