a scholastic method

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A stochastic method for stand-alone photovoltaic system sizing

Claudia Valeria Tavora Cabral a, Delly Oliveira Filho a,*, Antonia Sonia Alves C. Diniz b,Jose Helvecio Martins a, Olga Moraes Toledo a, Lauro de Vilhena B. Machado Neto b

a Department of Agricultural Engineering, Federal University of Vicosa, Av. P. H. Rolfs, s/n. 36570-000 Vicosa, Minas Gerais, Brazilb Group of Studies in Energy GREEN Solar, Pontifical Catholic University of Minas Gerais PUC Minas, Rua Dom Jose Gaspar no. 500, Predio 03,

Sala 218 Coracao Eucarstico 30535-610 Belo Horizonte Minas Gerais, Brazil

Received 28 October 2009; received in revised form 5 May 2010; accepted 17 June 2010Available online 16 July 2010

Communicated by: Associate Editor Elias Stefanakos

Abstract

Photovoltaic systems utilize solar energy to generate electrical energy to meet load demands. Optimal sizing of these systems includesthe characterization of solar radiation. Solar radiation at the Earths surface has random characteristics and has been the focus of variousacademic studies. The objective of this study was to stochastically analyze parameters involved in the sizing of photovoltaic generatorsand develop a methodology for sizing of stand-alone photovoltaic systems. Energy storage for isolated systems and solar radiation wereanalyzed stochastically due to their random behavior. For the development of the methodology proposed stochastic analysis were stud-ied including the Markov chain and beta probability density function. The obtained results were compared with those for sizing of standalone using from the Sandia method (deterministic), in which the stochastic model presented more reliable values. Both models presentadvantages and disadvantages, however, the stochastic one is more complex and provides more reliable and realistic results.

2010 Elsevier Ltd. All rights reserved.

Keywords: Deterministic sizing; Stochastic sizing; Solar energy; Stand-alone photovoltaic systems; Probabilistic modeling

1. Introduction

Due to concern about the environment and depletion offossil fuels, more attention has been given to studies thatmay lead to advances in the use of renewable energysources. In countries such as Germany and Spain, wherethere are fixed subsidies for use of renewable energy sources,

especially solar energy, the number of installed solar sys-tems has significantly increased. The main cause of thisgrowth is the reduction in photovoltaic modules cost inrecent years (Nemet, 2006). In Brazil, solar radiation mapsdemonstrate the great potential available for the use of solarenergy, even more so than other countries that presentlyhave many more of these systems installed (Martins et al.,2008a,b).

Accurate sizing of photovoltaic systems is necessary toimprove their reliability and economical feasibility. Sizingof stand-alone photovoltaic systems in isolated areas isan important area of interest of many researchers, present-ing different approaches (Arun et al., 2009; Borowy andSalameh, 1996; Kaushika et al., 2005; Markvart et al.,2006; Posadillo and Lopez Luque, 2008; Shrestha and

Goel, 1998; Sidrach-de-Cardona and Mora Lopez, 1998;Yang et al., 2008).Deterministic methods for system sizing utilize average

seasonal or annual values in their analyses, natural oscilla-tions in solar radiation and load demands are not consid-ered. A deterministic simulation model contains noprobabilistic or random components (Sandia NationalLab, 2009). Due to their random behavior, solar radiationand load demand must be treated as static to simplify theirrepresentation in these methods in which the output is deter-mined from a series of inputs, and their relations to the

0038-092X/$ - see front matter 2010 Elsevier Ltd. All rights reserved.

doi:10.1016/j.solener.2010.06.006

* Corresponding author. Tel.: +55 31 3899 1897; fax: +55 31 3899 2735.E-mail address: [email protected] (D.O. Filho).

www.elsevier.com/locate/solener

Available online at www.sciencedirect.com

Solar Energy 84 (2010) 16281636
http://dx.doi.org/10.1016/j.solener.2010.06.006mailto:[email protected]://dx.doi.org/10.1016/j.solener.2010.06.006http://dx.doi.org/10.1016/j.solener.2010.06.006mailto:[email protected]://dx.doi.org/10.1016/j.solener.2010.06.006


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model are specified, requiring extensive computational time(Maxwell, 1998).

