Sliding Mode Control of Induction-Motor-Pump Supplied by Photovoltaic Generator
[email protected],
[email protected] Unité de Recherche en Automatique et Informatique Industrielle, National Institute of Applied Sciences and Technology (INSAT) Centre Urbain Nord, BP 676, 1080 Tunis Cedex, Tunisia Abstract- In this paper, a sliding mode technique is used to control an indirect field oriented induction motor pump which is supplied by a photovoltaic generator. After the model of the system is described, the sliding mode control SMC rule applied to the system is given. Then, the maximum power point tracking algorithm MPPT which is used to map the illumination value to the speed of the induction motor is explained. Finally, simulation results are presented to show the effectiveness of the proposed method. Index Terms—Sliding mode control, indirect field oriented induction motor pump, Photovoltaic generator, Maximum-power-point tracking.
I.
INTRODUCTION
In order to cover the energy requirement, the researches are being made for renewable energy. Privately, photovoltaic energy is produced by direct transformation of solar illumination to electrical energy. One of the world wide applications is the water pumping systems driven by the induction motor, [1],[8]. This latter is distinguished by its rigidness, reliability and relatively low cost. However, the difficulty to control the induction motor is related to the fact that its mathematical model in Park configuration is nonlinear and highly coupled, [4],[6]. Due to the development of power electronics and microprocessors, the induction motor control is possible by applying field oriented techniques, [7],[10]. These techniques provide the decoupling stator and rotor machine frames that allow obtaining a dynamical model similar to that of DC machine. Nevertheless, a discontinuous behavior is imposed by the switching devices of the inverter supplying the induction machine. Therefore, it is suitable to look for some techniques which are appropriate to discontinuous operation of the switching devices. Among these techniques, one can choose a variable structure control method and its associated sliding modes. This latter allows a high performance of the control scheme and especially the robustness of the algorithm with regard to changing parameters and external disturbances, [12]. In this work, the photovoltaic water pumping system is constituted by a photovoltaic generator PVG, a condenser, a PWM inverter, an induction motor and a centrifugal pump. In order to force the PVG to operate at its maximum power point under different solar illumination conditions, we develop a maximum power point tracking algorithm MPPT which
978-1-4244-9066-0/11/$26.00 ©2011 IEEE
calculates the reference speed of the induction motor versus the solar illumination, [1],[3],[5]. The induction motor pump is controlled by the field oriented technique and its associated sliding mode. II. MODELING OF THE PHOTOVOLTAIC WATER PUMPING SYSTEM A. Modeling of the photovoltaic generator PVG In the case of an array with Ns series connected solar cells and NP parallel connected panels, the array current may be related to the array voltage by (1), as in [1-3]. I pv = Np I ph − Np Is ⎡⎣exp( (Vpv + Rs I pv ) NsVT ) −1⎤⎦ −( (Vpv + Rs I pv ) Rsh ) (1) Where: Ipv, Vpv and Ppv are respectively the photovoltaic (PV) current, voltage and power. Iph is the light-generated current and Is is the reverse saturation current. Rs and Rsh are respectively the PV array series and shunt resistances. A is the ideality factor, K is the constant of Boltzman, T is the cell temperature, q is the electronic charge and VT is the thermodynamic potential given by (2). VT = A K T / q (2)
In ideal conditions, Rs is much smaller and Rsh is much greater, [1]. So, the nonlinear characteristic of the PVG in ideal conditions is given by (3). I pv = N p I ph − N p I s ⎡ exp (V pv + Rs I pv ) N sVT − 1⎤ (3) ⎣ ⎦ Fig. 1, illustrates the influence of the solar illumination on the PVG maximum power points (MPP).
(
)
Ppv=f(Vpv) 2000 1 kw/m²
MPP 1500 P pv (W)
1
Faouzi Bacha1 and Moncef Gasmi2
0.9 kw/m² 0.7 kw/m²
1000
0.5 kw/m² 500
0
0
200
400 V pv (v)
600
800
Fig. 1. Ppv = f(Vpv) under different values of E.
