Integrating renewable energy sources into the power grid relies heavily on the performance of grid connected inverters. Among various filter topologies, the LCL filter is widely preferred for grid tied inverter interfaces due to its superior high-frequency harmonic attenuation capability with smaller size and lower cost compared to simple L filters. However, the power quality of the current injected by a grid tied inverter is susceptible to two major practical challenges: background harmonics present in the grid voltage and the variable grid impedance under weak grid conditions. The former leads to distorted output current, while the latter can degrade system stability, potentially causing oscillations or even instability. This work proposes a composite control strategy that synergistically combines a current disturbance observer (DOB) with an additional virtual impedance to address both issues simultaneously, ensuring high-quality current injection and enhanced robustness against grid impedance variations.
The conventional control structure for a single-phase LCL-type grid tied inverter often employs a dual-loop strategy with a PI controller in the outer current loop and capacitor current feedback in the inner active damping loop. While effective for stability and tracking, this structure offers limited inherent rejection of grid voltage disturbances. The output current \(i_g(s)\) can be expressed as the sum of a reference tracking term and a disturbance term induced by the grid voltage \(u_g(s)\):
$$
i_g(s) = G_1(s) i_{ref}(s) + G_2(s) u_g(s)
$$
where \(G_1(s)\) is the closed-loop transfer function from reference to output, and \(G_2(s)\) is the disturbance transfer function from grid voltage to output current. Even with a well-tuned PI controller, the disturbance term \(G_2(s) u_g(s)\) introduces steady-state error at the fundamental frequency and harmonic distortion if \(u_g(s)\) contains background harmonics. Therefore, suppressing the influence of \(G_2(s)\) is crucial for achieving high power quality in a grid tied inverter.
To mitigate the impact of grid voltage harmonics, we introduce a disturbance observer into the control loop. The fundamental principle of a DOB is to estimate lumped disturbances, including model uncertainties and external perturbations, in real-time based on the difference between the actual system output and the output of a nominal model. This estimated disturbance is then fed forward to the control input for cancellation. In our context, the grid voltage background harmonics are treated as a primary external disturbance. The block diagram of the proposed current disturbance observation method integrated into the LCL-type grid tied inverter control system is shown conceptually below. The plant \(P(s)\) represents the actual inverter with the LCL filter and PWM, while \(P_n(s)\) is its nominal model. The low-pass filter \(Q(s)\) is key to the DOB’s performance.
The designed DOB estimates the disturbance \(\hat{d}(s)\) as:
$$
\hat{d}(s) = Q(s)P_n^{-1}(s)y(s) – Q(s)u(s)
$$
where \(u(s)\) is the control signal and \(y(s)\) is the system output (grid current). With the estimated disturbance fed forward, the actual control signal becomes \(u(s) = u_c(s) – \hat{d}(s)\), where \(u_c(s)\) is the output of the primary PI controller. If the nominal model is accurate (\(P_n(s) \approx P(s)\)) and \(Q(s) \approx 1\) in the low-frequency disturbance band, the disturbance rejection transfer function from \(u_g(s)\) to \(i_g(s)\) becomes approximately zero, effectively eliminating the disturbance effect. The low-pass filter \(Q(s)\) is typically chosen as \(1/(1+\tau s)^m\), where \(m\) must be greater than or equal to the relative degree of \(P_n(s)\). For the third-order LCL system, we select \(m=3\). The time constant \(\tau\) determines the disturbance rejection bandwidth; a smaller \(\tau\) provides a wider bandwidth but may increase sensitivity to measurement noise.
With the DOB integrated, the new grid current expression is:
$$
i_g(s) = T_1(s) i_{ref}(s) + T_2(s) u_g(s)
$$
where \(T_1(s) = G_1(s)\) remains unchanged, preserving reference tracking performance. The new disturbance transfer function becomes \(T_2(s) = G_2(s)(1 – Q(s))\). The factor \((1 – Q(s))\) is the key improvement. Since \(Q(s) \approx 1\) at low frequencies, \(T_2(s)\) is greatly attenuated in the harmonic frequency range of interest, significantly improving the grid tied inverter’s immunity to background voltage harmonics.
