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DK-iteration robust control design of a wind turbine
Mirzaei, Mahmood; Niemann, Hans Henrik; Poulsen, Niels Kjølstad
Published in:2011 IEEE International Conference on Control Applications
Link to article, DOI:10.1109/CCA.2011.6044429
Publication date:2011
Document VersionPublisher's PDF, also known as Version of record
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Citation (APA):Mirzaei, M., Niemann, H. H., & Poulsen, N. K. (2011). DK-iteration robust control design of a wind turbine. In2011 IEEE International Conference on Control Applications IEEE. DOI: 10.1109/CCA.2011.6044429
DK-Iteration Robust Control Design of a Wind Turbine
Mahmood Mirzaei, Hans Henrik Niemann and Niels Kjølstad Poulsen
Abstract— The problem of robust control of a wind turbineis considered in this paper. A controller is designed based ona 2 degrees of freedom linearized model. An extended Kalmanfilter is used to estimate effective wind speed and the estimatedwind speed is used to find the operating point of the windturbine. Due to imprecise wind speed estimation, uncertaintyin the obtained linear model is considered. Uncertainties in thedrivetrain stiffness and damping parameters are also consideredas these values are lumped parameters of a distributed systemand therefore they include inherent uncertainties. We includethese uncertainties as parametric uncertainties in the modeland design a robust controller using DK-iteration method. Thecontroller is applied on a full complexity simulation model andsimulations are performed for wind speed step changes.
I. INTRODUCTION
There is an increasing interest in wind energy and windturbines are the most common wind energy conversionsystems (WECS). Control is an essential part of the windturbine system because control methods can decrease thecost of energy by increasing the power capture by keepingthe turbine close to its maximum efficiency and also byreducing structural loadings and therefore increasing lifetimeof the wind turbine. Wind turbines essentially have tworegions of operation, partial load and full load. In the partialload wind speed is not fast enough to produce rated power.In this region the main control objective is to track themaximum power coefficient(CPmax) and extract as muchpower as possible. Pitch is mostly fixed in this region andgenerator reaction torque is adjusted to control rotationalspeed and keep the operating point close to CPmax. Inthe full load region wind speed is above rated and windpower exceeds rated power of the generator, therefore bydecreasing aerodynamics efficiency of the rotor we try tocontrol the captured power and this is done by pitching theblades. There are several methods for wind turbine controlranging from classical control methods [1] which are themost used method in real applications to advanced andmodel based control methods which have been the focusof research in the past few years [2]. Gain scheduling [3],adaptive control [4], time invariant MIMO methods [5],nonlinear control [6], robust control [7], model predictive
This work is supported by the CASED Project funded by grant DSF-09-063197 of the Danish Council for Strategic Research.
M. Mirzaei is with department of Informatics and Mathematical Mod-eling, Technical University of Denmark, 2800 Kongens Lyngby, [email protected]
H. H. Niemann is with department of Electrical Engineering,Technical University of Denmark, 2800 Kongens Lyngby, [email protected]
N. K. Poulsen is with department of Informatics and Mathematical Mod-eling, Technical University of Denmark, 2800 Kongens Lyngby, [email protected]
Wind
AerodynamicsPitch Servo Structure
Drivetrain
GeneratorGen. Servo
vfw
veFTθ
vt
τ Pout
τr ωr
ωg τg
θref
τref
Fig. 1: Wind turbine subsystems
control [8] are to mention a few. Advanced control methodsare thought to be the future of wind turbine control as theycan employ new generations of sensors on wind turbines(e.g. LIDAR [9]), new generation of actuators (e.g. trailingedge flaps [10]) and also conveniently treat the turbine asa MIMO system. The last feature seems to become moreimportant than before as wind turbines become bigger andmore flexible which make decoupling different modes anddesigning controller for each mode more difficult. The windturbine in this paper is treated as a MIMO system withpitch reference(θref ) and generator reaction torque(Qref ) asinputs and rotor rotational speed(ωr), generator rotationalspeed(ωg) and generated power(Pe) as outputs. This paperis organized as follows: In the section II modeling of thewind turbine including modeling for wind speed estimation,linearization and uncertainty modeling is addressed. In thesection III-A controller design is explain and in the sectionIV robust performance problems is addressed. And finally inthe section V simulation results are presented.
