With the rapid development of the global economy, the demand for energy has significantly increased, making the energy crisis an unavoidable issue. To accelerate the energy transition, the widespread adoption of solar energy has become a common choice for countries worldwide. Solar power often needs to be transmitted over long-distance power lines and through multiple transformers to reach load centers, which increases grid impedance and causes the point of common coupling (PCC) voltage in photovoltaic grid-connected systems to exhibit weak grid characteristics. This limits the power absorption capability of solar inverters. Conducting research on the static power stability operation region and optimal control of solar inverters under weak grid conditions is crucial for stabilizing the transmission of solar power to the grid, enhancing the photovoltaic absorption capacity of power systems in weak grid environments, and holds significant importance for the construction of new power systems.
To improve the transmission capacity of solar power in the grid, current research primarily focuses on optimizing the reactive power compensation strategy of solar inverters and installing additional reactive power compensation devices. In terms of optimizing the reactive power compensation strategy of solar inverters, some studies have proposed power synchronization control strategies, which simplify the control structure and enhance the active power output capability of solar inverters, but their response speed is relatively slow. To address this, an adaptive reactive power droop control method has been proposed, which improves the active power output capability of solar inverters while considering both the power factor and the PCC voltage amplitude. It has been pointed out that under constant AC-side voltage control compared to constant reactive power control, the solar inverter system can operate at a lower short-circuit ratio (SCR). A segmented adaptive compensation control method for PCC voltage has been proposed, which enhances the active power output capability of solar inverters without sacrificing the power factor. Installing reactive power compensation devices can more directly and effectively improve the active power output capability of solar inverters. Studies have indicated that after connecting reactive power compensation devices, the active power output capability of solar inverters is related to the type of reactive power compensation device, SCR, and the capacity of the reactive power compensation device. For devices with the same capacity, static var generators (SVGs) have better compensation effects than static var compensators (SVCs). Although adding SVGs can effectively improve the active power output capability of solar inverters, it increases construction costs. In light of this, an improved SVG topology has been proposed, allowing SVGs to operate at lower DC-side voltages, thereby saving construction costs. While the aforementioned research has to some extent enhanced the active power output capability of solar inverters, the following issues remain: traditional constant AC-side voltage control typically stabilizes the PCC voltage at a fixed value, but in extremely weak grids, this control method may lead to an excessively low power factor or the non-existence of a real solution for the solar inverter operating point. Existing studies have not detailed the analysis of the static power stability operation region of solar inverters, and research combining power transmission with stability optimization control under weak grid conditions is rarely reported.
Therefore, this study investigates the static power stability operation region and optimal control of solar inverters under weak grid conditions. First, with the static power stability operation of solar inverters as the objective, and the output current amplitude and the allowable range of PCC voltage as constraints, the operation modes of solar inverters in different SCR intervals are determined, and the corresponding calculation method for current reference values is provided. Second, to improve the stability of solar inverter operation at low SCR, an impedance reshaping optimal control method is proposed. This method incorporates a positive-negative sequence separation link based on a quasi-proportional-resonant controller in the front stage of the phase-locked loop (PLL), which helps smooth the overall phase of the system’s mid-frequency band. Additionally, an amplitude-phase compensation link is introduced in the current loop, which further increases the overall phase of the system in the mid-frequency band. Finally, a simulation model is built, demonstrating that the proposed impedance reshaping optimal control method can ensure the stability of solar inverters even at an SCR of 1. Based on this method, the correctness of the static power stability operation region of solar inverters is verified, enabling solar inverters to select appropriate operation modes in different SCR intervals to ensure stable transmission of solar power to the grid.
Power Output Characteristics of Solar Inverters
Large-scale centralized photovoltaic power stations are mainly located in remote areas and need to be transmitted over long-distance high-voltage transmission lines to load centers for consumption. The line reactance value is much higher than the resistance value, so the influence of line resistance is neglected. For ease of analysis, relevant circuit parameters on the high-voltage side are converted to the low-voltage side. The typical topology of a renewable energy grid-connected system is shown below, where $U_{dc}$ is the DC-side voltage, $C_{dc}$ is the DC-side capacitor, $P_{out}$ is the active power output on the AC side, $L_{1f}$ and $L_{2f}$ are filter inductors, $C_f$ is the filter capacitor, $i_j$ (where $j = a, b, c$) is the output current of the solar inverter, $i_{cj}$ is the filter capacitor current, $u_{pccj}$ is the PCC voltage, $u_{gj}$ is the grid voltage, and $L_g$ is the grid equivalent inductance.
