# Robust [H.sub.[infinity]] control of a doubly fed asynchronous machine.

1 Introduction

From all the renewable energy electricity production systems, the wind turbine systems are the most used specially the doubly fed asynchronous machine based systems, the control of theses systems is particularly difficult because all of the uncertainties introduced such as: the wind speed variations, the electrical energy consumption variation, the system parameters variations, in this paper we focus on the robust control ([H.sub.[infinity]] controller design method) of the doubly fed asynchronous machine which is the most used in the wind turbine system due to its low cost, simplicity of construction and maintenance [1].

This paper is organised as follow:

Section 2 presents the wind turbine system equipped with the doubly fed asynchronous machine and then the mathematical electrical equations from what the system is modelled (in the state space form) are given.

The section 3 presents the H[H.sub.[infinity]] robust controller design method with the LMI's solution used to control our system.

The section 4 presents a numerical application and results in both the frequency and time plan are presented And finally a conclusion is given in section 5.

2 System presentation and modelling

The following figure represents the wind turbine system

[FIGURE 1 OMITTED]

The system use the wind power to drag the double fed asynchronous machine who acts as a generator, the output power produced must have the same high quality when it enters the electrical network, i.e.: 220 volts amplitude and 60 Hz frequency and the harmonics held to a low level in spite of wind speed changes and electrical energy consumption in active or reactive power form. References [4], [5], [6] describe detailed models of wind turbines for simulations, we use the model equipped with the doubly fed induction generators (asynchronous machine) (for more details see [7]), the system electrical equations are given in(d,q)frame orientation, then the stator voltage differential equations are:

[V.sub.ds] = [R.sub.s] x [I.sub.ds] + d/dt [[PHI].sub.ds] - [w.sub.s] x [[PHI].sub.qs] (1)

[V.sub.qs] = [R.sub.s] x [I.sub.qs] + d/dt [[PHI].sub.qs] - [w.sub.s] x [[PHI].sub.ds] (2)

The rotor voltage differential equations are:

[V.sub.dr] = [R.sub.r] x [I.sub.dr] + d/dt [[PHI].sub.dr] - [w.sub.r] x [[PHI].sub.qr] (3)

[V.sub.qr] = [R.sub.r] x [I.sub.qr] + d/dt [[PHI].sub.qr] - [w.sub.r] x [[PHI].sub.dr] (4)

The stator flux vectors equations are:

[[PHI].sub.ds] = [L.sub.s] x [I.sub.ds] + M x [I.sub.dr] (5)

[[PHI].sub.qs] = [L.sub.s] x [I.sub.qs] + M x [I.sub.qr] (6)

The rotor flux vectors equations:

[[PHI].sub.dr] = [L.sub.r] x [I.sub.dr] + M x [I.sub.ds] (7)

[[PHI].sub.qr] = [L.sub.r] x [I.sub.qr] + M x [I.sub.qs] (8)

The electromagnetic couple flux equation :

[C.sub.em] = p x M/[L.sub.s] ([[PHI].sub.ds] x [I.sub.qr] - [[PHI].sub.qs] x [I.sub.dr]) (9)

The electromagnetic couple mecanic equation :

[C.sub.em] = [C.sub.r] + J d[OMEGA]/dt + f[OMEGA] (10)

With:

[V.sub.ds], [V.sub.qs] : Statoric voltage vector components in 'd' and 'q' axes respectively.

[V.sub.dr], [V.sub.qr] : Rotoric voltage vector components in 'd' and 'q' axes respectively.

[I.sub.ds], [I.sub.qs] : Statoric current vector components in 'd' and 'q' axes respectively.

[I.sub.dr], [I.sub.qr] : Rotoric current vector components in 'd' and 'q' axes respectively.

[[PHI].sub.ds], [[PHI].sub.qs] : Statoric flux vector components in 'd' and 'q' axes respectively.

[[PHI].sub.dr], [[PHI].sub.qr] : Rotoric flux vector components in 'd' and 'q' axes respectively.

[R.sub.s], [R.sub.r] : Stator and rotor resistances (of one phase) respectively.

[L.sub.s],[L.sub.r] : Stator and rotor cyclic inductances respectively.

[w.sub.s],[w.sub.r] : Statoric and rotoric current pulsations respectively.

