Microgrid‐based parallel‐operated voltage‐source inverters: Stability analysis and enhancement in presence of active loads

This paper deals with the stability analysis and enhancement in the AC‐microgrid system comprising parallel‐operated voltage‐source inverters (VSIs) supplying an active‐load. The active‐load behaves such as a constant power load from the AC‐side. In the paper, two models for the active‐load modeling in the AC microgrid are considered: (i) simple model in which the active‐load is modeled as an ideal current source, (ii) detailed model in which all dynamics of the active‐load converter and related controllers are taken into account. The main findings and contribution of this paper are as follows. (a) All linearized state equations of the study system are extracted and transferred into a unified common dq‐reference frame, and then, small‐signal stability of the study system is carried out in the common reference frame by using the simple and detailed models of the active‐load. (b) It is shown that the simple model of the active‐load, as a current source, leads to more pessimistic results in comparison to the active‐load detailed model. (c) Impacts of the active‐load power and bandwidth of the active‐load voltage control loop on the system stability are examined. (d) Control of the generation‐side VSIs is modified and a stabilizing approach, called generation‐side virtual capacitor, is proposed for stabilizing the system under high active‐load penetration. Next, modal analysis and time‐domain simulations are used to investigate the study system responses with and without the proposed stabilizing approach.


| INTRODUCTION
A small power system consisting of a set of different generation sources, active and passive loads, power electronics converters, and transmission lines is called microgrid (MG). 1 The MG electrical power is usually generated by small-scale conventional power plants with synchronous machines or by generation sources employing power electronics converters. 2 Advanced control of a single voltage-source inverter (VSI) in the grid-connected or stand-alone applications has been well-established in the literature.In 3 a design of PWM-sliding mode control (SMC) controller using linearized model for three-phase grid-connected inverter with LCL filter is proposed.In 4 a cascaded control strategy is proposed for a grid-supporting inverter, where, in the external loop, a virtual synchronous generatorbased control strategy is adopted to increase the frequency stability, and in the inner voltage and current control loops, an SMC-based vector control strategy is used.
Parallel-operated VSIs employed in MGs are widely reported in the related literature.Parallel-operated VSIs in the grid-connected mode inject active and reactive power to the grid, 5 while appropriate voltage and frequency are provided by the main grid for proper operation of VSIs.The droop-based control approach, due to its advantages such as simplicity, not needing a communication link and autonomous operation, is widely used in islanded MGs with parallel-operated VSIs. 6In the islanded operation mode, VSIs are responsible for providing voltage and frequency in the allowable range in addition to supplying the demanded load. 5,7[10][11][12][13][14] The loads in MG systems are typically divided into passive or active loads.Passive loads such as incandescent lamps and heaters are modeled as constant impedance loads. 1 The "active-load" phrase can be used for describing a part of loads interfaced with power electronics converters, such as light-emitting diode lamps, closed-loop motor drives, active rectifiers, and front-end converters providing regulated voltage bus. 5 Active-loads with tightly closed-loop control behave such as constant power loads (CPLs) over a short time period. 2,15,16PLs from the small signal perspective show negative incremental resistance resulting in system instability. 12everal papers such as [17][18][19][20][21][22] have dealt with the subject of stability analysis and enhancement in DC MGs supplying CPLs.A few papers such as 15,[23][24][25] have also focused on the dynamic modeling, stability analysis and stability enhancement in converter-based AC MGs with activeloads.In 25 to avoid phase locked loop (PLL) drawbacks, a new method is proposed to track the MG reference voltage for active-load synchronization, and the impact of active-loads on the dynamic stability of the autonomous MG is investigated.Radwan et al. 15 address the stability analysis and active-impedance-based stabilization of active rectifier loads in grid-connected and isolated VSIbased MG systems.However, in most of these papers dealing with active loads, only one voltage-source converter in the generation-side is employed.
In 26 using small-signal analysis, the influence of integrating an active load in the AC MG on the system stability is investigated under different operating conditions and parameters variations.However, no solution is presented for the system stability enhancement.Lenz Cesar et al. 27 deal with the stability analysis of parallelconnected voltage-source converters with CPLs using a bifurcation analysis approach, in which, reduced and simple modes are considered for the CPLs.
Further, for the active-load modeling in the AC MGs, two models can be used: (1) simple model in which the active-load is modeled as an ideal current source, (2)  detailed model in which all dynamics of the active-load converter and related controllers are taken into account.There is one question that "Is the accuracy of the activeload simple model sufficient for stability analysis studies?"In the related literature, there is no comparison between the simple and detailed models of the activeload considering the stability issue.
The subject of this paper is regarding the stability analysis and enhancement in the AC MG system comprising parallel-operated VSIs supplying an activeload.The active-load in this study is an active rectifier regulating its DC-side voltage and behaving such as a CPL from the AC-side.The main findings of this paper are as follows: 1.In the paper, two models for the active-load, known as the simple and detailed models, are considered and the small-signal stability of the system is conducted by using these models.2. It is shown that in the case with modeling the activeload as an ideal current source, the instability occurs at the lower load and stability margin decreases.This means that the simple model of the active-load, as a current source, leads to more pessimistic results in comparison to the active-load detailed model.3. Impacts of the active-load power and bandwidth of the active-load voltage control loop on the system stability are examined.4. Control of the generation-side inverters (GSIs) is modified and a stabilizing approach, known as generation-side virtual capacitor, is proposed for stabilizing the system under high active-load penetration.Next, modal analysis and time-domain simulation results are used to investigate the study system responses with and without the proposed stabilizing approach.
The paper is organized as follows.In Section 2, local modeling and control loops of the generationside and load-side converters are given.In Section 3, all system state equations are transferred into a common reference frame, and then in Section 4, stability analysis of the whole system is conducted in the common reference frame, and next, the system stability is enhanced by modifying the control system of the generation-side VSIs.Finally, in Section 5, by using time-domain simulations, performance of the study system is investigated with and without the proposed stabilizing control approach.

