## 3 Operating Point Calculation for Pumps  
This section briefly describes the mathematical models of the examined pumping systems. In order to assess the characteristics of the operation mode of the pumps, the graphical analysis method is used. Figure 3a represents a single-pump system. Figure 3b represents a multi-pump system. Pumps \(P\) and \(P1\) are coupled with induction motors \(M\) and \(M1\), respectively. Frequency converters (FC) feed these motors. Pump \(P2\) is coupled with induction motor \(M2\). The throttle \(V\) is equipped with drive \(M3\) (throttle control) and when the pumps operate simultaneously, it is regulated together with the electric drive \(M1\). The rest of the valves shown in Figure 3 as well as non-return valves are used for protective purposes not for control. Figure 2: Pump characteristics [1] (**a**) model 1, \(Q\)-\(H\) curve; (**b**) model 2, \(Q\)-\(H\) curve; (**c**) model 1, mechanical power versus flowrate; (**d**) model 2, mechanical power versus flowrate; (**e**) model 1, efficiency versus flowrate; (**f**) model 2, efficiency versus flowrate. The values of \(Q\) and \(H\) of individual pumps are calculated according to their technical characteristics (Figure 2). During the development of a mathematical model for the pump system, polynomial interpolations of the technical characteristics of the pumps are used [8,23]. The mathematical model of a multi-pump system consisting of two similar pumps and a hydraulic load is described by the following equations [8,23]:  
\[H_{1}=a\cdot Q_{1}^{2}+b\cdot Q_{1}\cdot s_{1}+c\cdot s_{1}^{2} \tag{1}\]  
\[H_{2}=a\cdot Q_{2}^{2}+b\cdot Q_{2}\cdot s_{2}+c\cdot s_{2}^{2} \tag{2}\]  
\[H_{req}=H_{st}+k\cdot Q_{req}^{2} \tag{3}\]  
where \(Q_{1}\) and \(H_{1}\) are the flowrate and the head of the variable speed controlled pump; \(Q_{2}\) and \(H_{2}\) are the flowrate and the head of the non-adjustable speed pump; \(a=-0.02903\), \(b=0.15655\), and \(c=18.284\) are the coefficients of the interpolation polynomial obtained according to the \(Q\)-\(H\) characteristics of pump Calpeda--B--NM 32/12D at the rated rotational speed (Figure 2); \(s_{1}\) and \(s_{2}\) are the rotational speeds of the variable speed and non-adjustable speed pumps in relative units (\(s_{1}=n_{1}/n_{rate}\); \(s_{2}=n_{2}/n_{rate}\)); \(n_{1}\) and \(n_{2}\) are the rotational speeds in absolute values; \(n_{rate}\) = 2900 rpm; \(Q_{req}\) and \(H_{req}\) are the required values of the water supply and hydraulic head (hydraulic loads); \(H_{st}\) and \(k\) are the static head and the hydraulic friction coefficient of the hydraulic system. The flowrate of separate pumps \(Q_{1}\) and \(Q_{2}\) and the total flowrate of the pumping system \(Q_{req}\) are related according to Equation (4) [8]:  
\[Q_{req}=Q_{1}+Q_{2} \tag{4}\]  
The friction coefficient \(k\) is defined as:  
\[k=\frac{H_{\max}-H_{st}}{Q_{\max}^{2}} \tag{5}\]  
where \(H_{max}\) = 16 m, \(Q_{max}\) = 24 m\({}^{3}\)/h, \(H_{st}\) = 0.5-\(H_{max}\)[25]. Figure 3: Pumping system structure: (**a**) single-pump, power 1500 W; (**b**) multi-pump, each of the pump has a nominal power of 750 W.  
To calculate the variation of flow \(Q\), head \(H\), and power on the shaft \(P_{mch}\) at different rotational speeds, the affinity laws are applied [23]:  
\[\frac{Q_{N1}}{Q_{N2}}=\frac{N_{1}}{N_{2}};\;\frac{H_{N1}}{H_{N2}}=\left(\frac{N _{1}}{N_{2}}\right)^{2};\;\frac{P_{mch\,n1}}{P_{mch\,n2}}=\left(\frac{N_{1}}{N_ {2}}\right)^{3}, \tag{6}\]  
where \(Q_{N1}\), \(H_{N1}\) and \(P_{mchN1}\) are the flowrate, pump head, and power on the shaft at speed \(N_{1}<n_{rate}\); \(Q_{N2}\), \(H_{N2}\) and \(P_{mch}\) are the flowrate, pump head, and power on the shaft at speed \(N_{2}=n_{rate}\). During the operation of pumps in parallel without throttling the flow \(Q\) of pumps in the multi-pump system sums up (4) at the same hydraulic head value. Then, the intersection point of the resulting performance \(H\)-\(Q\) curve and system curve is obtained (Figure 4).