The increasing penetration of distributed energy resources into medium- and low-voltage distribution networks has made the integration of renewable sources through on-grid inverters ubiquitous. However, these distribution grids often exhibit weak grid characteristics, including uncertain impedance and voltage distortion. Under such weak grid conditions, on-grid inverters face significant challenges. On one hand, voltage harmonic disturbances—comprising both grid background harmonics and those introduced by inverter non-linearities like dead-time effects—severely degrade the inverter’s performance and power quality. On the other hand, substantial system uncertainties exist, stemming from hardware aging, variations in filter circuit parameters, and unpredictable grid impedance. These factors collectively heighten system complexity. Therefore, developing control strategies that effectively suppress harmonics while enhancing system robustness in weak grid environments has become a critical issue for on-grid inverter design.
Active Disturbance Rejection Control (ADRC), which is based on an Extended State Observer (ESO), offers notable robustness and a model-independent nature, making it particularly suitable for operation in weak grids. However, the conventional ESO struggles to precisely estimate the harmonic disturbances present in on-grid inverters due to its limited bandwidth. This limitation leads to a degradation in the steady-state performance of the inverter’s current control loop, resulting in distorted grid current. The theoretical root of this problem lies in the fact that while a traditional ESO can completely eliminate the estimation error for constant disturbances, it can only achieve bounded error estimation for time-varying disturbances. Consequently, a significant body of research focuses on improving the ESO’s accuracy in observing time-varying disturbances. A prominent approach is the Generalized ESO (GESO), which extends the traditional ESO by treating higher-order derivatives of the generalized disturbance as new state variables. In theory, if the nth derivative of a time-varying disturbance is zero, an nth-order GESO can completely eliminate the estimation error. However, harmonic disturbances in the grid are infinitely differentiable. Thus, a very high-order GESO would be required for high observation accuracy, and even then, the error cannot be fully eradicated. Furthermore, an excessively large ESO bandwidth amplifies high-frequency noise in the system, potentially causing control loop oscillations or instability, which further limits harmonic disturbance observation.
Recently, the Generalized Integrator-ESO (GI-ESO) has been proposed. By incorporating the mathematical model of periodic signals into the ESO, it aims to eliminate the observation error for specific frequency periodic disturbances. The GI-ESO partially decouples the relationship between the observed disturbance frequency and the observer bandwidth, allowing for accurate observation of relatively high-frequency specific disturbances at lower bandwidths. This addresses the inherent conflict between observation accuracy and noise amplification in conventional ESOs. Nevertheless, in on-grid inverter systems, harmonic disturbances occur at integer multiples of the fundamental grid frequency, meaning a multitude of different periodic harmonic disturbances coexist. A GI-ESO would need to incorporate numerous resonant terms at different frequencies, introducing a plethora of parameters into the controller without guaranteeing the convergence of the observer’s estimate for the generalized disturbance.
In the field of power conversion, where handling numerous periodic harmonic signals is essential, Repetitive Control (RC) based on the internal model principle is widely adopted. Due to its simple structure and reliable performance, RC can efficiently track or suppress periodic signals, making it highly effective at mitigating grid background harmonics and dead-time-induced harmonics in on-grid inverters. However, the stability and steady-state performance of an RC system heavily depend on the design of its compensator. Common design methods include zero-phase error tracking theory, fractional-order phase compensation, and low-pass filters. While these methods can theoretically ensure system stability, they rely on an accurate model of the plant. When model parameters change, the system’s steady-state performance can deteriorate significantly, potentially leading to instability. Thus, despite its superior harmonic suppression capability, RC suffers from poor robustness and is ill-equipped to handle uncertainties arising from system parameter variations.
To address these limitations, this paper integrates repetitive control into the ESO framework, proposing a Repetitive ESO (RESO). Subsequently, an ADRC strategy based on RESO (RESO-ADRC) is constructed. Compared to the GI-ESO, the proposed RESO can observe a wide spectrum of harmonic disturbances at different frequencies, features fewer observer parameters, has a simpler structure, and offers stronger convergence guarantees. Compared to standalone repetitive control, RESO-ADRC not only suppresses a broad range of periodic harmonic disturbances but also retains the model-independent advantage of traditional ADRC, thereby exhibiting stronger robustness. Consequently, RESO-ADRC ensures both high steady-state performance and strong robustness for on-grid inverters operating under harmonic disturbances and model uncertainties.
