Modeling of Multiple Master–Slave Control under Island Microgrid and Stability Analysis Based on Control Parameter Configuration

Abstract

The stable operation of a microgrid is crucial to the integration of renewable energy sources. However, with the expansion of scale in electronic devices applied in the microgrid, the interaction between voltage source converters poses a great threat to system stability. In this paper, the model of a three-source microgrid with a multi master–slave control method in islanded mode is built first of all. Two sources out of three use droop control as the main control source, and another is a subordinate one with constant power control which is also known as real and reactive power (PQ) control. Then, the small signal decoupling control model and its stability discriminant equation are established combined with “virtual impedance”. To delve deeper into the interaction between converters, mutual influence of paralleled converters of two main control micro sources and their effect on system stability is explored from the perspective of control parameters. Finally, simulation and analysis are launched and the study serves as a reference for parameter setting of converters in a microgrid.

1. Introduction

Increasing exhaustion of fossil fuel and concerns about environmental pollution have led to extensive exploration of alternative energy sources. Of special interest are renewable energy sources (RES) such as solar and wind energy generation. This has resulted in the emergence of distributed generators (DGs) and the concept of microgrid (MG) has been introduced for proper utilization of DGs [1,2]. As different DGs have different characteristics and natural uncertainty, their presence can have negative effects on MG. Deviations of voltage and frequency are larger when MG operates in islanded mode in comparison with grid-connected mode [3,4]. Therefore, research on the stability of MG in islanded mode is of vital importance.

Usually, voltage source converters (VSCs) [5] are used to interconnect different DGs, whose output is mostly in the form of direct or non-common frequency alternating current. Although the increasing number of power electronic converters improves the control speed, it brings fuzzy influence on stability owing to the mutual influence between them [6]. To address the abovementioned issue, various methods have been proposed in the literature. In [7,8], the sensitivity of load voltage and voltage compensation term are introduced to control the bus voltage as well as improve the response speed and accuracy. However, limitations in applicable scope of load and voltage reduce its practicality. In [9,10], energy storage devices are used not only to suppress the transient power fluctuations, but also to greatly improve MGs’ stability thanks to a super-capacitor. However, this method affects the entire system and declines its inertia. In [11,12], a model that contains converters, controllers and alternating current (AC) grids is built, and high order state space expressions are formulated to determine its stability. However, the mutual influence between converters is neglected and a large number of equations make stability analysis more complex. To improve the accuracy of analysis, the model of MG has been improved for higher accuracy [13,14,15,16], including a linear model of MG combined with a boost converters model [13]; a global model ignoring the change in rotating angles and coupling terms between DGs [14]; a dynamic model of multi-module comprising multi-class of DGs and loads [15]; and a new type of small-signal model containing secondary control that analyzed the stability of the system with eigenvalue [16]. However, the models established in the above literature didn’t take into account the impact of electronic devices in grids. In view of this problem, some research delves deeper into the impact of control parameters on system stability [17]. It employs the root locus method to explore the reasonable ranges of control parameters by a small signal model. When multi-sources are paralleled, droop control can achieve coordination control without communication and the master–slave control method considers the power tracking of PV and wind power and the system stability [18,19]. However, the influence in different control parameters on the system is not considered, for their parameters are partly or almost the same.

In view of the problem in [18], this paper builds a model of multi-source MG, emphasizing the exploration of the mutual influence between converters on stability. To satisfy the decoupling condition, “virtual impedance” is introduced and a three-source decoupling small-signal model is built in addition to the characteristic polynomial also being deducted. On this basis, the mutual influence between converters on stability is analyzed on the level of control parameters by means of simulation studies. This method and the result can provide a reference for parameters’ configuration of converters in MG with a large number of electronic devices plugged in.

2. The Structure of Multi-Source Microgrid and Analysis of Its Problems

The diversity of DGs is a common feature of MGs [15]. Therefore, adopting appropriate control methods is an accurate and fast way to achieve the stable operation of islanded MG.

2.1. Multi Master–Slave Control Structure of MG

The traditional master–slave control method consists of one master controller and several slave controllers, which forms the energy supply system (ESS), in order to take into account the power tracking the renewable energy and the system stability [19]. With the increase in the permeability and scale of MG, however, a single master controller can no longer maintain the regulation of voltage and frequency. Hence, several master control sources should be set up with a droop control method. This solution is adopted in DGs (e.g., diesel engine, micro gas turbine, fuel cell, etc.), which regulate bus voltage and frequency. Relatively, photovoltaic and wind turbines with PQ control methods run as slave sources. The use of droop control alone implements the peer-to-peer control method, and it realizes the peer-to-peer control method, which can achieve local control and coordination control without communication.

