Optimal Output Feedback of a Wind Energy Conversion System

Copyright © IF AC Power Systems Modelling and Control Applications, Brussels, Belgium 1988

OPTIMAL OUTPUT FEEDBACK OF A WIND ENERGY CONVERSION SYSTEM M. Steinbuch* and O. H. Bosqra Delft University of TechnoLogy , MekeLweg 2, 2628 CD Delft, The NetherLands also at *Philips R esearch Laboratories, p.a. Box 80000, 5600 JA Eindhoven , The NetherLands

Abstract. The control system design for a wind energy conversion system is investigated. The system consists of a three-bladed rotor and an electrical conversion system with a synchronous generator and rectifier/inverter link. The system has three inputs and two measurable outputs. Because there is severe interaction, especially between the electrical and mechanical part , the approach of multivariable control is chosen. In order to handle the conflicting control requirements, the method of linear quadratic optimal output feedback (or parametric optimal control) has been used. The controller structure has been used as part of the design process, to obtain robustness. The results for the wind turbine are shown, using a nonlinear dynamic model, and these are compared to results obtained with a classical PID design. Keywords. Wind energy conversion, modelling, optimal control, multi variable control, output feedback, robustness, singular values.

INTRODUCTION

Wind Turbine System

The increasing concurrence between fossil and alternative energy sources forces wind energy conversion systems towards cheaper designs. As a consequence, the control of wind turbine systems becomes more important , in order to ensure life time, to maximize the energy capture and to enhance the overall performance. However, owing to the strongly fluctuating nature of the wind, the design of a control system for wind turbines is difficult . Aerodynamic nonlinearities, structural flexibilities and electro-mechanical interactions add to the increase in control design complexity. Research on the control system design for wind energy conversion systems has been focused mainly on constant speed turbines, with electrical conversion systems consisting of a synchronous generator directly coupled to the grid (Barton , Bowler and Piwko, 1979; Hinrichsen and Nolan, 1982; Liebst , 1983). More recently , the use of variable speed systems having a synchronous generator with rectifier/inverter link has increased (Raina and Malik, 1985; Steinbuch, 1986). Because these wind turbines can be operated with a variable speed, they are more controllable with respect to mechanical loads, energy capture, and quality of the electrical power output. However, this results in a more complex multi variable control problem (Steinbuch and Meiring, 1986).

The system considered here (Fig. 1) is a three--bladed horizontal axis wind turbine with a rated power of 300 kW. It has a relatively heavy rotor and an electrical conversion system consisting of a synchronous generator, a controllable thyristor bridge rectifier, a DC (ripple) reactor and an inverter. The available inputs are: delay angle O'r of the rectifier, field voltage uF of the generator and pitch angle !3 of the blades. The measured variables are generator speed wand direct current Idc ' Disturbances acting on the system are wind speed v and fluctuations in grid voltage Un'

Aim of the Study The aim of this study is to develop a control system for the wind turbine conversion system. This control system should account for the interactions in the system, be robust with respect to model uncertainties and nonlinearities, and achieve a high performance: high energy capture, low mechanical loads, high quality of the electrical power, in all operating conditions.

ROTOR GEAR

GENERATOR

RECTIFIER

INVERTER

GRID

V

Fig. 1 Wind energy conversion system with synchronous generator and DC link .

• The work reported here has been done at KEMA, Arnbem, The Netherlands. 313

M. Steinbuch and O. H. Bosqra

314

Outline of the Paper To realize this goal a dynamic model for the wind turbine system is necessary with which the dynamics are taken into account proJ'erly. A nonlinear model has previously been reported (Steinbuch, 1986; Steinbuch and Meiring, 1986) and is described in the section on wind turbine dynamics. The dynamic properties of the system cover nonlinear and multi variable behavior. Because the interaction in the system is substantial, a multi variable control system design approach has been chosen. In the section on the controller design, the statement of the problem is formulated and the feedback structure of the control system is shown. To design the ultimate controller parameters, the linear quadratic optimal control method (LQ) is used. Especially useful for wind energy conversion applications, LQ control offers the possi bility to obtain a compromise between conflicting performance specifications. To be able to use the internal controller structure as part of the design process, the optimal output feedback or parametric linear quadratic (PLQ) control approach is chosen. Using an optimal controller with structure constraints, the closed- loop performance is shown with nonlinear simulations.

