The widespread integration of photovoltaic (PV) generation presents significant challenges to power system security and stability. A critical requirement, often mandated by grid codes such as the Chinese standard GB/T 19964-2012, is the Low Voltage Ride-Through (LVRT) capability. Solar inverters must remain connected to the grid and support it during temporary voltage dips caused by faults or large motor starts. In practice, these voltage sags are frequently unbalanced, characterized by the presence of negative-sequence voltage components. This condition severely impacts the operation of grid-connected solar inverters, leading to output power oscillations, distortion in three-phase currents, and, critically, a substantial increase in the peak output current. This elevated current can reach several times its nominal value, threatening the safety of power semiconductor switches and potentially triggering overcurrent protection, leading to an undesired inverter trip. Therefore, developing effective control strategies to manage and limit the peak output current of solar inverters during unbalanced grid faults is essential for reliable system operation and compliance with grid standards.
While research exists on general control of solar inverters under unbalanced conditions, focused strategies for actively limiting the current peak remain less explored. This paper delves into the control challenges for solar inverters during unbalanced voltage sags. We analyze prevalent control objectives, provide a rigorous derivation for calculating the three-phase peak currents and the absolute maximum peak, and propose a direct control method to constrain the output current within the inverter’s safe operating limits.
Mathematical Model of a Grid-Connected Solar Inverter
The typical topology of a three-phase, two-level voltage source inverter (VSI) interfacing a PV array to the grid is considered. The fundamental dynamic equations in the three-phase (abc) stationary frame are given by:
$$ u_{i} = R i_{i} + L \frac{d i_{i}}{dt} + e_{i}, \quad i \in \{a, b, c\} $$
where \( u_i \) and \( i_i \) are the inverter output voltage and current, \( e_i \) is the grid voltage, and \( R \) and \( L \) are the resistance and inductance of the output filter, respectively.
Applying the Clarke transformation, the model in the \( \alpha\beta \) stationary frame is:
$$
\begin{aligned}
u_{\alpha} &= R i_{\alpha} + L \frac{d i_{\alpha}}{dt} + e_{\alpha} \\
u_{\beta} &= R i_{\beta} + L \frac{d i_{\beta}}{dt} + e_{\beta}
\end{aligned}
$$
For control purposes, the system is often analyzed in a synchronous rotating (dq) frame. Under unbalanced grid conditions, it is necessary to decompose variables into positive-sequence (superscript ‘+’) and negative-sequence (superscript ‘-‘) components. Using dual synchronous reference frames rotating at \( +\omega t \) and \( -\omega t \) respectively, the complete model is:
$$
\begin{aligned}
u^{+}_{d} &= (R + sL) i^{+}_{d} – \omega L i^{+}_{q} + e^{+}_{d} \\
u^{+}_{q} &= (R + sL) i^{+}_{q} + \omega L i^{+}_{d} + e^{+}_{q} \\
u^{-}_{d} &= (R + sL) i^{-}_{d} + \omega L i^{-}_{q} + e^{-}_{d} \\
u^{-}_{q} &= (R + sL) i^{-}_{q} – \omega L i^{-}_{d} + e^{-}_{q}
\end{aligned}
$$
Here, \( s \) is the differential operator, and \( \omega \) is the grid angular frequency.
Control Objectives for Solar Inverters Under Unbalanced Voltage
Under unbalanced grid voltages, the instantaneous active (\(P\)) and reactive (\(Q\)) power output of a solar inverter contains a constant term and oscillating components at twice the grid frequency (2ω). The power can be expressed as:
$$
\begin{aligned}
P &= P_0 + P_{c2}\cos(2\omega t) + P_{s2}\sin(2\omega t) \\
Q &= Q_0 + Q_{c2}\cos(2\omega t) + Q_{s2}\sin(2\omega t)
\end{aligned}
$$
The coefficients \(P_0, Q_0, P_{c2}, P_{s2}, Q_{c2}, Q_{s2}\) are functions of the positive and negative sequence components of voltage and current in the dq frame. This inherent power pulsation creates several problems, including stress on the DC-link capacitor, interference with Maximum Power Point Tracking (MPPT) algorithms, and potential instability. Consequently, different control objectives for the current references of the solar inverter have been established, each prioritizing a specific aspect of performance. The three primary objectives are summarized in the table below.
