Abstract
The power grid in a typical micro distribution system is nonideal, presenting itself as a voltage source with significant impedance. Thus, gridconnected converters interact with each other via the nonideal grid. In this study, we consider the practical scenario of voltagesource converters connected to a threephase voltage source with significant impedance. We show that stability can be compromised in the interacting converters. Specifically, the stable operating regions in selected parameter space may be reduced when gridconnected converters interact under certain conditions. In this paper, we develop bifurcation boundaries in the parameter space with respect to Hopftype instability. A smallsignal model in the dqframe is adopted to analyze the system using an impedancebased approach. Moreover, results are presented in designoriented forms so as to facilitate the identification of variation trends of the parameter ranges that guarantee stable operation.
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1 Introduction
Threephase pulsewidthmodulated (PWM) voltagesource converters (VSCs) operating in rectifier mode have become a popular choice for acdc power conversion in many medium to high power applications. Due to the many desirable features that the VSC offers, e.g., low current harmonics, controllable power quality, and high efficiency, the VSC has been used in a variety of industrial and commercial applications, such as uninterruptible power supply systems, power supplies for telecommunication equipment, HVDC systems, distributed energy sources for renewable energy generation, battery storage systems, power conversion systems for process technology, and so on [1]. In most of these applications, the VSC does not only function as a highperformance power load to the ac power source, but also a reliable interface for many power conversion systems with the ac source.
As possibilities of applications of VSCs increase, it can be appreciated that multiple VSCs may connect to the power grid at a socalled point of common coupling (PCC), while each VSC works under its own specific condition and application, 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], micro grids [5], etc. As each converter is designed separately for its own load condition, the mutual coupling in practice via the grid may pose a stability 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 stability of the system should be considered with the coupled system model in mind rather than the unrealistic though simple model where each converter is assumed to behave independently.
For the purpose of illustrating the effect of mutual coupling, it suffices to consider two converters under the structure shown in Fig. 2. To maintain generality of our study, the VSCs are controlled under a typical feedback configuration that contains an outer voltage loop which operates in conjunction with a sinusoidal pulsewidthmodulated (SPWM) inner current loop for achieving a constant output dc voltage, as presented in Fig. 3. The two converters have been designed separately according to their respective application conditions. However, when they are connected to the nonideal grid, their interaction may have an impact on stability.
The stability issue of gridconnected VSCs in the presence of varying grid impedance has been discussed in several prior publications. For instance, Belkhayat [6] presented an impedancebased approach to evaluate the stability of ac interfaces in the dqframe by employing a generalized Nyquist stability criterion (GNC) based on smallsignal modeling. The stability of gridconnected inverters and rectifiers has also been discussed [7–9]. Moreover, a converter’s impedance modeling has been studied in some detail [10–12]. Some preliminary studies of the effect of grid impedance on gridconnected interacting converters have been reported in [13–15]. Furthermore, some bifurcation phenomena have been observed and reported in some earlier studies [16–18].
In this paper, the important issue of the effect of interaction between the gridconnected converters under nonideal power grid conditions is addressed, which is largely unexplored despite its practical importance. The 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 loss of stability corresponding to emergence of a lowfrequency oscillation as commonly studied under the broad class of Hopftype bifurcation phenomena [19]. We show that the bifurcation boundaries (stability regions from an engineering viewpoint) generally shrink when converters interact via the nonideal grid. In other words, analysis assuming independent (uncoupled) converters under ideal grid condition therefore provides overoptimistic stability prediction. In Sect. 2, we take a quick tour of the instability phenomenon arising from converters’ interaction. In Sect. 3, a detailed stability assessment employing an impedancebased approach is presented. Finally, in Sect. 4, we present our results in a designoriented format that reveals the way in which stability would be affected by variation of selected parameters, and hence facilitates the choice of parameters for stable operation [20].
2 A glimpse at the instability
We begin by taking a quick glimpse at the way in which a voltagesource converter loses stability. The circuit of Fig. 2 is studied in full circuit implementation using MATLAB. The values of the circuit components used in the simulation are summarized in Tables 1, 2 and 3. Converters 1 and 2 employ the same control method which is described in Fig. 3.
Figure 4a shows stable operation of the two converters with different load conditions when separately connected to the same nonideal grids with impedance L_{ g } = 1.2 mH. Thus, the converters are stable when analyzed separately. However, when the converters are connected at the same time to the nonideal grid having the same impedance at the point of common coupling (PCC), as shown in Fig. 2, the converters’ output voltages and input currents have manifested lowfrequency osci.
Figure 4b The dc voltage U_{dc} of both converters oscillate at 255 Hz around the regulated level, while all converters’ parameters have remained the same as in the case of separate uncoupled operation.
