RESEARCH ARTICLE

Optimal Feedback Control of Nonlinear Variable-Speed MarineCurrent Turbine Using a Two-Mass Model

Rajae Gaamouche1 & Abdelbari Redouane2 & Imad El harraki2 & Bouchra Belhorma3 & Abdennebi El Hasnaoui2

Received: 9 May 2019 /Accepted: 15 November 2019# The Author(s) 2020

AbstractThis paper presents a contribution related to the control of nonlinear variable-speed marine current turbine (MCT)without pitch operating below the rated marine current speed. Given that the operation of the MCT can be divided intoseveral operating zones on the basis of the marine current speed, the system control objectives are different for eachzone. To deal with this issue, we develop a new control approach based on a linear quadratic regulator with variablegenerator torque. Our proposed approach enables the optimization of the rotational speed of the turbine, which maxi-mizes the power extracted by the MCT and minimizes the transient loads on the drivetrain. The novelty of our study isthe use of a real profile of marine current speed from the northern coasts of Morocco. The simulation results obtainedusing MATLAB Simulink indicate the effectiveness and robustness of the proposed control approach on the electricaland mechanical parameters with the variations of marine current speed.

Keywords Marinecurrent turbine .Two-massmodel .Tipspeedratio .Linearization .Optimalcontrol .Linearquadratic regulator(LQR)

1 Introduction

The majority of renewable energy sources, such as wind andsolar energies, have a fluctuating character, which alters thequality of the power injected into the electrical grid (Kumar

et al. 2016; Anvari et al. 2016). In fact, the integration ofthese renewable sources into the electrical grid can lead tothe major need for the adjustment of voltage and frequencyto overcome eventual electrical disturbances (e.g., voltagedroops and short circuits) (Antonova et al. 2012). By con-trast, marine current energy is more stable than wind andsolar energies and represents a promising solution to dealwith the aforementioned issues (Melikoglu 2018; Chenet al. 2018). Indeed, marine current energy mainly originatesfrom the tides, which are a direct result of the combinationof the Earth’s rotation and the gravitational forces exerted ona water body by the Moon and Sun (Hodur 1997). Thus,predicting the particular location of the main component ofthese currents for long time periods is possible (Hodur 1997;Rourke et al. 2010).

In addition, according to Thiébaut and Sentchev(2015), marine current resources are abundant in such ascale that their use for energy production can satisfy theworld’s energy demand. Marine current turbines (MCTs)are evolving rapidly because more reliable oceanographicdata are available and all of the advanced techniques de-veloped, tested, and applied for wind turbines can beadapted to MCTs even if the dynamics of tides are totallydifferent from those of wind (Fox et al. 2018).

Article Highlights• A hydrodynamic model based on both a profile of the daily variation ofmarine current speed and the experimental results of Mason-Jones isproposed.

• A linearization technique to regulate the system around a specific operating point is applied.

• A new optimal control based on the LQR approach that achieves theoptimization of the operation of the turbine is developed.

• The effectiveness and robustness of the proposed control approach areinvestigated.

* Rajae Gaamoucheraja.escom@gmail.com

1 Electromechanics Department,Mohammadia School of Engineering,10000 Rabat, Morocco

2 Electromechanics Department, National Superior School of Mines,10000 Rabat, Morocco

3 National Centre for Energy Sciences and Nuclear Techniques,10000 Rabat, Morocco

https://doi.org/10.1007/s11804-020-00134-6

/ Published online: 30 June 2020

Journal of Marine Science and Application (2020) 19:83–95

In recent years, numerous studies of the hydrodynamicaspects (Mycek et al. 2014; Frost et al. 2015; Blackmoreet al. 2016) and the design of specific generators for themarine environment (Benelghali et al. 2012; Seck et al.2018; Chen et al. 2019) have been conducted. Meanwhile,other studies focus on the electronic part of power (Phamet al. 2017; Einrí et al. 2019; Omkar et al. 2019; Qianet al. 2019).

