**1. Introduction**

Decreased pressure reduces water leakage from pipes, lowers pipe burst frequency and may reduce water consumption [1]. The device that is by far the most widely used to reduce pressure in a water distribution system (WDS) is a pressure control valve (PCV): commonly a pressure reducing valve (PRV) [2,3].

A closed-loop technique uses measurements in the WDS, while an open-loop technique does not. Earlier advanced pressure management techniques, with the first being the simplest, include: (1) time modulation (open-loop) [1]; (2) flow modulation (closed-loop) [4,5]; and (3) remote node modulation, which is not real-time (closed-loop) [6]. These earlier techniques reduce the pressure better than a classical PCV with no controller, but do not reduce the pressure as low as possible.

A WDS node where the pressure is sensitive to PCV adjustment, and whenever possible, also has the lowest pressure, is called a critical node (CN) [7]. To keep the pressure at the CN continually constant [8], the PCV setting must be changed in real-time, i.e., not manually, intermittently or only at specific times. This is usually accomplished by adjusting the setting every time-step, where the time-step is typically of the order of minutes [7].

The remote real-time control (RRTC) technique strives to make the pressure throughout the WDS as low as possible [9,10], by attempting to set the pressure at a remote CN in real-time to a low and constant target set-point value [11,12]. This is made possible by recent advances in the availability of wireless technology [13–15].

A laboratory experiment demonstrated how RRTC with a PRV can be attained by use of a controller [16]. An example of a field demonstration is the one in the district of Benevento, Italy [10].

The controllers in [7,17,18] only use the pressure measurement at the CN. Controllers that also use the flow rate through the PCV, taking changes in WDS conditions into account, were subsequently developed [19–23]. The water flow rate in a pipe equipped with the PCV needs to be measured.

Consumption in a real WDS is stochastic in nature. Recently, several numerical RRTC studies take this into account: either approximating it as random fluctuations [18], or using a comprehensive bottom-up approach [22,23]. This paper reports numerical results on two closely related flow-dependent PCV controllers in the latter approach. One of these was formulated and studied for the first time with stochastic consumption, because this may critically affect the controller's viability.

#### **2. Head-Loss Controller**

In this and the next section, a derivation of various controllers, outlining assumptions made, is presented. This is done in an effort to bring them all together and to put them in the same rigorous framework. The aim is to emphasise that the controllers are important from the viewpoint of hydraulic theory. Particularly, the derivation is within the context of a WDS where there is not significant time-dependence on a time-scale shorter than the control time-step *Tc*, or on a time-scale of a few *Tc* (see Appendix A).

Let *H*˜ be the head-loss across the PCV, and *H* the head at the CN. It can be argued from the Newton–Raphson numerical method (see Appendix A) that an appropriate controller, called the "head-loss" controller [21], would calculate the adjusted head-loss

$$
\hat{H}\_{i+1} = \hat{H}\_i - \mathcal{S}\_i \left( H\_i - H\_{sp} \right) \tag{1}
$$

from the current head-loss *H*˜*i*. Here, *Hsp* is the target set-point head of the CN; and the notation and sensitivity *Si* are defined in Appendix A (see also [24]). The information at iteration *i* determines the next iteration *i* + 1. The iterations are separated by *Tc*. The value of *Si* varies for different iterations, and is impractical to determine for a real-world WDS without a hydraulic model [21]. When the CN head depends very sensitively on the PCV head-loss, *Si* = −1 for a PRV. Using this value in Equation (1) yields the controller employed in [9,11,20].

Equation (1) represents the choice of controller, from the viewpoint of theory [12,19,21,23]. However, the controller evaluates hydraulic quantities at a specific time, and hence is not sensible for quantities that exhibit significant time-dependence.

#### **3. Controllers Based on Known PCV Flow Rate**

A PCV is conventionally modelled by [17,19,20]:

$$H = \frac{\oint}{2g}v^2 \qquad v = \frac{Q}{A} \tag{2}$$

where *ξ* is the (dimension-less) PCV head-loss coefficient, *v* is the water velocity, *Q* is the flow rate through the PCV, *A* is the area of the port opening within the PCV, and *g* is the acceleration due to gravity. Substituting Equation (2) into Equation (1) implies that the adjusted head-loss coefficient can be calculated as

$$\xi\_{i+1}^{\mathbb{Z}} = \xi\_i^{\mathbb{Z}} \left( \frac{v\_i}{v\_{i+1}} \right)^2 - \frac{2\mathcal{g}S\_i}{v\_{i+1}^2} \left( H\_i - H\_{sp} \right) \tag{3}$$

from the current head-loss coefficient *ξi*. Equation (3) can be used as the very general form of a controller, as conceived in [23] for *Si* = −1. It is called the "**g**eneral parameter-less controller with known **v**ariable PCV **f**low" (GVF). It is parameter-less, because it contains no tunable parameter. Specifically, *Si* is not tunable.

The right hand side of Equation (3) is separated into two parts. The first part does not involve the future (does not depend on *vi*+1), and is important because it does not require modelling the future. The second part is the remainder (denoted by Φ and Ψ). Separating Equation (3) into these parts yields

$$\mathfrak{z}\_{i+1}^{\mathfrak{x}} = \mathfrak{z}\_i - \frac{2\mathfrak{g}S\_i}{\upsilon\_i^2} \left( H\_i - H\_{sp} \right) + \Phi\_i + \Psi\_i \tag{4}$$

where

$$\Phi\_i = -\xi\_i f\_i \qquad \Psi\_i = \frac{2gS\_i}{v\_i^2} f\_i \left( H\_i - H\_{sp} \right) \qquad f\_i = 1 - \frac{1}{\left( 1 + \frac{\Delta v\_i}{v\_i} \right)^2} \tag{5}$$

with Δ*vi* ≡ *vi*+<sup>1</sup> − *vi*. All dependence on the future in the remainder part is through the dimensionless *fi*, and hence through Δ*vi*, which needs to be modelled.

Neglecting Δ*vi* leads to a controller with

$$
\Psi\_{\dot{i}} \approx 0 \qquad \Psi\_{\dot{i}} \approx 0 \tag{6}
$$

Equations (4) and (6) define the "parameter-**l**ess proportional controller with known **c**onstant PCV **f**low" (LCF) [21]. With *Si* = −1, it is first derived in [19]; and is also called "valve resistance" (RES) control [11,20].

Another controller can be obtained by only keeping the dominant terms in Φ and Ψ, which are linear and up to first order in the difference terms. *fi* is proportional to Δ*vi*. Hence, Ψ*<sup>i</sup>* is the only term in Equation (4) that is proportional to *two* difference terms (Δ*vi* and *Hi* − *Hsp*) and can accordingly be neglected. *fi* can be evaluated to lowest order in the difference term Δ*vi*, leading to a controller with

$$
\Phi\_i \approx -\frac{2\xi\_i^\chi}{v\_i} \Delta v\_i \qquad \Psi\_i \approx 0 \tag{7}
$$

Equations (4) and (7) define the "parameter-**l**ess controller with known **v**ariable PCV **f**low" (LVF), first derived in [21].
