# Integration of a novel Vernier-DSPM generator in a grid connected hybrid renewable energy system with battery storage

Abderrahmane Redouane, Rachid Saou, Youcef Belkhier, Amrane Oukaour

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Abderrahmane Redouane, Rachid Saou, Youcef Belkhier, Amrane Oukaour. Integration of a novel Vernier-DSPM generator in a grid connected hybrid renewable energy system with battery storage. Results in engineering, 2025, 25, pp.103814. 10.1016/j.rineng.2024.103814. hal-05041394

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# Integration of a novel Vernier-DSPM generator in a grid connected hybrid renewable energy system with battery storage

Abderrahmane Redouane<sup>a</sup>, Rachid Saou<sup>a</sup>, Youcef Belkhier<sup>b,*</sup> Amrane Oukaour<sup>c</sup>

## ARTICLE INFO

### Keywords:

Renewable energy system

Grid-connected

Wind turbine system

Dspm generator

Battery storage system

Control of the sliding mode of super twist

## ABSTRACT

This research investigates a hybrid renewable energy system (HRESs) connected to the grid that uses three energy sources: solar energy, wind energy, and battery energy. Unlike conventional wind turbine systems, this paper uses a direct-drive high torque low speed generator called the Vernier Doubly Salient Permanent Magnet Generator (V-DSPMG). The generator shaft is directly connected to the turbine shaft without a gearbox, which enhances the performance of the wind power system. A mathematical model of the components of the grid-connected system is presented. An efficient Tip Speed Ratio (TSR) based sliding mode control (SMC) maximum power point tracking (MPPT) algorithm based on constant plus proportional rate reaching law is utilized to extract the optimal power from the wind turbine. A perturb and observe (P&O) MPPT algorithm is used in the photovoltaic system because of its high efficiency and ease of implementation. To control the bidirectional power flow in the battery storage system, a proportional-integral (PI) current control is employed. Furthermore, this paper proposes a new super-twisting sliding mode control (NSTSMC) method to regulate the DC-Link voltage. To demonstrate its performance, a comparison is made with Super-Twisting SMC (STSMC) and conventional PI control. A five-level Active Neutral Point Clamped (5L-ANPC) DC/AC converter is proposed for efficient power flow, better output voltage harmonic spectrum, and better power quality injected into the grid. The considered system is performed under MATLAB/Simulink and OPAL-RT real time validation. The results demonstrate that this configuration establishes a highly reliable and efficient energy system. Each energy source is skillfully engineered to autonomously meet power demands. If there is a shortfall in power from any single source, the system is designed to balance this using the other sources. This interconnected and supportive arrangement significantly amplifies the system's reliability and sustainability, making it an exemplary model in the realm of hybrid renewable energy systems for HRES.

## 1. Introduction

Recent years have witnessed an increase in the adoption of renewable energy sources (RESs), driven by the escalating global energy demand, the finite nature of conventional energy resources, and the adverse environmental impacts of greenhouse gas emissions. In particular, wind and photovoltaic energy conversion systems are among the cleanest forms of energy conversion. However, their intermittent nature, which is heavily dependent on climatic conditions, introduces challenges due to their nonlinear output [1,2]. This intermittency requires the integration of various RESs coupled with energy storage systems ESSs to ensure a consistent energy supply [3,4].

HRESs, which are more complex than single-source systems, combine multiple RESs with storage units to ensure continuous power delivery. The synergistic use of wind and solar energy, which individually do not provide a constant power output, achieves a more reliable power supply due to their complementary nature [5]. Energy storage systems play a pivotal role in these setups, ensuring energy availability even when renewable sources are not sufficient.

These systems operate in two primary modes: non-autonomous and autonomous. In non-autonomous mode, the system relies on the power grid to compensate for any power shortfalls. In contrast, the autonomous mode is designed to cater to isolated areas by independently providing sufficient power. Autonomous mode can be seen as a specific case of non-autonomous operation when the hybrid system can independently

<footnote><sup>a</sup> Faculté de Technologie, Laboratoire de Génie Électrique, Université de Bejaïa, Bejaïa 06000, Algeria</footnote>

<footnote><sup>b</sup> University of Brest, UMR CNRS 6027 IRDL, Brest 29238, France</footnote>

<footnote><sup>c</sup> Laboratoire de Recherche en Sciences du Numérique GREYC, Université de Caen Normandie, 6 Boulevard du Maréchal Juin, Caen 14032, France</footnote>

<footnote>* Corresponding author.
E-mail address: youcef.belkhier@univ-brest.fr (Y. Belkhier).</footnote>
Nomenclature

$\omega_e$ Electrical angular velocity

$\theta_e$ Electrical position

$V_{iabc}$ Phase Voltages

$I_{iabc}$ Phase Currents

$V_{idq}$ Direct & quadratic voltages

$I_{idq}$ Direct & quadratic currents

$C_{DC}$ DC-Link capacitor

$I_c$ DC-Link capacitor current

$I_{in}$ DC-Link input current

$I_{DC}$ DC-Link current

$V_{labc}$ Phase Voltages

$I_{labc}$ Phase currents

$I_{ldq}$ Direct & quadratic currents

$V_{gabc}$ Phase Voltages

$I_{gabc}$ Phase currents

$I_{gdq}$ Direct & quadratic currents

$\theta$ Electrical position

$Q_{V\_DSPMG}$ Reactive power

$P_{V\_DSPMG}$ Active power

$V_w$ Wind speed

$\lambda$ Tip Speed Ratio

$C_p$ Power coefficient

$P_t$ Turbine power

$\Omega_t$ Turbine speed

$\beta$ Turbine pitch angle

$v_{abc}$ Phase Voltages

$\dot{i}_{abc}$ Phase currents

$\phi_{mabc}$ Phase flux

$e_{mabc}$ Electromotive force

$L_{abc}$ Self-inductances

$M_{abc}$ Mutual inductances

$v_{dq}$ Direct and quadratic voltages

$\dot{i}_{dq}$ Direct and quadratic current

$\phi_{mdq}$ Direct and quadratic flux

$e_{mdq}$ Direct and quadratic electromotive force

$L_{dq}$ Direct and quadratic self-inductance

$M_{dq}$ Direct and Quadratic Mutual Inductances

$T_{em}$ Electromagnetic Torque

Abbreviations

PFC Power Feedback Control

OTC Optimal Torque Control

TSR tip speed ratio

HCSA Hill Climbing Search Algorithm

PI Proportional Integrator Control

FLC Fuzzy Logic Control

FO-PI fractional order proportional

FOSMC Fractional Order Sliding Mode Control

SMC Sliding Mode Control Integrator

WRIG wound-rotor induction generator

OR-DSPMG Outer Rotor Doubly Saleint Permanent Magnet Generator

SEIG Self-Excited Induction Generator

DFIG Doubly-fed Induction Generator

STSMC Super-Twisting Sliding Mode Control

NSTSMC New Super-Twisting Sliding Mode Control

MO Multi-Objective

GWO Grey Wolf Optimizer

DSPMG Doubly Salient Permanent Magnet Generator

V-DSPMG Vernier Doubly Salient Permanent Magnet Generator

RES Renewable Energy Sources

HRES Hybrid Renewable Energy Systems for

ESS Energy Storage Systems

PV Photovoltaic

MPPT Maximum Power Point Tracking

P&O Perturbation and Observation

SCIG Squirrel cage induction generator

ANPC Active Neutral Point Clamped

meet power demands without relying on the grid [6,7].

HRESs, have become a focal point in scientific research. Various studies have explored different configurations and architectures of these systems [6–9], along with multiple strategies to optimize the size of system components [10–14]. Research has addressed various aspects, including simpler control methods and component reduction in stand-alone hybrid systems for cost-effectiveness and improved power quality [15]. An innovative MPPT control methods are presented to enhance the power of the PV and wind systems in [16]. The suggested technique for the PV system is based on three methods: P&O, fuzzy logic control (FLC), and SMC. The wind energy system is based on the FLC and the hill climbing search algorithm (HCSA). Reference [17] proposes a standalone hybrid system where each renewable energy source has its own dedicated battery bank. The results show that the hybrid system enhances energy production without adding storage capacity. The performance of batteries and superconducting magnetic storage systems combined with a hybrid PV wind system is investigated in [18]. Both systems were tested under regular and extreme environmental conditions, as well as varying load scenarios. The results show the superiority of the superconducting magnetic storage system in terms of system stability and power quality. In [19], a novel architecture topology for a grid-connected hybrid wind and photovoltaic system is proposed. The wind system connects to the grid using a back-to-back configuration, while the PV system is directly connected to the DC-Link without the use of any auxiliary converter. The proposed architecture uses a reduced number of converters and improves the efficiency of the transferred

power. DC-Link voltage in [16–19] is regulated with a conventional PI controller, which drawbacks like low robustness, overshoot, and slow convergence. A New Robust P&O based Fractional Order Sliding Mode Control (RPO-FOSMC) is proposed to extract the maximum power from the wind turbine. This approach shows better dynamic performance than traditional MPPT [20]. In addition, a multi-objective optimization technique is included to enhance the resilience and dynamic performance of RPO-FOSMC.

Reference [21] proposes a double integral sliding mode control (DISMC) for MPPT in PV systems to enhance efficiency and reduce settling time under partial weather conditions.

Reference [21] proposes a double integral sliding mode control (DISMC) for MPPT in PV system to improve the efficiency and minimize the settling time under partial weather conditions. The DISMC control method outperforms conventional SMC and Integral SMC by achieving 99.10 % efficiency with a settling time of 0.035 second. However, DISMC has certain disadvantages, such as higher computational requirements, and complexity in the control algorithm. Reference [22] proposes a grid-connected wind turbine using an outer rotation OR-DSPMG generator. To allow the generator to operate in the overspeed zone, a flux weakening approach is used. DC-Link voltage is balanced by using FLC-PI control, which performs better than conventional PI but need complex design like fuzzy membership functions and rule bases. In [23], authors present a novel control strategy combining integral sliding mode control with fuzzy adaptive PI control to enhance voltage stability in DC microgrids. This method significantly improves
stability and precision but requires substantial computational resources and complex implementation. In [24], the Fractional-Order (FO-PI) to improve the robustness of the voltage source converter of an Autonomous Microgrid system was designed. However, the technique has a drawback: the fractional order parameters are sensitive, and small changes can affect the controller's performances and stability. Reference [25] addresses residential load energy management using renewable energies based on a multi-source system. A conventional SMC Sliding-Mode control is used for DC-link voltage regulation. This controller type has the benefit of being robust and simpler, but generally suffers from the sliding variable's high oscillation frequency around the sliding surface. Reference [26] introduces an advanced microgrid control integrating PV, wind, battery storage, EV, and a DC load, all connected to the DC-link. A fuzzy logic controller, optimized using Particle Swarm Optimization (PSO) is proposed to regulate the DC link voltage to enhance system stability and energy distribution. Reference [27] suggests a Fractional-Order sliding mode controller FOSMC for controlling active and reactive power of a doubly fed induction generator (DFIG) based wind turbine using a multi-objective grey wolf optimizer MOGWO. The FOSMC is designed utilizing the MOGWO approach to minimize chattering and address system uncertainties. In [28], an STSMC is proposed to control the direct and quadratic currents of the self-excited induction generator (SEIG), and to balance the DC-Link voltage. Moreover, a performance comparison between STSMC and FLC-PI was made, and the results show the superiority of STSMC in terms of robustness and dynamic response. Among the control methods mentioned above, the sliding mode control law surpasses other techniques and presents better performance in terms of tracking precision, resistance to external disturbances, uncertainty, and nonlinear uncertain systems [29]. Therefore, this paper proposes a new super-twisting sliding-mode control (NSTSMC) to improve the DC-Link voltage performance of the HRES in tracking accuracy, dynamic response, and disturbances robustness.

