The increasing penetration of distributed renewable energy sources, primarily photovoltaic (PV) and wind, presents both opportunities and challenges for modern power systems. A significant challenge arises from the widespread presence of non-linear loads in medium- and low-voltage distribution networks. These loads inject harmonic currents into the grid, which, when flowing through line impedances, cause harmonic voltage drops and lead to a degradation of power quality at the point of common coupling. Simultaneously, the inherent intermittency and variability of solar irradiance mean that PV inverters often operate below their rated capacity. This underutilization presents an opportunity to leverage their existing hardware for ancillary services. Given the topological similarity between PV inverters and active power filters, there is growing interest in developing control strategies that enable grid-tied solar inverters to perform multifunctional roles—specifically, transmitting active power while concurrently mitigating local harmonic pollution.
Traditional current control schemes for solar inverters, such as Proportional-Integral (PI) control in the synchronous rotating (dq) frame, perform well under ideal grid conditions. However, their performance deteriorates significantly when the grid voltage is distorted, leading to distorted inverter output currents. Proportional-Resonant (PR) controllers offer an alternative by providing theoretically infinite gain at selected harmonic frequencies. While cascading multiple PR controllers can improve harmonic rejection, this approach complicates the control system design due to the numerous parameters that require careful tuning.
Repetitive Control (RC), founded on the Internal Model Principle, provides an elegant solution for the zero steady-state error tracking of periodic signals. Its structure inherently contains models of all periodic disturbances and references with a fundamental period, making it exceptionally suitable for tracking sinusoidal references and rejecting harmonic disturbances present in the grid current of a solar inverter. However, designing a robust repetitive controller that maintains stability and performance in the face of system parameter variations (like LCL filter perturbations) and grid impedance changes is non-trivial.
This article presents a systematic design methodology for a robust current controller for a multifunctional PV inverter. The core of the method is the application of H∞ control theory to design the stabilizing compensator within a repetitive control framework. The design guarantees robustness against parameter uncertainties while achieving high-precision tracking of fundamental and low-order harmonic current references. We begin by deriving a state-space model of a single-phase LCL-filter-based inverter in the stationary (αβ) frame. Subsequently, we formulate the repetitive controller design as a standard H∞ optimization problem. The solution yields a stabilizing compensator that ensures system stability and meets desired robustness criteria. Finally, the effectiveness of the proposed H∞-based repetitive controller is validated through detailed time-domain simulations, demonstrating its capability for simultaneous active power injection and harmonic current compensation.
1. System Modeling and Control Structure
1.1 State-Space Model of the LCL Filter
Consider a three-phase three-wire grid-tied PV system with an LCL filter. Under the assumption of symmetrical three-phase parameters, the system can be equivalently analyzed in the stationary αβ-frame, where it decomposes into two identical, decoupled single-phase systems. The single-phase equivalent plant P for controller design is shown below.
The state variables are chosen as the inverter-side inductor current \( i_1 \), the grid-side inductor current \( i_2 \), and the capacitor voltage \( u_c \). The control input is the inverter bridge output voltage \( u \). The external inputs are the grid voltage \( u_g \) and the reference current \( i_{ref} \). The controlled output is the current tracking error \( e = i_{ref} – i_2 \). Applying Kirchhoff’s laws, the continuous-time state-space representation of the plant is:
$$
\begin{aligned}
\dot{\mathbf{x}} &= \mathbf{A}\mathbf{x} + \mathbf{B}_1\mathbf{w} + \mathbf{B}_2 u \\
y &= e = \mathbf{C}_1\mathbf{x} + \mathbf{D}_1\mathbf{w} + \mathbf{D}_2 u
\end{aligned}
$$
where:
$$
\mathbf{x} = \begin{bmatrix} i_1 & i_2 & u_c \end{bmatrix}^T, \quad \mathbf{w} = \begin{bmatrix} u_g & i_{ref} \end{bmatrix}^T
$$
$$
\mathbf{A} = \begin{bmatrix}
-\frac{R_d}{L_1} & \frac{R_d}{L_1} & -\frac{1}{L_1} \\
\frac{R_d}{L_2} & -\frac{R_d}{L_2} & \frac{1}{L_2} \\
\frac{1}{C} & -\frac{1}{C} & 0
\end{bmatrix}, \quad
\mathbf{B}_1 = \begin{bmatrix}
1/L_1 & 0 \\
-1/L_2 & 0 \\
0 & 0
\end{bmatrix}, \quad
\mathbf{B}_2 = \begin{bmatrix}
1/L_1 \\ 0 \\ 0
\end{bmatrix}
$$
$$
\mathbf{C}_1 = \begin{bmatrix} 0 & -1 & 0 \end{bmatrix}, \quad \mathbf{D}_1 = \begin{bmatrix} 0 & 1 \end{bmatrix}, \quad \mathbf{D}_2 = [0]
$$
The transfer function representation of the plant P(s) is given by:
$$
P(s) = \begin{bmatrix} \mathbf{D}_1 & \mathbf{D}_2 \end{bmatrix} + \mathbf{C}_1(s\mathbf{I} – \mathbf{A})^{-1} \begin{bmatrix} \mathbf{B}_1 & \mathbf{B}_2 \end{bmatrix}
$$
For conciseness, we denote this as:
$$
P(s) \sim \left[ \begin{array}{c|cc}
\mathbf{A} & \mathbf{B}_1 & \mathbf{B}_2 \\
\hline
\mathbf{C}_1 & \mathbf{D}_1 & \mathbf{D}_2
\end{array} \right]
$$
1.2 Reference Current Generation for Multifunctional Operation
The core objective of a multifunctional solar inverter is to inject active power from the PV array while compensating for harmonics generated by local non-linear loads. According to Kirchhoff’s current law at the point of common coupling (PCC):
$$
\mathbf{i}_{abc} + \mathbf{i}_{L,abc} = \mathbf{i}_{g,abc}
$$
where \( \mathbf{i}_{abc} \) is the inverter output current, \( \mathbf{i}_{L,abc} \) is the local load current, and \( \mathbf{i}_{g,abc} \) is the net current injected into the grid. If the load current contains harmonic components \( \mathbf{i}_{L,h} \), and the inverter can accurately track a reference composed of the fundamental active current \( \mathbf{i}_{p}^* \) (from the PV MPPT algorithm) and the negative of the load harmonics \( -\mathbf{i}_{L,h} \), then the grid current becomes harmonic-free:
$$
\mathbf{i}_{g} \approx \mathbf{i}_{p}^*
$$
The reference current for the inverter’s inner loop controller is thus:
$$
\mathbf{i}_{ref} = \mathbf{i}_{p}^* – \mathbf{i}_{L,h}
$$
The load harmonic component \( \mathbf{i}_{L,h} \) can be extracted using various harmonic detection methods, such as those based on instantaneous power theory or synchronous reference frame filtering.
2. H∞ Robust Repetitive Controller Design
2.1 Structure of the Repetitive Control Scheme
The proposed current control structure for the solar inverter integrates a repetitive control (RC) scheme within an H∞ robust design framework. The block diagram of the RC inner loop is shown below. The key elements are:
- Repetitive Internal Model: \( M(s) = \frac{1}{1 – W(s)e^{-\tau s}} \). This provides high gain at the fundamental frequency and its integer multiples, enabling perfect tracking/rejection of periodic signals.
- Low-Pass Filter \( W(s) \): Typically \( W(s) = \frac{\omega_c}{s + \omega_c} \). It enhances system robustness by attenuating the high-frequency gain of the RC, improving stability margins, especially at frequencies beyond its bandwidth.
- Delay \( e^{-\tau s} \): The delay time \( \tau \) is ideally equal to the fundamental period \( T \). A practical design rule to improve robustness against grid frequency variations is:
$$
\tau + \frac{1}{\omega_c} = T
$$
This relationship allows the gain at the fundamental frequency to remain high even if the grid frequency deviates slightly from its nominal value.
2.2 Formulation as a Standard H∞ Problem
To apply H∞ synthesis, we reformulate the design problem. We first “open” the positive feedback loop of the repetitive internal model. We then construct an augmented plant \( \tilde{P} \) that includes weighting functions to shape the performance and robustness objectives. The standard H∞ problem configuration is shown in the following figure.
Here, \( v \) is the output of the opened repetitive loop, and \( \xi \) is a weighting coefficient introduced to provide design flexibility for adjusting the trade-off between performance and robustness. The control input remains \( u \). The exogenous input vector is \( \tilde{w} = [v,\ w]^T = [v,\ u_g,\ i_{ref}]^T \). The generalized output vector \( \tilde{z} = [\tilde{z}_1,\ \tilde{z}_2]^T \) is constructed for optimization:
- \( \tilde{z}_1 = W(s)(e + \xi v) \): This channel penalizes the weighted tracking error, directly linking performance (small \( e \)) to the H∞ norm.
