Structured LPV Control of Wind...

Structured LPV Control of Wind Turbines

Fabiano Daher Adegas, Jakob Stoustrup, Torben [email protected]

Department of Electronic Systems,Aalborg University,

Denmark

November 29, 2011

mailto:[email protected]

25

Structured LPVControl of Wind

TurbinesFabiano Daher Adegas,

Jakob Stoustrup,Torben Knudsen

Introduction

Wind Turbine LPVModel

System and ControllerDescription

Iterative LMIAlgorithm

LPV PI Control forHigh Wind Speeds

Acknowledgement

Dept. of Electronic Systems,Aalborg University,

Denmark

Agenda

Introduction

Wind Turbine LPV Model

System and Controller Description

Iterative LMI Algorithm

LPV PI Control for High Wind Speeds

Acknowledgement

25




3 Introduction





Acknowledgement


Denmark

IntroductionMotivation

Main challenges for the application of wind turbine control:I Known parameter-dependencies (gain-scheduling);I Unknown parameter variations (robustness);I Faults (reconfiguration);I Simple implementation.

Structured Linear Parameter Varying Control of Wind Turbines 3

Wind turbine

Wind speedLPV controller

w(k)y(k)

u(k)

θop(t)

θf(t) estimator

systemFault diagnosis

Fig. 1: Block diagram of the controller structures. The black boxes are common tothe LPV controllers, while the red dashed box illustrates the fault diagnosis systemrequired by the AFTC.

• Prevent catastrophic failures and faults from deteriorating other parts of the windturbine, by early fault detection and accommodation.

• Reduce maintenance costs by providing remote diagnostic details and avoidingreplacement of functional parts, by applying condition-based maintenance in-stead of time-based maintenance.

• Increase energy production when a fault has occurred by means of fault-tolerantcontrol.

This chapter gives an overview of the most common faults that can be modelledas varying parameters. For a clear exposure, the fault-tolerant controller is designedto cope with the simple case of a single fault: altered dynamics of the hydraulicpitch system due to low hydraulic pressure. The fault is a gradual fault affectingthe control actions of the turbine. The method used also applies to fast parametervariations, i.e. abrupt faults in the extreme case [12].

Realizing advanced gain-scheduled controllers can in practice be difficult andmay lead to numerical challenges [20, 19]. Several plant and controller matricesmust be stored on the controller memory. Moreover, matrix factorizations and in-versions are among the operations that must be done online by the controller at eachsampling time [4, 6].

The synthesis procedures presented in this chapter are serious candidates forsolving a majority of practical wind turbine control problems, provided a sufficientlygood model of the wind turbine is available. We believe that the resulting controllercan also be easily implemented in practice due to the following reasons:

1. Structured controller: the controller structure can be chosen arbitrarily. De-centralized of any order, dynamic (full or reduced-order) output feedback, staticoutput, and full state feedback are among the possible structures. This is in linewith the current control philosophy within wind industry.

2. Low data storage: the required data to be stored in the control computer memoryis only the controller matrices, and scalar functions of the scheduling variablesrepresenting plant nonlinearities (basis functions).

25




4 Introduction





Acknowledgement


Denmark

IntroductionMotivation

Linear parameter-varying (LPV) modeling and control for practicalwind turbine control problems.

minimize ‖Tz→w (θ, α,K )‖i,2

K ∈ K

whereI K represents a structural constraint on the controller;I θ is a vector of time-varying scheduling parameters;I α is a vector of uncertain parameters.

25




Introduction

5 Wind Turbine LPVModel




Acknowledgement


Denmark

Wind Turbine LPV ModelNominal Model

6 Fabiano Daher Adegas, Christoffer Sloth, Jakob Stoustrup

Pitch Systemβref(t)

Vw(t)∑

Pg(t)

Qg,ref(t)Ωg(t)

Qg(t)Ωr(t)

Qa(t)

Ta(t)

β (t)

V (t)

q(t)

Tower

Generator

Converter&Drive trainAerodynamics

Fig. 3: Sub-model-level block diagram of a variable-speed variable-pitch WT.

