Sliding Mode Energy-Reliability Optimization of a Variable Speed ...

Proceedings of the 2006 International Workshop on Variable Structure Systems Alghero, Italy, June 5-7, 2006

MonD.1

Sliding Mode Energy-Reliability Optimization of a Variable Speed Wind Power System I. Munteanu*, J. Guiraud**, D. Roye**, S. Bacha**, and A. I. Bratcu* *

**

“Dunărea de Jos” University of GalaĠi, Romania 47, Domnească 800008-GalaĠi, Romania

Abstract— This paper concerns the design of a sliding control law for maximizing the power harvested from the wind by a variable speed induction generator based wind power system. The parameters of this law are tunable for limiting the mechanical stress. An appropriate sliding surface has been found in the speed-power plane, which allows the operation more or less close to the optimal regimes characteristic, thus implementing an energy-reliability optimization. The doubly-fed induction generator used here is torque controlled. The proposed control law is validated by both off-line and on-line, real time simulation, the latter on a dedicated experimental rig, showing that the control objective is fully accomplished.

The variable speed operation of WPSs allows various control strategies to be put into practice [2]: output power regulation and control of its parameters, improvement of the drive train dynamics compliance to alleviate the mechanical stress, tracking of an imposed power-speed trajectory, etc. In the partial load regime, the variable speed fixed pitch horizontal axis wind turbines (HAWT), considered in this paper, has a power coefficient (aerodynamic efficiency), Cp, depending on the tip speed ratio (i.e., the ratio between the blades’ peripheral speed and the wind speed, O : ˜ R v ) and having a maximum for Oopt, a well-determined value [1] (fig. 1b). The power characteristic of the wind turbine has a maximum for each wind speed. All these maxima form the so-called optimal regimes characteristic (ORC, fig.1c).

I. INTRODUCTION Wind power systems (WPSs) technology is now a certitude for the possibility of sustaining the continuously growing energy needs of the mankind using clean and inexhaustible sources. This rapidly growing domain is at the present a stable segment of the electrical energy market, based on a mature technology [1]. The area of the WPSs’ control reveals the unitary-systemic character of the wind energy conversion problem, expressed by multi-purpose optimization requirements, as imposed by the WPSs’ exploitation experience and grid integration. A global optimal control approach is needed, for ensuring the functionality and viability of the system as a whole, and also the provided energy’s quality. These are not trivial tasks, provided the erratic behaviour of the atmospheric conditions. The WPSs’ global performance, i.e., their economical efficiency and integration capacity into energy grids, is aimed at, as shown in the sequel. Then, the variable structure control of WPSs is briefly discussed. Depending on the values of the average (steady state) wind speed, four zones may be identified in the static operation of a WPS [1] (fig. 1a). Zones I and IV, where the provided power is zero, and the so called full load zone (III), where the power must be limited to the nominal one, are not concerned by this paper. The interest is here focused on zone II, called partial load zone, where the extracted power proportionally depends of the wind speed cubed.

1-4244-0208-5/06/$20 ©2006 IEEE

Laboratoire d’Électrotechnique de Grenoble BP 46 38402-Saint Martin d’Hères, France

Pn Pwt a) IV I II III vS vN vM v 0.5

C p (Ȝ) b)

0

Ȝ opt

Ȝ

4.5 4.0 3.5 3.0 2.50

Pwt > kW @ c)

ORC

2.0 1.5 1.0 0.5

9 m/s

7 m/s 5 m/s

: > rad/s@ 12 0 5 10 15 20 25 30 35 40 45

Fig. 1. a) Power captured by a WPS versus average wind speed; b) Typical aerodynamic performance curve of a WPS; c) The maximum power points forming the optimal regimes characteristic (ORC).

The studied class of WPSs can be controlled at variable speed for maximizing the captured power, i.e., for optimizing the aerodynamic efficiency, Cp, by tracking Oopt (fig. 1b). This means to drive the operating point around the ORC (fig. 1c) [3]; WPSs’ specificity requires an improved reliability that the control effort should not affect. In the literature, various methods are proposed for optimizing the WPSs’ behavior. If Oopt is known, the (optimal) control may be implemented using some linear approaches [4], by tracking the corresponding value of the shaft speed, :opt O opt ˜ v R (not always efficient because of

90 92

the modeling uncertainties). Nonlinear methods are often used, some of which having the main drawback of exclusively aiming at maximizing the energetic efficiency, while ignoring the large torque variations induced by tracking the shaft speed, :opt , related to the system’s reliability, e.g., the so-

II.

