## 2 Derivation of the Physical Model of the Set of Performance Curves
### Identification of the Tuning Parameters  
The tuning parameters can be determined utilizing the Levenberg-Marquardt method. This method is applied to address multidimensional, non-linear, and unconstrained optimization problems. The solution requires a function \(\Phi(t)\) that is twice continuously differentiable [23]:  
\[\Phi(t):=\frac{1}{2}\left\|z(t)\right\|^{2}. \tag{18}\]  
The objective of the optimization problem is to find an optimal vector \(t_{opt}\) that minimizes \(\Phi(t)\), thereby representing the optimal tuning parameters for the specific pump [23]:  
\[t_{opt}:=\underset{t\in\mathbb{R}}{\text{argmin}}\Phi(t). \tag{19}\]  
The vector \(z(t)\) defines the difference between the pump power calculated using the physical model (Equation (20)) and the measurements taken at the specific pump's data points. In the following, \(z(t)\) is designated as the objective function:  
\[z(t):=\big{(}P_{K,1}-P_{K,exp,1},P_{K,2}-P_{K,exp,2},\cdots,P_{K,n}-P_{K,exp, n}\big{)}^{T} \tag{20}\]  
The Levenberg-Marquardt method is iterative in nature, meaning a new vector \(t^{k+1}\) is calculated for each iteration step \(k\) by Equation (21) [24]. \[t^{k+1}:=t^{k}-\bigg{(}\big{(}k^{k}\big{)}^{T}k^{+}+\mu I\bigg{)}^{-1}\Big{(} k^{k}\Big{)}^{T}z(t)^{k} \tag{21}\]\(j^{k}\) denotes the Jacobian matrix at time step \(k\) and \(I\) the identity matrix. The Jacobian matrix is a \(n\times m\) matrix that represents the total differential of \(z(t)\) at \(n\) measurement points and \(m\) tuning parameters (Equation (22)). \[J:=Dz(t) \tag{22}\]  
In the Levenberg-Marquardt method, the objective is not only \(\Phi(t)\to min!\) but also \(\|t^{k+1}-t^{k}\|^{2}\to min!\). The weighting of the two objectives is determined by the parameter \(\mu\in[0,\infty]\). The matrix \((\,(J^{k})^{T}J^{k}+\mu I\,)^{-1}\) is always positive definite and thus not singular [24]. Physical interpretability must be taken into consideration when identifying the tuning parameters. To achieve this, it is necessary to consider inequality constraints. However, the Levenberg-Marquardt method operates without any constraints. For this reason, \(P_{K}(t)\) estimated by the physical model in \(z(t)\) was augmented with a logarithmic barrier term to \(b(t)\)[25]. The definition of \(b(t)\) is as follows:  
\[b(t):=P_{K}(t)-\lambda\sum LN\big{(}t_{i}-r_{i,j}\big{)}. \tag{23}\]  
Inequality restrictions are taken into account via \(r_{i,j}\in R\) in the right-hand term of \(b(t)\). The factor \(\lambda\) controls how close the tuning factor may be taken to the interval limits. During the identification process, a value of \(\lambda=1e^{-6}\) was consistently assumed. Algorithm 1 outlines the procedure for identifying the tuning parameters using the Levenberg-Marquardt method with the involvement of barrier terms. The method is based on that described in [24]. ```
\(N_{max}:=1000\) \(t^{1}:=[0\quad P_{K,exp}(Q^{*})\cdot 0.02\quad 9e-5\quad 6\quad 0]\) \(\mu:=0\) \(\beta_{0}:=0.5\ \land\ \beta_{1}:=0.9\) \(\varepsilon:=0\)
``` FOR \(k=1:N_{max}\) \(\text{WHILE}\,\varepsilon<\beta_{0}\) \(\text{Calculation of}\ b(t^{k})^{k}\) \(\text{Calculation of}\ b^{k}\) \(z(t^{k}):=P_{K,exp}-b(t^{k})^{k}\) \(\text{Calculation of}\ t^{k+1}\) \(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  
physical model. Two low-pass filters can be used to reduce the fluctuations of the input signals as well as the calculated flow rate. In addition to the parameters for the calculation of the physical model, the speed (\(n_{exp}\)) and the power consumption (\(P_{K,exp}\)) are required for the algorithm. As suggested in [5], these quantities can be taken from a frequency converter. In cases where the measured pump head is unavailable, an approximate function (\(H_{calc}(Q_{calc})\)) can be assumed. Algorithm 2 describes the flow rate calculation algorithm. The damping characteristics of the low-pass filters can be modified by adjusting the damping coefficients (\(k_{1},\ k_{2}\)). The BEP is taken as the initial operating point.