Dodo6/Topic_Modelling_using_LDA

1electronics2Article3 4Modeling and Stability Analysis of Parallel Inverters5in Island Microgrid6Xiaohuan Wang, Hongyang Qing *, Peng Huang and Chunjiang Zhang7Key Lab of Power Electronics for Energy Conservation and Motor Drive of Hebei Province, College of Electrical8Engineering, Yanshan University, Qinhuangdao 066004, China; wxh@ysu.edu.cn (X.W.);9hp779946584@163.com (P.H.); zhangcj@ysu.edu.cn (C.Z.)10* Correspondence: qinghongyang163@163.com11Received: 10 February 2020; Accepted: 7 March 2020; Published: 10 March 202012 13


14
15 16Abstract: The island microgrid is composed of a large number of inverters and various types of power17equipment, and the interaction between inverters with different control methods may cause system18instability, which will cause the power equipment to malfunction. Therefore, effective methods19for analyzing the stability of the microgrid system have become particularly important. Generally,20impedance modeling methods are used to analyze the stability of power electronic converter systems.21In this paper, the impedance models of a PQ-controlled inverter and droop-controlled inverter are22established in d-q frame. In view of the difference of output characteristics between the two control23methods, the island microgrid is equivalent to a double closed-loop system. The impedance model of24the parallel system is derived and the open loop transfer function of the system is extracted. Based on25the generalized Nyquist criterion (GNC), the stability of parallel system working in island microgrid26mode is analyzed using this proposed impedance model. The simulation and experiment results are27presented to verify the analysis.28Keywords: island microgrid; impedance model; stability analysis29 301. Introduction31At present, with the development of new energy sources such as photovoltaics and wind power,32the microgrid has attracted extensive attention from the society and academia [1]. The microgrid is33a single-control independent power generation system composed of distributed generations (DG),34loads, energy storage devices, and control devices. It can operate in grid-connected mode and island35mode [2]. Nowadays, the stability of power electronic networks has received widespread attention [3],36and the stability of island microgrid systems has gradually become the focus of research [4].37In island microgrid system, inverters with different control modes exist in each DG unit. These38inverters are connected to common AC bus through distribution cable. Distribution cable impedance39exists due to the problem of interconnection line distance. The interaction between distribution cable40impedance and different control modes of inverters will cause resonance and affect system stability [5].41Therefore, how to analyze the system stability of the island microgrid through theoretical research and42the accurate design of the system has become an important research direction.43There are many kinds of power electronic devices in microgrid, which are easily affected by44disturbance, so it is necessary to analyze the stability of small signal in microgrid. Small signal stability45refers to the ability that the power system in synchronization state has to return to synchronous46operation without spontaneous oscillation or non-periodic out of step after small interference [6].47There are many analysis methods using small signal modeling, while the traditional method for power48system stability analysis is the state space method, which can analyze the definite meaning of stability.49When analyzing the stability of the state space model, the certainty of the system is needed. However,50Electronics 2020, 9, 463; doi:10.3390/electronics903046351 52www.mdpi.com/journal/electronics53 54Electronics 2020, 9, 46355 562 of 2057 58for a flexible and variable microgrid system, its capacity is small and scattered, and the structure59of the constituent units is easy to change, the parameters need to be recalculated, and portability is60poor [7]. In addition, the state space matrix constructed by the inverter system has a higher order,61and the order of the parallel inverter will increase exponentially. The final model is more and more62complicated and difficult to implement [8]. Because the three-phase stationary coordinate system can63be transformed into the two-phase stationary coordinate system α-β frame and the two-phase rotating64coordinate system d-q frame through the Clark transformation