In recent years, the global scale of photovoltaic (PV) power generation has grown rapidly, with China contributing the largest increment, driving sustainable development in the energy and power sector and accelerating the adoption of renewable energy. The utility interactive inverter, also known as the grid-connected inverter, plays a crucial role in facilitating energy transfer between PV systems and the grid, ensuring the delivery of high-quality electrical power. As a key component, the utility interactive inverter must maintain output stability while mitigating harmonics introduced by switching operations. To address this, filters are essential, and among various types, the LCL filter offers superior harmonic suppression compared to L-type and LC-type filters, making it widely used in utility interactive inverter applications. However, the LCL filter is a third-order system prone to high-frequency resonance peaks and stability issues, necessitating damping methods for suppression. This paper explores the design and simulation of an LCL-type utility interactive inverter for PV systems, employing a dual-loop control strategy and passive damping via capacitor-series resistance to enhance performance. We will analyze the LCL filter characteristics, detail the control methodology, and present simulation results using MATLAB/Simulink, demonstrating the inverter’s ability to achieve unit power factor grid connection with robust stability.
The LCL filter topology, as shown in Figure 1, consists of inverter-side inductance \(L_1\), grid-side inductance \(L_2\), and a capacitor branch \(C\), with damping resistor \(R_d\) added in series with the capacitor to suppress resonance. This configuration effectively filters high-frequency harmonics generated by the utility interactive inverter’s switching actions. The transfer function relating the grid current \(i_2(s)\) to the inverter output voltage \(U_i(s)\) is derived from the circuit equations:
$$ G_{\text{LCL}}(s) = \frac{i_2(s)}{U_i(s)} = \frac{1}{L_1 C L_2 s^3 + (L_1 + L_2)s} = \frac{1}{s L_1 L_2 C} \cdot \frac{1}{s^2 + \omega_{\text{res}}^2} $$
where the resonant angular frequency \(\omega_{\text{res}}\) is given by:
$$ \omega_{\text{res}} = \sqrt{\frac{L_1 + L_2}{L_1 L_2 C}} $$
To understand the filter’s behavior, we analyze its frequency response. The LCL filter exhibits a resonance peak at \(\omega_{\text{res}}\), which can destabilize the utility interactive inverter system if not mitigated. Introducing damping modifies the transfer function to:
$$ G_{\text{LCL,damped}}(s) = \frac{1}{s L_1 L_2 C} \cdot \frac{1}{s^2 + 2\lambda \omega_{\text{res}} s + \omega_{\text{res}}^2} $$
where \(\lambda\) is the damping ratio. This addition suppresses the peak while preserving favorable low-frequency and high-frequency attenuation. Among passive damping methods, capacitor-series resistance is highly effective, as it directly adds a resistive term to the denominator. The revised transfer function with resistor \(R_d\) is:
$$ G_{\text{LCL,Rd}}(s) = \frac{i_2(s)}{V_r(s)} = \frac{s C R_d + 1}{L_1 C L_2 s^3 + C(L_1 + L_2) R_d s^2 + (L_1 + L_2)s} $$
This approach ensures stability by reducing the resonance peak below 0 dB, as illustrated in frequency response plots. To quantify the damping effect, we can compare different resistor values, as summarized in Table 1, which shows the impact on resonance suppression and phase margin for a utility interactive inverter.
| Resistor \(R_d\) (Ω) | Resonance Peak Magnitude (dB) | Phase Margin (°) | Stability Improvement |
|---|---|---|---|
| 0 | +20 | -180 | Unstable |
| 1 | +5 | -135 | Marginal |
| 3.73 | -10 | -90 | Stable |
| 10 | -25 | -45 | Over-damped |
In practice, selecting \(R_d\) around 5% of the capacitor’s reactance at resonance optimally balances damping and power loss. This design choice is critical for the utility interactive inverter to maintain efficiency while avoiding oscillations.
The control strategy for the LCL-type utility interactive inverter involves a dual-loop approach: an outer power loop (PQ control) and an inner current loop (PI control). This ensures precise regulation of active and reactive power injected into the grid. The system topology, depicted in Figure 2, includes a PV array represented by DC voltage \(U_{dc}\), DC-link capacitor \(C_{dc}\), a three-phase inverter with switches \(S_1\) to \(S_6\), LCL filter with components \(L_1\), \(L_2\), \(C\), and damping resistor \(R_d\), and the grid voltage \(U_g\). The utility interactive inverter’s output currents \(i_{1k}\) and grid currents \(i_{2k}\) (where \(k = a, b, c\)) are monitored for feedback control.
