A Microgrid Stability Improvement Method by Applying Virtual Adaptive Resistor Paralleling with a Grid-Connected Inverter

Abstract

An increase in renewable energy generation in the microgrid can cause voltage oscillation problems. To address this issue, an equivalent circuit of the microgrid was established, including a synchronous generator, grid-connected inverter, and constant power load. Then, the impact of different renewable energy generation ratios, different direct current (DC) voltage loops, and phase-locked loop control bandwidths of the grid-connected inverter on microgrid stability were analyzed. The results indicate that an increase in the renewable energy generation ratio leads to a decrease in the stability margin of the microgrid. A microgrid stability improvement method involving the parallel connection of a virtual resistor with the grid-connected inverter was proposed. The resistance value of the virtual resistor was obtained through an adaptive algorithm. This method ensures the stable operation of the microgrid under different renewable energy generation ratios.

1. Introduction

To address environmental pollution caused by the extensive extraction and use of fossil fuels, the proportion of renewable energy generation, such as wind energy and photovoltaics, is increasing in microgrids [1]. However, the large-scale integration of renewable energy poses potential risks to the stable operation of microgrids [2,3,4]. Therefore, it is necessary to analyze the stability of microgrids under conditions of a high proportion of renewable energy generation.

Current research on microgrid stability mainly includes small-signal stability and large-signal stability analyses. Since analysis of the stability mechanism primarily focuses on small-signal stability analysis [5,6,7,8,9], this article investigates the small-signal stability of microgrids. At present, there is a considerable difference in the small-signal stability analysis of synchronous generators (SGs) and grid-connected inverters (GCIs) in microgrids. The stability analysis of SGs focuses on analyzing their power angle stability [10,11], whereas the stability of GCIs is typically analyzed by establishing an impedance model [12,13,14,15,16]. As the concept of power angle does not apply to GCIs, this paper analyzes microgrid stability from an impedance perspective. Additionally, the Nyquist stability criterion (NSC) is a commonly used stability analysis method [17,18]. This paper employs this criterion to analyze the stability of microgrids under different renewable energy generation ratios.

In microgrids, GCIs serve as interface devices between renewable energy sources and the alternating current (AC) bus [19]. They convert direct current into high-quality alternating current, which is fed into the AC bus. Under different renewable energy generation ratios, the output power of the GCI varies, leading to a shift in the GCI output impedance, which affects the stability of the microgrid. Many studies have been conducted on the impedance characteristics of GCIs [20,21,22,23]. A mathematical model of the GCI that includes a phase-locked loop (PLL) has been established, and the impact of the PLL on the stability of the microgrid has been analyzed. The results show that the PLL introduces negative resistance in the output impedance of the GCI, which is considered the primary cause of system instability [22].

In order to improve the stability of the GCI, the main improvement methods are the passive resistance method and the active resistance method. The passive resistance method improves the stability of the system by introducing passive resistors. However, this method results in power loss and reduces energy transmission efficiency. In order to overcome this problem, system stability can be improved by enhancing the control strategy of the GCI to achieve the effect of virtual parallel/series resistance. This method is called the active resistance method. At present, the commonly used active resistance methods mainly include active dampers [23,24] and reshaping the impedance of the GCI [25,26]. Active dampers achieve the effect of parallel virtual resistance through control strategies [23]. In practical applications, the virtual resistance of an active damper is usually difficult to design. To solve this problem, in [24], an adaptive algorithm was proposed to determine the resistance of the parallel resistor. However, active dampers need to add new devices to the system. In [27,28], the control strategy of the existing devices in the system was reformed to improve the stability of the system. In [28], a stability improvement method based on voltage feedforward was proposed. By introducing a feedforward path from the PCC voltage to the current reference, the output impedance of the GCI is reshaped, and the stability of the GCI is improved under weak grid conditions without altering the control parameters and dynamic performance of the PLL. However, unlike weak grid conditions, the complex impedance characteristics of the external grid in microgrids complicate the design of the feedforward path-transfer function. To address this issue, this paper proposes a microgrid stability improvement method based on paralleling a virtual adaptive resistor via voltage feedforward. This method connects a virtual resistor in parallel with the GCI output port, and its resistance value is obtained through an adaptive algorithm. This method could enhance microgrid stability under different renewable energy generation ratios.

The structure of this paper is as follows: The first section derives the equivalent circuit of the microgrid that includes SGs, GCIs, and constant power loads (CPLs). The second section utilizes the NSC to analyze microgrid stability under different renewable energy generation ratios and control bandwidths of the PLL and DC voltage control (DCVC) loop of the GCI. The analysis results indicate that paralleling a resistor at the output port of the GCI can improve system stability. The third section proposes a microgrid stability improvement method based on paralleling a virtual adaptive resistor to enhance microgrid stability under different renewable energy generation ratios. This method adjusts the resistance value of the virtual resistor through an adaptive algorithm. The fourth section sets up an experimental platform based on a real-time digital simulator (RTDS) to validate the accuracy of the stability analysis results and the effectiveness of the proposed stability improvement method. Finally, the fifth section summarizes the entire paper.

