As the penetration of photovoltaic power generation increases, the short-circuit current contributed by solar inverters can no longer be neglected. In this study, I analyze the topology, mathematical model, and grid-connected control of a single-stage solar inverter. I propose a low-voltage ride-through (LVRT) control strategy based on reactive current injection. The symmetrical short-circuit current characteristics of the solar inverter are qualitatively examined. By designing current-loop control parameters and neglecting the dynamics of the DC‑link voltage, I simplify the solar inverter into a controlled current source. A detailed electromagnetic transient model and an equivalent controlled‑source model of a single‑stage photovoltaic system are built in DIgSILENT/PowerFactory. Simulation results show that when the grid‑connected point voltage experiences a mild or severe sag due to a symmetrical three‑phase short‑circuit fault, the dynamic response of the voltage loop can be ignored. For deeper voltage sags that do not yet cause overcurrent, the transient characteristics of the inverter output current are mainly governed by the active current component. When the control strategy switches, the phase of the inverter output current shifts. The case study verifies the effectiveness of the simplified model, laying the foundation for further research on the fault characteristics of inverter‑interfaced sources.
1. Introduction
The short‑circuit current behavior of a solar inverter depends heavily on its control strategy. A detailed dynamic model of the entire photovoltaic system is time‑consuming and cumbersome. Hence, developing an equivalent model suitable for short‑circuit analysis is essential. Several studies have addressed the fault characteristics of inverter‑based sources. Some have used time‑domain simulations to show that a solar inverter can contribute to short‑circuit currents, albeit with limited magnitude, and that the three‑phase currents under various fault types differ little. Other researchers have improved short‑circuit calculation methods by assuming constant power output during faults — an assumption that overlooks actual control changes. In practice, as the installed capacity grows, a solar inverter must possess low‑voltage ride‑through capability. During a fault, the control strategy is adjusted, and the fault current characteristics change accordingly. In this work, I propose a reactive‑current‑based low‑voltage control strategy and then derive a simplified equivalent model for symmetrical faults.
2. Topology, Control, and Low‑Voltage Ride‑Through of a Single‑Stage Solar Inverter
2.1 Topology and Grid‑Connected Control
The typical topology of a single‑stage solar inverter is shown in Figure 1. The inverter is connected to the grid via an LCL filter. In the d‑q reference frame, the mathematical model is:
$$ \begin{cases}
u_{gd} = -L_f p i_{gd} – R_f i_{gd} + \omega L_f i_{gq} + v_{gd} \\[4pt]
u_{gq} = -L_f p i_{gq} – R_f i_{gq} – \omega L_f i_{gd} + v_{gq}
\end{cases} $$
where \(u_{gd}, u_{gq}\) are the inverter output voltage components, \(v_{gd}, v_{gq}\) are the grid voltage components, \(i_{gd}, i_{gq}\) are the current components, \(L_f, R_f\) are the filter inductance and resistance, \(\omega\) is the grid angular frequency, and \(p\) denotes the differential operator.
The active and reactive powers at the grid side are:
$$ \begin{cases}
P_g = v_{gd} i_{gd} + v_{gq} i_{gq} \\[4pt]
Q_g = v_{gq} i_{gd} – v_{gd} i_{gq}
\end{cases} $$
Using grid‑voltage‑oriented vector control, the grid voltage is aligned with the d‑axis: \(v_{gd} = V_g\), \(v_{gq}=0\). Then
$$ \begin{cases}
P_g = V_g i_{gd} \\[4pt]
Q_g = -V_g i_{gq}
\end{cases} $$
In normal operation, a double‑loop control (outer DC‑voltage loop, inner current loop) is used with unity power factor. The DC‑voltage reference \(V_{dc,ref}\) is obtained from maximum power point tracking (MPPT), and the reactive current reference \(i_{gq,ref}\) is set to zero.
