Regular paper. Sliding mode control of a photovoltaic grid connected system. JES Proof

Transcription

1 Y. Weslati A. Sellami F. Bacha R. Andoulsi Regular paper Sliding mode control of a photovoltaic grid connected system This paper presents a sliding mode controller for a single-phase grid-connected photovoltaic system. The proposed system consists of maximum power point tracker (MPPT) based on sliding mode approach. The proposed MPPT controller generates current reference directly from the sensor board, and the current controller uses the sliding mode technique for the tight regulation of the injected grid current. The sliding mode controller has been designed using a time-varying sliding surface to control the sinusoidal current injected to the grid. The structure of a proposed system is simple and robust against modelling uncertainties and parameter variations. The mathematical modelling is developed and reliable simulations of the proposed controller are presented. Keywords: Photovoltaic power system, Grid connected, Sliding mode controller, Maximum power point tracking. NOMENCATURE C Nominal input dc link capacitor in µf P res Power injected to the grid in W C C4 Full-bridge inverter switch P PV Solar array power in W I res Current injected in A q 9 Electronic charge:.622. C I ph ight generated current in A R Output filter inductor resistance in Ω I ref Current reference in A R s Series resistance in Ω I ref Current reference in A R sh Shunt resistance in Ω I ref 2 Current reference in A R l oad resistance in Ω I ref 3 Current reference in A T Cell temperature in K u Control input, dimensionless I Cell reverse saturation current in A u eq Equivalent input, dimensionless I pt Optimum current in A u n Nonlinear control input I P Solar array current in A V co Open circuit voltage in V K Boltzmann s V 23 4 d Diode voltage in V constant.387. J / K V ond Output fundamental voltage K i Sliding mode controller gain inverter in V Nominal output filter inductor in mh V res Grid voltage in V n Number of parallel modules V RM Peak grid voltage in V N S Number of series modules V p Solar array voltage in V 2. INTRODUCTION Among the alternative energy sources used for the production of electricity, the wind and the photovoltaic solar energy became more attractive these last years. For sectors where electricity was not accessible, the solar systems were employed more and more. In some provinces, where the wind is not proportioned to produce required demand, a photovoltaic

2 generator (PVG) is composed of a certain number of solar cells connected like series and parallels according to the required power. Although their prices decrease, the systems of PVG always require expensive investments; consequently, it is very significant to extract the maximum of energy from PVG. This later can be only employed effectively when it functions at its optimum point of operation. Since the operation point of PVG depends on the load, the illumination and the temperature. It is almost impossible to use all solar energy available in variable atmospheric conditions when the system is not controlled. It is crucial to force the PVG system to operate at the optimum operation point. Consequently, various methods for maximum power point tracker (MPPT) are presented in the literature [-4]. The perturbation and observation method is an algorithm which is used in MPPT [5]. The neural network is another algorithm was trained with given data, then the optimal operating point is identified and the maximum power from the solar array is estimated [6]. In this paper we use a simple method which consists in serving the photocurrent of the PVG so as to have optimum current. This result is very important; in fact we can have the optimum value of the current thanks to sensor board placed close to a generator in the global system model. Single-stage grid connected photovoltaic systems have advantages such as simple topology, high efficiency. The typical Single-stage grid connected photovoltaic systems consists of a series-parallel connection arrangement of the available PV modules, inverters (converter DC-AC) and the utility grid. The inverter is the key-component for successful operation of the grid connected PV system. It utilize various control strategies: power control power-current controlled, [7], power controlled, [8], [9], and current controlled grid connected PV system, [5], []. In this paper the sliding mode controller has been designed to control the sinusoidal current injected to the grid. Section of the paper presents the dynamic model of the PV generator and the influence of illumination and temperature to the current voltage characteristic of the generator. In section 2, we present the structure and the dynamic model of the PV generator connected to the grid via a voltage inverter. Section 3 of the paper describes the background of sliding mode control. In section 4, the application of the sliding mode control to the grid connected photovoltaic system is presented. Finally, section 5 is reserved to present some simulation results and their comments. 3. DESCRIPTION OF THE PV GENERATOR 3. Generator modeling The electrical equivalent-circuit of a solar cell is shown in Fig.. It is composed of a light-generated current source, diode, series resistance and parallel resistance. The characteristic equation for the current and voltage of a solar cell is given as follows: q Vp + IpRS Ip = Iph I { exp ( Vp + IpRs) } nkt () RSH where I is the solar cell output current (A), V is the solar cell output voltage (V), I P is the light generated current (A), I is the cell reverse saturation current (A), q 9 =.622. C is the electronic charge, n is the dimensionless deviation factor from the ideal p n junction diode, k J K cell temperature (K), R is the series resistance (Ω), and s P 23 4 =.387. / is Boltzmann s constant, T is the R sh is the shunt resistance (Ω). The equivalent circuit for solar cells arranged in parallel and series. The mathematical equation relating the array current to the array voltage becomes: ph