Figure 1 shows a simplified structure of three-source MG in islanded mode. In Figure 1, V_odn + jV_oqn and R_n + jX_n are the output voltage of DGn and the equivalent impedance between bus and its converter; i_dn + ji_qn is the output current of DGn (n = 1, 2, 3); U_Gd + jU_Gq is the bus voltage of MG. DG1 and DG2 serve as master controllers adopting a droop control method, embodying the flexibility and redundancy of a peer-to-peer control method in power vacancy allocation; DG3 serves as a slave controller adopting the PQ control method, ensuring its constant output in islanded mode; Load is comprised of resistance and reactance. This system adopts a master–slave control method when three DGs work and it adopts a peer-to-peer control method when DG3 quits running.

2.2. The Problem under Study

Fully symmetrical structure is mostly used in research on multi-source MG, whose parameters are the same in the same control method, ignoring the independence of control parameters of different DGs. The goal of this paper is to evaluate the following aspects: (1) whether there are interactions between control parameters to maintain stability when several converters operate in parallel; (2) whether there are interactions between control parameters of paralleled master controllers when a system’s control strategy changes from single master–slave control to multiple master–slave control method; and (3) how these interactions influence the stability of MG.

This paper verified the influence of droop coefficient difference on a multi master–slave control method based on the system model as shown in Figure 1. Set up the active droop coefficients of DG1, DG2 as 5 × 10⁻⁵, 10 × 10⁻⁵; and the reactive droop coefficients of them as 0.003; other parameter settings are shown in Appendix A; the waveform of bus voltage under different working conditions is shown in Figure 2. Figure 2a indicates the public bus bar voltage waveform when it works with DG1 and DG3, which contains only one master micro source; similarly, Figure 2b indicates the bus voltage waveform when it works with DG2 and DG3. The bus voltage can be kept stable in the two cases. Under the same coefficient setting, when DG1, DG2 and DG3 run together, the bus voltage waveform is oscillating and the microgrid is unstable as shown in Figure 2c. The cause of this phenomenon may be that the Cooperative operation of DG1 and DG2 makes the two controllers interact with each other, so that the operation of the microgrid is deviated from its stable running point, and then the instability occurs.

Based on this conjecture, this paper delves deep into the interaction between control parameters under multiple master controllers’ conditions, combining the case with the theoretical model.

3. Theoretical Analysis of Control Methods

3.1. PQ Control Method

To improve the response speed and accuracy of power distribution, the PQ control method is adopted [20,21,22]. Double-loop control structure is employed: the inner loop uses current control to ensure response speed, thereby improving the operation characteristics of the system; the outer loop adopts power control [23].

The structure of inner current loop is shown in Figure 3 [24].

In Figure 3, V_cd and V_cq are the d,q axis voltage component of converters and V_od and V_oq are the d,q axis voltage components of the filter; L_f is the inductance of filter; K_ip and K_ii are the proportional and integral parameters of the proportional-integral (PI) controller.

To satisfy the decoupling condition of the d,q axis, “virtual impedance” is introduced in low-voltage AC MG [25,26]. After decoupling, structure of the q axis is chosen to analyze because d,q share the same structure. Considering the pulse-width modulation (PWM) and the sampling process, the control diagram is shown in Figure 4.

As shown above, T_s is the time constant of sampling and its value is pretty small. 1/(T_ss + 1) and 1/(0.5T_ss + 1) are the delay modules of sampling and PWM, and they can approximately merge into 1/(1.5T_ss + 1) [27]. Set up the switching frequency of the system as: f_s = 20 kHz, T_s = 1/f_s = 0.05 ms. The gain of the modulator is K_pwm = U_dc/2 = 400.

Then, the open-loop transfer function is as shown

$$G_{iq0}(s)=\frac{K_{ip}{K}_{pwm}}{(1.5T_{s}s+1)L_{f}s}.$$

(1a)

Similarly, the open-loop transfer function of the d-axis is

$$G_{id0}(s)=\frac{K_{ip}{K}_{pwm}}{(1.5T_{s}s+1)L_{f}s}.$$

(1b)

In order to impose that the cut-off frequency of control is 0.1 times the switching frequency, the following relations must be satisfied:

$$\{\begin{array}{l}{\omega }_x=\frac{2\pi f_s}{10}\\ 20\lg \left|G_{iq0}(j{\omega }_x)\right|=0\end{array},$$

(2)

where ω_x is the designed cut-off frequency.