MODELLING ASSUMPTIONS

Fig. 2 presents the block diagram of a grid connected asynchronous generator based WPS controlled to operate at variable speed. The rotor (low speed capturing device) drives the high speed shaft of the doubly-fed induction generator (DFIG) by means of a speed multiplier. A power electronics converter (rectifier-inverter pair) controls the electric generator and the power fed into the grid [17]. The rectifier, together with the DFIG, forms the electromagnetic subsystem (EMS), while the pair inverter – local grid represents the grid interface system (GIS). Generally, there are two local control blocks – i.e., the provided power regulation and conditioning (containing the GIS) and the generator torque/speed control (containing the EMS) – and a global behavior control block, which is of interest in this paper.

called “Maximum Power Point Tracking” (MPPT) method [5], using minimal information about the system, and the fuzzy control techniques [6], [7]. Ekelund [8] used an optimization criterion complying with the minimization of the generator torque variations, which induces mechanical fatigue. The system’s parameters depend on the average wind speed, so a gain scheduling procedure was used to tune the controller, and so it is in [9]. In [10] a control algorithm based upon Ekelund’s criterion, but without using any adaptive structure, is proposed: it uses the idea of separating the seasonal (low frequency), and the turbulence (high frequency) wind speed components from the Van der Hoven wind model [11]. The variable structure control (VSC) as nonlinear method is particularly suitable to variable speed WPSs’ control, provided the nonlinear nature of the system. This method is robust to disturbances and parametric uncertainties (so it does not need a precise knowledge about the system) and can be implemented by the power electronics already existing in the system. But for the applications using the electromagnetic torque as control input, the mechanical stress increases due to the chattering. VSC has been used on different configurations of WPSs, either for regulating the generated power [12], [13], or for multi-purpose optimization: trade-off between maximizing the harvested power and minimizing the control effort [14], [15]. In [16] a systematic VSC design method, combining the sliding mode techniques with passivity concepts, is applied to the power regulation of a solar/wind generation system. However, the sliding surface must be chosen and not systematically computed. It is also the case of the VSC strategies with integral compensation proposed in [15] for tracking a WPS’s optimum, irrespectively from the wind speed. In [12] the power regulation uses a Hf approach, based upon the quasi-linearization of the plant’s nonlinear model, to design a stable sliding surface. Such a method has the drawback of needing a quite precise modeling knowledge. This paper proposes a sliding mode approach for tracking the energetic optimum of a doubly-fed induction generator based variable speed WPS. The sliding surface is systematically derived from a desired reduced order dynamics and allows the turbine operation more or less close to the ORC, as imposed by a trade-off between the torque (control input) ripple and the optimum tracking. The rest of the paper is organized as follows. In the next section a model is obtained. The sliding surface and control law are synthesized in the third section. The fourth section contains simulation results, both off-line and real time, the latter performed on an experimental rig described in the same section. The last section lists some concluding remarks.

R O T O R

wind

Control Algorithm Gear Box Drive Train

3~

:

grid

3

v~

3

DFIG

Torque Control

AC/AC

PWM

Fig. 2. Block diagram of the controlled variable speed WPS.

The simplified model (a single mass) [18] of this system is obtained, by adopting the following assumptions: the wind speed is modeled as stationary random process, having a mean (low frequency) component and a turbulence component; the system operates only in the partial load region and the frequency domain of the wind speed is limited; the power coefficient is known and the structural dynamics are negligible; the electric generator is ideal (its constructive features do not practically influence its dynamics and its parameters are constant); the power loss induce a constant efficiency for the whole wind speed range; the wind power system is torque controlled by means of a vector control scheme (to implement the variable speed regime) and the EMS dynamics are of first order [17]. The following notations are used: : – rotational speed of the high speed shaft; * wt – wind torque; *G – electromagnetic torque; i – speed multiplier (gearbox) ratio; J t – inertia of the high speed shaft; Tg – time constant of the EMS. From the movement equation of the high speed shaft [18], the system’s equations write: ­˜ °:(t ) * wt (:, v)  i ˜ J t  *G J t (1) ® ˜ °* (t )  * T  u T G g g ¯ G So, the state vector is x

>:

T

*G @

and u

*G , the

electromagnetic torque reference, is the control input.

93

III.

B. Control law This subsection is dedicated to computation of the two components of the sliding mode control law: the equivalent control input, ueq , and the on-off component, u N [19].