Stochastic methods involve the data treatment, estima-tion of sizing parameters, and data availability in somesenses is considered to be more complex. Many systemshave the property that given the present state, the paststates have no influence on the future. This is called theMarkov property, and the systems which present this prop-erty are known as Markov chains (Ehnberg and Bollen,2005).

The loss of power supply probability (LPSP) determinesthe probability for which the system (photovoltaic genera-tors and batteries) is not able to supply for the load demand.It evaluates the performance of the system for an assumedor known load curve (Abouzahr and Ramakumar, 1991;Yang et al., 2007). The LPSP represents the probability ofpower generated by both the battery and the photovoltaicgenerator to be insufficient to attend the demand and whenthe storage is depleted and its voltage has fallen below theallowed values (Borowy and Salameh, 1996). A zeroLPSP means that load will be always supplied and a LPSPvalue of one indicates that the energy load will never be metby the PV stand-alone generation system. Energy is storedin batteries when the power generated by the photovoltaicgenerators is larger than that of the demand, and is utilized

when the energy provided by the modules is smaller than the

load demand. The instant battery charge ti, depends on thebattery charge in the previous instant, ti1. In order to pre-vent overcharges and complete discharge from the bank ofbatteries, a charge controller should be used. Before over-charges can occur, the control system interrupts chargingand also disconnects the charge before complete discharge.The capacity of the energy storage system is defined in termsof the amount of energy which can be extracted and not thetotal quantity which is stored. Charging efficiency of thebatteries is specified by the manufacturer and discharge effi-ciency was considered to be equal to one in this work.

A comparative study between deterministic and stochas-tic sizing of stand-alone photovoltaic systems was per-formed, analyzing both the stochastic model for solarradiation, as well as the LPSP, with the objective of verify-ing the best method for photovoltaic system modeling. Thiswork also aimed to identify the involved sizing parametersfor photovoltaic systems, considering that calculation ofthe optimal number of photovoltaic generators and batter-ies is based on the LPSP.

2. Methodology

The stochastic sizing program for stand-alone photovol-taic systems requires various variables which are utilized in

both the stochastic and deterministic analyses. For the

Nomenclature

C overall system cost ($)C0 system fixed costs ($)EB(t) energy stored in batteries at time t (kWh)

EB(t-1) energy stored in batteries at time (t1) (kWh)EBmax maximum level of energy permitted in the bat-

teries (Wh)EBmin minimum level of energy permitted in the batter-

ies (Wh)EG(t) energy generated by the photovoltaic array (kWh)EL(t) load demand at time t (Wh)EPV energy generated by the photovoltaic module

(kWh)f(P) output power probability density function at

each of the radiation levelsI circuit output current (A)Imp maximum power current (A)

ISC short-circuit current, (A)KT daily clearness index (decimal)LPS loss of power supply (decimal)LPSP loss of power supply probability (percent)Nbat number of batteries in the bank (decimal)NPV number of photovoltaic modules in the array

(decimal)P maximum power output produced by the photo-

voltaicgenerator at each of the radiation levels (W)

PV photovoltaicPmed average maximum power output from the

photovoltaic generator (W)

RS series resistance (X

)S total solar irradiance on the photovoltaicgenerator plane (W/m2)

Sref reference solar radiation (1000 W/m2)

T solar cell temperature (C)Tamb ambient temperature (C)Tref reference temperature of the solar cell (25 C)V circuit output voltage (V)Vmp maximum power voltage (V)Voc open circuit voltage (V)

Greek letters

a temperature coefficient of short-circuit current

at reference solar irradiance (A/C)am photovoltaic generator cost ($/module)b temperature coefficient of open circuit voltage at

reference solar irradiance (V/C)bb battery cost ($/un)gbat battery charging efficiencyginv inverter efficiency

C.V.T. Cabral et al. / Solar Energy 84 (2010) 16281636 1629


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deterministic analysis, the sizing worksheets of the SandiaNational Laboratory were utilized Sandia National Lab(2009); while in the case of the stochastic analysis, themethodology described by Borowy and Salameh (1994,1996) was adapted. Based on these studies and experimen-tal setup for the stochastic sizing of stand-alone photovol-

taic systems, a comparison could be made between the twomethods, verifying the shortcomings of each of the sizingprocesses.

