A Methodology for the Design of Photovoltaic Water Supply Systems

Transcription

1 PROGRESS IN PHOTOVOLTAICS: RESEARCH AND APPLICATIONS Prog. Photovolt: Res. Appl. 2001; 9:349±361 DOI: /pip.386) Applications A Methodology for the Design of Photovoltaic Water Supply Systems O. C. Vilela*,y and N. Fraidenraich Research Group on Alternative Energy Sources FAE)±Nuclear Energy Department DEN),Federal University of Pernambuco UFPE) Av. Prof. Luiz Freire 1000, ,Recife PE),Brazil Photovoltaic pumping systems are used nowadays as a valuable alternative to supply water to communities living in remote rural areas. Owing to the seasonal variation and the stochastic behavior of solar radiation, at certain times the supply of water may not be able to meet demand. A study has been made of the relationship between water pumping capacity, reservoir size and water demand, for a given water de cit. As a result, curves of equal water de cit iso-de cit lines) can be obtained for various combinations of PV pumping capacity and reservoir size. A methodology to generate those curves is described, using as its main tool the characteristic curve of the system, that is, the relationship between water ow and collected solar radiation. The characteristic curve represents the combined behavior of the water pumping system and the well. The in uence of the minimum collected solar radiation level, necessary to start the system's operation the critical radiation level I C ), is also analyzed. Results show that PV pumping systems with different characteristic curves, but with the same critical levels, yield the same set of iso-de cit lines. This drastically reduces the number of necessary solutions to those corresponding to a few values of I C. Iso-de cit lines, calculated for the locality of Recife PE), Brazil, are used to illustrate the sizing procedure PV water supply systems. Copyright # 2001 John Wiley & Sons, Ltd. 1. INTRODUCTION Photovoltaic pumping systems are used nowadays as a valuable alternative for water supply in rural communities, not connected to the conventional electricity grid. Water supply systems driven by PV generators are composed of a photovoltaic pumping system, a storage tank and a water source, most frequently, a well. Usually, the demand for water occurs during daylight hours and the pumping period varies according to the behavior of daily solar radiation and, in particular, to the minimum solar radiation required to get the pumping system into operation the critical radiation level I C ). Owing to the stochastic nature of solar radiation, during some periods, either the system is not pumping, because the solar radiation level is less than I C, or the amount of water pumped is not suf cient to meet requirements. The use of a storage tank is then necessary. The reservoir is * Correspondence to: O. C. Vilela, Research Group on Alternative Energy Sources FAE)±Nuclear Energy Department DEN), Federal University of Pernambuco UFPE), Av. Prof. Luiz Freire 1000, , Recife PE), Brazil. y ocv@npd.ufpe.br Published online 1 August 2001 Received 14 November 2000 Copyright # 2001 John Wiley & Sons, Ltd. Revised 10 April 2001

