With the rapid growth of grid-connected photovoltaic capacity, the impact of solar inverters on system safety and stable operation has become increasingly significant. In practical power grid operation, voltage sags are frequent events, often caused by short-circuit faults, large motor starting, and similar disturbances. Importantly, these voltage sags are predominantly unbalanced, meaning that the grid voltage contains a negative-sequence component. Under such conditions, solar inverters face serious challenges: the negative-sequence voltage leads to output power oscillations, current distortion, and a notable increase in the peak output current. This paper presents a comprehensive study on the peak output current behavior of solar inverters under unbalanced voltage sags and proposes a practical control strategy to limit the maximum current peak to a safe value.
Although solar inverters are designed to handle normal grid disturbances, unbalanced voltage sags can push their peak currents to several times the rated value. This excessive current threatens the power electronic switches and the entire photovoltaic system, potentially triggering overcurrent protection and disconnecting the solar inverter from the grid. Therefore, effective peak current control is essential for enhancing the low-voltage ride-through (LVRT) capability of solar inverters. This research focuses on the most widely adopted control methods for solar inverters under unbalanced voltage sags, analyzes the three-phase output currents in detail, derives exact expressions for the current peaks, and introduces a method to keep the maximum output current within the allowed limit.
Low-Voltage Ride-Through Requirements
In 2012, the Chinese national standard GB/T 19964—2012, titled “Technical Requirements for Connecting Photovoltaic Power Stations to Power Systems,” specified explicit LVRT requirements for solar inverters connected through 35 kV and above voltage levels. The key requirements are:
- When the voltage at the point of common coupling (PCC) drops to zero, the solar inverter must remain connected without tripping for at least 0.15 s.
- If the PCC voltage falls below the curve defined in the standard, the solar inverter may be disconnected from the grid.
Additionally, for large solar farms connected via 220 kV (or 330 kV) collection systems and then stepped up to 500 kV (or 750 kV), the solar inverters should provide dynamic reactive power support during grid faults.
Topology and Mathematical Model of Solar Inverters
A typical three-phase solar inverter topology is shown in the following figure. The solar inverter consists of a DC-link capacitor, three-phase bridge arms, and output filter inductors and resistors. The grid voltage is denoted as \(e_i\) (\(i=a,b,c\)), the inverter output voltage as \(u_i\), and the output current as \(i_i\). The DC-side voltage is \(U_{dc}\), and the output resistance and inductance are \(R\) and \(L\).
The mathematical model of the solar inverter in the \(abc\) frame is:
$$ u_i = Ri_i + L\frac{di_i}{dt} + e_i $$
Applying the Clarke transformation, the model in the \(\alpha\beta\) frame becomes:
$$ u_\alpha = Ri_\alpha + L\frac{di_\alpha}{dt} + e_\alpha $$
$$ u_\beta = Ri_\beta + L\frac{di_\beta}{dt} + e_\beta $$
Further transforming into the \(dq\) synchronous reference frame with positive- and negative-sequence components, the model is expressed as:
$$ u_d^+ = (R+sL)i_d^+ – \omega L i_q^+ + e_d^+ $$
$$ u_q^+ = (R+sL)i_q^+ + \omega L i_d^+ + e_q^+ $$
$$ u_d^- = (R+sL)i_d^- + \omega L i_q^- + e_d^- $$
$$ u_q^- = (R+sL)i_q^- – \omega L i_d^- + e_q^- $$
where superscripts \(+\) and \(-\) denote positive- and negative-sequence components respectively.
Control Strategies for Solar Inverters Under Unbalanced Voltage Sags
Research on solar inverter control under unbalanced grid conditions dates back to the 1990s. When the grid voltage becomes unbalanced, the negative-sequence component causes DC-link voltage oscillation, output power fluctuation, and current waveform distortion if the conventional balanced control algorithm is used. These issues seriously affect the stability of solar inverters. The instantaneous complex power at the grid side can be written as:
$$ S = P + jQ = 1.5 \left( u_{dq}^+ e^{j\omega t} + u_{dq}^- e^{-j\omega t} \right) \left( i_{dq}^+ e^{j\omega t} + i_{dq}^- e^{-j\omega t} \right)^* $$
Expanding this expression yields the instantaneous active and reactive power:
$$ P = P_0 + P_{c2}\cos(2\omega t) + P_{s2}\sin(2\omega t) $$
$$ Q = Q_0 + Q_{c2}\cos(2\omega t) + Q_{s2}\sin(2\omega t) $$
The coefficients are:
$$ P_0 = 1.5(u_d^+ i_d^+ + u_q^+ i_q^+ + u_d^- i_d^- + u_q^- i_q^-) $$
$$ P_{c2} = 1.5(u_d^+ i_d^- + u_q^+ i_q^- + u_d^- i_d^+ + u_q^- i_q^+) $$
$$ P_{s2} = 1.5(u_q^+ i_d^- – u_d^+ i_q^- – u_q^- i_d^+ + u_d^- i_q^+) $$
$$ Q_0 = 1.5(u_q^+ i_d^+ – u_d^+ i_q^+ + u_q^- i_d^- – u_d^- i_q^-) $$
$$ Q_{c2} = 1.5(u_q^+ i_d^- – u_d^+ i_q^- – u_q^- i_d^+ + u_d^- i_q^+) $$
$$ Q_{s2} = 1.5(u_d^+ i_d^- + u_q^+ i_q^- – u_d^- i_d^+ – u_q^- i_q^+) $$
By controlling the positive- and negative-sequence components of the grid-side current in the \(dq\) frame, different control objectives can be achieved. The three most common objectives for solar inverters are:
- Balanced three-phase output current (eliminate negative-sequence current).
