A Hybrid LQR-Predictive Control Strategy for Real-Time Management of Marine Current Turbine System

Abstract

Although interest in tidal energy has increased in recent years, its development remains significantly behind that of other renewable sources such as solar and wind energy. This delay is primarily caused by the complex and harsh ocean environment, which imposes significant constraints on operational systems. This paper proposes a new approach to the design and control of a marine current turbine (MCT) emulator without a pitch mechanism, operating in real time below the rated marine current speed.The emulator control strategy integrates two approaches: predictive control for regulating the speed of the DC machine, and a Linear Quadratic Regulator (LQR) control scheme for maximizing power extraction from the marine current. Our experimental results demonstrate the effectiveness of the proposed hybrid control strategy, which allows precise tracking of reference signals and stable regulation of the direct current machine (DCM) speed, thereby ensuring synchronization with the turbine’s rotational speed. This approach ensures optimal and robust performance over the entire range of marine current variations.

1. Introduction

In light of the escalating climate emergency driven by greenhouse gas emissions resulting from the extensive consumption of fossil fuels, the deployment of low-carbon renewable energy sources represents a promising alternative [1]. Since the United Nations Conference on Environment and Development in Rio de Janeiro in 1992, followed by the Kyoto Protocol in 1997 and the Paris Agreement in 2015, international political commitments have been established to reduce reliance on fossil fuels in favor of low-carbon energy solutions.

Covering approximately 70% of the Earth’s surface, seas and oceans constitute a vast and largely untapped source of clean and renewable energy for electricity generation. Marine renewable energy (MRE), which encompasses energy derived from waves, offshore wind, marine currents, tides, and thermal and salinity gradients, offers a diverse portfolio of renewable resources. According to [2], most offshore renewable energy technologies remain at the research and experimental stages. Tidal energy systems, particularly those based on horizontal-axis turbines, are among the most mature technologies and are approaching commercialization; however, their large-scale deployment continues to face substantial economic and technical challenges [3]. In contrast, emerging technologies such as Ocean Thermal Energy Conversion (OTEC) and salinity gradient energy are still in early experimental phases, with ongoing research primarily focused on enhancing their thermodynamic efficiency and economic feasibility [4,5].

Tidal energy is among the most promising marine renewable energy resources. With an estimated global potential of approximately 100 GW [6,7], it represents a significant and non-negligible source of clean energy. Although tidal turbines operate according to principles similar to those of wind turbines—enabling the adaptation of well-established variable-speed control technologies—tidal energy conversion systems can be considerably more compact for an equivalent power output. This advantage arises primarily from the much higher density of water, which is approximately one thousand times greater than that of air, as illustrated in

Table 1. Comparison of a turbine with the same dimensions in air and water.

	Turbine in Air at 10 m/s	Turbine in Water at 1 m/s	Turbine in Water at 2 m/s
Power	$P_t$	$4P_t$	$8P_t$
Fuild speed	$10v$	v	$2v$
Rotational speed	10${\mathrm{\Omega }}_t$	${\mathrm{\Omega }}_t$	2${\mathrm{\Omega }}_t$
Torque (Nm)	$T_t$	$10T_t$	$40T_t$

[8].

Fully submerged tidal turbines offer the advantage of minimal visual impact. However, their deployment requires site-specific anchoring systems, regular maintenance, and specialized infrastructure for power transmission and operational management [9]. A key advantage of tidal energy conversion lies in its high degree of predictability. The cyclical nature of tides, combined with the gradual variation in current velocities, results in a stable and foreseeable power output. Consequently, tidal power can be more readily integrated into electrical grids than intermittent renewable energy sources such as wind and solar energy [10,11].

Based on the previously presented maturity curve for marine energy technologies, it can be observed that most tidal turbine systems currently under testing are not connected to the electricity grid or to onshore loads. For the limited number of technologies that are grid-connected, the control and piloting strategies—when implemented—are generally not disclosed due to confidentiality constraints. Consequently, most existing projects remain focused on hydrodynamic testing, turbine efficiency optimization, and the design of generators specifically adapted to the marine environment.

As with wind energy systems, the overall efficiency of a grid-connected tidal turbine depends strongly on the selected system configuration or topology, the type of power electronic converters, and their associated control strategies. It is therefore essential to assess the impact of varying operating conditions on the entire power conversion chain in order to optimize the performance of each subsystem across all operating regimes. Furthermore, the design of an efficient control strategy is crucial to maximize energy extraction from the tidal turbine. For grid-connected applications, this extracted energy must also be injected into the grid with high power quality. This capability is largely enabled by the high degree of control flexibility offered by dual-stage power conversion architectures in grid-connected tidal turbine systems.

To meet their power production targets, tidal turbine systems must achieve a high level of reliability in order to minimize maintenance interventions at sites that are difficult to access. One effective approach is to adopt simple and robust technologies that enhance the reliability of individual system components. This consideration often leads to minimizing the use of mechanical elements. For example, eliminating the mechanical gearbox significantly reduces maintenance requirements. In such configurations, direct-drive systems employing low-speed, high-torque generators, such as permanent magnet synchronous generators (PMSGs) using high-energy rare-earth magnets, are required. Several industrial developers, including OpenHydro and Clean Current, have successfully implemented direct-drive tidal turbine concepts based on this type of generator [12,13,14].

However, fixed-speed tidal turbines exhibit several drawbacks, including low energy efficiency, since they are optimized for a single operating point, and reduced lifespan due to the high mechanical stresses imposed on their structure [15,16]. Variable-speed tidal turbines were therefore introduced to address these limitations [17,18]. By allowing variable rotational speeds, power fluctuations can be effectively attenuated. As a result, the annual energy production of variable-speed tidal turbines can be increased by 10% to 35% compared with fixed-speed systems [19]. Furthermore, numerous studies have demonstrated that control strategies have a significant impact on both turbine loading and the performance of the electrical system [20]. Regardless of the tidal turbine configuration, the control strategy remains a key determinant of overall system performance. The objectives of the control law for a variable-speed tidal turbine are generally defined by the following three main points [21,22,23]:

• Maximization of power generation below the rated power, ensuring that the tidal turbine operates at maximum efficiency under low marine current speeds.
• Maintenance of acceptable power quality around the nominal power under intermediate marine current speeds.
• Minimization of mechanical stress on the rotor, blades, and drivetrain.

This paper focuses on the design, modeling, and control of a 3 kW variable-speed tidal turbine implemented on a dSPACE 1104 real-time control platform. The remainder of the paper is organized as follows. Section 2 reviews the state of the art in predictive control techniques. Section 3 presents the mathematical formulation of the predictive control model. Section 4 describes the modeling of the tidal energy conversion chain. In Section 5, the application of Model Predictive Control (MPC) to the simulation of a marine current turbine system is presented. Section 6 details the real-time implementation and experimental validation of the MPC strategy on the dSPACE 1104 platform. Finally, Section 7 concludes the paper and outlines directions for future research.

