In my recent research, I have delved deeply into the stability and power quality challenges faced by grid-connected inverters, particularly when operating under weak grid conditions. The increasing penetration of renewable energy sources such as wind and solar power has led to a shift from traditional strong grids to weak grids characterized by significant grid impedance and abundant background harmonics. Among the various types of solar inverters employed in distributed generation systems, the LCL-filtered three-phase inverter is widely adopted due to its superior harmonic attenuation performance. However, the interaction between the inverter control system and the weak grid often leads to harmonic resonance issues that degrade the quality of injected current and can even cause system instability.
In many practical applications, the conventional point of common coupling (PCC) voltage feedforward strategy is used to suppress background harmonics. This method is simple and effective under ideal grid conditions. Yet, when the grid impedance becomes non-negligible, the feedforward loop introduces an undesirable positive feedback path that reduces the phase margin of the system, thereby provoking harmonic oscillations. To address this critical problem, I have proposed a weighted proportional-differential (PD) PCC voltage feedforward scheme. The core idea is to reshape the equivalent output impedance of the grid-connected inverter so that its magnitude and phase characteristics are improved in the low-to-medium frequency range, thus preserving harmonic suppression capability while ensuring system stability.
In this article, I will present a comprehensive analysis of my proposed strategy. I first establish an impedance-based model of the LCL-type grid-connected inverter, then examine the detrimental effects of the conventional feedforward approach using vector diagrams and Bode plots. Following that, I introduce the weighted PD feedforward controller and discuss the selection of optimal weighting coefficients. Simulation results obtained from MATLAB/Simulink confirm the effectiveness of the method. The remainder of the paper is organized as follows: Section I describes the mathematical modeling of the inverter system; Section II analyzes the stability impact of conventional feedforward; Section III presents the proposed weighted PD feedforward and its design; Section IV provides simulation verification; and Section V concludes the work.
I. Modeling of the LCL-Type Grid-Connected Inverter
I begin by deriving the control structure of a typical three-phase LCL-filtered grid-connected inverter, which is one of the most common types of solar inverters used in distributed generation. The inverter employs a dual-loop current control strategy: an inner capacitor current feedback loop for active damping of the LCL filter resonance, and an outer grid current loop for power injection. The system parameters are summarized in Table I.
| Parameter | Symbol | Value |
|---|---|---|
| DC-link voltage | Udc | 800 V |
| Rated power | P | 10 kW |
| Grid voltage (line-to-line RMS) | 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 current controller Gc(s) is implemented as a quasi-proportional-resonant (quasi-PR) controller:
$$
G_c(s) = K_p + \frac{2K_r\omega_c s}{s^2 + 2\omega_c s + \omega_0^2}
$$
where ω0 = 2π·50 rad/s is the fundamental angular frequency, and ωc = 3.14 rad/s is the bandwidth. The digital control delay Gd(s) is approximated by a first-order inertia element:
$$
G_d(s) = e^{-1.5sT_s} \approx \frac{1}{1.5sT_s + 1}
$$
with Ts = 1/fs. The inverter gain KPWM is defined as the ratio of the DC-link voltage to the carrier amplitude. After deriving the closed-loop transfer functions, I obtain the equivalent output impedance of the inverter Zo(s) as seen from the PCC:
$$
Z_o(s) = \frac{L_1 L_2 C s^3 + K_{PWM} G_d(s) H_{i1} L_2 C s^2 + (L_1+L_2)s + H_{i2}K_{PWM}G_d(s)G_c(s)}{s^2 L_1 C + s C H_{i1} K_{PWM} G_d(s) + 1}
$$
Using the impedance-based stability criterion, the system is stable if: (1) the inverter is stable when the grid impedance Zg = 0, and (2) the Nyquist plot of Zg/Zo does not encircle the critical point (-1, 0). For a weak grid with short-circuit ratio SCR = 2 (Lg = 7.2 mH), the phase margin at the intersection frequency must be positive.