2. The Equivalent Model of the Microgrid

This paper takes the microgrid shown in Figure 1 as an example for this study. In Figure 1, the GCI connects to the microgrid at the point of connection (PoC) and then links with the SG set and CPL at the point of common coupling (PCC).

2.1. The Impedance Model of SG Set

The SG is used to maintain a constant bus voltage in the microgrid. Due to the large inertia of the rotor, the speed of the SG changes very slowly. Therefore, when establishing the impedance model of the SG set, the rotor speed can be considered constant. SG can be seen as a voltage source, and its equivalent circuit model can be simplified as the equivalent circuit. The output voltage ${\widehat{{\displaystyle \mathit{v}}}}_{\mathrm{SGout}}(s)$ is shown as (1), which has been derived in [29].

$${\widehat{{\displaystyle \mathit{v}}}}_{\mathrm{SGout}}(s)={\widehat{{\displaystyle \mathit{v}}}}_{\mathrm{SG}}(s)-{\mathit{Z}}_{\mathrm{SGS}}(s){\widehat{{\displaystyle \mathit{i}}}}_{\mathrm{SGout}}(s)$$

(1)

where ${\widehat{{\displaystyle \mathit{v}}}}_{\mathrm{SG}}(s)$ is the SG output voltage command tracking component, which is the output signal of the SG voltage control loop [29]. Its expression is

$${\widehat{{\displaystyle \mathit{v}}}}_{\mathrm{SG}}(s)={\left[\mathit{E}+3\sqrt{3}{k}_T{G}_e(s){\mathit{G}}_f(s){\mathit{G}}_a(s)/\pi \right]}^{-1}3\sqrt{3}{k}_T{G}_e(s){\mathit{G}}_f(s)/\pi {\widehat{{\displaystyle \mathit{v}}}}_{\mathrm{SGoutm\_ref}}(s)$$

(2)

where k_T is the turn ratio of the transformer. G_e(s) is the excitation regulator, which is usually a proportional–integral (PI) regulator. G_f(s) is the transfer function from excitation voltage to output voltage. G_a(s) is the transfer function of the fundamental amplitude calculation. v_{SGoutm_ref} is the reference for the amplitude of the SG output voltage.

Z_SGS(s) is the output impedance of the SG set, and its expression is:

$${\mathit{Z}}_{\mathrm{SGS}}(s)={\left[\mathit{E}+3\sqrt{3}{k}_T{G}_e(s){\mathit{G}}_f(s){\mathit{G}}_a(s)/\pi \right]}^{-1}{\mathit{Z}}_{\mathrm{SG}}(s)$$

(3)

where Z_SG is the output impedance of the SG with the SG voltage control open loop, and its expression is given by (4).

$${\mathit{Z}}_{SG}=\left[\begin{array}{cc}s{L}_s-1.5M_f^2\cos \delta (s\cos \delta +{\omega }_m\sin \delta )/L_f& -{\omega }_m{L}_s+1.5M_f^2\sin \delta ({\omega }_m\sin \delta +s\cos \delta )/L_f\\ {\omega }_m{L}_s-1.5M_f^2\cos \delta ({\omega }_m\cos \delta -s\sin \delta )/L_f& s{L}_s-1.5M_f^2\sin \delta (s\sin \delta -{\omega }_m\cos \delta )/L_f\end{array}\right]$$

(4)

In (4), L_s is the inductance of the armature winding. L_f is the inductance of the excitation winding. M_f is the maximum mutual inductance between the excitation winding and the armature winding, and δ is the power angle of the SG.

2.2. The Admittance Model of GCI

The control structure of the GCI includes PLL, DC voltage control (DCVC), and current control loop. The PLL and DCVC are used to provide the phase and amplitude reference for the current control loop, respectively. The output current of GCI is shown in (5), which has been derived in [29].

$${\hat{\mathit{i}}}_{invout}(s)={\hat{\mathit{i}}}_{inv}(s)-{\mathit{Y}}_{inv}(s){\hat{\mathit{v}}}_{PoC}(s)$$

(5)

where ${\hat{\mathit{i}}}_{inv}(s)$ is the grid-connected current tracking component.

