Fuzzy Based Output Feedback Control for wind energy conversion System: An LMI Approach A. H. Besheer, H. M. Emara, and M. M. Abdel_Aziz, Senior Member, IEEE DC link current α R Converter firing angle (0 ≤ α R ≤ 90 o ) α I Inverter firing angle (90 o ≤ α I ≤ 180o ) B, J Net friction and inertia of the rotating parts of the system i DC
Abstract— This paper addresses the problem of regulating wind energy conversion system by using fuzzy output feedback controller. First, a Takagi-Sugeno fuzzy model is employed to represent the nonlinear dynamics of the wind energy conversion system. Then, based on the fuzzy model and utilizing the concept of parallel distributed compensation, a fuzzy observer based fuzzy controller is developed to stabilize the nonlinear system. Sufficient condition for stability of WECS fuzzy model using fuzzy output feedback controller are derived. The controller design problem is caste as a linear matrix inequality (LMI) which can be solved very efficiently using convex optimization techniques. The design technique is applied to a dynamic model of wind energy conversion system to illustrate the feasibility of the proposed solution. The fuzzy observer and fuzzy controller are capable of disturbance rejection. Index Terms—fuzzy observer based fuzzy controller, T-S fuzzy model, wind energy conversion system and linear matrix inequality.
I. NOMENCLATURE i ds , i qs
Peak stator d and q axes currents
i ds , i qs
Peak rotor d and q axes currents Rms stator current
v ds , v qs
Peak stator d and q axes voltages
Magnetizing inductance Lr , Ls Rotor and stator self inductances rr , rs Rotor and stator resistances ω r Shaft speed (rad/sec) ω e Electrical frequency (rad/sec) N Number of poles idc , iqc Peak d and q axes capacitor currents. Lm
i1 Peak magnitude of ac line current flowing into the converter
id 1, iq1 Peak d and q axes currents flowing into the converter
Self excitation capacitance DC link inductance and resistance VR Converter dc voltage VRM Maximum converter dc voltage C
LDC , R DC
A. H. Besheer is with Desert Environment Research Institute, Mounoufia University,Sadat City, Sixth Zone, Egypt(corresponding author to provide phone: 002020101110744; e-mail:
[email protected]) H. M. Emara is with Faculty of Engineering, Cairo University, 12613 P.O.Box, Giza, Egypt (e-mail:
[email protected]). M. M. Abdel_Aziz is with Faculty of Engineering, Cairo University, 12613 P.O.Box, Giza, Egypt (e-mail:
[email protected]).
142440178X/06/$20.00 ©2006 IEEE
II. INTRODUCTION
T
HE re-emergence of the wind as a significant source of the world energy must rank as one of the significant development of the late 20th century. In the world of today there is a need for alternatives to the large coal and oil fired power plants. Renewable energy is an attractive choice, and in particular wind turbines have proven to be a solution [1]. In Egypt, ambitious targets of wind energy program to reach 8200 MW by the year 2017 producing 25 Tera-watt-hour annually, this leads to fuel saving of 6 MTOE [2]. The control of grid connected wind turbines contrasts strongly with that of conventional power stations. In the latter, the source of energy -whether nuclear or fossil- is tightly controlled and the operation of the generator is controlled in response to demand from the grid. The generator is sufficiently large (in the range of 1000MW) that is not completely dominated by the grid and only small disturbances need to be regulated. In grid connected wind turbines, the source of energy undergoes large and rapid fluctuations. Moreover, the generator is completely dominated by the grid, since the generator is relatively small (less than 1 MW). Large transient disturbances result and the control task for wind turbines is inherently more demanding than for conventional power generation. Other features which make wind turbine control an interesting application area for control theory and engineering is the distinct non-linear dynamics nature of such systems. Finally, control must be achieved at minimum expense and hence both control measurement and control actuator hardware is severely restricted [3]. A prerequisite to any control system design is the specification of the design objectives. Optimizing the performance (i.e. the efficiency) of the main components of variable speed wind turbine is considered the essential objective which dominates the literature on control of wind energy conversion systems especially those connected to the grid. This optimization process generally implies two control targets. The first one is maximization the output power below rated wind speed i.e. maximizing the energy capture from the rotor blades. The second one is limiting the output power above rated wind speed where the goal of the controller is to track the nominal (rated) power of the wind turbine. The first target is always desirable and it can be reached by maximizing the power efficiency for the wind turbine in variable speed wind energy conversion schemes (VSWECS).