- Constant reactive power (eliminate reactive power oscillation).
- Constant active power (eliminate active power oscillation).
Given the importance of stable active power for the DC-link voltage, MPPT performance, and overall system efficiency, this paper selects the constant active power objective (Objective III) as the basis for analyzing peak current behavior and developing the control strategy.
Reference Current for Constant Active Power Control
Setting \(P_{c2}=P_{s2}=0\) and neglecting the reactive power oscillations, the current references are obtained from the power expressions as:
$$ i_d^{+*} = \frac{2P_0}{3D_1} u_d^+ + \frac{2Q_0}{3D_2} u_q^+ $$
$$ i_q^{+*} = \frac{2P_0}{3D_1} u_q^+ – \frac{2Q_0}{3D_2} u_d^+ $$
$$ i_d^{-*} = \frac{2P_0}{3D_1} u_d^- – \frac{2Q_0}{3D_2} u_q^- $$
$$ i_q^{-*} = \frac{2P_0}{3D_1} u_q^- + \frac{2Q_0}{3D_2} u_d^- $$
where
$$ D_1 = (u_d^+)^2 + (u_q^+)^2 – (u_d^-)^2 – (u_q^-)^2 $$
$$ D_2 = (u_d^+)^2 + (u_q^+)^2 + (u_d^-)^2 + (u_q^-)^2 $$
These reference currents are transformed back to the \(abc\) frame to obtain the instantaneous three-phase current expressions. The positive- and negative-sequence current components in the \(abc\) frame are:
$$ i_a^+ = (K_1 u_d^+ + K_2 u_q^+)\cos\theta^+ + (K_1 u_q^+ – K_2 u_d^+)\sin\theta^+ $$
$$ i_b^+ = (K_1 u_d^+ + K_2 u_q^+)\cos(\theta^+ – \frac{2\pi}{3}) + (K_1 u_q^+ – K_2 u_d^+)\sin(\theta^+ – \frac{2\pi}{3}) $$
$$ i_c^+ = (K_1 u_d^+ + K_2 u_q^+)\cos(\theta^+ + \frac{2\pi}{3}) + (K_1 u_q^+ – K_2 u_d^+)\sin(\theta^+ + \frac{2\pi}{3}) $$
and
$$ i_a^- = (K_1 u_d^- – K_2 u_q^-)\cos\theta^- + (K_1 u_q^- + K_2 u_d^-)\sin\theta^- $$
with similar expressions for \(i_b^-\) and \(i_c^-\). Here \(K_1 = 2P_0/(3D_1)\), \(K_2 = 2Q_0/(3D_2)\), \(\theta^+ = \omega t + \phi^+\), and \(\theta^- = -\omega t + \phi^-\).
Peak Output Current Calculation
To derive the peak current, we align the \(dq\) axes with the grid voltage vectors. Under this alignment:
$$ u_d^+ = U^+, \quad u_q^+ = 0, \quad u_d^- = U^-, \quad u_q^- = 0 $$
where \(U^+\) and \(U^-\) are the amplitudes of the positive- and negative-sequence voltage components. The unbalanced factor is defined as:
$$ \varepsilon = \frac{U^-}{U^+} $$
Substituting these into the current expressions and simplifying, the three-phase currents can be expressed as:
$$ i_a = m \sqrt{1 + \varepsilon^2 + 2\varepsilon \cos(\Delta\phi)} \cos(\theta^+ + \phi_a) $$
$$ i_b = m \sqrt{1 + \varepsilon^2 + 2\varepsilon \cos(\Delta\phi – \frac{2\pi}{3})} \cos(\theta^+ + \phi_b) $$
$$ i_c = m \sqrt{1 + \varepsilon^2 + 2\varepsilon \cos(\Delta\phi + \frac{2\pi}{3})} \cos(\theta^+ + \phi_c) $$
where
$$ m = \frac{2}{3(1-\varepsilon^2)} \sqrt{ P_0^2 + Q_0^2 } / U^+ $$
and \(\Delta\phi = \phi^+ + \phi^-\) is the phase-angle difference between the positive- and negative-sequence voltage vectors. The detailed derivation of the phase angles \(\phi_a\), \(\phi_b\), and \(\phi_c\) is omitted for brevity but is available in the full analysis.
