In my extensive work with power electronic systems, ensuring the high-quality output of a grid connected inverter under non-ideal grid conditions remains a paramount challenge. The proliferation of renewable energy sources has made the grid connected inverter a ubiquitous interface, yet its performance is critically dependent on the stability and purity of the grid voltage. Among various disturbances, grid frequency fluctuations pose a significant threat to the efficacy of conventional harmonic suppression techniques. When the grid frequency deviates from its nominal value, the carefully tuned resonant peaks of standard controllers lose their alignment with the actual harmonic spectrum, leading to a severe deterioration in output current quality. This work presents my comprehensive analysis and solution: a frequency-adaptive repetitive control strategy designed to maintain superior performance for a grid connected inverter even in the presence of substantial grid frequency disturbances.
The core objective of any current control loop in a grid connected inverter is to track a sinusoidal reference with minimal error and distortion. Classical Proportional-Resonant (PR) controllers offer infinite gain at selected frequencies, making them effective for mitigating specific low-order harmonics. However, their performance is intrinsically tied to a fixed frequency. Similarly, repetitive controllers (RC), which are inspired by the internal model principle and iterative learning, embed a model of periodic signals to eliminate steady-state error at all harmonics of the fundamental frequency. The transfer function of a standard “odd-harmonic” repetitive controller in the discrete domain is given by:
$$G_{RC}(z) = \frac{k_r \cdot C_f(z) \cdot z^{-m}}{1 – Q(z)z^{-N/2}}$$
Here, \(k_r\) is the controller gain, \(C_f(z)\) is a compensation filter (often a low-pass or phase-lead filter), \(Q(z)\) is a zero-phase filter for robustness, \(m\) is a phase lead step, and \(N\) is the number of samples per fundamental period (\(N = f_s / f_g\), where \(f_s\) is the sampling frequency and \(f_g\) is the grid frequency). The term \(z^{-N/2}\) represents the delay line for odd-harmonic suppression. The undeniable weakness of this powerful controller is its reliance on the precise value of \(N\). A change in the actual grid frequency \(f_g\) causes a mismatch between the delay line’s period and the true period of the grid disturbance, rendering the controller ineffective and often unstable.
To ground this discussion, let me first establish the mathematical model of the system under consideration. I focus on a three-phase grid connected inverter with an LCL output filter, as this topology is standard for its superior high-frequency attenuation. The power stage, excluding the grid voltage as a disturbance, can be described by a transfer function from the inverter bridge voltage \(V_{inv}\) to the grid current \(I_g\). For controller design, the model is often discretized considering computational and pulse-width modulation (PWM) delays. The plant transfer function \(G_p(z)\) is derived using a Zero-Order Hold (ZOH) equivalent:
$$G_p(z) = \mathcal{Z}\left\{ \frac{1 – e^{-sT_s}}{s} \cdot e^{-sT_d} \cdot G(s) \right\}$$
where \(T_s\) is the sampling period, \(T_d\) is the total computation delay, and \(G(s)\) is the continuous-time transfer function of the LCL filter with active damping. For the inner loop, the capacitor current is typically fed back with gain \(k_c\) to damp the LCL resonance. The outer loop controller for the grid current is a combination of a proportional gain \(K\) and the repetitive controller \(G_{RC}(z)\). A grid voltage feedforward path is also added to improve dynamic response. The stability of the overall system can be analyzed through an equivalent error block diagram. The error signal \(E(z)\) in response to a reference \(R(z)\) and a grid-induced disturbance \(D_s(z)\) can be derived as:
$$E(z) = \frac{[1 – Q(z)z^{-N/2}] \cdot [R(z) + D_s(z)G_p(z)]}{[1 + KG_p(z)][1 – z^{-N/2}H(z)]}$$
where \(H(z)\) is a function encompassing \(K\), \(k_r\), \(C_f(z)\), and \(G_p(z)\). According to the small-gain theorem, a sufficient condition for stability is:
$$|Q(z) – k_r C_f(z) G_{p,eq}(z) z^{-m}|_{\infty} < 1$$
where \(G_{p,eq}(z) = G_p(z)/(1 + KG_p(z))\). This inequality guides the design of the five critical repetitive controller parameters: \(k_r\), \(C_f(z)\), \(Q(z)\), \(m\), and fundamentally, \(N\).