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PSCE 2006
It can be achieved by keeping the tip speed ratio of the turbine at its optimum value despite wind variation. In [4], a speed control problem to achieve this target is discussed. A direct adaptive control strategy is proposed. It is based on the combination of two control actions; a radial basis function network based adaptive controller, which drives the tracking error to zero with user specified dynamics and a supervisory controller based on crude bounds of the system nonlinearities. This technique -in spite of its succession- has main problem that, the supervisory control action has a discontinuous policy which is undesirable in practice. Above rated wind speed, the captured power need to be limited. Although there are both mechanical and electrical constraints, the more severe ones are commonly on the electric machine and the electronic converter. Output power regulation may also be desired even below rated wind speed. For instance, when a power demand has to be tracked such as in some autonomous systems [5] or when the supplied power is restricted by power quality problems of weak grids [6]. In this case, the system is usually operated in an output power regulation mode. Output power regulation is discussed in many papers, in [5] where a dead beat control of output power was proposed. The knowledge of wind velocity is needed for the controller implementation. So, it is estimated from the power-speed characteristic of the turbine. However, a precise knowledge of the turbine aerodynamics is needed for estimation purposes. On the other hand, the turbine aerodynamics is not uniquely related with wind velocity, i.e., the wind cannot be uniquely determined from speed and power measurement. There are other control attempts concerning indirect regulation of the output power. They are based on controlling the aerodynamic torque to track a given constant power curve, but aerodynamic torque or shaft acceleration estimators are required. In addition, the aerodynamic power may differ substantially from output power during transients because of large turbine inertia [7]. A dynamical sliding mode control for power regulation of VSWECS using induction generators is developed in [7] and [8]. The discontinuous control policy may leads to chattering and this phenomenon results in low control accuracy hence high heat loss in electrical power circuits is inevitable in the sliding mode control. It may also excite un-modeled high frequency dynamics, which degrades the performance of the system and may even lead to instability. In [9] the power transfer through the DC link of WECS is regulated indirectly by regulating DC link current. A simple PI controller is designed using standard frequency domain optimization. Since the PI controller is tuned for a specific linearized model about an operating point, it may provide insufficient damping for any different operating point caused by disturbances and parameter uncertainties. This may lead to undesirable and sustained oscillation in power and voltage. Furthermore, large wind speed deviation may even lead to an unstable system [10]. The past decade have witnessed rapidly growing interest and the emerging of fuzzy control as one of the most efficient and successful techniques for system and control applications. Generally speaking, fuzzy models have advantages of excellent capability to describe a given system and intuitive
persuasion toward human operators over linear models. Some of these models are based on the pattern-recognition technique [11], [12] and others are based on system programming theory [13]–[15]. In addition, other methods such as neural-networkbased methods are suggested [16]–[18]. One of the most outstanding models among them is the model suggested by Takagi and Sugeno in 1985 [19]. The model is based on the system programming method, has excellent capability to describe a given unknown system, and is very suitable for model-based control [20]–[21]. This technique allows modeling the nonlinear dynamics by means of a suitable “blending” of linear subsystems; each one of them corresponds to a different operation point. For each local linear model, a linear feedback control is designed. The resulting overall controller, which is nonlinear in general, is again a fuzzy