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Solar Energy 98 (2013) 190–195

Developing a discrete harmony search algorithm for size optimizationof wind–photovoltaic hybrid energy system

Alireza Askarzadeh ⇑

Department of Energy Management and Optimization, Institute of Science and High Technology and Environmental Sciences, Graduate University

of Advanced Technology, Kerman, Iran

Received 9 June 2013; received in revised form 4 September 2013; accepted 7 October 2013Available online 6 November 2013

Communicated by: Associate Editor Mukund R. Patel

Abstract

This paper tries to develop an efficient methodology for optimizing size of wind–photovoltaic hybrid energy systems. Since findingoptimal size belongs to discrete optimization problems, optimization algorithm must be able to effectively handle this kind of problem.For this aim, a Discrete Harmony Search (DHS) algorithm is developed. The proposed methodology is easy to implement, can efficientlyhandle the discrete problem and can quickly find a good solution. The obtained results show that the proposed methodology finds prom-ising results in less than one second.� 2013 Elsevier Ltd. All rights reserved.

Keywords: Discrete harmony search algorithm; Size optimization; Hybrid energy systems

1. Introduction

Owing to increasing cost, resultant pollution and prob-able depletion of fossil fuels, renewable energy sourcesare being used worldwide to contribute in meeting theenergy demand. Wind turbines and solar panels are prom-ising renewable energy sources that have considerablyattracted the attention of many researchers and communi-ties. Because of seasonal and periodical variations, neithera standalone wind nor solar energy system can continu-ously supply a load demand. As a result, for continuouslyproducing electricity in remote areas, hybrid energy sys-tems must be used by combining the wind and solar ener-gies along with battery storage.

To supply annual load demand and minimize totalannual cost, hybrid generation system must be optimally

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http://dx.doi.org/10.1016/j.solener.2013.10.008

⇑ Tel./fax: +98 342 6233176.E-mail addresses: a.askarzadeh@kgut.ac.ir, askarzadeh_a@yahoo.

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sized. Size optimization problem has become the heart ofvarious investigations (Erken and Erken, 2009; Erkenand Erken, 2008; Erken et al., 2009; Geem, 2012; Kaabecheet al., 2011; Yang et al., 2007). The main focus of theseresearches is to introduce efficient methodologies to findoptimal size. Literature review shows that four types ofmethodologies have been applied to size optimization ofhybrid energy systems: probabilistic, analytical, heuristicand hybrid methods. A detailed review of these methodol-ogies can be found in (Luna-Rubio et al., 2012). Probabilis-tic and analytical methods are the simplest sizingmethodologies, but the obtained results are not the best.These methods not only need high computational timebut also conduct local search and may find local solutions.Heuristic methods have proved themselves as good candi-dates to find the optimum solution of hard optimizationproblems in a reasonable amount of computational time.By using stochastic rules, heuristic techniques have morechance to find optimal solution. Hybrid methods try tocombine optimization techniques for improving theobtained results.

Nomenclature

CCpt annual capital cost ($)CMtn annual maintenance cost ($)CT total annual cost ($)i interest raten life span of the system (years)Tp turbine price ($)Tif turbine installation fee ($)Pp panel price ($)Pif panel installation fee ($)CBatt cost of battery ($)CBackup cost of the backup generator ($)SBatt rated capacity of battery (kW h)Dt time between the samples (h)LSBatt battery’s life span (years)

CSolMnt solar panel’s maintenance cost ($/kW h)

CWindMnt wind turbine’s maintenance cost ($/kW h)

WGen total generated energy (J)WDmd total demanded energy (J)PGen total generated power (W)PDmd total demanded power (W)

NmaxWind maximum number of wind turbines

NmaxPV maximum number of PV panels

NmaxBatt maximum number of batteries

SReq required storage capacity (kW h)NBatt number of batteriesNPV number of PV panelsNWind number of wind turbinesP t

Wind power generated by the wind turbines (W)P t

Wind-Each the power generated by each wind turbine (W)P t

Sol power generated by the solar panels (W)P t

Sol-Each power generated by each solar panel (W)Nh number of harmoniesitermax maximum number of iterationsiter iteration indexHMCR harmony memory considering ratePAR pitch adjusting ratebw bandwidth of generationPARmax maximum pitch adjusting ratePARmin minimum pitch adjusting ratebwmax maximum bandwidthbwmin minimum bandwidth