| MODELING AND CONTROL OF THE STUDY SYSTEM COMPRISING PARALLEL-CONNECTED VSCs AND ACTIVE-LOAD
The study system, shown in Figure 1, comprises two parallel-connected VSCs supplying an active rectifier as an active-load.
F I G U R E 1 Study system comprising two parallel-connected GSIs.GSI, generation-side inverter.
In Figure 1, two VSIs, comprising LC filters, are connected to the common bus via transmission lines, that is Lines 1 and 2. In this study, two paralleled VSIs, known as generation-side inverters (GSIs), are controlled based on droop relations in per unit, as given below: where subscript i stands for the ith GSI, ω * i and V * i are references of the angular frequency and voltage amplitude related to the ith GSI in pu, P i and Q i are the active and reactive power output of the ith GSI in pu, P ¯i and Q ¯i are filtered values of the active and reactive power, T i denotes the low-pass filter time constant related to measurement filter of P i and Q i , and m i and n i are droop coefficients related to the GSI frequency and voltage, respectively.The rated values of the GSI frequency and voltage, that is, ω ref and V ref , are equal to 1 pu.According to Equations ( 1) and ( 2), the control model of each VSC is obtained as depicted in Figure 2.

| Control of parallel GSIs
Each GSI is controlled in its local rotating dq frame in which the d-axis is aligned with the local voltage space vector.For instance, the first GSI is controlled in the rotating frame with Park angle argument of θ 1 and dqframe rotating speed of ω 1 , in which the d-axis is aligned with the space vector voltage of voltage v c1 .Hence, the dq components of reference voltages for the ith GSI in the local dq frame are v V * = * cid ci and v * = 0 ciq , where V * ci is obtained according to Equation (2).The control structure of each GSI comprises the inner current and outer voltage control loops.Figure 2 shows the control model and overall control structure of the first GSI, comprising inner current and outer voltage control loops.

| Control loops of GSI
From Figure 1, the relationship between the current and voltage of the fisrt GSI, in pu, can be given as where v inv1 is the voltage generated by the GSI switching and v c1 is the output voltage of the first GSI.By setting Overall control structure of the first GSI.GSI, generation-side inverter.
KHALOOEI and RAHIMI , the closed loop control of the GSI current can be obtained as given in Figure 3. 1 , are achieved from the outer voltage control loops.From Figure 1, the relation between i L1 and v c1 in pu and in the dq frame can be given as From Equation (4), the voltage control loop of the first GSI is obtained as given in Figure 4, where α s α /( + ) i i denotes the inner current control loop.

| Active-load modeling and control
There are two approaches for active-load modeling, (i) ideal modeling, in which, the active-load is modeled as an ideal current source, (ii) detailed modeling, in which, all dynamics of the active-load converter and related controllers are taken into account.