$${\hat{\mathit{i}}}_{inv}(s)={\left\{\mathit{E}+{\mathit{G}}_2(s){\mathit{G}}_1(s)\left[\mathit{E}+{\mathit{G}}_{fb\_dcvc\_inv}(s)\right]\right\}}^{-1}{\displaystyle \frac{{\mathit{G}}_2(s){\mathit{G}}_1(s)G_{\mathrm{inv}\_\mathrm{vr}}(s)}{s{C}_{\mathrm{dc}}{V}_{\mathrm{dc}\_\mathrm{inv}}}}{\widehat{p}}_{\mathrm{MPPT}}(s)$$

(6)

where G_{inv_vr}(s) is the DCVC loop regulator, which is a PI controller. G₁(s) and G₂(s) are

$${\mathit{G}}_1(s)={\left\{\mathit{E}+\left[{\mathit{Z}}_{\mathit{L}}(s)+H_{ic}{K}_{\mathrm{PWM}}\right]{\mathit{Z}}_{\mathit{C}}^{-1}(s)\right\}}^{-1}{K}_{\mathrm{PWM}}{\mathit{G}}_{inv\_cr}(s)$$

(7)

$${\mathit{G}}_2(s)={\{[{\mathit{Z}}_{\mathit{L}}(s)+H_{ic}{K}_{\mathrm{PWM}}]{\mathit{Z}}_{\mathit{L}\mathit{g}}(s){\mathit{Z}}_{\mathit{C}}^{-1}(s)+{\mathit{Z}}_{\mathit{L}}(s)+{\mathit{Z}}_{\mathit{L}\mathit{g}}(s)\}}^{-1}\{\mathit{E}+[{\mathit{Z}}_{\mathit{L}}(s)+H_{ic}{K}_{\mathrm{PWM}}]{\mathit{Z}}_{\mathit{C}}^{-1}(s)\}$$

(8)

where G_{inv_cr}(s) is the current loop regulator, which is a PI controller; Z_L(s), Z_C(s), and Z_Lg(s) are the impedances of the inverter-side inductance, filter capacitor, and grid-side inductance of the LCL filter, respectively. K_PWM is the transfer function from the modulation wave to the output voltage of GCI.

G_{fb_dcvc_inv}(s) is the transfer function of the feedback path introduced by the DCVC loop,

$${\mathit{G}}_{fb\_dcvc\_inv}\left(s\right)={\displaystyle \frac{G_{\mathrm{inv}\_\mathrm{vr}}\left(s\right)}{s{C}_{\mathrm{dc}}{V}_{\mathrm{dc}\_\mathrm{inv}}}}\left[\begin{array}{cc}{V}_{\mathrm{PoC}\_d}& V_{\mathrm{PoC}\_q}\\ 0& 0\end{array}\right].$$

(9)

where V_{P°C_d} and V_{P°C_q} are the steady-state values of the PoC voltage in the dq-frame.

In (5), ${\mathit{Y}}_{inv}(s)$ is the output admittance of the GCI

$$\begin{array}{ll}{\mathit{Y}}_{inv}(s)=& {\left\{\mathit{E}+{\mathit{G}}_2(s){\mathit{G}}_1(s)\left[\mathit{E}+{\mathit{G}}_{fb\_dcvc\_inv}(s)\right]\right\}}^{-1}{\mathit{G}}_2(s)\cdot \\ & \left\{\mathit{E}+{\mathit{G}}_1(s)\left[{\mathit{G}}_{ff\_pll\_inv}(s)+{\mathit{G}}_{ff\_dcvc\_inv}(s)\right]\right\}\end{array}$$

(10)

G_{ff_dcvc_inv}(s) is the transfer function of the forward path introduced by the DCVC loop,

$${\mathit{G}}_{ff\_dcvc\_inv}\left(s\right)={\displaystyle \frac{G_{\mathrm{inv}\_\mathrm{vr}}\left(s\right)}{s{C}_{\mathrm{dc}}{V}_{\mathrm{dc}\_\mathrm{inv}}}}\left[\begin{array}{cc}{I}_{\mathrm{invout}\_d}& I_{\mathrm{invout}\_q}\\ 0& 0\end{array}\right]$$

(11)

where I_{invout_d} and I_{invout_q} are the steady-state values of the d-axis and q-axis components of the output current of the GCI, respectively.

G_{ff_pll_inv}(s) is the forward path-transfer function introduced by the PLL, shown as

$${\mathit{G}}_{ff\_pll\_inv}(s)={\displaystyle \frac{G_{\mathrm{inv}\_\mathrm{pllr}}(s)}{s+V_{\mathrm{PoC}\_d}{G}_{\mathrm{inv}\_\mathrm{pllr}}(s)}}\left[\begin{array}{cc}0& I_{\mathrm{invout}\_q}+(H_{ic}{I}_{C\_q}+V_{\mathrm{M}\_q})/G_{\mathrm{inv}\_\mathrm{cr}}(s)\\ 0& -I_{\mathrm{invout}\_d}-(H_{ic}{I}_{C\_d}+V_{\mathrm{M}\_d})/G_{\mathrm{inv}\_\mathrm{cr}}(s)\end{array}\right]$$

(12)

where G_{inv_pllr}(s) is the PLL regulator.