A. Askarzadeh / Solar Energy 98 (2013) 190–195 191

This paper attempts to find the optimum size of a wind–photovoltaic hybrid energy system. The ultimate aim ofthe size optimization is to determine the optimum numberof wind turbines, solar panels and batteries. Because theproblem under consideration consists of integer decisionvariables, a discrete optimization algorithm could effectivelyhandle this problem. For this aim, a recently invented heuris-tic technique, harmony search (HS), is considered and a dis-crete variant of this algorithm, Discrete Harmony Search(DHS), is developed to effectively solve the optimizationproblem. By inspiration from the improvisation process ofmusicians (Geem et al., 2001), harmony search (HS) is ametaheuristic optimization technique which has simple con-cept, is easy to implement, does not need gradient informa-tion, requires reasonable amount of computational time tofind a good solution, and is stochastic. The capabilities ofHS have led to its applications in different areas (Askarzadehand Rezazadeh, 2011, 2012a,b). One of the most importantfeatures of HS is the ability of handling discrete variables.So, HS could be a good candidate to solve discrete optimiza-tion problems.

2. Optimization formulation

The ultimate aim of the size optimization problem is tominimize the total annual cost (CT). Total annual cost issum of annual capital cost (CCpt) and annual maintenancecost (CMtn). So, the problem is mathematically formulatedas follows:

Minimize CT ¼ CCpt þ CMtn ð1ÞMaintenance cost occurs during the project life while

capital cost occurs at the beginning of a project.In order to convert initial capital cost to annual capital

cost, capital recovery factor (CRF) defined by Eq. (2) isused.

CRF ¼ ið1þ iÞn

ð1þ iÞn � 1ð2Þ

where i is the interest rate and n denotes the life span of thesystem.

By breaking up capital cost into annual costs of windturbine, solar panel, battery and backup generator, Eq.(3) is obtained.

CCpt ¼ið1þ iÞn

ð1þ iÞn � 1

�NWind � CWind þ NSol � CSol

þ nLSBatt

� �� NBatt � CBatt þ CBackup

�ð3Þ

where NWind is the number of wind turbines, CWind definedby sum of turbine price (Tp) and turbine installation fee(Tif) is unit cost of wind turbine, NSol is the number of solarpanels, Cwind defined by sum of panel price (Pp) and panelinstallation fee (Pif) is unit cost of solar panel, LSBatt is bat-tery’s life span because battery is vulnerable in the renew-able generation system, NBatt is the number batteries,CBatt is unit cost of battery and CBackup is the cost of thebackup generator for the use when there are no sufficientwind and solar energies and the storage batteries are low.

192 A. Askarzadeh / Solar Energy 98 (2013) 190–195

The required storage capacity for the system can beobtained by using energy curve (DW) defined by:

DW ¼ W Gen � W Dmd ¼Z

DP dt ¼ZðP Gen � P DmdÞdt ð4Þ

where WGen is total generated energy, WDmd denotes totaldemanded energy, PGen is total generated power and PDmd

is total demanded power.Battery must cycle between the positive and negative

peaks of the energy curve and at least should have a capac-ity equal to the difference between the positive and negativepeaks of the energy curve. It must be noted that to guaran-tee the battery’s life span, the battery should not be cycledthrough more than a specific percentage of its rated capac-ity (g). Therefore, number of batteries can be determinedby the following expression:

NBatt ¼ RoundupSReq

g� SBatt

� �ð5Þ

where Roundup returns the rounded up value, SBatt is therated capacity of each battery and SReq is the required stor-age capacity defined by:

SReq ¼ MaxZ

DP dt� �

�MinZ

DP dt� �

ð6Þ

The total power generated by the wind turbines at time t

is obtained by:

P tWind ¼ NWind � P t

Wind-Each ð7Þ

where P tWind is the power generated by the wind turbines

and P tWind-Each is the power generated by each wind turbine.

In the same way, for solar panels the total power gener-ated is obtained by:

P tSol ¼ N Sol � P t

Sol-Each ð8Þ

where P tSol is the power generated by the solar panels and

P tSol-Each is the power generated by each solar panel.

For the annual maintenance cost the following equationis used:

CMnt ¼ CWindMnt �

X24

t¼1

P tWind � Dt

� �þ CSol

Mnt �X24

t¼1

P tSol � Dt

� �" #

� 365

ð9Þ

where CWindMnt is the wind turbine’s maintenance cost per

kW h, CSolMnt is the solar panel’s maintenance cost per

kW h and Dt denotes the time between the samples.Eqs. (1)–(9) show the description of the objective func-

tion in detail. In addition to these equations, some con-straints need to be regarded during the optimizationprocess. These constraints are as follows:

X24

t¼1

P tWind � Dt

� �þX24

t¼1

P tSol � Dt

� �PX24

t¼1

P tDmd � Dt

� �ð10Þ

NWind ¼ Integer; N Wind P 0 ð11ÞNSol ¼ Integer; NSol P 0 ð12Þ

In addition to the above-mentioned constraints, theoptional constraints of resource limitations can be alsoadded to the problem.

