Simplified Model for Optimal Sizing of the Off-Grid PV System Regarding Value of Loss of Load Probability

In this paper, a simplified model for optimal sizing of the off-grid PV system regarding value of loss of load probability is described. The model gives optimal size of system in terms of required number of PV modules, peak power, number of batteries and cost of system regarding the defined value of loss of load probability, load curve and period for which optimal size will be determined. The model is applied for determination of optimal size of the off-grid PV system for the city of Osijek. Based on measured load curve, optimal size of the system is determined for values of loss of load probability from 0.00 to 0.10 in steps of 0.01, and additionally for 0.15.


INTRODUCTION
Because of the limited reserves of fossil and nuclear fuels and their negative impact on environment, usage of renewable energy sources for electricity production is increasing every year.According to [1] total renewable power capacity (excluding large hydro power plants) in 2015 was 785 GW and at the end of 2016 was 921 GW.Total installed capacity of photovoltaic (PV) systems is also increasing every year the same as PV cells global production.At the end of 2016 total installed capacity of PV systems was 303 GW (at the end of 2015 it was 228 GW) (Ren21,n.d.).According to [2] production data for global PV cell production in 2015 varied between 56 GW and 61 GW for 2016 and estimates for 2016 are in the 65 to 75 GW range.These data are mainly related to gridconnected PV systems, because total installed capacity of off-grid PV systems is very difficult to track.However, that does not mean that off-grid PV systems are not important for reduction of fossil fuels consumption.
Off-grid PV systems (and other renewable off-grid systems as well) can be very helpful for reduction of fossil fuels and CO2 emissions, for power supply of rural areas where there is no electrical grid, for power supply of cottages etc.For such purposes, it is important to find optimal system depending on consumers' needs.Rossi, Toppano and Brunelli in [3] presented real time optimization of battery banks in hybrid residential electrical systems with the aim of maximizing lifetime of the battery banks and to reduce energy bills by managing variability of PV production in price-varying scenarios.Bhandari et al.,in [4] presented a review of trends of optimization techniques of hybrid renewable energy power systems.Furthermore, in [5] authors presented a case study for PV micro-grid for rural electrification in India, while Hassan et al.,in [6] presented usage of the off-grid PV systems as a solution for the ambient pollution avoidance in rural areas of Iraq.Abdulateef, in [7] presented a simulation model of off-grid PV system which is designed for system sizing in case of maximum solar radiation.Merei et al. in [8] introduced optimization model of the off-grid hybrid power system for supply of telecommunication base station where they investigated combination of three different battery technologies.Huneke et al.,in [9] used linear programming for optimization of hybrid off-grid energy systems.There are many other researches related to optimization and sizing of PV systems such as [10][11][12][13][14][15] and [16].In this paper, simplified model for optimal sizing of the off-grid PV system regarding value of loss of load probability (LOLP) is presented.This model enables to find optimal size of the off-grid PV system according to desired value of the LOLP.The advantage of this model in comparison with the previously described papers is that it enables to find optimal off-grid PV system size for desirable period of year.
This paper consists of 4 chapters.First chapter presents introduction.In the second chapter, description of the simplified model is shown.In the third chapter the results and discussion description of the off-grid PV system, case scenario description and results of the case study are presented.The fourth chapter presents conclusions of the paper.

DESCRIPTION OF THE SIMPLIFIED MODEL
This model can be used for finding optimal size of the off-grid PV system in a way to determine optimal number of PV modules, number of batteries and peak power for defined load profile curve in respect to minimum cost of the system and defined value of LOLP.Another advantage of this model is it that can determine optimal size of the system regarding different periods.E.g. it can determine optimal system for usage period of one year, season, month, week or any other period.In Fig. 1 block diagram of simplified model for optimal sizing of the off-grid PV system in respect to value of LOLP is shown.
According to the block diagram presented in Fig. 1, mathematical description of the model is presented as follows.For each hour t, electricity production of PV modules is calculated according (1): Where: G t -average insolation in hour t for the given module angle, A -surface size of the observed module, inumber of modules, η r -rated efficiency of the observed PV module.Observe that value of the average output power of PV modules in particular hour t equals the electricity production in the same hour t.
Maximum and minimum capacity of batteries is defined as ( 2) and (3): Where: C BMAX -average insolation in hour t for the given module angle, δ -coefficient setting the minimum remaining battery capacity, j -number of batteries.Difference between output power of the PV modules and load in hour t is calculated using (4): Then, for a given combination of number of the PV modules i and number of batteries j, in each hour t variable Z i,j,t is calculated according (5): Where: C BAT .j,t−1 -battery capacity for a number of batteries j at the end of previous hour, for the first hour C BAT .j,t−1 equals C START .j , i.e. level of battery capacity at the begging of the observed period.
Then three logical tests are examined: ) Where: C BAT j,t -battery capacity for a number of batteries j at the end of hour t, Diff i,j,t -power deficit in hour t, for a given combination of a number of PV modules i and number of batteries j. b) if , , and , , At the end of observed period, loss of load probability LOLP i,j is calculated for a given combination of the number of PV modules i and number of batteries j as the following (10): , , 1 , In a double for loop for the given range of i and j, is calculated for each pair of i and j for the observed period n using expression (10) and taking into account expressions (1)-( 9).The obtained matrix of LOLP i,j related to tested i and j is searched for values that are smaller than requested the price of the associated system is calculated as: Where: P R -rated power of the PV module (W), C R -rated capacity of the battery (Wh), PV COST -capital cost of PV system per W (includes installation and the cost of the inverter), BAT COST -capital cost of battery per Wh (includes installation and the cost of the charging regulator).
Minimal COST i,j reveals the cheapest system consisting of a number of PV modules i and number of batteries j which satisfied requested value of LOLP.