- \( \tilde{z}_2 = \mu u \): This channel penalizes the control effort, with \( \mu \) as a weighting coefficient. This helps to constrain controller gain and improve robustness.
The measurement output for the controller \( C(s) \) is \( \tilde{y} = e + \xi v \).
The state-space realization of the low-pass filter \( W(s) \) is:
$$
W(s) \sim \left[ \begin{array}{c|c}
A_w & B_w \\
\hline
C_w & 0
\end{array} \right] = \left[ \begin{array}{c|c}
-\omega_c & \omega_c \\
\hline
1 & 0
\end{array} \right]
$$
By combining the state-space models of the plant \( P(s) \) and the filter \( W(s) \) according to the interconnection structure, we derive the augmented generalized plant \( \tilde{P}(s) \):
$$
\tilde{P}(s) \sim
\left[ \begin{array}{ccc|cc}
\mathbf{A} & \mathbf{0} & \mathbf{0} & \mathbf{B}_1 & \mathbf{B}_2 \\
B_w\mathbf{C}_1 & A_w & B_w\xi & B_w\mathbf{D}_1 & B_w\mathbf{D}_2 \\
\hline
\mathbf{0} & C_w & 0 & 0 & 0 \\
\mathbf{0} & 0 & 0 & 0 & \mu \\
\hline
\mathbf{C}_1 & \mathbf{0} & \xi & \mathbf{D}_1 & \mathbf{D}_2
\end{array} \right]
$$
The corresponding partitions are:
$$
\tilde{P}(s) = \begin{bmatrix}
\tilde{P}_{11}(s) & \tilde{P}_{12}(s) \\
\tilde{P}_{21}(s) & \tilde{P}_{22}(s)
\end{bmatrix}
$$
where the input-output relationship is:
$$
\begin{bmatrix} \tilde{z} \\ \tilde{y} \end{bmatrix} = \tilde{P}(s) \begin{bmatrix} \tilde{w} \\ u \end{bmatrix}, \quad u = C(s) \tilde{y}
$$
The standard H∞ control problem is to find a stabilizing controller \( C(s) \) that minimizes the \( \mathcal{H}_\infty \) norm of the closed-loop transfer function \( T_{\tilde{z}\tilde{w}} \) from the exogenous inputs \( \tilde{w} \) to the regulated outputs \( \tilde{z} \):
$$
\min_{C(s) \text{ stabilizing}} \| T_{\tilde{z}\tilde{w}}(s) \|_\infty
$$
This minimization ensures that the worst-case energy gain from disturbances (\( u_g, i_{ref} \)) and the internal signal \( v \) to the weighted error and control effort is bounded, guaranteeing robust performance and stability.
2.3 Stability Criterion for the Repetitive Control System
After obtaining the stabilizing compensator \( C(s) \) from the H∞ synthesis, we must verify the stability of the final repetitive control system with the internal model closed. A sufficient condition for exponential stability of the system in Fig. 3 is that the \( \mathcal{H}_\infty \) norm of the transfer function from point \( a \) to point \( b \) (with external input \( w=0 \)) is less than 1:
$$
\| T_{ba}(s) \|_\infty < 1
$$
where \( T_{ba}(s) \) is given by:
$$
T_{ba}(s) = \left[ 1 – P_{22}(s)C(s) \right]^{-1} W(s)
$$
and \( P_{22}(s) = \mathbf{C}_1(s\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}_2 + \mathbf{D}_2 \). Given a state-space realization of \( C(s) \), \( T_{ba}(s) \) can be computed, and its \( \mathcal{H}_\infty \) norm can be evaluated directly to confirm the closed-loop repetitive system’s stability.
