## 3 Pump Operating Cycle  
According to reference [59], in this study, a duty cycle (flowrate versus time) was assumed for the system that is typical for centrifugal pumps with an uncontrolled speed and approximately constant flowrate. In practice, even in the case of fixed-speed pumps, the required flowrate rarely remains constant. For instance, even for the simplest pumping application, when a pump moves fluid from one tank to another, the pump operating point will vary as the level of water in the tank also changes. Therefore, the pump does not operate at the best efficiency point all the time. Figure 4 shows the assumed operating cycle for a pump with approximately constant flowrate, according to [59].  
Figure 4: Pump loading diagram.  
Figure 3: Motor characteristics vs. motor shaft power: (**a**) efficiency; (**b**) power factor; (**c**) current.  
Since the electric motor is powered directly from the electric grid and the rotational speed cannot be adjusted, it is assumed that the flow is regulated by throttling. In this case, the pump operating point will move along the \(Q\)-\(H\) curve from the catalogue until the intersection with the hydraulic load curve. Figure 5a shows three intersection points for the considered loading conditions (75%, 100% and 110% of \(Q_{BEP}\)) for the \(Q\)-\(H\) curve according to the catalogue data for the selected pump [53].  
The mechanical (input) power curve of the pump as a function of the flow is also specified in the catalogue (Figure 5b). The mechanical power \(P_{mech}\) loading the motor was obtained from this curve for three considered operation points (75%, 100% and 110% of \(Q_{BEP}\)). The flow corresponding to 100% of the maximum flow in the pump operating cycle was chosen at the best efficiency point (maximum efficiency of the pump, \(Q\) = 60 m\({}^{3}\)/h, \(\eta_{pump}\) = 0.755) and determined based on the pump efficiency curve (Figure 5c) [53]. The efficiencies of the electric motors under the three considered loading conditions were found using polynomial interpolation of the efficiency data of Table 3. The obtained interpolated efficiencies are demonstrated in Table 4. Table 4 also shows the following characteristics for each considered operating point: flow, hydraulic head, mechanical power, motor torque, and pump efficiency.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(Q\), \%} & \multirow{2}{*}{\(Q\), \(\mathbf{m^{3}/h}\)} & \multirow{2}{*}{\(H\), \(\mathbf{m}\)} & \multirow{2}{*}{\(P_{mech}\), \(\mathbf{W}\)} & \multirow{2}{*}{\(T\), \(\mathbf{N}\)-m} & \multicolumn{3}{c}{**Pump**} & \multicolumn{3}{c}{**Motor Efficiency**} \\ \cline{5-10}  & & & & & **Efficiency,** & **DOL** & **DOL** & **I2E 1M** & **I2E 1M** & **I2E 1M** \\ \hline
110 & 66 & 14.4 & 3453 & 22.0 & 75 & 0.924 & 0.915 & 0.867 & 0.89 & 0.91 \\
100 & 60 & 15.4 & 3335 & 21.2 & 75.5 & 0.925 & 0.914 & 0.868 & 0.891 & 0.91 \\
75 & 45 & 17.25 & 2962 & 18.9 & 71.4 & 0.925 & 0.909 & 0.868 & 0.891 & 0.908 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Characteristics of the pump duty cycle.  
Figure 5: Pump performances: (**a**) \(Q\)-\(H\) curve; (**b**) pump mechanical power; (**c**) pump efficiency.