B. Modeling of the PWM inverter associated to the PVG The three-phase inverter consists of three independent arms. Each one includes two switches which are complementary and controlled by the Pulse Width Modulation PWM, [2],[3].
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To avoid the submission of the PVG to the over-voltages coming from the inverter, we insert a condenser C as it is shown in Fig.4. The induction motor stator voltages (vsa, vsb, vsc) are expressed in terms of the upper switches as follows: ⎡vsa ⎤ ⎡ 2 −1 −1⎤ ⎡ K1 ⎤ ⎢v ⎥ = V pv ⎢ −1 2 −1⎥ ⎢ K ⎥ (4) ⎢ sb ⎥ ⎥⎢ 2⎥ 3 ⎢ ⎢⎣ vsc ⎥⎦ ⎢⎣ −1 −1 2 ⎥⎦ ⎢⎣ K 3 ⎥⎦ K1, K2 and K3 are the controller signals applied to the switches. The photovoltaic current is given by: I pv = K1isa + K 2 isb + K 3isc + I c (5) Where (isa, isb, isc) are the induction motor stator currents of the inverter and Ic is given by (6). I c = CdV pv / dt (6) C. Modeling of the induction motor By considering the stator voltages (vds, vqs) as control inputs, the stator currents (ids, iqs), the rotor flux (Φdr,Φqr) and the speed (Ω) as state variables, the electrical model of the induction machine in the d-q referential axis linked to rotating field is given by (6), as in [1], [4],[8].
⎧ ⎡ −Rsm ids + σ Ls ω s iqs + ( M / Lr )((Φ dr / Tr ) ⎤ ⎪ ⎪ ⎥ dids / dt = (1/ σ Ls ) ⎢ ⎪ ⎢ + ωr Φ qr ) + vds ⎥ ⎪ ⎪ ⎣ ⎦ ⎪ ⎪ ⎪ ⎡ ⎤ R i L i M L T ( / )(( / ) σ ω − − + Φ sm qs s s ds r qr r ⎪ ⎥ ⎨ dids / dt = (1/ σ Ls ) ⎢⎢ ⎥ ⎪ v ) − Φ + ω r dr qs ⎪ ⎣ ⎦ ⎪ ⎪ ⎪ d dt M T i T / ( / ) ( ) / Φ = + ω − ω Φ − Φ dr r ds s r qr dr r ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ d Φ qr / dt = ( M / Tr )iqs − (ω s − ωr )Φ dr − Φqr / Tr (7) Where: Rsm = Rs + Rr M 2 / L2r
(8)
Rs and Rr are respectively the stator and rotor resistances. Ls and Lr : are respectively the stator and rotor inductances. Tr : is the rotor time constant (Tr =Lr/ Rr). M : is the mutual inductance. ωs : is the stator angular frequency. ωr : is the rotor speed. σ : is the total leakage coefficient (σ =1-M²/LsLr). The mechanical modeling part of the system is given by: Jd Ω dt = ( Cem − Cr − f Ω ) (9) Where: J is the total inertia of the machine, f is the coefficient of friction, Cr is the load torque and Ω is the mechanical speed of the machine. The electromagnetic torque is given by: Cem = ( 3 pM 2 Lr ) ⋅ ⎡⎣Φ dr iqs − Φ qr ids ⎤⎦ (10) Where p is the number of pole pairs.
