The proliferation of renewable energy generation has led to the large-scale integration of distributed power sources into the grid, driving significant development in utility interactive inverters. The output voltage on the machine side of a utility interactive inverter inevitably contains a substantial amount of high-frequency harmonics. Compared to L-filters, LCL filters offer superior harmonic suppression, smaller size, and lower cost, making them more cost-effective for achieving the same filtering requirements. However, the LCL filter introduces an inherent resonance problem.
Existing methods to suppress resonance in LCL-type utility interactive inverters primarily include passive damping, active damping, and order-reduction techniques. Among the order-reduction methods, the Weighted Average Current (WAC) control is favored for its simple parameter design, high bandwidth, and loop order-reduction characteristics.
As distributed generation permeates the grid, grid impedance experiences wide-range fluctuations, causing the grid to exhibit weak grid characteristics. The order-reduction property of WAC can be compromised due to the influence of grid impedance on the weighting coefficient, leading to incomplete system order reduction. To suppress low-order current harmonics caused by distorted grid voltage, grid voltage feedforward control is widely adopted. Combining WAC control with Point of Common Coupling (PCC) voltage feedforward can not only mitigate the adverse effects of grid voltage harmonics on the grid current quality but also achieve complete reduction of the weighted current loop.
Furthermore, control delay in digital control systems introduces non-negligible phase lag. This causes the traditional combined strategy of WAC and PCC voltage feedforward to lose its order-reduction property, degrading the stability and grid current quality of the utility interactive inverter system. This article first analyzes the system destabilization caused by the loss of the order-reduction property in conventional WAC under the combined influence of control delay and weak grid conditions. The analysis shifts from the perspective of loop order-reduction to the analysis of system output impedance based on an equivalent impedance model. Then, building upon proportional voltage feedforward and focusing on enhancing harmonic immunity and stability, a voltage feedforward control strategy based on multiple second-order filters is proposed. This strategy reshapes the system’s output impedance, thereby improving the phase margin of the equivalent output impedance in the mid-to-low frequency range and enhancing the system’s adaptability to wide variations in grid impedance. Simulation and experimental results confirm that this strategy exhibits strong adaptability to wide-range weak grid impedance variations and can output high-quality grid-injected current.
1. Principles and Challenges of WAC Control
1.1 Mathematical Model of WAC Control
The topology of a single-phase LCL-type utility interactive inverter is shown in the figure below. In the main circuit, L1, L2, C, and Rd represent the inverter-side inductor, grid-side inductor, filter capacitor, and damping resistor, respectively. The parasitic resistances of L1 and L2 are neglected. The grid inductance is Lg (considering the grid impedance as purely inductive). Vdc, Vin, and Vg are the DC-link voltage, inverter output voltage, and equivalent grid voltage, respectively. VPCC is the voltage at the Point of Common Coupling. i1, i2, and ic denote the inverter-side current, grid current, and capacitor current, respectively. In the control section, the weighting coefficients (1-β) and β are multiplied with i1 and i2, respectively, and summed to obtain the weighted value iWAC. The current reference iref is generated by combining the phase information from the Phase-Locked Loop (PLL) with the setpoint I. Gf(s) is the VPCC feedforward term, and Gc(s) is the PI current regulator.
The control block diagram can be derived as shown. In the diagram, KPWM is the transfer function of the inverter bridge, given by $$K_{PWM} = \frac{V_{dc}}{V_{tri}}$$, where Vtri is the triangular carrier amplitude.
From the control block diagram, the system’s open-loop transfer function can be derived as:
$$T_O(s) = \frac{K_{PWM}G_c(s)\left(1 + sR_dC + s^2(1-\beta)(L_2+L_g)C\right)} {s(L_1+L_2)\left[(1+sR_dC)\left(1+\frac{L_g(1-G_f(s)K_{PWM})}{L_1+L_2}\right) + s^2\frac{L_1}{L_1+L_2}(L_2+L_g)C\right]}$$
If the voltage feedforward function Gf(s) and the weighting coefficient β are designed as: $$G_f(s) = \frac{1}{K_{PWM}}, \quad \beta = \frac{L_2}{L_1+L_2}$$, then Eq. (2) simplifies to:
$$T_O(s) = \frac{K_{PWM}G_c(s)}{s(L_1+L_2)}$$
This demonstrates that the WAC control strategy reduces the system from third-order to first-order, effectively making the LCL-type utility interactive inverter equivalent to an L-type inverter.