The parameters of the grid-connected system are summarized in the following table:
| Parameter | Value |
|---|---|
| Rated DC-side power $P_N$ (kW) | 50 |
| Rated solar inverter current $I_N$ (A) | 107.18 |
| DC-side voltage $U_{dc}$ (V) | 800 |
| DC-side capacitor $C_{dc}$ (μF) | 5000 |
| Filter inductor $L_{1f}$ (mH) | 2 |
| Filter inductor $L_{2f}$ (mH) | 0.5 |
| Filter capacitor $C_f$ (μF) | 10 |
| Rated voltage $U_N$ (V) | 311 |
Based on the topology, the expressions for the PCC voltage amplitude $U_{pcc}$ and the output active power $P_{out}$ are derived as follows:
$$U_{pcc} = \sqrt{(U_N)^2 – (I_d X_g)^2} – I_q X_g$$
$$P_{out} = 1.5 I_d U_{pcc} = 1.5 I_d \left[ \sqrt{(U_N)^2 – (I_d X_g)^2} – I_q X_g \right]$$
where $U_N$ is the grid rated voltage amplitude, $X_g$ is the grid inductive reactance, $I_d$ and $I_q$ are the d-axis and q-axis currents in the dq rotating coordinate system, respectively. According to the voltage orientation principle, the q-axis component of the PCC voltage is zero, so $I_d$ and $I_q$ can be referred to as the active current and reactive current, respectively, and $I_d \leq U_N / X_g$.
The short-circuit ratio (SCR), denoted as $R_{SC}$, is defined as:
$$R_{SC} = \frac{1.5 U_N^2}{P_N X_g} = \frac{U_N}{I_N X_g}$$
The output characteristic curve of a solar inverter operating at unit power factor is illustrated in the following analysis. At unit power factor, $U_{pcc}$ decreases as $I_d$ increases, and the smaller the $R_{SC}$, the faster $U_{pcc}$ decreases. Due to this characteristic of $U_{pcc}$, $P_{out}$ has a maximum value $P_{out,max}$ at unit power factor. As $R_{SC}$ continuously decreases, $P_{out,max}$ also shows a trend of continuous decrease. When $R_{SC}$ decreases to a certain extent, $P_{out,max} < P_N$ may occur, which is not conducive to static power stability. Setting $I_q = 0$, from the equation for $P_{out}$, we obtain:
$$P_{out,max} = \frac{3 U_N^2}{4 X_g}$$
Substituting $P_{out,max} = P_N$ into the equation for $R_{SC}$ yields the ideal static power stability critical $R_{SC}$ of 2. However, in practice, $U_{pcc}$ needs to meet certain range requirements, and the issue of solar inverter output current amplitude limit must be considered. Therefore, further analysis is required.
Static Power Stability Operation Region of Solar Inverters
Static Power Stability Operation Region at Unit Power Factor
Grid codes specify that the variation range of $U_{pcc}$ should not exceed 10% of $U_N$, i.e., $U_{pcc} \in [0.9 U_N, 1.1 U_N]$. At unit power factor, $U_{pcc}$ decreases gradually as $I_d$ increases. Substituting $U_{pcc} = 0.9 U_N$, $P_{out} = P_N$, and $I_q = 0$ into the power equation yields the corresponding $I_d$:
$$I_d = \frac{P_N}{1.35 U_N}$$
To prevent solar inverter overload, the output current amplitude $I$ must satisfy:
$$I = \sqrt{I_d^2 + I_q^2} < I_{lim}$$
where $I_{lim}$ is the solar inverter output current limit. The long-term overload capability of solar inverters is generally 1.1 times the rated current, so $I_{lim} = 1.1 I_N$.
