# Enhanced Linear State Observer-Based PLL-Less Vector-Oriented Control Method for a Three-Phase PWM Rectifier

Mabrouk Defdaf, Djamel Ziane, Ali Chebabhi, Said Barkat, Mohamed Fouad Benkhoris, Abdelhalim Kessal

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

## ► To cite this version:

Mabrouk Defdaf, Djamel Ziane, Ali Chebabhi, Said Barkat, Mohamed Fouad Benkhoris, et al. Enhanced Linear State Observer-Based PLL-Less Vector-Oriented Control Method for a Three-Phase PWM Rectifier. International Transactions on Electrical Energy Systems, 2024, 2024 (1), 10.1155/etep/7191464. hal-05026947

HAL Id: hal-05026947

https://hal.science/hal-05026947v1

Submitted on 4 Jun 2025

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 Attribution 4.0 International License
Research Article

# Enhanced Linear State Observer-Based PLL-Less Vector-Oriented Control Method for a Three-Phase PWM Rectifier

Mabrouk Defdaf ID,<fnref n="1" /> Djamel Ziane ID,<fnref n="2" /> Ali Chebabhi ID,<fnref n="1" /> Said Barkat ID,<fnref n="1" />
Mohamed Fouad Benkhoris ID,<fnref n="2" /> and Abdelhalim Kessal ID<fnref n="3" />

Correspondence should be addressed to Djamel Ziane; djamel.ziane@univ-nantes.fr

Received 23 February 2024; Revised 19 September 2024; Accepted 25 October 2024

Academic Editor: Hamed Jafari Kaleybar

Copyright © 2024 Mabrouk Defdaf et al. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.

In practical three-phase PWM rectifiers, external disturbances, such as changes in load and grid voltage, and internal disturbances, such as DC capacitor uncertainty, have caused major problems, including DC-link voltage fluctuations, harmonic distortion, and power factor degradation. They can also cause problems with power balance, stability, system dynamics, reduced efficiency, and increased stress. This paper proposes an enhanced linear state observer (ELSO) with a PLL-less voltage-oriented control (PLL-less VOC) method for a three-phase PWM rectifier in the synchronous coordinate frame (dq coordinate), aiming to simultaneously improve the DC-link voltage dynamics, disturbance rejection ability, and the grid current control loop's transient response. The ELSO is suggested instead of the load current sensor not only to estimate the disturbance but also to estimate the overall switching losses of the PWM rectifier, resulting in good DC-link voltage disturbance rejection and transient fluctuation suppression, high robustness against DC capacitance change, low size and cost, and high reliability. Furthermore, contrary to the traditional VOC method, where the PLL and Park transforms are important parts for controlling the rectifier in the dq coordinate, the PLL-less VOC method can control the grid currents directly in the dq coordinate without needing PLL and Park transforms, which can offer a fast transient response and low computational burden. The frequency domain analysis of the ELSO is used to identify its gains and enhance its stability and disturbance rejection capability. Finally, hardware-in-the-loop (HIL) simulation using the OPAL-RT 4510 real-time simulator is also given to confirm the superiority and efficacy of the suggested control method.

**Keywords:** disturbance observer; enhanced linear state observer; PLL-less voltage-oriented control method; three-phase PWM rectifier

# 1. Introduction

Recently, three-phase PWM rectifiers have been extensively utilized in an array of modern industrial power applications, including DC microgrids, EV charging systems, renewable energy sources, and grid power factor correction [1-5]. This topology can provide sinusoidal currents with a unity power factor (UPF) and high-quality DC-link voltage for all these applications.

Various control methods have been designed and proposed with respect to their control strategies. Despite the different ideas behind these methods, they are all developed with the same goal, which is to improve the control flexibility of grid currents or powers, achieve a UPF with minimal grid current harmonics, and preserve a stable DC-link voltage [6]. The most commonly used method is referred to as the direct power control (DPC) method, which is predicated on controlling the active and reactive powers through power

<footnote n="1">¹Electrical Engineering Laboratory, Faculty of Technology, University of M'sila, M'sila 28000, Algeria</footnote>

<footnote n="2">²IREENA Laboratory, University of Nantes, Saint-Nazaire, France</footnote>

<footnote n="3">³LPMRN Laboratory, University Mohamed El Bachir El Ibrahimi of Bordj Bou Arreridj, El-Anasser, Algeria</footnote>
loops while controlling the DC-link voltage in the outer loop [7, 8]. This method results in a good transient response, but it has poor steady-state performance and also causes high ripples in the grid currents [9].

Contrary to the DPC method, the voltage-oriented control (VOC) method is developed in the synchronous coordinate frame (dq coordinate) based on the orientation of the grid voltage vector to its active component while the reactive grid voltage component is set to zero. The key benefit of this method is that it uses the Park transformation based on the grid voltage phase angle to convert the three-phase grid currents and voltages from AC components to DC active and reactive components in the dq coordinate, which changes the reference tracking issue into a regulation issue. Furthermore, using the VOC method, the dynamics of the PWM rectifier in the dq coordinate are transformed into a linear time-invariant (LTI) system, which makes it easy to analyze and control [10, 11].

However, to obtain an excellent performance by using this method, the grid voltage phase angle used in the Park transforms must be extracted accurately [12]. In recent years, the phase-locked loop (PLL) has emerged as the most popular way for obtaining the grid voltage phase angle in the VOC method for PWM rectifier applications. The PLL, however, will result in a slower response in transient states. Furthermore, it may contribute to the instability issue in certain scenarios [13, 14]. In addition, the PLL with Park transforms will result in an additional computational burden. Therefore, an improved and straightforward VOC method is vitally required to attain higher transient and steady-state performances.

To simultaneously improve the transient responses and avoid the instability problem caused by the PLL, the authors in [10] developed a new control method without PLL in a virtual dq coordinate. The problems of PLL under grid voltage disturbance were solved by introducing a virtual phase angle using two low-pass filters (LPFs) instead of the PLL. Active and reactive grid currents and voltages in the dq coordinate are then derived using park transformations based on this virtual phase angle, while modulation voltages are obtained by applying inverse Park transformations to the outputs of the grid currents' control loop also based on the same phase angle. Consequently, direct control of active and reactive grid currents in the dq reference frame without the PLL can be achieved. However, the inherent complexity of Park and its inverse transforms and the LPFs used in this method make it very complex and have a large computational burden [13].

Recently, Gui et al. [15] proposed a new PLL-less VOC method for the control of grid-connected inverters in the dq coordinate to suppress the effects of the PLL and Park transformations and, at the same time, achieve enhanced performances under different conditions. Unlike traditional VOC methods, this PLL-less VOC method requires no PLL and Park transforms, and it has the same structure and efficiency as the traditional VOC when its PLL is properly operating. This approach not only controls the PWM converters but also uses the ideas of the DPC method to develop the dynamic model of the PWM converters in the dq coordinate. Furthermore, this method combines the benefits of traditional DPC and traditional VOC, which makes it

a relatively low computational burden and widely used for three-phase power converters [14–19].

On the other hand, to improve the PWM rectifier dynamics, improve the flow of the active grid power to the load, and overcome the system's instability caused by the DC-link voltage fluctuations occurring due to disturbances of active grid power or DC load and uncertainties or changes of the DC capacitor, several techniques and methods were proposed to regulate the DC-link voltage under load disturbance. In [20–22], a load current feedforward based on the sensor control method in both traditional VOC and DPC methods was added to the reference of active current derived from the DC-link voltage control loop. This can simultaneously enhance the DC-link voltage dynamics and response of the inner loop. However, this method requires an additional number of sensors. As it is well known, a high number of sensors leads to higher costs and power losses. It also increases the possibility of measurement mistakes or sensor malfunctions, which in turn negatively impacts the system's reliability [23–25].

