The increasing integration of distributed generation, particularly from renewable sources like photovoltaics (PV), presents both opportunities and challenges for modern power systems. While contributing to clean energy goals, the intermittent nature of PV generation often leaves the associated power electronic interface—the solar inverter—underutilized. Concurrently, the proliferation of non-linear loads in distribution networks injects harmonic currents, leading to significant power quality issues such as voltage distortion. This work explores a promising synergy: leveraging the inherent capacity and topology of grid-connected solar inverters to perform ancillary power quality functions, specifically local harmonic current compensation, alongside their primary role of active power injection. The core challenge lies in designing a current control strategy for the inverter that achieves precise tracking of both fundamental and harmonic reference signals while maintaining robustness against system parameter variations and grid disturbances. This article presents a comprehensive design methodology for the inner-loop current controller of a three-phase LCL-filter-based solar inverter, employing an H∞-based repetitive control structure to meet these demanding performance criteria.
System Configuration and Modeling
The studied system centers on a three-phase, three-wire voltage source inverter (VSI) connected to the grid through an LCL filter. The primary function is to inject active power from the PV array. The multifunctional capability is realized by superimposing a harmonic current reference, typically extracted from local non-linear load currents using detection algorithms like those based on instantaneous power theory, onto the fundamental active current reference. The control objective for the solar inverter is to accurately synthesize this combined current reference at its output terminals.
The heart of the control system is the current controller. Traditional methods like Proportional-Integral (PI) control in the synchronous rotating (dq) frame offer good performance under ideal grid conditions but suffer from steady-state error and coupling when tracking harmonic signals or under distorted grid voltages. Proportional-Resonant (PR) controllers can provide high gain at specific frequencies but require parallel structures for multiple harmonics, complicating parameter tuning. Repetitive Control (RC), founded on the internal model principle, is inherently suited for tracking or rejecting periodic signals, such as harmonics. It embeds a model of the periodicity (a delay line within a positive feedback loop) to achieve zero steady-state error for all periodic content up to its bandwidth. However, designing a robust stabilizing compensator for the RC to work with a complex plant like an LCL filter is non-trivial.
We adopt an H∞ robust control framework to design this compensator. H∞ synthesis aims to find a stabilizing controller that minimizes the worst-case gain (the H∞ norm) of a closed-loop transfer function, thereby ensuring robust performance and stability against model uncertainties and disturbances. This approach systematically handles the trade-offs between performance, robustness, and control effort.
State-Space Model of the LCL-Filter Plant
The first step is to derive a control-oriented model. For a balanced three-phase system, the dynamics in the stationary (αβ) frame are equivalent to two independent, identical single-phase systems. We therefore focus on the single-phase equivalent model of the LCL filter, as shown below, where the inverter bridge is represented by its average model, yielding the control input voltage \(u\).
Choosing the inverter-side inductor current \(i_1\), the grid-side inductor current \(i_2\) (which is also the output current \(i_g\)), and the capacitor voltage \(u_c\) as state variables, the state-space equations are derived from Kirchhoff’s laws. Defining the state vector \(\mathbf{x} = [i_1, i_2, u_c]^T\), the external input vector \(\mathbf{w} = [u_g, i_{ref}]^T\) (containing the grid voltage disturbance and the reference current), and the control input \(u\), the model is:
$$
\begin{aligned}
\dot{\mathbf{x}} &= \mathbf{A}\mathbf{x} + \mathbf{B}_1\mathbf{w} + \mathbf{B}_2 u \\
y &= e = i_{ref} – i_2 = \mathbf{C}_1\mathbf{x} + \mathbf{D}_1\mathbf{w} + \mathbf{D}_2 u
\end{aligned}
$$
The matrices are defined as follows, where \(L_1\), \(L_2\), \(C\), and \(R_d\) are the inverter-side inductance, grid-side inductance, filter capacitance, and passive damping resistance, respectively:
$$
\mathbf{A} = \begin{bmatrix}
-\frac{R_d}{L_1} & \frac{R_d}{L_1} & -\frac{1}{L_1} \\
\frac{R_d}{L_2} & -\frac{R_d}{L_2} & \frac{1}{L_2} \\
\frac{1}{C} & -\frac{1}{C} & 0
\end{bmatrix}, \quad
\mathbf{B}_1 = \begin{bmatrix}
0 & 0 \\
0 & 0 \\
0 & 0
\end{bmatrix}, \quad
\mathbf{B}_2 = \begin{bmatrix}
\frac{1}{L_1} \\
0 \\
0
\end{bmatrix}
$$
$$
\mathbf{C}_1 = \begin{bmatrix}
0 & -1 & 0
\end{bmatrix}, \quad
\mathbf{D}_1 = \begin{bmatrix}
0 & 1
\end{bmatrix}, \quad
\mathbf{D}_2 = \begin{bmatrix}
0
\end{bmatrix}
$$
The plant \(P(s)\) from inputs \([u, w^T]^T\) to output \(e\) has the compact state-space representation:
$$
P(s) \triangleq \left[ \begin{array}{c|c}
\mathbf{A} & \mathbf{B}_1 & \mathbf{B}_2 \\
\hline
\mathbf{C}_1 & \mathbf{D}_1 & \mathbf{D}_2
\end{array} \right]
$$
H∞-Based Repetitive Controller Design
The proposed control structure integrates a repetitive control (RC) scheme within an H∞ optimization framework. The standard RC has the form \(M(s) = 1/(1 – Q(s)e^{-sT})\), where \(T\) is the fundamental period and \(Q(s)\) is typically a low-pass filter to enhance robustness. We select a first-order low-pass filter \(Q(s) = W(s) = \omega_c / (s + \omega_c)\). To improve robustness against grid frequency variations, the delay time \(\tau\) is chosen slightly less than \(T\), satisfying \(\tau + 1/\omega_c = T\).
