The Doubly Fed Induction Generator Robust Vector Control Based on ...

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Transactions on Systems, Signals & Devices Vol. 4, No. 2, pp. 237-249

The Doubly Fed Induction Generator Robust Vector Control Based on Lyapunov Method S. Drid,1 A. Makouf,1 M.-S. Nait-Said1 and M. Tadjine2 1

L.S.P.I.E Research Laboratory, Electrical engineering department, University of Batna, Algeria 2

Process Control Laboratory, Electrical engineering department, ENP Algiers, Algeria

Abstract

This paper deals with a robust control intended to a doubly fed induction generator (DFIG). The control system is established on the all-flux induction machine model using the Lyapunov linearisation approach. The obtained decoupling control between active and reactive stator power keeps the power factor close to unity. Associated with sliding-mode control, this solution shows good robustness against parameter deviations, measurement errors and noises. The global asymptotic stability of the overall system is proven theoretically and the experimental results largely confirm the effectiveness of the proposed DFIG system control.

Keywords: Doubly-fed induction generator, Wound rotor, Vector control, Lyapunov function, Power factor unity, Active and reactive power, Variable speed.

1. Introduction The first use of wind power was to sail ships in the Nile some 5000 years ago. Today, it is one of the most important sources of renewable energy in the world; it knew an extraordinary growth during the last decade, because this energy is recognized to be ecological and economic to produce electricity. At the same time, there was a fast development

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relating to the wind turbine technology [1–3]. In the area of wind power generation systems, where the input power varies considerably, variablespeed generation (VSG) is more interesting than fixed-speed systems. In these systems, a maximum power point tracker (MPPT) adjusts a system quantity to maximize turbine power output [1–7]. The generator operating directly at the variable speed drive is extremely attractive particularly for small machines where the rotor speed is high. The direct drive eliminates altogether the mechanical gear. This, result in multiple benefits like: lower weight, reduced noise and vibration and lower power losses by several percent [8]. This can be carried out using the synchronous generators provided that a static frequency converter is used to connect the machine to the grid. Another solution uses a double fed induction generator (DFIG) where the rotor is fed by a variable ac voltage sources which can be controlled in frequencies according to variable speed of the rotor shaft due to the variation of speed wind. Then the electric power at constant frequency is simply provided from the stator of the machine. Consequently, the use of the DFIG can give the increasing attention for the wind power generation. Particularly, the DFIG advantage is that the rotor is controlled from a reduced power converter, while the simpler vector control can be utilized to control the stator power-factor and the power flow. Capacities to produce electricity with power-factors close to unit which would reduce the system cost comparatively to using the condensers [2, 3]. A main advantage of the doubly fed induction machine is the accessibility of its both armatures from which the power flow control can be easily occurred between machine and grid [9]. Hence, the DFIG can operate likely in motor or generator inducing four characteristic operation modes summarized in Fig. 1. In the latter, Ps, Pr and Pm indicate respectively the stator, the rotor and mechanics powers. DFIM, in Fig. 1, means the doubly fed induction machine. When the DFIM operates like motor at sub-synchronous speed, Fig. 1-a, the power Pr is provided by the rotor, this is generally known like slip power recuperation mode. If speed increases so that the motor operate into super-synchronous speed, Fig. 1-b, the power Pr is absorbed by the rotor. While DFIM operates in generator at sub-synchronous speed, Fig. 1-c, the power Pr is then absorbed by the rotor. If speed increases so that the generator operates at super-synchronous speed, Fig. 1-d, the power Pr changes direction and the rotor provides a possible recovery power. For this kind of machines, both stator and rotor currents are accessible to physical measurement. So the vector fluxes estimation (stator or rotor) is simply deduced from simple non differential fluxcurrent equations, only depending on inductive parameter variations.

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The estimator robustness could be guaranteed using the sliding mode theory. The nonlinear feedback based on Lyapunov theory is adopted to control the active and reactive stator power flow.

Fig. 1. Operational modes characteristic of the DFIM.

The main of this present paper is the vector control for DFIG designed from the nonlinear control associated to the sliding mode control using Lyapunov theory from which it will be guaranteed global system stability and control robustness. The results will show that the proposed DFIG system control, operating at the variable speed, demonstrates its efficiency in renewable energy area.

2. The DFIM Model It is expressed, in the synchronous reference frame, by the following equations • Voltage equations: us = Rs is + dΦs + jωs Φs dt dΦ r + jω Φ ur = Rr ir + r r dt

(1)

• Current-Flux equations: is = λΦr + γΦs ir = χΦr + λΦs

(2)

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1 1 −M , γ = σL , χ = σL . where, λ = σL s Lr s s From (1) and (2), the all flux state model is done like

us = γ1 Φs − γ2 Φr + dΦs + jωs Φs dt dΦ r + jω Φ ur = −γ3 Φs + γ4 Φr + r r dt where: γ1 = σT1 s , γ2 = σTM , γ3 = σTM ,γ4 = s Lr r Ls The electromagnetic torque is given by Ce =

(3)

1 σTr .

