New Approaches in Finite Control Set Model Predictive Control for Grid-Connected Photovoltaic Inverters: State of the Art

Abstract

Grid-connected PV inverters require sophisticated control procedures for smooth integration with the modern electrical grid. The ability of FCS-MPC to manage the discrete character of power electronic devices is highly acknowledged, since it enables direct manipulation of switching states without requiring modulation techniques. This review discusses the latest approaches in FCS-MPC methods for PV-based grid-connected inverter systems. It also classifies these methods according to control objectives, such as active and reactive power control, harmonic suppression, and voltage regulation. The application of FCS-MPC particularly emphasizing its benefits, including quick response times, resistance to changes in parameters, and the capacity to manage restrictions and nonlinearities in the system without the requirement for modulators, has been investigated in this review. Recent developments in robust and adaptive MPC strategies, which enhance system performance despite distorted grid settings and parametric uncertainties, are emphasized. This analysis classifies FCS-MPC techniques based on their control goals, optimal parameters and cost function, this paper also identifies drawbacks in these existing control methods and provide recommendation for future research in FCS-MPC for grid-connected PV-inverter systems.

1. Introduction

The large integration of renewable energy resources (REs) particularly PVs in the modern electrical grid system is causing a rapid advancement in power converter control. The control approaches are needed to be fast, resilient, and robust against the changing grid conditions. Finite control set model predictive control (FCS MPC) has become very power in power electronics [1,2] due to its simple yet efficient design. The MPC is regarded as one of the most significant advancements and were initially used in the process industry [3,4]. The latest developments in microcontrollers have enabled the deployment of such model-based predictive control approaches. The predictive control allows the inclusion of system constraints and nonlinearities, it predicts the future behaviors of variables by using the discrete mathematical model of the system. After considering the model’s requirements and constraints, a cost function is then designed, which is optimized over a prediction horizon for known combinations of control outputs o this entire process is repeated during each sampling time [5]. MPC has emerged as a significant and promising technique for controlling power electronics systems. MPC is a control strategy that uses a mathematical model of the system being controlled to predict its behavior over a future time horizon, and then optimizes a control action based on those predictions. This predictive capability allows MPC to optimize the control actions in real-time, considering system constraints, objectives, and disturbances, leading to superior performance compared to traditional control methods. The significance of MPC in power electronics lies in its ability to address several challenges associated with modern power electronics systems. One of the key challenges is the increasing complexity of power electronics systems, such as multilevel converters, grid-connected inverters, and electric vehicle charging stations, which require sophisticated control strategies to achieve optimal performance. MPC provides a systematic and flexible approach to tackle this complexity by leveraging mathematical models and optimization techniques to handle nonlinearities, uncertainties, and constraints, making it suitable for a wide range of power electronics applications. The MPC has many advantages over traditional linear controllers, including faster response, high robustness to parameter variation, explicit multivariable control accounting for the process and actuator, and the ability to handle system complexities and nonlinearities [6]. MPC is suitable for microgrids due to its features that have low complexity, no modulator used, variable switching frequency, online optimization, nonlinearity, and constant switching frequency [7]. MPC is a model-based, optimal, predictive, iterative, and feedback-based control method that can handle complex and nonlinear system characteristics. MPC can provide fast and accurate tracking of reference signals, reduce the impact of disturbances and uncertainties on the system, and ensure the stability of the system.

MPC for power electronics are subcategorized as continuous control set (CCS-MPC) and finite control set (FSC-MPC) [8]. In continuous control set-type MPC, the model along with the control inputs are continuous, which are applied through a modulating signal; however, this type of approach is not suitable for power electronics [9]. On the other hand, in finite control set-type MPC, the model is discretized with a constant sampling time. Power electronics devices have finite sets of switching states with corresponding known outputs and the discrete nature of power electronics suits best with the concept of FCS-MPC as it offers the switches to be manipulated exclusively by the MPC controller without the need for the modulator [10]. In FCS-MPC for power electronics, the cost function is optimized during each sampling time, the future behavior of power electronics is predicted for defined samples of time also called the predictions horizon, and the optimal switching state is saved to be applied in the next sampling time.

MPC forecasts future behavior using a system model, then optimizes control actions based on these forecasts. To find the optimal control inputs over a future time horizon, it solves an optimization problem at each time step. In this approach, the modeling complexity is higher as a result of the requirement for system modeling and real-time optimization. The controller solves a quadratic or nonlinear optimization problem by evaluating several control strategies and choosing the best one. On the other hand, traditional controllers like PID and PI use feedback to control the system. A PID controller adds a derivative term to predict future errors, whereas a PI controller modifies control actions based on the proportional and integral of the error signal. When compared to conventional controllers, which are easier to build and design, FCS-MPC’s dynamic performance is generally more effective, particularly when managing complex dynamics and constraints. When compared to conventional and nonlinear control techniques, MPC’s faster dynamic response, increased stability margin, and decreased power oscillations during steady-state operation have made it a widely preferred control algorithm for DC–AC converters. Classical controllers, though helpful in establishing steady-state performance, frequently result in a longer time to achieve it. A comprehensive analysis of performance-based comparison between FCS-MPC and a conventional proportional integral (PI) controller is provided in [11,12,13].