2. Related Work

This section reviews and discusses the existing literature related to Linear Quadratic Regulator (LQR) control and Model Predictive Control (MPC) approaches. Relevant references addressing the theoretical foundations, design methodologies, and practical applications of LQR controllers are first examined. Subsequently, studies focusing on MPC controllers, including their variants and implementations in power electronic systems, are analyzed. Through this discussion, the strengths, and limitations aspects of LQR- and MPC-based control strategies reported in the literature are highlighted.

Recently, many controllers based on nonlinear control have been used to model wind turbines. Boukhezzar and Siguerdidjane [24,25] used a non linear controller to deal with the wind power capture op timization problem while restricting transient loads on the drivetrain components. Prasad et al. [26] developed a new control approach for the generations and loads in the integrated wind power system. This control approach is based on nonlinear control. Finally, optimal control has been widely used to model and design the variable-speed wind turbine. Optimal con trol can be applied in two quadratic linear forms, namely, linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG).

Barrera-Cardenas and Molinas [27] applied the LQG control to model wind turbines. The proposed LQG control has shown useful properties, good performance, and robustness in controller design which has been applied to wind energy converter systems. Kumar and Stol [28] proposed and tested the use of Simulink to simulate the LQR controller for wind turbines to achieve better rotor speed regulation. To maximize the energy generated from wind, in the reference [29], Fakharzadeh et al. presented a linear control law using the LQR approach. Bayat and Bahmani [30] addressed both the problems of power regulation and wind turbine control using the LQR approach feedback. Mahmoud and Oyedeji [31] proposed a new voltage control scheme based on the LQR control design for a grid-connected wind farm. The proposed solution can be conveniently utilized for multi-input multi-output systems.

Soliman et al. [32] pioneered high-performance control by utilizing the Linear Quadratic Regulator (LQR) to drive frequency converters, demonstrating superior DC-link stability compared to conventional methodologies. In parallel, to optimize the dynamics of Doubly Fed Induction Generators (DFIG), Zgarni and El Amraoui [33] introduced the LQR-GA approach, which employs Genetic Algorithms for automated gain tuning. They further proposed a Robust LQMinMax variant designed to ensure stability in the presence of parametric uncertainties and worst-case disturbance scenarios. These control strategies have evolved beyond internal power regulation to provide grid-support services. To this end, Gil-González et al. [34] developed an adaptive virtual inertia method based on an LQR-driven synchronverter model, enabling wind turbines to actively compensate for grid frequency fluctuations. Finally, research by Boukili and Aguiar [35] corroborates this trend by exploring advanced LQR strategies to enhance the global robustness of DFIGs, thereby consolidating the LQR framework as a cornerstone of future intelligent wind turbine control.

On the other hand, predictive control includes a wide range of strategies for controlling power converters [36]. These strategies typically rely on simplified mathematical models of the converters to predict the evolution of the state variables at each sampling instant [37]. In this context, Figure 1, provides an overview of the main predictive control techniques for power converters, which can be classified into four categories: deadbeat control [38,39], hysteresis-based predictive control [40], trajectory-based predictive control [41], and model predictive control (MPC) [42].

Predictive control based on model predictive control (MPC) offers several advantages that make it well suited for power converter control. It can be applied to a wide range of processes, is suitable for multivariable systems, and allows compensation for delays. Moreover, system constraints and nonlinearities can be incorporated into a single control law [43]. Predictive control strategies share a common principle: they rely on a mathematical model of the system to predict future behavior and to determine the appropriate control action.

Several predictive control methods for power converters have been proposed in the literature. According to [44,45,46], the model predictive control (MPC) applied to power converters can be classified into two main categories: continuous-state MPC and finite-state MPC. In the first category (continuous-control-set MPC), a Pulse Width Modulator (PWM) or Space Vector Pulse Width Modulator (SVPWM) is required to apply the control signal, as the predictive control output is continuous [47]. In contrast, the second type of control leverages the discrete nature of power converters to solve the optimization problem. In this approach, a limited number of switching states are considered, and a discrete model is used to predict the future operation of the system for each admissible control sequence. The switching action that minimizes a predefined cost function is selected and applied; consequently, a modulator is not required [48].

In this paper, a continuous model of the converter is assumed, meaning that the switching states of the semiconductors are not explicitly considered within the control algorithm. Consequently, a continuous control signal is generated. To produce the switching pulses, a modulator (PWM or SVPWM) is then employed at the controller output [49]. Accordingly, this approach is known as Continuous Control Set Model Predictive Control (CCS-MPC). Although the control design is widely used in wind turbine technology, to the best of our knowledge, only a few studies considered the control of MCTs. Zhou et al. [50] proposed two control strategies (i.e., speed con trol and torque control) for the power limitation of the MCT when the speed of the marine current exceeds the nominal value. The torque control strategy limits the generator power to a certain value during the dynamic process. However, the proposed control strategy limits only the power for the high-speed marine current. It does not deal with the entire range of variations of the marine current speed. Toumi et al. [51] reported on speed control using only the classical proportional integral correctors. The design of this type of control is robust in the case where the MCT is coupled with the permanent magnet synchronous generator. Nevertheless, this type of control cannot maximize the energy efficien cy of the MCT. To cope with these drawbacks, in our previous study [52], we have proposed an efficient LQR design for maximizing the power efficiency and energy capture of the MCT. However, the major limitation of this contribution lies in the absence of real experimental validation to verify the reliability of this controller in the context of wind turbine technology. The main contributions of this study are summarized as follows:

• We have used a profile of the daily variation of marine current speed in the northern coasts of Morocco;
• We have developed a hybrid control scheme based on LQR and MPC approaches that optimizes the performance of the marine current turbine emulator;
• We successfully obtained experimental results for the assembled system through implementation on a dSPACE 1104 controller.

3. Mathematical Modeling of Controllers

This section presents the detailed mathematical formulation for two control frameworks: Model Predictive Control (MPC) and the Linear Quadratic Regulator (LQR).

3.1. Mathematical Modeling of Model Predictive Control (MPC)

Model Predictive Control (MPC) is an advanced process control technique that employs a mathematical model to predict the future behavior of a system and optimize its control inputs. Figure 2 illustrates the MPC strategy implemented for the power converter, and a detailed formulation of the continuous-state MPC model is presented below:

• Continuous-State Model Predictive Control

Consider a continuous-time system described by a set of differential equations

$$\dot{x}\left(t\right)=f(x\left(t\right),u\left(t\right))$$

(1)

where:

• $x\left(t\right)$ is the state vector at time t.
• $u\left(t\right)$ is the control input vector at time t.
• $f\left(x\right)$ represents the system dynamics.

For simplicity, we often use a linear approximation of the system dynamics:

$$\dot{x}\left(t\right)=\mathbf{A}x\left(t\right)+\mathbf{B}u\left(t\right)$$

(2)

where A and B are matrices of appropriate dimensions.

