# High-Performance Tuning for Model Predictive Control for a Renewable Energy Grid-Interface Converter with LCL filter

JEFFERSON S. COSTA<fnref n="1" />,<fnref n="2" />, (Student Member, IEEE), ANGELO LUNARDI<fnref n="1" />, ALESSIO IOVINE<fnref n="3" />, (Member, IEEE), DARLAN A. FERNANDES<fnref n="4" />, (Senior Member, IEEE), DANIEL ALBIERO<fnref n="1" />,<fnref n="5" />, and Alfeu J. Squarezi Filho<fnref n="1" />,<fnref n="5" /> (Senior Member, IEEE)

Corresponding author: Jefferson S. Costa (e-mail: jeffersonsc@ufpa.br).

This work was supported in part by the National Council for Scientific and Technological Development (CNPq) under Grants 407867/2022-8 and 444259/2024-4, and by the São Paulo Research Foundation (FAPESP) under Grants 2022/00323-3, 2023/00682-6, and 2025/02648-5. Additionally, the CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil) is acknowledged for providing financial support for the publication of this article.

**ABSTRACT** Model predictive control (MPC) has emerged as a highly regarded control strategy in power electronics for renewable energy applications. While it minimizes tracking errors and control effort, a significant challenge is the lack of systematic tuning strategies to meet these systems' energy quality performance requirements. This paper proposes a comprehensive MPC tuning methodology for grid-integrating converters with LCL filters, incorporating modulation and delay compensation. We conduct a stability analysis to define precise constraints for cost function weights. The fine-tuning strategy systematically maps a Figure of Merit (FoM) for system performance using an advanced computational model, revealing that optimal tunings reside in narrow parameter regions. Experimental validation on a 2 kW workbench confirmed that all proposed MPC tunings met IEEE Std. 519-2014 power quality criteria and consistently outperformed a two-sample deadbeat controller, exhibiting enhanced dynamic response and power quality.

**INDEX TERMS** Grid-connected inverter, Model predictive control (MPC), LCL filter, Renewable Energy Sources (RES), Stability.

## I. INTRODUCTION

The global capacity of Renewable Energy Sources (RES) is expanding rapidly in pursuit of the ambitious goal of reducing reliance on fossil fuels and lowering CO₂ emissions to address climate change [1]. However, integrating high levels of variable RES, such as wind and solar photovoltaic (PV) power, presents considerable challenges for power system operators worldwide [2]. The inherent intermittency of these sources can be mitigated through energy storage systems (ESSs), which help stabilize power output by smoothing fluctuations. Since wind, PV, and ESSs operate primarily in Direct Current (DC), power converters are essential to facilitate their integration into the local

Alternating Current (AC) grid [3].

The two-level Voltage Source Converter (VSC) with an inductive-capacitive-inductive (LCL) filter is the most prominent power converter topology for interfacing RES with the grid [1]. This topology is bidirectional, enabling the integration of RES, ESSs, and applications such as battery chargers for electric vehicles. The LCL filter offers advantages over the simpler L-filter in size, cost, and harmonic attenuation [4]. However, its inherent resonance complicates control design, making damping essential. This research adopts a passive resistor for effective resonance damping.

The reliable and efficient integration of RES into power systems relies on the power converter's control technique,

<footnote n="1">¹Center for Engineering, Modeling and Applied Social Sciences of Federal University of ABC, Santo André, São Paulo 09210-580 Brazil</footnote>

<footnote n="2">²Department of Electrical Engineering of Federal University of Pará, Tucuruí, Pará 68455-901 Brazil</footnote>

<footnote n="3">³CNRS and the Laboratoire des Signaux et Systèmes (L2S), CentraleSupélec, Université Paris-Saclay, 91192 Gif-sur-Yvette, France</footnote>

<footnote n="4">⁴Department of Renewable Energies Engineering of Federal University of Paraíba, João Pessoa, Paraíba 58051-900 Brazil</footnote>

<footnote n="5">⁵School of Agricultural Engineering, University of Campinas, FEAGRI, Campus Zeferino Vaz, Campinas, São Paulo 13083-875, Brazil</footnote>
which must effectively handle synchronization, power and power factor regulation, harmonic distortion, and disturbances in grid voltage or the DC bus [2]. Common control methods include the classical PI current controller, proportional-resonant (PR) control, nonlinear controls such as sliding mode and fuzzy logic, and Model predictive control (MPC).

MPC integrates key features of advanced control strategies, including closed-loop control, minimization of error and control effort, ease of application to multivariable systems, and the ability to impose physical constraints [5]. In power electronics, MPCs present a computational edge compared to controllers that rely on nonlinear models. For example, the Generalized Predictive Control (GPC) operates using a control law derived from the Controlled Auto-Regressive Integrated Moving Average (CARIMA) model [6]. The Finite Control Set (FCS) leverages the discrete states of the converter to identify the optimal input required to meet the control objectives, albeit with a variable switching frequency [7]. Conversely, the Continuous Control Set (CCS) minimizes a linear matrix cost function to generate the control signal, facilitating extended prediction horizons while maintaining a fixed switching frequency through a PWM modulator [8].

This paper presents a tuning methodology for CCS-MPC with delay compensation, designed explicitly for a RES grid-connected converter with an LCL filter. The primary goal is to advance systematic MPC design in power electronics—a significant scientific challenge due to multiple degrees of freedom, including the predictive time horizons and the cost function weight factors, which influence system stability and performance [5]. Given this complexity, most designs rely on empirical methods, as in [9].