1.2 Stability Analysis Considering Digital Control Delay
Beyond the weak grid environment, digital control delay is a critical issue in modern digital control systems. The entire control process introduces one sampling period of computation delay and 0.5 sampling periods of PWM update delay, totaling 1.5 sampling periods of control delay. In the continuous domain, this digital control delay is often approximated by a Padé approximation. A second-order Padé approximation for a 1.5Ts delay is:
$$G_d(s) = e^{-s1.5T_s} \approx \frac{1 – 0.75sT_s + 0.083(1.5sT_s)^2}{1 + 0.75sT_s + 0.083(1.5sT_s)^2}$$
where Ts is the sampling period. Considering this control delay, the open-loop transfer function becomes:
$$T_{O\_D}(s) = \frac{K_{PWM}G_c(s)G_d(s)\left(1 + sR_dC + s^2(1-\beta)(L_2+L_g)C\right)} {s(L_1+L_2)\left[(1+sR_dC)\left(1+\frac{L_g(1-G_f(s)K_{PWM}G_d(s))}{L_1+L_2}\right) + s^2\frac{L_1}{L_1+L_2}(L_2+L_g)C\right]}$$
With the previous ideal design for Gf(s) and β, the denominator does not simplify completely due to the presence of Gd(s) in the term containing Lg. This means the WAC control loses its perfect order-reduction property. Analysis of the Bode plot of TO_D(s) under different grid impedances (Lg = 0, 1.6, 3.2, 6.4 mH) reveals a reverse resonance peak. This peak causes the phase curve to drop below -120°, and as Lg increases, the peak moves toward lower frequencies, continuously reducing the system’s phase margin. The most direct compensation would involve a lead function of e^{s1.5Ts}, which is impractical to implement.
2. Instability Analysis Based on the Equivalent Impedance Model
Given that traditional WAC struggles to balance stability and order-reduction under digital delay, this analysis moves away from examining the order-reduction property. Instead, it employs an equivalent impedance model to assess stability based on the interaction between the inverter output impedance and the grid impedance.
2.1 Establishing the Equivalent Impedance Model
The impedance model divides the power electronic grid interface system at the PCC into source and load subsystems. The utility interactive inverter subsystem is typically modeled as a current source I(s) in parallel with an output impedance Zout(s). The weak grid subsystem is modeled as a voltage source Vg(s) in series with a grid impedance Zg(s).
By manipulating the control block diagram, the expressions for I(s) and Zout(s) can be derived. First, the original control block is equivalently transformed. Defining intermediate transfer functions G1(s) and G2(s):
$$G_1(s) = \frac{G_d(s)K_{PWM}(R_dCs+1)}{s^2L_1C + R_dCs + 1 + G_c(s)G_d(s)s(1-\beta)K_{PWM}C}$$
$$G_2(s) = \frac{s^2L_1C + R_dCs + 1 + G_c(s)G_d(s)s(1-\beta)K_{PWM}C}{s^3L_1L_2C + s(L_1+L_2)(R_dCs+1) + G_c(s)G_d(s)(1-\beta)K_{PWM}(s^2L_2C+R_dCs+1)}$$
The reference current generator I(s) and the output impedance Zout(s) are then:
$$I(s) = \frac{G_c(s)G_1(s)G_2(s)}{1 + \beta G_c(s)G_1(s)G_2(s)} i_{ref}(s)$$
$$Z_{out}(s) = -\frac{V_{PCC}(s)}{i_2(s)} = \frac{1 + \beta G_c(s)G_1(s)G_2(s)}{G_2(s)[1 – G_1(s)G_f(s)]}$$
According to impedance-based stability theory, the dynamic interaction between Zout(s) and Zg(s) directly determines system stability. The phase margin (PM) of the interconnected system can be evaluated from the impedance ratio at the crossover frequency fc (where |Zout(j2πfc)| = |Zg(j2πfc)|):
$$PM = 180^\circ – \left[ \arg\left(Z_g(j2\pi f_c)\right) – \arg\left(Z_{out}(j2\pi f_c)\right) \right] = 90^\circ + \arg\left(Z_{out}(j2\pi f_c)\right)$$
The system is stable if PM > 0°, which requires arg(Zout(j2πfc)) > -90°.
Analyzing the Bode plot of Zout(s) for increasing Lg (0.8, 1.6, 3.2, 6.4 mH) shows that the crossover frequency fc moves to lower frequencies as Lg increases. The phase of Zout(s) at these lower fc values drops significantly below -90°, reducing the phase margin and eventually leading to instability. This aligns with the previous open-loop analysis.
