Modeling and Optimal Sizing of Hybrid Renewable Energy System

Rachid Belfkira*, Cristian Nichita, Pascal Reghem, Georges Barakat

GREAH, Groupe de Recherche en Electrotechnique et Automatique du Havre University of Le Havre, 25 rue Philippe Lebon, BP 540

76058 Le Havre, France Tel.: +33 / (0) 232744331 Fax: +33 / (0) 232744348

* e-mail: rachid.belfkira@univ-lehavre.fr e-mail: georges.barakat@univ-lehavre.fr

Abstract This paper presents a new methodology of sizing optimization of a stand-alone hybrid renewable energy system. The developed approach makes use of a deterministic algorithm to minimize the life cycle cost of the system while guaranteeing the availability of the energy. Firstly, the mathematical modeling of the principal elements of the hybrid wind/PV system is exposed showing the main sizing variables. Then, the deterministic algorithm is presented and implemented to minimize the objective function which is equal to the life cycle cost of the hybrid system and finally, the obtained results are exposed and discussed.

Keywords Renewable energy systems, energy storage, power supply, modeling.

I. INTRODUCTIONAround two billion people world-wide do not have

access to electricity services, of which the main share in rural areas in developing countries. Renewable energy resources are a favorable alternative for rural energy supply [1].

Renewable energy sources essentially have unpredictable random behaviors. However, some of them, like solar radiation and wind speed, have complementary profiles. Stand-alone hybrid power systems (Fig. 1) usually take advantage of this particular characteristic combining photovoltaic (PV) panels and wind turbines (WT).

Because of the intermittent solar irradiation and wind speed characteristics, which highly influence the resulting energy production, the major aspects in the design of PV and wind generator (WG) power generation systems are the reliable power supply of the consumer under varying atmospheric conditions and the corresponding total system cost. Then it is essential to select the number of PV modules, WGs and batteries, and their installation details such that power is uninterruptedly supplied to the load and simultaneously the minimum system cost is achieved [2].

The use of renewable energy technology to meet the energy demands has been steadily increasing over the years.

Several research tasks concerning the design and the sizing of the hybrid systems were carried out. In [3], based on the available hourly average data of wind speed,

insolation, and the power demand, the generation capacity is determined to best match the power demand by minimizing the difference between generation and load (P) over a 24-hour period. The objective function to be minimized is the sum of the annual cost of the capital over the life of the generating system and its annual maintenance cost. The iterative procedure is adopted for selecting the wind turbine size and the number of PV panels needed for a stand-alone system to meet a specific load.

An alternative methodology for the optimal sizing of stand-alone PV/WG systems has been proposed by Koutroulis et al. [2], which the purpose is to suggest, among a list of commercially available system devices, the optimal number and type of units ensuring that the 20-year round total system cost is minimized subject to the constraint that the load energy requirements are completely covered, resulting in zero load rejection. The 20-year round total system cost is equal to the sum of the respective components capital and maintenance costs. The decision variables included in the optimization process are the number and type of PV modules, WGs and battery chargers, the PV modules tilt angle, the installation height of the WGs and the battery type and nominal capacity. The minimization of the cost (objective) function is implemented employing a genetic algorithms approach.

In [4] the authors have developed the HOGA program (Hybrid Optimization by Genetic Algorithms) to calculate the optimal configuration of the hybrid PV-Diesel system. This optimal configuration is described very precisely: the number and the type of PV panels, the number and the type of batteries, the inverter power, the Diesel generator power, the optimal control strategy of the system with its parameters, the Total Net Present Value (cost of the investments plus the discounted present values of all future costs) of the system and finally, the number of running hours for the Diesel generator per year.

Chedid and Rahman have used linear programming techniques to determine the optimal sizes of the PV and WG power sources and the batteries by minimizing the system total cost function which consists of both initial cost and yearly operation and maintenance costs [5].