In three-phase PWM rectifiers, measurement errors or sensor malfunctions may cause the rectifier to operate incorrectly, causing an additional ripple in the DC-link voltage. Another issue is the uncertainties or changes in the DC capacitor, resulting from the integration or abandonment of renewable energy sources, PWM inverters, or DC-DC PWM converters connected to the DC-link. These changes impact the regulation of the DC-link voltage, and the feedforward based on the sensor control method is unable to adequately handle them. Consequently, a number of sensorless-based DC-link voltage control methods have been suggested to address the DC-link voltage or load current sensor faults, as well as to improve the robustness of the DC-link voltage against changes in the DC capacitor, assure a zero steady-state DC-link voltage error, increase system reliability, and lower both the cost and size [23–27].

Nonlinear and sliding mode observers based on traditional VOC methods were suggested in [23, 24] for the regulation of DC-link voltage. These observers perform well in terms of DC-link voltage regulation and transient fluctuation suppression; however, in addition to the traditional VOC problems, the analysis, tuning, and design of these observers are highly complicated due to their nonlinearity. In order to overcome these problems, an enhanced linear state observer (ELSO) for load current feedforward-based DPC method under load disturbance and DC capacitor uncertainties was proposed in [25–27]. This approach can decrease the size and cost, increase the reliability of the system, and enhance the DC-link voltage dynamics, but it has not addressed the drawback of grid current ripples inherent in the DPC method.

In this paper, an ELSO for disturbance feedforward-based PLL-less VOC method of a three-phase PWM rectifier is suggested to solve the major inherent problems related to the aforementioned control methods. The key contribution consists of combining the benefits of the two methods, the ELSO and the PLL-less VOC. The PLL-less VOC method is adopted instead of the traditional VOC and DPC methods for its ability to model and control the three-phase PWM
rectifier in the dq coordinate without needing any PLL and Park transformation, resulting in a fast transient response and low computational burden. Meanwhile, the ELSO is used instead of the sensor for its ability to estimate the total disturbance that includes the load current disturbance and the overall switching power losses, which is added to the output of the DC-link voltage controller not only to provide the inner loop active current reference but also to simultaneously improve the DC-link voltage dynamics and transient response of the grid current control loop, reject the total disturbance, improve the robustness of the DC-link voltage control loop against the change of the DC capacitor, ensure a zero steady-state DC-link voltage error, improve the reliability of the system, and reduce both the cost and size. The significant benefits of the suggested control method are simple design, low computational burden, fast transient response, good disturbance rejection, and it is independent of the type of load connected to the DC-link.

The remaining parts of the paper are arranged as follows. The dynamics of the DC-link voltage and grid currents are presented in Section 2. The inner and outer loop controllers for the suggested ELSO-based PLL-less control approach are each thoroughly designed in the first and second subsections of Section 3, which is devoted to this subject. Subsequently, the third subsection delves into the detailed design of the suggested ELSO, including its model, stability, and tuning parameters. The suggested ELSO control approach is compared in Section 4 with the traditional PI controller and the PI with feedforward based on the sensor. This paper is finally concluded in Section 5.

## 2. Modeling of the PWM Rectifier

The structure of the three-phase PWM rectifier is depicted in Figure 1. The main objective of this system is to transfer the power from the grid (AC side) to the DC load (DC side) with high quality and good performance.

The PWM rectifier input voltages are denoted as $v_{iabc}$, the grid voltages are denoted as $v_{gabc}$, and the grid currents are denoted as $i_{gabc}$. $L_g$ and $C_{dc}$ denote the equivalent inductances of the line filter and the DC capacitor, respectively. $I_{dc}$, $V_{dc}$, and $I_C$ are the output DC current, DC-link voltage, and DC capacitor current, respectively. $R$ and $I_o$ denote the resistance and load current, respectively. $R_s$ represents the rectifier's overall switching losses.

The rectifier model expressing the grid current dynamics in the three-phase abc coordinate frame is given by

$$ \left\{ \begin{aligned} \frac{di_{ga}}{dt} &= \frac{-R_g}{L_g}i_{ga} + \frac{v_{ga}}{L_g} - \frac{v_{ia}}{L_g}, \\ \frac{di_{gb}}{dt} &= \frac{-R_g}{L_g}i_{gb} + \frac{v_{gb}}{L_g} - \frac{v_{ib}}{L_g}, \\ \frac{di_{gc}}{dt} &= \frac{-R_g}{L_g}i_{gc} + \frac{v_{gc}}{L_g} - \frac{v_{ic}}{L_g}. \end{aligned} \right. \quad (1) $$

By transforming equation (1) into the dq coordinate using the concepts of the PLL-less VOC method presented in [19], the grid current dynamics can be expressed as

$$ \left\{ \begin{aligned} \frac{di_{gd}}{dt} &= \frac{-R_g}{L_g}i_{gd} + \omega i_{gq} + \frac{V_{gmax}}{L_g} - \frac{v_{id}}{L_g}, \\ \frac{di_{gq}}{dt} &= \frac{-R_g}{L_g}i_{gq} - \omega i_{gd} - \frac{v_{iq}}{L_g}, \end{aligned} \right. \quad (2) $$

where $v_{idq}$ and $i_{gdq}$ are the PWM rectifier input voltages and grid currents in the dq coordinate, respectively. $V_{gmax}$ and $\omega$ are the grid voltage magnitude and angular frequency, respectively.

As shown in Figure 1, the dynamic of $V_{dc}$ is expressed as

$$ \frac{dV_{dc}}{dt} = \frac{1}{C_{dc}}I_{c} = \frac{1}{C_{dc}}(I_{dc} - I_{dis}), \quad (3) $$

where $I_{dis}$ is the total disturbance current that consists of the load current and current through the resistance modeling the overall switching losses, which is expressed according to Figure 1 as

$$ I_{dis} = \frac{V_{dc}}{R_s} + I_L. \quad (4) $$

When the power losses of the rectifier switches and the inductor filter are not taken into account, the relation between both sides of the PWM rectifier can be expressed as

$$ P_g = P_{dc}. \quad (5) $$

Therefore, the grid powers can be described using the VOC concept in the dq coordinate as [16]

$$ \left\{ \begin{aligned} P_g &= i_{gd}V_{gmax}, \\ Q_g &= -i_{gq}V_{gmax}. \end{aligned} \right. \quad (6) $$

From equations (5) and (6), the relationship between the grid active current $i_{gd}$ and the rectifier output current $I_{dc}$ is given by

$$ V_{gmax}i_{gd} = V_{dc}I_{dc}. \quad (7) $$

This results in

$$ i_{gd} = \frac{V_{dc}}{V_{gmax}}I_{dc}. \quad (8) $$

Therefore, the dynamic of $V_{dc}$ in equation (3) becomes

$$ \frac{dV_{dc}}{dt} = \frac{1}{C_{dc}} \left( \frac{V_{gmax}}{V_{dc}} i_{gd} - I_{dis} \right). \quad (9) $$

This equation implies that the DC-link voltage control cannot be performed only by its controller but also by controlling the active grid current and rejecting disturbances. Furthermore, if the DC-link voltage is not effectively regulated, the current control loop of the VOC method will be adversely impacted. Thus, for a three-phase PWM rectifier, to provide superior performance in controlling the
<figure><img src="image_3.png" /><figcaption>FIGURE 1: Power circuit of the three-phase PWM rectifier.</figcaption></figure>

grid currents and DC-link voltage as well as high disturbance rejection, an enhanced control method is absolutely necessary.

