The increasing integration of solar photovoltaic (PV) systems into the power grid necessitates a robust understanding of their behavior under grid disturbances. Grid codes worldwide, including the Chinese standard GB/T 19964-2012, mandate Low Voltage Ride-Through (LVRT) capability, requiring solar inverters to remain connected and support the grid during voltage sags. In practical operation, voltage sags are frequently caused by asymmetrical faults, leading to unbalanced grid conditions characterized by the presence of negative-sequence voltage components. This asymmetry poses significant challenges for grid-connected solar inverters. The negative-sequence voltage induces double-frequency oscillations in the output active and reactive power, causes distortion and asymmetry in the three-phase output currents, and critically, leads to a substantial increase in the peak amplitude of these currents. This peak current can reach several times its rated value during symmetrical operation, threatening the safety of the semiconductor switches within the inverter, potentially triggering overcurrent protection, and jeopardizing the entire PV system’s stability. Therefore, developing effective control strategies to manage and limit the output current peak of solar inverters during unbalanced voltage sags is of paramount importance for reliable and grid-code-compliant operation. This article delves into the modeling, analysis, and proposed control methodologies to address this critical issue.
The core of a grid-connected PV system is the power electronic interface, typically a voltage-source inverter. The standard three-phase two-level inverter topology, connected to the grid via an L or LCL filter, forms the basis of our analysis. The mathematical model of the solar inverter in the stationary αβ-frame is given by:
$$ u_{\alpha} = Ri_{\alpha} + L\frac{di_{\alpha}}{dt} + e_{\alpha} $$
$$ u_{\beta} = Ri_{\beta} + L\frac{di_{\beta}}{dt} + e_{\beta} $$
where \( u_{\alpha}, u_{\beta} \), \( i_{\alpha}, i_{\beta} \), and \( e_{\alpha}, e_{\beta} \) are the inverter output voltage, output current, and grid voltage components in the αβ-frame, respectively, and \( R \) and \( L \) represent the equivalent resistance and inductance of the output filter.
For control design, it is convenient to transform the model into a synchronous rotating dq-frame. Under unbalanced grid conditions, both positive- and negative-sequence components exist. Using decoupled double synchronous reference frame (DDSRF) theory, the model can be separated into positive-sequence (rotating at \( +\omega \)) and negative-sequence (rotating at \( -\omega \)) components:
$$ u_{d}^{+} = (R + sL)i_{d}^{+} – \omega L i_{q}^{+} + e_{d}^{+} $$
$$ u_{q}^{+} = (R + sL)i_{q}^{+} + \omega L i_{d}^{+} + e_{q}^{+} $$
$$ u_{d}^{-} = (R + sL)i_{d}^{-} + \omega L i_{q}^{-} + e_{d}^{-} $$
$$ u_{q}^{-} = (R + sL)i_{q}^{-} – \omega L i_{d}^{-} + e_{q}^{-} $$
Here, the superscripts ‘+’ and ‘-‘ denote positive- and negative-sequence components, respectively.
Under unbalanced voltages, the instantaneous active power \( P \) and reactive power \( Q \) injected by the solar inverter contain constant terms and double-frequency oscillating terms. The complex power can be expressed in terms of the dq-sequence components as:
$$ S = P + jQ = 1.5 (\mathbf{u}_{dq}^{+}e^{j\omega t} + \mathbf{u}_{dq}^{-}e^{-j\omega t})(\mathbf{i}_{dq}^{+}e^{j\omega t} + \mathbf{i}_{dq}^{-}e^{-j\omega t})^* $$
This leads to the expressions:
$$ P = P_0 + P_{c2}\cos(2\omega t) + P_{s2}\sin(2\omega t) $$
$$ Q = Q_0 + Q_{c2}\cos(2\omega t) + Q_{s2}\sin(2\omega t) $$
where \( P_0, Q_0 \) are the average powers, and \( P_{c2}, P_{s2}, Q_{c2}, Q_{s2} \) are the amplitudes of the double-frequency oscillations. These oscillatory coefficients are functions of both voltage and current sequence components.