The mathematical model of the utility interactive inverter in the abc-frame is given by:
$$ \begin{aligned} L_1 \frac{di_{1k}}{dt} + u_{ck} &= u_{ik} \\ C \frac{du_{ck}}{dt} &= i_{1k} – i_{2k} \\ L_2 \frac{di_{2k}}{dt} + u_{gk} &= u_{ck} \end{aligned} $$
where \(u_{ik}\) is the inverter output voltage, \(u_{ck}\) is the capacitor voltage, and \(u_{gk}\) is the grid voltage. Transforming to the dq-frame simplifies control by converting AC quantities to DC references. Using Park transformation with angle \(\theta\) from a phase-locked loop (PLL), the power equations become:
$$ \begin{aligned} P &= \frac{3}{2} (u_{gd} i_d + u_{gq} i_q) \\ Q &= \frac{3}{2} (u_{gd} i_q – u_{gq} i_d) \end{aligned} $$
Aligning the d-axis with the grid voltage vector sets \(u_{gq} = 0\) and \(u_{gd} = |U_{g,m}|\), where \(U_{g,m}\) is the grid voltage amplitude. Thus, active and reactive power are directly proportional to \(i_d\) and \(i_q\), respectively:
$$ P = \frac{3}{2} u_{gd} i_d, \quad Q = \frac{3}{2} u_{gd} i_q $$
This allows independent control of P and Q via current references \(i_{d,\text{ref}}\) and \(i_{q,\text{ref}}\). The outer PQ loop generates these references based on desired power setpoints \(P_{\text{ref}}\) and \(Q_{\text{ref}}\). The inner current loop employs PI controllers to regulate \(i_d\) and \(i_q\). Neglecting the capacitor for simplicity, the dq-model of the utility interactive inverter is:
$$ \begin{aligned} L \frac{di_{gd}}{dt} &= u_d – u_{gd} + \omega L i_{gq} \\ L \frac{di_{gq}}{dt} &= u_q – u_{gq} – \omega L i_{gd} \end{aligned} $$
where \(L = L_1 + L_2\) is the total inductance, and \(\omega\) is the grid angular frequency. To achieve decoupled control, we introduce cross-coupling and feedforward terms. The PI controllers output voltage references:
$$ \begin{aligned} \Delta u_d &= \left(K_{ip} + \frac{K_{ii}}{s}\right) (i_{d,\text{ref}} – i_d) \\ \Delta u_q &= \left(K_{ip} + \frac{K_{ii}}{s}\right) (i_{q,\text{ref}} – i_q) \end{aligned} $$
The final control voltages for the utility interactive inverter are:
$$ \begin{aligned} u_d &= \Delta u_d – \omega L i_q + u_{gd} \\ u_q &= \Delta u_q + \omega L i_d + u_{gq} \end{aligned} $$
This ensures independent d and q-axis current control, enabling the utility interactive inverter to track power commands accurately. The control structure integrates these loops, as shown in Figure 3, with SVPWM modulation generating gate signals for the inverter switches.
To validate the design, we developed a simulation model in MATLAB/Simulink. The parameters, listed in Table 2, reflect a typical utility interactive inverter setup for PV applications. The simulation evaluates performance under various conditions, focusing on stability, harmonic distortion, and power factor.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Grid Voltage | \(U_g\) | 380 | V (line-to-line) |
| DC Link Voltage | \(U_{dc}\) | 800 | V |
| Grid Frequency | \(f\) | 50 | Hz |
| Inverter-side Inductance | \(L_1\) | 3.2 | mH |
| Grid-side Inductance | \(L_2\) | 1.62 | mH |
| Filter Capacitance | \(C\) | 10.02 | μF |
| Damping Resistor | \(R_d\) | 3.73 | Ω |
| Switching Frequency | \(f_s\) | 10 | kHz |
| PI Controller (Kip, Kii) | — | 0.5, 100 | — |
First, we examine the system without damping (\(R_d = 0\)). The resonance at \(\omega_{\text{res}} \approx 2\pi \times 1250\) rad/s causes instability, as seen in the divergent grid current waveform (Figure 4). This underscores the necessity of damping in the utility interactive inverter. Next, with damping enabled and setpoints \(P_{\text{ref}} = 10\) kW and \(Q_{\text{ref}} = 0\), the utility interactive inverter achieves stable operation. The phase-A grid current and voltage are plotted in Figure 5, showing sinusoidal waveforms that are in phase, indicating unit power factor. The current THD is below 3%, meeting grid standards. The power response, shown in Figure 6, demonstrates fast tracking of references with minimal overshoot. Active power \(P\) settles at 10 kW within 0.05 s, while reactive power \(Q\) remains near zero, confirming effective PQ control for the utility interactive inverter.