II. Impact of Conventional PCC Voltage Feedforward on Stability
The conventional grid voltage feedforward strategy adds a term Gf(s) = 1/KPWM to the modulation signal, effectively placing a virtual impedance in parallel with the inverter’s output impedance. The equivalent output impedance becomes:
$$
Z_{oeq}(s) = \frac{Z_o(s) \cdot Z_p(s)}{Z_o(s) + Z_p(s)}
$$
where Zp(s) represents the feedforward-induced impedance. After algebraic manipulation, I define the ratio K1(s) = Zoeq(s)/Zo(s) as:
$$
K_1(s) = \frac{1}{1 – \frac{K_{PWM} G_d(s) G_f(s)}{s^2 L_1 C + s H_{i1} C K_{PWM} G_d(s) + 1}} = \frac{1}{1 – H(s)}
$$
In the low-to-medium frequency range, H(s) is approximately equal to Gd(s). Therefore, K1(jω) ≈ 1/[1 – Gd(jω)]. Figure 6 in my analysis (not displayed here) shows that K1(jω) introduces approximately -90° of phase lag at low frequencies. To visualize this effect, I have drawn a vector diagram (Figure 7) showing the relationship between 1, H(jω), and the resultant vector F(jω) = 1 – H(jω). At low ω, H(jω) has a magnitude near 1 and a lag of about 1.5ωTs, causing F(jω) to lead the unit vector by an angle δ, which translates into a phase lag in the overall impedance. As ω increases, δ decreases, and beyond a certain frequency f2, δ becomes negative, meaning the feedforward creates a phase lead instead. However, in the critical low-frequency region, the phase lag severely reduces the phase margin.
The Bode plot of the equivalent output impedance (Figure 8) confirms that under weak grid conditions, the magnitude of Zoeq is higher than Zo at very low frequencies (beneficial for harmonic rejection), but the phase drops by about 90°. At the intersection frequency fc where |Zoeq| = |Zg|, the phase margin becomes negative, leading to instability. The impact is particularly severe for types of solar inverters that rely on high-bandwidth current control.
To quantify the problem, I list in Table II the key characteristics of the conventional feedforward and the proposed method for comparison.
| Feedforward Strategy | Transfer Function | Effect on Low-Freq Phase | Effect on Low-Freq Magnitude | Stability for SCR=2 |
|---|---|---|---|---|
| No feedforward | Gf(s)=0 | 0° | Baseline | Stable |
| Conventional proportional (λp=1) | Gf(s)=1/KPWM | ~ -90° | Increased | Unstable |
| Simple weighted (λp<1) | Gf(s)=λp/KPWM | Reduced lag but still negative | Reduced (weaker harmonic rejection) | Marginally stable |
| Proposed weighted PD | Gf‘(s)=(λp+λds)/KPWM | ~ 0° at design frequency | Increased at fundamental and above | Stable with high margin |
III. Proposed Weighted Proportional-Differential PCC Voltage Feedforward
Motivated by the vector diagram analysis, I propose to introduce a derivative term in the feedforward path to generate a vector M(jω) that is perpendicular to H(jω) (see Figure 10). The new feedforward function is:
$$
G’_f(s) = \frac{\lambda_p + \lambda_d s}{K_{PWM}}
$$
where λp and λd are the proportional and differential weighting coefficients, respectively. The new ratio K2(s) becomes:
$$
K_2(s) = \frac{Z’_{oeq}(s)}{Z_o(s)} = \frac{1}{1 – (\lambda_p + \lambda_d s)H(s)}
$$
The ideal condition would be to make the synthetic vector H1(jω) = (λp + jωλd)H(jω) exactly equal to the unit vector, so that F1(jω) = 0, yielding infinite impedance magnitude and zero phase lag. This ideal occurs when:
$$
\lambda_p = \cos(1.5\omega T_s), \quad \lambda_d = \frac{\sin(1.5\omega T_s)}{\omega}
$$
However, since the condition is frequency-dependent, I must select a single frequency to optimize the performance. By evaluating Bode plots of K2(s) for different design frequencies (Figure 11), I find that choosing ω = 2π·750 rad/s (i.e., 750 Hz) provides a good trade-off. At this frequency, the coefficients are calculated as: λp = 0.938 and λd = 7.34×10-5. Thus, the proposed feedforward function is:
$$
G’_f(s) = \frac{7.34\times10^{-5}s + 0.938}{K_{PWM}}
$$
The resulting equivalent output impedance is:
$$
Z’_{oeq}(s) = \frac{L_1L_2C s^3 + K_{PWM}G_d(s)H_{i1}L_2C s^2 + (L_1+L_2)s + H_{i2}K_{PWM}G_d(s)G_c(s)}{s^2L_1C + sCH_{i1}K_{PWM}G_d(s) + 1 – (\lambda_d s + \lambda_p)G_d(s)}
$$
Figure 12 compares the Bode plots of Zoeq for three strategies: conventional proportional, simple weighted (λp=0.5, λd=0), and the proposed weighted PD. The results clearly show that the proposed method achieves nearly zero phase lag at low frequencies, while maintaining a higher magnitude than the simple weighted approach. This ensures that for all types of solar inverters operating under weak grids, both stability and harmonic rejection are enhanced.
