uncertainty assessment and robust model predictive control...

Uncertainty Assessment and Robust ModelPredictive Control of a Wind Turbine

Mahmood Mirzaei, Niels K. Poulsen, Hans H. Niemann

DTU (Informatics, Electro)

November 27, 2011

(Technical University of Denmark) November 27, 2011 1 / 30

Outline

Wind turbine design as a constrained optimizationAero-Servo-Elastic design

I Aerodynamics spaceI Structural spaceI Control space

Limits of performance in control design: one of the constraints(How to push the constraints imposed by control)

Classical control methods vs. advanced control methods

Model uncertainty as one of the limits (limits of performance)

Robust MPC synthesis

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How big wind turbines should be?So far an exponential trend!

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Wind Turbine DesignAn optimization problem (roughly speaking):

max .

∫ T0 Pe(t)dt

Total costSubject to: Some constraints

Maximize produced energy per $!

Maximize Pe and T

Total cost: Manufacturing, installation, operation andmaintenanceConstraints:

I Aerodynamics, structural and manufacturing feasibilityI Limits of performance on control systems (Noisy sensors, actuator

bandwidth and limits, delays and non-min. phase systems)

Control can achieve what’s achievable!“even the best control system cannot make a Ferrari out of a

Volkswagen” [SP05]

I etc!(Technical University of Denmark) November 27, 2011 4 / 30

One of the constraints

Apparently control is one of the constraints!

1 What limits control performance? (we need to be aware of ourlimits)

2 Our goal is to choose a method and design controllers that canachieve what’s achievable:

I Classical control methods (being used!)I Advanced control methods (being researched!)

(Technical University of Denmark) November 27, 2011 5 / 30

Control systems: Limits of performance

Input-Output ControllabilityBigger turbines =⇒ more flexible structures =⇒ less I/Ocontrollability (with the same sensors and actuators)

Limitations imposed by RHP-zerosBigger turbines =⇒ more flexible structures =⇒ more interactions=⇒ more prominent RHP-zeros!

Limitations imposed by input constraints (inputs saturate, limited slewrates, etc.)

Limitations imposed by uncertaintyOur modeling is based on approximating PDEs with ODEs. Bigger windturbines =⇒ less accuracy in the approximation.

(Technical University of Denmark) November 27, 2011 6 / 30

Classical control methods

Based on:

1 Pairing inputs and outputs

2 Designing individual SISO controllers for each pair

3 Removing cross couplings (filters, feedforward etc.)

And different control (some times conflicting) objectives are achievedby different controllers.

Speed control

Power control

Load reduction (e.g. active tower damping)

However as wind turbines become bigger, designing individualcontrollers for each objective is getting more and more complex!

We might need one MIMO controller that can achieve all theobjectives together.

(Technical University of Denmark) November 27, 2011 7 / 30

Model Based Controllers

In order to push the constraints on performance of the control system,we need:

1 A MIMO controller

2 Better I/O controllability(by employing new sensors and actuators)

3 Handling input and state constraints

4 Taking into account uncertainty in the model

For the first three: Model Predictive ControlAnd to include the last one: Robust Model Predictive Control

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Model Predictive Control (MPC)

Steps:

1 Measurement and estimation

2 Solve the following optimization problem:

minU=uk+i|kN−1

i=0

`(xN ) +

N−1∑i=0

`(xk+i|k, uk+i|k)

Subject to xk+1 = f(xk, uk, dk)

uk+i|k ∈ Uxk+i|k ∈ X

3 Apply the first element of the input sequence.

4 Set k = k + 1 and go back to the first step.

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Model Predictive Control (MPC)

One important shortcoming of MPC:

It assumes a perfect model, which is always incorrect!

Why does nominal MPC work then?

Nominal MPC is based on feedback. And feedback is used tocounteract the effect of uncertainty!

We tune the controller to get it to work.

Question: How could we include imperfection in our modeling?Robust control design!

