# Simplified sensorless predictive control of grid-side converter in wind energy conversion systems under distorted grid conditions

Adel Rahoui, Koussaila Mesbah, Boussad Boukais, Noumidia Amoura, Abdelhakim Saim, Azeddine Houari

<figure><img src="image_1.png" /></figure>

## To cite this version:

Adel Rahoui, Koussaila Mesbah, Boussad Boukais, Noumidia Amoura, Abdelhakim Saim, et al. Simplified sensorless predictive control of grid-side converter in wind energy conversion systems under distorted grid conditions. Energy Reports, 2025, 14, pp.1156-1168. 10.1016/j.egyr.2025.06.054. hal-05521800

HAL Id: hal-05521800

https://hal.science/hal-05521800v1

Submitted on 21 Feb 2026

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Research paper

# Simplified sensorless predictive control of grid-side converter in wind energy conversion systems under distorted grid conditions

Adel Rahoui<sup>a,b,∗</sup>, Koussaila Mesbah<sup>b,†</sup>, Boussad Boukais<sup>b,‡</sup>, Noumidia Amoura<sup>b,§</sup>, Abdelhakim Saim<sup>c,¶</sup>, Azeddine Houari<sup>c</sup>

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

### Keywords:

Adaptive linear neuron (ADALINE)

Grid-connected pulse width modulation (PWM) converter

Model predictive direct power control (MPDPC)

Sensorless control

Virtual flux (VF) estimator

## ABSTRACT

This paper proposes a new direct power control (DPC) strategy without voltage sensors for grid-connected pulse width modulation converters in wind energy conversion systems. This method relies on model predictive DPC (MPDPC) combined with virtual flux (VF) estimation. A novel VF estimator, based on an adaptive linear neuron (ADALINE) configured as a quadrature signal generator, is designed to eliminate the need for grid voltage sensors. In parallel, a simplified MPDPC approach is introduced, optimizing the selection process of the optimal control vector by reducing the number of equations involved. This simplification lowers computational demands while maintaining high control performance. The combination of the simplified MPDPC and the proposed VF estimator results in a sensorless voltage control strategy, referred to as VF-SMPDPC. The proposed strategy is validated through real-time simulations using the OPAL-RT system. A comprehensive comparative analysis demonstrates the superior performance of the ADALINE-based estimator over other VF estimation methods. Furthermore, the VF-SMPDPC strategy significantly outperforms conventional MPDPC, particularly under distorted grid conditions, proving its effectiveness and robustness in practical applications.

## 1. Introduction

Electrical energy generation from renewable sources, such as micro-hydropower plants and wind energy conversion systems (WECS), has seen steady growth in recent years (Abu-Rub et al., 2014). Among the different wind turbine technologies, type-IV wind power systems, also known as full-scale AC-DC-AC converter interfaced wind turbines, have gained significant attention due to their superior controllability and grid integration capabilities. These systems fully decouple the machine from the grid using a machine-side converter (MSC), a DC link, and a grid-side converter (GSC), as illustrated in Fig. 1 (Boukais et al., 2024; Rajendran et al., 2023). This architecture enables variable-speed operation of the generator, thereby maximizing wind energy extraction across a wide range of wind speeds. It also ensures continuous power injection during voltage sags or grid faults (fault ride-through), while providing precise control of voltage and frequency to meet the requirements of modern grid codes (Mahvash et al., 2024).

Traditionally, GSCs have been controlled using vector-oriented

control (VOC), which has dominated the field for many years (Malinowski et al., 2003). In this approach, active and reactive powers are indirectly regulated through current control loops. This method requires the current components decoupling along the *dq*-axes, as well as precise grid synchronization using a phase-locked loop. Furthermore, accurate tuning of controllers is essential to achieve acceptable dynamic performance. Over the past two decades, direct power control (DPC) has emerged as a promising alternative (Noguchi et al., 1998). Unlike VOC, DPC directly controls the active and reactive powers, offering several advantages, such as a simple structure, excellent dynamic response, and robustness against parameter variations (Guo and Ren, 2023). However, conventional DPC with predefined switching table presents several limitations (Noguchi et al., 1998). One of the primary challenges is its limited control precision. Indeed, the predefined switching table is based on fixed sector divisions and does not account for finer variations in power demands or operating conditions (Tao et al., 2016). Additionally, this method has limited capability to simultaneously control active and reactive power, often resulting in power ripples during steady-state

<footnote><sup>a</sup> Preparatory Classes Department, Ecole Nationale Supérieure des Travaux Publics, Rue Sidi Garidi BP 32, Vieux Kouba, Algiers 16051, Algeria</footnote>

<footnote><sup>b</sup> L2CSP Laboratory, Mouloud Mammeri University of Tizi-Ouzou, BP 17 RP, Tizi-Ouzou 15000, Algeria</footnote>

<footnote><sup>c</sup> Institut de Recherche en Énergie Électrique de Nantes Atlantique (IREENA), Nantes Université, UR 4642, Saint-Nazaire F-44600, France</footnote>

<footnote>* Corresponding author.
E-mail address: abdelhakim.saim@univ-nantes.fr (A. Saim).</footnote>
operation. Finally, a high sampling frequency is required to achieve satisfactory control performance (Malinowski et al., 2003; Tao et al., 2016).

To address the limitations of conventional DPC with predefined switching table, several strategies have been proposed. To reduce power ripples, some authors have developed optimized switching tables (Razali et al., 2015; Jabbarnejad et al., 2020), others have proposed a DPC with space vector modulation (Malinowski et al., 2004). In (Bouafia et al., 2009; Ahmad et al., 2021), a fuzzy-logic-based DPC was introduced to eliminate the need for predefined switching table. Authors in (Rath et al., 2024) propose a deadbeat DPC with online inductance identification to reduce sensitivity to parameter variations. More recently, model predictive direct power control (MPDPC) has been suggested as an alternative to conventional DPC (Guo and Ren, 2023; Nguyen et al., 2024; Rath and Srungavarapu, 2024). This technique relies on selecting the optimal control vector by minimizing a cost function that incorporates the errors in both active and reactive powers. As a result, improved steady-state performance with reduced power ripples can be achieved. MPDPC also offers excellent dynamic performance and a simplified control structure. However, the cost function minimization induces significant computational demands. Additionally, the performance of MPDPC may degrade under grid voltage disturbances.

On the other hand, sensorless grid voltage control can enhance the performance of the GSC. Replacing voltage sensors with estimators is a relevant approach that reduces system cost and improves reliability (Mesbah et al., 2024). Sensorless control, operating alongside conventional methods, is often employed to ensure continuity of service in the event of sensors failure (Mesbah et al., 2024). Several observer-based strategies have been developed to enable grid voltage estimation. In (Jukić et al., 2024), a sliding mode observer exploiting the discontinuous operation mode of the converter is proposed. A recursive least squares algorithm has been proposed in (Achlerkar and Ketan Panigrahi, 2021) for estimation of voltage/current parameters. An augmented state observer-based approach, enabling low voltage ride through capability without voltage sensors has been proposed in (Benyamina et al., 2021). While these approaches yield satisfactory results, they often involve complex tuning procedures and high computational burden, limiting their practical implementation. Due to its simplicity, many studies have focused on virtual flux (VF)-based approaches. In these methods, the grid voltage sensors are replaced by a VF estimator, derived by integrating the grid voltages. This integration is usually performed using a first-order low-pass filter (LPF) to avoid saturation and offset issues commonly associated with pure integrators (Norniella et al., 2014; Suul et al., 2012). However, the first-order LPF has two major drawbacks: phase shift and amplitude attenuation. Although these issues can be minimized by reducing the filter's cutoff frequency, such reduction degrades the estimator's dynamic performance. To overcome these limitations, more advanced strategies have been proposed, including

adaptive first-order LPF (Tao et al., 2016; Razali et al., 2015), second-order LPF (Norniella et al., 2014; Kulka, 2008), adaptive second-order generalized integrators (SOGI) (Suul et al., 2012; Ketzer and Jacobina, 2016; Roslan et al., 2018), band-pass filters (BPFs) (Pöllänen, 2003), and VF estimator with initial bias compensation (Zhang et al., 2015; Sun et al., 2022). While these strategies deliver satisfactory results, some have only been validated under ideal grid voltage conditions (Zhang et al., 2015), and others suffer from delays introduced by the LPF stages or cascading filter structures (Tao et al., 2016; Razali et al., 2015; Norniella et al., 2014). Furthermore, the resonant structure of SOGI makes its performance sensitive to the discretization method used (Suul et al., 2012; Ketzer and Jacobina, 2016; Roslan et al., 2018).

