1、IEEE TRANIONS ON POWER ELECTRONICS, VOL. 30, NO. 7, JULY 20153589Effects of Interaction of Power Converters Coupled via Power Grid: A Design-Oriented StudyCheng Wan, Meng Huang, Member, IEEE, Chi K. Tse, Fellow, IEEE, and Xinbo Ruan, Senior Member, IEEEAbstractVoltage-source converters are commonly

2、employed as rectifiers for providing a regulated dc voltage from an ac power source. In a typical microdistribution system, the power grid is nonideal and often presents itself as a voltage source with signif- ic mpedance. Thus, power converters connected to the grid interact with each other via the

3、 nonideal grid. In this paper, we study how stability can be compromised in a system of interacting grid-connected converters, which are used typically as rectifiers. Specifically, the stable operating regions in the selected parameter space may shrink when grid-connected converters interact under c

4、ertain conditions. We consider the effect of both source (grid) impedance and transmission line impedance between converters, and derive bifurcation boundaries in the parameter space. A small- signal m in the dq-frame is adopted to analyze the interacting system using an impedance-based approach. It

5、 is shown that the system of interacting converters can become unstable. Moreover, results are presented in design-oriented forms so as to facilitate the identification of variation trends of stable operation boundaries. Experimental results verify the instability phenomenon.Index TermsGrid-connecte

6、d converters, interacting systems, power converters, stability analysis.Fig. 1. Multiple VSCs connected to power grid.energy sources for renewable energy generation, battery stor- age systems, power conversion systems for process technology, and so on 1. In most of these applications, the VSC does n

7、ot only function as a high-performance power load to the ac power source, but also a reliable interface for many power conversion systems with the ac source.As the spectrum of applications of VSCs widens, it can be appreciated that multiple VSCs may connect to the power grid at a so-called common po

8、int of coupling (PCC), while each VSC works under its own specific condition, as shown in the usual representation of Fig. 1 2. The same structure is also employed in some specific application scenarios, including aircraft power systems 3, shipboard power systems 4, microgrids 5, etc. As each conver

9、ter is designed separately for its own load condi- tion, the mutual coupling in practice via the grid may pose a sta- bility concern as the grid is nonideal in reality and presents at the point of common coupling a significant amount of impedance, which is not always predictable. As a result, the st

10、ability of the system should be considered with the coupled system mI. INTRODUCTIONHREE-PHASE pulsewidth-modulatedT() voltage-source converters (VSCs) operating in rectifier mode havebecome a popular choice for acdc power conversion in many medium- to high-power applications. Due to the many desirab

11、lefeatures that the VSC offers, e.g., low current harmonics, con- trollable power quality, and high efficiency, the VSC has been used in a variety of industrial and commercial applications,such as uninterruptible power supply systems, powersfor telecommunication equipment, HVDC systems, distributedM

12、anuscript received March 16, 2014; revised June 27, 2014; accepted August12, 2014. Date of publication August 21, 2014; date of current version February13, 2015. The work was supported in part by Council under GRF Grant 5267/12E, andResearch Grants Polytechnic UniversityGrant G-YJ62. Recommended for

13、 publication by Associate Editor Y. Sozer.C. Wan was with the Department of Electronic and Information Engineer-in mind rather than the unrealistic though simple m each converter is assumed to behave independently., whereing,Polytechnic University,. He is nowwith the State Key Laboratory of Advanced

14、 Electromagnetic Engineering andTechnology, Huazhong University of Science and Technology, Wuhan 430074,For the purpose of illustrating the effect of mutual coupling,(: wchghust).it suffices to consider two converters under the structure shown in Fig. 2. To maintain generality of our study, the VSCs

15、 are controlled under a typical feedback configuration that contains an outer voltage loop, which operates in conjunction with aM. Huang was with the Department of Electronic and Information Engi-neering,Polytechnic University,. He is nowwith the School of Electronic Engineering, Wuhan University, W

16、uhan 430072,(: meng.huangconnect.polyu.hk).C. K. Tse is with the Department of Electronic and Information Engi-neering,Polytechnic University,(:sinusoidalinner current loop for achieving a .hk).output dc voltage, as presented in Fig. 3. The two convertersX. Ruan is with t