The critical operating point for static power stability at unit power factor can be expressed as:
$$
\begin{cases}
U_{pcc} = \frac{P_N}{1.5 I_{lim}} \\
I_d = I_{lim}
\end{cases}
$$
From the equations for $U_{pcc}$, $R_{SC}$, and the critical operating point, the critical short-circuit ratio for static power stability at unit power factor is calculated as $R_{SC1} = 2.641$. When $R_{SC} \geq R_{SC1}$, solar inverters can adopt unit power factor control, and the current reference values can be set as:
$$
\begin{cases}
I_{d,ref} = \frac{1.5 U_N^2 – \sqrt{2.25 U_N^4 – 4 (X_g P_N)^2}}{3 X_g^2} \\
I_{q,ref} = 0
\end{cases}
$$
where $I_{d,ref}$ is the active current reference value and $I_{q,ref}$ is the reactive current reference value.
Static Power Stability Operation Region at Non-Unit Power Factor
When $R_{SC} < R_{SC1}$, certain reactive power compensation is required to achieve static power stability. To minimize reactive power capacity waste, the remaining capacity of solar inverters is prioritized for reactive power compensation. Setting $P_{out} = P_N$, the $(-I_q)$-$I_d$ curves for $R_{SC}$ values of 1.3, 1.2, 0.96, and 0.85 are analyzed. The $(-I_q)$-$I_d$ curve intersects with $I = 1.1 I_N$, and as $R_{SC}$ decreases, the intersection point gradually shifts left. Therefore, $I = 1.1 I_N$, $U_{pcc} = 1.1 U_N$, and $P_{out} = P_N$ represent the critical operating point for static power stability when relying solely on solar inverters for reactive power compensation. Substituting these values into the power, $R_{SC}$, and current equations yields the critical short-circuit ratio for static power stability when relying solely on solar inverters for reactive power compensation as $R_{SC2} = 1.062$.
When $R_{SC} \in [R_{SC2}, R_{SC1})$, static power stability can be achieved by relying solely on solar inverters for reactive power compensation. Substituting $I = 1.1 I_N$ and $P_{out} = P_N$ into the power and current equations, the intersection point $(I_{dr}, -I_{qr})$ of the $(-I_q)$-$I_d$ curve with $I = 1.1 I_N$ within $U_{pcc} \in [0.9 U_N, 1.1 U_N]$ is given by:
$$
\begin{cases}
I_{dr} = \frac{-B + \sqrt{B^2 – 4AC}}{2A} \\
I_{qr} = -\sqrt{(1.1 I_N)^2 – (I_{dr})^2}
\end{cases}
$$
where:
$$
\begin{cases}
A = 3.555 (P_N X_g)^2 + (1.5 U_N)^4 + (1.65 I_N X_g)^4 \\
B = -4.5 (P_N U_N)^2 – 5.445 (P_N I_N X_g)^2 \\
C = (P_N)^4
\end{cases}
$$
The operating range of solar inverters is $U_{pcc} \in [P_N / (1.5 I_{dr}), 1.1 U_N]$. To improve the power factor, the operating point with the highest power factor within this range can be found. The power factor $F_P$ is expressed as:
$$F_P = \frac{1}{\sqrt{1 + \left( \frac{-I_q}{I_d} \right)^2}}$$
The active current $I_{d0}$ and PCC voltage $U_{pcc0}$ corresponding to the maximum value of $F_P$ are:
$$
\begin{cases}
I_{d0} = \frac{2 R_{SC} I_N}{R_{SC}^2 + 4} \\
U_{pcc0} = \frac{P_N \sqrt{R_{SC}^2 + 4}}{3 R_{SC} I_N}
\end{cases}
$$
The operating point $U_{PF,max}$ with the highest power factor is determined as:
$$
U_{PF,max} =
\begin{cases}
\frac{P_N}{1.5 I_{dr}}, & U_{pcc0} < \frac{P_N}{1.5 I_{dr}} \\
U_{pcc0}, & \frac{P_N}{1.5 I_{dr}} \leq U_{pcc0} \leq 1.1 U_N \\
1.1 U_N, & U_{pcc0} > 1.1 U_N
\end{cases}
$$
The current reference values corresponding to $U_{PF,max}$ are:
$$
\begin{cases}
I_{d,ref} = \frac{P_N}{1.5 U_{PF,max}} \\
I_{q,ref} = -\frac{1}{X_g} \left[ U_{PF,max} – \sqrt{(U_N)^2 – (I_{d,ref} X_g)^2} \right]
\end{cases}
$$