In the case of wind turbines, the electrical machines used are generally mechanically coupled to a gearbox [30,31]. Speed reducers are bulky, noisy, costly, and increase the overall inertia of the system, which reduces control efficiency. Therefore, eliminating speed reducers becomes very advantageous. This can be achieved by using machines that operate directly at low speed, called slow or direct-drive machines. These electrical machines have been the subject of numerous studies, and several manufacturers have been producing turbines with slow-speed generators without gearboxes for many years [32]. These direct-drive machines are often synchronous, with either wound or permanent magnet rotors, and feature a very high number of poles as well as high torque. Several machine topologies have been studied and proposed in the literature to meet these needs: radial flux, axial flux, and transverse flux machines. Other special configurations include variable reluctance machines, whether excited or not.

Among these unconventional machines, the doubly salient permanent magnet (DSPM) machine, with toothed poles and a toothed rotor, is particularly suited for direct-drive low-speed applications. Its rotational speed is inversely proportional to the number of teeth on the rotor. This machine has a doubly salient structure and uses concentrated windings. Its excitation field is provided by non-rotating permanent magnets located in the stator, with no magnets or windings in the toothed rotor.

The statoric tooth pitch of this machine, $\tau_s$, is identical to that of the rotor, $\tau_r$ [31–33]. However, this DSPM suffers from non-sinusoidal electromotive force (e.m.f) and significant electromagnetic torque ripples, leading to vibrations and noise [31,34,35]. Addressing these concerns, a Vernier-Doubly Salient Permanent Magnet machine (V-DSPM) has been proposed in [36]. This innovation utilizes the Vernier effect by setting the stator's tooth pitch, $\tau_s$, differently from the rotor's, $\tau_r$, mitigating torque ripples and improving overall performance. This is the special machine we'll be using in this work.

This work, in summary, investigates the design of grid-connected systems based on hybrid renewable sources associated with a battery

storage system. The objective is to improve the energy generation efficiency of the different sources while ensuring stable output power and continuous supply for the demand side. A novel direct driven generator, the Vernier doubly salient permanent magnet V-DSPMG is employed to enhance wind power systems performance. In addition, a new NSTSMC is proposed to improve the stability of the system's DC-Link. The present work provided the following contributions:

* The wind energy conversion system utilizes a novel low-speed high-torque direct-driven generator, which can significantly improve the produced power by eliminating energy losses due to the gearbox.

* The proposed NSTSMC allows one to improve the performances of the DC-Link voltage. This suggested control offers two major benefits compared to the conventional super-twisting sliding-mode control:

* Ø Fast convergence capability: it can drive the system state to the desired trajectory quickly. This fast convergence enhances the transient response and tracking performance.

* Ø Robust performance under parameter variations and disturbances.

* A five-level ANPC DC/AC converter is proposed to ensure the power flow between energy sources and the grid-side. This topology improves the output voltage harmonic spectrum of the grid side.

The presented paper is organized as follows: Section 2 describes the proposed system. Section 3 and Section 4 presents the mathematical model of the system components and their control methods. The results of the performed simulation are presented and discussed in Section 5. Section 6 shows the experimental results using OPAL-RT. Finally, Section 7 concludes this paper.

## 2. Description of the proposed system

Microgrid systems are categorized into three architectural configurations: DC-Link, AC-Link, and a combination of both. This study focuses primarily on the DC-Link configuration as illustrated in Fig. 1.

The DC microgrid comprises two wind energy conversion systems connected in parallel, each rated at 10 kW, to achieve a total wind energy system capacity of 20 kW. Each wind turbine is coupled to a novel 10 kW Vernier Doubly Salient Permanent magnet generator (V-DSPMG). The windings of each generator are connected to a three-phase AC/DC rectifier, which then links to a DC/DC boost converter, ensuring efficient power conversion. A TSR-MPPT (Tip Speed Ratio) is employed to extract the maximum power from wind turbines, and a sliding mode current controller to control the DC/DC boost converters.

A 30 kW PV solar system is made up of 17 parallel strings, each string containing five modules. Each module produces 350 W. The PV solar system is similarly connected to a separate DC/DC boost converter. A P&O MPPT algorithm is used to maximize power extraction from the photovoltaic array and to control the DC/DC boost converter.

Eight lithium-ion batteries with individual ratings of 24 V and 100 Ah are connected in series to provide a total storage energy system capacity of 20 kWh. A bidirectional DC/DC converter is used with a single PI current control loop to ensure the charging and discharging of the battery storage system.

The three conversion systems are interconnected through a common DC Link. A three-phase, five-level ANPC DC/AC converter distributes power from the renewable energy systems and the storage system to the variable load and the main grid.

## 3. Mathematical modeling of system components

### 3.1. Wind turbine system model

The captured wind turbine power is defined as [37–39]:
<figure><img src="image_3.png" /><figcaption>Fig. 1. Proposed configuration of the HRES system connected to the grid.</figcaption></figure>

$$P_t = \frac{1}{2} \rho \pi R^2 C_p(\lambda, \beta) V_w^3 \quad (1)$$

Where: $C_p$ is the wind turbine power coefficient, $\rho$ is the air density, $R$ is the turbine blades radius, $\lambda$ is the tip speed ratio, $V_w$ is the wind velocity, $\beta$ is the blade pitch angle.

The wind turbine tip speed ratio is defined as follows:

$$\lambda = \frac{\Omega_t R}{V_w} \quad (2)$$

Where $\Omega_t$ is the turbine angular speed rad/s.

The power coefficient $C_p(\lambda, \beta)$ is given form [40] as follow:

$$C_p(\lambda, \beta) = 0.5176 \left( \frac{116}{\lambda_t} - 0.4\beta - 5 \right) e^{-\frac{21}{\lambda_t}} + 0.0068\lambda \quad (3)$$

With: $\frac{1}{\lambda_t} = \frac{1}{\lambda+0.08\beta} - \frac{0.035}{\beta^{3+1}}$

In the study presented in this paper, the pitch angle $\beta$, is set to a fixed value of zero. Consequently, the power coefficient curve $C_p(\lambda, \beta)$ exhibits a single maximum point. This maximum efficiency, identified as $C_{p(max)}(\lambda, \beta)$, is quantified at 0.428 and occurs at an optimal tip speed ratio $\lambda_{opt}$ of 2.25, as illustrated in the corresponding Fig. 2. To maximize the power extraction of the wind turbine, it is essential to regulate the rotor speed to align with the optimal value of the tip speed ratio, denoted as $\lambda_{opt}$. This adjustment ensures that the turbine operates at its highest efficiency, harnessing the maximum available wind energy.

The generator utilized in this paper is a low-speed direct drive generator. That means that the generator does not require a gearbox system to connect with the wind turbine, in other words, the generator speed is equal to the turbine speed $\Omega_t = \Omega_g$. Thus, the energy losses due to the gearbox are eliminated.

Since the generator speed and the turbine speed are equal, then the total inertia $J_T$ of the system is equal to the sum of the generator inertia $J_g$ and the turbine inertia $J_t$:

$$J_T = J_g + J_t \quad (4)$$

<figure><img src="image_4.png" /><figcaption>Fig. 2. Wind turbine power coefficient curve $C_p(\lambda)$.</figcaption></figure>

The mechanical shaft equation is given as follows [41]:

$$J_T \frac{d\Omega_g}{dt} = T_m - T_{em} - f\Omega_g \quad (5)$$

Where: $T_m$, $T_{em}$, and $f$ are the mechanical torque, electromagnetic torque, and generator friction.

The aerodynamic model of the wind turbine system is presented in Fig. 3. As shown in the scheme, there is no gearbox system; the wind turbine shaft is connected directly to the generator shaft. Furthermore, the scheme shows that the generator speed $\Omega_g$ can be controlled by the electromagnetic torque $T_{em}$.

## 3.2. V-DSPMG generator model

Extensive research within the field of wind energy conversion systems has highlighted a range of electrical generators, as delineated in the literature [30,42]. Typically, these generators are mechanically coupled to the turbine shaft via a gearbox to achieve the required rotational
<figure><img src="image_5.png" /><figcaption>Fig. 3. Mathematical model of the wind turbine system.</figcaption></figure>

speed. However, this speed adaptation mechanism is not without drawbacks: it is often bulky, costly, noisy, prone to energy losses, and requires regular maintenance.

Addressing these limitations, a novel approach is proposed in [33], which introduces a direct-driven, high-torque, low-speed generator known as the double salient permanent magnet generator DSPMG. This generator represents a type of variable reluctance generator, featuring 64 rotor teeth $N_r$ and 48 stator teeth $N_s$ distributed across 12 stator slots $N_p$, and 4 teeth per slot $N_{dp}$. In particular, it incorporates four permanent magnets 2P embedded within the stator yoke. The winding configuration is uniquely designed, with each of the three phases comprising four series-connected concentric coils. While the rotor is passive, without any electrical windings [31,34,43]. The number of rotor teeth $N_r$ is fixed to 64 in order to achieve a typical electrical frequency of 50 Hz at the nominal speed of 50 rpm. The rotation speed of the DSPM is inversely proportional to the number of rotor teeth.

The V-DSPMG (see Fig. 4) is a conventional DSPMG that exploits the Vernier effect, achieved by having a rotor tooth pitch $\tau_r$ different from the stator tooth pitch $\tau_s$.

$$ \tau_s = \frac{2\pi}{N_{seq}}, \quad \tau_r = \frac{2\pi}{N_r} \qquad (6) $$

$N_{seq}$, different from $N_s$, represents the number of stator teeth the generator would have if the stator armature were distributed-toothed and not slotted.