Some studies have aimed to improve MPC tuning in this application. For instance, in [10], the authors propose a tuning approach focused only on the weight factors of the MPC cost function, using an artificial neural network applied to a VSC with a simple L filter. Similarly, in [11], a method is introduced for designing the MPC weighting matrices based on the fake algebraic Riccati equation (FARE), applied to wind grid integration with a doubly fed induction generator (DFIG). Xue et al. presented a MPC for LCL-filtered inverters, considering state variable tracking errors and employing an observer for delay compensation [12]. However, their approach did not offer the same flexibility in defining prediction and control horizons or the granular control effort weighting addressed here. In previous work [8], a performance mapping of MPC with modulation in the synchronous reference frame was proposed for a VSC with an L filter. However, that study did not consider the digital control loop delay or the impact of the PLL dynamic.

The contributions of this work are as follows, distinguishing it from previous approaches, such as that discussed in:

* Proposal of a Systematic and Comprehensive Tuning Methodology for MPC: Unlike empirical approaches or those that restrict the flexibility of the cost function,

these work proposes a detailed performance mapping that considers the interaction between prediction and control horizons ($n_y, n_i$) and the cost function weighting ratio ($\gamma_i/\gamma_y$). This allows for the precise identification of isolated or narrow optimal tuning regions, which is non-trivial in high-order systems with multiple degrees of freedom.

* In-depth Stability Analysis for Cost Function Weighting: We present a detailed stability analysis based on the discrete system root locus criterion. This analysis not only establishes minimum and maximum limits for the $\gamma_i/\gamma_y$ ratio to ensure system stability but also show how this ratio can be pre-tuned based on nominal converter parameters, providing a theoretical foundation for initial tuning.

* Advanced Computational Model and Performance Metric (FoM): We introduce an advanced computational model that integrates PWM modulator delay compensation and PLL dynamics. Based on this model, we propose a Figure of Merit (FoM) to objectively evaluate and optimize system performance, combining power tracking accuracy (ISE) with power quality (THD). This provides an objective tool for fine-tuning, overcoming reliance on trial-and-error methods.

* Detailed Investigation of Control Effort: While the cost function in focuses on tracking errors of state variables, our approach explores the weighting of control effort. This weighting in our method can increase or decrease, even without variable constraints, the control effort applied in driving the inverter, as previously investigated in another research group's article [8]. This characteristic allows for a more refined trade-off between dynamic response and transient attenuation, which is crucial for practical RES integration systems.

* Comprehensive Experimental Validation and Benchmark Comparison: The identified optimal tunings were rigorously validated on a 2 kW experimental workbench, demonstrating compliance with IEEE Std. 519-2014 power quality criteria. Furthermore, the MPC's performance was compared to, and consistently demonstrated superiority over, a two-sample deadbeat controller. Although the deadbeat controller is an established and low-complexity technique, it serves as a clear benchmark to validate the dynamic and power quality improvements achieved with our tuning methodology.

The structure of this paper is organized as follows: Section II presents the model of the RES grid-integrating converter with the LCL filter. Section III describes the proposed MPC strategy with delay compensation in the asynchronous frame. Section IV outlines the tuning proposal for the MPC. Experimental results are presented in Section V, and the conclusions of this study are summarized in Section VI.
## II. MODEL OF THE RES CONVERTER WITH LCL FILTER.

The block diagram in Fig. 1 illustrates the elements of the proposed MPC for the RES grid-interfaced converter. The RES is expected to provide the DC bus with a constant voltage $v_{dc}$. The resistor connected in series with the capacitor in the LCL filter serves as a passive method of attenuation for the resonance dynamics [4].

<figure><img src="image_1.png" /><figcaption>FIGURE 1. Proposed MPC for the RES grid-integrating converter with LCL-filter.</figcaption></figure>

The model (1) represents the LCL-filter dynamics considering the voltages of the converter ($\vec{v}_i$), in the capacitor ($\vec{v}_c$), and in the grid connection point ($\vec{v}_g$), as well as the currents on the converter ($\vec{i}_i$) and on the grid side ($\vec{i}_g$) [7]. The variables are expressed in a vector asynchronous frame, where the components are represented by $\alpha$ (real) and $\beta$ (imaginary) components. The parameters ($L_1, R_1$) and ($L_2, R_2$) represent the converter and grid-side inductors, respectively. The capacitance and resistance in the LCL filter branch are represented by $C_f$ and $R_f$.

$$ \begin{bmatrix} \frac{\vec{d}_{i, \alpha \beta}}{dt} \\ \frac{\vec{d}_{g, \alpha \beta}}{dt} \\ \frac{\vec{d}_{c, \alpha \beta}}{dt} \end{bmatrix} = \underbrace{\begin{bmatrix} -\frac{(R_f + R_1)}{L_1} & \frac{R_f}{L_1} & -\frac{1}{L_1} \\ \frac{R_f}{L_2} & -\frac{(R_f + R_2)}{L_2} & \frac{1}{L_2} \\ \frac{1}{C} & -\frac{1}{C} & 0 \end{bmatrix}}_{B_c} \begin{bmatrix} \vec{x}(t) \\ \vec{i}_{i, \alpha \beta} \\ \vec{i}_{g, \alpha \beta} \\ \vec{v}_{c, \alpha \beta} \end{bmatrix} + \underbrace{\begin{bmatrix} \frac{1}{L_1} \\ 0 \\ 0 \end{bmatrix} \vec{v}_{i, \alpha \beta} + \begin{bmatrix} 0 \\ -\frac{1}{L_2} \\ 0 \end{bmatrix} \vec{v}_{g, \alpha \beta}}_{G_c} \quad (1) $$

