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March 26-29

Sliding Mode Control of a Variable Speed Wind Energy Conversion System with DFIG

Mohamed Machmoum, Frederic Poitiers Nantes Atlantique Electrical Engineering and Electronics Research Institute Polytechnic school of Nantes University, rue Christian Pauc, La Chantrerie, BP 60601, 44306 Nantes Cedex 3 E-mail: [email protected]

Copyright © 2009 MC2D & MITI

Abstract: This paper presents the study of a variable speed wind energy conversion system based on a Doubly Fed Induction Generator. Appropriate state space model of the DFIG is deduced. An original control strategy based on a variable structure control theory, called also sliding mode control is applied to achieve control of active and reactive powers exchanged between the stator of the DFIG and the grid. Simulation results and improvement of the behavior of the DFIG are presented and discussed to validate the proposed control strategy. Keywords: Doubly Fed Induction Generator, Sliding Mode Control, State space model, Wind Energy. 1. Introduction The first works about variable structure systems using sliding mode control have been realized during the 50 years by Emelyanov's team. This approach was first developed for a linear second order system and has now been extended to linear, non-linear, discrete and multi-variable systems. Variable-structure control is nowadays frequently used because of its simplicity and efficiency. By using variable-structure control in a system it is possible to bring the running point of the system in a sliding hypersurface. When this hypersurface is reached the sliding mode occurs. Sliding-mode control can be very interesting because it is not sensible to parameters variations and perturbations, its behavior only depends on the sliding hypersurface parameters. Nevertheless, problems such as commutation delays and hysteresis can involve oscillations around

sliding surface called chattering phenomenon. Many solutions have been proposed to resolve these problems. One of these solutions consists to approximate the discontinuous function by a continuous function near the sliding surface, the chattering is then reduced but the precision is altered. This paper proposes to apply variable structure control to a doubly-fed induction generators (DFIG) used in wind-energy conversion systems. The DFIG consists of a wound rotor induction generator directly connected to the grid on the stator side and via a back-to-back power converter on the rotor side as shown in fig. 1. This system has recently become very popular as a generator for variable speed wind turbines due to its flexibility, stability in power generation and to the fact that the power electronic converter only has to handle a fraction (20-30% rating) of the total power [16]. The grid-side converter is usually used to regulate the DC bus between the two converters

regardless of the magnitude and direction of the rotor power, while keeping sinusoidal grid currents. It can be used for controlling reactive power flow between the grid and the grid side converter ( the power factor is usually set to unity). The rotor-side converter is used to control the machine’s behavior in both sub. and super-synchronous modes as well tracking the maximum power output characteristic of the wind turbine. The vector control for this converter ensures decoupling control of stator active and reactive power drawn from the grid [7-9]. Many papers have presented control schemes of DFIG , generally based on vector control concept (with stator flux or voltage orientation) associated with classical PI regulators as proposed by Pena in [10], or based on direct power or torque control [11-13]. The aim of this paper is to apply a Variable Structure Control (VSC), called also the sliding mode control to a DFIG [14-16]. The use of this control mode can be justified by the high performances required by DFIG and the robustness of the controller. The parameters of the DFIG can change during operation, the load can vary and the wind speed is not a predictable source. This fact can be dangerous for the DFIG without appropriate and robust controllers. The proposed strategy is used to control active and reactive powers between the stator of and the grid. A variable structure controller is first calculated to evaluate performances of the system under varying wind speed conditions. The different steps of the controller synthesis and original ways allowing improving the behavior for power references tracking and DC bus voltage variations are then analyzed. Simulation results are presented and discussed to validate the proposed control strategy.

problem of this control is to determinate the parameters of each structure and the commutation functions. Switching from one structure to another permits to have the benefits of each structure. The system can then have interesting properties as becoming stable while it is elaborated with unstable structures.

Figure 2 : System behavior around sliding surface

Let us consider the non-linear system below:

xɺ = f ( x, u )

(2.1)

where x is the system error and u the delivered signal of the controller. The sliding surface is given by :

σ ( x) = 0

(2.2)

As the control is applied, the system will take one of the two forms presented below :

(

)

x = f − x, U − if σ ( x) < 0 (2.3)

(

)

x = f + x, U + if σ ( x) > 0

(2.4)

This can be illustrated by Fig. 2. By considering a non-linear system given by:

Xɺ = f ( X , t ) + g ( X , t ) .U

(2.5)

Where:

X ∈ ℜn , U ∈ℜ m f ( X , t ) ∈ ℜn g ( X , t ) ∈ ℜn×m

f and g are continuous and their derivative are also continuous.

