Proceedings of the Seventh International Conference on Machine Learning and Cybernetics, Kunming, 12-15 July 2008

MULTI-OBJECTIVE POWER CONTROL OF A VARIABLE SPEED WIND TURBINE BASED ON THEORY H∞

JI-HONG LIU, DA-PING XU, XI-YUN YANG

Dept. of Automation, North China Electric Power University, Beijing, 102206, China E-MAIL: ljh_mail@sina.com

Abstract: A power control strategy for a real variable speed wind

turbine system is presented in this paper. The H∞-based mixed sensitivity method is applied to realize multi-objective control such as the tracking characteristics for the rated power, poles placement and robustness to disturbances. LMI Toolbox in MATLAB is used to optimize the parameters of H∞controller. First in this paper, a physical model of the fundamental drive-system dynamics of a 300 kW horizontal axis variable speed wind turbine (VWT) is derived. By linearizing this physical model, we can get its linearized state-space form. Second, H∞ -based S/T/R algorithm is introduced in brief. Finally, the H∞ controller of mixed sensitivity is designed on the basis of VWT model and simulation is made. The Simulation results show the better performance of the proposed control strategy compared with the conventional PI control.

Keywords： Power control; Wind turbine; H∞-based robust control;

Mixed Sensitivity (S/T/R) ; Multi-objective control ; LMI; pole placement

1. Introduction

Variable speed wind turbine (VWT) is a rather new area and the advantage is that VWT can increase aerodynamic efficiency and maximize energy capture, so nowadays, VWT is used widely. In practice, to effectively extract wind power, the variable speed wind turbine operates in two regions as shown in the Fig.1, depending on the wind speed, maximum allowable rotor speed and the rated power [1]. Below-rated wind speed (partial load)

Where 1 2ν ν ν< < Above-rated wind speed (full load)

Where 2 3ν ν ν< <

Figure 1. Wind turbine aerodynamic power

ν is the mean wind speed. The wind turbine is

stopped for 1ν ν< or 3ν ν> . Where 1ν is the cut-in

wind speed, 2ν is the wind speed that leads to rated power

and 3ν is the wind speed at which the turbine needs to be shut down for protection. The turbine is normally operated between 1ν and 3ν limit of wind speed, typically 5 m/s to 25 m/s. As the wind speed increases, the energy available for capture increases as roughly the cube of the wind speed. At rated wind speed the power input to the wind turbine will have reached the limit (rated power). Above this wind speed , the excess power in the wind must be discarded by the rotor to prevent the turbine overloading. The power is maintained at its rated value until a maximum wind speed is reached when the turbine is shut down. On varying the pitch of the blade, the power derived from the wind is reduced.

Classical techniques such as PID and PI controllers of blade pitch are typically used to limit power and speed for turbine operating above rated wind speed [2][3][4]. Many researchers have also developed other methods (e.g.,

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Proceedings of the Seventh International Conference on Machine Learning and Cybernetics, Kunming, 12-15 July 2008

reducing loads [5], using adaptive control method [6][7]). LQ and LQG control techniques have also been demonstrated in [8][9].

The controller presented in this paper is designed for variable speed wind turbines operating above the rated wind speed. In this area, the control system objective shifts from maximizing power capture to maintaining output power to its rated value, being robust against disturbances of wind speed, grid voltage and measurement noise. In order to get better tracking dynamics, the poles of close-loop system are restricted in a special region. This multi-objective design tries to satisfy several targets by limiting the norm of the closed-loop transfer function and it can be formulated as a mixed sensitivity problem(S/T/R) based on the H∞ control theory. In this method, the selection of weighing function is a nodus. In this paper, the rules for the selection of weighing functions are introduced. The simulation results show that the designed controller has better performances in tracking dynamics, restraining disturbances, and robustness compared with the conventional PI controller.

2. Variable Speed Wind Turbine System

2.1. General functional module

Generally, the wind turbine model consists of : Wind speed model to generate a wind speed that can

be applied to the turbine. Rotor model that converts the kinetic energy in the

wind into mechanical power. Generator model that converts the mechanical power

into electrical power. Control system including rotor speed controller and

power controller.

turbimutuconsis drplace

change the driving torque generated by the rotor. The angle of attack (the direction of the wind as felt by the rotor blades) also influences the driving torque. Pitch angle of wind turbine blade directly influences this angle and thereby opens the possibility to influence the aerodynamic energy capture by the rotor. The rotor is connected with a gearbox to the generator that is supposed to convert the mechanical energy supplied by the rotating shafts into electrical energy and feed this to the public grid. All these parts are contained in the nacelle.