2.3. The Admittance Model of the CPL

The control structure of the CPL used in this paper is similar to that of the GCI, which includes PLL, DCVC, and the current control loop. Therefore, the impedance expressions of the CPL and GCI are similar. The model of the CPL is shown in (13), which has been derived in [29].

$${\hat{\mathit{i}}}_{cplin}(s)={\mathit{Y}}_{cpl}(s){\hat{\mathit{v}}}_{cpl}(s)$$

(13)

where Y_cpl(s) is the input admittance of the CPL,

$${\mathit{Y}}_{cpl}(s)={\{{\mathit{Z}}_{\mathit{L}}(s)+K_{\mathrm{PWM}}{G}_{\mathrm{cpl}\_\mathrm{cr}}(s)[\mathit{E}+{\mathit{G}}_{fb\_dcvc\_cpl}(s)]\}}^{-1}\{\mathit{E}+K_{\mathrm{PWM}}{G}_{\mathrm{cpl}\_\mathrm{cr}}(s)[{\mathit{G}}_{ff\_pll\_cpl}(s)+{\mathit{G}}_{ff\_dcvc\_cpl}(s)]\}.$$

(14)

where Z_L(s) is the impedance of the L filter, and G_{cpl_cr}(s) is the current loop PI controller.

G_{ff_pll_cpl}(s) is the feedforward path-transfer function introduced by the PLL.

$${\mathit{G}}_{ff\_pll\_cpl}\left(s\right)={\displaystyle \frac{G_{\mathrm{cpl}\_\mathrm{pllr}}(s)}{s+V_{\mathrm{cpl}\_d}{G}_{\mathrm{cpl}\_\mathrm{pllr}}(s)}}\left[\begin{array}{cc}0& -I_{\mathrm{cplin}\_q}+V_{\mathrm{M}q\_\mathrm{cpl}}/G_{i\_\mathrm{cpl}}\left(s\right)\\ 0& I_{\mathrm{cplin}\_d}-V_{\mathrm{M}d\_\mathrm{cpl}}/G_{i\_\mathrm{cpl}}\left(s\right)\end{array}\right]$$

(15)

where I_{cplin_d} and I_{cplin_q} are the steady-state values of the d-axis and q-axis components of the input current, respectively; V_{Md_cpl} and V_{Mq_cpl} are the steady-state values of the d-axis and q-axis components of the rectifier modulation wave, respectively. G_{cpl_pllr}(s) is the PI controller of the PLL.

G_{ff_dcvc_cpl}(s) is the feedforward path-transfer function introduced by the DCVC loop of the CPL.

$${\mathit{G}}_{ff\_dcvc\_cpl}\left(s\right)={\displaystyle \frac{R_{\mathrm{lo}}{G}_{\mathrm{cpl}\_\mathrm{vr}}\left(s\right)}{V_{\mathrm{dc}\_\mathrm{cpl}}\left(2+sC{R}_{\mathrm{lo}}\right)}}\left[\begin{array}{cc}-I_{\mathrm{cplin}\_d}& -I_{\mathrm{cplin}\_q}\\ 0& 0\end{array}\right]$$

(16)

where G_{cpl_vr}(s) is the PI controller of the DCVC loop.

G_{fb_dcvc_cpl}(s) is the transfer function of the current feedback path introduced by the DCVC loop,

$${\mathit{G}}_{\mathrm{fb}\_\mathrm{dcvc}\_\mathrm{cpl}}\left(\mathit{s}\right)={\displaystyle \frac{R_{\mathrm{lo}}{G}_{\mathrm{cpl}\_\mathrm{vr}}\left(s\right)}{V_{\mathrm{dc}\_\mathrm{cpl}}\left(2+sC{R}_{\mathrm{lo}}\right)}}\left[\begin{array}{cc}{V}_{\mathrm{cpl}\_d}& V_{\mathrm{cpl}\_q}\\ 0& 0\end{array}\right].$$

(17)

where V_{cpl_d} and V_{cpl_q} are the steady-state values of the d-axis and q-axis components of the CPL input voltage, respectively.

2.4. The Equivalent Circuit of the Microgrid

Based on (1), (5), and (13), the equivalent circuit of the microgrid shown in Figure 1 can be derived as depicted in Figure 2.