The methodology applied for the stochastic sizing wasdivided in the following parts:

(i) solar radiation: simulation model for global, diffuse,ground reflected, sky, and direct solar radiation fromthe monthly average daily global radiation data;

(ii) photovoltaic generator: calculation of average powerphotovoltaic modules supply;

(iii) energy storage: calculation of battery charging anddischarging;

(iv) loss of power supply: Calculation of system reliabilityby the loss of power supply probability (LPSP);

(v) sizing: optimal sizing of batteries and photovoltaicmodules based on the economical and reliabilityaspects.

For the proposed stochastic sizing, the following inputparameters are needed:

(i) Equipment characteristic data provided by industry:batteries, photovoltaic modules and inverters;

(ii) climate characterization: average temperature, solar

radiation; and(iii) load behavior.

2.1. Solar radiation

The simulation model for global solar radiation isintroduced in this section. Borowy and Salameh (1996)used a series of radiation data, with data recorded atall hours of all days for a period of 30 years. Since it isdifficult to obtain such extensive meteorological data inBrazil, monthly average daily global radiation valuesderived from typical meteorological year over a period

of 20 years provided by the electrical energy Brazilianutility CEMIG were used to generate hourly data to beused in the simulation.

Estimation of the following parameters was doneaccording to Iqbal (1983) and (Duffie and Beckman(2006): (i) the hourly global radiation on a inclined surface;(ii) the hourly global radiation on a horizontal surface; (iii)the daily global radiation on a horizontal surface; (iv)hourly diffuse radiation on a horizontal surface calculatedfrom hourly global radiation and extraterrestrial hourlyradiation; (v) hourly sky diffuse radiation based on ananisotropic model; and (vi) hourly ground reflected diffuseradiation based on anisotropic reflection model.

The daily clearness index, KT, was estimated based onMarkov transition matrices (Lorenzo, 1994). This estimatewas made using historic data from many meteorologicalstations spread around the world.

Each Markov matrix is associated with a range of KTvalues defined by their maximum and minimum values.

Each of these intervals is divided into 10 subintervals, withthe same number of rows and columns. To generate thevalues of KT, from the Markov chains, the monthly clear-ness index was required as an input. For this purpose, cal-culations were performed from the monthly average dailyglobal radiation.

The average power output from the photovoltaic mod-ules was estimated from the solar radiation data. The betaprobability distribution function was used to describe theradiation behavior, since it represents the best fit amongthose studied for synthetic data.

2.2. Photovoltaic generator

For calculation of the average power output, it was con-sidered that the system has a maximum power pointtracker. The manufactures provide data of voltage and cur-rent for the photovoltaic modules at the maximum powerpoint, at reference temperature and radiation.

The power produced by the photovoltaic module at eachradiation level is the product of output voltage and current(Salameh et al., 1995). In the calculation of the outputpower from the photovoltaic modules, it was consideredthat the maximum power would be utilized. Average poweroutput shown in Eq. (1), is the maximum power produced

by the photovoltaic generator at each radiation level mul-tiplied by the power probability density function integratedfor the time interval considered.

PAvg

ZPmax;maxPmin;max

P fP dP 1

The photovoltaic generator output current calculationwas done considering the mathematic model used todescribe the solar cell based on its one-diode model equiv-alent circuit (Villalva et al., 2009).

Based on the model, an equation could be deduced forthe characterization of the photovoltaic generator perfor-

mance, relating voltage, current, radiation and temperature(Salameh et al., 1995). Eq. (2) shows the current producedby the photovoltaic generator. Eqs. (3)(8) provide furtherdetails of the relationships among the variables and param-eters that appear in Eq. (2).