When $R_{SC} < R_{SC2}$, reactive power compensation devices need to be installed to achieve static power stability. Achieving static power stability at $U_{pcc} = 1.1 U_N$ corresponds to $I_d = P_N / (1.65 U_N)$. In the $(-I_q)$-$I_d$ curve, $I_{q1}$ and $I_{q2}$ represent the reactive currents output by the solar inverter and the reactive power compensation device, respectively. As $R_{SC}$ further decreases, $U_N / X_g$ may become less than $P_N / (1.65 U_N)$. Substituting $U_N / X_g = P_N / (1.65 U_N)$ into the $R_{SC}$ equation yields $R_{SC} = 0.91$. Since the short-circuit ratio is generally greater than or equal to 1, when $R_{SC} \in [1, R_{SC2})$, reactive power compensation devices need to be installed to achieve static power stability. Common reactive power compensation devices are composed of switching devices, and reducing the output current of these devices can save construction costs. Solving simultaneously with $R_{SC} = 0.91$, the $R_{SC}$ equation, and the reactive current equation, it is found that the $I_d$ corresponding to the minimum point of the $(-I_q)$-$I_d$ curve is less than $P_N / (1.65 U_N)$. Therefore, when $R_{SC} \in [1, R_{SC2})$, controlling $U_{pcc}$ at $1.1 U_N$ can reduce the output current of reactive power compensation devices, saving construction costs. Substituting $I_d = P_N / (1.65 U_N)$ into the reactive current equation yields the current reference values for the solar inverter and the reactive power compensation device:
$$
\begin{cases}
I_{d,ref} = \frac{P_N}{1.65 U_N} \\
I_{q1,ref} = -\frac{464}{1110} I_N \\
I_{q2,ref} = -\frac{1.21 U_N – \sqrt{(1.1 U_N)^2 – (I_N X_g)^2}}{1.1 X_g} – I_{q1,ref}
\end{cases}
$$
where $I_{q1,ref}$ is the reactive current reference value for the solar inverter, and $I_{q2,ref}$ is the reactive current reference value for the reactive power compensation device.
When $R_{SC} \in [1, R_{SC1})$, reactive power compensation devices can also be used to improve the grid power factor. In this case, $U_{PF,max}$ is expressed as:
$$
U_{PF,max} =
\begin{cases}
\max \left( \frac{P_N}{1.65 U_N}, \frac{P_N X_g}{1.5 U_N} \right), & U_{pcc0} < \max \left( \frac{P_N}{1.65 U_N}, \frac{P_N X_g}{1.5 U_N} \right) \\
U_{pcc0}, & \max \left( \frac{P_N}{1.65 U_N}, \frac{P_N X_g}{1.5 U_N} \right) \leq U_{pcc0} \leq 1.1 U_N \\
1.1 U_N, & U_{pcc0} > 1.1 U_N
\end{cases}
$$
Substituting $U_{PF,max}$ into the current reference equations yields $I_{d,ref}$ and $I_{q,ref}$. Denoting the current amplitude corresponding to $I_{d,ref}$ and $I_{q,ref}$ as $I_{ref}$, $I_{q1,ref}$ and $I_{q2,ref}$ are further determined as:
$$
\begin{cases}
I_{q1,ref} = I_{q,ref} \\
I_{q2,ref} = 0
\end{cases}
\quad \text{if } I_{ref} \leq 1.1 I_N
$$
$$
\begin{cases}
I_{q1,ref} = -\sqrt{1.21 I_N^2 – (I_{d,ref})^2} \\
I_{q2,ref} = I_{q,ref} – I_{q1,ref}
\end{cases}
\quad \text{if } I_{ref} > 1.1 I_N
$$
Optimal Control of Solar Inverters
Impedance Model of Solar Inverters
In weak grids, increased grid impedance can adversely affect the stability of solar inverters. Without optimal control, it is difficult to achieve static power stability based on the theory in Section 2. The impedance models of both the phase-locked loop (PLL) and the current loop can be reshaped simultaneously to alter the equivalent output impedance of solar inverters and enhance system stability. The control structure and mathematical block diagram model of the solar inverter control system are described below. Variables with the subscript $a\beta$ represent electrical quantities in the $a\beta$ stationary coordinate system, and variables with the subscript $dq$ represent electrical quantities in the $dq$ rotating coordinate system. QPR denotes the quasi-proportional-resonant controller, PI denotes the proportional-integral controller, $H_i$ is the filter capacitor current feedback coefficient, $K_{PWM}$ is the pulse width modulation gain, $\omega_N$ is the grid rated angular frequency, $\omega_c$ is the PLL output angular frequency, $\theta_c$ is the PLL output phase angle, $G_{QPR}(s)$ is the transfer function of the quasi-proportional-resonant controller, $G_{PLL}(s)$ is the transfer function from $i_{ref}$ to $u_{pcc}$, $G_{ic}(s)$ is the current loop compensation环节, and $G_{Pc}(s)$ is the PLL compensation环节.
