## 2 Methodology
### Mathematical Modeling  
#### 2.2.1 Solar PV Design  
Mathematical modeling of the system starts with the rated power of the refrigerator. This rated AC power of the machine is obtained from the manufacturer datasheet and is denoted as \(P_{nted}\). The power needed is provided by the solar PV and battery system which is connected to the refrigerator through inverter of efficiency \(\eta_{inv}\). With this efficiency, the maximum power needed in DC from the solar system is calculated using Equation (1). Inverter efficiencies of up to 99% can be found thanks to the research and development in power electronics. Inverter used in this analysis has a DC-to-AC ratio of 1.0 and an efficiency of 97% [46]. \[\text{Maximum power required in }DC\left(P_{max,\ DC}\right)=\ \frac{P_{nted}}{\eta_{inv}\ \times DC-AC\ ratio} \tag{1}\]  
This power is consumed by the refrigeration machine throughout a day to generate 200 tonne of ice. Consequently, the total energy required in kWh by the system per day is calculated by using Equation (2). \[\text{Total energy demand per day in }DC\left(E_{total}\right)=\ P_{max,DC}\times 24 \tag{2}\]This is the total amount of energy to be generated using solar PV each day. The solar system is designed to be a stand-alone solar power system. Solar PV module chosen for this application is a 400 W poly-crystalline module manufactured by Canadian solar with a rated efficiency of 19.2% [40]. Electrical data including optimum operating voltage and optimum operating current are obtained from the datasheet available from the manufacturer. To calculate the design operating voltage, Equation (3), a voltage efficiency of 90% is used. This analysis is performed to ensure acceptable module output current which is less than optimum operating current. \[\text{PV operating voltage}\left(V_{operating}\right)\ =\ V_{\ opt.\ power\ voltage}\times\eta_{voltage} \tag{3}\]  
The guaranteed power output from a single PV module at STC is obtained from relation (4) with a safety factor of 1.10. This number is also verified from the I-V curve available from the manufacturer's datasheet. \[\text{Gurrented power}\left(P_{gurrented}\right)\ =\ \frac{V_{operating}\times I_{ operating}}{S.F.} \tag{4}\]  
Output power from a solar PV also depends on ambient operating temperature. So, thermal efficiency is accounted for obtaining an output power form PV by using Equation (5). An equation to determine the thermal efficiency of the module is given by Equation (6). Proposed by [PERSON] [47], \(\beta_{ref}\) in the equation represents the temperature coefficient of the module, while \(T_{c}\) and \(T_{ref}\) are the solar cell temperature and reference temperature respectively. Since, the system is not designed for specific mine location, no ambient temperature data is available to calculate exact thermal efficiency of the system. Thermal efficiency considered for this design analysis is 90%. \[P_{operating}\ =\ P_{gurrented}\times\eta_{thermal} \tag{5}\]  
\[\eta_{thermal}\ =\ \left[1\ -\ \beta_{ref}\Big{(}T_{c}\ -\ T_{ref}\Big{)}\right] \tag{6}\]  
Total energy output from each module per day is calculated using peak sun hours for the desired location using Equation (7) with a safety factor of 1.10. Peak sun hours (\(SH_{Pank}\) ) is the number of hours during which the intensity of sunlight is 1000 W/m\({}^{2}\) for that specific place per day. The code is developed in such a way that; with the input of sun peak hours for any mine location, a system feasibility result can be concluded. \[E_{per\ module\ per\ day}\ =\ \frac{P_{operating}\times SH_{peak}}{S.F.} \tag{7}\]  
Using this number, the total number of modules required to meet the energy demand is calculated using Equation (8)  
\[N_{total}\ =\ \left(\frac{E_{total}}{E_{per\ module\ per\ day}}\right) \tag{8}\]  
Equation (9) gives the number of modules required per string. These are the numbers of solar panels that should be connected in series to provide an adequate voltage to the battery supply. This is also equal to the inverter input DC voltage supply.