With the rapid development of the global economy, the demand for energy has been increasing, leading to an inevitable energy crisis. To accelerate the energy transition, the widespread adoption of solar energy has become a common choice worldwide. Photovoltaic power must be transmitted to load centers through long transmission lines and multiple transformers, which increases grid impedance and results in the point of common coupling (PCC) voltage exhibiting weak grid characteristics, thereby limiting the power absorption capacity of photovoltaic systems. Research on the static power stability operation region and optimal control of grid-connected inverters (GCIs), also referred to as utility interactive inverters, is essential for ensuring stable power transmission from photovoltaic sources to the grid, improving the photovoltaic absorption capability of power systems under weak grid conditions, and contributing to the construction of new power systems.
To enhance the power transmission capability of photovoltaic systems in grids, current research primarily focuses on optimizing the reactive power compensation strategies of utility interactive inverters and installing additional reactive power compensation devices. In terms of optimizing GCI reactive power compensation strategies, power synchronization control strategies have been proposed to simplify control structures and improve the active power output capability of utility interactive inverters, but their response speed is slow. To address this, an adaptive reactive power droop control method has been introduced, which balances power factor and PCC voltage amplitude while enhancing the active power output capability of the utility interactive inverter. It has been pointed out that constant AC-side voltage control, compared to constant reactive power control, allows the GCI system to operate at lower short-circuit ratios (SCRs). A segmented adaptive compensation control method for PCC voltage has been proposed to improve the active power output capability of the utility interactive inverter without sacrificing power factor. Installing reactive power compensation devices can more directly and effectively enhance the active power output capability of utility interactive inverters. Studies indicate that after connecting reactive power compensation devices, the active power output capability of the utility interactive inverter is related to the type of compensation device, SCR, and the capacity of the compensation device. For 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 utility interactive inverters, it increases construction costs. To mitigate this, an improved SVG topology has been proposed, allowing SVGs to operate at lower DC-side voltages and reducing construction costs. While these studies have improved the active power output capability of utility interactive inverters to some extent, several issues remain: traditional constant AC-side voltage control typically stabilizes PCC voltage at a fixed value, but in extremely weak grids, this approach may lead to excessively low power factors or the non-existence of real solutions for GCI operating points; existing research lacks detailed analysis of the static power stability operation region of utility interactive inverters, and studies combining power transmission with stability optimization control under weak grid conditions are rarely reported.
Therefore, this study investigates the static power stability operation region and optimal control of utility interactive inverters under weak grid conditions. First, with the static power stable operation of the utility interactive inverter as the objective, and the output current amplitude and the permissible range of PCC voltage as constraints, the operation mode of the utility interactive inverter is determined in different SCR intervals, and the corresponding calculation method for current reference values is provided. Second, an impedance reshaping optimal control method is proposed to improve the stability of utility interactive inverter operation at low SCRs. The method incorporates a positive-negative sequence separation link based on a quasi-proportional-resonant controller in the phase-locked loop (PLL) front stage, which helps to 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, and the proposed impedance reshaping optimal control method is shown to ensure the stability of the utility interactive inverter even at an SCR of 1. Based on this method, the correctness of the static power stability operation region of the utility interactive inverter is verified, enabling the utility interactive inverter to select appropriate operation modes in different SCR intervals to ensure stable power transmission from photovoltaic sources to the grid.
