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  Wind Turbine Pitch Optimization Benjamin Biegel Morten Juelsgaard Matt Kraning Stephen Boyd Jakob Stoustrup  Abstract —We consider a static wind model for athree-bladed, horizontal-axis, pitch-controlled wind tur-bine. When placed in a wind field, the turbine experiencesseveral mechanical loads, which generate power but alsocreate structural fatigue. We address the problem of find-ing blade pitch profiles for maximizing power productionwhile simultaneously minimizing fatigue loads. In this pa-per, we show how this problem can be approximately solvedusing convex optimization. When there is full knowledgeof the wind field, numerical simulations show that forceand torque RMS variation can be reduced by over 96%compared to any constant pitch profile while sacrificing atmost 7% of the maximum attainable output power. Usingiterative learning, we show that very similar performancecan be achieved by using only load measurements, with noknowledge of the wind field or wind turbine model. I. I NTRODUCTION Wind turbines are expensive to build and maintain.The wind field from which they generate power is alsothe source of large fatigue loads on the turbine, whichcreate structural wear and tear, increasing maintenancecosts and decreasing the operational lifetime of theturbine. These costs are significant, and dramaticallyimpact the profitability of the turbine. Many studies [1],[2], [3], [4], [5] have been performed which attemptto reduce fatigue loads while also generating sufficientpower by dynamically controlling blade pitching. In thispaper, we present a general blade pitching approach forfatigue load minimization based on convex optimization.We focus on pitch controlled wind turbines, andinvestigate the use of pitching to obtain the maximumpower output of a wind turbine. We then use this numberto bound how much power must be sacrificed to achievea given level of fatigue load reduction. We concludeby showing how iterative learning, using only loadmeasurements, can achieve performance very close tothat of a controller with perfect knowledge of the windfield and wind turbine model. The first, second, and fifth authors are with the Department of Automation and Control, University of Aalborg, Aalborg, Denmark.Email:  { biegel, juels, jakob } @es.aau.dk. The third and fourth authorsare with the Information Systems Laboratory, Electrical EngineeringDepartment, Stanford University, Stanford, CA 94305-9510, USA.Email:  { mkraning, boyd } @stanford.edu. II. W IND  T URBINE  P ITCH  O PTIMIZATION  A. Description We consider a three-bladed, horizontal-axis, pitch-controlled wind turbine, as illustrated in figure 1. Weassume that the wind turbine is affected by a windfield that is constant over time, but varying over thearea swept by the blades. The blades of the rotor arenumbered 1, 2, 3 and the angle  θ  ∈  [0 , 2 π/ 3]  definesthe position of the rotor. θ 123 x yzf τ  x τ  y τ  z Fig. 1. Perspective view of a wind turbine. The wind turbine is controlled via the pitch of thethree blades. As the wind is assumed constant over time,the blade pitching is a periodic function of the angle  θ .We let  p ( θ ) =  β   = ( β  1 ,β  2 ,β  3 )  ∈  R 3 ,  0  ≤  θ  ≤  2 π 3 denote the pitching angles of all three blades at angle θ , where we suppress the explicit dependence of   β   onthe angle  θ . These pitch angles affect the generation of torques,  τ   = ( τ  x ,τ  y ,τ  z )  ∈  R 3 , around the three mainaxes and a net force,  f   ∈  R , on the whole structure. Thecoordinate system is defined such that the  y - and  z -axes 2011 IEEE International Conference on Control Applications (CCA)Part of 2011 IEEE Multi-Conference on Systems and ControlDenver, CO, USA. September 28-30, 2011978-1-4577-1061-2/11/$26.00 ©2011 IEEE1327  span the rotor plane while the  x -axis is perpendicularto the rotor plane, as in figure 1. The origin of ourcoordinate system is located at the intersection of therotor plane with the axis of rotation.During operation,  