PM ∗ ℑm(Φs ⊗ Φr ) σLs Lr

(4)

The stator power equations are: ∗

Ps = ℑm(us ⊗ is ) ∗ Qs = ℜe(us ⊗ is )

(5)

3. Robust control strategy 3.1 Stator voltage constraint To simplify calculations, let us consider the stator voltage constraint given as follows in dq-axis. usd = 0 , usq = us

(6)

3.2 Power law control Separating the real-part and the imaginary-part of (3), one will have.      dΦsd = f1 + usd  dΦrd = f3 + urd dt dt , (7)    dΦsq = f + u  dΦrq = f + u 2 sq 4 rq dt dt −f1 = γ1 Φsd − γ2 Φrd − ωs Φsq −f2 = γ1 Φsq − γ2 Φrq + ωs Φsd −f3 = −γ3 Φsd + γ4 Φrd − ωr Φrq −f4 = −γ3 Φsq + γ4 Φrq + ωr Φrd

(8)

The active and reactive powers, according to (6), are given respectively as: Ps = us (λΦrq + γΦsq ) (9) Qs = us (λΦrd + γΦsd )

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Let us formulate a Lyapunov function as follows V1 = (Ps − Psref )2 + (Qs − Qsref )2

(10)

Its derivative function is V˙1 = (Ps − Psref )(P˙s − P˙sref ) + (Qs − Qsref )(Q˙ s − Q˙ sref )

(11)

Substituting (7) and (9) in (11) gives V˙1 = (Ps − Psref )(α1 + λus urq − P˙sref )+ (Qs − Qsref )(α2 + λus urd − Q˙ sref )

(12)

where α1 = λus f4 + γ(f2 + us ) and α2 = λus f3 + γf1 (12) can become negative definite, if we define the following control law: urd = urq =

1 λus (−α2 1 λus (−α1

+ Q˙ sref − K2 (Qs − Qsref )) + Q˙ sref − K1 (Ps − Psref ))

(13)

Indeed, (13) replaced in (12) gives the required result as: V˙ 1 = −K1 (Ps − Psref )2 + −K2 (Qs − Qsref )2 < 0 Then (12) is asymptotically stable. Hence, using the Lyapunov theorem [10], one can conclude as follows lim (Ps − Psref ) = 0

t→∞

lim (Qs − Qsref ) = 0

(14)

t→∞

3.3 Power robust control The robust control is designed in order to solve the large model uncertainties due to parameter variations, errors measurement and noises. For feedback control, the model uncertainties are more globally related to the nonlinear function, fi with (i = 1, 2, 3 and 4) in (13), with the parameter drifts. In fact, these nonlinear feedback functions can be strongly affected by the conventional effect of induction motor (IM) such as temperature, saturation and skin associated to the different non linearities caused by harmonic pollution and the noise measurements. Globally we can write: fi = fbi + ∆fi (15)

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thus, αi defined above becomes: αi = α bi + ∆αi . The fbi is the result of the whole parameters and state variations as previously mentioned. Replacing (14) in (7), we obtain:      dΦsd = fb1 + ∆f1 + usd  dΦrd = fb3 + ∆f3 + urd dt dt , (16)    dΦsq = fb2 + ∆f2 + usq  dΦrq = fb4 + ∆f4 + urq dt dt

Tacking into accountfbi the new law control can be deduced as follows. h i   urd = 1 −α2 + Q˙ sref − K2 (Qs − Qsref ) − K22 sgn (Qs − Qsref ) λus h i  u = 1 −α + Q˙ − K (P − P ) − K sgn (P − P ) rq 1 sref 1 s sref 11 s sref λus (17) Then the analogue derivative Lyapunov function, established from (11), using (15) and (16), becomes as follows V˙ 1 = (Ps − Psref ) [∆α1 − K1 (Ps − Psref ) − K11 sgn (Ps − Psref )] + (Qs − Qsref ) [∆α2 − K2 (Qs − Qsref ) − K22 sgn (Qs − Qsref )] (18) Hence the fbi variations can be absorbed when system stability is increased if we choose: K11 > |∆α1 | , K22 > |∆α2 | Then we can write [10]:

V˙ 2 < V˙ 1 < 0

(19) (20)

We can conclude that the control law giving by (16) to end at the convergent process stability for any αbi i Figure 2 illustrates a general block diagram of the suggested DFIG control scheme. As it is shown, one can see that the stator powers are controlled. Note that the placement of the estimator-block which estimate from the armatures terminal measurements, firstly the stator voltage in term of modulus and position, respectively us, s and r, and secondly the feedback functions α1 , α2 .