My design procedure for the repetitive controller in a fixed-frequency scenario is meticulous. First, the inner active damping gain \(k_c\) and the outer proportional gain \(K\) are chosen to stabilize the plant \(G_p(z)\) with ample phase margin (e.g., >50 degrees) and reasonable damping. This forms a stable baseline system. Next, the zero-phase low-pass filter \(Q(z)\) is designed. Its purpose is to roll off the high-frequency gain of the repetitive controller to ensure robustness against model uncertainties. A standard second-order form is:
$$Q(z) = \frac{a_1 z + a_0 + a_1 z^{-1}}{a_0 + 2a_1}$$
Setting \(a_0=0.5\) and \(a_1=0.25\) provides unity gain at DC and zero gain at the Nyquist frequency, effectively stabilizing the resonant peaks. The compensator \(C_f(z)\) is typically a simple phase-lead element \(z^m\). The value of \(m\) and the gain \(k_r\) are co-designed by analyzing the magnitude of the critical transfer function \(H(z)\) across frequency. The goal is to maximize \(k_r\) for fast error convergence while minimizing the peak of \(|H(z)|\) for stability. Through graphical or numerical search, optimal values (e.g., \(m=3\), \(k_r=2.8\)) are identified. This designed repetitive controller, when integrated with the grid connected inverter, can reduce the output current Total Harmonic Distortion (THD) to below 1% under a perfectly nominal 50Hz grid.
The fatal flaw emerges with grid frequency drift. If \(f_g\) changes to \(f_g + \Delta f\), the actual required delay line length becomes \(N’ = f_s / (f_g + \Delta f)\), which is not an integer and differs from the fixed \(N\) in the controller. The controller’s internal model no longer matches the external disturbance period. The resonant peaks in the controller’s frequency response are now misaligned from the actual harmonic frequencies present in the grid. This mismatch drastically reduces the gain at the true harmonic frequencies, allowing distortion to pass through, while potentially introducing gain at other frequencies that can cause instability. For a grid connected inverter, this translates to a rapid increase in output current THD, as shown in the table below which summarizes simulation results for different frequency deviations with a fixed RC.
| Grid Frequency (Hz) | Output Current THD (%) with Fixed RC | 5th Harmonic Magnitude (% of Fundamental) |
|---|---|---|
| 50.0 (Nominal) | 0.8 | 0.1 |
| 50.2 | 1.8 | 1.5 |
| 50.4 | 3.5 | 3.2 |
| 50.6 | 5.1 | 4.8 |
To solve this, I developed a frequency-adaptive mechanism. The core idea is to keep the ratio \(n = N/2 = f_s / (2f_g)\) constant by dynamically adjusting the sampling and switching frequency \(f_s\) in response to changes in \(f_g\). This ensures the delay line \(z^{-N/2}\) always corresponds exactly to one half of the actual grid period, perfectly aligning the controller’s internal model with the disturbance period. The implementation involves a dedicated frequency-locked loop (FLL) or a precise period measurement circuit that continuously estimates the real-time grid period \(T_g\). The required PWM counter value \(N_{pwm}\) for the microcontroller, which determines the switching period \(T_{sw} = 1/f_s\), is given by:
$$N_{pwm} = \frac{T_{sw}}{T_{cpu}} = \frac{1}{f_s \cdot T_{cpu}}$$
where \(T_{cpu}\) is the CPU clock period. Since \(n = f_s / (2f_g) = \text{constant}\), we have \(f_s = 2n \cdot f_g\). Substituting this into the equation above yields the relationship between the measured grid frequency and the required hardware counter value:
$$N_{pwm, req} = \frac{1}{2n \cdot f_g \cdot T_{cpu}}$$
A PI controller processes the error between this required \(N_{pwm, req}\) and the current \(N_{pwm}\) value to generate an adjustment. The output of this PI controller continuously updates the PWM period register, thereby modifying \(f_s\) and locking it to the grid frequency. The design of this adaptive PI controller must account for the maximum slew rate of grid frequency changes (e.g., 1 Hz/s per grid codes). Using the final value theorem for a ramp input, the integral gain \(K_i\) can be sized to guarantee that the steady-state period error is within one CPU count, achieving extremely fine resolution. This adaptive scheme essentially makes the grid connected inverter a variable-frequency system, with its control clock seamlessly tracking the grid.
