Effects of Reliability Index on Optimal Configuration of Hybrid Solar/Battery Energy System by Optimization Approach: A Case Study

Stand-alone hybrid energy systems based on solar and energy storage are an effective option for rural areas to meet the load demand. The objective of the current work is to the optimal configuration of a stand-alone hybrid photovoltaic/battery energy storage system with the help of an efficient metaheuristic algorithm, improved harmony search, to supply electrical of a residential load in Iran. The objective function is a minimization of total life cycle cost (TLCC) subject to the reliability index (loss of load probability). The optimal configurations of hybrid systems are compared in respect of different losses of load probability (0 to 20%). Sensitivity analysis and effects of economic parameters based on photovoltaic and battery prices are carried out to study the possibility of the suggested scheme. The results show that, by increasing the reliability index from 0 to 20%, the optimal number of panels and batteries decreases by 52 and 1202. Also, it is found that the TLCC of the system and cost of system components are increased by decreasing of the reliability index value.


Introduction
The provision of electricity is one of the key elements for the economic growth of a country. Around 17% of people in the countries, specifically those living in isolated regions, still have no contact with electricity [1,2]. Due to the high cost of network transmission to remote areas, mainly, diesel generators are used to supply load demands to remote locations. Due to the high cost of fuel, environmental pollutants, and the shortage of fossil fuels, diesel power generation is not always a beneficial and cost-effective solution. To solve this problem, the use of renewable energy has been considered by many researchers in recent years. Renewable energy systems, especially solar PV systems, are an effective solution for stand-alone locations [3].
However, uncertainty in solar radiation and the dependence of solar systems on the climate is a problem for pro-viding continuous load in remote areas. So it is a viable solution for stand-alone locations to consider a PV system with an energy storage unit initially. Among storage systems, the battery is one of the most popular [4][5][6][7]. In this regard, it is necessary to define the optimal configuration of power scheme components to remote areas to supply the load demand with the minimum cost and maximum reliability. Therefore, efficient modeling and a powerful optimization method to solve these problems are essential.
Several studies in the literature have focused on investigating mathematical modeling, optimal sizing, and technoeconomic analysis of hybrid energy schemes based on solar energy. Javed et al. [2] used a genetic algorithm to optimize a hybrid solar/wind system with storage for an isolated island. The results were compared with that of the HOMER (hybrid optimization of multiple energy resources) software. Das et al. [8] obtained a techno-economic optimal design of a stand-alone hybrid solar/biogas/energy storage scheme for a radio transmitter station in India by metaheuristic optimization techniques. Saedpanah et al. [9] used a multiobjective optimization model for designing an off-grid photovoltaic system in remote areas of Iran. Asrami et al. [10] used three scenarios based on a genetic algorithm to find the optimal solution to utilize PV systems for residential buildings in urban areas. Nagapurkar and Smith [11] developed a methodology for optimizing small-scale microgrids based on solar/wind/battery/biodiesel/hydrogen systems located in the US cities of Tucson using a genetic algorithm. Yu et al. [12] presented an optimization method based on an adaptive marine predator algorithm to optimize a hybrid solar/diesel/storage system to meet the load demand of a remote location in Hoxtolgay, China. Ashraf et al. [13] proposed an optimization method for optimal configuration of a hybrid PV/diesel/battery energy system to a practical case study in Gobi Desert, China. Cho and Valenzuela [14] proposed an algorithm based on integer Nelder-Mead for designing the capacities of a residential stand-alone hybrid solar/battery system. Mukhopadhyay and Das [15] used particle swarm optimization for optimal sizing of stand-alone hybrid PV/battery systems in remote areas. Jiang et al. [16] proposed an optimization method to design the photovoltaic/battery energy storage systems with multiple types of batteries to minimize the total cost. Anoune et al. [17] presented a heuristic approach based on a genetic algorithm to the sizing-optimization of a photovoltaic-wind-battery based on hybrid system to minimize the total cost. Ajiwiguna et al. [18] developed an algorithm for optimizing the capacities of the stand-alone battery-less PV-reverse osmosis system and their capacity combination to find the lowest water cost.
Previously mentioned researches have mainly focused on optimization hybrid energy systems based on solar energy with the lowest total cost. Some studies have also examined the impact of the reliability index (RI) on the hybrid energy system. Previously, studies usually used the HOMER software tool to perform techno-economic analyses based on the input information of hybrid systems. The HOMER software tools allow a quick hybrid energy system assessment, but changes in the modeling of hybrid system components are limited. The ability to modify mathematical models and input information for different renewable energy technologies is restricted in the HOMER tool. Also, in previous studies, a comprehensive analysis of the effects of critical economic parameters and reliability index on the optimization of hybrid systems by an efficient metaheuristic algorithm is rarely seen.
In this paper, an optimization model of a stand-alone PV/battery energy storage scheme to the optimal configuration of the hybrid system to supply electrical load demands is presented. The methodology followed in this study considered a remote area in Iran residential communities, namely, Rafsanjan. For optimal sizing of hybrid system components, an improved metaheuristic algorithm based on harmony search is presented. The objective function is a minimization of total life cycle cost (TLCC) and loss of load probability as a reliability index. To study the possibility of the suggested hybrid system, effects of economic parameters based on photovoltaic and battery prices are presented. Also, the optimal configurations of the hybrid system are compared in respect of different reliability indexes.