blending of each individual linear controller. However, this design procedure may guarantees the stability of the fuzzy system using fuzzy blending state feedback controller but in some practical application such as the case of asynchronous WECS, some of the states is often not readily available. Under such circumstances, an introduction of a fuzzy observer is considered a solution for such problem [22]. The design procedure in this paper aims at designing stable fuzzy observer based fuzzy controllers for nonlinear wind energy conversion system. Lyapunov stability theory is used to design the state observer based controller for linear time varying or nonlinear systems. More significantly, the stability analysis and control design problems are reduced to linear matrix inequality (LMI) problem [23]. Numerically, the LMI problems can be solved very efficiently by means of the mathematical programming literature. Several standard software packages for solving LMI condition have been developed such as Matlab LMI toolbox [24]. In this paper, we will adopt and extend these two useful methodologies (i.e., Lyapunov stability theory and LMI approach) to a fuzzy case for designing the fuzzy observer based fuzzy control for WECS. The control and observer gains could be found from the LMI formulation. This paper is organized as follows: Section III presents a dynamic modeling of WECS. The Fuzzy modeling, fuzzy observer based control design and the condition for stability are presented for WECS in section IV. In Section V simulation results illustrate the effectiveness of the proposed control method for wind systems. A conclusion is drawn in section VI. III. DYNAMIC MODELING OF WECS Variable speed Wind energy conversion system is basically composed of wind turbine coupled to synchronous or asynchronous electric generator which connected to the electric grid by means of modern converter system. The converter itself may consist of a rectifier, a DC interconnected circuit and grid commutated inverter which allows the turbine rotating speed to be decoupled from the grid frequency. These are becoming more and more important because they reduce both of the strains on the mechanical parts and also the electrical power variation [25]. In this section the dynamic modeling of each subsystem for the proposed WECS is presented. The objective of the
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modeling excludes the self excitation process of the induction generator and is aimed only at the behavior after self excitation. The WECS adopted in this work is shown in Fig.1. It consists of three bladed horizontal axis wind turbine (HAWT) that drives self excited induction generator which is connected to the grid via AC-DC-AC link scheme. The ACDC-AC link scheme used in this paper consists of a controlled rectifier, a DC link reactor and a controlled line commutated inverter. A. Horizontal Axis Wind Turbine A simplified model for HAWT is used for the aerodynamics with wind speed as input. The torque at the turbine shaft neglecting losses in the drive train is given by [26]: (1) Tm = 0.5πρC t R 3V w2 where Tm is the turbine mechanical torque, ρ is the air density, R is the turbine radius, Ct is the turbine torque coefficient and Vw is the wind velocity.
i1 = iq21 + id21 =
pids = ( we + k 2 Lm wr ) iqs − k1rs ids + k1Lm wr iqr + k 2 rr idr − k1vds
(3)
piqr = k2 rs iqs + Ls k2 wr ids − [(rr + Lm k2 rr ) / Lr ]iqr + ( Ls k1wr − we )idr
(4)
piqr = − Ls k 2 wr iqs + k 2 rs ids − ( Ls k1wr − we )iqr + −( rr + Lm k 2 rr ) / Lr ]idr + k 2 v ds (5)
(6)
where k1 = Lr /( Ls Lr − L2m ) and k 2 = Lm /( Ls Lr − L2m ) The above equations were derived assuming that the initial orientation of the q-d synchronously rotating reference frame is such that the d-axis is aligned with stator terminal voltage phasor (i.e. vqs= 0). C. Self excitation Capacitor The dynamical equations of self excitation capacitance shown in Fig. 2 can be described as follows: i (7) pvds = dc + ω e Vqs pvqs =
C
− ω e Vds
(8)
as Vqs=0, (7) and (8) can be rewritten as: i pvds = dc C iqc we = CVds
iDC
(12)
π
iq1 = −
R
2 3
π
iDC sin α R
(14)
Thus from (11), we have VRM =
3 3
π
Vds
(15)
