In recent years, the rapid development of photovoltaic (PV) generation has led to its increasing penetration in power systems. Traditional solar inverters are designed under ideal grid conditions; however, when the grid becomes unbalanced, their performance can be severely affected. Issues such as uncontrolled second-order fluctuations in active and reactive power, increased grid current distortion, and DC-link voltage oscillations may arise, potentially exceeding limits and degrading power quality. To enhance the operational performance of solar inverters under unbalanced grid conditions, this paper proposes a multi-objective coordinated control strategy based on a repetitive controller in the stationary α-β frame. This strategy aims to balance power quality and current harmonic suppression without requiring phase-locked loops (PLLs) or sequence separation modules, simplifying the control structure and improving robustness.
The core of this approach lies in a weighted power reference distribution method that allows for flexible trade-offs between constant power injection and low current distortion. By adjusting distribution factors, the strategy can prioritize either power stability or current quality based on grid requirements. Additionally, a compound controller combining PI and repetitive control is designed to achieve accurate current tracking. The linear superposition theorem is applied to derive an equivalent controlled object model, facilitating the design of a stabilization compensator. Simulation results using PSCAD software validate the effectiveness of the proposed strategy. Throughout this paper, the term ‘solar inverters’ will be frequently used to emphasize the focus on grid-connected PV systems.
The topology of a typical three-phase solar inverter is shown below. It consists of a DC voltage source representing the PV array, a voltage-source inverter, and an LCL filter for grid connection to attenuate switching harmonics. The LCL filter parameters are critical for stability and performance, and their design involves a trade-off between filtering effectiveness and system dynamics.
Under ideal grid conditions, solar inverters typically employ P/Q control in the synchronous rotating d-q frame to regulate active and reactive power injection. The control structure involves transforming grid voltages and currents to the d-q frame, using PI controllers to generate reference voltages, and pulse-width modulation (PWM) to drive the inverter. However, under unbalanced grid voltages, this conventional method fails to maintain desired performance because negative-sequence components introduce double-frequency oscillations in power and currents. Therefore, developing advanced control strategies for solar inverters in unbalanced grids is essential for reliable renewable energy integration.
To address this, I first analyze the power and current relationships in the stationary α-β frame. According to instantaneous power theory, the active power \(P\) and reactive power \(Q\) injected by the solar inverter can be expressed as:
$$ \begin{bmatrix} P \\ Q \end{bmatrix} = \frac{3}{2} \begin{bmatrix} u_{\alpha} & u_{\beta} \\ u_{\beta} & -u_{\alpha} \end{bmatrix} \begin{bmatrix} i_{\alpha} \\ i_{\beta} \end{bmatrix} $$
where \(u_{\alpha}\), \(u_{\beta}\), \(i_{\alpha}\), and \(i_{\beta}\) are the instantaneous voltage and current components in the α-β frame. For P/Q control, the current references in the α-β frame are derived from the power references \(P^*\) and \(Q^*\):
$$ \begin{bmatrix} i_{\alpha}^* \\ i_{\beta}^* \end{bmatrix} = \frac{2}{3} \begin{bmatrix} u_{\alpha} & u_{\beta} \\ u_{\beta} & -u_{\alpha} \end{bmatrix}^{-1} \begin{bmatrix} P^* \\ Q^* \end{bmatrix} = \frac{2}{3} \frac{1}{u_{\alpha}^2 + u_{\beta}^2} \begin{bmatrix} P^* u_{\alpha} + Q^* u_{\beta} \\ P^* u_{\beta} – Q^* u_{\alpha} \end{bmatrix} $$
Under unbalanced grid conditions, the grid voltages contain both positive-sequence and negative-sequence components. Let \(u_{\alpha}^+\), \(u_{\beta}^+\) and \(u_{\alpha}^-\), \(u_{\beta}^-\) denote the positive and negative sequence components, respectively, with magnitudes \(U^+\) and \(U^-\). The denominator \(u_{\alpha}^2 + u_{\beta}^2\) becomes:
$$ u_{\alpha}^2 + u_{\beta}^2 = (U^+)^2 + (U^-)^2 – 2U^+U^- \cos(2\omega t) $$
where \(\omega\) is the grid angular frequency. This double-frequency term causes the current references to contain harmonics, leading to distorted grid currents if tracked perfectly. To mitigate this, I propose a weighted power reference distribution method. Define constant power references \(P_c^*\) and \(Q_c^*\) and fluctuating power references \(\tilde{P}^*\) and \(\tilde{Q}^*\) as:
$$ P_c^* = (1 – \lambda) P^*, \quad Q_c^* = (1 – \mu) Q^* $$
$$ \tilde{P}^* = \lambda P^*, \quad \tilde{Q}^* = \mu Q^* $$
where \(\lambda\) and \(\mu\) are distribution factors for active and reactive power, respectively, with \(0 \leq \lambda \leq 1\) and \(0 \leq \mu \leq 1\). The current references are then modified to:
$$ i_{\alpha}^* = \frac{2}{3} \left( \frac{\tilde{P}^* u_{\alpha} + \tilde{Q}^* u_{\beta}}{u_f} + \frac{P_c^* u_{\alpha} + Q_c^* u_{\beta}}{u_{\alpha}^2 + u_{\beta}^2} \right) $$
$$ i_{\beta}^* = \frac{2}{3} \left( \frac{\tilde{P}^* u_{\beta} – \tilde{Q}^* u_{\alpha}}{u_f} + \frac{P_c^* u_{\beta} – Q_c^* u_{\alpha}}{u_{\alpha}^2 + u_{\beta}^2} \right) $$
Here, \(u_f\) is a filtered version of \(u_{\alpha}^2 + u_{\beta}^2\) with the double-frequency component removed, achieved using a notch filter tuned at \(2\omega\). This filtering reduces current harmonics but introduces power fluctuations. By adjusting \(\lambda\) and \(\mu\), the strategy can prioritize either low current distortion (when \(\lambda = \mu = 1\)) or constant power injection (when \(\lambda = \mu = 0\)), or achieve a compromise. For instance, if the solar inverter is primarily used for reactive power support, \(\mu\) can be set low to minimize reactive power fluctuations while keeping current distortion within limits.
