optimum sizing of photovoltaic battery systems incorporating uncertainty through design space...

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Solar Energy 83 (2009) 1013–1025

Optimum sizing of photovoltaic battery systemsincorporating uncertainty through design space approach

P. Arun, Rangan Banerjee, Santanu Bandyopadhyay *

Department of Energy Science and Engineering, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India

Received 11 October 2008; received in revised form 18 December 2008; accepted 8 January 2009Available online 11 February 2009

Communicated by: Associate editor Arturo Morales-Acevedo

Abstract

Photovoltaic-battery system is an option for decentralized power generation for isolated locations receiving abundant sunshine. Amethodology for the optimum sizing of photovoltaic-battery system for remote electrification incorporating the uncertainty associatedwith solar insolation is proposed in this paper. The proposed methodology is based on the design space approach involving a time seriessimulation of the entire system. The design space approach was originally proposed for sizing of the system with deterministic resourceand demand. In the present paper, chance constrained programming approach has been utilized for incorporating the resource uncer-tainty in the system sizing and the concept of design space is extended to incorporate resource uncertainty. The set of all feasible designconfigurations is represented by a sizing curve. The sizing curve for a given confidence level, connects the combinations of the photovol-taic array ratings and the corresponding minimum battery capacities capable of meeting the specified load, plotted on an array rating vs.battery capacity diagram.The methodology is validated using a sequential Monte Carlo simulation approach with illustrative examples.It is shown that for the case of constant coefficient of variation of solar insolation, the set of sizing curves for different confidence levelsmay be represented by a generalized curve. Selection of optimum system configuration for different reliability levels based on the min-imum cost of energy is also presented. The effect of ambient temperature on sizing a stand-alone photovoltaic-battery system is also illus-trated through a representative example.� 2009 Elsevier Ltd. All rights reserved.

Keywords: Photovoltaic-battery system; Optimum system sizing; Chance constrained programming; Sizing curve; Design space; Uncertainty

1. Introduction

Photovoltaic-battery system is a sustainable option forisolated electrical power generation for locations receivingabundant sunshine. The design of such a remote electrifica-tion unit requires estimation of the capacities of the photo-

0038-092X/$ - see front matter � 2009 Elsevier Ltd. All rights reserved.

doi:10.1016/j.solener.2009.01.003

Abbreviations: ACC, annualized capital cost, Rs./y; ALCC, annualized lifecycle cost, Rs./y; AOM, annual operation and maintenance cost, Rs./y;COE, cost of energy, Rs./kWh; CRF, capital recovery factor; DOD, depthof discharge; LOLE, loss of load expectation; LOLP, loss of load prob-ability; Rs., Indian rupees (Rs. 45 � US$1 in 2008).

* Corresponding author. Tel.: +91 22 25767894; fax: +91 22 25726875.E-mail address: [email protected] (S. Bandyopadhyay).

voltaic generator and the battery bank to satisfy a givendemand. Incorporation of the variability associated withthe solar insolation is important in the system sizing as itaffects the performance and reliability of the overall sys-tem. In this paper, a methodology is proposed for the opti-mum sizing of a photovoltaic-battery system for apredefined system reliability level. It enables the construc-tion of a sizing curve. The sizing curve connects the combi-nations of the photovoltaic array ratings and thecorresponding minimum battery capacities capable ofmeeting the specified load, plotted on an array rating vs.battery capacity diagram. This helps in identifying theentire range of feasible system configurations or the designspace satisfying a specified reliability level. Generation of

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Nomenclature

A photovoltaic array area, m2

A* array area variable for the generalized sizingcurve, m2

Amin minimum photovoltaic array area, m2

B battery bank capacity, WhBmin minimum battery energy value, WhBr maximum battery energy value, WhC0 capital cost, Rs.d discount rateD demand, WDactual minimum deterministic demand considered in

the chance constraint, WEdem energy delivered, Whf net charging/discharging efficiencyh loss of load duration in the estimation of hourly

confidence levelH total duration considered in the estimation of

hourly confidence levelIg global solar radiation intensity, W/m2

Id diffuse radiation intensity, W/m2

IT solar radiation intensity on tilted surface, W/m2

n life, yearsP power generated by the excess power generated

by photovoltaic array, WPdu photovoltaic array, Wrb tilt factor for hourly beam radiationt loss of load duration in the estimation of LOLE

T maximum time considered in the time seriesTa ambient temperature, K

Tc cell temperature, KTr reference cell temperature, KUPV heat transfer coefficient of the module, W/m2 KxP coefficient of variation of hourly solar insolation

(and power)za Standard normal variate with a cumulative

probability of a

Greek symbols

a confidence levelah hourly confidence level(as) transmissivity–absorptivity coefficient of the

moduleb tilt of photovoltaic arraybT temperature coefficient of the photovoltaic ar-

ray, K–1

g0 photovoltaic system efficiencyg0r photovoltaic system efficiency at reference cell

temperaturegc charging efficiencygd discharging efficiencylP mean photovoltaic power, Wlp mean of loss of load expectationq reflectivity of ground surfacerP standard deviation of photovoltaic power, Wrp standard deviation of loss of load expectation� coefficient of variation of loss of load expecta-

tion

1014 P. Arun et al. / Solar Energy 83 (2009) 1013–1025

the design space helps in selecting and optimizing anappropriate system configuration based on the desiredobjective. Kulkarni et al. (2007, 2008, 2009) introducedthe concept of design space for optimum sizing of solarhot water systems. The representation of the design spacehas been illustrated for diesel generator-battery systems(Arun et al., 2008), photovoltaic-battery systems (Arunet al., 2007), as well as wind-PV hybrid systems (Royet al., 2007). The objective of the present work is to includethe uncertainty associated with solar insolation in design-ing a photovoltaic-battery system. It is presented as anextension to the design space methodology. System optimi-zation is also illustrated based on the criterion of minimumcost of energy for a given reliability level.