Thus, it can be inferred that the converters could interact with each other and lose stability via a Hopftype bifurcation when the values of parameters are selected beyond a specific region in the parameter space (referred to as stability region here).
3 Analysis
For an individual gridconnected VSC, control models and their closedloop stability analysis have been studied [7], employing techniques like root locus analysis and Bode diagrams. Moreover, the statespace approach is also used to determine stability in the time domain [15]. For gridconnected converters, prior studies [6, 21] have pointed out that the impedancebased approach is more advantageous and flexible. For the case of the gridconnected system under study, the impedancebased analysis is highly suitable, as will be demonstrated in the subsequent subsections.
3.1 Basic system model: the uncoupled case
Following the formulation of Sun [22], the threephase VSC can be represented by an averaged model in the dq rotating coordinate. Since converters 1 and 2 share the same model, we provide, for brevity, the equations for converter 1 as follows:
where subscript 1 denotes converter 1 (likewise, subscript 2 denotes converter 2 in the subsequent analysis); ω_{ l } = 2πf_{ l }; d_{1d} = u_{1d}/u_{dc1} and d_{1q} = u_{1q}/u_{dc1} are the duty cycles; i_{1d} and i_{1q} are the input currents of converter 1 in the dqframe; u_{ gd } and u_{ gq } are the voltages at the point of common coupling (PCC); u_{1d} and u_{1q} are the leg voltages in the dqframe; and i_{o1} is the load current. For simplicity, we ignore the inductor resistance r in the following analysis. From (1), we can obtain the smallsignal representation via the usual linearization procedure, i.e.,
where D_{1d}, D_{1q}, I_{1d}, I_{1q} and U_{dc1} are the system’s steadystate values which are given by
Omitting the standard smallsignal derivation, the statespace representation of the closedloop system can be obtained as
where
and
For brevity of presentation, we define
The impedance function of the closed loop system can be written as
Therefore, the characteristic polynomial that determines the stability of the closed loop system can be found as
Thus, we can assess the stability of the system by inspecting the roots of Φ_{ CL }(s) = 0, i.e., eigenvalues of T(s). Stability requires that all eigenvalues have negative real parts. Putting the parameter values in (8), we can compute the eigenvalues as given in Table 4, which indicate stable operation of the uncoupled converters.
3.2 Analysis of the interacting converters
The foregoing subsection derives the conventional converter’s closedloop model and assesses the stability of the converters when working independently (uncoupled). In this subsection, the converters are connected to the power grid at a common coupling point and thus interact with each other. As the two converters have identical control loops, from (7), we can derive the converters’ impedance as
Combining (9) and I_{ o } = U_{ dc }/R_{ L }, we obtain the input admittance of each converter as seen at the coupling point, i.e.,
where Y_{ VSCk } (k = 1, 2) is the converter’s input admittance and Y(s) is the total input admittance of the two converters.
In a likewise manner, the grid impedance can be found from
where
Thus, the system’s impedance can be considered in the usual Thevenin form, as shown in Fig. 5.
Based on this smallsignal model, the current I_{ s } from power grid to the coupled converters is
Since the converters are stable when operating independently, both U_{ s } and Y(s) are stable and the stability of the connected system can be assessed by applying Nyquist criterion [23] to (Fig. 6)
The Nyquist plot of det(I + Y(s)Z_{ s }), as given in Fig. 7, shows encirclement of the origin which asserts that the system is unstable. Numerical calculations of the poles of the closedloop system also indicate unstable operation, as shown in Table 5. This result is in full agreement with the simulations presented earlier.
For a general system with multiconverters connected to a nonideal power grid at the same PCC as shown in Fig. 1, we can track the trajectory to determine the overall system’s stability. The total input admittance of the connected system is Y_{ VSC } = Y_{VSC1} + Y_{VSC2} + Y_{VSC3} + … + Y_{ VSCn }, and the impedance in dqframe could be obtained as shown in Fig. 8. The stability could be assessed by examining the root locus of det(I + Y_{ VSC }Z_{ s }).
The above analysis clearly exposes that the interacting converters can be unstable while the individual separately operated converters are stable.
4 Stability regions in designoriented forms
In the foregoing analysis, the system’s stability has been determined precisely using the interface impedance. However, for engineering design purposes, it is more important to identify the system parameters and their variation trends that have significant impact on the stable region of operation [20, 24]. In this section, we derive stability boundaries using the foregoing analysis and present them in designoriented forms. Results from simulations of the actual switching circuits will also be presented for comparison and verification.