The state-space representation in (2) represents the continuous-time model defined in (1), whose over-set terms are: the state vector $\vec{x} \in \mathbb{C}^3$, $\vec{x} = [\vec{i}_{i, \alpha \beta} \ \vec{i}_{g, \alpha \beta} \ \vec{v}_{c, \alpha \beta}]^\top$, the state matrix $A_c \in \mathbb{R}^{3 \times 3}$, the input matrix for the inverter voltage $B_c \in \mathbb{R}^3$, and the input matrix for the grid voltage $G_c \in \mathbb{R}^3$:

$$ \frac{d\vec{x}}{dt} = A_c\vec{x} + B_c\vec{v}_{i, \alpha \beta} + G_c\vec{v}_{g, \alpha \beta}. \quad (2) $$

The discretization of the continuous model (2) using a zero-order-hold (ZOH) and the sampling time $T_s$ produces

the discrete model (3). The state equation matrices in discrete time are $A_d = \exp(A_c T_s)$, $B_d = \int_0^{T_s} \exp(A_c t) B_c dt$, and $G_d = \int_0^{T_s} \exp(A_c t) G_c dt$, as defined in [7]. The output matrix $C_d = [0 \ 1 \ 0]$ is defined so that the system output is $\vec{y} = \vec{i}_{g, \alpha \beta}$. The single-sampling-time delay in the control signal $\vec{v}_{i, \alpha \beta}$ represents the MPC delay compensation strategy utilized in this study, as detailed in [13].

$$ \begin{aligned} \vec{x}(k+1) &= A_d \vec{x}(k) + B_d \vec{v}_{i, \alpha \beta}(k-1) + G_d \vec{v}_{g, \alpha \beta}(k) \\ \vec{y}(k+1) &= C_d \vec{x}(k+1) \end{aligned} \quad (3) $$

In power electronics, the voltage produced by the SV-PWM modulator is represented by the equation (4), accounting for a total transport delay of 1.5 times the sampling time $T_s$ [14].

$$ \vec{v}_{i, \alpha \beta}(s) = \vec{v}_{i, \alpha \beta}^*(s) e^{-1.5T_s} \quad (4) $$

where $\vec{v}_{i, \alpha \beta}^*$ is optimal MPC control signal and $s$ is the Laplace transform variable.

The augmented model of the system used for the design of the MPC with delay compensation is:

$$ \begin{aligned} \vec{x}_a(k+1) &= A_a \vec{x}_a(k) + B_a \vec{v}_{i, \alpha \beta}(k) + G_a \vec{v}_{g, \alpha \beta}(k) \\ \vec{y}(k+1) &= C_a \vec{x}_a(k+1) \end{aligned} \quad (5) $$

where the new state vector is represented as $\vec{x}_a = [\vec{x}(k) \ \vec{v}_{i, \alpha \beta}(k-1)]$, and the matrices $A_a, B_a, G_a$, and $C_a$ are as follows:

$$ \begin{aligned} A_a &= \begin{bmatrix} A_d & B_d \\ 0_{3 \times 3} & 0 \end{bmatrix}, & B_a &= \begin{bmatrix} 0_{3 \times 1} & 1 \end{bmatrix}^\top \\ G_a &= \begin{bmatrix} G_d & 0 \end{bmatrix}^\top, & C_a &= \begin{bmatrix} C_d & 0 \end{bmatrix}. \end{aligned} \quad (6) $$

### A. GRID SYNCHRONIZATION

The power quality performance for the proposed MPC in the presence of distorted grid voltage relies on a reference signal synchronized with the phase $\theta_g$ of the fundamental positive sequence component of the voltage at the grid connection point ($V_g = \vec{v}_{g,dq+1}$) [15]. The reference $\vec{i}_{g, \alpha \beta}^{ref}$ is calculated as a function of the desired level of the power to be injected into the grid $P^{ref}$ and $Q^{ref}$:

$$ \begin{bmatrix} i_{g, \alpha \beta}^{ref} \\ i_{g, \beta}^{ref} \end{bmatrix} = \frac{2}{3V_g} \begin{bmatrix} \cos \theta_g & \sin \theta_g \\ \sin \theta_g & -\cos \theta_g \end{bmatrix} \begin{bmatrix} P^{ref} \\ Q^{ref} \end{bmatrix}. \quad (7) $$

The Phase-Locked Loop (PLL) must reject negative sequence and harmonics components for proper grid synchronization. This article uses the reliable and robust Decoupled Double Synchronous Reference Frame (DDSRF) PLL algorithm, well-established in grid-connected power converter applications [5]. The DDSRF-PLL utilizes a dual decomposition strategy to separate the positive and negative sequence components, low-pass filters to attenuate harmonic components, and a synchronous reference frame (SRF) PLL to estimate the instantaneous phase $\theta_g$.

The small-signal disturbance model for DDSRF-PLL is equivalent to the dual-second-order-generalized-integrator (DSOGI)-PLL model, according to [14]. For a control loop
operating in the synchronous reference frame, the disturbances in the grid voltage components, $\Delta v_{g,d}$ and $\Delta v_{g,q}$, contribute to the disturbance signals in the current reference, $\Delta i_d^{ref}$ and $\Delta i_q^{ref}$, through the matrix transference function $P(s)$ detailed in [14], in the format:

$$ \begin{bmatrix} \Delta i_d^{ref} \\ \Delta i_q^{ref} \end{bmatrix} = P(s) \begin{bmatrix} \Delta v_{g,d} \\ \Delta v_{g,q} \end{bmatrix}. \quad (8) $$

Where $P(s)$ incorporates the target power values $P^{ref}$ and $Q^{ref}$, which are intended for injection into the grid [14]. Additionally, it includes the gains of the proportional-integral (PI) controller used in the SRF-PLL, the DSOGI feedback parameter, and the grid's angular frequency $\omega_g$.