3 q Vp IpRs Np Vp IpRs Ip = NsIph I exp ( + ) ( + ) nkt Ns N p RSH Ns N p (2) Where N P represents the number of parallel modules, and each module is composed of N s cells connected in series. Placing solar cells in series allows the heightening of voltage, placing solar cells in parallel enable us to reach what is needed by the charging operation. Mixed grouping will deliver a current NP Ip under the voltage Ns Vp ( Ip, V p being the current and the voltage of the solar cell). Fig.: Equivalent Circuit for a PV cell or module 3.2 Influence of illumination on PV generator ight-generated current I is practically proportional to the illumination and the space of ph junction exposed the solar rays, out the contrary, open voltage circuit doesn t depend on the illumination; it is rather linked to the type of substance and the type of the concerned junction. Ip E = W/m2 E = 888 W/m2 E2 = 87.5 W/m2 E3 = W/m Vp Fig. 2: The effect of the irradiation on PV generator 3.3 Influence of temperature on PV generator Temperature is a very important parameter in the characteristics of solar cells. In fact, if temperature rises, tight generated current goes up slightly. However, the current I = f( V ) very rapidly rises with temperature rise. It engenders drop of open circuit d d voltagev co. Iph Vd I D Rs Rsh Ip Rl Vp

4 k 34k ip.5 32k 3k Fig. 3: Influence off cell temperature of the PV generator The characteristic Ip = f( Vp) depends on solar illumination and temperature. This climatic variations lead to the motion of the maximum power point, and to control this motion; we place one or several inverters between the generator and the receiver. Theses inverters, known as MPPT, ensure the link between the generator and the receiver. As a consequence, they force the generator to deliver its maximum power. 4. MODEING OF THE SINGE PHASE GRID CONNECTED PV SYSTEM 4. Presentation of the structure PV I p C V p C C2 vp C3 C4 I res V ond Fig.4: Typical configuration of single stage grid connected PV system The configuration of a single-phase grid-connected photovoltaic system is shown in Fig. 4. It consists of solar array, input capacitor C, single-phase inverter, filter inductor, and grid voltage V res. The solar cells are connected in a series parallel configuration to match the required solar voltage and power rating. The direct current (DC) link capacitor maintains the solar-array voltage at a certain level for the voltage source inverter. The single-phase inverter with filter inductor converts a DC input voltage into an AC sinusoidal voltage by means of appropriate switch signals to make the output current in phase with the utility voltage and so obtain a power factor of unity. 4.2 System modelling We consider the configuration of Fig.4. where V p and I p are the solar-array voltage and inductor current, respectively, and i res is the current injected in the grid. The grid voltage is given as follows: Vres() t = VRM sinωt (3) Where ω is the angular frequency given by 2πf, and f is the grid frequency (f=5 Hz). R V res