Chord line

RotorplaneV (t)(1−a)

W (r, t)

rΩr(t)(1+a′)

ϕ

αϕ

β

fT

fQ

Fig. 4: Forces on a blade element.

fQ =12

ρc(r)W 2(r, t)(

CL(r,α(r, t))sinϕ(r, t)−CD(r,α(r, t))cosϕ(r, t))

[N]

fT =12

ρc(r)W 2(r, t)(

CL(r,α(r, t))sinϕ(r, t)+CD(r,α(r, t))cosϕ(r, t))

[N]

with the squared local inflow velocity W 2(r, t), local angle of attack α(r, t) and localinflow angle ϕ(r, t) described as,

W 2(r, t) = (V (t)(1−a(r)))2 +(rΩr(t)(1+a′(r))

)2, [m2/s2]

α(r, t) = ϕ(r, t)−φ(r)−β (t), []

ϕ(r, t) = tan−1(

V (t)(1−a(r))(rΩr(t)(1+a′(r))

)−1)

[].

In the above expressions, ρ is the air density, c(r) is the local chord length, CL(r,α)and CD(r,α) are the local steady-state lift and drag coefficients, V (t) is a mean windspeed over the rotor disk, Ωr(t) is the rotor speed, a(r) and a′(r) are the axial and

I BEM aerodynamics (static);I Flexible two-mass drive-train;I Fore-aft tower translation (first bending mode);I Second-order pitch system;I First order torque delay.

25




Introduction





Acknowledgement


Denmark

Wind Turbine LPV ModelAerodynamics

Linearized torque and thrust equations,

Qa(t) ≈ Qθop +∂Qa∂V

∣∣∣∣θop

V (t) +∂Qa∂Ωr

∣∣∣∣θop

Ωr(t) +∂Qa∂β

∣∣∣∣θop

β(t)

Ta(t) ≈ T θop +∂Ta∂V

∣∣∣∣θop

V (t) +∂Ta∂Ωr

∣∣∣∣θop

Ωr(t) +∂Ta∂β

∣∣∣∣θop

β(t)

5 10 15 20 25−6

−4

−2

0x 10

6

V [m/s]

∂Q

a/∂Ω

r[kNm/rad/s]

5 10 15 20 251

2

3

4

5x 10

5

V [m/s]

∂Q

a/∂V

[kNm/rad]

5 10 15 20 25−3

−2

−1

0

1x 10

7

V [m/s]

∂Q

a/∂β

[kNm/rad]

5 10 15 20 25−4

−2

0

2x 10

5

V [m/s]

∂Ta/∂Ω

r[kN/rad/s]

5 10 15 20 252

2.5

3

3.5

4

4.5x 10

4

V [m/s]

∂Ta/∂V

[kNm/m/s]

5 10 15 20 25−2.5

−2

−1.5

−1

−0.5

0x 10

6

V [m/s]

∂Ta/∂β

[kNm/rad]

25




Introduction





Acknowledgement


Denmark

Wind Turbine LPV ModelAerodynamic Uncertainty

Simplification of the aerodynamic phenomena:I Blade Element Momentum (BEM) codes;I Neglected dynamics (e.g. dynamic inflow);I Deviations from the operating points.

∂Q∂β

(θ, α) :=∂Q∂β

∣∣∣∣θop

+ fl (α), fl (α) := al + blα

where al , bl characterizes the additive uncertainty for the l-thaerodynamic gain and α is an uncertainty parameter.

Λ = α : α ≤ α ≤ α .

12 14 16 18 20 22 24 26−4

−3

−2

−1

0x 10

7

V [m/s]

∂Q

∂β[N

m/rad]

α = 0α = α

α = α

25




Introduction





Acknowledgement


Denmark

Wind Turbine LPV ModelFaults

Failures that gradually change system’s dynamics:I Bias and proportional error in sensors: pitch angle, generator

speed;I Offset of the generated torque due to an offset in the internal

power converter control loops;I Reduction in conversion efficiency;I Altered dynamics of pitch system (Pressure drop, pump wear,

high air content in the oil);A comprehensive list of wind turbine faults is given in Sloth et al,2009, Odgaard and Stoustrup, 2009, Adegas et al, 2011.