If the switching frequency of the PWM modulator is 20 kHz, set up the inductor-capacitor (LC) passive filter’s inductance L_f as 1.5 mH. K_ip ≈ 0.065 can be obtained. The design principles of the converter output LC passive filter is

$$\{\begin{array}{l}10f_n\le f_c\le f_s/5\\ 2\pi f_c{L}_f=1/(2\pi f_c{C}_f)\end{array},$$

(3)

where f_c is the designed resonant frequency of filter, and f_n is the fundamental frequency. Setting up L_f is 1.5 mH and C_f is 20 μF.

Based on the above analysis, the control structure of the PQ control method is shown in Figure 5 [28].

As shown, K_pp and K_pi are the proportional and integral parameters of the PI controller for active power, respectively; K_Op and K_Oi are appropriate parameters for reactive power. Due to the symmetry of the two parts, it makes K_Pp = K_Qp and K_Pi = K_Qi. In addition, P^*, Q^* are the reference values of active and reference power, respectively. After decoupling, the open-loop transfer function is shown

$$\{\begin{array}{l}{G}_Q(s)=\frac{1.5U_{pcc}(K_{Qp}s+K_{Qi})}{C_{f}X{s}^2}{G}_{id}(s)\\ G_P(s)=\frac{1.5U_{pcc}(K_{Pp}s+K_{Pi})}{C_{f}X{s}^2}{G}_{iq}(s)\end{array}.$$

(4)

3.2. Inductive Droop Control

The DGs with droop control can independently adjust the balance of frequency and voltage and control the operation of MG as the main control micro source. Its control structure still adopts voltage-current double loop structure, where the principle of its current control loop is similar to Section 2.1. Likewise, the q-axis structure of voltage control loop after decoupling is shown in Figure 6.

The open-loop transfer function in this structure is [29]

$$G_{vq0}(s)=\frac{G_{iq}(s)(K_{vp}s+K_{vi})}{C_f(T_{s}s+1)s^2}.$$

(5)

It is necessary to install an LC passive filter at the converter outlet to filter high order harmonics, which can be designed as Equation (3).

This paper adopts: f_n = 50 Hz, f_s = 20 kHz, f_c = 1000 Hz. Considering the voltage drop on filter inductance, L_f = 1.5 mH, and C_f = 16.89 μF can be obtained. At this point, the PI parameters of outer loop must be satisfied:

$$20\lg \left|G_{vq0}(j{\omega }_x)\right|=0.$$

(6)

Taking the stability and rapidity into account, K_vp = 0.1 and K_vi = 407.65 can be obtained; then, K_vi = 400.

The converter is connected to the point of common coupling (PCC). The voltage of this point is a planned constant and selected as the reference voltage, that is: U_pcc < 0 = U_pcc + j0, the output voltage of converter filter is: V₀ < θ = V_0d + jV_0q. Supposing that the impedance of low-voltage MG is R_l + jX_l, where X_l = ωL_l, ω is the AC signal frequency, L_l is the equivalent inductance of transmission line; then, the voltage and current relation of the d and q axes are

$$\{\begin{array}{l}{V}_{0d}-U_{pcc}=(R_l+L_{l}s)I_{0d}-X_l{I}_{0q}\\ V_{0q}-0=(R_l+L_{l}s)I_{0q}+X_l{I}_{0d}\end{array}.$$

(7)

The expression of small signal quantity of the current as shown is

$$\{\begin{array}{l}\Delta I_{0q}=\Delta I_{0q1}+\Delta I_{0q2}=-H_a\times \Delta V_{0d}+H_b\times \Delta V_{0q}\\ \Delta I_{0d}=\Delta I_{0d1}+\Delta I_{0d2}=H_b\times \Delta V_{0d}+H_a\times \Delta V_{0q}\end{array},$$

(8)

where H_a and H_b can be expressed as

$$H_a=\frac{X_l}{X_l^2+{(R_l+L_{l}s)}^2};H_b=\frac{(R_l+L_{l}s)}{X_l^2+{(R_l+L_{l}s)}^2}.$$

(9)

Actually, I_0d and I_0q are DC variables, and their differential results are negligible for L_ls and much smaller than X_l.