To ensure energy conversion and torque production, the studied structure must satisfy the following conditions [31,36]:

$$ \left\{ \begin{array}{l} \pm N_{seq} \pm N_r = \pm P \pm P_e \\ \pm N_{seq} \neq 2P \pm 2P_e \\ \pm N_r \neq 2P \pm 2P_e \end{array} \right. \qquad (7) $$

$P_e$ and $P$ denote the number of magnet pairs and slot pairs per phase, respectively. These conditions are met, for machine, when considering $N_{seq}=68$ and $N_{seq}=60$ [36]. The machine with $N_{seq}=68$ demonstrates better performances according, which is why it was chosen for this study.

Table 1 shows a performance comparison between V-DSPMG and different conventional electrical generators used in wind energy conversion systems.

The V-DSPMG used in this work was designed for a power of 10 kW. The optimization, focused on the maximization of the machine mass to

<table><caption>**Table 1**

Comparison of performance between electrical generators used in wind energy systems.</caption><thead><tr><th></th><th>DFIG</th><th>SCIG</th><th>WRIG</th><th>V-DSPMG</th></tr></thead><tbody><tr><td>Gearbox</td><td>Required</td><td>Required</td><td>Required</td><td>Not required</td></tr><tr><td>Maintenance</td><td>Required</td><td>Required</td><td>Required</td><td>Not required</td></tr><tr><td>Torque density</td><td>Medium</td><td>Low</td><td>Medium</td><td>High</td></tr><tr><td>Power quality</td><td>High</td><td>Low</td><td>Low</td><td>High</td></tr><tr><td>Efficiency</td><td>High</td><td>Low</td><td>Low</td><td>High</td></tr><tr><td>Reliability</td><td>Low</td><td>High</td><td>Low</td><td>High</td></tr></tbody></table>

Where: wound rotor induction generator (WRIG), doubly fed induction generator (DFIG), squirrel cage induction generator (SCIG).

<figure><img src="image_6.png" /><figcaption>Fig. 4. Structure of: (a) Rotor and stator teeth of the machine; (b) cross section of the Vernier-DSPMG.</figcaption></figure>
torque ratio, is done by a genetic algorithm (GA) combined with finite element method (FEM). The optimization applies to the parameters of the small stator and rotor teeth, the dimensions of the magnets, as well as to global parameters such as: the yokes of the rotor and stator, winding height, rotor radius, etc. Its self- and mutual inductances, and the flux per phase generated by the two pairs of magnets, obtained using finite element analysis, are shown in the following Fig. 5.

To streamline the dynamic modeling of the V-DSPMG, a focused simplification approach has been adopted. This approach involves disregarding the Fourier terms of order two and higher as shown in Table 2. As a result of this simplification, the mathematical representations of key parameters such as self-inductances, mutual inductances, and magnetic flux components can be articulated in a more straightforward manner.

These parameters are detailed and defined as follows in the literature [44]:

$$ 
\left\{ 
\begin{array}{l}
L_a = L_0 + L_1 \cos \theta_e \\
L_b = L_0 + L_1 \cos \left( \theta_e - \frac{2}{3} \pi \right) \\
L_c = L_0 + L_1 \cos \left( \theta_e - \frac{4}{3} \pi \right)
\end{array}
\right. 
\qquad (8) 
$$

$$ 
\left\{ 
\begin{array}{l}
M_{ab} = M_{ba} = M_0 + M_1 \cos \left( \theta_e - \frac{4}{3} \pi \right) \\
M_{ac} = M_{ca} = M_0 + M_1 \cos \left( \theta_e - \frac{2}{3} \pi \right) \\
M_{bc} = M_{cb} = M_0 + M_1 \cos \theta_e
\end{array}
\right. 
\qquad (9) 
$$

$$ 
\left\{ 
\begin{array}{l}
\varphi_{ma} = \varphi_0 + \varphi_1 \cos \theta_e \\
\varphi_{mb} = \varphi_0 + \varphi_1 \cos \left( \theta_e - \frac{2}{3} \pi \right) \\
\varphi_{mc} = \varphi_0 + \varphi_1 \cos \left( \theta_e - \frac{4}{3} \pi \right)
\end{array}
\right. 
\qquad (10) 
$$

Where the details of the above parameters and the values are given in

<table><caption>**Table 2**

Harmonics order of: PM flux, self and mutual inductances.</caption><thead><tr><th>Harmonic order</th><th>PM flux (Wb)</th><th>Self-inductance (mH)</th><th>Mutual inductance (mH)</th></tr></thead><tbody><tr><td>0</td><td>0.6814</td><td>30.5</td><td>-15.5</td></tr><tr><td>1</td><td>0.4147</td><td>14</td><td>6.7</td></tr><tr><td>2</td><td>0.0354</td><td>1.54</td><td>1.5</td></tr><tr><td>3</td><td>0.0081</td><td>0.9</td><td>0.4</td></tr><tr><td>4</td><td>0.0003</td><td>0.1</td><td>0</td></tr><tr><td>5</td><td>0.0012</td><td>0</td><td>0</td></tr><tr><td>THD (%)</td><td>14.06</td><td>13.04</td><td>17.3</td></tr></tbody></table>

annex A2, $\theta_e$ is the electrical angle (rad) with: $\theta_e = N_r \Omega_g$.

The total flux matrix is given as follows:

$$ 
\begin{bmatrix} \varphi_a \\ \varphi_b \\ \varphi_c \end{bmatrix} = \begin{bmatrix} \varphi_{ma} \\ \varphi_{mb} \\ \varphi_{mc} \end{bmatrix} + \begin{bmatrix} L_a & M_{ab} & M_{ac} \\ M_{ba} & L_b & M_{bc} \\ M_{ca} & M_{cb} & L_c \end{bmatrix} \begin{bmatrix} i_a \\ i_b \\ i_c \end{bmatrix} \qquad (11) 
$$

The expression of the V-DSPMG voltages in the abc frame:

$$ 
\begin{aligned}
- \begin{bmatrix} v_a \\ v_b \\ v_c \end{bmatrix} = & \begin{bmatrix} e_{ma} \\ e_{mb} \\ e_{mc} \end{bmatrix} + \begin{bmatrix} R_s & 0 & 0 \\ 0 & R_s & 0 \\ 0 & 0 & R_s \end{bmatrix} \begin{bmatrix} i_a \\ i_b \\ i_c \end{bmatrix} \\
& + \frac{d}{dt} \left\{ \begin{bmatrix} L_a & M_{ab} & M_{ac} \\ M_{ba} & L_b & M_{bc} \\ M_{ca} & M_{cb} & L_c \end{bmatrix} \begin{bmatrix} i_a \\ i_b \\ i_c \end{bmatrix} \right\}
\end{aligned} 
\qquad (12) 
$$

Where: $e_m$ is the electromotive force and $R_s$ is the phase resistance.

$$ [e_m]_{abc} = \frac{d}{dt}[\varphi_m]_{abc} \qquad (13) $$

The expression of the electromagnetic torque:

$$ T_{em} = - \left( \frac{N_r}{2} [i_s]^t \left[ \frac{\partial L_s}{\partial \theta_e} \right] [i_s] + N_r [i_s]^t \left[ \frac{\partial \varphi_m}{\partial \theta_e} \right] \right) \qquad (14) $$

The instantaneous active power $P_{V-DSPMG}$ and reactive power $Q_{V-DSPMG}$ are expressed by the following equation:

$$ P_{V-DSPMG} = v_a i_a + v_b i_b + v_c i_c \qquad (15) $$

<figure><img src="image_7.png" /><figcaption>**Fig. 5.** V-DSPMG: (a) Self-inductances; (b) mutual inductances; (c) PM linkage flux.</figcaption></figure>
$$P_{V-DSPMG} = -\left(i_a \frac{d\phi_{ma}}{dt} + i_b \frac{d\phi_{mb}}{dt} + i_c \frac{d\phi_{mc}}{dt}\right) - R_s(i_a^2 + i_b^2 + i_c^2) - \left(i_a^2 \frac{dL_a}{dt} + i_b^2 \frac{dL_b}{dt} + i_c^2 \frac{dL_c}{dt}\right) 2\left(i_a i_b \frac{dM_{ab}}{dt} + i_a i_c \frac{dM_{ac}}{dt} + i_b i_c \frac{dM_{bc}}{dt}\right) - \frac{1}{2}\left(L_a \frac{d i_a^2}{dt} + L_b \frac{d i_b^2}{dt} + L_c \frac{d i_c^2}{dt}\right) - M_{ab} \frac{d}{dt}(i_a i_b) - M_{ac} \frac{d}{dt}(i_a i_c) - M_{bc} \frac{d}{dt}(i_b i_c) \quad (16)$$

$$Q_{V-DSPMG} = -\frac{1}{\sqrt{3}}((v_b - v_c)i_a + (v_c - v_a)i_b + (v_a - v_b)i_c) \quad (17)$$

$$Q_{V-DSPMG} = w_e \varphi_1 \left( i_a \cos\theta_e + i_b \cos\left(\theta_e - \frac{2\pi}{3}\right) + i_c \cos\left(\theta_e - \frac{4\pi}{3}\right) \right) + \frac{1}{\sqrt{3}} \left( (i_c - i_b) \frac{d}{dt} L_a i_a + (i_a - i_c) \frac{d}{dt} L_b i_b + (i_b - i_a) \frac{d}{dt} L_c i_c \right) + \frac{1}{\sqrt{3}} i_a \frac{d}{dt} (M_{bc}(i_c - i_b) + i_a(M_{ab} - M_{ac})) + \frac{1}{\sqrt{3}} i_b \frac{d}{dt} (M_{ac}(i_a - i_c) + i_b(M_{bc} - M_{ab})) + \frac{1}{\sqrt{3}} i_c \frac{d}{dt} (M_{ab}(i_b - i_a) + i_c(M_{ac} - M_{bc})) \quad (18)$$

The dynamical model of the V-DSPMG generator in the dq frame:

$$P(\theta) = \sqrt{\frac{2}{3}} \begin{bmatrix} \cos(\theta) & \cos\left(\theta - \frac{2\pi}{3}\right) & \cos\left(\theta - \frac{4\pi}{3}\right) \\ -\sin(\theta) & -\sin\left(\theta - \frac{2\pi}{3}\right) & -\sin\left(\theta - \frac{4\pi}{3}\right) \end{bmatrix} \quad (19)$$

$$\left\{ \begin{aligned} L_d &= L_0 - M_0 + \frac{1}{2}(L_1 + 2M_1) \cos 3\theta_e \\ L_q &= L_0 - M_0 - \frac{1}{2}(L_1 + 2M_1) \cos 3\theta_e \\ M_{dq} &= -\frac{1}{2}(L_1 + 2M_1) \sin 3\theta_e \end{aligned} \right. \quad (20)$$