System Modeling
The stand-alone hybrid energy systems include a solar photovoltaic panel, storage unite based on battery, an inverter/converter system, and other devices and cables. In this system, first, the power generated by solar panels satisfies the required load. After satisfying the load demand, the power generated from PV panels is used to charge the battery bank to supply the load when sunlight is unavailable. Then, the extra battery charge level is dumped. To optimize the scheme, all of the system components must first be exclusively modeled and then their optimal sizing calculated to meet the load demand. The full model of the stand-alone hybrid solar/battery scheme is shown in Figure 1.
2.1. The Model of Photovoltaic (PV). The proposed model for generating power of PV panel (p PV ) based on the solar radiation (R) and ambient temperature (T air ) has been written as follows [41]: where P R,PV refers to the rated power of the utilized panel. N T , T ref , and R ref refer to the temperature coefficient (here −3:7 × 10 −3 (1/°C)), reference temperature (here 25°C), and reference solar radiation (here 1000 W/m 2 ), respectively [42]. The temperature of the cell can be formulated as follows [41]: Here, NOCT refers to the normal operating cell temperature (°C). The total produced power by PV panels is P PV ðtÞ = N PV × p PV ðtÞ, based on number of panels (N PV ).

The Model of Battery Storage.
To increase the reliability of the hybrid system and save the surplus output power of PV, an energy storage unit (battery bank) has been used. When the output power of the PV system is higher than 2 International Journal of Photoenergy the required load, the energy storage unit will start charging. Energy storage is used, when the output power of the PV panels is insufficient to supply the load demand. At this time, the energy storage unit will start discharging. The state of charge of the battery storage unit during the time from t − 1 to t is given in Equations (3) and (4) [43,44].
Charging state: Discharging state: Here, ω, S BAT ðtÞ, η INV , η BC , η BDC , and E L refer to the rate of hourly self-discharge, state of charge of the battery at time t, the efficiency of the inverter, charging, and discharging of the battery (here 1), demand of load, respectively, and t is the hourly time (1 h) [43][44][45][46].

The Model of Inverter.
According to the output current of solar panels, a converter must be used to convert the direct current to alternating current. The output power of an inverter (P INV ) is obtained by the following Equation (12): where P D is the hourly demand.

The Objective Function and Constraints
According to the capital cost and replacement cost, which occur in the beginning and during the project lifetime, the capital recover factor (CRF) based on the interest rate (r) and system life span (n) is mathematically modeled as follows [47]: Due to the lifetime of the batteries and inverters (here 5 and 10 years, respectively), their replacement should be considered in the optimization during the project lifetime. The present worth of battery (C&R BAT ) and inverter/converter (C&R INV ) achieved as follows:    According to the capital and replacement cost of the components (PV, battery, and inverter), the total annual capital and replacement cost is mathematically achieved as follows: Here, N INV and N BAT represent the numbers of inverter and battery, respectively.
The annual O&M cost of the components (PV, battery, and inverter) is mathematically achieved as follows: 3.2. Constraints. To have a highly reliable system, the concept of the loss of load probability (LLP) must be consid-ered in the optimization, which is mathematically modeled as follows: Here, LLS denotes to loss of load supply which is achieved based on generated energy (E Gen ) as follows: and LLP ≤ RIwhere RI stands the highest allowable value of LLP. The numbers of PV panel and battery constraints and storage units are as follows:

Results and Discussion
The presented optimization model is considered to achieve a case study in Rafsanjan. Rafsanjan is a city in the north-west of Kerman province, Iran, with an altitude of 1,460 m which is placed at 30°24′ 24″ N 55°59′ 38″ E. The winters in Rafsanjan are cold and freezing as well as hot and dried in the summers. The ambient temperature usually differs from −17°C to 43°C. The case study includes around five homes located in a remote area of Rafsanjan, Iran. Actual data of solar insolation, ambient temperature, and typical load demand of the case study based on hourly distribution during a year are used in this study (8760 h). The residential load profile of the case study for five households is shown in Figure 2, in which the minimum and the maximum load demands of the system are 1.6 and 7.5 kW. The ambient temperature and solar insolation profiles of the studied area are presented in Figure 3.
The specifications of hybrid system components are given in Table 1.
In this study, the MATLAB software is used to implement the modified harmony search (HS) algorithm. The harmony search algorithm was first introduced in 2001 [54]. HS is a type of emerging metaheuristic optimization algorithm based on three operators, namely, pitch adjusting rule, harmony memory considering rule, and random search. The HS is trying to mimic the process of the musicians' improvisation. Because of exploitation and ease of application, the HS has drawn worldwide attention mainly. The employed HS in the present study is denoted in the previous studies [55]. In a modified harmony search algorithm, the parameters of pitch adjusting rate and bandwidth of generation are improved for adjusting the convergence rate of the method to the optimal solution, while them constant in the original algorithm. The modified harmony search algorithm used in this paper is the same as that proposed in [41]. The parameters of the modified proposed optimization method used in this paper as harmony memory considering rate and the maximum number of iterations are considered 0.9 and 3000, respectively. Also, the maximum and minimum of pitch adjusting rate are 1 and 0.1, respectively. The maximum and minimum of generation bandwidth are 1 and 0.01, respectively, which are determined by the atrial-and-error method. To provide valid results of the suggested algorithm, 30 independent runs are executed, and the optimal results are determined. In this optimal configuration of the hybrid solar/battery energy system, the control variables are numbers of PV panels (260-watt monocrystalline solar panel) and battery storage units (VMAX SLR155 12 V 155 Ah AGM deep cycle solar battery) that the minimum bound of these variables are set to 0 and also the maximum the bound of them are set to 200 and 20000, respectively. Figure 4 shows the flowchart of the suggested process. The optimal result of the suggested algorithm, in RI = 2%, shows that the optimal number of PV panels and battery storage is 172 and 1137, respectively. The best choice of TLCC and LLP is $111,280 and 1.8516%, respectively. Since changes in the reliability index affect the cost and the optimal number of components, it is necessary to examine the effects of the reliability index on the optimization of the system.  Figure 9: (a) The convergence characteristic of different reliability indexes on optimal hybrid solar/battery system; (b) zoom part. 6 International Journal of Photoenergy 4.1. Effects of the Reliability Index. The optimal configurations of hybrid solar/battery energy system for different reliability indexes (0 to 20%) are shown in Table 2. In RI = 0%, the optimal number of PV panels, battery storage, and TLCC is 184, 1511, and $146,070, respectively. It is found that by reducing the reliability index from 2 to 0%, the values of numbers of PV panels, battery storage, and TLCC increase to 12, 374, and $34,790, respectively. It can be seen that the LCC of the batteries and panels are $138,675 and $6530 for in this case, and also, the annual operation and maintenance cost and annual capital and replacement cost of the optimal hybrid system are $17,363 and $128,661, respectively. The O&M costs of the batteries and panels are $15,110 and $2,208, respectively. In RI = 5%, the optimal values of N PV , N BAT , TLCC, and LLP are 160, 858, $85,250, and 4.5051%, respectively. It is found that by increasing RI from 2 to 5%, the values of N PV , N BAT , and TLCC decrease to 7%, 25%, and 23%, respectively, and also show that the values of LCC PV , LC C BAT , C&R, and O&M of optimal system decrease to $425, $25,606, $23,098, and $2,934, respectively. In this case, the optimal values of O&M of the batteries and panels are $8,580 and $1,920, respectively. For RI equal to 10% and 12%, the optimal value of TLCC is decreased to $70,310 and $60,190, respectively, and the optimal numbers of PV and batteries are 153 and 147 and 698 and 590, respectively. In RI = 15%, the minimal TLCC is $45,260, and the optimal values of N PV , N BAT , and LLP are 140, 430, and 14.6732%, respectively. For RI equal to 18% and 20%, the optimal values of TLCC are decrease to $36,910 and $33,860, respectively, and the optimal C&R and O&M of system are 31,830 and 29,143$ and 5,075 and 4,719$, respectively. It can be seen that by increasing the reliability index, the optimal components of the hybrid system and TLCC are decreased, as shown in Figures 5 and 6. Variation in operation and maintenance cost and capital and replacement cost of the hybrid schemes in optimized situations versus reliability index is presented in Figure 7. In Figure 8, variation in cost criteria of the optimal system component (LCC of the batteries and panels and O&M of the batteries and panels) versus RI is presented. It is found that the cost criteria reduce by increment in the value of the RI. The convergence characteristic of different reliability indexes on optimal hybrid solar/battery scheme is presented in Figure 9, which shows that by reducing reliability index from 20 to 0%, the objective function value is increase from $33,860 to $146,070, respectively.

