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Model Predictive Control of a Wind Turbine Nikola Hure Department of Control and Computer Engineering Faculty of Electrical Engineering and Computing of University of Zagreb [email protected]

Abstract—This work gives a short overview of the role that model predictive control has in the development of the advanced wind turbine control algorithms. The advantages of model predictive control compared to conventional controllers involved in wind turbine control are outlined. Wind turbine model predictive speed controller based on identified piecewise affine discrete-time statespace model is designed. Its performance is compared to baseline linear controller through simulations for both below and above the nominal wind speed. The designed MPC controller showed better performance.

I. I NTRODUCTION Wind energy has become fastest growing renewable source considering the installed capacity per year [1]. Unlike the other alternative sources, wind power industry has reached mature commercial phase. Nevertheless, wind turbines are continuously increasing size and nominal power capacity in order to achieve more competitive cost of energy compared to conventional sources. Among the large wind turbines, variable-speed and pitchcontrolled turbines are predominant. The efficiency that they achieve depends mostly on the control algorithm used for turbine operation. Besides, in order to make the turbine profitable, controller has to take care for structural loads mitigation. Only considering both of the forementioned control objectives can lead to optimal lifetime and production combination that will lead to maximal profit. It is high intermittence of wind power and high nonlinearity of multiple input-multiple output (MIMO) wind turbine dynamics that make this task a challenge. Most of the research on the development of wind turbine controllers is mainly based on linear controllers (e.g. [2], [3]). The main drawback they have in the wind turbine control is impossibility to take into account system constraints. Besides advanced control algorithms which have already been developed, interesting role in the future of large wind turbines control will for sure be reserved for model predictive control (MPC) [4]. Good mathematical wind turbine model together with measuring instruments of upcoming wind or short-term meteorological service data can be used to reduce the loads of a wind turbine along with maintaining high electricity production demands. MPC inherently handles multiple objective MIMO systems what makes it particularly interesting for the wind turbine control. This work is organized in five sections. In Section II basics of wind turbine modelling and control are presented. This is followed with a short introduction to model predictive control. In Section III is given a short review of existing research fields in application of MPC paradigm to a wind turbine control. In Section IV is presented a design of a model predictive speed

controller for a 1 [MW] wind turbine. Obtained simulation results are discussed in Section V. II. P RELIMINARY A. Mathematical Model of a Wind Turbine Aerodinamical properties of wind turbine blades allow converting airflow energy into the rotational energy of the rotor which is then converted to the electrical energy. Power stored in air cylinder with the radius R, the density ρ and the wind velocity vw is given with 1 2 3 R πρvw . (1) 2 The amount of wind power that is extracted by a wind turbine depends on the power coefficient Cp Pw =

1 2 3 R πρvw Cp (λ, β), (2) 2 where β refers to the blade pitch angle and λ is a dimensionless quantity known as the tip speed ratio Pwt =

λ=

|ωR| , vw

(3)

where ω denotes the rotational speed of the rotor and R is the rotor radius. To obtain a dynamical model of the wind turbine, aerodynamic forces acting on wind turbine blades have to be modeled. Precise mathematical model which describes forming of the lift and drag forces along the blades due to the airflow through the swept area of the rotor is based on implicit equations [5]. For that reason it is impractical for the controller design. The model used in this work is a simplified nonlinear model of the wind turbine dynamics. It describes all significant physical phenomenas experienced by the wind turbine with rated power 1 [MW]. Therefore, it is a good starting point for the design of a controller with objectives to optimize power production and reduce fatigue of the wind turbine’s tower. The assumptions of the model are: (i) rigid blades, (ii) the wind tower has fore-aft deflections that can be well modeled using first mode, (iii) the wind is uniform over the wind field. The dynamics of the rotor is given with Jt ω˙ = Ma (β, ω, vw ) − Mg ,

(4)

where Ma holds for the aerodynamic torque and Mg is the generator torque. Fore-aft deflections xt of the tower top are modeled with the second order diferential equation Mt x¨t + Dt x˙t + Ct xt = Ft (β, ω, vw ),

(5)

2

(6)

where Tβ is the time constant of the system and βref the pitch angle reference. 1) Discrete-time PieceWise Affine Wind Turbine Model: To overcome high nonlinearities of the wind turbine dynamics, nonlinear model is approximated with multiple affine models that are valid on a certain segment Di of the system state-space D = ∪si=1 Di , x(k + 1) = Ai x(k) + Bi u(k) + fi iff x(k) ∈ Di .