P + jQ = U_pcc(I_0d − jI_0q) is the power equation in the dq0 coordinate system, and the power small signal model can be obtained:

$$\{\begin{array}{l}\Delta Q=-1.5U_{pcc}(\Delta I_{0q1}+\Delta I_{0q2})\\ \Delta P=1.5U_{pcc}(\Delta I_{0d1}+\Delta I_{0d2})\end{array}.$$

(10)

For double loop structure, the virtual impedance is equivalent to introducing a logical inductive feedback link in essence, which corrects the reference voltage of dual-loop control and controls the output power further.

Supposing that θ is quite small, then V_0q^*= E^*sinθ ≈ E^*θ, V^*_0d ≈ E^*, where E^* is the reference voltage from reactive droop loop. Furthermore, it can be considered that ΔV^*_0q ≈ EΔθ because changes of E^* are much smaller than that of θ, where E is the reference voltage before reactive droop loop. The control structure is shown in Figure 7 [26].

After reasonable setting of virtual impedance, it is equivalent that the previous line impedance connects in series with a reactance X_v, which satisfies the following relationship X_v + X_v >> R_l, and then R_l in H_a and H_b can be neglected, that is H_b = 0, indicating the realization of decoupling in droop control.

4. Small Signal Modeling of Three-Source MG

Theoretical analysis of a three-source MG with all DGs working at the same time is performed on the basis of model built in Section 1.1. To help analyze the stability of system, the electrical quantities are expressed in steady state small signal equations [30,31], according to the relationship of voltage and current in dq0 coordinate system:

$$\{\begin{array}{l}\Delta V_{odn}-\Delta U_{Gd}=(R_n+\frac{X_n}{\omega }s)\Delta i_{dn}-X_n\Delta i_{qn}\\ \Delta V_{oqn}-\Delta U_{Gq}=(R_n+\frac{X_n}{\omega }s)\Delta i_{qn}+X_n\Delta i_{dn}\end{array}.$$

(11)

Generally, the differential terms in Equation (11) has less influence and can be ignored. Therefore, the system model satisfies Equation (12):

$$\{\begin{array}{l}\Delta V_{od1}-\Delta U_{Gd}=R_1\Delta i_{d1}-X_1\Delta i_{q1}\\ \Delta V_{oq1}-\Delta U_{Gq}=R_1\Delta i_{q1}+X_1\Delta i_{d1}\\ \Delta V_{od2}-\Delta U_{Gd}=R_2\Delta i_{d2}-X_2\Delta i_{q2}\\ \Delta V_{oq2}-\Delta U_{Gq}=R_2\Delta i_{q2}+X_2\Delta i_{d2}\\ \Delta V_{od3}-\Delta U_{Gd}=R_3\Delta i_{d3}-X_3\Delta i_{q3}\\ \Delta V_{oq3}-\Delta U_{Gq}=R_1\Delta i_{q3}+X_1\Delta i_{d3}\\ 0-\Delta U_{Gd}=R_4\Delta i_{d4}-X_4\Delta i_{q4}\\ 0-\Delta U_{Gq}=R_2\Delta i_{q4}+X_4\Delta i_{d4}\\ \Delta i_{d1}+\Delta i_{d2}+\Delta i_{d3}+\Delta i_{d4}=0\\ \Delta i_{q1}+\Delta i_{q2}+\Delta i_{q3}+\Delta i_{q4}=0\end{array}.$$

(12)

Assuming that ΔV_od1, ΔV_oq1, ΔV_od2, ΔV_oq2, ΔV_od3, ΔV_oq3, R₁, X₁, R₂, X₂, R₃, X₃, R₄, X₄ are known, Δi_d1, Δi_q1, Δi_d2, Δi_q2, Δi_d3, Δi_q3, Δi_d4, Δi_q4 can be expressed with the above quantities by solving Equation (12), ΔU_Gd, ΔjU_Gq by Equation (13):

$$\{\begin{array}{l}\Delta U_{Gd}=J_1\Delta V_{od1}+J_2\Delta V_{od2}+J_3\Delta V_{od3}+J_4\Delta V_{oq1}+J_5\Delta V_{oq2}+J_6\Delta V_{oq3}\\ \Delta U_{Gq}=J_7\Delta V_{od1}+J_8\Delta V_{od2}+J_9\Delta V_{od3}+J_{10}\Delta V_{oq1}+J_{11}\Delta V_{oq2}+J_{12}\Delta V_{oq3}\end{array}.$$