To accurately track these current references, which may contain harmonic components, I employ a repetitive controller based on the internal model principle. A repetitive controller incorporates a model of periodic signals, enabling zero steady-state error for periodic references or disturbances. The continuous-time transfer function of a repetitive controller is:
$$ G_R(s) = \frac{1}{1 – e^{-Ls}} $$
where \(L = 1/f_0\) is the period of the fundamental frequency \(f_0\). In discrete-time implementation, this becomes:
$$ G_R(z) = \frac{1}{1 – Q(z) z^{-N}} $$
where \(N = T_0 / T_s\) is the number of samples per fundamental period, \(T_s\) is the sampling time, and \(Q(z)\) is a low-pass filter to enhance robustness, typically chosen as a constant slightly less than 1, e.g., \(Q(z) = 0.96\). The controller output \(c(k)\) at step \(k\) is given by:
$$ c(k) = e(k) + Q(z) c(k – N) $$
where \(e(k)\) is the current tracking error. To improve dynamic response and stability, I combine the repetitive controller with a PI controller in parallel, forming a compound controller. The structure is shown in the block diagram below, where \(G(z)\) represents the generalized plant including the solar inverter, LCL filter, and delays, and \(S(z)\) is a stabilization compensator.
Using linear superposition, the equivalent controlled object for the repetitive controller is derived as the closed-loop system of the PI controller and the plant. The transfer function of this equivalent object \(G_e(z)\) is:
$$ G_e(z) = \frac{G(z)}{1 + G_{PI}(z) G(z)} $$
where \(G_{PI}(z)\) is the discrete-time PI controller. The design procedure involves first tuning the PI parameters to make \(G_e(z)\) have near-zero gain and phase shift at low frequencies, simplifying the compensator design. Then, \(S(z)\) is designed to compensate the mid- and high-frequency characteristics of \(G_e(z)\), ensuring stability and performance. The stability criterion based on the small gain theorem is:
$$ |Q(z) – S(z) G_e(z)| < 1 $$
for all frequencies. A common choice for \(S(z)\) is a phase-lead compensator or a low-pass filter that inverts the phase of \(G_e(z)\) at critical frequencies. The overall system block diagram with the compound controller is depicted, illustrating the interaction between the power reference generation, current control, and modulation stages for solar inverters.
To validate the proposed strategy, I conducted simulations in PSCAD/EMTDC. The system parameters are summarized in Table 1.
| Parameter | Value | Unit |
|---|---|---|
| DC-link voltage | 100 | V |
| Grid voltage (nominal) | 30 (phase) | V |
| LCL filter: \(L_1\) | 0.5 | mH |
| LCL filter: \(L_2\) | 0.1 | mH |
| LCL filter: \(C\) | 10 | μF |
| Damping resistor \(R\) | 1 | Ω |
| Switching frequency | 10 | kHz |
| Sampling frequency | 20 | kHz |
| Fundamental frequency | 50 | Hz |
An unbalanced grid condition is emulated by setting phase voltages: \(V_a = 20 \, \text{V}\), \(V_b = 30 \, \text{V}\), \(V_c = 50 \, \text{V}\). The power references are \(P^* = 2 \, \text{kW}\) and \(Q^* = 0.6 \, \text{kvar}\). The distribution factors \(\lambda\) and \(\mu\) are varied to demonstrate the multi-objective capability. The simulation results are analyzed for three cases:
Case 1: \(\lambda = 1, \mu = 1\) (Current Harmonic Suppression Priority)
In this case, the power references are fully allocated to the fluctuating components. The active and reactive powers exhibit significant double-frequency oscillations, with the ripple magnitude relative to the DC component being approximately 49.25% for active power and 45.13% for reactive power. However, the grid currents have low total harmonic distortion (THD), below 1.08% per phase, indicating excellent current quality for solar inverters.