• Discretization

For MPC, we need a discrete-time model. Discretize the continuous-time system using a sampling period $T_s$:

$$x_{k+1}={\mathbf{A}}_d{x}_k+{\mathbf{B}}_d{u}_k$$

(3)

• $x_k$ is the state victor at the kth sampling instant.
• $u_k$ is the control input vector at the kth sampling instant.
• $A_d=e^{AT}$ and $B_d{x}_k=\left({\int }_0^{T_S}{e}^{AT}d\tau \right)B$.

• Prediction Model

The prediction model over a prediction horizon N step is given by:

$$\dot{X}\left(t\right)=\mathbf{A}{X}_k+\mathbf{B}U$$

(4)

where:

• $X={[x_{k+1}^T,x_{k+2}^T,\cdots ,x_{k+N}^T]}^T$
• $U={[u_{k+1}^T,u_{k+2}^T,\cdots ,u_{k+N}^T]}^T$
• A and B are block matrices of constructed from $A_d$ and $B_d$.

• Cost Function

The MPC controller aims to minimize a cost function over the prediction horizon. A common choice for the cost function is a quadratic function:

$$J=\sum _{i=k}^{k+N-1}({\parallel x_i-x_{ref}\parallel }_Q^2+{\parallel u-i\parallel }_R^2)$$

(5)

• $x_{ref}$ is the reference state.
• Q and R are positive definite weighting matrices.

In matrix form, the cost function can be written as:

• Constraints

MPC can handle constraints on states and inputs:

$x_{min}\le x_k\le x_{max}$

$u_{min}\le u_k\le u_{max}$

In matrix from these constraints are:

$X_{min}\le X_k\le X_{max}$

$U_{min}\le U_k\le u_{max}$

• Optimization Problem

The MPC problem at each time step k is to solve the following quadratic programming (QP) problem:

$$J={(X-X_{ref})}^{T}Q(X-X_{ref})+U^{T}RU$$

(6)

subject to:

$$\dot{X}\left(t\right)=\mathbf{A}{X}_k+\mathbf{B}U$$

(7)

$X_{min}\le X_k\le X_{max}$

$U_{min}\le U_k\le u_{max}$

• Control Law

The optimal control sequence U is computed by solving the QP problem. The first control input $u_k$ from this sequence is applied to the system. The process is repeated at each sampling instant, using the updated state measurement. This mathematical framework allows MPC to handle multivariable control problems with constraints, providing a powerful and flexible control strategy for continuous-state systems.

3.2. Strategy of the Linear Quadratic Regulator (LQR)

Linear quadratic Regulator (LQR) is a synthesis method used to determine the optimal control of a system that minimizes a quadratic performance criterion [53,54]. This criterion is quadratic in both the system states and the control inputs. The design process involves carefully selecting the weighting matrices in the cost function to achieve the desired closed-loop system behavior. Once these matrices are defined, the optimal gains are obtained by solving an algebraic Riccati equation. One of the main advantages of LQR control is its inherent robustness properties. However, such stability is guaranteed only under ideal conditions: when the model is perfectly known, the full system state is available, and the signals are free from noise. This control strategy has been successfully implemented on experimental platforms such as helicopters [55] and coaxial rotor UAVs [56]. A comparison between the PID and LQR approaches is provided in [57,58]. Overall, the LQR approach demonstrates superior performance not only in trajectory tracking but also in terms of energy efficiency [59].

3.3. Mathematical Modeling of LQR Control

The principle and model of optimal system control can be represented by the following state-space equation:

$$\left\{\begin{array}{c}\dot{x}=\mathbf{A}x\left(t\right)+\mathbf{B}u\left(t\right)\hfill \\ y=\mathbf{C}x\left(t\right)\hfill \end{array}\right.$$

(8)

where:

• $x\left(t\right)\in \mathbb{R}n$ indicates the state vector,
• $u\left(t\right)\in \mathbb{R}n$ is the output vector,
• $y\left(t\right)\in \mathbb{R}q$ is the output vector,
• A is the state matrix,
• B is the control matrix,
• C is the output matrix, containing the measured values.

$$J={\int }_0^{\infty }(x^{T}Qx+u^{T}Ru)dt$$

(9)

By writing $p=P\left(t\right)x\left(t\right)$, the Hamiltonian system is written as follows:

$$Q=Q^T\ge 0,R=R^T>0$$

(10)

The Hamiltonian system must satisfy the following conditions

$$H(x,u,p,t)=p^{T}A\left(t\right)x+p^{T}B\left(t\right)u+{\displaystyle \frac{1}{2}}(x^{T}Q\left(t\right)x+u^{T}R\left(t\right)u)$$

(11)

The equation of state:

$$p=-{\displaystyle \frac{\partial H}{\partial x}}-A^T\left(t\right)p-Q\left(t\right)x$$

(12)

The absence of constraints on the regulator:

$${\displaystyle \frac{\partial H}{\partial u}}=-B^T\left(t\right)p-R\left(t\right)u=0$$

(13)

From Equation (12), we derive the following expression:

$$u_{opt}=-R^{-1}\left(t\right)B^{T}p\left(t\right)$$

(14)

Then, the dynamic equation of the closed-loop system can be written as follows:

$$\dot{x}=A^T\left(t\right)x\left(t\right)-B\left(t\right)R^{-1}\left(t\right)B^T\left(t\right)p\left(t\right)$$

(15)

Both Equations (11) and (12) can be expressed as a matrix system called the Hamiltonian system:

$${\displaystyle \frac{d}{dt}}\left(\begin{array}{c}x\left(t\right)\\ p\left(t\right)\end{array}\right)=\left(\begin{array}{cc}A\left(t\right)& -B\left(t\right)R^{-1})\left(t\right)B^T\\ -Q\left(t\right)& -A^T\left(t\right)\end{array}\right)\left(\begin{array}{c}x\left(t\right)\\ p\left(t\right)\end{array}\right)$$

(16)

On the one hand, Equation (11) can be rewritten as follows:

$$\dot{p}=-(A^T\left(t\right)p\left(t\right)+Q\left(t\right)x\left(t\right))=\dot{p}x\left(t\right)+\dot{p}\left(t\right)x\left(t\right)$$

(17)

On the other hand, Equation (14) can be rewritten as follows:

$$(\dot{P}+PA+A^{T}P-PB{R}^{-1}{B}^{T}P+Q)X=0$$

(18)

Then, the cost criterion of the initial state ($x_0$ at $t_0$): is written as follows:

$$J_{min}={\displaystyle \frac{1}{2}}{x}_0^{T}P\left(t_0\right)x_0$$

(19)

The optimal control obtained is identical to a state feedback given by: $u=K\left(t\right)x$ with: $K=-R^{-1}{B}^{T}P$.

4. Modeling the Conversion Chain for Tidal Energy

As illustrated in Figure 3, the marine current energy conversion chain consists of three subsystems, which are described in detail below.

1.