The transfer functions $G_1(s)$, $G_2(s)$, $G_{PLL}(s)$, and $G_{QPR}(s)$ are defined as:
$$
G_1(s) = \frac{K_{PWM} G_{QPR}(s)}{s^2 L_{1f} C_f + s K_{PWM} H_i C_f + 1}
$$
$$
G_2(s) = \frac{s^2 L_{1f} C_f + s K_{PWM} H_i C_f + 1}{s^3 L_{1f} L_{2f} C_f + s^2 K_{PWM} H_i L_{2f} C_f + s L_f’}
$$
where $L_f’ = L_{1f} + L_{2f}$.
$$
G_{QPR}(s) = k_p + \frac{2 k_r \omega_i s}{s^2 + 2 \omega_i s + \omega_N^2}
$$
$$
G_{PLL}(s) = \frac{\sqrt{(I_{d,ref})^2 + (I_{q1,ref})^2} \left[ k_{p-PLL} H_s + k_{i-PLL} \right]}{2 H_s^2 + 2 U_{pcc} \left[ k_{p-PLL} H_s + k_{i-PLL} \right]}
$$
where $H_s = s – j \omega_c$, $k_{p-PLL}$ is the proportional coefficient of the PI controller, $k_{i-PLL}$ is the integral coefficient of the PI controller, $k_p$ is the proportional coefficient of the QPR controller, $k_r$ is the resonant coefficient of the QPR controller, and $\omega_i$ is the bandwidth angular frequency of the QPR controller.
Based on the control system model, the Norton equivalent circuit of the solar inverter is derived. The equivalent impedances $Z_1(s)$, $Z_2(s)$, and $Z_{inv}(s)$ are given by:
$$
Z_1(s) = -\frac{1 + G_{ic}(s) G_1(s) G_2(s)}{G_{ic}(s) G_1(s) G_2(s) G_{Pc}(s) G_{PLL}(s)}
$$
$$
Z_2(s) = \frac{1 + G_{ic}(s) G_1(s) G_2(s)}{G_2(s)}
$$
$$
Z_{inv}(s) = \frac{1 + G_{ic}(s) G_1(s) G_2(s)}{G_2(s) – G_{ic}(s) G_1(s) G_2(s) G_{Pc}(s) G_{PLL}(s)}
$$
Design of Impedance Reshaping Control
In the presence of grid harmonics and three-phase unbalance, the quasi-reduced order resonant (QROR) controller can effectively extract the fundamental positive sequence component of the grid. Therefore, adding a positive-negative sequence separation link based on the QROR controller in the front stage of the PLL is beneficial for the PLL to accurately track the fundamental frequency and improve the system’s adaptability in weak grids. The positive-negative sequence separation structure based on the QROR controller can be mathematically represented as:
$$
\begin{cases}
u^+_\alpha = \frac{k_1}{s – j \omega_N + \omega_r} (u_\alpha – u^+_\alpha – u^-_\alpha) \\
u^-_\alpha = \frac{k_1}{s + j \omega_N + \omega_r} (u_\alpha – u^+_\alpha – u^-_\alpha)
\end{cases}
$$
where $u^+_\alpha$ is the positive sequence voltage, $u^-_\alpha$ is the negative sequence voltage, $\omega_r$ is the bandwidth angular frequency of the QROR controller, and $k_1$ is the gain of the QROR controller.
