## 3 Results: Optimization Workflow
### Numerical Results  
Executing the optimization workflow (cf. Figure 2) produces numerical results, which are presented below. The visualization of the resulting MDAO problem is given as an extended design structure matrix (XDSM) in Figure 15. A Matlab implementation of the genetic algorithm NSGA-II is used as the optimization algorithm, which can handle discrete and real-valued design variables and consider constraints and multiple objectives. The algorithm is suitable for optimizing various engineering problems ranging from gearboxes [42; 43] to wind farms [44; 45]. A pump is designed based on five design variables. With this design, a mechanical shaft power must be provided to the pump for a given flow rate cycle. The connected electrical motor provides this shaft power and consumes an amount of electrical energy in the process. Four variables (head coefficient \(\psi_{Th}\), maximum rotational speed \(n_{max}\), maximum impeller width \(b_{2}\) and diameter \(D_{2}\)) are constrained by five nonlinear constraints  
Figure 14: Executable optimization workflow. (\(g\)), which are given by the system requirements (Figure 6). The installation space of the pump \(V_{Pump}\) (\(f_{1}\)) and the required amount of electrical energy \(E_{EI}\) (\(f_{2}\)) are objectives. The design of the electric coolant pump is based on the procedure in [39]. The design determines the flow characteristics (described by the pump characteristic curve), efficiency, and spatial dimensions of the pump. In the design process under consideration, five design variables are available for this purpose: pump rotational speed \(n_{Opt}\), flow rate \(Q_{Opt}\) and head \(H_{opt}\) in the best efficiency point as well as impeller vane exit angle \(\beta_{2}\) and number of impeller blades \(z_{U}\). The mathematical optimization problem is formulated as follows (Equation (1)):  
\[ \begin{array}{c}\text{minimize }f(x)=\left(V_{Pump},E_{EI}\right)\\ \text{by varying }x=\left(n_{Opt},\ Q_{Opt},\ H_{opt},\ \beta_{2},\ z_{U}\right)\\ \text{subject to }g_{1}(x)=D_{2}-D_{2,max}<0\\ g_{2}(x)=b_{2}-b_{2,max}<0\\ g_{3}(x)=n_{cycle,max}-n_{max}<0\\ g_{4}(x)=\psi_{Th}-\psi_{Th,max}<0\\ g_{5}(x)=\psi_{Th,min}-\psi_{Th}<0\end{array}  \tag{1}\]  
The pump characteristic curve (Equation (2)) defines the head \(H\) and flow rate \(Q\) behavior of the pump at constant rotational speed. The coefficients \(a\), \(b\), \(c\), and \(d\) are functions of \(\psi_{Th}\), \(Q_{Opt}\), \(H_{Opt}\) and \(n_{q}\) and determined using the design workflow. \[H=a\cdot Q^{3}+b\cdot Q^{2}+c\cdot Q+d \tag{2}\]  
The system characteristic curve describes the head and flow rate behavior of the cooling circuit. An operating point consisting of head and flow rate is set as the intersection of the pump characteristic curve and the system characteristic curve. In the considered pump application, the necessary flow rate depends on the amount of heat to be dissipated by the cooling circuit. The operating point is therefore changed by adjusting the pump rotational speed. This operating point shift is modeled by scaling the pump characteristic curve according to the affinity laws in Equations (3) and (4). Index I indicates the unscaled, and index II the scaled operating point. The system characteristic curve remains constant. \[\frac{Q_{II}}{Q_{I}}=\left(\frac{n_{II}}{n_{I}}\right) \tag{3}\]  
\[\frac{H_{II}}{H_{I}}=\left(\frac{n_{II}}{n_{I}}\right)^{2} \tag{4}\]  
Figure 15: XDSM of the optimization problem. These relationships are shown in Figure 16. Changing the pump speed to adjust the flow rate to the demand generally results in a decrease in pump efficiency \(\eta\). In this paper, the coolant flow rate requirement of an internal combustion engine for a mid-size vehicle in the Worldwide harmonized Light-duty vehicles Test Cycle (WLTC) is considered. In the driving cycle, the engine power and, therefore, heat generation varies over time. This results in a varying coolant flow rate. Due to the varying flow rate, the pump is not operated continuously at its best efficiency point.