To overcome the aforementioned limitations, this paper proposes a new sensorless predictive control scheme for the GSC. This scheme relies on a novel VF estimator based on an adaptive linear neuron (ADALINE), configured as a quadrature signal generator (ADALINE-QSG) to emulate the properties of a pure integrator. This estimator offers several advantages, including a self-learning capability that prevents amplitude and phase errors commonly observed with LPF-based estimators, a discrete-time design that enhances precision and simplifies practical implementation, and an inherent harmonics and noise rejection ability. The proposed estimator is combined with a new simplified MPDPC strategy (SMPDPC), which reduces the computational burden required for minimizing the cost function. The resulting sensorless control scheme, referred to as VF-SMPDPC, simplifies the computation process while maintaining high performance, offering a more robust solution particularly suited to disturbed grid conditions. To validate the proposed VF estimator and the VF-SMPDPC strategy, real-time simulations were conducted on the OPAL-RT system. First, comparisons were made to evaluate the performance of the proposed estimator against other VF estimation methods. Then, the VF-SMPDPC strategy was compared to conventional MPDPC.

## 2. Overview of VF estimators

The concept of VF is based on defining the flux $\psi$ as the integral of the voltage $e$, as shown in (1).

$$ \psi = \psi_0 + \int e \, dt. \qquad (1) $$

Initially, this definition was applied to improve control in electric drive systems (Takahashi and Noguchi, 1986). Later, it was adopted for grid-connected converters, drawing an analogy with AC machines, where flux is derived from voltage measurements (Duarte et al., 1999). With the growing interest in sensorless control of grid-connected converters, the VF-based approach gained widespread use due to its simplicity. For instance, in (Malinowski et al., 2004), this technique was

<figure><img src="image_4.png" /><figcaption>Fig. 1. Topology of a grid-connected wind energy conversion system.</figcaption></figure>
used to calculate instantaneous power without the need for voltage sensors.

The basic principle of VF estimation involves calculating the grid voltages by integrating the converter input voltages $v_{c,abc}$ and considering the voltage drop across the input filter (Fig. 1). To avoid direct voltage measurements that require additional sensors, the required voltages $v_{c,abc}$ are typically reconstructed from the DC bus voltage and the switching states of the converter.

## 2.1. VF estimation using pure integrator

In the two-level converter configuration shown in Fig. 2, which represents the GSC of Fig. 1, VF estimation eliminates the need to measure grid voltages.

Here in Fig. 2, the grid voltages and input filter are treated as components of a virtual AC machine. The inductance $L$ and resistance $R$ represent the stator winding of the virtual machine, and the grid voltages are induced by the VF. The model of the GSC in the $\alpha\beta$ reference frame is given by:

$$ \mathbf{e}_g = \mathbf{v}_c + \mathbf{Ri}_g + L \frac{\mathbf{di}_g}{dt} \qquad (2) $$

where $v_c = [v_{c\alpha} v_{c\beta}]^T$, $e_g = [e_{g\alpha} e_{g\beta}]^T$, and $i_g = [i_{g\alpha} i_{g\beta}]^T$ denote the converter voltage, grid voltage, and grid current vectors, respectively.

Integrating these voltages produces the VF vector, given by:

$$ \psi_g = \int \mathbf{e}_g dt = \int \mathbf{v} dt + \mathbf{Li}_g \qquad (3) $$

where $\psi_g = [\psi_{g\alpha} \psi_{g\beta}]^T$ is the VF, and $v = v_c + Ri_g$ represents the sum of the converter voltage and the resistive voltage drop.

The voltage drops across $R$ is typically small and can often be neglected, but for general accuracy, it will be included in this paper. The converter voltage $v_c$ in $\alpha\beta$ coordinates, derived from the DC bus voltage $V_{dc}$ and the switching signals $S_a, S_b$, and $S_c$, is given by:

$$ \begin{cases} v_{c\alpha} = \frac{2}{3} V_{dc} \left( S_a - \frac{1}{2} (S_b + S_c) \right) \\ v_{c\beta} = \frac{2}{3} V_{dc} \left( \frac{\sqrt{3}}{2} S_b - \frac{\sqrt{3}}{2} S_c \right). \end{cases} \qquad (4) $$

Since VF is the integral of voltage (see Fig. 3), its instantaneous angle $\theta_{\psi}$ lags the grid voltage angle $\theta$ by 90°. The instantaneous angle of the grid voltage in a three-phase system can be obtained using:

$$ \theta = \tan^{-1} \left( \frac{\psi_{g\beta}}{\psi_{g\alpha}} \right) + 90^{\circ}. \qquad (5) $$

VF estimation yields accurate instantaneous angle values even when the grid voltage is disturbed. However, when used in a sensorless control scheme, this method requires knowledge of the initial flux value, necessitating a proper startup procedure.

## 2.2. Practical implementation of the VF estimator

The pure integrator used in (3) for VF estimation cannot be applied directly in practice due to two major drawbacks (Cirrincione et al., 2004):

<figure><img src="image_5.png" /><figcaption>Fig. 2. Studied three-phase GSC.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 3. Basic concept of VF estimation.</figcaption></figure>

a. Measured signals always contain a DC offset caused by sensors, measurement noise, or the limited resolution of digital signal processors. Even small offsets cause flux drift, leading to integrator saturation and, eventually, system instability.

b. During integration, the initial condition is assumed to be zero, which is rarely the case in practice. This results in an offset at the integrator output. For sinusoidal input signals, the initial condition will be zero only if integration starts at the signal's peak value.

The effects of a pure integrator on the estimated flux are shown in Fig. 4. Fig. 4a illustrates the result when a 2% offset is introduced into the input voltage, with flux initialized around the x-axis. Fig. 4b shows the flux estimate when the integrator is not initialized. In both cases, flux drift is evident, leading to noticeable estimation errors.

Several methods have been proposed to practically implement VF estimation, aiming to prevent signal drift and saturation. The simplest methods use analog or digital filters that emulate the behavior of a pure integrator (Malinowski et al., 2003, 2004). More advanced and adaptive integration methods have also been proposed (Pöllänen, 2003; Roslan et al., 2018; Cirrincione et al., 2004; Idris and Yatim, 2002).

### 2.2.1. Filter-based VF estimation

The most straightforward alternative to the pure integrator is a first-order LPF. This approach is widely used in sensorless control strategies based on VF estimation. The cutoff frequency of the filter is set much lower than the grid frequency, typically at 10% of the nominal grid frequency (Malinowski et al., 2004; Pöllänen, 2003). In this frequency range, the phase lag is close to 90°, yielding results similar to those of a pure integrator. The transfer function of the first-order LPF-based VF estimator is shown in Fig. 5 and is expressed as:

$$ \psi = \frac{\omega_{LPF}}{s + \omega_{LPF}} v \qquad (6) $$

where $\omega_{LPF} = \omega/10$, with $\omega_{LPF}$ as the filter's cutoff frequency and $\omega$ as the grid's frequency.

A gain is typically used to enhance the estimator's convergence speed. While this solution eliminates the offset and saturation issues of a pure integrator, it introduces significant amplitude attenuation and phase delay, which can degrade system performance.

Due to the limitations of the first-order LPF, various BPFs topologies have been employed to eliminate the offset in the estimated VF (Pöllänen, 2003; Duarte et al., 1999; Bhattacharya et al., 1996). These filters offer more flexibility in selecting the cutoff frequency but still exhibit a slight phase lag and significant amplitude attenuation in the estimated flux. Two similar BPF structures are described in (Duarte et al., 1999; Bhattacharya et al., 1996), with the following transfer function:

$$ \psi = \frac{s \omega_2 \omega_3}{\omega_1 (\omega_2 + s) (\omega_3 + s)} v \qquad (7) $$
<figure><img src="image_7.png" /><figcaption>Fig. 4. Flux estimation using a pure integrator; (a) initialization with incorrect initial conditions, (b) input signal with a 2% offset.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5. Block diagram of a VF estimator based on a first-order LPF.</figcaption></figure>

where $\omega_1 = \omega/10000$, $\omega_2 = \omega/1000$, and $\omega_3 = \omega/100$.