I ISC 1 C1 expV DV

C2 Voc

1

DI 2

C2 Vmp=Voc 1

ln1 Imp=ISC3

C1 1 Imp

ISC

exp

Vmp

C2 Voc

4

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DI aS

Sref

DT

S

Sref 1

ISC 5

DV b DT RS DI 6

DT T Tref 7

T Tamb 0:02S 8

Using Eq. (1), the monthly average hourly power outputfrom the photovoltaic module could be calculated. Thestand-alone photovoltaic system sizing considers not onlythe power generated on the photovoltaic modules but theenergy storage capacity in the batteries.

Eq. (9) shows the energy generated by the photovoltaicarray (EG(t)) at time t.

EGt NPV EPVt 9

2.3. Energy storage

Since it was assumed that the battery charging efficiencyis equal to that reported by the manufacturer and the dis-charge efficiency was equal to one (100%), two possibilitieswere considered to express the energy storage in the batteryduring a period of t hours: (i) the amount of energy gener-ated by the photovoltaic generator exceeds the loaddemand, and (ii) the energy demands of the load aregreater than the available amount of generated energy. Inthe second case, the battery will be discharged to meetthe necessary load demands.

The inverter was evaluated in terms of peak loaddemand, and its efficiency, ginv, is a function of the DC

input voltage and the load type. This methodology consid-ers the variation of the inverter efficiency according to theload demand.

The battery capacity is defined as a function of theamount of energy that can be extracted and not the amountstored in a given time interval (Borowy and Salameh,1996). The amount of energy stored in the battery duringthe period of t hours, should be between the maximumand minimum allowed for the energy levels. This procedureprotects the battery from damage and avoids drastic reduc-tion of its life cycle.

2.3.1. Case 1: charging of the batteries

In this case, energy generated by the photovoltaic arrayexceeds that of the load demand. From Eq. (10) it canobserved that the amount of stored energy stored in thebatteries is the excess energy during the period of t hours(Borowy and Salameh, 1996).

EBt EBt1 EGt ELt

ginv

gbat 10

2.3.2. Case 2: discharging of the batteries

In this case, the load demand is greater than the avail-able amount of generated energy. It is shown that the

amount of energy discharge from the batteries can be

calculated from Eq. (11), since a portion of the storedenergy is required to meet the energy needs of the load(Borowy and Salameh, 1996).

EBt EBt1 ELt

ginv EGt

11

The amount of energy stored in the batteries at time t issubject to restriction (Borowy and Salameh, 1996) given byEq. (12). This restriction means that the batteries will notbe overcharged or completely discharged to prevent themfrom being damaged.

EBmin 6 EBt 6 EBmax 12

After the calculations of the solar radiation, power gen-erated, and energy storage, reliability should be calculatedin order to evaluate the loss of power supply.

2.4. Loss of power supply

When energy generated by the photovoltaic modulesand energy stored in the batteries is insufficient to meetthe demands of the system at time t, this deficit is knownas the loss of power supply (LPS) and is expressed as Eq.(13), (Borowy and Salameh, 1996). The loss of power sup-ply probability (LPSP) function was then determined for agiven time frame as calculated in Eq. (14).

LPSt ELt EGt EBt1 EBminginv 13

LPSP

PTt1LPSt

PT

t1ELt14

The number of photovoltaic modules and batteriesneeded for a determined LPSP, for energy generation andstorage, can now be analyzed for each hour of a typicalday in each month based on previous calculations. Stochas-tic sizing includes economical aspects that can be per-formed considered the desired LPSP.

2.5. Sizing

For a given LPSP there are several suitable combina-tions of number of batteries and photovoltaic modules.The procedure is schematically shown in Fig. 1. The simu-

lation calculated the monthly and annual LPSP for a givencombination of number of batteries (Nbat) and photovol-taic modules (NPV). Optimal sizing is a function of eco-nomical analysis.

It is necessary to determine the best combination of Nbatand NPV while maintaining a minimum cost. The functionthat describes this purpose is given by (Borowy andSalameh, 1996):

C am NPV bb Nbat C0 15

The optimal solution of Eq. (15) is given by:

@NPV

@Nbat

bb

am16



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The minimum cost solution was calculated based on thelife cycle analysis of the present value. Replacement of thebatteries was planned when their lives ended.