To improve the energy conversion performance ofMCTs and ensure that tidal energy resources are cost-effective, rotor speed regulation, which can be classi-fied into two types, i.e., fixed and variable speeds, isinvestigated in this study. Jena and Rajendran (2015)and Kettache (Khettache 2019) conducted a compara-tive study between fixed-speed and variable-speedwind generators. The variable-speed wind turbine ismore advantageous than the fixed-speed wind turbinebecause it increases the energy efficiency, improves thequality of the kinetic energy produced, and stabilizesthe fluctuations of the voltage and power of the elec-trical grid.

In the context of wind turbine technologies, manycommand algorithms have been designed and developedin the past decades to optimize the operation of turbinesto maximize the energy efficiency. These controllers canbe classified into four categories, namely, optimal con-trol, nonlinear control, predictive control, and fuzzylogic.

Liu et al. (2016) proposed a fuzzy logic controller forthe wind turbine system to achieve the objective of max-imum power extraction based on a two-mass model. Bassiet al. (Bassi and Mobarak 2017) examined the variable-speed wind turbine in full and partial load segmentsthrough predictive control. This approach considers themaximum power point, with the goal of maximizingthe power generated by the wind turbine.

In the litrature, many controllers based on nonlinearcontrol have been used to model wind turbines.Boukhezzar and Siguerdidjane (2010, 2011) used a non-linear controller to deal with the wind power capture op-timization problem while restricting transient loads on thedrivetrain components. Prasad et al. (2019) developed anew control approach for the generations and loads in theintegrated wind power system. This control approach isbased on nonlinear control.

Finally, optimal control has been widely used to modeland design the variable-speed wind turbine. Optimal con-trol can be applied in two quadratic linear forms, namely,linear quadratic regulator (LQR) and linear quadraticGaussian (LQG).

Barrera-Cardenas and Molinas (2012) applied theLQG control to model wind turbines. The proposedLQG control has shown useful properties, good

performance, and robustness in controller design whichhas been applied to wind energy converter systems.Kumar and Stol (2010) proposed and tested the use ofSimulink to simulate the LQR controller for wind tur-bines to achieve better rotor speed regulation. To maxi-mize the energy generated from wind, Fakharzadeh et al.2013) presented a linear control law using the LQR ap-proach. Bayat and Bahmani (2017) addressed both theproblems of power regulation and wind turbine controlusing the LQR approach feedback. Mahmoud andOyedeji (2016) proposed a new voltage control schemebased on the LQR control design for a grid-connectedwind farm. The proposed solution can be convenientlyutilized for multi-input multi-output systems.

Although the control design is widely used in windturbine technology, to the best of our knowledge, only afew studies considered the control of MCTs. Zhou et al.(2013) proposed two control strategies (i.e., speed con-trol and torque control) for the power limitation of theMCT when the speed of the marine current exceeds thenominal value. The torque control strategy limits thegenerator power to a certain value during the dynamicprocess. However, the proposed control strategy limitsonly the power for the high-speed marine current. Itdoes not deal with the entire range of variations of themarine current speed. Toumi et al. (2017) reported onspeed control using only the classical proportional–integral correctors. The design of this type of control isrobust in the case where the MCT is coupled with thepermanent magnet synchronous generator. Nevertheless,this type of control cannot maximize the energy efficien-cy of the MCT. To cope with these drawbacks, in thisstudy, we propose an efficient LQR design for maximiz-ing the power efficiency and energy capture of the MCT.Our decision to use this controller is motivated by theresults obtained in the context of wind turbinetechnology.

The main contributions of this study are summarized asfollows:

& We have used a profile of the daily variation of marinecurrent speed in the northern coasts of Morocco;

& We have developed a hydrodynamic model based on theexperimental results of Mason-Jones et al.;

& We have modeled flexible transmission using the two-mass model;

& We have applied a linearization technique to regulate thesystem around a specific operating point;

& We have developed a new optimal control based on theLQR approach that achieves the optimization of the oper-ation of the turbine;

& We have obtained the simulation results of the assembledsystem using MATLAB Simulink.