Sizing of a photovoltaic-battery system involves thedetermination of the array rating (in terms of array areaor the peak power, Wp) required and the capacity of thebattery bank. The existing methods range from simple cor-relations to detailed mathematical models. Broadly thesemethods can be grouped into deterministic and stochasticapproaches. In the deterministic approach, the uncertaintyassociated with the solar insolation is not considered. Sim-

ple procedures to obtain an independent design of the pho-tovoltaic array and battery bank have been proposed(Bhuiyan and Asgar, 2003; Sandia National Lab, 2008).Correlations between the array capacity and the batterysize have been reported (Barra et al., 1984; Bartoli et al.,1984). Egido and Lorenzo (1992) proposed a correlationbetween the normalized array capacity, storage size andloss of load probability (LOLP) for different locations inSpain. Such a correlation is site specific. Tsalides andThanailakis (1986) used a simulation-based design ofstand-alone photovoltaic systems for a remote village innorthern Greece. Notton et al. (1996) studied the effect oftime step, input and output power profile on the sizing ofautonomous photovoltaic systems based on a simulationprocedure. Sidrach-de-Cardona and Lopez (1998) pro-posed a simple simulation model for sizing photovoltaicsystems. Techno-economic analysis of photovoltaic sys-tems for various locations in Greece was reported by Kal-dellis (2004). Hontoria et al. (2005) used an artificial neuralnetwork based methodology to obtain the sizing curve forstand-alone photovoltaic systems. Markvart et al. (2006)presented the sizing curve using superposition of contribu-

Photovoltaic array

Battery bank

Charge controller Inverter

AC load

DC bus AC bus

Dump load

Fig. 1. Schematic of a photovoltaic-battery system for remoteelectrification.

P. Arun et al. / Solar Energy 83 (2009) 1013–1025 1015

tions from individual climatic cycles of low daily solar radi-ation for a location in the south east of England.

Probabilistic approaches of sizing photovoltaic systemaccount the effect of the solar insolation variability in thesystem design. Bucciarelli (1984) proposed a sizing methodtreating storage energy variation as a random walk. Theprobability density for daily increment or decrement ofstorage level was approximated by a two-event probabilitydistribution. The method was further extended to accountfor the effect of correlation between day to day insolationvalues (Bucciarelli, 1986). Bucciarelli’s method was modi-fied by Gordon (1987) and Bagul et al. (1996) where thestorage energy transitions were approximated by three-events as compared to the two-event probability. Bucciar-elli’s (1984) method and the three-event probabilityapproaches (Gordon, 1987; Bagul et al., 1996) relate thearray size and battery capacity for constant loss of loadprobability (LOLP) through closed form equations. How-ever, it is observed that for normalized array size greaterthan about 1.2, both array and battery size tends toincrease for the same LOLP, which restricts its usabilityto a specific range (Egido and Lorenzo, 1992). Groumposand Papageorgiou (1987) proposed an optimal sizingscheme for stand-alone photovoltaic battery system for agiven value of LOLP considering the daily insolation andits standard deviation and had illustrated the method fora village power system in Arizona, USA.

Several software tools are also available for the design ofphotovoltaic-battery systems. RETScreen (CETC, 2004) isuseful for preliminary system sizing of isolated power sys-tems. Hybrid2 (RERL, 2007) developed by NationalRenewable Energy Laboratory, USA is capable of per-forming the detailed time series simulation of photovoltaicpower systems. The Hybrid Optimization Model for Elec-tric Renewables or HOMER, in short (NREL, 2007), useshourly simulations for arriving at optimal system sizing ofisolated power systems. However, most of the availablesoftware tools identify and simulate a single design option.They generally do not generate and evaluate the range ofpossible design options. Also, the effect of non-linearityin the system modeling and the randomness associated withthe design variables needs to be accounted in such tools.

Simulation models are data intensive but provide aframework for performing parametric studies and systemoptimization. It is beneficial to incorporate the uncertaintyeffects on design variables in such a framework. Chanceconstrained programming is a useful tool applicable forstudying mathematical models with random variables.The approach was introduced by Charnes and Cooper(1959) based on the basic concept of constraints in themathematical model complying with specified values ofprobability. The methodology has been applied to dealwith uncertainty in several disciplines of engineering. Ithas been used to account the effect of uncertainty in prob-lems of structural optimization (Rao, 1980), hydro systemdesign (Changchit and Terrell, 1993; Azaiez et al., 2005),financial risk management (Aseeri and Bagajewicz, 2004),

process engineering (Li et al., 2008) etc. In this paper,chance constrained programming approach is combinedwith the design space approach to design a stand-alonephotovoltaic system.

In this paper, a generic methodology is presented to gen-erate the design space for different reliability levels for pho-tovoltaic-battery systems using chance constrainedprogramming. It is developed as an extension to the deter-ministic sizing approach based on the time series simula-tion of the entire system. Comparisons with existingmethods are also given for the deterministic and stochasticapproaches. The system reliability is validated throughsequential Monte Carlo simulation of the system. It isobserved that the sizing offered by the proposed schemeoffers a conservative system design. Under the simplifiedassumption of a constant coefficient of variation of thesolar insolation, it is shown that a single generalized sizingcurve may be obtained. It enables system sizing for variousconfidence levels using a single curve. The proposed meth-odology is demonstrated through illustrative examples.

2. Design space generation with deterministic approach

The system sizing methodology for photovoltaic-batterysystems for a given load and solar insolation profile follow-ing a deterministic approach is discussed in this section.The schematic of a photovoltaic-battery system configura-tion is shown in Fig. 1. The components include photovol-taic modules, battery bank, charge controller and inverter.The proposed methodology employs a time series simula-tion based on the energy balance of the overall system.The power generated by the photovoltaic array at anygiven time t is given by:

P ðtÞ ¼ g0AIT ð1Þ

where g0 is the photovoltaic system efficiency, A is the totalarray area (m2) and IT the total instantaneous radiationincident on the array (W/m2). Efficiency of a photovoltaicarray depends on the material of construction, cell config-uration, processing technology, operating temperature, so-


lar insolation, operating current and voltage of the array,etc. Solar radiation incident on the array is calculated usingthe equation (Sukhatme, 1997):

IT ¼ ðIg � IdÞrb þ Id1þ cos b

2

� �þ Ig

1� cos b2

� �q ð2Þ

The net power flow across the storage is accounted con-sidering the efficiencies for the power conversion duringcharging and discharging processes. The rate of changeof energy stored (dQB=dt) in the battery bank is propor-tional to the net power generated. The power generated isrepresented as the difference between the power generatedby the photovoltaic array (P) and the power supplied tothe load (D).

dQB

dt¼ ðP ðtÞ � DðtÞÞf ðtÞ ð3Þ

where f(t) represents the efficiencies associated with thecharging and discharging processes at the time step.

f ðtÞ ¼ gc whenever P ðtÞP DðtÞ

f ðtÞ ¼ 1

gdwhenever P ðtÞ < DðtÞ

ð4Þ

These equations relate the rate of change of energy of thestorage with the input power, demand power and thepower conversion efficiencies during charging/discharging.The inverter efficiency is accounted in the estimation ofthe demand. The change in stored energy over a time per-iod of Dt, may be expressed as follows:

QBðt þ DtÞ ¼ QBðtÞ þZ tþDt

tðPðtÞ � DðtÞÞf ðtÞdt ð5Þ

For a relatively small time period, Eq. (5) may be approx-imated as:

QBðt þ DtÞ ¼ QBðtÞ þ ðP ðtÞ � DðtÞÞf ðtÞDt ð6Þ

During the system operation over the time period Dt,whenever the energy supplied by the array is greater thanthe demand (P(t) > D(t)), the surplus energy is used tocharge the battery bank. Depending on the solar radiationinput, if the energy delivered by the photovoltaic array islower than the load, the battery meets the deficit in energydemand. The load can be met if the battery has not reachedits depth of discharge and in that case, the stored chemicalenergy is converted into electrical energy. It is assumed thatthe charging and discharging takes place with a constantefficiency. The self-discharge loss for the battery is assumedto be negligible. The minimum array area required and thecorresponding storage capacity for meeting the specifiedload may be obtained by solving Eq. (6) over the entireduration. To solve Eq. (6), required inputs are the expectedload curve and the solar radiation profile over a specifictime interval (e.g., daily, seasonal or yearly data), photo-voltaic array conversion efficiency and the nominal charg-ing and discharging efficiencies of the battery bank-converter system. In the generalized methodology, to

obtain the minimum array area, a numerical search is per-formed that satisfies the energy balance (6) with the follow-ing additional constraints:

QBðtÞP 0 8t ð7ÞQBðt ¼ 0Þ ¼ QBðt ¼ T Þ ð8Þ

Eq. (7) ensures that the battery energy level is always non-negative, while Eq. (8) represents the repeatability of thebattery state of energy over the time period. The repeatabil-ity condition implies that there is no net energy supplied toor drawn from the battery bank over the time horizon. It isassumed that the load is recurring in the same pattern aftertime T. The required battery bank capacity (Br) is obtainedas:

Br ¼maxfQBðtÞg

DODð9Þ

where DOD is the allowable depth of discharge of thebattery.

Hence the value of the minimum array area (A = Amin)and the corresponding capacity of the battery bank (Br)may be obtained. Any array area, higher than the mini-mum, is capable of supplying the load. However, thecapacity of the battery bank would be lower due to theincrease in power input. It is important from the designers’perspective to identify all the feasible combinations for thearray area and the corresponding storage capacity. Toidentify the minimum storage capacity for a given arrayarea, the energy balance Eq. (3) may be modified asfollows:

dQB

dt¼ ðP ðtÞ � P duðtÞ � DðtÞÞf ðtÞ ð10Þ

Here Pdu(t) represents the excess power from the array thatto be dumped. In reality, dump loads are not usually in-stalled for stand-alone PV systems. It mathematically rep-resents the excess power for the photovoltaic array whichis not really generated as the charge regulator disconnectsthe photovoltaic generator from the DC bus. It may benoted that Pdu(t) is a non-negative variable introduced inEq. (3) when the array area is greater than the minimumrequired.

The simulations to obtain the minimum storage capacityare carried out for different values of the array area(A > Amin). For each value of array area (A) considered,the corresponding minimum battery bank capacity isobtained by minimizing the required storage capacity Eq.(9). The optimization variables are the initial batteryenergy, QB(t = 0) and the excess power Pdu(t). The combi-nations of the different array areas and the correspondingminimum storage capacities may be plotted on an arrayarea (or array rating) vs. battery bank capacity diagramwhich is defined as the sizing curve of the system. The siz-ing curve represents the minimum storage capacityrequired for a given array rating. The sizing curve dividesthe entire space into feasible and infeasible regions. Theregion above the sizing curve represents the feasible region

Array rating (Wp)

Bat

tery

cap

acity

(W

h)

Infeasible design region

Minimum arraysize

Feasible design region(design space)

sizing curve

Battery capacity corresponding to the minimum array size

Fig. 2. Typical sizing curve and design space for a photovoltaic batterysystem.

0

200

400

600

800

1 3 5 7 9 11 13 15 17 19 21 23

Time(hour of the day)

Inso

latio

n (W

/m2 )

Fig. 3. Variation in the mean (solid line) and standard deviation (dashedline) of the hourly solar insolation on the array surface for an averagedday.

0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8

Array rating (kWp)

Bat

tery

cap

acity

(kW

h)

Sandia design

Based on one year

Based on an averaged day

Fig. 4. Sizing curves and design space for example 1.

Table 1Input parameters for the system sizing and optimization.

Photovoltaic array efficiency, % 10Inverter efficiency, % 90Net charging efficiency, % 85Net discharging efficiency, % 85Battery depth of discharge, % 70Temperature coefficient of module, K–1 0.0044Reference temperature, �C 25Heat transfer coefficient, W/m2K 28.8Transmissivity–absorptivity coefficient 0.9


as any combination of array area and battery capacity rep-resents a feasible design option. The entire feasible regionincluding the sizing curve is the design space for a givenproblem. A typical sizing curve and associated design spaceare shown in Fig. 2.

2.1. Example 1: deterministic system sizing

The system sizing methodology is illustrated with anexample in this section. For a systematic comparison, thesizing results reported by Notton et al. (1996) have beenutilized. The proposed methodology helps in identifyingthe minimum system configurations as well as the set ofall feasible design configurations, called the design space.

A constant load of 42 W prevailing over the entire day isconsidered for the analysis (Notton et al., 1996). The sizingcurves are generated for Ajaccio (41�550N, 8�390E) usingthe same input parameters as reported by Notton et al.(1996). For illustration, the sizing curve is generated ini-tially based on a representative day. The data for the rep-resentative day is obtained as the averaged value of the18 years hourly data recorded for the site. The variationof solar insolation in terms of the hourly mean and stan-dard deviation for the averaged day is shown in Fig. 3. Fur-thermore, the sizing curve is also generated for a period ofone year to account and illustrate the hour by hour varia-tion and seasonal effects in the solar insolation. Followingthe proposed procedure, the system sizing curves are gener-ated and presented in Fig. 4. The various input parametersused in the sizing are given in Table 1.