II. MODELING OF THE WIND TURBINE
For modeling purposes, the whole wind turbine can bedivided into 4 subsystems: Aerodynamics subsystem, struc-tural subsystem, electrical subsystem and actuator subsystem.Figure 1 shows the basic subsystems and their interactions.The dominant dynamics of the wind turbine come fromits flexible structure. Several degrees of freedom could beconsidered to model the flexible structure, but for controldesign mostly just a few important degrees of freedom areconsidered. Mostly the degrees of freedom whose eigenfrequncies lie inside actuator bandwidth are considered oth-erwise including them into the design model is uselessand makes the design model unnecessarily complicated.In this work we only consider two degrees of freedom,
2011 IEEE International Conference on Control Applications (CCA)Part of 2011 IEEE Multi-Conference on Systems and ControlDenver, CO, USA. September 28-30, 2011
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namely the rotational degree of freedom(DOF) and drivetraintorsion. The aerodynamics subsystem in the model getseffective wind speed(ve), pitch angle(θ) and rotational speedof the rotor(ωr) and returns aerodynamic torque (Tr) andthrust(FT ). This subsystem is responsible for the nonlinearityin the wind turbine model. More details are presented in thesection II-B.
A. Wind Model
Wind model can be modeled as a complicated nonlinearstochastic process, however for practical purposes it couldbe approximated based on a linear model. In this model thewind has two elements, mean value term(vm) and turbulentterm(vt):
ve = vm + vt
The turbulent term could be modeled by the following statespace model:[
vtvt
]=
[0 1
− 1p1(vm)p2(vm) −p1(vm)+p2(vm)
p1(vm)p2(vm)
] [vtvt
]+[
0k(vm)
p1(vm)p2(vm)
]e, e ∈ N(0, 1)
(1)
The parameters k(vm), p1(vm) and p2(vm) are estimated byapproximating wind power distribution and as it is indicated,they are dependent on the mean wind speed (vm).
B. Nonlinear Model
Blade element momentum(BEM) theory [11] is used tocalculate aerodynamic torque and thrust on the wind turbine.This theory explains how torque and thrust are related towind speed, blade pitch angle and rotational speed of therotor with the following formulas:
Qr =1
2
1
ωrρπR2v3
eCP (θ, ω, ve)
Qt =1
2ρπR2v2
eCT (θ, ω, ve)
In which Qr and Qt are aerodynamic torque and thrust, ρis air density, ωr is rotor rotational speed, ve is effectivewind speed, CP is the power coefficient and CT is thethrust force coefficient. For the sake of simplicity, insteadof presenting these two coefficients as functions of threevariables ω, ve and θ, they are presented as a function oftwo variables λ and θ in which λ = Rω
veand it is called tip
speed ratio. As we have not used individual pitch in this workabsolute angular position of the rotor and generator are of nointerest to us, therefore we use ψ = θr−θg instead which isthe drivetrain torsion. Having aerodynamic torque the wholesystem equation with 2 degrees of freedom becomes:
Jrωr = Qr − c(ωr −ωgNg
)− kψ
(NgJg)ωg = c(ωr −ωgNg
) + kψ −NgQg(2)
In which Jr and Jg are rotor and generator moments ofinertia, ψ is the drivetrain torsion, c and k are the drivetrain
damping and stiffness factors respectively lumped in thelow speed side of the shaft. For numerical values of theseparameters and other parameters given in this paper, werefer the reader to [12]. These equations give us a nonlinearmodel however our control design method is based on linearmodels, therefore we need to linearize the nonlinear modelof the system which could be easily achieve using Taylorexpansions around the operating points.