| Objective | Goal | Current Reference Focus | Consequence |
|---|---|---|---|
| I: Balanced Currents | Eliminate negative-sequence current. | Set \( i_d^- = i_q^- = 0 \). | Outputs symmetrical three-phase currents. However, both active and reactive power exhibit large 2ω oscillations. |
| II: Constant Reactive Power | Eliminate 2ω oscillations in Q. | Set \( Q_{c2} = Q_{s2} = 0 \). | Provides stable reactive power support, beneficial for grid voltage recovery. Active power oscillates, and currents are unbalanced. |
| III: Constant Active Power | Eliminate 2ω oscillations in P. | Set \( P_{c2} = P_{s2} = 0 \). | Maintains stable active power flow, crucial for DC-link stability and MPPT. Reactive power oscillates, and currents are unbalanced. |
For distributed PV systems connected at lower voltage levels (e.g., to the distribution network), grid codes often emphasize LVRT capability without explicit stringent requirements for dynamic reactive support during faults. In this context, Objective III (Constant Active Power) is highly critical. Oscillations in active power can cause significant DC-link voltage fluctuations, potentially driving the operating point away from the MPPT and reducing efficiency. Therefore, stabilizing active power is paramount for the internal stability and performance of the solar inverter system itself. This analysis will therefore focus on the control strategy derived from Objective III.
Peak Output Current Analysis Under Objective III Control
Adopting Objective III to eliminate active power oscillations, the reference currents in the positive and negative sequence dq frames can be derived. Assuming the grid voltage vector is aligned such that \( u_q^+ = 0 \) and \( u_q^- = 0 \), and defining \( U^+ \) and \( U^- \) as the magnitudes of the positive and negative sequence voltages, the current references are given by:
$$
\begin{aligned}
i_d^{+*} &= \frac{2P_0}{3D_1}U^+ \\
i_q^{+*} &= -\frac{2Q_0}{3D_2}U^+ \\
i_d^{-*} &= -\frac{2P_0}{3D_1}U^- \\
i_q^{-*} &= -\frac{2Q_0}{3D_2}U^-
\end{aligned}
$$
where \( D_1 = (U^+)^2 – (U^-)^2 = (U^+)^2(1-\varepsilon^2) \) and \( D_2 = (U^+)^2 + (U^-)^2 = (U^+)^2(1+\varepsilon^2) \), with \( \varepsilon = U^-/U^+ \) defined as the voltage unbalance factor.
Transforming these references back to the three-phase (abc) domain yields the instantaneous phase currents. Through rigorous derivation, the phase currents can be expressed in a compact form:
$$
\begin{aligned}
i_a &= m \left[ \sqrt{1+\varepsilon^2+2\varepsilon\cos(\Delta\phi)} \sin(\omega t + \phi_a) \right] \\
i_b &= m \left[ \sqrt{1+\varepsilon^2+2\varepsilon\cos(\Delta\phi – 2\pi/3)} \sin(\omega t + \phi_b) \right] \\
i_c &= m \left[ \sqrt{1+\varepsilon^2+2\varepsilon\cos(\Delta\phi + 2\pi/3)} \sin(\omega t + \phi_c) \right]
\end{aligned}
$$
where:
$$ m = \frac{2\sqrt{(P_0^2(1+\varepsilon^2)^2 + Q_0^2(1-\varepsilon^2)^2)}}{3(U^+)^2(1-\varepsilon^4)} $$
\( \Delta\phi = \phi^+ + \phi^- \) is the phase angle difference between the positive and negative sequence voltage vectors, and \( \phi_a, \phi_b, \phi_c \) are phase shifts dependent on \( P_0, Q_0, \varepsilon, \) and \( \Delta\phi \).