Many of the common sizing methods utilize daily ormonthly mean values for solar radiation and load. TheSandia National Lab (2008) guidelines follow a determinis-tic approach for the system design taking monthly meanvalues of the daily insolation and daily load data to arriveat the system sizing. It follows a worst-scenario approachwith respect to the resource and provides a unique designpoint. The solar radiation data corresponding to the mini-mum insolation month is used for sizing the array. The sys-tem sizing obtained using the Sandia guidelines (2008) isrepresented as a single design point in the feasible designspace for comparison. It is observed that Sandia guidelines(2008) provide a conservative estimate for the batterycapacity as it is based on the number of days of autonomy.For the present example, for the same array rating, the bat-tery capacity given by Sandia guidelines (2008) comes outto be about 9.2 times the battery capacity given by the pro-posed approach based on an averaged day. Though Sandiaguidelines (2008) allow a quick estimation, it provides asingle solution which may lead to sub-optimal capacity uti-lization as the array and battery capacity are sized indepen-dently. As the Sandia guidelines (2008) utilize the worst-case design methodology, it results in an extremely conser-vative design with very high reliability of the overall sys-tem. In the proposed methodology, a time stepsimulation of the entire system has been performed toobtain system sizing close to actual requirement. However,


the major improvement offered by the proposed methodol-ogy is a set of feasible solutions, compared to a single pointsolution obtained using Sandia guidelines (2008). It mayalso be noted that the over sizing the components doesnot permit understanding of the trade-offs. As the pro-posed methodology identifies the entire range of feasiblesolution, it is possible to optimize the system configurationbased on the relative cost of each component.

It is seen that the sizing curve obtained using the oneyear hourly solar insolation data provides a higher systemcapacity as compared to that obtained with the averagedday. The system sizing obtained in this case accounts forall the minimum insolation periods which prevail in theyear. It is desirable to use the yearly data for the resourceand the demand as the seasonality associated with thesevariables is properly captured in the sizing process. How-ever, it may be noted that in many situations such extensivedatabase of solar insolation and load may not be availableand the system planning has to be based on the availabledata for the site. It is illustrated with this example thatthe proposed sizing methodology is not limited by theavailability of the resource and demand data.

To compare the proposed methodology with the resultspresented by Notton et al. (1996), the same input data setas in the original work is used for system simulation usingthe proposed model equations. It may be noted that thesystem sizing differs from that obtained considering anaveraged day as a detailed data set for the solar insolationis now used. In Fig. 5, the sizing curves obtained using theproposed system equations taking one year solar radiationdata are plotted along with the results reported by Nottonet al. (1996).

The designer has the option of selecting an appropriateconfiguration from the feasible design region (or the designspace) according to the requirements of additional capacityreserve and days of autonomy. Any configuration locatedon the sizing curve as well as inside the design space wouldbe capable of meeting the demand. The design space whichcontains all the feasible configurations thus provides aframework for system optimization based on an appropri-ate objective function.

0

10

20

30

40

2 4 6 8

Array area (m2)

Sto

rage

cap

acity

(kW

h)

Results by Notton et al.

Proposed method

Fig. 5. Comparison of the proposed method with results reported byNotton et al. (1996).

3. Generation of the design space with probabilistic approach

The design space methodology is extended to incorpo-rate the effects of uncertainty in solar insolation. Thehourly solar input is treated as a stochastic variable. Thesystem is sized for a specified reliability level. Chance con-strained programming approach has been used to solve thisoptimization problem under uncertainty.

The hourly solar input is assumed to follow a normaldistribution with known mean and standard deviation.This is a simplifying assumption for illustrating the pro-posed methodology. It is possible to evaluate the errorinvolved in the analysis under such an assumption. Theload is assumed to be deterministic over the time step.The source uncertainty is incorporated through a probabi-listic constraint. The system has to cater to at least thespecified deterministic demand depending on the probabil-ity of power generated by the photovoltaic array. Thechance constraint relating the probability of the demand,D(t), being met by the photovoltaic system is given as:

Prob½DðtÞP DactualðtÞ�P a ð11Þ

where Dactual is the deterministic demand to be met over thetime step and a the specified reliability of compliance of theconstraint or the confidence level. The demand met by thesystem, D(t), in Eq. (11) can be replaced by the system en-ergy balance (6). After algebraic manipulation, the chanceconstraint for the demand can be written as:

ProbQBðt þ DtÞ

f ðtÞDt� QBðtÞ

f ðtÞDtþ DactualðtÞ 6 PðtÞ

� �P a ð12Þ

Eq. (12) is a chance constraint incorporating the randomvariable. The deterministic equivalent for the energy bal-ance in terms of the demand (Dactual), mean array powerlPðtÞ and standard deviation of array power rP ðtÞ is obtainedas:

QBðt þ DtÞf ðtÞDt

� QBðtÞf ðtÞDt

6 ðlPðtÞ � DactualðtÞ � rPðtÞzaÞ ð13Þ

where za is the inverse of the cumulative normal probabilitydistribution corresponding to the required confidence levela with zero mean and unity standard deviation. Expressingthe deterministic equivalent in terms of the battery energyvalues for any time step considering the excess powerPdu(t), we get:

QBðt þ DtÞ ¼ QBðtÞ þ ½lPðtÞ � DactualðtÞ � rPðtÞza

� P duðtÞ�f ðtÞDt ð14Þ

The storage capacity for a given array area capable ofmeeting the specified load and meeting the reliabilityrequirements may be obtained by solving Eq. (14) overthe entire duration. The required battery capacity for a gi-ven array size is obtained using Eq. (9) following the pro-cedure described in the previous section.

It may be noted that za represents the inverse of thecumulative normal probability distribution corresponding


to the required confidence level a with zero mean and unitystandard deviation. The normal distribution is symmetricwith respect to its mean. The mean of the standard normaldistribution is obtained with a cumulative probability of0.5. Therefore, it may be observed from Eq. (14) that thesystem sizing obtained for a = 0.5 corresponds to the deter-ministic case, as za = 0 in this case. The combinations ofthe different array ratings and the corresponding minimumstorage requirements may be plotted on an array rating vs.battery capacity diagram which is the sizing curve for theconfidence level (a). The confidence level and the loss ofload expectation (LOLE) are approximately related bythe following expression:

LOLE � ð1� aÞDtT

ð15Þ

where Dt is the time step for the simulation and T is the to-tal time frame considered for the analysis. It may be notedthat the above relation is accurate for resource profiles witha sharp peak.

3.1. Validation through Monte Carlo simulation

Sequential Monte Carlo method helps in simulating theoccurrence of random events over time, recognizing the

Normally distributed solar insolation and uniformly distributed initial battery level used as simulation

input

Count of total hours of loss of load and the time-step wise distribution of the hours of loss of load whenever

equation (7) is not satisfied

Calculation of the battery energy level at each time step through equation (6)

Estimation of the system confidence level and LOLEusing equations (16) and (17)

Estimated values of system confidence level and LOLE using (16) and (17)

Coefficient of variation of LOLE

(17) attained a steady value?