The aerodynamic subsystem is modeled by the nonlinear wind torque characteristics.

2 3 ( , )12

pr

CT R

λ βπρ ν=

Ω (1)

Where is rotor shaft torque,rT ρ is air density, R is

blade length,ν is wind speed, λ is tip speed ratio, is the low-speed shaft rotational speed,

Ωβ is Pitch angle.

( , )PC λ β is the power coefficient of the turbine. The electromechanical subsystem yields the

electromagnetic torque . This subsystem interacts with the turbine rotor through the transmission device; the dynamics of this latter is expressed by:

eT

t rdJ TdtΩ = − eT (2)

Where expresses the total inertia of the turbine. tJThe dynamics of the asynchronous generator can be

approximated by a first-order subsystem. The wind model is added to the general model of the

system. The wind can be modeled as a stochastic process with two components: the seasonal, slowly variable component ν , and the turbulence, rapidly variable component ν∆ .

( ) ( )t tν ν= + ∆ν (3)

The low-frequency component, ν , determines the average position of the operation point on the wind turbine characteristic, and ν∆ generates the high frequency noises transmission generator

νWT

gWrT eP

β

Figure.2 schematically represents the flexible wind ne under consideration. Each module of wind turbine is ally connected by interaction variables. The rotor, isting of a number of blades (in most cases 2 or 3), iven by the wind. The wind speed is not constant at any and any time under natural conditions that will

around this point. The two components may be identified on the wide-band (six decades) wind model of Van der Hoven[10](Figure.3)

rW eT

Figure 2. Block scheme of wind turbine

iU

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Proceedings of the Seventh International Conference on Machine Learning and Cybernetics, Kunming, 12-15 July 2008

Figure 3. Van der Hoven wind mode

2.2. The linear model

In this paper we consider a 300kw horizontal axis variable speed turbine with the following main characteristics: Blade radius 15m Number of blades 3 Gearbox ratio 38.06 Air density 31.25 /kg mRated aerodynamic driving torque 80.7KN.m Rated wind speed 12m/s Rated power 300kw Output voltage 220V Inertia coefficient 350000Kg. 2m

In order to design a controller using H∞ theory, a simple linear model is required. As considering the transmission torsion and some additional assumptions, we can obtained the complete linearization state equation of the wind turbine system as follows[11]. All the values in the equations are the relative values at the selected operating point.(OP is chose at wind speed and voltage

). 12 /v m= s

220iu V=X AX Bu EwY CX Du Fw

= + += + +

(4)

With：

[ ]

[ ]

[ ][ ]

Tg gm

r

Ti

e

X

u

w uy P

β ξ ξ ω ωβ

ν

=

=

==

Where 5X R∈ is the state vector,u is the control variable, is the measured vector, is the disturbance vector.

R∈y R∈ 2w R∈

, , , , ,A B C D E F are known constant matrices of appropriate dimensions.

β Pitch angle of wind turbine blades

rβ Reference input of pitch angle

ξ Relative angle in secondary shaft

v Wind speed

eP Active power

gω Generator speed

gmω Generator measurement speed

iu Grid voltage

5.0 0 0 0 00 0 1.0 0 0

10.5229 1066.67 3.38028 23.5107 00 993.804 3.125 23.5107 00 0 0 10.0 10.0

A

−

= − − −−

−

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

[ ]5.0 0 0 0 0TB =

[ ]0 0 0 1.2231 5 0C e=

[ ]0D =

0 00 0

0.9261 0.60040 0.6000 0

E4

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦

[ ]0 3.074 3F e=

3. LMI Approach to H∞-based S/T/R Algorithm with Pole Placement Restriction

3.1. Multi-objective control using LMI

This section offers a brief overview of multi-objective controller design in terms of LMI. The design problem treated in this work is to construct an internally stable

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Proceedings of the Seventh International Conference on Machine Learning and Cybernetics, Kunming, 12-15 July 2008

controller which satisfies H ∞ performances and pole placement.

3.1.1 S/T/R sub-optimal problem

Figure 4 depicts the mixed sensitivity configuration where G is the nominal plant, P indicates the generalized plant, K represents the controller to be

designed, sW , and are weighting functions respectively shaping characteristics of transfer functions

， and

tW uW

1( )S I GK −= + 1( )T GK I GK −= +1( )R K I GK −= + .Transfer function R is considered for

reducing actuator saturation.