The dynamics introduced by the DC link is expressed as (16) LDC piDC + RDC iDC = vR − vI The detailed of these above equation and its parameters can be found in [9] and [27]. Hence, the overall system has a seventh order nonlinear model. Local linear models can be derived by applying Taylor linearization technique. A fuzzy controller based on local models is derived in the following section. IV. PROPOSED CONTROL STRATEGY
The nonlinear dynamical model of the induction machine in the d-q reference frame for the proposed WECS when generating can be stated as follows, the entire symbols for WECS used in this paper are given in the nomenclature. (2) piqs = −k1rs iqs − (we + k2 Lm wr ) ids + k2 rr iqr − k1Lm wr idr
C iqc
π
The peak d and q axes currents flowing into converter can be deduced with the assumption of lossless converter 2 3 (13) id1 = iDC cos α
B. Self Excited Induction Generator
pwr = −( B / j )wr + (3N 2 Lm / 8 j )(iqs idr − idsiqr ) + ( N / 2 j)Tm
2 3
(9)
For nonlinear systems, a good approximation is provided by so called Takagi-Sugeno fuzzy model [19]. This fuzzy modeling is simple and natural. It is based on the suitable choice of a set of linear subsystems according to rules associated with some physical knowledge and some linguistic characterization of the properties of the system. These linear subsystems properly describe at least locally the behavior of the nonlinear system for a predefined region of the state space (i.e. the system dynamics is captured by a set of fuzzy implications which characterize local relations in the state space). The overall fuzzy model of the system is achieved by fuzzy “blending” of the linear system models. The general T-S fuzzy system is of the following form: Plant Rule i: IF x1(t) is Fi1 and . . . and xn(t) is Fin (17) THEN x (t) = Ai x(t) + Bi u(t) for i = 1,2,. .r where x(t) = [x1(t),x2(t),…,xn(t)]T ∈ Rn denotes the state vector, u(t)= [u1(t),u2(t),…,um(t)]T ∈ Rm denotes the control input, Fij is the fuzzy set. Ai ∈ Rnxn , Bi ∈ Rnxm , Ci ∈ R pxn and r is the number of IF-THEN rules. In this study, the objective is to design a fuzzy controller for both the rectifier and inverter firing angle so as to regulate the power flow in the DC link. In other words, the objective is to design a fuzzy controller so as to stabilize the WECS around a predefined reference point xr irrespective to a perturbation in the initial states (i.e. tracking problem).
(10)
A. The Problem of regulation around set point for Fuzzy Controller
D. Asynchronous AC-DC-AC Link The AC-DC-AC link scheme used in this paper consists of a controlled rectifier, a DC link reactor and a controlled line commutated inverter. Assuming that the converter is lossless, the instantaneous power balance equation is written as (11) 1.5Vdsid1 = VRiDC = VRM iDC cosα R The ac and dc currents of the converter are related by
In this section, the fuzzy controller design for tracking control problem of nonlinear dynamic system is considered. Generally, let a nonlinear dynamic system be: (18) x = f ( x(t ), u (t ), t ) where x(t ) ∈ Rn is the state of the dynamic system, u (t ) ∈ R m is the control input and let f ( x, u ) be continuous differentiable. Using Taylor linearization technique, the nonlinear dynamic system can be linearized by
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§ ∂f · § ∂f · x = ( x − xr ) ¨ ¸ + ( u − ur ) ¨ ¸ + f h.o.t ( x, u ) © ∂x ¹ x = xr , u = u r © ∂u ¹ x = xr , u = u r
(19)
where fh.o.t is the higher order terms of f and f (xr , ur ) = 0 . Then the local linearized system which represents the behavior of the nonlinear dynamic system at the neighborhood of the reference state (xr, ur) is: ~ x = A~ x + Bu~ § ∂f · § ∂f · ,B = ¨ ¸ ,~ x = x − xr , u~ = u − ur A=¨ ¸ © ∂u ¹ x = xr , u = u r © ∂x ¹ x = xr ,u =ur
(20)
Given a pair of ( ~x (t ), u~ (t )) , the final output of the fuzzy system is inferred as follows [28],[19]: r
¦ µi ( x(t ))( Ai x~ (t ) + Biu~ (t ))
~ x (t ) = i =1
r
¦ µi ( x (t ))
i =1
x~ (t ) =
r
(21)
¦ hi ( x(t ))( Ai ~x (t ) + Biu~(t ))
i =1
Where
j =1
r
µi ( x(t )) = ∏ Fij ( x(t ))
where Kj (for j = 1,2,…,r) are the controller gains. If, instead of the actual state, the estimated state is available, the control low with PDC technique becomes r (30) u (t ) = ¦ h j ( x(t )) (K j ~ xˆ (t ) + u r )
j =1
µi ( x(t ))
hi ( x(t )) =
r
¦ µi ( x(t ))
i =1
j =1
(22) Fij(xj(t)) is the grade of membership of xj(t) in Fij . In this paper, we assume µi (x(t)) 0, for i = 1,2,. . . , r and ¦r µi ( x(t )) > 0 for all t .