| Parameter | Symbol | Value |
|---|---|---|
| Rated Power | \(P_0\) | 6 kW |
| DC Link Voltage | \(U_{dc}\) | 360 V |
| Grid Voltage (RMS) | \(U_g\) | 220 V |
| Switching Frequency | \(f_s\) | 10 kHz |
| Fundamental Frequency | \(f_0\) | 50 Hz |
| Inverter-side Inductor | \(L_1\) | 600 µH |
| Grid-side Inductor | \(L_2\) | 360 µH |
| Filter Capacitor | \(C\) | 10 µF |
| PI Controller (Kp, Ki) | \(G_{PI}(s)\) | 0.45, 1400 |
| Capacitor Current Feedback | \(H_{i1}\) | 0.10 |
| Grid Current Feedback | \(H_{i2}\) | 0.15 |
| DOB Time Constant | \(\tau\) | 0.016 ms |
While the DOB effectively suppresses harmonic disturbances, it can exacerbate stability concerns in weak grids. In a weak grid, the Thevenin equivalent grid impedance \(Z_g(s)\) is no longer negligible. The system stability can be analyzed using the impedance-based method, where the grid tied inverter is modeled as a Norton equivalent circuit with an output impedance \(Z_o(s)\). The interaction between the inverter output impedance \(Z_o(s)\) and the grid impedance \(Z_g(s)\) determines the system’s small-signal stability at the point of common coupling (PCC). The stability criterion requires that the ratio \(Z_g(s)/Z_o(s)\) satisfies the Nyquist stability criterion. With the DOB, the inverter’s output impedance is derived as \(Z_o(s) = -1/T_2(s)\).
As the grid inductance \(L_g\) increases (weaker grid), the magnitude of \(Z_g(s) = sL_g\) rises. The intersection frequency \(f_c\) between \(|Z_o(j\omega)|\) and \(|Z_g(j\omega)|\) decreases, and the phase margin (PM) at this crossover, defined as \(PM = 180^\circ + \angle(Z_g(j2\pi f_c)/Z_o(j2\pi f_c))\), may become insufficient. If the PM drops below a critical threshold (e.g., 45°), the system risks instability. This is a critical robustness issue for a grid tied inverter operating in varying grid conditions.
To enhance robustness, we propose augmenting the current disturbance observation method with an additional virtual impedance \(Z_v(s)\). The concept is to reshape the inverter’s equivalent output impedance to be more inductive across a wide frequency range, thereby increasing the phase margin at the impedance intersection frequency for a larger range of \(Z_g(s)\). The virtual impedance is added in series with the original output impedance \(Z_o(s)\), resulting in a new equivalent output impedance \(Z_{eq}(s) = Z_o(s) + Z_v(s)\).
To effectively increase the phase of \(Z_{eq}(s)\) without altering the disturbance rejection properties of the DOB, we design the virtual impedance as:
$$
Z_v(s) = \frac{L_v s}{1 – Q(s)}
$$
where \(L_v\) is the virtual inductance coefficient. The term \((1 – Q(s))\) in the denominator aligns with the structure of \(Z_o(s)\) from the DOB, simplifying the implementation. This virtual impedance is realized not as a physical component but as an additional feedback term in the control law. From the circuit perspective, it is equivalent to placing an inductor \(L_v\) in series between the LCL filter and the grid. Through block diagram manipulation, this is implemented by adding a feedforward term to the current controller’s output:
$$
G_v(s) = \frac{L_v s}{K_{PWM}}
$$
where \(K_{PWM}\) is the inverter bridge gain.
The selection of \(L_v\) involves a trade-off. A larger \(L_v\) increases the phase of \(Z_{eq}(s)\), improving the phase margin against grid impedance and thus the robustness of the grid tied inverter. However, an excessively large \(L_v\) can degrade the stability of the current control loop itself, as it affects the term \(1/(1+Z_v(s)/Z_o(s))\). Therefore, the optimal \(L_v\) must be chosen within a bounded range \([L_{v,min}, L_{v,max}]\) to ensure overall system stability with sufficient margin. Using impedance plots and the Nyquist criterion, we can graphically determine this range. For the given system parameters and a target short-circuit ratio (SCR) corresponding to \(L_g = 2.6 mH\), requiring \(PM_1 \ge 45^\circ\) for \(Z_g/Z_{eq}\) and \(PM_2 \ge 60^\circ\) for \(Z_v/Z_o\), the viable range is found to be \(L_v \in (0.86 mH, 0.93 mH)\). We select \(L_v = 0.9 mH\) for the subsequent analysis.