From these expressions, the peak values of the three-phase currents are directly obtained:
$$
\begin{aligned}
I_{a}^{peak} &= m \sqrt{1+\varepsilon^2+2\varepsilon\cos(\Delta\phi)} \\
I_{b}^{peak} &= m \sqrt{1+\varepsilon^2+2\varepsilon\cos(\Delta\phi – 2\pi/3)} \\
I_{c}^{peak} &= m \sqrt{1+\varepsilon^2+2\varepsilon\cos(\Delta\phi + 2\pi/3)}
\end{aligned}
$$
The absolute maximum possible peak current (\(I_{max}^{peak}\)) that the solar inverter might produce occurs when the cosine term in the corresponding phase reaches its maximum value of +1. This happens for one of the phases depending on \( \Delta\phi \). The worst-case peak current is:
$$ I_{max}^{peak} = m (1 + \varepsilon) = \frac{2(1+\varepsilon)\sqrt{(P_0^2(1+\varepsilon^2)^2 + Q_0^2(1-\varepsilon^2)^2)}}{3(U^+)^2(1-\varepsilon^4)} $$
This equation reveals the critical factors influencing the current stress on the solar inverter: the power references (\(P_0, Q_0\)), the positive-sequence voltage magnitude (\(U^+\)), the unbalance factor (\(\varepsilon\)), and the phase angle difference (\(\Delta\phi\)). Compared to the balanced condition (\(\varepsilon=0\)), the peak current can be significantly higher during an unbalanced sag.
Proposed Control Strategy for Limiting Current Peaks
During a grid fault, the parameters \(U^+\), \(\varepsilon\), and \(\Delta\phi\) are determined by the external grid condition and cannot be controlled by the solar inverter. The only available degrees of freedom to manage the output current are the active and reactive power references, \(P_0\) and \(Q_0\). The proposed strategy is to operate the solar inverter at its maximum allowable current limit during the voltage sag to prevent overcurrent trips while providing the maximum possible support to the grid.
Let \(I_{limit}\) be the maximum permissible peak current of the solar inverter’s power devices. To ensure safe operation, we require \(I_{max}^{peak} \leq I_{limit}\). This imposes the following constraint on the power references:
$$ P_0^2(1+\varepsilon^2)^2 + Q_0^2(1-\varepsilon^2)^2 \leq \frac{9 (I_{limit})^2 (U^+)^4 (1-\varepsilon^4)^2}{4(1+\varepsilon)^2} $$
During a voltage sag, grid codes may require or it may be beneficial to provide reactive current support. A common practice is to inject reactive current proportional to the voltage drop. For the purpose of defining a setpoint, we assume a fixed ratio between reactive and active power during the fault, for instance, \(Q_0 = k P_0\), where a typical value could be \(k = 0.5\). Substituting this relationship into the inequality above allows us to solve for the maximum allowable active power reference \(P_{0}^{max}\):
$$ P_0 \leq P_{0}^{max} = \frac{\sqrt{3} I_{limit} (U^+)^2 (1-\varepsilon^4)}{2(1+\varepsilon)\sqrt{(1+\varepsilon^2)^2 + k^2(1-\varepsilon^2)^2}} $$
The proposed control algorithm for the solar inverter is then:
- Normal Operation: Operate with \(P_0 = P_{MPPT}\) (from the MPPT algorithm) and \(Q_0 = 0\) (or as per scheduled power factor).
- Fault Detection: Continuously monitor grid voltages. Detect an unbalanced voltage sag (e.g., significant negative-sequence component appears).
- Reference Calculation & Limiting: Upon fault detection, calculate the positive and negative sequence voltages (\(U^+\), \(\varepsilon\), \(\Delta\phi\)). Set the reactive power reference \(Q_0 = k P_{0}^{max}\). Calculate \(P_{0}^{max}\) from the equation above. The new power references become \(P_0 = P_{0}^{max}\) and \(Q_0 = k P_{0}^{max}\). Generate the corresponding current references using the Objective III formulas.