D. Modeling of the centrifugal pump The hydrodynamic load torque of the centrifugal pump is given by (9), as in [1]. Cr = Apωr2 (11) Where Ap is the torque constant (Ap =Pn/ωrn3), Pn is the nominal power of the induction-motor and ωrn is the rotor nominal speed. The centrifugal pump is described by the lows of similarity which are given by (10). ⎧Q ′ = ( N ′ / N ) ⋅ Q (12) ⎨ 2 ⎩ H ′ = ( N ′ / N ) .H Where: Q’ and Q are respectively the flow and the nominal flow of the pump, H’ and H are respectively his height and total height; N’ and N are respectively his speed and nominal speed. III. THE ROTOR FLUX FIELD ORIENTED CONTROL Using the electrical model of the induction machine given by (6), one can remark the interaction of both inputs, which makes the control design more difficult. So, the first step of our work is to obtain a decoupled system in order to control the electromagnetic torque via the stator quadrature current iqs with a similar manner of a DC machine. The field orientation is obtained by (13) as in [6],[7],[10]. ⎪⎧Φ dr = Φ r (13) ⎨ ⎪⎩Φ qr = 0 The rotor flux Φr and its position θs are estimated by means of stator current and speed and are given as follows: Φ r = ( M /(1 + s.Tr )).ids (14)
θ s = ∫ (( M .iqs /(Tr .Φ r )) + pΩ)dt
(15)
Where s is the differential operator (s=d/dt). Then, the stator equations on d,q-axis become, [4]: ⎧ dids / dt = (1/ σ Ls ) [ vds − Rsm ids + σ Ls ω s iqs + ( M / Lr Tr )Φ r ] ⎪ ⎪ (16) ⎨ ⎪ ⎪ dids / dt = (1/ σ Ls ) [ vqs − Rsm iqs − σ Ls ω s ids + ( M / Lr )ωr Φr ] ⎩ IV. SLIDING MODE CONTROL APPLIED TO THE INDUCTION MOTOR A. General concept The variable structure systems and their associated sliding regimes are characterized by a discontinuous nature of the control action with which a desired dynamic of the system is obtained by choosing appropriate sliding surfaces. The control actions provide the switching between subsystems which give a desired behavior of the closed loop system, [9],[11],[12]. Let us consider a class of nonlinear system described by the following equation, [4]: ⎧⎪ x = f ( x ) + g ( x ) ⋅ u (17) ⎨ ⎪⎩ y = h ( x ) Where x(t)∈Rn is the state vector, u(t)∈Rm is the control action and y(t)∈Rp is the output. Assuming that the system is controllable and observable, the sliding mode control objectives consist of the following steps:
183
•
•
Design of the switching surface S(x)∈Rm so that the state trajectories of the plant restricted to the equilibrium surface have a desired behavior such as tracking, regulation and stability. Determinate a switching control strategy, u(x) so that to drive the state trajectory to the equilibrium surface and maintain it on the surface. This strategy has the form: ⎪⎧u if S ( x ) > 0 (18) u = ⎨ max ⎪⎩ umin if S ( x ) < 0
Where: S(x) is the switching manifold. • Reduce the chattering phenomenon due to discontinuous nature of the control. A well-known surface chosen to obtain a sliding mode regime which guarantees the convergence of the state x to its reference xref is given as follows: S ( x ) = ( xref − x ) (19) Two parts have to be distinguished in the control design procedure. The first one interests the attractivity of the state trajectory to the sliding surface and the second represents the dynamic response of the representative point in sliding mode. This latter is very important in terms of application of nonlinear control techniques, because it eliminates the uncertain effect of the model and external perturbation. Among the strategies of the sliding mode control available in the literature, one can chose for the controller the following expression: u = ueq + un (20)