This paper presents a summary of the novel FCS-MPC approaches for grid-connected inverters. It presents a detailed review of the most recent advances in finite control set MPC (FCS-MPC), focusing on approaches. The performance of FCS-MPC is limited by the finite nature of power electronics devices, although it outperforms traditional proportional controllers in terms of tracking efficiency and Total Harmonic Distortion (THD). However, the performance of FCS-MPC can be limited in the case of three-phase inverters, which contain only eight discrete switching states with no transition states in between two discrete states. Most of the research that is currently available on FCS-MPC concentrates on including active and reactive power in the cost function. Nevertheless, only a small portion of research has examined the incorporation of resonant characteristics for improved grid frequency tracking and power quality. Another feature that has also become an essential component of the modern grid-connected inverter is providing inertial support other than peak demand sharing. However, the integration of advanced features such as grid-forming capabilities and inertial support in FCS-MPC remains underexplored. Maintaining grid frequency during load changes or grid-transients requires the PV-based inverter resources to act as synchronous generator and support the grid to return in normal conditions. There is a notable gap in the current literature on FCS-MPC in terms of advancements in implementing such features, which makes FCS-MPC less capable of supporting grid stability and resilience under dynamic operating situations.

2. Modelling of MPC

The modeling of MPC comprises two different steps. The first step is the formulation of the grid inverter model that includes determination of finite switching states of the inverter and the output voltage corresponding to each switching state. The second step is the formulation of the predictive model for the grid inverter. Depending on the preference, the predictive model can either be based on current, voltage or power. These two steps are discussed in detail in the next sections.

2.1. Grid-Connected Inverter

The grid-connected inverters are used to connect the DC sources like renewable energy sources (REs) and energy storage systems (ESS) with the electrical power system. The grid-connected inverters play an important role in ensuring quality and continuity of power as they have to follow the grid standard codes and load-side demands. Integrating a huge number of inverter-based REs with the grid is causing the introduction of instability and power quality issues [14,15]. Therefore, the controller should be equipped with dedicated mechanisms to overcome issues like voltage fluctuations and harmonics. Thus, the performance of the inverter and the controller become very critical in this regard, and a fast, stable controller with appropriate features is needed [16].

The grid-connected inverter system consists of a three-phase, two-level voltage source inverter tied with the dc bus along with the RES as shown in Figure 1. This topology has been frequently used in studies on MPC design; therefore, this type of inverter will be used to explain the MPC designs in this study. Usually, a grid-connected converter is designed within the stationary reference frame (SRF) and very few studies have been reported in the rotating reference frame (RRF). The inverter is connected with the grid through filtering inductor ${\mathrm{L}}_{\mathrm{g}}$, while the two switches on same leg operate in complement to each other. There are eight different switching states with output voltages mentioned in

Table 1. Switching states and output corresponding voltage.

Switching States			Voltage Space Vector
${\mathit{S}}_{\mathit{a}}$	${\mathit{S}}_{\mathit{b}}$	${\mathit{S}}_{\mathit{c}}$	V
0	0	0	$0$
0	0	1	$V_{DC}\left({\displaystyle \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-j{\displaystyle \raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)$
0	1	0	$V_{DC}\left({\displaystyle \raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+j{\displaystyle \raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)$
0	1	1	$V_{DC}\left({\displaystyle \raisebox{1ex}{$-2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)$
1	0	0	$V_{DC}\left({\displaystyle \raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)$
1	0	1	$V_{DC}\left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-j{\displaystyle \raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)$
1	1	0	$V_{DC}\left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}+j{\displaystyle \raisebox{1ex}{$\sqrt{3}$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)$
1	1	1	$0$

. The output voltage corresponding to the switching states can be expressed as follows.

$$V={\displaystyle \frac{2}{3}}{V}_{DC}\left(S_a+a{S}_b+a^2{S}_c\right)$$

(1)

where $a=e^{j\left({\displaystyle \frac{2\pi }{3}}\right)}=\mathit{cos}\left({\displaystyle \frac{2\pi }{3}}\right)+jsin\left({\displaystyle \frac{2\pi }{3}}\right)$.

The inductance acting as a filter at the output of inverters is the linking medium between the inverter and the grid. The grid inverter current model having resistance $R_g$ and grid voltage $V_{\alpha }^g$ can be described with the following equations within the SRF [17].

$${\displaystyle \frac{{di}_{\alpha }\left(t\right)}{dt}}={\displaystyle \frac{1}{L_g}}{V}_{\alpha }\left(t\right)-{\displaystyle \frac{R_g}{L_g}}{i}_{\alpha }\left(t\right)-{\displaystyle \frac{1}{L_g}}{V}_{\alpha }^g\left(t\right)$$

(2)

$${\displaystyle \frac{{di}_{\beta }\left(t\right)}{dt}}={\displaystyle \frac{1}{L_g}}{V}_{\beta }\left(t\right)-{\displaystyle \frac{R_g}{L_g}}{i}_{\beta }\left(t\right)-{\displaystyle \frac{1}{L_g}}{V}_{\beta }^g\left(t\right)$$

(3)