and Park transformation, the model65has a new representation method. Therefore, scholars have proposed an impedance-based modeling66method [9] which takes the input and output of the inverters as impedance models, greatly simplifying67the difficulty of modeling.68Impedance modeling methods are divided into harmonic linearization modeling [10,11] and d-q69frame modeling [12]. In harmonic linearization modeling, complex phase sequence transformation70and algebraic operations are required according to the circuit structure, which increases the difficulty71of modeling and lacks of research on parallel systems. And the small-signal impedance model of the72three-phase inverter established by this method still has coupling impedance components between73positive and negative sequences [13–15]. In [16], a partial impedance model is established in the α-β74frame. The impedance model can be analyzed in the unified coordinate system, but the overall inverter75model with a power loop cannot be established in the α-β frame. In [17], the three-phase voltage-source76inverter is equivalent to two coupled DC systems by using d-q frame modeling method, and the d-q77frame impedance model of inverter is derived by linearization method. However, this impedance78model is generally a two-dimensional matrix model, whereby the traditional Nyquist criterion is not79applicable, so the generalized Nyquist criterion (GNC) [13,18,19] is used to measure impedances in80either d-q or sequence frame to analyze the impedance stability. The d-q frame impedance matrices81of an active front end (AFE), a voltage-source inverter (VSI), and the grid-tied inverters (GTI) are82discussed in [20], which verified the applicability and effectiveness of GNC. The stability of the system83from the perspective of the influence between the parallel inverter and the distribution cable impedance84is discussed in [21]. The source-load system of three-phase inverter can be presented as Norton and85Thevenin equivalents to establish the impedance model of the parallel inverter system [22]. The method86of establishing the impedance model of the parallel converter system and how to analyze the stability87of the impedance model are given in [23]. The circuit analysis methods of the inverter circuit are given88in [24,25], which provides the basis for modeling in this paper. According to the above, the existing89models shown above mainly study the parallel connection of inverters in the α-β frame or the same90control method. This article will complement this research deficiency and establish inverter parallel91models of different control methods in the d-q frame. In addition, the influence of the distribution92cable impedance is also considered. This model can not only study the influence of parameter changes93of a single inverter, but also analyze the parallel stability of multiple units, which has certain novelty.94The rest of this paper is organized as follows. In Section 2, the common output impedance95models of PQ-controlled inverter and droop-controlled inverter are established firstly. In Section 3, the96whole model of island microgrid system is built and the open loop transfer function of the system is97extracted. In Section 4, the stability of the system under different conditions is analyzed by GNC, and98the correctness of the previous theoretical analysis is verified by simulation. In Section 5, the correctness99of the analysis results is verified by experiments, which proves the accuracy and effectiveness of the100model. Finally, the paper summarizes and points out the problems that should be paid attention to in101system design in Section 6.102 103iLa104 105uia106Electronics 2020, 9, 463107 108ioa109uob iob110uoc ioc111 112RL uoa113 114iLb115iLc116 117uib118 119Udc120 121L122 123uic124 125AC BUS126 127Z line128 1293 of 20130 131ic ( a ,b ,c )132 1332. Output Impedance Model of Inverter134 135Rc136 1372.1. Open-Loop Output Impedance Model138 139C140 141Figure 1 is the schematic diagram of the main circuit of the three-phase inverter with a LC filter.142According to the circuit principle, the small signal mathematical model [24,25] of the inverters in d-q143Figure 1. Three-phase inverter topology.144frame can be obtained as shown in Equation (1).145Electronics 2020, 9, x FOR PEER146 REVIEW147 1483 of 20149 150151152153154155156RL + sL - ωL   i Ld  uid  uod 157158159160161=162163164165 ωL RL + sL          166167168169170 i Lq  uiq  uoq 171172173174175-1176177u   i   