τ   and  f   depend on the angle of therotor, the pitch of the blades, the wind field present at theswept area, and the angular velocity of the rotor itself.We neglect the last dependency by assuming a constantrotor angular velocity and we let the wind field at theswept area be defined by the parameter vector  η . Then τ   and  f   can be expressed by the functionals τ   = Ψ(  p ( θ ) ,θ,η ) , f   = Υ(  p ( θ ) ,θ,η ) . In this work we use an overline to denote the mean valueof a function of   θ  in the interval  0  ≤  θ  ≤  2 π/ 3 ,  e.g. g  = 32 π    2 π/ 30 g ( θ ) d θ, and we overload this notation for vector valued functionscomponentwise.The mean values  τ   and  f   can be interpreted as DC-terms around which the torques and force vary. Wedefine the AC-terms of the torques and force by δτ   =  τ   − τ, δf   =  f   − f, respectively. At a given angle  θ ,  δτ   and  δf   dependnot only on the pitch at this angle, but on the entirepitch profile  p ( θ )  in the interval  0  ≤  θ  ≤  2 π 3  , throughthe mean values  τ   and  f  . Using the root mean square(RMS) values of the AC-terms, we define the variationof the torques and force by  J   = ( J  x ,J  y ,J  z ,J  f  )  ∈  R 4+ ,where J  x  =  δτ  2 x  1 / 2 , J  y  =  δτ  2 y  1 / 2 ,J  z  =  δτ  2 z  1 / 2 , J  f   =  δf  2  1 / 2 .  B. Objectives Wind turbine operation is a multiobjective optimiza-tion problem. First, we desire a large, even power output.As instantaneous power output is proportional to  τ  x ,this corresponds to a large mean output torque,  τ  x , withonly small variations as measured by  J  x . In addition, wewant low mechanical fatigue on our structure in order tolengthen its operational lifetime and reduce maintenancecosts. The torques  τ  y  and  τ  z  and force  f   describe variousmechanical loads experienced by the turbine structure.Their DC-terms are regarded as specifications which thewind turbine structure should be designed to handle andare therefore not treated in this work. The AC-termsof the mechanical loads cause structural fatigue. Wetherefore also want small RMS values of   δτ  y ,  δτ  z , and δf  , which are given by  J  y ,  J  z , and  J  f  . C. Constraints The pitch profiles are controlled by mechanical pitchactuators, which have limits in both range and pitchspeed. We also assume that the wind turbine is notdesigned to operate in stall mode. These constraints canbe expressed as β  min  ≤  p ( θ )  ≤  β  max ,  dp ( θ ) dt  ≤  β  slew , where  β  min ,  β  max  are, respectively, the minimum andmaximum pitch angles that are operationally possibleand at which the rotor will not stall, and  β  slew  is themaximum pitch speed. We use  P   to denote the entireset of these constraints, which is a convex set.III. M ODEL  D ESCRIPTION In this section we consider the torque and forcefunctionals,  Ψ(  p ( θ ) ,θ,η )  and  Υ(  p ( θ ) ,θ,η ) . We showhow to find expressions for these functionals by forminga static model of the wind field and the wind turbine.  A. Wind Model Let the vector field  V  in ( y,z )  ∈  R 3 describe theincoming wind field at the area  A  swept by the turbineblades, where A  =  { ( x,y,z )  ∈  R 3 |  x  = 0 , z 2 +  y 2 ≤  R 2 } and  R  is the length of each blade. We assume that thewind field has no  z -component, and that the direction of the wind field is the same over the entire swept area. Wefurther assume that the magnitude of the wind field canbe described as a sum of wind phenomena contributions.The wind field is then given by V  in ( y,z ) = ( v bl  +  v vs ( z ) +  v hs ( y ) +  v ts ( y,z ))   Γ where    Γ = (cos γ, sin γ, 0) ,  γ   is the direction of thewind,  v bl  is the baseline wind speed,  v vs  and  v hs  arerespectively the vertical and horizontal shear, and  v ts is the tower shadow. In the following, we describe thecharacteristics of the four wind phenomena. a) Baseline Wind Speed:  The baseline wind speedparameter  v bl  ∈  R  describes the wind speed at thesrcin of our coordinate system. All other wind termsare deviations from