2.2. MPC and the Working Principle

There are eight vector states in the stationary reference frame as shown in Figure 2; only six vectors have been mentioned as two vectors are zero. This figure shows the output voltage V(k) and V(k + 1) for switching states during the current and next sampling time, respectively. In this example, the reference voltage signal for the next sampling time is $V^{*}$(k + 1), MPC selects the optimal switching state, which results in output voltage $V_3$(k + 1) for the next sample. The MPC uses a mathematical model of the system to predict the control variables, and to optimize a user-defined cost function by incorporating each output voltage option for the next sampling time, and the optimal voltage and its corresponding switching states are saved and then applied in the next sampling time. The control variables can either be the output current or the active/reactive power being injected to the grid; in both cases, the predicted current is obtained by using a discretization method like forward Euler’s discretization [18,19] with a sampling time $T_s$. The predicted current for the next sampling time is obtained as.

$$i_{\alpha }\left(k+1\right)=i_{\alpha }\left(k\right)\left(1-{\displaystyle \frac{R_g{T}_s}{L_g}}\right)+{\displaystyle \frac{T_s}{L_g}}{V}_{\alpha }\left(k\right)-{\displaystyle \frac{T_s}{L_g}}{V}_{\alpha }^g\left(k\right)$$

(4)

$$i_{\beta }\left(k+1\right)=i_{\beta }\left(k\right)\left(1-{\displaystyle \frac{R_g{T}_s}{L_g}}\right)+{\displaystyle \frac{T_s}{L_g}}{V}_{\beta }\left(k\right)-{\displaystyle \frac{T_s}{L_g}}{V}_{\beta }^g\left(k\right)$$

(5)

where $V_{\alpha ,\beta }\left(k\right)$ is the output voltage of the inverter.

2.3. Cost Function and the Control Law

MPC-based controllers have many advantages over conventional linear controllers, such as the involvement of cost function in the design of the overall control law. Modulation techniques are necessary components of linear controllers, which limits their performance and applicability [20]. However, the FCS-MPC explicitly manipulates the switches [21] by using mathematical models of the grid inverter system while incorporating the systems requirements and constraints. The objective function forms the basis of the controller and governs the control law of MPC. In most studies, the cost function is selected as the sum of square errors of current [22,23,24], power [25,26,27], or voltage [28,29]—this type cost function reduces ripples and allows the reference to be tracked very closely. For more precise tracking and harmonic suppressions, a resonant or repetitive feature is added in the control law, which allows the generation of large gain for the MPC at the grid frequency and suppresses harmonics [30,31,32,33]. Such controllers are inspired from the Internal Model Principal [34]. When the current is the optimizing parameter, the cost function has a form similar to the following.

$$J={\sum \left[{i_{\alpha ,\beta }\left(k\right)}^{*}-i_{\alpha ,\beta }\left(k+1\right)\right]}^2$$

(6)

The cost function in (6) is solved for optimal value of the inverter’s output voltage and it is used in the voltage optimizing cost function similar to the following.

$$J={\sum \left[{V_{\alpha ,\beta }\left(k\right)}^{opt}-V_{\alpha ,\beta }\left(k\right)\right]}^2$$

(7)

The optimal value of voltage is calculated using the equation obtained after solving (6), the sets of $V_{\alpha ,\beta }\left(k\right)$ are used to optimize (7) and the corresponding switching is applied to the inverter. A typical FCS-MPC block diagram is shown in Figure 3.

3. Model Predictive Control Strategies

MPC has been widely applied to various fields, such as power systems, robotics, automotive, and aerospace engineering. Grid-connected inverters need to regulate the output voltage and current to match the grid frequency and phase, as well as to maximize the power extraction from the renewable sources. MPC can achieve these objectives by using a discrete-time model of the inverter and the grid to predict the future values of the output current and power, and selecting the optimal switching state of the inverter from a finite set of voltage vectors. MPC can handle multiple control objectives and constraints in a unified framework, such as power quality, maximum power point tracking (MPPT), grid synchronization, and fault ride-through capability. MPC can improve the dynamic performance and tracking accuracy of the inverter, as well as reduce the switching frequency and harmonic distortion. Moreover, MPC can adapt to the changes in the system parameters and operating conditions, such as irradiance, temperature, load, and grid faults, by using online measurements and model updates. The working principle of FCS-MPC for grid connected inverters is depicted in Figure 4.

The authors in [35] presented a comprehensive model predictive control approach with multiple modes aiming to reduce losses through predictive modeling and control of switching frequency and higher voltage/current switching. The research work in [36] explores the smooth transition of inverters from islanded to grid-connected mode and suggested adaptive model predictive method, and offers valuable insights into enhancing system inertia and strength.

3.1. Active and Reactive Power Control

The MPC-based controllers are new and have gained the attention of researchers. Various MPC methods have been proposed with different cost functions and objectives. Active and reactive power can be regulated by simply including the reference power commands in the cost function and compiling the MPC to predict powers $P_{k+1},Q_{k+1}$ in regards with the $P_{ref},Q_{ref}$ [25,37,38].

$$J={{(P}_{ref}-P_{k+1})}^2+{{(Q}_{ref}-Q_{k+1})}^2$$

(8)

The prediction horizon can be increased like in the literature [27], where authors have used one more sample ahead prediction $P_{k+2},Q_{k+2}$in the cost function. In many studies where only active power support is required, $Q_{ref}$ = 0; in the case of reactive power injection for voltage support, the $Q_{ref}$ can be changed using a suitable control approach, like power flow control [25]. Figure 5a shows block diagram of this approach. These approaches have limited features included in the control law and can be regarded as the foundation of FCS-MPC for grid-connected inverters. Since most of the literature explicitly controls the active power, i.e., $Q_{ref}$ = 0, they provide active power support to the grid and lack the auxiliary support in terms of voltage regulation during intermittent PV power generation and frequency fluctuation due to load variations [39].