25




Introduction





Acknowledgement


Denmark

Wind Turbine LPV ModelFaults

Example: Pitch systemDamping ratio and natural frequency from their nominal values ζ0and ωn,0 to their faulty values ζf and ωn,f. Convex combination ofthe vertices of the parameter sets,

β(t) = -2ζ(θf)ωn(θf)β(t)− ω2n(θf)β(t) + ω2

n(θf)βref(t)

ω2n(θf) = (1− θf)ω

2n,0 + θfω

2n,lp

-2ζ(θf)ωn(θf) = -2(1− θf)ζ0ωn,0 − 2θfζlpωn,lp

where θf ∈ [0, 1] is an scheduled indicator for the fault.

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

Time [s]

Pitch

angle,β[]

25




Introduction


10 System and ControllerDescription



Acknowledgement


Denmark

System and Controller DescriptionAugmented System

Discrete-time LPV system obtained by discretization (Bilinear) ofcontinous-time counterpart,

x(k + 1) = A(θ, α)x(k) + Bw (θ, α)w(k) + Bu(θ, α)u(k)

z(k) = Cz (θ, α)x(k) + Dzw (θ, α)w(k) + Dzu(θ, α)u(k)

y(k) = Cy (θ, α)x(k) + Dyw (θ, α)w(k).

Affine in scalar functions ρi (θ) known as basis functions and θf ,m. A Bw BuCz Dzw DzuCy Dyw 0

0

+∑

i

A Bw BuCz Dzw DzuCy Dyw 0

i

(ρi (θ) + fi (α))

+∑

m

A Bw BuCz Dzw DzuCy Dyw 0

m

θf ,m, i = 1, . . . , nρ, m = 1, . . . , nθf .

25




Introduction





Acknowledgement


Denmark

System and Controller DescriptionAugmented System

The aerodynamic gains are natural candidates for ρi (θ),

ρ1(θ) :=1Jr

∂Qa∂Ω

∣∣∣∣θop

ρ4(θ) :=1

Mt

∂Ta∂Ω

∣∣∣∣θop

ρ2(θ) :=1Jr

∂Qa∂V

∣∣∣∣θop

ρ5(θ) :=1

Mt

∂Ta∂V

∣∣∣∣θop

ρ3(θ) :=1Jr

∂Qa∂β

∣∣∣∣θop

ρ6(θ) :=1

Mt

∂Ta∂β

∣∣∣∣θop

where the division by Jr and Mt is adopted to improve numericalconditioning.

25




Introduction





Acknowledgement


Denmark

System and Controller DescriptionLPV Controller

The LPV controller has the form,xc(k + 1) = Ac(θ)xc(k) + Bc(θ)y(k)

u(k) = Cc(θ)xc(k) + Dc(θ)y(k),

Controller matrices are continuous functions of θ with similar typeof dependence,

Ac(θ) = Ac,0 +

nθ∑i=1

ρi (θ)Ac,i +

nθf∑i=1

θf,iAc,nρ+i ,

Bc(θ) = Bc,0 +

nθ∑i=1

ρi (θ)Bc,i +

nθf∑i=1

θf,iBc,nρ+i ,

Cc(θ) = Cc,0 +

nθ∑i=1

ρi (θ)Cc,i +

nθf∑i=1

θf,iCc,nρ+i ,

Dc(θ) = Dc,0 +

nθ∑i=1

ρi (θ)Dc,i +

nθf∑i=1

θf,iDc,nρ+i .

25




Introduction





Acknowledgement


Denmark

System and Controller DescriptionClosed-Loop LPV System

The controller matrices can be represented in a compact way,

K (θ) :=

[Dc(θ) Cc(θ)Bc(θ) Ac(θ)

].