The power output characteristics of photovoltaic utility interactive inverters are critical for understanding their behavior in weak grids. Large-scale centralized photovoltaic power stations are mainly located in remote areas and require long-distance high-voltage transmission lines to deliver power to load centers. The line reactance value is much higher than the resistance value, so the line resistance is neglected in analysis. For convenience, high-side circuit parameters are converted to the low side. The typical topology of a renewable energy grid-connected system is shown in the figure, 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 utility interactive 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 inductor. The parameters of the grid-connected system are summarized in Table 1.
| Parameter | Value |
|---|---|
| Rated DC-side power $P_N$ (kW) | 50 |
| Rated current of utility interactive inverter $I_N$ (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 |
From the system topology, the PCC voltage amplitude $U_{pcc}$ and active power output $P_{out}$ can be expressed as:
$$U_{pcc} = \sqrt{U_N^2 – (I_d X_g)^2} – I_q X_g$$
$$P_{out} = 1.5 I_d U_{pcc} = 1.5 I_d \left( \sqrt{U_N^2 – (I_d X_g)^2} – I_q X_g \right)$$
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 PCC voltage is zero, so $I_d$ and $I_q$ are referred to as active and reactive currents, with $I_d \leq U_N / X_g$. 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 the utility interactive inverter operating at unit power factor is shown in Figure 2. At unit power factor, $U_{pcc}$ decreases as $I_d$ increases, and the smaller the $R_{SC}$, the faster $U_{pcc}$ drops. Due to this characteristic, $P_{out}$ has a maximum value $P_{out,max}$ at unit power factor. As $R_{SC}$ decreases, $P_{out,max}$ also decreases. When $R_{SC}$ drops to a certain level, $P_{out,max} < P_N$ may occur, which is detrimental to static power stability. Setting $I_q = 0$, from Equation (2), we obtain:
$$P_{out,max} = \frac{3U_N^2}{4X_g}$$
Substituting $P_{out,max} = P_N$ into Equation (3), the ideal static power stability critical $R_{SC}$ is 2. However, in practice, $U_{pcc}$ must satisfy a permissible range, and the utility interactive inverter output current amplitude limit must be considered. Therefore, further analysis is required.
The static power stability operation region of the utility interactive inverter is analyzed under two scenarios: unit power factor and non-unit power factor. Grid codes specify that the variation range of $U_{pcc}$ should not exceed 10% of $U_N$, i.e., $U_{pcc} \in [0.9U_N, 1.1U_N]$. At unit power factor, $U_{pcc}$ decreases as $I_d$ increases. Substituting $U_{pcc} = 0.9U_N$, $P_{out} = P_N$, and $I_q = 0$ into Equation (2), the corresponding $I_d$ is:
$$I_d = \frac{P_N}{1.35U_N}$$
To prevent overloading of the utility interactive inverter, the output current amplitude $I$ must satisfy:
$$I = \sqrt{I_d^2 + I_q^2} < I_{lim}$$
where $I_{lim}$ is the utility interactive inverter output current limit. The long-term overload capability of utility interactive inverters is typically 1.1 times the rated current, so $I_{lim} = 1.1I_N$. The critical operating point for static power stability at unit power factor can be expressed as:
$$\begin{cases}
U_{pcc} = \frac{P_N}{1.5I_{lim}} \\
I_d = I_{lim}
\end{cases}$$
From Equations (1), (3), and (7), the critical $R_{SC}$ for static power stability at unit power factor is $R_{SC1} = 2.641$. When $R_{SC} \geq R_{SC1}$, the utility interactive inverter can adopt unit power factor control, and the current reference values are set as:
$$\begin{cases}
I_{d,ref} = \frac{1.5U_N^2 – \sqrt{2.25U_N^4 – 4(X_g P_N)^2}}{3X_g^2} \\
I_{q,ref} = 0
\end{cases}$$
When $R_{SC} < R_{SC1}$, reactive power compensation is required to achieve static power stability. To minimize reactive power waste, the remaining capacity of the utility interactive inverter is prioritized for compensation. Setting $P_{out} = P_N$, the $(-I_q)$-$I_d$ curves for $R_{SC} = 1.3, 1.2, 0.96,$ and $0.85$ are shown in Figure 3. The intersection of the $(-I_q)$-$I_d$ curve and $I = 1.1I_N$ gradually shifts left as $R_{SC}$ decreases. Thus, $I = 1.1I_N$, $U_{pcc} = 1.1U_N$, and $P_{out} = P_N$ represent the critical operating point for static power stability when relying solely on the utility interactive inverter for reactive power compensation. Substituting these values into Equations (2), (3), and (6), the critical $R_{SC}$ for static power stability with only utility interactive inverter compensation is $R_{SC2} = 1.062$.