3.2. Increased Switching States

One significant restriction in MPC approaches for inverters is the eight-vector limit on switching states. Because of this limitation, control actions must be discrete, with no substates or transitions between two specified states. As a result, the variety of possible states that lie between the maximum and lowest values are ignored by this traditional method. By not using the intermediate states that may maximize the inverter’s performance, this mistake may seriously compromise power quality. The result is often the introduction of high THD, which can degrade the efficiency and reliability of power delivery. Furthermore, the inability to explore these intermediary states limits the controller’s flexibility and responsiveness to dynamic grid conditions, leading to suboptimal performance [20,36]. Innovative approaches are needed to overcome this restriction, such as the use of virtual or transition vectors, which may reduce the gap between current states and improve power quality overall by lowering THD. These developments would enable more precise control decisions, enhancing the inverter’s capacity to manage grid disruptions and nonlinearities. By utilizing these intermediate states, it is also possible to improve grid standards compliance and track reference signals more precisely, which will strengthen and stabilize the electricity system as a whole [37].

To overcome this limitation, the authors in [23] proposed a method that is shown in Figure 5b. This approach uses a high-quality model predictive controller to introduce transition vectors, or virtual vectors, between two states. These transition vectors are inserted into the basic vectors in a deliberate manner within a single switching cycle. This novel method uses transition vectors to discover and take use of the unexplored space between the current states, so attempting to address some of the fundamental drawbacks of the conventional eight-vector restriction.

These extra states allow the controller to tune the inverter’s output more accurately, improving performance and lowering THD. To be more precise, the process involves employing an orthogonal digital signal generator (ODSG) to convert single-phase power into two phases [40,41]. By supplying extra intermediate states that may be utilized to adjust the voltage and current outputs, this transformation enables more precise and adaptable control over the inverter’s output. This method attempts to improve the overall power quality by introducing the idea of virtual vectors and strategically deploying them in conjunction with basic vectors. It also broadens the range of accessible states. The model predictive controller can better regulate the inverter’s output by including virtual vectors into each switching cycle. This results in smoother state transitions and a lower chance of sudden changes that might result in THD. By strategically using transition vectors, the inverter can maintain high power quality and adapt more effectively to changing grid situations by providing a finer level of control. This method also enhances the inverter’s precision in tracking reference signals, guaranteeing that the power supplied to the grid satisfies strict quality requirements. SRF: stationary reference frame, RRF: rotational reference frame, and SP: single phase.

3.3. Robust MPC Approaches

The system’s mathematical modeling serves as the foundation for the performance of model predictive control (MPC). However, MPC performance becomes vulnerable to fluctuations and changes in parameters when it explicitly relies on a model to optimize future responses. Grid parameters are frequently unstable in real-world applications, which can have a big impact on how accurate the model predicts the future and, in turn, how well the control system works. As a result, in order for MPC to be genuinely robust and dependable, it has to be outfitted with methods for anticipating and optimizing future behavior even in the face of unknown system factors. To overcome this issue, researchers investigated incorporating robust approaches into MPC frameworks to account for these uncertainties. This integration is especially important since uncertainties can significantly affect predicted accuracy and control system performance as a whole. In order to deal with such variations, robust MPC techniques include mechanisms that modify the control actions in response to the observed deviations from the predicted parameter values. Adaptive algorithms, for example, can greatly improve the MPC system’s resilience by dynamically updating model parameters depending on real-time input.

In [11,42,43], the parametric uncertainties are included in the predicted control model (4), (5). The new model with uncertain parameters added up with actual model is given as below.

$${\check{i}}_{\alpha ,\beta }\left(k+1\right)=i_{\alpha ,\beta }\left(k\right)\left(1-{\displaystyle \frac{\left(R_g+\check{R}\right)T_s}{L_g+\check{L}}}\right)+{\displaystyle \frac{T_s}{L_g+\check{L}}}{V}_{\alpha ,\beta }\left(k\right)-{\displaystyle \frac{T_s}{L_g+\check{L}}}{\check{V}}_{\alpha ,\beta }^g\left(k\right)$$

(9)

where ${\check{i}}_{\alpha ,\beta }\left(k+1\right)$ is the predicted current considering the parameter uncertainties, the output voltage of the inverter only depends on the switching states therefore parameter changes have no influence on it. This enhancement involves creating a new model that incorporates uncertain parameters. This augmented model, when combined with the actual model, provides a more comprehensive representation of the system. The idea is to make the MPC not only aware of the expected behavior based on the ideal model but also capable of adapting and optimizing its control strategy in the face of uncertainties in the parameters that might deviate from the predicted values.

The error in prediction is the difference between the model (4), (5) and the model that includes uncertainties (9), this error is incorporated in the design of cost function to select the optimal switching state, which minimizes the prediction error. The prediction error, essentially the difference between these two models, becomes a crucial factor in the MPC design. This error signifies the disparity between what the model predicts and what might actually happen due to uncertainties in the parameters. This error occurs when the current prediction is influenced by the error in the previous prediction, and it progresses as a ripple effect: if the initial prediction is slightly off, it can lead the MPC system to implement a switching state that might not be necessary.