Description of the Off-Grid PV System
In this section off-grid PV system is described.Offgrid PV system consists of: PV modules, batteries, inverter/charge controller and loads.In Fig. 2 simplified scheme of the system is shown.
In this off-grid PV system mono crystalline PV modules with the following characteristics are assumed: peak power 250 W and efficiency 15.In Tab. 1 specific costs for different components of off-grid PV system are shown.Costs are calculated based on realistic price for different PV system components available in [17].Data from table 1 will be used in model for finding optimal size of the off-grid PV system.PV COST includes cost of the all system components except batteries and it is expressed in EUR per peak kW of system.BAT COST represents cost of the batteries and it is expressed in EUR/kWh.

Case scenario description
The model described in chapter 2 is used to find optimal size of the off-grid system for a house in Osijek, Croatia.Data for solar irradiation are taken for the city of Osijek according to PV GIS ("PV GIS", n.d.).Mean hourly data of solar radiation for one year are used.Characteristics of the components in the system are assumed as it is described in chapter 3.1.
In Fig. 3, weekly load diagram is shown.Weekly load diagram is measured for 7 days and based on this measurement yearly load diagram will be generated.To get yearly load diagram, for each week weekly diagram from Fig. 3 is assumed.Maximum load power is 2286.5 W and the minimum load is 14.8 W.
In this case scenario, model is used to find optimal size of the system for power supply for period from 1 st June to 31 st August.Different values of LOLP in this case scenarios are assumed.Optimal size of the off-grid system for the following values of LOLP is determined: 0.00, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10, and 0.15, respectively.

Results of Case Study
In Tab. 2 results for different values of LOLP are presented.Results are presented with optimal size of the off-grid PV system which includes number of PV modules, number of batteries, peak power of the system and cost of the off-grid system for different values of LOLP and for the period from 1 st June to 31 st August.Optimal size of the offgrid PV system for value of LOLP 0.00, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10, and 0.15, respectively is presented.
In Fig. 4, savings in investment cost in respect to different values of LOLP are shown.As can be seen in Fig. 4, a considerable part of the investment cost (34% in respect to scenario when LOLP equals 0) can be saved if investor is willing to accept LOLP value of 0.03.Further increase in LOLP value does not have justification in substantial investment cost reduction.Due to the excessive number of data and the clarity in the result presentation, in Figs.5-7 results are presented only for period of one week from 1 st July to 7 th July.In Fig. 4 load power, PV generation and battery capacity for the period from 1 st July to 7 th July for value of LOLP = 0 are shown.As it can be seen, the battery capacity does not reach the minimum value of 20% of total capacity which implies that there is no shortage in powering the load.This is the most expensive system consisting of 6 PV modules and 12 batteries.
In Fig. 6 load power, PV generation and battery capacity for period from 1 st July to 7 th July for value of LOLP = 0.1.As it can be seen, the battery capacity reaches minimum value of 20% of total capacity in hours 61, 62 representing 11 h and 12 h of 3 rd July, in hour 79 representing 7 h of 4 th July and in hour 129 representing 9 h of 6 th July.In Fig. 7 load power, PV generation and battery capacity are shown for period from 1 st July to 7 th July for value of LOLP = 0.15.In this case, the additional period of shortage in powering the load can be seen in respect to the case of LOLP = 0.1.

CONCLUSION
Simplified model for optimal sizing of the off-grid PV system in respect to value of loss of load probability is described.The model is applied to find optimal size of the off-grid PV system for the city of Osijek.Optimal size of the off-grid PV system which includes the number of PV modules, number of batteries, peak power of the system and cost of the off-grid system in respect to values of LOLP and for the period from 1st June to 31st August is determined.Optimal size of the off-grid PV system for values of LOLP 0.00, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10, and 0.15, respectively is determined.Savings in investment cost in respect to different values of LOLP are shown.A considerable part of the investment cost (34% in respect to scenario when LOLP equals 0) can be saved if investor is willing to accept LOLP value of 0.03.Further increase in LOLP value does not have justification in substantial investment cost reduction.

Figure 1
Figure 1 Block diagram of simplified model 3 %.Deep cycle batteries with the following characteristics are assumed: voltage 12 V, capacity 160 Ah and charging depth 20%.

Figure 2
Figure 2 Scheme of the off-grid PV system

Figure 3
Figure 3 Weekly load diagram

Figure 4 Figure 5 Figure 6
Figure 4 Savings in investment cost in respect to different values of LOLP

Figure 7
Figure 7 Load power, PV generation and battery capacity for period 1.7 to 7.7 for value of LOLP = 0.15

Table 1
Costs of PV system elements

Table 2
Costs of PV system elements