84 Journal of Marine Science and Application

The remainder of this paper is organized as follows: inSection 2, we describe the MCT system and the aerody-namic and two-mass models. In Section 3, we present thelinearization model for the determination of a state space.In Section 4, we develop a new optimal control based onthe LQR approach to find a compromise between theoptimization of the energy generated by the turbine andthe reduction of the load at the level of two-mass me-chanical shaft. In Section 5, we provide an overview ofthe system control objective implementation of the MCTsystem in the low-speed and high-speed areas of the ma-rine current. In Section 6, we validate our optimal controlperformance using MATLAB Simulink. Finally, inSection 7, we conclude this paper and indicate the direc-tion of future work.

2 Marine Current Turbine System Modeling

The marine current energy conversion chain, which is a com-bination of three subsystems, is modeled in Figure 1:

1) The aerodynamic subsystem (Figure 1(a)) consists of sev-eral blades (mostly three blades) and a hub. The blades ofthe MCT extract the kinetic energy from the marine cur-rent and convert it into mechanical energy.

2) The mechanical subsystem (Figure 1(b)) is composed ofthe gearbox (also called the drivetrain). This subsystemtransforms the rotation rate of the shaft from low rotation-al speed at the rotor side into high rotational speed at thegenerator side.

3) The electromechanical subsystem (Figure 1(c)) is composedof the generator and a power electronics module, whichconverts the mechanical energy at the turbine into electricalenergy. This subsystem will not be modeled in this work. Infact, the dynamics of electrical machines and power elec-tronics systems are faster than that of the other parts of theMCT. Consequently, the system will be considered a me-chanical structure. Therefore, the dynamics of the generatorcan be disregarded. The Tem value of the generator is

considered equal to the reference value Tem-ref. Pe is equalto the product of Tem and Ωg:

Pe ¼ Ωg � T em ð1Þ

We describe the model of each subsystem in the followingsubsections:

2.1 Model of the Marine Current and AerodynamicSubsystem

2.1.1 Model of the Resource

Before installing the MCT, it is necessary to model theresources of the considered site. For this, the speed ofthe marine current can be predicted using two methods(Gaamouche et al. 2018):

1) Direct measurement: This method can be performedusing conventional devices (e.g., helical current metersand acoustic doppler current profilers), a model proposedby the SHOM (French Naval Hydrographic andOceanographic Service), or marine ships.

2) Modelization: Several techniques, such as harmonic anal-ysis method, double cosine method, TideSim, and Tide2D, can be used to model marine currents.

The lack of data of marine current speed in Morocco leadsto difficulties in utilizing this resource. Figure 2 presents thevariation of marine current speed for a certain day in April2018 in the northern coasts of the Moroccan kingdom nearSpanish borders. The data measured by the European MarineObservation and Data Network were utilized in this study.This figure shows that we can launch such project in thisregion.

2.1.2 Aerodynamic Modeling

The aerodynamic power recovered by the rotor of the MCTcan be expressed as follows:

P ¼ 1

2ρSV3

m ð2Þ

where S=π ·R2 is the surface of turbine rotation and ρ is ap-proximately equal to 1024 kg/m3 for seawater.

Despite the tremendous technological advances, a tidal tur-bine can extract only a fraction of this power, as shown in thefollowing equation:

Pt ¼ P � Cp ¼ 1

2ρSCpV3

m ð3Þ

Figure 1 MCT model. (a) Aerodynamic system. (b) Drive train. (c)Generator

85R. Gaamouche et al.: Optimal Feedback Control of Nonlinear Variable-Speed Marine Current Turbine Using a Two-Mass Model

In this study, we considered that the MCT is not pitched.Moreover, Cp can be approximated by an equation dependingonly on λ, which is defined as the ratio between Ωt and Vm:

λ ¼ Ωt � Rt

Vmð4Þ

Mason-Jones et al. (2012) provided a relevant set of exper-imental data to validate the theoretical and numerical methodsfor MCT. In this study, we focused on the Cp curve accordingto λ, which is obtained from these experimental data when theturbine rotates at a uniform speed of 3.08 m/s over a diameterof 20 m. We notice that the data presented in Table 1 areextracted from this curve. On the basis of this table, we pro-posed the interpolation function of Cp to model the turbine.This function can be expressed as follows:

Cp ¼ a0 þ a1cos λ� 0:174ð Þ þ b1sin λ� 0:174ð Þþ a2cos 2� λ� 0:174ð Þ þ b2sin 2λ� 0:174ð Þþ a3cos 3� λ� 0:174ð Þ þ b3sin 3� λ� 0:174ð Þþ a4cos 4� λ� 0:174ð Þ þ b4sin 4� λ� 0:174ð Þ ð5Þ

Cp can be considered a strong nonlinear function of λ.Afterward, we implemented our interpolation function for

modeling the turbine using the MATLAB Simulink environ-ment. To validate our proposed model, in this work, we se-lected the turbine “Guinard Energies P66 3.5 kW.” The ob-tained curve of Cp is shown in Figure 3. We notice that themaximum Cp value is 0.39, which corresponds to a tip speedratio of 3.67 to a marine current speed of 3 m/s. This value isconsidered the optimal tip speed ratio (λopt) to achieve maxi-mum power point tracking under rated marine current speeds.

2.2 Model of the Mechanical Subsystem

The mechanical subsystem (Figure 1(b)) is composed of thegearbox (drivetrain). The gearbox adjusts the speed of the

turbine to that of the generator through two shafts, namely,the slow shaft on the turbine side and the fast shaft on thegenerator side. In the literature, the two types of mechanicaltransmission models are rigid transmission and flexible trans-mission. The flexible transmission model attracted our atten-tion because the mechanical coupling between the turbine andthe electric machine is modeled using the two-mass model, asshown in Figure 1(b). The two masses are connected to aflexible shaft characterized by the elasticity coefficient of thedrive shaft of the blades k and the coefficient of friction of theshaft relative to the gearbox d. For wind turbines, this two-mass model is sufficient to properly represent the dynamicbehavior of the turbine (Fakharzadeh et al. 2013). For thispurpose, the MCT drive system will be modeled using thetwo-mass model. Then, we write the following equations forthe low-speed drivetrain:

J tdΩt

dt¼ T aer−Tmec

JGdΩG

dt¼ Tmec−Gg � T em

dTmec

dt¼ k Ωt−Ωg

� �þ ddΩt

dt−dΩG

dt

� �

8>>>>><>>>>>:

ð6Þ

Table 1 Experimentaldata of Cp and λ Cp λ Cp λ

0.035 0.65 0.39 3.88

0.135 1.3 0.38 4.2

0.2 1.62 0.364 4.51

0.275 1.93 0.31 5.15

0.355 2.59 0.27 5.51

0.374 2.91 0.16 6.15

0.4 3.6 0.08 6.5

Figure 2 Marine current speed profile Figure 3 Cp curve of the MCT

86 Journal of Marine Science and Application

where Jt and Ωt are the inertia and rotational speed of theturbine, respectively; Gg is the gain of gearbox; and Ωg andJg are the rotational speed and inertia of the generator broughtback to the low-speed shaft, respectively, which are defined asfollows:

ΩG ¼ Ωg

Gg

JG ¼ G2g � J g

8<: ð7Þ

Taer is the aerodynamic torque extracted by the turbine,which is expressed as follows:

T aer ¼ k � Ω2t ð8Þ

where K is derived as follows:

K ¼ ρπR5t Cp

2λ3 ð9Þ

Then, the nonlinear system (Eq. (6)) of the MCT can beexpressed as follows:

J tdΩt

dt¼ K �Ω2

t −Tmec

JGdΩG

dt¼ Tmec−Gg � T em

dTmec

dt¼ k Ωt−Ωg

� �þ dkΩ2

t −Tmec

J t−Tmec þ GgT em

JG

� �

8>>>>><>>>>>:

ð10Þ

3 Linearized and State Representationof the MCT System

Given that determining the optimal control for nonlinear sys-tems is difficult, deriving a solution for an equivalentlinear control system is necessary. For the MCT system,nonlinearity is detected in the aerodynamic torque Taer.For this, we need to adopt a linearization approach forTaer (Eq. (8)) according to Ωt.

