## 4 Determination of Pump Characteristics and Mechanical Power during the Operation Cycle  
To determine the performances of the above-mentioned pumping systems, first, mechanical powers \(P_{mech}\) are calculated, for ten different operation modes shown in Figure 1. Table 1 represents the calculated characteristics of the single pump system at ten different water flowrates, according to Figure 1. In the case of the single-pump system with one variable speed pump, the working points of the pumping system move along the system curve and are defined as intersection points of \(Q\)-\(H\) pump characteristics at a certain rotation speed \(n\) and the hydraulic system curve. The curve \(Q\)-\(H\) and the mechanical power at a decrease in the pump rotational speed are calculated using affinity laws (6), (Figure 6). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(Q_{req}\), \% & **10** & **20** & **30** & **40** & **50** & **60** & **70** & **80** & **90** & **100** \\ \hline \(n\), rpm & 1918 & 1940 & 1991 & 2068.7 & 2181.4 & 2288 & 2424 & 2572 & 2731 & 2900 \\ \(Q_{req}\), m\({}^{3}\)/h & 2.4 & 4.8 & 7.2 & 9.6 & 12 & 14.4 & 16.8 & 19.2 & 21.6 & 24 \\ \(H\), m & 8.07 & 8.28 & 8.63 & 9.13 & 9.89 & 10.53 & 12.39 & 12.51 & 13.7 & 15 \\ \(P_{mech}\), W & 234 & 290 & 356 & 437 & 547 & 663 & 817 & 1002 & 12,223 & 14,856 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Operating cycle data for the single-pump system. Figure 5: Pump working points in parallel operation. In the case of the joint operation, two similar pumps operate in parallel, and the total flowrate equals the sum of the flowrates of the individual pumps \(Q=Q_{1}+Q_{2}\). The first pump has an adjustable speed drive, and the rotation speed of the second pump is constant. The range of the output flowrate can be divided into two parts:  
* Water flowrate regulation in the range from 0 to 60% is achieved by speed variation of the variable speed pump. A non-adjustable pump in this range of flows is not switched on and is closed by a return valve to prevent water from flowing through it in the backward direction;
* Water flowrate regulation in the range from 60 to 100% is achieved by the operation of both pumps. Water is supplied by joint control of the rotational speed of the 1 st pump according to the regulation law (10) and of throttling the 2 nd pump. In a dynamic mode, this type of regulation can be achieved with the help of proportional-integral (PI) controllers. In the case when the operation of the pumps in parallel is provided only by the rotation speed control of one of the pumps without throttling of a non-adjustable pump, the pump loading caused by flowrate will be different. The pump with a non-adjustable drive operating at nominal rotation speed will be overloaded and a variable speed drive will be underloaded. As the value of \(Q_{req}\) decreases, the load difference will increase. If the flow rate is such that for a given speed \(n_{1}\) the head \(H_{2}\) approximately equals or is greater than the shut-off head of the adjustable pump, then the flowrate of a variable speed pump will be close to zero (deadheading), or even negative (reverse flow). The reduction of the total flow of pumps operating in parallel by increasing the reverse flow of one of the pumps is identical to bypass-control. However, continuous operation of the pump in such modes is unacceptable due to low energy efficiency and can lead to deterioration of the pump equipment [2]. To avoid these operation modes, it is necessary to reduce the pressure in the pipeline connected to a non-adjustable pump with the help of throttling. Based on the considerations given above, in the case of calculating the joint operation of two pumps in multi-pump system, the working points of the variable speed pump are determined in the same way as in the case of operation of one pump with a flow \(Q_{req}\)/2 according to Equations (1)-(3).