The topology of a single-phase LCL-type on-grid inverter system is considered for analysis. The system includes a DC bus, a full-bridge inverter, an LCL filter with passive damping (a resistor in series with the capacitor), and the grid with its impedance. A Phase-Locked Loop (PLL) provides the grid phase for synchronization. The primary control objective is to force the grid current \( i_g(t) \) to track a sinusoidal reference \( i_{ref}(t) \) that is in phase with the grid voltage.
To simplify controller design, the high-order LCL filter model is often reduced. In ADRC, the ignored dynamics can be treated as part of the total disturbance to be observed and compensated. Applying Padé approximation, the transfer function from the inverter output voltage \( u_i(s) \) to the grid current \( i_g(s) \) for an LCL filter with series RC damping can be derived. After inverse Laplace transformation and rearrangement, the dynamic equation can be expressed with the grid current derivative on the left side. By examining the coefficients, it becomes evident that the terms associated with higher-order derivatives of currents and voltages are several orders of magnitude smaller than the dominant term under proper damping conditions. These can be lumped into a generalized disturbance. Furthermore, the precise values of the inverter-side inductance \( L_1 \) and grid-side inductance \( L_2 \) are often unknown, leading to an inaccurate estimate \( b_0 \) of the true input gain \( b = 1/(L_1 + L_2) \). The discrepancy \( (b – b_0)u_i(t) \), along with the dead-time-induced voltage harmonic \( u_\sigma(t) \), is also included in the generalized disturbance \( f(t) \).
The complete expression for the generalized disturbance \( f(t) \) encapsulates the effects of grid voltage and its derivatives, grid current derivatives, the dead-time effect, and the input gain mismatch. Critically, the spectral content of \( f(t) \) is primarily concentrated at the fundamental grid frequency and its integer harmonics. This is because \( i_g(t) \) is ideally a 50/60 Hz sinusoid, dead-time introduces odd harmonics, and \( u_i(t) \) responds linearly to these inputs. After consolidating these high-order and non-ideal terms, the system model is effectively reduced to a first-order form:
$$ \dot{i}_g(t) = f(t) + b_0 u_i(t) $$
where \( b_0 \) is the nominal (design) input gain. The accuracy of the ESO in observing \( f(t) \) is paramount for the performance of the ADRC strategy.
Discretizing the model and treating \( f(t) \) as an extended state yields the extended state equation. The conventional discrete-time ESO is then given by:
$$ \begin{aligned}
e_{1,k} &= i_{g,k} – \hat{i}_{g,k} \\
\hat{i}_{g,k+1} &= \hat{i}_{g,k} + T_s \left( b_0 u_{k} + \hat{f}_k \right) + T_s h_1 e_{1,k} \\
\hat{f}_{k+1} &= \hat{f}_k + T_s h_2 e_{1,k}
\end{aligned} $$
where \( T_s \) is the sampling period, \( \hat{i}_{g,k} \) and \( \hat{f}_k \) are the estimates, and \( h_1, h_2 \) are observer gains typically set based on the observer bandwidth \( \omega_0 \) as \( h_1 = 2\omega_0 \), \( h_2 = \omega_0^2 \). The integral action on the error \( e_{1,k} \) for disturbance estimation, based on the internal model principle, only contains the model for a DC signal (integrator). Therefore, it cannot precisely observe the multitude of periodic harmonic signals present in the on-grid inverter’s disturbance.
The core idea of the Repetitive ESO (RESO) is to embed a Proportional-Repetitive Control (PRC) block as the internal model for disturbances within the observer. The structure modifies the disturbance estimate update law to:
$$ \hat{f}_k = G_{prc}(z) \left( i_{g,k} – \hat{i}_{g,k} \right) $$
where \( G_{prc}(z) \) is the transfer function of the PRC controller:
$$ G_{prc}(z) = k_p + \frac{S(z) Q(z) z^{-N}}{1 – Q(z) z^{-N}} $$
Here, \( k_p \) is a proportional gain, \( S(z) \) is a compensator designed to ensure convergence, \( Q(z) \) is a zero-phase low-pass filter that shapes the observer bandwidth, and \( N = f_s / f_1 \) is the ratio of sampling frequency to fundamental frequency.