In the following section, a new control method that provides an effective control of the grid currents and DC-link voltage with low computational complexity and good disturbance rejection is proposed.

## 3. Proposed ELSO-Based PLL-Less VOC Method

The PLL-less VOC method is a new control method in the dq coordinate that has been recently investigated [14–16] in order to get around the limitations of both methods. This approach does not require the use of PLL or Park transformations, and it is capable of providing decoupled control, a fast transient response, and low computational complexity. Figure 2 shows the schematic of the proposed ELSO-based PLL-less VOC method used for the control of the considered rectifier system. As in the traditional PLL-based VOC method, the PLL-less VOC method is also composed of inner and outer control loops. The inner loop is established in the dq coordinate using two independent PI controllers for controlling the grid currents $i_{gdq}$ provided using the PLL-less VOC method to obtain the PWM rectifier active and reactive input voltage references $v_{idq}^*$, while the outer loop is designed using the suggested PI-ELSO control method with the objective of controlling the DC-link voltage while rejecting the total disturbance, as well as to obtain the active grid current reference $i_{gd}^*$.

**3.1. Inner Loop Controller Design.** As shown in Figure 2, the PLL-less VOC method is utilized in the inner loop control in which the active and reactive grid currents $i_{gdq}$ are provided using the concepts of both the instantaneous power theory and the VOC method using the following steps.

First, using the concept of instantaneous power theory, P and Q can be written in the abc coordinate frame as [14, 15]

$$ egin{cases} P = v_{ga}i_{ga} + v_{gb}i_{gb} + v_{gc}i_{gc}, \\ Q = \sqrt{\frac{1}{3}}(V_{gbc}i_{ga} + V_{gca}i_{gb} + V_{gab}i_{gc}), \end{cases} \quad (10) $$

where $V_{gbc} = (v_{gb} - v_{gc})$, $V_{gca} = (v_{gc} - v_{ga})$, and $V_{gab} = (v_{ga} - v_{gb})$ are line-to-line grid voltage.

In the dq coordinate, the grid powers P and Q are expressed by [28]

$$ \begin{cases} P = v_{gd}i_{gd} + v_{gq}i_{gq}, \\ Q = -v_{gd}i_{gq} + v_{gq}i_{gd}. \end{cases} \quad (11) $$

Using the PLL-less VOC concept [17, 19], the active grid voltage component $v_{gd}$ in the dq coordinate is aligned to the grid voltage vector resulting in a constant active grid voltage ($v_{gd} = V_{gmax}$) and zero reactive grid voltage ($v_{gq} = 0$). In this case, the powers P and Q in equation (11) become

$$ \begin{cases} P = V_{gmax}i_{gd}, \\ Q = -V_{gmax}i_{gq}. \end{cases} \quad (12) $$

From this equation, the active and reactive grid powers can be regulated indirectly through the regulation of active and reactive grid currents.

From equation (12), the currents $i_{gdq}$ can be obtained without using Park transform as follows:

$$ \begin{cases} i_{gd} = \frac{P}{V_{gmax}}, \\ i_{gq} = \frac{-Q}{V_{gmax}}. \end{cases} \quad (13) $$

It is clear from equation (2) that both active and reactive grid currents $i_{gd}$ and $i_{gq}$ are coupled between them by the terms $\omega L_g i_{gq}$ and $\omega L_g i_{gd}$. This coupling is addressed by introducing the two control variables $u_d$ and $u_q$ in the manner described as follows[17]:

$$ \begin{cases} u_d = -\left(L_g \frac{di_{gd}}{dt} + R_g i_{gd}\right), \\ u_q = -\left(L_g \frac{di_{gq}}{dt} + R_g i_{gq}\right). \end{cases} \quad (14) $$
<figure><img src="image_4.png" /><figcaption>FIGURE 2: Proposed ELSO with PLL-less VOC circuit schematic of the considered PWM rectifier.</figcaption></figure>

These variables are also obtained as shown in Figure 2 from the regulation of the active and reactive currents $i_{gd}$ and $i_{fq}$ in the inner loop using two independent PI controllers as:

$$ \left\{ \begin{aligned} u_d &= \left( k_{pi} + \frac{k_{ii}}{s} \right) (i_{gd}^* - i_{gd}), \\ u_q &= \left( k_{pi} + \frac{k_{ii}}{s} \right) (i_{gq}^* - i_{gq}). \end{aligned} \right. \quad (15) $$

From equations (2), (14), and (15), the PWM rectifier active and reactive input voltage references $v_{idq}^*$ are expressed by

$$ \left\{ \begin{aligned} v_{id}^* &= \left( k_{pi} + \frac{k_{ii}}{s} \right) (i_{gd}^* - i_{gd}) + \omega L_g i_{gq} + V_{gmax}, \\ v_{iq}^* &= \left( k_{pi} + \frac{k_{ii}}{s} \right) (i_{gq}^* - i_{gq}) - \omega L_g i_{gd}, \end{aligned} \right. \quad (16) $$

where $k_{pi}$ and $k_{ii}$ denote the gains of current controllers that are expressed according to the pole placement theory as [28]

$$ \left\{ \begin{aligned} k_{pi} &= 2\xi_i\omega_{0i}L_g - R_g, \\ k_{ii} &= L_g\omega_{0i}^2, \end{aligned} \right. \quad (17) $$

where $\omega_{0i}$ and $\xi_i$ are the bandwidth and the damping factor of the grid current controllers, respectively.

Finally, the active and reactive modulation references in the $\alpha\beta$ coordinate frame $v_{idq}^*$, which are the input of the PWM generator (SVPWM), can be calculated without using inverse Park transformation as [13, 14]

$$ \left\{ \begin{aligned} v_{id}^* &= \frac{1}{V_{gmax}} (v_{g\alpha}^* v_{id}^* + v_{g\beta}^* v_{iq}^*), \\ v_{iq}^* &= \frac{1}{V_{gmax}} (v_{g\beta}^* v_{id}^* - v_{g\alpha}^* v_{iq}^*). \end{aligned} \right. \quad (18) $$

3.2. *Outer Loop Controller Design.* As illustrated in Figure 2, the outer loop is reserved for the regulation of the DC-link voltage, which consists of the suggested PI-ELSO controller. The PI controller is used to provide the capacitor current reference $I_C^*$, while the ELSO is used in the feedforward method to estimate the total disturbance current $I_{dis}$.
According to equation (2) and Figure 2, $I_C^*$ is provided from the regulation of the estimated DC-link voltage ($\hat{V}_{dc}$) using PI controller as follows:

$$I_C^* = \left( k_{pdc} + \frac{k_{idc}}{s} \right) (V_{dc}^* - \hat{V}_{dc}). \quad (19)$$

The estimated $\hat{V}_{dc}$ is used as input of the DC-link voltage controller rather than the measured value $V_{dc}$ to enhance its control performance. Here, $k_{pdc}$ and $k_{idc}$ are the gains of the DC-link voltage controller, which are expressed as

$$\left\{ \begin{array}{l} k_{pdc} = 2\xi_{dc}\omega_{0dc}C_{dc}, \\ k_{idc} = C_{dc}\omega_{0dc}^2, \end{array} \right. \quad (20)$$

where $\omega_{0dc}$ and $\xi_{dc}$ are the bandwidth and the damping factor of the DC-link voltage control loop, respectively.