i 1781791801181+182sR183C184ωR185C186sC187ωC188189190191192193c194c195196 od  =  Ld  -  od 197198199200201202203204   -ωRcC 1 + sRcC   ωC sC          205206207208uoq   i Lq   i oq 209 210Electronics2112020, 9,to212x FOR213PEER REVIEW214According215Equation216(1), the217 218frame can be obtained in Figure 2.219220 221(1)(1)222 223of 20224open-loop transfer function block diagram of the inverter in3 d-q225 226227 228u idq229 230231 232233 234inverter235u odqFigure 1. Three-phase236i Ldq topology.237i odq238-1239GLL240 241242∧   ∧   ∧ 243  RL + sL - ωL   i Ld  = uid  - uod 244  ωL RL + sL  ∧ G245 CC ∧   ∧ 246247 i Lq  uiq  uoq 248249Figure 2. Block diagram of open-loop250transfer251 ∧ function252  ∧ for ∧inverter.253254 1 + sR C - ωR C  -1  sC - ωC255 uod   i Ld   i od 256c257c258259=260  ωC sC   ∧   ∧   ∧ 261The small-signal variable262relationship263ofthe264inverter is set as Equation (2).265  -ωR266Figure267inverter268topology.269 open-loop270 uoqfor271c C 1 +1.sR272cC273Figure 1.Three-phase274Three-phase275inverter276 topology.277  i Lq   i oq 278279280 281282 283(1)284 285286 287Gfunction288I289LL290According291d-q292i odqopen-loop293=294uidqtransfer295+296ublock297odq298According to299to Equation300Equation (1),301(1), the302the303open-loop304transfer305function306block diagram307diagram of308of the309the inverter310inverter in311in(2)312d-q313∧314∧+ G ∧315G316I317G318frame319can320be321obtained322in323Figure3242.325326LL327LL328CC329330331332333334335frame can be obtained in Figure 2.336where337 338GLL339 340  RL + sL - ωL   i Ld  = uid  - uod 341  ωL RL + sL  ∧   ∧   ∧ 342343 i Lq  uiq  uoq 344-1345 RL + sL - ωL 3461-1+ sRcC - ωRcC∧    sC347 ∧i - ωC348 ∧i , I = 1 0349 1 + sR C, -GωR350=351=352Ld353CC c C  sC - ωC  uod354355356c357cC   =ωC358 sC359 -  od   0 1360 ωL R L + sL 361ωRcC 1 + sR362363364365∧366∧367  -ωRcC 1 + sRcC   ωC sC  uoq   i Lq   ∧i oq 368     369370 371(1)372 3732.2. Impedance374Model375of PQ-Controlled376According377to Equation378(1), the Inverter379open-loop transfer function block diagram of the inverter in d-q380 381frame382can 3beisobtained383in Figure3842.385Figure386the control387block diagram388of the PQ-controlled inverter.389Figure 2.3902. Block391of open-loop392open-loop transfer393transfer function394function for395for inverter.396inverter.397Figure398diagram of399*400 401P402 403K p + Ki / s404 405*406 407iod of408The409relationship410The small-signal411small-signal variable412variable1.5413relationship414ofthe415theopen-loop416open-loopdfor417inverterisisset418setas419asEquation420Equation(2).421(2).422dforinverter423u424SPWM425 426iod427 428od429 430∧431GLL4321.5uod∧∧433ioqII ∧∧434∧435G436LL437+438odq =439idq+440odq441uuidq442ii odq443uuodq444*=445*446d447Q448ioq GLL449II +450G qCC451+ GLL452LL453LL GCC454 455(2)456(2)457 458K p + Ki / s459 460where461Figure4623. Block463diagram464of the PQ-controlled465Figure4662. Block467transfer function468# diagram469" of open-loop470#−1inverter.471" for inverter.#472 473"474#475-1476sL - ωL 477RL + sL RL +−ωL4781 +479sC  −ωC  1 0  1 04801 +sRsR481ωRcCc C  sC - ωC482cC483c C -−ωR484GLL =485, GCC486=487,488I489=490GLL = 491, GCC = 492, I=493C inverter494ωL  ωL495RL + R496sLL relationship497ωC is set498sC as 0Equation4991500The small-signal501variable502ofωR503the504ccC505cfor506+ sL507ωR508Copen-loop50911+ +510sRsR5111 0 (2).512c C   ωC sC 513"514 515516 517518 519∧520 521∧522I ∧523G LL524i odq =525u idq +526u odq527G LL528I + G LL G CC5292.2. Impedance Model of PQ-Controlled Inverter530 531Figure 3 is the control block diagram of the PQ-controlled inverter.532where533-1534 RL + sL - ωL 5351 + sRcC - ωRcC   sC - ωC 5361 0 537GLL = 538G539=540 , CC 541  ωC sC  , I = 542543ωL544R545+546sL547ωR548C5491550+551sR552C553554555556L557c  558 c559 0 1560 561(2)562 563G LL564 565I + G LL G CC566 567where568-1569 RL + sL - ωL 5701 + sRcC - ωRcC   sC - ωC 5711 0 572GLL = 573G574=575,576, I=577578CC579580581582583584Electronics 2020, 9, 463585 ωL RL + sL586 ωRcC 1 + sRcC   ωC sC 587 0 1588 5894 of 20590 5912.2.5922.2. Impedance593Impedance Model594ModelofofPQ-Controlled595PQ-ControlledInverter596Inverter597Figure598Figure33isisthe599thecontrol600controlblock601blockdiagram602diagramofofthe603thePQ-controlled604PQ-controlledinverter.605inverter.606 607Figure3.3. Block608Block diagram609diagramof610ofthe611thePQ-controlled612PQ-controlledinverter.613inverter.614Figure615 616According to Figure 3, the expression between the duty ratio and the output current can