The interconnection of system and controller leads to the followingclosed-loop LPV system denoted Scl ,

Scl : xcl(k + 1) = A(θ, α,K (θ))xcl(k) + B(θ, α,K (θ))w(k)

z(k) = C(θ, α,K (θ))xcl(k) +D(θ, α,K (θ))w(k).

θ ranges over a hyperrectangle denoted Θ,

Θ =θ : θi ≤ θi ≤ θi , i = 1, . . . , nθ

.

Rate of variation ∆θ = θ(k + 1)− θ(k) belongs to a hypercubedenoted V,

V = ∆θ : |∆θi | ≤ vi , i = 1, . . . , nθ .

25

Iterative LMI AlgorithmInduced L2-gain

Proposition (L2-gain)If there exist K (θ), P(θ, α) = P(θ, α)T and Q(θ) satisfying,

r2P(θ + ∆θ, α) A(θ, α,K (θ))Q(θ) B(θ, α,K (θ)) 0? −P(θ, α) +Q(θ)T + ? 0 Q(θ)TC(θ, α,K (θ))T

? ? γI D(θ, α,K (θ))T

? ? ? γI

> 0

with r = 1, ∀ (θ,∆θ, α) ∈ Θ× V × Λ, then the system Scl isexponentially stable and ‖Tzw (θ, α)‖2 < γ.

25




Introduction



15 Iterative LMIAlgorithm


Acknowledgement


Denmark

Iterative LMI AlgorithmLyapunov and Slack Matrices

The Lyapunov and slack variables are here defined affine functionsof the basis functions,

P(θ, α) = P0 +

nθ∑i=1

(ρi (θ) + fi (α)) Pi +

nθf∑i=1

θf ,iPnρ+i

Q(θ) = Q0 +

nθ∑i=1

ρi (θ)Qi +

nθf∑i=1

θf ,iQnρ+i

The Lyapunov function at θ+ := θ + ∆θ can be described as,

P(θ+, α) = P0 +

nθ∑i=1

(ρi (θ

+) + fi (α))

Pi +

nθf∑i=1

(θ+

f ,i

)Pnθ+i .

Conveniently, the basis functions at θ+ are approximated by alinear function of ρ(θ) and ∆θ,

ρi (θ+) := ρi (θ) +

∂ρi (θ)

∂θ∆θ,

25




Introduction



16 Iterative LMIAlgorithm


Acknowledgement


Denmark

Iterative LMI AlgorithmIteration Scheme

I Sequence of LMI problems:Q(θ)j = P(θ)j−1;

I Gridded parameter spacesubset denoted Θg ⊂ Θ. LMIchecked atΘg × Vert(V)× Vert(Λ) ateach iteration;

I Minimization of performancelevel γ;

I Feasibility phase creates aconvergent sequence rj thattries to tend 1.

0 10 20 30 40 50 60 70 80 90 100 1100.5

1

1.5

2

r0 10 20 30 40 50 60 70 80 90 100 110

0

5

10

15

γ

20 30 40 50 60 70 80 90 100 1102.5

3

3.5

4

4.5

5

Iteration (j)

γ

Feasibility Cost Optimization

25




Introduction




17 LPV PI Control forHigh Wind Speeds

Acknowledgement


Denmark

Numerical ExampleFault-Tolerant PI LPV Control for High Wind Speeds

G (s, )p

k (p )

v

ref

g

k (q

Qg

k 2 s (1 + s )dt dt dt

s + 2 s +2

dt dt dt

¶Qg

¶g

q

(s + z )1

s

k2

k (s + z )3 3

(s + p )3

z

w= s + 2 s +2

2

(s + z )(s + )1 2

k(i

u=

y

^^

^

^

LPV Controller

Classical Design

WeightingFunctions

25




Introduction





Acknowledgement


Denmark


Considering Gp augmented with the integrator filter as the plantfor synthesis purposes, the LPV controller structure reduces to aparameter-dependent static output feedback of the form,

K (θ) = Dc,0 +6∑

i=1ρi (θ)Dc,i + θfDc,7,

Dc,n :=[Dp,n Di,n Dq,n

], n = 0, 1, . . . , 7.