According to equation (8), the active grid current reference $i_{gd}^*$ is expressed as

$$i_{gd}^* = \frac{\hat{V}_{dc}}{V_{gmax}} (I_{dc}^*), \quad (21)$$

where

$$I_{dc}^* = I_C^* + I_{dis}. \quad (22)$$

Substituting equations (19) and (22) into (21) gives

$$i_{gd}^* = \frac{\hat{V}_{dc}}{V_{gmax}} \left( \left( k_{pdc} + \frac{k_{idc}}{s} \right) (V_{dc}^* - \hat{V}_{dc}) + I_{dis} \right). \quad (23)$$

### 3.3. Proposed ELSO Design.
This section describes the design of the proposed ELSO for the outer loop control to eliminate the need for a load current sensor. Specifically, ELSO estimates the DC-link voltage $V_{dc}$ and the overall disturbance $I_{dis}$, as illustrated in Figure 2. The disturbance $I_{dis}$ acts as feedforward to the DC-link voltage controller to make up for the disturbance and simultaneously generate the active grid current reference $i_{gd}^*$ as in equation (23), while the estimated $\hat{V}_{dc}$ is used as input of the DC-link voltage controller rather than the measured value $V_{dc}$.

The dynamic of $V_{dc}$ in equation (3) can be expressed by

$$\frac{dV_{dc}}{dt} = \frac{1}{C}I_{dc} - \frac{1}{C}I_{dis}. \quad (24)$$

To design the proposed ELSO, the DC load and the overall switching losses represented by $R_s$ are considered as disturbances. In equation (24), $V_{dc}$ is taken as the first state variable $x_1 = V_{dc}$ and $-(1/C)I_{dis}$ as the second state variable $x_2 = -(1/C)I_{dis}$, while $I_{dc}$ is considered as the observer input $u$. Furthermore, the dynamic of $x_2$ is designated as $\dot{x}_2 = \Gamma$. Accordingly, the state-space model of equation (24) is written as

$$\left\{ \begin{array}{l} \dot{x}_1 = x_2 + \frac{1}{C_{dc}}u, \\ \dot{x}_2 = \Gamma. \end{array} \right. \quad (25)$$

To estimate the states $x_1$ and $x_2$ in real time, ELSO is constructed using the new estimated state variable $\hat{x}_1$ and $\hat{x}_2$ based on the state-space model in equation (25) as

$$\left\{ \begin{array}{l} \hat{x}_1 = \hat{x}_2 + \frac{1}{C_{dc}}u + \rho_1(x_1 - \hat{x}_1), \\ \hat{x}_2 = \rho_2(x_1 - \hat{x}_1), \\ y_1 = \hat{x}_1, \\ y_2 = \hat{x}_2, \end{array} \right. \quad (26)$$

where $(x_1 - \hat{x}_1)$ is the ELSO error, $\rho_1$ and $\rho_2$ are the ELSO gains, and $y_1$ and $y_2$ are the ELSO outputs. The ELSO gains are adjusted using the pole placement approach detailed in the following subsection.

### 3.4. Bandwidth Tuning and Stability Analysis of ELSO.
According to equations (25) and (26), the ELSO error dynamics can be expressed as

$$\left[ \begin{matrix} \dot{\xi}_1 \\ \dot{\xi}_2 \end{matrix} \right] = - \frac{\left[ \begin{matrix} \rho_1 & -1 \\ \rho_2 & 0 \end{matrix} \right]}{A} \left[ \begin{matrix} \varepsilon_1 \\ \varepsilon_2 \end{matrix} \right] + \left[ \begin{matrix} 0 \\ 1 \end{matrix} \right] \Gamma, \quad (27)$$

where $\varepsilon_1 = x_1 - \hat{x}_1$ and $\varepsilon_2 = x_2 - \hat{x}_2$ are the errors of state estimation and disturbance estimation, respectively.

The characteristic polynomial of A can be expressed as

$$P_A(s) = s^2 + \rho_1 s + \rho_2. \quad (28)$$

The characteristic polynomial of a stable second-order closed loop system $P(s)$ is expressed as

$$P_A(s) = s^2 + 2\xi_{\text{elso}}\omega_{0\text{elso}}s + \omega_{0\text{elso}}^2, \quad (29)$$

where $\zeta_{\text{elso}}$ and $\omega_{0\text{elso}}$ are the damping factor and bandwidth of the ELSO, respectively. To guarantee that the polynomial (28) is Hurwitz stable [29, 30], $\rho_1$ and $\rho_2$ are calculated by identifying equations (28) and (29), which results in

$$\left\{ \begin{array}{l} \rho_1 = 2\xi_{\text{elso}}\omega_{0\text{elso}}, \\ \rho_2 = \omega_{0\text{elso}}^2. \end{array} \right. \quad (30)$$

It is clear from this equation that the bandwidth $\omega_{0\text{elso}}$ is the main parameter that needs to be chosen in order to improve performance and ensure that the suggested ELSO operates effectively. This bandwidth is selected between 1 and 5 times of the DC-link voltage regulator's bandwidth ($\omega_{0\text{elso}} = 1 \sim 15 \omega_{0dc}$) to guarantee the fast tracking of the estimated state dynamics under any change in the real state dynamics [31, 32]. Also, the bandwidth of the ELSO cannot be chosen very large to avoid degrading its immunity against noise and not impacting the performance of the grid current controllers [33, 34]. So, in order to offer the best control performance while taking the requirements criteria of response speed, noise immunity, stability, and robustness, the grid current controller's bandwidth is selected to equal
4000 rad/s. The bandwidth $\omega_{0elso}$ is chosen to be 1500 rad/s and the bandwidth $\omega_{0dc}$ is set to 150 rad/s. In the three loops, the damping factor is selected so that it causes an adequate overshoot under any transient process, with $\xi_i = \xi_{dc} = \xi_{elso} = 1$. Figure 3 displays the ELSO schematic diagram.

By substituting equation (30) into equation (27), the new state-space model of the estimated state variable $\hat{x}_1$ and $\hat{x}_2$ is described as

$$ \begin{bmatrix} \hat{x}_1 \\ \hat{x}_2 \end{bmatrix} = \frac{\begin{bmatrix} -2\omega_{0elso} & 1 \\ -\omega_{0elso}^2 & 0 \end{bmatrix} \begin{bmatrix} \hat{x}_1 \\ \hat{x}_2 \end{bmatrix} + \begin{bmatrix} \frac{1}{C_{dc}} & 2\omega_{0elso} \\ 0 & \omega_{0elso}^2 \end{bmatrix} \begin{bmatrix} u \\ x_1 \end{bmatrix}}{A_{elso}} \quad (31) $$