Control strategies for solar inverters under unbalanced sags aim to regulate the four current degrees of freedom (\( i_{d}^{+}, i_{q}^{+}, i_{d}^{-}, i_{q}^{-} \)) to achieve specific objectives. The three most common control targets are summarized in the table below:
| Control Target | Objective | Key Current Constraints | Power Quality Impact |
|---|---|---|---|
| Target I: Balanced Currents | Eliminate negative-sequence current output. | \( i_{d}^{-}=0, i_{q}^{-}=0 \) | Produces balanced three-phase currents but results in significant 2ω oscillations in both P and Q. |
| Target II: Constant Reactive Power | Eliminate double-frequency oscillation in Q. | \( Q_{c2}=0, Q_{s2}=0 \) | Provides stable reactive power support (beneficial for voltage recovery) but causes large oscillations in P and unbalanced currents. |
| Target III: Constant Active Power | Eliminate double-frequency oscillation in P. | \( P_{c2}=0, P_{s2}=0 \) | Maintains stable active power (crucial for DC-link stability and MPPT) but causes oscillations in Q and unbalanced currents. |
For distributed PV systems where DC-link stability and MPPT efficiency are prioritized, Target III (Constant Active Power) is often the preferred strategy. The reference currents for this target are derived by setting \( P_{c2}=0 \) and \( P_{s2}=0 \), and solving for the current components given references for \( P_0 \) and \( Q_0 \):
$$ \begin{bmatrix} i_{d}^{+*} \\ i_{q}^{+*} \\ i_{d}^{-*} \\ i_{q}^{-*} \end{bmatrix} = \frac{2}{3} \begin{bmatrix} \frac{P_0 u_{d}^{+}}{D_1} + \frac{Q_0 u_{q}^{+}}{D_2} \\ \frac{P_0 u_{q}^{+}}{D_1} – \frac{Q_0 u_{d}^{+}}{D_2} \\ -\frac{P_0 u_{d}^{-}}{D_1} + \frac{Q_0 u_{q}^{-}}{D_2} \\ -\frac{P_0 u_{q}^{-}}{D_1} – \frac{Q_0 u_{d}^{-}}{D_2} \end{bmatrix} $$
where \( D_1 = (u_{d}^{+})^2+(u_{q}^{+})^2 – (u_{d}^{-})^2-(u_{q}^{-})^2 \) and \( D_2 = (u_{d}^{+})^2+(u_{q}^{+})^2 + (u_{d}^{-})^2+(u_{q}^{-})^2 \).
The critical challenge under this control strategy is the potential for excessive peak phase currents. To analyze this, we express the three-phase currents in the time domain. Assuming the grid voltage positive- and negative-sequence components are:
$$ \mathbf{e}^{+} = U^{+} e^{j(\omega t + \phi^{+})}, \quad \mathbf{e}^{-} = U^{-} e^{j(-\omega t + \phi^{-})} $$
and defining the voltage unbalance factor \( \varepsilon = U^{-}/U^{+} \) and the phase angle difference \( \Delta\phi = \phi^{+} + \phi^{-} \), the phase currents can be derived through coordinate transformations and substitutions. After extensive algebraic manipulation, the instantaneous phase currents can be expressed in a compact form:
$$ i_a = m[\sqrt{1+\varepsilon^2+2\varepsilon\cos(\Delta\phi)} \sin(\omega t + \phi_a)] $$
$$ i_b = m[\sqrt{1+\varepsilon^2+2\varepsilon\cos(\Delta\phi – 2\pi/3)} \sin(\omega t + \phi_b)] $$
$$ i_c = m[\sqrt{1+\varepsilon^2+2\varepsilon\cos(\Delta\phi + 2\pi/3)} \sin(\omega t + \phi_c)] $$
where \( m \) is a magnitude factor dependent on \( P_0, Q_0, U^{+}, \) and \( \varepsilon \), and \( \phi_a, \phi_b, \phi_c \) are phase angles.
From these expressions, the peak value of each phase current is directly obtained:
$$ I_{a}^{peak} = m\sqrt{1+\varepsilon^2+2\varepsilon\cos(\Delta\phi)} $$
$$ I_{b}^{peak} = m\sqrt{1+\varepsilon^2+2\varepsilon\cos(\Delta\phi – 2\pi/3)} $$
$$ I_{c}^{peak} = m\sqrt{1+\varepsilon^2+2\varepsilon\cos(\Delta\phi + 2\pi/3)} $$
The maximum possible peak current across all phases, \( I_{max}^{peak} \), occurs when the cosine term in the square root reaches its maximum value of +1. This happens for one phase when \( \Delta\phi = 0, -2\pi/3, \) or \( +2\pi/3 \). Therefore, the worst-case peak current is:
$$ I_{max}^{peak} = m (1 + \varepsilon) $$
Substituting the expression for \( m \), which is derived as \( m = \frac{2\sqrt{P_0^2(1+\varepsilon^2)+Q_0^2(1-\varepsilon^2)}}{3U^{+}(1-\varepsilon^2)} \), we get the final formula for the maximum peak current:
$$ I_{max}^{peak} = \frac{2(1+\varepsilon)\sqrt{P_0^2(1+\varepsilon^2)+Q_0^2(1-\varepsilon^2)}}{3U^{+}(1-\varepsilon^2)} $$
This equation is fundamental for assessing the stress on the solar inverter during an unbalanced sag.