The implications of varying \(f_s\) must be considered. The LCL filter’s resonance frequency \(f_{res} = \frac{1}{2\pi}\sqrt{(L_1+L_2)/(L_1 L_2 C)}\) remains fixed, but the ratio \(f_{sw}/f_{res}\) changes. However, since the variation in \(f_s\) is only ±2% (e.g., 15.68 kHz to 16.32 kHz for a 16 kHz nominal), the impact on filter attenuation and controller stability margins is negligible and can be accommodated by the robustness of the original \(Q(z)\) filter design. The primary digital control transfer functions, now expressed in terms of the constant ratio \(n\), become invariant to grid frequency changes:
$$G_{RC, adaptive}(z) = \frac{k_r \cdot C_f(z) \cdot z^{-m}}{1 – Q(z)z^{-n}} \quad \text{where } n = \text{constant}$$
This represents a significant advancement. The grid connected inverter with this adaptive controller maintains its high harmonic impedance at the exact harmonic frequencies of the actual grid voltage, regardless of the fundamental frequency’s value within a specified range.
Experimental validation on a 15A, 3-phase grid connected inverter prototype confirms the theory. The system parameters were: DC link voltage \(V_{dc} = 700V\), grid voltage \(V_g = 220V_{rms}\), nominal frequency \(f_g = 50Hz\), LCL filter with \(L_1 = 100\mu H\), \(L_2 = 60\mu H\), \(C = 70\mu F\). The controller parameters were as designed earlier: \(K=3\), \(k_c=5\), \(k_r=2.8\), \(m=3\), \(a_0=0.5\), \(a_1=0.25\). The performance was compared against a multi-resonant PR controller tuned for the 3rd, 5th, 7th, and 9th harmonics.
Under a nominal 50Hz grid with 3% voltage THD (primarily 5th harmonic), both the fixed repetitive and the adaptive repetitive controllers achieved an output current THD below 1%, outperforming the PR controller which yielded around 2.3% THD. The critical test involved a dynamic frequency change. The grid frequency was ramped from 50Hz to 50.2Hz at a rate of 1 Hz/s. The results, summarized in the table below, are conclusive.
| Control Strategy | THD during Ramp (Peak) | Steady-State THD at 50.2Hz | THD over 49-51Hz Range (Max) |
|---|---|---|---|
| Proportional + Multi-resonant PR | 3.5% | 2.8% | 5.3% |
| Fixed Repetitive Control (RC) | 3.2% | 1.8% | 8.1% |
| Proposed Adaptive RC | 0.8% | 0.8% | 0.8% |
The adaptive controller maintained a consistently low THD of 0.8% throughout the disturbance and across the wide frequency range, while the performance of both fixed controllers degraded significantly. The slight THD increase for the fixed RC over a wide range can be attributed to the growing mismatch causing not just loss of harmonic attenuation but also minor instability.
In conclusion, the integration of a frequency-adaptive mechanism into the repetitive control framework presents a robust and high-performance solution for the modern grid connected inverter. The method directly addresses the fundamental limitation of periodic controllers by ensuring the internal model period perpetually matches the grid disturbance period. This is achieved through a seamless adjustment of the inverter’s switching clock, a task elegantly handled by modern digital signal processors. The mathematical modeling, parameter design procedure, and stability analysis I have detailed provide a complete roadmap for implementation. The experimental evidence unequivocally demonstrates that this strategy enables a grid connected inverter to achieve and maintain exceptional output current quality, with THD consistently below 1%, irrespective of practical grid frequency fluctuations. This enhances grid compatibility, improves overall system reliability, and is a critical step towards ensuring the stable operation of power networks with high penetration of inverter-based resources.