### III. MPC WITH DELAY COMPENSATION

The proposed MPC tuning starts with the definition of the predictive model (Fig. 1), based on the discrete model of the converter (3) [9], [16]. The prediction of the future current in the grid side up to the sliding horizon $n_y$ can be expressed in (9). Here, $\tilde{\mathcal{Y}} \in \mathbb{C}^{n_y}$ represents the prediction vector (10); $\vec{x}_a(k)$ denotes the actual measurements of the state variables; $V_I \in \mathbb{C}^{n_i}$ is the prediction vector of the control signal up to the horizon $n_i$ (11); $V_G \in \mathbb{C}^{n_i}$ signifies the prediction vector for the grid voltage (12). Additionally, $\Psi \in \mathbb{R}^{n_y \times 3}$ stands for the predictive state matrix (14), and the input matrices $\mathcal{H} \in \mathbb{R}^{n_y \times n_i}$ and $\mathcal{D} \in \mathbb{R}^{n_y \times n_i}$ are defined in (14) and (15), respectively. In equations (11) and (12), the subscriptions $\alpha\beta$ were omitted for simplicity.

$$ \tilde{\mathcal{Y}} = \Psi \vec{x}_a(k) + \mathcal{H} V_I + \mathcal{D} V_G \quad (9) $$

$$ \tilde{\mathcal{Y}} = [\vec{y}(k+1) \ \vec{y}(k+2) \ \dots \ \vec{y}(k+n_y)]^\top \quad (10) $$

$$ V_I = [\vec{v}_i(k) \ \vec{v}_i(k+1) \ \dots \ \vec{v}_i(k+n_i)]^\top \quad (11) $$

$$ V_G = [\vec{v}_g(k) \ \vec{v}_g(k+1) \ \dots \ \vec{v}_g(k+n_i)]^\top \quad (12) $$

$$ \Psi = [C_a A_a \ C_a A_a^2 \ C_a A_a^3 \ \dots \ C_a A_a^{n_y}]^\top \quad (13) $$

$$ \mathcal{H} = \begin{bmatrix} C_a B_a & 0 & \dots & 0 \\ C_a A_a B_a & C_a B_a & \dots & 0 \\ C_a A_a^2 B_a & C_a A_a B_a & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ C_a A_a^{n_y-1} B_a & C_a A_a^{n_y-2} B_a & \dots & C_a B_a \end{bmatrix} \quad (14) $$

$$ \mathcal{D} = \begin{bmatrix} C_a G_a & 0 & \dots & 0 \\ C_a A_a G_a & C_a G_a & \dots & 0 \\ C_a A_a^2 G_a & C_a A_a G_a & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ C_a A_a^{n_y-1} G_a & C_a A_a^{n_y-2} G_a & \dots & C_a G_a \end{bmatrix} \quad (15) $$

### A. COST FUNCTION

The cost function (16) is formulated to minimize control effort $V_I$ while maintaining the desired grid current levels $\tilde{\mathcal{Y}}^r$ [17]. In the equation (16), the symbols $\Gamma_y$ and $\Gamma_i$ in

(16) denote diagonal weighting matrices associated with the tracking error and the control effort, respectively [9].

$$ \mathcal{J} = (\tilde{\mathcal{Y}}^r - \tilde{\mathcal{Y}})^\top \Gamma_y (\tilde{\mathcal{Y}}^r - \tilde{\mathcal{Y}}) + V_I^\top \Gamma_i V_I \quad (16) $$

The vector $\tilde{\mathcal{Y}}^r$ is defined by the reference signal $\vec{y}^{ref}(k) = [i_{g,\alpha}^{ref}(k), i_{g,\beta}^{ref}(k)]^\top$, which is repeated $n_y$ times in (17) through multiplication by $F = [1 \ 1 \ \dots \ 1]^\top$ [5].

$$ \tilde{\mathcal{Y}}^r = F_{[n_y \times 1]} \vec{y}^{ref}(k) \quad (17) $$

To simplify the degrees of freedom associated with the MPC, the positive definite block diagonal matrices $\Gamma_y \in \mathbb{R}^{n_y \times n_y}$ and $\Gamma_i \in \mathcal{R}^{n_i \times n_i}$ are redefined as diagonal matrices with equal terms $\gamma_y$ and $\gamma_i$, respectively. They take the form $\Gamma_y = \gamma_y \text{diag}(1 \ 1 \ 1 \ \dots \ 1)$ and $\Gamma_i = \gamma_i \text{diag}(1 \ 1 \ 1 \ \dots \ 1)$. This reduction results in four essential parameters for CCS-MPC performance: $\gamma_y, \gamma_i, n_y$, and $n_i$.

### B. MINIMIZATION OF THE COST FUNCTION

The optimization of the cost function (16) can be achieved by solving the partial derivative $\partial \mathcal{J} / \partial V_I = 0$, and considering no variable constraints [17]. The resulting solution $V_I$ represents the predicted control signal vector in (18), where the matrix $\mathcal{H} = (\mathcal{H}^\top \Gamma_y \mathcal{H} - \Gamma_i)^{-1} \mathcal{H}^\top \Gamma_y$ is independent of system measurements and can be computed outside the real-time control loop [5].

$$ V_I = \mathcal{H} [\tilde{\mathcal{Y}}^r - \Psi \vec{x}(k) - \mathcal{D} V_G] \quad (18) $$

The first element of $V_I$ represents the control signal $\vec{v}_{i,\alpha\beta}^*(k)$ for driving the converter and can be expressed as:

$$ \vec{v}_{i,\alpha\beta}^*(k) = \mathcal{W} V_I, \quad (19) $$

$$ \text{where } \mathcal{W} = [1 \ 0 \ \dots \ 0]_{[1 \times n_i]} \quad [5]. $$

### IV. TUNING PROPOSAL FOR THE MPC

In this section, we discuss the tuning of the MPC for the converter with LCL filter for integrating RES with the power grid. In the analysis, we are considering the block diagram of the closed-loop MPC, as shown in Fig. 2. The matrices $\mathcal{H}$, $F$, $\Psi$, and $\mathcal{D}$ represent the MPC, $M_d(z)$ represents the modulator delay, and $\vec{v}_g$ is the grid voltage that influences the reference of the control loop through the PLL dynamics $P(z)$ [14].