M : Cyclic mutual inductance.

p : Number of pair of the machine poles.

[C.sub.r]: Gherbi et al.

[C.sub.r] : Resistant torque.

f : Viscous rubbing coefficient.

J : Inertia moment.

2.1 State space model

In order to apply the robust controller design method, we have to put the system model in the state space from; we consider the rotoric voltage [V.sub.dr], [V.sub.qr] as the inputs and the statoric voltage [V.sub.ds], [V.sub.qs] as the outputs, i.e. we have to design a controller who acts on the rotoric voltages to keep the output statoric voltages at 220 volts and 50Hz frequency in spite of the electric network perturbations (demand variations ... etc) and the wind speed variations (see figure.2).

[FIGURE 2 OMITTED]

Where: u, y and e are the rotoric voltage vector (control vector), statoric output voltage vector and the error signal between the input reference and the output system respectively. K, G are the controller and the wind turbine system respectively. R: is the statoric voltage references vector and perturbations are the electric energy demand variations, wind speed variations ...etc.

Let us consider x = [[[[PHI].sub.dr] [[PHI].sub.qr]].sup.T] as a state vector, and u = [[[I.sub.ds] [I.sub.qs] [V.sub.ds] [V.sub.qs]].sup.T] the command vector, the stator flux vector is oriented in d axis of Parks reference frame then : [[PHI].sub.qs] = 0 and [I.sub.ds], [I.sub.qs] are considered constant in the steady state i.e.: [[??].sub.ds] = [[??].sub.qs] = 0.

We use the following doubly fed asynchronous machine parameters:

[R.sub.s] = 5[OMEGA] ;[R.sub.r] = 1.0113[OMEGA] ; M = 0.1346H

[L.sub.s] 0.3409H ;[L.sub.r] =0.605H ;[w.sub.r] = 146.6Hz ;

[w.sub.s] = 2[pi] x 50Hz

Let w = [w.sub.s] - [w.sub.r] and [sigma] = 1 - [M.sup.2]/[L.sub.s] x [L.sub.r].

The state space (11) can be obtained by the combining of the equations (1) to (8) as follow:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Where:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

And:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3 The [H.sub.[infinity]] controller design method

It is necessary to recall the basics of a control loop (figure. 3). With G': the perturbed system.

[FIGURE 3 OMITTED]

Figure 3: The control loop with the output multiplicative uncertainties

The multiplicative uncertainties at the process output which include all the perturbations that act in the system are then: [[DELTA].sub.m] = (G'-G) x [G.sup.-1], with G'=G(I+ [[DELTA].sub.m]) :is the perturbed system, figure. 4 show the singular values plot at the frequency plan of [[DELTA].sub.m], we can see that the uncertainties are smaller at low frequencies and grow at the medium and high frequencies, this mean a strong perturbation at high frequencies (the transient phase), we also note a pick at: [omega] = 260rad/s, this is due to the fact that the system is highly coupled at this pulsation.

We can bound the system uncertainties by the following weighting matrix function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

The figure. 5 show that the singular values of [W.sub.t](jw) bounds the maximum singular values of the uncertainties in the entire frequency plan.

The robust stability condition [11] is then:

[bar.[sigma]][T (jw) x [W.sub.t] (jw)] < 1 (13)

Or:

[bar.[sigma]][T(jw)]< [bar.[sigma]][[W.sub.t](jw)].sup.-1] (14)

Where: [bar.[sigma]] is the maximum singular value and T(jw)is the nominal closed loop transfer matrix defined by:

T(jw)=G(jw) x K(jw) x [[I + G(jw) x K(jw)].sup.-1] (15)

The equations (13) allow us to guaranty the stability robustness, in other hand we most guaranty satisfying performances (no overshoot, time response ... etc) in the closed loop (performances robustness), this can by done by the performance robustness condition [8]:

[bar.[sigma]] [S(jw) x [W.sub.p] (jw)] < 1 (16)

Or:

[bar.[sigma]][S(jw)] < [bar.[sigma]][[[W.sub.p](jw)].sup.-1] (17)

Where:

S(jw) is the sensitivity matrix given by:

S(jw)=[[I + G(jw) x K(jw)].sup.-1] (18)

[W.sub.P](jw) is a weighting matrix function designed to meet the performance specifications desired in the frequency plan, we choose the following matrix function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

The figure. 6 represent the singular values of [W.sub.p] (jw) in the frequency plan, one notice that the specifications on the performances are bigger in low frequencies (integrator frequency behaviour), and this guaranty no static error.

