The large-scale integration of renewable energy sources, primarily interfaced through power electronic converters like grid-connected inverters, has become a cornerstone of modern power systems. While this shift is essential for sustainability, it introduces a critical challenge: the significant reduction of system inertia and damping traditionally provided by rotating masses in synchronous generators. This decline deteriorates system frequency stability and dynamic performance, making the grid more vulnerable to disturbances and threatening power quality and security. To address this inherent weakness of inverter-dominated grids, the Virtual Synchronous Generator (VSG) control strategy has been proposed as a promising solution. By emulating the swing equation of a synchronous generator, a grid-connected inverter under VSG control can provide virtual inertia and damping, thus supporting system frequency.
However, the conventional VSG implementation employs fixed virtual inertia (J) and damping (D) coefficients. This rigidity often leads to a trade-off between rapid frequency deviation suppression and steady-state oscillation mitigation. During transient events, fixed parameters may result in excessive frequency excursions or poor dynamic response. Adaptive VSG strategies that adjust J and D in real-time have been developed to overcome this limitation. Yet, many of these adaptive methods rely heavily on accurate mathematical models of the system, which are difficult to obtain in practice and may lack robustness against model uncertainties and varying operating conditions.
This article proposes a novel hierarchical control strategy for grid-connected inverters to achieve superior frequency regulation. The core innovation lies in the synergistic combination of a Fuzzy Rule-Based (Fuzzy) adaptive parameter tuner and a Weight-Adaptive Model Predictive Controller (WAMPC). The Fuzzy logic module operates as a slow, proactive adjuster. It dynamically updates the VSG’s J and D coefficients based on real-time measurements of frequency deviation ($\Delta \omega$) and its rate-of-change ($d\omega/dt$), without requiring an explicit system model. This provides a foundational “inertial tone” to the system. Building upon this, the WAMPC acts as a fast, reactive compensator. It predicts future frequency deviations and dynamically adjusts the weighting coefficients in its cost function to optimally compensate the VSG’s active power reference. This dual-layer approach ensures robust frequency stability during both slow load variations and severe transient faults.
The performance of the proposed WAMPC-Fuzzy-VSG strategy is rigorously validated against traditional VSG, Fuzzy-VSG, MPC-VSG, MPC-Fuzzy-VSG, and a referenced WAMPC-IDA-VSG strategy through comprehensive simulations in MATLAB/Simulink under various disturbance scenarios, including load switching, three-phase short-circuit faults, and grid voltage/frequency variations. The results demonstrate a significant improvement in minimizing frequency deviation, reducing overshoot, and enhancing the overall dynamic response of the grid-connected inverter system.
Fundamental Principles and Limitations of Conventional VSG Control
The primary objective of VSG control is to endow a grid-connected inverter with the external characteristics of a synchronous generator. The core of this emulation is the implementation of the swing equation and droop control. The active power-frequency loop is governed by the following equations, which are central to the operation of any grid-connected inverter aiming to provide grid support:
$$
J \frac{d \omega}{d t} = P_m – P_e – D(\omega – \omega_g)
$$
$$
P_{ref} + k_p (\omega_g – \omega) = P_m
$$
Here, $\omega$ is the VSG’s output angular frequency, $\omega_g$ is the grid’s rated angular frequency, $P_m$ is the virtual mechanical power, $P_e$ is the measured inverter output active power, and $P_{ref}$ is the active power reference. The parameters $J$, $D$, and $k_p$ represent the virtual inertia, virtual damping, and frequency droop coefficient, respectively. The reactive power-voltage loop is typically managed by a complementary droop equation: $E = E_0 + k_q (Q_{ref} – Q_e)$, where $E$ is the output voltage amplitude, $E_0$ is the rated voltage, $k_q$ is the voltage droop coefficient, $Q_{ref}$ is the reactive power reference, and $Q_e$ is the output reactive power.