### A. STABILITY ANALYSIS

The stability analysis of the CCS-MPC utilizes root-locus criteria for discrete systems to place constraints on controller parameters, ensuring that the poles remain inside the unit circle in the z-plane [18]. Similar to [19], the closed-loop poles of the CCS-MPC can be shown to correspond to the eigenvalues provided by equation (20), by substituting the control law from (19) into the discrete model described in
<figure><img src="image_2.png" /><figcaption>FIGURE 2. Block diagram for stability analysis of the proposed MPC.</figcaption></figure>

(5). The nominal configuration of the RES grid-integration converter with LCL filter is presented in Table 1.

$$|A_d - B_d \mathcal{H} \Psi - zI| = 0 \quad (20)$$

<table><caption>TABLE 1. Nominal parameters of the RES converter with LCL filter.</caption><thead><tr><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>Converter Power</td><td>2 kVA</td></tr><tr><td>Nominal Current</td><td>12 A</td></tr><tr><td>Grid Voltage (V<sub>g</sub>)</td><td>110 V</td></tr><tr><td>Grid Frequency f<sub>g</sub></td><td>60 Hz</td></tr><tr><td>DC link Voltage, V<sub>dc</sub></td><td>300 V</td></tr><tr><td>Switching Frequency, f<sub>s</sub></td><td>20 kHz</td></tr><tr><td>Sampling Time, T<sub>s</sub></td><td>50 µs</td></tr><tr><td>L<sub>1</sub>; R<sub>1</sub></td><td>10 mH; 0.2 Ω</td></tr><tr><td>L<sub>2</sub>; R<sub>2</sub></td><td>150 µH; 0.05 Ω</td></tr><tr><td>C<sub>f</sub></td><td>15 µF</td></tr><tr><td>R<sub>f</sub></td><td>1.5 Ω</td></tr><tr><td>k; k<sub>p,SRF</sub>; k<sub>i,SRF</sub></td><td>0.5; 180; 3200</td></tr></tbody></table>

As noted in [20], the key factor in tuning the cost function of the CCS-MPC is the ratio between the weight factors $γ_y$ and $γ_i$, regardless of their individual values. In this work, we adopt a tuning range of $γ_i/γ_y$ from $10^{-5}$ to $10^{-2}$, with time horizons $n_y = 2, 3, 4, 5$ and $n_i = n_y - 1$. The value of $γ_y$ is arbitrarily set to $10^{-2}$. Figure 3 shows the root locus of the closed-loop CCS-MPC in the z-plane for various time horizons.

The initial proposed tuning $γ_i/γ_y = 10^{-2}$ normalizes the tracking error and control effort terms in the cost function, following a similar approach used in FSC-MPC for torque and flux control of the tree phase Induction Machine [5], [21]. The value is approximately the square of the ratio between the nominal current and the grid voltage of the converter (Table 1):

$$\frac{I_n^2}{v_g^2} = \frac{12^2}{110^2} = 0.012 \approx 1 \times 10^{-2}. \quad (21)$$

The results in Fig. 3 show that improper tuning of the proposed MPC can destabilize the RES grid-integrating converter. Our analysis indicates that for $n_y = 5$, for example, the lowest tuning value of $γ_i/γ_y$ to ensure MPC stability is

<figure><img src="image_3.png" /><figcaption>FIGURE 3. Root locus for the proposed MPC. The brown markers represent the poles for the initial tuning $γ_i/γ_y = 10^{-2}$. The arrows indicate the poles' trajectories as the tuning ratio $γ_i/γ_y$ decreases, with the black markers denoting the poles at the final tuning of $γ_i/γ_y = 10^{-5}$. The green markers represent the DDSRF-PLL poles.</figcaption></figure>

$γ_i/γ_y = 1.18 \times 10^{-5}$. Table 2 presents the minimum tuning to guarantee control system stability for the tested time horizons. In order to achieve stability over different time periods, it is advisable to adjust the weight factors of the cost function to a minimum of $γ_i/γ_y = 1.51 \times 10^{-5}$.

<table><caption>TABLE 2. Tuning limits for $γ_i/γ_y$ to guarantee stability.</caption><thead><tr><th>Time Horizon</th><th>Min. γ<sub>i</sub>/γ<sub>y</sub></th></tr></thead><tbody><tr><td>n<sub>y</sub> = 2</td><td>γ<sub>i</sub>/γ<sub>y</sub> = 1.22 × 10<sup>-5</sup></td></tr><tr><td>n<sub>y</sub> = 3</td><td>γ<sub>i</sub>/γ<sub>y</sub> = 1.51 × 10<sup>-5</sup></td></tr><tr><td>n<sub>y</sub> = 4</td><td>γ<sub>i</sub>/γ<sub>y</sub> = 1.15 × 10<sup>-5</sup></td></tr><tr><td>n<sub>y</sub> = 5</td><td>γ<sub>i</sub>/γ<sub>y</sub> = 1.18 × 10<sup>-5</sup></td></tr></tbody></table>