In Figure 2, Z_{TL_SG}, Z_{TL_inv}, and Z_{TL_cpl} are the transmission line impedance of the SG set, the GCI, and the CPL, respectively.

$${\mathit{Z}}_{\mathbf{TL\_SG}}(s)=L_{\mathrm{TL}\_\mathrm{SG}}\left[\begin{array}{cc}s& -{\omega }_o\\ {\omega }_o& s\end{array}\right]$$

(18)

$${\mathit{Z}}_{\mathbf{TL\_inv}}(s)=L_{\mathrm{TL}\_\mathrm{inv}}\left[\begin{array}{cc}s& -{\omega }_o\\ {\omega }_o& s\end{array}\right]$$

(19)

$${\mathit{Z}}_{\mathbf{TL\_cpl}}(s)=L_{\mathrm{TL}\_\mathrm{cpl}}\left[\begin{array}{cc}s& -{\omega }_o\\ {\omega }_o& s\end{array}\right]$$

(20)

where L_{TL_SG}, L_{TL_inv} and L_{TL_cpl} are the inductance of the transmission lines corresponding to Z_{TL_SG}(s), Z_{TL_inv}(s), and Z_{TL_cpl}(s), respectively.

As for the GCI, the output of the DCVC loop serves as the d-axis component of the current reference. Thus, the DCVC loop primarily affects the dd element of the GCI output admittance. By substituting (9), (10), and (12) into (10), it can be concluded that the DCVC loop causes the dd element to exhibit positive resistance characteristics, and its magnitude increases with the output current of the GCI rising. The PLL tracks the voltage phase by controlling the q-axis component of the grid voltage to zero, thus primarily affecting the qq element of the GCI output admittance.

Figure 3 shows the Bode plot of the GCI output admittance under different output power of the GCI. The parameters of the GCI used in Figure 3 are shown in

Table 1. Parameters of GCI.

Parameter	Symbol	Value
Rated power	PNinv	20 kW
RMS value of output voltage	VPoC_rms	220 V
DC-side voltage	Vdc	750 V
Switching frequency	fs	10 kHz
DC-side capacitor	Cdc	1 mF
Inverter-side inductance	L1	1 mH
Filter capacitor	Cf	20 μF
Grid-side inductance	L2	0.5 mH

.

In Figure 3, as the output power of the GCI increases, the magnitude of the dd element of the GCI output admittance increases in the low-frequency range and exhibits positive resistance characteristics. Meanwhile, the magnitude of the GCI output admittance qq element also increases in the low-frequency range, but it exhibits negative resistance characteristics. This could potentially lead to microgrid instability.

3. Stability Analysis of the Microgrid

To simplify the stability analysis process, Figure 3 can be equivalently transformed into Figure 4.

Where ${\mathit{i}}_{inv\_l}(s)$ and Y_{inv_l}(s) are the equivalent current source and parallel admittance of the GCI paralleling its transmission line impedance. Z_{SGS_l}(s) is the equivalent impedance of the SG set series and its transmission line impedance. Y_{cpl_l}(s) is the equivalent admittance of the CPL series and its transmission line impedance. They can be expressed as:

$${\mathit{Y}}_{inv\_l}(s)={[\mathit{E}+{\mathit{Y}}_{inv}(s){\mathit{Z}}_{TL\_inv}(s)]}^{-1}{\mathit{Y}}_{inv}(s)$$

(21)

$$\begin{array}{l}{\mathit{i}}_{inv\_l}(s)={[\mathit{E}+{\mathit{Y}}_{inv}(s){\mathit{Z}}_{TL\_inv}(s)]}^{-1}{\mathit{i}}_{inv}(s)\\ P_{\mathrm{VR}}\le P_{\mathrm{VR\_max}}={\displaystyle \frac{3V_{\lim }}{R_{\min }}}\\ R_{\min }={\displaystyle \frac{3V_{\lim }^2}{R_{\mathrm{VR\_max}}}}={\displaystyle \frac{3V_{\lim }^2}{P_{\mathrm{Ninv}}-P_{\mathrm{inv}}}}\end{array}$$

(22)

$${\mathit{Z}}_{SGS\_l}(s)={\mathit{Z}}_{SGS}(s)+{\mathit{Z}}_{TL\_SG}(s)$$

(23)

$${\mathit{Y}}_{cpl\_l}(s)=[\mathit{E}+{\mathit{Y}}_{cpl}(s){\mathit{Z}}_{TL\_cpl}(s){]}^{-1}{\mathit{Y}}_{cpl}(s)$$

(24)

