## 1 Introduction  
Pumping systems are among the main electrical energy consumers in industry and household applications. These systems account for approximately 22% of the energy consumed by electrical motors [1]. The problem of the energy efficiency of pumping systems is topical among the researchers [2, 3]. Although most of the pumping units are still using drives without variable speed control [3], in some countries almost 20-30% of pumping systems are already supplied with variable speed drives (VSDs), due to their high efficiency [4]. Often, instead of a high-power single pumping system, a multi-pump system with two or more pumps with a lower power is used [5, 6, 7]. This type of solution can provide cheap operation costs, easier maintenance, and a longer lifespan of a pumping unit. In addition, there is a possibility of flowrate control in a multi-pump system without a pressure drop and significant efficiency drop of pumping units. It is almost impossible to achieve by means of either throttling or speed control in a single-pump pumping system [8]. Maintaining the pressure in the system at flowrate control is crucial in applications with high static pressure [6; 7]. The possibility of energy loss reduction for the multi-pump system is achieved with the help of switching off some of the pumps at low flowrates, as well as by optimizing the operation of separate pumps, in the case where they are supplied with variable speed drives [6; 7; 8]. A wide variety of articles concern the energy efficiency issues of multi-pump systems. In [8] for example, the optimization of multi-pump system operation is considered in order to increase reliability and improve the efficiency of the system. The efficiency of the two systems is compared depending on the fluid flowrate. For pumps operating in parallel mode, the possibility of the optimal control strategy is studied with the help of a genetic algorithm. The obtained results show that the highest efficiency is achieved by ensuring the same flowrate of two pumps. This article also studies the comparison of a system consisting of two pumps with one frequency converter and the same system with two separate frequency converters for each pumping unit. It is shown that the latter system has better efficiency. In [9], the solution for energy efficiency boosting in a multi-pump system with variable speed drives is proposed. The concept does not include any additional flow meters or measuring instruments for flow-head characteristics. An original strategy for multi-pump system control using throttling and speed control methods based on flow rate estimation is proposed. In [10], an original algorithm is proposed for reliable prediction of the operation of parallel controlled pumps in the multi-pump system and the control of the number of working pumps that is optimal from the energy efficiency point of view for variable speed drives without throttling implementation. In [11], the parallel operation of two pumps with the same parameters of the nominal pressure and different values of the nominal flow is investigated. It is shown that the overall efficiency of the system decreases with an increase in the ratio of the nominal flow rates for individual pumps. For comparison, 28 various combinations of pump system parameters were investigated. In [12; 13], the parallel operation of the multi-pump system consisting of four pumps is considered, where the flowrate is controlled by three different methods (throttling, bypass, and by the speed regulation). A genetic algorithm is used to find the proper energy-efficient management strategy. In [14], a system with five parallel pumps is investigated. The power consumption is compared for different cases. For instance, when only one frequency converter is used and in the case of using five separate frequency converters for each particular pump. It is shown that in the case of application of variable speed drives for each pump instead of one variable speed drive leads to a reduction of power consumption from the grid by 2.5%.