## B. CONTROLLER TUNING

In this section, we present the performance mapping for fine-tuning the proposed MPC. This analysis was carried out using the SimPowerSystems toolbox of Matlab/Simulink®, with a focus on tuning the $γ_i/γ_y$ ratio over various time horizons. To comprehensively assess the grid-integrating converter's performance, covering both accurate power regulation and high power quality, we introduce a Figure of Merit (FoM). This FoM is calculated by summing the integral square error (ISE) as defined in equation (22) [22] and the total harmonic distortion (THD) of the current injected into the grid. This combined metric is crucial as, for RES integration systems, achieving precise power tracking (low ISE) and maintaining grid compliance through low harmonic distortion (low THD) are both critical and often involve inherent trade-offs. The proposed FoM provides
a single-value assessment that reflects this dual requirement for optimal system operation. In (22), the vector $\vec{e} = [P - P_{ref}, Q - Q_{ref}]^T$ represents the error between the actual and reference values of active and reactive power.

$$ISE = \|\vec{e}(t)\|_2 = \sqrt{\int_0^\infty |\vec{e}(\tau)|^2 d\tau} \quad (22)$$

To describe the proposed FoM, the typical results for tuning $n_y = 5$ and $\gamma_i/\gamma_y = 3.56 \times 10^{-5}$ are shown in Fig. 4. In this scenario, the ISE performance in power regulation is 13.36 dB (Fig. 4-a), and the THD performance in energy quality is 3.95% (Fig. 4-c). Consequently, the FoM in this case is 17.3.

<figure><img src="image_4.png" /><figcaption>FIGURE 4. Simulated results: (a) tracking active and reactive power references; (b) comparison of the grid voltage and current component $\alpha$ with the current reference; (c) spectral components of the grid current for the references $P_{ref} = 750$ W and $Q_{ref} = 0$ var.</figcaption></figure>

### C. EFFECT OF PREDICTION AND CONTROL HORIZONS

In this section, we analyzed the impact of $n_y$ and $n_i$ on the performance of the proposed controller while keeping the cost function gain $\gamma_i/\gamma_y$ fixed at $10^{-4}$. We evaluated the performance FoM for different pairs of $n_y$ and $n_i$, and the results are presented as color maps in Fig. 5. The analysis revealed that increasing both $n_y$ and $n_i$ simultaneously contributes to improved performance, as it enhances the prediction of the system variables [21]. To avoid the low-performance regions indicated by red areas in Fig. 5, we advise selecting time horizons where $n_y \ge 3$ and $n_i = n_y - 1$.

<figure><img src="image_5.png" /><figcaption>FIGURE 5. Performance FoM for tuning the time horizons of the proposed MPC.</figcaption></figure>

### D. EFFECT OF COST FUNCTION WEIGHT FACTORS

In Fig. 6, we analyze the impact of $\gamma_y$ and $\gamma_i$ on the performance of the proposed controller, with fixed time horizons of $n_y = 3$ and $n_i = 2$. The performance FoM is nearly uniform along diagonal lines where the ratio $\gamma_i/\gamma_y$ remains constant. Unlike time horizons, which have a tuning range, the findings in Fig. 6 indicate that the best closed-loop performance is achieved with a narrow tuning range of the cost function gain. Specifically, the optimal performance was achieved when tuning $\gamma_i/\gamma_y = 3.56 \times 10^{-5}$.

<figure><img src="image_6.png" /><figcaption>FIGURE 6. Performance FoM for tuning the cost function weight factors $\gamma_i$ and $\gamma_y$.</figcaption></figure>

### E. GLOBAL PERFORMANCE MAP FOR THE PROPOSED MPC

In this section, we are concurrently mapping the performance FoM by considering the gain $\gamma_i/\gamma_y$ of the cost function and the time horizons of the predictive model. The performance map in Fig. 7 depends on $n_y$ and $\gamma_i/\gamma_y = c$, with
the other controller parameters calculated as: $n_i = n_y - 1$, $\gamma_y = 10^3$ and $\gamma_i = c\gamma_y$.

<figure><img src="image_7.png" /><figcaption>FIGURE 7. Performance map for the proposed MPC. The white lines show constant performance contours, and the gray marks the best performance points tested in section V.</figcaption></figure>

The results in Fig. 7 show that the fine-tuning of the proposed MPC occurs in islands or only in a few narrow regions defined for the pairs of gain $\gamma_i/\gamma_y$ and prediction horizon $n_y$ (gray markers). One example is the tuning $\gamma_i/\gamma_y = 4.1 \times 10^{-5}$ and $n_y = 12$, for which the merit figure is 17.16, with a THD of 3.74% and an ISE of 13.42 dB. The highlighted points in Fig. 7 will be experimentally evaluated in section V to validate the fine-tuning for the proposed MPC.

## V. EXPERIMENTAL RESULTS FOR THE MPC TUNING

The experimental validation for the tuning proposal of the MPC was performed in the 2 kW/3-phase workbench presented in Fig. 8. The main components of the workbench include electronic boards for conditioning and measurements of the three-phase voltage and current, a DSP Texas Instruments model TMS320F28335, a three-phase/two-level Semikron converter model "SKS20F (B6CI)2P+E1CIF+B6U 14V12", the LCL filter components, and an AC programmable power supply model SUPPLIER FCATHQ 4500VA/380V/500Hz to emulate the power grid conditions [23]. The parameters for the grid-integrating converter workbench are the same as described in Table 1.

This section presents experimental tests conducted to validate the fine-tuning of the proposed MPC, as emphasized in Fig. 7. The results illustrated in Figures 9 and 10 were obtained using a tuning $\gamma_i/\gamma_y = 3.56 \times 10^{-5}$ and $n_y = 5$. Figure 9 shows the power injected by the RES converter into the grid compared to the desired values. Figure 10 shows the measured AC current and voltage at the grid connection point during the power step response. Additionally, the amplitude spectrum of the current in the grid, illustrated in Fig. 11, is particularly significant. The experimental performance presented an ISE of 12.04 dB and a THD of

<figure><img src="image_8.png" /><figcaption>FIGURE 8. Workbench for the RES grid-integrating converter.</figcaption></figure>

2.04%. This THD value, being well below the 5% limit for individual harmonic distortion specified by IEEE Std. 519-2014 for point of common coupling in low voltage systems, confirms the excellent power quality. These results exceed the theoretical expectations outlined in section IV-E and confirm that the proposed MPC tuning effectively maintains power quality performance with a faster transitory response, ensuring THD remains below 5% while the system operates at 20% of its nominal capacity.