In a standard VSG implementation for a grid-connected inverter, these parameters ($J$, $D$, $k_p$, $k_q$) are constant. The control structure integrates these equations with inner voltage and current loops to generate PWM signals for the inverter bridge. While effective in providing basic inertia support, this fixed-parameter approach has clear drawbacks. A large, fixed $J$ can slow down the frequency response, allowing larger deviations before corrective action, while a small $J$ may lead to excessive frequency changes. Similarly, a fixed $D$ may not optimally damp oscillations across different disturbance types and sizes. This limitation motivates the development of adaptive strategies for grid-connected inverters to dynamically adjust these critical parameters.
Design of the Fuzzy Rule-Based Adaptive Module
To overcome the model-dependency of many adaptive VSG methods, a Fuzzy Logic Controller (FLC) is designed to adjust $J$ and $D$ in real-time. The FLC uses the measurable system states—frequency deviation ($\Delta \omega = \omega_g – \omega$) and its derivative ($d\omega/dt$)—as inputs. This makes it highly suitable for the complex, nonlinear environment of a grid-connected inverter without needing an explicit analytical model. The output of the FLC is the incremental change for inertia ($\Delta J$) and damping ($\Delta D$), which are then added to their nominal values: $J = J_0 + \Delta J$, $D = D_0 + \Delta D$.
The design process involves fuzzification, rule base definition, and defuzzification. The inputs $\Delta \omega$ and $d\omega/dt$ are normalized and mapped onto fuzzy sets using membership functions. Five linguistic variables are defined for each: Negative Large (NL), Negative Small (NS), Zero (ZO), Positive Small (PS), and Positive Large (PL). The shapes of the membership functions are tuned to provide smooth transitions; for instance, the “ZO” region for $\Delta \omega$ is wider to promote steady-state stability, while the “PL/NL” regions for $d\omega/dt$ are narrower to enable fast response to rapid frequency changes.
The rule base is the core of the FLC’s intelligence, encoding the desired relationship between system state and parameter adjustment for the grid-connected inverter. The rules are formulated based on the physical understanding of $J$ and $D$:
- Virtual Inertia (J): Primarily counteracts the rate of frequency change. Therefore, when the frequency is deviating ($\Delta \omega \times d\omega/dt > 0$), $J$ should be increased to slow down the deviation. During recovery ($\Delta \omega \times d\omega/dt < 0$), $J$ should be decreased to allow faster return to nominal frequency.
- Virtual Damping (D): Primarily dissipates oscillation energy related to the frequency deviation itself. A positive $D$ is generally needed when there is a significant frequency deviation, regardless of the direction of change, to damp the oscillations.
The rule tables for $\Delta J$ and $\Delta D$ are constructed accordingly. Table 1 and Table 2 summarize the fuzzy rules, where the rows and columns correspond to $d\omega/dt$ and $\Delta \omega$, respectively, and the cells contain the output linguistic variable.
| $\Delta J$ | $\Delta \omega$ | ||||
|---|---|---|---|---|---|
| NL | NS | ZO | PS | PL | |
| NL | PL | PL | PS | ZO | NS |
| NS | PS | PS | ZO | ZO | ZO |
| ZO | ZO | ZO | ZO | ZO | PS |
| PS | ZO | ZO | ZO | ZO | PL |
| PL | NS | ZO | PS | PL | PL |
| $\Delta D$ | $\Delta \omega$ | ||||
|---|---|---|---|---|---|
| NL | NS | ZO | PS | PL | |
| NL | PL | PS | ZO | ZO | PS |
| NS | PS | PS | ZO | ZO | PS |
| ZO | PS | PS | ZO | ZO | PS |
| PS | PS | ZO | ZO | PS | PS |
| PL | PS | ZO | ZO | PS | PL |
The final step is defuzzification, where the fuzzy output sets are converted into crisp numerical values for $\Delta J$ and $\Delta D$. The centroid method is used for this purpose. The resulting control surfaces, showing $\Delta J$ and $\Delta D$ as functions of the two inputs, are smooth and continuous, ensuring gradual parameter changes that avoid abrupt transitions in the grid-connected inverter’s behavior.