Although this structure can effectively emulate a pure integrator, it still results in significant amplitude attenuation of the estimated VF. Another simple method for VF estimation, based on a second-order LPF, was proposed in (Norniella et al., 2014; Kulka, 2008). This method employs two cascaded first-order LPFs with a cutoff frequency $\omega_{LPF2}$ set to the grid's nominal frequency. While this approach offers a faster response time, the offset in the input signal is amplified by a factor of two (Kulka, 2008).

$$ \psi = \frac{2\omega_{LPF2}^2}{(s + \omega_{LPF2})^2} v. \qquad (8) $$

The filtering methods presented in this section, along with their variations, provide good results for VF estimation under ideal grid conditions. However, in non-ideal grid conditions, their performance can significantly degrade. The following section will introduce more advanced techniques proposed for VF estimation.

### 2.2.2. Adaptive integration and advanced estimation techniques

As discussed in the previous section, VF estimators based on filters (first-order LPF, second-order LPF, and BPF) exhibit limitations, including phase lag and amplitude attenuation in the output signals. To address these shortcomings, more advanced strategies have been developed. Initially, these estimators were proposed for AC machine control applications, ensuring system stability even under severe operating conditions, such as unbalances, harmonic distortion, or abrupt frequency changes. These techniques have since been adapted for VF estimation in grid-connected PWM converters.

A method for compensating the phase lag and amplitude attenuation of the estimated flux using a first-order LPF was proposed by (Idris and Yatim, 2002) for electric drive applications. The compensation terms are calculated using the ratio between the grid frequency $\omega$ and the filter's cutoff frequency $\omega_{LPF}$, as follows:

$$ \left\{ \begin{array}{l} \psi_a = \psi_{a,LPF} + \frac{\omega_{LPF}}{\omega} \psi_{\beta,LPF} \\ \psi_{\beta} = \psi_{\beta,LPF} - \frac{\omega_{LPF}}{\omega} \psi_{a,LPF} \end{array} \right. \qquad (9) $$

This method was later employed by several authors (Tao et al., 2016; Razali et al., 2015) for VF estimation in grid-connected converters (Fig. 6). A slight modification to the structure proposed by (Idris and Yatim, 2002) was made in (Dai et al., 2011) to improve the estimator's accuracy.

To overcome the limitations of the LPF, authors in (Suul et al., 2012; Ketzer and Jacobina, 2016; Roslan et al., 2018) have introduced an estimator based on an adaptive SOGI for VF estimation. This approach is similar to the synchronization technique proposed by (Luna et al., 2015), which uses grid voltage measurements to separate positive and negative sequences. The SOGI is configured as a BPF with a cutoff frequency equal to the grid's frequency $\omega$, allowing simultaneous integration and filtering of the input voltage $v$. The transfer functions are given by:

$$ \left\{ \begin{aligned} \frac{v(s)}{v(s)} &= \frac{k_e \omega s}{s^2 + k_e \omega s + \omega^2} \\ \frac{\omega \psi(s)}{v(s)} &= \frac{k_e \omega^2}{s^2 + k_e \omega s + \omega^2} \end{aligned} \right. \qquad (10) $$

where $\psi$ represents the flux obtained by integrating the input voltage $v$, $v'$ is the filtered input voltage, and $k_e$ is a gain that influences the convergence speed of the SOGI.

While this strategy demonstrates good performance, it remains sensitive to any offset present in the input voltage (Ketzer and Jacobina, 2016). Furthermore, the SOGI is vulnerable to grid frequency variations, necessitating the use of a frequency-locked loop (Luna et al., 2015; Rahoui et al., 2020a) which increases the computational load due to the complexity of the filter's structure (Fig. 7).

To eliminate offsets in the measured signals before and after integration, caused by the initial flux condition, a technique based on ADALINE type neural networks was developed in (Cirrincione et al., 2004) for flux estimation in electric drive systems. To adaptively eliminate offsets in both the input and output signals, two ADALINE filters are employed. The system's structure is shown in Fig. 8.

The two ADALINE filters operate as adaptive notch filters (ANFs), receiving a constant input value $x(k) = -1$. Each filter's weight ($w_1$ and $w_2$) converges toward the offset value present in the input signal. This value is then subtracted from the input, producing an offset-free output. The learning rule for the two ADALINE filters is given by:

$$ w_i(k+1) = w_i(k) + 2\mu\epsilon(k) \qquad (11) $$

<figure><img src="image_9.png" /><figcaption>Fig. 6. Block diagram of a first-order LPF-based VF estimator with adaptive phase and amplitude compensation.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 7. Block diagram of a SOGI-based filter.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 8. Block diagram of a VF estimator with two cascaded ADALINE filters.</figcaption></figure>

where $\mu$ is the learning coefficient, and $\epsilon(k)$ is the error between the input signal and the estimated signal at time $k$.

This method is straightforward to implement, as only two weights need to be updated in real-time to estimate the flux.

Other neural network-based techniques have also been used for flux estimation in AC machine control (Sevcík et al., 2023; Lee et al., 2025). However, few neural network-based VF estimation techniques have been implemented for controlling grid-connected PWM converters.

### 3. New ADALINE-QSG for VF estimation

A novel approach, utilizing an ADALINE filter configured as a QSG for VF estimation, is presented in (Rahoui et al., 2020b). The ADALINE-QSG structure, illustrated in Fig. 9, generates two output vectors: the first output, $v'$, is a filtered version of the input vector $v$, while the second output, $qv'$, is a 90° phase-shifted version of $v'$. The key concept is to utilize the quadrature signal, which exhibits the necessary phase characteristics for accurate VF estimation.

#### 3.1. Design of the proposed VF estimator

To formulate the estimation problem, the input vector $v$ is first expressed in the discrete time domain using a Fourier series expansion as follows:

$$ \begin{bmatrix} v_{\alpha}(k) \\ v_{\beta}(k) \end{bmatrix} = \underbrace{\begin{bmatrix} V_1 \cos(\omega_1 k T_s + \varphi_1) \\ V_1 \sin(\omega_1 k T_s + \varphi_1) \end{bmatrix}}_{\text{Fundamental components}} + \underbrace{\sum_{n=2}^{\infty} \left( \begin{bmatrix} V_n \cos(\omega_n k T_s + \varphi_n) \\ V_n \sin(\omega_n k T_s + \varphi_n) \end{bmatrix} \right)}_{\text{Harmonic components + offset}} + \begin{bmatrix} A \\ B \end{bmatrix} \quad (12) $$

<figure><img src="image_12.png" /><figcaption>Fig. 9. Structure of the proposed ADALINE-VF Estimator.</figcaption></figure>

where $V_1$, $\omega_1$, and $\varphi_1$ represent the amplitude, frequency, and initial phase angle of the fundamental component, respectively, while $V_n$, $\omega_n$, and $\varphi_n$ correspond to the n-th harmonic terms. $T_s$ is the sampling period, and $A$ and $B$ denote the offsets for $v_\alpha$ and $v_\beta$, respectively.

From (12), the fundamental component $v_1(k) = [v_{1\alpha}(k) \ v_{1\beta}(k)]^T$ can be expressed as:

$$ \begin{cases} v_{1\alpha}(k) = V_1 \cos(\omega_1 k T_s + \varphi_1) \\ v_{1\beta}(k) = V_1 \sin(\omega_1 k T_s + \varphi_1) \end{cases} \quad (13) $$

Eq. (13) can be rewritten more explicitly as:

$$ \begin{cases} v_{1\alpha}(k) = V_1 \cos(\varphi_1) \cos(\omega_1 k T_s) - V_1 \sin(\varphi_1) \sin(\omega_1 k T_s) \\ v_{1\beta}(k) = V_1 \sin(\varphi_1) \cos(\omega_1 k T_s) + V_1 \cos(\varphi_1) \sin(\omega_1 k T_s) \end{cases} \quad (14) $$

Finally, the fundamental components of $v$ can be expressed in matrix form as:

$$ v_1(k) = \underbrace{\begin{bmatrix} V_1 \cos(\varphi_1) & -V_1 \sin(\varphi_1) \\ V_1 \sin(\varphi_1) & V_1 \cos(\varphi_1) \end{bmatrix}}_{X} \begin{bmatrix} \cos(\omega_1 k T_s) \\ \sin(\omega_1 k T_s) \end{bmatrix} \quad (15) $$

Eq. (15) can be learned by an adaptive neural network filter with two weights, updated online. These weights are represented by the vector $W = [w_1 \ w_2]$. Fig. 9 shows the VF estimator based on the ADALINE filter. The input vector $d(k)$ consists of two unit-amplitude sinusoids with frequencies matching the grid's fundamental frequency $\omega_1$. During the learning process, the weight vector $W$ is adaptively updated to converge toward $X$. The least mean square algorithm (Rahoui et al., 2020b) with a learning rate $\eta$ is used to recursively update the weight vector $W$ as follows:

$$ W(k+1) = W(k) + \eta \lambda(k) d(k) \quad (16) $$

where $W = [w_1 \ w_2] = \begin{bmatrix} w_{1\alpha} & w_{2\alpha} \\ w_{1\beta} & w_{2\beta} \end{bmatrix}$, $\lambda(k) = v(k) - v'(k)$ is the voltage estimation error, and $\eta$ is the learning rate.

