## 3 Optimal OBC Cooling Plate Design and Results  
There are several different optimization methods available in the literature; however, it is important to select a suitable objective function to solve the optimization problem [24]. In this study, product and error distance functions were introduced to improve optimization accuracy, and hence, overall system efficiency. When a random function \(P(R_{i})\) can be expressed as a product of the individual responses \(R_{i}\), and it is assumed that the sum of the individual constraint conditions (\(Q_{t}\)) is constant, then, the product function can be defined as follows [25; 26; 27]: \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Channel** & **Pressure [kPa]** & **Velocity [m/s]** & \begin{tabular}{c} **Volume Flow** \\ **[m\({}^{\frac{1}{3}}\)/s]** \\ \end{tabular} & \begin{tabular}{c} **Volume Flow** \\ **Distribution [\%]** \\ \end{tabular} & \
\begin{tabular}{c} **Normalized Volume** \\ **FlowDistribution N [\%]** \\ \end{tabular} \\ \hline \(R_{1}\) & 16.198 & 0.396 & 0.020 & 13.187 & 0.791 \\ \(R_{2}\) & 17.200 & 0.667 & 0.033 & 22.223 & 1.333 \\ \(R_{3}\) & 17.676 & 0.579 & 0.029 & 19.315 & 1.159 \\ \(R_{4}\) & 17.932 & 0.493 & 0.025 & 16.442 & 0.987 \\ \(R_{5}\) & 18.074 & 0.442 & 0.022 & 14.739 & 0.884 \\ \(R_{6}\) & 18.108 & 0.423 & 0.021 & 14.094 & 0.846 \\ \(\sigma\) & 0.734 & 0.104 & 0.005 & 3.475 & 0.208 \\ MM & 1.910 & 0.271 & 0.014 & 9.036 & 0.542 \\ MM Ratio [\%] & 10.95 & 54.200 & 54.200 & - & - \\ \hline \hline \end{tabular}
\begin{tabular}{c} \(\sigma\) : Standard Deviation, _MM_: Max-Min, _MM Ratio_\([\%]_{o}=\frac{MM}{Average(K)}\). \\ \end{tabular} \\ \hline \end{tabular}
\end{table} 
Table 2: Characteristic fluid dynamics metrics for initial cooling model obtained via CFD analysis. Figure 4: Design variables for the OBC/LDC flow multi-channel model. Figure 3: Velocity and pressure distributions for initial cooling model obtained via CFD: (**a**) velocity (m/s), (**b**) pressure (kPa). (The channel numbers are \(R_{1}\), \(R_{2}\),..., \(R_{6}\) from the inlet.). Objective function:  
\[P(R_{i})=\ \prod_{i=1}^{n}R_{i}\] (5) Constraint condition: \[(R_{i})=\ \sum_{i=1}^{n}R_{i}-Q_{t}=0 \tag{6}\]  
If the Lagrange multiplier is used to determine the conditions for the maximum/minimum values of \(P(R_{i})\), we obtain the following function:  
\[L(R_{i},\ \lambda)=\ \prod_{i=1}^{n}R_{i}-\lambda\left(\ \sum_{i=1}^{n}R_{i}-Q_{t}\right) \tag{7}\]  
Thus, the maximum/minimum value of Equation (7) can be obtained by setting its first-order differentiation to zero:  
\[\frac{\partial\text{L}(\text{R},\lambda)}{\partial R_{i}}=0 \tag{8}\]  
On solving the first-order differential equation in Equation (8), we obtain the following condition:  
\[R_{1}=R_{1}\cdots\ =\ R_{n} \tag{9}\]  
Thus, it is clear that Equation (9) yields the target value of each response \(R_{i}\). In addition, when \(R_{i}\), i.e., the flow rate of each channel, is normalized to 1, we get \(R_{i}^{\prime}=\ \frac{R_{i}\_response}{R_{i}\_target}\), and assuming that the same flow rate flows in each flow path \(R_{i}^{\prime}=\ \frac{n\cdot R_{i}}{Q_{t}}\); then, the optimization solution can be obtained as \(P(R_{i}^{\prime})=1-\ \prod_{i=1}^{n}R_{i}^{\prime}\ =0\). Thus, the following equation can be introduced as the objective function based on the product function:  
\[E(R_{i})_{dist}=\left(\sum_{i=1}^{n}\left(1-\ \frac{nR_{i}}{Q_{t}}\right)^{2} \right)^{1/2} \tag{10}\]  
Furthermore, using the error of the response value compared with the reference value, another objective function can be set based on the error distance as follows [27, 28]:  
\[E(R_{i})_{max}=\left(1-\ \prod_{i=1}^{n}\frac{nR_{i}}{Q_{t}}\right)^{2} \tag{11}\]  
The constraint condition for the objective functions in Equations (10) and (11) is the constant total flow rate, which can be expressed as follows:  
Figure 4:0  
Thus, Equations (10) and (11) are functions that represent the characteristics of the cooling system.