(7)

The discrete-time PWA models are obtained using clustering based PWA approximation of nonlinear aerodynamical functions and incorporating those functions into the dynamical model of the turbine (4,5). The complexity of the identified model is measured in the number of submodels s. The aim of the model identification is to obtain the best model with the least complexity. Details about the wind turbine PWA model identification can be found in [6]. B. The Wind Turbine Control Strategy Due to the high system nonlinearity and large span of a wind power (1) for wind turbine operation, the control system has to be designed for wind powers both below and above the rated power of the wind turbine generator (Fig. 1). The result is completely different control strategy applied below and above the nominal wind speed. Nominal wind speed is minimal velocity which allows producing nominal power of the wind turbine. When the effective wind speed is below nominal, the controller aims to operate with the wind turbine in a way that will assure maximal power production. It can be achieved by holding pitch angle at the optimal position βopt and controlling the generator torque to keep the turbine speed closest to the one that will assure optimal tip speed ratio λopt . In that way the wind turbine operates around the maximum of the power coefficient Cp max . Theoretical maximum efficiency of the wind turbine is called Betz coefficient with value around 59.3%. Most of the modern wind turbines run at 40 to 50% capacity. Above the nominal wind speed wind power exceeds the rated power of the wind turbine with blades set to the optimal angle. To restrain the power that won’t damage the generator, the blades are turned to the angle which will pass through the excess of wind power. The torque of the generator is then held at the nominal value and the pitch controller keeps the rotational speed of the generator around it’s nominal value. Request to the control system that is opposed to the maximal power production but equally important is reducing fatigue. The wind turbine structure is subject to great loads caused by the nonuniform wind field atacking the turbine. Furthermore, wind gusts can initiate wind turbine’s tower oscillations of large amplitude. The pitch controller has to mitigate the

Followed production curve l

Pwt,nom co nt ro

Tβ β˙ + β = βref ,

Optimal production curve Pwt

Pitch control

To rq ue

where Mt denotes the modal mass, Dt is the damping coefficient and Ct the spring constant of the wind turbine tower. Ft is the effective thrust force experienced by the rotor. Blade pitching servo system can be well modeled with

vw,cut−in vw,nom Fig. 1.

vw,cut−out

vw

Wind turbine power curve

forementioned effects and operate in a way that will not impose additional loads. More about the wind turbine control algorithms can be found e.g. in [7] and [5].

C. Model Predictive Control Due to a heavy computational burden was model predictive control initially applied only for control of slow processes (e.g. in petrochemical industry). Today it is a widely used advanced control methodology with a significant impact on industrial control engineering. The most important benefits of MPC are: (i) handling of multiple-input and multiple-output control problems, (ii) taking into account actuator limitations and constraints of the system variables, (iii) inherent optimization of a specified system criterion, (iv) possibility of handling large classes of non-linear systems. Model predictive control is based on a receding horizon idea. In time step k = 0, based on the (i) current value of the system states x0 , (ii) known mathematical model of the process and (iii) the cost function J(x0 , U ) and (iv) constraints, the prediction of control inputs U = [u(k), u(k + 1), . . . , u(k + N − 1)] is calculated in order to minimize the cost function J, where N ∈ N is prediction horizon length . The optimal input vector U ∗ will lead states to the desired values if a mathematical model of the process is ideal and future disturbances are completely known. Since that is never the case because mathematical model inherently contains uncertanties and disturbances can not always be measured, some kind of a feedback has to be introduced. It is done with a receding horizon approach: the calculation is performed in every time step k ∈ {1, 2, . . .} taking into account newest measurement of the system states and only the first control input u(k) is applied to the process. 1) Constrained Finite Time Optimal Control: CFTOC is the MPC used for the controller design in this work. It considers linear constraints and limited prediction horizon. Given the discrete-time linear time-invariant model of the system x(k + 1) = Ax(k) + Bu(k),