The peak value of each phase current is therefore:
$$ i_{a,\max} = m \sqrt{1 + \varepsilon^2 + 2\varepsilon \cos(\Delta\phi)} $$
$$ i_{b,\max} = m \sqrt{1 + \varepsilon^2 + 2\varepsilon \cos(\Delta\phi – \frac{2\pi}{3})} $$
$$ i_{c,\max} = m \sqrt{1 + \varepsilon^2 + 2\varepsilon \cos(\Delta\phi + \frac{2\pi}{3})} $$
The maximum possible peak current among all phases occurs when the cosine term reaches its maximum value. This happens when \(\Delta\phi\) equals \(0\), \(2\pi/3\), or \(-2\pi/3\) for phases A, B, and C respectively. The resulting maximum peak current is:
$$ i_{\max} = \frac{2}{3} \frac{\sqrt{P_0^2 + Q_0^2}}{U^+} \sqrt{\frac{1+\varepsilon}{1-\varepsilon}} $$
This expression reveals that the maximum peak current depends on the active and reactive power references, the positive-sequence voltage amplitude, and the unbalanced factor. It is clear that the peak current increases significantly with the unbalanced factor and with the injection of reactive power.
To provide a clearer comparison, the following table summarizes the three control objectives and their impact on the output current peaks:
| Control Objective | Negative-sequence current | Active power oscillation | Reactive power oscillation | Peak current behavior |
|---|---|---|---|---|
| Balanced current (Objective I) | Eliminated | Large | Large | Peak current is lower than other objectives but still influenced by positive-sequence voltage drop |
| Constant reactive power (Objective II) | Present | Large | Eliminated | Peak current depends on both active and reactive power; reactive injection increases peak |
| Constant active power (Objective III) | Present | Eliminated | Large | Peak current calculated by our derived formula; reactive power increases peak |
Peak Current Control Strategy
From the derived expression for \(i_{\max}\), it is evident that the peak current of solar inverters under unbalanced voltage sags is influenced by the active power reference \(P_0\), the reactive power reference \(Q_0\), the positive-sequence voltage amplitude \(U^+\), the unbalanced factor \(\varepsilon\), and the phase-angle difference \(\Delta\phi\). Among these parameters, the grid voltage characteristics (\(U^+\), \(\varepsilon\), \(\Delta\phi\)) are determined by the external grid fault and cannot be adjusted by the solar inverter controller. Therefore, the only practical way to limit the peak current is to regulate the active and reactive power references.
Let \(I_{\max}\) be the maximum allowable current peak of the solar inverter. To ensure safe operation, the following inequality must hold:
$$ \frac{2}{3} \frac{\sqrt{P_0^2 + Q_0^2}}{U^+} \sqrt{\frac{1+\varepsilon}{1-\varepsilon}} \le I_{\max} $$
Squaring both sides and rearranging gives the constraint on the power references:
$$ P_0^2 + Q_0^2 \le \frac{9}{4} (I_{\max} U^+)^2 \frac{1-\varepsilon}{1+\varepsilon} $$
In practice, during grid faults, solar inverters are often required to provide reactive power support. According to current grid codes, the exact amount of reactive power is not always specified for distributed PV systems. In this study, we set the reactive power reference to 50% of the active power reference:
$$ Q_0 = 0.5 P_0 $$
Substituting this into the constraint yields:
$$ P_0^2 + (0.5P_0)^2 \le \frac{9}{4} (I_{\max} U^+)^2 \frac{1-\varepsilon}{1+\varepsilon} $$
$$ P_0 \le \frac{3 I_{\max} U^+}{\sqrt{5}} \sqrt{\frac{1-\varepsilon}{1+\varepsilon}} $$
Thus, if the solar inverter operates at its maximum allowable current, the optimal power references are:
$$ P_0^* = \frac{3 I_{\max} U^+}{\sqrt{5}} \sqrt{\frac{1-\varepsilon}{1+\varepsilon}} $$
$$ Q_0^* = 0.5 P_0^* $$
This control strategy ensures that the output current peak of solar inverters never exceeds the specified limit \(I_{\max}\), even during severe unbalanced voltage sags. The controller monitors the positive- and negative-sequence voltage components in real time, calculates the unbalanced factor \(\varepsilon\), and adjusts the power references accordingly. When the grid voltage is balanced (\(\varepsilon = 0\)), the references revert to the nominal values, allowing the solar inverter to deliver maximum power with unity power factor.
