The proliferation of renewable energy sources and distributed generation has placed grid-tied inverters at the forefront of modern power systems. These critical power electronic interfaces are responsible for converting direct current (DC) from sources like photovoltaic panels or batteries into high-quality alternating current (AC) synchronized with the utility grid. However, the inherent switching operation of a grid tied inverter introduces significant harmonic distortion into the output current. This distortion primarily manifests as high-frequency switching harmonics, generated by the pulse-width modulation (PWM) process, and low-frequency harmonics, often induced by grid voltage distortions, nonlinear loads, or control imperfections. Excessive harmonics degrade power quality, increase losses, cause equipment overheating, and can lead to protective device malfunctions or even system instability. Therefore, developing effective harmonic suppression strategies is paramount for the reliable and efficient operation of any grid tied inverter system. This study presents a comprehensive methodology addressing both high and low-frequency harmonic challenges through an optimized LCL filter design and a sophisticated composite current control strategy.
For high-frequency harmonic attenuation, the LCL filter is universally recognized as superior to simple L or LC filters due to its steeper high-frequency roll-off characteristic, which allows for smaller inductance values and reduced size and cost. The basic topology of a single-phase full-bridge grid tied inverter with an LCL filter is shown below. The inverter bridge, driven by sinusoidal PWM (SPWM), produces a voltage \(v_{inv}\) rich in switching harmonics. The LCL filter, comprising inverter-side inductor \(L_1\), filter capacitor \(C\), and grid-side inductor \(L_2\), attenuates these harmonics before the current \(i_2\) is injected into the grid voltage \(v_g\).
However, the LCL filter’s third-order nature introduces a resonant peak that can cause system instability. This necessitates a systematic and optimized parameter design process. The design is governed by multiple constraints derived from system specifications and performance requirements. First, the total inductance \(L_{total} = L_1 + L_2\) is constrained. Its upper limit is determined by the allowable voltage drop across the inductors at the fundamental frequency to maintain a sufficient modulation index without requiring excessive DC-link voltage. A common design rule limits this drop to 5% of the grid voltage:
$$ \omega L_{total} I_L \leq 0.05 V_g \Rightarrow L_{total} \leq \frac{0.05 V_g}{\omega I_L} $$
Conversely, the lower limit of \(L_{total}\) is set by the maximum allowable current ripple \(\Delta I_{L,max}\) through the inverter-side inductor. For a full-bridge inverter with bipolar SPWM, the worst-case ripple occurs at zero modulation index. The constraint is given by:
$$ L_{total} \geq \frac{V_{dc} T_s}{2 \lambda_L I_L} $$
where \( \lambda_L = \Delta I_{L,max} / I_L \) is the current ripple ratio, typically chosen between 30% and 50%. Furthermore, the ratio \( L_1 / L_2 \) impacts the overall filter volume and performance. A ratio between 3 and 7 is often selected, as the grid-side impedance (from cables, transformers, etc.) can partially compensate for a smaller \(L_2\). The filter capacitor \(C\) is primarily constrained by the maximum permissible reactive power it generates at the fundamental frequency, usually limited to about 5% of the rated output power:
$$ C \leq \frac{\lambda_C P_o}{\omega V_g^2} $$
The resonant peak of the LCL filter’s transfer function \(G_{LCL}(s) = i_2(s)/v_{inv}(s)\) is a major concern:
$$ G_{LCL}(s) = \frac{1}{s L_1 L_2 C (s^2 + \omega_r^2)} $$
$$ \omega_r = \sqrt{\frac{L_1 + L_2}{L_1 L_2 C}} $$
To damp this resonance passively and robustly, a resistor \(R_c\) is placed in series with the filter capacitor. This modifies the transfer function to:
$$ G_{LCL\_R}(s) = \frac{C R_c s + 1}{s^3 L_1 L_2 C + s^2 L_1 L_2 R_c C + s(L_1+L_2)} $$
The value of \(R_c\) is chosen to introduce a desired damping ratio \(\zeta\) (e.g., \(\zeta=0.28\) for a critically flat magnitude response) to the resonant poles:
$$ R_c = \frac{2 \zeta \sqrt{L_1 L_2 C (L_1+L_2)}}{C (L_1+L_2)} \approx \frac{0.56 \sqrt{L_1 L_2 C}}{C \sqrt{L_1+L_2}} $$
While effective for damping, \(R_c\) incurs power losses due to the circulation of harmonic currents. Therefore, a final optimization step is performed. Within the bounded parameter space defined by the constraints above, the set \(\{L_1, L_2, C\}\) that minimizes the total power loss \(P_{loss}\) in the damping branch is sought. The loss comprises fundamental and switching frequency components:
$$ P_{loss} = \frac{R_c (\omega C V_g)^2}{1+(\omega C R_c)^2} + \frac{R_c (\omega_s C V_{inv} L_2)^2}{(L_1 L_2 (\omega_s^2 – \omega_r^2))^2 + (\omega_s L_1 L_2 C R_c)^2} $$
An enumeration search within the feasible region yields the optimal parameters. For a 1 kW, 220V/50Hz grid tied inverter with a 20 kHz switching frequency and a 400V DC link, the optimal parameters and their constraints are summarized below.
