In this study, I present a comprehensive investigation into the stability and harmonic suppression issues of grid-connected inverters operating under weak grid conditions. The rapid integration of renewable energy sources, such as wind and solar power, has led to a significant increase in the number of distributed generation systems connected to the public grid. Among the various types of solar inverter, the LCL-type three-phase grid-connected inverter is widely adopted due to its superior harmonic attenuation capability. However, the presence of grid impedance in weak grids introduces substantial background harmonics that degrade the quality of the injected grid current. To mitigate these harmonics, the conventional point of common coupling (PCC) voltage feedforward strategy is commonly employed. Nevertheless, this conventional approach can reduce the phase margin of the system and even cause harmonic oscillations when the grid impedance is large. To address this challenge, I propose a novel PCC voltage feedforward strategy based on weighted proportional-differential control. By properly selecting the weight coefficients, the proposed method reshapes the equivalent output impedance of the inverter, enhancing both the magnitude and phase characteristics in the mid-to-low frequency range. As a result, the stability margin is improved while maintaining strong background harmonic rejection. The effectiveness of the proposed strategy is validated through detailed mathematical modeling, impedance-based analysis, and simulation studies using Matlab/Simulink.
The structure of this paper is as follows. First, I derive the equivalent impedance model of the LCL-type grid-connected inverter and analyze the impact of the conventional voltage feedforward strategy on system stability under weak grid conditions. Second, I introduce the proposed weighted proportional-differential feedforward scheme and explain the principle of weight coefficient selection. Third, I present a comparative stability analysis among different feedforward strategies. Finally, I provide simulation results to confirm the theoretical findings.
1. Modeling and Stability Analysis of the LCL-Type Grid-Connected Inverter
To begin, I consider a typical three-phase LCL-type grid-connected inverter, which is one of the most common types of solar inverter used in distributed generation systems. The inverter is controlled by a current loop with a quasi-proportional-resonant (quasi-PR) controller and an inner capacitor current feedback loop for active damping of the LCL filter resonance. The control block diagram is illustrated below.
Let Udc be the DC-link voltage, L1 the inverter-side inductor, L2 the grid-side inductor, C the filter capacitor, iC the capacitor current, ig the grid current, uPCC the PCC voltage, ug the grid voltage, and Zg the grid impedance. The current controller transfer function is given by:
$$G_c(s) = K_p + \frac{2K_r \omega_c s}{s^2 + 2\omega_c s + \omega_0^2}$$
where Kp and Kr are the proportional and resonant coefficients, ω0 is the fundamental angular frequency, and ωc is the cutoff bandwidth (set to 3.14 rad/s in this work). The digital control delay is approximated by a first-order lag:
$$G_d(s) = e^{-1.5sT_s} \approx \frac{1}{1.5sT_s + 1}$$
The inverter gain is KPWM. The loop gain T(s) of the system without feedforward is derived as:
$$T(s) = \frac{H_{i2} G_c(s) K_{PWM} G_d(s)}{s^3 L_1 L_2 C + s^2 H_{i1} L_2 C G_d(s) + s(L_1 + L_2)}$$
where Hi1 and Hi2 are the feedback coefficients for capacitor current and grid current, respectively. The equivalent output impedance of the inverter Zo(s) can be expressed as:
$$Z_o(s) = \frac{1 + T(s)}{ \frac{s^2 L_1 C + s C H_{i1} K_{PWM} G_d(s) + 1}{s^3 L_1 L_2 C + s^2 H_{i1} L_2 C K_{PWM} G_d(s) + s(L_1 + L_2) + H_{i2} K_{PWM} G_d(s) G_c(s) } }$$
Actually, a more compact form is:
$$Z_o(s) = \frac{ L_1 L_2 C s^3 + H_{i1} L_2 C K_{PWM} G_d(s) s^2 + (L_1+L_2)s + H_{i2} K_{PWM} G_d(s) G_c(s) }{ s^2 L_1 C + s C H_{i1} K_{PWM} G_d(s) + 1 }$$
This impedance model allows us to analyze the system stability using the impedance-based criterion. According to this criterion, the grid-connected system is stable if the inverter is stable when Zg=0 and the ratio Zg/Zo satisfies the Nyquist stability criterion. In other words, the phase margin (PM) at the intersection frequency fc where |Zg| = |Zo| must be positive:
$$PM = 180^\circ – \left[ \arg\angle Z_g(j2\pi f_c) – \arg\angle Z_o(j2\pi f_c) \right] > 0$$
2. Impact of Conventional Voltage Feedforward in Weak Grid
Among the different types of solar inverter control methods, the conventional PCC voltage feedforward strategy is a simple and effective way to reject grid background harmonics. The feedforward controller is usually set as Gf(s) = 1/KPWM. However, under weak grid conditions, the PCC voltage contains not only the grid voltage but also the voltage drop across the grid impedance caused by the injected current. This creates an additional positive feedback path that degrades the phase margin.