In this paper, the proposed optimization procedure is based on a dynamic evaluation of the wind and solar energetic potential based on statistical models of wind speed and solar radiation of the site of production. This

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978-1-4244-1742-1/08/$25.00 c 2008 IEEE

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dynamic evaluation of the energetic potential of the site permits the introduction of new constraints making the optimization procedure more flexible like the maximum acceptable time of energy unavailability and the minimum power level authorised regarding the power demand. Consequently, this approach results in a more realistic optimization.

II. HYBRID SYSTEM MODELING

A. Wind Turbine Model Using the wind speed at a reference height hr from the

database, the velocity at a specific hub height for the location is estimated on an hourly basis throughout the specified period through the following expression [2]

( ) ( )

=

rr h

h.tvtv (1)

where: v is the wind speed at projected height h,vr is wind speed at reference height hr , is the power-law exponent (~1/7 for open land). In function of this wind speed, the model used to

calculate the output power, PWT(t) (W), generated by the wind turbine generator is as follows:

( )( )3 <

< 0

R ci r

WT R r co

a.v t b.P v v vP t P v v v

otherwise

0) and discharging process (P(t)=

(11)

where fiRn are the objective functions and hkRp, gjRqare respectively the equality and the inequality constraints.

One of the major steps of the optimization process consists in the minimization of the objective functions. The optimization algorithms are generally divided into two groups: deterministic and stochastic. Many researchers have recently proved that the DIRECT algorithm is an effective deterministic algorithm to find the global optimum of the problem (11).

Developed by Jones et al. [13] and acronym for DIviding RECTangles, the DIRECT algorithm is a deterministic global optimization technique that is used to find the minimum of a Lipschitz continuous function without knowing the Lipschitz constant. The objective function and constraints must be Lipschitz-continuous in the research space , satisfying

( ) ( ) 212121 x,xxxLxfxf (12) This assumption means that the rates-of-change of the

objective function and constraints are bounded. Traditionally, when this assumption (12) is satisfied, the global optimization problem was solved by the Lipschitz optimization method, which had been considered as a practical and deterministic approach to many science and engineering problems for several decades.

DIRECT evolved from the one-dimensional Piyavskii-Shubert algorithm and was further extended from one dimension to multiple dimensions by adopting a center-sampling strategy. Its corresponding 1-D description contrasted with Piyavskii-Shuberts algorithm can be found in [14]. Here, only the multidimensional DIRECT algorithm, which is of more interest for our application, is described. DIRECTs behavior in multiple dimensions can be viewed as taking steps in potentially optimal directions within the entire design space. The potentially optimal directions are determined through evaluating the objective function at center points of the subdivided boxes. The multivariate DIRECT algorithm can be described by the following steps [14] 1) Normalize the search space to unit hypercube. 2) Sample the center point c1 of the hypercube; Evaluate f

(c1). Set fmin = f (c1), m = 1 (evaluation counter), and t= 0 (iteration counter).

3) Identify the set S of potentially optimal boxes. 4) Select any box j S.5) Divide the box j as follows:

a) Identify the set I of dimensions with the maximum side length . Let equal one-third of this maximum side length ( = 1/3 ).

b) Sample the function at the points c ei, for all iI, where c is the center of the box and ei is the ithunit vector.

c) Divide the box j containing c into thirds along the dimensions in I, starting with the dimension with the lowest value of wi = min{f(c+ei), f(cei)}, and continuing to the dimension with the highest wi. Update fmin, xmin and m.

1836 2008 13th International Power Electronics and Motion Control Conference (EPE-PEMC 2008)

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6) Set S = S {j}. If S = GO TO Step 4. 7) Set t = t + 1. If iteration limit or evaluation limit has

been reached, stop. Otherwise, GO TO Step 3.

This algorithm has been applied to optimize a hybrid wind/PV system. In the developed method, the DIRECT optimal sizing methodology outputs the optimum numbers and the types of the components of the hybrid wind/PV system, ensuring that the system total cost is minimized subject to the constraint that the load energy demand is completely covered.