The direct and quadratic flux matrix is given as follows:

$$\begin{bmatrix} \varphi_d \\ \varphi_q \end{bmatrix} = \begin{bmatrix} L_d & M_{dq} \\ M_{dq} & L_q \end{bmatrix} \begin{bmatrix} i_d \\ i_q \end{bmatrix} + \sqrt{\frac{3}{2}} \begin{bmatrix} \varphi_1 \\ 0 \end{bmatrix} \quad (21)$$

The expression of the V-DSPMG voltages in a dq frame:

$$-\begin{bmatrix} v_d \\ v_q \end{bmatrix} = \begin{bmatrix} R_s + 2w_e M_{dq} & -\frac{1}{2}w_e(3L_d - L_q) \\ \frac{1}{2}w_e(3L_q - L_d) & R_s - 2w_e M_{dq} \end{bmatrix} \begin{bmatrix} i_d \\ i_q \end{bmatrix} + \begin{bmatrix} L_d & M_{dq} \\ M_{dq} & L_q \end{bmatrix} \frac{d}{dt} \begin{bmatrix} i_d \\ i_q \end{bmatrix} + \begin{bmatrix} e_{md} \\ e_{mq} \end{bmatrix} \quad (22)$$

Where: $e_{md}$ and $e_{mq}$ are the direct and quadratic electromotive force.

$$\begin{bmatrix} e_{md} \\ e_{mq} \end{bmatrix} = \sqrt{\frac{3}{2}} w_e \varphi_1 \begin{bmatrix} 0 \\ 1 \end{bmatrix} \quad (23)$$

Electromagnetic torque and powers in the dq frame:

$$T_{em} = - \left[ \sqrt{\frac{3}{2}} N_r \varphi_1 i_q - \frac{1}{2} N_r (L_d - L_q) i_d i_q + \frac{1}{2} N_r M_{dq} (i_d^2 - i_q^2) \right] \quad (24)$$

The instantaneous active and reactive power can be expressed as follows:

$$\left\{ \begin{array}{l} P_{V-DSPMG} = v_d i_d + v_q i_q \\ Q_{V-DSPMG} = v_d i_q - v_q i_d \end{array} \right. \quad (25)$$

$$P_{V-DSPMG} = -e_{mq} i_q - \left(R_s i_d^2 + R_s i_q^2\right) - w_e \left[2M_{dq} \left(i_d^2 - i_q^2\right) - 2(L_d - L_q)i_d i_q\right] - \left[\frac{1}{2}L_d \frac{di_d^2}{dt} + \frac{1}{2}L_q \frac{di_q^2}{dt} + M_{dq} \frac{d(i_d i_q)}{dt}\right] \quad (26)$$

$$Q_{V-DSPMG} = e_{mq}i_d + w_e(L_q i_d^2 + L_d i_q^2) - 4w_e M_{dq}i_d i_q - w_e \frac{1}{2}(L_d - L_q)(i_d^2 - i_q^2) - \left(L_d i_q \frac{di_d}{dt} - L_q i_d \frac{di_q}{dt}\right) + \frac{1}{2}M_{dq}\left(\frac{di_d^2}{dt} - \frac{di_q^2}{dt}\right) \quad (27)$$

### 3.3. Photovoltaic model

Solar cells operate on the photovoltaic principle, directly converting solar energy into electrical energy. When solar radiation impinges on a photovoltaic cell, it is converted into photon energy, which is then harnessed to power electrical appliances [45]. The photovoltaic cell model proposed in this study adheres to the single-diode SDM framework. This model encompasses an ideal direct current DC source, denoted as $I_{ph}$, in conjunction with a parallel resistor $R_p$, a series resistor $R_s$, and a single diode $D$, as depicted in Fig. 6a. Additionally, a photovoltaic array within this model is configured as multiple parallel strings of PV modules $N_{pc}$, with each string comprising a series connection of multiple PV cells $N_{sc}$ as illustrated in Fig. 6b.

The PV cell output current $I_{cell}$ can be expressed by [46]:

$$I_c = I_{ph,c} - I_D - I_p \quad (28)$$

Where:

$$\left\{ \begin{aligned} I_D &= I_o \left[ e^{\frac{V_c+R_s I_c}{A K_z T}} - 1 \right] \\ I_p &= \frac{V_c + R_s I_c}{R_p} \end{aligned} \right. \quad (29)$$

$$I_c = I_{ph,c} - I_o \left[ e^{\frac{V_c+R_s I_c}{A K_z T}} - 1 \right] - \frac{V_c + R_s I_c}{R_p} \quad (30)$$

Where: $I_o$ represents the diode's reverse saturation current. $q$ denotes the charge of an electron. $A$ is the diode ideality factor. $K_z$ represents Boltzmann's constant. Finally, $T$ represents the temperature of the PV cell.

The PV module is fabricated by interconnecting multiple PV cells in series. Thus, the PV module output current $I_m$ is given as follows:

$$I_m = I_{ph,m} - I_o \left[ e^{\frac{V_m+R_s I_m}{A N_{pc} A K_z T}} - 1 \right] - \frac{V_m + R_s I_m}{R_p} \quad (31)$$

Where: $I_{ph,m}$ is the current generated by PV module.

A PV array consists of multiple PV modules configured in multiple parallel strings. The output current of a PV array $I_{PV}$, is defined as follows [47]:

$$I_a = N_{pc} I_{ph,a} - N_{pc} I_o \left[ e^{\frac{V_{PV} + \frac{N_{pc} R_s}{N_{pc} R_p}}{A K_z T}} - 1 \right] - \frac{V_{PV} + \frac{N_{pc} R_s}{N_{pc} R_p}}{\frac{N_{pc}}{N_{pc} R_p}} \quad (32)$$

Where: $I_{ph,a}$ is the current generated by PV array.

The PV array output power $P_{PV}$ can be given as follows:

$$P_{PV} = V_{PV} I_{PV} \quad (33)$$
<figure><img src="image_8.png" /><figcaption>Fig. 6. (a) equivalent circuit of PV cells, (b) equivalent circuit of PV array.</figcaption></figure>

Fig. 7 provides a clear graphical representation of the power/voltage and current/voltage characteristics of a photovoltaic PV array under the condition of constant temperature and varying levels of solar irradiance. The graph effectively illustrates the direct correlation between solar irradiation and the power output of the photovoltaic array. It clear that as solar irradiance increases, there is a corresponding increase in the power generated by the PV array. On the contrary, a decrease in irradiation results in a proportional minimizing in the output power. This relationship highlights the sensitivity of the photovoltaic array's performance to changes in solar irradiance, underscoring the importance of irradiance as a key factor in determining the efficiency of solar energy conversion.

## 3.4. Battery model

Batteries, which function as electrochemical devices, play a pivotal role in the realm of energy conversion. They are designed to store electrical energy as chemical energy during charging and convert it back to electrical energy during discharging. In HRES, the battery storage system holds a vital position. It effectively manages the energy surplus when the power generated by wind turbines or solar arrays exceeds the demand for power. In addition, it ensures power equilibrium when the generation from these sources is insufficient to meet the load requirements.

The literature offers a spectrum of battery models for various applications [48,49]. In this research, the $R_{int}$ model is employed to model the lithium-ion battery, as illustrated in Fig. 8. This model characterizes the dynamic behavior of the battery using an ideal voltage source $E_0$ and an internal resistance $R_{int}$ in series. The $R_{int}$ model is selected due to its fundamental structure and ease of implementation, which makes it an

<figure><img src="image_9.png" /><figcaption>Fig. 8. Equivalent circuit of the lithium-ion battery model.</figcaption></figure>

efficient tool for analyzing battery behavior in hybrid systems [50].

According to the meshes law, the battery voltage can be given as follows:

$$V_{bat} = E_{bat} \pm R_{int} I_{bat} \quad (34)$$

Where: $I_{bat}$ and $V_{bat}$ are the battery current and voltage, respectively. The state of charge (SOC) is given by [51]:

$$SOC(t) = SOC_{init} - \frac{1}{Q_{max}} \int_{0}^{t} I_{bat} dt \quad (35)$$

Where: $SOC_{init}$ is the initial state of charge of the battery, and $Q_{max}$ is the battery maximum capacity.

The following equations describe the behavior of the battery based on its charge and discharge regime [52]:

<figure><img src="image_10.png" /><figcaption>Fig. 7. (a) PV panel power characteristic; (b) PV panel current characteristic.</figcaption></figure>
$$E_{bat\_disch} = E_0 - K_b \left( \frac{Q_{max}}{Q_{max} - It} \right) I_{bat} - K_b \left( \frac{Q_{max}}{Q_{max} - It} \right) It + A_b e^{(-B \cdot It)} \quad (36)$$

$$E_{bat\_ch} = E_0 - K_b \left( \frac{Q_{max}}{It + 0.1 \cdot Q_{max}} \right) I_{bat}^* - K_b \left( \frac{Q_{max}}{Q_{max} - It} \right) It + A_b e^{(-B \cdot It)} \quad (37)$$

Where: $K_b$ is the polarization coefficient of the resistance (Ω); $I_{bat}^*$ is the low-frequency current dynamics (A). $It = \int I_{bat} dt$ is the extracted capacity (Ah); $A_b$ is the exponential voltage (V) and $B$ is the exponential capacity ($Ah^{-1}$).

# 4. System components control

## 4.1. Wind turbine control

The generator utilized in this paper is a low-speed direct drive generator. That means that the generator does not require a gearbox system to connect with the wind turbine, in other words, the generator speed is equal to the turbine speed $\Omega_t = \Omega_g$. Thus, the energy losses due to the gearbox are eliminated.

Since the generator speed and the turbine speed are equal, then the total inertia $J_T$ of the system is equal to the sum of the generator inertia $J_g$ and the turbine inertia $J_t$:

$$J_T = J_g + J_t \quad (38)$$

The mechanical shaft equation is given as follows [41]:

$$J_T \frac{d\Omega_g}{dt} = T_m - T_{em} - f\Omega_g \quad (39)$$

Where: $T_m$, $T_{em}$, and $f$ are the mechanical torque, electromagnetic torque, and generator friction.

The aerodynamic model of the wind turbine system is presented in Fig. 9. As shown in the scheme, there is no gearbox system; the wind turbine shaft is connected directly to the generator shaft. Furthermore, the scheme shows that the generator speed $\Omega_g$ can be controlled by the electromagnetic torque $T_{em}$.