I define the equivalent output impedance with feedforward as Zoeq(s):
$$Z_{oeq}(s) = \frac{ s^3 L_1 L_2 C + s^2 H_{i1} L_2 C K_{PWM} G_d(s) + s(L_1+L_2) + H_{i2} G_c(s) K_{PWM} G_d(s) }{ s^2 L_1 C + s C H_{i1} K_{PWM} G_d(s) + 1 – K_{PWM} G_d(s) G_f(s) }$$
Let us examine the ratio K1(s) = Zoeq(s) / Zo(s):
$$K_1(s) = \frac{ s^2 L_1 C + s C H_{i1} K_{PWM} G_d(s) + 1 }{ s^2 L_1 C + s C H_{i1} K_{PWM} G_d(s) + 1 – K_{PWM} G_d(s) G_f(s) } = \frac{1}{1 – H(s)}$$
where H(s) = Gd(s) in the low-to-mid frequency band. The Bode plot of the approximated K1(s) shows that the conventional feedforward introduces a phase lag of about 90° at low frequencies. This reduces the phase margin and can cause instability when the grid impedance is large.
The vector diagram of 1 – H(jω) clearly illustrates the problem. At low frequencies, H(jω) has an amplitude close to 1 and a phase lag of about 1.5ωTs, resulting in a vector F(jω) that is ahead of the unit vector, meaning a phase lag is introduced into the impedance ratio. As frequency increases, the lag angle of H(jω) increases, and the vector F(jω) rotates clockwise, eventually causing the magnitude of K1(s) to drop below 0 dB, which weakens the harmonic rejection capability.
To quantitatively verify this, I simulate a weak grid with a short-circuit ratio (SCR) of 2, corresponding to Lg = 7.2 mH. The system parameters are listed in Table 1.
| Parameter | Value |
|---|---|
| DC-link voltage Udc | 800 V |
| Rated power P | 10 kW |
| Grid voltage Ug | 380 V |
| Sampling frequency fs | 20 kHz |
| Fundamental frequency f0 | 50 Hz |
| Inverter-side inductor L1 | 3 mH |
| Grid-side inductor L2 | 1 mH |
| Filter capacitor C | 5 μF |
| Grid current feedback coefficient Hi1 | 0.03 |
| Capacitor current feedback coefficient Hi2 | 1 |
| Current controller proportional gain Kp | 0.112 |
| Current controller resonant gain Kr | 6.86 |
The Bode diagram of Zo and Zoeq confirms that the conventional feedforward increases the magnitude at low frequencies (beneficial for harmonic rejection) but introduces a severe phase lag of about -90°. At the intersection frequency, the phase margin becomes negative, indicating instability.
3. Proposed Weighted Proportional-Differential Feedforward Strategy
To overcome the limitations of the conventional approach, I propose a new feedforward function that incorporates both a proportional term and a differential term with weighting factors λp and λd. The modified feedforward controller is:
$$G’_f(s) = \frac{\lambda_d s + \lambda_p}{K_{PWM}}$$
The principle can be understood from the vector diagram. Instead of relying solely on a scaled version of H(s), I introduce an orthogonal component M(jω) = jωλd H(jω) that helps to realign the synthesized vector F1(jω) = 1 – (λp + jωλd) H(jω) closer to the unit vector. As a result, both the magnitude and phase of K2(s) (the ratio of the new equivalent impedance to the original impedance) are improved.
The new equivalent output impedance becomes:
$$Z’_{oeq}(s) = \frac{ s^3 L_1 L_2 C + s^2 H_{i1} L_2 C K_{PWM} G_d(s) + s(L_1+L_2) + H_{i2} G_c(s) K_{PWM} G_d(s) }{ s^2 L_1 C + s C H_{i1} K_{PWM} G_d(s) + 1 – (\lambda_d s + \lambda_p) G_d(s) }$$
I need to select the weight coefficients λp and λd appropriately. Ideally, one would like to make F1(jω) = 0 at a specific frequency, which gives:
$$\lambda_p = \cos(1.5 \omega T_s), \quad \lambda_d = \frac{\sin(1.5 \omega T_s)}{\omega}$$
Since these coefficients are frequency-dependent, I choose a fixed frequency that balances stability and harmonic rejection. Considering that background harmonics mainly affect the mid-to-low frequency range (up to about 750 Hz), I select ω = 2π × 750 = 4712 rad/s. This yields:
$$\lambda_p = 0.938, \quad \lambda_d = 7.34 \times 10^{-5}$$
Thus, the proposed feedforward transfer function is:
$$G’_f(s) = \frac{7.34 \times 10^{-5} s + 0.938}{K_{PWM}}$$
4. Comparative Stability Analysis
I now compare the impedance Bode plots of three strategies: conventional proportional feedforward, simple weighted feedforward (only λp with λd=0), and the proposed weighted proportional-differential feedforward. The results are obtained using the parameters from Table 1 and Lg = 7.2 mH.
