INTRODUCTION
Global environmental concerns and the everincreasing need for energy, coupled with a steady progress in renewable energy technologies are opening up new opportunities for utilization of renewable energy resources. Hybrid wind/photovoltaic (PV) power systems are one important type of standalone renewable energy power systems. The hybrid combination of PV panels and Wind Turbine Generators (WTGs) improves overall energy output and reduces energy storage requirements (Habib et al., 1999).
Proper design of standalone renewable energy power systems is a challenging
task, as the coordination among renewable energy resources, generators, energy
storages and loads is very complicated. Generally the main objectives of the
optimization design are maximization of power reliability and minimization of
cost, so this is a multiobjective optimization problem. There are a few singleobjective
optimization methods, which convert one objective into constraint, for sizing
standalone renewable energy power systems (Habib et al., 1999; Borowy
and Salameh, 1996; Willis and Scott, 2000; Kaldellis, 2004; McGowan and Manwell,
1999; Ai et al., 2003; Elhadidy and Shaahid, 1999; AlAshwal and Moghram,
1997; Kellogg et al., 1998). One representative method, the tangency
method, is to optimize the size of PV panels and the capacity of batteries for
given type and size of WTGs and tilt angle of PV panels (Borowy and Salameh,
1996; Willis and Scott, 2000). But the problems remained are that how to determine
the optimal type and size of WTGs and the optimal tilt angle of the PV panels
and how to solve the multiobjective optimization problems.
We propose a method in this study, which can not only take the type and size of WTGs, the tilt angle and size of PV panels and the capacity of batteries as decision variables, but also give the Paretooptimal solutions for higherlevel decision. We employ one of the multiobjective evolutionary algorithms (MOEAs), the elitist nondominated sorting genetic algorithm (NSGAII) (Deb, 2001), to size standalone renewable energy power systems.
LOSS OF POWER SUPPLY PROBABILITY
System configuration: The configuration of standalone hybrid wind/PV power systems is shown in Fig. 1. The present study, we investigated the case that a system has only one type of WTGs.