Figure 1: wind turbine with DFIG

2. Variable Structure Systems Unlike state feedback, where the control doesn't change during system operation, variable structure control can switch at any time from one structure to another. The

The two steps necessary to synthesize a variable structure controller presented on Fig. 3 are: - define commutation surface σ ( x) = 0 whose order is smaller than the system order and which represent the desired dynamics. - calculate the control u(x,t) with the objective that every state which is outside the

commutation surface must join in in a finite time. The system takes the dynamics of this surface and the system evolution joins the equilibrium point.

vector is consequently in quadrature advance in comparison with the stator flux vector: V = 0 and V = V = ω ψ ds qs s s s (2.9) The stator active and reactive power can be expressed as functions of rotor currents:

M   P = Vs I qs = −Vs L I qr  s  Vψ V M Q = V I = s s − s I  s ds dr L L s s 

Figure 3 : Variable-structure controller.

3. Application of Variable Structure Control to Doubly-Fed Induction Generator A State Model of the DFIG The general electrical state model of the induction machine obtained using Park transformation is given by the following equations:

Considering the state variables of equation (2.8) ,the electrical model of the induction machine can be written as follows:     M dψ Vqs  − ωsψ qs + ds Rs.aψ ds − Rs.c LrσIdr +  Lsωs  dt     Vds   dψ qs   M  Vqs  + ωsψ ds + V  Rs.aψ qs − Rs.c LrσIdr +   ω L dt qs  s s   =  Vdr         dI M M   Rr.b L σI + Vqs  − Rr.cψ ds −ωr  LrσIqr + Vds  + Lrσ dr  r dr  ω Lsωs  L dt Vqr    s s    dIqr       M M Vds  − Rr.cψ qs + ωr  LrσIdr + Vqs  + Lr * Rr.b LrσIqr +  Lsωs  Lsωs  dt    

(2.11)

With:

Stator and rotor voltages: d  ɺ Vdqs = R s I dqs + dt ψ dqs ∓ θ sψ qds  d V =R I + ψ ∓ θɺ ψ  dqr r dqr dt dqr r qdr

(2.10)

σ = 1−

M2 1 1 M ;a= ;b= ;c= σ Ls σ Lr σ Ls Lr Ls Lr

The state model can then be written as :

(2.6)

Xɺ = f ( X , t ) + g ( X , t )U dq

Stator and rotor fluxes:

With:

= L I + MI ψ s dqs dqr  dqs  ψ dqr = L r I dqr + MI dqs

(2.7) The state variable vector is then :

ψ ds  ψ qs  X =   I dr  I   qr  (2.8) By choosing a reference frame linked to the stator flux and if the per phase stator resistance is neglected, which is a realist approximation for medium and high power machines used in wind energy conversion, the stator voltage

 dψ ds   dt   dψ   ds   dt  Xɺ =  dI   dr   dt   dI qr     dt 

(2.12)

Udq = [Vds Vqs Vdr Vqr ] 1 0   g ( x, t ) = 0   0 

0

0

1

0 1 0 Lrσ 0

0

0  0   0   1   Lrσ 

(2.13)

(2.14)

    M Vqs  + ωsψ qs  − Rs.aψ ds + Rs.c LrσI dr +  Ls *ωs          − Rs.aψ qs + Rs.c LrσI qr + M Vds  − ωsψ ds      Ls *ωs   f ( X , t) =         1  M M  L σ  − Rr.b LrσI dr + L ω Vqs  + Rr.cψ ds + ωr  LrσI qr + L ω Vds   s s s s       r   1      M M * − Rr.b LrσI qr + Vds  + Rr.cψ qs − ωr  LrσI dr + Vqs   Lsωs  Lsωs     Lrσ 

(2.15)

The rotor currents (which are linked to active and reactive powers by equation (2.10)) have to track appropriate current references, so, a sliding mode control based on the above Park reference frame is used. The sliding surfaces representing the error between the measured and references rotor currents are given by this relation:

σ d = λ ( I drref − I dr )  σ q = λ ( I qrref − I qr )

(2.16)

control vector used to constraint the system to σ sdq = 0 converge to . The control vector U dq eq σɺ = 0 is obtain by imposing dq so the equivalent control components are given by the following relation:

U eq dq

(2.17)

To obtain good performances, dynamic and commutations around the surfaces, the control vector is imposed as follows: (2.18)

The sliding mode will exist only if the following condition is met:

σɺ sdqσ sdq < 0 The product

σɺ sdqσ sdq

Simulation of the whole system has been realized using Matlab-Simulink. The blockdiagram of this simulation is presented on Fig. 4. The DFIG model is associated with a turbine emulator which is controlled with MPPT (Maximum Power Point Tracking) strategy. In this control mode, we can distinguish three running areas (Fig. 5): - For low wind speeds, the DFIG is controlled at variable speed in order to maintain the

B The "sign" function

U dq = U eq dq + Ksign(σ dq )

4. Simulations results A System configuration

Vdr and Vqr will be the two components of the

       M M Vds    Vqs  + Rr.cψ ds + ωr  Lr σI qr +  −  − Rr.b Lr σI dr + Ls ωs Ls ωs        =       M M −  − Rr.b Lr σI qr +     ψ ω σ + − + V Rr . c L I * V ds  qs r r dr qs     Ls ωs Ls ωs       

Figure 4 : Block-diagram of the whole system

(2.19) can be written as follows for

λ=

Ωg R

V

wind relative speed and so, the power coefficient of the wind turbine to their optimal value. The turbine speed Ωg is then variable and the pitch angle β of the wind turbine is kept to its optimal value and is not modified in this area.

