## 3 Analysis and Validation of the Physical Model  
### Parameter Identification and Regression Analysis  
The tuning parameters of the three test objects were identified using Algorithm 1. The restriction matrix for the parameter identification is defined by the following:  
\[R:= \begin{bmatrix}-0.2&0.2\\ P_{K,exp}(Q^{*})\cdot 0.005&P_{K,exp}(Q^{*})\cdot 0.2\\ 1e-6&1e-2\\ 5&15\\ -0.5&15\end{bmatrix} . \tag{25}\]  
C\({}^{2}\)-splines were fitted to the original measurement data. From these, a varying number of equidistantly distributed measurement points were extracted and fed into Algorithm 1 to identify the tuning parameters. Figure 3 shows the ratio of the tuning factors to their average value with the varying number of measuring points (\(N\)) for the three PO. It is conspicuous that \(t_{4}\) for PO001 and PO003 deviate from the average value as the number of measurement points increase. Figure 4a illustrates the graph of recirculation power (\(P_{Rez}\)) for the number of measurement points \(N\in[9;20]\) of PO001. The parameter affects the position of \(Q_{Rez,tc}\) and \(Q_{Rez,min}\). However, the changes are negligible. Figure 3: Ratio of tuning parameters (\(t_{i}\in t,\ 0\leq i\leq 4\)) to their mean value as a function of the number of measuring points during parameterization for PO001, PO002, and PO003. Figure 4: (**a**) Curve of recirculation power (\(P_{Rez}\)) versus flow rate (\(Q\)) for varying tuning parameters \(t_{4}\) of PO001; (**b**) pump power of the measuring points (Data) and course of the physical map model (Physically), as well as an approximation polynomial of the fourth degree (Polynom 4 th) versus the flow rate for PO001; (**c**) comparison of the quadratic norm of the objective functional for the three test objects for the third- and fourth-degree polynomials and the physical model; (**d**) comparison of the quadratic norm of the objective function for the third- and fourth-degree polynomials and the physical model through cross-validation for PO001; (**e**) comparison of the quadratic norm of the objective function for the third- and fourth-degree polynomials and the physical model through cross-validation for PO002; (**f**) comparison of the quadratic norm of the objective function for the third- and fourth-degree polynomials and the physical model through cross-validation for PO003. In addition to the tuning parameters, the power consumption values of the PO001, PO002, and PO003 were approximated by using a third- and fourth-degree polynomial. Figure 4b shows the measurement points (Data), the physical model (Physically), and the fourth-degree approximation polynomial (Polynom 4 th) for PO001. The measured points are well presented by the physical model as well as the polynomial. There is a 0.3% deviation between the physical model and the measurement points. The deviation of the polynomial is 0.6%. Section 1 demonstrated the common utilization of approximation functions in the flow rate calculation. For assessing the physical model against the third- and fourth-degree polynomials, the quadratic norm of the objective function (\(\|z(t)\|^{2}\)) is taken into account, following the optimization criterion outlined in Equation (19). Both the polynomials and the physical model were parameterized using ten data points. As depicted in Figure 4c, the physical model replicates the measured data with comparable accuracy to the fourth-degree polynomial. In the case of PO002, even greater accuracy is achieved. Furthermore, it is evident that for both PO001 and PO002, the accuracy using the fourth-degree polynomial and the physical model can be notably improved compared to the third-degree polynomial. In contrast to polynomial approximations, the physical model offers an advantage in terms of its physical interpretability. Consequently, data points located between the measured values can also be mapped with minimal error. Two sets, each comprising five measurement points, were established for PO001, PO002, and PO003.