DESIGN OF THE SLIDING MODE OPTIMAL CONTROL LAW

A. Sliding surface Here, the goal is to find a sliding surface allowing the turbine’s operation more or less close to the ORC. The image in the ( : ,P) plane of the sought surface must have a nonempty intersection with the ORC for each value of the wind speed and also a controllable slope for adjusting the sliding mode dynamic. Equations (1) may be written as: x f (x, t )  B(x, t ) ˜ u ,

The equivalent control input is computed as: ueq

T

a2 @ ˜ > : *G @ ,

0 results V1 (:)  a3 ˜ *G

torque coefficient and wV w:

(3)

V

a1 ˜ J t ˜ :  *G ˜ 1  a2 ˜ J t  * wt i

V

i ˜  OC

' p (O )  C p ( O )



O2

,(11)

 KvR i ˜  OC 2

' p ( O )  C p (O )



O2

(12)

Let G(O )

 OC

' p ( O )  C p (O )



O2 .

(13)

Note that G(O) is practically constant around the ORC, having an a priori known value. Using wV w*G 1  a2 ˜ J t first in (10) and then in (9) and replacing (11) in (9), one obtains after some algebra: ueq  Tg 1  a2 ˜ J t ˜  * wt i  *G ˜  a1  A(O, v)  *G

(5) (6)

Provided that the dynamics on the sliding surface are ˜

Jt ˜ :

(7)

ueq

* wt i  * G { a1 ˜ J t ˜ :  a2 ˜ J t ˜ *G , results that: *G  Tg 1  a2 J t ˜  a1 J t :  a2 J t *G ˜  a1  A(O, v)

(14)

As to (14), the control law has torque dimensions. Parameter a1 imposes the convergence speed to the sliding mode regime, a1 1 Tsm , where Tsm ! 0 is time constant. The value of a2 results from imposing the steady state “target”, i.e. the (optimal) operating point (OOP) – at O opt :

˜

a1 ˜ J t ˜ :  *G ˜ a2 ˜ J t   * wt i  *G   

˜



a1 ˜ J t  KvR 2

A(O, v)

J t ˜ : and:

J t ˜:

being the power coefficient’s

where:

Parameter a1 represents a measure of the sliding mode dynamic. The steady state regime (equilibrium) is imposed by choosing parameter a2. Here, the equilibrium point is set to the optimal one (i.e., on the ORC). From the movement equation it holds that * wt i  *G

C p (O ) O being the

a1 ˜ J t  A(O, v)

1  a2 ˜ J t . From (4), (5) and (6), one obtains:

Let a3

C 'p (O)

derivative in relation to O . One obtains:

0 and, so:

V1 (:) a3 ˜ *G From (3) one can obtain: *G 1 1  a2 ˜ J t ˜  * wt (:, v) i  a1 ˜ J t ˜ :

(10)



0.5S ˜ U ˜ R 2 invariant, C* (O)

with K

where a1 and a2 correspond to the first order dynamics on the sliding surface, as detailed later. Equation (2) and condition wV wx ˜ B(x, t ) z 0 (existence of the equivalent control input [19], [20]), require that wV w*G a3 z 0 . A first form of the commutation surface may thus be written: V(:, *G ) V1 (:)  a3 ˜ *G (4) From V(:, *G )

wV w*G Tg



The sliding mode dynamics (that is, on the sliding surface) may be imposed as equivalent to some linear ones: * wt (:, v) i  *G J t { > a1

T

relation to the speed, wV w: a1 ˜ J t  1 i ˜ w* wt w: , may be deduced using the partial derivative of the wind torque: w* wt w: 0.5 ˜ S ˜ U ˜ R 3 ˜ v 2 ˜ wC* (O ) w: , Kv 2 R ˜ wC* (O ) wO ˜ wO w: KvR 2 ˜ OC 'p (O )  C p (O ) O 2

T

˜

wV w*G @ ˜ ª¬0 1 Tg º¼

The sliding surface does not explicitly depend on time, so wV wt 0 . The partial derivative of the sliding surface in

ª¬0 1 Tg º¼ . The state equation is already in the regular form [19], so the state equation and the reduced order dynamics are those circled in the relation below: ª ˜ º « : » ª * wt (:, v)  i ˜ J t  *G J t º  ª 0 º ˜ u (2) » « » « ˜ » ««  *G Tg ¼» ¬1 Tg ¼ «¬ *G »¼ ¬

:

>wV w:

wV wx ˜ B

ª¬* wt (:, v)  J t ˜ i  *G J t  *G Tg º¼ is nonlinear because of the wind torque, * wt (:, v) , and B(x, t )

(9)

Next, the expressions involved will be computed.

T

f (x, t )

where

1

 > wV wx ˜ B @ ˜ > wV wt  wV wx ˜ f (x, t )@

˜

:

(8)

˜

a1 ˜ :opt  a2 ˜ * opt

 a1 ˜ :opt * opt .