2.6. Validation

Validation was performed for each of the separate partsof the stochastically sized stand-alone photovoltaic system.This was done at the Green Laboratory at the PontificalCatholic University of Minas Gerais (PUCMINAS) and

at the Energy Laboratory of the Agricultural Engineering

Department of the Federal University of Vicosa (UFV),both in Brazil.

A stand-alone photovoltaic system prototype consistedof six photovoltaic modules, six batteries, one load control-ler and one sinusoidal inverter and the load. The load anddemand characteristics were: (i) load: eight 32 W fluores-cent lamps and four 16 W fluorescent lamps; (ii) operation:5 days a week (MondayFriday), daily utilization of 4 h(6:0010:00 pm); and (iii) voltage: 24 V. The stand-alonephotovoltaic system was assembled at the Green Labora-tory, at latitude of 195505700 South and longitude of

43560

3200

West, with surface albedo of 74%.

Npv


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The photovoltaic system considered had the followingcharacteristics: (i) photovoltaic modules with 100 W and12 V nominal voltage, each; (ii) 85 Ah batteries nominalcapacity, 0.95 discharge efficiency, maximum depth of dis-charge (DOD) of 0.8 and 12 V nominal voltage; (iii)1000 W inverter with a nominal 24 V DC, 120 V AC,

60 Hz and its efficiency was allowed to vary with the loadsince its characteristic efficiency curve was provided.Attached to the photovoltaic modules, two pyranome-

ters were installed, one at the horizontal position and oneat the modules plane. The global radiation data were mea-sured each minute and compared with the simulation data.

Other system parameters measured were: output voltageand current of the modules, batteries, and inverter. Thesedata acquisitions were done using an automated system(PCI-DAS 1001 board) and processed by the Labview

program.In order to compare the simulated and real data, a sta-

tistic analysis was performed by using the root mean square

error, mean bias error and correlation coefficient.

3. Results and discussion

A comparison between the average measured and simu-lated monthly average daily global radiation on an inclinedplane (module plane) was performed for the month of Sep-tember 2005, located at Belo Horizonte, Brazil (Fig. 2). Thestatistic analysis of these data showed: (i) mean bias errorof 0.044; (ii) root mean square error of 0.118 and (iii) cor-relation coefficient of 0.996.

Another evaluation was the assessment of measured andsimulated output power for the photovoltaic modules(Fig. 3). The statistic analysis of these data was also per-formed and the results were: (i) mean bias error of0.034; (ii) root mean square error of 0.070 and (iii) corre-lation coefficient of 0.998.

The monthly average hourly storage energy in the bat-teries was measured for one month and it was supposedthat the batteries started completely full and the charge effi-ciency was considered to be 0.9 and the discharge efficiencyis 1.0. Fig. 4 shows the comparison among measured andsimulated results for monthly average hourly storageenergy. The analysis shows a: (i) mean bias error of0.018; (ii) root mean square error of 0.141 and (iii) corre-lation coefficient of 0.965.

Once sure that the measured and simulated data pre-sented a correlation greater than 0.960, which is consideredacceptable, a simulation was performed for the prototypestochastic sizing of a stand-alone photovoltaic system tobe installed at the research laboratory.

The Sandia sizing method (deterministic method) calcu-lated a system demand of 8 PV modules and eight batteriesfor the considered load. For the same load, the stochasticsizing methodology proposed was simulated with the resultsof the deterministic sizing. The simulation shows the deter-

ministic method to be oversized. This can be verified due to

the fact that the batteries were nearly always fully chargedwhen simulating a complete year. Fig. 5 presents the simu-lation for operation (monthly average hourly) of the photo-voltaic generators and batteries along 1 year. The green andred lines on the graph of energy stored in the batteries arethe maximum and minimum energy permitted in the batterybank, respectively. The calculated LPSP for this number ofphotovoltaic generators and batteries is 0.00% in this case,confirming over sizing of the deterministic method. If aLPSP of 0.48% was considered (loss of energy for 2 daysper year), the number of modules and batteries requiredwere 5 and 3, respectively. This demonstrates that the sys-tem, using the deterministic sizing, was truly is oversized.Even if considering an LPSP of 0%, the number of photo-voltaic generators and batteries could be as low as 5 and4, respectively. These figures are the results found in the sto-chastic method.

