Using state-space representation for the modelisation of photovoltaic systems

Applied Mathematics and Computation 124 (2001) 129±138 www.elsevier.com/locate/amc

Using state-space representation for the modelisation of photovoltaic systems R. Adjakou *, C. Lishou, N. Dieye, L. Protin LER-Groupe de Traitements Informatiques J.E.R., AUPELF-UREF No. 6003,  Ecole Sup erieure Polytechnique, B.P. 5085 Dakar-Fann, S en egal

Abstract This paper is about the research carried out for a better knowledge of the micro¯uctuations of functioning parameters following external perturbations among photovoltaic systems. In several cases studied, it is proven that in site conditions, photovoltaic systems are submitted to many external parameters. To establish relations which can determine the importance of the ¯uctuations of energetic parameters of these systems, we have adopted a state-space representation. This method is useful when one deals with all parameters of a given studied system. As a result, the use of MATLAB software makes it possible to present the evolution curves of the functioning point of the pvgenerator when external perturbations happen. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Pv-system; Modelisation; Simulation; State space

1. Introduction The photovoltaic generator's maximum power point varies according to the irradiation value and the global system functioning conditions. Indeed, during the day, variations, as a function of time, of this functioning point depend not only on the ¯uctuations of the incident solar irradiation but also on the pro®le of consumption. When one desires to maintain this functioning point to the

*

Corresponding author. E-mail address: [email protected] (R. Adjakou).

0096-3003/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 0 ) 0 0 0 8 9 - 8

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maximum power, one is brought to elaborate devices that correct all sources of perturbations. The study we have led supposes that all dynamic regimes are succession of several permanent regimes even if these had only a short instant [1,2]. When considering a photovoltaic system, one can note that there are several sources of perturbations, which make the system multivariable. The case is all the more interesting that we recommend a state-space study.

2. Studied system The system in which we are interested is composed of the following. 3. Problem statement Implantation sites are isolated and equipment supports severe conditions of operation. To apprehend the behaviour of installations better we recommend an automatic study of the system by the state-space method. The good functioning of a photovoltaic system requires its good sizing but also an optimisation of energy transfer. The ®rst condition to respect is of course, maintaining of the maximum power point. On site, several factors disturb the system: · solar irradiation, · temporal variation of the pro®le of consumption, · quasi instantaneous variation of a duty ratio. A variation of the solar irradiation for example brings to act on the MPPT device. This generates in the system other types of perturbations that will be interesting to study [3]. The best manner to lead a complete study is a study in the state-space. Indeed, we recommend a global study of the system for the following two reasons: · nowadays, computer tools are reliable and well adapted to such mathematic manipulations, · studying multivariable systems (therefore complex) enables us to surround completely monovariable systems (who can more, can less) [1]. Taking into account the system represented below, the output s…t† is written according to the input u…t† and initial conditions X …t0 †. It is necessary therefore to have a minimal knowledge of the system at the instant t0 . To have this, we will choose X, the state vector, so as to easily know its components.

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4. State-space modelling The main elements of the system are: · a photovoltaic generator, · an MPPT device, · a battery of electrochemical accumulators, · two DC/DC converters. Fig. 1 shows the interaction between these elements. The synoptic diagram shows two static converters which we are going to analyse and of which we will de®ne the structures. The structural de®nition of a static converter is based on a certain number of electrotechnical rules. These rules are relative to the branches compatibility and to the non discontinuity of state variables (inductance current, capacity voltage). The ®rst DC/DC converter ensures the connection between the generator and battery of accumulators. The battery voltage is 48 V while the generator is cabled for 180 V of nominal voltage. The constraint is to reduce the generator voltage and to adapt it to the battery. This is why it appears to us to judiciously transform the nature of sources in the presence. Thus, to confer to the photovoltaic generator the characteristic of a voltage source, a capacity is put in parallel with it. Because of the same reasons, an inductance is inserted in the battery branch so as to transform it into a current branch (Fig. 2). Filtering elements are inserted between the DC/DC converter and battery so as to reduce undulations issued from input and output parameters of the DC/ DC converter (inductance at the output, capacity at the input). A complete diagram of the installation is presented in Fig. 3, with study parameters. The following parameters are identi®ed: · Ip ; Vp : generator parameters,

Fig. 1. pv-System synoptic.

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Fig. 2. Conncection between pv-generator and battery.

Fig. 3. Details of pv-system.

· · · · · · · · ·

Vf ; If : output voltage and current of the ®rst DC/DC converter, Vs : output voltage of the ®lter, L1 ; C: ®lter, Vb ; Ib : battery voltage and current, Iu : input current of the second DC/DC converter, Vch ; Ich : load voltage and current, R1 and R2 : duty ratio of DC/DC converters, the photovoltaic generator charging the battery, the battery powering the load.

4.1. Generator-battery [3] In the diagram block representation, relations between parameters are shown in Fig. 4.

Fig. 4. Relations between functioning parameters studied.

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Fig. 5. Fluctuations studied parameters.