4. Laboratory setup based on dSPASE DS1103 The basic structure of the laboratory setup is depicted in Fig. 3. The DFIG setup consists of induction machine with wound rotor and DC motor, which is used as a driven.

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Fig. 2. The block diagram of DFIG control scheme.

Fig. 3. Structure of the laboratory setup.

The Stator of the DFIG is connected to the grid. The generator rotor is fed by the inverter controlled directly by the DS1103 board. The dSPACE DS1103 PPC is plugged in the host PC. The DC motor is supplied by a rectifier. The encoder is used for the measure mechanical speed. The sensors used for the currents and voltages measure are

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respectively LA-25NP and LV-25P. The Interface to provide galvanic isolation to all signals connected to the DS1103 PPC controller. In Fig. 4 view of the laboratory setup is shown. All parts of the laboratory setup can be seen in this picture.

Fig. 4. Laboratory setup.

5. Experimental Tests 5.1 Timetable reference profiles In order to validate our approach, experimental tests were carried out using the proposed control scheme. The testing conditions were as follows. Figure 5 represents the imposed speed to the generator. The speed changed from 167rd/s to 137 rd/s at 4s. Figure 6 illustrates the reference profiles of the stator active power. It varied between 0W, 800W and 400W. The reactive power is fixed at 0VAR. For introducing the effects of parameter variations, the functions 1 and 2 are increased of 50% compared to their normal values at 2s.

5.2 Experimental results Figures 7 and 9 show, respectively, the active and reactive powers responses versus time according to the profile described above. Figure 8 present the active power errors versus time.

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Fig. 5. Generator speed (rd/s).

Fig. 6. Power reference profile Ps (W).

Fig. 7. Active power response.

Fig. 8. Active power error.

Figure 10 shows the power factor versus time which is easily maintained to unity. Figures 11 and 12 present respectively stator current and zooming, at t = [3 − 3.1]s, of the stator current and voltage respectively corresponding to each appropriate scale. Figure 13 present rotor current. Figure 14 presents the variations of the functions versus time.

Fig. 9. Reactive power response.

Fig. 10. Stator power factor.

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Fig. 11. Stator current (A).

Fig. 12. Stator voltage and current.

Fig. 13. Rotor current (A).

Fig. 14. Functions variations.

6. Conclusion The double accessibility is an important advantage of the doubly fed induction Generator. This induces to good control of the power flow between machine and grid permitting to inject the power such that the power factor is closed to unity. In this paper, one has investigated a robust vector control intended for doubly fed induction generator (DFIG). The stability of the robust control has been proven using the Lyapunov theory. The robustness against parameters variations (internal or external) has been achieved by the introduced sliding mode controller. The active and reactive powers track respectively their desired references without any recorded effect due to speed and/or all parameters variations. The obtained results demonstrate that the proposed DFIG system control which operates at the variable speed may be considered as an interesting solution in renewable energy area.

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Appendix Rated Data of the Induction machine with wound rotor: • Rated values: 800W; 220/380V; 50Hz; 3.8/2.2 A; 1420rpm; 50Hz. • Rotor: 3*120V; 4.1A.

Nomenclature s, r : Rotor and stator indices. d, q : Direct and quadrate indices for orthogonal components. √ x ¯ : Complex Variable such as:¯ x = ℜe(¯ x) + jℑm(¯ x) with j = −1 . ¯ x ¯ : It can be a voltage as u ¯ , a current as ¯i or a flux as φ. ∗ x ¯ : Complex conjugate. Rs : Stator and rotor resistances. Ls : Stator and rotor inductances. Ts , Tr : Stator and rotor time-constants (Ts,r = Ls /Rs,r ). σ : Total leakage flux coefficient σ = 1 − M 2 /(Lr Ls ). M : Mutual inductance. ρ : Absolute rotor position. P : Number of pairs poles. δ : Torque angle. ρs ,ρr : Stator and rotor fluxes absolute positions. ω : Mechanical rotor frequency (rd/s). Ω : Rotor speed (rd/s). ωs : Stator current frequency (rd/s). ωr : Induced rotor current frequency (rd/s). Ce : Electromagnetic torque. ∼ : Symbol indicating measured value. ∧ : Symbol indicating estimated value. ∗ : Symbol indicating command value. DFIG : Doubly Fed Induction Generator.