It is worth noting that the proposed method is derived based on the constant active power control objective. For large solar farms that must provide dynamic reactive power support, a similar derivation can be carried out using the constant reactive power control objective. The resulting peak current expression and limiting strategy would follow the same analytical steps but with the roles of active and reactive power interchanged.
Simulation Verification
To validate the effectiveness of the proposed peak current control strategy, a detailed simulation model of a grid-connected photovoltaic system was built using PSCAD/EMTDC. The solar inverter employs dual synchronous reference frame PI control to independently regulate the positive- and negative-sequence currents. The system parameters are listed in the following table:
| Parameter | Value |
|---|---|
| Rated power | 0.5 MW |
| DC-side voltage | 750 V |
| DC-link capacitance | 5000 μF |
| Filter inductance | 1.5 mH |
| Switching frequency | 4 kHz |
| Maximum allowable current peak \(I_{\max}\) | 1.5 kA |
During normal grid operation (balanced voltage), the solar inverter operates at unity power factor and delivers rated active power \(P = 0.5\) MW, resulting in a symmetrical three-phase current with a peak of 1 kA. In the simulation, an unbalanced voltage sag occurs at \(t = 3\) s. The behavior is examined for two different unbalanced factors: \(\varepsilon = 30\%\) and \(\varepsilon = 40\%\).
Case 1: Unbalanced Factor \(\varepsilon = 30\%\)
If the solar inverter continues to operate at the rated active power (without peak current control), the output waveforms show that the peak current reaches 1.62 kA, which exceeds the allowed limit of 1.5 kA. When the proposed control strategy is applied, the power references are recalculated using the derived formulas:
$$ P_0^* = 0.42 \text{ MW}, \quad Q_0^* = 0.21 \text{ Mvar} $$
With these references, the maximum output current peak is suppressed to 1.47 kA, complying with the safety requirement. The measured active and reactive powers are approximately 0.41 MW and 0.19 Mvar, closely matching the calculated references. The small deviation is due to switching transients and control dynamics.
Case 2: Unbalanced Factor \(\varepsilon = 40\%\)
Under a more severe unbalance, if the solar inverter still outputs rated active power, the peak current rises to 2.05 kA, posing a serious threat to the equipment. Applying the proposed strategy, the power references become:
$$ P_0^* = 0.34 \text{ MW}, \quad Q_0^* = 0.17 \text{ Mvar} $$
The resulting maximum output current peak is 1.49 kA, safely below the 1.5 kA limit. The measured powers are 0.33 MW and 0.15 Mvar, again in good agreement with the analytical values.
The simulation results confirm that solar inverters equipped with the proposed peak current control can effectively maintain their output current peaks within the allowable range during unbalanced voltage sags. The control strategy adapts the power references in real time based on the voltage unbalance severity, ensuring safe operation of the photovoltaic system.
Discussion
The proposed strategy offers several advantages for solar inverters:
- It provides a precise analytical expression for the maximum peak current, which can be used online to compute safe power reference limits.
- It is computationally simple and can be readily implemented in existing solar inverter controllers.
- It is applicable to both distributed and utility-scale photovoltaic systems, with appropriate modifications to the reactive power requirement.
One limitation is that the constant active power control objective inevitably introduces reactive power oscillations, which may be undesirable in certain grid conditions. However, during voltage sags, the main priority is to protect the solar inverter while maintaining grid connection. Future work could extend the method to a multi-objective optimization framework that balances peak current limitation, power quality, and reactive power support.
Conclusion
This paper presented a comprehensive analysis of the peak output current of solar inverters under unbalanced voltage sags. A detailed derivation led to a closed-form expression for the maximum phase current peak as a function of active/reactive power references, positive-sequence voltage amplitude, and the voltage unbalanced factor. Based on this expression, a practical control strategy was proposed to limit the peak current to a safe value by adjusting the power references. The strategy was validated through simulation under different unbalanced factors (30% and 40%), showing that the output current peak of solar inverters can be kept below the maximum allowable limit while still providing reactive power support. The proposed method significantly improves the safety and robustness of solar inverters during grid faults, contributing to the stable operation of photovoltaic systems.