Ueq is the control function noted equivalent control for which the trajectory response remains on the sliding surface, [4]. In this case, the invariance condition is expressed as: ⎪⎧ S ( x ) = 0 (21) ⎨ ⎪⎩ S ( x ) = 0 The equivalent control can be interpreted as the average value of the control switching representing the successive commutation in the range [umin , umax], [4]. Let us consider the system described by (17). When the sliding mode regime arise, the dynamic of the system in sliding mode is subjected to the condition S ( x ) = 0 thus for the ideal sliding mode we have also S ( x ) = 0 . δS δx δS δS ⎡⎣ f ( x ) + g ( x ) ueq ⎤⎦ + S ( x ) = = ⎡ g ( x ) un ⎤⎦ (22) δ x δt δ x δx ⎣ For S ( x ) = 0 and un=0 (on S ( x ) = 0 ), we obtain: −1
ueq = − ⎡⎣(δ S δ x ) g ( x ) ⎤⎦ ⋅ ⎡⎣(δ S δ x ) f ( x ) ⎤⎦ By replacing ueq in (22), we obtain: S ( x ) = (δ S δ x ) g ( x ) un
(24)
(26)
A simple form of the control action using sliding mode theory is a relay function which is given by (27). un = k ⋅ sgn ( S ( x ) ) (27) Replacing Un in (26), we obtain S ( x ) ⋅ S ( x ) = (δ S δ x ) g ( x ) k S ( x ) < 0
(28)
The term (δS /δ x)g(x) is negative for the class of the system considered, whereas the gain k is chosen positive to satisfy attractivity and stability conditions. In this context, we can verify the stability of the sliding surface with using theorem of Lyapunov, [4]. Let us chose the following positive function (V(x)>0) such as: V ( x) = S 2 ( x) / 2 (29)
Its derivative is given by: V ( x ) = S ( x ) ⋅ S ( x )
(30)
One must verify the decreasing of the Lyapunov function to zero. For this purpose, it is sufficient to assure that its derivative is negative. In order to reduce the chattering phenomenon due to the discontinuous nature of the controller, a smooth function is defined to replace the discontinuous part of the control action. Thus, the controller becomes: ⎧⎪(k / ε ) ⋅ S ( x ) if S ( x ) < ε ≠ 0 un = ⎨ (31) ⎪⎩ k ⋅ sgn ( S ( x ) ) if S ( x ) > ε B. Application of the SMC to the induction machine In this study, the sliding mode control theory is applied to the rotor field oriented induction motor in such away to obtain simple surfaces. The proposed control scheme is a cascade structure as shown in Fig. 2, in which two surfaces for each axis are required. The internal loops allow to control the stator current components (ids , iqs); whereas the external loops provide the speed and the rotor flux (Ω , Φr ) regulations, [4],[9]. ( Φr )ref Ω ref
Fig. 2. The inner and outer loops of the SMC.
The sliding surfaces for each axis are chosen as follows: ⎧⎪ S ( Φ r ) = ( Φ r ) ref − Φ r (32) d -axis: ⎨ ⎪⎩ S ( ids ) = ( ids ) ref − ids
(23)
The term un is added to the global function of the controller in order to guarantee the attractiveness of the chosen sliding surface. This latter is achieved by the condition: S ( x ) ⋅ S ( x ) < 0 (25) Then:
S ( x ) ⋅ S ( x ) = S ( x )(δ S δ x ) g ( x ) un < 0
⎧ S ( Ω ) = ( Ω ) ref − Ω ⎪ q -axis: ⎨ (33) ⎪⎩ S ( iqs ) = ( iqs )ref − iqs The (ids)ref and (ids)ref references are determined by the outer loops and take respectively the values of (ids)c and (iqs)c . Using (15, 16, 21, and 31), it follows:
184
•
On d-axis: for the rotor flux regulation: )/ M S ( Φ r ) = 0 ⇒ ( ids )eq = (Φ r + Tr ⋅ Φ rref
(34)
S ( Φ r ) ⋅ S ( Φ r ) < 0 ⇒ ⎧ (kφ / ε φ ) ⋅ S ( Φ r ) if S ( Φ r ) < ε φ
( ids )n = ⎪⎨
⎪⎩ kφ ⋅ sgn ( S ( Φ r ) ) if S ( Φ r ) > ε φ Thus, the controller is: ( ids )c = ( ids )eq + ( ids )n And for direct current regulation, it follows: S ( ids ) = 0 ⇒
( vds )eq = σ Ls ( ids )ref
+ Rsm ids − σ Lsωs iqs − ( M / Lr Tr )Φ r
(35)
(36)
(37)
⎡( vsa )c ⎤ ⎡ cos (θ s ) − sin (θ s ) ⎤ ⎡( vds ) ⎤ ⎢ ⎥ 2⎢ v = cos θ − 2 π / 3 − sin θ − 2 π / 3 ) ( s ) ⎥⎥ ⎢ c ⎥ ⎢( sb )c ⎥ 3 ⎢ ( s v ⎢( vsc ) ⎥ ⎢⎣cos (θ s + 2π / 3) − sin (θ s + 2π / 3) ⎥⎦ ⎣⎢( qs )c ⎦⎥ c⎦ ⎣
⎧ (k d / ε d ) ⋅ S ( ids ) if S ( ids ) < ε d ( vds )n = ⎪⎨ ⎪⎩ kd ⋅ sgn ( S ( ids ) ) if S ( ids ) > ε d The controller on d-axis is given by: ( vds )c = ( vds )eq + ( vds )n
(38)
C. Calculation of the reference variables The reference flux value is (Φr)ref =1wb. The reference speed Ωref is calculated by the MPPT algorithm analyzed in section IV.