Sensitivity Analysis of the Economic Parameters.
The results of the variations of the PV panel unit cost on the optimal variables and cost criteria (in $) of the hybrid solar/battery scheme for RI = 2% are given in Table 3. It is observed that the value of LLP and TLCC of the hybrid system, for PV cost equal to 250 $/m 2 , is 1.9608% and $115,640, respectively. In this case, the optimal values of N PV and N BAT are 171 and 1136; also, the LCC of the panels and batteries are $10,287 and $104,259, respectively, and the optimal C&R and O&M of the system are $101,907 and $13,457, respectively.
In PV cost = 50 $/m 2 , the optimal number of PV panels, battery storage, and TLCC is 171, 1114, and $106,760, respectively. It is found that by reducing PV cost from 250 to 50$/m 2 , the values of TLCC, C&R, and LCC PV decrease to $8,880, $8,387, and $6,588, respectively. It can be seen that the TLCC, C&R, and LCC PV are $106,760, $93,520,  7 International Journal of Photoenergy and $3,699, for in this case, and also, the annual O&M costs of the optimal hybrid system and PV panels are $13,237 and $2052, respectively. In PV cost = 500 $/m 2 , the optimal values of TLCC and LLP are $126,570 and 1.7103%. It is found that by increasing PV cost from 200 to 500 $/m 2 , the values of TLCC and C&R increase to 9.5% and 10.7%, respectively. It can be seen that by increasing the PV panel unit cost, the TLCC and C&R of the hybrid system, and the LCC of PV panels are increased, as shown in Figure 10.
The results of the variations of the battery unit cost on the optimal variables and cost criteria (in $) of the hybrid solar/battery scheme for RI = 2% are given in Table 4. It is shown that the value of LLP and TLCC of the hybrid system, for battery cost equal to $300, is 1.9927% and $106190, respectively. In this case, the optimal values of numbers of panels and batteries are 171 and 1114; also, the LCC of the panels and batteries are $6,070 and $99,301, respectively, and the optimal C&R and O&M of the system are $92,951 and $13,237, respectively.
In battery cost = $50, the optimal number of PV panels, battery storage, and TLCC is 173, 1193, and $34640, respec-tively. It can be seen that by reducing battery cost from 300 to $50, the values of TLCC, C&R, and LCC BAT decrease to $71,550, $72,378, and $71,635, respectively. It can be seen that the TLCC, C&R, and LCC BAT are $34640, $20,573, and $27,666, for in this case, and also, the annual O&M costs of the optimal hybrid system and batteries are $14,051 and $11,930, respectively. In battery cost = $550, the optimal values of TLCC and LLP are $184,620 and 1.9481%, respectively. It is found that by increasing battery cost from 300 to $550, the values of TLCC and C&R increase to 74% and 84%, respectively. In Figure 11, variation in TLCC, LCC BAT , and C&R of the optimal hybrid system versus battery cost is presented. It can be seen that by increasing the battery unit cost, the TLCC and C&R of the hybrid system and the LCC of batteries are increased.

Conclusion
In this paper, an optimal configuration of a stand-alone hybrid photovoltaic (PV)/battery energy storage system is investigated to provide the load demand in Iran. The objective function is a minimization of total life cycle cost (TLCC) subject to the reliability index (loss of load probability) by a modified harmony search algorithm. Also, sensitivity analysis and effects of reliability index (0 to 20%) and economic parameters based on photovoltaic and battery prices are presented, and the results are analyzed. The results show that, by reducing reliability index from 2 to 0%, the values of number of PV panels, battery storage, and TLCC increase to 7%, 33%, and 31%, respectively, and by increasing the reliability index from 2 to 5%, the values of numbers of PV panels, batteries, and TLCC decrease to 7%, 25%, and 23%, respectively, and also show that by increasing PV cost from 200 to 500 $/m 2 , the values of TLCC and C&R increase to 9.5% and 10.7%, respectively, and by reducing PV cost from 250 to 50 $/m 2 , the values of TLCC and C&R decrease to 7.7% and 8.2%, respectively. It is found that by increasing battery cost from 300 to $550, the values of TLCC and C&R increase to 74% and 84%, respectively, and by reducing battery cost from 300 to $50, the values of TLCC and C&R decrease to 67% and 78%, respectively. Future work will be investigated different stand-alone hybrid energy systems  Total life cycle cost T air : Ambient temperature (°C) T ref : Reference temperature (°C) T c : Temperature of the cell ω: Rate of hourly self-discharge η INV : Efficiency of the inverter (%) η BC : Battery charging efficiency (%) η BDC : Battery discharging efficiency (%).

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.