(13)

It is clear that X₁ >> R₁, X₂ >> R₂ [25,26] when DG1 and DG2 adopt the droop control method, and then the influence of line resistance can be ignored, that is: R₁ = R₂ = 0. Suppose that the load is mostly active and then: X₄ = 0. Therefore, the coefficients in Equation (13) can be expressed as

$$\{\begin{array}{l}{J}_1=J_{10}=[R_3^2{R}_4^2(X_1{X}_2+X_2^2)+R_4^2{X}_2{X}_3(X_1{X}_2+X_1{X}_3+X_2{X}_3)]/D_d\\ J_2=J_{11}=[R_3^2{R}_4^2(X_1{X}_2+X_1^2)+R_4^2{X}_1{X}_3(X_1{X}_2+X_1{X}_3+X_2{X}_3)]/D_d\\ J_3=J_{12}=[R_3{R}_4{X}_1^2{X}_2^2+R_4^2{X}_1{X}_2(X_1{X}_2+X_1{X}_3+X_2{X}_3)]/D_d\\ J_4=-J_7=[R_4{X}_1{X}_2^2(R_3^2+R_3{R}_4+X_3^2)]/D_d\\ J_5=-J_8=[R_4{X}_1^2{X}_2(R_3^2+R_3{R}_4+X_3^2)]/D_d\\ J_6=-J_9=[-R_3{R}_4^2{X}_1{X}_2(X_1+X_2)+R_4{X}_1^2{X}_2^2{X}_3]/D_d\\ D_d=R_3^2{R}_4^2{(X_1+X_2)}^2+{(R_1+R_2)}^2{X}_1^2{X}_2^2+X_1^2{X}_2^2{X}_3^2+2R_4^2{X}_1{X}_2{X}_3(X_1+X_2+X_3)+R_4^2{X}_3^2(X_1^2+X_3^2)\end{array}.$$

(14)

After ignoring the differential terms, model of droop control is shown by Equations (13) and (14) is obtained from model of PQ control:

$$\{\begin{array}{l}[\Delta E_1^{*}-\frac{3n_1{U}_{pcc}}{2X_1}(\Delta V_{od1}-\Delta U_{Gd})]G_{vd1}(s)=\Delta V_{od1}\\ [\Delta {\omega }_{01}^{*}-\frac{3m_1{U}_{pcc}}{2X_1}(\Delta V_{oq1}-\Delta U_{Gq})]G_{vq1}(s)=\Delta V_{oq1}\\ [\Delta E_2^{*}-\frac{3n_2{U}_{pcc}}{2X_2}(\Delta V_{od2}-\Delta U_{Gd})]G_{vd2}(s)=\Delta V_{od2}\\ [\Delta {\omega }_{02}^{*}-\frac{3m_2{U}_{pcc}}{2X_2}(\Delta V_{oq2}-\Delta U_{Gq})]G_{vq2}(s)=\Delta V_{oq2}\end{array},$$

(15)

$$\{\begin{array}{l}{G}_Q(s)\left\{\Delta Q_3^{*}+1.5U_{pcc}[-X_{3}M+R_{3}N]\right\}=\Delta V_{od3}\\ G_P(s)\left\{\Delta P_3^{*}-1.5U_{pcc}[R_{3}M+X_{3}N]\right\}=\Delta V_{od3}\\ M=\frac{(\Delta V_{od3}-\Delta U_{Gd})}{X_3^2+R_3^2}\\ N=\frac{(\Delta V_{oq3}-\Delta U_{Gq})}{X_3^2+R_3^2}\end{array}.$$

(16)

To simplify the equations, setting up H₁ = 1.5n₁U_pcc/X₁, H₂ = 1.5m₁U_pcc/X₁, H₃ = 1.5n₂U_pcc/X₂, H₄ = 1.5m₂U_pcc/X₂, H₅ = 1.5U_pccX₃/($X_3^2$ + $R_3^2$), H₆ = 1.5U_pccR₃/($X_3^2$ + $R_3^2$), by solving simultaneously Equations (15) and (16), Equation (17) is obtained.