0 ; so, a2

One may attempt to dynamically modify the (static) gain, a2, using the following expression:

a1 ˜ J t ˜ :  a2 ˜ J t ˜ *G  J t ˜ : So, the switching surface depends on the derivative of the rotational speed, which is an inconvenient for the real time implementation. A first order high pass filter with time constant of 0.1 s may be used to estimate this derivative.

a2





a1 ˜ :opt * opt ˜ 1  k ˜ :  :opt





:opt ,

(15)

with kt0, to reduce the operating point’s variations around the OOP. Expression (15) is not valuable but around the OOP,

94

otherwise parameter a2 can take sufficiently large values for the system leave the normal operating regime. The next section shows that the larger the value of k is, the more quickly the optimal steady state, (:opt , * opt ) , is reached, so

IV.

the tracking has a better quality. The system is thus forced to follow more accurately the control goal, i.e. the energetic optimization, but the control input variations are more significant, affecting the reliability. This parameter adjusts the control effort and can be used to design a desired energyreliability trade-off. As k increases, the slope of sliding surface’s image in the ( : ,P) plane is increased to follow the ORC. As (:opt , * opt ) pair depends on the wind speed, then

A. Description of the experimental rig used for real time simulation The functional diagram of the experimental rig for real time simulation (fig. 3) essentially instantiates the block structure from fig. 2. The wind turbine rotor and the mechanical transmission are replaced by a wind turbine emulator (WTE), reproducing the dynamical mechanical characteristics of the turbine’s high speed shaft [10]. Effectively, this is realized by means of a torque controlled DC motor, whose reference is computed in the DS1005 dSPACE system (connected to PC1 computer in fig. 3). A second fixed point TMS320F240 DSP (connected to PC2 computer) is used to effectively control and supervise the DC machine.

so does the sliding surface. The on-off component of the control law, u N , is obtained by choosing as Lyapunov (energy) function the square of the obtained sliding surface. One may thus deduce: u N D ˜ sgn h (V) , (16) where sgn h (˜) is a hysteretic sign function, of width h. The total sliding mode control law is the sum of the equivalent component and on-off component: (17) u ueq  u N

WTE 3

EMS

torque control

high speed shaft characteristics

iMCC

DSP torque TMS320F240 reference

GIS

DFIG

MCC

local grid

NUMERICAL AND REAL TIME SIMULATION RESULTS

Off-line simulations have been performed under Matlab/Simulink© software, using Simulink diagrams. Real time experiments have been also performed on a 1:1 test rig composed of a wind turbine emulator and a torque controlled DFIG, described in subsection A. The appendix contains the parameters used in both simulation cases, the two sets of results are compared in subsection B.

compute high speed torque

v

i~

3

:

3

filter

filter

i~

v~

real time computations

local grid

generator torque control

electrical power control

v~

SLIDING MODE CONTROLLER PC2 Testpoint Interface

User

PC1

Simulink&ControlDesk Interface DS1005 (dSPACE) Fig. 3. Structure of the experimental rig.

The EMS and the GIS exist as they are in the real system (fig. 2). Their controls are implemented in the dSPACE system, which also supports the sliding mode control. The control systems have been implemented in Matlab/Simulink© and translated into executable code for DS1005 by using the Real Time Workshop toolbox. The global evolution is supervised by a ControlDesk interface.

Fig. 4c shows the relative position of the ORC versus the sliding surface’s image in the ORC plane for different wind speeds.The second and the third rows contain simulations for a minute sized time horizon, for a wind speed of medium turbulence intensity of I=0.17, obtained by the von Karman spectrum in the IEC standard (fig. 4d), and k=5. Figs. 4g and 4h show the turbine’s variable operation regime. Figs. 4e and 4f show how the control law maintains the optimal conversion regime, the better as the wind speed is larger (fig. 4i), suggesting to adaptively adjust the k parameter. The original form of the control law had to suffer some changes for real time testing, because of the physical limitations and non unit efficiency, initially neglected in the modeling phase.

A. Real time versus off-line simulation results Figure 4 presents two kinds of simulation results, namely evolutions in response to step changes in the wind speed (fig. 4a..4c) for trade-off parameter k=0, and to pseudo-random sequences of wind speed respectively (fig. 4d..4i). Figs. 4a and 4b show that the operating point is attracted to the new sliding surface and then evolves in sliding mode to the OOP.