3. Controller Synthesis and Reduction
We now demonstrate the design process with a specific set of parameters for the LCL filter and the solar inverter system.
| Parameter | Symbol | Value |
|---|---|---|
| Inverter-side Inductance | \( L_1 \) | 0.74 mH |
| Grid-side Inductance | \( L_2 \) | 0.055 mH |
| Filter Capacitance | \( C \) | 6.6 µF |
| Passive Damping Resistor | \( R_d \) | 0.5 Ω |
| Low-Pass Filter Cut-off Freq. | \( \omega_c \) | 6283 rad/s (1 kHz) |
| Repetitive Control Delay | \( \tau \) | 0.0198 s |
| Grid Fundamental Frequency | \( f \) | 50 Hz |
| Switching Frequency | \( f_{sw} \) | 10 kHz |
Selecting the weighting coefficients as \( \xi = 12 \) and \( \mu = 0.1 \), the generalized plant \( \tilde{P}(s) \) is constructed using Eq. (14). Employing the MATLAB Robust Control Toolbox function `hinfsyn`, a stabilizing H∞ controller \( C(s) \) is computed. The resulting controller is of relatively high order:
$$
C(s) = \frac{9.2756 \times 10^6 s (s^2 + 9758 s + 2.959 \times 10^9)}{(s + 1.861 \times 10^6)(s + 6283)(s^2 + 1.657 \times 10^4 s + 2.936 \times 10^9)}
$$
For practical digital implementation, a reduced-order approximation is desirable. Noting that the pole at \( s \approx -1.861 \times 10^6 \) is far into the high-frequency region, its contribution in the low- and mid-frequency range (relevant for current control) is negligible. Furthermore, the complex zero/pole pairs approximately cancel each other. After simplification, a first-order compensator is obtained:
$$
C'(s) = \frac{4.984 s}{s + 6283}
$$
The frequency responses of the full-order \( C(s) \) and reduced-order \( C'(s) \) compensators are compared. The Bode plots reveal that their magnitude and phase characteristics are nearly identical in the frequency range of interest (below 2 kHz), confirming the validity of the reduction. The \( \mathcal{H}_\infty \) norm of the stability transfer function \( T_{ba}(s) \) is calculated for both controllers. The results are summarized below:
| Controller | Order | \(\| T_{ba}(s) \|_\infty\) | Stability Condition (\(\|T_{ba}\|_\infty < 1\))? |
|---|---|---|---|
| Full-order \( C(s) \) | 5th | 0.49853 | Yes |
| Reduced-order \( C'(s) \) | 1st | 0.52055 | Yes |
Both controllers satisfy the stability criterion \( \| T_{ba} \|_\infty < 1 \), guaranteeing the exponential stability of the overall repetitive control system. The slight increase in the norm for the reduced-order controller is acceptable, as it remains well below 1.
4. Simulation Analysis and Verification
The proposed control strategy is verified using a detailed switching model of the three-phase solar inverter in a simulation environment. The system parameters are as listed in Table 1. The DC-link voltage is assumed to be well-regulated, and the focus is on the performance of the AC current control loop.
4.1 Robustness to Distorted Grid Voltage
This test evaluates the controller’s ability to maintain high-quality current injection even when the grid voltage is significantly distorted—a critical requirement for any practical solar inverter. A distorted grid voltage is simulated with a Total Harmonic Distortion (THD) of approximately 3.64%, containing significant 5th, 7th, 11th, and 13th harmonic components. The PV inverter is commanded to inject 18.664 kW of active power (corresponding to a fundamental current amplitude of about 40 A).
The simulation results demonstrate the effectiveness of the H∞ repetitive controller. The inverter output current waveform remains nearly sinusoidal despite the highly distorted grid voltage. Analysis shows the grid current THD is only about 2.0%, which complies with stringent grid codes (e.g., IEEE 1547). The fundamental current amplitude is tracked accurately at 39.59 A, confirming excellent steady-state tracking precision. This performance highlights the controller’s inherent harmonic disturbance rejection capability, a direct consequence of the repetitive control’s internal model and the robust design that maintains performance under non-ideal grid conditions.
4.2 Verification of Multifunctional Harmonic Compensation
This test validates the core multifunctional capability: the solar inverter providing active power while simultaneously compensating for harmonic currents from a local non-linear load. A three-phase diode bridge rectifier with an RL load is connected at the PCC. Initially, the inverter only injects active power. At t = 0.5 s, the harmonic compensation function is enabled by adding the detected load harmonic current \( i_{L,h} \) to the reference current \( i_{ref} \).