Based on Figure 4, the PCC voltage is given as

$${\widehat{{\displaystyle \mathit{v}}}}_{\mathrm{PCC}}(s)={\displaystyle \frac{adj\left\{\mathit{E}+{\mathit{Z}}_{\mathrm{SGS\_l}}(s)\left[{\mathit{Y}}_{\mathrm{inv\_l}}(s)+{\mathit{Y}}_{\mathrm{cpl\_l}}(s)\right]\right\}〈{\widehat{{\displaystyle \mathit{v}}}}_{SG}(s)+{\mathit{Z}}_{\mathrm{SGS\_l}}(s){\widehat{{\displaystyle \mathit{i}}}}_{\mathrm{inv\_l}}(s)〉}{1+\underset{T(s)}{\underbrace{\left|\mathit{E}+{\mathit{Z}}_{\mathrm{SGS\_l}}(s)\left[{\mathit{Y}}_{\mathrm{inv\_l}}(s)+{\mathit{Y}}_{\mathrm{cpl\_l}}(s)\right]\right|-1}}}}$$

(25)

If the three subsystems paralleled at the PCC are designed to be stable individually, the Z_{SGS_l}(s), Y_{inv_l}(s), and Y_{cpl_l}(s) do not contain right-half-plane poles. At this point, it is only necessary to determine whether the denominator of (25) contains right-half-plane zeros, i.e., whether T(s) satisfies the NSC, to ascertain the stability of the microgrid.

3.1. Analysis of Microgrid Stability Under Different Photovoltaic Generation Ratios

To investigate the impact of varying photovoltaic generation ratios on microgrid stability, the stability of the system is analyzed under the following three scenarios.

(1)

One GCI operates in parallel with one SG and supplies power to two CPLs (full load).

(2)

Two GCIs operate in parallel with one SG and supply power to two CPLs (full load).

(3)

Three GCIs operate in parallel with one SG and supply power to two CPLs (full load).

In all three scenarios, the input power of the CPL remains unchanged. The parameters for the GCIs, SG, and CPL are provided in Table 1,

Table 2. Parameters of SG.

Parameter	Symbol	Value	Parameter	Symbol	Value
Rated power	PNinv	60 kW	Fundamental frequency	fo	50 Hz
Rated power factor	PF	0.85	Rated output voltage	VSG	220 V

and

Table 3. Parameters of CPL.

Parameter	Symbol	Value	Parameter	Symbol	Value
Rated power	PNcpl	40 kW	DC-side capacitor	Cdc	1 mF
Switching frequency	fs	10 kHz	RMS value of output voltage	Vcpl_rms	220 V
DC-side voltage	Vdc	1000 V	Filter inductance	L	5 mH

.

Figure 5 shows the Bode plot of T(s) under different photovoltaic generation ratios.

When there is only one GCI (corresponding to the blue curve in Figure 5), T(s) effectively crosses −180° positively once (phase increase) and negatively once (phase decrease), satisfying the NSC, indicating that the GCI is stable. However, when there are two and three GCIs, corresponding to the red and green curve in Figure 5, respectively, T(s) effectively crosses −180° positively once and negatively twice, failing to satisfy the NSC, leading to instability of the system. Additionally, in Figure 5, the microgrid with three GCIs has a smaller gain margin (−25.2 dB) compared to that of the microgrid with two GCIs (−7.7 dB). That means the system stability deteriorates as the number of GCIs increases.

3.2. Analysis of Microgrid Stability Under Different PLL and DCVC Loop Parameters

The following section explores the impact of different control parameters of GCI on microgrid stability. The analysis uses the system with two GCIs operating in parallel with one SG as an example. Five different cases for the GCI are applied for comparative analysis, as shown in

Table 4. GCI control parameter.

	Control Bandwidth of PLL		Control Bandwidth of DCVC Loop
Case 1	fPLL_inv	30 Hz	fDCVC_inv	10 Hz
Case 2	fPLL_inv	20 Hz	fDCVC_inv	10 Hz
Case 3	fPLL_inv	10 Hz	fDCVC_inv	10 Hz
Case 4	fPLL_inv	30 Hz	fDCVC_inv	30 Hz
Case 5	fPLL_inv	30 Hz	fDCVC_inv	50 Hz

.

First, Cases 1, 2, and 3 are selected to analyze the effect of different PLL control bandwidths on microgrid stability. The corresponding Bode plots are presented in Figure 6. When the PLL bandwidth is 30 Hz and 20 Hz, corresponding to the green and red curve in Figure 6, respectively, T(s) effectively crosses −180° positively once and negatively twice, which does not satisfy the NSC. When the PLL bandwidth of the GCI is reduced to 10 Hz (corresponding to the blue curve in Figure 6), T(s) effectively crosses −180° positively once and negatively once, satisfying the NSC. In addition, it is worth noting that as the PLL bandwidth decreases, the stability margin of the system increases. When the PLL bandwidth is 30 Hz, 20 Hz, and 10 Hz, the gain margin of the microgrid is −12.4 dB, −7.7 dB, and 4.9 dB, respectively. This indicates that reducing the PLL control bandwidth of the GCI can improve the stability of the microgrid.