The quadrature vector $qv'(k) = [qv_\alpha'(k) \ qv_\beta'(k)]^T$ can be computed using trigonometric identities as follows:

$$ qv'(k) = [w_1 \ w_2][\sin(\omega_1 k T_s) - \cos(\omega_1 k T_s)]^T \quad (17) $$

Since the quadrature vector $qv'$ lags the direct vector $v'$ by 90°, it corresponds to the integral of $v'$, scaled by the frequency $\omega_1$. Therefore, the VF $\psi$ corresponding to $v'$ can be obtained as:

$$ qv' = \omega_1 \int v dt = \omega_1 \psi. \quad (18) $$

Consequently, the VF $\psi_g$ in (3) can be expressed in terms of $\psi$ as:

$$ \psi_g = \frac{1}{\omega_1} qv' + Li_g = \psi + Li_g. \quad (19) $$

#### 3.2. Stability analysis and learning rate tuning

The stability of the proposed estimator is a critical requirement for ensuring reliable operation. As the ADALINE-QSG performs real-time learning without prior initialization, its stability is primarily governed by the appropriate choice of the learning rate $\eta$. Accordingly, a precise range of $\eta$ values is established based on the Lyapunov's convergence theory (Mesbah et al., 2024).

The estimation error, defined as the difference between the actual coefficient vector $X(k)$ and its estimate $W(k)$, is expressed as:

$$ \tilde{W}(k) \equiv X(k) - W(k). \quad (20) $$

A Lyapunov candidate function is then defined as:

$$ v(k) = \tilde{W}^T(k) \tilde{W}(k). \quad (21) $$

This function is positive definite and represents the instantaneous
energy of the estimation error. The variation of the Lyapunov function between successive iterations is given by:

$$ \Delta \mathbf{v}(k) = \mathbf{v}(\tilde{\mathbf{W}}(k+1)) - \mathbf{v}(\tilde{\mathbf{W}}(k)) < 0. \quad (22) $$

The Lyapunov's convergence criterion must be satisfied, such that

$$ \mathbf{v}(k)\Delta\mathbf{v}(k) < 0. \quad (23) $$

By substituting the adaptive update law (16) into (20), the variation becomes:

$$ \begin{aligned} \Delta \mathbf{v}(k) &= \|\tilde{\mathbf{W}}(k) - \eta \mathbf{d}^T(k) \mathbf{d}(k) \tilde{\mathbf{W}}(k)\|^2 - \tilde{\mathbf{W}}^T(k) \tilde{\mathbf{W}}(k) \\ &= \eta [\tilde{\mathbf{W}}(k) \mathbf{d}(k)]^2 [-2 + \eta \mathbf{d}^T(k) \mathbf{d}(k)]. \end{aligned} \quad (24) $$

Assuming normalized inputs such that $||d(k)||^2 = 1$, stability is ensured if:

$$ 0 < \eta < 2. \quad (25) $$

This inequality guarantees that the Lyapunov function decreases over time, leading to asymptotic convergence of the estimation process. The choice of $\eta$ thus represents a trade-off between estimation accuracy and convergence speed. Lower values of $\eta$ enhance noise rejection and reduce overshoot but result in slower adaptation, while higher values improve responsiveness at the cost of possible oscillations. In this work, a learning rate of $\eta = 0.004$ was selected to ensure fast convergence and robust operation under various grid conditions.

# 4. Predictive control of GSC

## 4.1. Conventional MPDPC

MPDPC operates by minimizing a cost function to determine the optimal control vector for each switching period. The method uses the converter model in (2) to establish a predictive framework for regulating instantaneous active and reactive power. Initially, the instantaneous power theory (Akagi et al., 2017) is applied to define the complex power $S$ as follows:

$$ \mathbf{s} = \frac{3}{2} (\mathbf{i}_g^* \mathbf{e}_g) \quad (26) $$

where $*$ denotes the conjugate of a complex vector.

The instantaneous active $P$ and reactive $Q$ powers are then given by:

$$ \mathbf{P} = \Re e(\mathbf{S}) = \frac{3}{2} \Re e(\mathbf{i}_g^* \mathbf{e}_g) \quad (27) $$

$$ \mathbf{Q} = \Im m(\mathbf{S}) = \frac{3}{2} \Im m(\mathbf{i}_g^* \mathbf{e}_g). \quad (28) $$

Assuming balanced and sinusoidal grid voltages, $e_g = |e_g|e^{j\omega t}$, the derivative of the complex power can be obtained as:

$$ \frac{d\mathbf{S}}{dt} = \frac{1}{L} \left[ \frac{3}{2} \left( |\mathbf{e}_g|^2 - \mathbf{v}_c^*(k) \mathbf{e}_g(k) \right) - (R - j\omega L)\mathbf{S} \right]. \quad (29) $$

The continuous-time model is discretized with a sampling period $T_s$, using the first-order forward Euler method, which provides an acceptable approximation for small $T_s$ values (e.g., 10 µs) (Zhang et al., 2015). The variation of the complex power between two successive sampling instants is given by:

$$ \frac{\mathbf{S}(k+1) - \mathbf{S}(k)}{T_s} = \frac{1}{L} \left[ \frac{3}{2} \left( |\mathbf{e}_g(k)|^2 - \mathbf{v}_c^*(k) \mathbf{e}_g(k) \right) - (R - j\omega L) \mathbf{S}(k) \right]. \quad (30) $$

By separating the real and imaginary components, the converter predictive model for active and reactive power is derived as follows:

$$ \begin{bmatrix} P(k+1) \\ Q(k+1) \end{bmatrix} = \begin{bmatrix} P(k) \\ Q(k) \end{bmatrix} + \frac{3}{2} \frac{T_s}{L} \begin{bmatrix} |e_g(k)|^2 \\ 0 \end{bmatrix} - \frac{3}{2} \begin{bmatrix} \mathbf{v}_{ca}(k) & \mathbf{v}_{c\beta}(k) \\ -\mathbf{v}_{c\beta}(k) & \mathbf{v}_{cat}(k) \end{bmatrix} \quad (31) \\ \begin{bmatrix} e_{ga}(k) \\ e_{g\beta}(k) \end{bmatrix} - \frac{T_s}{L} \begin{bmatrix} R & \omega L \\ -\omega L & R \end{bmatrix} \begin{bmatrix} P(k) \\ Q(k) \end{bmatrix}. $$

The input voltage vector $\mathbf{v}_c$ can take seven distinct values based on the states of the switches in the three-phase converter. Thus, the predicted active and reactive powers $P_i(k+1)$ and $Q_i(k+1)$ should be computed for each of the seven control vectors $\mathbf{v}_{ci}$ ($i = 0, 1, ..., 6$). The predictive powers equations are then written as:

$$ \begin{bmatrix} P_i(k+1) \\ Q_i(k+1) \end{bmatrix} = \begin{bmatrix} P(k) \\ Q(k) \end{bmatrix} + \frac{3}{2} \frac{T_s}{L} \begin{bmatrix} |e_g(k)|^2 \\ 0 \end{bmatrix} - \frac{3}{2} \begin{bmatrix} \mathbf{v}_{cat}(k) & \mathbf{v}_{c\beta}(k) \\ -\mathbf{v}_{c\beta}(k) & \mathbf{v}_{cat}(k) \end{bmatrix} \quad (32) \\ \begin{bmatrix} e_{ga}(k) \\ e_{g\beta}(k) \end{bmatrix} - \frac{T_s}{L} \begin{bmatrix} R & \omega L \\ -\omega L & R \end{bmatrix} \begin{bmatrix} P(k) \\ Q(k) \end{bmatrix}. $$

A cost function $g_i$ is used to evaluate the performance of each control vector, aiming to minimize the error between the predicted and reference powers:

$$ g_i = |P_{\text{ref}}(k+1) - P_i(k+1)| + |Q_{\text{ref}}(k+1) - Q_i(k+1)|. \quad (33) $$

where $P_{\text{ref}}(k+1)$ and $Q_{\text{ref}}(k+1)$ are the reference active and reactive powers predicted for the next sampling period.