C. Uncertain Model
As it was mentioned, for control design we need to havea linear model of the system and the following model of thewind turbine is used: xy∆
y
=
A B1 B2
C1 D11 D12
C2 D21 D22
xu∆
u
In which states, inputs and outputs are:
x =[ωr ωg ψ θ Qg ve ve
]Tu =
[θref Qref
]Ty =
[ωr ωg Pe
]Tωr is rotational speed of the rotor, ωg is rotational speed ofthe generator, ψ is drivetrain deflection, θ pitch of the blade,Qg is the generator reaction torque, ve and ve are wind modelstates, θref is the reference value for pitch angle and Qrefis the reference value for the generator reaction torque andPe is the electrical power. System equations are:
ωr =a− cJr
ωr +c
Jrωg −
k
Jrψ + b1θ + b2ve
ωg =c
NgJgωr −
c
N2g Jg
ωg +k
NgJgψ − Qg
Jg
ψ = ωr −ωgNg
θ = − 1
τθθ +
1
τθθref
Qg = − 1
τgQg +
1
τgQref Pe = Qg0ωg + ωg0Qg
ve = − 1
p1p2ve −
p1 + p2
p1p2ve +
k
p1p2e
There are always discrepancies between real system andmathematical models, which lead to uncertain models. Inthis work, sources of uncertainties are taken to be:• Uncertainty in the drivetrain stiffness and damping
parameters.• Uncertainty in the linearized model.
Uncertainty in the linearized model could be a result ofapproximate CP curve calculations, wrong wind speed esti-mation which results in picking the wrong operating point oraerodynamic changes due to blade flexibility or ice coatingson the blades. Multiplicative uncertainty is used to representthe uncertain parameters. The uncertainty matrix becomes:
uaub1ukuc
=
δa 0 0 00 δb1 0 00 0 δk 00 0 0 δc
yayb1ykyc
1494
A B1 B2
C1 D11 D12
C2 D21 D22
∫∫
ωrωg
y∆u∆
ve
ω∗gθrefTref
Pe
P ∗e
P
Fig. 2: System interconnections
y∆ =[ya yb1 yk yc
]Tis the uncertainty output, y is
the output, u∆ =[ua ub1 uk uc
]Tis the uncertainty
input and u is the input. Now having system equations, wecan make the interconnection matrix P (see figure 2):y∆
zy
= P
u∆
du
D. Simulation Model
The FAST (Fatigue, Aerodynamics, Structures, and Tur-bulence) code [13] is used as the simulation model and the5MW reference wind turbine is used as the plant [12]. In thesimulation model 10 degrees of freedom are enabled whichare: generator, drivetrain torsion, 1st and 2nd tower fore-aft,1st and 2nd tower side-side, 1st and 2nd blade flapwise, 1stblade edgewise degrees of freedom.
E. Wind Speed Estimation
Based on the nonlinear model given in (2) and the windmodel given in (1) an extended Kalman filter is designed toestimate the effective wind speed. This wind speed is usedto find the operating point of the wind turbine (θ∗, λ∗ andC∗p ) and linearize the nonlinear model.