Initial K (θ) based on analytical pole placement (tower fore aftmode neglected).

kp(θ) =

2ξΩωΩ

(Jr + N2

g Jg)− Ng

∂Qg∂Ωg

+ ρ1(θ)

−Ngρ3(θ),

ki(θ) =ω2

Ω

(1 + ξ2

Ω

) (Jr + N2

g Jg)

−Ngρ3(θ)

The tower feedback gain of the initial controller is kq(θ) = 0,meaning no active tower damping.

25




Introduction





Acknowledgement


Denmark


0

2

4γj

0.4

0.6

0.8

kj

p

0.2

0.3

0.4

kj

i

0 10 20 30 40 50 60 70 80 900

0.05

0.1

j (Iterations)

kj

q

Figure: Evolution of performance level γ and controller gains kp, ki, kqduring the iterative LMI synthesis. Controller gains computed atθop = 15 m/s, θf = 0.

25




Introduction





Acknowledgement


Denmark


PI and Tower Feedback Gains

0

0.20.4

0.60.8

1

10

15

20

25

0.2

0.4

0.6

0.8

1

1.2

Vθf

kp

0

0.20.4

0.60.8

1

10

15

20

25

0

0.1

0.2

0.3

0.4

0.5

Vθf

ki

0

0.20.4

0.60.8

1

10

15

20

25

0.04

0.06

0.08

0.1

0.12

Vθf

kq

25




Introduction





Acknowledgement


Denmark


0 100 200 300 400 500 6008

10

12

14

16

18

20

22

24

26

Time [s]

v hub[m

/s]

(d)

100 200 300 400 500 6001.48

1.5

1.52

1.54

1.56

1.58

1.6

1.62

1.64

Time [s]

Ωr

[rad/s]

Pole Placement LPV PI w/ tower damping

(e)

100 200 300 400 500 6000.1

0.15

0.2

0.25

0.3

0.35

0.4

Time [s]

q[m

]


(f)

100 200 300 400 500 600

−0.1

−0.05

0

0.05

0.1

0.15

Time [s]

q[rad/s]


(g)

25




Introduction





Acknowledgement


Denmark


100 200 300 400 500 6008

10

12

14

16

18

20

22

Time [s]

β[deg]


(h)

100 200 300 400 500 600−3

−2

−1

0

1

2

Time [s]

β[deg/s]


(i)

100 200 300 400 500 6001.94

1.95

1.96

1.97

1.98

1.99

2

2.01

2.02x 10

6

Time [s]

Pg

[kW

]


(j)

25




Introduction





Acknowledgement


Denmark


0 100 200 300 400 500 6008

10

12

14

16

18

20

22

24

26

Time [s]

v hub[m

/s]

50 100 150 200 250 300 350 400 450 500 550 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [s]

θ f[−

]

50 100 150 200 250 300 350 400 450 500 550 600

1.5

1.55

1.6

1.65

1.7

Time [s]

ωr

[rad/s]

PI LPV PI LPV (Fault−Tolerant)

50 100 150 200 250 300 350 400 450 500 550 6001.85

1.9

1.95

2

2.05

2.1x 106

Time [s]

Pg[kW

]


25




Introduction





Acknowledgement


Denmark


50 100 150 200 250 300 350 400 450 500 550 6000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Time [s]

q[m

]


50 100 150 200 250 300 350 400 450 500 550 600−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Time [s]

q[rad/s]


50 100 150 200 250 300 350 400 450 500 550 6006

8

10

12

14

16

18

20

22

Time [s]

β[deg]


50 100 150 200 250 300 350 400 450 500 550 600−8

−6

−4

−2

0

2

4

6

8

Time [s]

β[deg

/s]

PI LPV

PI LPV (Fault−Tolerant)

25




Introduction





25 Acknowledgement


Denmark

Acknowledgement

Thank you !

Work sponsored by the Danish Ministry of Science, Technologyand Innovation under Project CASED (Concurrent

Aeroservoelastic Analysis and Design of Wind Turbines).

Structured LPV Control of Wind...

Documents

Transcript of Structured LPV Control of Wind...