Table 2. Summary of the latest literature on FCS-MPC for a grid-connected inverter.

Optimizing Parameter	Control Objective	Characteristic	Reference Frame	Cost Function	Reference
Power	To control active/reactive power	Long prediction horizon	SRF	${{(P}_{ref}-P_{k+2})}^2+{{(Q}_{ref}-Q_{k+2})}^2$	[27]
		Harmonic suppression, Kalman filtering	SRF	$\mathrm{J}=\mathsf{\Delta }{\mathrm{P}}^2+\mathsf{\Delta }{Q}^2$	[25,37,38]
	LCL filter Resonance suppression	Optional current feedback	SRF	$\mathrm{J}=({{\mathrm{P}}^{*}-P_{k+1})}^2+{{(\mathrm{Q}}^{*}-Q_{k+1})}^2$	[44]
Current	Minimizing harmonics	Increased switching states, long prediction horizon	SP, SRF	$C^j={\displaystyle \frac{1}{N_p^j}}\sum _{l=0}^{N_p^j-1}‖u_{abc}^j\left(k+l\right)-u_{abc}^j\left(k+l-1\right)‖$	[26,30]
	To regulate output current/voltage	Model predictive power control, MPPT-MPC	SRF	${G={(\mathrm{i}}_{\alpha }-{{\mathrm{i}}_{\alpha }}^{*})}^2+{(\mathrm{i}}_{\beta }-{{\mathrm{i}}_{\beta }}^{*}{)}^2$	[45]
	Regulate reactive power for LVRT	Priority-based weighting for P and Q injection,	RRF	$g=f\left(i^{*}\left(k+1\right), i\left(k+1\right)\right)+h\left(v_{p,n}\right(k+1\left)\right)$	[17,46]
	To track ac reference signal,	Long prediction horizon, multilevel	SRF, SP, RRF	${g={(\mathrm{i}}_{i\alpha ,k+2}-{{\mathrm{i}}_{i\alpha , k+2}}^{*})}^2+{(\mathrm{i}}_{i\beta ,k+2}-{{\mathrm{i}}_{i\beta ,k+2}}^{*}{)}^2$	[26,47,48,49,50,51]
		Reduced computational burden, robust control against parameter changes	SRF, RRF	$g=\left|{(\mathrm{i}}_{\alpha }(k+1)-{{\mathrm{i}}_{\alpha }}^{*}(\mathrm{k}+1)\right|+\left|{(\mathrm{i}}_{\beta }(k+1)-{{\mathrm{i}}_{\beta }}^{*}(\mathrm{k}+1)\right|+\lambda \left(\left|V_p\left(k+1\right)-V_n(k+1)\right|\right)$	[42,43,52,53,54,55]
		Multivariable	SRF	$J\left(k\right)=\sum \left|{\mathrm{i}}_{2x}^{*}(k+1)-{{\mathrm{i}}_{2x}}^p(\mathrm{k}+1)\right|+0.01\left|{{∆\mathrm{u}}_{dc}}^{*}(k+1)-{{∆\mathrm{u}}_{dc}}^P(\mathrm{k}+1)\right|$	[56]
Voltage	To track voltage command signal	Multiple vectors, a hybrid predictive controller, modulated switching	SRF	$J={\mathrm{K}}_c\left|V^{ref}+V^k\right|$	[57,58,59]
		Lyapunov function, robust, multilayer	SRF	$J=\left|V^{ref}(k+1)+V^{opt}(k+1)\right|$	[11,60,61,62,63]
		Long prediction horizon	SRF	$J=\sum _{j=1}^N\widehat{y}{\left(\left(t+j|t\right)-w(t+j)\right)}^2+\sum _{j=1}^{N_u}{\left(\lambda \left(j\right)(∆u\left(t+j-1|t\right))\right)}^2$	[64]
		Internal model based	SRF	$J={\int }_0^T(e{\left(t+j|t\right)}^{T}Qe{\left(t+j|t\right)}^{T}dj+{\eta }^T{R}_L\eta$	[30]
		Quasi-resonant MPC	SRF	$$\begin{gathered} G={\displaystyle \frac{{T_s}^2}{{L_g}^2}}{\left[\begin{array}{c}\left(V_{\alpha }\left(k\right)-{V_{\alpha }}^{Opt}\left(k\right)\right)\\ \left(V_{\beta }\left(k\right)-{V_{\beta }}^{Opt}\left(k\right)\right)\end{array}\right]}^T\times \\ \hspace{1em}\hspace{1em}\left[\begin{array}{c}\left(V_{\alpha }\left(k\right)-{V_{\alpha }}^{Opt}\left(k\right)\right)\\ \left(V_{\beta }\left(k\right)-{V_{\beta }}^{Opt}\left(k\right)\right)\end{array}\right] \end{gathered}$$	[65]

presents comparison of various FCS-MPC techniques for grid connected inverter on the cost function and optimizing parameter.