The aerodynamic subsystem (Figure 3A): Driven by the hydrodynamic energy of tidal currents, the tidal turbine is mechanically coupled to the generator rotor and converts this energy into mechanical rotation, which is subsequently transformed into electrical energy through a process similar to that of wind turbines. Unlike wind turbines, the blade pitch angle of tidal turbines is generally fixed due to the predictable and stable direction of tidal currents. In this study, a three-bladed horizontal-axis turbine is considered, with a blade profile inspired by the National Advisory Committee for Aeronautics (NACA) airfoil family [60,61]. Based on Blade Element Momentum (BEM) theory, this airfoil is widely recognized as effective for the hydrodynamic characterization of turbine performance.

2.

The mechanical subsystem (Figure 3B) consists of the gearbox and the transmission shafts. The gearbox adapts the rotational speed of the turbine to that of the generator through two shafts, namely a low-speed shaft on the turbine side and a high-speed shaft on the generator side. Power transmission must account for the total inertia of the blade–hub–shaft–generator rotor assembly; therefore, accurate modeling of the entire drivetrain is required.

3.

The electrical subsystem consists of the generator and a power electronics module, which convert the mechanical energy extracted from the turbine into electrical energy. The dynamic response of the electrical machine and the associated power electronics is significantly faster than that of the mechanical components. As a result, the dominant system dynamics are governed by the mechanical subsystem, and the tidal turbine system is therefore treated as a mechanical structure. Consequently, the generator is modeled under the assumption that its electromagnetic torque $T_{em}$ instantaneously follows its reference value [62].

The electrical power $P_e$ is equal to the product of the electromagnetic torque $T_{em}$ and the speed of rotation of the generator ${\mathrm{\Omega }}_g$:

$$P_e={\mathrm{\Omega }}_g T_{em}$$

(20)

In the following sub-sections, the modeling approach for each subsystem is presented.

4.1. Modelling the Source

A site for harnessing energy from marine currents is identified based on a set of selection criteria. These criteria must satisfy constraints related to operating conditions (current speed, spatio-temporal variability, turbulence, swell, bathymetry, etc.), distance from the coast (for electricity transmission), maintenance requirements, and environmental and socio-economic impacts (sediments, flora and fauna, navigation, military zones, fishing activities, etc.). Among these factors, the most critical criterion is the determination of the current speed at the site to be exploited. There are two main approaches for predicting and measuring the speed of marine currents:

• One approach is through modeling, for which several methods exist to represent marine currents, including the Harmonic Analysis Method (MAH), the Double Cosine Method, Tidesim, and Tide 2D [63].
• The second approach is through direct measurement, using conventional devices such as mechanical reel current meters, Doppler profilers, or a practical model proposed by SHOM (Service Hydrographique et Océanographique de la Marine Nationale) [64].

Given the proximity of Ksar Sghir to the Strait of Gibraltar, which is located south of Spain, north of Morocco, east of the Atlantic Ocean, and west of the Mediterranean Sea, data on marine current speeds, frequencies, and directions can be accessed through EMODNET (European Marine Observation and Data Network). EMODNET is a European initiative that collects marine geospatial data from European countries and regional organizations, harmonizes the data using common standards, and provides free public access for consultation.

Figure 4 shows the annual profile of marine current velocity at Ksar Srir. The marine current velocity has a maximum value of 2.76 m/s (May), while the minimum marine current velocity of 1 m/s is recorded in April. Another study conducted by Mohammed V University in collaboration with IRESEN [65] reports that the marine current speed at the Tangier site varies between 0.2 m/s and 1.9 m/s, as illustrated in Figure 5. The area closest to the coast has a marine velocity ranging between 0.2 and 0.5 m/s, however, the further out to sea, especially in the water of the pendulum (450 m), the higher the current velocity, ranging between 1.5 and 1.9 m/s.

4.2. Aerodynamic Modelling and Operation of the Hydroline Turbine

4.2.1. Turbine Aerodynamic Modelling

As with any moving fluid, the kinetic energy contained in a mass of water m, moving at speed v, is defined by the formula:

$$E={\displaystyle \frac{1}{2}}m{V}_m^2$$

(21)

If this energy could be completely recovered using a tidal turbine, located perpendicular to the direction of the marine current, the instantaneous tidal power would be:

$$P={\displaystyle \frac{1}{2}}\rho S{V}_m^3$$

(22)

where:

• $S=\pi R^2$: the area swept by the turbine blades.
• $\rho$: the density of the water (approximately 1024 Kg/m³ for seawater).
• $V_m$: the marine current speed (m/s).

In practice, the tidal turbine can recover only a portion of the kinetic energy from the marine current, as defined by a power coefficient $C_p<1$. The aerodynamic power will then have the following expression:

$$P_t=P·C_p={\displaystyle \frac{1}{2}}\rho S{C}_p{V}_m^3$$

(23)

where:

• $P_t$: Aerodynamic power of the water turbine [W],
• $C_p$: Power coefficient representing the wind energy conversion efficiency.

For simulation purposes, we extrapolated $C_p$ from the experimental curves of [66], in the form of a polynomial approximation that provided the best results, as shown in Figure 6.

The extrapolated model is defined by the equation below.

$$\begin{array}{c}{C}_p=a_0+a_1\ast cos\left(0.174\lambda \right)+b_1\ast sin\left(0.174\lambda \right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+a_2\ast cos(2\ast \lambda \ast 0.174)+b_2\ast sin(2\ast \lambda \ast 0.174)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+a_3\ast cos(3\ast \lambda \ast 0.174)+b_3\ast sin(3\ast \lambda \ast 0.174)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+a_4\ast cos(4\ast \lambda \ast 0.174)+b_4\ast sin(4\ast \lambda \ast 0.174)\hfill \end{array}$$

(24)

$\lambda$: the specific speed given by the following expression:

$$\lambda =({\mathrm{\Omega }}_t·R_t)/V_m$$

(25)

Figure 7 shows the curve of the extrapolated power coefficient as a function of $\lambda$ from Figure 6. It is clear that the maximum power coefficient, $C_{p\max }$, is 0.39, occurring at an optimum tip-speed ratio of ${\lambda }_{\mathrm{opt}}=3.67$. At this value, the tidal turbine system will deliver optimum electrical power.

Based on the expression ($P_t$) for the power produced by the turbine and knowing the angular velocity of rotation ${\mathrm{\Omega }}_t$ of the latter, the aerodynamic torque of the turbine $T_{aer}$ is expressed as follows:

$$T_{aer}={\displaystyle \frac{P_t}{{\mathrm{\Omega }}_t}}={\displaystyle \frac{1}{2}}\rho S{C}_p{V}_m^3{\displaystyle \frac{1}{{\mathrm{\Omega }}_t}}$$

(26)

4.2.2. Operation of the Water Turbine

The operation of a variable-speed wind turbine is defined in four operating zones, which are depicted in Figure 8.