Fig. 1: 
Schematic of standalone hybrid wind/PV power systems 
Calculation of wind and solar energy: The wind speed to a particular hub height is calculated by the equation in (Borowy and Salameh, 1996). It is necessary to estimate the solar radiation incident on a tilted solar panel surface when only the total radiation on a horizontal surface is known. The HayDaviesKlucherReindl (HDKR) anisotropic model described by Duffie and Beckman (1991) is employed to calculate the incident radiation on the tilted PV panel surface.
Models of system components: The power curve is represented by a piecewise cubic spline interpolation in the calculation. The output power from a PV panel can be calculated by an analytical model given by Lasnier and Ang (1990), in which a Maximum Power Point Tracker (MPPT) is considered.
Lead acid batteries are main energy storage devices in standalone power systems. The battery charge efficiency is set equal to the roundtrip efficiency and the discharge efficiency is set equal to 1. The maximum battery life can be obtained if the Depth Of Discharge (DOD) is set equal to 3050%.
The MPPT, the battery controller, the inverter and distribution lines are assumed to have constant efficiencies. Assume the efficiencies of the MPPT, the battery controller and distribution lines as 1 and that of the inverter as 0.9.
Loss of power supply probability: The Loss of Power Supply Probability (LPSP) (Borowy and Salameh, 1996), which is defined in terms of the battery State Of Charge (SOC), is the power reliability index of a system. The LPSP can be defined as the longterm average fraction of the load that is not supplied by the standalone power system. In terms of the SOC, the LPSP can be defined as:
where E_{B,t} is energy stored in batteries at any hour t and E_{Bmin} is battery minimum allowable energy level.
Operation simulation: The logistic model and time series method (Manwell et al., 1998) are employed for simulation studies. Logistical models are used primarily for longterm performance predictions, for component sizing and for providing input to economic analyses. An essential feature of the time series method is that it employs an energy balance approach within each time step. This assures that energy is conserved throughout the entire simulation and that the model is internally consistent. In particular the sum of all energy sources must equal the sum of all sinks.
The simulation period is 1 year and the time step is 1 h. The load, wind energy and solar energy are assumed to be constant during a time step. When the available energy generated and stored in batteries is insufficient to satisfy the load demand E_{L,t} for hour t, the deficit is called Loss of Power Supply (LPS). The LPSP for a considered period T is the ratio of all LPS_{t} values for that period to the sum of the load demand, as defined by:
MULTIOBJECTIVE OPTIMIZATION USING NSGAII
Problem description: Maximization of power reliability and minimization of cost are two conflicting objectives of sizing standalone hybrid wind/PV power systems. The total capital cost of WTGs, PV panels and batteries, C_{WPB}, can be taken as the cost index (Borowy and Salameh, 1996; Duffie and Beckman, 1991; Lasnier and Ang, 1990). The type and size of WTGs, the tilt angle and size of PV panels and the capacity of batteries have a considerable influence upon the power reliability and capital cost and can be optimized. The multiobjective optimization problem is as follows:
where:
C_{bat} 
Cost of the battery, 
C_{PV} 
Cost of the PV panel, 
C_{WTG} 
Cost of the WTG, 
N_{bat} 
Number of the batteries, 
N_{bat_p} 
Number of the batteries in parallel, 
N_{bat_s} 
Number of the batteries in series, 
N_{PV} 
Number of the PV panels, 
N_{PV_P} 
Number of the PV panels in parallel, 
N_{PV_s} 
Number of the PV panels in series, 
N_{WTG} 
Number of the WTGs, 
Type 
Type of WTGs. 
In the optimization model, the tilt angle is a variable of the HDKR model that is used in computing LPSP. The fixed tilt angle (Hartley et al., 1999) β is set as an integer in degrees. The optimal fixed tilt angle rests on geographical and meteorological conditions of the location and the load characteristics as well as the system configuration. The range of optimization of β is [0°, 90°] for southfacing panels.
NSGAII: Although the field of research and application on multiobjective optimization is not new, the use of MOEAs in various engineering and business applications is a recent phenomenon (Deb et al., 2004). MOEAs have an edge over the classical methods in that they can find multiple Paretooptimal solutions in one single simulation run. Deb and his students suggested NSGAII in 2000, which has been shown to outperform other current elitist MOEAs on a number of difficult test problems (Deb et al., 2000). NSGAII uses (i) a faster nondominated sorting approach, (ii) an elitist strategy and (iii) no niching parameter. Diversity is preserved by the use of crowded comparison criterion in the tournament selection and in the phase of population reduction (Deb, 2001).
APPLICATION EXAMPLE
The location of Boston, Massachusetts with Latitude 42°22'N is chosen. The load curve of a typical house (Borowy and Salameh, 1996) is shown in Fig. 2. The typical meteorological year data sets (TMY2s) contain hourly values of solar radiation and meteorological elements for a oneyear period (Marion and Urban, 1995). The data of extraterrestrial horizontal radiation, global horizontal radiation, diffuse horizontal radiation, temperature and wind speed of station Boston are utilized. Generally, the ground reflectance is 0.2.
The FD series WTGs with rated power of 1, 3, 5, 7.5 and 10 kW made by Tianfeng
Green Energy Company of China were considered. The power curves of the WTG are
shown in Fig. 3.

Fig. 2: 
A typical load profile in Boston 

Fig. 3: 
The power curves of the WTGs (The symbols represent data sampled
from the power curve graphs given by the manufacturer) 
A 50 W_{peak} PV panel made by Yunnan Semiconductor Device Factory
in China was used for this simulation study. The capacity of a single battery
used was 200 Ah. That battery has a roundtrip efficiency of 0.7 and DOD = 50%.
The form of the individual of the population is [Type N_{WTG }β N_{PV_p} N_{bat_p}], a integervalued vector of 5 values. The LPSP of every individual is calculated by simulation of 8760 h. For the realcoded NSGAII, the discrete version of Simulated Binary Crossover (SBX) operator and the realparameter mutation operator are used (Deb et al., 2000). When a prespecified iteration count (N = N_{max}) is reached, NSGAII is terminated. N_{max} = 200 and a population size of N_{pop} = 50 are used. The crossover and mutation probability of p_{c} = 0.9 and p_{m} = 0.1 are used for the realcoded NSGAII.

Fig. 4: 
Obtained NSGAII solutions for the C_{WPB} and LPSP
optimization are compared with the constraint method (Epsilon) solutions.
Initial solutions of NSGAII and the benchmark solution are also shown.
(b) is part of (a) 
According to the voltage, let NPV_s = 3 and Nbat_s = 24. The solutions termed
as Initial in Fig. 4 denote the objective vectors with which
the NSGAII search process is started. The figure indicates that the random
solutions (within the chosen variable bounds) are far from being close to the
optimized front (marked as NAGAII). The figure clearly shows that a wide range
of distribution in LPSP and C_{WPB} values are obtained. Part of the
resulting LPSPC_{WPB} tradeoff is nonconvex and it can be inferred
that compared to the benchmark solution (marked as ‘Benchmark’ in
the figure) there exist better solutions.
In order to verify whether the obtained NAGAII solutions are actually close
to the true Paretooptimal front of this problem, we use the tangency method.
The method also belongs to theconstraint method that can find Paretooptimal
solutions whether the objective space is convex or nonconvex.

