In the context of accelerating the achievement of carbon peak and carbon neutrality goals, the large-scale development of renewable energy generation technologies and the construction of new power systems have become paramount. The proportion of power electronic devices in the grid is increasing rapidly, posing significant challenges to the stable operation of these systems. The grid-connected inverter, as the interface for integrating new energy sources into the grid, is evolving from current-source grid-following control to voltage-source grid-forming control. Among voltage-type inverters, droop control and virtual synchronous generator control are predominant. While most research on voltage-type grid-connected inverters focuses on refined modeling, small-signal stability, and transient response stability, these studies often assume the existence of a steady-state operating point without theoretically analyzing the conditions under which such a point exists. As grid strength weakens, the impedance characteristics of the grid become more pronounced, and the existence of a steady-state operating point is no longer guaranteed. Therefore, deriving a criterion to determine the existence of a steady-state operating point is of practical significance. Note that the steady-state operating point discussed here refers to a possible operating state of the grid-connected system, irrespective of the stability issues caused by control loops. Even if a steady-state operating point exists, the control loop may prevent the system from converging to it; however, if the steady-state operating point does not exist, no modification of the control loop can achieve the desired operating state.
This article is structured as follows: In Section 2, based on the topology of a single-phase voltage-type grid-connected inverter, the steady-state power transmission model expression is derived, and the rated steady-state operating conditions are given. In Section 3, based on the power model and operating conditions, the existence criterion for the steady-state operating point is derived. According to typical operating conditions, the criterion is simplified, and the feasible region is plotted. In Section 4, simulation and experiment verify the effectiveness of the derived criterion. In Section 5, conclusions are drawn.
Steady-State Power Transmission Model and Operating Conditions
Topology of Voltage-Type Grid-Connected Inverter
Taking a single-phase voltage-type grid-connected inverter as an example, its topology when connected to the grid is shown in the figure below. It consists of a single-phase voltage-type inverter, the point of common coupling, and an equivalent grid.
In the figure, \( L_f \) and \( C_f \) are the AC filter inductor and capacitor of the single-phase voltage-type inverter, respectively; \( E \) and \( I \) are the voltage and current injected by the inverter into the grid through the point of common coupling; \( R \) and \( L \) are the resistance and inductance of the equivalent grid, respectively; \( U \) is the voltage source of the equivalent grid, generally considered constant.
Steady-State Power Transmission Expression
To study whether the voltage-type grid-connected inverter shown in the figure has a feasible steady-state operating point under given conditions, the steady-state components of the system need to be extracted.
When only the steady state is considered, the dynamic response characteristics of the single-phase voltage-type inverter can be ignored, and it is considered to stably inject voltage and current into the grid, so the inverter can be equivalently represented as a steady-state voltage source \( E \). Based on the same consideration, ignoring the transient characteristics of the equivalent grid impedance, the steady-state grid reactance is:
$$X = \omega_n L$$
where \( \omega_n \) is the steady-state angular frequency of the inverter system.
Based on AC phasor theory, the steady-state electrical phasor diagram is easily obtained. Let \( \delta \) be the phase difference between the grid voltage \( E \) and the grid voltage \( U \). The phasor expression for the grid current is:
$$\dot{I} = \frac{\dot{E} – \dot{U}}{R + jX}$$
The steady-state transmitted apparent power is:
$$S = P + jQ = \dot{E} \dot{I}^* = \frac{1}{Z^2} \left\{ R[E^2 – EU(\cos \delta + j \sin \delta)] + jX[E^2 – EU(\cos \delta – j \sin \delta)] \right\}$$
where \( Z = \sqrt{R^2 + X^2} \) is the magnitude of the equivalent grid impedance.
