As a clean and pollution-free renewable energy source, photovoltaic (PV) power generation has garnered widespread attention globally. However, the integration of large-scale wind and solar power into the grid poses challenges due to the intermittent, stochastic, and uncontrollable nature of these resources. The stability and security of power systems are significantly tested by the high penetration of such variable generation. Moreover, in many regions, wind and solar installations are located in remote areas far from load centers, necessitating long-distance transmission. Under these conditions, grid line impedance cannot be neglected, leading to a weak grid characteristic. In weak grids, voltage and current harmonics at the point of common coupling (PCC) can be amplified, exacerbating power quality issues. To comply with standards such as GB/T 14549-1993, which sets limits on harmonic content for renewable energy integration, it is crucial to mitigate harmonic distortion in grid currents. This mitigation primarily relies on the control strategies of grid-tied inverters. Therefore, an in-depth study of control strategies for grid-tied inverters in weak grids is essential to enhance current quality and stability.
Various controllers have been employed for grid-tied inverters, including proportional-integral (PI) controllers, proportional-resonant (PR) controllers, and repetitive controllers (RC). PI controllers, limited by their control gain, exhibit low tracking accuracy and significant steady-state error in the αβ stationary reference frame. To address this, researchers have proposed PI control in the rotating dq reference frame, which improves tracking precision. However, this approach requires complex decoupling transformations, increasing the complexity of control strategy design. PR controllers offer high gain at their resonant frequency, enabling accurate tracking and suppression of specific harmonics. But when multiple harmonics need to be suppressed, multiple resonant controllers must be connected in parallel, leading to a proportional multi-resonant control strategy. The main drawback of this method is the increased difficulty in digital implementation and parameter design as control precision rises.
RC, based on the internal model principle, provides extremely high gain at the fundamental frequency and its integer multiples, theoretically achieving zero steady-state error for periodic signals. Its simple structure, ease of digital implementation, and straightforward parameter design make it suitable for controlling grid-tied inverters in weak grid environments. However, RC inherently has a delay of one fundamental period, resulting in slow dynamic response. To improve dynamic performance, RC is often combined with other controllers in a composite control strategy.
In this research, we propose a composite control strategy combining PR and RC (PRRC) for LCL-type grid-tied inverters used in PV systems. The parallel structure reduces the impact of RC’s inherent delay, enhancing system dynamics. By leveraging RC’s infinite gain at harmonic frequencies, zero steady-state error tracking of grid current harmonics is achieved, suppressing harmonic distortion and improving current quality. We derive stability conditions for the system and provide a detailed design methodology for the composite control parameters. Theoretical analysis of stability under weak grid conditions is conducted. Finally, simulations verify the feasibility and effectiveness of the proposed strategy.
The structure of a typical grid-tied inverter system is illustrated below. It consists of a DC voltage source \( V_{dc} \), an LCL filter with inductors \( L_1 \) and \( L_2 \), capacitor \( C \), and grid impedance \( Z_g \). The currents \( i_L \), \( i_g \), and \( i_C \) represent the inverter-side current, grid current, and capacitor current, respectively. Voltages \( v_C \), \( v_{pcc} \), and \( v_g \) denote the capacitor voltage, PCC voltage, and grid voltage. A phase-locked loop (PLL) extracts the phase angle synchronized with \( v_{pcc} \). The reference grid current amplitude is \( I_{ref} \), and the reference signal is \( i_{ref} \). Grid current feedback active damping (GCFAD) is employed to attenuate the resonance peak of the LCL filter.
The control block diagram of the grid-tied inverter in the s-domain is shown, where \( G_i(s) \) is the current controller transfer function, \( k_{pwm} \) is the equivalent gain of the inverter bridge, and \( H(s) \) is the GCFAD coefficient. The GCFAD in the s-domain is expressed as:
$$ H(s) = -\frac{k_c s}{s + \omega_c} $$
where \( k_c \) is the gain and \( \omega_c \) is the cutoff frequency. The LCL filter transfer function \( G_{LCL}(s) \) is:
$$ G_{LCL}(s) = \frac{1}{L_1 L_2 C s^3 + (L_1 + L_2) s} $$
In the z-domain, the control block diagram is transformed, and the composite controller consists of \( G_{pr}(z) \) for PR and \( G_{rc}(z) \) for RC. The error signal is \( E(z) \), \( Q(z) \) is a term introduced to enhance RC stability, \( N = f_s / f \) is the order of RC (with \( f_s \) as sampling frequency and \( f \) as grid frequency), and \( G_c(z) \) is the RC compensator. The RC transfer function in discrete domain is:
$$ G_{rc}(z) = \frac{z^{-N} G_c(z)}{1 – z^{-N} Q(z)} $$
The equivalent plant for both PR and RC in the z-domain is \( P_0(z) \), given by:
$$ P_0(z) = \frac{K_{pwm} G_d(z) G_{LCL}(z)}{1 + K_{pwm} G_d(z) G_{LCL}(z) H(z)} $$
where \( G_d(z) \) is the discretized version of the plant. The composite control strategy aims to leverage the fast dynamics of PR and the high-precision tracking of RC.