(8)

which is subject to the constraints Hx(k) + Lu(k) ≤ K,

(9)

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the following problem has to be solved ∗

J (x0 ) := min J(x0 , U ) (10) U  xk+1 = Axk + Buk , k ≥ 0, subj. to Hxk + Luk ≤ K, k = 0, . . . , N − 1, where xk = x(k), uk = u(k) for simplicity, U is the optimization vector of control inputs and J ∗ is the value function. The cost function J can be given either as a linear or a quadratic funtion [8]. 2) Explicit Control Law Computation: Solving the CFTOC problem can be very demanding, especially if it is done in real-time. To minimize the computational demands, optimization can be performed offline with system states as an additional parameter. This leads to the so-called parametric programming. The CFTOC problem is subdivided into smaller optimization problems backwards in time using a dynamic programming approach [9]. The resulting control law is a PWA function of current states: uk = Fi xk + Gi iff xk ∈ Pi ,

(11)

R where P = ∪N i=1 Pi is a polytope collection [10] in the statespace containing all control regions. Nevertheless, it should be noted that the explicit control law computation is limited only for systems with relatively small number of system variables and shorter prediction horizons due to its large complexity. The wind turbine model used in this work meets those requirements.

III. M AIN R ESEARCH F IELDS The European Technology Platform for Wind Energy identified main research priorities in the field of the wind turbine system control. According to the published strategic research agenda for the period from year 2008 to 2030 [11], there are several important chalenges that have to be prevailed in order to achieve optimal balance between power production and loads. These are briefly discussed in the sequel, together with a possible enrollment of model predictive control and an overview of existing literature where model predictive control is used to address control problems. A. Optimization of the electricity output and capacity factor, both for the individual wind turbine and the wind farm With the increase of installed wind power go together stringent requirements in quality of supplied electrical energy. Moreover, intermittence of wind power can cause oscillatory behavior of power generated with an individual wind turbine. In order to make wind turbines more competitive to conventional energy sources, the control system must ensure smooth power generation and greater reliability of the produced power. Information about current wind field potential and shortterm predictions can be used for the distribution of the power reference to each wind turbine in a wind farm in order to achieve minimal deviations from the desired total production [12]. The existence of a battery or of another kind of a local storage in a wind farm can also be used for improving the suppression of the output power fluctuations [13]. Both of

forementioned examples contain optimization problems which can be sistematically aproached and handled using model predictive control. The information about wind conditions can be used to adjust cost weightining factors of the value function in order to achieve optimal balance between pursuing maximal power and electricity quality [14]. The net capacity factor of a power plant is the ratio of the actual output of a power plant over a period of time and its potential output if it had operated at full nameplate capacity the entire time. For the improvement of the capacity factor multiple maximum power point tracking algorithms have been developed, e.g. [15]. Maximizing power capture can be also achieved with optimal rotor speed tracking: Z Z λopt vw dτ. (12) J ∗ = min ω − ωopt dτ = min ω − u u R Optimization problem (12) must also include constraints on system variables. Shaft torsional torque together with the maximal change rate of a generator torque reference are usually incorporated as constraints in the optimization problem [16]. It must be pointed out that the power production maximization has to take into account mechanical loads reduction which is discussed in the next subsection.

B. Reduction of mechanical loads on the wind turbine structure The significant part of the wind turbine structural loads are caused by the rotational sampling of turbulence, the tower shadow and wind shear. In the analysis, the most important are loads experienced by the blade’s root [5]. It is shown that loads can be significantly reduced using individual pitch control (IPC) [17]. Model predictive control can be utilized for minimizing loads of higher frequency in the periodic loads power spectrum for its ability to predict disturbances and counteract with appropriate control inputs. The constraints that should be included in the optimization problem are blade pitch amplitude and rate plus constraint which will insure that minimizing loads does not interfere with the wind turbine speed control. In [18] it is shown that via MPC and IPC can loads caused by the wind effects be significantly reduced. The authors used load measurement in the blade’s root to estimate wind speed experienced by the blade together with the repetitive property of experienced loads to predict wind conditions for the next passing blade. The authors in [19] studied the influence of short-time wind predictions in structural loads mitigation. The results they presented confirmed the assumption that knowing wind in advance can help in reducing fatigue caused by the wind shear and the tower shadow. Structural loads mitigation can be counterproductive when wind speed estimation/prediction deviates from actual wind conditions. Therefore, it would be better to use robust model predictive control [20] and to include measure of reliability of the wind speed information for reducing fatigue.