The optimization procedure is achieved by minimizing the total cost function consisting of the sum of the individual system devices capital, the 20-year round maintenance costs and the installation costs

( )( )(

)(

1

1

1

20

20 +

20

PV

WT

BAT

C PV , p W T BAT , p

ni i i iPV PV PV I ,PV

in

j j j jWT WT WT I ,W T

j

j j jh hm I ,h

nk k kBAT BAT I ,BAT

kkBAT BAT

F N , N ,N

N . C .M C

N . C .M C

C .C C

N . C C

y C

=

=

=

= + +

+ +

+ + +

+ +

+

( )

( ) ) 20 1k k

I ,BAT

k kBAT BAT

C

y .M

+

+

(13)

where NPV,p, NWT and NBAT,p represent the sizing variables, where NPV,p is the total number of parallel PV strings, NWTis the total number of wind turbines and NBAT,p is the total number of parallel battery strings, nPV, nWT, nBAT are the total numbers of PV panel types, wind turbine types and battery units types, respectively, and CiPV, CjWT, CkBAT are the corresponding capital costs (), MiPV, MjWT, MkBAT are the corresponding maintenance costs per year (/year) and CiI,PV, CjI,WT and CkI,BAT are the corresponding installation costs (). Cjh is the WT tower capital cost (), Cjhm is the WT tower maintenance cost per year (/year), CjI,h is the WT tower installation cost () and ykBAT is the expected number of battery replacements during the 20-year system operation, because of limited battery lifetime. The costs of converters and of other components are included in the installation cost. NiPV = NiPV,pNiPV,s is the total number of PV panels of type i, and NkBAT = NkBAT,pNkBAT,s is the total number of batteries of type k in the battery bank.

The minimization of the objective function is subject to the constraints that the power produced by the system is equal to the power demanded by the load and the state of charge of the battery bank is limited between SOCmin and SOCmax as follows

( ) ( )( )

P L

min max

P t P t

SOC SOC t SOC

=

(14)

where PP(t) is the power produced by the system and it is calculated as follows

( ) ( ) ( )P reP t P t P t= (15) where Pre(t) is the power produced by the renewable resources as follows

( ) ( ) ( )1 1

PV W T

PV WT WT

n nj ji i

re PVi j

P t N .P t N .P t= =

= + (16)

and P(t) > 0 during the charging process of the battery and P(t) < 0 in the discharging process as calculated in the eq. (6).

Additional constraints to be imposed are

1

1

1

i iPV ,p PV , p max

j jWT W T maxk kBAT , p BAT , p max

N N

N N

N N

(17)

where NiPV,pmax, NjWTmax and NkBAT,pmax were calculated according to the nominal power of PV panel, wind turbine and nominal capacity of battery, respectively, and the peak of the load demand.

IV. OPTIMIZATION RESULTS AND DISCUSSIONThe optimization methodology developed above was

applied to sizing a hybrid energy system supplying a variable load. In fig. 3, the hourly power demand during a day is presented. This power reaches the maximum values between 13 h and 15 h and between 21 h and 23 h in the day; this is due to the utilization of the household electrical appliances in these periods.

For the site of Fecamp in the region of Haute-Normandie, in France, where the hybrid energy system is assumed to be installed, a long-term data of wind speed and ambient temperature were recorded for every hour of the day during the period of six months, are used for the calculation of the power produced by the hybrid system and are plotted in fig. 4. The wind speed was measured at a 40 meters height which is considered as the reference height for the site (cf. eq. (1)).

In this example, two types of each component of the hybrid wind/PV system have been used. The specifications and the related capital, maintenance and installation costs of each component type, which are input to the optimal sizing procedure, are listed in Tables I-III. The maintenance cost of each unit per year and the installation cost of each component have been set at 1% and 10% respectively of the corresponding capital cost.