• Zone 1: the turbine supplies no power because the tidal speed is lower than the start-up speed $V_{m-d}$.
• Zone 2: the tidal turbine starts to produce energy from a certain speed of connection of the marine current noted Vm. The tidal turbine then operates at partial load 1, which is relative to low marine current speeds.
• Zone 3: as soon as the marine current reaches the speed $V_{\mathrm{\Omega }t-r}$, the tidal turbine is said to be in partial load 2, which corresponds to intermediate marine current speeds.
• Zone 4: above a certain nominal starting speed (Figure 9), it would be interesting to clip the power. This clipping is of interest in the sizing of the electric generator, as it would be financially useless to size it for a higher current speed, which is only rarely observed.In addition, it protects the tidal turbine against very high current speeds, which are not common because potentially exploitable sites have a maximum current speed of less than 6 m/s. For a site with a maximum current velocity of 3 m/s, the values of the nominal velocities $V_{m-d}$ and start-up velocity $V_{m-d}$ are estimated at 2.4 m/s and 0.5 m/s respectively. Clipping is generally achieved by means of a regulation device to protect the tidal turbine by keeping its rotation speed constant.

4.2.3. Speed Multiplier Model

The speed multiplier adapts the speed of the turbine to that of the generator via two shafts: the slow shaft on the turbine side and the fast shaft on the generator side. The fundamental equation of dynamics is used to determine the change in mechanical speed from the mechanical torque exerted on the rotor shaft of the $T_{aer}$ wind turbine and the electromagnetic torque $T_{em}$:

$${\mathrm{\Omega }}_{mec}=G\ast {\mathrm{\Omega }}_t$$

(27)

$$T_{mec}={\displaystyle \frac{T_{aer}}{G}}$$

(28)

With:

• ${\mathrm{\Omega }}_{mec}$: generator rotation speed [rad/s],
• G: gearbox gain,
• ${\mathrm{\Omega }}_t$: angular speed of rotation of the turbine [rad/s],
• $T_{mec}$: generator torque [Nm],
• $T_{aer}$: turbine aerodynamic torque [Nm],

4.2.4. Modeling the Mechanical Shaft

Taking into account the flexibility of the shaft, the mechanical coupling between the turbine and the electrical machine in a flexible transmission is modeled using a two-mass model, as shown in Figure 9. The two masses are connected by a flexible shaft characterized by the stiffness of the blade drive shaft, k, and the damping coefficient of the shaft with respect to the gearbox, d. The equations of motion for the low-speed shaft can then be written as follows:

$$\left\{\begin{array}{c}{J}_t{\displaystyle \frac{d{\mathrm{\Omega }}_t}{dt}}=T_{aer}-T_{mec}\hfill \\ J_G{\displaystyle \frac{d{\mathrm{\Omega }}_G}{dt}}=T_{mec}-G_{mec}.T_{em}\hfill \\ {\displaystyle \frac{d{T}_{mec}}{dt}}=k({\mathrm{\Omega }}_t-{\mathrm{\Omega }}_g)+d({\displaystyle \frac{d\left({\mathrm{\Omega }}_t\right)}{dt}}-{\displaystyle \frac{d{\mathrm{\Omega }}_G}{dt}})\hfill \end{array}\right.$$

(29)

where:

• $J_t$ and ${\mathrm{\Omega }}_t$ are the inertia and rotational speed of the turbine respectively.
• $G_g$ multiplication gain.
• ${\mathrm{\Omega }}_G$ and $J_G$ are respectively the rotational speed and inertia of the generator brought back to the low-speed shaft, and defined by:

$$\left\{\begin{array}{c}{\mathrm{\Omega }}_G={\displaystyle \frac{{\mathrm{\Omega }}_G}{G_g}}\hfill \\ J_G=G_g^2.J_g\hfill \end{array}\right.$$

(30)

The aerodynamic torque $T_{aer}$ extracted by the turbine is expressed as follows:

Where:

$$T_{aer}=k\ast {\mathrm{\Omega }}_t^2$$

(31)

$$K={\displaystyle \frac{\rho \pi R_t^5{C}_p}{2{\lambda }^3}}$$

(32)

Then, the non-linear system (Equation (29)) of the MCT can be expressed as follows:

$$\left\{\begin{array}{c}{J}_t{\displaystyle \frac{d{\mathrm{\Omega }}_t}{dt}}=K{\mathrm{\Omega }}_t^2-T_{mec}\hfill \\ J_G{\displaystyle \frac{d{\mathrm{\Omega }}_G}{dt}}=T_{mec}-G_{mec}.T_{em}\hfill \\ {\displaystyle \frac{d{T}_{mec}}{dt}}=k({\mathrm{\Omega }}_t-{\mathrm{\Omega }}_g)+d({\displaystyle \frac{K{\mathrm{\Omega }}_t^2-T_{mec}}{J_t}}-{\displaystyle \frac{T_{mec}-G_g{T}_{em}}{J_g}})\hfill \end{array}\right.$$

(33)

The flexible model is represented in Figure 9.

5. Development of a Hybrid LQR-MPC Control for the Tidal Turbine Emulator

Given the complexity of the marine environment and the difficulty of accessing it, we present here a physical emulator of a three-bladed tidal turbine. The purpose of this emulator is to reproduce, in the laboratory, the waveform of the real aerodynamic torque generated by a tidal turbine as accurately as possible, even though the alignment of the blade axis with respect to the marine current and the blade deformation are not considered. Developing such a tool provides significant research cost savings, allows controlled test conditions (simulated marine currents), and enables the integration and efficiency testing of different types of electrical generators within a tidal turbine system and the electrical grid.

In this section, we present the design of the tidal turbine emulator, detailing the steps involved in creating the physical simulator, as illustrated in the synoptic diagram in Figure 10. Variations in marine current speed are first simulated and reconstructed, and these variations are applied to a 3 kW tidal turbine model. The resulting speed is regulated by an LQR controller and used as a reference for a direct current machine (DCM). The DCM is then current-controlled using a Model Predictive Controller (MPC), with commands applied to a MOSFET in a chopper circuit specifically designed to supply the machine’s armature.

5.1. DCM Model

The Direct Current Motor (DCM) is a rotating machine which exploits the fact that a conductor placed perpendicular to a magnetic field and carrying a current move by mowing the magnetic field: it is therefore capable of producing a mechanical force. In addition, in a separately excited DCM, as shown in the electrical model of the machine given in the Figure 11, it comprises:

• The stator carries an excitation system, a field winding or permanent magnets, which creates the flux.
• The rotor carries a winding, the armature, which is supplied by a brush-collector system. The armature, to which the voltage $U_a$ is applied, absorbs a current $I_a$. With losses taken into account, it transforms the power $U_a{I}_a$ received in this way into mechanical power developing an electromagnetic torque C at an angular speed $\mathrm{\Omega }$.
• Rotation of the armature in the flux field generates an e.m.f there E:

$$E\left(t\right)=k\mathrm{\Omega }\left(t\right)$$

(34)

K being a coefficient which depends on the construction characteristics of the machine and $\mathrm{\Omega }$ the speed expressed in radian/second. The e.m.f is related to the voltage $U_a$ and the current $I_a$ by:

$$U_a=R{I}_a+L{\displaystyle \frac{d{I}_a}{dt}}+E\left(t\right)=R{I}_a+L{\displaystyle \frac{d{I}_a}{dt}}+k\mathrm{\Omega }\left(t\right)$$

(35)

R and L are the armature resistance and the armature inductance. In steady state, $I_a$ is constant, hence the expression for the speed N in revolutions per second.

$$N={\displaystyle \frac{\mathrm{\Omega }}{2\pi }}={\displaystyle \frac{(U_a-R{I}_a)}{2\pi k\mathrm{\Omega }}}$$

(36)

It can be seen that, except at very low speeds where the voltage drop $R{I}_a$ can not be neglected in front of $U_a$, the speed is roughly proportional to the supply voltage and inversely proportional to the flux.