When $R_{SC} \in [R_{SC2}, R_{SC1})$, static power stability can be achieved solely through reactive power compensation from the utility interactive inverter. The operating region is $U_{pcc} \in [P_N/(1.5I_{dr}), 1.1U_N]$, where $(I_{dr}, -I_{qr})$ is the intersection of the $(-I_q)$-$I_d$ curve and $I = 1.1I_N$ within $U_{pcc} \in [0.9U_N, 1.1U_N]$. The values are calculated as:
$$\begin{cases}
I_{dr} = \frac{-B + \sqrt{B^2 – 4AC}}{2A} \\
I_{qr} = -\sqrt{(1.1I_N)^2 – I_{dr}^2}
\end{cases}$$
where:
$$\begin{cases}
A = 3.555(P_N X_g)^2 + (1.5U_N)^4 + (1.65I_N X_g)^4 \\
B = -4.5(P_N U_N)^2 – 5.445(P_N I_N X_g)^2 \\
C = P_N^4
\end{cases}$$
To maximize power factor, the operating point with the highest power factor within the permissible range is identified. The power factor $F_P$ is:
$$F_P = \frac{1}{\sqrt{1 + \left( \frac{-I_q}{I_d} \right)^2}}$$
The active current $I_{d0}$ and PCC voltage $U_{pcc0}$ corresponding to the maximum $F_P$ 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:
$$U_{PF,max} = \begin{cases}
\frac{P_N}{1.5 I_{dr}}, & U_{pcc0} < \frac{P_N}{1.5 I_{dr}} \\
U_{pcc0}, & \frac{P_N}{1.5 I_{dr}} \leq U_{pcc0} \leq 1.1U_N \\
1.1U_N, & U_{pcc0} > 1.1U_N
\end{cases}$$
The corresponding current reference values 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 $R_{SC} < R_{SC2}$, additional reactive power compensation devices are required for static power stability. Setting $U_{pcc} = 1.1U_N$ achieves static power stability with the corresponding $I_d = P_N/(1.65U_N)$. In Figure 3, $I_{q1}$ and $I_{q2}$ represent the reactive currents output by the utility interactive inverter and the compensation device, respectively. As $R_{SC}$ decreases further, $U_N/X_g$ may become less than $P_N/(1.65U_N)$. Substituting $U_N/X_g = P_N/(1.65U_N)$ into Equation (3) yields $R_{SC} = 0.91$. Since SCR is generally greater than or equal to 1, when $R_{SC} \in [1, R_{SC2})$, reactive power compensation devices are needed. Common reactive power compensation devices consist of switching devices, and reducing their output current can save construction costs. For $R_{SC} \in [1, R_{SC2})$, controlling $U_{pcc}$ at $1.1U_N$ minimizes the output current of the compensation device. The current reference values for the utility interactive inverter and the compensation device are:
$$\begin{cases}
I_{d,ref} = \frac{P_N}{1.65U_N} \\
I_{q1,ref} = -\frac{464}{1110} I_N \\
I_{q2,ref} = -\frac{1.21U_N – \sqrt{(1.1U_N)^2 – (I_N X_g)^2}}{1.1X_g} – I_{q1,ref}
\end{cases}$$
When $R_{SC} \in [1, R_{SC1})$, reactive power compensation devices can also be used to improve grid power factor. The operating point $U_{PF,max}$ is then:
$$U_{PF,max} = \begin{cases}
\max\left( \frac{P_N}{1.65U_N}, \frac{P_N X_g}{1.5U_N} \right), & U_{pcc0} < \max\left( \frac{P_N}{1.65U_N}, \frac{P_N X_g}{1.5U_N} \right) \\
U_{pcc0}, & \max\left( \frac{P_N}{1.65U_N}, \frac{P_N X_g}{1.5U_N} \right) \leq U_{pcc0} \leq 1.1U_N \\
1.1U_N, & U_{pcc0} > 1.1U_N
\end{cases}$$
Substituting Equation (17) into Equation (15) gives $I_{d,ref}$ and $I_{q,ref}$. Denoting the current amplitude corresponding to these values as $I_{ref}$, the reactive current reference values are:
$$\begin{cases}
I_{q1,ref} = I_{q,ref}, \quad I_{q2,ref} = 0, & I_{ref} \leq 1.1I_N \\
I_{q1,ref} = -\sqrt{1.21 I_N^2 – I_{d,ref}^2}, \quad I_{q2,ref} = I_{q,ref} – I_{q1,ref}, & I_{ref} > 1.1I_N
\end{cases}$$
The optimization control of the utility interactive inverter is crucial for stability in weak grids. Increased grid impedance can adversely affect the stability of utility interactive inverters. Without optimization, it is difficult to achieve static power stability based on the theoretical analysis in Section 2. By reshaping the impedance models of both the phase-locked loop (PLL) and the current loop, the equivalent output impedance of the utility interactive inverter can be altered to enhance system stability. The control structure and mathematical block diagram model of the utility interactive inverter control system are shown in Figures 4 and 5, respectively. Variables with subscript $αβ$ are in the $αβ$ stationary coordinate system, and those with subscript $dq$ are in the $dq$ rotating coordinate system. QPR denotes a quasi-proportional-resonant controller, PI denotes a proportional-integral controller, $H_i$ is the filter capacitor current feedback coefficient, $K_{PWM}$ is the pulse-width modulation gain, $ω_N$ is the grid rated angular frequency, $ω_c$ is the PLL output angular frequency, $θ_c$ 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 link, and $G_{Pc}(s)$ is the PLL compensation link.
The transfer functions $G_1(s)$, $G_2(s)$, $G_{PLL}(s)$, and $G_{QPR}(s)$ are:
$$G_1(s) = \frac{K_{PWM} G_{QPR}(s)}{s^2 L_{1f} C_f + s K_{PWM} H_i C_f + 1}$$
$$G_2(s) = \frac{s^2 L_{1f} C_f + s K_{PWM} H_i C_f + 1}{s^3 L_{1f} L_{2f} C_f + s^2 K_{PWM} H_i L_{2f} C_f + s L’_f}$$
where $L’_f = L_{1f} + L_{2f}$.
$$G_{QPR}(s) = k_p + \frac{2k_r ω_i s}{s^2 + 2ω_i s + ω_N^2}$$
$$G_{PLL}(s) = \frac{\sqrt{I_{d,ref}^2 + I_{q1,ref}^2} \left( k_{p-PLL} H_s + k_{i-PLL} \right)}{2H_s^2 + 2U_{pcc} \left( k_{p-PLL} H_s + k_{i-PLL} \right)}$$
where $H_s = s – jω_c$, $k_{p-PLL}$ 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 QPR, $k_r$ is the resonant coefficient of QPR, and $ω_i$ is the bandwidth angular frequency of QPR.