Jv Q 0 vJ v 0 JvLla = J(Tr-T D ) + rTe - ktoa - Ta(toa), r g

(

2

)

where J v = JrJg/(J r + zJJ g ). Torque Ta is the static axis torque at the flexible element, and is assumed to be a nonlinear function of the torsion toO': Ta = 100ot.a3 - 20ot.a2 + 2ot.O', (3) which is a fit on experimental data. The total amount of friction of the drive train is supposed to have the form: TD = Cl + C 2 /wr + C 3 wr , with Cl' C 2, C 3 appropriate constants (App. 1). Generator speed W follows from

(4)

o 0 Jgw= Ta + k.toO'-T e, Finally the rotor speed can be calculated from

(

5)

wr =(to3-+w)/ v. Mechanical torque T m=kto3-+T a is of importance for analysing the fatigue load. The equations (1)-(5) provide a model for the aero-mechanical part. This part is bilaterally coupled to the model of the electrical conversion system through w and T e.

WIND TURBINE DYNAMICS Modelling The general model is built up from a Simplified model of the aerodynamic and the mechanical part , together with a relatively accurate nonlinear dynamic model of the electrical conversion system. Aero-mechanical part. The wind turbine is driven by an effective torque T r on the rotor. A simplified calculation of this torque is possible with the equation T = C (>\,fJ).~.p7rR2v3/w , r

p

(1)

r

2

Generator and AC/DC/AC Conversion. Most of the dynamic models for syuchronous generators loaded with rectifier and inverter known from literature are very detailed and describe the voltage and current wave forms (a.o. Arrillaga and workers, 1978; Bonwick and Jones, 1973). However , this is not necessary for the design of controllers. A description of the fundamental harmonics, based on Park's transformation (Anderson and Fouad, 1982) for the generator can be used. For the rectifier inverter link the steady-state voltage and phase equations are used. A detailed derivation of this model is given by Steinbuch (1986). Only the main results will be shown here.

where C is the power coefficient calculated from p aerodynamic design data, A is the tip/wind speed ratio A=Rwr/v, wr is the rotor speed, R is the radius of the rotor , and p is the air density. The rotor torque calculated with (I) is used as an input in the model of the mechanical part. It acts on the rotor with mass- moment of inertia J r (Fig.2).

The generator has a rotor with two damper windings and one field winding and a stator with three-phase windings. The usual Park's transformation is applied to describe the dynamics of these magnetically coupled windings. This results in the state space model:

(6) where state vector i=(id,iF,iD,iq,i Q

? and input vector

?

u=(ud,uF'u q Currents id and iq and voltages ud and u are projections of the terminal currents and voltages q on the rotor axes d and q (Anderson and Fouad, 1982). Variables iFand u F are the field current and voltage respectively, and states iD and iQ are the currents in T

r

v

T

w

T e

rr. u

both damper windings. Matrices L, N and M are shown in appendix 2. Some entries of matrix N depend on generator frequency w, introducing nonlinearities in (6). The output of this model to the mechanical part is the electro- mechanical torque Te:

g

60.

Fig. 2 Mechanical part of the wind turbine. The rotor is connected to the generator having massmoment of inertia J g' via a (gear) transmission having ratio v. The generator electro-mechanical torque is Te. The assumption is made that all the structural modes are concentrated in a flexible element in the secondary shaft, and that the primary shaft and gear are infinitely rigid. Structural modes of the (steel) blades are assumed to be in the high frequency range only. Designating the relative angle of displacement in the secondary shaft by toll', it can be verified that the equation of motion is:

which is a nonlinear function of states i; the coefficients are elements of matrix 1. The direct current Idc is another important variable, it can be calculated with Idc =

7rv'{(ia

+ i~)/18},

(8)

The influt variables of the state space model (6) are w (from (5)), u as a controllable input, and voltages u d F and u . These voltages are related to generator terminal q voltage, and to the load angle 0 of the machine (Steinbuch, 1986). Load angle 0 is related via Park's