Note that the controller (30) is nonlinear in general. Let us denote the state estimation error as ~ e =~ x (t ) − ~ xˆ (t )
Therefore, we get hi (x(t)) 0 for i = 1,2,. . . , r and r
¦ hi ( x(t )) = 1
ˆ ~ e = ~ x ( t ) − ~ x (t )
(32) The augmented system can be written as the following form:
(23) (24)
i =1
In practice, all of states are not fully measurable (e.g., the rotor currents), it is necessary to design a fuzzy observer in order to implement the fuzzy controller. Definition 3.1: If the pairs (Ai ,Ci ), i= 1, 2,…, r are observable, the fuzzy system (21) is called locally observable. For the fuzzy observer design, it is assumed that the fuzzy system (21) is locally observable. First, the local state observers are designed as follows, based on the double (Ai , Ci ). Observer Rule i: IF x1(t) is Fi1 and . . . and xn(t) is Fin Then ~xˆ = Ai x~ˆ (t ) + Bi u~(t ) + Li ( ~y (t ) − ~yˆ (t )) ~ for i = 1,2,. .r (25) yˆ (t ) = Ci ~ xˆ (t) th where Li is the observer gain for the i observer rule. The overall fuzzy observer is represented as follows: ˆ r ~ (26) x = ¦ hi ( x(t)[ Ai ~ xˆ (t ) + Biu~(t) + Li ( ~ y (t ) − ~ yˆ (t ))] i =1
r
¦ hi (x(t )Ci x~ˆ (t )
i=1
B. Parallel distributed compensation technique
(31)
By differentiating (31), we get
i =1
~ yˆ (t) =
The concept of parallel distributed compensation (PDC) in [28] and [29] is utilized to design fuzzy controllers to stabilize fuzzy system (21). The idea is to design a compensator for each rule of the fuzzy model. For each rule, we can use linear control design techniques. The resulting overall fuzzy controller, which is nonlinear in general, is a fuzzy blending of each individual linear controller. The fuzzy controller shares the same fuzzy sets with the fuzzy system (21). Definition 3.2: If the pairs (Ai ,Bi ), i= 1, 2,…, r are stabilisable, the fuzzy system (21) is called locally stabilisable. For the fuzzy controller design, it is assumed that the fuzzy system (21) is locally stabilisable. The local state feedback controllers are designed as follows. Control rule j: IF x1 (t ) is Fi1 and . . . and xn (t ) is Fis Then u(t ) = K j ~x (t ) + ur for j = 1,2,…r (28) Hence, the fuzzy control decision is given by: r (29) u (t ) = ¦ h j ( x(t ))(K j ~ x (t ) + u r )
(27)
ª º § r · ¨ ¦ hi ( x(t ))( Ai ~ « » x (t ) + Biu~ (t )) ¸ ¨ ¸ « » = 1 i ¹ © « » r ·» ª x~ (t )º «§ r ~ ~ ~ ~ ˆ «~ » = «¨ ¦ hi ( x(t ))( Ai x (t ) + Biu (t )) − ¦ hi ( x(t ))( Ai x (t ) + Biu (t ) ¸» ¨ ¸ i =1 ¬«e (t ) ¼» «¨ i =1 ¸» «¨ r ¸» ~ «¨ + ¦ hi ( x(t ))( LiC j e ) ¸¸» «¬¨© i =1 ¹»¼
(33)
After manipulation, (33) can be expressed as follows: ª~ x (t ) º ª Ai + Bi K j «~ » = « 0 «¬e (t ) »¼ «¬
− Bi K j º ª x(t ) º » Ai − Li C j » «¬e(t ) »¼ ¼
(34)
Let us denote x (t ) º ª~ x (t ) = « ~ » , e ¬ (t ) ¼
− Bi K j º ª( Ai + Bi K j ) Aij = « 0 ( Ai − Li C j ) »»¼ «¬
(35)
Therefore, the augmented system defined in (35) can be expressed as the following form: r r (36) x = ¦ ¦ hi ( x(t ))h j ( x(t ))[ Aij x (t )] i = j =1