Case 2: \(\lambda\) varying from 1 to 0, \(\mu = 1\) (Trade-off for Active Power)
Here, \(\lambda\) is ramped down from 1 to 0 over time, while \(\mu\) remains 1. As \(\lambda\) decreases, the constant power component for active power increases, reducing the active power ripple. At steady state with \(\lambda = 0\), the active power ripple is minimized to about 54 W. However, the grid current distortion increases; when \(\lambda\) drops below a threshold, the current THD exceeds 5%, violating standards. At \(\lambda \approx 0.5\), the currents become more balanced with lower amplitude, showcasing a compromise. This illustrates the flexibility of the strategy for solar inverters in managing power-quality objectives.
Case 3: \(\mu\) varying from 1 to 0, \(\lambda = 1\) (Trade-off for Reactive Power)
Similarly, varying \(\mu\) affects reactive power fluctuations and current distortion. Reducing \(\mu\) suppresses reactive power oscillations but increases current harmonics. The trends mirror Case 2, emphasizing that the distribution factors allow independent tuning for active and reactive power based on application needs, such as when solar inverters provide ancillary services.
The effectiveness of the repetitive controller is evident in the accurate tracking of current references containing harmonics. The compound controller achieves fast transient response and zero steady-state error, outperforming conventional PI-based methods. The equivalent object model simplifies compensator design, ensuring stability even under severe grid imbalances. Further analysis includes sensitivity to parameter variations and comparison with other methods like dual synchronous reference frame control. The proposed strategy reduces computational burden by eliminating PLLs and sequence separation, making it suitable for practical implementation in solar inverters.
In conclusion, this paper presents a multi-objective coordinated control strategy for solar inverters under unbalanced grid conditions. The key contributions are: (1) a weighted power reference distribution method that enables flexible trade-offs between constant power injection and low current distortion; (2) a compound controller combining PI and repetitive control in the stationary α-β frame, eliminating the need for PLLs and sequence separation; and (3) an equivalent controlled object model based on linear superposition, simplifying stabilizer design. Simulations verify that the strategy can maintain desired performance across various operating points, enhancing the robustness and functionality of solar inverters in modern power grids. Future work may focus on hardware-in-the-loop testing and integration with energy storage systems for broader applications.
The mathematical formulations and control design are generalized for three-phase solar inverters, but the principles can be extended to single-phase systems or other distributed generation units. The repetitive controller’s ability to reject periodic disturbances makes it particularly effective for harmonic suppression in solar inverters connected to weak grids. Additionally, the weighted distribution approach can be adaptive, with \(\lambda\) and \(\mu\) dynamically adjusted based on real-time grid conditions or operator commands, paving the way for smarter grid support from solar inverters.
To further illustrate the control performance, consider the frequency response of the equivalent object \(G_e(z)\) and the compensator \(S(z)\). Using the parameters from Table 1, the PI controller is tuned to achieve a bandwidth of 500 Hz. The Bode plot of \(G_e(z)\) shows near-zero gain at low frequencies, as desired. The compensator \(S(z)\) is designed as:
$$ S(z) = k_c \frac{z – a}{z – b} $$
where \(k_c\), \(a\), and \(b\) are chosen to provide phase lead at the crossover frequency. For instance, with \(k_c = 1.2\), \(a = 0.8\), and \(b = 0.6\), the stability criterion is satisfied. The repetitive controller’s delay line length \(N\) is set to 400 for a 50 Hz grid with \(T_s = 50 \, \mu\text{s}\). The impact of the low-pass filter \(Q(z)\) on robustness is analyzed by varying its coefficient; a value of 0.96 offers a good balance between harmonic rejection and stability margin.
Table 2 summarizes the performance metrics for different distribution factors, highlighting the trade-offs. The metrics include total harmonic distortion (THD) of grid currents, power ripple factor (PRF) defined as the ratio of double-frequency amplitude to DC power, and current unbalance factor (CUF) based on symmetrical components. The data demonstrates that solar inverters using the proposed strategy can meet various grid codes by adjusting \(\lambda\) and \(\mu\).
| Case (\(\lambda, \mu\)) | THD (%) | PRF (P) (%) | PRF (Q) (%) | CUF (%) |
|---|---|---|---|---|
| (1, 1) | 1.08 | 49.25 | 45.13 | 15.2 |
| (0.5, 1) | 3.45 | 24.70 | 45.10 | 8.7 |
| (0, 1) | 6.89 | 2.70 | 45.05 | 12.3 |
| (1, 0.5) | 2.91 | 49.20 | 22.55 | 10.5 |
| (0, 0) | 8.12 | 2.65 | 2.25 | 18.4 |
In summary, the proposed control strategy offers a comprehensive solution for solar inverters operating in unbalanced grids. By leveraging repetitive control and power weighting, it addresses multiple objectives simultaneously, enhancing the reliability and power quality of PV systems. Future extensions could incorporate adaptive algorithms for automatic tuning of distribution factors based on real-time measurements, further optimizing the performance of solar inverters in dynamic grid environments.