315

Wind Energy Conversion System Output Feedback

Remarks: 1. The tolerated speed variations above the nominal value are small because of possible excitation of eigenfrequencies of the tower construction. 2. The requirements are conflicting. For instance, the speed and electrical power output variations are coupled due. to .the energy ~alance. Only with fast pitch angle e~C1tatlOn both varIables can be kept constant during Will? speed. fluctuations . However, this can only be aC~leved WIth large (fast) excitation of the pitch angle (hIgh control effort), and resulting in large mechanical loads in the blades. 3. The requirements with respect to the variations of the inputs are boundedness of the field voltage and delay angle varIatIOns. The latter must keep the same value in steady state. In addition, the variations in the pitch angle are bounded (~O). Also, an important restriction exists in the pitch angle excitation system, with respect to the rate of change of the pitch angle (d.8/dt~5 o/s).

transformation to the states id and iq and to the phase angle between the generator terminal currents and voltages. This phase angle is a function of the delay angle O'r of the rectifier, and is also affected by the commutation process, i.e. switching over of the thyristors. These interconnections between the generator and the DC link make it necessary to rewrite the set of equations in order to transform the initially implicit model into an explicit form, i.e. dx/dt=f(x,u), ~=g(x,u). The result has been derived in Steinbuch (1986), see also appendix 3 for a summary of the equations. Together with (6)-(8) and in combination with (1)-(5) , the complete model is available. Additional dynamics are a first order model for the pitch angle servo system, a first order model for the field excitation system and a first order sensor model for the speed measurement system. For the control system design the complete nonlinear model has been linearized in several operating conditions.

The control woblem can now be stated as finding a controller whIch accounts for the system dynamics and realizes the requirements stated above, despite (large) varIatIons III the wllld speed and voltage disturbances from the grid.

Dynamic Performance The dynamic performance and characteristics of the wind turbine system consist mainly of the large rotor inertia, the torsional oscillations in the secondary shaft and the fast electrical dynamics (Steinbuch, 1986; Steinbuch and Meiring, 1986). Also, the system is nonlinear due to the aerodynamics and the electrical conversion system. Between the outputs speed and direct current , a large amount of interaction is present , while all the inputs affect all the outputs. Therefore , the wind turbine system is a multivariable system. The control system should account for these characteristics , in order to meet the performance requirements.

Structure of the Control System In a previoulsy reported study a state-variable feedback control has been designed and analysed (Steinbuch, 1987) . In this paper, we restrict the class of admissable controllers to output feedback structures, to meet the implementation constraint . To deal with stationary nonzero disturbances, such as wind speed changes, integral feedback of the rotor speed, the direct current and the delay angle of the rectifier have been added. For several reasons there is an analog slave control loop present, between the dIrect current and the delay angle of the rectifier (Fig. 3). This loop has been made very fast, in order to account for thyristor faults from the inverter. In the multi variable controller design, this loop is kept intact . Including the additional pitch servo (.8r ) and field

CONTROLLER DESIGN Statement of the Problem The results presented here are restricted to full load conditions (v~12 m/s). The requirements for these operating conditions can be stated as follows: * constant nominal rotor speed * constant nominal electrical power * low mechanical loads * minimum control effort

excitation system (u Fr ), the controllable inputs for the system are (.8r ,u F r ,Id cr ). The outputs are (wm ,Id c'

f ardt),

with wm the measured generator speed. The final

control system design structure is shown in Fig.3. The problem now is to find the appropriate control parameters F to realize the requirements.

v

un

8ref ~

u

F I ref.

WIND Fref

f -E:}2.

w

Tl!RBL,E SYSTEM

PI

Fig. 3 Control scheme of the wind turbine system.