In the realm of wind energy conversion, particularly at lower wind speeds, the Maximum Power Point Tracking (MPPT) algorithm plays a crucial role in optimizing power extraction. The literature reveals a plethora of algorithms specifically tailored for wind energy systems, including power Signal Feedback Control (PFC), Optimal Torque Control (OTC), Tip Speed Ratio (TSR), and Perturbation and Observation (P&O) [53–55].

This study adopts the TSR method of tip speed ratio control, due to its several advantages such as improved efficiency, rapid convergence,

better accuracy, good robustness, and simplified operational complexity [54,55]. This approach ensures that the wind turbine operates at its maximum power coefficient $C_p$ by maintaining the tip speed ratio $\lambda$ at an optimal value $\lambda_{opt}$. To effectively control the generator's speed loop, SMC based on the constant plus proportional rate reaching law is employed. The sliding surface for this control system is defined as follows:

$$S(\Omega_g) = \Omega_g^* - \Omega_g \quad (40)$$

The equivalent control law appears when the sliding surface and its derivative are equal to zero:

$$S(\Omega_g) = \dot{S}(\Omega_g) = 0 \quad (41)$$

Substituting Eqs. (39) and (40) in (41). The equivalent control law is given as follows:

$$T_{em}^{eq} = T_m - f\Omega_g - J_t\dot{\Omega}_g^* \quad (42)$$

The sliding mode control function is given as follows [56]:

$$U_{SMC} = -Q_t \operatorname{sign}(S) - K_t S \quad (43)$$

Eq. (43) is used as a discontinuous control law:

$$T_{em}^d = -Q_t \operatorname{sign}(S) - K_t S \quad (44)$$

The proposed hybrid control law is obtained as follows:

$$T_{em}^{ref} = T_{em}^{eq} + T_{em}^{d} \quad (45)$$

$$T_{em}^{ref} = T_m - f\Omega_g - J_t\dot{\Omega}_g^* + J_t(Q_t\operatorname{sign}(S) + K_tS) \quad (46)$$

The diagram of the sliding mode control method is shown in Fig. 10.

Figs. 11a and 11b illustrate the efficiency of the MPPT-based tip speed ratio approach. This method ensures the achievement of the optimal tip speed ratio $\lambda_{opt}$ and the maximum wind power coefficient $C_{p(max)}(\lambda, \beta)$, demonstrating the efficacy of the system under varying environmental conditions.

## 4.2. V-DSPMG generator control

For optimal conversion of the electrical energy produced by the system, a dual-stage configuration is used. This system incorporates a three-phase diode rectifier connected to a DC/DC boost converter, as depicted in Fig. 12. The diode bridge plays a pivotal role in rectifying the output voltages generated by the system. Simultaneously, the boost converter is key to improving power extraction, achieved by adjusting the power switch, labeled $S_w$.

Wind energy systems present significant nonlinearities and un-

<figure><img src="image_11.png" /><figcaption>Fig. 9. Mathematical model of the wind turbine system.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig. 10. TSR based sliding mode control.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 11. (a) Tip Speed Ratio curve; (b) Wind power coefficient curve.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 12. Global Wind Turbine Control Scheme.</figcaption></figure>
certainties due to wind speed variations. To address these challenges, sliding mode control (SMC) proves to be particularly suitable, as it offers good robustness against uncertainties and external disturbances [57, 58]. Moreover, its ability to ensure stability and maintain optimal performance in nonlinear and dynamic environments which makes it a suitable approach for regulating wind energy systems. Consequently, a sliding mode control strategy is employed in this section, which is based on the Second Gao's Law [59]. The sliding surface is defined as follows:

$$S(I_L) = I_L^* - I_L \qquad (47)$$

The current inductor equation of the DC/DC boost converter is given as follows:

$$\frac{dI_L}{dt} = \frac{V_{in}}{L_w} - (1 - d_w) \frac{V_{DC}}{L_w} \qquad (48)$$

Where: $d_w$ is the duty cycle.

In the steady state, the sliding surface and its derivative can be expressed as follows:

$$S(I_L) = \dot{S}(I_L) = 0 \qquad (49)$$

The second Gao low is given as follows:

$$\dot{S}(t) = -Q_w \text{sign}(S) - K_w S \qquad (50)$$

By substituting Eqs. (48) and (50) into (49), the low control can be expressed as follows:

$$d_w = \frac{V_{DC} - V_{in}}{V_{DC}} + \frac{L_w(K_w S(I_L) + Q_w \text{sign}(S(I_L)))}{V_{DC}} + \frac{L_w \frac{dI_L^*}{dt}}{V_{DC}} \qquad (51)$$

With $Q_w$ and $K_w$ are parameters to be tuned.

### 4.3. Photovoltaic system control

In the conversion process of electrical energy generated by the photovoltaic PV system, a single-stage configuration is adopted, as depicted in Fig. 13. This configuration strategically employs a single DC/ DC boost converter. The rationale behind this choice is multifaceted: it significantly reduces the physical dimensions of the system, lowers overall costs, and simplifies the complexity inherent in the system's control mechanisms. This streamlined approach not only enhances the

efficiency of the energy conversion process but also contributes to the general practicality and feasibility of the system implementation.

A range of MPPT algorithms, extensively discussed in the literature [60–62], are utilized to optimize power extraction from photovoltaic arrays. For this particular study, the P&O algorithm has been selected, primarily for its high efficiency, low cost, straightforward implementation, and widespread adoption in photovoltaic systems. [60,61]. As shown in Fig. 14, the operational principle of the P&O technique involves a dynamic adjustment of the duty cycle $d$ of the DC/DC boost converter, based on changes in the power and voltage of the photovoltaic array.

When an increase in both PV power and voltage is observed, the disturbance step size $\Delta_d$ is added to the duty cycle. In contrast, if there is an increase in PV power accompanied by a decrease in PV voltage, the size of the perturbation step size $\Delta_d$ is subtracted from the duty cycle for the subsequent perturbation cycle. This process is iteratively executed until the maximum power point (MPP) is effectively reached. This approach enables precise tracking of the MPP, thus maximizing the energy harvested from the photovoltaic array.

### 4.4. Battery storage system control

In the domain of energy management within a storage system, the implementation of a bidirectional DC/DC converter is essential. This converter is ingeniously designed as a hybrid of two circuits: a step-down and a step-up circuit. It comprises an inductor $L_b$, a pair of power switches $S_b$ and $S'_b$, and a capacitor $C_b$. The unique configuration of these components enables the converter to operate in two distinct modes: charging mode and regenerating mode.

The control mechanism for the battery storage system has been thoroughly explored in several studies [63–66]. Batteries exhibit relatively slow dynamics and stable behavior. Therefore, the use of a PI controller is suitable due to its straightforward structure and ease of implementation. Consequently, in this section, a single current control loop is used to control the battery storage system. The reference value of the battery current $I_{bat}$ is calculated by dividing the reference power, provided by the energy management system (EMS), by the battery voltage $V_{bat}$. A comparison is conducted between the actual battery current $I_{bat}$ and the reference battery current $I_{bat}^*$, with the resulting error

<figure><img src="image_15.png" /><figcaption>Fig. 13. Photovoltaic global control scheme.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Fig. 14. Perturb & observe the MPPT flow chart.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 15. Global control scheme of the storage system.</figcaption></figure>

signal processed through another PI controller. The duty cycle of the converter is adjusted on the basis of the output of the PI controller. Fig. 15 presents a comprehensive control scheme of the battery storage

system that encapsulates the entire control process.

### 4.5. Grid side converter control

The grid side converter uses a three-phase 5L-ANPC DC/AC converter [67]. Fig. 16 illustrates the phase leg of the converter. It contains eight switches per phase, four line-frequency switches ($S_{x1}, S'_{x1}$) ($S_{x2}, S'_{x2}$), and four high-frequency switches ($S_{x3}, S'_{x3}$) ($S_{x4}, S'_{x4}$), one flying capacitor $C_{fx}$, and two capacitors on the DC link $C_1, C_2$. Consequently, this topology reduces the total size, complexity of the control, and cost. Additionally, it offers the benefits of increased flexibility and robustness, along with a smaller output filter size and reduced power losses [67]. The switching states of the 5L-ANPC converter details are shown in Table 3.

Various control methods for three-phase converters have been studied and presented in the literature, mainly, like PI, SMC, and FLC. The PI controller is one of the simplest controllers and is used only for linear systems. Similarly, SMC and FLC controllers are used for

<figure><img src="image_18.png" /><figcaption>Fig. 16. Phase leg of 5L-ANPC inverter.</figcaption></figure>
<table><caption>**Table 3**
Switching states of the 5L-ANPC converter.</caption><thead><tr><th>S<sub>x2</sub></th><th>S<sub>x3</sub></th><th>S<sub>x4</sub></th><th>V<sub>ox</sub></th><th>i<sub>fx</sub></th><th>i<sub>NPx</sub></th><th>Switching states</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>1</td><td>+ V<sub>DC</sub>/2</td><td>0</td><td>0</td><td>V<sub>0</sub></td></tr><tr><td>1</td><td>1</td><td>1</td><td>+ V<sub>DC</sub>/4</td><td>i<sub>ox</sub></td><td>0</td><td>V<sub>1</sub></td></tr><tr><td>1</td><td>1</td><td>0</td><td>+ V<sub>DC</sub>/4</td><td>- i<sub>ox</sub></td><td>i<sub>ox</sub></td><td>V<sub>2</sub></td></tr><tr><td>1</td><td>1</td><td>0</td><td>0</td><td>0</td><td>i<sub>ox</sub></td><td>V<sub>3</sub></td></tr><tr><td>0</td><td>0</td><td>1</td><td>1</td><td>0</td><td>i<sub>ox</sub></td><td>V<sub>4</sub></td></tr><tr><td>0</td><td>0</td><td>1</td><td>0</td><td>- V<sub>DC</sub>/4</td><td>i<sub>ox</sub></td><td>V<sub>5</sub></td></tr><tr><td>0</td><td>0</td><td>1</td><td>0</td><td>- V<sub>DC</sub>/4</td><td>- i<sub>ox</sub></td><td>V<sub>6</sub></td></tr><tr><td>0</td><td>0</td><td>0</td><td>1</td><td>- V<sub>DC</sub>/2</td><td>0</td><td>V<sub>7</sub></td></tr></tbody></table>

nonlinear systems. They offer serval advantages such as fast dynamic response, robustness, and high stability [68,69].