By neglecting frictions and losses, we can express the power of the induction motor as follows, [1]: P ≈ Cr Ω = Ap Ω3 (47) So, the reference speed Ωref can be expressed as: Ω ref = 3 Popt / Ap
(39)
In the same way: On q-axis: for the speed regulation: + k Ω + C ) /( pM Φ / L ) (40) S ( Ω ) = 0 ⇒ ( iqs ) = ( J Ω ref f r r r
TABLE I.
S ( Ω ) ⋅ S ( Ω ) < 0 ⇒
REFERENCE SPEED AND PV POWER VERSUS THE ILLUMINATION VALUE
⎧⎪ (kω / ε ω ) ⋅ S ( Ω ) if S ( Ω ) < ε ω qs n = ⎨ ⎪⎩ kω ⋅ sgn ( S ( Ω ) ) if S ( Ω ) > ε ω And the controller is given by: ( iqs )c = ( iqs )eq + ( iqs )n
(i )
And for the quadrature current regulation, it follows: S ( iqs ) = 0 ⇒ qs eq
= σ Ls (iqs )ref + Rsm iqs + σ Lsωs ids − ( M / Lr )ωr Φ r
(41)
(42)
(43)
S ( iqs ) ⋅ S ( iqs ) < 0 ⇒
(v )
qs n
⎧ (kq / ε q ) ⋅ S ( iqs ) if S ( iqs ) < ε q ⎪ =⎨ ⎪⎩kq ⋅ sgn S ( iqs ) if S ( iqs ) > ε q
(
(48)
The following table represents the reference speed and the optimal PV power for different values of the solar illumination.
eq
(v )
(46)
V. MPPT ALGORITHM
S ( ids ) ⋅ S ( ids ) < 0 ⇒
•
two times of stator quadrature current value (iqs)max admissible by the machine. Using Park transformation, the reference voltages in the (a,b,c) coordinates are given by (46), [13].
(44)
)
And the controller is given by: ( vqs )c = ( vqs )eq + ( vqs )n
(45)
To satisfy the stability condition of the system, all of the following gains (kd, kq, kΦ, kΩ) should be chosen positive. High performances may be obtained by choosing appropriate gains, [4]. kd and kq take the admissible values of transient stator voltage in direct and quadrature axis, then kd=(vds)max and kq=(vqs)max. kΦ takes the value of (ids)max=(Φr)ref/M. kΩ takes
E(w/m²)
Popt (W)
100 200 300 400 500 600 700 800 900 1000
132.9 299 473 654.9 841 1032 1226 1423 1622 1825
Ωref (rad/s) 67.14 87.97 102.5 114.25 124.18 132.95 140.81 147.98 154.58 160.77
We can determinate the relation between illumination E and the approximated reference speed Ω*app by using the curve fitting technique, as in [3],[5]. The approximated reference speed is given by (49). Ω*app = 36.325 + 0.3511 ⋅ E − 5.795.10−4 ⋅ E 2 + (49) 5.611.10−7 ⋅ E 3 − 2.088.10−10 ⋅ E 4 VI. ARCHITECTURE OF THE PHOTOVOLTAIC WATER-PUMPING SYSTEM The architecture of the photovoltaic water pumping system is described by Fig. 3. The subsystems of the proposed configuration are a PVG, a condenser, a PWM inverter, an induction-motor and a centrifugal pump.