The cost function is minimized at the beginning of each switching period to identify the optimal control vector that drives $P(k+1)$ and $Q(k+1)$ closest to their reference values by the end of the period. Since the evaluation is based on the future error, future references must be known. For reactive power, which is often constant or zero (unity power factor), the simple approximation $Q_{\text{ref}}(k+1) \approx Q_{\text{ref}}(k)$ is sufficient in most cases.

For active power reference, which is continuously adjusted by the DC-link voltage regulator, higher precision is required. An extrapolation method, such as Lagrange interpolation, is used to predict $P_{\text{ref}}(k+1)$:

$$ P_{\text{ref}}(k+1) = 2P_{\text{ref}}(k) - P_{\text{ref}}(k-1). \quad (34) $$

The MPDPC strategy applies only one optimal control vector during each switching period. Typically, the optimal vector applied during the interval $[kT_s(k+1)T_s]$ is the one that minimizes the cost function, ensuring that the predicted active and reactive powers $P_i(k+1)$ and $Q_i(k+1)$, are as close as possible to their reference values $P_{\text{ref}}(k+1)$ and $Q_{\text{ref}}(k+1)$. Since vectors $\mathbf{v}_0$ and $\mathbf{v}_7$ yield identical effects, only $\mathbf{v}_0$ is considered in this optimization process.

Fig. 10 illustrates the block diagram of the conventional MPDPC strategy. This approach requires calculating the predicted powers (32) and cost function (33) seven times, corresponding to the seven control vectors $\mathbf{v}_{ci}$. In the following section, a simplified approach to reduce computational requirements is presented.

<figure><img src="image_13.png" /><figcaption>Fig. 10. Block diagram of conventional MPDPC.</figcaption></figure>
## 4.2. Proposed VF-SMPDPC

To reduce the number of equations and computations involved in MPDPC, a simplified strategy (SMPDPC) is proposed, inspired by the works of (Zhang et al., 2017; Xia et al., 2014). Instead of predicting the instantaneous active and reactive powers at $k+1$, which includes calculating $P_i(k+1)$ and $Q_i(k+1)$ for each of the seven possible switching vectors, the converter voltage vector $v_c(k)$ is directly computed. This vector, applied at the converter input, requires only a single calculation. The reference converter voltage vector $v_{c,ref}(k)$ can be determined by substituting the predicted complex power $S(k+1)$ with the reference power $S_{ref}(k+1)$ in (30). Rearranging the equation yields:

$$v_{c,ref}(k) = e_g(k) - (R + j\omega L)\dot{i}_g(k) - \frac{2L}{3T_s} \left( \frac{S_{ref}(k+1) - S(k)}{e_g(k)} \right). \quad (35)$$

By separating (35) into its real and imaginary components, the control voltage vector is expressed by:

$$\begin{bmatrix} v_{c,ref\alpha}(k) \\ v_{c,ref\beta}(k) \end{bmatrix} = \begin{bmatrix} e_{ga}(k) \\ e_{gj}(k) \end{bmatrix} - \begin{bmatrix} R & -\omega L \\ \omega L & R \end{bmatrix} \begin{bmatrix} i_{ga}(k) \\ i_{gj}(k) \end{bmatrix} - \frac{2L}{3T_s} |e_g(k)|^2 \quad (36) \\ \begin{bmatrix} e_{ga}(k) & e_{gj}(k) \\ e_{gj}(k) & -e_{ga}(k) \end{bmatrix} \begin{bmatrix} P_{ref}(k+1) - P(k) \\ Q_{ref}(k+1) - Q(k) \end{bmatrix}.$$

This approach ensures that the predicted active and reactive powers align with their reference values at the next sampling instant. A new cost function $g_{ni}$ is introduced to assess how closely each of the seven possible switching vectors, $v_{cib}$, matches the calculated reference vector $v_c(k)$.

$$g_{ni} = |v_{c,ref\alpha}(k) - v_{cib}(k)| + |v_{c,ref\beta}(k) - v_{cib}(k)|. \quad (37)$$

This approach significantly reduces the computational requirement while maintaining performance similar to the conventional MPDPC. When combined with the VF estimator presented in 3, this results in a sensorless control scheme, referred to as VF-SMPDPC, which eliminates the need for grid voltage sensors without increasing computational burden.

To implement sensorless control, the estimated VF is correlated with the grid voltages through the following expressions:

$$e_{ga} = -\omega\psi_{gj} \quad (38)$$

$$e_{gj} = \omega\psi_{ga} \quad (39)$$

By substituting these relationships into the predictive control model (36), the resulting system can be expressed in terms of VF variables:

$$\begin{bmatrix} v_{c,ref\alpha}(k) \\ v_{c,ref\beta}(k) \end{bmatrix} = \omega \begin{bmatrix} -\psi_{g\beta}(k) \\ \psi_{ga}(k) \end{bmatrix} - \begin{bmatrix} R & -\omega L \\ \omega L & R \end{bmatrix} \begin{bmatrix} i_{g\alpha}(k) \\ i_{gj}(k) \end{bmatrix} - \frac{2L}{3T_s \omega} |\psi_g(k)|^2 \begin{bmatrix} -\psi_{gj}(k) & \psi_{ga}(k) \\ \psi_{ga}(k) & \psi_{gj}(k) \end{bmatrix} \begin{bmatrix} P_{ref}(k+1) - P(k) \\ Q_{ref}(k+1) - Q(k) \end{bmatrix}. \quad (40)$$

This formulation allows for selecting the optimal control vector $v_{cib}$ to

<figure><img src="image_14.png" /><figcaption>Fig. 11. Block diagram of the proposed VF-SMPDPC.</figcaption></figure>

be applied during each switching period by minimizing the cost function (37). The block diagram of the VF-SMPDPC control strategy is shown in Fig. 11.

## 5. Comparative studies and performance evaluation

To evaluate the performance of the proposed VF estimator and the VF-SMPDPC control strategy, the experiments were conducted in a real-time simulation environment using the OP4512-OPAL-RT system, as depicted in Fig. 12. All models were implemented in MATLAB/Simulink and optimized to comply with the RT-LAB software requirements. Subsequently, these MATLAB models were compiled into C code through RT-LAB for execution on the OP4512-OPAL-RT system. Both the controller and the power system models were deployed within the core processing units of the OPAL-RT CPU. The required signals (e.g. grid currents and voltages) were sent to four channels of the GW Instek GDS-2204 A Oscilloscope for monitoring through the analog input/output ports of OPAL-RT. For comparison, the performance of the proposed estimator was evaluated against the first-order LPF estimator and those from (Cirrincione et al., 2004; Idris and Yatim, 2002). Additionally, the VF-SMPDPC strategy was compared to conventional MPDPC. The setup parameters are summarized in Table 1. Unity power factor operation was ensured by setting reactive power $Q$ to zero, and the DC bus voltage $V_{dc}$ was regulated to 220 V.

### 5.1. VF estimators comparison

For the comparative study, the cutoff frequency of the first-order LPF estimator and the adaptive LPF estimator (Idris and Yatim, 2002) was set to 8 Hz. The learning rate $\mu$ of the ANF estimator was optimally set according to (Cirrincione et al., 2004). The learning rate $\eta$ of the proposed ADALINE-VF estimator was set to 0.004. Three tests were conducted: the first assesses startup performance without prior initialization, the second simulates a symmetrical voltage sag, and the third applies distorted grid voltages.

#### 5.1.1. Startup performance

Fig. 13 and Table 2 present the comparative results of the VF estimators during startup, with no prior initialization. The classical LPF estimator exhibits slow response, requiring 99 ms to stabilize, while the ANF-based approach shows a slightly longer convergence time of 101 ms. Although the ANF effectively mitigates the amplitude attenuation inherent to the LPF, the cascaded structure of two ANFs introduces a non-negligible computational delay, worsening its response speed. With the adaptive LPF, the flux estimation time improves significantly, reaching 61 ms, demonstrating its dynamical adjustment ability. However, despite this enhancement, a noticeable phase lag remains,

<figure><img src="image_15.png" /><figcaption>Fig. 12. Photography of OPAL-RT based real-time system for performance evaluation.</figcaption></figure>
<table><caption>**Table 1**
System parameters.</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Sampling frequency f<sub>s</sub> (kHz)</td><td>100</td></tr><tr><td>Grid voltage (V<sub>rms</sub>)</td><td>55</td></tr><tr><td>Grid frequency f (Hz)</td><td>50</td></tr><tr><td>Input filter inductance L (mH)</td><td>8</td></tr><tr><td>Input filter resistance R (Ω)</td><td>1</td></tr><tr><td>DC bus voltage reference V<sub>dc-ref</sub> (V)</td><td>220</td></tr><tr><td>DC bus capacitance C (mF)</td><td>3.3</td></tr></tbody></table>

particularly affecting transient conditions. In contrast, the proposed ADALINE-based VF estimator achieves precise flux estimation in just 8 ms, over ten times faster than the classical LPF. This rapid response is attributed to its self-learning capability, which allows real-time adaptation to variations without introducing phase shift or amplitude attenuation. Furthermore, while the LPF, adaptive LPF, and ANF-based methods exhibit notable overshoots of 24%, 17%, and 28%, respectively, the proposed estimator achieves a smooth and stable initialization with no noticeable overshoot, highlighting its superior transient behavior.