The operating point corresponding to the marine currentspeed Vm is variable. Consequently, linearizing the MCT sys-tem around different operating points corresponding to themarine current speed Vm is possible. Thus, we can obtain thederivative of the system from an operating point given themarine current speed Vm:

ΔCaer ¼ δCaer

δΩt

����op

ΔΩt ¼ β0ΔΩt ð11Þ

where

β0 ¼ρπR3

t V2m

2Ωt0

δCp λð Þδλ

−Cp0

λ0

� �ð12Þ

The symbol Δ designates the variation according to thechosen operating point (OP). New state variables correspond-ing to the variations around the operating point are defined asfollows:

ΔΩt ¼ Ωt−Ωt0

ΔΩG ¼ ΩG−ΩG0

ΔTmec ¼ Tmec−Tmec0

On the basis of the preceding expressions, the linearizationof the flexible model (Eq. (8)) around an operating point leadsto the following equations:

J tΔΩt ¼ β0ΔΩt−ΔTmec

JGΔΩG ¼ ΔTmec−GgΔT em

ΔTmec ¼ K þ dβ0

J t

� �ΔΩt−KΔΩG−d

1

J tþ 1

JG

� �ΔTmec

8><>: ð13Þ

The linearization of the nonlinear system around an oper-ating point, expressed in Eq. (11), enabled us to express thestate space model as follows:

ΔΩt

ΔΩG

ΔCmec

24

35 ¼

β0i

J t0

1

J t

0 01

Jg−BV

k þ dβ0

J t−k −d

1

J tþ 1

JG

� �

26666664

37777775

ΔΩt

ΔΩG

ΔCmec

24

35

þ

0Gg

JGdGg

JG

26664

37775ΔCem

ð14Þ

In a more compact form, the system expressed in Eq. (14)can be rewritten as follows:

x ¼ Axþ Bu ð15Þ

where

x ¼ΔΩt

ΔΩG

ΔCmec

24

35;A ¼

β0i

J t0

1

J t

0 01

J g−BV

k þ dβ0

J t−k −d

1

J tþ 1

JG

� �

26666664

37777775;

B ¼

0Gg

JGdGg

JG

26664

37775; and u ¼ ΔCem

87R. Gaamouche et al.: Optimal Feedback Control of Nonlinear Variable-Speed Marine Current Turbine Using a Two-Mass Model

In Figure 4, we can distinguish four operating areas of avariable-speed MCT, in which the index a1 of the matrix Ai,j

depends on the variation of the marine current:

1) Zones 1 and 4: The MCT does not provide any powerbecause the marine current speed is lower than the startspeed and higher than the rated speed. Thus, the MCTwill be stopped.

2) Zone 2: For the low marine current speed (Vm≤VΩt− r):

A11 ¼ −ρπR4t VmCpmax

2λ20 J t

3) Zone 3: When the marine current speed is betweenVΩt− r ≤Vm≤Vm− r:

A11 ¼ ρπR3t V

2mCpmax

2J t Ωt−r

∂CP

∂λ−Cp

λ

� �

4 LQR Optimal Control

The objective of optimal control has two main orientations.The first orientation aims to minimize energy, whereas thesecond orientation seeks to reduce the convergence time ofthe system. The general objective is to find the optimal controlthat minimizes the criterion that varies according to the orien-tation adopted. A large variety of optimal control techniqueshave been applied to the MCT in a permanent attempt toimprove its function and benefit as much as possible fromthe energy that it can produce. Optimal control is the statefeedback of a nonlinear time-invariant system with LQR thathas evolved significantly in recent years. The principle of theLQR command is shown in Figure 5 (Khargonekar et al.1990; Anderson et al. 2007; Athans and Falb 2013; Levine2018).