this value. b) Vertical Wind Shear:  The vertical wind shearparameter  ξ  vs  ∈  R  describes the variation of wind speedas a function of altitude. This wind phenomena is knownas wind shear [6] v vs ( z ) =  v bl  ξ  vs   zH    +  ξ  vs ( ξ  vs  − 1)2   zH   2 + ξ  vs ( ξ  vs  − 1)( ξ  vs  − 2)2   zH   3  , 1328  where  H   is the height of the turbine hub above theground. c) Horizontal Wind Shear:  The horizontal windshear parameter  ξ  hs  ∈  R  describes how the wind speedvaries horizontally across the area swept by the blades.We assume a linear dependency between horizontalposition and the horizontal wind shear, which is thengiven by v hs ( y ) =  v bl ξ  hs y. d) Tower Shadow:  The tower shadow parameter t s  ∈ { 0 , 1 }  determines if the effect of the turbine toweron the wind field is included in the wind field model.The tower shadow is described by [1] v ts ( y,z ) =   − t s  r td t − yd t + y  2 z  ≤  0 , 0 otherwise , where  r t  ∈  R  is the radius of the tower shaft, and  d t  ∈  R is the distance of the rotor plane from the tower mid-line.Using the static wind model above, we define thecomponents of the wind parameter vector  η  by η  = ( γ,v bl ,ξ  vs ,ξ  hs ,t s ) , which fully specifies the wind field  V  in ( y,z ) . Figure 2illustrates an example wind field  V  in ( y,z )  over the sweptarea.Horizontal position [m]      V   e   r    t     i   c   a     l   p   o   s     i    t     i   o   n     [   m     ] 00 − 20 − 20 − 40 − 40 2020404013141516 Fig. 2. Example of a wind field with wind parameter vector  η  =[ γ,v bl ,ξ vs ,ξ hs ,t s ] = [0 , 15 , 0 . 2 , 5 · 10 − 4 , 1]  for a wind turbine withblade radius  R  = 40  m. The colors indicate the wind velocity in therange 13 to 16 m/s.  B. Turbine Model The wind velocity experienced by a wind turbineblade is known as the effective wind velocity and isdefined as  V  eff   =  V  in  +  V  rot  ∈  R 3 , where  V  rot  =(0 ,ω r r sin θ,ω r r cos θ )  ∈  R 3 is the wind velocity dueto the rotation of the blade itself and  ω r  is the constantangular velocity of the rotor. When pitching a blade toan angle  β  , it is subjected to the forces [7] dF  t  =  ρ 2  V   eff   22 b ( C  l ( α )sin ψ  −  C  d ( α )cos ψ ) dr,dF  a  =  ρ 2  V   eff   22 b ( C  l ( α )cos ψ  +  C  d ( α )sin ψ ) dr, where  dF  t , dF  a  ∈  R  are the tangential and axial forces,respectively, acting on an infinitesimal blade elementof length  dr  and width  b . The functions  C  l ( α )  and C  d ( α )  are the lift and drag coefficients, respectively.They depend on the shape of the blade, and are functionsof the wind angle of attack   α . The parameter  ρ  is thedensity of air, while  ψ  is the angle between  V  rot  and V  eff  . The blade also has a static pitch along its length,denoted  β  t . Figure 3 illustrates these relations. ydF  t dF  a αβ   +  β  t ψV  in V  rot bγ V  eff  x Fig. 3. Relations between the incoming wind and the the axial andtangential forces generated. Once the axial and tangential forces acting on theblades are known, it is possible to form expressions for Ψ(  p ( θ ) ,θ,η )  and  Υ(  p ( θ ) ,θ,η )  for each blade [7] τ  ix ( θ ) =    Rr =0 Ξ i r ( C  l ( α i )sin ψ i  −  C  d ( α i )cos ψ i ) dr  (1) τ  iy ( θ ) =    Rr =0 Ξ i r sin θ ( C  l ( α i )cos ψ i  +  C  d ( α i )sin ψ i ) dr (2) τ  iz ( θ ) =    Rr =0 Ξ i r cos θ ( C  l ( α i )cos ψ i  +  C  d ( α i )sin ψ i ) dr (3) f  i ( θ ) =    Rr =0 Ξ i  ( C  l ( α i )sin ψ i  −  C  d ( α i )cos ψ i ) dr,  (4) where  Ξ i  =  ρ 2  V  eff  ,i  22 b , and  i  ∈ { 1 , 2 , 3 }  referencesquantities associated with blade  i . The net force expe-rienced by the turbine is given by  f  ( θ ) =  3 i =1  f  i ( θ ) ,with similar expressions for the net torques. 