- Fault Recovery: When grid voltages recover, smoothly transition back to normal MPPT and power factor control references.
This strategy ensures the solar inverter remains connected (LVRT compliance) while guaranteeing its output current never exceeds the safe hardware limit \(I_{limit}\), thus protecting the system.
Simulation Verification
The proposed strategy is validated through simulation of a 0.5 MW grid-connected solar inverter system. The key system parameters are listed below.
| Parameter | Value |
|---|---|
| Rated Power | 0.5 MW |
| DC-link Voltage | 750 V |
| Grid Filter Inductance (L) | 1.5 mH |
| Grid Filter Resistance (R) | 0.1 Ω |
| Switching Frequency | 4 kHz |
| Current Limit (\(I_{limit}\)) | 1.5 kA (peak) |
The control system uses a dual synchronous reference frame PI controller to independently regulate the positive and negative sequence currents as per the derived references. Two severe unbalanced voltage sag scenarios are tested, where the voltage unbalance factor \(\varepsilon\) is set to 30% and 40% respectively, occurring at t = 3 s. The positive sequence voltage \(U^+\) is 0.7 p.u. The reactive power ratio is set to \(k = 0.5\).
| Scenario (ε) | Control Mode | Power Setpoints (P0, Q0) | Max. Peak Current Observed | Remarks |
|---|---|---|---|---|
| 30% Unbalance | Nominal Power (No Limit) | 0.5 MW, 0 Mvar | 1.62 kA (Phase B) | Exceeds \(I_{limit}\) (1.5 kA). Active power constant, reactive power oscillates. |
| Proposed Current-Limiting | 0.42 MW, 0.21 Mvar | 1.47 kA (Phase B) | Current successfully limited below \(I_{limit}\). Power output matches calculated \(P_{0}^{max}\). | |
| 40% Unbalance | Nominal Power (No Limit) | 0.5 MW, 0 Mvar | 2.05 kA (Phase B) | Severe overcurrent. |
| Proposed Current-Limiting | 0.34 MW, 0.17 Mvar | 1.49 kA (Phase B) | Current effectively clamped at safe limit. Active power is reduced to 0.34 MW as per strategy. |
The simulation results confirm the analytical derivations. Operating the solar inverter at its rated power during an unbalanced sag leads to dangerous overcurrent conditions. In contrast, the proposed control strategy automatically calculates and applies reduced power references (\(P_{0}^{max}\), \(Q_{0}^{max}\)) based on the real-time grid voltage conditions (\(U^+\), \(\varepsilon\)). This successfully restricts the peak output currents to a safe value below the defined hardware limit of 1.5 kA, ensuring the inverter’s survival through the fault. The trade-off is a necessary reduction in active power output during the fault period, which is a prudent compromise to avoid tripping and maintain grid connection.
Conclusion
This paper addresses the critical issue of output current peaks in solar inverters subjected to unbalanced voltage sags. A detailed analysis of the inverter’s behavior under such conditions was conducted, with a focus on the constant active power control objective (Objective III). We derived precise analytical expressions for the three-phase peak currents and the absolute worst-case peak, identifying voltage unbalance (\(\varepsilon\)), positive-sequence voltage magnitude (\(U^+\)), and power setpoints (\(P_0, Q_0\)) as the key influencing factors.
Building on this analysis, a pragmatic control strategy was proposed. The method continuously evaluates the grid condition during a fault and dynamically adjusts the active and reactive power references to ensure the solar inverter’s output current never exceeds its maximum permissible limit. This guarantees the protection of the power electronics while fulfilling LVRT requirements. The strategy was successfully validated through simulation, demonstrating its effectiveness in preventing overcurrent during severe unbalanced sags.
The main trade-off of this approach is the reduction in power output during the fault. Future work could explore more advanced strategies that optimally balance current limiting with maximum feasible power or reactive support, potentially using model predictive control or other optimization-based techniques. Furthermore, the analysis framework can be extended to other control objectives, such as constant reactive power (Objective II), which is particularly relevant for large-scale PV plants required to provide dynamic grid voltage support during faults.