III. CONTROLLER DESIGN
A. Control Objectives
The most basic control objective of a wind turbine is tomaximize power capture in the turbine life time, which thisin turn means maximizing power captured from the windand prolonging life time of the wind turbine by minimizingthe fatigue loads. Generally maximizing power capture isconsidered in the partial load and minimizing fatigue loadsis mainly considered above rated. As we are operating inthe full load region in this work, we have considered thesecond objective. Control objectives are formulated in theform of weighting functions on input disturbances(d) andexogenous outputs(z). In order to avoid high frequencyactivity of the actuators, we have put high pass filter oncontrol signals to punish high frequency actions. Also wehave setup low pass filters to punish low frequency of thesystem outputs as their high frequency dynamics are outsideof our actuator bandwidth and we can not control them
PWi Wo
K
∆ W∆
∆P
z′d′u∆ y∆
yu
N(K)
Fig. 3: System setup for robust performance problem
anyway. For regulating power and rotational speed, Pe−P ∗eand
∫ωg − ω∗g and for minimizing fatigue loads on the
drivetrain ωg −Ngωr are punished. The resulting controlleris a dynamical system with measurements y as its inputs andcontrol signals u as its outputs:
xc = Acxc +Bcy
u = Ccxc
IV. ROBUST PERFORMANCE PROBLEM
A. Theory
Robust performance means that the performance objectiveis satisfied for all possible plants in the uncertainty set.The robust performance condition can be cast into a robuststability problem with an additional perturbation block thatdefines H∞ performance specifications [14]. The structuredsingular value µ is a very powerful tool for the analysis ofrobust performance with a given controller. However this isan analysis tool, in order to design a controller, we need asynthesis tool. A scaled version of the upper bound of µ isused for controller synthesis. The problem is formulated inthe following form:
µ∆(N(K)) ≤ minD∈D
σ(DN(K)D−1)
Now, the synthesis problem can be cast into the followingoptimization problem in which one tries to to find a controllerthat minimizes the peak value over frequency of this upperbound:
minK∈K
(minD∈D
‖ DN(K)D−1 ‖∞)
This problem is solved by an iterative approach which iscalled DK-iteration. For detailed explanations on the methodand notations the reader is referred to [14].
B. Implementation
We have used µ-Synthesis toolbox [15] to implement theDK-Iteration algorithm. W∆ is used to scale the ∆ matrix.We have taken uncertainty of 10% of the nominal values fordrivetrain stiffness and damping coefficients and 20% for
1495
100−22
−20
−18
−16
−14
−12
Mag
nitud
e (d
B)
Wi1
100−7.5
−7
−6.5
−6
−5.5
−5Wi2
100−80
−60
−40
−20
0
Mag
nitud
e (d
B)
Wo1
100−100
−50
0
50Wo2
100−5.5
−5
−4.5
−4
−3.5
−3Wo3
100−60
−50
−40
−30
−20
−10
Mag
nitud
e (d
B)
Wo4
100−20
0
20
40
Mag
nitud
e (d
B)
Wo5
Fig. 4: Bode plots for performance specifications(y-axis is indB and x-axis is in rad/s)
the linearization parameters therefore the weighting matrixbecomes:
W∆ = diag(0.2, 0.2, 0.1, 0.1)
∆P (scaled by Wi and Wo matrices) defines performance ofthe system in the form of a complex perturbation matrix.Wi and Wo are frequency dependent weight matrices ondisturbances and exogenous outputs respectively of the form:
Wo = diag(Wo1, . . . ,Wo5)
Wi = diag(Wi1,Wi2)
Bode plots of the weighting functions are given in the figure4. Figure 4 shows bode plots of weighting functions. Inputdisturbances (d) to the system are:
d =
[veω∗g
]Wind SpeedRotor rotation reference
And exogenous outputs (z) are:
z =
θrefQref
ω∗r −ωg
Ng∫ω∗g − ωg∫P ∗e − Pe
Pitch referenceGenerator reaction torque referenceDeflection of the drivetrainIntegral on rotational speed errorIntegral on generated power error
These weightings are used to specify performance of thesystem. As we have parametric uncertainties in the plant and
10−2
10−1
100
101
102
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1CLOSED−LOOP MIXED MU: CONTROLLER #3
FREQUENCY (rad/s)
Fig. 5: Closed loop mixed µ
Iteration number 1 2 3Controller Order 19 19 19γ Acheived 9682006.84 45.289 7.347Peak µ-Value 2482.23 0.865 0.808
TABLE I: DK-iteration summery
complex perturbation for performancs, mixed µ is used todesign the controller. The resulting mixed-µ is given in figure5 and the iteration summery is given in the table I. Theobtained controller is of the order 19, and has maximum gainof 15.86dB. As high order controllers are problematic in thereal implementations, we have used balanced order reduction[16] to reduce its order to 10. Hankel singular values of thecontroller are shown in figure 6 and the jump from order10 to 11 is found a reasonable place for controller orderreduction.