1329  IV. P ROBLEM  F ORMULATION  A. Power Maximization We formulate the problem of maximizing mean outputtorque subject to physical blade pitching constraints as aconvex optimization problem. Solutions to this problemwill be used to evaluate power output reduction whenreducing fatigue loads.The power maximization problem can be formulatedasmaximize  τ  x subject to  p  ∈ P   (5)with variables  p ( θ )  ∈  R 3 for  θ  ∈  [0 ,  2 π 3  ] . In the modelgiven by equations (1)-(4),  τ  x  only depends on  p ( θ ) through the lift and drag coefficients  C  l ( α )  and  C  d ( α ) ,an example of which are illustrated in figure 4 [8].   C  d C  l      C      l               (    α               )    a   n     d      C      d               (    α               )      [  -     ] α  [rad] − 1  − 0 . 5 0 0 . 5 1 − 2 − 1012 Fig. 4. Example lift and drag coefficient curves. The dotted linesindicate the interval in which the coefficients have been approximated. The allowed range for  α  is limited, as operation install mode is prohibited. Using these limits,  C  l ( α )  can beapproximated by a concave function in  α , while  C  d ( α ) can be approximated by convex function in  α . As  τ  x  isthe sum of a positive weighting of the concave function C  l ( α )  and a negative weighting of the convex function C  d ( α ) ,  τ  x  and its mean value  τ  x  are concave functionsof   α , and thus also of   p ( θ ) . Since the constraint set  P  is convex, problem 5 is a convex optimization problemwhich can be solved globally and efficiently [9].  B. Fatigue Load Minimization In this section we address the problem of maximizingthe mean output torque, while keeping it even and min-imizing fatigue loads. This is a non-convex multiobjec-tive optimization problem. We show how local solutionsto this problem can be found by using sequential convexprogramming (SCP) [10].Maximizing output power while minimizing fatigueloads and output power variation corresponds to maxi-mizing  τ  x  while simultaneously minimizing all compo-nents of   J  . We formulate this as the scalarized problemmaximize  Φ(  p ) =  τ  x  − λ T  J  subject to  p  ∈ P  ,  (6)with variables  p ( θ )  ∈  R 3 for  θ  ∈  [0 ,  2 π 3  ] , and scalar-ization parameters  λ  = ( λ x ,λ y ,λ z ,λ f  )  ∈  R 4+ . Forsimplicity, and to reflect the equal importance of bothfatigue load minimization and even power output, wewill use the scalarization parameters  λ  =  µ 1 ,  µ  ∈  R + ,for the remainder of this paper. By varying  µ , a trade-off curve between  τ  x  and  1 T  J   can be found. Unlike thepower maximization problem, the additional objectiveterm  1 T  J   is non-convex and thus problem 6 is not aconvex optimization problem. We therefore choose tosolve it locally by using SCP.The SCP method finds a local solution iteratively.At iteration  k , a convex approximation of problem 6is formed about a point  p ( k ) . This problem is formed byreplacing the non-convex term  J   by a convex approx-imation  ˆ J  ( k ) , which leads to the convex optimizationproblemmaximize  ˆΦ(  p ) =  τ  x  − µ 1 T   ˆ J  ( k ) subject to  p  ∈ P  , p  ∈ T   ( k ) , (7)with variables  p ( θ )  ∈  R 3 for  θ  ∈  [0 ,  2 π 3  ] . The con-straint set  T   ( k ) is a (convex) trust region around theapproximation point  p ( k ) in which  ˆ J  ( k ) is a sufficientlyaccurate approximation of   J  . The initial approximationpoint  p (0) is typically chosen to be a point with rea-sonable performance, such as a well chosen constantpitch profile, and for subsequent values of   k ,  p ( k ) is setequal to the solution of problem 7 at iteration  k − 1 . Byrunning a sufficient number of iterations,  p ( k ) convergesto a local optimum for problem 6 [11]. Although SCPis not guaranteed to find a globally optimal solution, itleverages the convex parts of the srcinal non-convexproblem, which often leads to a good solution.V. N UMERICAL  E XAMPLES We present two numerical examples which solvediscretized versions of the power maximization andfatigue load minimization problems. Starting from themodel presented in equations (1)-(4), and using the windfield depicted in figure 2, we break each blade into n  smaller blade elements, and divide the swept area 0  ≤  θ  ≤  2 π 3  into  m  discrete values. We approximate 1330
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