be617acquired in Equation (3).618"619 620dd621dq622 623#624 625"626 627=628 629Kp + ki /s63006310632Kp + ki /s633 634# "635 636i∗od637i∗oq638 639#640 641"642−643 644iod645ioq646 647#!648(3)649 650From Equations (2) and (3), the unfiltered output voltage of the inverter can be obtained through651small signal processing in Equation (4).652∧653 654∧655 656∧657 658udq = GPWM Gpi (i∗odq − iodq )659where660 661"662GPWM =663 664Gpwm6650666 667#668 6690670Gpwm671 672"673, Gpi =674 675(4)676 677Kp + ki /s67806790680Kp + ki /s681 682#683 684Through the above analysis, the overall current source controlled inverter system can be divided685into two parts: the main circuit and the control circuit. When the influence of the power loop is ignored,686the output reference value of the power loop is constant. The equivalent model of PQ-controlled687inverter is shown in Figure 4. The relationship between output current and output voltage and688input current reference value can be derived in Equations (5) and (6), the final expression of io under689PQ-controlled inverter can be acquired as Equation (7) by Equations (5) and (6).690GPWM Gpi691 692∧693 694i odq |∧695 696uodq =0697 698∧699i odq | ∗∧700iodq =0701 702=−703 704∧705i odq706 707where708Ki =709 710=711 712GLL + GPWM Gpi713 714∧715 716i∗odq717 7181 + GLL GCC ∧719u720GLL + GPWM Gpi odq721∧722 723∧724 725= Ki i∗odq − Yi uodq726 727GPWM Gpi728GLL + GPWM Gpi729 730, Yi =731 7321 + GLL GCC733GLL + GPWM Gpi734 735(5)736 737(6)738 739(7)740 741Through the above analysis, the overall current source controlled inverter system can be divided742into two parts: the main circuit and the control circuit. When the influence of the power loop is743ignored, the output reference value of the power loop is constant. The equivalent model of PQcontrolled inverter is shown in Figure 4. The relationship between output current and output voltage744and input745current746of20io747Electronics7482020,7499, 463 reference value can be derived in Equations (5) and (6), the final expression5 of750under PQ-controlled inverter can be acquired as Equation (7) by Equations (5) and (6).751 752Figure 4. Equivalent model of PQ-controlled inverter.753Figure 4. Equivalent model of PQ-controlled inverter.754 755The PQ-controlled inverter is equivalent to the current source, and the Norton equivalent circuit756Electronics 2020, 9, x FOR PEER REVIEW7575 of 20758is constructed. The Norton equivalent circuit of the PQ-controlled inverter is constructed in Figure 5a.759∧760∧761G762G763PWM when764pi765* is not zero, the Norton equivalent766Considering the influence of distribution767Ziline768|∧ cable= impedance,769(5)7705a. Considering the influence ofi odq771distribution772cable impedance,773when Zline is not zero, the Norton774odq775u odq = 0776G LL + G PWM G pi777circuit is constructed in Figure 5b.778equivalent circuit is constructed in Figure 5b.779∧780 781i odq | ∧782 783*784i odq785=0786 787=-788 7891 + G LLGCC ∧790u odq791G LL + G PWM G pi792∧793*794i odq795 796∧797 798∧799 800i odq = K i - Yi uodq801 802(a)803 804(6)805 806(b)807 808(7)809 810Figure5.5.Norton811Nortonequivalent812equivalentcircuit813circuitof814ofPQ-controlled815PQ-controlledinverter.816inverter. (a)817(a) No818Nodistribution819distribution cable820cable impedance.821impedance.822Figure823(b) Including distribution cable impedance.824 825where(b) Including distribution cable impedance.826 827G828 829G830 8311+G G832 833PWM834pi835LL836CC837Equation(8)838(8)can839canbe840beobtained841obtained842from843Figure844The845existence846ofdistribution847distribution848cableimpedance849impedancewill850will851=852Equation853Figure8545.5.The855of856cable857K i = from858, Yiexistence859G860+861G862G863G864+865G866G867LL868PWM869pi870LL871PWM872pi873affect874the875gain876of877closed-loop878transfer879function880of881PQ-controlled882inverter883and884the885equivalent886output887affect the gain of closed-loop transfer function of PQ-controlled inverter and the equivalent output888impedanceatatPCC,889PCC,which890which891indicates892that893distribution894cable895impedance896have897a certain898impact899on900impedance901indicates902that903distribution904cable905impedance906willwill907have908a certain909impact910on the911The PQ-controlled inverter is equivalent to the current source, and the Norton equivalent circuit912the stability913of PQ-controlled914inverter.915stability916of PQ-controlled917inverter.918is constructed. The Norton equivalent circuit of the PQ-controlled