Integrated WAMPC-Fuzzy-VSG Control Strategy
While the Fuzzy module optimally adjusts the VSG’s “plant” parameters (J and D), there remains a need for a fast, optimal control action to directly compensate the active power reference. This is where Model Predictive Control (MPC) excels. However, a standard MPC with fixed weights in its cost function cannot optimally balance the trade-off between frequency deviation minimization and control effort (power compensation) across different disturbance severities. Therefore, we propose a Weight-Adaptive MPC (WAMPC) that works in tandem with the Fuzzy module.
MPC Formulation for the Grid-Connected Inverter
The MPC is designed to act upon the active power loop. Assuming small frequency deviations ($\omega \approx \omega_g$), the swing equation can be linearized and expressed in a state-space form suitable for a grid-connected inverter:
$$
\dot{\omega} = -\frac{D}{J}\omega + \frac{1}{J}P_m – \frac{1}{J}P_e
$$
$$
\Delta \omega = \omega_g – \omega
$$
This continuous-time model is discretized using the zero-order hold (ZOH) method to obtain a discrete-time prediction model for the MPC algorithm:
$$
\Delta \omega(k+1) = A \Delta \omega(k) + B_m \Delta P_m(k) + B_e \Delta P_e(k)
$$
where $k$ is the discrete time step, $\Delta P_m$ is the control input (MPC’s compensation to the VSG’s power reference), and $\Delta P_e$ is a measurable disturbance (change in output power). The matrices $A$, $B_m$, and $B_e$ are derived from the system parameters $J$ and $D$, which are now time-varying outputs from the Fuzzy module.
Based on this model, a prediction equation over a horizon $N_p$ (prediction steps) is constructed:
$$
\mathbf{Y}(k+1|k) = \mathbf{S}_A \Delta \omega(k) + \mathbf{S}_m \Delta \mathbf{P}_m(k) + \mathbf{S}_e \Delta \mathbf{P}_e(k) + \mathbf{I} y(k)
$$
Here, $\mathbf{Y}$ is the vector of predicted frequency deviations, $\Delta \mathbf{P}_m$ is the vector of future control inputs, and $\mathbf{S}_A$, $\mathbf{S}_m$, $\mathbf{S}_e$ are matrices constructed from $A$, $B_m$, and $B_e$. The term $y(k)$ is the current measured output.
The core of the MPC is a rolling optimization problem solved at each time step $k$ for the grid-connected inverter:
$$
\min_{\Delta \mathbf{P}_m(k)} J(k) = \lVert \mathbf{\Gamma}_a ( \mathbf{Y}(k+1|k) – \mathbf{R}(k+1) ) \rVert^2 + \lVert \mathbf{\Gamma}_b \Delta \mathbf{P}_m(k) \rVert^2
$$
$$
\text{subject to: } \mathbf{Y}_{min} \leq \mathbf{Y}(k+1|k) \leq \mathbf{Y}_{max}
$$
The cost function $J(k)$ has two competing terms. The first term penalizes deviations of the predicted frequency from its reference (typically zero, so $\mathbf{R}(k+1)=0$). The second term penalizes the control effort, i.e., the magnitude of the power compensation. $\mathbf{\Gamma}_a = \text{diag}(a, a, …)$ and $\mathbf{\Gamma}_b = \text{diag}(b, b, …)$ are diagonal weighting matrices. The constraints $\mathbf{Y}_{min}$ and $\mathbf{Y}_{max}$ enforce frequency deviation limits (e.g., $\pm 0.2$ Hz as per grid codes). After solving this quadratic programming (QP) problem, only the first element of the optimal control sequence $\Delta \mathbf{P}_m^*(k)$ is applied to the VSG’s power reference: $P_{m,new} = P_m + \Delta P_m^*(k|k)$. This process repeats at every sampling instant.