No

Yes

Input: Time series of solar insolation mean and standard deviation, array and battery ratings and

system characteristics

Fig. 6. Monte Carlo simulation for system reliability estimation.

underlying statistical properties of the system. Using thisapproach it is possible to model the time dependence ofthe random variables involved in the analysis of a system.Monte Carlo simulation method is used for validating theresults obtained using the chance constrained approachfor the photovoltaic-battery system sizing. The methodol-ogy of Monte Carlo simulation is represented in Fig. 6.The estimated values of confidence level and LOLE

obtained by Monte Carlo simulation for different systemconfigurations from the design space are compared withthat given by the chance constrained approach.

The sizing curve and design space are generated follow-ing the proposed methodology, for specified confidence lev-els. For checking the system reliability predicted by thedesign space, specific array–battery bank configurationsare selected from the design space. The selected system issimulated using the system energy balance (6). However,the hourly insolation is made to randomly vary in the sim-ulation. The random insolation values used in the simula-tion are sampled from a normal distribution withspecified mean and standard deviation for the time step.For comparison, the data set used is such that the meanand standard deviation data of the hourly insolation corre-spond to the same values as used in the chance constrainedmodel for deriving the sizing curve and the design space.The initial battery level is assumed to follow a uniform dis-tribution based on the minimum and the maximum energycapacity of the battery. The hourly simulations are carriedout for the span of a year (using 8760 hourly data points),the battery energy level (QB) is checked with the minimumpermissible battery energy level (Bmin) at each hour. Thesystem confidence level (a) is estimated using the followingequation:

ah � 1�P

hH

ð16Þ

whereP

h corresponds to the total duration when there isa loss of load (i.e., QBðtÞ < Bmin) for the specified hour dur-ing the day. H is the total number of specified hours consid-ered in that interval (i.e., 365 h for a yearly simulation) forhourly simulation in the specified time horizon. The mini-mum value of (ah) obtained for each of the 24 h time periodwould correspond to the system confidence level (a). Fur-thermore, the system LOLE is estimated as:

LOLE �P

tT

ð17Þ

whereP

t corresponds to the total duration when there isloss of load (i.e., QBðtÞ < Bmin) over the entire time frameand T the total time horizon of the simulation (i.e.,8760 h). Fraction of time load is not met represents theLOLP of the system. Eq. (17) represents the LOLP ofthe power system by calculating the fraction of time loadis not met. It may be noted that this equation is differentfrom Eq. (15). In Eq. (15), the approximate relation be-tween the confidence level and LOLE is established. Theconfidence level and LOLE is computed at each iteration


by randomly varying the hourly insolation and the initialbattery level. The simulation is terminated when the esti-mated LOLE value for a system achieves a specified degreeof confidence. The system reliability is estimated as themean of the results obtained over the repeated simulations.The coefficient of variation (e) is used as the parameter fordeciding the stopping criterion,

e ¼ rp

lp

ð18Þ

where lp is the estimated mean of the LOLE and rp is theestimated standard deviation. The simulation is stoppedwhen the value of coefficient of variation attains a reason-ably steady value over different iterations.

3.2. Example 2: system sizing for specified reliability

In this section the application of the proposed method-ology for system sizing and generation of the sizing curvesfor a desired level of confidence is illustrated. The sameinput data set and system parameters as in example 1 areused in the analysis. The additional data set used is thehourly standard deviation of the solar insolation (Fig. 3).The sizing curves are obtained for a representative averageday for typical values of confidence levels (Fig. 7). Theresults obtained through the chance constrained modelare validated through the sequential Monte Carlo simula-tion. It may be noted that the hourly insolation inputs tothe Monte Carlo simulation are generated using the ran-dom number generation that follows the normal distribu-tion with the mean and standard deviation provided inFig. 3. Specific configurations selected from the sizing curveare simulated to estimate the system reliability in terms ofconfidence levels and approximate values of LOLE. Theconfidence level and LOLE values obtained for representa-tive configurations selected from the sizing curve based onthe Monte Carlo simulations are given in Tables 2–5.

For all cases, it is observed that the Monte Carlo simu-lation of the system predicts a higher reliability comparedto the chance constrained model. This indicates that theproposed model offers a conservative design for the system.

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5 6 7

Array rating (kWp)

Bat

tery

cap

acity

(kW

h)

α = 0.5

α = 0.7

α = 0.8

α = 0.95

Fig. 7. Sizing curves and the design space for various values of confidencelevels (example 2).

The system sizing obtained with the proposed method iscompared with the two-event probability (Bucciarelli, 1984,1986) method and three-event probability (Gordon, 1987;Bagul et al., 1996) method for a given reliability level.The sizing curve based on an averaged day for a confidencelevel of 0.5 (LOLE of 0.021) and the system sizing curveswith the two-event probability (Bucciarelli, 1984, 1986)and three-event probability (Gordon, 1987; Bagul et al.,1996) approaches are represented in Fig. 8. Compared tothe proposed sizing curve, two-event and three-event prob-ability based sizing curves have regions where the analyticequations are not valid giving a physically inconsistent siz-ing. It reconfirms the observations of Egido and Lorenzo(1992). It is observed that if the uncertainty of solar insola-tion at individual hour is incorporated in the design, theover sizing of the system can be avoided, maintaining adesired level of reliability. It may be noted that in two-event and three-event probability methods, the uncertaintyin solar insolation is taken over a daily time scale.

3.3. Generalized sizing curve

For locations where the standard deviation of the solarinsolation does not significantly vary with time a genericrepresentation of the sizing curve is possible. Under thecondition of constant coefficient of variation, the set ofall sizing curves for different confidence levels may be com-bined into a single generalized sizing curve. The coefficientof variation for the array power (xP) is given by:

xP ¼rP ðtÞ

lP ðtÞð19Þ

Whenever the coefficient of variation for the array power isconstant, Eq. (14) may be rewritten as:

QBðt þ DtÞ ¼ QBðtÞ þ ½g0lITðtÞA� � DactualðtÞ

� P duðtÞ�f ðtÞDt ð20Þ

where

A� ¼ Að1� xP zaÞ ð21Þ

Under this condition, a single curve may be plotted be-tween the modified area (A*) and the battery capacitywhich is the sizing curve corresponding to a = 0.5. The sys-tem sizing for different reliability levels may be quicklydetermined from such a plot. For a desired confidence le-vel, the corresponding array area (A) may be found fromEq. (21). Using the input data for example 2, the general-ized sizing curve is plotted in Fig. 9 for a constant coeffi-cient of variation value (xP) of 0.45 for the array power.As an example, for determining the minimum batterycapacity corresponding to an array area of 20 m2, meetinga confidence level of 0.8, the value of A* is 12.43 m2. Therequired minimum battery capacity may be found directlyfrom the battery capacity vs. the modified area plot(Fig. 9). Corresponding to the value of 12.43 m2, the re-quired minimum battery capacity is 882.2 Wh. The mini-

Table 2Comparison of system confidence level (a) with Monte Carlo simulationfor representative configurations from the design space with a = 0.5.