| Ideal model of the active-load
According to Figure 1, the active and reactive power absorbed by the active-load, in pu, can be given as AL od od oq oq AL oq od od oq (5)   where v cd , v cq , i cd and i cq are dq components related to AC-side of the active-load.From Equation (5), i cd and i cq , as a function of the active-load voltage and power, can be given as In the dq-frame aligned with the active-load voltage, we have The linearized form of Equation ( 7) around the operating point is where the symbol Δ denotes the deviation around the operating point, subscript 0 stands for the operating point.From Equation (8), the small signal model of the active-load is obtained according to Figure 5.According to Figure 5, in the ideal model, the active-load from the d-axis current point of view behaves such as a current source , in all frequency range, where the resistance −R 0 may reduce the system stability margin.Also, from Figure 5, the active-load from the qaxis current point of view behaves such as a positive incremental resistance

| Detailed modeling of active-load
The active-load in the study system of Figure 1 is indeed an active rectifier providing a constant voltage at the DCside with constant magnitude and frequency.As mentioned before, the active rectifier is controlled in the rotating dq frame with common bus voltage orientation, where the phase and frequency of the common bus voltage, that is v o in Figure 1, is extracted by the PLL. Figure 6 shows the synchronous reference frame PLL (SRF-PLL) used for extracting the phase and frequency of the common bus voltage v o .Similar to the control loops of the GSI, the closed loop transfer function of the active rectifier current can be given as where ω i represents the active-load current closed-loop bandwidth.
The power balance equation for the DC-link capacitor in the DC-side of the active rectifier can be given as where P AL , P loss , and P R load are active power imported to the active rectifier (in pu), converter loss (in pu) and active power of the resistance R load at the DC-side of the active rectifier, respectively.Also, s b is the base power, and C dc and V dc are the DC-side capacitor and voltage.The linearized form of Equation ( 10) can be given as Equation (11).
From Equation (11), the active rectifier voltage control loop can be given as depicted in Figure 7, where In Figure 7, the term ω s ω / + i i denotes the closedloop transfer function of the inner current control loop, and if the inner current loop is much faster than the outer DC-link voltage control loop, we can replace ω s ω / + i i with 1. From Figure 7, the closed-loop transfer function of v v / * dc dc is obtained as .As will be shown in Section 4, ω v dc as the closed loop bandwidth of the active-load has an important role on the stability of the study system and should be selected carefully.

| STATE-SPACE MODELING OF THE STUDY SYSTEM AT THE COMMON dq REFERENCE FRAME
In the study system of Figure 1, the local voltages related to GSIs with voltages of v c1 and v c2 are connected to the common bus with a voltage of v o through transmission lines.In Section 2, each GSI is independently controlled in its local rotating frame in which the d-axis is aliened with the converter output voltage.In other words, first and second GSIs are controlled in the dq frames with the d-axes aligned with vectors v̲ s c1 and v̲ s c2 , respectively, where v̲ s c1 and v̲ s c2 are the space vectors of voltages v c1 and v c2 in the stationary reference frame and can be given as Also, the active-load is controlled in the dq frame with the d-axis aligned with the vector of common bus voltage v̲ s o .For the control of active-load, the phase of the space vector v̲ s o , that is θ o , is extracted by the PLL, and thus θ pll is used instead of θ o for control purposes.Hence, the active- load is controlled in the PLL dq frame in which the d-axis is aligned with the vector of common bus voltage pll .Figure 8 shows two dq frames in which daxes d 1 and d o , are aligned with the space vectors of v̲ s c1 and v̲ s o , where θ 1 is the angle between the d 1 -axis and α-axis and θ pll is the angle between the d o -axis and α-axis.
Figure 9 shows the simplified model of the study system in which the phase angle of the common bus voltage is considered equal to zero, and the voltage of the i th GSI is assumed to lead the common bus voltage by the In this section, linearized state equations of the entire system comprising GSIs, transmission lines, and activeload are given in a common dq-frame with the d-axis aliened with v̲ s o .In the following, first, the state equations of the first GSI and related transmission line are given.
From Figure 9, the lineraized state equations related to d and q components of Line 1 current, in pu, are given as where the superscript c in the state equations denotes the common dq frame, the subscript 0 stands for operating point and the symbol Δ represents the deviation around the operating point.ω Δ pll in Equation ( 15) is obtained from Figure 6 as ω where x c pll is a state variable related to the PI controller of the PLL and can be given as Further, the state equation related to θ pll in Figure 6 is given by ( ) From Equation (1), the linearized state equation related to P ω − droop equation of the first GSI is given by where ω Δ = 0 ref .Also, from Equation (2), the linearized state equation related to reference voltage magnitude of first GSI, that is, where 19) and ( 20) can be given as According to Figure 8, the dq-frame with the d-axis aligned with v c s 1 leads the common dq-frame by δ 1 , and thus, the dq voltage references of the first GSI in the common dq frame, that is v* c d c 1 and v* c q c 1 , can be given as According to Figure 9, the linearized state equations related to v c d c 1 and v c q c 1 can be given as Also, state equations related to dq-axes voltage controllers of the first GSI can be given as Further, according to Sections 2.1.1,the linearized state equations related to i 1 can be given as where α i denotes the closed loop bandwidth of the GSI current control loop.i Δ *

