**Appendix A. Notation and Derivation of Head-Loss Controller**

Let *tc i* be a time period which differs from iteration to iteration; and *tc i* < *Tc*. At time *ti*, the PCV head-loss, velocity, flow rate and head-loss coefficient are, respectively, *H*˜*i*, *vi*, *Qi* and *ξi*; and the head at the CN is *Hi*. For all quantities *X* listed here, except *ξ*, *Xi* is defined as the quantity *X*(*t*) *evaluated* at *t* = *ti*. The PCV adjustment process commences soon after time *ti*, and continues until time *ti* + *tc i*, when the PCV is completely adjusted to the new coefficient *ξi*+1. At time *ti*+<sup>1</sup> ≡ *ti* + *Tc*, the coefficient is still *ξi*+1; and the head-loss, velocity and flow are denoted by *H*˜*i*+1, *vi*+<sup>1</sup> and *Qi*+1, respectively.

The Newton–Raphson numerical method has as its goal to find *z* such that *f*(*z*) = 0, i.e., find the root of a function of one variable. *z* is found by the iteration [29]

$$z\_{i+1} = z\_i - \frac{f(z\_i)}{f'(z\_i)}\tag{A1}$$

For the sake of argument, assume a WDS *with no* time-dependence, with only changes in *H*˜ allowed. Identifying *<sup>z</sup>* with *<sup>H</sup>*˜ and defining *<sup>f</sup>*(*H*˜ ) = *<sup>H</sup>*(*H*˜ ) <sup>−</sup> *Hsp* means the goal is to find *<sup>H</sup>*˜ such that *H*(*H*˜ ) = *Hsp*, as required [30]. Applying Equation (A1) leads to Equation (1). At this point, *i* is simply an iteration variable, with no notion of time attached to it. For the method to be applicable, *f* , and hence *H*, must be a continuous and differentiable function of *H*˜ . The iteration *i* can be chosen to refer to time *ti*, because the WDS has no time-dependence. Particularly, the sensitivity

$$\frac{1}{S\_i} \equiv \frac{dH}{d\tilde{H}}\tag{A2}$$

is evaluated at time *ti*.

In a general WDS *with* time-dependence, a head-loss controller can then be postulated (not derived) by applying Equation (1) even when there is time-dependence. The more there is time-dependence from one iteration to the next, over a few iterations, the less reliable the head-loss controller is expected to be. Such situations are when there is significant time-dependence on a time-scale shorter than *Tc*, or on a time-scale of a few *Tc*.

## **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