This system equation indicates that the estimation states $\hat{x}_1$ and $\hat{x}_2$ of the ELSO are related to input $u$ and the state variable $x_1$ at the same time, which implies that the ELSO used for the disturbance estimation is represented in the Laplace domain by two transfer functions, one $H_u$ relating $u = I_{dc}$ to the estimation of $\hat{x}_2 = -(1/C_{dc})\hat{I}_{dis}$ and the other $H_y$ relating $x_1 = V_{dc}$ to the estimation of $\hat{x}_2 = -(1/C_{dc})\hat{I}_{dis}$. Using the Laplace transform of equation (31), the transfer functions $H_u$ and $H_y$ are expressed as follows:

$$ \begin{aligned} H_u &= \frac{\hat{X}_2(s)}{U(s)} = \frac{\hat{I}_{dis}(s)}{-C_{dc}I_{dc}(s)} \\ &= \begin{bmatrix} 0 & 1 \end{bmatrix} [sI - A_{elso}]^{-1} \begin{bmatrix} \frac{1}{C_{dc}} \\ 0 \end{bmatrix} \quad (32) \\ &= \frac{-\omega_{0elso}^2}{C_{dc}(s + \omega_{0elso})^2}, \end{aligned} $$

$$ \begin{aligned} H_y &= \frac{\hat{X}_2(s)}{X_1(s)} = \frac{\hat{I}_{dis}(s)}{-C_{dc}V_{dc}(s)} \\ &= \begin{bmatrix} 0 & 1 \end{bmatrix} [sI - A_{elso}]^{-1} \begin{bmatrix} 2\omega_{0elso} \\ \omega_{0elso}^2 \end{bmatrix} \quad (33) \\ &= \frac{\omega_{0elso}^2 s}{(s + \omega_{0elso})^2}. \end{aligned} $$

<figure><img src="image_5.png" /><figcaption>FIGURE 3: Schematic diagram of the proposed ELSO.</figcaption></figure>

From the two equations (32) and (33), the transfer function describing the relationship between the inputs of ELSO ($I_{dc}$ and $V_{dc}$) and the estimated disturbance ($I_{dis}$) can be expressed as

$$ \hat{X}_2(s) = \frac{1}{C_{dc}} \hat{I}_{dis} = - \frac{1}{C_{dc}} \frac{\omega_{0elso}^2}{(s + \omega_{0elso})^2} U(s) + \frac{\omega_{0elso}^2 s}{(s + \omega_{0elso})^2} X_1(s). \quad (34) $$

Equation (34) can be rewritten as follows:

$$ \hat{I}_{dis} = \frac{\omega_{0elso}^2}{(s + \omega_{0elso})^2} I_{dc}(s) - \frac{C_{dc} \omega_{0elso}^2 s}{(s + \omega_{0elso})^2} U_{dc}(s). \quad (35) $$

When we assume during the design of the proposed ELSO-based DC-link voltage regulation that the load current is totally compensated by the ELSO and neglect the dynamics of the active grid current controller, the output DC current reference ($I_{dc}^*$) resulting from the PI-ELSO controller becomes the same as the DC output current ($I_{dc}$). This assumption can simplify the design of the ELSO-based DC-link voltage controller and cannot impact the accuracy of the overall control system. So, using this assumption and equations (32)–(34), the block diagram of the ELSO for the DC-link voltage controller is given in Figure 4.

The Laplace transform of equation (3) results in

$$ V_{dc}(s) = \frac{R_s}{C_{dc}R_s s + 1} I_{dc}(s) - \frac{R_s}{C_{dc}R_s s + 1} I_L(s). \quad (36) $$

When the load current is assumed to be totally compensated, the DC side plant transfer function $G_p(s)$ can be expressed as

$$ G_p(s) = \frac{V_{dc}(s)}{I_{dc}(s)} = \frac{R_s}{C_{dc}R_s s + 1}. \quad (37) $$

From Figure 4, the transfer function of the modified DC side plant $G_{Mp}$ relating the output of DC-link voltage regulator $I_C^*$ to $V_{dc}$ is expressed by

$$ \begin{aligned} G_{Mp} &= \frac{V_{dc}(s)}{I_C^*(s)} = \frac{G_p}{1 + C_{dc}H_{Idc} + C_{dc}G_pH_{Udc}} \\ &= \frac{G_p}{((s^2 + 2\omega_{0elso}s)/(s + \omega_{0elso})^2) + (\omega_{0elso}^2 s/(s + \omega_{0elso})^2)G_p}. \end{aligned} \quad (38) $$
<figure><img src="image_6.png" /><figcaption>FIGURE 4: Block diagram of the equivalent transfer function of ELSO-based DC-link voltage controller plant.</figcaption></figure>

As can be seen from equation (38), the $G_{Mp}(s)$ denominator is the sum of two filters: a LPF and a high-pass filter (HPF). The modified plant form is changed into an integral form when its frequency is varied inside the ELSO's bandwidth ($\omega \ll \omega_{0elso}$) by the LPF and into the original plant when its frequency exceeds the ELSO's bandwidth ($\omega \gg \omega_{0elso}$) by the HPF, which is expressed as

$$G_{Mp}(s) \approx \begin{cases} \frac{1}{C_{dc}s}, & \text{when } \omega \ll \omega_{0elso}, \\ G_p(s), & \text{when } \omega \gg \omega_{0elso}. \end{cases} \quad (39)$$

So, the ELSO's bandwidth has a great effect on the behavior of $G_{Mp}(s)$. Indeed, when the bandwidth changes, the poles and the frequency response of $G_{Mp}(s)$ change, which means that determining the poles' positions of $G_{Mp}(s)$ depends on the value of $\omega_{0elso}$.

The transfer function $G_{Mp}(s)$ has three poles; one of them must be almost at the origin even when the value of $\omega_{0elso}$ is rather high. However, what we want to know here is where the poles go when the bandwidth $\omega_{0elso}$ is varied from low to high. This can be done using the root locus method by taking $\omega_{0elso}$ as the design parameter. The root locus plot in Figure 5 depicts the movements of the three poles of $G_{Mp}(s)$ in equation (38) when $\omega_{0elso}$ varied from 20 rad/s to 300 rad/s.

As shown in Figure 5, when $\omega_{0elso} = 50$ rad/s, three poles start at -50, -50, and 0. The two poles at -50 move to the left as the value of $\omega_{0elso}$ increases, while the pole in the original stays in its place regardless of this increase. From this, the modified $G_{Mp}(s)$ behaves as an integral action, which is completely consistent with the analytical study in equation (39).

## 4. Hardware-in-the-Loop (HIL) Simulation Study and Results' Discussion

In order to examine the superiority and efficacy of the proposed ELSO-based PLL-less VOC method, multiple HIL simulations have been conducted using the OPAL-RT 4510 real-time simulator. The corresponding setup HIL simulation platform is depicted in Figure 6, which is a part of the IRENA laboratory's smart power platform. The parameters used for these HIL simulations are listed in Table 1.

To show the impact of the load disturbance, DC capacitor change, unbalanced and distorted grid voltages, and

<figure><img src="image_7.png" /><figcaption>FIGURE 5: Root loci of $G_{Mp}(s)$ under increasing bandwidth.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>FIGURE 6: Configuration of OPAL-RT real-time simulator setup.</figcaption></figure>

disturbance grid voltages on the dynamic responses of the rectifier and to examine the control performance of the proposed control method under these operating conditions, four scenarios are performed in the following subsections.