2 350 O. C. VILELA AND N. FRAIDENRAICH a simple way to match supply and demand, reducing the periods in which there is water de cit e.g., early in the morning). Analyses of the performance of PV water pumping systems, including well behavior, have been published in the technical literature. 1±3 For systems with no `memory' without storage), as considered in those papers, the sequence of events is irrelevant, only the statistical weight of solar radiation is important. Several con gurations of PV water supply systems, including the storage tank, have been considered by Hadj Arab et al. 4 The authors use a simulation program to analyze the in uence of several parameters on the system's performance, such as load pro le, water tank capacity, manometric head and PV array con guration. In our work we have developed a general procedure to predict the long-term behavior of PV pumping systems coupled to a well and backed up by a storage tank. The presence of a component with memory storage tank), and the stochastic nature of the system's input solar radiation)requires due consideration of the time sequence of events to give a proper account of the hourly matching between pumped water ow _V and water demand or of the water available in the reservoir, in case it is necessary to complement the water being pumped to satisfy demand. Photovoltaic pumping systems can be analyzed by studying the behavior of each component or group of components PV array, inverter/converter, motor pump, pipeline and well). Alternatively, and this is the approach used in this paper, their behavior can be described as a single relationship between water ow rate output)and collected solar radiation input), called the characteristic curve CC). It is important to mention that the CC, as de ned here, is considered as representing a large class of systems. Once the CC has been obtained we can model the speci c con guration with which we are dealing and consider that other systems with different con gurations can be represented by the same curve, as long as they satisfy the same input±output relationship. Since the presence of the storage tank requires the registration of a time sequence of events short memory), this component is considered separately. Regarding this, the methodology presented is similar to that developed for solar autonomous photovoltaic systems SAPVS)including solar home systems SHS), 5±9 analyzing the interaction of the system represented by the CC with the water reservoir. However, some important differences should be noted: a)the existence of a critical solar radiation level I C solar autonomous systems start to send energy to the battery, even at low solar radiation levels, which is not the case for PV pumping systems); b) nonlinear behavior of the relationship between water ow and collected solar radiation the relationship between electrical energy and collected solar radiation can be considered as linear for solar autonomous photovoltaic systems). The design procedure for PV water supply systems consists of estimating the size of the PV pumping system and storage tank which would be able to meet a water demand with a predetermined reliability level. The complement of the reliability level, the water de cit level, can be easily quanti ed as the relationship between water de cit and water demand. As expected, similarly to what happens in rural electri cation systems, there are a large number of combinations of pumping systems and water reservoirs leading to the same water de cit level. A key objective of this work is to obtain a graphical representation relating the size of the water pumping system to the storage capacity, for a given water de cit. The result of the procedure is a unique graphical relationship the iso-de cit line)between properly chosen parameters the water pumping capacity 10 and the size of the storage tank), both related to demand. We propose to use these lines as a basic tool for design purposes. Some remarks about the procedure developed in this work should be made: 1. As mentioned, the calculation procedure is based on the existence of a characteristic curve CC). We describe a method to obtain that curve, combining laboratory indoor tests basic manufacturers' information) or outdoor tests, with knowledge of the well behavior. Well and pipeline losses are described by a single curve the well curve). 2. For the sake of simplicity, it is assumed that the load is uniform. The in uence of load variability, for SHSs, has been studied by Narvarte and Lorenzo, 11 showing that daily variations on the load pro le can modify the level of reliability of the system's performance signi cantly. Even though similar behavior can be expected for water supply systems, the procedure developed here, assuming that the water demand is uniform, is a rst step, essential for future developments taking into account load variations. The complexity resulting from

3 PV WATER SUPPLY SYSTEMS 351 load variability effects can not be neglected since a detailed analysis will, very likely, involve consideration of a large class of load pro les characterized by the monthly average, standard deviation and eventually the persistence strength day-to-day correlation coef cient), if the load is of a stochastic nature. In the next section we describe how to obtain the CC for a PV pumping system confronted with various starting conditions: eld test results, indoor or outdoor laboratory information. We then present a methodology to generate iso-de cit lines for water supply systems, and the in uence of the critical level on the behavior of the iso-de cit lines is discussed. A case example for the design of a system is then given, followed by a summary and nal comments. 2. OPERATIONAL CHARACTERISTIC OF PV WATER PUMPING SYSTEMS The operational behavior of a PV pumping system coupled to a well is represented, as mentioned, by the curve relating water ow rate _V and collected solar radiation I coll Figure 1). It can be obtained directly through eld experiments, by measuring _V and I coll, under real conditions of operation. This CC yields a reliable representation of the system's behavior, dependent, however, on local conditions. Depending on the type of information available at the procedure's starting point, a variety of paths can be followed to obtain the `CC'. Figure 2). Four types of curves are shown in boxes I±IV in this gure: I) _V± electrical power input, parameterized by the water head; II) _V±I coll, the pumping system curve PSC), also parameterized by the water head; III) _V±I coll, the characteristic curve CC)itself; and IV)The plot of water pumping head against _V, describing the behavior of the well and pipeline. Curves expressing the relationship between water ow and electrical power input, parameterized by the water head, are obtained from indoor laboratory tests, and are sometimes available from the manufacturers. Outdoor laboratory tests yield the relationship between water ow and collected solar radiation, also with water head as a parameter. A description of the different procedures leading to the CC follows. 2.1 Initial information Water ow vs electrical power input obtained from indoor laboratory tests These curves, parameterized by the water head H box I in Figure 2), represent intrinsic properties of the pumping system, with no in uence from ambient conditions. By simulating the performance of the photovoltaic array coupled to the inverter±moto±pump system, we convert the input of electrical power to collected Figure 1. Characteristic curve of a PV water pumping system