The serial connection numbers of the two types 1 and 2 of the PV arrays and of the batteries which are determined by the operating voltage of the system which is chosen to be equal to a standard value of 48 V, take respectively the values: N1PV,S = 2, N2PV,S = 3, N1BAT,S = 4 and N2BAT,S = 4. The expected battery lifetime has been set at 3 years with proper maintenance resulting in ykBAT = 6. Since the tower heights of wind turbines affect the results significantly, 30 meter high tower is chosen.

Using all these data and parameters, the minimization of the system total cost is achieved by selecting an appropriate system configuration. The optimal sizing results, consisting of both the device types and their number, are shown in table IV. From these results, one can deduce that the rate of penetration of the wind power

2008 13th International Power Electronics and Motion Control Conference (EPE-PEMC 2008) 1837

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mohammadSticky Note :1- PV 2- 3-

mohammadSticky Note :1- 2- 20 3-

mohammadSticky NoteC----> capitalM----> maintenanceCi----> installation cost

mohammadSticky NoteM ---> . 20 20

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is higher than that of the PV power; this is due to the highly speed of the wind of the site of Fecamp compared to the solar radiation.

Fig. 5 presents the variation of the system total cost (fitness function) during the optimization procedure. It can be noted that a near optimal solution was derived during the early stages of the function evaluations.

Fig. 6 shows the state of charge (SOC) of the battery bank for the obtained configuration. One can deduce that

the SOC reaches the lower limit of discharge nearly between 300 h and 500 h and between 860 h and 2100 h, this is due to the low power produced by the renewable resources in these periods. Also, one can verify that the state of charge of the battery bank can never exceed the permissible maximum value, SOCmax (100% of SOC) and can never be below the permissible minimum value, SOCmin (20% of SOC).

TABLE IPHOTOVOLTAIC PANELS SPECIFICATIONS

Type 1 2 Voc (V) 32.6 21 Isc (A) 7.87 7.22

Vmax (V) 25.9 17 Imax (A) 6.95 6.47

NCOT (C) 45.9 43 Capital cost () 603 519.14

Installation cost () 60.3 51.9 Maintenance cost per year (/year) 6.03 5.19

TABLE IIWIND TURBINES SPECIFICATIONS

Type 1 2 Power rating (W) 10000 7500

vr (m/s) 13.8 13.8 vci (m/s) 3.1 3.1 vco (m/s) 25 25

Capital cost () 20682 16978 Installation cost () 2068.2 1697.8 Maintenance cost per year (/year) 206.82 169.78

Tower capital cost () 741 741

Tower installation cost () 7.41 7.41

Tower maintenance cost per year

(/year)74.1 74.1

TABLE IIIBATTERIES SPECIFICATIONS

Type 1 2 Nominal capacity

(Ah) 100 230

Voltage (V) 12 12

DOD (%) 80 80

Efficiency (%) 80 80

Capital cost () 126 264 Installation cost () 12.6 26.4 Maintenance cost per year (/year) 1.26 2.64

TABLE IVOPTIMAL SIZING RESULTS

Type 1 2

NPV,p 1 0

NWT 1 0

NBAT,p 1 1

Cost () 41242

0 500 1000 1500 2000 2500 3000 3500 40000

5

10

15

20

25

30

Win

d sp

eed

(m/s

ec)

Number of hour

0 500 1000 1500 2000 2500 3000 3500 4000-5

0

5

10

15

20

25

30

Ambi

ant t

empe

ratu

re [C

]

Number of hour

Fig. 4. Hourly mean values during a period of seven months of meteorological conditions: (a) wind speed and (b) ambient

temperature

5 10 15 200

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Number of hour

P loa

d [W

]

Fig. 3. Hourly demand power in a day

(a)

(b)

1838 2008 13th International Power Electronics and Motion Control Conference (EPE-PEMC 2008)

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V. CONCLUSIONIn this paper, a methodology of sizing a stand-alone

hybrid wind/PV system using the DIRECT algorithm has been explained. This developed methodology is based on the use of the collection of six months data of the wind speed and the ambient temperature on one hand and on the other hand on the estimation of the solar irradiation for the site under consideration.