L d c
1 and i Δ *

L q c
1 in Equation ( 25) are obtained according to Figure 4 and Equation (24) as given below F I G U R E 9 Simplified model of the study system.
Equations (14-26) describe the dynamics of the first GSI.In the same way, the linearized dynamics of the second GSI can be obtained similar to the first GSI, where the state variables of the first and second GSIs can be given as 3.1 | Active-load state-space modeling In the rest of this section, the linearized state-space modeling of the active-load in the ideal and detailed cases are given.According to Figure 1, the linearized state equations of v c od and v c oq are given as In the case with the ideal model of the active-load, i Δ c od and i Δ c oq in Equation (28) are obtained according to Equation (8) as functions of v c od and v c oq .The state variables of the active-load with the ideal model can be given as where x pll is the state variable related to the PLL controller obtained from Equation (17).
In the case with the detailed model of the active-load, the state equations related to i Δ c od and i Δ c oq and related controllers, DC-link voltage and corresponding controller, and PLL are added.
Similar to Equation ( 25), the state equations associated to i Δ c od and i Δ c oq are given as where ω i is the bandwidth of the active-load current control loop.i * c oq in Equation ( 30) is set to zero for achieving unity power factor, and i Δ * c od according to Figure 7 is given as The linearized state equations corresponding to the DC-side voltage of the active-load v dc and active-load voltage controller are given as Equations (16-17), (28), and (30-32) describe the linearized dynamics of the active-load in the detailed case with the following state variables The detailed linearized dynamics of the entire study system comprises 29 state variables , where the superscript T denotes the matrix transpose and x GSI 1 , x GSI 2 and x AL−detailed are obtained from Equations ( 27) and (33).

| SMALL SIGNAL STABILITY ANALYSIS
In this section, small-signal stability of the active-load considering the ideal and detailed models of the activeload are presented.It is noted that in the ideal model, the active load is considered as an ideal current source with a certain power value, and at all frequencies, the small signal model of the active load is a negative resistance.Thus, the active load with an ideal model consists of an ideal current source with a specified power value, without a control loop.Therefore, the discussion of the bandwidth of the control loop about the ideal active load model has no meaning, and as will be shown, the most important parameter affecting on the stability is the amount of active load power.But in the detailed model, the active load includes an ac to AC to DC PWM converter (or active rectifier) with a closed loop control system for the DC side voltage.In the detailed model, at all frequencies, the small signal model of the active load does not behave such as a negative resistance, and only in a limited frequency range, the small signal model of the active load is a negative resistance.As will be shown, in the detailed model, the amount of the active load power and the closed loop bandwidth of the active load voltage control have the greatest impacts on the stability.