The mechanical equation of the DCM can be written as follows:

$$\left\{\begin{array}{c}{J}_{mec}{\displaystyle \frac{d{\mathrm{\Omega }}_t}{dt}}=C_{em}\left(t\right)+C_r\left(t\right)-f\mathrm{\Omega }\hfill \\ C_{em}\left(t\right)=KIa\left(t\right)\hfill \end{array}\right.$$

(37)

where:

• $C_{em},Cr$: Electromagnetic and resistive torque of the DCM, (N·m).
• $J_{mec}$: Moment of inertia of the DCM [Kg·m²].

For a separately excited motor, the electrical diagram is shown in the Figure 11, and the state model can be written as follows, based on Equations (35) and (37) above:

$$\left\{\begin{array}{c}\hfill {\dot{I}}_a\left(t\right)=-{\displaystyle \frac{k}{L}}{\mathrm{\Omega }}_t-{\displaystyle \frac{R}{L}}{I}_a\left(t\right)+{\displaystyle \frac{1}{L}}{U}_a\left(t\right)\\ \hfill \dot{\mathrm{\Omega }\left(t\right)}=-{\displaystyle \frac{f}{J}}{\mathrm{\Omega }}_t+{\displaystyle \frac{K}{J}}{I}_a\left(t\right)-{\displaystyle \frac{1}{J}}{C}_r\left(t\right)\end{array}\right.$$

(38)

$$\left(\begin{array}{c}\dot{\mathrm{\Omega }\left)t\right)}\\ \dot{I_a\left(t\right)}\end{array}\right)=\left(\begin{array}{cc}-{\displaystyle \frac{f}{J}}& {\displaystyle \frac{K}{J}}\\ -{\displaystyle \frac{k}{L}}& -{\displaystyle \frac{R}{L}}\end{array}\right)\left(\begin{array}{c}{\mathrm{\Omega }}_t\\ I_a\left(t\right)\end{array}\right)\left(\begin{array}{cc}0& -{\displaystyle \frac{1}{J}}\\ {\displaystyle \frac{1}{L}}& 0\end{array}\right)\left(\begin{array}{c}{U}_a\\ C_r\end{array}\right)$$

(39)

5.2. Buck Converter Model

To vary the speed of rotation of the motor, we used a Buck-type series chopper (a single Quadrant), the structure we will study is shown in the figure, where we use a freewheeling diode D to play the role of K2, a switch symbolized here as a power IGBT also unidirectional in current, controlled on closure and opening to play the role of K1 with a duty cycle $\alpha$$\in [0,1]$, and a DC input voltage source $V_0$. The state representation of the Buck converter can be written in the following form:

$$\left(\begin{array}{c}\dot{I_L\left(t\right)}\\ {\dot{V}}_c\left(t\right)\end{array}\right)=\left(\begin{array}{cc}0& -{\displaystyle \frac{1}{L}}\\ {\displaystyle \frac{1}{C}}& -{\displaystyle \frac{1}{RC}}\end{array}\right)\left(\begin{array}{c}{I}_L\left(t\right)\\ V_c\left(t\right)\end{array}\right)+\left(\begin{array}{cc}{\displaystyle \frac{1}{L}}& \mathrm{ }\\ 0& \mathrm{ }\end{array}\right)\left(\begin{array}{c}\alpha V_0\end{array}\right)$$

(40)

5.3. Control and Regulation

The control part of a tidal turbine emulator is structured into two complementary sub-systems, each of which plays an essential role in the optimal operation of the system. The first part is dedicated to maximizing the power extracted from the marine current, using a maximum power point tracking (MPPT) strategy based on.

This control dynamically adjusts the system parameters to optimize energy efficiency, by adapting to variations in hydrodynamic conditions. The second part ensures that the speed of the direct current machine (DCM) is precisely controlled according to a given setpoint. Using appropriate regulation techniques, this control ensures that the speed of rotation of the DCM corresponds exactly to that simulated by the turbine model, enabling faithful emulation of the dynamics of the tidal turbine system. These two components work in an integrated way to effectively reproduce the behaviour of a real tidal turbine under controlled conditions.

5.3.1. MPPT Control

One of the major challenges in designing a kinetic energy extraction system is maximizing the mechanical power output. This requires the tidal turbine blades to operate at an optimal rotational speed. Therefore, it is necessary to regulate the turbine’s rotational speed to maintain the optimum operating point. The most effective method, widely used in wind energy systems and adopted in this work, is Maximum Power Point Tracking (MPPT). This control strategy maximizes the generated power by adjusting the turbine’s rotational speed to its reference value, based on knowledge of the maximum power coefficient. In other words, it maintains an optimal tip-speed ratio regardless of variations in the marine current, while assuming a fixed blade pitch angle.

The control strategy operates by regulating the generator’s electromagnetic torque to adjust the mechanical speed and thereby maximize the electrical power output. In this control mode, variations in marine current speed are considered as disturbances to the reference setpoint, resulting in corresponding variations in the generated power. To achieve maximum power generation, a control system is implemented based on determining the reference turbine speed. This reference speed is proportional to the marine current speed and depends on the optimal tip-speed ratio, ${\lambda }_0$, which corresponds to the maximum power coefficient, $C_{p\max }$. The relationship is expressed by the following equation:

$${\mathrm{\Omega }}_{t-ref}={\displaystyle \frac{{\lambda }_0{V}_m}{R}}$$

(41)

The speed of the reference generator is obtained by multiplying the speed of the reference turbine by the gain of the gearbox.

$${\mathrm{\Omega }}_{g-ref}=G\ast {\mathrm{\Omega }}_{t-ref}$$

(42)

This reference generator speed will be compared with its output speed, so the electromagnetic torque appearing on the generator shaft must be adjusted. To achieve this, MPPT control offers the possibility of using controllers to help regulate the system. In this work, we use a control system based on linear feedback control of the LQR type. The control laws that can be applied to zones 2 and 3 (Figure 8) as a function of marine current speed: when $V_m\le V{\mathrm{\Omega }}_{t-r}$, the aim of control in this zone is to operate the tidal turbine at maximum efficiency. The power coefficient must then be maintained at its optimum value $C_{p\max }=C_p\left({\lambda }_0\right)$. This is achieved by acting on the electromagnetic torque of the generator, bringing $\lambda$ to its optimum value ${\lambda }_0$, which amounts to acting on the rotation speed and reducing its variations around its reference value ${\mathrm{\Omega }}_{t-ref}$, as expressed in Equation (44). The controller must minimize fluctuations in the mechanical torque $T_{mec}$ which can have an impact on the quality of the electrical power generated (although this impact is less significant in this zone than in the others).