From the above equations, the closed-loop transfer function from $u^+_\alpha$ to $u_\alpha$, i.e., $G_{Pc}(s)$, is derived as:
$$G_{Pc}(s) = \frac{k_1 s + j k_1 \omega_N + k_1 \omega_r}{s^2 + 2 (\omega_r + k_1) s + \omega_N^2 + \omega_r^2 + 2 k_1 \omega_r}$$
The expression for $G_{ic}(s)$ is set as:
$$G_{ic}(s) = x_3 \frac{1 + x_1 s}{1 + x_2 s}$$
where $x_1$ and $x_2$ are phase compensation coefficients, and $x_3$ is the amplitude compensation coefficient.
When compensating $G_{ic}(s)$ at frequency $f_c$, to achieve the maximum compensation phase angle, $x_1$ and $x_2$ must satisfy the following conditions:
$$
\begin{cases}
x_2 = \frac{1}{x_1 (2 \pi f_c)^2} \\
x_1 = \frac{\tan \phi_{c max} + \sqrt{1 + (\tan \phi_{c max})^2}}{2 \pi f_c}
\end{cases}
$$
where $\phi_{c max}$ is the maximum compensation phase angle.
In this study, $f_c = 135$ Hz is chosen, and $\phi_{c max} = 45^\circ$ is set. From the above equations, $x_1 = 0.002846$ and $x_2 = 0.000488$ are calculated. Although $(1 + x_1 s)/(1 + x_2 s)$ can alter the phase-frequency characteristics of the system, it also affects the amplitude-frequency characteristics. Therefore, $x_3$ is set to balance the impact of $(1 + x_1 s)/(1 + x_2 s)$, and $x_3 = 0.3$ is chosen here.
The other control parameters of the system are set as follows: $k_{p-PLL} = 1.4$, $k_{i-PLL} = 300$, $H_i = 12.5$, $\omega_i = 2\pi$, $k_p = 6.14$, $k_r = 1600$, $\omega_r = 0.5\pi$, $k_1 = 100$. According to the impedance stability criterion, the stability margin of the system can be characterized by judging whether $X_g(s)/Z_{inv}(s)$ satisfies the Nyquist criterion. The impedance characteristics before and after compensation are analyzed.
After introducing the $G_{Pc}(s)$ compensation link, the amplitude of the equivalent output impedance of the solar inverter in the mid-frequency band is increased, and the phase in the mid-frequency band becomes overall smoother. However, as the grid impedance increases, most of the phase in the mid-frequency band remains below $-45^\circ$. After further introducing the $G_{ic}(s)$ compensation link, not only are the advantages of the $G_{Pc}(s)$ compensation link retained, but the phase in the mid-frequency band is overall increased, with most phases above $-45^\circ$, significantly enhancing the system’s adaptability to weak grids.
Simulation Analysis
To verify the correctness of the theoretical analysis, a simulation model based on the solar inverter control structure is built in simulation software. To validate the effectiveness of PLL impedance reshaping combined with current loop impedance reshaping, comparative simulations are conducted under two control modes. Control Mode 1: No compensation links are introduced. Control Mode 2: Both $G_{Pc}(s)$ and $G_{ic}(s)$ are introduced for compensation. The simulation results show that without introducing compensation links, the adaptability of solar inverters to weak grids is poor. After introducing both $G_{Pc}(s)$ and $G_{ic}(s)$ for compensation, even at $R_{SC} = 1.03$, solar inverters can still output current normally, proving the effectiveness of PLL impedance reshaping combined with current loop impedance reshaping.