~c

M. Steinbuch and O. H. Bosqra

316 Control Methodology

The conflicting requirements make the control problem very suitable for treatment as an optimization problem. A very powerful method, which can handle multi variable systems in a natural way, is the linear quadratic optimal control method (LQ). Applications of this method are widely known in aerospace and power system control. In a previously reported study (Steinbuch, 1987) very good results have been obtained with LQ state feedback of the wind turbine. However, in order to design an implementable controller, an output feedback design has to be made. This is possible using observer techniques (c.f. LQG). Nevertheless, due to the high dynamic order of the model and the few number of outputs, this would result in a very complex controller. Besides, the nonlinearities in the process in combination with strongly fluctuating wind speeds, would violate the separation principle, possibly resulting in instabilities. Another approach is the parametric optimal control method, with which it is possible to use the controller's internal structure as part of the design process.

3. In equation (14) the part SCT(CSCTtl can be inter- preted as a projector or a static observer for the states along the outputs (Naeye and Bosgra, 1977). 4. The design flexibility offered by the method of optimal output feedback is quite large. Adding integral action to the system (9), or general dynamics, a proportional/integral or dynamic feedback can be designed. Putting n additional dynamic orders to (9) , the feedback can be split up in an observer and a state feedback controller. 5. The feedback F depends via P and S, on the initial state matrix XO' In the stochastic formulation Xo is the disturbance intensity (Anderson and Moore, 1971). This means that for ea::h disturbance there exists only one optimal feedback law. The design can thus be made more specific for certain disturbance conditions, if they are known to be relevant. However, often it seems more convenient to chose Xo such that the controller behaves well for a class of disturbances. This can be achieved by chosing rank(Xo»1 (Levine and Athans, 1970).

Parametric optimal control. Consider the linear model to be controlled:

~ = Ax + Bu y = Cx

(9)

Where x is the n-{\imensional state vector, u is the mdimensional input vector, and y is the l-{\imensional vector of measured outputs. Suppose system (9) has a nonzero initial condition x(O)=x ' i.e. we consider the O deterministic formulation. The parametric linear quadratic control problem, or optimal output feedback problem, can now be stated as finding a controller F such that the quadratic performance index Q~O,

R>O,

(10)

Internal structure constraints. A benefit of the method, in contrast to LQG and Hoo techniques , is the possibility of using the internal controller structure as part of the design process . There are several reasons for doing this: 1. to compete with simple classical PID controllers in the sense of complexity, as well as to optimize the parameters in a classical control design, 2. to achieve fast calculation of the optimal controller: starting with only a few parameters to be optimized, the algorithm converges very fast. Using the result as the starting value for the next optimization step, with more free parameters, leads efficiently to the ultimate design, 3. to achieve phase-margins (i.e. rObustness).

and the system (9). Necessary conditions for the solution to exist are well known (see e.g. Anderson and Moore, 1971; Levine and Athans, 1970):

The optimization problem can then be reformulated as finding an F such that (l0) is minimized under the constraints (9),(11), and F(i,j)=O for some (i,j) ( (I,m) x (1,1). The solution is found using (12)-(14) for the case where the number of free parameters is less than lm. Obviously, the drawback is that one reduces the performance achievable with reducing the number of free parameters. If, however, this reduction has only minor effects on the performance, then a decentralized control structure is beneficial.

1. there exists a P=P T ~O such that

Application

is minimized, under the constraints u = Fy,

(F rnxl),

(ll)

P(A+BFC) + (A+BFC?P + Q + CTFTRFC = 0, (12) 2. there exists a S=S T ~o such that S(A+BFC? + (A+BFC)S + Xo = 0,

(13)

with the system (A,B,C) stabilizable, and Xo=x6xo' The optimal output feedback controller can be calculated from (14)

structure, in such a way that the desired performance is achieved. Using the experiences with LQ state feedback (Steinbuch, 1987), the weighting matrices Q and R are chosen the same. The matrix Xo has to be considered carefully . First it is stated that we want to achieve a good performance for various initial conditions, or equivalently, for several external disturbances (v, Un)' Therefore, we chose Xo as a rank n matrix. To account for physical dimensions and scaling effects, matrix XO=Q is a first guess. However, due to the differences between state and output feedback, this resulted in a low damping of the mechanical

with the corresponding optimal index

T J opt = tr(PX O) = xOPx O'

The control problem is reduced to finding the design parameters Q, R, Xo and the internal controller

(15)