Given the importance of the DC-Link in hybrid DC-configuration systems, in this study a Super-Twisting Sliding Mode Control (STSMC) is chosen. The super-twisting approach is used for SMC to accelerate the convergence of the system to its equilibrium point [29]. As a consequence, this study aims to enhance the design of the super-twisting method for better sliding mode convergence and robustness. The STSMC algorithm reaching law can be described by:

$$ \begin{cases} U_1 = -K |S|^{0.5} \operatorname{sign}(S) - U_2 \\ \dot{U}_2 = Q \operatorname{sign}(S) \end{cases} \qquad (52) $$

$$ \begin{cases} K > \frac{\mu}{\delta_u} \\ Q \ge \frac{4\mu\delta_u (K + \mu)}{\delta_L^3 (K - \mu)} \end{cases} \qquad (53) $$

$$ \begin{cases} \mu \ge \Theta \text{ and } \delta_u \ge \Phi \ge \delta_L \\ \ddot{S} = \Theta(x, t) + \Phi(x, t) \dot{U} \end{cases} \qquad (54) $$

Where: K, Q are positive constants. μ is the positive boundary of the uncertain function Θ, and δ_u and δ_L are the upper and lower positive boundaries respectively of the uncertain function Φ at the sliding manifold's second derivative [70].

### 4.5.1. Proposed new STSMC for the DC-bus voltage control

The exponential term K |S|<sup>0.5</sup> sign(S) of the conventional Super-Twisting Sliding Mode control improves the control reaching speed. However, the sliding surface is computed using a square root, which directly impacts the speed at which the control reaches the surface. To overcome this, an NSTSMC reaching law is designed to improve the convergence speed while maintaining the system's robustness and stability. This is achieved by incorporating an additional nonlinear integral term. This later term serves as a dynamic adjustment mechanism that enhances the convergence speed of the control law when the sliding variable S deviates from the desired surface. This new approach allows the state of the system to reach the sliding surface faster and with a smoother transition. The new STSMC reaching law is given by:

$$ \begin{cases} U_1 = -K_v |S|^{0.5} \operatorname{sign}(S) - \int \epsilon_v |S| S^{\alpha_v} \operatorname{sign}(|S|-1) - U_2 \\ \dot{U}_2 = Q_v \operatorname{sign}(S) \end{cases} \qquad (55) $$

Where: K_v, Q_v, ε_v, a_v are positive constants,

$$ \begin{cases} K_v > \frac{\mu_v}{\delta_{uv}} \\ Q_v \ge \frac{4\mu_v \delta_{uv} (K_v + \mu_v)}{\delta_{LV}^3 (K_v - \mu_v)} \\ \epsilon_v > 0, \quad 0 < a_v < 0.5 \end{cases} \qquad (56) $$

Where: μ_v is the positive boundary of the uncertain function Θ, and

δ<sub>uv</sub> and δ<sub>Lv</sub> are the upper and lower positive boundaries of the uncertain function Φ at the sliding manifold's second derivative [70], which in this case V<sub>DC</sub>.

$$ \left\{ \begin{array}{l} \mu_v \ge \Theta \text{ and } \delta_{uv} \ge \Phi \ge \delta_{Lv} \\ \dot{V}_{DC} = \Theta(x, t) + \Phi(x, t) \dot{U} \end{array} \right. \qquad (57) $$

Where: Θ and Φ are chosen as positive constant.

Analyzing the new STSMC reaching law in Eq. (55) leads to the following conclusions:

* When the state of the system is far from the sliding surface (case 1: S > 1), the switching function is equal to 1 sign(|S|-1) = 1, and the exponential term K_v |S|<sup>0.5</sup> sign(S) + ∫ ε_v |S|<sup>a</sup> sign(|S|-1)S becomes K_v |S|<sup>0.5</sup> sign(S) + ∫ ε_v |S|<sup>a</sup> S > K_v |S|<sup>0.5</sup> sign(S). Thus, the state of the system of the NSTSMC reaching law attains the sliding surface faster than the conventional one.

* If the state of the system is close to the sliding surface (case 2: S < 1), the switching function is equal to -1 sign(|S|-1) = -1, and the exponential term K_v |S|<sup>0.5</sup> sign(S) - ∫ ε_v |S|<sup>a</sup> sign(|S|-1)S becomes K_v |S|<sup>0.5</sup> sign(S) + ∫ ε_v |S|<sup>-a</sup> S > K_v |S|<sup>0.5</sup> sign(S). Therefore, when the state of the NSTSMC reaching law system is close to the sliding surface, the reaching speed increases.

* Based on the comparative analysis of the NSTSMC and conventional STSMC reaching laws, it is clear that the NSTSMC law achieves faster convergence than the conventional one.

For efficient power transfer, NSTSMC controls the DC Link voltage control loop as shown in Fig. 17. The mathematical model of the DC-Link voltage can be expressed as follows:

$$ I_e = C_{DC} \frac{dV_{DC}}{dt} = I_{in} - I_{DC} \qquad (58) $$

The chosen sliding surface and its derivative for V<sub>DC</sub> are:

$$ S = V_{DC}^* - V_{DC} \qquad (59) $$

$$ \dot{S} = \dot{V}_{DC}^* - \dot{V}_{DC} \qquad (60) $$

Assume that the grid-connected power converter is ideal:

$$ P_{DC} \approx P_{grid} \qquad (61) $$

$$ V_{DC} I_{DC} = \frac{3}{2} V_{gd} I_{gd} \qquad (62) $$

By substituting Eq. (58) and Eq. (62) into Eq. (60):

$$ \dot{S} = -\frac{I_{in}}{C_{DC}} + \frac{3V_{gd}I_{gd}}{2C_{DC}V_{DC}} \qquad (63) $$

By using Eq. (55):

$$ I_{gd}^* = \frac{2V_{DC}}{3V_{gd}} \left[ I_{in} - C_{DC} \left( -K_v |S|^{0.5} \operatorname{sign}(S) - \int \epsilon_v |S| S^{\alpha_v} \operatorname{sign}(|S|-1) - Q_v \int \operatorname{sign}(S) \right) \right] \qquad (64) $$

In order to analyze the stability of the new reaching law. Examine the following Lyapunov function, V<sub>NSTSMC</sub>:
<figure><img src="image_19.png" /><figcaption>Fig. 17. Proposed new super-twisting sliding-mode control flowchart.</figcaption></figure>

$$V_{\text{NSTSMC}} = \frac{1}{2} S^2 \qquad (65)$$

The Lyapunov stability condition of the system is as follows:

$$\dot{V}_{\text{NSTSMC}} = S \dot{S} \le 0 \qquad (66)$$

By substituting Eq (64) into Eq (63) and the result into Eq (66). The derivative of Lyapunov function becomes:

$$\left\{ \begin{aligned} V_{id} &= R_f I_{gd} + L_f \frac{dI_{id}}{dt} - \omega L_f I_{gq} + V_{gd} \\ V_{iq} &= R_f I_{gq} + L_f \frac{dI_{iq}}{dt} + \omega L_f I_{gd} + V_{gq} \end{aligned} \right. \qquad (69)$$

The open-loop transfer function of the $R_f L_f$ filter with PI corrector is given by:

$$\dot{V}_{\text{NSTSMC}} = S \left[ -\frac{I_{in}}{C_{DC}} + \frac{3V_{gd} \frac{2V_{DC}}{3V_{gd}} \left[ I_{in} - C_{DC} \left( -K_v |S|^{0.5} \text{sign}(S) - \int \epsilon_v |S| S^{a_v} \text{sign}(|S|-1) - Q_v \int \text{sign}(S) \right) \right]}{2C_{DC}V_{DC}} \right] \qquad (67)$$

Simplifying Eq. (67), the Lyapunov function becomes:

$$\dot{V}_{\text{NSTSMC}} = -S K_v |S|^{0.5} \text{sign}(S) - S \epsilon_v \int |S| S^{a_v} \text{sign}(|S|-1) - S Q_v \int \text{sign}(S) \le 0 \qquad (68)$$

For a large value of $Q_v$, the inequality condition in Eq. (65) can be satisfied. Therefore, according to the Lyapunov stability condition, the system is stable.

### 4.5.2. Grid side active and reactive power control

The active and reactive power can be independently controlled through the direct current $I_{gd}$ and quadrature current $I_{gq}$. A Voltage Oriented Control (VOC) strategy, incorporating a PI controller, is well-suited for this type of control, as depicted in Fig. 17. The expressions of the $dq$ components of the output voltages of the inverter are:

$$H_{OL}(s) = K_p \frac{T_i s + 1}{T_i s} \frac{1}{1 + T_{PWM} s} \cdot \frac{1/R_f}{1 + (L_f/R_f)s} \qquad (70)$$

Where $T_{PWM} = 1,5T_s$, with $T_s$ is the sampling period [71].

The time constant of the PI corrector is selected to compensate for the dominating pole $T_i = L_f/R_f$. Therefore, the closed-loop transfer function can be expressed as follows:

$$H_{CL}(s) = \frac{\frac{K_p}{T_i T_{PWM} R_f}}{S^2 + \frac{1}{T_{PWM}} s + \frac{K_p}{T_i T_{PWM} R_f}} \qquad (71)$$

With:

$$\left\{ \begin{aligned} K_p &= \omega_n^2 T_i T_{PWM} R_f \\ T_i &= \frac{L_f}{R_f} \end{aligned} \right. \qquad (72)$$

<caption>Fig. 18</caption>
<figure><img src="image_20.png" /><figcaption>Fig. 18. Global control scheme of the grid side converter.</figcaption></figure>

## 4.6. Energy management for grid-connected hybrid renewable energy system

The primary objective of the energy management system (EMS) is to ensure a balance between the energy generated by the renewable sources, the energy required by the loads, and the energy managed by the storage system. Additionally, the EMS safeguards the storage units from overcharging or deep discharging while accommodating potential energy exchanges with the utility grid.

In this study, minimizing grid energy consumption is a priority due to its associated costs. Therefore, power generated by renewable energy

sources is given precedence over grid power to meet load demand, positioning grid power as a last resort option within the energy management hierarchy.

Referring to the system architecture depicted in Fig. 1, the power balance equation governing the energy management control, is formulated as follows:

$$P_{PV} + P_{wind} \pm P_{bat} \pm P_{grid} - P_{load} = 0 \quad (73)$$

* Case 1: Surplus power

<figure><img src="image_21.png" /><figcaption>Fig. 19. Energy management flowchart.</figcaption></figure>
If the total power generated by the wind and photovoltaic systems exceeds the load demand ($P_{PV} + P_{wind} > P_{load}$), the surplus power ($P_{sur} = P_{PV} + P_{wind} - P_{load}$) is first used to charge the battery ($P_{bat} = P_{sur}$). The battery continues charging until it reaches its full capacity. If the battery is completely charged, the remaining surplus power is delivered to the grid ($P_{grid} = P_{sur}$). Since the battery can only charge up to its maximum allowable power $P_{bat}^{max}$, any surplus power exceeding ($P_{sur} > P_{bat}^{max}$) is managed as follows: the battery is charged at its maximum power, and the excess power is exported to the utility grid ($P_{grid} = P_{sur} - P_{bat}^{max}$).