Then the standard problem of Hoe Control theory is then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

i.e.: to find a stabilising controller K that minimise the norm (20).

With: [parallel][[parallel].sub.[infinity]] is The Hinfinity norm.

4 Application

The minimisation problem (20) is solved by using two Riccati equations [9] or with the linear matrix inequalities approach. For our system, we use the linear matrix inequalities solution (for more details see [10]). The solution (controller) can be obtained via the Matlab instruction hinflmi available at 'LMI Toolbox' of Matlab[R] Math works Inc [11].

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

The figure 7 and the figure 8 show the satisfaction of the stability and performances robustness conditions (14) and (17).

The figure. 9 show the step responses step responses of the closed loop controlled nominal system with:

R = ([V.sub.ds_ref] = 1/0 [V.sub.qs_ref] = 0/1) respectively.

The Outputs [V.sub.ds] and [V.sub.qs] follow the references with a good time response and no overshoot.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

5 Conclusion

In this paper we deal with the control problem of a wind turbine equipped with a doubly fed asynchronous machine subject to various perturbations and system uncertainties (wind speed variations, electrical energy consumption, system parameters variations ... etc), we show that the H[infinity] controller design method can be successfully applied to this kind of systems keeping stability and good performances in spite of the perturbations and system uncertainties.

References

[1] G. L. Johnson (2006), 'Wind energy systems: Electronic Edition', Manhattan, KS, October 10.

[2] 'AWEA Electrical Guide to Utility Scale Wind Turbines', (2005), The American Wind Energy Association, available at http://www.awea.org.

[3] P. Gahinet, P. Akparian (1994), 'A linear Matrix Inequality Approach to [H.sub.[infinity]] Control ', Int. J. of Robust & Nonlinear Control", vol. 4, pp. 421-448.

[4] J. Soens, J. Driesen, R. Belmans (2005), ' Equivalent Transfer Function for a Variable-speed Wind Turbine in Power System Dynamic Simulations ', International Journal of Distributed Energy Resources, Vol.1 No 2, pp. 111-133.

[5] 'Dynamic Modelling of Doubly-Fed Induction Machine Wind-Generators' (2003), Dig Silent GmbH Technical Documentation, available at http://www.digsilent.de.

[6] J. Soens, J. Driesen, R. Belmans (2004), ' Wind turbine modelling approaches for dynamic power system simulations ', IEEE Young Researchers Symposium in Electrical Power Engineering--Intelligent Energy Conversion, (CD-Rom), Delft, The Netherlands.

[7] J. Soens, V. Van Thong, J. Driesen, R. Belmans (2003), ' Modelling wind turbine generators for power system simulations ', European Wind Energy Conference EWEC.

[8] Sigurd Skogestad, Ian Postlethwaite (1996), 'Multivariable Feedback Control Analysis and Design', John Wiley and Sons. pp: 72 to 75

[9] J. C. Doyle, K. Glover, P. P. Khargonekar and Bruce A. Francis (1989), 'State-Space Solution to Standard [H.sub.2] and [H.sub.[infinity]] Control Problems', IEEE Transactions on Automatic Control, Vol. 34, No. 8.

[10] D.-W. Gu, P. Hr. Petkov and M. M. Konstantinov (2005), 'Robust Control Design with MATLAB[R]', [c] Springer-Verlag London Limited.pp:27 to 29

[11] P. Gahinet, A. Nemirovski, A. J. Laub, M. Chilali (1995). "LMI Control Toolbox for Use with MATLAB[R]", User's Guide Version 1, The Math Works, and Inc.

Gherbi Sofiane

Department of electrical engineering, Faculty of Science of the engineer

20 August 1956 University, Skikda, Algeria

E-mail: sgherbi@gmail.com

Yahmedi Said

Department of electronic, Faculty of Science of the engineer

E-mail: sais.yahmedi@carmail.com

Sedraoui Moussa

Department of electronics, Faculty of Science of the engineer

Constantine University, road of A1N EL BEY Constantine, Algeria

E-mail: msedraoui@gmail.com