In the context of increasing global renewable energy integration, photovoltaic (PV) systems have become a cornerstone of modern power grids. As a researcher deeply involved in power electronics and grid stability, I have focused on understanding the behavior of solar inverters during grid faults, particularly symmetrical short-circuit events. The penetration of solar power is rising rapidly, and the short-circuit current characteristics of solar inverters can no longer be ignored. Traditional power systems rely on synchronous generators with well-defined fault current contributions, but solar inverters, being power electronic interfaces, exhibit distinct dynamic responses that challenge conventional protection schemes. My work aims to elucidate these characteristics and develop simplified equivalent models to facilitate power system analysis and design. This article presents a comprehensive study on the symmetrical short-circuit current behavior of single-stage solar inverters, incorporating low-voltage ride-through (LVRT) capabilities, and proposes an equivalent controlled current source model validated through detailed simulations.
The topology of a single-stage solar inverter is fundamental to its operation. Typically, it consists of a PV array, a DC-link capacitor, and a voltage source inverter (VSI) connected to the grid through an LCL filter. The inverter converts DC power from the PV panels to AC power synchronized with the grid. In my research, I consider a standard configuration where the inverter uses pulse-width modulation (PWM) for control. The mathematical model in the d-q reference frame, aligned with the grid voltage, is essential for decoupling active and reactive power control. The equations governing the inverter are:
$$ \begin{cases} u_{gd} = -L_f p i_{gd} – R_f i_{gd} + \omega L_f i_{gq} + v_{gd} \\ u_{gq} = -L_f p i_{gq} – R_f i_{gq} – \omega L_f i_{gd} + v_{gq} \end{cases} $$
Here, \( u_{gd} \) and \( u_{gq} \) are the d- and q-axis components of the inverter output voltage, \( v_{gd} \) and \( v_{gq} \) are the grid voltage components, \( i_{gd} \) and \( i_{gq} \) are the grid current components, \( L_f \) and \( R_f \) are the filter inductance and resistance, \( \omega \) is the grid angular frequency, and \( p \) is the differential operator. With grid voltage orientation (\( v_{gd} = V_g \), \( v_{gq} = 0 \)), the active and reactive power outputs simplify to \( P_g = V_g i_{gd} \) and \( Q_g = -V_g i_{gq} \), enabling independent control via current references. Under normal conditions, the solar inverter operates with unit power factor, maximizing power extraction through maximum power point tracking (MPPT) while maintaining DC-link voltage stability via outer loop control.
To enhance grid support during faults, solar inverters must incorporate LVRT strategies. In my study, I adopt a dynamic reactive current injection method based on grid codes, such as those outlined in national standards. When a symmetrical fault causes voltage sag at the point of interconnection, the solar inverter adjusts its reactive current reference \( i_{gq}^{ref} \) according to the sag depth. The relationship is defined as:
$$ i_{gq}^{ref} = \begin{cases} 1.5(0.9 – U_T) I_N & \text{for } 0.2 \leq U_T \leq 0.9 \\ 1.05 I_N & \text{for } U_T < 0.2 \\ 0 & \text{for } U_T > 0.9 \end{cases} $$
where \( U_T \) is the per-unit voltage at the interconnection point, and \( I_N \) is the rated current. Simultaneously, the active current reference \( i_{gd}^{ref} \) is limited to prevent overcurrent, considering the inverter’s maximum current \( i_{max} \). If the current exceeds limits, the voltage outer loop is blocked, and references are set directly; otherwise, MPPT-based control continues. This dual approach ensures that the solar inverter remains connected during faults while providing reactive support to stabilize the grid.
The design of current control loops is critical for dynamic performance. For the solar inverter, I design the PI controllers in the d- and q-axis current loops using zero-pole cancellation techniques. The open-loop transfer function for the d-axis current control, including PWM delay, is:
$$ G_o(s) = \left( K_p \frac{1 + T_i s}{T_i s} \right) \left( \frac{K_{PWM}}{1 + 1.5 T_{PWM} s} \right) \times U_N \times \left( \frac{1}{R_f + L_f s} \right) \times \frac{3U_N}{S_N} $$
By setting \( T_i = L_f / R_f \) to cancel the filter pole, the system simplifies to a standard second-order form. For optimal damping (\( \xi = \sqrt{2}/2 \)), the proportional gain \( K_p \) is derived as:
$$ K_p = \frac{S_N L_f}{3\sqrt{3} U_N^2 K_{PWM} T_{PWM}} $$
This parameterization ensures fast and stable current tracking, which is vital for fault response. The closed-loop transfer function approximates a first-order lag when PWM switching frequency is high:
$$ G(s) \approx \frac{1}{3 T_{PWM} s + 1} $$
This simplification underpins the equivalent model development, as it captures the current loop dynamics without excessive complexity.