### Case 2: Power deficit

When the total power generated by the wind and photovoltaic systems is insufficient to meet the load demand ($P_{PV} + P_{wind} < P_{load}$), a power deficit ($P_{def} = P_{load} - P_{PV} - P_{wind}$) arises. In this situation, the battery discharges to prioritize the load demand. If the battery is completely discharged, the remaining deficit is supplied entirely by the utility grid. Since the battery can only discharge up to its maximum allowable power $P_{bat}^{max}$, any deficit exceeding this limit $P_{bat}^{max}$ is handled by discharging the battery at its maximum capacity, with the remaining deficit supplied by the grid ($P_{grid} = P_{def} - P_{bat}^{max}$).

Fig. 19 illustrates the global flow chart of the energy management system for the grid-connected renewable energy system.

## 5. Simulation results and discussion

The simulation of the proposed grid connected was conducted in a MATLAB/Simulink environment as depicted in Fig. 20. A key feature of the wind power system within this setup is the employment of a direct-driven generator, specifically, the Vernier Doubly Saliant Permanent Magnet Generator V-DSPMG. Furthermore, a new super-twisting sliding mode control NSTSMC method is proposed for managing the DC-Link voltage. For our simulations, we assume that 1 s represents 1 hour of time.

The simulation is carried out in two distinct parts to comprehensively evaluate the system's performance:

* **Normal operating conditions:** This phase examines the system's behavior under typical operation scenarios, including:

* **Ø Load variations:** Changes in load demand.

* **Ø Meteorological variations:** Gradual changes in solar and wind conditions.

* **Critical operating conditions:** This phase assesses the robustness of the DC-Link voltage control strategy under extreme scenarios including:

* **Ø Grid faults:** Simulations involving voltage sags, overvoltage and phase-loss conditions.

* **Ø DC-Link parameter variations:** Changes in the capacitance values of the DC-Link, such as asymmetric degradation or increase.

* **Ø Meteorological variations:** Sudden and abrupt change in solar irradiance and wind speed.

### 5.1. Simulation results under critical operating condition

The profiles of irradiance and wind speed, which fluctuate over a period of 0 to 10 s, are shown in Figs. 21a and 21b. Figs. 21 to 23 present the simulation results for the wind power system equipped with the V-DSPMG generator.

In the simulation, the angular speed $\Omega_g$ of the Vernier Double Saliant Permanent Magnet Generator V-DSPMG is observed to closely follow the variations in wind speeds. This correlation is further exemplified in Fig. 22, which depicts the generator speed adjusted to match its optimal reference speed. Fig. 23 highlights the high electromagnetic torque of the generator, a direct consequence of its low rotational speed. These characteristic underscores the efficiency of the V-DSPMG in harnessing wind energy. Additionally, Fig. 24 shows the direct and quadratic currents of the V-DSPMG, providing insight into the electrical performance of the generator.

<figure><img src="image_22.png" /><figcaption>Fig. 20. Simulated grid-connected hybrid renewable energy system under MATALB/Simulink.</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>Fig. 21. (a) Solar irradiance profile; (b) Wind speed profile.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Fig. 22. V-DSPMG generator speed $\Omega_g$.</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>Fig. 23. V-DSPMG electromagnetic torque $T_{em}$.</figcaption></figure>

Fig. 25 illustrates the stability of the DC-Link voltage within the system. Despite weather and load fluctuations, the DC-Link voltage remains consistently balanced, effectively maintaining the required operational value of 500. These simulation results demonstrate the robustness, reliability, batter tracking performance, and fast convergence of the proposed STSMC under varying weather and load conditions compared to the STSMC and the PI controller. Table 4 shows a comparison of different control strategies.

### Performance Evaluation of Control Strategies:

To evaluate the performance of three control strategies (PI, STSMC, STSMC), a phase-plane trajectory analysis is performed. This analysis enables the assessment of each control strategy's stability, convergence speed, and accuracy. Fig. 26 illustrates a comparative performance analysis of the three control strategies: PI, STSMC, and STSMC.

<figure><img src="image_26.png" /><figcaption>Fig. 24. V-DSPMG direct $i_d$ and quadratic $i_q$ current.</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>Fig. 25. Waveform of the VDC DC-Link voltage $V_{DC}$.</figcaption></figure>

<table><caption>Table 4
Comparison of different control strategies.</caption><thead><tr><td>Method</td><td>Response time (s)</td><td>Overshoot (%)</td><td>Undershoot (%)</td><td>Error (%)</td></tr></thead><tbody><tr><td>Proposed</td><td>0.05</td><td>0.06</td><td>0.34</td><td>0.002</td></tr><tr><td>STSMC</td><td>0.1</td><td>0.76</td><td>0.8</td><td>0.002</td></tr><tr><td>PI</td><td>0.4</td><td>3</td><td>2.28</td><td>0.16</td></tr></tbody></table>

* PI controller
<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /></figure>

<figure><img src="image_30.png" /><figcaption>Fig. 26. Phase-plane trajectories of the different controllers.</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>Fig. 27. Simulation results of (a) waveform of input voltage waveform $V_{ia}$; (b) harmonic spectrum.</figcaption></figure>

The PI controller exhibits an extended convergence trajectory, indicating weak robustness and slow convergence to the desired state. This slow response presents challenges in practical applications where fast settling times and a rapid dynamic response are essential for maintaining system performance and stability.

* STSMC Controller:

* The trajectory of the STSMC controller is considerably shorter, reflecting good robustness and rapid convergence to the desired state. This demonstrates the controller's effectiveness in stabilizing the system and highlights its efficiency and reliability in dynamic environments.
<figure><img src="image_32.png" /><figcaption>Fig. 28. Direct and quadratic current of: (a) Inverter; (b) Grid.</figcaption></figure>

<figure><img src="image_33.png" /><figcaption>Fig. 29. simulation results of the current $I_{ga}$ grid: (a) Receiving mode; (b) Generating mode.</figcaption></figure>

<figure><img src="image_34.png" /><figcaption>Fig. 30. Harmonic spectrums of the current $I_{ga}$ grid: (a) Receive mode; (b) Generation mode.</figcaption></figure>

## • NSTSMC Controller:

The NSTSMC outperforms both the PI and STSMC controllers, exhibiting a superior trajectory. It achieves enhanced robustness and faster convergence while maintaining system stabilization. This control strategy significantly improves convergence speed and robustness ensuring superior effectiveness efficiency, and reliability in various operating conditions.

The inverter within the hybrid system plays a crucial role in generating the necessary output voltages. This is effectively demonstrated in Fig. 27, which illustrates the output voltage waveforms of the inverter and its harmonic spectrum.

Fig. 28, shows the direct and quadratic inverter and grid currents. The purity of the grid currents, which exhibit a sinusoidal nature, is evident in Fig. 29. Figs. 30 offer a comprehensive analysis of the quality of waveforms and the harmonic spectrums of the grid current. This
<figure><img src="image_35.png" /><figcaption>Fig. 31. Waveforms of: $P_{PV}$, $P_{wind}$, $P_{bat}$, $P_{grid}$, $P_{load}$.</figcaption></figure>

characteristic is essential to ensure the smooth operation of electrical appliances and equipment connected to the system.

Fig. 31 delineates the intricate dynamics of the power flow between the components of the DC microgrid system and the main grid. In intervals from 0 to 1.5 s, the combined output of the photovoltaic and wind power systems ($P_{PV} + P_{wind}$) falls short of the load demand ($P_{load}$), the battery system compensates for the deficit by discharging ($P_{bat} > 0$). On the contrary, from 1.5 to 2.5 s, the photovoltaic and wind systems collectively produce power that exceeds the load demand ($P_{PV} + P_{wind} > P_{load}$), enabling the excess energy to charge the battery system ($P_{bat} > 0$). During these periods, the power of the grid is kept to zero ($P_{grid} = 0$).

From 2.5 to 5 s, with the load demand set to zero, the total power generated by the photovoltaic and wind systems exceeds the load demand ($P_{PV} + P_{wind} > P_{load}$). Therefore, the surplus power is used to charge the battery to its rated capacity ($P_{bat} < 0$). The remaining excess power is fed back into the grid ($P_{grid} > 0$).

From 5 to 7.5 s, the power flow remains well balanced, similar to the period 0 to 2.5 s, with the grid power maintained at zero. From 7.5 to 10 s, the power consumption $P_{load}$ is higher than the power generated by the solar and wind systems ($P_{PV} + P_{wind} < P_{load}$). Consequently, the battery compensates for the power shortage operating at its rated capacity ($P_{bat} > 0$), and the remaining deficit is compensated by drawing power from the main grid ($P_{grid} < 0$).

## 5.2. Simulation results under critical operating condition

### Moderate voltage sag:

A moderate voltage sag of –35 % was introduced on a single phase of the grid for a duration of 0.5 s.

### DC-Link Voltage Response:

<figure><img src="image_36.png" /><figcaption>Fig. 32. (a) DC-Link voltage $V_{DC}$, (b) Phase-plane trajectories.</figcaption></figure>

Fig. 32a shows the DC-link voltage response, where the STSMC exhibits noticeable oscillations during the fault period.

### Phase Trajectory Analysis:

Fig. 32b illustrates the phase trajectory, revealing significant oscillations in both the error and its derivative under the STSMC control, particularly during the fault duration. In contrast, the STSMC exhibits a more stable and controlled response significantly reducing oscillations in both the error and its derivative. These findings confirm the superior performance of the proposed STSMC compared to the conventional STSMC. The STSMC demonstrates enhanced robustness, improved stability, and faster time convergence validating its effectiveness under critical operating conditions.

### Severe Overvoltage:

A severe overvoltage of +30 % was applied to a single phase of the grid for a duration of 0.5 s.

* Fig. 33a depicts the DC-link voltage response

* The STMC phase trajectory (Fig. 33b) exhibits significant oscillations during the fault period.

* In contrast, the STSMC achieves smoother convergence with reduced oscillations demonstrating faster stabilization, which is critical for ensuring reliable DC-Link voltage.

### Phase Loss:

A phase-loss is simulated by disconnecting one grid phase 0.5 s. Fig. 34a shows the response of the DC-link voltage response. The proposed STSMC shows a more stable trajectory with minimal oscillations (Fig. 34) whereas the STSMC exhibits larger oscillations, highlighting its limitations.