The Norton equivalent circuit of the utility interactive inverter is shown in Figure 6. The equivalent impedances $Z_1(s)$, $Z_2(s)$, and $Z_{inv}(s)$ are:
$$Z_1(s) = -\frac{1 + G_{ic}(s) G_1(s) G_2(s)}{G_{ic}(s) G_1(s) G_2(s) G_{Pc}(s) G_{PLL}(s)}$$
$$Z_2(s) = \frac{1 + G_{ic}(s) G_1(s) G_2(s)}{G_2(s)}$$
$$Z_{inv}(s) = \frac{1 + G_{ic}(s) G_1(s) G_2(s)}{G_2(s) – G_{ic}(s) G_1(s) G_2(s) G_{Pc}(s) G_{PLL}(s)}$$
The design of impedance reshaping control is essential for improving stability. When the grid contains harmonics and three-phase unbalance, a 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 a QROR controller in the PLL front stage helps the PLL accurately track the fundamental frequency, enhancing the system’s adaptability to weak grids. The positive-negative sequence separation structure using a QROR controller can be mathematically expressed as:
$$\begin{cases}
u^+_α = \frac{k_1}{s – jω_N + ω_r} (u_α – u^+_α – u^-_α) \\
u^-_α = \frac{k_1}{s + jω_N + ω_r} (u_α – u^+_α – u^-_α)
\end{cases}$$
where $u^+_α$ is the positive-sequence voltage, $u^-_α$ is the negative-sequence voltage, $ω_r$ is the bandwidth angular frequency of the QROR controller, and $k_1$ is the gain of the QROR controller. The closed-loop transfer function from $u^+_α$ to $u_α$, i.e., $G_{Pc}(s)$, is:
$$G_{Pc}(s) = \frac{k_1 s + j k_1 ω_N + k_1 ω_r}{s^2 + 2(ω_r + k_1)s + ω_N^2 + ω_r^2 + 2k_1 ω_r}$$
The current loop compensation link $G_{ic}(s)$ 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 at frequency $f_c$, to achieve the maximum compensation phase angle, $x_1$ and $x_2$ must satisfy:
$$\begin{cases}
x_2 = \frac{1}{x_1 (2π f_c)^2} \\
x_1 = \frac{\tan φ_{c max} + \sqrt{1 + (\tan φ_{c max})^2}}{2π f_c}
\end{cases}$$
where $φ_{c max}$ is the maximum compensation phase angle. Here, $f_c = 135$ Hz and $φ_{c max} = 45°$ are chosen, yielding $x_1 = 0.002846$ and $x_2 = 0.000488$. Although $(1 + x_1 s)/(1 + x_2 s)$ alters the phase-frequency characteristics, it also affects the magnitude-frequency characteristics. Therefore, $x_3$ is set to balance this effect, with $x_3 = 0.3$.
Other control parameters are: $k_{p-PLL} = 1.4$, $k_{i-PLL} = 300$, $H_i = 12.5$, $ω_i = 2π$, $k_p = 6.14$, $k_r = 1600$, $ω_r = 0.5π$, $k_1 = 100$. According to the impedance stability criterion, the system stability margin can be characterized by checking whether $X_g(s)/Z_{inv}(s)$ satisfies the Nyquist criterion. The impedance characteristics before and after compensation are shown in Figure 7. After introducing the $G_{Pc}(s)$ compensation link, the magnitude of the utility interactive inverter equivalent output impedance in the mid-frequency band increases, and the phase in this band becomes smoother. However, as grid impedance increases, most of the mid-frequency phase remains below -45°. Further introducing the $G_{ic}(s)$ compensation link not only retains the advantages of $G_{Pc}(s)$ but also increases the overall phase in the mid-frequency band, with most phases above -45°, greatly enhancing the system’s adaptability to weak grids.
Simulation analysis is conducted to verify the correctness of the theoretical analysis. A simulation model based on Figure 4 is built. To validate the effectiveness of PLL impedance reshaping combined with current loop impedance reshaping, comparative simulations are performed under two control modes: Control Mode 1 without compensation links, and Control Mode 2 with both $G_{Pc}(s)$ and $G_{ic}(s)$ compensation. The results are shown in Figure 8. Without compensation, the utility interactive inverter has poor adaptability to weak grids. With both $G_{Pc}(s)$ and $G_{ic}(s)$ compensation, the utility interactive inverter can output current normally even at $R_{SC} = 1.03$, proving the effectiveness of the combined impedance reshaping approach.
