## 4 Models and Algorithm Design
### Representation of the Renewable Resources
#### 4.1.1 Solar Power Plants  
The mathematical model for a solar power plant is called a \"solar power output model\". This model calculates solar power output based on solar irradiation, temperature, and other parameters. This model thus respects the effect of temperature on the efficiency of solar panels. The variable \(P_{s_{i}}\) represents the power output of the \(i\)-th solar power plant based on photovoltaic (PV) panels number [27; 50; 59]:  
\[P_{s_{i}}=3.24\cdot M_{PV_{i}}\cdot\left[1-0.0041\cdot\left(T_{t_{i}}-8\right) \right]\cdot S_{t_{i}}, \tag{12}\]where \(M_{PV_{i}}\) represents the capacity of one PV panel whose value is subsequently multiplied by the number of panels in the given solar power plant. Term \(T_{t_{i}}\) denoted outdoor temperature and \(S_{t_{i}}\) means solar irradiation at concrete time \(t\) in the \(i\)-th solar power plant. Part \(\left[1-0.0041\cdot\left(T_{t_{i}}-8\right)\right]\) adjusts the efficiency of the solar panel based on the deviation of the outdoor temperature (\(T_{t_{i}}\)) from the reference temperature (8 \({}^{\circ}\)C). Constant 0.0041 is often determined empirically based on the specific characteristics of the solar panels being used. It represents the temperature coefficient of the efficiency of the solar panel. Different solar panels may have different temperature coefficients. This is the temperature at which the solar panel efficiency is defined or measured under standard test conditions. The constant 3.24 in Equation (12) is a scaling factor that converts solar irradiation and other parameters into the desired output unit, such as megawatts. In practice, the value of this scaling factor may vary depending on the system, location, and solar panel characteristics, so it could be adjusted accordingly to match the actual performance of the solar panel system. Using a scaling factor helps ensure that the equation is consistent with the desired units for the output power. The scaling factor itself is dimensionless. If it is necessary to calculate several calculations in a row, the index \(k\) can be used--\(P_{S_{i}}(k)\).  
The price function for the solar power plant can be defined as [10,26,28]:  
\[C_{s_{i}}(P_{s_{i}})=\alpha_{s_{i}}P_{s_{i}}, \tag{13}\]  
where the \(\alpha_{s_{i}}\) symbolizes the optional parameter representing the price of the \(i\)-th solar power plant, operation and maintenance costs etc. [10].  
Another shape that can be encountered is more focus on the business case price function. This shape thus considers the return on the entire investment [26,55]:  
\[C_{s_{i}}(P_{s_{i}})=a_{s}I^{P_{s}}P_{s_{i}}+G^{E_{s}}P_{s_{i}}, \tag{14}\]  
where variable \(P_{s_{i}}\) denotes \(i\)-th solar power plant output power. Next, parameter \(I^{P_{s}}\) represents investment cost per energy unit (often in USD/kW). Parameter \(G^{E_{s}}\) indicate operations and maintenance cost (often in USD/kW). Variable \(a_{s}\) in (14) can have following form [26]:  
\[a_{s}=\frac{r}{\left[1-\left(1+r\right)^{-IN}\right]}. \tag{15}\]  
Variable \(a_{s}\) denotes the annuitization (annuitization represents converting an annuity investment into a series of periodic income payments) coefficient, which is dimensionless. Next, \(r\) stands for interest rate (often described in units of percentages). Parameter \(IN\) denotes an investment lifetime (often in years). A typical value for \(IN\) encountered is 20-30 years.  
But sometimes, these shapes of the \(C_{s_{i}}\) are unnecessarily complicated. One can also encounter cases where the price function \(C_{s_{i}}\) is equal to zero [8]:  
\[C_{s_{i}}(P_{s_{i}})=0. \tag{16}\]  
Such simplification can be made because solar irradiation has zero cost [54,57].