The dynamic and steady-state results are compelling. Upon enabling compensation, the inverter current quickly adapts within one fundamental cycle to include the harmonic components opposite to those of the load. The grid current, which is the sum of the load and inverter currents, becomes significantly cleaner. Quantitative analysis reveals that the grid current THD drops from 3.71% (without compensation) to 1.96% (with compensation). A harmonic spectrum analysis confirms a substantial reduction in the magnitudes of low-order harmonics (5th, 7th, 11th, 13th, etc.). The tracking error for the fundamental active current component remains minimal, proving that the harmonic compensation function does not interfere with the primary task of active power delivery. The slight increase in higher-order harmonics (>20th) in the grid current is attributed to the limited gain of the repetitive controller at very high frequencies and the switching noise from the inverter.
| Condition | Grid Current THD | Fundamental Current Tracking Error | Key Observation |
|---|---|---|---|
| Without Compensation (Inverter in pure active power mode) | 3.71% | < 1.5% | Grid current distortion reflects load harmonics. |
| With Harmonic Compensation Enabled | 1.96% | < 1.5% | Low-order harmonics (≤19th) are significantly suppressed. Active power delivery is maintained. |
5. Discussion and Comparative Analysis
The proposed H∞-based repetitive control offers distinct advantages for multifunctional solar inverter applications over conventional methods. The following table provides a qualitative comparison.
| Control Strategy | Design Complexity | Steady-State Accuracy | Harmonic Rejection | Robustness to Parameter Variation | Suitability for Multifunction |
|---|---|---|---|---|---|
| PI in dq-frame | Low | High (for DC only) | Poor under distorted grid | Moderate | Low |
| Multi-Resonant (PR) | High (tuning many resonant peaks) | High at selected freqs. | Good for specific harmonics | Low (sensitive to freq. shift) | Medium |
| Basic Repetitive Control | Medium | Very High (all harmonics) | Excellent | Low (needs careful stabilization) | High |
| Proposed H∞ Repetitive Control | Medium-High (systematic) | Very High | Excellent | High (explicitly designed for) | Very High |
Design Philosophy and Advantages:
The H∞ synthesis framework provides a systematic, mathematical approach to designing the stabilizing compensator \( C(s) \). Unlike ad-hoc tuning of lead-lag networks for a basic RC, the H∞ method automatically finds a controller that optimally balances performance (tracking error) and robustness (stability against model uncertainty, such as LCL filter parameter drifts or grid impedance variations) in a defined sense. The weighting coefficients \( \xi \) and \( \mu \) offer clear, intuitive knobs for this trade-off: increasing \( \xi \) emphasizes error reduction, while increasing \( \mu \) penalizes control effort and leads to a more robust, conservative controller.
Performance Characteristics:
The internal model of the repetitive controller ensures asymptotic rejection of all periodic disturbances with the fundamental period. This makes it inherently superior to PR control for compensating unknown harmonic spectra, as it does not require prior knowledge of the exact harmonic orders present. The inclusion of the low-pass filter \( W(s) \) and the robust compensator \( C(s) \) ensures that this high performance is maintained without sacrificing stability margins. The simulation results under distorted grid conditions (Section 4.1) are a direct testament to this robust performance.
Implementation Considerations for the Solar Inverter:
The reduced-order compensator \( C'(s) \) is a simple first-order high-pass filter, which is trivial to implement in a digital signal processor. The repetitive control algorithm itself requires a delay line of length corresponding to \( \tau \) (approximately one fundamental period in samples), which is a standard memory requirement for digital RC. The computational burden is therefore manageable for modern microcontrollers or DSPs used in solar inverter applications.
6. Conclusion
This article has presented a comprehensive methodology for designing a robust, multifunctional current controller for grid-connected solar inverters. The core contribution is the integration of repetitive control theory with H∞ robust synthesis techniques. By deriving a state-space model of the LCL-filtered inverter and formulating the repetitive controller design as a standard H∞ optimization problem, we obtained a stabilizing compensator that guarantees system stability and robustness against parameter variations.
The designed controller enables the PV inverter to perform dual functions seamlessly: (1) high-fidelity injection of active power from the PV array, and (2) effective compensation of low-order harmonic currents generated by local non-linear loads. Time-domain simulation studies confirmed the controller’s key features: high steady-state tracking accuracy, excellent rejection of grid voltage harmonics, and fast dynamic response when the harmonic compensation mode is activated. The control strategy successfully transforms the solar inverter from a simple power conversion unit into an active grid-supportive device, enhancing the utilization of its hardware and improving the power quality at the point of common coupling. This approach provides a viable and systematic solution for the next generation of smart, multifunctional photovoltaic energy conversion systems.
Future Work: Potential extensions of this work include experimental validation on a laboratory-scale prototype, extension to single-phase systems or unbalanced grid conditions, and the integration of this current control loop with an outer loop responsible for DC-link voltage regulation and maximum power point tracking (MPPT) under multifunctional operation.