To analyze short-circuit currents efficiently, I propose a simplified equivalent model for the solar inverter. During symmetrical faults, the inverter’s output is dominated by current control, and if voltage sags are severe or mild, the DC-link voltage dynamics can be neglected. Thus, the solar inverter is modeled as a controlled current source whose outputs depend on the grid voltage. The active current reference accounts for pre-fault power \( P \) and current limits:
$$ i_{gd}^{ref} = \min\left( \sqrt{i_{max}^2 – (i_{gq}^{ref})^2}, \frac{P}{U_T} \right) $$
while the reactive current reference follows the LVRT curve. The current source dynamics are represented by the first-order lag from the current loop. This model significantly reduces computational burden in system studies while maintaining accuracy for specific sag conditions. It is particularly valid for mild sags where DC voltage remains steady, or deep sags where current limiting activates swiftly, bypassing voltage loop transients.
My simulation analysis validates this equivalent model using detailed electromagnetic transient models in a professional software environment. I consider a 1 MVA solar inverter system connected to a grid with parameters summarized in Table 1. Three symmetrical fault scenarios are tested: mild sag (voltage to 0.88 p.u.), deeper sag (0.68 p.u.), and severe sag (0.28 p.u.). The solar inverter operates at full power pre-fault, and faults are applied at 2 seconds. Results compare the detailed model and the equivalent controlled current source model.
| Parameter | Symbol | Value |
|---|---|---|
| Rated Power | \( S_N \) | 1 MVA |
| Grid Voltage (L-L) | \( U_T \) | 0.38 kV |
| Filter Inductance | \( L_f \) | 0.1 mH |
| Filter Resistance | \( R_f \) | 1 mΩ |
| DC-Link Capacitance | \( C_{dc} \) | 5000 μF |
| PWM Switching Frequency | \( f_{PWM} \) | 1500 Hz |
| Current Limit | \( i_{max} \) | 1.5 p.u. |
| Current Loop PI Gain | \( K_p \) | 0.4 |
| Current Loop Integral Time | \( T_i \) | 0.1 s |
For the mild sag case, the equivalent model closely matches the detailed model, with current amplitude errors below 6%. This confirms that DC voltage variations are negligible, and the solar inverter behaves as a voltage-controlled current source. The reactive current injection shifts the current phase angle, as calculated by \( \Delta \phi_i = \arctan(i_{gq} / i_{gd}) \). In the deeper sag scenario, the equivalent model shows higher initial currents due to ignored DC voltage transients, but steady-state values converge after the voltage loop settles. The error peaks at 24%, highlighting that for moderate sags without overcurrent, voltage loop dynamics affect transient response. However, the model still predicts steady-state behavior accurately. For severe sags, the equivalent model excels, with errors under 3%, as current limiting dominates and voltage loop effects are minimal. Table 2 summarizes these findings, emphasizing the model’s applicability.
| Voltage Sag Depth (p.u.) | Steady-State Current Amplitude (p.u.) | Maximum Relative Error in Current Amplitude | Phase Shift (degrees) | Model Validity |
|---|---|---|---|---|
| 0.88 (Mild) | 1.13 | 5.8% | 1.0 | High |
| 0.68 (Deeper) | 1.49 | 24.1% | 12.5 | Moderate (steady-state only) |
| 0.28 (Severe) | 1.50 (limited) | 2.4% | 42.4 | High |
These simulations underscore key insights into solar inverter behavior. First, during mild or severe voltage sags, the dynamic response of the voltage loop is negligible, allowing simplification to a controlled current source. Second, for deeper sags without overcurrent, transient characteristics are influenced by active current dynamics, necessitating caution in model application. Third, the transition from normal to LVRT control always induces a phase shift in output currents, which must be considered in protection coordination. The equivalent model proves effective for fault analysis, laying groundwork for integrating solar inverters into power system studies.
In conclusion, my research on symmetrical short-circuit current characteristics of solar inverters demonstrates that their fault response is highly controllable and dependent on LVRT strategies. By designing current loops appropriately and simplifying dynamics, the solar inverter can be modeled as a voltage-controlled current source, valid for a range of sag conditions. This equivalent model enhances computational efficiency without sacrificing critical accuracy, supporting grid planning and stability assessments. Future work could extend to asymmetric faults and interactions with other inverter-based resources, further refining models for the evolving power landscape. As solar penetration grows, understanding and modeling solar inverter fault behavior remains paramount for reliable and resilient energy systems.