**5. Numerical Study in the Example WDS**

Controllers LCF and LVF were applied to the RRTC of a network in northern Italy (see skeletonised layout in Figure 1), which caters to about 30,000 inhabitants. This network has already been used for investigations in the area of pressure management [19,22,25].

**Figure 1.** Example water distribution system.

The network consists of a single source node (node 27), although networks with more sources could also be considered. There are 26 demanding nodes with ground elevation of 0 m a.s.l. and 32 pipes. The network can be considered as a single pressure zone, because of its size and the uniform ground elevation.

The references above give further details about the characteristics of the network. The source node has a head varying around 40 m a.s.l. [22]. A single DN300 plunger valve is the PRV, located at the end of pipe 26-11. The PRV has the head-loss coefficient *ξ*(*α*) given by

$$\xi^{\mathfrak{x}} = 10^{c\_1 - c\_2 \log\_{10}(1 - a)}\tag{9}$$

where data from the valve manufacturer allow the coefficients *c*<sup>1</sup> = 1.5 and *c*<sup>2</sup> = 2.8 to be calculated. The setting *α* is adjusted by the controller; and is constrained to range from 0 (completely open) to 0.95 (nearly completely closed). The maximum value *α* = 0.95 was chosen consistent with the real use of control valves, the objective of which is to modulate flow, rather than to interrupt it. However, it must be noted that this upper boundary does not affect the results of the simulations, as shown below.

The lowest pressure values during the day are found at node 1, which was hence selected as the CN where pressure control was applied. The RRTC of the PCV was performed to enable the pressure head at the CN to be near the target set-point head of *Hsp* = 25 m.

The bottom-up approach detailed in [22,25] was used to obtain the consumption for each node. This approach is based on consumption pulse generation through the Poisson model [26], considering pulse duration and intensity to be dependent random variables, both of which are expressed through the beta distribution. The pulse arrival rate at each node was calculated to obtain the expected average nodal demand, while considering the pattern of total demand observed in the WDS in a single day. More details of the bottom-up approach for demand generation are presented in [22], along with the parameter values used for demand generation.

To describe the hydraulics of the WDS, the model described in [22] was used. This model enables the unsteady flow modelling of the WDS and the accurate reproduction of the hydraulic behaviour of the valve. Compared to other software available in the market, this proprietary model has the advantage of considering unsteady flow pipe resistances, thus yielding more realistic results.