NWind 6 NmaxWind ð13Þ

NSol 6 N maxSol ð14Þ

where NmaxWind , Nmax

Sol are the maximum available number ofwind turbines and solar panels, respectively.

The decision variables for optimizing the size of wind–photovoltaic hybrid energy system are NWind and NSol.These variables must be optimally determined so that thetotal annual cost of the system to be minimized subjectto the constraints. From Eq. (5), it is seen that because SReq

is a function of NWind and NSol, the number of storage bat-teries is dependent on NWind and NSol and hence, there is noneed to consider NBatt is the optimization problem since itis adjusted by Eq. (5).

3. Discrete Harmony Search (DHS)

Improvisation in music is a process in which each musi-cian sounds a note by his (or her) instrument in the allowedrange. The played notes together make a harmony by aquality and the musicians attempt to increase the qualityof the current harmony by practice. To sound a note, amusician may use one of the following strategies: (1)sounding a note from his (or her) memory, (2) soundinga note near a memorized note and (3) sounding a randomnote from the allowed range. The above rules make theconcept of HS algorithm. The key parameters which playimportant role in the convergence of HS algorithm are har-mony memory considering rate (HMCR), pitch adjustingrate (PAR) and bandwidth of generation (bw). Theseparameters can be potentially useful in adjusting conver-gence rate of the algorithm to the optimal solution. HMCR

varying between 0 and 1 is the rate of choosing one valuefrom the HM. PAR and bw are defined as follows:

PARðiterÞ ¼ PARmin þPARmax � PARmin

itermax

� iter ð15Þ

bwðiterÞ ¼ bwmax exp Lnðbwmin=bwmaxÞ �iter

itermax

� �ð16Þ

where PARmax and PARmin are maximum and minimumpitch adjusting rates, iter denotes iteration index, itermax

is maximum number of iterations, and bwmax, bwmin aremaximum and minimum bandwidths, respectively.

In order to implement the proposed DHS algorithm forfinding optimum size of wind–photovoltaic hybrid energysystem, the following steps are used.

Table 1The dataset of the wind–photovoltaic hybrid energy system.

t P tDmd (kW) P t

Wind-Each (kW) P tSol-Each (W)

1 1.37 0.58 02 1.23 0.49 03 1.17 0.48 04 1.2 0.53 05 1.32 0.47 06 1.78 0.51 07 2.63 0.46 1.68 2.86 0.46 3.49 2.49 0.61 10.3

A. Askarzadeh / Solar Energy 98 (2013) 190–195 193

Step 1: Nh feasible solutions are generated where eachone is named harmony. Each harmony consists of twointeger values. The first one denotes number of wind tur-bines and the second specifies number of solar panels.The harmonies are memorized in the harmony memory(HM).Step 2: For each harmony, the value of the total annualcost is calculated.Step 3: The algorithm’s adjustable parameters are set.Step 4: A new harmony is produced by the followingpseudocode:

10 2.18 0.76 24.611 2.02 1.1 31.712 1.92 1.53 35.313 1.8 1.67 36.614 1.69 1.89 37.415 1.6 2.43 36.816 1.63 2.45 33.517 1.85 1.91 24.218 2.26 1.76 13.419 2.55 1.57 5.620 2.57 1.16 1.521 2.51 0.87 022 2.46 0.76 023 2.25 0.74 024 1.77 0.7 0

Fig. 1. Average hourly electrical demand in a day.

for k = 1:2if r1 > HMCRxnew(k) = a feasible random integer number;elsexnew(k) = the value corresponding to a randomselected harmony from HM;

if r2 < PAR

xnew(k) = xnew(k) + rw;end

endend

where xnew is the improvised harmony and r1 as well as r2

are uniformly distributed random numbers between 0 and1. The parameter rw is obtained as follows:

rw ¼1 r3 < 0:5

�1 otherwise

�ð17Þ

where r3 is a uniformly distributed random number be-tween 0 and 1.

Step 5: If the improvised harmony is a feasible solutionwith better quality than the worst harmony memorizedin the HM, it is included in the HM and the existingworst harmony is excluded from the HM.Step 6: Steps 3–5 are repeated until the stopping crite-rion is reached.Step 7: The values corresponding to the best harmony ofthe HM are returned as the optimum number of windturbines and solar panels.