inverter is constructed in Figure919 920∧∧921i oi o==922 923where924where925 926∧∧927∧∧92811929∗930((K931pcc )932i ioi *−-Y933i uu934K935Y9361 + Yi Zline937i o938i pcc )939 940(8)941(8)942 9431 + Yi Zline944 945Zline = Rline + sLline946 947Z948 949= R lin e + sL lin e950 951lin e9522.3. Impedance Model of Droop-Controlled Inverter953 954Figure 6 isModel955the block956diagram of theInverter957droop-controlled inverter, which also ignores the influence9582.3. Impedance959of Droop-Controlled960of the power loop, i.e., the constant output reference value of power loop. The control model of the961Figure 6 is the block diagram of the droop-controlled inverter, which also ignores the influence962voltage and current loop is derived according to Figure 6. The expressions for obtaining the reference963of the power loop, i.e., the constant output reference value of power loop. The control model of the964value of the inductor current in the d-q frame system are listed in Equation (9).965voltage and current loop is derived according to Figure. 6. The expressions for obtaining the reference966value of the inductor current in the d-q frame967system968are listed969in Equation (9).970∧971∧972∧973i∗Ldq = Gv u∗odq − Gv iodq974(9)975where976 977"978Gu =979 980Kpu + kiu /s98109820983Kpu + kiu /s984 985#986 987Figure 6. Block diagram of the droop-controlled inverter.988∧989 990∧991 992∧993 994i*Ldq = Gv u*odq - Gv iodq995 996(9)997 9982.3. Impedance Model of Droop-Controlled Inverter999Figure 6 is the block diagram of the droop-controlled inverter, which also ignores the influence1000of the power loop, i.e., the constant output reference value of power loop. The control model of the1001Electronics10022020,current10039, 463 loop is derived according to Figure. 6. The expressions for obtaining the reference10046 of 201005voltage and1006value of the inductor current in the d-q frame system are listed in Equation (9).1007 1008Figure 6. Block diagram of the droop-controlled inverter.1009 1010Then construct an expression for the duty ratio in Equation (10).1011∧1012 1013∧1014 1015∧1016 1017i*Ldq = Gv u*odq1018∧ - Gv∧iodq1019∧1020ddq = Gi (i∗Ldq − iLdq )1021 1022where1023where1024 1025(9)1026(10)1027 1028"1029 1030#1031Kpi1032+ k+ii /s10330 1034K1035k1036/1037s1038010391040pu1041iu1042Gi =1043Gu =  01044Kpi + kii/s1045 0 K pu + kiu / s1046The droop control inverter takes the output voltage as the control amount, and can express the1047Then construct1048expression1049for variables1050the duty ratio1051in Equation1052(10). (11).1053relationship1054betweenan1055the1056small signal1057as follows1058in Equation1059 ∧∧1060∧∧1061∧∧1062106310641065=G1066Giii(ii *oLdq+-G1067iLdq1068 didqL =1069id d)10701071∧1072∧10731074∧10751076 u1077o = Gui i o + G d1078 1079(10)1080(11)1081 1082ud1083 1084The relationship between the inductor current and the output current in the three-phase stationary1085coordinate system can be obtained in Equation (12).1086∧1087iL1088∧1089io1090 1091=1092 1093sL + RL +109411095sC1096 109711098sC1099 1100+ RC1101 1102+ RC1103 1104(12)1105 1106The relationship between the inductor current and the output current in d-q frame can be obtained1107in Equation (13):1108 ∧ 1109 ∧ 1110 i od 1111 i Ld 11121113111411151116(13)1117 ∧  = Gii  ∧ 1118i Lq1119i oq1120where1121Gii =1122 1123I1124I + GCC GLL1125 1126In a similar way, the small-signal relationship between duty cycle and inductor current can be1127obtained in Equation (14).1128∧1129iL1130∧1131 1132=1133 1134Gpwm1135sL + RL +1136 113711138sC1139 1140+ RC1141 1142(14)1143 1144d1145The transfer relationship between the duty ratio and the inductor current can be obtained in1146Equation (15).1147GPWM GCC1148Gid =1149(15)1150I + GCC GLL1151The relationship between output current and output voltage can be obtained in Equation (16) and1152the corresponding relationship between output voltage and duty cycle can be obtained in Equation (17).1153Gui =1154 1155GLL1156I + GLL GCC1157 1158(16)1159 1160Electronics 2020, 9, x FOR PEER REVIEW1161 11627 of 201163 1164G LL1165Gui =1166G LLGCC1167Gui = I + G LL1168I + G LL GCC1169 1170Electronics 2020, 9, 4631171 1172(16)1173(16)11747 of 201175 1176G PWM1177Gud =1178G1179PWM1180I +GG1181PWM1182CC G LL1183GG1184udud==1185I I++GGCC GGLL1186CC LL1187 1188(17)1189(17)1190(17)1191 1192The1193Theclosed-loop1194closed-loopcontrol1195controlblock1196blockdiagram1197diagramafter1198afterthe1199thepower1200powerloop

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