### 4.1. Performance of the Proposed Control Method When the DC Load Changes Under Ideal Grid Voltage.
In this scenario, a disturbance on the system DC side is introduced by changing the load from 300 to 100 Ω at 0.05 s under
<table><caption>TABLE 1: PWM rectifier system and HIL simulation parameters.</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>Rated power</td><td>378 Watt</td></tr><tr><td>RMS of phase-to-neutral grid voltage</td><td>110 V</td></tr><tr><td>Grid frequency</td><td>50 Hz</td></tr><tr><td>DC-link voltage V<sub>dc</sub></td><td>300 V</td></tr><tr><td>Grid inductance and resistance L<sub>g</sub> and R<sub>g</sub></td><td>6 mH, 0.18 Ω</td></tr><tr><td>DC capacitor C<sub>dc</sub></td><td>1100 μF</td></tr><tr><td>DC load R</td><td>300 Ω</td></tr><tr><td>Switching frequency f<sub>s</sub></td><td>16 kHz</td></tr><tr><td>Switching loss resistor R<sub>s</sub></td><td>1000 Ω</td></tr><tr><td>Sampling time</td><td>10 μs</td></tr></tbody></table>

a nominal value of the DC capacitor 1100 µF and ideal grid voltage. Figures 7, 8, and 9 show the results of the three control methods: the PI controller, PI with feedforward based on sensor, and proposed PI-ELSO control method, respectively. The objective is to verify the performances of these control methods in terms of load disturbance rejection.

The capability of disturbance rejection can be examined via the maximum undershoot voltage caused by this disturbance and its corresponding voltage response time. So, from the three figures, it can be seen that the three control methods can converge the DC-link voltage to the initial state after rejecting the total disturbance. Also, the grid current increases to offset this disturbance and the grid current control loop-based PLL-less VOC method reacts rapidly to this change showing a fast transient response while maintaining UPF operation. However, the traditional PI controller provides the poorest DC voltage control performance at the DC load change in terms of undershoot of 7 V and response time of 0.2 s, as shown in Figure 7(a). While both the PI with feedforward based on sensor approach and the suggested PI-ELSO control method outperform the traditional PI controller in terms of tracking performance and load disturbance rejection, as shown in Figures 8(a) and 9(a), the PI with feedforward based on sensor control presents 3.6 V as undershoot and 0.06 s as response time (Figure 8(a)). The proposed PI-ELSO control method has the same undershoot as that of the PI with feedforward based on the sensor (Figure 9(a)), but its response time is somewhat longer than that of the PI with feedforward based on the sensor due to the small lag introduced by the ELSO. The results of this test highlight the benefits of the suggested PI-ELSO control method in terms of DC voltage regulation and disturbance rejection.

## 4.2. Performance of the Proposed Control Method When the Load Changes and the Capacitor Increase Under Ideal Grid Voltages.
In contrast to the first test, in this test, the DC capacitor has been varied from 1100 to 2200 µF. Figures 10, 11, and 12 display the obtained results with the PI controller, PI with feedforward based on sensor, and the proposed PI-ELSO control method, respectively. As in the first test, the results indicate clearly that the PI controller presents the

lowest DC voltage control performance in terms of undershoot and response time. Indeed, it performs an undershoot of 6.2 V and needs 0.2 s to compensate for it as shown in Figure 10(a). On the contrary, as shown in Figure 11(a), the PI with feedforward based on the sensor presents enhanced performance during load disturbance, which presents the undershoot of 4 V and needs approximately 0.1 s to compensate it. Figure 12 shows the results with the proposed PI-ELSO control method. It is clear from these results that the proposed PI-ELSO has similar performance in terms of undershoot as that of the PI with feedforward based on sensor, but it has lower overshoot and response time than the PI with feedforward based on sensor (Figure 11(a)). This outcome highlights the benefits of the proposed PI-ELSO control method in terms of DC voltage regulation, disturbance rejection, and robustness against parameter change.

## 4.3. Performance of the Proposed Control Method When the Load Changes Under Unbalanced and Distorted Grid Voltages.
To further confirm the ability of the suggested control method in rejecting the disturbances under unbalanced/distorted grid voltages, two tests are performed under a load change of 300-100 Ω at 0.05 s. The first one is performed under a 10% voltage sag in the first-phase grid voltage (Figure 13(a)). The second is performed under distorted grid voltages that contain 3% harmonics consisting of 2.8% of the fifth and 1% of the 7th (Figure 13(b)). The results of both tests with the PI with feedforward based on the sensor and the proposed PI-ELSO control method are illustrated in Figures 14, 15, 16, and 17. It can be observed from these figures that both control methods have the same performance, which confirms that the suggested PI-ELSO control method-based three-phase PWM rectifier control has good robustness against disturbances under distorted and unbalanced grid voltages. Both tests are performed under the nominal value of the DC capacitor ($C_{dc}$ = 1100 µF). It can also be noticed that the inner loop current control-based PLL-less VOC method has rapidly responded to the increment of the grid current used to compensate for the DC voltage disturbance, as well as maintaining the UPF operation of the PWM rectifier under unbalanced/distorted grid voltages.

## 4.4. Performance of the Proposed Control Method Under Disturbance Grid Voltage.
Finally, the efficacy of the suggested PI-ELSO control method is examined under disturbance grid voltage conditions. In this case, the magnitude of the grid voltage is subjected to 20% voltage sag at 0.5 s as shown in Figure 18. In this test, the DC load and capacitor are set to their nominal values. As shown in Figure 19 illustrating the test results, the disturbance grid voltages affect the DC-link voltage dynamic response and performance. However, the proposed PI-ELSO control
<figure><img src="image_9.png" /><figcaption>FIGURE 7: Results with traditional PI when the load is varied from 300 to 100 Ω with $C_{dc}$ = 1100 μF: (a) DC-link voltage $V_{dc}$, (b) load current $I_o$, (c) active and reactive grid currents $i_{gd}$ and $i_{gq}$ and, (d) first-phase grid voltage and its related current $v_{ga}$ and $i_{ga}$.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>FIGURE 8: Results with PI with feedforward based on the sensor when the load is varied from 300 to 100 Ω with $C_{dc}$ = 1100 μF: (a) DC-link voltage $V_{dc}$, (b) load current $I_o$, (c) active and reactive grid currents $i_{gd}$ and $i_{gq}$, and (d) first-phase grid voltage and its related current $v_{ga}$ and $i_{ga}$.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>FIGURE 9: Results with the proposed PI-ELSO control method when the load is varied from 300 to 100 Ω with $C_{dc}$ = 1100 μF: (a) DC-link voltage $V_{dc}$, (b) load current $I_o$, (c) active and reactive grid currents $i_{gd}$ and $i_{gq}$, and (d) first-phase grid voltage and its related current $v_{ga}$ and $i_{ga}$.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>FIGURE 10: Results with PI controller when the load is varied from 300 to 100 Ω with $C_{dc}$ = 2200 μF: (a) DC-link voltage $V_{dc}$, (b) load current $I_o$, (c) active and reactive grid currents $i_{gd}$ and $i_{gq}$, and (d) first-phase grid voltage and its related current $v_{ga}$ and $i_{ga}$.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>FIGURE 11: Results using PI with feedforward based on the sensor when the load is varied from 300 to 100 Ω with $C_{dc}$ = 2200 μF: (a) DC-link voltage $V_{dc}$, (b) load current $I_o$, (c) active and reactive grid currents $i_{gd}$ and $i_{gq}$, and (d) first-phase grid voltage and its related current $v_{ga}$ and $i_{ga}$.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>FIGURE 12: Results using the proposed PI-ELSO control method when the load is varied from 300 to 100 Ω with $C_{dc}$ = 2200 μF: (a) DC-link voltage $V_{dc}$, (b) load current $I_o$, (c) active and reactive grid currents $i_{gd}$ and $i_{gq}$, and (d) first-phase grid voltage and its related current $v_{ga}$ and $i_{ga}$.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>FIGURE 13: Grid voltages, that is, (a) unbalanced voltages with 10% voltage sag in the first phase has (b) distorted grid voltages with 3% harmonics.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>FIGURE 14: Results using PI with feedforward based on the sensor under the first-phase 10% voltage sag: (a) DC-link voltage $V_{dc}$, (b) load current $I_o$, (c) active and reactive grid currents $i_{gd}$ and $i_{gq}$, and (d) first-phase grid voltage and its related current $v_{ga}$ and $i_{ga}$.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>FIGURE 15: Continued.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>FIGURE 15: Results using the proposed PI-ELSO control method under the first-phase 10% voltage sag: (a) DC-link voltage $V_{dc}$, (b) load current $I_o$, (c) active and reactive grid currents $i_{gd}$ and $i_{gq}$, and (d) first-phase grid voltage and its related current $v_{ga}$ and $i_{ga}$.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>FIGURE 16: Results using PI with feedforward based on the sensor under distorted grid voltages with 3% harmonics: (a) DC-link voltage $V_{dc}$, (b) load current $I_o$, (c) active and reactive grid currents $i_{gd}$ and $i_{gq}$, and (d) first-phase grid voltage and its related current $v_{ga}$ and $i_{ga}$.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>FIGURE 17: Continued.</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>FIGURE 17: Results using the proposed PI-ELSO control method under distorted grid voltages with 3% harmonics: (a) DC-link voltage $V_{dc}$, (b) load current $I_o$, (c) active and reactive grid currents $i_{gd}$ and $i_{gq}$, and (d) first-phase grid voltage and its related current $v_{ga}$ and $i_{ga}$.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>FIGURE 18: Disturbance grid voltages.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>FIGURE 19: Results using the proposed PI-ELSO control method under 20% voltage sag of the three-phase grid voltages: (a) DC-link voltage $V_{dc}$, (b) load current $I_o$, (c) active and reactive grid currents $i_{gd}$ and $i_{gq}$, and (d) first-phase grid voltage and its related current $v_{ga}$ and $i_{ga}$.</figcaption></figure>
method with the PLL-less VOC method has rejected the
disturbance of the DC-link voltage, in which the grid cur-
rents are increased to offset both the disturbance at the DC-
link voltage and the grid voltage sag, and the inner loop
current control has rapidly responded to this increment
while also maintaining the UPF operation of the PWM
rectifier.