The core design task is to find a robust stabilizing compensator \(C(s)\) for this RC structure. This is formulated as a standard H∞ problem. We open the positive feedback loop of the internal model and define an augmented plant \(\tilde{P}\) that includes the original plant \(P(s)\), the filter \(W(s)\), and weighting coefficients \(\xi\) and \(\mu\). These weights provide design knobs to balance tracking performance against control effort and robustness.
The generalized plant \(\tilde{P}\) maps the exogenous inputs \(\tilde{w} = [v, w^T]^T\) (where \(v\) is the output of the opened delay line) and the control input \(u\) to the regulated outputs \(\tilde{z} = [z_1, z_2]^T\) (where \(z_1 = W(s)(e+\xi v)\) penalizes tracking error and \(z_2 = \mu u\) penalizes control effort) and the measured output \(\tilde{y} = e + \xi v\). The state-space realization of \(\tilde{P}\) is constructed as:
$$
\tilde{P} \triangleq
\left[ \begin{array}{ccc|cc}
\mathbf{A} & \mathbf{0} & \mathbf{0} & \mathbf{B}_1 & \mathbf{B}_2 \\
\mathbf{B}_w\mathbf{C}_1 & \mathbf{A}_w & \mathbf{B}_w\xi & \mathbf{B}_w\mathbf{D}_1 & \mathbf{B}_w\mathbf{D}_2 \\
\hline
\mathbf{0} & \mathbf{C}_w & 0 & 0 & 0 \\
\mathbf{0} & \mathbf{0} & 0 & 0 & \mu \\
\hline
\mathbf{C}_1 & \mathbf{0} & \xi & \mathbf{D}_1 & \mathbf{D}_2
\end{array} \right]
$$
where \(W(s) \triangleq \left[ \begin{array}{c|c} \mathbf{A}_w & \mathbf{B}_w \\ \hline \mathbf{C}_w & 0 \end{array} \right] = \left[ \begin{array}{c|c} -\omega_c & \omega_c \\ \hline 1 & 0 \end{array} \right]\).
The H∞ control synthesis problem is then: Find a stabilizing controller \(C(s)\) that minimizes the H∞ norm of the closed-loop transfer function \(T_{\tilde{z}\tilde{w}}\) from \(\tilde{w}\) to \(\tilde{z}\). This minimization is performed numerically using standard software tools (e.g., the `hinfsyn` function in MATLAB). A key stability criterion for the final repetitive control system is that the transfer function \(T_{ba}(s)\) from point \(a\) to point \(b\) in the opened loop must have an H∞ norm less than 1: \(\| T_{ba} \|_\infty < 1\).