- For medium wind speeds, the DFIG is controlled in order to maintain the rotational speed Ωg to its nominal value. - For high wind speeds, the DFIG is controlled in order to produce its nominal power. The power extracted from the wind turbine is then limited by adjusting the blade's pitch angle.

d-axis:

σ d σɺ d = − g ( x, t ) Ksign(σ d )σ d

(2.20)

If this product must be negative, knowing that

g ( x, t ) is a positive matrix and that sign(σ d )σ d is positive, the value of K must be positive. Figure 5 : Running areas for MPPT

C Effects of “integral” action

Figure 6 : Block-diagram of variable structure control applied to the DFIG with "sign" function.

An integral action is now added to the control in order to reduce the gain of the "sign" function and then reduce the chattering effect. This action will also improve the controller's performances in terms of reference tracking. The integral surface form is given by: t

B Variable structure control with "sign" function The block-diagram of the variable structure control of the DFIG is presented on Fig. 6. The stator active and reactive powers are controlled. Simulation results with "sign" function are presented on Fig. 7 where the DFIG is used at sub-synchronous and super-synchronous speeds due to MPPT control (synchronous speed is 157 rad/s and speed variations go from 65 to 180 rad/s) and the rotor current phase is consequently modified during these variations. The results show the evolution of stator active and reactive powers and DC-bus voltage. It can be seen that these three signals present important ripple. This is due to the use of hysteresis band and "sign" function which induces a high commutation level. To reduce the ripple and improve precision in reference tracking it is then necessary to add an integral action to the control.

ηdq = ξ ∫ σ dq dτ 0

(2.21)

The integral function is added during the positive phase of the sliding surface, the equivalent control will have the following form:

U dq = U dq + Ksign(σ dq ) + ηdq eq

(2.22)

And the block-diagram of the control is modified as shown on Fig. 8.

Figure 8 : Block-diagram of variable structure control applied to the DFIG with "sign" function and integral action

Simulations results are presented on Fig. 9 and show that ripple on active and reactive powers has decreased because the integral action has reduced the gain of the "sign" function. However, ripple on DC-bus voltage is always present. To reduce this ripple, there are two usual solutions: - increasing the value of the capacitance but the cost is not negligible. - increasing the hysteresis band but it will also increase the chattering effect. As these solutions are not totally satisfying, the next part presents a third solution where another discontinuous component is introduced in the control.

D The “sat” function Figure 7 : Simulation results with "sign" function

Another discontinuous component is then added in the control with the following form:

considerably reduced and the DC-bus voltage has no longer ripple. 5. Conclusion

Figure 9 : Simulation results with "sign" function and integral action

∆U = Ksat (σ )

(2.23)

and:

σ  sat (σ ) =  φ  sign(σ ) 

if σ ≤ φ

(2.24)

if σ ≥ φ

A variable structure control of a doubly-fed induction generator has been presented. This structure has been used for reference tracking of active and reactive powers exchanged between the stator and the grid by controlling the rotor converter. DFIG is often used in wind energy conversion system, that's why the model of the DFIG has been associated with a wind turbine model controlled with MPPT strategy. The whole system thus constituted permits to control the DFIG at sub. and super synchronous speeds. The first sliding-mode control with sign function presented show interesting results but the important ripple level obtained on active and reactive powers and on the DC-bus voltage is not satisfying. The integral function is then introduced to improve performances in reference tracking and to limit the gain of the sign function. Ripple on active and reactive powers is then reduced but the DC-bus still presents important ripple. Another discontinuous function called "sat" function is then added in the control to reduce the commutation level. Ripple level is then considerably reduced on active and reactive powers and the DC-bus voltage becomes very stable. These results show the possibility to use variable structure control for DFIG used in wind energy conversion systems, indeed, this structure can be numerically synthesized easily and give as good results as linear controllers.

References

Figure 10 : Simulation results with "sign" function, integral action and "sat" function

The simulation results obtained after adding the "sat" function are presented on Fig. 10. The association of this function (which reduces commutation in the control's discontinuous component) with the integral action permits to obtain good results. Indeed, ripple on active and reactive powers is

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