2.2 Low‑Voltage Dynamic Reactive Control
Large‑scale photovoltaic stations are required to provide dynamic reactive support during voltage sags. According to the Chinese national standard for PV station grid connection, the inverter should inject reactive current proportional to the voltage drop. The reactive current reference is:
$$
i_{gq,ref} =
\begin{cases}
1.5 (0.9 – U_T) I_N, & 0.2 \le U_T \le 0.9 \\[4pt]
1.05 I_N, & U_T < 0.2 \\[4pt]
0, & U_T > 0.9
\end{cases}
$$
where \(U_T\) is the per‑unit voltage at the point of common coupling (PCC), and \(I_N\) is the rated current of the solar inverter. Considering the current limit, the active current reference is:
$$
i_{gd,ref} = \min\left( \sqrt{i_{\max}^2 – i_{gq,ref}^2}, \; K_{P,dc}(V_{dc,ref}-V_{dc}) + K_{I,dc}\int (V_{dc,ref}-V_{dc}) dt \right)
$$
where \(i_{\max}\) is the maximum allowed current. If the inverter does not exceed its limit, the normal MPPT outer loop remains active; otherwise, the outer loop is blocked and the active current is set by the limit.
3. Current‑Loop Parameter Design and Simplified Modeling
3.1 Current‑Loop Parameter Design
The current inner loop uses d‑q decoupled control. Taking the d‑axis as an example, the block diagram is shown in Figure 2 of the original paper. The open‑loop transfer function approximates a first‑order inertia and a proportional‑integral (PI) controller:
$$
G_o(s) = \left(K_p \frac{1+T_i s}{T_i s}\right) \left(\frac{K_{PWM}}{1+1.5T_{PWM} s}\right) \times U_N \times \left(\frac{1}{R_f + L_f s}\right) \times \frac{3U_N}{S_N}
$$
where \(K_p, T_i\) are PI parameters, \(K_{PWM}\) is the inverter gain, \(T_{PWM}\) is the switching period, and \(S_N\) is the rated power. By zero‑pole cancellation (\(T_i = L_f/R_f\)), the closed‑loop transfer function becomes:
$$
G(s) = \frac{\omega_n^2}{s^2 + 2\xi \omega_n s + \omega_n^2}
$$
with
$$
\omega_n^2 = \frac{2\sqrt{3}U_N^2 K_p K_{PWM}}{3 S_N L_f T_{PWM}}, \quad \xi = \frac{3 T_{PWM}}{2} \omega_n
$$
For optimal damping (\(\xi = \sqrt{2}/2\)), the proportional gain is:
$$
K_p = \frac{S_N L_f}{3\sqrt{3} U_N^2 K_{PWM} T_{PWM}}
$$
Using the system parameters, one can calculate initial values and fine‑tune them. Similarly, the q‑axis PI parameters are obtained.
3.2 Simplified Equivalent Model of the Solar Inverter
With the designed PI parameters, the closed‑loop transfer function of the current loop simplifies to:
$$
G(s) = \frac{1}{\frac{9}{2} T_{PWM}^2 s^2 + 3 T_{PWM} s + 1}
$$
For high switching frequencies, \(T_{PWM}\) is small, so the second‑order term can be neglected, yielding a first‑order inertia:
$$
G(s) \approx \frac{1}{3 T_{PWM} s + 1}
$$
Now, I focus on the symmetrical three‑phase short‑circuit fault. The inverter output current is entirely controlled. Under the proposed LVRT strategy, the reactive current reference depends solely on the PCC voltage. The active current reference has two modes:
- Mode A: If the current does not exceed the limit, the active current follows the MPPT outer loop.
- Mode B: If the limit is reached, the active current is determined by \(\sqrt{i_{\max}^2 – i_{gq,ref}^2}\).
To simplify, I assume that the environmental conditions (temperature, irradiance) change slowly, so the DC‑side power input is constant. I also assume that the DC‑link capacitor is large enough to keep the DC voltage near the MPPT reference, thus ignoring its dynamics. Consequently, the active current reference becomes:
$$
i_{gd,ref}^* = \min\left( \sqrt{i_{\max}^2 – i_{gq,ref}^2}, \; \frac{P}{U_T} \right)
$$
where \(P\) is the prefault active power. The reactive current reference is given by the voltage‑dependent equation. Thus, the solar inverter behaves as a controlled current source that depends on the PCC voltage, with the inner‑loop dynamics approximated by the first‑order lag \(1/(3T_{PWM}s+1)\). When the control strategy switches from normal unity‑power‑factor mode to LVRT mode, the output current phase shifts by:
$$
\Delta \phi_i = \arctan\left(\frac{i_{gq}}{i_{gd}}\right)
$$
This simplified modeling neglects the DC‑voltage dynamic. For mild sags, the DC voltage remains almost constant; for severe sags, the outer loop is quickly blocked. In both cases, the simplification is valid. For deeper sags that do not trigger the current limit, the DC‑voltage dynamic becomes noticeable, and the simplified model can only predict the steady‑state current.