Dynamic Weight Adaptation Law
The key innovation is making the weights $a$ and $b$ adaptive. The goal is to dynamically prioritize frequency recovery during large disturbances and minimize control effort during steady-state or small deviations for the grid-connected inverter. The adaptation rules are designed as follows:
- Frequency Deviation Weight ($a$): This weight should increase with the absolute frequency deviation to prioritize frequency restoration. An exponential function provides a suitable smooth and saturating characteristic:
$$ a = \begin{cases}
a_0, & \text{if } |\Delta \omega| \cdot (d\omega/dt) \leq 0 \\
a_0 + k_1 (1 – e^{-k_2 |\Delta \omega|} ), & \text{if } |\Delta \omega| \cdot (d\omega/dt) > 0
\end{cases} $$
Here, $a_0$ is the nominal steady-state weight. $k_1$ sets the maximum increase, and $k_2$ determines the sensitivity to $\Delta \omega$. - Control Effort Weight ($b$): This weight should decrease during large and rapid frequency changes to allow the MPC to apply stronger control actions. A hyperbolic tangent function based on the rate-of-change is used to ensure a smooth and bounded decrease:
$$ b = \begin{cases}
b_0, & \text{if } |\Delta \omega| \cdot (d\omega/dt) \leq 0 \\
b_0 – k_3 \cdot \tanh\left( \frac{|d\omega/dt|}{\alpha} \right), & \text{if } |\Delta \omega| \cdot (d\omega/dt) > 0
\end{cases} $$
where $b_0$ is the nominal weight, $k_3 < b_0$ ensures $b$ remains positive, and $\alpha$ is a smoothing factor. The condition $|\Delta \omega| \cdot (d\omega/dt) > 0$ ensures adaptation only when the frequency is actively deviating from nominal.
This WAMPC module receives the updated $J$ and $D$ from the Fuzzy block to refresh its prediction model. It then uses the measured $\Delta \omega$ and $d\omega/dt$ to compute the adaptive weights $a$ and $b$. Finally, it solves the optimization problem to generate the optimal power compensation $\Delta P_m$ for the grid-connected inverter. This creates a powerful synergy: the Fuzzy logic sets favorable system dynamics (inertia/damping), and the WAMPC provides an optimized, context-aware control action to exploit those dynamics for the best possible frequency response.
Simulation Analysis and Performance Validation
A detailed simulation model of a grid-connected inverter system was built in MATLAB/Simulink to validate the proposed WAMPC-Fuzzy-VSG strategy. The system consists of a DC source, a three-phase inverter with an LCL filter, the proposed control blocks, and a connection to an infinite bus (grid) through a transformer and local loads. The performance was compared against five other control strategies: 1) Conventional VSG (Fixed J, D), 2) Fuzzy-VSG, 3) MPC-VSG (MPC with fixed weights, fixed J, D), 4) MPC-Fuzzy-VSG (MPC with fixed weights, fuzzy-based J, D), and 5) a referenced WAMPC-IDA-VSG strategy from literature (with model-based inertia adaptation and single-weight adaptation). Key system parameters are summarized in Table 3.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| DC Link Voltage ($U_{dc}$) | 800 V | Rated Ang. Freq. ($\omega_g$) | 314 rad/s |
| Filter Resistance ($R_f$) | 0.1 $\Omega$ | Droop Coeff. ($k_p$) | 1×10⁻⁴ |
| Filter Inductance ($L_f$) | 2 mH | Droop Coeff. ($k_q$) | 1.5×10⁻⁴ |
| Filter Capacitance ($C_f$) | 30 µF | Nominal Active Power ($P_{ref}$) | 10 kW |
Scenario 1: Step Load Change. Initially, the grid-connected inverter supplies a 10 kW load. At t=2 s, the load is stepped up to 25 kW, and at t=4.5 s, it is stepped back down to 10 kW. Figure X (conceptual) shows the system frequency response. The proposed WAMPC-Fuzzy-VSG strategy demonstrates the best performance. It not only achieves the smallest frequency deviation (0.041 Hz vs. 0.069 Hz for conventional VSG, an improvement of ~40.6%) but also shows the fastest recovery and the least overshoot when the load is removed. The analysis of the active power output reveals that the WAMPC effectively provides additional power compensation on top of the VSG’s inherent droop response, and the adaptive weights enable a more aggressive yet stable compensation during the transient.