4. Simulation results

MATLAB environment is used to implement the pro-posed methodology. The DHS’s parameters are adjustedas follows: Nh = 5; itermax = 500; HMCR = 0.95; PARmax =0.7; PARmin = 0.1; bwmax = 1; bwmin = 0.01. The test systemused here is similar to that proposed by Kellogg et al.(1998).The whole dataset of this system has been extracted andreported by Geem (2012) to provide a benchmark problemfor other researchers. Table 1 represents annual averagehourly demand P t

Dmd

� �, power generated by each wind

turbine P tWind-Each

� �and power generated by each solar panel

P tSol-Each

� �. Figs. 1–3 illustrate annual average hourly

demand, annual power generated by each wind turbineand annual power generated by each solar panel.

Table 2 shows the values of the design variables for thetest system. Because life span of each battery is regarded4 years, 5 installations are needed during the system’s lifespan (20 years).

In order to find the optimum size, the proposed DHSalgorithm is run for the benchmark system by consideringNmax

Wind ¼ N maxSol ¼ 100. The optimum solution found is NWind =

2 and NSol = 0 with the optimal cost of 5652.66 $. Bythese values the optimum number of batteries is obtained11. The results are same as those obtained by Branch-

Fig. 2. Average hourly power generated by a solar panel in a day.

Fig. 3. Average hourly power generated by a solar panel in a day.

Table 2Design parameters of the benchmarkproblem.

Parameter Value

i 0.06n 20 yearsTp 20,000 $/turbineTif 0.25 � Tp

Pp 350 $/panelPif 0.5 � Pp

CBatt 170 $CBackup 2000 $g 0.8SBatt 2.1 kW hDt 1 hBls 4 yearsCSol

Mnt 0.005 $/kW hCWind

Mnt 0.02 $/kW h

Fig. 4. Convergence of DHS to find the optimum size.

Fig. 5. Convergence of DHS to find the optimum size (NWind = 0 andNWind = 1).

194 A. Askarzadeh / Solar Energy 98 (2013) 190–195

and-Bound (B&B) and Generalized Reduced Gradient(GRG) methods (Geem, 2012).

Fig. 4 shows the convergence process of DHS algorithmwhich illustrates the minimum value of the total annual

cost during the iterations. It is seen that after 150 itera-tions, DHS converges to the optimal solution. It is worth-while to mention that DHS finds the optimum solution inless than 1 s (0.25 s) which means that the proposed meth-odology reaches the best solution very quickly.

As can be seen, the optimization algorithm only selectsthe wind turbines because the unit cost of wind turbinesis cheaper than that of solar panels. For using some solarpanels, the following constraints can be considered in theproblem:

NWind ¼ 1 ð18ÞNWind ¼ 0 ð19Þ

By considering Eq. (18), the optimum solution found isNWind = 1, NSol = 72 and NBatt = 11 with the optimal costof 6692.61 $. By taking into account Eq. (19) andNmax

Sol ¼ 200, the algorithm yields NWind = 0, NSol = 160and NBatt = 17 with the optimal cost of 8844.09 $. Thesesolutions are found in 0.20 s. Fig. 5 indicates the conver-gence process of the algorithm in these cases.

Fig 6 shows the difference between the generated anddemanded powers DP t ¼ P t

Wind þ P tSol � P t

Dmd

� �for the

hybrid (NWind = 1, NSol = 72), PV alone (NWind = 0,NSol = 160) and wind alone (NWind = 2, NSol = 0) systems.

Fig. 6. Difference between the generated and demanded powers.

A. Askarzadeh / Solar Energy 98 (2013) 190–195 195

As another investigation, for providing a more realisticproblem, P t

Wind-Each reduces to 70% of its original value asfollows:

P tWind-Each 0:7� P t

Wind-Each ð20ÞBy considering N max

Wind ¼ NmaxSol ¼ 100, DHS finds NWind = 2,

NSol = 37 and NBatt = 9 with the optimal cost of 7178.70$; by considering Eq. (18) and N max

Sol ¼ 100, DHS findsNWind = 1, NSol = 98 and NBatt = 13 with the optimal costof 7988.24 $. This problem can be solved by DHS algo-rithm by considering different constraints.

5. Conclusion

This paper develops a discrete harmony search-basedmethod for finding the optimum size of a wind–photovol-taic hybrid energy system. In the proposed methodology,the integer decision variables can be efficiently handled.On a benchmark problem, it is seen that DHS not only pro-duces promising results but also finds the optimum solu-tion in less than one second. It can be concluded that

DHS could be an efficient candidate for optimizing the sizeof hybrid energy systems.

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