# System Dynamics Modeling for Estimating the Locations of Road Icing Using GIS
## 2 Materials and Methods
### Modeling Road Icing
#### 2.4.1 Road Temperature  
Road temperature is one of the important factors in road icing occurrence. This is because the moisture present on roads freezes at sub-zero temperatures [41]. For traditional road temperature calculation, a large number of input factors are required, including \"weather variables, such as solar radiation energy, air temperature, atmospheric pressure, and wind, as well as thermal characteristic values according to the material of the road surface\" [23]. This, however, is not appropriate due to the nature of simulation, which must exhibit the highest efficiency with limited input. In this study, a method of producing the highest estimation performance with a small number of input factors through an evolution algorithm was devised [42; 43; 44; 45]. The input factors used were hillshade, air temperature, cloudiness, and bridge location as shown in Figure 5 and Table 3.  
A multiple linear regression equation was constructed using these four factors as shown in Equation (1), and the parameters (A, B, C, D, and E) were optimized using the evolution algorithm [42; 43; 44; 45; 46]. In the case of parameter D, the initial value was set to \(-1\) to \(-2\) over time through literature survey because its optimization through the evolution algorithm was estimated to be insignificant. \[\mathrm{T_{r}}{=}\mathrm{f(T_{a},~{}H,~{}V_{c},L_{b})}{=}\mathrm{A*T_{a}}{+} \mathrm{B*H*C*}\{(\mathrm{V_{c}-10})*-1\}{+}\mathrm{D*L_{b}}{+}\mathrm{E} \tag{1}\]  
The air temperature in Equation (1) is an integral value obtained by adding the differential value over time to the initial value (T\({}_{a}\))\({}_{t=0}\). The air temperature is calculated using Equation (2). \[\mathrm{T_{a}}=\int{(T_{a}{}^{\prime})_{t}\mathrm{dt}}+{(T_{a})_{t=0}} \tag{2}\]  
Hillshade in Equation (1) is an integral value obtained by adding the differential value over time to the initial value H\({}_{t=0}\). Hillshade is calculated using Equation (3). \[\mathrm{H=}\int{H_{t}\mathrm{dt}}+{H_{t=0}} \tag{3}\]  
Cloudiness in Equation (1) is an integral value obtained by adding the differential value over time to the initial value (V\({}_{c}\))\({}_{t=0}\). Cloudiness is calculated using Equation (4). \[\mathrm{V_{c}}=\int{(V_{c}{}^{\prime})_{t}\mathrm{dt}}+{(V_{c})_{t=0}} \tag{4}\]  
The evolution algorithm mainly goes through the processes of initial set setting, crossbreeding or mutation, fitness calculation, roulette model, and termination condition checking. In this study, the termination condition of the evolution algorithm is when the fitness function F(t) has the minimum value [42; 46]. F(t) is the formula for the mean \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Factor** & **Category** & **Symbol** & **Value Range** & **Unit** \\ \hline Hillshade (from DEM) & & H & 0\(\cdot\)254 & - \\ River System & & RS & 0\(\cdot\)1 & - \\ Bridge Location & & L\({}_{b}\) & 0 or 1 & - \\ Air Temp & Input & T\({}_{a}\) & \(-\)15\(\cdot\)10 & \(\cdot\)C \\ Cloudy & (t = 0\(\cdot\)13 h) & V\({}_{c}\) & 0\(\cdot\)10 & - \\ Vapor Pressure & & P\({}_{v}\) & 0 or more & Pa \\ Wind Speed & & S\({}_{w}\) & 0 or more & m/s \\ Precipitation & & P & 0 or more & mm \\ \hline Evaporation & Intermediate & E & 0 or more & g/m\({}^{3}\) \\ Condensation & (t = 0\(\cdot\)13 h) & C & 0 or more & g/m\({}^{2}\) \\ \hline Road Temperature & Output & T\({}_{r}\) & \(-\)15\(\cdot\)10 & \(\cdot\)\(\cdot\)C \\ Road Moisture & (t = 0\(\cdot\)13 h) & M\({}_{r}\) & 0 or more & g/m\({}^{2}\) \\ Road Icing & & RI & 0 or more & g/m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Main input and output factors of the model. absolute percentage error value within modeling and expressed as Equation (5). At is the actual value and Ft is the estimated value. \[\text{F(t)}=\frac{100\%}{n}\sum_{t=0}^{t=13}\left|\frac{A_{t}-F_{t}}{A_{t}}\right| \tag{5}\]  
Genes whose fitness was calculated are selected through the roulette function.