In general, the system model can be expressed as the fol-lowing state space equation:

x ¼ Ax tð Þ þ Bu tð Þy ¼ Cx tð Þ

ð16Þ

where x(t) € ℝn denotes the state vector, u(t) € ℝn denotes thecontrol vector, y(t) € ℝq denotes the output vector, A is thematrix of evolution or state, B is the control or input matrix,and C is the output matrix or measured value. Moreover, thepair (A, B) is assumed to denote that the system iscontrollable. The process of the feedback regulatorneeds to follow the state space equation. The simplifiedproblem of the LQR is to find the matrix of the corrector K,which minimizes the function of the cost (or the criterion ofperformance), as follows:

J ¼ 1

2∫∞0 xTQxþ uTRu� �

dt ð17Þ

where the weighting matrices Q and R satisfy the followingexpression:

Q ¼ QT≥0;R ¼ RT > 0 ð18Þ

The Hamiltonian system is written as follows:

H x; u; p; tð Þ ¼ pTA tð Þxþ pTB tð Þu

þ 1

2xTQ tð Þxþ uTR tð Þu� � ð19Þ

The Hamiltonian system must satisfy the followingconditions:

& The state equation:

p ¼ −∂H∂x

¼ −AT tð Þp−Q tð Þx ð20Þ

& The absence of a constraint on the command:

∂H∂u

¼ −BT tð Þp−R tð Þu ¼ 0 ð21ÞFigure 4 Control regions for a turbine controller

Figure 5 Principle of the LQR command

88 Journal of Marine Science and Application

From Eq. (21), we deduce the following expression:

uopt ¼ −R−1 tð ÞBTp tð Þ ð22Þ

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

x ¼ AT tð Þx tð Þ−B tð ÞR−1 tð ÞBT tð Þp tð Þ ð23Þ

Equations (20) and (21) can be written in the form of amatrix system, which is also called the Hamiltonian system:

d

dtx tð Þp tð Þ

� �¼ A tð Þ −B tð ÞR−1 tð ÞBT

−Q tð Þ −AT tð Þ� �

x tð Þp tð Þ

� �ð24Þ

With p(t) =P(t)x(t), Eq. (20) can be rewritten as follows:

p ¼ − AT tð ÞP tð Þ þ Q tð Þ� �x tð Þ ¼ Px tð Þ þ P tð Þx tð Þ ð25Þ

Equation (23) can be rewritten as follows:

P þ PAþ ATP−PBR−1BTP þ Q �

X ¼ 0 ð26Þ

P is the positive solution (symmetric) to the algebraicRiccati equation (Eq. (26)):

P þ PAþ ATP−PBR−1BTP þ Q ¼ 0 ð27Þ

Then, we derive the minimum criterion of the initial state(x0 at t0):

Jmin ¼ 1

2xT0P t0ð Þx0 ð28Þ

Notably, the optimal control obtained can be written asstate feedback u= −K(t) x, where:

K ¼ −R−1BTP ð29Þ

5 Control Objectives

In this part, we developed the control laws that can be applied tozones 2 and 3 (Figure 4) on the basis of the marine currentspeed:

& When Vm≤VΩt−r , the main control objectives are as fol-lows: The MCT starts to generate energy at a certain ma-rine current speed. As result, the operation of the MCT isrelated to the low marine current speed. In this area, thecontrol objective is to operate the turbine at maximumefficiency. To ensure that the power coefficient is main-tained at the optimal value Cpmax =Cp(λ0), λ must reachits optimum value λ0 (Figure 3). This means that the

control in this area acts on the electromagnetic torque ofthe generator, which in turn acts on the rotational speedand reduces its variations from the reference value Ωt−ref,as expressed in Eq. (30).

The controller must minimize the fluctuations of the me-chanical torque Tmec, which affects the quality of the electri-cal power generated (although this effect is less important inthis area than in other areas). The reference equation formaximizing energy conversion is expressed as follows:

ΩT−ref ¼ λ0Vm

Rt; for Vm≤VΩt−r

Taero−ref ¼ 1

2ρπR5

t Ω2T−ref

Cpmax

λ0; for Vm≤VΩt−r

8><>: ð30Þ

& When VΩt−r ≤ Vm ≤ Vm− r, the main control objectives areas follows: When the marine current reaches the value ofVΩt−r , the MCT corresponds to the intermediate marinecurrent speeds. In this area, the rotational speed of theturbine reaches its nominal value. The main control objec-tive is to reduce the variations of Ωt from the nominalvalue Ωt−r while acting on the electromagnetic torque.The reference equation for this area is expressed as fol-lows:

ΩT−ref ¼ Ωt−r; VΩt−r ≤Vm≤Vm−r

Caero−ref ¼ 1

2ρπR5

tΩ2T−ref

Cp−I

λ0; VΩt−r ≤Vm≤Vm−r

8<:

ð31Þwith:

λ0 ¼ Ωt−rRt

Vm;

Cpmax ¼ Cp λ0ð Þ;

8<:

VΩt−r ≤Vm≤Vm−rVΩt−r ≤Vm≤Vm−r

ð32Þ

The reference values for the electromagnetic and mechan-ical torques in zones 1 and 2 are derived as follows:

T em−ref ¼ T aero−ref

Gg

Tmec−ref ¼ T em−ref

8<: ð33Þ

The linearization of the aerodynamic torque, as presentedin Section 3, makes it possible to rewrite Eq. (14) in thefollowing form:

x ¼ Axþ BΔT em

y ¼ x

ð34Þ

The proposed control strategy aims to minimize the LQR:

J ¼ 1

2∫∞0 xTQxþ ΔTT

emRΔT em

� �dt ð35Þ

89R. Gaamouche et al.: Optimal Feedback Control of Nonlinear Variable-Speed Marine Current Turbine Using a Two-Mass Model

Figure 6 presents the block diagram of the system con-trolled during operation.

6 Simulation Results

To validate our proposed control strategy based on the LQRapproach, we conducted simulations using the MATLABSimulink software. The simulation results of the MCT aredepicted in Figure 7.

The considered system is a variable-speed, fixed-pitch MCT with a nominal power rating of 3.5 kW.

Table 2 lists the values of the parameters used in thesimulation.

The aim of the proposed control law is to minimize thequadratic criterion previously expressed in Eq. (35). Theweight coefficients used in the quadratic criterion J are de-rived as follows:

Q ¼1 0 01 1000 01 0 1

24

35

R ¼ 100½ �

8>><>>:

ð36Þ

The resolution of Riccatti’s equation (i.e., Eq. (34)) for theinvestigated system defines the control gain K, which is pre-viously expressed in Eq. (29). The result is the optimal statefeedback, which is expressed in Eq. (37) or (38) in the follow-ing form:

Δu ¼ −K tð ÞΔx ð37Þ

Figure 6 Block diagram of the system controlled during operation

Figure 7 Simulation of the MCT system with LQR regulator using MATLAB Simulink

Table 2 Simulationparameters Parameter Value

ρ (kg/m3) 1000

Rt (m) 0.45

Jt (kg m2) 0.1

Jg (kg m2) 0.00196

G 1.5

d 0.02

k 20

90 Journal of Marine Science and Application

(a) Vm

(b) Rotor speed

(c) Genreator speed

(d) Tem

(e) Tmec

(f) Pe

Fig. 8 MCT speed variation and state and output variables in the partial load regime (zone 2)

91R. Gaamouche et al.: Optimal Feedback Control of Nonlinear Variable-Speed Marine Current Turbine Using a Two-Mass Model

(a) Vm

(b) Rotor speed

(c) Genreator speed

(d) Tem

(e) Tmec

(f) Pe

Figure 9 MCT speed variation and state and output variables in the partial load regime (zone 3)

92 Journal of Marine Science and Application

or

Δu ¼ u−uref ; u ¼ T em

Δx ¼ x−xref ; x ¼Ωt

ΩG

Tmec

24

35

8><>: ð38Þ

From this equation, we able to derive the following expres-sion:

T em ¼ −K � Δxþ T em−ref ð39Þ

To illustrate the effectiveness of our proposed controlsystem for a variable-speed MCT, we conducted two sim-ulations of zones 2 and 3. The results are presented inFigure 4.

6.1 Partial Load Operation (Zone 2)

In this simulation, we considered the low-speed MCT thatcorresponds to Vm≤VΩt−r . Figure 8 shows the different param-eters that we have optimized to maximize the energy collectedby the turbine. Figure 8 (a) presents the variation of the marinecurrent velocity profile used to determine in which regioncontrol will be applied. Moreover, Figure 8 (d) shows thatthe proposed LQR control follows the reference values nearlyperfectly, with a static error of 0. Moreover, the gain providedby this control is equal toK= [0.2079− 02896− 0.0353] whenVm = 1 m/s.