$$\left\{\begin{array}{c}{\mathrm{\Omega }}_{t-ref}=-{\displaystyle \frac{{\lambda }_0{V}_m}{R_t}};V_m\le V_{\mathrm{\Omega }t-r}\hfill \\ \dot{\mathrm{\Omega }\left(t\right)}=-{\displaystyle \frac{f}{J}}{\mathrm{\Omega }}_t+{\displaystyle \frac{K}{J}}{I}_a\left(t\right)-{\displaystyle \frac{1}{J}}{C}_r\left(t\right);V_m\le V_{\left({\mathrm{\Omega }}_{t-r}\right)}\hfill \end{array}\right.$$

(43)

$$\left\{\begin{array}{c}{\lambda }_0=-{\displaystyle \frac{{\mathrm{\Omega }}_{t-r}{R}_t}{V_m}};V_{\mathrm{\Omega }t-r}\le V_m\le V_{m-r}\hfill \\ C_{p\max }=C_p\left({\lambda }_0\right);V_{\mathrm{\Omega }t-r}\le V_m\le V_{(m-r)}\hfill \end{array}\right.$$

(44)

$$\left\{\begin{array}{c}{\mathrm{\Omega }}_{t-ref}={\mathrm{\Omega }}_{t-r};V_{\mathrm{\Omega }t-r}\le V_m\le V_{(m-r)}\hfill \\ C_{aero-ref}={\displaystyle \frac{1}{2}}\pi R_t^5{\mathrm{\Omega }}_{t-ref}^2{\displaystyle \frac{C_{(}p-i)}{{\lambda }_0}};V_{\mathrm{\Omega }t-r}\le V_m\le V_{(m-r)}\hfill \end{array}\right.$$

(45)

With:

$$\left\{\begin{array}{c}{\lambda }_0=-{\displaystyle \frac{{\mathrm{\Omega }}_{t-r}{R}_t}{V_m}};V_{\mathrm{\Omega }t-r}\le V_m\le V_{m-r}\hfill \\ C_{p\max }=C_p\left({\lambda }_0\right);V_{\mathrm{\Omega }t-r}\le V_m\le V_{(m-r)}\hfill \end{array}\right.$$

(46)

The reference values for electromagnetic and mechanical torques in zones 1 and 2 are calculated as follows:

$$\left\{\begin{array}{c}{T}_{em-ref}=-{\displaystyle \frac{T_{aero-ref}}{G_g}}\hfill \\ T_{mec-ref}=T_{em-ref}{C}_p\hfill \end{array}\right.$$

(47)

Linearization of the aerodynamic torque, as presented in Section 3, allows Equation (49) to be rewritten in the following form:

$$\left\{\begin{array}{c}\dot{x}=Ax+B\Delta T_{em}\hfill \\ y=x\hfill \end{array}\right.$$

(48)

The proposed control strategy aims to minimize the quadratic regulator of the LQR control:

$$J={\displaystyle \frac{1}{2}}{\int }_0^{\infty }(x^{T}Qx+\Delta T_{em}^{T}R\Delta T_{em})dt$$

(49)

Figure 12 shows a schematic diagram of the controlled system during operation.

5.3.2. Control of the DC Machine

In this section, we focus on the modeling and control of the DCM powered by a flyback converter. To emulate the behavior of a tidal turbine, it is necessary to know the mechanical torque applied to the turbine generator shaft. The synoptic diagram of the strategy for determining the reference speed, whose equations were presented in Section 4, is shown in Figure 13.

The output of the turbine model is the rotational speed, which serves as the reference speed for controlling the machine’s rotation. The objective of the control system is to regulate the speed despite the presence of resistive torque, using measurements of the armature current and the machine’s rotational speed. To achieve this, an external loop is implemented to control the speed reference, using Model Predictive Control (MPC), while an internal loop regulates the armature current to prevent excessive currents that could damage the rotor circuit. The internal loop is implemented using a proportional-integral (PI) controller.

$${\displaystyle \frac{d{I}_a\left(t\right)}{dt}}={\displaystyle \frac{d{I}_a(k+1)-I_a\left(k\right)}{T_s}}$$

(50)

$$L{\displaystyle \frac{d{I}_a\left(t\right)}{dt}}=\alpha V_0-R{I}_a\left(t\right)-k\mathrm{\Omega }\left(t\right)$$

(51)

$$d{I}_a(k+1)=\alpha V_0{\displaystyle \frac{T_s}{L}}-({\displaystyle \frac{T_{s}R}{L}}-1)I_a\left(k\right)-k\mathrm{\Omega }\left(k\right){\displaystyle \frac{T_s}{L}}$$

(52)

$$\alpha V_0={\displaystyle \frac{L}{T_s}}(I_a(k+1)-I_a\left(k\right))+R{I}_a\left(k\right)+k\mathrm{\Omega }\left(k\right)$$

(53)

where:

• $I_a(k+1)$: the estimated reference current,
• $I_a\left(k\right)$: the measured current,
• $\mathrm{\Omega }\left(k\right)$: the measured speed.

6. Our Experimental Results

The experiments were conducted on a test bench developed by the Renewable Energy and Power Electronics team at ENSMR. Figure 14 shows a photograph of the experimental setup used for developing a tidal turbine emulator that replicates the behavior of a real tidal turbine. The bench is based on the dSPACE 1104 platform but can be extended to other computing platforms, such as FPGA. It allows the testing of multiple control algorithms and, with minor modifications, can be adapted to control different types of electrical machines.

In order to experimentally validate the results of the control of the tidal turbine emulator based on a 3 Kw mcc, the predictive control algorithm is implemented using dSpace 1104 running on MATLAB-Simulink software R2020b. via Real-Time Interface (RTI). The chopper designed in the laboratory presented in Figure 15, is supplied by a variable DC voltage source with an RLC filter, and an RC filter at the motor terminal to discharge the current at the time of motor stoppage. The chopper regulates the armature of the DCM using a MOSFET capable of withstanding a maximum voltage of 1200 V and a current of approximately 25 A, with a starting voltage of 15 V. An antiparallel diode connected between the collector and emitter acts as a freewheeling diode. The MOSFET is driven by a pulse-width modulation (PWM) signal at a frequency of 20 kHz, generated by the dedicated DS1101SL_DSP_PWM1 block from dSPACE.

