**1. Introduction**

The sustainable managemen<sup>t</sup> of the water supply is becoming a major challenge within the framework of the water–energy–food nexus [1,2]. A reduction in water leakage and an efficient use of energy are considered essential in order to contain greenhouse gas emissions and to limit the impacts of climate change [3–5]. In the last few years, a number of research papers in literature have focused on new methodologies for the efficient managemen<sup>t</sup> of a WDN [6]. Mini hydro power plants are a common practice in water transmission lines [7,8], where a stable discharge and pressure drop are present and can be exploited to produce energy. Pressure reducing valves (PRV) were first proposed to obtain optimal pressure values in water distribution lines, with the benefit of reducing the amount of water leakage [9,10]. Several methods for the optimal location of PRVs have been developed [11,12]. Recently, the substitution of PRVs with micro hydro power plants has been proposed [13–15]. In the presence of pumping systems, the strategy for energy cost savings and energy efficiency could be based on the increase in the hydraulic efficiency of the pumps and the electrical efficiency of motors; on the introduction of performance standards to which the pumps on the market must comply; on the use of variable frequency drivers to increase the efficiency performance of pumping groups working in variable conditions; and on the energetic assessment of energy use in the network [16,17].

Compensation tanks, which fix the pressure in a particular water district, are commonly inserted between water transmission and water distribution systems. At the tank intake, a residual pressure of the flow can be present and it is usually merely dissipated by a valve. In the presence of a large variability of ground elevation between the parts of the system, a second tank positioned at a higher elevation may be necessary in order to guarantee sufficient pressure to the users of this higher district. The hydraulic energy dissipated at the lower tank could be comparable with the energy to be supplied to the pump to deliver water to the higher tank. Therefore, a turbine could be placed at the tank inflow in order to both reduce pressure and recover energy to supply the pump. Three main problems limit the convenience of the installation of a turbine governed generator to power the pumping system, namely (i) the small amount of power available; (ii) the high cost of traditional turbines; and (iii) the low combined efficiency of the two electromechanical devices (the turbine generator and the pump motor) [18–20]. PATs could be employed instead of classic turbines, since they are considered a viable and flexible solution for energy production in a WDN, due to their lower cost and their generally acceptable efficiency [21–24]. In order to reduce the investment cost and increase the efficiency of the system, the PAT and a pump can be directly coupled and mounted on the same shaft without any intermediate generator or motor. The resulting PAT-pump turbocharger should be a viable solution. However, an appropriate P&P theory for water systems has, to the best of the authors' knowledge, not ye<sup>t</sup> been developed. The main difficulty in the design of such a device derives from the variability of the rotational speed. While an asynchronous generator defines the rotational speed of the turbine, according to the electric grid frequency, the rotational speed of a turbocharger is unknown and influences the hydraulic behaviour of the whole system. In this paper, a complete P&P formulation is developed, and a complete mathematical model is given for the design of the P&P plant when the boundary conditions are assigned. In addition, a non-dimensional solution is presented to discuss the general behaviour of the P&P plant. The solutions show that the performance of the system depends on the number of stages of both the pump and the PAT and on the pump–PAT discharge ratio. Furthermore, the model has been applied to four different scenarios based on the experimental discharge pattern of a pressure control station in a water distribution network in southern Italy. The results are investigated in terms of pressure reduction and pumping discharge and head, to show the hydraulic potential of the P&P device. The annual energy saving is also calculated for each scenario.