To further illustrate the advantage, I summarize the phase margin values for different grid impedances in Table III.
| Feedforward Strategy | Phase Margin (degrees) | Stability Status |
|---|---|---|
| No feedforward | 42.3° | Stable |
| Conventional proportional | -15.7° | Unstable (oscillatory) |
| Simple weighted (λp=0.5) | 12.1° | Marginally stable |
| Proposed weighted PD | 38.9° | Stable |
IV. Simulation Validation
I have constructed a detailed simulation model in MATLAB/Simulink using the parameters from Table I. The grid impedance is varied to represent different weak grid conditions (Lg = 1.5 mH, 3.6 mH, and 7.2 mH). I first test the conventional proportional feedforward. As shown in Figure 13 (not reproduced here), when Lg = 7.2 mH, the grid current becomes highly distorted and oscillatory, confirming instability. In contrast, with the proposed weighted PD feedforward (Figure 14), the current remains sinusoidal and well-regulated even under the same weak grid condition. The total harmonic distortion (THD) of the grid current is measured and listed in Table IV for the worst-case scenario (Lg = 7.2 mH with background harmonics).
| Feedforward Strategy | THD (%) |
|---|---|
| Conventional proportional | 3.31 |
| Simple weighted (λp=0.5) | 4.85 |
| Proposed weighted PD | 2.01 |
To comprehensively evaluate background harmonic rejection, I inject 3rd (10%), 5th (8%), 7th (5%), and 11th (3%) harmonics at the PCC. Figure 17 (not shown) illustrates the individual harmonic content in the grid current for the three strategies. The proposed weighted PD feedforward achieves the lowest harmonic amplitude at every frequency, demonstrating superior harmonic suppression while maintaining system stability. This is particularly important for modern types of solar inverters that must comply with strict grid codes such as IEEE 1547 and IEC 61727.
Furthermore, I examine the dynamic response of the system under a step change in the current reference. The proposed feedforward maintains a fast settling time with negligible overshoot, whereas the conventional method leads to sustained oscillations. The robustness of the proposed strategy is also verified under varying grid impedance (from Lg = 1 mH to 10 mH), confirming that the inverter remains stable across a wide range of weak grid conditions.
In addition to the time-domain results, I perform impedance-based stability analysis using Nyquist plots of Zg/Zoeq. For the conventional feedforward, the Nyquist contour encircles the critical point (-1, 0), indicating instability. For the proposed weighted PD feedforward, the contour stays away from the critical point, confirming a robust stability margin. This analysis is consistent with the Bode plot observations and the simulation waveforms.
The weighted proportional-differential PCC voltage feedforward strategy I have developed not only resolves the phase lag problem inherent in conventional feedforward but also enhances the overall impedance characteristics of the inverter. This makes it an ideal solution for various types of solar inverters operating in weak grids with high harmonic distortion. The design is simple to implement, requiring only the addition of a derivative term with a single tuning parameter λd, and the computational burden is minimal. Compared to other advanced methods such as adaptive filters or resonant controllers, the proposed approach offers a good balance between performance and complexity.
V. Conclusion
Through systematic modeling, vector analysis, and simulation, I have demonstrated that the conventional PCC voltage feedforward strategy, while effective under ideal grids, introduces a low-frequency phase lag that destabilizes the grid-connected inverter under weak grid conditions. To overcome this limitation, I have proposed a weighted proportional-differential feedforward controller. By selecting appropriate coefficients λp and λd based on the vector diagram analysis, the proposed method effectively cancels the phase lag and maintains a high impedance magnitude in the low-to-medium frequency range. The resulting inverter exhibits robust stability, excellent harmonic rejection, and fast dynamic response, making it highly suitable for modern types of solar inverters deployed in weak grids. The simulation results confirm that the proposed strategy outperforms both conventional proportional feedforward and simple weighted feedforward, achieving a THD as low as 2.01% under severe background harmonics, while ensuring positive phase margin across a wide range of grid impedances. Future work will focus on extending the technique to single-phase inverters and including the effects of phase-locked loop dynamics.