4 352 O. C. VILELA AND N. FRAIDENRAICH Figure 2. Different routes to obtain the characteristic curve solar radiation. The result is a new family of curves relating water ow to collected solar radiation _V±I coll, box II)also parameterized by the water head, depending now on local ambient conditions solar radiation and temperature). Geometrically, the family of curves just described can be thought of as being a single surface, where the water ow is expressed as a function of two variables, collected solar radiation and water head _V ˆG I coll, H), Figure 3). The well curve can also be considered as a surface HˆH _V), parallel to the I coll axis. Its intersection with the surface _V ˆG I coll, H), projected onto the plane [ _V, I coll ], yields the `CC'. Analytically, the intersection of the two surfaces can be obtained by substituting HˆH _V)into the equation _V ˆG I coll, H), yielding the CC _V ˆV I coll )). For illustrative purposes, we describe a practical procedure. Let us consider one of the constant water head curves, e. g., _V ˆ G 1) I coll, H 1 ), with parameter H 1, and the well curve HˆH _V) Figure 4). Determine, by numerical or graphical methods, the value _V 1 which corresponds, in the well curve, to the water head H 1ˆH _V 1 ). To obtain the rst point on the CC, it is suf cient to nd, for example graphically, the intersection of the horizontal line _V ˆ _V 1 with the curve _V ˆG 1) I coll, H 1 ). We then determine the solar radiation level I 1, coll which satis es _V 1ˆG 1) I 1, coll, H 1 ), where the pair of values _V 1, H 1 )belong to the well curve. Extending this procedure to the whole set of constant water head curves with parameters H 1, H 2, H 3,..., H n we nd I 1,coll, I 2,coll, I 3,coll,..., I n,call, which correspond to the water ow values _V 1, _V 2, _V 3,..., _V n. The relationship between _V i and I i,coll describes the CC of the system, which `travels' across the whole family of constant water head lines Figure 4). In particular, its starting point, at _V ˆ0, corresponds to the critical solar radiation level I C and the static water head H static. If none of the elements of the set of parameters H i is equal to H static, this point can be determined by interpolation Water ow vs collected solar radiation obtained from by outdoor laboratory tests Naturally, the curves depend on the local climate. Once obtained, they can be converted into the CC by nding the intersection between the surface _V ˆG I coll, H)and the well equation HˆH _V), as mentioned before.

5 PV WATER SUPPLY SYSTEMS 353 Figure 3. Characteristic curve obtained by the intersection of the surface representing the pumping system behavior _V ˆ G I coll, H )and the well surface HˆH _V ), parallel to the I coll axis Figure 4. Simple procedure to determine points _V i, I i,coll )belonging to the characteristic curve Summarizing, the method described for the indoor tests section 2.1.1)can be considered as a two-step procedure. The rst step converts electrical power input into collected solar radiation, yielding the relationship _V ˆG I coll, H), the PSC. The second step introduces the well curve, transforming the set of PSCs into a single curve, representing the behavior of the whole system _V ˆV I coll ). If well and pipeline losses are negligible HˆH static, only the rst step is necessary. With regard to the procedure for the outdoor tests, just the second step is required. A practical example of how to generate a characteristic curve from manufacturers data is described in the Appendix.