2.2 Root Cause of Instability
To understand the impact of Zout(s) on stability, it is decomposed into two parts: Zout(s) = Z1(s) / Z2(s), where
$$Z_1(s) = \frac{1 + \beta G_c(s)G_1(s)G_2(s)}{G_2(s)}, \quad Z_2(s) = \frac{1}{1 – G_1(s)G_f(s)}$$
Analysis of the Bode plots for Z1(s) and Z2(s) reveals that around the critical crossover frequency fc, the phase of Z1(s) remains above -90° and does not majorly threaten stability. The phase of Z2(s), however, decays rapidly, contributing significantly to the poor phase of Zout(s). The term -G1(s)Gf(s) in the denominator of Z2(s) originates from the PCC voltage feedforward path. This indicates that under traditional proportional feedforward (Gf(s)=1/KPWM), the inherent control delay Gd(s) within G1(s) causes the phase lag in Zout(s) at mid-low frequencies, destabilizing the utility interactive inverter in weak grids.
3. Proposed Voltage Feedforward Strategy Based on Multiple Second-Order Filters
3.1 Principle of the Improved WAC Control Strategy
The analysis shows that the proportional PCC voltage feedforward under WAC leads to a lagging phase in the inverter’s equivalent output impedance, drastically reducing stability margin. Additionally, real grids contain abundant low-frequency background harmonics. These harmonics can be amplified near the crossover frequency, severely degrading grid current quality. Therefore, to enhance both harmonic immunity and stability, Zout(s) should present high impedance at background harmonic frequencies while maintaining sufficient phase margin at fc.
To improve the utility interactive inverter‘s adaptability to wide grid impedance variations, the output impedance needs to be reshaped. The traditional proportional feedforward coefficient sacrifices weak-grid stability for harmonic rejection. Since harmonics below the 13th order dominate in practical grids, the feedforward can be designed to target specific harmonic frequencies. This is achieved by employing multiple Second-Order Generalized Integrators (SOGIs) in the feedforward path to extract and feed forward specific harmonic components from VPCC, while attenuating signals at other frequencies. This maintains high output impedance magnitude at major harmonic frequencies (for rejection) while improving the phase characteristics around fc for robustness.
The modified feedforward path becomes G’_f(s) = Gf(s) * GSOGI_n(s), where GSOGI_n(s) is the transfer function for the nth harmonic SOGI:
$$G_{SOGI\_n}(s) = \frac{\omega_v s}{s^2 + \omega_v s + (n\omega_0)^2}, \quad n = 3, 5, 7, 9, …$$
Here, ωv is the bandwidth coefficient, ω0 is the fundamental grid angular frequency, and n is the harmonic order.
3.2 Filter Parameter Design
The reshaped output impedance Z’_out(s) is obtained by substituting G’_f(s) into the expression for Zout(s). The parameters n and ωv influence the stability and performance. Analysis of Bode plots for Z’_out(s) with varying parameters leads to the following design guidelines:
- Harmonic Order (n): As n increases, the phase margin improves. However, higher n requires more parallel SOGI units, increasing complexity. A compromise is to target the most dominant low-order harmonics (e.g., 3rd, 5th, 7th). In this design, n=7 is selected for significant phase boost.
- Bandwidth Coefficient (ωv): A larger ωv provides a narrower bandwidth, better harmonic extraction, but slower dynamic response and less phase boost in the critical frequency region. A smaller ωv gives wider bandwidth and more phase compensation but may reduce harmonic selectivity. ωv = 30π rad/s is chosen as a balance.
To ensure zero steady-state error at the fundamental frequency under grid frequency variations, a Quasi-Proportional Resonant (Q-PR) controller is used instead of a PI controller:
$$G_c(s) = k_p + \frac{2k_r\omega_c s}{s^2 + 2\omega_c s + \omega_0^2}$$
where kp is the proportional gain, kr is the resonant gain, ωc is the controller bandwidth, and ω0 is the fundamental angular frequency. ωc is set to 3.14 rad/s (corresponding to a maximum grid frequency deviation Δf of 0.5 Hz) to ensure adequate gain around the fundamental frequency.
The Bode plot of the reshaped Z’_out(s) using the Q-PR controller and the multi-SOGI feedforward (with n=7, ωv=30π) shows a remarkable phase improvement at the crossover frequency. For Lg = 6.4 mH, the phase margin increases to 25.6°, which is 40.6° higher than before compensation. Furthermore, while the magnitude of Z’_out(s) is slightly reduced at very low frequencies, it is increased at the targeted harmonic frequencies, enhancing harmonic rejection capability.