Further analysis involves varying operating conditions to test robustness. Table 3 summarizes performance metrics under different power levels for the utility interactive inverter. The results highlight the system’s ability to maintain stability and power quality across a range of outputs.
| Power Setpoint \(P_{\text{ref}}\) (kW) | Current THD (%) | Power Factor | Settling Time (s) | Stability |
|---|---|---|---|---|
| 5 | 2.8 | 0.999 | 0.04 | Stable |
| 10 | 2.5 | 0.998 | 0.05 | Stable |
| 15 | 2.7 | 0.997 | 0.06 | Stable |
| 20 | 3.0 | 0.995 | 0.07 | Stable |
The dual-loop control strategy proves effective for the utility interactive inverter, offering strong anti-interference capabilities. Even with grid voltage disturbances (e.g., 10% sag), the utility interactive inverter maintains synchronized operation, as shown by the dynamic response in Figure 7. The current loop’s bandwidth, derived from PI tuning, ensures rapid correction of deviations. The closed-loop transfer function of the current control system can be expressed as:
$$ G_{\text{cl}}(s) = \frac{i_d(s)}{i_{d,\text{ref}}(s)} = \frac{K_{ip} s + K_{ii}}{L s^2 + (K_{ip} + R_d) s + K_{ii}} $$
where \(R_d\) represents the damping resistor’s effect. This second-order system provides adequate phase margin (> 45°) for stability. Additionally, the utility interactive inverter’s efficiency is evaluated by calculating losses in the damping resistor and switches. For \(R_d = 3.73 \Omega\), the power loss is approximately 0.5% of rated output, which is acceptable for most PV applications.
In conclusion, this study presents a comprehensive design and simulation of an LCL-type utility interactive inverter for photovoltaic power generation. By incorporating capacitor-series resistance for passive damping, the resonance peak is effectively suppressed, ensuring system stability. The dual-loop control strategy, combining outer PQ control and inner current PI control with decoupling, enables precise power regulation and unit power factor operation. Simulation results validate the utility interactive inverter’s performance, demonstrating low THD, fast dynamic response, and robustness against disturbances. Future work could explore active damping methods or advanced control techniques to further enhance the utility interactive inverter’s efficiency and adaptability in varying grid conditions. Overall, this design offers a reliable solution for integrating PV systems into the grid, contributing to the growth of renewable energy.
To deepen the analysis, we can derive additional formulas for key aspects of the utility interactive inverter. For instance, the design of LCL filter parameters often involves trade-offs between size, cost, and performance. The inductance and capacitance values are selected to limit current ripple and harmonic distortion. The ripple current on the inverter side can be approximated as:
$$ \Delta i_1 \approx \frac{U_{dc}}{8 f_s L_1} $$
where \(f_s\) is the switching frequency. Similarly, the capacitor value is chosen to absorb high-frequency harmonics while minimizing reactive power consumption. The reactive power drawn by the capacitor at fundamental frequency is:
$$ Q_C = 3 \omega C U_g^2 $$
This should be kept below 5% of the rated power to maintain efficiency. Another critical aspect is the tuning of PI controllers. Using the symmetrical optimum method for the current loop, the parameters can be calculated as:
$$ K_{ip} = \frac{L}{2 T_s}, \quad K_{ii} = \frac{1}{2 T_s} $$
where \(T_s\) is the sampling period. For the utility interactive inverter, this ensures a balance between response speed and stability. Furthermore, the impact of grid impedance on system performance must be considered. With grid inductance \(L_g\), the total inductance becomes \(L_2 + L_g\), altering the resonant frequency to:
$$ \omega_{\text{res,new}} = \sqrt{\frac{L_1 + L_2 + L_g}{L_1 (L_2 + L_g) C}} $$
This may require adaptive damping or control adjustments in practical utility interactive inverter installations. In summary, the utility interactive inverter design involves multidisciplinary considerations, from power electronics to control theory, all aimed at achieving reliable grid integration. Through continuous optimization, utility interactive inverters will play an increasingly vital role in the renewable energy landscape.