$$\{\begin{array}{l}[\Delta E_1^{*}-H_1(\Delta V_{od1}-\Delta U_{Gd})]G_{vd1}(s)=\Delta V_{od1}\\ [\Delta {\omega }_{01}^{*}-H_2(\Delta V_{oq1}-\Delta U_{Gq})]G_{vq1}(s)=\Delta V_{oq1}\\ [\Delta E_2^{*}-H_3(\Delta V_{od2}-\Delta U_{Gd})]G_{vd2}(s)=\Delta V_{od2}\\ [\Delta {\omega }_{02}^{*}-H_4(\Delta V_{oq2}-\Delta U_{Gq})]G_{vq2}(s)=\Delta V_{oq2}\\ G_Q(s)[\Delta Q_3^{*}-H_5(\Delta V_{od3}-\Delta U_{Gd})+\\ H_6(\Delta V_{oq3}-\Delta U_{Gq})]=\Delta V_{od3}\\ G_P(s)[\Delta P_3^{*}-H_6(\Delta V_{od3}-\Delta U_{Gd})+\\ H_5(\Delta V_{oq3}-\Delta U_{Gq})]=\Delta V_{od3}\end{array},$$

(17)

where $\Delta E_1^{*}$ and Δω_0i^* are the reference voltage and reference frequency of DGi (I = 1, 2), which adopts droop control.

By solving Equations (13) and (17), Equation (18) can be obtained:

$${\left[\begin{array}{cc}\Delta U_{Gd}& \Delta U_{Gq}\end{array}\right]}^T=H_G^{2\times 6}{\left[\begin{array}{cccccc}\Delta E_1^{*}& \Delta {\omega }_{01}^{*}& \Delta E_2^{*}& \Delta {\omega }_{02}^{*}& \Delta Q_3^{*}& \Delta P_3^{*}\end{array}\right]}^T.$$

(18)

On the premise of guaranteeing the stability of the inner loop, G_id(s) = G_iq(s) = 1 is reasonable [21,32]; therefore, G_vd(s) = G_vq(s) = 1. In addition, H_G is a matrix of 2 × 6, whose simplified one is shown in Equation (19):

$$\begin{array}{cc}\hfill \left[\begin{array}{c}\Delta U_{Gd}\\ \Delta U_{Gq}\end{array}\right]=& \left[\begin{array}{cccccc}\frac{C_1(s)}{D(s)}& \frac{C_2(s)}{D(s)}& \frac{C_3(s)}{D(s)}& \frac{C_4(s)}{D(s)}& \frac{C_5(s)}{D(s)}& \frac{C_6(s)}{D(s)}\\ \frac{F_1(s)}{D(s)}& \frac{F_2(s)}{D(s)}& \frac{F_3(s)}{D(s)}& \frac{F_4(s)}{D(s)}& \frac{F_5(s)}{D(s)}& \frac{F_6(s)}{D(s)}\end{array}\right]\hfill \\ & \times {\left[\begin{array}{cccccc}\Delta E_1^{*}& \Delta {\omega }_{01}^{*}& \Delta E_2^{*}& \Delta {\omega }_{02}^{*}& \Delta Q_3^{*}& \Delta P_3^{*}\end{array}\right]}^T.\hfill \end{array}$$

(19)

As shown above, Equation (19) is the small-signal control model of the islanded MG, elements of whose coefficient matrix share the same denominator named D(s). By analyzing the characteristic equation D(s) = 0, the stability of the system can be determined.

5. Simulation and Stability Analysis

5.1. Parameter Setting and Calculation

Based on Figure 1, DG1 and DG2 should satisfy: P₁ = 30,000 − (f − 50)/m₁, P₂ = 30,000 − (f − 50)/m₂, Q₁ = 5000 − (E₁ − 311)/n₁, Q₂ = 5000 − (E₂ − 311)/n₂.

Using the parameters in Appendix A, H₁–H₆ can be obtained: H₁ = 466.5n₁, H₂ = 466.5m₁, H₃ = 466.5n₂, H₄ = 466.5m₂, H₅ = 208.6 and H₆ = 417.2.

5.2. Influence of Active Droop Coefficient m₁ and m₂ on Stability

To explore the impact of single variable m₁ on system stability, a priority assignment is first implemented: n₁ = n₂ = 0.001 V/var, K_Pp = K_Qp = 0.0005, K_Pi = K_Qi = 0.5.