### 5.1.2. Symmetrical voltage sag

The performance of the VF estimators was further evaluated under a 50% symmetrical voltage sag, as depicted in Fig. 14. The classical LPF estimator struggles to adapt to the sudden voltage dip, requiring 81 ms to stabilize. Similarly, the adaptive LPF and ANF-based estimators show response times of 51 ms and 106 ms, respectively, highlighting their sensitivity to transient conditions. As illustrated in Table 2, the proposed approach, demonstrates superior dynamic response, stabilizing in just 9 ms after the disturbance.

<figure><img src="image_16.png" /><figcaption>**Fig. 13.** Estimated VF $\psi_g$ (20 Wb/div) during startup: (a) LPF estimator, (b) Adaptive LPF estimator, (c) ANF estimator, and (d) Proposed estimator.</figcaption></figure>

### 5.1.3. Harmonic distortion

Fig. 15 compares the performance of the four VF estimators under distorted grid conditions, where 30% of both the 5th and 7th harmonics were injected into the grid voltage. The results reveal that the proposed estimator effectively extracts only the fundamental flux component, yielding a nearly sinusoidal VF waveform. Conversely, the LPF, adaptive LPF, and ANF-based estimators exhibit pronounced harmonic contamination, resulting in highly distorted flux waveforms. Although the ANF utilizes ADALINE filtering, it is designed exclusively for offset cancellation and lacks the capability to suppress harmonic distortions, unlike the proposed estimator, which precisely isolates the fundamental VF component. A quantitative comparison in Table 2 highlights this advantage: the THD values for the LPF, adaptive LPF, and ANF estimators are 7.30%, 7.32%, and 7.28%, respectively, while the proposed estimator reduces THD to just 1.73%. This substantial improvement demonstrates the estimator's enhanced harmonic rejection capability, making it particularly suited for grid-connected power converters operating in distorted environments.

<table><caption>**Table 2**
Comparison summary of the compared VF estimators.</caption><thead><tr><th></th><th>LPF</th><th>Adaptive LPF</th><th>ANF</th><th>Proposed</th></tr></thead><tbody><tr><td>Response time (t<sub>r-5 %</sub>) during startup (ms)</td><td>99</td><td>61</td><td>101</td><td>8</td></tr><tr><td>Overshoot during startup (%)</td><td>24</td><td>17</td><td>28</td><td>≈ 0</td></tr><tr><td>Response time (t<sub>r-5 %</sub>) during sag type A (ms)</td><td>81</td><td>51</td><td>106</td><td>9</td></tr><tr><td>Estimated VF THD under distorted grid conditions (ms)</td><td>7.30</td><td>7.32</td><td>7.28</td><td>1.73</td></tr></tbody></table>

<figure><img src="image_17.png" /></figure>
<figure><img src="image_18.png" /><figcaption>Fig. 14. Estimated VF $\psi_g$ (20 Wb/div) during a voltage sag type A: (a) LPF estimator, (b) Adaptive LPF estimator, (c) ANF estimator, and (d) Proposed estimator.</figcaption></figure>

## 5.2. Comparison between conventional MPDPC and VF-SMPDPC

To evaluate the performance of VF-SMPDPC, it was compared with conventional MPDPC under the same conditions. The parameters used for the simulation are listed in Table 1. Throughout the tests, reactive power $Q$ was kept at zero to maintain unity power factor.

### 5.2.1. Dynamic performance comparison

Fig. 16 presents the simulation results for ideal sinusoidal grid voltages. To compare the performance of the inner control loops, the outer DC bus voltage regulation loop was disabled, and the reference active power was externally imposed. In this test, the reference active power $P_{ref}$ was initially set to 600 W, and a step change of 400 W was applied (increasing $P_{ref}$ from 600 W to 1000 W).

Fig. 16 shows that both MPDPC and VF-SMPDPC achieve similar current waveforms under steady-state conditions, with THD values below 1 %. The results in Fig. 16b demonstrate that the proposed strategy provides accurate control of both active and reactive power. During the step change, the active power follows the new reference with high accuracy, and the reactive power remains at zero during the transient. These results are similar to those obtained with conventional MPDPC (Fig. 16a). The transient responses of both algorithms are also similar, with current amplitudes increasing from 5 A to 8.5 A. Furthermore, a closer look at the grid current waveforms (Figs. 16c and 16d) reveals that both strategies have very short response times. Based on these results, it can be concluded that the proposed VF-SMPDPC strategy achieves performance comparable to conventional MPDPC while significantly reducing computational complexity and minimizing sensor requirements.

### 5.2.2. Comparison under distorted grid conditions

The results depicted in Fig. 17 illustrate grid voltages distorted by

5th and 7th harmonic components, accounting for 10 % of the 5th harmonic and 10 % of the 7th harmonic. During this test, the reference DC bus voltage was maintained at 220 V. Fig. 17a illustrates that the absorbed current under conventional MPDPC strategy is significantly distorted, exhibiting a THD of 14.71 %, which exceeds the IEEE Standard 519 recommendation of a maximum 5 % THD for current (IEEE Standard for Harmonic Control in Electric Power Systems, 2022). In contrast, the proposed control strategy maintains the absorbed current at a nearly sinusoidal level, achieving a THD of only 1.23 % (see Fig. 17b). This enhancement is attributed to the ADALINE-VF estimator, which focuses on estimating the fundamental VF components, effectively reducing the harmonics present in the grid voltages. In both cases, the DC bus voltage remains at its reference value of 220 V.

To further complement this analysis, a detailed comparison of current THD as a function of grid voltage distortion was performed. The results are presented in Fig. 18. Under purely sinusoidal grid voltages (THD = 0 %), both control strategies achieve comparable current quality, with low THD levels. However, as the harmonic distortion in the grid voltage increases, the performance of conventional MPDPC deteriorates rapidly. Once the grid voltage THD exceeds 5 %, the resulting current THD surpasses the acceptable limit and reaches as high as 33 % for a 30 % grid voltage THD. In contrast, the proposed VF-SMPDPC strategy demonstrates remarkable robustness. Even under severe voltage distortion levels, the current THD remains consistently below the 5 % threshold. This clearly demonstrates the superior harmonic immunity of the proposed VF estimator, which ensures that only the fundamental VF components are retained for control purposes.

## 5.3. Robustness tests of the VF-SMPDPC

Fig. 19 illustrates the system’s transient behavior following a step change in the reactive power reference $Q_{ref}$. The test consists of a step
<figure><img src="image_19.png" /><figcaption>**Fig. 15.** Estimated VF $\psi_g$ (20 Wb/div) for distorted grid voltages with 30 % 5th and 7th harmonics: (a) LPF estimator, (b) Adaptive LPF estimator, (c) ANF estimator, and (d) Proposed estimator.</figcaption></figure>

from 0 VAr to + 400 VAr, while the active power reference $P_{ref}$ is held constant at 600 W. Initially, the grid current $i_{ga}$ is in phase with the grid voltage $e_{ga}$, which corresponds to unity power factor operation (i.e., zero reactive power). After the step is applied, $i_{ga}$ lags $e_{ga}$, which corresponds to the expected phase shift for positive reactive power flow. Besides, the currents vary quickly without significant overshoots when the converter passes from unity to lagging power factor operation. From these results, correct operation of the VF-SMPDPC with the proposed VF estimator at non-zero reactive power operation is verified.