Figure 8 (b), (c), and (e) present the variations of threeparameters, namely, rotor speed, generator speed, and mechan-ical torque, respectively. The LQR controller ensures that thesystem maintains the power coefficient at the optimal valueCpmax. As a result, the parametres can be optimized in zone 2.Figure 8 (f) shows the variation of the electrical power accord-ing to generator speed and electromagnetic torque. Figure 8 (f)clearly shows that LQR maximizes the electrical power.

6.2 Partial Load Operation (Zone 3)

In this simulation, the rotational speed of the MCT reaches itsrated value that corresponds to VΩt−r ≤ Vm ≤ Vm− r. Figure 9shows the different parameters that we have optimized to main-tain the turbine at its optimum speed. In addition, these param-eters enable the generator to turn at a speed that does not fluc-tuate rapidly (minimize the transient load). Figure 9 (a) presentsthe variation of the marine current velocity profile used to de-termine in which region control will be applied. In this case, amarine current profile with the average value ofVm = 3m/s wasimposed.

Figure 9 (d) shows that our proposed LQR control can beapplied to zone 3, with a good performance in terms of elec-tromagnetic torque. Furthermore, from this figure, we noticethat the static error is always equal to 0 and the gain provided

by this control is equal to K= [0.1851−0.2899−0.0337] at Vm= 3 m/s.

Figure 9 (b), (c), and (e) show the variations of three pa-rameters, namely, rotor speed, generator speed, and mechan-ical torque, respectively. We notice that the LQR controllercan ensure that the systemmaintains the rotational speed at thenominal value. Consequently, the simulation results indicatethat these three parameters have been optimized with the useof LQR control. In addition, Figure 9 (f) presents the variationof the output power of the MCT. We notice that the value ofthis parameter decreases from the nominal value to 3.5 kWbecause we used the LQR control.

On the basis of the simulation results obtained for zones 2and 3, we confirm that our proposed LQR control can be setand used to act only on the electromagnetic torque to automat-ically regulate the different parameters of the MCT system,such as rotor speed, generator speed, and mechanical torque.

7 Conclusions

In this study, we proposed the LQRoptimal controller approachcoupled with the reference model of the outputs for thevariable-speed MCT. Our solution maximizes power capturewhere the tidal speed is greater or less than the nominal speed.

In the first step, we introduced the differential equations formodeling the state space representation, which connects theoutputs and controls to ensure better precision, and the refer-ence model of the outputs whose main role is to impose thedesired dynamic on the response of the system’s outputs, in-cluding the power produced, whose response should be rapid.

In the second step, the LQR command, which involvesminimizing the quadratic criterion that takes into consider-ation all of the variables of the system, was presented.Furthermore, the LQR command can ensure the best compro-mise between the desired performance and the feedback of thecommand. This principle of criterion minimization leads us todetermine the optimal control structure of the state feedbackfor the linearized MCT system around different operatingpoints.

The obtained simulation results enabled us to conclude thatour linear control can provide a satisfactory performance interms of following the reference values for any marine currentspeed nearly perfectly. In future work, we will focus on theapplication of more sophisticated control strategies to dealwith the influence of any marine current speed by consideringthe entire system.

Nomenclature Vm, marine current speed, m/s; ρ, water density, kg/m3;Rt, rotor radius, m; P, aerodynamic power, W; Taer, aerodynamic torque,N·m; Pe, electrical power, W; λ, tip speed ratio;Cp, power coefficient;Ωt,rotor speed, rad/s; Ωg, generator speed, rad/s; Ωtt, low-speed shaft, rad/s;Jt, rotor inertia, kg·m2; Jg, generator inertia, kg·m2; d, damping

93R. Gaamouche et al.: Optimal Feedback Control of Nonlinear Variable-Speed Marine Current Turbine Using a Two-Mass Model

coefficient, N·m/(rad·s); k, stiffness coefficient, N·m/rad;Ωt − r, rotor ratedspeed; G, gearbox gain; Tem, generator torque, N·m; Tmec, mechanicaltorque, N·m; MCT, marine current turbine; LQR, linear quadraticregulator

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