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Robust Control Design

Classical robust control methods:

H∞ loop-shaping

LQG and loop transfer recovery

etc

Robust MPC:

1 Uncertainty description2 Robust MPC synthesis:

I Nominal performance, robust constraints satisfactionI Robust performance

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Uncertainty Description-1

Impulse/Step-Response

Σ : y(t) =

N∑k=0

h(t)u(t− k)

S = Σ : h−t ≤ h(t) ≤ h+t

Structured Feedback Uncertainty

LTI system with uncertainty matrix in the feedback loop

Multi-Plant

Σ ∈ Σ1,Σ2, . . . ,Σn

(Technical University of Denmark) November 27, 2011 12 / 30

Uncertainty Description-2

Polytopic Uncertainty

xk+1 = A(θ)xk +B(θ)uk

yk = Cxk[A(θ) B(θ)

]∈ Ω

Bounded Input Disturbances

xk+1 = Axk +Buk + wk

yk = Cxk

wk ∈W

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Sources of uncertainty in our model

Our model (called WT1):

Rotational DOF + Drive train torsional DOF

Sources of uncertainty in modeling WT1

Uncertainty in parameters of the linearized modelImportant!

Uncertainty in the drivetrain parametersNot very important! Very small changes in the dynamical model

(Technical University of Denmark) November 27, 2011 14 / 30

Nonlinear model and linearization

Nonlinear model:

Jrωr = Qr − c(ωr −ωgNg

)− kψ

(NgJg)ωg = c(ωr −ωgNg

) + kψ −NgQg

ψ = ωr −ωgNg

Pe = Qgωg

Linearization:

∆Qr(ω, θ, ve) =∂Qr∂ω︸︷︷︸a(ve)

∆ω +∂Qr∂θ︸︷︷︸

b1(ve)

∆θ +∂Qr∂ve︸︷︷︸

b2(ve)

∆ve

∆Pe =∂Pe∂ωg︸︷︷︸Qg0

∆ωg +∂Pe∂Qg︸︷︷︸ωg0

∆Qg

(Technical University of Denmark) November 27, 2011 15 / 30

Linearized model using estimated wind speed

ωr =a(ve)− c

Jrωr +

c

Jrωg −

k

Jrψ + b1(ve)θ + b2(ve)ve (1)

ωg =c

NgJgωr −

c

N2g Jg

ωg +k

NgJgψ − Qg

Jg(2)

ψ = ωr −ωgNg

(3)

Pe = Qg0ωg + ωg0Qg (4)

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Parameters as a function of wind speed

Parameters of the linear system as functions of wind speed:

0 5 10 15 20 25−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0x 10

6

(a) a(ve)

0 5 10 15 20 25−20

−15

−10

−5

0

5x 10

4

(b) b(ve)

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Norm-bounded Uncertain Model(with uncertainty only in the gain matrix)

a(δ1) =a(1 + p1δ1) |δ1| ≤ 1 (5)

b1(δ2) =b1(1 + p2δ2) |δ2| ≤ 1 (6)

b2(δ3) =b2(1 + p3δ3) |δ3| ≤ 1 (7)

Mapping upper bound of confidence interval of ve on the parametersgives small p1. So we can consider the uncertainties only in b1 and b2.

xk+1 = Axk +B(∆k)uk (8)

yk = Cxk +Duk (9)

B(∆k) = B0 +Bp∆kCp, ∆k ∈∆ (10)

∆ = ∆ : ‖∆‖ ≤ 1 (11)

(Technical University of Denmark) November 27, 2011 18 / 30

Including robustness in MPC

Now that we have uncertain model, how do we formulate the controlproblem?

Define a nominal model and optimize nominal performancesubject to robust constraint satisfaction (i.e. constraints must besatisfied for any possible realization of uncertainty)

Optimize the worst-case performance (minimize on control inputs,maximize on uncertainty) which is called minimax problem.

Considering uncertainty only in the B matrix helps us a lot informulating a minimax MPC with tractable computation.

Good news! Because we might be able to solve the optimizationproblem in the sampling time!