(1) When m₂ = 5 × 10⁻⁵ Hz/W, the characteristic equation D(s) = 0 is

$$\begin{array}{l}(m_1+0.00242)s^5+(234m_1+0.542)s^4+(125m_1+0.392)s^3\\ +(9860m_1+42.3)s^2+(3790m_1+15.7)s+(\mathrm{58,000}{m}_1+122)=0\end{array}.$$

(20)

The root locus equation as Equation (20) whose gain is m₁ can be acquired by processing Equation (21):

$$m_1{W}_1(s)+1=0,$$

(21)

where W₁(s) and the similar expressions hereafter in this paper can be found in Appendix B.

The root locus of D(s) when m₁ varies from 0 to +∞ is shown in Figure 8.

There are five branches, S₁ starts from (−233.8x, 0) to (−225.3, 0) and always stays in left half-plane and far away from the imaginary axis, so it is not shown in the picture; S₂ and S₃ locate in left half-plane and have no impact on stability; S₄ and S₅ move towards the positive direction with the change of m₁, leading to the decline in stability, they finally cross the imaginary axis (m₁ = 3.2 × 10⁻⁵) and enter the right half-plane indicating that the system is no longer stable. Therefore, the proper range of m₁ to ensure system stability is [0, 3.2 × 10⁻⁵) in this condition.

Theoretically, the smaller m₁ is, the weaker the impact of change in DG1’s output power on system’s frequency is, especially when m₁ = 0, DG1 works in constant frequency control mode and has the strongest stability. At this point, DG1 has theoretically infinite power supply, which owns the absolute frequency stability and peak regulating ability. However, this assumption is not consistent with the energy attributes of MG. That is, MG’s output power is limited and relies on other MGs’ cooperation, so m₁ cannot be too small. On the other hand, a large value of m₁ also results in the fast oscillation of system frequency under small power fluctuation, which is harmful to system stability and this situation is shown in Figure 8 when m₁ > 3.2 × 10⁻⁵.

When m₁ = 2.5 × 10⁻⁵ Hz/W and m₂ = 5 × 10⁻⁵ Hz/W, the bus voltage waveform is shown in Figure 9. Because 2.5 × 10⁻⁵ ∈ [0, 3.2 × 10⁻⁵) and the bus voltage is stable, it means that the m₁ in this range ensures microgrid stable operation, which can verify the correctness of above analysis.

(2) When m₂ = 10 × 10⁻⁵ Hz/W, the root locus equation whose gain is m₁ is

$$m_1{W}_2\left(s\right)+1=0.$$

(22)

The root locus of D(s) when m₁ varies from 0 to +∞ is shown in Figure 10.

The locus is so similar to Figure 8 that it won’t be described again. The proper range of m₁ is [0, 2.9 × 10⁻⁵), and system stability declines with the increase of m₁.

When m₁ = 2 × 10⁻⁵ Hz/W and m₂ = 10 × 10⁻⁵ Hz/W, the bus voltage waveform is shown in Figure 11. The waveform is similar to Figure 9, without more detailed analysis.

(3) When m₂ = 2.5 × 10⁻⁵ Hz/W, the root locus equation whose gain is m₁ is

$$m_1{W}_3\left(s\right)+1=0.$$

(23)

The root locus of D(s) when m₁ varies from 0 to +∞ is shown in Figure 12.

The locus is so similar to Figure 8 that it won’t be described again. The proper range of m₁ is [0, 7.21 × 10⁻⁵), and system stability declines with the increase of m₁.

When m₁ = 5 × 10⁻⁵ Hz/W and m₂ = 2.5 × 10⁻⁵ Hz/W, the bus voltage waveform is shown in Figure 13. The waveform is similar to Figure 9, without more detailed analysis.

In summary, the comparison of range for m₁ when m₂ is different is shown in

Table 1. The influence of active droop coefficients between DG1 and DG2.

m2 (Hz/W)	Range of m1 (Hz/W)
2.5 × 10−5	[0, 7.21 × 10−5)
5 × 10−5	[0, 3.2 × 10−5)
10 × 10−5	[0, 2.9 × 10−5)

. In Table 1, m₁ is less than 2.9 × 10⁻⁵ when m₂ is 10 × 10⁻⁵. It is the reason why bus voltage is unstable in Section 1.2. The allowable range of m₁ narrows with the increase of m₂. From the perspective of system stability, the active droop coefficients of two DGs are supposed to be coordinated and restricted in a lower range so that the unit power loss won’t cause large change in frequency and system frequency collapse.