Array area (m2) Battery capacity(Wh)

Confidence level(Monte Carlo)

6 952 0.62748 906 0.5616

10 892 0.583612 884 0.649314 875 0.616416 867 0.671218 859 0.687720 851 0.6849




6 1041 0.96168 1025 0.9699

10 1009 0.983612 993 0.975314 977 0.978116 961 0.980818 945 0.967120 930 0.9753




6 1133 0.99738 1096 0.9973

10 1079 0.994512 1076 0.997314 1074 0.997316 1072 0.997318 1070 0.997320 1068 0.9973




16 1265 0.997318 1257 0.997320 1250 0.997322 1245 0.997324 1245 0.997326 1245 128 1245 130 1245 132 1245 1


mum battery capacity corresponding to array area of 20 m2

and confidence level of 0.95 is also indicated on Fig. 9.

4. System optimization

The sizing curve identifies the design space which helpsthe designer in selecting an optimum system configurationbased on the desired objective. In this section, the proce-dure for selecting an optimum system configuration is dis-cussed. The cost of energy (COE), which accounts for thecapital as well as the operating costs associated with thesystem, is chosen as an appropriate economic parameterto evaluate and to optimize the system configuration. It iscalculated as:

COE ¼ ALCCEdem

ð22Þ

The annualized life cycle cost (ALCC) includes the annual-ized capital cost and the annual operating and maintenancecosts. The annual operating and maintenance cost is takenas 1% of the total capital cost. Edem is the total annual en-ergy to be delivered by the system. The annualized capitalcost (ACC) is calculated as

ACC ¼X

i

C0i� CRF i ð23Þ

where, the capital recovery factor (CRF) is a function ofthe life (n) and the discount rate (d).

CRF i ¼dð1þ dÞni

ð1þ dÞni � 1ð24Þ

C0i is the capital cost of the ith system component corre-sponding to the photovoltaic module, battery bank, inver-ter, and balance of system. CRFi is the capital recoveryfactor for the ith component and it is a function of the dis-count rate (d) and life of the component (ni).

4.1. Example 3: optimal system sizing incorporatinguncertainty

To illustrate the system optimization, example 2 hasbeen considered. The cost data considered for the economicanalysis are given in Table 6 (Kolhe et al., 2002; NREL,2006). Based on the minimum cost of energy for the differ-ent configurations, the optimum configuration is identified

Table 6Economic parameters considered for system optimization.

Discount rate, d% 10Photovoltaic array cost, Rs./kWp 150,000Photovoltaic array life, years 20Inverter cost, Rs./kW 18,000Inverter life, years 10Battery bank cost, Rs./kWh 4000Battery bank life, years 5Balance of system cost, % of the hardware cost 10Balance of system life, years 10Annual operation and maintenance cost, % of project cost 1

0

4

8

12

16

20

24

28

0.2 0.4 0.6 0.8 1

Array rating (kWp)

Bat

tery

cap

acity

(kW

h)

Bucciarelli's method

Three event probabilitymethodProposed method

Fig. 8. Comparison of the proposed method with two-event and three-event probability methods for LOLE value of 0.021 (example 2).

0

200

400

600

800

1000

1200

0 10 20 30 40 50 60

Modified array area, A* (m2)

Batte

ry c

apac

ity (W

h) A = 20 m2, Br = 882.2 Wh , α = 0.8

A = 20 m2, Br = 970.4 Wh , α = 0.95

Fig. 9. Generalized sizing curve.

Table 7Optimum cost of energy for different confidence levels (example 3).

Confidencelevel

Array rating(kWp)

Battery capacity(kWh)

Cost ofenergy(Rs/kWh)

0.50 0.29 1.02 19.850.70 0.40 1.08 26.480.80 0.52 1.15 33.730.95 1.56 1.27 95.26

0

5

10

15

20

0 2 4 6 8 10 12 14 16 18 20 22 24

Time (hour of the day)

Dem

and

(kW

)

Fig. 10. Load curve for the rural location in Kerala, India (example 4).


and represented on the sizing curves for various values ofconfidence levels. The results are similar to those reportedin Fig. 7 and not shown for brevity. The optimum cost of

energy for different values of confidence levels are listed inTable 7. It is observed that for ensuring higher levels of reli-ability, with increase in the system capacities the associatedcost of energy also increases. Depending on the reliabilityrequirements and the availability of capital any appropri-ate configuration may be selected from the design space.It may be noted that the minimum cost of energy may becalculated by searching for the minimum value of cost ofenergy along the sizing curve for a specified confidencelevel. In absence of any discrete variables, optimum config-uration must lie on the sizing curve. This is principle opti-mization approach adopted to calculate the optimumsystem configuration. The entire set of system equationsis also optimized numerically using non-linear optimizationtechnique to validate these results.

4.2. Example 4: optimal sizing for a remote indian location

The optimal system sizing, based on the estimated loadprofile and solar radiation profile for an isolated rural loca-tion in Western Ghat region of Kerala, India (Ashok,2007), is presented here. The load curve shows a varyingdemand pattern over the day (Fig. 10). In previous exam-ples, the effect of ambient temperature has not been consid-ered explicitly. However, the proposed methodology is notlimited by such an assumption. The effect of the ambienttemperature of the sizing curve is demonstrated with thisexample. The solar insolation profiles as well as the ambi-ent temperature used for the system sizing are representedin Fig. 11. The module efficiency has been modeled as afunction of the cell temperature (Mattei et al., 2006):

g0 ¼ g0r½1� bT ðT c � T rÞ� ð25Þ

where g0r is the module efficiency at the reference tempera-ture (Tr), Tc is the cell temperature, and bT is the tempera-ture coefficient of the module. The cell temperature can becalculated based on the thermal energy balance of the mod-ule (Mattei et al., 2006).

T c ¼U PV T a þ IT ½ðasÞ � g0r � bT g0rT r�

U PV � bT g0rITð26Þ

where UPV is the heat transfer coefficient of the module, Ta

is the ambient temperature, and (as) is the effective trans-missivity–absorptivity coefficient of the module. The inputparameters considered for the system sizing and optimiza-tion, are as given in Tables 1 and 6.

