## 5 Pump Unit Lifetime Energy Costs  
Using the results obtained in the previous sections, the energy-saving indicators are calculated for various cases: excluding losses in the cable and the transformer (Figure 8a), taking into account losses in the cable only (Figure 8b), and taking into account the cable and transformer losses (Figure 8c). The daily energy consumption of an electric motor for the whole operating cycle (see Figure 4) can be found as the following:  
\[E_{\mathit{day}}=t_{\Sigma}\cdot\sum_{i=1}^{3}(P_{1\,i}\cdot t_{i}/t_{\Sigma}). \tag{7}\]  
Figure 6: Cable and transformer losses depending on the load current. Figure 7: Power losses at three points of the pump duty cycle with various flows: (**a**) motors; (**b**) cable + transformer; (**c**) cable; (**d**) transformer. where \(i=1\dots\) 3 is the index of a loading point; \(P_{1\,i}\) is the eclectic power \(P_{1}\) in \(i\)-th loading point; \(t_{i}\) is the operation time of a loading point; \(t_{\Sigma}\) is the whole time period (24 h). Then the annual energy consumption can be obtained according (8):  
\[E_{year}=E_{day}\cdot 365. \tag{8}\]  
The cost of electricity consumed (in Euro), considering the applied grid tariffs \(GT=0.2036\) \(\epsilon\)/kW-h for non-household consumers [65] for Germany in the second half of 2019, was calculated as follows:  
\[C_{year}=E_{year}\cdot GT. \tag{9}\]  
The expected lifetime of a pump is often evaluated to be about 20 years [66; 67]. In this section, the energy cost is estimated for a service life of \(n=20\) years, excluding maintenance costs and the initial cost of the motors. The net present value (NPV) of the lifecycle cost was obtained as follows:  
\[C_{LCC}=\sum_{j=1}^{n}\Bigl{(}C_{year\,j}/\bigl{[}1+(y-p)\bigr{]}^{j}\Bigr{)}, \tag{10}\]  
where \(C_{year\,j}\) is the energy cost of \(j\)th year; \(y\) is the interest rate (\(y=0.04\)); \(p\) is the expected annual inflation (\(p=0.02\)); \(n\) is the lifetime of the pump unit (\(n=20\) years) [67]. Life cycle cost savings \(S_{LCC}\) for a given motor is calculated according to (11):  
\[S_{LCC}=C_{LCC}-C_{LCC\,IE\,2}, \tag{11}\]  
where \(C_{LCC}\) is the lifecycle electricity cost of the considered motor; \(C_{LCC\,IE\,2}\) is the lifecycle electricity cost of the IE2 IM. \(S_{LCC}\) percentage is calculated according to (12):  
\[S_{LCC}=100\%\cdot(C_{LCC}-C_{LCC\,IE\,2})/C_{LCC\,IE\,2}, \tag{12}\]  
Results based on Equations (1)-(12), for three different cases (Figure 8) are presented in Tables 6-8. For ease of comparison, in all cases, the results of the energy consumption of the pumping station consisting of three motors (denoted as \"x3\") are presented. The structure shown in Figure 8c illustrates the power supply of a typical pumping station [68]. The number of the pump units in one pumping station can be different, usually it is 3 or more. The structures in Figure 8a,b are simplified models of the structure in Figure 8c without taking into account the losses in the supply cable and transformer (Figure 8a) and taking into account only the losses in the cable, without taking into account the losses in the transformer (Figure 8b). The simplified structures are considered to assess the impact of the cable and transformer losses on the results of the comparison of electric motors of various types. Figure 8: Cont.