17、he College of Automation Engineering, Nanjing Universityhave been designed separately according to their respective ap- plication conditions. However, when they are connected to the grid, their interaction may have an impact on stability. Further- more, in practice, the effect of transmission line i

18、mpedance LLof Aeronautics and Astronautics, Nanjing 210016, ).(: ruanxbColor versions of one or more of the figures in this paper are available online.Digital Object Identifier 10.1109/TPEL.2014.2349936at0885-8993 2014 IEEE.Seeal use is permitted, but republication/redistribution requires IEEE permi

19、ssion. standards/publications/rights/index.html for more information.3590IEEE TRANIONS ON POWER ELECTRONICS, VOL. 30, NO. 7, JULY 2015Fig. 2.Basic mof two VSCs connected to the nonideal grid.Fig. 3.Controller schematic of the three-phase VSC with filter in a rotating dq reference frame.between conve

20、rters should also be considered 6, as represented in Fig. 2.The stability issue of grid-connected VSCs in the presence of varying grid impedance has been discussed in several prior pub- lications. For instance, Belkhayat 7 presented an impedance- based approach to evaluate the stability of ac interf

21、aces in the dq-frame by employing a generalized Nyquist stability criterionstability significantly has rarely been discussed and identified. Furthermore, Chandrasekaran et al. 18, Wan et al. 19, and Huang et al. 20 have reported the effects of grid impedance on grid-connected interacting converters,

22、 but these works focused on the stability of one converter and avoided the complication of analyzing the more realistic situation where two or more con- verters interact in the presence of significant grid impedance. While literature abounds with studies of three-phase rectifiers 2326, in-depth stud

23、y of variation of control parameters on stability of interacting three-phase rectifiers is still glaringly insufficient. Furthermore, from previous study of nonlinear be- havior of power converters, some bifurcation phenomena have been observed and reported 2729, which provide effective analytical t

24、ools for some in-depth study of stability of interacting converters.based on small-signal ming, and for three-phase rectifiers,the analysis assumed an ideal power grid. The stability of grid-connected inverters and rectifiers have also been discussed by Sun 9, Burgos et al. 10, Belkhayat et al. 12,

25、and Vesti et al.11. Moreover, a converters impedance ming has beenstudied in someby Harnefors et al. 13, Cespedes andSun 14, 15, He et al. 16, and Huang et al. 17. However,the key parameters of converters that would affect the systemsWAN et al.: EFFECTS OF INTERACTION OF POWER CONVERTERS COUPLED VIA

26、 POWER GRID: A DESIGN-ORIENTED STUDY3591TABLE IPARAMETERS OF NONIDEAL POWER GRIDFig. 4(a) shows stable operation of the two converters with different load conditions when separately connected to the same nonideal grids with impedance Ls = 1.2 mH. Thus, the con- verters are stable when analyzed separ

27、ately. However, when the converters are connected at the same time to the nonideal grid having the same impedance at the PCC, i.e., LL = 0, as shown in Fig. 2, the converters output voltages have manifested low- frequency oscillations that are typical Hopf-type instability, as shown in Fig. 4(b). Th

28、e dc voltage Udc of both converters oscil- late at 255 Hz around the regulated level, while all converters parameters have remained the same as in the case of indepen- dent (uncoupled) operation. Thus, it can be inferred that the converters could interact with each other and lose stability via a Hop

29、f-type bifurcation when the values of parameters are se- lected beyond a specific region in the parameter space (referred to as stability region here).Moreover, the two converters may not be connected at theusa , s b, sc flLsLL110 Vrms50 Hz0.11.5 mH0100 HTABLE IICIRCUIT PARAMETERS OF CONVERTER 1U dc

30、 , re fC dc L 1RL 1rfs360 V1200 F3 mH3075 0.01 10 kHzParameters are as Shown in Figs. 2 and 3TABLE IIICIRCUIT PARAMETERS OF CONVERTER 2same PCC in practice. Thus, LL = 0. When the convertersUCLRrfdc , re fdc 2L 2are connected to the same grid at different points of coupling, the presence of nonzero