The dynamic performance of the proposed VF-SMPDPC strategy is further validated through a step change in the DC bus voltage reference. As shown in Fig. 20, the reference voltage $V_{dc-ref}$ is increased from 160 V to 220 V. In this configuration, the active power reference $P_{ref}$ is generated by an outer control loop responsible for regulating the DC bus voltage. In response to the step in $V_{dc-ref}$, the controller reacts promptly, causing the active power $P$ to increase from approximately 500 W to 1000 W in order to supply the additional energy demand and maintain the new voltage level. The waveform of $V_{dc}$ closely tracks the reference, both before and after the step change, without exhibiting overshoot or instability. This confirms the fast transient response and accurate steady-state regulation capabilities of the proposed control method. Moreover, the reactive power $Q$ remains tightly regulated around zero during transient, indicating that the controller successfully decouples active and reactive power control, even during abrupt changes in DC voltage demand.

## 5.4. Synthesis of the performances comparison

A consolidated comparison of the two implemented control schemes is presented in Table 3, highlighting key performance aspects such as harmonic immunity, transient response, sensor requirements, and

computational load.

The conventional MPDPC exhibits excellent dynamic behavior during transients, with a fast current response time. However, its performance significantly deteriorates under distorted grid voltage conditions due to its design based on the assumption of ideal sinusoidal voltages. This limitation results in poor harmonic rejection capability, as demonstrated in 5.2. In contrast, the proposed VF-SMPDPC maintains strong transient performance while offering significant improvements in harmonic immunity. Thanks to the integration of the ADALINE-based VF estimator, the strategy accurately filters out voltage harmonics and operates effectively even under highly distorted grid conditions. The estimator also enables sensorless control, eliminating the need for direct grid voltage measurements—a key advantage in terms of hardware simplification and system reliability. From a computational standpoint, the VF-SMPDPC achieves a lower execution time (5.02 µs) compared to the conventional MPDPC (7.14 µs), despite incorporating an additional estimation stage. This gain is primarily attributed to the streamlined structure of the predictive control algorithm, particularly the proposed simplified optimal voltage vector selection process. This optimization not only compensates for the computational cost introduced by the estimator but also reduces the overall processing burden relative to the conventional approach.

## 6. Conclusion

This paper proposed a sensorless control scheme for grid-connected converters by combining a simplified predictive controller with a novel VF estimation method. The VF estimator, based on an ADALINE-type neural network, effectively replaced grid voltage sensors by providing accurate, robust, and real-time VF estimation. Its dynamic performance was extensively evaluated against other VF estimation
<figure><img src="image_20.png" /><figcaption>**Fig. 16.** Active power *P* and *P*<sub>ref</sub> (100 W/div), reactive power *Q* (100 VAr/div), grid current *i*<sub>ga</sub> (10 V/div), and zoom in grid currents *i*<sub>g,abc</sub> (5 A/div) under active power step change: (a), (c) Conventional MPDPC, and (b), (d) VF-SMPDPC.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>**Fig. 17.** Grid voltage *e*<sub>ga</sub> (100 V/div), grid current *i*<sub>ga</sub> (10 V/div), and DC bus voltage *V*<sub>dc</sub> (100 V/div) under distorted grid voltages: (a) Conventional MPDPC, and (b) proposed VF-SMPDPC.</figcaption></figure>

methods such as first-order LPF, adaptive LPF and ANF within a real-
time simulation environment using the OP4512-OPAL-RT system. Ob-
tained results highlighted the superior performance of the proposed
estimator, characterized by rapid response, minimal sensitivity to grid
voltage harmonic distortion, and precise estimation. Additionally, a
simplified predictive control strategy was developed to reduce compu-
tational complexity while ensuring effective power regulation. Inte-
grating this predictive control with the new VF estimator led to the
realization of a sensorless VF-SMPDPC control scheme. Comparative
evaluations demonstrated that the proposed VF-SMPDPC significantly
outperformed the conventional MPDPC method that relies on direct grid

voltage sensing. Even under distorted grid conditions, the proposed
scheme maintained high control accuracy and low harmonic distortion,
ensuring reliable and efficient operation.

CRediT authorship contribution statement

**Azeddine Houari:** Writing – review & editing, Supervision. **Abdel-hakim Saim:** Writing – review & editing, Supervision. **Noumidia Amoura:** Validation, Software. **Boussad Boukais:** Writing – review & editing, Validation, Conceptualization. **Koussaila Mesbah:** Writing – review & editing, Software, Resources, Methodology,
<figure><img src="image_22.png" /><figcaption>**Fig. 18.** Comparison of current THD for conventional MPDPC and proposed VF-SMPDPC under different grid voltage distortion levels.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>**Fig. 19.** Grid voltage $e_{ga}$ (50 V/div), grid current $i_{ga}$ (10 V/div), and reactive power $Q$ and $Q_{ref}$ (220 VAr/div) under reactive power step change.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>**Fig. 20.** Active power $P$ (300 VAr/div), reactive power $Q$ (300 VAr/div), and DC bus voltage $V_{dc}$ and $V_{dc-ref}$ (75 V/div) during DC bus voltage step change.</figcaption></figure>

<table><caption>**Table 3**

Comparison Summary of the Implemented Control Schemes.</caption><thead><tr><th></th><th>Harmonic immunity</th><th>Transient response</th><th>Grid voltage sensors</th><th>Execution time</th></tr></thead><tbody><tr><td>Conventional MPDPC</td><td>Poor</td><td>Very fast</td><td>Required</td><td>7.14 µs</td></tr><tr><td>Proposed VF-SMPDPC</td><td>Very good</td><td>Very fast</td><td>Not required</td><td>5.02 µs</td></tr></tbody></table>

Conceptualization. Adel Rahoui: Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Resources, Methodology, Investigation, Formal analysis, Conceptualization.

## Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

## Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Data Availability

Data will be made available on request.

## References

Abu-Rub, H., Malinowski, M., Al-Haddad, K., 2014. Power Electronics for Renewable Energy Systems, Transportation and Industrial Applications. Wiley. (https://books.google.dz/books?id=TWq8AwAAQBAJ) ([Online]. Available).

Achlerkar, P.D., Ketan Panigrahi, B., 2021. Recursive least squares-based adaptive parameter estimation scheme for signal transformation and grid synchronization. IEEE J. Emerg. Sel. Top. Power Electron 9 (2), 2427–2439. https://doi.org/10.1109/JESTPE.2020.2970445.

Ahmad, S., et al., 2021. Fuzzy logic-based direct power control method for pv inverter of grid-tied AC microgrid without phase-locked loop. Art. no. 24 Electronics 10 (24). https://doi.org/10.3390/electronics10243095.

H. Akagi, E. Watanabe, and M. Aredes, “Instantaneous Power Theory and Applications to Power Conditioning,” pp. 37–109, Feb. 2017, doi: 10.1002/9781119307181.ch3.

Benyamina, F., Benrabah, A., Khoucha, F., Zia, M.F., Achour, Y., Benbouzid, M., 2021. An augmented state observer-based sensorless control of grid-connected inverters under grid faults. Int. J. Electr. Power Energy Syst. 133, 107222. https://doi.org/10.1016/j.ijepes.2021.107222.

Bhattacharya, S., Veltman, A., Divan, D.M., Lorenz, R.D., Jun. 1996. Flux-based active filter controller. IEEE Trans. Ind. Appl. 32 (3), 491–502. https://doi.org/10.1109/28.502159.

Bouafia, A., Krim, F., Gaubert, J.-P., 2009. Fuzzy-logic-based switching state selection for direct power control of three-phase PWM rectifier. Ind. Electron. IEEE Trans. 56, 1984–1992. https://doi.org/10.1109/TIE.2009.2014746.

Boukais, B., Mesbah, K., Rahoui, A., Saim, A., Houari, A., Benkhoris, M.F., 2024. Development of a 3 kW wind energy conversion system emulator using a grid-connected doubly-fed induction generator. Art. no. 12 Actuators 13 (12). https://doi.org/10.3390/act13120487.

Cirrincione, M., Pucci, M., Cirrincione, G., Capolino, G.-A., 2004. A new adaptive integration methodology for estimating flux in induction machine drives. IEEE Trans. Power Electron 19 (1), 25–34. https://doi.org/10.1109/TPEL.2003.820565.

Dai, P., Dong, S., Fu, X., Li, Y., 2011. Vector control of PWM rectifier based on a novel virtual flux observer. 2011 IEEE Int. Conf. Mechatron. Autom. 1641–1645. https://doi.org/10.1109/ICMA.2011.5985960.

Duarte, J.L., Van Zwam, A., Wijnands, C., Vandenput, A., 1999. Reference frames fit for controlling PWM rectifiers. IEEE Trans. Ind. Electron 46 (3), 628–630. https://doi.org/10.1109/41.767071.