C. Stability Analysis via Lyapunov approach A sufficient stability condition derived by K.Tanaka et al., in [30] for ensuring stability of (21) using a control law (30) is given as follows: Theorem 1: The equilibrium of a fuzzy control system (21) is asymptotically stable in the large using the control law (30) if there exists a common positive definite matrix P such that (37) ( Aij )T P + P ( Aij ) ¢ 0 for i, j = 1, 2, ..., r
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The proof can be found in [22] and [30]. Its basic idea is the use of a quadratic Lyapunov function V ( x) = ~x (t )T P ~x (t ) . Theorem 1 thus presents a sufficient condition for quadratic stability of the system (21) using the control law (30). According to theorem 1, the most important step in designing the fuzzy observer based fuzzy controller is the solution of (37) for a common P = PT ² 0 , a suitable set of controller gains and observer gains Kj and Li (i, j=1,2,…r). Equation (37) forms a set of bilinear matrix inequalities (BMI's). The BMI in (37) should be transformed into pure LMI as follows: 1) For the convenience of design, assume 0 º ªP (38) P = « 11 » ¬ 0
P22 ¼
This choice is suitable for simplifying the design of fuzzy controller and fuzzy observer. 2) Define W as in (39) and apply congruence transformation (i.e. multiply both side of inequality (37) by W) [31]. −1 0 º ªW11 0º ª P11 (39) W =« » »=« ¬ 0
I¼
¬« 0
I ¼»
where W11 = P11−1 3) With Y j = K jW11 and
Zi = P22 Li
Since four parameters
, apply the change of variables.
P11, P22 , K j , Li should
be determined
from (37) after applying the above mentioned procedures which results in the following LMI's conditions: (40) W11 AiT + AiW11 + ( BiY j )T + ( BiY j ) ¢ 0 (41) The inequality in (40) and (41) are linear matrix inequality feasibility problems (LMIP's) in W11, Yj, P22 and Zi which can be solved very efficiently by the convex optimization technique such as interior point algorithm [23]. Software packages such as LMI optimization toolbox in Matlab [24] have been developed for this purpose and can be employed to easily solve the LMIP. By solving (40) and (41) the controller gain ( K j = Y jW11−1 ) and the observer gain ( Li = P22−1Z i ) can be easily determined. K. Tanaka and H. Wang, in [12] gave detailed description for the above procedures. According to the analysis above, the tracking control via fuzzy observer based state feedback is summarized as follows: Ai P22 + P22 Ai − ( Z iC j )T − ( Z i C j ) ¢ 0
Design procedures: Step 1) Select membership functions and fuzzy plant rules in (17). Step 2) Solve LMIP in (40) to obtain W11, Yj, and −1 K j = Y jW11
Step 3) Solve LMIP in (41) to obtain P22, Zi , and −1 Li = P22 Zi
Step 4) Construct the fuzzy observer (26). Step 5) Construct the fuzzy controller (29). Step 6) To check the validaty of the fuzzy observer based fuzzy controller, apply the constructed observer and controlleron the original nonlinear plant.