Sizing curves have been plotted considering (i) only thenominal efficiency and (ii) by considering the temperatureeffect for confidence levels of 0.5 and 0.95, respectively(Fig. 12). It is observed that the effect of temperature onthe sizing curve is pronounced for the lower array ratings.Further the sizing requirements are found to be higher forhigher levels of system confidence when the temperatureeffect is accounted. Higher the ambient temperature, morearea is required at higher confidence level. The comparisonof system confidence levels and LOLE with those obtainedwith Monte Carlo simulation approach is presented in


Table 8. It is observed that the proposed methodologyoffers a conservative design.

Optimum system configurations for these confidencelevels are also determined based on the minimum cost ofenergy. The optimum cost of energy for a confidence levelof 0.5 (LOLE of 0.02) is Rs. 21.26/kWh while for meeting aconfidence level of 0.95 (LOLE of 0.002) it comes to be Rs.47.37/kWh. When the temperature effect is accounted, thecost of energy for meeting a confidence level of 0.5 is Rs.

0

200

400

600

800

1 3 5 7 9 11 13 15 17 19 21 23

Time(hour of the day)

Inso

latio

n (W

/m2 )

0

5

10

15

20

25

30

35

Am

bien

t Tem

pera

ture

(o C

)

Fig. 11. Variation in the mean (thin curve) and standard deviation (boldcurve) of the hourly solar insolation as well as the ambient temperature(dashed curve) for a representative day for the rural location in Kerala,India (example 4).

325

350

375

400

425

50 100 150 200 250 300 350 400 450 500 550 600 650 700

Array power (kWp)

Bat

tery

cap

acity

(kW

h)

α = 0.5

α = 0.95Considering temperature effect

Considering temperature effect

α = 0.95

Fig. 12. Optimum system configurations for the rural location in Kerala,India (example 4).

Table 8The system confidence levels (a) and corresponding LOLE compared with thespace (example 4).

a = 0.5 a

Arrayarea(m2)

Without the effect of ambienttemperature

Considering the effect ofambient temperature

Aa(Battery

capacity(kWh)

Confidencelevel (MonteCarlo)

Batterycapacity(kWh)


1000 374 0.5534 – – 21200 372 0.6632 374 0.6438 21500 369 0.7205 371 0.7589 32000 365 0.8164 367 0.8301 33000 356 0.8438 359 0.8712 34000 352 0.9041 352 0.8904 4

25.12/kWh while for meeting a confidence level of 0.95 itcomes to be Rs. 65.66/kWh. For higher confidence level,the effect of temperature variation cannot be neglected.

The system performance can be compared consideringthe ratio of the catered load power to the maximum photo-voltaic power that the generator could generate for theactual irradiance and temperature conditions. This is alsoknown as the performance ratio. The average performanceratio of the optimum system for the confidence level of 0.5is found to be 0.67. For the confidence level of 0.95, thevalue of performance ratio drops to 0.4 because of thehigher system rating. Typically, as the system reliabilityincreases the performance ratio of the PV system getsreduced.

5. Conclusions

The concept of design space for the optimum system siz-ing of photovoltaic-battery bank systems incorporatingsolar resource uncertainty is presented in this paper. Basedon the proposed approach, for given resource, demand andsystem characteristics, a sizing curve may be plotted on thephotovoltaic array rating vs. storage capacity diagram fora desired confidence level. The proposed methodology isthus capable of handling the uncertainty associated withthe solar insolation at the design stage following a chanceconstrained programming approach. Generation of thedesign space offers flexibility in the overall design processas it identifies all the feasible configurations for meeting agiven load and reliability. The methodology for obtainingthe design space is based on the energy balance of the over-all system. In deriving the discretised solution for theenergy balance, the interval of integration has been consid-ered to be finitely small so that demand and resource valuesare constant during that time and therefore the conversionefficiency values also remain constant over the time period.In the energy balance, discrete hourly time steps have beenconsidered in all the illustrations as the data for resourceand demand are generally available for hourly intervals.However, the proposed methodology is flexible to takeup any time step. It may also be noted that though, forillustrating the method, averaged day profile for the solar

Monte Carlo simulation for representative configurations from the design

= 0.95

rrayream2)

Without the effect of ambienttemperature

Considering the effect ofambient temperature





100 387 1 – –500 383 1 – –000 378 1 408 0.9973600 374 1 404 1800 373 1 403 1000 373 1 402 1


resource and demand have been considered, the method isnot limited by the extent of data used. Depending on theavailability of data for the resource and the demand accu-rate design space may be generated. The approach can alsobe adopted incorporating alternate models for the systemcomponents as the basic methodology is simulation based.

In the chance constrained model, the hourly solar inso-lation is treated as a normally distributed random variable.This is a simplifying assumption but alternate models forthe probability distribution may be considered for the anal-ysis. It is shown that under the condition of constant coef-ficient of variation of hourly solar insolation (and hencepower), the set of sizing curves for different confidence lev-els may be represented by a single curve. This helps in pro-viding a quick estimation of the system sizing for anydesired value of reliability.

Through examples, identification of the design space fortypical demand-resource profiles has been presented. Sys-tematic comparison is made with the existing results andcurrent methodologies adopted for system design. The reli-ability levels given by the design space approach is vali-dated through the sequential Monte Carlo simulationapproach. It is seen that the generation of the design spacefor a specified reliability level would prevent the over sizingof systems. Larger capacity systems may be planned only insituations where very low values of LOLE are desired.Selection of optimum system based on minimum cost ofenergy is also discussed for different values of reliability.The methodology can be also used in the design of inte-grated isolated systems involving other sources of energyand storage. Thus, the design space approach is a usefultool for the system planning of isolated power systems asit provides the trade-offs between the cost of energy andthe system reliability.

The effect of the ambient temperature of the sizing curveis demonstrated through an illustrative example. It isobserved that the effect of temperature on the sizing curveis pronounced for the lower array ratings. As the array rat-ing increases, the effect of ambient temperature on systemsizing reduces for deterministic cases. For a high systemreliability, the effect of ambient temperature on system siz-ing and the optimum cost of energy cannot be neglected.

Acknowledgements

The authors are thankful to Mr. G. Notton for providingthe input solar radiation data for the examples presented inthe paper. The first author is grateful to the Ministry ofNew and Renewable Energy, Government of India for pro-viding the financial support for the research work.

References

Arun, P., Banerjee, R., Bandyopadhyay, S., 2007. Sizing curve for designof isolated power systems. Energy for Sustainable Development 11 (4),21–28.

Arun, P., Banerjee, R., Bandyopadhyay, S., 2008. Optimum sizing ofbattery integrated diesel generator for remote electrification throughdesign-space approach. Energy 33, 1155–1168.