**Asymmetric Degradation of DC-Link Capacitance:** A 30 % reduction is applied to one of the parallel-connected capacitors in the DC-Link, while the other capacitor retains its nominal value.

* Fig. 35a shows the response of the DC-link voltage response.

* The STSMC phase trajectory (Fig. 35b) shows noticeable oscillations during this condition

* Conversely, the STSMC ensures smoother convergence with reducer oscillations achieving faster stabilization of the DC-Link voltage.

**Asymmetric Increase of DC-Link Capacitance:** A 50 % increase is applied to one of the parallel-connected capacitors in the DC-Link, while the other maintains its nominal value.
<figure><img src="image_37.png" /><figcaption>Fig. 33. (a) DC-Link voltage $V_{DC}$, (b) Phase-plane trajectories.</figcaption></figure>

<figure><img src="image_38.png" /><figcaption>Fig. 34. (a) DC-Link voltage $V_{DC}$, (b) Phase-plane trajectories.</figcaption></figure>

<figure><img src="image_39.png" /><figcaption>Fig. 35. (a) DC-Link voltage $V_{DC}$, (b) Phase-plane trajectories.</figcaption></figure>

* Fig. 36a illustrates the response of the DC-link voltage.

* As with the case of asymmetric degradation, the STSMC phase trajectory, (Fig. 36b), demonstrates oscillations.

* The NSTSMC provides smoother convergence and significantly reduces oscillations, underlining its superior capability to rapidly stabilize the DC-link voltage.
<figure><img src="image_40.png" /><figcaption>Fig. 36. (a) DC-Link voltage $V_{DC}$, (b) Phase-plane trajectories.</figcaption></figure>

<figure><img src="image_41.png" /><figcaption>Fig. 37. (a) Solar irradiance profile; (b) Wind speed profile.</figcaption></figure>

<figure><img src="image_42.png" /><figcaption>Fig. 38. Waveform of the VDC DC-Link voltage $V_{DC}$.</figcaption></figure>
<figure><img src="image_43.png" /><figcaption>Fig. 39. Phase-plane trajectories of the STSMC and NSTSMC.</figcaption></figure>

## Response to Sudden Weather Variations

Simulations of sudden weather changes included:

* Wind speed variations at 1 and 2 s.

* Solar irradiation variations at 3 and 7 s, as shown in Fig. 37.

Fig. 38 depicts the system's DC-link voltage response to these disturbances. The enhanced trajectory of the proposed NSTSMC, illustrated in Fig. 38, demonstrates its superior robustness and stability compared to the STSMC.

The enhanced trajectory of the proposed NSTSMC, illustrated in Fig. 39, demonstrates its superior robustness and stability compared to the STSMC.

In conclusion, under all critical conditions tested, the NSTSMC outperforms the STSMC by exhibiting enhanced robustness, reduced oscillations, and faster stabilization, ensuring reliable and efficient DC-link voltage control.

## 6. Experimental results

To validate the performance and effectiveness of our proposed system, real-time simulations were conducted using OPAL-RT's advanced simulation platform. OPAL-RT's hardware-in-the-loop (HIL) testing environment was utilized to replicate real-world conditions and evaluate the system's response under various operating scenarios. The platform and real-time simulation steps are illustrated in Fig. 40. The steps involved in the real-time validation process included model export from Simulink to RT-LAB (step 1), configuring the real-time simulation environment (step 2), and executing the model on OPAL-RT hardware (step 3). This setup enabled us to test and validate key components of the system in a controlled and dynamic environment, providing insights into its behavior under both normal and fault conditions. The use of OPAL-RT allowed for precise measurement of critical parameters, ensuring that our design met the required specifications and performance standards.
<figure><img src="image_44.png" /><figcaption>Fig. 40. OPAL-RT real time validation platform.</figcaption></figure>

<figure><img src="image_45.png" /><figcaption>Fig. 41. Real time DC-voltage response.</figcaption></figure>

<figure><img src="image_46.png" /><figcaption>Fig. 42. Real time grid voltage and current responses.</figcaption></figure>

The obtained experimental results which are shown in Figs. 41, 42, and 43, represent the DC-link voltage, grid voltage and current, and the hybrid power output, respectively, closely match the simulated results, as expected. This alignment of experimental and simulated data validates the accuracy and reliability of our system design.

## 7. Conclusion

This research conducts a comprehensive investigation and simulation analysis of grid-connected system based on hybrid renewable energy sources, operating under varying weather and load conditions, using a MATLAB/Simulink environment and real time validation using OPAL-RT platform. The system comprises a solar panel array, a wind turbine with a direct-driven low-speed Vernier Doubly Saliant
<figure><img src="image_47.png" /><figcaption>Fig. 43. Real time grid voltage and current responses.</figcaption></figure>

Permanent Magnet Generator V-DSPMG, and a battery energy storage system. The results affirm the system's ability to consistently meet the required load demand power, even under fluctuating environmental conditions.

Implementation of an MPPT algorithm based on SMC to increase the performance of the wind turbine system. This approach ensures optimal extraction of power with rapid dynamics. The wind system benefits significantly from the integration of the direct-attack low-speed V-DSPMG generator. This generator addresses the limitations commonly associated with conventional generators such as high cost, large size, noise, and frequent maintenance by eliminating the need for a gearbox system.

The photovoltaic system employs an MPPT algorithm based on the P&O method, enabling it to effectively track maximum power under varying solar irradiance. The battery storage system is regulated using a PI control method applied to its DC/DC bidirectional converter. A novel control method, the New Super-Twisting Sliding-Mode Control NSTSMC, is proposed to regulate the DC-Link voltage, ensure efficient power flow management, and enhance the performance of the DC/AC converter, particularly in terms of fast dynamic response and robustness.

This grid-connected system based on hybrid renewable energy sources is adept at providing continuous power supply to the load, irrespective of low wind speeds or reduced solar irradiation. The excess power generated by solar panels or wind turbines is efficiently stored in the battery system or sent to the main grid. On the contrary, in situations where a power deficit occurs, the battery system plays a crucial role in compensating for the shortfall. The integrated approach of this

microgrid system underscores its potential as a robust and sustainable solution for renewable energy applications.

## Funding statement

This research did not receive external funding.

## CRediT authorship contribution statement

**Abderrahmane Redouane:** Writing – original draft, Visualization, Validation, Software, Resources, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. **Rachid Saou:** Writing – review & editing, Visualization, Validation, Supervision, Software, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Youcef Belkhier:** Writing – review & editing, Visualization, Validation, Supervision, Resources, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Amrane Oukaour:** Writing – review & editing, Visualization, Validation, Supervision, Software, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization.

## 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.

# Appendix A

<table><caption>**Table A1**
Wind turbine variables and parameters.</caption><thead><tr><th>Parameters</th></tr></thead><tbody><tr><td>V<sub>w(max)</sub>: Wind speed (9.2 m/s)</td></tr><tr><td>C<sub>p(max)</sub>: maximum power coefficient (0.428)</td></tr><tr><td>λ<sub>opt</sub>: Optimal tip speed ratio (2.25)</td></tr><tr><td>ρ: Air density (1.225 Kg/m<sup>3</sup>)</td></tr><tr><td>R: Wind turbine radius (3.951 m)</td></tr><tr><td>P<sub>t</sub>: Rated wind turbine power (10 kW)</td></tr></tbody></table>
<table><caption>**Table A2**
V-DSPM generator parameters.</caption><thead><tr><th colspan="2">Parameters</th></tr></thead><tbody><tr><td>R<sub>s</sub></td><td>Stator resistance (87.37 mΩ)</td></tr><tr><td>L<sub>0</sub></td><td>Continuous self-inductance (30.5 mH)</td></tr><tr><td>L<sub>1</sub></td><td>First-harmonic self-inductance (14 mH)</td></tr><tr><td>M<sub>0</sub></td><td>Continuous mutual-inductance (–15.5 mH)</td></tr><tr><td>M<sub>1</sub></td><td>First harmonic mutual-inductance (6.7 mH)</td></tr><tr><td>φ<sub>0</sub></td><td>Continuous permanent magnet flux (0.6814 Wb)</td></tr><tr><td>φ<sub>1</sub></td><td>First-harmonic permanent magnet flux (0.4147 Wb)</td></tr><tr><td>N<sub>r</sub></td><td>Number of teeth in the rotor (64)</td></tr><tr><td>N<sub>s</sub></td><td>Number of teeth in the stator (48)</td></tr><tr><td>N<sub>seq</sub></td><td>Number of equivalent teeth in the stator (68)</td></tr><tr><td>N<sub>ps</sub></td><td>Stator pole (12)</td></tr><tr><td>P</td><td>Number of magnet pairs (2)</td></tr><tr><td>τ<sub>r</sub>, τ<sub>s</sub></td><td>Rotor and stator tooth pitch</td></tr><tr><td>J<sub>g</sub></td><td>Rotor inertia (Nm s<sup>-1</sup>)</td></tr><tr><td>f</td><td>Viscous friction coefficient</td></tr><tr><td>P<sub>g,max</sub></td><td>Rated power (10 kW)</td></tr><tr><td>Ω<sub>g,max</sub></td><td>Rated rotor speed (50 rpm)</td></tr></tbody></table>

<table><caption>**Table A3**
Grid-side inverter parameters.</caption><thead><tr><th colspan="2">Parameters</th></tr></thead><tbody><tr><td>R<sub>f</sub></td><td>Filter resistance (0.01 Ω)</td></tr><tr><td>L<sub>f</sub></td><td>Filter inductance (3.25 mH)</td></tr><tr><td>V<sub>DC</sub></td><td>DC-link voltage (500 V)</td></tr><tr><td>C<sub>DC</sub></td><td>DC-link capacitance (12 mF)</td></tr></tbody></table>

<table><caption>**Table A4**
Simulation parameters.</caption><thead><tr><th>System</th><th>Parameter</th></tr></thead><tbody><tr><td>MPPT based TSR</td><td>Q<sub>s</sub> = 100<br>K<sub>i</sub> = 0.1</td></tr><tr><td>DC/DC boost converter</td><td>Q<sub>w</sub> = 10,000<br>K<sub>w</sub> = 10</td></tr><tr><td>Three phase 5L-ANPC converter</td><td>Q<sub>v</sub> = 4500<br>K<sub>v</sub> = 250<br>ε<sub>v</sub> = 600<br>a<sub>v</sub> = 0.3<br>K<sub>p</sub> = 17.5<br>T<sub>i</sub> = 0.35</td></tr></tbody></table>

## Data availability

No data was used for the research described in the article.

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