adaptive passivity–based control for maximum power...

CENTRE NATIONAL DE LA RECHERCHESCIENTIFIQUE

Adaptive Passivity–based Control for Maximum PowerExtraction of Stand–alone Windmill Systems

Romeo Ortega†, Fernando Mancilla–David‡ and Fernando Jaramillo†

†LSS, Supelec, France‡University of Colorado Denver, USA

Layout

Mathematical model of the small scale windmill system

Wind Speed estimator

Passivity–based controller

Simulation results

Conclusions and future work

Aalborg, November 28, 2011 – p. 1/19


Model of the System

Battery charging windmill system



PM Synchronous Generator and Wind Turbine

In the rotor (dq) reference frame

Lid = −Rid + LiqP2ωm − vd,

Liq = −Riq − LidP2ωm + φωe − vq,

where id, iq , vd, vq , are the currents and voltages in the dq reference frame, L and R

are the stator winding’s inductance and resistance, ωm is the mechanical speed, φ isthe permanent magnetic flux, and P is the number of pole pairs.

The mechanical dynamics, with J the rotor inertia,

Jωm =Pw

ωm︸︷︷︸

Tm

−3

2

P

2φiq

︸︷︷︸

Te

.

The mechanical power at the windmill shaft

Pw =1

2ρACp(λ)v

3w, λ :=

rωm

vw.

with vw the unknown wind speed and Cp(λ), the power coefficient.



Rectifier, dc/dc Converter and Full Model

dq model

vd = id√

i2d+i2q

MvbD,

vq =iq

√

i2d+i2q

MvbD.(1)

where is the vb battery voltage, M = π

3√

3is the gain of the passive diode rectifier and

D the duty ratio.

Overall system

Lx1 = −Rx1 + L1x2x3 − C1x1u

Lx2 = −Rx2 − L1x1x3 + φ1x3 − C1x2u (FM)

J1x3 = −φ1x2 +Φ(x3, vw),

with x := col(id, iq, rωm), the input u := D√

x2

1+x2

2

, and the function

Φ(x3, vw) := C2

v3w

x3

Cp

(x3

vw

)

. (2)



Control Problem

Operate the system at the point of maximum power extraction λ⋆ := argmaxCp(λ),

which is typically known.

If vw is known the control task is the regulation of ωm around the reference speed

ω∗m =

λ⋆vw

r=:

1

rx3⋆.

An on–line wind speed estimator is added to generate

ω∗m =

λ⋆vw

r.

The problem is translated into asymptotic stabilization of the desired equilibriumx⋆ = (xs

1⋆, x2⋆, x3⋆)

Remark Nonlinearly parameterized nonlinear system, hence linear control and standardestimation are not applicable. Solved using passivity–based control (PBC) (Ortega, et al.,Springer Book ’98) and immersion and invariance (I&I) adaptation principles (Astolfi andOrtega, TAC’03, Springer Book ’07).



Assumptions for Wind Speed Estimation

Assumption 1 The power coefficient is aknown, smooth, function

Cp : [0, λM ] → R+,

which verifies

C′p(λ)

> 0 for λ ∈ [0, λ⋆)

= 0 for λ = λ⋆

< 0 for λ ∈ (λ⋆, λM ],

where λ⋆ := argmaxCp(λ).

λ

Cp

λ∗

Cp∗

Assumption 2 The wind speed vw is an unknown positive constant.

Assumption 3 The electrical torque Te and the motor speed wm are measurable.

Assumption 4 For all λ ∈ (0, λ⋆), the power coefficient verifies

3

λCp(λ) > C′

p(λ).



Some Remarks

Cp(λ) can be easily obtained from experimental data, and the algorithm implementedfrom a table look–up.

Constant wind speed assumption only needed for the theory. An on–line estimator isable to track slowly–varying parameters, assumption justified by the time scaleseparation between the wind dynamics and the mechanical and electrical signals.

On–line estimators average the noise—in contrast with differentiator–based orextended Kalman filter schemes currently used.

Measuring wm and Te is standard practice in windmill systems.

Theory applicable also if blade pitch β is included, i.e., Cp(λ, β), or for more completedescriptions of the mechanical dynamics.

Assumption 4 is satisfied in normal operating range (for Region 2), where the torquecoefficient has negative slope. Indeed,

CT (λ) :=1

λCp(λ),

satisfies

C′T (λ) ≤ 0 ⇒ Assumption 4.



Main Estimation Result

Proposition (Ortega, et al., IJACSP’12) Consider the system (FM), verifying Assumptions1–4. The I&I estimator

˙vIw = γ

[

Te −ρA

2

(vIw + γωm)3

ωm

Cp

(rωm

vIw + γωm

)]

vw = vIw + γωm,

where γ > 0, is an adaptation gain, is asymptotically consistent, that is,

limt→∞

vw(t) = vw.



Simulation Results: Periodic Wind

Done in Vestas professional software, with the full model, look–up table for Cp(λ), and realwind data.