5.3. Influence of Reactive Droop Coefficient n₁ and n₂ on Stability

To explore the impact of single variable n₁ on system stability, a priority assignment is first implemented: m₁ = m₂ = 2.5 × 10⁻⁵ Hz/W, K_Pp = K_Qp = 0.0005, K_Pi = K_Qi = 0.5.

(1) When n₂ = 0.002 V/var, the root locus of the equation D(s) = 0 whose gain is n₁ is

$$n_1{W}_4\left(s\right)+1=0.$$

(24)

The root locus when n₁ varies from 0 to +∞ is shown in Figure 14.

There are six branches, S₁ first moves towards left and then turns right and always stays in the left half-plane, indicating that the system stability first enhances and then degrades; S₂ and S₃ are located in the left half-plane and have no impact on stability; S₄ and S₅ move towards the positive direction with the change of n₁, leading to the decline in stability, they finally cross the imaginary axis (n₁ = 0.00147) and enter the right half-plane indicating that the system is no longer stable. Therefore, the proper range of n₁ to ensure system stability is [0, 0.00147) in this condition.

Theoretically, the smaller n₁ is, the weaker the impact of change in DG1’s output reactive power on system’s voltage is. Especially when n₁ = 0, DG1 works in constant voltage control mode and has the strongest stability. At this point, DG1 has theoretically infinite reactive power supply, which owns the absolute voltage stability and reactive power compensation ability. However, this assumption is not consistent with the energy attributes of MG. That is, MG’s output reactive power is limited and relies on other MGs’ cooperation, so n₁ cannot be too small. On the other hand, a large value of n₁ also results in voltage collapse under small reactive power fluctuation, and this instable situation is shown in picture when n₁ > 0.00147.

When n₁ = 0.001 V/var and n₂ = 0.002 V/var, the bus voltage waveform is shown in Figure 15. Because 0.001 ∈ [0, 0.00147) and the bus voltage is stable, it means that the n₁ in this range ensures microgrid stable operation, which can verify the correctness of the above analysis.

(2) When n₂ = 0.001 V/var, the root locus equation is

$$n_1{W}_5\left(s\right)+1=0.$$

(25)

The root locus of D(s) when n₁ varies from 0 to +∞ is shown in Figure 16.

Because this locus is similar to Figure 14, it won’t be described again. The proper range of n₁ is [0, 0.00169), and system stability declines with the increase of n₁.

When n₁ = 0.001 V/var and n₂ = 0.001 V/var, the bus voltage waveform is shown in Figure 17. The waveform is similar to Figure 15, without more detailed analysis.

(3) When n₂ = 0.004 V/var, the root locus equation is

$$n_1{W}_6\left(s\right)+1=0.$$

(26)

The root locus of D(s) when n₁ varies from 0 to +∞ is shown in Figure 18.

Because this locus is similar to Figure 14, it won’t be described again. The proper range of n₁ is [0, 0.00133), and system stability declines with the increase of n₁.

When n₁ = 0.001 V/var and n₂ = 0.004 V/var, the bus voltage waveform is shown in Figure 19. The waveform is similar to Figure 15, without more detailed analysis.

In summary, the comparison of range of n₁ when n₂ is different is shown in

Table 2. The influence of reactive droop coefficients between DG1 and DG2.

n2 (V/var)	Range of n1 (V/var)
0.001	[0, 0.00169)
0.002	[0, 0.00147)
0.004	[0, 0.00133)

. The allowable range of n₁ narrows with the increase of n₂. From the perspective of system stability, the reactive droop coefficients of two DGs are supposed to be coordinated because it is restricted in such a low value range that the unit reactive power loss won’t cause large change in voltage—thus avoiding the vicious cycle: “change in voltage-reactive power loss-enlargement change in voltage—larger reactive power loss”.

6. Conclusions

In this paper, research on influence of the wide application of electronic devices on system stability is done by a three-source MG with a multiple master-slave control method.

Different from the traditional master–slave control model, this paper established a model where several master controllers coordinate and control the system in islanded mode. Then, the “virtual impedance” was introduced to satisfy the decoupling condition and the small-signal decoupling control model, which contains two main control sources and subordinate one, was also developed on the basis of theory. Finally, the interaction between various DGs with droop control from the perspective of control parameters was explored with the aid of the root locus method. The results show that the value range of the droop coefficient for the two main DGs has mutual influence and the coordinated allocation of parameters also affects the stability of microgrid operation. This research could serve as a reference for parameter setting and contributed to study on stability of multi-source MG.