Fig. 5: 
The optimal configuration of [N_{PV_p} N_{bat_p}]
= [4 6] for the given conditions as Type = 1, N_{WTG} = 6, β
= 28° and LPSP_{epsilon} = 0.0096 

Fig. 6: 
β = 28° is optimal for the configuration [Type N_{WTG
}N_{PV_p} N_{bat_p}] = [I 6 4 6] 
The method is to convert the first objective (Minimization of LPSP) into an
additional constraint as LPSP = LPSP_{epsilon} and minimize only the
second objective. The procedure is:
We take an NSGAII solution [Type N_{WTG }β N_{PV_p} N_{bat_p}]
= [I 6 28° 4 6], LPSP = 0.0096, C_{WPB} = 143160 (Yuan) for example.
Let Type = 1, N_{WTG} = 6, β = 28° and LPSP_{epsilon}
= 0.0096, we get [N_{PV_p} N_{bat_p}] = [4 6], as shown in Fig.
5.

Fig. 7: 
The LPS of the configuration [Type N_{WTG }β
N_{PV_p} N_{bat_p}] = [I 6 28° 4 6] 

Fig. 8: 
The SOC of the battery bank of the configuration [Type N_{WTG
}β N_{PV_p} N_{bat_p}] = [I 6 28° 4 6] 
The minimum cost is at the tangent point of the cost line and the curve that
represents the relationship between the size of PV panels and capacity of batteries.
The slope of the cost line is:
Enumerate the cases of every Type, N_{WTG} and β, we get the optimal configuration [Type N_{WTG }β N_{PV_p} N_{bat_p}] = [I 6 28° 4 6], C_{WPB} = 143160 (Yuan) for LPSP_{epsilon} = 0.0096. So [Type N_{WTG }β N_{PV_p} N_{bat_p}] = [I 6 28° 4 6] is a Paretooptimal solution to this optimization problem. Figure 4 marks the 5 solutions as ‘Epsilon’ solutions obtained by 5 independent runs of the constraint method, each performed with a different LPSP_{epsilon} value. Since these solutions are found to lie on or near the nondominated front obtained by NSGAII, it can be stated that the nondominated front found by NSGAII is the true Paretooptimal front.
For the configuration [Type N_{WTG } N_{PV_p} N_{bat_p}] = [I 6 4 6], the optimal tilt angle is 28°, as shown in Fig. 6. The LPS of the configuration [Type N_{WTG }β N_{PV_p} N_{bat_p}] = [I 6 28° 4 6] is shown in Fig. 7 and the SOC of the battery bank is shown in Fig. 8. The optimal tile angle is less than the Latitude value means that the PV panels can generate more power to complement the insufficient power output of the WTGs in summer.
Usually, one larger WTG is chosen. We select one FD3KW WTG with rated power 3 kW and use the tangency method to size the power system. Let β = 42°. The result is [N_{PV_p} N_{bat_p}] = [8 7], LPSP = 0.0091, C_{WPB} = 167460 (Yuan). This configuration, which is taken as the benchmark, is not a Paretooptimal solution to this optimal design because the FD3KW WTG is not as economical as the FD1KW WTG.
CONCLUSIONS
Sizing of standalone hybrid wind/PV power systems is a multiobjective optimization problem with two objectives being maximization of power reliability and minimization of cost. The LPSP is obtained by operation simulation of the system. A multiobjective evolutionary algorithm, NSGAII, is utilized and it can find solutions on or near the true nondominated front of the problem. This has been validated by solving the multiobjective problem with a constraint method. When using NSGAII, the decision variables are the type and size of WTGs, the tilt angle and size of PV panels and the capacity of batteries. The advantage of using NSGAII is that it can find multiple optimized solutions in a single run.
ACKNOWLEDGMENTS
The authors would like to acknowledge Prof. Kalyanmoy Deb and his lab (KanGAL), for the NSGAII source code and acknowledge National Renewable Energy Laboratory (NREL) for the Typical Meteorological Year (TMY) data sets. This research was supported by the Research Project (No. 2004EA105003) of China.