−100 0 100 200 300 400 500 6008

10

12

14

16

18

20

22

Time (s)

Win

d S

peed

(m

/s)

WSEstimated WS



Simulation Results: Turbulent Wind

−100 0 100 200 300 400 500 60010

12

14

16

18

20

22

Time (s)

Win

d S

peed

(m

/s)

WSEstimated WS



Simulation Results: Gust

140 160 180 200 220 240 260 280

12

13

14

15

16

17

18

19

Time (s)

Win

d S

peed

(m

/s)

WSEstimated WS



An Asymptotically Stable PBC with Known Wind Speed

Proposition Consider the system (FM), with known wind speed, in closed–loop with thePBC

Lx1d = −Rx1d +L1

φ1

Φ(x3⋆, vw)x3 − C1x1u

J1x3d = −Φ(x3⋆, vw) + Φ(x3, vw)−R3a(x3 − x3d)

u = −1

C1x2

[R

φ1

Φ(x3⋆, vw) + L1x1dx3 − φ1x3d], (3)

where R3a > 0 is a damping injection. The equilibrium x⋆ is asymptotically stable.

Remarks

The scheme is made adaptive replacing Φ(x3⋆, vw) by Φ(x3⋆, vw).

An integral action can be added

J1x3d = −Φ(x3⋆, vw) + Φ(x3, vw)−R3a(x3 − x3d) + ξ, ξ = −KIe3,

with KI > 0 an integral gain.



Sketch of the Proof

(FM) written in Euler–Lagrange form (to reveal workless forces):

Dx+ [C(x) +R]x = G(x)u+ b(x),

where

D := diagL,L, J1 > 0, C(x) = −C⊤(x) :=

0 −L1x3 0

L1x3 0 −φ1

0 φ1 0

,

R := diagR,R, 0 ≥ 0, b(x) :=

0

0

Φ(x3, vw)

, G(x) :=

−C1x1

−C1x2

0

.

The systems energy function H(x) = 1

2xTDx, satisfies the power–balance equation

H = −R|idq|2

︸︷︷︸

dissipation

− |vdq||idq|︸︷︷︸

elec. power

+2

3Pw,

︸︷︷︸

mech. power



Energy Shaping and Damping Injection

Idea is to assign a new energy function

W (e) :=1

2e⊤De,

where e := x− xd and xd := col(x1d, x2∗, x3d).The controller can be written as

Dxd + [C(x) +R]xd = b(x) +G(x)u+ udi

where udi is an additional damping injection signal: udi = −Rae. This yields the errorequation

De+ C(x)e+ (R+Ra)e = 0

Taking the derivative of W yields

W = −e⊤(R+Ra)e ≤ −2minR,R3a

maxL, J1W

establishing that e(t) → 0, exponentially fast. Proof completed with some signal chasing toprove that xd(t) → x⋆.



Simulation Results

Power coefficient given by the function

Cp(λ) = e−cp1λ

( cp2

λ− cp3

)

+ cp4λ,

where cp1 = 21.0000, cp2 = 125.229, cp3 = 9.7803, and cp4 = 0.0068—from Matlabpackage. This yields λ∗ = 8.1 and Cp∗ = 0.48.

Windmill/battery system parameters

Item Value

Pole pairs P = 28

Synchronous resistance R = 0.3676 (Ω)

Synchronous reactance L = 3.55 (mH)

Flux φ = 0.2867 (Wb)

Inertia J = 7.856 (kg m2)

Blades radius r = 1.84 (m)

Battery voltage vb = 48 (V )



Wind Speed Estimator

The estimator and controller parameters γ = 0.1, R3a = 0.01 and KI = 1.

0 20 40 60 80 100 1207

8

9

10

11

12

13

t (sec)

Win

dSpee

d(m

/se

c)

Actual: vw

Estimated: vw (γ = 0.1)



Duty Ratio, Power Coefficient and Power Capture

40

60

80

100D

(%)

0.47

0.48

0.49

Cp∗ = 0.48

Cp

(-)

0 20 40 60 80 100 1200

2000

4000

6000

Pw

(W)

t (sec)



Evolution of all States

0

2

4

x1

(A)

x1d

(A)

0

10

20

x2

(A)

x2∗

(A)

0 20 40 60 80 100 12060

80

100

x3

(m/se

c)

t (sec)0 20 40 60 80 100 120

x3d

(m/se

c)

t (sec)



Future Work

Analysis of the adaptive scheme:

Tough without persistency of excitation!

Consider torsional modes:

Derive I&I estimator.

PBC.

Incorporate torsion minimization considerations.

The stability result is only local, because stable invertibility of the system is needed.

New PBC’s, for Hamiltonian models, obviate this assumption.

But a PDE needs to be solved.

Comparison of adaptive PBC with ω2m–controller. Partial results for adaptive PI.

Can Assumption 4 be relaxed?

It is not (globally) satisfied for several known turbines.

Analysis when it does not hold?


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