| Small signal stability analysis of the study system considering the activeload as an ideal current source
By modeling the active-load as an ideal current source, the linearized dynamics of the entire system comprises 25 state variables , where x x , T T GSI GSI 1 2 and x T AL−ideal as state-variables of GSIs and active-load are according to Equations ( 27) and (29).
The linearized state equations of the study system, extracted in Section 3, can be written in the form of Ax = dx dt , where A is the 25 × 25 state matrix.By using matrix A and doing modal analysis, the eigenvalues of the study system are extracted.Modal analyses conducted on the study system shows that in the case with ideal model of the active-load, the value of active-load power P load has a significant impact on the study system stability.Hence, in this section, the impacts of P load on the stability of the study system of Figure 1 with parameters of Appendix A are examined.In the study system, the rated power of each GSI is 28 KVA and the rated load power is 50 kW.Table 1 shows the eigenvalues of the study system and corresponding dominant state variables at different values of the active-load power.Figure 10 shows the locations of the system modes at different values of the load power.
According to Table 1 and Figure 10, since, the modes λ 13,14 play an important role on the system stability, the root locus of the modes λ 13,14 against the load power change (from 5 to 32.5 kW) is shown in Figure 11.It is clear that for the load power values around the 10 kW and greater than 10 kW, the modes λ 13,14 are unstable.From Table 1 4.2 | Small-signal stability analysis of the study system considering the case with detailed modeling of the active-load By using the detailed model of the active-load, the linearized dynamics of the entire system comprises 29 state variables , where x x , GSI GSI 1 2 and x AL−detailed as state-variables of GSIs and active-load are according to Equations ( 27) and (33).
The linearized state equations of the study system with detailed model of the active-load, extracted in Section 3, can be given in the form of , where A is the 29 × 29 state matrix.By using state matrix A and doing modal analysis, the eigenvalues of the study system are extracted.Modal analysis conducted on the study system shows that in the case with a detailed model of the active-load, the value of the active-load power P load and the bandwidth of the active-load voltage control loop ω v dc have significant impacts on the study system stability.Note: The bold values are dominant modes of the system playing important roles on system stability.

KHALOOEI and RAHIMI
| 3545 Hence, in this section, the impacts of P load and ω v dc on the stability of the study system of Figure 1 with parameters of Appendix A are examined.As stated before, in the study system, the rated power of each GSI is 28 KVA and the rated load power is 50 kW.Δ v dc and v Δ dc belong to the active-load.This means that the there is coupling between the dynamics of the GSIs and active-load in the unstable modes λ 17,18 .According to Table 2, the natural frequency of unstable modes is around the 67 Hz.
Table 3 and Figure 13 show the system modes under different values of ω v -and at P = 29 load kW, where ω v dc is the bandwidth of the active-load voltage control loop.According to Table 3, at P = 29 load kW, the system is stable at ω π = 2 × 44 v dc rad/s, and at ω π = 2 × 55 v dc rad/s, the modes λ 17,18 and thus the system are unstable.This means that increasing the bandwidth of the active-load voltage control loop ω v dc even at light load conditions may result in an unstable system.
Comparing Tables 1 and 2 shows that in the case with modeling the active-load as an ideal current source, the instability occurs at the lower load and stability margin decreases.This means that the simple model of the activeload as a current source leads to more pessimistic results in comparison to the active-load detailed model.

| Stabilizing the study system through modifying the GSIs control
According to Table 1, the state variables related to the output voltages of the GSIs, and the common bus voltage have the most contribution in the unstable modes.Hence, one approach to stabilize the system at presence of CPL is to add a virtual capacitance at the output of each GSI through modifying the GSI control loop.Figure 14 shows the improved control structure of the first GSI in which the dq F I G U R E 11 Root locus of the modes λ 13,14 with increasing load power from 7 to 20 kW.components of the converter reference current are modified by the feedback from the GSI output voltages.In Figure 14, the dq components of the GSI output voltage, passed through the high pass filters and multiplied by gain k v , are used for modifying the GSI reference currents.In Tables 2  and 3, depicting the modal analysis results in the case with detailed model of the active-load, the frequency of unstable modes λ 17,18 is around the 67 Hz, and then in Figure 14, the high-pass filter cut-off frequency should be selected below 67 Hz.In this study, the high-pass filter cut-off frequency is selected equal to 40 Hz, and the gain k v is chosen as k v = 0.55.
λ 24,25 −11.32 ± 11.62i −11.41 ± 11.66i Table 4 shows the eigenvalues of the study system at the rated load power of 50 kW and at the bandwidth of 44 Hz for the active-load voltage control loop once the proposed stabilizing approach is employed.According to Table 4, by applying the proposed control approach, the study system becomes stable for all values of the load power from zero to the rated value (50 kW).