This unnecessary decision, based on a slightly flawed prediction, can then accumulate over time as the system keeps making decisions at each sampling time. To make the control strategy robust and effective, this prediction error is factored into the design of the cost function. which leads toward the implementation of an unnecessary switching state in the next sampling time, and the error will accumulate as the sampling time moves ahead. The study in [66] presents a Linear Quadratic Regulator (LQR)-based prediction model that employs a Linear Matrix Inequality (LMI)-based model predictive control (MPC) method. The goal of this method is to attain zero-reference tracking error even when parametric uncertainties and grid voltage imbalance are present. In order to successfully compensate for distorted grid harmonics during grid frequency fluctuations, the method integrates resonant controllers with grid frequency-adaptive capacity. Robust MPC approaches typically use a more traditional control strategy in order to achieve robustness against uncertainties. This caution may lead to reduced performance or more restrained control actions, which may cause the inverter output to react more slowly or less aggressively to the grid changes.

3.4. Resonant MPC

Resonant model predictive control (RMPC) is an advanced control method that combines the advantages of model predictive control with resonant control to manage resonant systems—like resonant power converters—more precisely and efficiently [67]. The main idea behind RMPC is to predict future behavior of the resonant system by using a mathematical model as shown in Figure 6, and then optimize the control input to get the desired output. The Internal Model Principle (IMP) theory governs this method, which states that the controller’s design should include the external model of the system to be regulated [34]. RMPC optimizes in the resonance domain, as opposed to the time domain, as is the case with standard model predictive control. With this method, RMPC may take use of the intrinsic qualities of resonant systems, namely their capacity to manage periodic signals and efficiently suppress harmonics. Studies have shown that, in spite of its benefits, developing a digitally stable resonant MPC system has some difficulties [31,65,68]. Due to the difficulty of precisely modeling and forecasting the behavior of resonant systems, which frequently display highly dynamic and nonlinear properties, stability problems might occur. By adding resonant control components to the FCS-MPC design, this quasi-resonant technique enables the controller to better suppress harmonics and manage periodic disturbances. Resonant characteristics can be added to the predictive model to help the RMPC perform better in terms of lowering THD and preserving power quality. This technique increases the range of controlled states and improves the inverter’s precision in tracking reference signals, guaranteeing the grid receives high-quality power supply.

The proposed controller is designed by first formulating the transfer function and then embedding the grid frequency in the predictive model within the stationary reference frame. The quasi-resonant feature allows tuning of controller parameters for improved performance. The cost function is derived using the conventional model predictive control approach and a quasi-resonant feature is introduced in the predicted current model. The quasi-resonant feature in the cost function (10) allows the controller to utilize the previously optimal states of last two sample time in the prediction of next optimal switching combination.

$$G={\displaystyle \frac{{T_s}^2}{{L_g}^2}}{\left[\begin{array}{c}\left(V_{\alpha }\left(k\right)-{V_{\alpha }}^{Opt}\left(k\right)\right)\\ \left(V_{\beta }\left(k\right)-{V_{\beta }}^{Opt}\left(k\right)\right)\end{array}\right]}^T\left[\begin{array}{c}\left(V_{\alpha }\left(k\right)-{V_{\alpha }}^{Opt}\left(k\right)\right)\\ \left(V_{\beta }\left(k\right)-{V_{\beta }}^{Opt}\left(k\right)\right)\end{array}\right]$$

(10)

The RMPC method has been depicted in Figure 7. These methods undoubtedly provide efficient reference tracking and better harmonic suppression; however, the inclusion of a resonant nature introduces instability in the control law. The grid parameters and controller tunning parameters greatly influence the performance of the whole controller, and that is why in the event of adverse load fluctuations or high intermittency of PVs, the optimization of such approaches results in divergence. Therefore, operating the controller at the optimal grid and tunning parameters is the key requirement of such methods to utilize the maximum efficient performance.

3.5. Lyapunov Function MPC

Lyapunov-function-based model predictive control (MPC) is a powerful control technique that has been developed over the past few decades to solve complex control problems. The basic idea is to construct a scalar function, called a Lyapunov function, that decreases along the trajectories of the system and reaches a minimum at the equilibrium point. This method imposes a constraint on the MPC optimization problem that the derivative of the Lyapunov function of the closed-loop system is limited to be less than that under an auxiliary Lyapunov controller. This ensures that the MPC controller is at least as stable as the auxiliary controller. One of the key advantages of Lyapunov function-based MPC is its ability to handle complex nonlinear systems that are difficult to control using traditional control techniques. Lyapunov-based model predictive control has been applied in various fields, including economic control, adaptive control, robotic systems control, and nonlinear dynamical systems control [69,70,71,72]. By using a predictive control approach, this technique can predict the future behavior of the system and optimize the control inputs to ensure that the system remains stable and achieves the desired performance. The authors in [60] proposed a weighting factor free Lyapunov-function-based model predictive control (MPC) strategy for single-phase T-type rectifiers. The proposed approach determines the optimum switching states that force the error variables to zero from the derivative of the Lyapunov function. The negative definiteness of the Lyapunov function assures the stability of the system. The coefficient used in the formulation of the Lyapunov function does not have any effect on the performance, which makes the cost function weighting factor free; therefore, the proposed method does not require any weighting factors, which eliminates the need for determining their optimum values by trial and error. Simulation results are presented to validate the proposed control technique. The method proposed in [11] utilizes the Lyapunov function and discrete model of the PV converter and predicts the future active and reactive power of the system by calculating a cost function for all voltage vectors. The Lyapunov function used in this paper is a quadratic function of the state variables of the system. The function is used to ensure that the system remains stable and converges to a desired operating point. The proposed controller utilizes a Lyapunov-function-based approach to design the cost function for model predictive control. The cost function is designed such that it minimizes the error between the predicted and actual values of active and reactive power, while ensuring that the Lyapunov function remains positive definite. This ensures that the system remains stable and converges to a desired operating point. In [73], an N-step ahead Lyapunov-based MPC is proposed, the approach uses first-order linear interpolation to design the N step ahead predictive model followed by the calculation of N-step samples ahead Lyapunov function. The optimal switching combination is selected corresponding to minimum value of Lyapunov function. As presented in

Table 3. Prediction steps and improvement in THD [73].