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C. Development of control algorithms to ensure the aeroelastic stability of the wind turbine Aeroelasticity is the combination of elastic deformations with the aerodynamic loading. It has become an important factor in the wind turbine stability analysis [21]. This is due the fact that the wind turbines get bigger and materials used for construction lighter every year so aeroelastic effects must be taken into account during the controller design process. E.g. for the 5 [MW] wind turbine the simplified model used by authors in [19] incorporates first modes of the tower fore-aft deflection, side-side deflection and blade flapwise deflection for each blade in the model. Each of them is very important and ommiting only one mode may result with unwanted behavior of the entire control system. As blade length gets bigger, it becomes hard to tain modes of the blade. Therefore more sophisticated actuators have to be used to achieve desired aeroelastic stability. In [22] authors studied possibilities that can offer trailing edge flaps together with MPC controller in improving aeroelastic stability of the blades. The results show great reduction of loads experienced by the blade’s root. In order to validate aerolastic stability of the obtained controller, extensive simulations in widely approved wind turbine simulators (e.g. Bladed or FAST) have to be performed prior to the control system implementation on the real wind turbine. D. Development of new control sensors, such as Lidar, in order to forecast the flow in the rotor plane and the integration of this forecast into control strategies The wind flow is highly variable both in spatial and time domain. Development of new technologies for measurement of the wind field experienced by the rotor is of great importance for obtaining better balance between optimal power production and disturbance rejection. Moreover, MPC controller requires full-state fedback, which means that wind speed needs to be either measured or estimated. Many authors studied positive effects that can be achieved e.g. using Lidar or other sensor for wind power prediction [23]. For its intrinsic property of including disturbance in the optimization problem MPC is considered as the best control strategy for exploiting such measurements. E. Development of integrated control and maintenance strategies incorporating condition monitoring systems Together to the possible risk of damage, which is far greater for the larger wind turbines, goes aside importance of fault detection and with that related control algorithms. Condition monitoring is used for early detection of faults in order to minimize downtime and maximize productivity of wind turbines[24]. MPC can be used to asure that stricter constraints will be respected by the control system in order to prevent/delay further faults by the time when service of the system is done. Of great importance is also possible overspeed detection. Overspeed means overstepping maximal allowable speed of

the wind turbine rotor. When overspeed is detected, the blades turn to the maximal possible pitch angle. The consequence is instant drop of aerodynamic torque and a great stress experienced by the wind turbine structure. In-time reaction of the wind turbine actuators could significantly reduce the possibility of overspeed ocurrence. MPC can assure that all constraints are satisfied with the application of the calculated reference signals to the actuators which will lead the wind turbine into the safe state. There are two main advantages of MPC over other control strategies that make it more appropriate in the wind turbine system control: • straightforward and systematic inclusion of disturbance predictions in the optimization problem (e.g. wind profile predictions); • constraints, such as actuator limitations (e.g. pitch rate limit), are included in the optimization problem; controller which does not take into account actuator constraints may request excessive pitch rates; ommiting constraints during the controller design can therefore deteriorate the control system stability. IV. M ODEL P REDICTIVE S PEED C ONTROLLER D ESIGN FOR A 1 [MW] W IND T URBINE The MPC design that is presented in the sequel concentrates on a speed control of a wind turbine which will optimize power production in the region below the nominal wind speed and hold the nominal rotor speed in turbulent wind conditions above the nominal wind speed. The model that is considered in the controller synthesis is based on a discrete-time PWA state-space model of the 1 [M W ] wind turbine with pitch angle, rotational speed of the rotor, effective wind speed and generator torque as system states. Wind turbine states, such as pitch angle and effective wind speed are supposed to be reliably measured. On the contrary, wind turbine rotational speed measurements are superposed with Band-Limited White Noise signal. The generator torque state is used to penalize sudden changes of the generator torque reference. The following procedure is used for the wind turbine speed controller design. First step includes the wind turbine model identification and tuning the cost function weightining factors that will achieve best performance of the control system. This step is performed iteratively until satisfactory model is obtained and good weightining factors are found. The final step includes explicit control law computation which can then be incorporated in the microprocessor for the real-time control. The operating mode selection is performed with respect to effective wind speed acting on the turbine. Objectives and constraints used for the speed controller in both modes of operation are described in the following subsections. Discrete time step used for the control is Ts = 0.5[s] and the prediction horizon is N = 4. A. MPC Design Below the Nominal Wind Speed The most important objective of the speed controller below the nominal wind speed is optimizing power production. There