| SIMULATION RESULTS
In this section, the performance of the study system with and without the proposed control approach is examined at different values of active-load power P load and ω v dc , where ω v dc denotes the bandwidth of the load voltage control loop.The study system of    Figure 15 shows the times responses of the study system at P load = 29 kW and ω π = 2 × 44 v dc rad/s.It is clear that in Figure 15 and at P load = 29 kW and ω π 2 × 44 v dc rad/s, the systems responses are stable and this is in agreement with modal analyses results of Table 2.
Also, comparing results of Figure 15 and Table 1 shows that in the case with modeling the active-load as an ideal current source, the instability occurs at the lower load and this means that the simple model of active-load, as a current source, leads to more pessimistic results in comparison to the full detailed model.
Figure 16 shows the times responses of the study system at P load = 44.5 kW and ω π = 2 × 44 v dc rad/s.It is clear that in Figure 12 and at P load = 44.5 kW and ω π = 2 × 44 v dc rad/s, the systems responses are unstable and this confirms the modal analysis results of Table 2.In Figure 16, the active-load converter can not set its DC-side voltage at the reference value of 700 v, and the active power of each GSI is relatively low.Comparing Figures 15 and 16 shows that setting the ω v dc at π 2 × 44 rad/s, and increasing the active- load power from 29 to 44.5 kW moves some modes of the system to the right half s-plane resulting in an unstable system.

| Simulation results by employing the proposed control approach
Figure 17 shows time responses of (a) active-load DCside voltage, v dc , (b) active power output related to first and second GSIs, P 1 and P 2 , (c) system frequency f, and (d) active-load three-phase AC voltages, v o-abc .Figure 17 shows the times responses of the study system at the   17 by employing the proposed stabilizing approach and at the rated load, the systems responses become stable and this is in agreement with the modal analysis results of Table 4.In Figure 18, time responses of the study system with the proposed control approach against the load power change from 30 to 45 kW are examined.In Figure 19, time responses of the study system with the proposed control approach against the step change of the load reference voltage are depicted, in which the reference of the load voltage at the DC-side is changed from the 700 to 800 v at t = 0.2 s.According to Figures 18 and 19, it is clear that by using the proposed control approach, the dynamic time responses of the study system against the changes of the load power and active load reference voltage are stable.
In Figure 20, time responses of the study system with the proposed control approach is examined in presence of both the active load and nonlinear load, where the nonlinear load is indeed a three-phase diode-rectifier, and active powers of the active load and nonlinear load are identically equal to 25 kW.According to Figure 20, it is clear that due to existence of the nonlinear load, the three-phase AC voltages of the load and the active powers of the GSIs have a slight distortion.But the active power sharing between the GSIs is done well, and the voltage at the DC side of the active load is properly stabilized at 700 V.

| CONCLUSION
The subject of this paper is regarding the stability analysis and enhancement in the AC MG system comprising parallel-operated VSIs supplying an activeload.The active-load in this study is an active rectifier regulating its DC-side voltage and behaving such as a CPL from the AC-side.For the active-load modeling in the AC MGs, two models can be used: (1) simple model in which the active-load is modeled as an ideal current source, (2) detailed model in which all dynamics of the T A B L E 4 Eigenvalues of the study system, by using the proposed stabilizing approach, at the rated load power of 50 kW and at the bandwidth of 44 Hz for the active-load voltage control loop.ω ω , , , v q v q vc q vc q c q c q 1 2 1 2  active-load converter and related controllers are taken into account.There is one question that "is the accuracy of the active-load simple model is sufficient for stability analysis studies?"It is shown that in the case with modeling the active-load as an ideal current source, the instability occurs at the lower load and stability margin decreases.This means that the simple model of the active-load, as a current source, leads to more pessimistic results in comparison to the active-load detailed model.
Modal analysis conducted on the study system shows that in the case with a detailed model of the active-load, the value of the active-load power and the bandwidth of the active-load voltage control loop have significant impacts on the study system stability.Then, control of F I G U R E 20 Time responses of the study system with the proposed control approach in presence of both the active load and nonlinear load, (A) active-load DC-side voltage, v odc , (B) active power output related to first and second generation-side inverters, P 1 and P 2 , (C) system frequency f, (D) active-load three-phase AC voltages, v o-abc .
the GSIs is modified and a stabilizing approach, known as generation-side virtual capacitor, is proposed for stabilizing the system under high active-load penetration.Modal analysis and time-domain simulations are used to investigate the study system responses with and without the proposed stabilizing approach.

3
Current control loop of the first GSI.GSI, generation-side inverter.F I G U R E 4 Voltage control loop of the first GSI.GSI, generation-side inverter.