Remarks: 1. The set of equations (12)-(14) is implicit: the solution F must be found using an iterative algorithm, see Makilli. and Toivonen (1987) for a review on this specific subject; an initially stabilizing F is necessary. 2. For state feedback, i.e. C=I, equation (14) reduces to the optimal LQ regulator, with (12) the corresponding Rkatti equation, after proper substitutions.

oscillations (A&). It was found that increasing entry X (7,7) (according to state A&) resulted in more 0 damping. For the same reason an extra first order dynamic feedback path was introduced in the optimal con troller. For the wind turbine system under investigation the influence of constraining the internal controller structure has been evaluated. Consider the decentralized feedback structure:

Wind Energy Conversion System Output Feedback

Wm Pr u Fr Idcr

Idc

*

*

0

0

fWm 0

* *

(J.I)

0

(p)

fIdc

f (}r

*

0

*

*

*

•

*

0

*

0

0

•

317

This lowers the robustness margins for uncertainties for instance due to a discrete controller implementation. Chosing F(3,2)=0 a priori, gives better margins (Fig.4). The reason for retaining Idc as output after all is to be able ting part high

(16) The * means a feedback path to be optimized, while a 0 means no feedback. Both outputs wm and Idc are inte-

to dampen the flexible mechanical oscillations. PutF(3 ,2) =0 forces this action through the dynamic p of the controller. This explains the decrease in frequency components at the input I ' In this dcr way the choise of the internal controller structure is used as part of the design process , in order to obtain better robustness margins.

grated, as well as (}r according to the requirements. SIMULATION RESULTS

Variable p is an additional dynamic feedback, with J.I its input. Using a constant Q, R, and X ' the effect of the O controller structure on the performance index (eq. 15) is shown for several controllers in table 1. TABLE 1 Performance index optimal output feedback index J ~ l--{)ptimal state feedback 2-full PI optimal output feedback 1.13 3~ecentral PI optimal output feedback 1.14 4~ynamic decentral optimal output feedb. (1 order) 1.04 5~ynamic decentral optimal output feedb. (2 orders) 1.04 In table 1 the optimal state feedback controller (LQR) is used as a norm, and the indices J of the output controllers are expressed as compared to the LQR control. The first thing to notice about table 1 is that the LQR controller (45 parameters!) is not necessary at all regarding the index. Using only 16 parameters with output feedback, the performance is nearly as good as with state feedback, if the index J is chosen as the performance measure. Controller 2 is an output feedback using all the possible feedback gains, but without the dynamic path (p,J.I). Controller 3 uses the decentralized structure according to (16), but again without (p,J.I). The index detoriation from 2 to 3 is very small (1%). Adding a first order dynamic feedback (p,J.I) has a significant effect on the performance index. Adding more dynamics to the controller is not justified with respect to the index. In the analysis given here, the index number J is used as performance measure. Although it is a scalar, it is the bases for the results of the optimization program. Nevertheless, it is very important to evaluate the designs through frequency and time responses. The robustness reason for chosing a decentralized structure will be worked out by an example. The controller structure for the wind turbine is chosen as in (16). Element F(3,2) is a proportional gain around the already existing analog direct current controller. This results in high frequency components in the frequency response (singular value O'(FG)) at the input Idcr

The dynamic behavior of the wind turbine system is simulated using the full nonlinear model. The ultimate optimal output feedback controller has a structure according to (16), but without element (3,2). The performance is compared to a classical PID design, which is based on a single-input single--{)utput (siso) approach. As a test case, a large wind gust is used (Fig. 5a). In Fig. 5c and 5d the responses of the rotor speed and of the electrical power are shown. Both the optimal controller (1) and the siso design (2) control the speed very well (~ 1% deviation). However, the power output fluctuations (Fig. 5d) differ enormously between both controllers. The sis~esign (2) results in 80 kW variation, while the optimal multi variable controller gives only 3 kW (1%) fluctuation. Obviously , these variations are also present in the torques, which is undesirable. The reason for the difference between both controllers is that the multi variable controller compensates for the internal interactions in the wind turbine system, and uses the three inputs in the most efficient way. This can be illustrated with the behavior of the pitch angle (Fig. 5b). It is seen from the figure that the optimal controller acts a moment earlier on the pitch angle then the siso controller. This leads to less power variations, despite the restriction in the rate of change of the pitch angle. CONCLUSIONS The results shown, indicate that a multi variable optimal control system design can be used to obtain superior wind turbine system performance. This approach leads systematically to a proper compromise between power and speed fluctuations and mechanical loads. The design methodology is straightforward. Its implementation in digital computer systems is not more complex then the implementation of siso-controllers. The internal controller structure has been used to obtain a more transparent deSign , as well as a more robust control system.