Relevant to our present study is the region of stable operation bounded by the bifurcation boundary corresponding to the loss of stability via a Hopftype bifurcation (practically known as lowfrequency oscillation). For the system of two converters connected to a nonideal power grid, we focus on the following parameters:

1)
the grid impedance L_{ g },

2)
the dc gains of the two converters’ voltage loop k_{vp1}, k_{vp2},

3)
the converters’ load resistance R_{L1}, R_{L2}.
Moreover, for the purpose of comparison, the stability regions of converters coupled via the power grid and those of the independently operating (uncoupled) converters are presented. Our results are generated from simulations of the complete switching model which provides viable verification of the actual physical system.
In the sequel, we will present a series of designoriented stability charts which are computed from the analysis of Sect. 3 and also from full circuit simulations. Each chart contains two sets of stability boundaries, shown in red and blue, corresponding to two specific parameter values as labelled in the chart. In each chart, for the red or blue set of boundaries, the parameter plane is divided into regions I, II and III.

Region I (stable) is the parameter region where the system is stable for both cases of independently operated converters and gridcoupled converters.

Region II (intermediate) is the parameter region where the independently operated converters are stable, but the system of gridcoupled converters is unstable.

Region III (unstable) is the parameter region where the system is unstable for both cases of independently operated converters and gridcoupled converters
Thus, the boundary separating regions I and II is the bifurcation boundary or stability threshold for the system of coupled converters, and the boundary separating regions II and III is the bifurcation boundary for the independently operated converter. In each chart, analytically calculated stability boundaries are plotted in asterisk (*), and the corresponding full circuit simulated boundaries are shown in solid curves.
The dc gains of voltage loop k_{vp1}, k_{vp2} are the key parameters that affect stability with respect to Hopf bifurcation. Since the inner current loop gain is usually chosen to be sufficiently high to ensure that the current tracks the reference closely, we will ignore the inner loop gain in our study. For the system of converters connected at the same PCC, the stability regions in the k_{vp1}–k_{vp2} plane is shown in Fig. 9, for both the uncoupled and coupled systems for different values of dc loads and grid impedance. These charts reveal how variation of these parameters would affect the stability regions, and specifically we see that the stable operating regions are reduced when either dc load current, line impedance or dc link capacitor is increased.
The dc load resistance of the converters can affect stability, as shown in the designoriented charts of Fig. 10. From the charts presented in Fig. 10, when one converter is more heavily loaded, the system moves closer to the boundary of instability. Moreover, we observe that the region of stability shrinks when the converters are coupled via the onideal grid.
From the charts presented above, we see that the analytically computed boundary points are in perfect agreement with the fullcircuit simulated results. The effects of different parameters on the overall system stability can be summarized qualitatively in Table 6. Specifically, the system would alter the stability margin when parameters vary. In the table, the ‘‘ + ’’ and ‘‘−’’ signs indicate increasing and decreasing values of the parameters, respectively. Correspondingly, the sign ‘‘↑’’ means enlarging the overall system stability margin, whereas ‘‘↓’’ represents reducing the stability margin. For instance, when the load decreases (−), the overall system stability region shrinks (↓), and vice versa.
5 Conclusion
Power converters connected to a nonideal power grid are studied in this paper. The emphasis is on the mutual interaction of two or more converters via the grid and how such interaction may cause instability of the overall system. The study is important as converters are being increasingly deployed for applications involving power conversion functions that require interfacing with power grid which is often nonideal. The significant impedance present in the grid poses an issue deserving attention as converters’ stability is no longer a standalone problem. Through mutual interaction, converters become more prone to instability under certain conditions. This paper points out that the overall system stability should be reconsidered in the light of a more complete model that takes into account the interaction of converters connected to the grid at a common point of coupling. The analysis has been developed using the impedance approach, and stability boundaries are derived in various parameter planes. Findings reported in this paper would facilitate engineers in making design choices related to the selection of parameter values that would guarantee stability in a sufficiently wide parameter range.