31、transmission line impedance LL be- tween the two converters connection points could affect sta- bility, as shown in Fig. 5. We see that the two converters connected to the nonideal grid at the same PCC are origi- nally stable, as shown in Fig. 5(a), but can become unstable when the coupling points a

32、re separated by nonzero transmis- sion line impedance, as shown in Fig. 5(b). Thus, the convert-s360 V1200 F3 mH3075 0.01 10 kHzParameters are as Shown in Figs. 2 and 3In this paper, we attempt to address the importssue of theeffect of interaction between the grid-connected converters un-der nonidea

33、l power grid conditions, which has remained largely unexplored despite its practical importance. Our study will probe into the key control parameters and examine how variation of these parameters alters the stability regions under the effect of converters interaction. Specifically, we focus on the l

34、oss of stability corresponding to emergence of a low-frequency oscilla- tion as commonly studied under the broad class of Hopf-type bi- furcation phenomena 32. We show that the bifurcation bound- aries (stability regions from an engineering viewpoint) generally shrink when converters interact via th

35、e nonideal grid. In other words, results obtained from analysis that assumes indepen- dent (uncoupled) converters under ideal grid condition, there-ers connected to a nonideal grid can beunstable by thepresence of transmission line impedance separating the points of coupling.III. ANALYSISLiterature

36、abounds with conventional methods for analyz- ing converters stability 9. For an individual grid-connectedVSC, control ms and theird-loop stability analy-sis have been studied, employing techniques like root locusanalysis and Bode diagrams. Moreover, the state-space ap- proach is also used to determ

37、ine stability in the time do- main 20. For grid-connected converters, prior studies 7,22 have pointed out that the impedance-based approach isfore, proveroptimistic stability prediction. In Section II,we present several instability phenomena arising from convert-ers interaction. In Section III, in t

38、he presence and absencemovantageous and flexible. For the case of the grid-of transmission line impedance, aed stability assess-connected system under study, the impedance-based analysis is highly suitable, as will be demonstrated in the subsequent sections.ment employing an impedance-based approach

39、 is presented. In Section IV, we present our results in a design-oriented format that reveals the way in which stability would be affected by vari- ation of selected parameters, and hence, facilitates the choice of parameters for stable operation 33. Moreover, Section V presents the observed instabi

40、lity phenomenon experimentally. Finally, Section VI concludes the paper.A. Basic M: The Standalone ConverterFollowing the formulation of Sun 21, the three-phase VSCcan be represented by aaged min the dq rotatingcoordinate. Since converters 1 and 2 share the same m, weprovide, for brevity, the equati

41、ons for converter 1 as follows:II. GLIMPSE AT THE INSTABILITYWe begin by taking a quick glimpse at the way in which adi1dL1= L1 i1 + u r1 i1 u1lqgd dddtVSs stability. The circuit of Fig. 2 is studied in full circuitdi1qL1implementation using MATALAB. The values of the circuit= L i+ u r1 i1 u1l 1 1dg

42、q qqcomponents used in the simulation are summarized in Tables I, II, and III. Converters 1 and 2 employ the same control method that is described in Fig. 3.dt3dudc1 dt=2 (d1d i1d + d1q i1q ) io1(1)Cdc13592IEEE TRANIONS ON POWER ELECTRONICS, VOL. 30, NO. 7, JULY 2015Waveforms of the two converters o

43、utput voltages with RL 1 = 22.5 and RL 2 = 27 with the same set of control parameters kvp = 2.4, kvi =Fig. 4.20, kip = 24, kii = 100. (a) Two converters work independently (with no interaction) under same grid impedance Ls = 1.2 mH, showing stable operation; (b)two converters interact via connecting

44、 to the nonideal grid with Ls = 1.2 mH, showing unstable operation or Hopf-type instability.Fig. 5. Waveforms of the two converters output voltages with RL 1 = 22.5 and RL 2 = 30 with the same set of control parameters kvp = 2.4, kvi = 20, kip = 24, kii = 100. (a) Two converters interact at the same