<figure><img src="image_9.png" /><figcaption>FIGURE 9. Experimental measurements for the MPC: power injected into the grid compared to the references.</figcaption></figure>

The results in Fig. 12 compare the performance of the power step response, as well as the ISE and THD performance of the power quality, across the six best tunings for the MPC highlighted in Fig. 7. The performance metrics for all the evaluated tunings are relatively similar, with the tuning corresponding to the horizon $n_y = 5$ exhibiting clearly worse performance. This observation aligns with the theoretical analysis of the performance FoM, indicated by the red line in Fig. 12. Specifically, the FoM for the horizon $n_y = 5$ is elevated, suggesting a suboptimal dynamic performance relative to other tunings. This reinforces the validity of the theoretical analysis present in this paper, which was valuable in identifying these precise tunings for the proposed MPC.
<figure><img src="image_10.png" /><figcaption>FIGURE 10. Measurements for the MPC: AC current and voltage in the grid phase A compared to the power step response.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>FIGURE 11. Amplitude spectrum for the grid current: $P_{ref} = 750$ W and unit power factor.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>FIGURE 12. Performance comparison for the best MPC tunings: (a) Power step response; (b) Power quality ISE and THD.</figcaption></figure>

## A. PERFORMANCE COMPARISON WITH OTHER CONTROL TECHNIQUES

This section compares the proposed MPC tuning performance with the classical deadbeat controller. Here, we apply

the strategies of two sample deadbeat controller defined in [24]. The control strategy implemented here relies solely on the discrete model of the system and operates without the need for a state estimator. The main advantage is the absence of control parameters for tuning the deadbeat; however, it depends on the plant's operating conditions. To drive the converter, a PWM modulator with the same frequency (20 kHz) as the proposed MPC is utilized.

In the context of the discrete model featuring a one-time sample delay as described in section II,

$$ 
\begin{aligned}
\vec{x}(k+1) &= A_d\vec{x}(k) + B_d\vec{v}_i(k-1) + G_d\vec{v}_g(k) \\
\vec{y}(k+1) &= C_d\vec{x}(k+1).
\end{aligned} 
\quad (23) $$

The time sample advancement is implemented, and the prediction computed at time $k+2$ is utilized as the reference signal, resulting in the following two sample deadbeat control law,

$$ \vec{v}_i(k) = B_d^{-1} [C_d^{-1} \vec{y}^{ref}(k) - A_d^2 \vec{x}(k) - A_d B_d \vec{v}_i(k-1) - A_d G_d \vec{v}_g(k) - G_d \vec{v}_g(k+1)] . \quad (24) $$

In a stationary reference frame, it is simple to estimate the value of the grid voltage $\vec{v}_g(k+1)$ by considering the synchronous phase shift equivalent to a sample time in the format:

$$ \vec{v}_g(k+1) = \vec{v}_g(k) \times e^{-2\pi f_g T_s}. \quad (25) $$

The experimental results for the deadbeat controller are presented in Figures 13 and 14, which present the injected and target power levels, along with the AC grid current and voltage during the power step response. The deadbeat control exhibits valuable performance, effectively maintaining power quality by keeping the THD below 5%, even at a current level of 3 A, which corresponds to 20% of the converter's nominal power. Additionally, it meets the specified power demands with an average error approaching zero.

<figure><img src="image_13.png" /><figcaption>FIGURE 13. Experimental measurements for the Deadbeat controller: power injected into the grid compared to the references.</figcaption></figure>

Figure 15 summarizes the performance comparison with the best tuning proposed for the MPC ($\gamma_i/\gamma_y = 3.56 \times 10^{-5}$
<figure><img src="image_14.png" /><figcaption>FIGURE 14. Measurements for the Deadbeat: AC current and voltage in the grid phase A compared to the power step response.</figcaption></figure>

and $n_y = 5$) and the deadbeat controller. The proposed tuning for the MPC exhibited superior dynamic performance compared to the deadbeat controller across all evaluated power quality metrics. Notably, the MPC excelled in the transient response, exhibiting reduced settling time and lower overshoot, which enhances system stability and responsiveness. The findings confirm that the proposed tuning methodology effectively optimizes the MPC parameters for the RES converter integrated into the grid.

## VI. CONCLUSION

This article proposes a comprehensive and systematic tuning methodology for the MPC with modulation applied in a RES grid-integration converter with an LCL filter. The tuning of key parameters, including time horizons and cost function weights, is achieved through the detailed mapping of a proposed FoM for the system's performance, leveraging an advanced computational model. Our stability analysis reveals that specific constraints on the cost function weights are crucial for maintaining system stability. Specifically, we recommend a maximum threshold of $10^{-2}$ for the tuning ratio $\gamma_i/\gamma_y$, adaptable to application requirements as detailed in equation (21). Furthermore, a critical minimum value of $1.51 \times 10^{-5}$ for this ratio is necessary to ensure stability across various time horizons.