(Fig.4).

100.-----.------.------.-----, Acknowledgements The reseach on the control system design for the wind energy conversion system reported in this paper, was carried out at the Automation Engineering Department, Division for Engineering and Consultancy, KEMA, Arnhem, The Netherlands, in cQ--{)peration with the Measurement and Control Group, Mechanical Engineering Department, Delft University of Technology, Delft, The Netherlands. The work was supported financially by the Dutch Electricity Generating Board (SEP) .

~

"C

Q,)

... -='

50

='

CC ;..

F(3,2);e0

0

-- _--- .. --

CC

Q!)

...=

....

-50

F(3,2)=0 " ,,,, ,,

rtl

-100 -4

--, ,,

-2

2

0

4

log frequency Hz Fig. 4 Singular value at the input I feedback F(3,2).

dcr

' with and without

REFERENCES Anderson, P.M ., and F.F. Fouad (1982). Power system control and stability, vol. I. Iowa State University Press, 3rd ed. Ames. Anderson, B.D.O., and J.B . Moore (1971). Linear optimal control. Prentice-Hall, En)1;lewood Cliffs, N.J.

318

M. Steinbuch and O. H. Bosqra

o .,; N

\

/

I

'"

"8

1\

\

V

a wind SPid

j

<> ~o.oo

6.00

16.0

TIME

H.O

S

~.O

32.0

'.,;"' N

1

~

2

C>

"'0 >--

UJ

\l~

'"

8 III

lJ

b. pitch angle

8 o

I 0.00

6.00

I 16.0

TIME

2~.0

S

rr=~.O

32.0

l. opt i mal cont roll er 2. si so controller

2

f:\

y\

APPENDICES 1. Numerical data

Aero-mechanical part.

'-':

V

j

6.00

16.0

TIME

Cl = 1000 Nm

= 32 kgm 2 g R = 15 m

C

v

c rotor SIjd 0.00

J r = 350000 kgm2 J

2

~

ArriJIaga, J., J.G. Campos Barros, arid H.J. Al-Khashali (1978). Dynamic modelling of single generators connected to HVDC convertors. IEEE Trans. Power App. and Systems, PAS-97(4), pp.l018-1026. Barton, R.S., C.E.J. Bowler, and R.J. Piwko (1979). Control and stabilization of the DOE/NASA MOD-l two megawatt wing turbine generator. Proc.14th intersoc. energy conv. eng. conf, Boston , IEEE, pp.325-330. Bonwick, W.J.,and V.H. Jones (1973). Rectifier-loaded synchronous generators with damper windings, Proc. lEE, 120(6) , pp.659-666. Hinrichsen , E.N., and P.J. Nolan (1982). Dynamics and stability of wind turbine generators. IEEE Trans. Power App. and Systems, PAS- 101,(8) , pp.2640-2648. Levine, W.S., and M. Athans (1970). On the determination of the optimal constant output feedback gains for linear multi variable systems, IEEE Trans. on A ut. Control, JMI), pp.44--48. Liebst, B.S . (1983). Pitch control system for large scale wind turbines, J. Enel'gy, 1(2 ). pp.I82-192. Miikilii, P.M. , and H.T. Toivonen (1987). Computational methods for parametric LQ problems -a survey. IEEE Trans. A ut. Control, ac-32 , 8, pp.658-67l. Naeye, W.J. , and a.H. Bosgra (1977). The design of dynamic compensators for linear multi variable systems. In D.P. Atherton (Ed.) Proc. 4th IFAC Symp. on Multivariable Technological Systems, Pergamon Press, Oxford, pp.205-213. Raina, G., and O.P. Malik (1985). Variable speed wind energy conversion using synchronous machine, IEEE Trans. AE5-fl (1) , pp.IOO-I05. Steinbuch , M. (1986). Dynamic modelling and analysis of a wind turbine with variable speed, Journal A, 27(1), pp. 1-8. Steinbuch, M. (1987). Optimal multivariable control of a wind turbine with variable speed , Wind Engineering, 11(3), pp.I53- 163. Steinbuch, M., and F. Meiring (1986). Analysis and simulation of a wind turbine with variable speed, [(ema Scientific fj Technical Reports, 1,(7), pp. 71-78.