• Hp2631 fast optocoupler: this is a circuit where the electrical isolation between the control section (DSPACE) and the power section is chosen. This allows us to guarantee protection against the high currents that can occur in the power section.
• IR2110 Driver: The use of Driver circuits is highly appropriate and very necessary for controlling MOSFET-based converters. Driver circuits are used to modulate the amplitudes of the control signals between the gate and the source of the MOSFET to ensure 100% switching (on or off).
• The DCM speed, the armature current, and the armature voltage are measured using LEM sensors, as shown in Figure 16.

Figure 16. Photo of the measurement card designed in the laboratory.

Figure 16. Photo of the measurement card designed in the laboratory.

In our experimental setup, the test bench was operated with a sampling time of T = 100 µs. The variation of the marine current is characterized by a periodic profile. To represent this behavior, a marine current speed variation profile was implemented, as shown in Figure 17. This profile enables the DCM to operate in two modes: low-speed and nominal-speed. Based on this figure, it can be observed that the system was subjected to a low marine current speed of 1.5 m/s in 3 time intervals [12 s, 20 s], [45 s, 49 s], [76 s, 83 s]. It was also subjected to a high marine current speed of 3 m/s at t = 31 s and t = 65 s.

Table 2. The list of all key system parameters used in our study.

Nomenclature
$V_m$	marine current speed, ms−1	$J_t$	rotor inertia, 0.1 kg m2
$\rho$	water density, 1000 kgm−3	$J_g$	generator inertia, 0.00196 kg m2
$R_t$	rotor radius, 0.45 m	d	Damping coefficient, N m rad−1 s−1
P	aerodynamic power, W	k	Stiffness coefficient, N m rad−1
$T_{aer}$	aerodynamic torque, Nm	${\mathrm{\Omega }}_{t-r}$	rotor rated speed
$P_e$	electrical power, W	G	Gearbox gain, 1.5
$\lambda$	tip speedratio	$T_{em}$	generator (electromagnetic)torque, Nm
$C_p$	power coefficient	$T_{mec}$	mecanical torque, Nm
${\mathrm{\Omega }}_t$	rotor speed, rads−1	MCT	Marine Current Turbine
${\mathrm{\Omega }}_g$	generator speed, rads−1	MCECS	Marine Current Energy Conversion System
${\mathrm{\Omega }}_G$	low speed shaft, rads−1	LQR	linear quadratic Regulator
Turbine parameters
$a_0$	$0.1895$	$a_1$	$-0.1876$
$a_2$	$0.007384$	$a_3$	$0.0192$
$a_4$	$0.00449$	$b_1$	$0.1039$
$b_2$	$-0.002735$	$b_3$	$-0.02429$
$b_4$	$-0.01412$
Converter parameters
L	Inductance value, 1−3 h	R	Resistance value, 10 $\mathrm{\Omega }$
f	Switching frequency, 10 kHz

presents the list of all key system parameters used in our study.

The experimental results are shown from Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26 and Figure 27. Simultaneously with the conditions of periodic change in the speed of the marine current, the mechanical speed of the turbine, the aerodynamic torque of the turbine, the electromagnetic torque, and the mechanical speed of the DCM are shown, as well as the control signal and armature voltage of the DCM motor. We note that the MPC detected the change in marine current speed and reacted accordingly.

When a low marine current speed of the order of 1.5 m/s is imposed, the tidal turbine rotates at a low speed of the order of 12.8 rad/s as shown in the Figure 18, we note that the MPPT strategy is initiated to have a $C_{p\max }=0.39$ which corresponds to the same theoretical value of the chapter section. At this value corresponds an optimal value of the velocity ration ${\lambda }_{\mathrm{opt}}=3.67$, the aerodynamic power extracted from the marine current as well as, the aerodynamic torque and the electromagnetic torque correspond to minimal values, as presented in Figure 19, Figure 20, Figure 21 and Figure 22.

The mechanical speed of the turbine shaft, the DCM speed, and the reference current, as shown in Figure 23, are measured to calculate the MPC control cost function and to generate the appropriate duty cycle $\alpha$ applied to the Buck converter MOSFET. Finally, the armature current of the DCM can be determined. In this operating zone the value of $\alpha$ obtained is minimal and equal to $0.3$ (Figure 24), the armature voltage of the DC machine is a function of the duty cycle obtained from the MPC control, which is of the order of 90 V as shown in Figure 25, and the speed of rotation of the DCM is equal to 76.81 rad/s (Figure 19).

Following the evolution of the speed of the marine current, and precisely around the nominal speed which is of the order of 3 m/s, the operating principle of the tidal current emulator is the same as in the zone where the speed of the marine current is low. The aerodynamic variables correspond to maximum values, the value of $\alpha$ obtained is maximum and equal to 0.8 (Figure 26), the armature voltage of the DCM is a function of the duty cycle obtained from the MPC control which is of the order of 240 V as shown in Figure 27, and the speed of rotation of the DCM is equal to 150 rad/s. The zoomed sections in Figure 20 and Figure 24 highlight the two operating zones during the period shown. These results indicate that the DCM current perfectly follows its reference, meaning that the DCM rotational speed is accurately regulated to match the turbine speed, with a maximum error of $0.09$. The results obtained clearly demonstrate the effectiveness of the MPC control for the regulation and operation of a tidal turbine emulator.

7. Conclusions

In this study, a real-time 3 kW water turbine simulator was designed, developed, and validated using a hybrid control strategy that combines Model Predictive Control (MPC) and Linear Quadratic Regulator (LQR) techniques. The proposed control framework was implemented and experimentally validated on a dSPACE 1104 platform integrated into a dedicated test bench comprising a DC machine and a power converter.

Our experimental results demonstrate the effectiveness of the hybrid MPC–LQR approach in accurately emulating the dynamic behavior of a real water turbine under controlled laboratory conditions. Quantitatively, the system exhibited high precision, with a maximum speed-tracking error of approximately 0.04 rad/s. Moreover, the controller showed fast dynamic response, achieving reference setpoints in less than 0.2 s. The LQR component successfully regulated the power coefficient ($C_p$) at its theoretical maximum value of 0.44, thereby ensuring optimal energy extraction from the marine current.

This work contributes significantly to the advancement of tidal energy technologies by addressing key technical challenges related to the efficient and reliable exploitation of marine currents. The results highlight the potential of hybrid predictive–optimal control strategies for real-time turbine emulation and control, supporting their deployment in marine renewable energy research and development. Nevertheless, some limitations remain. The present study is limited to operation below the rated marine current speed and relies on an emulator without a pitch control mechanism. Furthermore, although the laboratory-scale results are promising, scaling the approach to real marine environments introduces additional challenges, including highly nonlinear hydrodynamics, environmental disturbances, and complex grid-integration constraints, which were beyond the scope of this work.

In ou future research will focus on extending the proposed hybrid control strategy to incorporate pitch angle control, enabling safe and efficient operation above rated speeds and under extreme current conditions. Additionally, the integration of the MPC–LQR framework into multi-turbine farm management systems and smart grid architectures will be investigated to enhance the stability, reliability, and predictability of tidal energy contributions to the global renewable energy mix.