Aseeri, A., Bagajewicz, M.J., 2004. New measures and procedures tomanage financial risk with applications to the planning of gascommercialization in Asia. Computers and Chemical Engineering 28,2791–2821.

Ashok, S., 2007. Optimised model for community based hybrid energysystem. Renewable Energy 32, 1155–1164.

Azaiez, M.N., Hariga, M., Al-Harkan, I., 2005. A chance-constrainedmulti-period model for a special multi-reservoir system. Computersand Operations Research 32, 1337–1351.

Bagul, A.D., Salameh, Z.M., Borowy, B., 1996. Sizing of a stand-alonehybrid wind photovoltaic system using a three-event probabilitydensity approximation. Solar Energy 56 (4), 323–335.

Barra, L., Catalanotti, S., Fontana, F., Lavorante, F., 1984. An analyticalmethod to determine the optimal size of a photovoltaic plant. SolarEnergy 33 (6), 509–514.

Bartoli, B., Cuomo, V., Fontana, F., Serio, C., Silvestrini, V., 1984. Thedesign of photovoltaic plants: an optimisation procedure. AppliedEnergy 18 (1), 37–47.

Bhuiyan, M.M.H., Asgar, A.M., 2003. Sizing of a stand-alone photovol-taic power system at Dhaka. Renewable Energy 28 (6), 929–938.

Bucciarelli Jr., L.L., 1984. Estimating loss-of-power probabilities ofstandalone photovoltaic solar energy systems. Solar Energy 32 (2),205–209.

Bucciarelli Jr., L.L., 1986. The effect of day-to-day correlation in solarradiation on the probability of loss-of-power in a stand-alonephotovoltaic energy system. Solar Energy 36 (1), 11–14.

CETC, 2004. Available from: <http://www.retscreen.net/ang/t_software.php>.

Changchit, C., Terrell, M.P., 1993. A multiobjective reservoir operationmodel with stochastic inflows. Computers and Industrial Engineering24 (2), 303–313.

Charnes, A., Cooper, W.W., 1959. Chance-constrained programming.Management Science 6, 73–79.

Egido, M., Lorenzo, E., 1992. The sizing of stand alone PV-systems: areview and a proposed new method. Solar Energy Materials and SolarCells 26, 51–69.

Gordon, J.M., 1987. Optimal sizing of stand-alone photovoltaic solarpower systems. Solar Cells 20 (4), 295–313.

Groumpos, P.P., Papageorgiou, G., 1987. An optimal sizing method forstand-alone photovoltaic power systems. Solar Energy 38 (5), 341–351.

Hontoria, L., Aguilera, J., Zufiria, P., 2005. A new approach for sizingstand alone photovoltaic systems based in neural networks. SolarEnergy 78 (2), 313–319.

Kaldellis, J.K., 2004. Optimum technoeconomic energy autonomousphotovoltaic solution for remote consumers throughout Greece.Energy Conversion and Management 45, 2745–2760.

Kolhe, M., Kolhe, S., Joshi, J.C., 2002. Economic viability of stand-alonesolar photovoltaic system in comparison with diesel-powered systemfor India. Energy Economics 24 (2), 155–165.

Kulkarni, G.N., Kedare, S.B., Bandyopadhyay, S., 2007. Determinationof design space and optimization of solar water heating systems. SolarEnergy 81 (8), 958–968.

Kulkarni, G.N., Kedare, S.B., Bandyopadhyay, S., 2008. Design of solarthermal systems utilizing pressurized hot water storage for industrialapplications. Solar Energy 82, 686–699.

Kulkarni, G.N., Kedare, S.B., Bandyopadhyay, S., 2009. Optimization ofSolar Water Heating Systems through Water Replenishment. EnergyConversion & Management 50 (3), 837–846.

Li, P., Arellano-Garcia, H., Wozny, G., 2008. Chance constrainedprogramming approach to process optimization under uncertainty.Computers and Chemical Engineering 32 (1–2), 25–45.

Markvart, T., Fragaki, A., Ross, J.N., 2006. PV system sizing usingobserved time series of solar radiation. Solar Energy 80, 46–50.

http://www.retscreen.net/ang/t_software.php

http://www.retscreen.net/ang/t_software.php


Mattei, M., Notton, G., Cristofari, C., Muselli, M., Poggi, P., 2006.Calculation of the polycrystalline PV module temperature using asimple method of energy balance. Renewable Energy 31 (4), 553–567.

Notton, G., Muselli, M., Poggi, P., Louche, A., 1996. Autonomousphotovoltaic systems: influences of some parameters on the sizing:simulation timestep, input and output power profile. RenewableEnergy 7, 353–369.

NREL, 2006. A review of PV inverter technology cost and performanceprojections. Subcontract Report-NREL/SR-620-38771. NationalRenewable Energy Laboratory, Golden, CO.

NREL, 2007. Available from: <http://www.nrel.gov/homer/includes/downloads/HOMERBrochure_English.pdf>.

Rao, S.S., 1980. Structural optimisation by chance constrained program-ming techniques. Computers and Structures 12 (6), 777–781.

RERL, 2007. Available from: <http://www.ceere.org/rerl/projects/soft-ware/hybrid2/Hy2_users_manual.pdf>.

Roy, A., Arun, P., Bandyopadhyay, S., 2007. Design and Optimization ofRenewable Energy Based Isolated Power Systems. SESI Journal 17,54–69.

Sandia National Lab, 2008. Available from: <http://photovoltaics.san-dia.gov/docs/Wkshts1-5.html>.

Sidrach-de-Cardona, M., Lopez, L.M., 1998. A simple model for sizingstand alone photovoltaic systems. Solar Energy Materials and SolarCells 55 (3), 199–214.

Sukhatme, S.P., 1997. Solar Energy – Principles of Thermal Collectionsand Storage, second ed. Tata-McGraw Hill, New Delhi.

Tsalides, P.H., Thanailakis, A., 1986. Loss-of-load probability and relatedparameters in optimum computer-aided design of stand-alone photo-voltaic systems. Solar Cells 18 (2), 115–127.

http://www.nrel.gov/homer/includes/downloads/HOMERBrochure_English.pdf

http://www.nrel.gov/homer/includes/downloads/HOMERBrochure_English.pdf

http://www.ceere.org/rerl/projects/software/hybrid2/Hy2_users_manual.pdf

http://www.ceere.org/rerl/projects/software/hybrid2/Hy2_users_manual.pdf

http://photovoltaics.sandia.gov/docs/Wkshts1-5.html

http://photovoltaics.sandia.gov/docs/Wkshts1-5.html

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