| Parameter | Symbol | Constraint / Optimal Value |
|---|---|---|
| Total Inductance | \(L_1 + L_2\) | \(3.6 \, \text{mH} \leq L_{total} \leq 7.7 \, \text{mH}\) |
| Inductor Ratio | \(L_1 / L_2\) | \(3 \leq L_1/L_2 \leq 7\) |
| Filter Capacitor | \(C\) | \(C \leq 3.29 \, \mu\text{F}\) |
| Optimal Inverter-side Inductor | \(L_1\) | \(3.0 \, \text{mH}\) |
| Optimal Grid-side Inductor | \(L_2\) | \(1.0 \, \text{mH}\) |
| Optimal Filter Capacitor | \(C\) | \(2.2 \, \mu\text{F}\) |
| Damping Resistor | \(R_c\) | \(10.0 \, \Omega\) |
The optimized LCL filter effectively attenuates switching harmonics. Spectral analysis shows the dominant switching harmonic at 20 kHz in the inverter-side current \(i_1\) is attenuated from approximately 14% of the fundamental to about 1.2% in the grid current \(i_2\), demonstrating excellent high-frequency suppression.
While the LCL filter handles high-frequency noise, low-frequency harmonics (e.g., 3rd, 5th, 7th) caused by grid background distortion and nonlinearities require a control-based solution. The primary control objective for a grid tied inverter is to force the output current \(i_2\) to track a sinusoidal reference \(i_2^*\) that is phase-locked to the grid voltage \(v_g\). A simple proportional-integral (PI) controller in the synchronous reference frame is commonplace but suffers from steady-state error and poor disturbance rejection for AC signals. Proportional-Resonant (PR) control offers an elegant solution in the stationary frame by providing theoretically infinite gain at a specific resonant frequency \(\omega_n\). A practical non-ideal PR controller transfer function is:
$$ G_{PR}(s) = K_p + \frac{2 K_i \omega_c s}{s^2 + 2\omega_c s + \omega_n^2} $$
where \(K_p\) is the proportional gain, \(K_i\) is the resonant gain, and \(\omega_c\) is the cutoff frequency that broadens the resonant peak for robustness against frequency variations. The PR controller can be tuned to provide high gain at the fundamental frequency (e.g., 50 Hz) and selected low-order harmonic frequencies (150 Hz, 250 Hz, etc.). However, implementing multiple resonant blocks for numerous harmonics increases computational burden and complexity.
Repetitive Control (RC) provides an alternative and powerful approach. Inspired by the internal model principle, a repetitive controller embeds a model of periodic signals into the loop, enabling zero steady-state error for all periodic disturbances with a period equal to the fundamental cycle. The basic digital RC law in the z-domain is:
$$ G_{RC}(z) = \frac{k_r z^{-N} Q(z) S(z)}{1 – z^{-N} Q(z)} $$
where \(N = f_s / f_1\) is the number of samples per fundamental period, \(k_r\) is a gain less than 1 for stability, \(Q(z)\) is typically a constant or low-pass filter slightly less than 1 to enhance stability margins, and \(S(z)\) is a compensator (often a lead filter like \(z^k\)) designed to phase-correct the plant within the bandwidth of interest. While RC offers exceptional harmonic rejection across a wide spectrum, its inherent one-period delay results in slow transient response to sudden changes or disturbances.