Simulations for static power stability are conducted under Control Mode 2. For $R_{SC} = 20, 2.641,$ and $2.1$, the utility interactive inverter outputs rated active power at unit power factor. The results are shown in Figure 9. For $R_{SC} = 20$, $U_{pcc} = 310.60$ V and $I < 1.1I_N$; for $R_{SC} = 2.641$, $U_{pcc} = 282.73$ V $> 0.9U_N$ and $I = 1.1I_N$; for $R_{SC} = 2.1$, $U_{pcc} = 251.23$ V $< 0.9U_N$ and $I > 1.1I_N$. This confirms that at unit power factor, smaller $R_{SC}$ requires larger current amplitude for the same active power output, and $R_{SC} = 2.641$ is the critical value for static power stability at unit power factor, consistent with theoretical analysis.
For $R_{SC} = 1.2$, static power stability can be achieved solely through reactive power compensation from the utility interactive inverter. The operating region is $U_{pcc} \in [1.0376U_N, 1.1U_N]$, and the operating point with the highest power factor is $U_{pcc} = 1.0376U_N$. When adding a reactive power compensation device, the operating point with the highest power factor is $U_{pcc} = 0.9718U_N$. The simulation settings are: from 0.3 to 0.4 s, control $U_{pcc}$ at $1.1U_N$; from 0.4 to 0.5 s, control at $1.0688U_N$; from 0.5 to 0.6 s, control at $1.0376U_N$; from 0.6 to 0.7 s, control at $U_N$; from 0.7 to 0.8 s, control at $0.9718U_N$; from 0.8 to 0.9 s, control at $0.9436U_N$. The results are shown in Figure 10. For $R_{SC} = 1.2$, the utility interactive inverter can achieve static power stability within $U_{pcc} \in [1.0376U_N, 1.1U_N]$ using only its own capacity. Controlling $U_{pcc}$ at $1.0376U_N$ results in $I$ just reaching $1.1I_N$, operating at the point with the highest power factor ($F_P = 0.8773$) when relying solely on the utility interactive inverter. With a compensation device controlling $U_{pcc}$ at $0.9718U_N$, $F_P$ is further improved to $0.8836$. Controlling $U_{pcc}$ at $0.9U_N$ as suggested in some literature would cause $I_d$ to exceed $1.1I_N$, leading to current limit violation. This demonstrates that $U_{pcc}$ should not be fixed at a constant value but adjusted based on the actual $R_{SC}$.
It is worth noting that for $R_{SC} \in [1, 1.062)$, setting the current reference values for the utility interactive inverter and the compensation device according to Equation (16) not only achieves static power stability but also reduces the output current of the compensation device, saving construction costs. This aspect is not further simulated here.
In conclusion, this study addresses the limitation of power output capability of utility interactive inverters in weak grid environments by analyzing the static power stability operation region and proposing an impedance reshaping optimal control method. Theoretical analysis and simulation verification lead to the following conclusions: First, the proposed PLL impedance reshaping combined with current loop impedance reshaping optimization control method not only increases the phase in the system’s mid-frequency band but also smooths the overall phase in this band, greatly enhancing the system’s adaptability to weak grids. Even at $R_{SC}$ close to 1, this method ensures the stability of utility interactive inverter operation. Second, when $R_{SC} \geq 2.641$, no reactive power compensation is needed, and the utility interactive inverter can achieve static power stability with unit power factor control. When $R_{SC} \in [1.062, 2.641)$, static power stability can be achieved solely through reactive power compensation from the utility interactive inverter, and additional compensation devices can be used to maximize grid power factor. When $R_{SC} \in [1, 1.062)$, reactive power compensation devices are required for static power stability, and controlling PCC voltage at $1.1$ times the rated voltage reduces the output current of the compensation device, saving construction costs.
Although this study investigates the static power stability operation region and optimal control of utility interactive inverters, it only considers pure inductance scenarios. In cases where resistance cannot be neglected, further research is needed. Additionally, optimization control factors are not comprehensive, and scenarios with grid background harmonics require further consideration.