**5. Conclusion**

In this paper, a simple PI-ELSO-based DC-link voltage control method in the outer loop associated with the PLL-less VOC method in the inner loop was proposed for a three-phase PWM rectifier system. The PLL-less VOC method was adopted instead of the traditional VOC and DPC methods due to its ability to model and control the three-phase PWM rectifier in the dq coordinate without needing any PLL and Park transformations, resulting in a fast transient response and low computational burden as well as avoiding the PLL problems. Furthermore, an ELSO-based PI controller instead of the load current sensor is adopted in the DC-link voltage control loop for its capability to estimate the total disturbance that includes the load disturbance and the overall switching power losses. Not only that, this approach improves the robustness of the DC-link voltage control against changes in the DC capacitor, guarantees a zero steady-state DC-link voltage error, increases system reliability, and lowers both the cost and size. HIL simulation results using OPAL-RT 4510 real-time simulator show that under different grid voltage conditions, the suggested PI-ELSO can reject the DC-link voltage disturbance with the same effectiveness as the PI with feedforward based on the sensor control approach. Moreover, it also successfully lowers response time and overshoots in certain situations.

This work is the initial step in developing an ELSO-based
PLL-less VOC method for a three-phase PWM rectifier. In
the future, we will improve this control method by
employing advanced grid voltage and load current observers
and an advanced control technique to regulate the grid
currents and DC-link voltage in the three-phase four-leg
three-level rectifier applications.

**Data Availability Statement**

The data used to support the findings of this study are in-
cluded within the article.

Ethics Statement

This paper does not contain any studies with human par-
ticipants or animals performed by any of the authors.

Conflicts of Interest

The authors declare no conflicts of interest.

Funding

Open Access funding was enabled and organized by
COUPERIN CY23.

**Acknowledgments**

This work was funded by the corresponding author's in-
stitution. Open Access funding was enabled and organized
by COUPERIN CY23.

**References**

[1] T. Song, P. Wang, Y. Zhang, F. Gao, Y. Tang, and S. Pholboon, "Suppression Method of Current Harmonic for Three-Phase PWM Rectifier in EV Charging System," *IEEE Transactions on Vehicular Technology* 69, no. 9 (2020): 9634-9642, https://doi.org/10.1109/tvt.2020.3005173.

[2] R. J. Wai and Y. Yang, "Design of Backstepping Direct Power Control for Three-Phase PWM Rectifier," *IEEE Transactions on Industry Applications* 55, no. 3 (2019): 3160-3173, https://doi.org/10.1109/tia.2019.2893832.

[3] M. R. Estabragh, A. Dastfan, and M. Rahimiyan, "Grid-Tied Hybrid AC-DC Microgrid: Finding Optimal Number of Parallel-Connected AC-DC Bidirectional Interfacing Converters," *International Transactions on Electrical Energy Systems* 2022 (2022): 1-14, https://doi.org/10.1155/2022/1932818.

[4] S. Ansari, A. Chandel, and M. Tariq, "A Comprehensive Review on Power Converters Control and Control Strategies of AC/DC Microgrid," *IEEE Access* 9 (2021): 17998-18015, https://doi.org/10.1109/access.2020.3020035.

[5] S. Gao, H. Zhao, Y. Gui, D. Zhou, and F. Blaabjerg, "An Improved Direct Power Control for Doubly Fed Induction Generator," *IEEE Transactions on Power Electronics* 36, no. 4 (2021): 4672-4685, https://doi.org/10.1109/tpel.2020.3024620.

[6] E. C. Santos, C. B. Jacobina, E. R. Silva, and N. Rocha, "Single-Phase to Three-Phase Power Converters: State of the Art," *IEEE Transactions on Industrial Electronics* 27, no. 5 (2012): 2437-2452.

[7] J. A. Restrepo, J. M. Aller, A. Bueno, et al., "Direct Power Control of a Dual Converter Operating as a Synchronous Rectifier," *IEEE Transactions on Power Electronics* 26, no. 5 (2011): 1410-1417, https://doi.org/10.1109/tpel.2010.2084106.

[8] M. N. Tasnim, T. Ahmed, M. A. Dorothi, et al., "Voltage-Oriented Control-Based Three-Phase, Three-Leg Bidirectional AC-DC Converter With Improved Power Quality for Microgrids," *Energies* 16, no. 17 (2023): 6188, https://doi.org/10.3390/en16176188.

[9] M. Zarif and M. Monfared, "Step-by-Step Design and Tuning of VOC Control Loops for Grid Connected Rectifiers," *International Journal of Electrical Power and Energy Systems* 64 (2015): 708-713, https://doi.org/10.1016/j.ijepes.2014.07.078.

[10] X. Muchao, L. Zaixun, L. Manjia, C. Yibo, and J. Chen, "Voltage Oriented Control Strategy of Three Phase PWM Rectifier in Future-Oriented Smart Charging Stations," in *Proceedings of the IEEE Sustainable Power and Energy Conference (ISPEC)* (2021), 3483-3490.

[11] F. Blaabjerg, R. Teodorescu, M. Liserre, and A. V. Timbus, "Overview of Control and Grid Synchronization for Distributed Power Generation Systems," *IEEE Transactions on Industrial Electronics* 53, no. 5 (2006): 1398-1409, https://doi.org/10.1109/tie.2006.881997.