Designing the parameters for the PRRC composite control is critical for performance. We first list the system parameters and non-RC controller designs in Table 1.
| Parameter | Value |
|---|---|
| Rated Power \( P \) (kW) | 9.1 |
| Grid Voltage \( v_g \) (V) | 110 |
| Grid-side Inductor \( L_2 \) (mH) | 1 |
| Inverter-side Inductor \( L_1 \) (mH) | 3 |
| Filter Capacitor \( C \) (μF) | 10 |
| DC Bus Voltage \( V_{DC} \) (V) | 200 |
| Grid Frequency \( f \) (Hz) | 50 |
| Sampling Frequency \( f_s \) (kHz) | 12 |
| Switching Frequency \( f_{sw} \) (kHz) | 12 |
| GCFAD Gain \( k_c \) | 30 |
| GCFAD Cutoff Frequency \( \omega_c \) (rad/s) | 12165 |
| PR Proportional Gain \( k_p \) | 6 |
| PR Integral Gain \( k_i \) | 2500 |
The PR controller transfer function in the s-domain is:
$$ G_{pr}(s) = k_p + \frac{2 k_i \omega_c s}{s^2 + 2\omega_c s + \omega_0^2} $$
where \( \omega_0 = 2\pi f \) is the fundamental angular frequency. For digital implementation, this is discretized using the Tustin method.
For RC parameter design, we start with stability analysis. The closed-loop transfer function from reference to error is derived as:
$$ e(z) = \frac{1 – Q(z) z^{-N}}{1 + [G_{pr}(z) + G_{rc}(z)] P_0(z)} i_{ref}(z) $$
Substituting \( G_{rc}(z) \) and simplifying, we get the characteristic equation:
$$ 1 – z^{-N} Q(z) – k_r z^m S(z) P(z) = 0 $$
where \( P(z) = \frac{P_0(z)}{1 + G_{pr}(z) P_0(z)} \) is the equivalent plant for RC, \( G_c(z) = k_r z^m S(z) \), with \( k_r \) as the RC gain, \( m \) as the phase lead compensation order, and \( S(z) \) as a low-pass filter. Using the small-gain theorem, a sufficient condition for stability is:
$$ |Q(e^{j\omega T_s}) – k_r e^{j m \omega T_s} S(e^{j\omega T_s}) P(e^{j\omega T_s})| < 1 \quad \text{for} \quad \omega \in [0, \pi/T_s] $$
where \( T_s = 1/f_s \) is the sampling period.
Based on this, we design RC parameters step by step. First, \( Q(z) \) is chosen as a constant slightly less than 1 to trade off stability and tracking accuracy; we set \( Q(z) = 0.98 \). Second, \( S(z) \) is designed as a fourth-order Butterworth low-pass filter with a cutoff frequency of 1 kHz to attenuate high-frequency noise while minimizing phase lag. Its transfer function is:
$$ S(z) = \frac{0.002576z^4 + 0.01031z^3 + 0.01546z^2 + 0.01031z + 0.002576}{z^4 – 2.639z^3 + 2.769z^2 – 1.339z + 0.2498} $$
Third, the phase compensation \( z^m \) is determined by analyzing the phase of \( z^m S(z) P(z) \). We find that \( m = 10 \) provides near-zero phase shift at the crossover frequency, ensuring adequate phase margin. Fourth, the gain \( k_r \) is selected to satisfy the stability condition; through Nyquist analysis, \( k_r = 0.8 \) is chosen to balance robustness and performance.
To validate the composite control strategy, we perform simulations in MATLAB/Simulink. The grid-tied inverter model uses the parameters from Table 1. A weak grid is emulated by adding an inductance \( L_g = 3 \) mH in series with the grid. The simulation results demonstrate the effectiveness of the PRRC strategy.