Prediction Steps	1	2	3	4	5
Switching frequency (kHz)	35.175	30.425	26.925	26.050	25.375
THD (%)	1.13	1.09	1.07	1.02	1.02

, when the prediction steps is increased the harmonic suppression performance of this controller is improved.

Although the goal of Lyapunov-based MPC is to provide stability, it can be difficult to accomplish so under all operational circumstances, especially when there are uncertainties or disturbances. It could prove challenging to choose the right Lyapunov functions and tuning parameters, and improper tuning could lead to instability or massive performance deviation.

4. Optimization Techniques

Cost function optimization plays a crucial role in the effectiveness and performance of finite control set model predictive control (FCS-MPC) for grid-connected PV inverters. Using thorough cost function design, FCS-MPC can accurately determine and choose the best switching state for power electronic equipment. This improves overall system performance by enabling better control over factors including current, voltage, and power. The authors in [74] created a convex optimization framework for FCS-MPC of grid-tied inverters with LCL. The cost function is derived from the squared Euclidean norm of the tracking error and is expressed in a quadratic form. The paper incorporates the system constraints directly into the optimization problem using the Karush–Kuhn–Tucker (KKT) conditions. By comparing the costs of each sector and choosing the one with the lowest cost, the optimization method incorporates space vector modulation (SVM) into the MPC framework. Most FCS-MPC techniques do not use modulation for switching the inverter, this causes the inverter to have a variable switching frequency which in return causes the generation of harmonics. One may utilize the Modulated MPC (M2PC) controller to apply different restrictions on the control issue while maintaining a fixed switching frequency [7]. By using this method, the optimization process includes the modulation step, allowing the converter to run at a steady frequency. Another optimization technique presented in [75] uses redundant switching states to optimize the cost function. This technique allows optimizing the cost function by choosing the least total switching states and significantly decreases the switching transition frequency. In [76], as a criteria to optimize the duty ratios of the inverters, a current tracking error oriented cost function is employed. To compensate for the time delay in the optimization process, a delay compensation approach (MPC-DC) is also presented to mitigate the impacts of sampling delay of the measuring parameters. The optimization technique usually involves following steps:

• The prediction model is formulated by incorporating the grid inverter model. Novel approaches like Lyapunov function [73] and resonance [65] are used to include more stability and optimal performance in the control law.
• The optimization technique is implemented on the predicted model obtained in first step and the cost function is designed. For example. if the optimization technique is based on the increased switching state as discussed in [23], the corresponding output voltage for each newly added switching states is calculated and used in the next step to obtain the optimal switching state.
• Each switching combination is applied on the prediction model, and the output behavior of the inverter is predicted for each switching combination.
• The combination with least predicted error is opted and applied for the next sampling time.

The main objective of optimization in FCS-MPC is to enhance the inverter system’s efficacy. The control algorithm can more precisely forecast future states and modify control actions in response by improving the prediction model. As a result, the inverter performs better overall, tracks reference signals more accurately, and has less error. The article [23] uses increased switching states, i.e., it incorporates the transition states to accurately predict the future behavior of the inverter system. This allows less accumulated error and accurate prediction performance. The authors in [65] introduce resonant nature in the FCS-MPC prediction model to accurately predict and track the sinusoidally varying gird referential parameters. The inclusion of a resonant feature introduces a high gain at the grid frequency, which greatly increased the reference tracking ability of FCS-MPC. The result suggests that a 0.6-fold reduction in the tracking error can be obtained by introducing a resonant feature in the prediction model.

5. Discussion and Recommendations for Future Development of MPC for Grid Connected Inverters

Future model predictive controller (MPC) development for grid-connected inverters should concentrate on a few important areas to address current research gaps and improve efficacy. First, real-time processing requires improving computational efficiency through the use of sophisticated algorithms and hardware acceleration. Every control step in FCS-MPC requires solving an optimization problem, which can be computationally demanding, particularly for large-scale systems. Excessive computational complexity may result from the requirement to assess various control operations and their effects. Simplified or lower-order models are two examples of approximation techniques that could be useful in effort to reduce the computational burden. Model order reduction approaches, for example, can reduce the size of the problem while keeping a level of accuracy within acceptable limits.