this example is output power regulation for WECS indirectly by regulating the DC link current. For this example, after manipulation, the state equations of the WECS can be stated as: x1 = −k1rs x1 − we x2 − k2 Lm wr x2 + k2 x3rr − k1Lm wr x4 x 2 = we x1 + k 2 Lm wr x1 − k1rs x2 + k1Lm wr x3 + k 2 x4 rr − k1x6
r + Lm k2 rr x3 = k2 rs x1 + k2 Ls wr x2 − ( r ) x3 + k1Ls wr x4 − we x4 Lr r + Lm k 2 rr x 4 = k 2 Ls wr x1 + k 2 rs x2 − k1Ls wr x3 + we x3 − ( r ) x4 + k2 x6 Lr x5 =
3N 2 Lm 3N 2 Lm B N x1x4 − x2 x3 − wr + Tm 8j 8j j 2j
x 2 3 π x7 cos u1 x6 = 2 + C C
R 3 3 π x7 cos u1 VIM x7 = − DC x7 + + cos u2 LDC LDC LDC
(42)
Where x + 2 3 π x7 sin u1 , x , x , x , x , , x and x denotes i , i , i we = 1 1 2 3 4 x5 6 7 qs ds qr Cx6
The control action are u1 and u2 denoting and . To minimize the design effort and complexity, the αR αI number of fuzzy rules should be small. The Takagi-Sugeno fuzzy model that approximates the dynamics of the nonlinear WEC plant (42) can be represented by the following two-rule fuzzy model: Rule 1: IF x7 is about 2.5 A THEN x = A1x + B1u Rule 2: IF x7 is about 3.2 A THEN x = A2 x + B2u where Ai and Bi can be obtained by linearizing the nonlinear system in (42) around some suitable predefined points taken from [9]. In this example, we uses MatLab/Simulink software package to get the linearized system matrices. Using theorem (1) and the LMI optimization toolbox in Matlab [24], we obtain the common solution for W11 and P22 as follows: idr , wr , vds and iDC .
ª 0.2170 « « 0.0705 « − 0.2203 « W11 = «− 0.0950 « 0.0449 « « 0.0760 « − 0.0321 ¬ ª -0.0770 « « − 0.0151 « 0.1122 « P22 = « − 0.0170 « − 0.0236 « « − 0.0028 « − 0.0770 ¬
0.0705
-0.2203
-0.0950
0.0449
0.0760
0.2922
− 0.0564 − 0.3020
0.0557
− 0.1119
− 0.0564
0.2256
0.0823
− 0.0321 − 0.0670
− 0.3020
0.0823
0.3291
− 0.0621
0.2055
0.6554 0.3054
0.3054 3.6131
− 0.0370
0.0476
− 0.0559
0.1122
-0.0170
-0.0236
0.0557 − 0.0321 − 0.0621 − 0.1119 − 0.0670 0.2055 0.0110 0.1104 0.0706 − 0.0163 0.0721 0.0040 0.0003 − 0.0183
0.0443 -0.0151
− 0.0163 0.0721 0.1195 − 0.0174 − 0.0174 0.0807 − 0.0248 0.0045 − 0.0029 0.0001 − 0.0956 0.0121
0.0040 0.0003 − 0.0248 − 0.0029 0.0045 0.0001 0.0598 0.0004 0.0004 0.0006 − 0.0107 − 0.0155
The controller parameters are found to be: -13.05 3.29 0.17 -0.03 0.99º ª-12.96 3.92 3 K1 = « » × 10 ¬ − 709 401.73 − 698.26 325.03 0.73 21.91 1.37 ¼ 3.9 -13.1 3.3 0.2 -0.1 1.0º ª -13.0 3 K2 = « » × 10 ¬− 1044.4 918.7 − 1064.8 809.6 17.0 1.6 4.0 ¼
The observer parameters are found to be: V. WECS SIMULATION EXAMPLE To illustrate the proposed fuzzy control approach, the WECS shown in Fig.1 is considered. The control objective in
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-0.0321 º » 0.0110 » 0.0443 » » − 0.0370» 0.0476 » » − 0.0559» 7.4347 »¼ -0.0028 º » − 0.0183» − 0.0956» » 0.0121 » × 10 6 − 0.0107» » − 0.0155» 2.3519 »¼