Guo, X., Ren, H.-P., 2023. Robust model predictive power control for three-phase VSRs under unbalanced grid. ISA Trans. 133, 450–462. https://doi.org/10.1016/j.isatra.2022.07.021.

Idris, N.R.N., Yatim, A.H.M., 2002. An improved stator flux estimation in steady-state operation for direct torque control of induction machines. IEEE Trans. Ind. Appl. 38 (1), 110–116. https://doi.org/10.1109/29.8980364.

IEEE Standard for Harmonic Control in Electric Power Systems, 2022. IEEE Std 519-2022 Revis. IEEE Std 519-2014, pp. 1–31, doi: 10.1109/IEEESTD.2022.9848440.

Jabbarnejad, A., Vaez-Zadeh, S., Jahanpour-Dehkordi, M., 2020. Combined control of grid connected converters based on a flexible switching table for fast dynamic and reduced harmonics. IEEE Trans. Energy Convers. 35 (1), 77–84. https://doi.org/10.1109/TEC.2019.2944994.

Jukić, F., Pravica, L., Kovačić, M., Stipetić, S., 2024. Voltage sensorless synchronization of a grid-side converter based on a discontinuous operating mode and a sliding mode observer. IEEE Trans. Ind. Appl. 60 (4), 6314–6325. https://doi.org/10.1109/TIA.2024.3379952.

Ketzer, M.B., Jacobina, C.B., 2016. Virtual flux sensorless control for shunt active power filters with quasi-resonant compensators. IEEE Trans. Power Electron 31 (7), 4818–4830. https://doi.org/10.1109/TPEL.2015.2487298.

Kulka, A., 2008. Sensorless Digital Control of Grid Connected Three Phase Converters for Renewable Sources.

Lee, J., Lee, Y., Lee, M., Jin, D., Yoon, Y., 2025. Estimation of flux saturation model for SynRMs using artificial neural network. IEEE Trans. Ind. Appl. 1–9. https://doi.org/10.1109/TIA.2025.3532412.
Luna, A., et al., 2015. Grid voltage synchronization for distributed generation systems under grid fault conditions. IEEE Trans. Ind. Appl. 51 (4), 3414–3425. https://doi.org/10.1109/TIA.2015.2391436.

Mahvash, H., Taher, S.A., Guerrero, J.M., 2024. Mitigation of severe false data injection attacks (FDIAs) in marine current turbine (MCT) type 4 synchronous generator renewable energy using promoted backstepping method. Renew. Energy 222, 120008. https://doi.org/10.1016/j.renene.2024.120008.

Malinowski, M., Kazmierkowski, M.P., Trzynadlowski, A.M., 2003. A comparative study of control techniques for PWM rectifiers in AC adjustable speed drives. IEEE Trans. Power Electron 18 (6), 1390–1396. https://doi.org/10.1109/TPEL.2003.818871.

Malinowski, M., Jasinski, M., Kazmierkowski, M.P., 2004. Simple direct power control of three-phase PWM rectifier using space-vector modulation (DPC-SVM). IEEE Trans. Ind. Electron 51 (2), 447–454. https://doi.org/10.1109/TIE.2004.825278.

Mesbah, K., Rahoui, A., Boukais, B., 2024. Neural network quadrature signal generator-based virtual flux estimation for predictive control of PWM converters. IEEE Trans. Ind. Electron 71 (5), 4785–4794. https://doi.org/10.1109/TIE.2023.3281704.

Nguyen, M.H., Kwak, S., Choi, S., 2024. Voltage injection based MPDPC technique for individual phase loss reduction in active front-end rectifier. IEEE Access 12, 51193–51207. https://doi.org/10.1109/ACCESS.2024.3382127.

Noguchi, T., Tomiki, H., Kondo, S., Takahashi, I., 1998. Direct power control of PWM converter without power-source voltage sensors. IEEE Trans. Ind. Appl. 34 (3), 473–479. https://doi.org/10.1109/28.673716.

Norniella, J.G., et al., 2014. Improving the dynamics of virtual-flux-based control of three-phase active rectifiers. IEEE Trans. Ind. Electron 61 (1), 177–187. https://doi.org/10.1109/TIE.2013.2245614.

Pöllänen, R., 2003. Converter-flux-based current control of voltage source PWM rectifiers-analysis and implementation.

Rahoui, A., Mesbah, K., Boukais, B., Otmane-Cherif, T., 2020b. Frequency-adaptive neural network based virtual flux estimation for sensorless control of PWM converters under unbalanced conditions. 2020 Int. Conf. Electr. Eng. (ICEE) 1–6. https://doi.org/10.1109/ICEE49691.2020.9249854.

Rahoui, A., Boukais, B., Mesbah, K., Otmane-Cherif, T., 2020a. Neural networks based frequency-locked loop for grid synchronization under unbalanced and distorted conditions. 2020 Int. Conf. Electr. Eng. (ICEE) 1–5. https://doi.org/10.1109/ICEE49691.2020.9249923.

Rajendran, S., Diaz, M., Devi, V.S.K., Jena, D., Travieso, J.C., Rodriguez, J., 2023. Wind turbine emulators—A Review. Art. no. 3 Processes 11 (3). https://doi.org/10.3390/pr11030747.

Rath, A., Srlungavarapu, G., 2024. Reduced complexity model predictive direct power control for unbalanced grid. Electr. Power Syst. Res. 234, 110563. https://doi.org/10.1016/j.epsr.2024.110563.

Rath, A., Gopalakrishna, S., Pattnaik, M., 2024. Improved sensorless DPC with constant switching frequency for non-ideal grid. IEEE Trans. Consum. Electron 70 (3), 5167–5176. https://doi.org/10.1109/TCE.2024.3412233.

Razali, A.M., Rahman, M.A., George, G., Rahim, N.A., 2015. Analysis and design of new switching lookup table for virtual flux direct power control of grid-connected three-phase PWM AC-DC converter. IEEE Trans. Ind. Appl. 51 (2), 1189–1200. https://doi.org/10.1109/TIA.2014.2344503.

Roslan, N.F., Luna, A., Rocabert, J., Candela, J.I., Rodriguez, P., 2018. Remote power control injection of grid-connected power converters based on virtual flux. Art. no. 3 Energies 11 (3). https://doi.org/10.3390/en11030488.

Sevécký, J., Smídl, V., Glac, A., Peroutka, Z., 2023. Neural ODE for estimation of flux linkage models of synchronous machines. IECON 2023 49th Annu. Conf. IEEE Ind. Electron. Soc. 1–6. https://doi.org/10.1109/IECON51785.2023.10312498.

Sun, Y., Li, Z., Zhang, Y., Li, Y., Zhang, Z., 2022. A time-domain virtual-flux based predictive control of modular multilevel converters for offshore wind energy integration. IEEE Trans. Energy Convers. 37 (3), 1803–1814. https://doi.org/10.1109/TEC.2021.3137411.

Suul, J.A., Luna, A., Rodriguez, P., Undeland, T., 2012. Voltage-sensor-less synchronization to unbalanced grids by frequency-adaptive virtual flux estimation. IEEE Trans. Ind. Electron 59 (7), 2910–2923. https://doi.org/10.1109/TIE.2011.2168793.

Takahashi, I., Noguchi, T., 1986. A new quick-response and high-efficiency control strategy of an induction motor. IEEE Trans. Ind. Appl. IA-22 (5), 820–827. https://doi.org/10.1109/TIA.1986.4504799.

Tao, Y., Wu, Q., Wang, L., Tang, W., 2016. Voltage sensorless predictive direct power control of three-phase PWM converters. IET Power Electron 9 (5), 1009–1018. https://doi.org/10.1049/iet-pel.2014.0713.

Xia, C., Liu, T., Shi, T., Song, Z., 2014. A simplified finite-control-set model-predictive control for power converters. IEEE Trans. Ind. Inf. 10 (2), 991–1002. https://doi.org/10.1109/TII.2013.2284558.

Zhang, Z., et al., 2015. Predictive control with novel virtual-flux estimation for back-to-back power converters. IEEE Trans. Ind. Electron 62 (5), 2823–2834. https://doi.org/10.1109/TIE.2014.2361802.

Zhang, Z., Hackl, C.M., Kennel, R., 2017. Computationally efficient DMPC for three-level NPC back-to-back converters in wind turbine systems with PMSG. IEEE Trans. Power Electron 32 (10), 8018–8034. https://doi.org/10.1109/TPEL.2016.2637081.