4. Simulation Validation
I built a detailed electromagnetic transient model of a single‑stage photovoltaic system and an equivalent controlled‑current‑source model in DIgSILENT/PowerFactory. The system parameters are:
| Parameter | Value |
|---|---|
| Rated power \(S_N\) | 1 MVA |
| PCC voltage \(U_T\) | 0.38 kV |
| Filter resistance \(R_f\) | 1 mΩ |
| Filter inductance \(L_f\) | 0.1 mH |
| DC capacitor \(C\) | 5000 μF |
| Current limit \(i_{\max}\) | 1.5 \(I_N\) |
| Switching frequency \(f_{PWM}\) | 1500 Hz |
| Current‑loop PI: \(K_p\) | 0.4 |
| Current‑loop PI: \(T_i\) | 0.1 s |
| Voltage‑loop PI: \(K_{P,dc}\) | 0.002 |
| Voltage‑loop PI: \(K_{I,dc}\) | 0.05 |
A three‑phase symmetrical fault is applied at 2 s on the 0.38 kV bus. Three cases are studied:
Case 1: Mild Voltage Sag (UT ≈ 0.88 p.u.)
Fault impedance: resistance 15 mΩ, reactance 25 mΩ. Figure 4 (not shown) compares the detailed model and the equivalent model. The two currents are almost identical; the maximum relative error of the current magnitude is 5.8%. The DC voltage stays nearly constant, confirming that the voltage loop dynamics can be neglected.
Case 2: Deeper Sag (UT ≈ 0.68 p.u.)
Fault impedance: 4.2 mΩ + j7 mΩ. The voltage drops to 0.68 p.u. The equivalent model initially overestimates the current magnitude (maximum error 24.1%) because the DC voltage rises due to active power imbalance, causing the MPPT point to drift. However, after about 0.15 s, the DC voltage stabilizes and the currents converge. In this case, the DC‑voltage dynamic cannot be ignored during the transient, but the simplified model correctly predicts the steady‑state current (1.49 p.u., still below the limit).
Case 3: Severe Sag (UT ≈ 0.28 p.u.)
Fault impedance: 0.6 mΩ + j1 mΩ. The equivalent model matches the detailed model extremely well, with a maximum relative error of 2.4%. The DC voltage rises quickly, and the outer voltage loop is blocked almost instantly. The inverter current reaches its limit (1.5 p.u.) in a very short time. The phase shift of the output current is about 42.4°, consistent with the theoretical prediction.
In all cases, before the fault, the solar inverter operates at unity power factor (current in phase with voltage). After the fault, reactive current is injected, causing a phase lag: 1.0°, 12.5°, and 42.4° for the three cases, respectively.
5. Conclusion
The short‑circuit current characteristics of a solar inverter are determined by its control strategy, which changes under fault conditions. By combining the LVRT reactive‑current injection and proper current‑loop design, I have developed a simplified controlled‑current‑source model for the solar inverter. Simulation results show that:
- For mild sags (voltage above ~0.9 p.u.) and severe sags (voltage below ~0.3 p.u.), the DC‑voltage dynamics are negligible, and the equivalent model achieves high accuracy.
- For moderate sags where the inverter current does not reach the limit, the active‑current transient, governed by the voltage outer loop, plays a significant role; the simplified model is accurate only in the steady state.
- The phase of the inverter output current shifts when the control transitions from unity power factor to reactive‑current injection.
The proposed equivalent model is effective for symmetrical fault analysis and offers a practical tool for studying the fault behavior of grid‑connected solar inverters.