## 1 Introduction  
The flow rate of centrifugal pumps plays an important role as a command variable for controlling pumping systems. Pumps are used to generate a flow rate, which needs to be adapted to the requirements of the specific application using one of various possible control methods. In particular, the flow rate is considered the main control variable in many applications (e.g., in the process industry) [1; 2]. The flow rate (\(Q\)) can be determined either by direct measurements or by using numerical algorithms. Various algorithms to determine the flow rate are known in the literature. Those algorithms require the characteristic curve of the pump, which is typically published by the manufactures at a constant pump speed. The application of speed-controlled pumps has increased significantly. Therefore, the set of performance curves is essential for determining the flow rate. Depending on the kind of algorithm, the measured values, and their approximation functions, either of the delivery head \(H(Q)\) or the power of the pump \(P_{K}(Q)\) are required. A frequently employed algorithm involves measuring either the pump head (\(H\)) or power (\(P_{K}\)) then determining the flow rate by applying the inverse of the \(H(Q)\) or \(P_{K}(Q)\) approximation functions. [PERSON] et al., utilized this algorithm for both pumps and fans [3]. They amalgamated flow rate calculations from both approximation functions to estimate the overall flow rate. [PERSON] & [PERSON] have also investigated this algorithm [4]. [PERSON] & [PERSON], along with [PERSON] et al., computed flow rates across various speeds. Therefore, they apply the following affinity laws: \(Q\propto n\), \(H\propto n^{2}\), and \(P_{K}\propto n^{3}\). The relative deviations between the measured and calculated flow rates in [4] range from 0% to 4%. [PERSON] (2015) provided an overview of the most common algorithms to determine the flow rate [2]. These algorithms can be divided into static and dynamic methods. [PERSON] & [PERSON] (2013) introduced the boundary curve method, wherein the stator current of the drive motor is measured as a function of the flow rate [5]. Two boundary curves are obtained by measuring the stator current characteristic diagram at various speeds for \(Q=0\) and \(Q=\max(Q)\). This algorithm closely resembles the approach in [PERSON] et al. 's patent, where they also utilized a boundary curve measured across different speeds [6]. Both algorithms require the manufacturer's published performance curve as the initial data. Even more recent publications, such as that of [PERSON] et al., or [PERSON] et al., require the manufacturer's performance curve [7,8]. [PERSON] et al., adjusted the pump characteristics at variable speeds by using the affinity laws [8]. They achieved an average error of about 2% in estimating the flow rate. [PERSON] et al. 's algorithm employs data about the flow rate and head at the best efficiency point (BEP) of the pump (\(Q^{*},~{}H^{*}\)) during nominal speed (\(n_{N}\)) to compute the pump characteristics [9]. Hence, the specific speed (\(n_{q}\)), as determined by Equation (1), is required. \[n_{q}=n_{N}\frac{\sqrt{Q_{opt}}}{H_{opt}}^{3/4} \tag{1}\]  
Through using the generic pump characteristic curves described by [PERSON] and [PERSON] [10,11], one can calculate the pump curves for specific speeds. Subsequently, the flow rate estimation is achieved through interpolation and polynomial approximation utilizing the \(H(Q)\) or \(P_{K}(Q)\) characteristic curve. The flow rate can be calculated with an error of 7-15% through using this approach. Variable speeds also involve the utilization of the affinity laws. [PERSON] et al., showed that substantial deviations arise when applying the affinity laws for calculations, particularly with a significant reduction in rotational speed [12]. They enhanced the approximation functions, leading to a more accurate reproduction of the data.