V. SIMULATION RESULTS
In this section simulation results for the obtained controllerare presented. The controller is implemented in MATLABand tested on full complexity FAST model of the referencewind turbine [12]. As it is mentioned in section II-E we haveaugmented model of wind turbine with a stochastic windmodel, however in order to make evaluation of the controlleron nominal and worst case, we have used simulations withstep changes in the wind speed.
0 5 10 15 2010
−10
10−5
100
105
Fig. 6: Hankel singular values of the controller
1496
600 700 800 900 1000 1100 1200 130013
14
15
16
17Wi
nd S
peed
(m/
s)
time(seconds)
Fig. 7: Wind speed
600 700 800 900 1000 1100 1200 13007
8
9
10
11
12
13
14
θ (d
egre
es)
time(seconds)
Fig. 8: Blade-pitch reference
A. Robust performance simulations
In this section simulation results of a step change in windspeed is presented. Control inputs which are pitch referenceθref and generator reaction torque reference Tref alongwith system outputs which are rotor rotational speed ωr andelectrical power Pe are plotted in figues 7-11.
B. Simulation for the worst case
In this section worst case scenarios, in which all the uncer-tainties are taken to be the maximum values, are presented.To do so, wind speed is taken to be 2m/s away fromthe linearization point and nominal values of the drivetrainstiffness and damping are replaced by the following values:
k = k(1 + Pkδk) for δk = ±1 & Pk = 0.1
c = c(1 + Pcδc) for δc = ±1 & Pc = 0.1
As it is seen in figures 12 and 13, in the worst cases thesystem becomes oscillatory but it maintains a reasonableperformance.
VI. CONCLUSION
In this paper we solved the problem of robust control ofa wind turbine using DK-iteration technique. The controlleris designed for the full load region, and an extension of thiswork would be to solve the problem for partial load too.Parametric uncertainty is considered in the uncertain modeland then we have used µ-synthesis method to design the
600 700 800 900 1000 1100 1200 1300
4.07
4.14x 10
4
T g (
NM)
time(seconds)
Fig. 9: Generator-torque reference
600 700 800 900 1000 1100 1200 130011.9
12.1
12.3
ω r (r
pm)
time(seconds)
Fig. 10: Rotor rotational speed(ωr)
600 700 800 900 1000 1100 1200 13004.9
5
5.1x 10
6
P e (
Watt
)
time(seconds)
Fig. 11: Electrical power
controller. The full model with augmented wind model is ofthe order 8 and the resulting controller is of the order 19,however balanced truncation model order reduction is usedto reduce order of the controller to 10. The final controller isimplemented on a FAST simulation model with 10 degreesof freedom and simulations with wind speed step changesare done for nominal plant and worst case plant. The resultssuggest that the controller can handle nominal case prettywell and the worst case with a little loss of performance.
1497
300 400 500 60012
13.5
15
Wind Speed (m/s)
time(seconds)
(a) Wind speed
300 400 500 6006
7
8
9
10
θ (degrees)
time(seconds)
(b) Pitch reference
300 400 500 600
4.07
4.14x 104
Tg (NM)
time(seconds)
(c) Generator reaction torque refer-ence
300 400 500 60011.9
12.1
12.3
ωr (rpm)
time(seconds)
(d) Rotor rotational speed
300 400 500 6004.9
5
5.1x 106
Pe (Watt)
time(seconds)
(e) Electrical power
Fig. 12: Worst case scenario with +2m/s wind speedestimation error
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300 400 500 60015
16.5
18
Wind Speed (m/s)
time(seconds)
(a) Wind speed
300 400 500 60010
11
12
13
14
15
θ (degrees)
time(seconds)
(b) Pitch reference
300 400 500 600
4.07
4.14x 104
Tg (NM)
time(seconds)
(c) Generator reaction torque refer-ence
300 400 500 60011.9
12.1
12.3
ωr (rpm)
time(seconds)
(d) Rotor rotational speed
300 400 500 6004.9
5
5.1x 106
Pe (Watt)
time(seconds)
(e) Electrical power
Fig. 13: Worst case scenario with −2m/s wind speedestimation error
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