The control strategy for limiting the current peak is now evident. The solar inverter has a maximum allowable current, \( I_{limit} \), dictated by its semiconductor ratings and protection thresholds. During normal operation, \( P_0 \) is set by the MPPT algorithm and \( Q_0 \) is typically zero (unity power factor). During a detected unbalanced voltage sag (\( \varepsilon > 0 \), \( U^{+} < 1 \) pu), the references \( P_0 \) and \( Q_0 \) must be dynamically adjusted to ensure that the calculated \( I_{max}^{peak} \) from the equation above does not exceed \( I_{limit} \). This creates a constraint:
$$ P_0^2(1+\varepsilon^2)+Q_0^2(1-\varepsilon^2) \leq \left[ \frac{3 U^{+} I_{limit} (1-\varepsilon^2)}{2(1+\varepsilon)} \right]^2 $$
The inverter must operate within this elliptical constraint in the P-Q plane. A common and prudent approach is to prioritize current limiting and operate at the boundary of this constraint. Assuming a fixed ratio between reactive and active power support during faults, such as \( Q_0 = 0.5 P_0 \) (0.5 lagging power factor during support), we can solve for the maximum permissible active power reference \( P_{0,max} \):
$$ P_{0,max} = \frac{3 U^{+} I_{limit} (1-\varepsilon^2)}{2(1+\varepsilon)\sqrt{(1+\varepsilon^2) + 0.25(1-\varepsilon^2)}} $$
The control system must, therefore, include a block that continuously monitors the positive-sequence voltage magnitude \( U^{+} \) and the unbalance factor \( \varepsilon \), calculates the safe operating \( P_{0,max} \) and corresponding \( Q_{0} \), and dynamically limits the power references sent to the current regulator. This ensures the solar inverter remains connected, provides supportive active and reactive power to the grid, and most importantly, protects itself from dangerous overcurrent conditions.
To validate the analysis and the proposed strategy, a detailed simulation model of a 0.5 MW solar inverter system was developed. The system parameters included a DC-link voltage of 750 V, a filter inductance of 1.5 mH, and a switching frequency of 4 kHz. The inverter’s current limit \( I_{limit} \) was set to 1.5 kA (1.5 pu). Two unbalanced sag scenarios were tested: 30% and 40% voltage unbalance (\( \varepsilon = 0.3 \) and \( 0.4 \)) with a positive-sequence voltage dip.
In the first case (\( \varepsilon=0.3 \)), if the inverter attempted to maintain its pre-fault active power output of 0.5 MW (with \( Q_0=0 \)), the simulated peak phase current reached 1.62 kA, exceeding the safe limit. When the proposed current-limiting strategy was activated, it calculated new power references: \( P_0 \approx 0.42 \) MW and \( Q_0 \approx 0.21 \) Mvar. The simulation results confirmed these values, showing an output of 0.41 MW and 0.19 Mvar, while successfully constraining the maximum phase current to 1.47 kA, safely below the 1.5 kA limit. The active power was constant, while the reactive power exhibited the expected double-frequency ripple, as per Target III strategy.
The second, more severe case (\( \varepsilon=0.4 \)) demonstrated the strategy’s necessity even more clearly. Without current limiting, the peak current soared to 2.05 kA for a 0.5 MW output. The control law recalculated the allowable power to \( P_0 \approx 0.34 \) MW and \( Q_0 \approx 0.17 \) Mvar. The solar inverter operated stably at these levels, outputting 0.33 MW and 0.15 Mvar, and critically, limiting the peak current to 1.49 kA. The transition of power references during the sag entry caused a transient in the current, which was effectively damped by the controller, demonstrating the practical viability of the approach. These simulation results conclusively verify the accuracy of the derived peak current formulas and the effectiveness of the proposed dynamic power reference limiting strategy for safeguarding solar inverters during unbalanced grid faults.
In conclusion, the operation of solar inverters under unbalanced voltage sags requires careful control to prevent hazardous overcurrent conditions. This analysis has detailed the mathematical relationship between output power, grid voltage unbalance, and the resulting phase current peaks. By deriving the exact expression for the maximum possible peak current, a clear and implementable constraint for safe operation is established. The proposed control strategy, which dynamically adjusts active and reactive power references based on real-time measurements of voltage magnitude and unbalance, ensures that the solar inverter remains within its current-carrying capacity. This enables reliable Low Voltage Ride-Through, provides supportive grid interaction, and protects the inverter hardware. Future work may explore optimizing the P-Q ratio during faults for better grid support or integrating these strategies with more advanced fault detection and islanding management schemes for solar inverters in modern power systems.