95

-16

-16

*G > Nm @

-17 -18

sliding surface

* opt , :opt

-19

v

-20

v

non sliding trajectory

-22

-23 116 12

118

120

124

8m/s

-20

* opt , :opt

-21

126

128

130

134

136

wind speed evolution

* opt , O opt m v 6

6.5

7

7.5

8m/s O > W @

: > rad/s @

100

120

140

160

180

0

35

ORC tracking k 5 D 0.3

ORC

1000

: > rad/s @

k=5 0.3

D

ORC 0

5

: / i > rad/s @

10

15

20

25

-5

30

35

*G > Nm @

40

k=5 0.3

D

-10 -15 -20 -25

variable speed operation

40 20

300

10

0

60

t >s @ 200

5

2000

state space trajectory

80

k=5 0.3

: / i > rad/s @ 0

4000

200

O > s @

100

0

8.5

k=5 0.3

v

1000

6000

*G > Nm @

-30

7

0

2000

-25

10

8

3000

-20

2

commutation ORC surface k=0 V˜:

4000

-15

4

Pwt > W @

5000

trajectory in ( *G  O ) plane

-10

6

9

-23

-5

Von Karman IEC

8

0

O opt

-22

0

v > m/s @

10

132

7m/s

-18

: > rad/s @

122

* opt , O opt m v

-19

7m/s

-21

6000

*G > Nm @

-17

0

50

100

150

trajectory in ( *G  O ) plane

-30

t >s @ 200

250

300

-35 -40

4

5

6

O > rad/s @

> Nm@

P>W@

b)

c) :

> Nm@

> rad/s @

O

*G

v

> m/s @

d)

t >s@

O

> Nm@

e) t >s@

f)

O

Fig. 5. Real time simulation results to be compared with the similar off-line simulation results from fig. 4.

a)

*G

b)

*G

> Nm@ k

P>W@

1

> Nm@

t >s @

d) : > rad/s @

c)

*G

> Nm@ k

P>W@

5

t >s @

e) : > rad/s @

k

P>W@

10

t >s @

f) :

> rad/s @

Fig. 6. Real time simulation results, showing the energy-control effort trade-off as k parameter increases.

96

transmission of ratio i=6, DFIG Leroy Somer FSLB 160M, 50 Hz, 380 V, 7.5 kW, 1440 rpm, *Gmax 40 Nm , 10 kW

Figs. 6, the final ones, show the influence of the trade-off parameter, k, on the control goal. Indeed, the ORC tracking precision increases with the value of k (fig 6d..6f). But so does the control input (electromagnetic torque) variations (fig. 5a..5c) and the mechanical stress induced. V.

inverters, EMS time constant: Tg = 20 ms. REFERENCES [1]

CONCLUSION AND FUTURE DEVELOPMENTS

[2]

This paper proposes a sliding mode control law for maximizing the power harvested from the wind, while limiting the electromagnetic torque variations. An appropriate sliding surface, having a non empty intersection with the ORC, variant with the wind speed and depending on a desired energy-reliability trade-off, has been found in the (:,P) plane. Thus, it is possible to drive the operating point more or less close to the ORC, by imposing some desired reduced order dynamics. The proposed control law has some drawbacks: a) the optimal operating point’s coordinates referred to the high speed shaft,

 :opt , *opt ,

[3]

[4]

[5]

[6]

must be computed using the

[7]

turbine’s parameters and the drive train efficiency; b) parameter A(O, v) values around the OOP must be estimated; c) the control input needs the gradient of the low speed shaft’s rotational speed; d) the effectiveness of the control law is not the same for all the operating range. Next, it is shown how points a)..d) can be surpassed: a) using a multiplying correction of value *opt with an

[8]

[9]

[10]

estimation of the electromechanical efficiency of the wind power system, to which one can add the integral of the tip speed ratio static error, O  O opt ; b) a consistent estimation of

[11]

the instantaneous value of A(O, v) is possible if the operating point is sufficiently close to the ORC; c) using a non noisy measure of the rotational speed; d) a wind speed dependent adaptation law for parameter k (the principal degree of freedom of the control law) can be found. Points b) and c) can have a smaller dependence of the functional and constructive parameters if increasing the value of parameter D of the onoff component of the control law, u N . Two future issues are of interest: a) study of the closed loop robustness; b) design of a sliding control law (using the same framework) for harvested power regulation.

[12]

[13]

[14]

[15]

[16]

APPENDIX The system used for validations has the following features: - type: fixed pitch two blades HAWT turbine based (high speed): P=5 kW at v=9.5 m/s, R=2.5 m; - energetic performance: maximal value of the power coefficient C pmax 0.47 at optimal tip speed ratio Oopt=7;

[17]

[18]

- wind site properties: constant air density: U=1.25 Kg/m3, constant statistical parameters (IEC standard, I=0.17), normal to the rotor surface; - electromechanical features: Jt=0.1 Kg˜m2, rigid

[19] [20]

97

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