# Selection of a Photovoltaic Panel Cooling Technique Using Multi-Criteria Decision Analysis
## 3 MCDM Approach to Evaluate PV Panel Cooling Methods Evaluation
### Structure of the Decision Matrix and its Standardization  
Assuming there are a PV panel cooling method m and evaluation criteria n for the PV panel cooling method, \(\rm P_{\rm ij}\) is the value of the criterion j and PV panel cooling method i. In order to eliminate the influence of the size of the criteria on immeasurability, it is necessary to standardize the criteria using equations of relative optimum membership degree. For the maximization criterion, the attribute value of criterion j in the method of cooling photovoltaic panels can be standardized using Equation (6), whereas for the minimization criterion, the same can be standardized by using Equation (7).  
\[\rm S_{\rm ij}=\left[\frac{P_{\rm ij}-minP_{\rm ij}}{maxP_{\rm ij}-minP_{\rm ij }}\right] \tag{6}\]  
\[\rm S_{\rm ij}=\left[\frac{maxP_{\rm ij}-P_{\rm ij}}{maxP_{\rm ij}-minP_{\rm ij }}\right] \tag{7}\]  
In Equations (6) and (7), \(\rm S_{\rm ij}\) is the standardized criterion value for the jth criterion of the ith alternative PV panel cooling method; \(\rm P_{\rm ij}\) is the jth criterion's value for the ith alternative PV panel cooling method; and (i = 1,..., m) and (j = 1,..., n). Supposing that the evaluation set of the multi-attribute decision making problem with the jth criterion's value in the ith PV panel cooling method is \(\rm P_{\rm ij}\), then the decision matrix is \(\rm P=[P_{\rm ij}]_{m}\times n\); refer to Table 6 below.  
After the standardization of all evaluation criteria for the given set of PV panel cooling method, the decision matrix is as below:  
\[\rm S^{\prime}_{\rm ij}=\left[S_{11}\ S_{12}\ \ldots\ S_{1n}\ ::::::\ S_{m1}\ S_{m2}\ \ldots\ S_{mn}\ \right] \tag{8}\]  
After the standardization of all evaluation criteria for all given PV panel cooling methods (refer Equations (6) to (8)), the decision matrix is expressed for each PV panel cooling method type as shown below in Table 7.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Alternative PV Panel Cooling Method (i)**} & \multicolumn{4}{c}{**Evaluation Criterion (j)**} \\ \cline{2-5}  & **1** & **2** &. & **n** \\ \hline
1 & \(\rm P_{\rm 11}\) & \(\rm P_{\rm 12}\) &. & \(\rm P_{\rm 1n}\) \\ \hline
2 & \(\rm P_{\rm 21}\) & \(\rm P_{\rm 22}\) &. & \(\rm P_{\rm 2n}\) \\ \hline
. &. &. &. &. \\ \hline. &. &. &. &. \. &. &. &. &. \. &. &. &. &. \. &. &. &. &. \\ \hline M & \(\rm P_{\rm m1}\) & \(\rm P_{\rm m2}\) &. & \(\rm P_{\rm 2n}\) \\ \hline Criterion Weight \(\rightarrow\) & \(\rm W_{\rm 1}\) & \(\rm W_{\rm 2}\) &. & \(\rm W_{\rm a}\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Decision matrix P for the jth criterion’s value for the ith alternative PV panel cooling method.  
\begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline \multirow{2}{*}{**Evaluation Criterion (j)**} & \multicolumn{4}{c}{**Alternative PV Panel Cooling Methods (i)**} \\ \cline{2-9}  & **AFC*** & **HPC** & **PCC** & **TCC** & **TPC** & **FFC** & **WSC** & **ESC** \\ \hline C1 \$ & 0.04 & 0.08 & 0.1 & 0.08 & 0.1 & 0.07 & 0.15 & 0.15 \\ \hline C2 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 & 0.02 \\ \hline C3 & 0.03 & 0.06 & 0.08 & 0.06 & 0.08 & 0.01 & 0.12 & 0.12 \\ \hline C4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline C5 & \(-1\) & \(-4\) & \(-2\) & \(-4\) & \(-5\) & \(-3\) & \(-5\) & \(-5\) \\ \hline C6 & 58 & 168 & 1125 & 25 & 25 & 68 & 75 & 75 \\ \hline C7 & Good & Acceptable & Poor & Poor & Good & Good & Good & Good \\ \hline C8 & 1 & 5 & 3 & 4 & 6 & 2 & 7 & 8 \\ \hline \hline \end{tabular} Note: $ Refer to Table 5.  
\end{table}
Table 7: Standardized decision matrix \(\rm S^{\prime}_{\rm ij}\) for alternative PV panel cooling methods.