Expanding this, the active and reactive power are:
$$P = \frac{1}{Z^2} \left[ R(E^2 – EU \cos \delta) + X EU \sin \delta \right]$$
$$Q = \frac{1}{Z^2} \left[ X(E^2 – EU \cos \delta) – R EU \sin \delta \right]$$
Rated Operating Constraints
Denote the rated capacity of the inverter as \( S_n \), and the grid power factor angle as \( \phi \in [-\pi/5, \pi/5] \), i.e., only considering grid power factors \( \cos \phi \in [0.8, 1] \) or leading/lagging within this range. When the grid-connected inverter outputs power at rated capacity, the constraints are:
$$P = S_n \cos \phi, \quad Q = S_n \sin \phi$$
The short-circuit ratio (SCR) is used to express the strength of the equivalent grid, defined as:
$$K = \frac{S_{sc}}{S_n} = \frac{U^2}{Z S_n}$$
where \( S_{sc} \) is the short-circuit capacity of the equivalent grid. The study limits the short-circuit ratio range to \( K \in [1, 100] \).
Existence Analysis of Steady-State Operating Point
Existence Criterion
Integrating the power equations and constraints, and eliminating the phase difference \( \delta \) which is difficult to measure during system operation, while using the short-circuit ratio definition to cancel the rated capacity \( S_n \) and the magnitude of the equivalent grid impedance \( Z \), the following constraint equation is obtained:
$$K^2 E^4 – 2K^2 U^2 \cos(\alpha – \phi) E^2 + K^4 U^4 = 0$$
where \( \alpha \in [0, \pi/2] \) is the impedance angle of the grid, satisfying:
$$R = Z \cos \alpha, \quad X = Z \sin \alpha$$
Equation (6) can be viewed as a quadratic equation in \( E^2 \). The discriminant determines whether real solutions exist:
$$\Delta = 4K^4 U^4 \left[ \cos^2(\alpha – \phi) – 1 \right] + 4K^2 U^4 \left[ K^2 \sin^2(\alpha – \phi) \right]$$
Simplifying, since \( K^4 U^4 > 0 \), the discriminant reduces to:
$$\Delta’ = K^2 + 2K \cos(\alpha – \phi) – \sin^2(\alpha – \phi) \geq 0$$
When the voltage-type inverter injects power into the grid at rated capacity, the interaction system must satisfy discriminant (9) to have a feasible steady-state operating point. Therefore, equation (9) is called the existence criterion for the steady-state operating point. Observing equation (9), the existence of a steady-state operating point depends only on the grid short-circuit ratio \( K \), grid impedance angle \( \alpha \), and grid power factor angle \( \phi \). When selecting operating conditions for a voltage-type grid-connected inverter, specific parameter values can be substituted into the equation to determine the existence of the steady-state operating point.
Feasible Region of Steady-State Operating Points
Criterion (9) contains three independent variables, making analysis inconvenient. Therefore, based on typical operating conditions, we fix the value of one variable and study the existence of steady-state operating points as the other two variables change.
Case 1: \( K = 1 \) (Weak Grid Condition)
The discriminant becomes:
$$\Delta’ = 4\cos^2(\alpha – \phi) – 4\cos(\alpha – \phi) – 3 \geq 0$$
Combining the value ranges of \( \alpha \) and \( \phi \), the simplified criterion for the existence of a steady-state operating point is:
$$|\alpha – \phi| \leq \frac{\pi}{3}$$
The feasible region is summarized in Table 1.
| Variable Range | Existence Condition | Feasible Region Description |
|---|---|---|
| \( \alpha \in [0, \pi/2] \), \( \phi \in [-\pi/5, \pi/5] \) | \( |\alpha – \phi| \leq \pi/3 \) | Region where the difference between grid impedance angle and power factor angle is within \( \pi/3 \). As grid inductance increases, the infeasible region expands. |
Case 2: \( \alpha = \pi/2 \) (Purely Inductive Grid)
The discriminant becomes:
$$\Delta’ = K^2 + 2K \sin \phi – \cos^2 \phi \geq 0$$
The simplified criterion is:
$$K^2 + 2K \sin \phi – 1 \geq 0$$
For a given \( \phi \), the minimum \( K \) required is found by solving the quadratic. Table 2 summarizes the feasible region.