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are different mathematical formulations of the cost function to ensure that objective. Linear 1-norm cost function of the optimal tip-speed ratio tracking is used in this work. Remark I: Maximal power production objective can be achieved with optimal tip-speed ratio tracking, but can as well be stated as input wind power maximization. In latter case, power is explicitely taken into account and would surely be better option in the case of the variable optimal tip-speed ratio. The optimal tip-speed ratio can be dependent on the rotational wind turbine speed in the case that drive train friction forces can not be neglected. The following cost function is used in the optimization: J(x0 , U ) =

k=N X−1

[Kλ |ωk+1 R − vw λopt |+

(13)

k=0

K∆Mg (k)|Mg k − Mg k−1 | + f (Mg k ) + f (wk+1 )], with the constraints:

where f (ωk+1 ) is an absolute function with dead zone length of 2ωtol around the nominal rotational speed value:   ∆ωk+1 − ωtol , iff ∆ωk+1 > ωtol , −ωtol − ∆ωk+1 , iff ∆ωk+1 ≤ −ωtol , (15) f (ωk+1 ) =  0, otherwise, where ∆ω[rad/s] is deviation from the nominal rotational speed of the turbine, ωk+1 − ωnom . The constraints included in the optimization problem are: Mg ≤ 1.2Mg,nom , |Mg k − Mg k−1 | ≤ 0.03Mg,nom , βmin ≤ βref ≤ βmax , ˙ ≤ β˙ max , |β| ωmin ≤ ω ≤ ωmax , vwmin ≤ vw ≤ vwmax . The reason for using the time dependent weightining factor K∆β in (14) is alike to usage of K∆Mg in (13).

0 ≤ Mg ≤ 1.2Mg,nom ,

V. S IMULATION R ESULTS

|Mg k − Mg k−1 | ≤ 0.03Mg,nom ,

The simulation tests are performed under turbulent wind conditions in Matlab Simulink. In order to validate the resulting MPC, its behavior is compared to the conventional linear controller [5]. The results are discussed in the sequel.

βref = βopt , βmin ≤ β ≤ βmax , ωmin ≤ ω ≤ ωmax , vwmin ≤ vw ≤ vwmax .

A. Below the nominal wind speed

The first term of the cost function (13) is used to perform optimal tip-speed ratio tracking. Then follows the penalization of the instant changes of the generator torque. It should be noted that the weightining factor K∆Mg is time dependent. Simulations tests have shown that noise in wind speed measurement cause increased activity of the generator torque reference which can have a negative effect on the generator. Therefore, the weightining factor used for penalizing the first torque reference in the horizon, which is calculated based on the noisy signal of the rotational speed, is larger than the factor used for subsequent iterations of the control input prediction. The last two terms of the equation are used to penalize torque reference overstepping the nameplate value and rotational speed overstepping minimal and maximal allowable generator speed, respectively.