185
( Φ r )ref
Φr
( Ω )ref Ω
S ( ids ) S ( iqs )
( vds )c
(v ) qs
c
Fig. 3. Architecture of the photovoltaic water pumping system controlled by the sliding mode control.
The main components of the sliding mode control block are there four sliding surfaces S(Φr) , S(Ω) , S(ids) and S(iqs). In order to calculate the reference speed, we use a sensor in order to measure the illumination value and a calculator to deduce the reference speed via the MPPT.
= f(t) (a)
150 100 5
6
5
6
15 10 5
VII. SIMULATION RESULTS In order to demonstrate the effectiveness of the proposed control technique applied to the photovoltaic water-pumping system, some simulations have been carried out. The proposed design scheme which is described by Fig. 3, is implemented in Matlab/Simulink software using parameters given in Appendix. a- Variation of the solar illumination value E: In a first step, we choose to vary the solar illumination value E as it is shown in Fig. 4, and to see its impact on the performances of the photovoltaic water pumping.
7 t (s) 8 Cem = f(t) (b)
7 t (s)
9
8
9
8
9
Q = f(t) (c)
15 10
5
6
7 t (s)
Ppv (W)
Fig. 5. Response of the system to the variation of E.
Fig. 4. The solar illumination waveform.
Fig. 5a, illustrates the waveform of the mechanical speed of the induction motor which is closed to its optimal value. The same remark is given to the electromagnetic torque shown by Fig. 5b. It is clearly shown that the induction motor is operating at its optimal conditions. Fig. 5c, represents the waveform of the centrifugal pump flow which is closed to its optimal value for each value of E. So, we can note the utility of the MPPT algorithm in optimization of the PVG performances. Fig. 6, illustrates the waveform of the PV power trajectory. Then we can conclude the important role of the SMC and the developed MPPT to make the photovoltaic water pumping system operating at its optimal conditions.
Fig. 6. Waveform of the PV power trajectory.
b- Response of the system to the parameters variation: In a second step, we choose to vary the rotor time constant Tr at 20% of its initial value at it is shown in Fig. 7, in order to prove the robustness of the system at the disturbances and the drifts of the motor-driven pump parameters.
Fig. 7. The variation of the rotor time constant.
186
Fig. 8a, 8b, and 8c, shows respectively the waveform of the motor speed, the electromagnetic torque and the pump flow. 200
Ls=Lr=0.462(H), M=0.44(H) Rs = 5.72 (Ω), Rr=4.2(Ω) J=0.0049(Kg.m²) f=1.5.10-4(Nm/rds-1) p=2 pole pairs P=1.5(Kw), N=1450(tr/min) Centrifugal pump parameters Q=15(m3/h) H=20(m)
150
REFERENCES
= f(t) (a)
100
4
5
4
5
6 t (s) 7 Cem = f(t) (b)
8
9
8
9
[1]
20 10 0
6 t (s)
[2]
7
Q = f(t) (c)
20
[3]
15 10
4
5
6 t (s)
7
8
9
Fig. 8. Study of the robustness of the system.