| Power Factor Angle \( \phi \) | Minimum \( K \) Required | Notes |
|---|---|---|
| \( \phi = 0 \) (unity power factor) | \( K \geq 1 \) | For purely inductive grid, at unity power factor, short-circuit ratio must be at least 1. |
| \( \phi = \pi/5 \) (lagging) | \( K \geq 0.618 \) (approx.) | Lagging power factor reduces required grid strength. |
| \( \phi = -\pi/5 \) (leading) | \( K \geq 1.618 \) (approx.) | Leading power factor demands higher grid strength. |
Case 3: \( \phi = 0 \) (Active Power Injection Only)
The discriminant becomes:
$$\Delta’ = K^2 + 2K \cos \alpha – \sin^2 \alpha \geq 0$$
The simplified criterion is:
$$K^2 + 2K \cos \alpha – 1 \geq 0$$
Table 3 summarizes the feasible region for this case.
| Grid Impedance Angle \( \alpha \) | Minimum \( K \) Required | Notes |
|---|---|---|
| \( \alpha = 0 \) (purely resistive) | \( K \geq \sqrt{2} – 1 \approx 0.414 \) | Purely resistive grid requires lowest short-circuit ratio. |
| \( \alpha = \pi/3 \) (mixed) | \( K \geq 1 \) | Moderate inductance increases required strength. |
| \( \alpha = \pi/2 \) (purely inductive) | \( K \geq 1 \) | Purely inductive grid requires \( K \geq 1 \) for active power injection. |
These tables illustrate that as the short-circuit ratio decreases (i.e., grid impedance increases), the infeasible region expands, reflecting the adverse effect of weak grid conditions on the existence of steady-state operating points.
Simulation and Experiment
To verify the proposed existence criterion, a single-phase voltage-type grid-connected inverter model is built in Matlab/Simulink for simulation, and a hardware-in-the-loop experimental platform is established using TMS320F280049 as the DSP controller and RT-Box. Key system parameters are listed in Table 4.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Rated Capacity \( S_n \) | 6.6 kVA | Filter Inductor \( L_f \) | 1 mH |
| Rated Frequency \( \omega_n \) | 314 rad/s | Filter Capacitor \( C_f \) | 20 µF |
| Grid Voltage \( U \) | 220 V | Switching Frequency \( f_s \) | 10 kHz |
Simulation Conditions and Results
Under the premise that the control method of the voltage-type grid-connected inverter is globally stabilizing within the feasible region, the critical steady-state operating points at the boundaries of the feasible regions (as identified in Tables 1-3) are selected as simulation conditions. During stable operation, at time \( t = 0 \) s, the operating point setpoint is slightly shifted toward the infeasible region, and whether the grid-connected inverter system can stabilize at the new operating state is observed.
The simulation results show that when the setpoint of the voltage-type grid-connected inverter is within the feasible region of the steady-state operating point, the grid-connected system maintains stable operation. If the setpoint shifts into the infeasible region, the system deviates from the original stable state and does not converge to a new operating point; instead, the operating condition exhibits periodic variations. This confirms the criterion: the existence of a steady-state operating point is determined solely by the parameters \( K \), \( \alpha \), and \( \phi \), independent of control loop adjustments.
Experimental Verification
Experimental conditions are set according to the simulation conditions. At a set time, the operating point setpoint of the grid-connected inverter system is shifted from the feasible region boundary into the infeasible region. The experimental results, showing grid voltage \( E \) and current \( I \), match the simulation results. Therefore, the conclusions drawn from theoretical analysis and simulation are validated.
Conclusion
In this study, by solving the power transmission equation of a voltage-type grid-connected inverter under rated operating constraints, a theoretical existence criterion for the steady-state operating point of such inverters in grid-connected conditions is obtained. Based on typical operating conditions, the feasible region of the steady-state operating point is plotted, and simplified criteria for corresponding conditions are derived. Simulation and experiment verify the effectiveness of these criteria and the feasible regions. The research provides a theoretical basis for selecting operating conditions of voltage-type grid-connected inverters, especially in weak grid scenarios where the existence of a steady-state operating point is not guaranteed. Future work may extend the analysis to three-phase systems, consider unbalanced or harmonic conditions, and integrate the existence criterion into adaptive control strategies for grid-connected inverters.
The key takeaway is that the existence of a steady-state operating point for a voltage-type grid-connected inverter depends critically on the short-circuit ratio, grid impedance angle, and power factor angle. Engineers must check these parameters against the derived criteria to ensure feasible operation before designing control loops. This insight is crucial for the reliable integration of renewable energy sources via grid-forming inverters into modern power systems.