6 354 O. C. VILELA AND N. FRAIDENRAICH 3. GENERATION OF WATER ISO-DEFICIT LINES The procedure developed to calculate equal de cit lines analyzes the relation existing among the pumped water volume, reservoir size and water demand. These relations depend on the temporal matching between pumped water ow and water demand, here considered as uniformly distributed during the daylight period. The mismatch between supply and demand results in de cit or wastage of water. Both, de cit and wastage depend, for a given water demand, on the relationship between the following dimensionless parameters storage and pumping capacity parameters) C S ˆ C W =L 1 and C P ˆ V M =L where C W is the water tank capacity, L the water demand, and V M the water pumping capacity, that is, the water volume which the system is able to pump considering operation as continuous). L and V M are expressed on a daily basis, averaged over the whole period of calculation. The analysis is made through a water balance calculated on an hourly basis, using the relation between pumped water ow and collected solar radiation CC), _V ˆV I coll ). Owing to the lack of real hourly solar radiation data, we rst generate daily values of solar radiation, using the procedure developed by Aguiar et al. 12 With these values, we calculate hourly solar radiation on the PV generator plane using the Collares-Pereira and Rabl model. 13 A water balance is then set up on an hourly basis Water balance The balance of water for the whole period considered can be written as: V M ˆ L D V EX 3 where D is the water de cit and V EX the water excess. All quantities are expressed on a daily basis, averaged over the complete computational period. It should be mentioned that the excess of water V EX in equation 3), is not effectively pumped, because of the existence of controlling devices that switch off the pump. In fact, the water volume actually pumped is given by: V M;EFF ˆ L D 4 Dividing equation 3)by the load L, we obtain the following dimensionless equation: D L ˆ 1 C P V EX L 5 where D L is the water de cit level D/L. For a given CC, the water balance is repeated for various combinations of reservoir size and water demand, generating pairs of parameters C P, C S )for different values of water de cit level D L. The iso-de cit lines are obtained by selecting the pairs of values C P, C S )which correspond to a water de cit level D L, within a very narrow interval D L D L. Long-term averages of daily irradiation are usually available in terms of monthly means. Each parameter Cp can be associated with a monthly value: C P;m ˆ VM;m 6 L where V M,m is the long-term water pumping capacity, monthly average, expressed on a daily basis. Thus, an alternative way to build the iso-de cit lines is to look for the relationship between the parameters C P,m and C S,

7 PV WATER SUPPLY SYSTEMS 355 for a given water de cit level D L. Furthermore, if the parameter C P,m is chosen for the month with the lowest pumping capacity V M,m, time-saving bene ts during the design process can arise, since the behavior of the system during that month will, very likely, determine the system size. The methodology described above was applied to generate iso-de cit curves for the city of Recife-PE, Brazil south, west). The water load L is uniformly distributed between 06:00 and 18:00 h. We consider that this period represents fairly well the pro le of water demand of rural communities living in the Brazilian Northeast region. 3.2 Solar radiation Starting with long-term averages of daily solar radiation for the locality of Recife, we calculated average values of monthly clearness index K t. Using Markov matrixes to generate solar radiation time series, 12 we obtained values of K t for 365 days, for ten years. The hourly solar radiation incident on the PV collection plane for Recife, the tilt angle yielding the maximum solar collected radiation for the worst month, June, is approximately 20 ), has been calculated from the Collares-Pereira and Rabl model. 13 Average values of horizontal solar radiation meteorological data)and collected solar radiation generated with that procedure, are shown in Table I. 3.3 Calculation procedure of iso-de cit lines The example that follows generates iso-de cit lines for a pumping system operating with two constant water heads no drawdown, 24 and 40 m). The CCs used Figure 5) were obtained with the experimental facility in our Laboratory, consisting of a closed-cycle water loop a brief description of the system is given in the Appendix). Iso-de cit lines for water de cit fractions D L equal to 1 and 2% were then generated Figures 6 and 7), with critical levels I C equal to 138 and 247 W/m 2, respectively, and C P,m calculated for the month of June. We can see that the curves come closer to each other for large values of C s. For large reservoirs much larger than the load), the parameter C P,m remains almost constant and tends, asymptotically, to 1 D L ). At the other extreme of the iso-de cit lines, the pump capacity of the system is much larger than the load, and small variations of C s result in large variations of the system capacity C P,m. In this region, the storage tank capacity needed to supply the water load is small, but nite. Each iso-de cit line has a region where the product of the storage tank capacity and the pump capacity is a minimum. These values, shown in Table II, are an indication of the smallest reservoir and pump capacity Table I. Daily solar radiation, monthly average, on the horizontal plane meteorological data)and collected solar radiation on tilted plane at angle ˆ 20 ), calculated by the time series procedure, Recife-PE, Brazil Month Solar radiation on the Collected solar radiation on horizontal plane Wh/m 2 )tilted plane ˆ 20 ) Wh/m 2 ) Annual average