References
Sanjuan S (2010) Voltage oriented control of threephase boost PWM converters. Master Thesis, Chalmers University of Technology, Gothenburg, Sweden
Wang XF, Guerrero JM, Chen Z et al (2010) Distributed energy resources in grid interactive AC microgrids. In: Proceedings of the 2nd IEEE international symposium on power electronics for distributed generation systems (PEDG’10), 16–18 June 2010, Heifei, China, pp 806–812
Areerak KN, Bozhko SV, Asher GM et al (2012) Stability study for a hybrid ac–dc moreelectric aircraft power system. IEEE Trans Aerosp Electr Syst 48(1):329–347
Ciezki JG, Ashton RW (2000) Selection and stability issues associated with a navy shipboard dc zonal electric distribution system. IEEE Trans Power Deliv 15(2):665–669
Ariyasinghe DP, Vilathgamuwa DM (2008) Stability analysis of microgrids with constant power loads. In: Proceedings of the IEEE international conference on sustainable energy technology (ICSET’08), 24–27 Nov 2008, Singapore, pp 279–284
Belkhayat M (1997) Stability criteria for ac power systems with regulated loads. Ph.D. Dissertation, Purdue University, West Lafayette, IN, USA
Sun J (2011) Impedancebased stability criterion for gridconnected inverters. IEEE Trans Power Electron 26(11):3075–3078
Burgos R, Boroyevich D, Wang F et al (2010) On the ac stability of high power factor threephase rectifiers. In: Proceedings of the IEEE energy conversion congress and exposition (ECCE’10), 12–16 Sept 2010, Atlanta, GA, USA, pp 3075–3078
Belkhayat M, Cooley R, Abed EH (1995) Stability and dynamics of power systems with regulated converters. In: Proceedings of the IEEE international symposium on circuits and systems (ISCAS’95), vol 1, 30 Apr–3 May 1995, Seattle, WA, USA, pp 143–145
Harnefors L, Bongiorno M, Lundberg S (2007) Inputadmittance calculation and shaping for controlled voltagesource converters. IEEE Trans Power Electron 54(6):3323–3334
Cespedes M, Sun J (2012) Impedance shaping of threephase gridparallel voltagesource converters. In: Proceedings of the 27th annual IEEE applied power electronics conference and exposition (APEC’12), 5–9 Feb 2012, Orlando, FL, USA, pp 754–760
Huang J, Corzine K, Belkhayat M (2007) Singlephase ac impedance modeling for stability of integrated power systems. In: Proceedings of the 2007 IEEE electric ship technologies symposium (ESTS’07), 21–23 May 2007, Arlington, VA, USA, pp 483–489
Chandrasekaran S, Boroyevich D, Lindner DK (1999) Input filter interactions in three phase ac–dc converters. In: Proceedings of the 30th annual IEEE power electronics specialists conference (PESC’99), vol 2, 26 June–2 July 1999, Charleston, SC, USA, pp 987–992
Wan C, Huang M, Tse CK et al (2013) Nonlinear behavior and instability in a threephase boost rectifier connected to a nonideal power grid with an interacting load. IEEE Trans Power Electron 28(7):3255–3265
Huang M, Tse CK, Wong SC et al (2013) Lowfrequency Hopf bifurcation and its effects on stability margin in threephase PFC power supplies connected to nonideal power grid. IEEE Trans Circ Syst I Regul Papers. doi:10.1109/TCSI.2013.2264698
Suto Z, Nagy I (2003) Analysis of nonlinear phenomena and design aspects of threephase spacevectormodulated converters. IEEE Trans Circ Syst I: Fundam Theory Appl 50(8):1064–1071
Dai D, Li S, Ma X et al (2007) Slowscale instability of singlestage powerfactorcorrection power supplies. IEEE Trans Circ Syst I Regul Papers 54(8):1724–1735
Xiong X, Tse CK, Ruan X (2013) Bifurcation analysis of standalone photovoltaicbattery hybrid power system. IEEE Trans Circ Syst I Regul Papers 60(5):1354–1365
Tse CK (2003) Complex behavior of switching power converters. CRC Press, Boca Raton, FL, USA
Tse CK, Li M (2011) Designoriented bifurcation analysis of power electronics systems. Int J Bifurc Chaos 21(6):1523–1539
Harnefors L, Bongiorno M, Lundberg S (2007) Inputadmittance calculation and shaping for controlled voltagesource converters. IEEE Trans Ind Electron 54(6):3323–3334
Sun J (2009) Smallsignal methods for ac distributed power systems—a review. IEEE Trans Power Electron 24(11):2545–2554
Macfarlane AGJ, Postlethwaite I (1977) The generalized Nyquist stability criterion and multivariable root loci. Int J Control 25(1):81–127
Rodriguez E, El Aroudi A, Guinjoan F et al (2012) A ripplebased designoriented approach for predicting fastscale instability in DCDC switching power supplies. IEEE Trans Circ Syst I Regul Papers 59(1):215–227
Acknowledgment
The work was supported by Hong Kong Polytechnic University Grants GU866 and GYJ32.
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Wan, C., Huang, M., Tse, C.K. et al. Stability of interacting gridconnected power converters. J. Mod. Power Syst. Clean Energy 1, 249–257 (2013). https://doi.org/10.1007/s405650130034y
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DOI: https://doi.org/10.1007/s405650130034y