[12] M. Davari and Y. A.-R. I. Mohamed, "Robust Vector Control of a Very Weak-Grid-Connected Voltage-Source Converter Considering the Phase-Locked Loop Dynamics," *IEEE Transactions on Power Electronics* 32, no. 2 (2017): 977-994, https://doi.org/10.1109/tpel.2016.2546341.

[13] S. Ahmad, H. Mubarak, T. Ahmed, and S. Mekhilef, "Point of Common Coupling Voltage Modulated Direct Power Control
of Grid-Tied Photovoltaic Inverter for AC Microgrid Application," *International Transactions on Electrical Energy Systems* 2023 (2023): 3641907.

[14] Y. Gui, X. Wang, and F. Blaabjerg, "Vector Current Control Derived From Direct Power Control for Grid-Connected Inverters," *IEEE Transactions on Power Electronics* 34, no. 9 (2019): 9224-9235, https://doi.org/10.1109/tpel.2018.2883507.

[15] Y. Gui, X. Wang, F. Blaabjerg, and D. Pan, "Control of Grid-Connected Voltage-Source Converters: The Relationship Between Direct-Power Control and Vector-Current Control," *IEEE Industrial Electronics Magazine* 13, no. 2 (2019): 31-40, https://doi.org/10.1109/mie.2019.2898012.

[16] A. Chebabhi, A. A. M. Al-dwa, M. Defdaf, and A. Kessal, "New Modeling and Enhanced Control Strategy for Grid-Connected Four-Leg Inverter Without Phase-Locked Loop and Park's Transformation," *Revue Roumaine des Sciences Techniques-Serie Électrotechnique et Énergétique* 68, no. 2 (2023): 121-126, https://doi.org/10.59277/rrst-ee.2023.68.2.1.

[17] Z. Gong, C. Liu, Y. Gui, F. F. da Silva, and C. L. Bak, "Power Decoupling Method for Voltage Source Inverters Using Grid Voltage Modulated Direct Power Control in Unbalanced System," *IEEE Transactions on Power Electronics* 38, no. 3 (2023): 3084-3099, https://doi.org/10.1109/tpel.2022.3220436.

[18] R. I. Amirnaser Yazdani, *Voltage-Sourced Converters in Power Systems: Modeling, Control, and Applications* (Wiley, 2010).

[19] A. Chebabhi, M. Defdaf, and A. Kessal, "Design and PIL Implementation of a New Robust Backstepping Based Low-Complexity Voltage-Oriented Control Method for Four-Leg Rectifiers," *International Journal of Electrical Power & Energy Systems* 155 (2024): 109676, https://doi.org/10.1016/j.ijepes.2023.109676.

[20] M. Davari and Y. A.-R. I. Mohamed, "Dynamics and Robust Control of a Grid-Connected VSC in Multiterminal DC Grids Considering the Instantaneous Power of DC-and AC-Side Filters and DC Grid Uncertainty," *IEEE Transactions on Power Electronics* 31, no. 3 (2016): 1942-1958, https://doi.org/10.1109/tpel.2015.2439645.

[21] M. Merai, M. W. Naouar, and I. Slama-Belkhodja, "An Improved DC Link Voltage Control Strategy for Grid Connected Converters," *IEEE Transactions on Power Electronics* 33, no. 4 (2018): 3575-3582, https://doi.org/10.1109/tpel.2017.2707398.

[22] S. Bayhan and H. Komurcugil, "A Current Sensorless Control Method for Multilevel Active Front-End Rectifiers With LCL Filter," *IET Power Electronics* 16, no. 5 (2023): 715-727, https://doi.org/10.1049/pel2.12416.

[23] C. Wang, X. Li, L. Guo, and Y. W. Li, "A Nonlinear-Disturbance-Observer-Based DC-Bus Voltage Control for a Hybrid AC/DC Microgrid," *IEEE Transactions on Power Electronics* 29, no. 11 (2014): 6162-6177, https://doi.org/10.1109/tpel.2013.2297376.

[24] J. Liu, S. Vazquez, L. Wu, A. Marquez, H. Gao, and L. G. Franquelo, "Extended State Observer-Based Sliding-Mode Control for Three-Phase Power Converters," *IEEE Transactions on Industrial Electronics* 64, no. 1 (2017): 22-31, https://doi.org/10.1109/tie.2016.2610400.

[25] S. C. Kim, T. H. Nguyen, D. C. Lee, K. B. Lee, and J. M. Kim, "Fault Tolerant Control of DC-Link Voltage Sensor for Three-Phase AC/DC/AC PWM Converters," *Journal of Power Electronics* 14, no. 4 (2014): 695-703, https://doi.org/10.6113/jpe.2014.14.4.695.

[26] J. Lu, S. Golestan, M. Savaghebi, J. C. Vasquez, J. M. Guerrero, and A. Marzabal, "An Enhanced State Observer for DC-Link Voltage Control of Three-Phase AC/DC Converters," *IEEE Transactions on Power Electronics* 33, no. 2 (2018): 936-942, https://doi.org/10.1109/tpel.2017.2726110.

[27] Y. Gui, F. Blaabjerg, X. Wang, J. D. Bendtsen, D. Yang, and J. Stoustrup, "Improved DC-Link Voltage Regulation Strategy for Grid-Connected Converters," *IEEE Transactions on Industrial Electronics* 68, no. 6 (2021): 4977-4987, https://doi.org/10.1109/tie.2020.2989720.

[28] A. Chebabhi, S. Barkat, and A. Kessal, "Combined Voltage-Oriented Control and Direct Power Control Based on Backstepping Control for Four-Leg PWM Rectifier Under Unbalanced Conditions," *Engineering Review* 42, no. 3 (2022): https://doi.org/10.30765/er.2020.

[29] B. Guo, S. Bacha, M. Alamir, A. Hably, and C. Boudinet, "Generalized Integrator-Extended State Observer With Applications to Grid-Connected Converters in the Presence of Disturbances," *IEEE Transactions on Control Systems Technology* 29, no. 2 (2021): 744-755, https://doi.org/10.1109/tcst.2020.2981571.

[30] X. Zhou, Q. Liu, Y. Ma, and B. Xie, "DC-Link Voltage Research of Photovoltaic Grid-Connected Inverter Using Improved Active Disturbance Rejection Control," *IEEE Access* 9 (2021): 9884-9894, https://doi.org/10.1109/access.2021.3050191.

[31] D. Yuan, X. J. Ma, Q.-H. Zeng, and X. Qiu, "Research on Frequency-Band Characteristics and Parameters Configuration of Linear Active Disturbance Rejection Control for Second-Order Systems," *Control Theory and Applications* 30, no. 12 (2013): 1630-1640.

[32] W. Xue and Y. Huang, "Performance Analysis of Active Disturbance Rejection Tracking Control for a Class of Uncertain LTI Systems," *ISA Transactions* 58 (2015): 133-154, https://doi.org/10.1016/j.isatra.2015.05.001.

[33] S. A. Khan, Y. Guo, Y. P. Siwakoti, D. D. C. Lu, and J. Zhu, "A Disturbance Rejection-Based Control Strategy for Five-Level T-Type Hybrid Power Converters With Ripple Voltage Estimation Capability," *IEEE Transactions on Industrial Electronics* 67, no. 9 (2020): 7364-7374, https://doi.org/10.1109/tie.2019.2942550.

[34] J. Tatsumi and Z. Gao, "On the Enhanced ADRC Design With a Low Observer Bandwidth," in *Proceedings of the 32nd Chinese Control Conference* (2013), 297-302.