So, a rapid response is obtained. The different variables are lightly affected by the Tr change at a short transient, and then, they return to their optimal values. The same remark is given to the centrifugal pump flow which proves that the water pumping system operating at its optimal conditions. Then, the robustness of the controlled system is achieved by the appropriate type controller which is a sliding mode control and the cascade structure used. VIII. CONCLUSION We have proved the utility of the developed algorithm MPPT in resolution of the problem of degradation of the performances of the PVG following the variation of the power according to climatic factors and its role to minimize the total cost since it replaces electronic devices like converter used for tracking of the maximum power point. Furthermore, the sliding mode control applied to the field oriented induction motor pump has succeeded in the regulation of the motor speed and then in optimization of the performances of the system. Such a control scheme provides protection of the connected inverter and machine with regards to stator current since these latter are controlled. Besides, the robustness quality of the proposed controllers appears clearly in the test results by changing machine parameters. Finally, this installation provides the minimization of the total cost since we choose to stock the water and not the energy which requires the use of electrical batteries.
[4]
[5]
[6]
R. Marouani, and F. Bacha, “A maximum-power-point tracking algorithm applied to a photovoltaic water-pumping system,” 8th International Symposium on Advanced Electromechanical Motion Systems (Electromotion - EPE Chapter ‘Electric Drives’), Lille, France, July-1-3, 2009. Y. Weslati, F. Bacha, A. Sellami, and R. Andoulsi, “Sliding mode control of a photovoltaic three-phase grid connected system,” 2nd International Conference on Electrical Systems Design and Technologies (ICEEDT), Hammamet, Tunisia, November 2008. Y. Weslati, A. Sallemi, F. Bacha, and R. Andoulsi, “Sliding mode control of a photovoltaic grid connected system,” Journal of Electrical Systems ISSN 1112-5209, vol. 4, issue 3, September 2008. M.O. Mahmoudi, N. Madani, M.F. Benkhoris, and F. Boudjema, “Cascade sliding mode control of a field oriented induction machine drive,” Eur, Phys. J. AP, vol. 7, pp. 217-225, 1999. H. Tarik Duru, “A maximum power tracking algorithm based on Impp=f(Pmax) function for matching passive and active loads to a photovoltaic generator,” Solar Energy, vol. 80, pp. 812-822, July 2006. Si Zhe Chen, Norbert C. Cheung Ka Chung Wong, and Jie Wu, ‘‘Integral Sliding-Mode Direct Torque Control of Doubly-Fed Induction Generators Under Unbalanced Grid Voltage,’’ IEEE Trans on Energy Conversion, VOL. 25, NO. 2, pp.356-368, JUNE 2010
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M. Arrouf, and N. Bouguechal, “Vector control of an induction motor fed by a photovoltaic generator,” Applied Energy, vol. 74, pp. 159-167, 2003. [8] M. Arrouf, and S. Ghabrour, “Modelling and simulation of a pumping system fed by photovoltaic generator within the Matlab/Simulink programming environment,” Desalination, vol. 209, pp. 23-30, April 2007. [9] S. R. Bhat, Andre Pittet, and B. S. Sonde,”Performance Optimization of Induction Motor-Pump System Using Photovoltaic Energy Source,” IEEE Trans on Ind App, VOL. IA-23, NO. 6, November/December 1987. [10] A. Saadi, A. Moussi “Optimisation of Chopping ratio of Back-Boost Converter by MPPT technique with a variable reference voltage applied to the Photovoltaic Water Pumping System,” Conference, IEEE ISIE 2006, pp-1717-1720. July 9-12, 2006, Montreal, Quebec, Canada. [11] A. Mezouar, M.K. Fellah, and S. Hadjeri, “Adaptative sliding mode observer for induction motor using two-time-scale approach,” Electric Power Systems Research, vol. 77, pp. 604-618, 2007. [12] V. I. Utkin, “Sliding mode control design principles and applications to electric drives,” IEEE Trans. Indus. Elec, vol. 40, pp. 23-36, February 1993.
APPENDIX TABLE II. PVG PARAMETERS PVG Parameters Ppv = 50 (Wc) Vpv = 17,2 (V) Ipv = 2.9 (A) Ns = 36 Np = 36 C=2000 (µF) Induction motor Parameters
187