| Control Method | Grid Condition | Current THD | Stability (PM at f_c) | Key Feature |
|---|---|---|---|---|
| Conventional PI + Damping | Strong Grid (Clean) | ~3.2% | High | Baseline |
| Conventional PI + Damping | Strong Grid (Distorted) | 5.85% | High | Poor harmonic rejection |
| With DOB only | Strong Grid (Distorted) | 0.82% | High | Excellent harmonic rejection |
| With DOB only | Weak Grid (\(L_g=1.8 mH\)) | 16.03% (Oscillatory) | ~0° (Near instability) | Poor robustness |
| DOB + Virtual Impedance | Weak Grid (\(L_g=2.6 mH\)) | 0.97% | >45° | Robust, high quality |
| DOB + Virtual Impedance | Very Weak Grid (\(L_g=3.6 mH\)) | 1.09% | Acceptable | Maintained robustness |
The effectiveness of the proposed composite method—current disturbance observation with additional virtual impedance—is validated through detailed simulation studies. The simulation model is built with the parameters listed in Table 1. The grid voltage is distorted with typical background harmonics: 10% 3rd and 5th, 5% 7th and 9th, and 2% 11th and 13th harmonics. First, the performance under harmonic distortion in a strong grid is evaluated. The conventional dual-loop control yields a grid current with a total harmonic distortion (THD) of 5.85%, which fails to meet grid codes (typically THD < 5%). In contrast, integrating the DOB alone drastically reduces the THD to 0.82%, demonstrating superb harmonic rejection capability for the grid tied inverter.
Next, the robustness under weak grid conditions is tested by inserting a series inductance \(L_g\) between the inverter and the ideal grid voltage source. With only the DOB, when \(L_g\) increases to 1.8 mH, the system approaches its stability limit. The grid current exhibits severe oscillations with a THD of 16.03%, confirming the vulnerability introduced by the DOB under model mismatch caused by grid impedance. This underscores the need for the virtual impedance enhancement. Applying the proposed composite method with \(L_v = 0.9 mH\), the system remains stable even with \(L_g = 2.6 mH\) and \(L_g = 3.6 mH\), delivering grid currents with THDs of 0.97% and 1.09%, respectively. The virtual impedance successfully reshapes the output impedance, ensuring adequate phase margin and thus robust stability for the grid tied inverter, while preserving the high current quality achieved by the DOB.
The stability can be further analyzed by examining the Nyquist plots of the critical impedance ratios. For the system with the proposed method, the Nyquist plot of \(Z_g(s)/Z_{eq}(s)\) for a weak grid (\(L_g=2.6 mH\)) does not encircle the (-1, j0) point and exhibits a large phase margin, confirming stability. Similarly, the plot of \(Z_v(s)/Z_o(s)\) also indicates a stable inner loop with sufficient margin. This analytical result aligns perfectly with the stable, low-THD simulation waveforms.
The design of the DOB’s low-pass filter \(Q(s)\) involves an important compromise. The filter’s bandwidth, inversely related to \(\tau\), must be wide enough to cover the significant harmonic frequencies (e.g., up to the 13th or 25th harmonic). However, an excessively wide bandwidth (very small \(\tau\)) makes the observer sensitive to high-frequency noise and unmodeled dynamics, potentially degrading performance. Furthermore, the order \(m\) of \(Q(s)\) must be at least the relative order of the nominal plant \(P_n(s)\) to ensure the transfer function \(Q(s)P_n^{-1}(s)\) is proper and physically realizable. For an LCL filter model considering only the fundamental dynamics, the relative order is 3, justifying our choice of \(m=3\). The final performance of any grid tied inverter employing this method depends on the accurate identification of the nominal plant parameters \(L_1, L_2, C,\) and \(K_{PWM}\).
In conclusion, this work presents a comprehensive control solution for LCL-type grid tied inverters facing the dual challenges of grid voltage harmonics and weak grid conditions. The proposed method integrates a disturbance observer for precise harmonic suppression and a virtual impedance loop for robust stability. The DOB effectively estimates and cancels the disturbance caused by background harmonics without altering the reference tracking dynamics, ensuring high-quality current injection from the grid tied inverter. The virtual impedance strategically reshapes the inverter’s output impedance characteristic, significantly improving the phase margin at the impedance intersection frequency and thus enhancing the system’s robustness against wide variations in grid impedance. This approach provides a valuable and practical control framework for ensuring reliable and high-performance operation of grid tied inverters in increasingly complex and weak power grid environments. The principles can be extended to three-phase systems and other distributed generation interfaces, contributing to the stable integration of renewable energy.