8 356 O. C. VILELA AND N. FRAIDENRAICH Figure 5. Characteristic curves of a system operating with constant water heads of 24 and 40 m. Figure 6. Iso-de cit lines for I Cˆ138 W/m 2 ; values of C P,m calculated for the month of June combination, able to supply the water load with a given D L value. For design purposes, one can analyze how the cost of the system varies for pairs of values C P,m, C s )around the region where their product is a minimum. The iso-de cit curves presented are valid for the locality of Recife, or other places in which the solar radiation, calculated at the collection plane of the photovoltaic generator, has a similar behavior. With regard to the

9 PV WATER SUPPLY SYSTEMS 357 Figure 7. Iso-de cit lines for I Cˆ247 W/m 2 ; values of C P,m calculated for the month of June Table II. Minimum product of parameters C P,m )and C S ) D Lˆ1% H m D Lˆ2% C P,m ) C S ) C P,m C S ) H m calculation of the monthly water pumping capacity V M,m, the utilizability method can be used as a simple and accurate way to obtain it INFLUENCE OF THE CRITICAL LEVEL ON THE ISO-DEFICIT LINES In this section we show that, in addition to the water de cit level D L, a second parameter, the critical radiation level I C, is also required to specify the iso-de cit lines. Results obtained up to now show that for a given pumping system, represented by a CC, we nd a set of wellde ned iso-de cit lines. A different CC will be associated, in principle, with another set of lines. However, it will be shown that, if the pumping systems are represented by different CCs, but have the same critical radiation level I C, they generate the same set of iso-de cit lines. For that purpose we generate lines with D L equal to 1 and 2% for six CC curves, each group of three deliberately chosen to have the same value of I C. The CCs with critical levels of approximately 138 and 247 W/m 2 are shown in Figure 8 a)and b), respectively. Although the water volume pumped by the systems represented by those curves are, on average, very different, the results for the 1 and 2% iso-de cit lines Figures 9 and 10), show that no signi cant difference can be observed among curves with equal I C values. Thus, we verify that the behavior of water supply systems is strongly dependent upon the critical radiation level I C. This parameter determines at what time of the day the pumping system starts working and at what time it stops. According to the proposed water load pro le, it is expected that some water demand will be present in the morning and in the evening, when the system has not started pumping or is no longer working. Systems with larger I C values will require, for example, larger water tanks to reach the same reliability level.

10 358 O. C. VILELA AND N. FRAIDENRAICH Figure 8. Set of characteristic curves: a) I Cˆ138 W/m 2 ; b) I Cˆ247 W/m 2 Figure 9. Iso-de cit lines obtained with characteristic curves A, B and C see Figure 8a) I Cˆ138 W/m 2 5. SIZING EXAMPLE In order to design a PV pumping system to be installed in Recife, supplying a demand of 8 m 3 of water per day, we might rst choose a photovoltaic pumping system that satis es the relations of energy balance to meet the speci ed water demand and water head, and of input to output power to guarantee that the maximum water ow does not exceed the maximum pumping rate of the well. In this way we obtain a rst indication of the size of the pump able to meet the speci ed demand. Next, we can select from catalogs a water pumping system having approximately the estimated parameters, power of the motor pump and PV array area). The CC is then generated, see section 2), using the manufacturer's curves for the PV pumping system and the experimental data of the well. Let us assume, for example, that the CC of the water pumping system is represented by the following polynomial relationship: _V ˆ I 2 coll I coll