The optimum numbers of wind turbines, PV panels and batteries depend on the particular site, load profile and the specifications and the related cost of each component of the hybrid system.

REFERENCES[1] T. Gul, Integrated analysis of hybrid systems for rural

electrification in developing countries, TRITA-LWR Master Thesis, Stockholm 2004.

[2] E. Koutroulis, D. Kolokotsa, A. Potirakis, K. Kalaitzakis, Methodology for optimal sizing of stand-alone photovoltaic/wind-generator systems using genetic algorithms, Solar Energy, 2006.

[3] W. D. Kellog, M. H. Nehir, G. Venkataramanan, V. Gerez, Generation unit sizing and cost analysis for stand-alone wind, photovoltaic, and hybrid wind/PV systems, IEEE Trans. on Energy Conversion, vol. 13, no. 1, March 1998.

[4] R. Dufo-Lopez, J. L. Bernal-Agustin, Design and control strategies of PV-Diesel systems using genetic algorithms, Solar Energy, vol. 79, pp. 33-46, 2005.

[5] R. Chedid, S. Rahman, Unit sizing and control for hybrid wind-solar power systems, IEEE Trans. on Energy Conversion, vol. 12, no. 1, pp. 79-85, 1997.

[6] F. Lasnier, T. G. Ang, Photovoltaic engineering handbook.Bristol, England: A. Hilger, 1990.

[7] T. Markvar, Solar Electricity, 2nd ed, J. Wiley & Sons, 2000. [8] H. Yang, L. Lu, W. Zhou, A novel optimization sizing model for

hybrid solar-wind power generation system, Solar Energy, vol. 81, pp 76-84, 2007.

[9] J. Bernard, Energie solaire: calculs et optimisation, Ellipses-Paris, 2004, pp. 53-93.

[10] G. Seeling-Hochmuth, Optimisation of hybrid energy systems sizing and operation control, Dissertation in Candidacy for the Degree of Dr.-Ing, University of Kassel, October 1998.

[11] B. S. Borowy, Z. M. Salameh, Methodology for optimally sizing the combination of a battery bank and PV array in a Wind/PV hybrid system, IEEE Trans. on Energy Conversion, vol. 11, no. 2, June 1996.

[12] J. Azzouzi, R. Belfkira, N. Abdel-Karim, G. Barakat, B. Dakyo Design optimization of an axial flux PM synchronous machine: comparison between DIRECT method and GAs method, EPE-PEMC, August 30September 1, 2006.

[13] D. R. Jones, C. C. Perttunen, B. E. Stuckman, Lipschitzian optimization without the Lipschitz constant, J. Optim. Theory Appl, vol. 79, pp. 157181, 1993.

[14] M. Bjrkman, K. Holmstrm, Global optimization using the DIRECT algorithm in Matlab, AMO - Advanced Modeling and Optimization, vol. 1 no. 2, 1999.

0 500 1000 1500 2000 2500 3000 3500 4000

20

30

40

50

60

70

80

90

100

Number of hour

SOC

[%

]

Fig. 6. Hourly variation of SOC of battery bank

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 110000

0.5

1

1.5

2

2.5

3

3.5x 105

Number of Function Evaluations

Tota

l Co

st ()

Fig. 5. The system total cost during the DIRECT optimization

2008 13th International Power Electronics and Motion Control Conference (EPE-PEMC 2008) 1839

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mohammadSticky Notethe SOC reaches the lower limit of discharge nearly between 300 h and 500 h and between 860 h and 2100 h, this is due to the low power produced by the renewable resources in these periods.

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