标题

P

G +

K

Wu

Ws

Wt

r

u e

1Z

3Z

2Z

_+

Figure 4. The mixed sensitivity configuration

When weighting functions have been selected, the

multi-objective design can be formulated as s

t

u

W SW TW R

γ

∞

< ,

a S/T/R sub-optimal problem. From figure4, the close-loop system can be represented:

cl cl cl cl

cl cl cl

X A X B WZ C X D W

= += +

(5)

In the LMI formulation, the objective of H∞control is achieved in the sub-optimal sense if and only if there exists a symmetric matrix P>0 such that:

0

T Tcl cl cl cl

Tcl cl

cl cl

A PA PB CB P I DC D I

γγ

⎡ ⎤+⎢ ⎥−⎢⎢ ⎥−⎣ ⎦

T <⎥ (6)

3.1.2. Poles Placement problem

The transient response of a linear system is related to the location of its poles. An acceptable transient response can be achieved by placing all closed-loop poles in a prescribed region, depicted in Figure5. When the closed-loop poles are in this region, it ensures minimum

damping ratio cosζ θ= . All of the poles of the state

matrix clA inside this conical sector if and only if there exists P>0 such that [12]:

sin ( ) cos ( )

0cos ( ) sin ( )

T Tcl cl cl cl

T Tcl cl cl cl

A P PA A P PAPA A P A P PA

θ θθ θ

⎡ ⎤+ −<⎢ ⎥− +⎣ ⎦

eR

mI

θ

Figure 5. Poles placement region

3.2. Design rules of the weighting matrices

The determination of weighting function matrices Ws(s), Wt(s) and Wu(s) is a crucial step in the controller design. In general, the weighting matrix Ws(s) is chosen as a low-pass filter to emphasize the tracking accuracy at low frequencies (small steady-state error) and restrain noises. Wt(s) is usually selected as a high-pass filter to reduce the control effort in high-frequency range. Wu(s) is usually designed as a high-pass filter or a constant matrix due to the control constraint (actuator saturation and rate limit).

4. Application to the Power Control of a Variable Speed Wind Turbine

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Proceedings of the Seventh International Conference on Machine Learning and Cybernetics, Kunming, 12-15 July 2008

In this section, we will design the power controller based on the H∞-based S/T/R Algorithm. The control objective is to maintain the output power to its rated value, to restrict the close-loop poles in special area and restrain the disturbances of wind speed, grid voltage and measurement noise. Consider a 300kw horizontal axis variable speed wind turbine which state-space model is described as system (4). The input variable of controller is active power deflection P∆ , the output variable is the

reference pitch angle rβ . The control block is shown as Figure.6.

K

WG

WTG*

eP

W

eP−

P∆+

rβ

Figure 6 . C ontrol b lock

7 2 6 6

5 4 3 2

2 10 6597 10 63954 10

42 1526 19148 79331 85657WT

S S

G

S S S S S

× − × − ×

+ + + + +

=

Applying the LMI method introduced above, we

obtained the following Controller.

2

/ / 2

10270.5505(s+5) (s+1.662)(s + 25.23s + 1031)

s (s+4545) (s+318) (s + 7749s + 1.518e007)S R T

K =

For the need of contrast, we also design the PI

controller as follows:

5( 0 .3 2 9 6 0 . 4 0 0 8 ) 1 0P I

sKs

−− − ×=

Rated power*

eP of 300KW is supplied as reference, at the same time the disturbances of 4m/s wind speed and 2V voltages are also added on output . Figure.7 shows the simulation results of S/T/R and PI control strategy. Figure.8 shows the pole-zero maps of S/T/R close-loop system.

eP

Figure 7. Comparison between S/T/R and PI controller

Figure 8. The pole-zero maps of S/T/R system

5. Conclusion

It can be seen from Figure7 that the PI control system has about 7% overshoot and 1.93s setting time; The S/T/R control system has no overshoot and about 1.31s setting time. Both of the controllers have no steady state error. From Figure8, we can see that the poles location of the S/T/R close-loop system is reasonable. So, we can draw a conclusion that S/T/R controller satisfies the proposed control objects .The S/T/R controller can cope with the disturbances acting on the system effectively and has better

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Proceedings of the Seventh International Conference on Machine Learning and Cybernetics, Kunming, 12-15 July 2008

performances and robustness compared with the conventional PI controller.

Acknowledgements

This paper is supported by the ministry of education key project (105049) and the national natural science fund project (50677021)

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