Under weak grid conditions, the PCC voltage and grid current waveforms are analyzed. Without the composite control, the total harmonic distortion (THD) of the PCC voltage is 5.85%, and the grid current THD is 4.96%. With the PRRC control, these values are reduced to 2.88% and 1.49%, respectively. The improvement is summarized in Table 2.
| Condition | PCC Voltage THD (%) | Grid Current THD (%) |
|---|---|---|
| Without PRRC (L_g = 3 mH) | 5.85 | 4.96 |
| With PRRC (L_g = 3 mH) | 2.88 | 1.49 |
The mathematical formulation for THD calculation is:
$$ \text{THD} = \frac{\sqrt{\sum_{h=2}^{H} I_h^2}}{I_1} \times 100\% $$
where \( I_h \) is the RMS value of the \( h \)-th harmonic component, and \( I_1 \) is the fundamental component RMS value. The reduction in THD confirms the harmonic suppression capability of the composite control.
Further analysis involves the frequency response of the system. The open-loop transfer function with PRRC is:
$$ T_{ol}(z) = [G_{pr}(z) + G_{rc}(z)] P_0(z) $$
Its Bode plot shows increased gain at harmonic frequencies, enhancing rejection. The stability margins are evaluated using the Nyquist criterion. For the designed parameters, the phase margin is over 65°, and the gain margin is sufficient, indicating robust stability even in weak grids.
The dynamic performance is tested by applying a step change in the reference current. The settling time with PRRC is significantly shorter than with RC alone, thanks to the parallel PR controller. The rise time and overshoot are within acceptable limits, as quantified in Table 3.
| Controller | Rise Time (ms) | Settling Time (ms) | Overshoot (%) |
|---|---|---|---|
| RC alone | 15.2 | 50.1 | 5.3 |
| PRRC composite | 8.7 | 22.4 | 2.1 |
The equations governing the step response are based on the second-order approximation:
$$ i_g(t) = I_{ref} \left(1 – e^{-\zeta \omega_n t} \cos(\omega_d t + \phi)\right) $$
where \( \zeta \) is the damping ratio, \( \omega_n \) is the natural frequency, and \( \omega_d = \omega_n \sqrt{1-\zeta^2} \). The composite control yields higher \( \omega_n \) and optimal \( \zeta \), improving dynamics.
In weak grids, the grid impedance varies, affecting system stability. We analyze the impedance interaction using the Middlebrook criterion. The output impedance of the grid-tied inverter with PRRC is:
$$ Z_{out}(s) = \frac{v_{pcc}(s)}{i_g(s)} $$
And the grid impedance is \( Z_g(s) = s L_g \). For stability, the ratio \( Z_g(s) / Z_{out}(s) \) should satisfy the Nyquist criterion. With PRRC, the phase of \( Z_{out}(s) \) is shaped to avoid encirclements, ensuring stability for \( L_g \) up to 5 mH.
To further illustrate, we derive the sensitivity function \( S(s) \), which indicates disturbance rejection:
$$ S(s) = \frac{1}{1 + T_{ol}(s)} $$
With PRRC, \( |S(j\omega)| \) is minimized at harmonic frequencies, reducing sensitivity to grid voltage distortions.
The design of the grid-tied inverter control involves discrete-time implementation. Using the bilinear transform, continuous-time transfer functions are converted to z-domain. For example, the PR controller in z-domain is:
$$ G_{pr}(z) = k_p + \frac{2 k_i \omega_c (1 – z^{-1})}{(1 + z^{-1})(\omega_0^2 T_s^2 + 2\omega_c T_s) + \cdots} $$
Practical considerations include computational delay and quantization effects. We assume a one-sample delay in the control loop, modeled as \( z^{-1} \). The modified plant becomes \( P_0(z) z^{-1} \), and the stability condition is adjusted accordingly.
In conclusion, the PRRC composite control strategy effectively addresses harmonic distortion and dynamic performance issues in grid-tied inverters operating in weak grids. The parallel structure mitigates RC’s slow response, while RC ensures zero steady-state error for harmonics. Parameter design based on stability conditions yields robust performance. Simulations confirm significant THD reduction and improved dynamics. Future work may explore adaptive tuning for varying grid conditions and hardware-in-the-loop validation.
This research underscores the importance of advanced control strategies for grid-tied inverters in modern power systems. As renewable penetration increases, such techniques will be vital for maintaining power quality and stability. The composite approach offers a balanced solution, combining the strengths of multiple controllers. We hope this contribution aids in the development of more resilient grid-tied inverter systems.
Throughout this discussion, the term ‘grid tied inverter’ has been emphasized to highlight its central role in renewable energy integration. The control of grid tied inverters is a key area for innovation, and the PRRC strategy represents a step forward in addressing weak grid challenges. By focusing on harmonic suppression and dynamic response, grid tied inverters can better support grid stability. Future studies might investigate other composite architectures or machine learning-based controls for grid tied inverters. Ultimately, the goal is to enable seamless integration of renewables through advanced grid tied inverter technologies.