Cases 1, 4, and 5 are selected to analyze the effect of different DC voltage loop control bandwidths on microgrid stability. The corresponding Bode plots are presented in Figure 7.

When the DCVC loop bandwidth is 10 Hz and 30 Hz, corresponding to the blue and red curve in Figure 7, respectively, T(s) effectively crosses −180° positively once and negatively twice, which does not satisfy the NSC. However, when the DCVC loop bandwidth of the GCI system is increased to 50 Hz (corresponding to the green curve in Figure 7), T(s) effectively crosses −180° positively once and negatively once, satisfying the NSC, and the microgrid is stable. From Figure 7, it can be observed that as the DCVC loop bandwidth increases, the gain margin of T(s) also increases. When the DCVC loop bandwidth is 10 Hz and 30 Hz, the gain margin of the microgrid is −7.7 dB and −3.3 dB, respectively. It indicates that increasing the DCVC loop control bandwidth of the GCI can enhance microgrid stability.

Based on the analysis results, the DCVC loop and PLL cause the dd and qq elements of the GCI output admittance to exhibit positive and negative resistance characteristics, respectively. Increasing the DCVC loop control bandwidth effectively widens the frequency range where the dd element exhibits positive resistance characteristics. On the other hand, decreasing the PLL control bandwidth reduces the frequency range where the qq element exhibits negative resistance characteristics. Therefore, it can be concluded that paralleling a resistor at the output port of the GCI can improve microgrid stability effectively.

4. Microgrid Stability Improvement Method by Paralleling Virtual Adaptive Resistor

The circuit diagram of the GCI paralleled with the resistor is shown in Figure 8.

If the resistance of the paralleled resistor is designed appropriately, it can enhance the stability of the microgrid. However, designing this resistance value usually requires complete information about the microgrid, which makes the design process challenging. Moreover, using actual resistors increases the energy losses. To address these issues, this paper proposes a microgrid stability improvement method using a virtual resistor. This approach achieves the effect of a parallel resistor by enhancing the control method of the GCI, and the resistance value is determined through the adaptive algorithm.

Figure 9a shows the control structure block diagram for the GCI with a parallel resistor. Figure 9a can be equivalently transformed into Figure 9b. In Figure 9b, T_{i_inv}(s) represents the closed-loop transfer function of the GCI current loop, and its expression is:

$${\mathit{T}}_{\mathrm{inv\_l}}(s)={[\mathit{E}+{\mathit{G}}_{x2}(s){\mathit{G}}_{x1}(s)]}^{-1}{\mathit{G}}_{x2}(s){\mathit{G}}_{x1}(s)$$

(26)

Based on Figure 9b, the paralleled resistor can be equivalently represented as a feedforward path from the PoC voltage to the current reference. Therefore, adding such a feedforward path can achieve the effect of a paralleled resistor.

The control structure diagram for the GCI with a parallel virtual resistor is shown in Figure 10. The PoC voltage v_PoC__abc(s) is sampled and transformed into the dq-frame using the Park transformation. A high-pass filter G_HPF(s) is used to filter out the steady-state component of v_PoC__dq(s). Then, divided by the virtual resistor R_v, the result is subtracted with the grid current reference, serving as the new current reference for grid current control.

However, an excessively large resistance value of the parallel virtual resistor cannot enhance the stability of the microgrid, and a small resistance value can introduce harmonics to the grid current. Therefore, to ensure stable operation of the microgrid under varying ratios of renewable energy generation, this paper employs an adaptive algorithm to determine the appropriate resistance value of the virtual resistor. The control structure block diagram of the adaptive algorithm is shown in Figure 11.

In Figure 11, the root-mean-square (RMS) values of ${\widehat{v}}_{\mathrm{PoC\_d}}$ and ${\widehat{v}}_{\mathrm{PoC}\text{\_}q}$ are sent to a low-pass filter G_LPF(s) to obtain $V_{\mathrm{PoC}}^2$, then compared with $V_{\lim }^2$, and the error signal is sent to a PI regulator G_RA(s). The output of G_RA(s) is used as the parallel virtual resistor 1/R_v. According to [30], the THD of the PoC voltage needs to be less than 3%. To meet the grid standards, V_lim needs to be less than 3% of the PoC voltage RMS value V_{PoC_rms}, and in this paper, V_lim is set to 1% of V_{PoC_rms}. Since an excessively small parallel virtual resistor can introduce harmonic currents to the grid current of the GCI, a limit R_min must be set for the parallel virtual resistor. The power P_VR of the parallel virtual resistor and its resistance value satisfy Equation (27).

$$P_{\mathrm{VR}}\le P_{\mathrm{VR\_max}}={\displaystyle \frac{3V_{\lim }}{R_{\min }}}$$

(27)

Based on (27), the minimum value of the parallel virtual resistance can be obtained as:

$$R_{\min }={\displaystyle \frac{3V_{\lim }^2}{R_{\mathrm{VR\_max}}}}={\displaystyle \frac{3{\mathrm{V}}_{\lim }^2}{P_{\mathrm{Ninv}}-P_{\mathrm{inv}}}}$$

(28)

where P_{VR_max} is designed as the difference between the actual output power of the GCI P_inv and the rated power P_N.