45、 PCC with Ls = 1.2 mH and transmission line impedance LL = 0, showing stable operation;(b) two converters interact via connecting to different points of coupling with Ls = 1.2 mH and transmission line impedance LL = 70 H, showing unstable operation or Hopf-type instability.where subscript 1 denotes

46、converter 1 (likewise, subscript 2 denotes converter 2 in the subsequent analysis); l = 2fl ;From (1), we can obtain the small-signal representation via the usual linearization procedure, i.e.d1d = u1d /udc1= u1q /udc1and d1qare the duty cycles; i1dD+ d U+ ugdand i1q are the input currents of conver

47、ter 1 in the dq-frame;di1ddt di1qdtu idc1 1d1d dc1=l 1qugd and ugq are the voltages at the PCC; u1d and u1q are the leg voltages in the dq-frame; and io1 is the load current. For simplic-L1L1D+ d U+ ugqudc1 1q1q dc1= il 1dity, we ignore the inductorr in the following analysis.L1L1WAN et al.: EFFECTS

48、 OF INTERACTION OF POWER CONVERTERS COUPLED VIA POWER GRID: A DESIGN-ORIENTED STUDY3593dudc1dt32Cdc1TABLE IVEIGENVALUES OF THE UNCOUPLED CONVERTERS (INDEPENDENTLY OPERATED)(D+ D+ d Iii=1d 1d1q 1q1d 1dio1Cdc1+ d I ) (2)1q 1qSystemEigenvaluesStabilitywhere D1d , D1q , I1d , I1q , and Udc1 are the syst

49、ems steady-state values, which are given byConverter 11524.885 j 2889.817,Stable4.168, 4.199, 7995.83, 8.029Converter 21936.866 j 2622.119,Stable 4.168, 4.199, 7995.83, 8.0672U 2 Usd Udc1l L1 I1d, I1d =dc1 D1d =, D1q = 3RL 1 UsdUdc1I1q = I1q ref , and Udc1 = Udc1ref .(3)and; the controlledHere, the

50、state variables are i1d , i1q , and udc1 1L101L10001Cdc1variables are d1d and d1q ; and the disturbances are usd , usq , andio . The small-signal mcan be represented in the s-domain as follows:B = 0 .(7)0D1dL1 D0 lI1d (s)I1d (s)For brevity of presentation, we defines 1q(s)=(s)03D1q 2CI1I1qlq L1 1L 1

51、0(s)(s)(s)01L 1I1dUgd UUdc1 (s)3D1d 2CUdc1 (s)I =, U =0, Y 11igI1 (s)dc1dc1qgq Udc1Gi1 L0D1Gv 1 G 10di L1D1d (s)G1id =1, A=L1Udc1q U1idc1G+ 0i1D 01 (s)LD1qL 11L13I3I1q1d 2C2Cdc1003D1d3Gi1 I1d3D1q3I1d l L1dc101L 10A=+1v2C2C2C2C 1 L 100UUdc1dc1 dc1dc1dc1 dc1(s)(s)Ugd +(4)U. 1Cdc13I G Ggq G1idA1vA1i Z1

52、d1 v 1 i11Cdc1T (s)= , Z= , Z =.1o1Io1 (s)2Cdc1 Udc1The impedance function of the ten as follows:d-loop system can be writ-The typical control configuration for VSCs consists of two loops, as shown in Fig. 3. The inner loop is used to minimize the errors between the reference signals (idref , iqref

53、) and the sensed currents (id , iq ), and the outer voltage loop is responsible for I1YOU1ig1= (s1 T (s)(8).Udc1OZ1oIo1minimizing the error between udcref and udc . The duty cyclescan be derived aswhere 1 is the unit matrix. Therefore, the characteristic polyno- mial that determines the stability of the be found asd-loop system canI1d (s)Gi1Udc1l L1G G v 1 i1D (s)1dUdc1 Gi1 Udc1Udc10 =(s)(5)I1qD1q (s) LCL (s) = det(s1 T (s).l 1(9)Udc1 (s)Udc1Thus, we can assess the stability of the system by inspecting the roots of CL (s)