The results highlight that the fine-tuning of the proposed MPC occurs in isolated or narrowly defined regions for the combined pairs of gain $\gamma_i/\gamma_y$ and prediction horizon, emphasizing the non-trivial nature of optimal parameter selection. The six best tunings were experimentally validated on a 2 kW experimental workbench. The evaluation results indicated that all the proposed MPC tuning effectively meets the established criteria for power quality performance. Notably, they outperformed the two-sample deadbeat controller across all evaluative parameters. Additionally, the experimental results corroborated the FoM performance analysis by accurately indicating a lower performance in one of the

<figure><img src="image_15.png" /><figcaption>FIGURE 15. Comparison between the MPC and Deadbeat: (a) Power step response; (b) Dynamic performance metrics; and (c) Amplitude spectrum for the AC grid current in phase A.</figcaption></figure>

tests with a higher FoM. In this way, the strategy proposed here can assist in fine-tuning other MPC applications.

## ACKNOWLEDGMENT

Following IEEE policy, the authors improved the text's readability by using Grammaly's Generative artificial intelligence (AI).

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JEFFERSON S. COSTA (Student Member, IEEE) received the B.S. and M.S. degrees in Electrical Engineering from the Federal University of Pará (UFPA), Tucuruí, Brazil, in 2013 and 2016, respectively. In 2025, he received the Ph.D. degree in Renewable Energy from the Federal University of ABC (UFABC), Santo André, Brazil. In 2024, he was a visiting Ph.D. student at the Laboratoire des Signaux et Systèmes (L2S), CentraleSupélec, Université Paris-Saclay, Gif-sur-Yvette, France.

He is currently an Associate Professor at UFPA, where he teaches electrical materials science, power electronics, and electromagnetism. His research interests include control applications in machine drives, power converters, electric vehicles, and photovoltaic and wind energy systems.

<figure><img src="image_17.png" /></figure>

ANGELO LUNARDI received the degree in electronic engineering from the Instituto Mauá Tecnologia, in 2015, the master's degree in electrical engineering from the Universidade Federal do ABC (UFABC), with a focus on control research applied to wind power generation, in 2017, and the Ph.D. degree in electrical engineering from the University of São Paulo (USP) with a thesis titled robust predictive control applied to the converter connected to the grid, in 2022. Currently,

he is a researcher at the UFABC and works with control systems for renewable energy.

<figure><img src="image_18.png" /></figure>

ALESSIO IOVINE (Member, IEEE) received the B.Sc. and M.Sc. degrees in electrical engineering and computer science from the University of L'Aquila, L'Aquila, Italy, in 2010 and 2012, respectively, and the European Doctorate degree in information science and engineering from the University of L'Aquila, in collaboration with CentraleSupélec, Paris-Saclay University, Gif-sur-Yvette, France, in 2016.

From 2016 to 2020, he held Post-Doctoral positions at University of L'Aquila, Efficacity Research Center (France), University of California at Berkeley, USA, and CentraleSupélec. In 2020, he joined the Centre National de la Recherche Scientifique (CNRS) as permanent Researcher, and is currently with the Laboratory of Signal and Systems (L2S), CentraleSupélec and Paris-Saclay University. He has co-authored more than 50 peer-reviewed papers covering advanced control methods for power and energy systems as well as traffic control ones, with smart grids and autonomous vehicles as core applications. His research interests focus on control methods for cyber-physical systems, particularly in the modeling and control of large-scale systems and optimal multi-level information management. Recently, he has concentrated on developing optimal control strategies to improve the integration and utilization of renewables and storage devices in power systems, as well as reducing consumption and maximizing traffic throughput in cooperative intelligent transportation systems.

Dr. Iovine is a member of the IFAC Technical Committee TC 6.3 Power and Energy Systems since 2020.
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DARLAN ALEXANDRIA FERNANDES (S'07-M'08-SM'25) received the B.S. degree in electrical engineering from the Federal University of Paraíba, Brazil, in 2002, and the M.S. and Ph.D. degrees in electrical engineering from the Federal University of Campina Grande, Brazil, in 2004 and 2008, respectively. He was a Visiting Scholar at the Center for Power Electronics Systems (CPES) at the Virginia Polytechnic Institute and State University (Virginia Tech), Blacksburg, United States, from 2018 to 2019. From 2007 to 2011, he was a Professor with the Industry Department in the Federal Center of Technological Education of Rio Grande do Norte. He is currently an Associate Professor with the Department of Electrical Engineering, Federal University of Paraíba, Brazil. His research interests are in the applications of power electronics in distribution systems, power quality, photovoltaic systems, and impedance-based control design techniques for static converters.

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DANIEL ALBIERO, Graduated at Agricultural Engineering from College of Agricultural Engineering from Unicamp (2001); incomplete degree in Physics from Unicamp (1996), master's degree in Agricultural Engineering from Unicamp (2005), doctorate at Agricultural Engineering from College of Agricultural Engineering from Unicamp (2009), Associate Professor (2023) for the FEAGRI-UNICAMP. He is currently Professor Ph.D. Machines Design and Robotics of School of Agricultural Engineering (FEAGRI) of University of Campinas (UNICAMP). Have experience in agricultural engineering, with an emphasis on dynamic soil, design and evaluation of agricultural machinery, agro-ecological machinery, equipment for family farms, optics applied to agricultural engineering, rural buildings with alternative material, physical and mechanical properties of biological materials, agricultural applications of biodiesel, agricultural aviation, quality management in agriculture, energy in agriculture and robotics.

<figure><img src="image_21.png" /></figure>

ALFEU J. SGUAREZI FILHO (Senior Member, IEEE) received his Master and Doctor degrees in Electrical Engineering from the Faculty of Electrical and Computer Engineering of the University of Campinas (Unicamp) in 2007 and 2010, respectively. Currently, he is Associate Professor at the Federal University of ABC (UFABC). He is a Senior Member of the IEEE and author of several articles in national and international scientific journals and book chapters in the areas of electrical machines, machine drives, electric vehicles, power electronics, wind and photovoltaic energies.

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