| Parameter | Symbol | Value |
|---|---|---|
| Grid Voltage (RMS) | Vg | 110 V |
| DC-Link Voltage | Vdc | 200 V |
| Switching Frequency | fsw | 15 kHz |
| Sampling Frequency | fs | 30 kHz |
| Inverter-side Inductor | L1 | 0.4 mH |
| Damping Resistor | Rd | 0.8 Ω |
| Filter Capacitor | C | 9.2 μF |
| Grid-side Inductor | L2 | 0.3 mH |
| Carrier Wave Amplitude | Vtri | 1.694 V |
| Q-PR Proportional Gain | kp | 0.1 |
| Q-PR Resonant Gain | kr | 80 |
| SOGI Bandwidth | ωv | 30π rad/s |
| Q-PR Bandwidth | ωc | 3.14 rad/s |
| Fundamental Frequency | ω0 | 100π rad/s |
4. Stability Analysis of the Improved Grid Current
The quality of the grid current i2 is a key indicator of control performance. The closed-loop transfer function from iref to i2 for the traditional WAC system with delay is:
$$T_{C\_D}(s) = \frac{K_{PWM}G_c(s)G_d(s)(sR_dC+1)}{s^3L_1(L_2+L_g)C + s(sR_dC+1)[L_1+L_2+L_g – L_gG_f(s)K_{PWM}G_d(s)] + K_{PWM}G_d(s)G_c(s)[s^2(1-\beta)(L_2+L_g)C + sR_dC+1]}$$
Plotting the dominant closed-loop poles of TC_D(s) as Lg varies from 0 to 6.4 mH for the traditional method shows poles moving towards and eventually into the right-half plane, indicating instability. In contrast, for the system with the proposed multi-SOGI feedforward strategy, all dominant closed-loop poles remain in the left-half plane for the entire Lg range, confirming robust stability.
5. Simulation and Experimental Verification
To validate the proposed improved WAC control strategy for the utility interactive inverter, a simulation model was built in Matlab/Simulink using the parameters from Table 1. The grid voltage was intentionally distorted with 10% 3rd, 5% 5th, and 3% 7th harmonic components to emulate a realistic weak grid.
5.1 Simulation Results
First, the traditional WAC control (with proportional feedforward) was tested. With Lg = 1 mH, the grid current THD was 2.26%. When Lg increased to 1.6 mH, the system became unstable, and the current THD soared to 34.93%, far exceeding grid codes.
The proposed strategy was then applied. The grid current waveforms remained sinusoidal and stable even under severe weak grid conditions. The results are summarized below:
| Control Strategy | Grid Inductance Lg | Grid Current THD | Stability |
|---|---|---|---|
| Traditional WAC | 1.0 mH | 2.26% | Stable |
| Traditional WAC | 1.6 mH | 34.93% | Unstable |
| Proposed WAC | 1.6 mH | 1.48% | Stable |
| Proposed WAC | 3.2 mH | 1.49% | Stable |
| Proposed WAC | 6.4 mH | 1.34% | Stable |
The simulation results clearly demonstrate that the proposed strategy maintains excellent current quality (THD < 4%) and stability across a wide range of weak grid impedances, showcasing superior harmonic immunity and robustness compared to the traditional approach.
5.2 Experimental Results
A 3 kW single-phase LCL utility interactive inverter prototype was built to validate the findings experimentally. The parameters matched those in Table 1. The experimental results corroborated the simulation findings. Under traditional WAC control with increased Lg, the grid current became severely distorted. With the proposed multi-SOGI feedforward control strategy applied, the grid current remained stable and sinusoidal with low distortion even under weak grid conditions corresponding to Lg = 6.4 mH, confirming the practical effectiveness of the method.
6. Conclusion
This article has addressed the stability challenge of LCL-type utility interactive inverters employing Weighted Average Current control in weak grids with digital control delay. The traditional WAC control loses its beneficial order-reduction property due to the phase lag introduced by digital delay, manifesting as a destabilizing reverse resonance peak in the frequency response that is exacerbated by increasing grid impedance.
The instability mechanism was thoroughly analyzed using an equivalent impedance model, pinpointing the detrimental phase contribution of the conventional proportional voltage feedforward path under delay. To simultaneously enhance harmonic rejection and stability robustness, a novel voltage feedforward strategy based on multiple second-order generalized integrators was proposed. This strategy selectively feeds forward major harmonic components, thereby reshaping the inverter’s output impedance. The reshaped impedance exhibits significantly improved phase margin at the impedance crossover frequency and maintains high impedance at targeted harmonic frequencies.
Both theoretical analysis and simulation/experimental results from a 3 kW prototype confirm that the proposed improved WAC control strategy endows the utility interactive inverter with strong adaptability to wide-range grid impedance variations. It ensures system stability while delivering high-quality, low-THD grid current that meets stringent interconnection standards, making it a viable solution for reliable integration of distributed generation into weak grids.