Scenario 2: Three-Phase Short-Circuit Fault. A severe transient disturbance was tested. A three-phase-to-ground fault occurs at the Point of Common Coupling (PCC) at t=2 s and is cleared after 0.5 seconds. During the fault, the voltage collapses, and the inverter’s output power drops to near zero. The critical metrics are the maximum frequency dip during the fault and the overshoot during the post-fault recovery. As shown in the results, the WAMPC-Fuzzy-VSG controller significantly outperforms all others. It limits the maximum frequency deviation to 0.06 Hz, a 53.5% improvement over conventional VSG (0.129 Hz). More importantly, during the recovery, it minimizes the frequency overshoot to 0.077 Hz, a 51.6% reduction compared to conventional VSG. This highlights the superior dynamic resilience provided by the combined fuzzy and adaptive-MPC action for the grid-connected inverter.
Scenario 3 & 4: Grid Voltage and Frequency Disturbances. The robustness of the controller was further tested against grid-side disturbances. In one test, the grid voltage experienced a temporary rise. In another, the grid frequency itself was perturbed. In both cases, the proposed strategy consistently maintained the tightest frequency regulation around the nominal 50 Hz, with the smallest deviation and fastest settling time compared to all benchmark strategies, including WAMPC-IDA-VSG. This demonstrates the generality and robustness of the fuzzy-based parameter adaptation over model-based methods.
Power Quality Assessment. A critical concern with advanced control strategies is their potential impact on the output waveform quality of the grid-connected inverter. Simulations confirmed that during load switching events, the WAMPC-Fuzzy-VSG control does not introduce voltage distortion. The current waveform maintains low Total Harmonic Distortion (THD), with values well below 2% (e.g., 1.038%, 0.965%, 1.034% for phases A, B, C respectively) after transients settle, complying with standard power quality requirements.
Conclusion and Future Perspective
This article has presented a comprehensive and effective frequency control strategy for grid-connected inverters, addressing the critical challenge of low inertia in renewable-rich power systems. The proposed WAMPC-Fuzzy-VSG method leverages the complementary strengths of two intelligent control paradigms. The fuzzy logic system provides a model-free, robust mechanism for the real-time adaptation of virtual inertia and damping, establishing a dynamic foundation for system response. The weight-adaptive model predictive controller builds upon this foundation, performing a multi-step optimized compensation of the active power reference. The dynamic adjustment of the cost function weights allows the controller to intelligently prioritize frequency recovery during severe disturbances while minimizing control effort during steady-state operation.
Extensive simulation studies under various disturbance scenarios—including load changes, symmetrical faults, and grid anomalies—validate the superior performance of the proposed strategy. It consistently achieves smaller frequency deviations, reduced overshoot, and faster settling times compared to conventional VSG and other advanced control strategies like Fuzzy-VSG, MPC-VSG, and even a referenced WAMPC-IDA-VSG method. The strategy proves to be a robust and practical solution for enhancing the frequency support capability of grid-connected inverters.
Future research will focus on scaling this approach to systems with multiple parallel grid-connected inverters operating in a microgrid or cluster. The core challenge is coordinated control. A promising direction is to embed the local WAMPC-Fuzzy-VSG controllers within a distributed cooperative framework based on consensus protocols. In such a framework, individual inverters would communicate limited information (e.g., local frequency measurements or control intentions) with their neighbors. Through consensus algorithms, they could collectively determine an optimal system-wide frequency response trajectory. Each local WAMPC controller would then use this consensus-based reference or constraint in its optimization, ensuring that its actions are not only locally optimal but also globally supportive of the overall network stability. This would extend the benefits of the proposed strategy from a single grid-connected inverter to a collaborative network of inverters, paving the way for truly resilient and self-organizing inverter-based power systems.