Fig.3 and 4 present the simulation results for the proposed fuzzy observer based fuzzy controller system. A perturbation in the initial states from the first operating point is applied for iqs, iqr, iDC. The initial condition is assumed to be: (x1 (0), x2 (0), x3 (0), x4 (0), x5 (0), x6 (0), x7 (0) )T = [11.59,1.77,-0.153, - 2.37,280.78,143.44,3]T Fig. 3 shows the trajectory of x7 (IDC), the output of WECS to be regulated, for fuzzy output feedback controller applied on WECS. Fig. 4 shows the error in the observed current for the chosen states. In this example the rotor resistance rr is assumed to be changed by 10% from its nominal value. Using the same designed controller, Fig. 5 reflects that the designed fuzzy controller exhibits robust performance despite of plant parameter variation. A small tracking error or the steady state error is appeared in Fig. 5 and this is due to the difference between the approximating steady state control input ¦r hi ( x(t ))ur and the actual steady state control input at the
Figure 2 Self excitation capacitor and current flow direction for d equivalent circuit 3.1
3
2.9
Current (A)
ª -17.90 º ª -20.26 º » » « « «− 72 .59 » « − 73.68» « 22.53 » « 25.16 » and » » « « L1 = « 61.11 » × 10 2 L 2 = « 60.94 » × 10 2 « 0.37 » « 0.36 » » » « « « 301.24 » « 343.05 » « 0.03 » « 0.39 » ¼ ¼ ¬ ¬
2.8
2.7
2.6
2.5
2.4
0.05
0.1
0.15 0.2 Time (s)
0.25
0.3
0.3
Fig. 3. DC link current 1
i =1
i
qs
iqr
0.8
iDC 0.6 Error in observed currents (A)
new operating point corresponding to rotor resistance change [32]. From the simulation results, the fuzzy output feedback controller can regulate the WECS with external disturbance that is appeared in this study as a perturbation in the initial condition of the generator state vector of the WECS and the desired performance can be achieved. Furthermore, the designed fuzzy controller exhibits robust performance with plant parameter variation.
0
0.4 0.2 0 -0.2 -0.4 -0.6 -0.8
To Utility
A +
0
0.05
0.1
0.15 0.2 Time (s)
0.25
0.3
0.3
Fig. 4. Error in observed current (iqs, iqr, iDC, A)
+
B
DC Link
3.1
-
C -
3
Asynchronous Machine Wind turbine
C A
B
Controlled rectifier
Inverter
2.9
Current (A)
Self excitation capacitors
Fig. 1. Wind energy conversion system ids
2.8
2.7
2.6
2.5
idl
2.4
0
0.05
0.1
0.15 0.2 Time (s)
0.25
0.3
0.35
Figure 5 DC link current with rotor resistance variation
Vds
C
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VI. CONCLUSION In this paper, fuzzy observer and fuzzy state feedback controller techniques are combined to solve practically the problem of regulating the WECS. The Takagi Sugeno fuzzy model is employed to approximate the nonlinear model of WECS. Based on the wind energy conversion fuzzy model, a fuzzy observer based fuzzy controller is developed to guarantee the stability of fuzzy model and fuzzy control system for the WECS. The proposed design method is conceptually simple and natural and it is suggested to cope with the intrinsic nonlinear behavior of WECS. Moreover, the stability analysis and control design problems are reduced to LMI problems. Therefore, they can be solved very efficiently in practice by convex programming techniques for LMI’s. Simulation results indicate that the desired fuzzy output feed back performance for nonlinear WECS can be achieves using the proposed method. REFERENCES [1]
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