11 PV WATER SUPPLY SYSTEMS 359 Figure 10. Iso-de cit lines obtained with characteristic curves D, E and F see Figure 8b) I Cˆ247 W/m 2 where I C < I coll < I coll max, and I coll,max is the maximum local collected solar radiation empirical expressions relate this value to the extraterrestrial collected solar radiation 14 ). Units for _V and I coll are m 3 /h and W/m 2, respectively. The smallest value of I coll, which makes _V ˆ0 denoted as I C, the critical radiation level), is 247 W/m 2.We can then use the iso-de cit lines shown in Figure 7. The water pumping capacity, for the city of Recife, in June, calculated with the utilizability method, 10 is V M,mˆ1602 m 3. Thus, the parameter C P,m associated with the water pumping capacity is equal to: C P;m ˆ Assuming that we require a water de cit fraction of 1%, from Figure 7 we obtain the reservoir size of 23 times the load, or 184 m 3. A smaller reservoir size, equal to 17 times the water demand, of volume 136 m 3, would supply the load with a water de cit fraction of 2%. This procedure can be repeated for PV pumping systems with different con gurations. Finally, by estimating the investment cost of the pumping system and the reservoir, we can identify the con guration with the best cost-bene t ratio. ' 2 6. SUMMARY AND COMMENTS The behavior of PV water pumping systems, based on the use of a characteristic curve, is described. These curves express the relationship between water ow rate and collected solar radiation for a given PV pumping system coupled to a well. The various routes to generate the characteristic curve are described and practical procedures to obtain it are explained and exempli ed. The use of a characteristic curve, to represent the system behavior, enables us to obtain the iso-de cit lines for any location and for different con gurations of PV pumping systems in a fairly simple way. The iso-de cit lines allow us to calculate the size of the water pumping system and storage tank that is able to meet a speci ed water demand with a given reliability level. The utilizability method can be used as a simple and useful tool to calculate the water pumping capacity. This water volume has to be determined when designing water supply systems by means of iso-de cit lines.

12 360 O. C. VILELA AND N. FRAIDENRAICH In addition to the water de cit level, the iso-de cit lines depend on a second parameter, the critical solar radiation level. Results show that PV pumping systems with different characteristic curves, but with the same critical levels, yield the same set of iso-de cit lines. This, drastically reduces the number of necessary solutions to those corresponding to a few values of the critical radiation level. APPENDIX: CASE EXAMPLE FOR GENERATING A CHARACTERISTIC CURVE A system composed of a photovoltaic generator with 21 C±Si modules, of nominal peak power 1113 W, a 1500 W inverter, with maximum power point tracking device, coupled to an asynchronous three-phase motor and a ten-stage centrifugal water pump can be characterized by the following equation: _V ˆ A H P 2 DC B H P DC C H A1 with P DC,crit P DC P DC,max, where P DC,crit is the value of P DC which makes _V ˆ 0 and P DC,max is the maximum power developed by the equipment. The units of _V and P DC are m 3 /h and W, respectively. The coef cients A H), B H)and C H)depend on the pumping head H Table A1)and can be obtained from curves _V ˆF P DC, H), given by the manufacturer. Simulation of the PV array behavior coupled to the pumping system and operating at the maximum power point of the array MPP), gives the relationship between collected solar radiation and DC power generated. As a result, a new family of curves PSC, Equation A2)is obtained. _V ˆ a H I 2 coll b H I coll c H A2 with I C I coll I coll,max ; units to _V and I coll are m 3 /h and W/m 2. From Table A2 we can determine the coef cients a H),b H)and c H). In the speci c case of the pumping system described, the best ts are second-degree polynomials, which can be written as follows: a ˆ a 0 H 2 a 00 H a 000 b ˆ b 0 H 2 b 00 H b 000 c ˆ c 0 H 2 c 00 H c 000 A3 A4 A5 where a 0 ˆ ; a 00 ˆ ; a 000 ˆ ; b 0 ˆ ; b 00 ˆ ; b 000 ˆ ; c 0 ˆ ; c 00 ˆ ; and c 000 ˆ Units for these coef cients can be easily obtained from Equations A3), A4) and A5), consistent with m for H, m 3 /h for water ow and W/m 2 for solar radiation. Table A1. Polynomial coef cients A H), B H)and C H)for Equation A1) H m) A 10 6 m 3 h 1 W 2 ) B 10 3 m 3 h 1 W 1 ) C m 3 /h)