F I G U R E 5
Small-signal model of the active-load considering the ideal model of the load, (A) d-axis model, (B) q-axis model.F I G U R E 6 SRF-PLL for extracting the phase and frequency of the common bus voltage v o .SRF, synchronous reference frame.
load angle δ i , where δ θ θ = − i ipll .The load angles δ 1 and δ 2 can be considered as state variables for the stability studies, and thus we can write

F
I G U R E 7 DC-side voltage control loop of the active rectifier.F I G U R E 8 Local and common dq frames in which d-axes are aligned with space vectors v̲ s c1 and v̲ s o .

F
I G U R E 10 Locations of the system modes at different values of the load power.
The bold values are dominant modes of the system playing important roles on system stability.FI G U R E 12 Locations of the system modes at two different values of the load power.KHALOOEI and RAHIMI| 3547

F I G U R E 13
Locations of the system modes at two different values of the active load control bandwidth.

Figure 1 , 5 . 1 |
Figure 1, with parameters of Appendix A, comprises to GSIs supplying the active-load.The rated voltage, frequency, and power of each GSI 380 V, 50 Hz and 28 kVA, and the rated active-load power is 50 kW.The results of the study system are given in pu where the base values for the voltage and power are V b = 380 V and S b = 25 kVA.

Figures 15 and 16
Figures 15 and 16 show time responses of (a) activeload DC-side voltage, v dc , (b) active power output related to first and second GSIs, P 1 and P 2 , (c) system frequency f, and (d) active-load three-phase AC voltages, v o-abc .Figure15shows the times responses of the study system at P load = 29 kW and ω π = 2 × 44

F
I G U R E 14 Improved control structure of the first GSI.GSI, generation-side inverter.
KHALOOEI and RAHIMI | 3549 rated load power P load = 50 kW and at ω π = 2 × 44 v dc rad/s.It is clear that in Figure

F I G U R E 15
Times responses of the study system without the proposed control approach at P load = 29 kW and ω π = 2 × 44 vdc rad/s, (A) active-load DC -side voltage, v odc , (B) active power output related to first and second generation-side inverters, P 1 and P 2 , (C) system frequency f, (D) active-load three-phase AC voltages, v o-abc .F I G U R E 16 Times responses of the study system without the proposed control approach at P load = 44.5 kW and ω π = 2 × 44 vdc rad/s, (A) active-load DC-side voltage, v odc , (B) active power output related to first and second generation-side inverters, P 1 and P 2 , (C) system frequency f, (D) active-load three-phase AC voltages, v o-abc .F I G U R E 17 Times responses of the study system with the proposed control approach at P load = 50 kW and ω π = 2 × 44 vdc rad/s, (A) active-load DC -side voltage, v odc , (B) active power output related to first and second generationside inverters, P 1 and P 2 , (C) system frequency f, (D) active-load three-phase AC voltages, v o-abc .F I G U R E 18 Times responses of the study system with the proposed control against the load power change from 30 to 45 kW, (A) active-load DC-side voltage, v odc , (B) active power output related to first and second generation-side inverters, P 1 and P 2 , (C) system frequency f, (D) active-load three-phase AC voltages, v o-abc .

F
I G U R E19 Times responses of the study system with the proposed control against the step change of the load reference voltage from the 700 to 800 v, (A) active-load DC -side voltage, v odc , (B) active power output related to first and second generation-side inverters, P 1 and P 2 , (C) system frequency f, (D) active-load three-phase AC voltages, v o-abc .
, the state variables x v , od related to the d-axis output voltages of the GSIs, and dcomponent of the common bus voltage have the most contribution in the unstable modes λ 13,14 .
Eigenvalues of the study system at different values of the load power in the case with ideal modeling of active-load.
T A B L E 1

Table 2 and
Figure 12 depict the eigenvalues of the study system and corresponding dominant state variables at two different values of the active-load power (i.e., 29 and 44.5 kW) and at the bandwidth of 44 Hz for the active-load voltage control loop, that is, ω π = 2 × 44 load kW, the modes λ 17,18 are unstable.From Table 2, the state variables x Δ v d 1 , x Δ v d 2 , x Δ v dc and v Δ dc , related to the d-axis output voltages controllers of the GSIs, DC-side voltage controller and DC-side voltage of the active rectifier, have the most contribution in the unstable modes λ 17,18 , where the state variables x Δ v d 1 and x Δ v d 2 belong to the GSIs, and the state variables x T A B L E 2 Eigenvalues of the study system at different values of the load power and at the bandwidth of 44 Hz for the active-load voltage control loop in the case with the detailed model of the active-load.