The compensator \( S(z) \) is designed using the perfect tracking controller principle. For the RESO’s internal loop, the plant from the repetitive control output \( y_{rc} \) to the estimated current \( \hat{i}_g \) is:
$$ P(z) = \frac{T_s}{z^2 – z + k_p T_s} $$
The compensator is chosen as the inverse of this nominal plant: \( S(z) = P^{-1}(z) = f_s^2 z^{-2} – f_s z^{-1} + k_p \). This non-causal compensator can be realized by combining the non-causal terms \( z^{+1}, z^{+2} \) with the inherent delay \( z^{-N} \) of the repetitive controller. With this compensator, the PRC transfer function becomes:
$$ G_{prc}(z) = k_p + \frac{Q(z) z^{-N} (f_s^2 z^{-2} – f_s z^{-1} + k_p)}{1 – Q(z) z^{-N}} $$
Crucially, \( S(z) \) acts inside the RESO and is based on the nominal, parameter-independent dynamics of the observer’s own structure, not the actual on-grid inverter plant. This makes the compensator inherently robust to variations in the inverter’s LCL filter parameters.
The bandwidth of the RESO is regulated by the zero-phase low-pass filter \( Q(z) \), which attenuates the high-frequency gain of the repetitive control action, limiting noise amplification. A simple first-order zero-phase filter can be designed as \( Q(z) = \alpha_0 z + (1-2\alpha_0) + \alpha_0 z^{-1} \), where \( \alpha_0 > 0 \). The parameter \( \alpha_0 \) adjusts the cutoff frequency, effectively setting the observation bandwidth for periodic disturbances. This allows the RESO to focus on estimating low and mid-frequency harmonics critical for on-grid inverter performance while rejecting high-frequency noise.
The convergence of the RESO is analyzed by examining the error dynamics. The transfer function from the actual disturbance \( f_k \) to the disturbance estimation error \( e_{2,k} = f_k – \hat{f}_k \) is derived. Substituting the designed \( G_{prc}(z) \) and \( S(z) \) leads to the characteristic equation of the RESO. A key finding is that when the compensator is perfectly designed (\( S(z) = P^{-1}(z) \)), the poles introduced by the repetitive control internal model \( (1 – Q(z)z^{-N}) \) are canceled from the characteristic equation. Consequently, the convergence of the RESO depends solely on the parameter \( k_p \), with a simple stability range of \( 0 < k_p < 1/T_s \). This demonstrates that the repetitive control mechanism does not compromise the observer’s convergence, which remains model-independent.
The RESO-ADRC control law combines proportional feedback with disturbance compensation and reference feedforward:
$$ u_{k+1} = \frac{1}{b_0} \left[ k_c (i_{ref,k+1} – \hat{i}_{g,k+1}) – \hat{f}_{k+1} + \frac{i_{ref,k+1} – i_{ref,k}}{T_s} \right] $$
The derivative feedforward of the reference signal helps reduce phase lag in tracking the sinusoidal current reference. The closed-loop system can be represented as a two-degree-of-freedom controller. Assuming perfect input gain matching (\( b_0 = b \)), the transfer functions from reference to current and from disturbance to current are derived.
The stability of the RESO-ADRC system is analyzed via its characteristic equation. The parameters \( k_p \) and \( k_c \) primarily determine the closed-loop pole locations. Their stable ranges are found to be:
$$ 0 < k_p < f_s, \quad 0 < k_c < 2f_s $$
These ranges are conveniently dependent only on the sampling frequency \( f_s \), allowing for straightforward tuning when the plant parameters are roughly known.
The disturbance rejection capability is evaluated by the sensitivity function. Compared to traditional ADRC, where increasing the ESO bandwidth \( \omega_0 \) uniformly improves disturbance attenuation but amplifies noise, RESO-ADRC provides exceptionally high gain (and thus strong attenuation) specifically at the harmonic frequencies embedded in the repetitive controller’s internal model. This targeted suppression significantly improves steady-state current quality in the on-grid inverter.
The tracking performance for the sinusoidal reference is determined by the complementary sensitivity function. Increasing \( k_c \) pushes the magnitude gain at the fundamental frequency closer to 1 and reduces phase lag, improving reference tracking accuracy.
Robustness to input gain mismatch (\( b_0 \neq b \)) is crucial. The analysis shows that stability can be maintained for a range of the gain ratio \( K = b_0 / b \). A critical stability surface \( K_s(k_c, k_p) \) can be determined numerically. The system remains stable if \( b_0 > K_s(k_c, k_p) \cdot b \). This implies that to enhance robustness against model uncertainty (e.g., unknown \( L_1, L_2 \)), one should choose a conservatively large value for \( b_0 \). While a larger \( b_0 \) increases the initial model mismatch and may slightly slow down the dynamic response as the RESO needs to learn a larger disturbance component, it trades off for greater robust stability. The parameter \( k_p \) has a more pronounced effect on robustness than \( k_c \); reducing \( k_p \) can improve robustness.