For the photovoltaic generator, a regulation around a voltage consign is chosen. This implies that Vp is the photocell block input and Ip the output of this block. The directions of interactions in the global system are deduced. The functioning of the DC/DC converter generates nonlinear equations between the di€erent input parameters (Ip ; Vp ; R1 ) and output parameters (If ; Vf ). One models the interactions on variations of these parameters around the functioning point. This functioning point is a minimal knowledge of the system at this instant t0 . These considerations result from the study of parameter ¯uctuations so that the new studied parameters are shown in Fig. 5. We present a panorama of models which we have ended. 4.1.1. Photocells We identify two functioning zones represented by two lines D1 and D2 (Fig. 6). In the current source zone …D1 †, one can write Ip ˆ Icc …1=a1 †Vp . In the voltage source zone …D2 †, one can write Vp ˆ Vco a2 Ip . In variations, whatever the functioning zone, DIp ˆ KDVp . 4.1.2. First DC/DC converter The following equations Vf ˆ R 1 Vp ; I p ˆ R1 I f : give in variations DVf ˆ R10 DVp ‡ Vp0 DR1 ; DIp ˆ R10 DIf ‡ If 0 DR1 : 4.1.3. Filter-battery The following equations: dIf =dt ˆ 1=L1 …Vf Vs †; dIb =dt ˆ 1=L2 …Vs Vb †; dVs =dt ˆ 1=C…If Ib †

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Fig. 6. Graphic photocells modelling.

give in variations 

DIf ˆ 1=L1 …DVf

DVs †;

DIb ˆ 1=L2 …DVs

DVb †;





DVs ˆ 1=C…DIf

DIb †:

We suppose that the battery imposes a constant voltage value in a point of the system. This enables the absence of Vb variations DVb ˆ 0. The equations we have elaborated above as well as the set of considerations have ended to the following expression. 2 3 0 1=L2 0  6 7 0 1=C X ˆ 4 1=C 5X 0 2 6 ‡4

1=L1

1=…K:L1 R21 † 0 0

1=…L1 R1 † Ip0 =…KR1 † ‡ Vf 0

3


7 5DR1

2

DIf

3

6 7 with X ˆ 4 DIb 5 DVs

4.2. Battery-load We study the case where the battery debits (cloudy sky period and night) on an R, L load. The synoptic is represented in Fig. 7. A piece-by-piece modelling gives the following equations: · to the level of the battery: dVs Iu ˆ Ib C ; dt dIb ; Vb ˆ Vs ‡ L2 dt · to the level of the converter:

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Fig. 7. Synoptic of load powered by the battery (only).

Iu ˆ R2 Ich ; Vch ˆ R2 Vs ; · to the level of the load: Vch ˆ RIch ‡ L

dIch : dt

When we consider small variations around a functioning point, we end the equations: DVb ˆ 0; 

DVb ˆ L2 DIb ‡ DVs ; DVs ˆ

DIb C

DIu ; C

DVch ˆ R20 DVs ‡ Vs0 DR2 ; DIu ˆ R20 DIch ‡ Ich0 DR2 ;  DIch ˆ

DVch L

R DIch : L

3 DIb The chosen state vector is X ˆ 4 DVs 5. The state-space representation derived is DIch ; 2

0  6 X ˆ 4 1=C 0

1=L2 0 R20 =L

2

0

3

2

0

3

7 6 7 R20 =C 5X ‡ 4 Ich0 =C 5DR2 Vs0 =L R=L

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5. Simulation results Simulations are based on the case studied in Section 4.1 and the block diagram (Fig. 8). The initial conditions are summarised on the functioning point: Vp ˆ 21 V ; 0 K ˆ 0:02; R1 ˆ 0:5; M0 Vs0 ˆ 12V ; If ˆ 5A 0 Cf ˆ 4:7 nF L1 ˆ L2 ˆ 0:6 mH : SIMULINK with MATLAB software used for the time response simulation. The global system input is ˆ 0 at t0 DCp ˆ 0:01 for t0 ‡ : In an open buckle, no fact is convincing. Indeed, we observe a unidirectional linearity between functioning parameters (Fig. 9). By initiating a state-space study, we aim to surround at best all complex facts which are dicultly observable from the exterior of the system. As the input of the automatic system can be a duty ratio perturbation, it can be similar for the photovoltaic generator (temperature, irradiation, etc...). This is why, following a ®rst duty ratio perturbation, it is interesting to observe the propagation of ¯uctuations to the level of the other parameters. These parameters are then considered as the input of the global system. By bucking the system, the obtained results are shown in Figs. 10 and 11. There is a perfect linearity between Ip and R1 so that the curves are of the same evolution. By bucking the system, we observe oscillations in the time response of the output considered here. This means that a perturbation

Fig. 8. SIMULINK Model synoptic.

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Fig. 9. DIp time response in an open buckle.

Fig. 10. DIp and DR1 time response in closed buckle.

Fig. 11. DVp time response in closed buckle and functiong point before and after perturbation.

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generates ¯uctuations around the functioning point before the system stabilises itself. When considering the evolution of functioning point after perturbation (Mf ), we note that the new functioning point is on a higher irradiation I±V curve. 6. Conclusion Engineering systems are rarely so simple that they can be described solely by a single equation. In this paper we combine two ways to establish models: · black box modelling, · physical modelling based on physical law applications. The result is a set of equations which enables us to model energy system in state-space environment. This resolves complex problems and all ¯uctuations have been studied by exploiting the same formulation. Based on this state-space model, we can simulate several cases: short-circuit on one component, cloudy sky, lost of load, etc. References [1] M. Andersson, Object oriented modeling and simulation of hybrid systems, Doctoral Dissertation, Department of Automatic Control, Lund Institute of Technology, 1994. [2] T.C. Yang, A case study of magnetic suspension control with consideration of controller design for a class of non linear unstable systems, Control and Computers 24 (2) (1996) 36±47. [3] V. V orperian, Simpli®ed analysis of PWM converter using Model of PWM switch, IEEE Transactions on Aerospace and Electronic Systems 26 (3) (1990) 490±505.