B. MPC Design Above the Nominal Wind Speed Above the nominal wind speed, the most important objective of the speed controller is to keep the rotational speed closest to the nominal with the least stress on the generator and pitch control systems. In that order of control objectives follows the resulting mathematical formulation of the cost function: J(x0 , U ) =

k=N X−1

[Kω opt f (ωk+1 ) + KMg |Mg k − Mg,nom |

k=0

+ K∆β (k)|βref,k − βk |],

(14)

Simulation results of the wind turbine operation during the turbulent wind below the nominal wind speed are shown in Figure 2. On the first subfigure from above is shown wind speed used in the simulation. Subfigure in the middle shows achieved tip speed ratio responses and on the third subfigure are generator torque references. MPC controller achieved better tip-speed ratio tracking compared to the baseline controller at the expense of the higher activity of the generator. B. Above the nominal wind speed The results obtained during the operation above the nominal wind speed are shown in Figure 3. Rotational speed of the rotor is shown in the second subfigure from above, below are generator torque reference and pitch angle responses, respectively. It can be noticed that slightly better performance of the nominal rotational speed tracking is achieved compared to the baseline controller. The loads experienced by the generator and pitch control system are similar for both controllers. C. Around the nominal wind speed Test of the sudden wind speed jump over the nominal wind speed is performed to show the superiority of the receding horizon paradigm over other system control approaches. Thanks to the prediction property, the MPC controller starts with the maximal increase rate of the pitch angle and forcing the torque of the generator by the time in which the rotational speed of the rotor overpasses the nominal speed. The result is lower overspeed achieved with the MPC (Fig. 4).

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9

vw [m/s]

8 7 6 5

0

100

200

300

400

500

600

700

Baseline controller, σ2λ=0.036626

9

MPC, σ2λ=0.028768

8.5 λ

8 7.5 7

2.5

100 x 10

200

300

400

500

600

700

5

Baseline controller MPC

Mg [Nm]

2 1.5 1 0.5 0

100

200

300

Fig. 2.

400 t[s]

500

600

700

Simulation results below the nominal wind speed

vw [m/s]

17 16 15 14 13

0

100

200

300

400

500

600

700

Baseline controller,

ω [rpm]

27.5

σ2ωnom=0.0064672

MPC, σ2ωnom=0.0056689

27

26.5

Mg [Nm]

4

100 x 10

200

300

400

500

600

700

Baseline controller, σ2Mg/M2nom=0.00079659

5

MPC, σ2Mg/M2g,nom=0.00079975

3.5

3

100

200

300

400 t[s]

500

600

700

16 Baseline controller MPC

β [°]

14 12 10 8

100

200

300

Fig. 3.

400 t[s]

500

Simulation results above the nominal wind speed

600

700

7

vw [m/s]

14 12 10 8

10

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ω [rpm]

30

Baseline controller MPC

25

20

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6

10 x 10

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Baseline controller MPC

4 2 0

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t[s] Baseline controller MPC

β [°]

15 10 5 0

10

20

30

40

50

60

70

80

90

100

t[s]

Fig. 4.

Simulation results for the sudden wind speed jump over the nominal wind speed

VI. C ONCLUSION The literature overview shows that MPC is highly involved in the research of the advanced wind turbine control algorithms. The main advantage over the other control approaches is predictive property and explicit integration of the constraints in the control problem. Moreover, the resulting control law is optimal with respect to the chosen objective. In this work, high nonlinearity of the wind turbine dynamics is approximated with a PWA model. The resulting model is utilized in the design of the MPC. The simulation results obtained with the MPC show better performance compared to the baseline controller. It should be noticed that much better results with the MPC would be achieved using longer prediction horizon and incorporating wind speed prediction in the optimization problem. Nevertheless, it would result with a more complicated optimization problem. R EFERENCES [1] “Wind in power - 2011 European statistics“, The European Wind Energy Association, 2012. [2] E. Van der Hooft, P. Schaak and T. Van Engelen, “Wind turbine control algorithms“, ECN, Technical Report, 2003. [3] F. D. Bianchi, H. D. Battista and R. J. Mantz, Wind turbine control systems. Advances in Industrial Control, Springer Verlag, 2007. [4] J. Maciejowsky, Predictive Control with Constraints, Prentice Hall, 2000. [5] M. Jelavi´c, ”Wind turbine control for structural dynamic loads reduction”, Ph.D. dissertation, Faculty of Electrical Engineering and Computing, University of Zagreb, 2009. [6] M. Vaˇsak, N. Hure and N. Peri´c, ”Identification of a discrete-time piecewise affine model of a pitch-controlled wind turbine”, in Proceedings of the 34th International Convention on Information and Communication Technology, Electronics and Microelectronics, Opatija, Croatia, 2011, pp. 104-109. [7] T. Burton, D. Sharpe, N. Jenkins, and E. Bossany, Wind Energy Handbook. John Wiley & Sons, 2001.