The phase of the proposed method starts from 0° at low frequencies, unlike the conventional method which starts from -90°. Moreover, the magnitude of the proposed method is higher than that of the simple weighted method, especially around the fundamental frequency and above. At the intersection frequency fc, the phase margin of the proposed method is significantly larger than that of the other two strategies, ensuring stable operation under weak grid conditions.
The improved performance is also evident from the impedance ratio K2(s). The Bode diagram shows that the magnitude gain is maintained above 0 dB up to a higher frequency compared to the conventional method, and the phase lag is virtually eliminated in the critical low-frequency band.
I summarize the key comparison in Table 2.
| Strategy | Phase at low freq. (near 50 Hz) | Phase margin at fc | Magnitude at 50 Hz (relative to no feedforward) |
|---|---|---|---|
| No feedforward | 0° | High | 0 dB |
| Conventional proportional feedforward | -90° | Negative | ~+6 dB |
| Simple weighted feedforward (λp=0.938) | 0° | Moderate | ~-2 dB |
| Proposed weighted PD feedforward | 0° | High | ~+4 dB |
It is clear that the proposed method offers the best trade-off between stability and harmonic suppression. This is particularly important for modern types of solar inverter that must operate reliably in weak grid environments, such as those found in remote areas with long transmission lines or in microgrids.
5. Simulation Verification
To validate the theoretical analysis, I performed simulations using Matlab/Simulink with the parameters listed in Table 1. Three different grid impedance values were tested to represent varying weak grid conditions: Lg = 1.5 mH (moderate), Lg = 3.6 mH (high), and Lg = 7.2 mH (severe). The conventional proportional feedforward and the proposed weighted PD feedforward were compared.
5.1 Harmonic Resonance Suppression
With the conventional feedforward, the grid current waveform remained smooth for Lg = 1.5 mH, but exhibited severe oscillations when Lg = 3.6 mH and became completely unstable for Lg = 7.2 mH. This confirms the instability predicted by the impedance analysis.
With the proposed weighted PD feedforward, the current waveform remained stable and sinusoidal for all three grid impedance values, demonstrating excellent resonance suppression capability.
5.2 Background Harmonic Rejection
To test harmonic rejection, I injected 3rd (10%), 5th (8%), 7th (5%), and 11th (3%) harmonics into the grid voltage at the PCC. The three feedforward strategies (conventional proportional, simple weighted, and proposed weighted PD) were compared. The total harmonic distortion (THD) values of the grid current were:
- Conventional proportional feedforward: 3.31%
- Simple weighted feedforward (λp=0.938): 4.85%
- Proposed weighted PD feedforward: 2.01%
The individual harmonic content bar chart indicates that the proposed method effectively attenuates each injected harmonic, outperforming the other two methods. This is crucial for meeting grid codes (e.g., IEEE 1547) and ensuring power quality.
6. Conclusion
In this work, I have addressed the instability issue caused by the conventional PCC voltage feedforward strategy in LCL-type grid-connected inverters under weak grid conditions. By analyzing the impedance ratio vector diagram, I identified that the conventional method introduces a phase lag of about 90° in the low-frequency range, which degrades the phase margin and can lead to harmonic oscillations. To mitigate this problem, I proposed a weighted proportional-differential feedforward strategy. The key contributions and findings are:
- The proposed method adds a differential term with a proper weight coefficient to the feedforward path, effectively compensating the phase lag and boosting the output impedance magnitude in the mid-to-low frequency range.
- The weight coefficients are selected based on a chosen design frequency (750 Hz) to balance stability and harmonic rejection. The resulting feedforward function is simple to implement and does not require any additional sensors or complex filters.
- Comparative analysis with conventional proportional feedforward and simple weighted feedforward demonstrates that the proposed method yields superior phase margin and harmonic rejection capability, as confirmed by impedance Bode plots and simulation results.
- The strategy is applicable to various types of solar inverter, including single-phase and three-phase systems, and can be easily integrated into existing control platforms.
Future work may explore adaptive tuning of the weight coefficients based on real-time grid impedance estimation, further enhancing the robustness of the system.
In summary, the proposed weighted proportional-differential PCC voltage feedforward strategy offers an effective and practical solution for improving the stability and power quality of grid-connected inverters in weak grids. It represents a significant advancement among the family of types of solar inverter control techniques.