In Figure 12, there are three GCIs, each GCI operates at full load condition. The Bode plot of T(s) is depicted before and after adding the parallel virtual adaptive resistor. It can be observed that before adding the parallel virtual resistor, T(s) crosses −180° positively twice and negatively once, which does not satisfy the NSC, and the microgrid is unstable (gain margin is −25.2 dB). After adding the parallel virtual resistor, T(s) crosses −180° positively once and negatively once, satisfying the NSC, and the microgrid is stable (gain margin is 0.47 dB). This demonstrates that using the proposed stability improvement method can improve the microgrid stability.

5. Experimental Verification

To verify the accuracy of the proposed stability improvement method, validation is conducted using a hardware-in-the-loop (HIL) testing platform based on RTDS. Figure 13 shows the photograph of the experimental platform.

The experimental results of the microgrid under different photovoltaic generation ratios are shown in Figure 14. When there is only one GCI (corresponding to the blue curve in Figure 5), the NSC is satisfied, and the GCI is stable, as seen in Figure 14a. When there are two and three GCIs (corresponding to the red and green curves in Figure 5, respectively), the system does not meet the NSC, leading to instability of the system. This result verifies the correctness of the theoretical analysis.

Figure 15 presents the experimental waveforms of the microgrid after applying the parallel virtual adaptive resistor method. According to Figure 5, when the system includes one, two, and three GCIs, respectively, the stability margin of the microgrid reaches the minimum value with three GCIs (−25.2 dB), and the microgrid is unstable. After adding the parallel adaptive resistor, the stability margin of the system increases to 0.47 dB, as seen in Figure 12. The experiments demonstrate that using the adaptive resistor can ensure stable operation of the microgrid under varying photovoltaic generation conditions.

To validate the necessity of the parallel adaptive resistor proposed in this paper, a comparative experiment is conducted with a parallel virtual fixed-value resistor, as shown in Figure 16. As seen, using a parallel fixed-value resistor can also improve microgrid stability, but the system still faces a risk of instability as photovoltaic generation further increases. As seen in Figure 16b, the microgrid is still unstable (the gain margin is −23.8 dB, as seen in Figure 12), paralleled with the virtual fixed-value resistor.

Figure 17 shows the output power waveforms of the GCI using the parallel virtual fixed-value resistor method and the parallel virtual adaptive resistor method. As seen in Figure 17a, after the output power of the GCI increases, the system still faces a risk of instability with a 20 Ω parallel virtual fixed-value resistor. As seen in Figure 17b, after the output power of the GCI increases, the adaptive algorithm adjusts the resistance value of the parallel virtual resistor from 20 Ω to 10 Ω, allowing the microgrid to maintain stability under varying photovoltaic generation conditions.

6. Conclusions

This paper analyzed the impact of renewable energy generation ratios and the control parameters of the GCI on the stability of microgrids. The main conclusions are:

(1)

Impedance/admittance models for each device within the microgrid are established. The output admittance characteristics of GCI in the different output power conditions are analyzed. The results indicated that the DCVC loop of GCI causes the dd element of the admittance matrix to exhibit positive resistance characteristics at the low-frequency range. The PLL causes the qq element of the admittance matrix to exhibit negative resistance characteristics at the low-frequency range. As the output power of the GCI increases, the magnitude of the qq element increases at the low-frequency range, which may lead to microgrid instability.

(2)

Based on the impedance/admittance models of each device, an equivalent circuit of the microgrid is derived. Based on this equivalent circuit, the NSC is employed to analyze the stability of the microgrid under different control bandwidths of the DCVC loop and the PLL. The results showed that increasing the DCVC loop control bandwidth and decreasing the PLL control bandwidth can effectively improve the stability of the microgrid.

(3)

Parallel resistors at the output port of the GCI can improve microgrid stability. Considering that real resistors would introduce energy losses, a method using a parallel virtual adaptive resistor is proposed. This method can effectively enhance microgrid stability under different ratios of renewable energy generation conditions.

The proposed stability improvement method can effectively improve system stability. However, this method is aimed at reforming the control strategy of the existing GCI in the system. The position of the virtual parallel resistance in the system is limited by the installation position of the GCI. In addition, is it necessary to reform all GCIs in the system? At which nodes do the GCIs need to be reformed to benefit the system more? These problems still need further study.