Fig. 6 shows the energy storage system (four batteries)

and photovoltaic array (five PV modules) functioning over

Fig. 2. Measured and simulated data of monthly average hourlyirradiance incident on an inclined plane on September 2005.

Fig. 3. Measured and simulated monthly average hourly data of powersupplied by the photovoltaic generator.



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the period of 1 year. It was observed that the batteries didnot suffer complete discharge and the load was stillattended during the entire simulation period (LPSP = 0%).

As one may observe, the designer could choose theLPSP that he/she desires and the calculation will informthe optimal number of batteries and photovoltaic genera-tors with minimum cost. This calculation may also be donein order to calculate the LPSP for a given number of bat-teries and modules chosen by the designer.

Analysis of the involved sizing parameters for the stand-alone photovoltaic system, at both the deterministic level(Sandia method) and stochastic level (developed model),made this comparison possible. Various observationsshould be made, including:

(a) Deterministic models used simple analytical equa-tions to predict operation of the photovoltaic sys-tems, that is, it does not consider their stochasticbehavior. Their reliability is limited to the consis-tency of the hourly, daily or monthly data used, pre-senting equations which are simpler to work withand simulate. On the other hand, stochastic methodsare more complex and involve statistical processingwhich requires greater computational time for theirsimulation but are more reliable. In general, solarradiation levels, load demand and battery state ofcharge are better characterized by a stochasticdescription.

(b) Often when working with stochastic models, a seriesof historical data of the studied event are neededand many times are not available at the majority ofmeteorological stations, depending on their complex-ity. This problem is not encountered with determinis-tic models since they use average values.

(c) Deterministic methods generally consider the capac-ity of the storage system as determined by the consec-utive number of days in which the demand can be metusing only the storage system, without consideringpossible provisions from the photovoltaic generators.In stochastic methods, all possible uses for energy

Fig. 4. Simulated and measured data for the energy storage in the batterybank during 1 month, considering a charge efficiency of 0.90 and adischarge efficiency of 1.00.

Fig. 5. Simulation of system operation (8 PV modules and eight batteries) during a period of 1 year for the analyzed stand-alone photovoltaic system: (a)

PV power supply (W); (0b) available energy in the batteries (Wh).



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provided by the generator are considered for thecharging of the storage system and direct use by theconsumer.

(d) For stochastic sizing, battery charge and solar radia-tion for each state of the system are analyzed, as inthe model which used the Markov chain, developedby Safie (1989). This model identifies the states in

which the load demands are not satisfied. In deter-ministic sizing, battery performance is not analyzedduring operation.

(e) Stochastic sizing predicts system operation betterthan the deterministic method, being more realisticand, as a consequence, more often economicallyfeasible.

4. Conclusions

This work portrayed the study of parameters involved inthe stochastic sizing of stand-alone photovoltaic systems. Astochastic program was developed for the calculation ofphotovoltaic generators and batteries necessary for a givenload. Comparisons were made between the results calculatedin this study and those of Sandia deterministic method. Itwas verified that the best method for photovoltaic sizingwas the stochastic method, since the generated results aremore economical and better forecasted. It was demonstratedthat for a determined loss of power supply probability, dif-ferent combinations of modules and batteries can be foundto meet the load demand. Both methods showed advantagesand disadvantages, forcing the designer to choose a moreappropriate model for the available knowledge, data, timeand equipment. Preference is given to the stochastic model

due to its ability to better portray the operation of the pho-

tovoltaic system. The number of modules and batteries islinked to the location in which the system will be installed,load profile and desired reliability.

Acknowledgments

The authors would like to thank Energy Company of

Minas Gerais, Brazil, (CEMIG) and Brazilian NationalElectrical Energy Agency (ANEEL) for their financialsupport.

References

Abouzahr, I., Ramakumar, R., 1991. Loss of power supply probability ofstand-alone photovoltaic systems: a closed form solution approach.Energy Conversion, IEEE Transactions on Energy Conversion 6, 111.