13 PV WATER SUPPLY SYSTEMS 361 Table A2. Polynomial coef cients for pumping system curves H m) a 10 6 m 3 h 1 W 2 m 4 ) b 10 3 m 3 h 1 W 1 m 2 ) c m 3 /h) Let us assume that the system will be installed in a well that can be expressed well plus pipeline)by Equation A6), obtained by eld tests. H ˆ 0 15 _V _V 30 A6 Water head H is in m and _V in m 3 /h. Substituting Equation A6)in A2), we obtain the relationship _V ˆV I coll ), describing the CC of the speci c system installed in the well: _V ˆ Icoll I coll A7 _V is in m 3 /h and I coll in W/m 2. REFERENCES 1. Mayer O, Baumeister A, Festl T. Design, simulation and diagnosis of photovoltaic pumping systems with DASTPVPS. Proceedings of the 13th European Photovoltaic Solar Energy Conference, Nice, 1995; 1915± Vilela OC, and Fraidenraich N. Methodology of analysis of photovoltaic pumping systems. Example for a system installed in the Northeast of Brazil. Proceedings of the 14th European Photovoltaic Solar Energy Conference, Barcelona,1997; 2284± Narvarte L, Lorenzo E, CaamanÄo E. PV pumping analytical design and characteristics of boreholes. Solar Energy, 2000; 68: 49± Hadj Arab A, Chenlo F, Lorenzo E. Analysis of different parameters of PV water pumping systems. Proceedings of the 14th European Photovoltaic Solar Energy Conference, Barcelona, 1997; 2538± Bucciarelli LL Jr. Estimating loss-of-power probabilities of stand-alone photovoltaic solar energy systems. Solar Energy, 1984; 32: 205± Bucciarelli LL Jr. The effect of day-to-day correlation in solar radiation on the probability of loss-of-power in a standalone photovoltaic energy system. Solar Energy 1986; 36: 11± Gordon JM. Optimal sizing of stand-alone photovoltaic solar power systems. Solar Cells 1987; 20: 295± Egido M, Lorenzo E. The sizing of stand alone PV-systems: a review and a proposed new method. Solar Energy Materials and Solar Cells 1992; 26: 51± Notton G, Muselli M, Poggi P, Louche A. Autonomous photovoltaic systems: In uences of some parameters on the sizing: simulation timestep, input and output power pro le. Renewable Energy 1996; 7: 353± Fraidenraich N, Vilela OC. Performance of solar systems with non linear behavior calculated by the utilizability method. Application to PV solar pumps. Solar Energy 2000; 69: 131± Narvarte L, Lorenzo E. LLP and random loads. Proceedings of the 14th European Photovoltaic Solar Energy Conference, Barcelona, 1997; 1048± Aguiar RJ, Collares-Pereira M, Conde JP. Simple procedure for generating sequences of daily radiation values using a library of Markov transition matrices. Solar Energy 1988; 40: 269± Collares-Pereira M, Rabl A. The average distribution of solar radiation. Correlations between diffuse and hemispherical and between hourly and insolations values. Solar Energy 1979; 22: 155± Holland KGT, Huget RG. A probability density function for the clearness index with applications. Solar Energy 1982; 30: 195±209.