Controller Synthesis and Reduction
The design process is applied to a solar inverter system with the parameters listed in Table 1.
| Parameter | Symbol | Value |
|---|---|---|
| Inverter-side Inductor | \(L_1\) | 0.74 mH |
| Grid-side Inductor | \(L_2\) | 0.055 mH |
| Filter Capacitor | \(C\) | 6.6 μF |
| Damping Resistor | \(R_d\) | 0.5 Ω |
| Low-pass Filter Cut-off Freq. | \(\omega_c\) | 6283 rad/s |
| Repetitive Control Delay | \(\tau\) | 0.0198 s |
| Switching Frequency | \(f_{sw}\) | 10 kHz |
| Grid Voltage (RMS) | \(U_g\) | 220 V |
With weighting coefficients chosen as \(\xi=12\) and \(\mu=0.1\), the H∞ synthesis yields a 5th-order compensator \(C(s)\). For practical digital implementation, a reduced-order model is desirable. Noting that the dynamics of one very fast pole can be approximated as a constant gain, and after canceling near pole-zero pairs, a simple yet effective first-order compensator is obtained:
$$
C'(s) = \frac{4.984s}{s + 6283}
$$
The Bode plots of the original and reduced controllers show nearly identical frequency responses in the low-to-mid frequency range, which is critical for harmonic control. Furthermore, the H∞ norm of \(T_{ba}(s)\) using the reduced controller computes to approximately 0.52, which satisfies the stability condition \(\| T_{ba} \|_\infty < 1\), confirming the robustness of the design.
Simulation Analysis and Performance Verification
The performance of the solar inverter with the designed H∞ repetitive controller is validated through detailed time-domain simulations. The following key operational scenarios are examined.
1. Operation Under Distorted Grid Voltage
A critical requirement for a grid-following solar inverter is to maintain high-quality current injection even when the point of common coupling (PCC) voltage is distorted. This tests the controller’s harmonic rejection capability. The grid voltage is intentionally distorted with significant 5th, 7th, and 11th harmonic components, resulting in a Total Harmonic Distortion (THD) of 3.64%. The solar inverter is commanded to inject active power corresponding to 18.7 kW.
The results, summarized in Table 2, demonstrate the controller’s effectiveness. Despite the severely distorted grid voltage, the solar inverter output current maintains a low THD of 2.0%, well within typical grid codes (e.g., below 5%). The fundamental current magnitude tracks its reference (40 A peak) with high accuracy at 39.59 A peak, proving that the power injection function remains precise.
| Metric | Grid Voltage | Solar Inverter Current |
|---|---|---|
| THD | 3.64% | 2.00% |
| Fundamental (Peak) | 311.1 V | 39.59 A (Ref: 40 A) |
| Key Harmonics | 5th, 7th, 11th prominent | All low-order harmonics suppressed |
2. Multifunctional Operation: Harmonic Compensation
This scenario validates the core multifunctional capability. A local non-linear load draws a distorted current. Initially, the solar inverter only injects active power. At \(t = 0.5 \,s\), the harmonic compensation mode is activated. The harmonic content of the load current is detected and added to the solar inverter’s current reference. The inverter must now track this composite reference containing both fundamental active current and harmonic components.
The dynamic response is excellent. Within one fundamental cycle after mode switching, the solar inverter’s output current successfully incorporates the required harmonic components. Crucially, as shown in Table 3, the active power transfer is unaffected—the fundamental component of the inverter current remains stable. The primary benefit is seen in the grid current: its THD is reduced from 3.71% (without compensation) to 1.96% (with compensation). A frequency spectrum analysis confirms substantial reduction in harmonics up to the 19th order. The performance for very high-order harmonics (above 20th) is limited, as expected, due to the finite bandwidth of the repetitive controller and the low-pass filter \(W(s)\).
| Condition | Solar Inverter Current THD | Grid Current THD | Fundamental Current Tracking |
|---|---|---|---|
| Power Injection Only | < 2.0% | 3.71% | Accurate |
| Power Injection + Harmonic Compensation | Increased (contains harmonics) | 1.96% | Accurate (Unaffected) |
The controller’s robustness is further evidenced by its consistent performance despite the changing dynamics introduced by the harmonic reference signals. The H∞ design ensures stability and performance are maintained across this operational shift.
Conclusion
This article has detailed the complete design and analysis of a robust current control strategy for multifunctional solar inverters. By formulating the compensator design for a repetitive control scheme within an H∞ framework, we achieve a system with dual strengths: exceptional steady-state accuracy for tracking periodic signals (fundamental and harmonics) and inherent robustness against model uncertainties and grid disturbances. The systematic design procedure yields a relatively simple, reducible controller suitable for practical implementation.
The simulation studies confirm that the solar inverter equipped with this controller excels in its primary role of high-fidelity active power injection, even under adverse grid voltage conditions. More importantly, it successfully demonstrates the ancillary function of local harmonic current compensation, significantly improving the quality of current drawn from the grid. This multifunctionality enhances the utilization and value proposition of PV systems, allowing them to contribute actively to power quality management in distributed networks. The proposed H∞-based repetitive control presents a powerful and systematic approach for developing the next generation of intelligent and grid-supportive solar inverters.