Arun, P., Banerjee, R., Bandyopadhyay, S., 2009. Optimum sizing ofphotovoltaic battery systems incorporating uncertainty through designspace approach. Solar Energy 83, 10131025.

Borowy, B.S., Salameh, Z.M., 1994. Optimum photovoltaic array size for

a hybrid wind/PV system. IEEE Transactions on Energy Conversion 9,482488.

Borowy, B.S., Salameh, Z.M., 1996. Methodology for optimally sizingthe combination of a battery bank and PV array in a wind/PVhybrid system. IEEE Transactions on Energy Conversion 11, 367375.

Duffie, J.A., Beckman, W.A., 2006. Solar Engineering of Thermal Process,third ed. Wiley, John Wiley & Sons.

Ehnberg, J.S.G., Bollen, M.H.J., 2005. Simulation of global solarradiation based on cloud observations. Solar Energy 78, 157162.

Iqbal, M., 1983. An introduction to solar radiation. Department ofMechanical Engineering, The University of British Columbia. Aca-demic Press, Vancouver, Canada, pp. 390.

Kaushika, N.D., Gautam, N.K., Kaushik, K., 2005. Simulation model forsizing of stand-alone solar PV system with interconnected array. Solar

Energy Materials and Solar Cells 85, 499519.

Fig. 6. Simulation of system operation (five modules and four batteries) during the period of 1 year for the analyzed stand-alone photovoltaic system: (a)PV power supply (W); (b) available energy in the batteries (Wh).



9/9

Lorenzo, E., 1994. Electricidad solar ingenieria de los sistemas fotovol-taicos, first ed. Instituto de Energia Solar, Universidade Politecnica deMadrid, pp. 338.

Markvart, T., Fragaki, A., Ross, J.N., 2006. PV system sizing usingobserved time series of solar radiation. Solar Energy 80, 4650.

Martins, F.R., Pereira, E.B., Silva, S.A.B., Abreu, S.L., Colle, S., 2008a.Solar energy scenarios in Brazil, Part one: resource assessment. EnergyPolicy 36, 28532864.

Martins, F.R., Ruther, R., Pereira, E.B., Abreu, S.L., 2008b. Solar energyscenarios in Brazil Part two: photovoltaics applications. Energy Policy36, 28652877.

Maxwell, E.L., 1998. METSTAT the solar radiation model used in theproduction of the national solar radiation data base (NSRDB). SolarEnergy 62, 263279.

Nemet, G.F., 2006. Beyond the learning curve: factors influencing costreductions in photovoltaic. Energy Policy 34, 32183232.

Posadillo, R., Lopez Luque, R., 2008. A sizing method for stand-alone PVinstallations with variable demand. Renewable Energy 33, 10491055.

Safie, F.M., 1989. Probabilistic modeling of solar power systems. In:Proceedings of the Reliability and Maintainability Symposium, 1989,pp. 425430.

Salameh, Z.M., Borowy, B.S., Amin, A.R.A., 1995. Photovoltaic module-site matching based on the capacity factors. IEEE Transactions onEnergy Conversion 10, 326332.

Sandia National Lab, 2009. General Sizing Instructions, Sandia Lab, S.N.,National Lab.

Shrestha, G.B., Goel, L., 1998. A study on optimal sizing of stand-alonephotovoltaic stations. IEEE Transactions on Energy Conversion 13,373378.

Sidrach-de-Cardona, M., Mora Lopez, L., 1998. A simple model for sizingstand-alone photovoltaic systems. Solar Energy Materials and SolarCells 55, 199214.

Villalva, M.G., Gazoli, J.R., Filho, E.R., 2009. Comprehensive approachto modeling and simulation of photovoltaic arrays. Power Electronics,IEEE Transactions on Energy Conversion 24, 11981208.

Yang, H., Lu, L., Zhou, W., 2007. A novel optimization sizing model forhybrid solar-wind power generation system. Solar Energy 81, 7684.

Yang, H., Zhou, W., Lu, L., Fang, Z., 2008. Optimal sizing method forstand-alone hybrid solar-wind system with LPSP technology by usinggenetic algorithm. Solar Energy 82, 354367.


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