It is necessary to provide robust and adaptive control techniques that include robust optimization to manage uncertainties like fluctuating voltages and frequencies and adaptive MPC to dynamically modify control parameters in response to changing grid circumstances. FCS-MPC must manage parameter uncertainties and grid disturbances, performance can be impacted by external disturbances and system parameter variability. To overcome such issues, robust FCS-MPC techniques can be formulated that address constraints and uncertainty, for instance, can take parameter uncertainty into consideration using adaptive techniques to modify control parameters in response to shifting operating conditions or system dynamics. To ensure grid stability and power quality, advanced frequency-adaptive capabilities and voltage control approaches are essential. Emphasizing harmonic suppression through resonant controllers and active filtering can further improve power quality. Multiobjective optimization frameworks should balance power loss minimization, THD reduction, and power factor optimization, with priority-based control schemes for dynamic adaptability. it is important to guarantee scalability and flexibility by utilizing modular designs. Furthermore, comprehensive practical testing and pilot projects are necessary to authenticate the efficacy of these sophisticated MPC methodologies in upholding elevated power quality inside contemporary grids.

By focusing on harmonic suppression through the use of resonant controllers and active filtering, power quality can be improved considerably, which is evident through Figure 8 that the existing FCS-MPC technology can significantly contribute towards the improvement of grid power quality even in during the distorted grid voltage scenario.

THD suppression in MPC can be greatly improved by increasing the number of switching states. Figure 8a shows the improvement in harmonic suppression achieved with the increased number of switching vectors compared to the conventional eight switching vectors. This improvement can be attributed to the control law’s availability to switch the inverter with better accuracy. Moreover, reference signal tracking becomes more accurate as the number of switching vectors increases because the optimization method can match the predicted output to the actual reference more precisely. The use of advanced optimization methods improves MPC performance even further, as seen in Figure 8b, particularly in situations when the grid is distorted. The THD can be significantly decreased by incorporating optimization techniques such inserting resonant components inside the control law. To be more precise, the optimization strategy can reduce THD to as low as 2.25%. Resonant control components improve the MPC’s ability to handle frequency-specific distortions, which in turn strengthens the control system’s resistance to grid disruptions and guarantees higher waveform quality.

In order to balance several control objectives, such as minimizing power losses, lowering THD, and maximizing power factor, multiobjective optimization frameworks should be created. It is also essential to have priority-based control systems that can adjust dynamically to the operational requirements and grid conditions at the moment. One area of current research necessity is improving MPC’s grid-forming capabilities, which are essential for preserving grid stability and enabling isolated microgrids. These features include the capacity to independently control frequency and voltage, simulating the operation of conventional synchronous generators. Further research is also needed in the field of ancillary grid support, which includes reactive power support, fault ride-through capabilities, and voltage and frequency management. Future MPCs can greatly improve the performance, reliability, dependability, and power quality of grid-connected inverters by addressing these recommendations and research gaps, assisting in the shift to modern, resilient, and sustainable power grids.

Finite control set model predictive control (FCS-MPC) has a lot of potential and benefits for grid-connected PV inverters, as indicated by the literature. Fast and accurate control actions are made possible by FCS-MPC’s direct control of switching states without the need for modulation methods, which makes it a very effective tool for preserving grid stability and power quality. The literature discussed in this paper emphasize how well FCS-MPC handles a variety of control objectives, such as voltage regulation, harmonic suppression, and active and reactive power control. These qualities are essential for modern electrical grids, which need to react quickly and precisely in order to maintain operational stability. Additionally, the advantage of FCS-MPC over conventional control systems is highlighted by its inherent ability to adapt to changing grid circumstances and its resilience to disturbances. In dynamic and unpredictable grid situations, FCS-MPC is positioned as a potential control approach due to its ability to handle nonlinearities and system restrictions.

Nevertheless, the existing literature also highlights specific constraints and domains requiring more investigation within the context of FCS-MPC. A significant obstacle that may confine the efficacy of FCS-MPC is the restricted ability of power electronic devices, especially in three-phase inverters that have limited discrete switching states. According to the literature, adding sophisticated features like inertial support and grid-forming capabilities might improve the stability and resilience of the grid, particularly in the face of load fluctuations and transient situations, this feature will add more significance to the FCS-MPC-based PV inverter system as the grid inertial support is an important requirement of modern grids. Furthermore, even though active and reactive power regulation have received a lot of attention, further research is required to fully understand how to include more robust resonant features to enhance overall power quality and grid frequency tracking. It will be essential to address these gaps by means of more research and development if FCS-MPC is to fulfill the modern grid requirement of supporting and stabilizing modern power grid systems.

6. Conclusions

This work has explored the latest developments and cutting-edge methods in FCS-MPC for grid-connected PV inverters. As renewable energy resources become more integrated into electrical grids, effective control mechanisms are increasingly essential for maintaining stability, power quality, and flexibility in response to dynamic grid conditions. FCS-MPC has emerged as a promising solution due to its ability to directly manipulate switching states without relying on modulators, making it particularly suitable for discrete power electronic systems.

This review categorizes and critically analyzes various FCS-MPC methodologies based on their control objectives, including active and reactive power control, harmonic suppression, and voltage regulation. Despite significant advancements, several challenges persist. Improving the computing efficiency of FCS-MPC is crucial for enabling real-time implementation in larger and more complex systems. Additionally, incorporating sophisticated sensor-less control methodologies could enhance system reliability and reduce costs. Furthermore, balancing multiple performance objectives in dynamic and unpredictable grid environments necessitates ongoing innovation in multiobjective optimization techniques. Addressing these challenges will be vital for advancing FCS-MPC and ensuring its effective deployment in future grid-connected inverter systems.