5 The switch status of the single-phase inverter can be represented by the input β, defined as follows: + C, C 4 : on, C2, C 3 : off ß = (4) C, C4 : off, C2, C3 : on When the switch input β =, the state equation can be written as: V p = ( ires + ip ) C i res = ( Vp R ires Vres ) (5) When the switch input β =, the state equation is given as: V p = ( ires + ip ) C i res = ( Vp R ires Vres ) (6) Combining (5) and (6), the state-space averaged model of the single-phase gridconnected photovoltaic system shown in Fig.4 can be derived as follows: V p = ( ires u ( t ) + ip ) i res = ( Vp u ( t ) R ires Vres ) C (7) Where ut () is defined as the average value of β for the switching period. The state equation given in (7) is a non-linear system and can be expressed in the general form as: x = A(,) x t + B() x u() t (8) Where x, Ax (,) t and Bx ( ) are given as follows: ip Vp x = ires Ax (,) t = C Ri. res V res ires c Bx ( ) = V p The circuit parameters C and correspond to their nominal values which are known exactly. In a photovoltaic grid connected system, the capacitor and the inductance have a variation from their nominal value. The classic controller requires exact information on the circuit parameters for the desired performance when it controls the non linear system. inear controllers are known not to perform well under the presence of disturbance and unknown parameter variations. For this reason, we choose to use a robust and non linear control strategy based on sliding mode approach. 5. BACKGROUND ON SIDNG MODE CONTRO Sliding mode control is a kind of non-linear control which is robust in the presence of parameter uncertainties and disturbances. It is able to constrain the system status to follow trajectories which lie on a suitable surface in the sliding surface. The main steps for sliding mode controller design can be summarized, by using an equivalent control concept, as follows: -Selecting a switching surface sxt (,) = (where x is the system s state vector) that provides the desired asymptotic behavior in steady state.

6 -Obtaining the equivalent control u eq by applying the invariance condition. sxt (,) = andsxt (,) = ut () = u () t (9) eq The existence of the equivalent control u eq assures the feasibility of a sliding motion over the switching surface sxt (,) = -Finally, selecting a non-linear control input u n to ensure that yapunov stability criteria. ss () 5. Sliding surface design Consider a non linear system of the form: x = Ax (,) t + B(,) x t u() t () The sliding surface is defined as follows: T sx ( ) = [ s( x) s2( xs ) 3( x)... sm( x) ] = (2) To develop the equivalent control, we consider the system (). If the system is at the surface of the sliding mode then sxt (,) =, t tr weret R, is the reaching time. Differentiating sxt (,) = give: s s s S = x = A + Bu = x x x (3) s B Is non-singular all over the space x The equivalent control can be given by: s s ueq = ( B) ( A) A (4) x x The dynamics of the system in the reaching space are given as follows s s x = [ I B( B) ( )] A (5) x x 5.2 control low selection Once the problem of existence solved by fixing the desired dynamics of the system in the sliding mode and that with the choice of a switching surface desired, we must resolve the second phase of the design, called reaching problem. It can be summed in u = f( x) conducting the trajectory state on the sliding surface in a defined time, constraining it to stay on that surface. The nonlinear switching input can be chosen [-2] as follows: un() t = Ki s α sgn( si) α, i =... m (6) The nonlinear switching input increases the reaching speed when the state is far away from the switching manifold, but reduces the rate when the state is near the manifold. Thus control law that provide a finite reaching time is given by:. Ti = si ( α), i =... m (7) ( α) k i 6. APPICATION TO A GRID PV SYSTEM The design of the sliding mode controller starts from the design of the sliding surface.

7 Usually, the sliding surface is constructed by the linear combination of state variable errors that are defined as the differences between the state variables and their references. Therefore, in this case, the sliding surface can be designed with errors of the solar-array voltage and inductor current. In a single-phase grid-connected photovoltaic system, the solar-array voltage is oscillating due to the sinusoidal inductor current, which results in an undesirable sliding mode performance. For this reason, no attempt has been made to apply the sliding mode control for a grid-connected photo-voltaic system. The main purpose of a grid-connected photovoltaic system is to transfer the maximum solar-array power into the grid with a power factor of unity. Therefore, the sliding surface should be designed to control the inductor current and solar-array power simultaneously. This requirement can be achieved by selecting a sliding surface only using the errors of the inductor current. The proposed time-varying sliding surface is defined by: sxt (,) = i i = i i sinω t (8) res ref res opt The current of generator statement, corresponds to the maximum power, noted I opt (optimal current), and is proportional to the photo current which is proportional to illumination following the relation [3] according to Iopt =.85Iph (9) The control input is given as follows: ut () = ueq () t + un () t (2) The equivalent control u eq can be obtained applying the invariance condition. sxt (,) = and sxt (,) = ut () = u () t (2) eq The existence of the equivalent control u eq assures the feasibility of a sliding motion over the switching surface sxt (,) =. Differentiating Bx () = give: s s sxt (,) ( Axt (,) + Bx (). ueq () t ) + = x t (22) The necessary condition for the existence of a sliding motion of sxt (,) is represented by the transversality condition and expressed as follows: s Bx ( ) (23) t s Where Bx ( ) denotes the directional derivative of the scalar function r with t respect to the vector field Bx ( ) if the transversality condition is satisfied, then substituting (8) into gives a result as follows (22). sxt (,) = ( Vp ut () R Ires Vres ) ioptωcosωt = (24) Therefore, the equivalent control-input is given as: R ires + Vres + iopt ω cos ωt/ VRM ueq () t = (25) Vp Once the desired time varying sliding surface was fixed, the nonlinear switching input can be chosen as follows: un() t = Ki s α sgn( si) α, i =... m (26) If (25) and (26) are substituted into (2), the range of k i ensuring ss < can be determined as follows:

8 Vp Vres ss = s( i res ioptωcos ωt) = s( ( ueq ( t) + un( t)) ioptωcos ωt) (27) Vres = s( Ki s α sgn( si)) From (25), (26) and (27) the control input ut () = u () t + u () t is given as follows: Rires + Vres + iopt ωcos ωt / VRM Ki s sgn (s) ut () = V p α Then, the control input is compared with the pulse-width modulation (PWM) ramp voltage and generates the appropriate switching pattern of the inverter. 7. SIMUATION RESUTS The proposed design scheme is implemented in Matlab/Simulink software for a PVG grid connected system with parameters given in Table. We tested the design of the PV generator connected to the single phase grid, and we run the global system for step change in PV generator current reference. Simulation results have been investigated in each case to view the influence of the sliding mode controller in power and current injected to the grid. Symbol Table : Parameters value R.2Ω.2H C V RM eq 2.e-3F 2V ki. α.9 In the first case, the PV generator system operates at the optimum power point. In this particular point, the reference current is I = 3.35A. ref Figure 5 represent respectively, (a) the reference current I ref, (b) the current injected in to grid I res and (c) is the switch surface. From this figure, it s clear that the current injected to the grid tracks the reference one. n (28)

9 5 iref [A] ires [A] S = iref - ires Fig.5: Simulation waveform: (a) the reference current I ref, (b) the current injected in the grid I res and (c) the switching surface. Figure 6 depicts, respectively, the PVG current I P (Fig.6.a), the solar array voltage V P (Fig.6.b), and the power generator P PV (Fig.6.c). All variables oscillate at the frequency of Hz which is twice of the grid voltage and current frequency. The average of solar array voltage and current is about 7.3V and.22a respectively. The power delivered by the photovoltaic system is 2W. 2 ip [A] vp [V] P vp [W] Fig. 6: Simulation Waveform : (a) the current I P, (b) the voltage V P and (c) the power P of the PV generator. PV Figure 7. depicts, respectively, (Fig.7.a) the equivalent control-inputu eq, (Fig.7.b) the non linear switching input u n and (Fig.7.c) the control inputu.

10 U eq U n U = U eq + U n Fig.7: Simulation waveform: (a) the equivalent control-input u eq, (b) the non linear switching input u n and (c) the control inputu. Figure 8 illustrates, respectively, (Fig.8.a) the current injected I res, (Fig.8.b) the grid voltage V res and (Fig.8.c) the inverter voltage V ond. The waveform of the injected current to the grid is sinusoidal and is in phase with grid voltage giving a power factor of unity. Hence, the angle between fundamental output voltage inverter and the grid voltage is small V res [V] V ond [V] i res [A] Fig.8: Simulation Waveform: (a) the current injected I res, (b) the grid voltage Vres s and (c) the inverter voltagev ond. phenomenon. Figure (9) shows the simulation results of the proposed controller structure compared to the one given in reference [5]. The switching input shows a high chattering value in the controller of reference [5]. However, in our new controller there is a reduction of chattering