Parameter design guidelines are summarized as follows:
- Observer Bandwidth (\( \alpha_0 \)): Select based on the highest harmonic frequency to be suppressed and noise considerations. A typical \( \alpha_0 = 0.5 \) offers a balanced bandwidth.
- RESO Proportional Gain (\( k_p \)): For a well-damped response without overshoot in the observer’s estimation, set \( k_p = f_s / 4 \).
- Controller Gain (\( k_c \)): For good reference tracking and stability margin, set \( k_c = f_s \).
- Nominal Input Gain (\( b_0 \)): Choose a value larger than the best estimate of \( b = 1/(L_1+L_2) \) to ensure robustness. The dynamic response will be acceptable as long as the RESO can converge.
The proposed RESO-ADRC strategy was validated on a 1.5 kW single-phase three-level on-grid inverter experimental platform with a switching/sampling frequency of 10 kHz. The controller parameters were set as: \( M=1, \alpha_0=0.5, k_p=2500, k_c=10000, b_0=1000 \). Two different LCL filter parameter sets (Filter A and Filter B) were used to test robustness.
The steady-state performance under Filter A was compared for RESO-ADRC, traditional RC, and traditional ADRC. While both RESO-ADRC and RC achieved low current THD (0.7% and 0.9% respectively), traditional ADRC resulted in significantly higher THD (11.2%) due to its poor harmonic suppression. This demonstrates the superior steady-state accuracy of the proposed method for on-grid inverters.
The robustness was tested by switching the physical LCL filter from set A to set B without changing any controller parameters. RESO-ADRC and traditional ADRC remained stable with minimal performance change (RESO-ADRC THD stayed at 0.8%, ADRC at 9.7%). However, the traditional RC system, whose compensator was designed specifically for Filter A, became unstable when connected to Filter B. This experiment clearly highlights the model-dependent fragility of RC and the model-independent robustness of both ADRC and the proposed RESO-ADRC for on-grid inverters.
Further tests under severely distorted weak grid voltage (THD 13.3%) showed that RESO-ADRC maintained a grid current THD of only 1.7%, confirming its excellent ability to reject grid voltage harmonics.
The impact of the input gain \( b_0 \) on dynamic performance was investigated. With the true \( b = 333 \) (for Filter A), dynamic responses for step changes in reference current were recorded for \( b_0 = 333, 650, \) and \( 1000 \). As \( b_0 \) increased, the settling time increased slightly (40ms, 60ms, 86ms), confirming the trade-off between robustness (larger \( b_0 \)) and dynamic speed. However, stability and good tracking were maintained in all cases.
Finally, the stability boundaries for parameters \( k_p \), \( k_c \), and \( b_0 \) were experimentally verified. Instability occurred when \( k_p \) exceeded \( f_s \), when \( k_c \) exceeded \( 2f_s \), and when \( b_0 \) was reduced below the critical value \( K_s \cdot b \), aligning perfectly with the theoretical analysis.
Additional tests with varying grid impedance \( L_g \) from 0 to 4 mH showed that RESO-ADRC maintained current THD below 1.0% in all cases, demonstrating its robustness to grid impedance uncertainty, a common challenge for on-grid inverters in weak networks.
| Grid Impedance \( L_g \) (mH) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Current THD (%) | 0.7 | 0.8 | 0.8 | 0.7 | 0.6 |
In summary, to address the limitation of traditional ADRC in suppressing periodic harmonics in on-grid inverters, this paper proposed a novel Active Disturbance Rejection Control strategy based on a Repetitive Extended State Observer (RESO-ADRC). The method integrates a Repetitive Control internal model into the ESO, endowing it with the capability to accurately observe harmonic disturbances. A key design feature is the compensator, which is based on the RESO’s own nominal dynamics rather than the uncertain plant model, ensuring convergence and robustness. The RESO’s bandwidth is conveniently shaped by a zero-phase low-pass filter. The resulting RESO-ADRC strategy exhibits high steady-state accuracy (low THD) due to precise harmonic observation and strong robustness to parameter variations due to its model-independent design core. Comprehensive stability, performance, and robustness analyses were provided, along with clear parameter design guidelines. Experimental results on a 1.5 kW on-grid inverter platform confirmed the effectiveness of the proposed method. It demonstrated superior harmonic suppression compared to traditional ADRC and significantly better robustness against plant parameter changes compared to traditional repetitive control, making it a highly promising solution for high-performance on-grid inverters operating in challenging weak grid environments.