To harness the strengths of both methods—fast dynamic response from PR control and superior steady-state harmonic rejection from RC—a composite control strategy is adopted. The two controllers are connected in parallel. The current error is processed simultaneously by both. The PR controller reacts immediately to disturbances, providing rapid transient correction. The repetitive controller, after a one-cycle learning period, applies precise corrective actions to eliminate periodic steady-state errors. The combined control law is:
$$ G_i(z) = G_{PR}(z) + G_{RC}(z) $$
The design of the composite controller involves careful parameter selection. The PR controller parameters (\(K_p, K_i, \omega_c\)) are designed based on the desired bandwidth and phase margin, considering the plant model which includes the LCL filter and PWM delay. The RC parameters are designed as follows: \(N=400\) for a 20 kHz sampling/switch frequency and 50 Hz grid, \(k_r\) is set close to 1 (e.g., 0.95) for fast convergence, \(Q(z)=0.95\) for stability, and \(S(z)\) is designed as a phase-lead compensator, often a low-order filter like \(z^3\) or a more sophisticated filter to match the plant phase around the crossover frequency.
| Control Parameter | Symbol | Designed Value / Purpose |
|---|---|---|
| PR Proportional Gain | \(K_p\) | 48.78 (Set for desired crossover frequency) |
| PR Resonant Gain | \(K_i\) | Tuned for harmonic rejection at 50, 150, 250 Hz… |
| Repetitive Control Gain | \(k_r\) | 0.95 (High for fast learning, <1 for stability) |
| Delay Samples | \(N\) | 400 (fs=20kHz / f1=50Hz) |
| Low-Pass / Constant | \(Q(z)\) | 0.95 (Improves stability margin) |
| Phase Compensator | \(S(z)\) | \(z^3\) or a 2nd-order Butterworth filter (Phase lead) |
The performance of the proposed harmonic suppression strategy—combining the optimized LCL filter and the PR+RC composite controller—is validated through detailed simulation of the grid tied inverter system. The system parameters used in the simulation are consistent with the design example provided earlier.
The effectiveness of the LCL filter is first confirmed. The inverter-side current \(i_1\) shows significant switching ripple at the 20 kHz component. After filtering, the grid-injected current \(i_2\) is a clean sinusoid with the switching noise attenuated by over 40 dB. The current ripple ratio aligns with the design target, and the power factor of the injected current is nearly unity (e.g., 0.997).
The robustness and harmonic rejection capability of the composite controller are tested under distorted grid voltage conditions. According to standards like IEEE 519 or GB/T 14549, grid voltage total harmonic distortion (THD) can be up to 5% at the point of common coupling. To simulate this, the grid voltage source is programmed with a series of low-order odd harmonics (3rd, 5th, 7th, 9th, 11th, 13th, 15th) with varying magnitudes, creating a voltage THD of approximately 3-4%. The controller’s reference current is generated from a Phase-Locked Loop (PLL) synchronized to this distorted voltage.
The key result is that despite the distorted grid voltage, the output current waveform closely tracks the sinusoidal reference with minimal distortion. The current THD is consistently maintained around 2.0-2.1%, well below the typical limit of 5% for grid-connected current. This demonstrates the composite controller’s exceptional ability to reject grid-voltage-induced harmonics. The PR controller quickly compensates for deviations, while the repetitive controller learns and cancels the periodic error patterns caused by the harmonic distortion.
The dynamic performance is evaluated by subjecting the system to a step change in the current reference magnitude, simulating a sudden change in commanded power output (e.g., from full load to half load). The grid current rapidly settles to the new reference value. Metrics such as overshoot (approximately 36.7%) and settling time (approximately 2 ms) confirm that the system maintains a good balance between fast dynamic response and steady-state precision, a direct benefit of the parallel PR+RC structure. The PR controller dominates the initial transient, preventing the sluggish response that a standalone repetitive controller would exhibit.
In conclusion, this study presents a holistic and effective approach to harmonic suppression for LCL-type grid tied inverter systems. The method addresses the full spectrum of harmonic challenges: high-frequency switching noise is mitigated through an LCL filter whose parameters are systematically optimized considering performance constraints and damping branch losses; low-frequency harmonics induced by the grid and system nonlinearities are suppressed by a sophisticated composite current controller combining Proportional-Resonant and Repetitive Control strategies. The PR controller ensures fast dynamic response, while the RC controller guarantees excellent steady-state tracking and periodic disturbance rejection. Simulation results validate the design, showing that the output current meets stringent harmonic standards (THD ~2%) even under significantly distorted grid voltage conditions, while maintaining robust dynamic performance. This combined filter-and-control methodology provides a reliable and high-performance solution for modern renewable energy integration and power quality-sensitive applications using grid tied inverter technology.