[8] M. Baoti´c, ”Optimal Control of Piecewise Affine Systems – a Multiparametric Approach”, Ph.D. dissertation, Swiss Federal Institute of Technology Zurich, 2005. [9] A. Bemporad, M. Morari, V. Dua and E. N. Pistikopoulos, ”The Explicit Solution of Model Predictive Control via Multiparametric Quadratic Programming”, in Proceedings of the American Control Conference, Chicago, Illinois, 2000. [10] G. M. Ziegler, Lectures on Polytopes. Springer, 1995. [11] “Strategic Research Agenda“, The European Technology Platform for Wind Energy, 2008. [12] V. Spudi´c, ”Wind turbine power control for coordinated control of wind farms”, in Proceedings of the 18th Internation Conference on Process Control, Tatranska Lomnica, Slovakia, 2011. [13] M. Khalid and A. V. Savkin, ”Model Predictive Control of Distributed and Aggregated Battery Energy Storage System for Capacity Optimization”, in Proceedings of 9th International Conference on Control and Automation (ICCA), Santiago, Chile, 2011. [14] A. Kusiak, W. Li, and Z. Song, ”Dynamic Control of Wind Turbines”, Renewable Energy, Vol. 35, No. 2, pp. 456-463, 2010. [15] I. Munteanu, A. I. Bratcu, N. A. Cutululis, E. Ceang, Optimal Control of Wind Energy Systems Towards a Global Approach, Springer-Verlag, London, 2008. [16] R. Burkart, K. Margellos, and J. Lygeros, ”Nonlinear control of wind turbines: An approach based on switched linear systems and feedback linearization”, ;in Proc. CDC-ECE, 2011, pp.5485-5490. [17] E. A. Bossanyi, ”Individual Blade Pitch Control for Load Reduction”, Wind Energy, Vol. 6, pp. 119–128, 2003. [18] J. Friis, E. Nielsen, J. Bonding, F. D. Adegas, J. Stoustrup and P. F. Odgaard, ”Repetitive Model Predictive Approach to Individual Pitch Control of Wind Turbines”, in 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), Orlando, Florida, 2011. [19] M. D. Spencer, K. A. Stol, C. P. Unsworth, J. E. Cater and S. E. Norris, ”Model predictive control of a wind turbine using short-term wind field predictions”, Wind Energy, Published online in Wiley Online Library (wileyonlinelibrary.com), 2012. [20] A. Bemporad and M. Morari, ”Robust Model Predictive Control: A Survey”, Robustness in Identification and Control, Vol. 245, pp. 207– 226, 1999. [21] M. O. L. Hansen, J. N. Sørensen, S. Voutsinas, N. Sørensen, H. Aa.Madsen, ”State of the art in wind turbine aerodynamics and aeroelasticity”, Progress in Aerospace Sciences, Vol. 42, pp. 285–330, 2006.

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[22] D. Castaignet, N. K. Poulsen, T. Buhl and J. J. Wedel-Heinen, ”Model Predictive Control of Trailing Edge Flaps on a Wind Turbine blade”, in American Control Conference, 2011. [23] M. Soltani, R. Wisniewski, P. Brath, S. Boyd, ”Load reduction of wind turbines using receding horizon control”, in Proceedings IEEE Multiconference on Systems and Control, Denver, USA, 2011, pp. 852-857. [24] F. P. G. M´arquez, A. M. Tobias, J. M. P. P`erez, M. Papaelias, ”Condition monitoring of wind turbines: Techniques and methods”, Renewable Energy, Vol. 46, pp. 169–178, 2012.