Simulations for verifying static power stability are conducted under Control Mode 2. $R_{SC}$ is set to 20, 2.641, and 2.1, and solar inverters output rated active power at unit power factor under these $R_{SC}$ values. The simulation results indicate that when $R_{SC} = 20$, $U_{pcc} = 310.60$ V and $I < 1.1 I_N$; when $R_{SC} = 2.641$, $U_{pcc} = 282.73$ V $> 0.9 U_N$ and $I = 1.1 I_N$; when $R_{SC} = 2.1$, $U_{pcc} = 251.23$ V $< 0.9 U_N$ and $I > 1.1 I_N$. This confirms that at unit power factor, the smaller the $R_{SC}$, the larger the current amplitude required to output the same active power. Moreover, 2.641 is the critical $R_{SC}$ for static power stability at unit power factor, consistent with theoretical analysis. Some previous studies only limit $U_{pcc}$ and conclude that static power stability can be achieved at unit power factor when $R_{SC} > 2.5$, indicating that such conclusions are limited.
Setting $R_{SC} = 1.2$, static power stability can be achieved by relying solely on solar inverters for reactive power compensation. The operating region is $U_{pcc} \in [1.0376 U_N, 1.1 U_N]$, and the operating point with the highest power factor is $U_{pcc} = 1.0376 U_N$. When reactive power compensation devices are added, the operating point with the highest power factor is $U_{pcc} = 0.9718 U_N$. In the simulation, from 0.3 to 0.4 s, $U_{pcc}$ is controlled at $1.1 U_N$; from 0.4 to 0.5 s, $U_{pcc}$ is controlled at $1.0688 U_N$; from 0.5 to 0.6 s, $U_{pcc}$ is controlled at $1.0376 U_N$; from 0.6 to 0.7 s, $U_{pcc}$ is controlled at $U_N$; from 0.7 to 0.8 s, $U_{pcc}$ is controlled at $0.9718 U_N$; from 0.8 to 0.9 s, $U_{pcc}$ is controlled at $0.9436 U_N$. The simulation results show that when $R_{SC} = 1.2$, solar inverters can achieve static power stability solely by their own capacity when operating within $U_{pcc} \in [1.0376 U_N, 1.1 U_N]$. When $U_{pcc}$ is controlled at $1.0376 U_N$, $I$ just reaches $1.1 I_N$, and the system operates at the point with the highest power factor when relying solely on solar inverters for reactive power compensation, with $F_P = 0.8773$. By using reactive power compensation devices to control $U_{pcc}$ at $0.9718 U_N$, $F_P$ can be further increased to $0.8836$. According to some previous methods, controlling $U_{pcc}$ at $0.9 U_N$ would result in a theoretical value of $I_d$ greater than $1.1 I_N$, causing $I$ to exceed the limit. This proves that $U_{pcc}$ cannot be controlled at a fixed value but needs to be adjusted according to the actual $R_{SC}$.
It should be noted that when $R_{SC} \in [1, 1.062)$, setting the current reference values for solar inverters and reactive power compensation devices according to the derived equations can not only achieve static power stability but also reduce the output current of reactive power compensation devices, saving construction costs. This part is not simulated here for brevity.
Conclusion
Addressing the issue of limited power output capability of solar inverters in weak grid environments, this study analyzes the static power stability operation region of solar inverters and proposes an impedance reshaping optimal control method. Through theoretical analysis and simulation verification, the following conclusions are drawn:
1. The proposed optimal control method combining PLL impedance reshaping and current loop impedance reshaping not only increases the phase of the system in the mid-frequency band but also makes the phase in the mid-frequency band overall smoother, significantly enhancing the system’s adaptability to weak grids. Even when the SCR is close to 1, this method ensures the stability of solar inverter operation.
2. When $R_{SC} \geq 2.641$, no reactive power compensation is needed, and solar inverters can achieve static power stability by adopting unit power factor control. When $R_{SC} \in [1.062, 2.641)$, static power stability can be achieved by relying solely on solar inverters for reactive power compensation, and reactive power compensation devices can be added to maximize the grid power factor. When $R_{SC} \in [1, 1.062)$, reactive power compensation devices need to be installed to achieve static power stability. Controlling the PCC voltage at 1.1 times the rated voltage can reduce the output current of reactive power compensation devices, saving construction costs.
Although this study investigates the static power stability operation region and optimal control of solar inverters, it only considers the pure inductance case. In scenarios where resistance cannot be neglected, the research content needs to be improved. Moreover, the optimization control does not consider comprehensive factors, and further consideration is needed for situations where the grid has background harmonics.