(Technical University of Denmark) November 27, 2011 19 / 30

Minimax MPC(with soft constraints on inputs and input rates)

minu

max∆

∑N−1

j=0‖yk+j|k‖2Q + ‖uk+j|k‖2R+ (12)

‖υk+j|k‖2S1+ ‖ξk+j|k‖2S2

(13)

subject to xk+1|k = Axk|k +B(∆k)uk (14)

yk|k = Cxk|k +Duk (15)

uk+j|k ≤ Umax + υk+j|k (16)

uk+j|k ≥ Umin − υk+j|k (17)

∆uk+j|k ≤ ∆Umax + ξk+j|k (18)

∆uk+j|k ≥ ∆Umin − ξk+j|k (19)

ηk+j|k ≥ 0 (20)

ξk+j|k ≥ 0 (21)

(Technical University of Denmark) November 27, 2011 20 / 30

Stacked version of the minimax MPC

minU

max∆N

Y TQY+UTRU + ΥTS1Υ + ΞTS2Ξ (22)

subject to U ≤ Umax + Υ (23)

U ≥ Umin −Υ (24)

∆U ≤ ∆Umax + Ξ (25)

∆U ≥ ∆Umin − Ξ (26)

Υ ≥ 0 (27)

Ξ ≥ 0 (28)

In which:

Y = Φxxk|k−1 + ΓU (29)

Γ = Γ0 + Γ∆(∆N ) (30)

∆N =(∆1 ∆2 . . . ∆N

)T(31)

(Technical University of Denmark) November 27, 2011 21 / 30

Final optimization problem

mint,τ,U

tx + tu + tυ +∑N−1

j=0tj (32)

subject to

(tx xTk|k−1ΦT

x + UTΓT0? Q−1 −

∑N−1j=0 τjVjV

Tj

) 0 (33)(

tu UT

? R−1

) 0

(tυ ΥT

? S−1

) 0

(tj UTW T

j

? τjI

) 0 (34)

I−IΨ−Ψ

U −

Umax + Υ−Umin + Υ

∆Umax + I0uk−1 + Ξ−∆Umin − I0uk−1 + Ξ

≤ 0 (35)

τj ≥ 0 Υ ≥ 0 Ξ ≥ 0 (36)

(Technical University of Denmark) November 27, 2011 22 / 30

Control system configuration

So far only regulation around operating points! We need to solve theproblem of offset between desired output and measured output.Solution is the offset free reference tracking (OFRT block).

Simulation model

+ Robust MPC ve estimator

OFRT State estimator

uik

uk

yk

ve

x

xusk

x =(x d

)T(Technical University of Denmark) November 27, 2011 23 / 30

Simulations

Parameters RMPC PI

ωr standard deviation (RMP) 0.389 0.728Pe standard deviation (Watts) 6.598× 104 9.050× 104

Pe mean value(Watts) 4.997× 106 4.999× 106

Pitch standard deviation (degrees) 10.261 8.623Shaft moment standard deviation (N.M.) 0.840× 103 2.376× 103

Table: RMPC and PI performance comparison

(Technical University of Denmark) November 27, 2011 24 / 30

Simulations

0 200 400 600 8000

5

10

15

20

25

(c) Blade-pitch reference (degrees)

0 200 400 600 80038

39

40

41

42

43

(d) Generator-torque reference (kiloN.M.)

(Technical University of Denmark) November 27, 2011 25 / 30

Simulations

0 200 400 600 80011

11.5

12

12.5

13

(e) Rotor rotational speed (ωr) (rpm)

0 200 400 600 8004.85

4.9

4.95

5

5.05

5.1

5.15

(f) Electrical power (mega watts)

(Technical University of Denmark) November 27, 2011 26 / 30

Simulations

0 200 400 600 80010

15

20

25

(g) Wind speed (blue-solid), Estimated windspeed (red-dashed) (m/s)

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Thank you very muchfor your attention!

(Technical University of Denmark) November 27, 2011 28 / 30

I/O Controllability

(Input-output) controllability is the ability to achieve acceptablecontrol performance; that is, to keep the outputs (y) within specifiedbounds or displacements from their references (r), in spite of unknownbut bounded variations, such as disturbances (d) and plant changes(including uncertainty), using available inputs (u) and availablemeasurements (ym or dm).

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References

Sigurd Skogestad and Ian Postlethwaite.Multivariable Feedback Control Analysis and design.John Wiley & Sons, Second Edition, 2005.

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