11 previous controller proposed controller Fig.9: Comparison to the prior controller: (a) the controller structure in reference [5] and (b) the proposed controller. In the second case, we perform simulations with step change of the current reference. Initially, the PV generator system operates at the current referenceiref 2 = 2.5A. Then, at t =.2s the current reference is instantaneously changes to Iref 3 = 4.5A. The Figure shows that the current injected to the grid tracks the variation of the reference current. 5 iref [A] ires [A] S = iref - ires Fig.: Simulation waveform of the variation of the reference current. Figure and 2 show that the power is proportional to the injected current; therefore a variation of reference current entails a variation of power injected to the grid. There is a small difference between the power injected and the power generator that is equal to the output power of the inverter, this small difference is due to the loss in the filter resistance.

12 P ond [W] P res [W] Fig.: Waveform of the inverter output power P ond and the power injected to grid P res for a step change of Iref from 2.5A to 4.5A. 2 ip [A] vp [V] P pv [W] Fig.2:. Waveform of the current, the voltage and the power of the PV generator for a step change of I from 2.5A to 4.5A. ref To test the robustness of the proposed control structure, we perform a simulation with parameter variation. Initially, the PV generator system operates at nominal inductor parameter =.2H. Then, at t =.s it is instantaneously changes to.3h. Figure 3 shows that the response of the active power injected to the grid was not affected by the variation of the inductor parameter. This confirms the robustness of the used sliding mode control.

13 25 actif power [5W/div] reactif power [5 V.AR/div].5..5 Fig.3: Waveforms of the active and reactive power for a change of parameter from.2 H to.3 H. at t=.s. 8. CONCUSION A Sliding mode controller for photovoltaic system connected to the grid has been proposed. The current reference is directly generated from the sensor board. The sliding mode controller was used as a current controller for the tight regulation of the current injected to the grid. The proposed control system is simple, efficient and the control signals are chattering free. In this new controller there is an importing reduction of the chattering phenomenon. The simulation results show the good performance of the proposed controller. References [] Chiang SJ, Chang KT, Yen CY. Residential photovoltaic energy storage system. IEEE Transactions on industrial electronics. vol. 45, no 3. pp , June. 998 [2] Chihchiang Hua a, Jongrong in b"a modified tracking algorithm for maximum power tracking of solar array, Energy Conversion and Management. pp April. 24. [3] G. Brandli and M. Dick, "Alternating current fed power supply," U.S. Patent , Nov. 4, 978. [4] 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. 8, pp82-822, July 26. [5] II. Song. Kim, "Sliding mode controller for the single-phase grid connected photovoltaic system," Applied Energy. vol. 83, pp. -5, October. 26. [6] A. B. G. Bahgat, N. H. Helwab, G. E. Ahmad, E. T. El Shenwyb, "Maximum power point tracking controller for PV systems using neural networks," Renewable Energy. Vol. 3, pp , 25. [7] M. Dai, M. N. Marwali, J. W. Jung and A. kheni, power flow control of a single distributed. IEEE power systems conference and exposition. Vol.. pp Octobre 24. [8] F.. Albuquerque, A. J. Moraes, G. C. Guimarks, S. M. R. Sanhueza, A. R. Vaz, optimisation of a photovoltaic system connected to electric power grid, IEEE transmission and distribution conference and exposition, pp atin American, 24.

14 [9] F. Scapino, F. Spertino, oad curves at DC side: a useful tool to predict behaviour and the design to grid-connected photovoltaic systems, IEEE International symposition on industrial electronics, Vol. 3, pp May 22. [] Sachin. Jain, Vivek. Agarwal " New current control based MPPT technique for single stage grid connected PV systems" Energy Conversion and Management vol. 48, pp , [] Vadim I. Utkin, "Sliding mode control Design Principles and Application to Electric Drives" IEEE Trans. Power Delivery, vol. 4, pp , Apr [2] Webbing. GAO, James C. Hung, "Variable Structure Control of Nonlinear Systems: A new approach" IEEE Trans. Power Delivery, vol. 3, pp , Apr [3] S.M. Alghuwainem, Matching of a DC motor to a photovoltaic generator using a step converter with a current looked loop IEEE transactions on Energy Conversion, Vol 9, N, 994, pp