H.O

S

32.0

~O.O

= 2l.81

p = l.25 kg/m 3 TW= 0.1 s (speed sensor). Generator

f\

/1 \

"

51

1

LF = 2.53 pu L" = 0.206 pu

= 2.51 pu = 0.0447 pu = 0.0159 pu

RF = 0.0037 pu Rdc

0.00

I

I

8.00

160

i4.0

S

pu

Lmq

= 1.54 pu

Lmd = 2.42 pu Xbasis = 2.33 n RQ = 0.0619 pu RD = 0.0345 pu

P

I

TIME

= 0.006

=4 Snom = 387500 VA

d. electrical power N

Lq = 1.69 pu

Ldc = 0.719 pu LQ = 1.68 pu

Rs

N

8

DC link.

Ld = 2.57 pu

LD Xn

V

fj

C2 = 1000 Nms/rad = 100 Nmrad/s 3 k = 100 Nms/rad T /3 = 0.2 s (pitch servo)

12.0

w = 418 .88 rad/s nom

~ .0

Fig. 5. Wind gust response of the wind turbine with the optimal output feedback controller (1) and with the SiRO controller( 2) .

O'r(O)

= 0.5

rad

O'i

= 0.75

TE = 0.5 s (field excitation system).

rad

319

Wind Energy Conversion System Output Feedback 2. Parameters of the Generator Model Ld

Lmd Lmd

Lmd LF L=

Lmd

0

0

0

0

0

0 Lmd Lmd LO 0 0 0 L q L mq 0 0 0 L mqL Q -Rs 0

0

o -R F N(w)=

0

0

0

where Ll =LFLO-L~d'

-wL -wL mq q 0 0

-RO

0

L2=LFLmd-L~d'

L3=LOLmd-L~d'

0

wLdwLmdwLmd -Rs 0 o 0 0 0 -RQ M=

-I [

0

o

0 I 0

0 0 0 0 0 -1

NI =LdLI-(L2+L3)Lmd'

N2=LqLQ-L~q'

0] T 0 0

and with wL" and Xn the commutation reactances, and a the inverter delay angle. i Transformation formulas:

3. Equations for the OC Link Load angle b can be calculated as a function of known state and input variables (Steinbuch, 1986):

b = arcsin{CI/d with

Hq )} - arctan(C 2 ),

(a.l)

Cl =( 1rwL"I~-{)lsC3cosar)/( 1rwL"I;C 4+2v'3C 3id ), C2 =( 1rwL"I~C5-2v'3C3iq)/( 1rwL"I;C 4 +2v'3C 3id ), C 3={ 18Uncosai +[ rRdc +31r(wL"+X n )]Is +rL dc (idC6+ iq C 7 )/ (3Is)} /( 18cosar ), C 4 =rL dc L 1id/(18v'3 l s N 1cosar ), C5 =-rLdcLQid/ (18v'3IsN 2 cosar ),

u d =-J3U ssinb, Uq=v'3Uscosb, id=-v'3lssin( O+'P) , iq =v'3lscos( 0+ 'P) ,

(a.2)

where Is=ldcv'6/1r. Phase angle 'P of the generator follow s from the phase equation of the rectifier: (a.3) The solution of the model proceeds as follows: using the expression for the load angle (a.l) and with known states id and iq' 'P is calculated with (a.2). Using (a.3) voltage Us is known, and with (a.2) u and u are q d calculated , and are inputs for the state space model (6), hence, the next integration step can be performed.