Wind Turbine Pitch Optimization
Benjamin Biegel Morten Juelsgaard Matt Kraning Stephen Boyd Jakob Stoustrup
Abstract
—We consider a static wind model for athreebladed, horizontalaxis, pitchcontrolled wind turbine. When placed in a wind ﬁeld, the turbine experiencesseveral mechanical loads, which generate power but alsocreate structural fatigue. We address the problem of ﬁnding blade pitch proﬁles for maximizing power productionwhile simultaneously minimizing fatigue loads. In this paper, 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 943059510, USA.Email:
{
mkraning, boyd
}
@stanford.edu.
II. W
IND
T
URBINE
P
ITCH
O
PTIMIZATION
A. Description
We consider a threebladed, horizontalaxis, pitchcontrolled 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 MultiConference on Systems and ControlDenver, CO, USA. September 2830, 20119781457710612/11/$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 DCterms around which the torques and force vary. Wedeﬁne the ACterms 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 ACterms, 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 optimization 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 DCterms are regarded as speciﬁcations which thewind turbine structure should be designed to handle andare therefore not treated in this work. The ACtermsof 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 midline.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 experienced 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 minimizing fatigue loads. This is a nonconvex multiobjective 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 maximizing
τ
x
while simultaneously minimizing all components 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 scalarization 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 tradeoff curve between
τ
x
and
1
T
J
can be found. Unlike thepower maximization problem, the additional objectiveterm
1
T
J
is nonconvex 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 nonconvex term
J
by a convex approximation
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 constraint 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 reasonable 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 nonconvexproblem, 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