References [1] J. B. Ekanayake, L. Holdsworth, X. G. Wu, and N. Jenkins. Dynamic Modeling of Doubly Fed Induction Generator Wind Turbines. IEEE Trans. Power system, 18(2):803–809, May 2003. [2] R. Datta and V. T. Ranganathan. Variable-Speed Wind Power Generation Using Doubly Fed Wound Rotor Induction Machine-A Comparison With Alternative Schemes. IEEE Trans. energy conversion, 17(3):414–421, September 2002.

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[3] A. Tapia, G. Tapia, J. X. Ostolaza, and J. R. S´ aenz. Modeling and Control of a Wind Turbine Driven Doubly Fed Induction Generator. IEEE Trans. energy conversion, 18(2):194–204, June 2003. [4] M. Djurovic, G. Joksimovic, R. Saveljic and I. Maricic. Double Fed Induction Generator with Two Pair of Poles. In Seventh International Conference on Electrical Machines and Drives, pp.449-452, September 1995. [5] C. Keleber and W. Schumacher. Adjustable Speed Constant Frequency Energy Generation with Doubly Fed Induction Machines. In European Conference Variable Speed in Small Hydro, Proceedings, Grenoble, France, 2000. [6] C. Keleber and W. Schumacher. Control of Doubly fed induction Machine as an Adjustable Motor/Generator. In European Conference Variable Speed in Small Hydro, Proceedings, Grenoble, France 2000. [7] W. Leonhard. Control Electrical Drive. Springer-Verlag, Berlin, Heidelberg, Germany, 1997. [8] Mukund R. Patel. Wind and Solar Power Systems. CRC Press LLC, New York, USA, 1999. [9] P. Debiprasad, E. L. Benedict, G. Venkataramanan and T. A. Lipo. A Novel Control Strategy for the Rotor Side Control of a Doubly-Fed Induction Machine. In Thirty-Sixth IAS Annual Meeting Conference IEEE, Proceedings, 3(30):1695–1702, 2001. [10] H. K. Khalil. Nonlinear Systems. Prentice-Hall, 2ed edition, printed in USA, 1996.

Biographies Sa¨ıd Drid was born in Batna, Algeria, in 1969. He received his B.Sc., M.Sc. and Ph.D. degrees in Electrical Engineering from the University of Batna, Algeria, respectively in 1994, 2000, and 2005. Currently, he is an Associate Professor at the Electrical Engineering Institute, University of Batna, Algeria. He is the head of the “Energy Saving and Renewable Energy” team in the Research Laboratory of Electromagnetic Induction and Propulsion Systems of Batna University. His research interests include electric machines and drives, field theory and computational electromagnetism. He is also a reviewer for IET Theory Control & Applications journal, IET Renewable Power Generation journal and Journal of Physical and Chemical News.

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Abdesslam Makouf was born in Batna, Algeria, in 1958. He received his B.Sc. in Electrical Engineering from the National Polytechnic Institute of Algiers, Algeria, in 1983, and his M.Sc. in Electrical and Computer Engineering from the Electrical Engineering Institute of Constantine University, Algeria, in 1992. He received his Ph.D. in Electrical Engineering from the University of Batna in 2001. Currently, he is a Full Professor at the Electrical Engineering Institute, University of Batna. He is also the head of the Research Laboratory of Electromagnetic Induction and Propulsion Systems of Batna. His research interests include electric machines and robust control.

Mohamed-Sa¨ıd Nait-Said was born in Batna, Algeria, in 1958. He received his B.Sc. in Electrical Engineering from the National Polytechnic Institute of Algiers, Algeria, in 1983, and his M.Sc. in Electrical and Computer Engineering from the Electrical Engineering Institute of Constantine University, Algeria, in 1992. He received his Ph.D. degree in Electrical and Computer Engineering from the University of Batna in 1999. Currently, he is a Full Professor at the Electrical Engineering Institute, University of Batna. From 2000 to 2005, he was the head of the Research Laboratory of Electromagnetic Induction and Propulsion Systems of Batna. His research interests include electric machines, drives control, and diagnosis.

Mohamed Tadjine was born in Algiers Algeria, in 1966. He received his B.Eng. degree in Automatic Control from the National Polytechnic Institute of Algiers, Algeria, in 1990, and his Ph.D. degree in Control Systems froe Institute National Polytechnic of Grenoble, France, in 1994. During 1995, he held teaching and research positions at the University of Haute Savi, Annecy, France. From 1996 to 1997, he was a Researcher in the Automatic Systems Laboratory, University of Picardie Jules Verne at Amiens, France. He is currently a Full Professor at the Automatic Control Department, National Polytechnic Institute of Algiers, Algeria. His research interest is robust nonlinear control.