**6. Results**

Since preliminary investigations proved that *Si* = −1 gave good results, this value was kept throughout all the calculations. As an example of the results, Figure 2 reports calculations over a day for LCF with *Tc* = 3 min. The instantaneous and averaged flow-rate *Qav* through the PCV, valve setting and instantaneous pressure at the CN are shown. As for the flow-rates, these values include the WDS pulsed demand [22] (Figure 4) and leakage, which added up on average to approximately 20% of the total output from the source. As for the valve setting, it must be noted that it always stayed far from the lower and upper boundaries, attesting to the proper regulation behaviour of the valve.

**Figure 2.** (**a**) Flow-rate *Q* every second and its value *Qav* averaged over 3 min; (**b**) valve setting *α* (evaluated every 3 min); and (**c**) ressure head *H* every second.

The LVF controller models the future from the past. All LVF*n* controllers that use velocities for a period 2*nTc* ≤ 42 min into the past were investigated. A different method from Equation (8), which uses a regression fit of past *Qav* values to predict the future flow, has also been proposed [23].

Let |*e*|*mean* and *emean* denote the average of |*H* − *Hsp*| and *H* − *Hsp*, respectively, where *H* is evaluated every second. These two performance measures determine the deviation of the pressure at the CN from the target set-point. In accordance with [22], the primary measure was |*e*|*mean*. In addition, let the performance measure Σ|Δ*α*| be the sum of the actuator setting absolute corrections evaluated at each iteration. This is a measurement of the wear and tear on the PCV due to setting changes. Performance is the best when the performance measures are as low as possible.

Undesirable behaviour due to PRV self-interactions was observed for *Tc* ≤ 1 min [22], thus *Tc* = 3, 5 and 10 min were considered. Of the time-steps studied, *Tc* = 3 min gave the lowest |*e*|*mean* for both LCF and LVF. The results for the full 24-h period with this time-step are now discussed.

For LVF*n*, |*e*|*mean* and Σ|Δ*α*| decreased monotonically as *n* increased (Figure 3). Evidently, decreases became insignificant nearing *n* = 7. It was also found that this monotonic decrease happened in each individual hourly period. The best performing LVF*n* was hence LVF7, which uses velocities for a period 42 min into the past. However, |*e*|*mean* and Σ|Δ*α*| were perfectly reasonable for LVF3, if a controller looking less into the past were desired.

**Figure 3.** Results for LVF*n* with *Tc* = 3 min over one day: (**a**) |*e*|*mean*; and (**b**) Σ|Δ*α*|. For *n* = 1, 2, the values are out of range at 11.1 and 7.4, respectively.

For the day, |*e*|*mean*(LVF)−|*e*|*mean*(LCF) = 0.016 m, thus LCF outperformed insignificantly. Evaluating a similar difference for Σ|Δ*α*|, it was found that LCF outperformed LVF7 by a tiny 0.85%.

It is interesting to determine the variation of the pressure deviation during the day. Comparison with the consumption pattern can more easily be done by evaluating hourly averages, in order to reduce the effect of stochastic fluctuation in consumption. In Figure 4, |*e*|*mean* is shown as a function of time. It is noticeable that the results for LCF and LVF7 are very similar at a certain time, and that there is apparently random variation from hour to hour. This suggests that |*e*|*mean* has significant stochastic fluctuation. On the other hand, *emean* as a function of time shows clear patterns (especially for LCF), and appears to be less dependent on stochastic fluctuation (Figure 5a).

**Figure 4.** |*e*|*mean* evaluated during an hour period preceding the time of the datum shown *Tc* = 3 min.

(**a**)

**Figure 5.** *Cont*.

**Figure 5.** *emean* evaluated during an hour period preceding the time of the datum shown (*Tc* = 3 min): (**a**) *emean*; and (**b**) out-performance of LVF7 over LCF, defined using *emean*. Out-performance is positive if the *emean* of LVF7 is nearer to 0 than the *emean* of LCF.

Assume that the flow rate through the PCV in Figure 2a is fitted by a smoothly varying *Qtrend*, which indicates the trend in the flow rate (the hourly variation). The results for LCF in Figure 5a have an interesting pattern. *Qtrend* increased noticeably during 5–8 h and 17–20 h. Accordingly, *emean* was negative for the points in Figure 5a representing these hours. In addition, the deviation of *emean* from zero was largest for the period 6–7 h when the rate of change of flow was the largest of the entire day. The flow decreased noticeably during 0–1 h and 20–24 h. Fittingly, *emean* was positive for the points in Figure 5a representing these hours.

All of these observations for LCF were consistent with a related study predicting that the deviation of the pressure is approximately proportional to −*Q* /*Q* in the context of non-stochastic consumption [27], where *Q* is the rate of change of *Q*.

For non-stochastic consumption, numerical results for LCF were previously pointed out to suggest that the deviation is driven by −*Q* [21]. This can be confirmed by Figures 7, 10, 12 and 13 of [11], and in the pressure shown in Figure 7 (7-RES) [12]. For stochastic consumption, this behaviour can also be seen in Figure 7b of [22].

Figure 5a shows that LVF7 was less prone than LCF to deviate significantly from the target set-point pressure. Figure 5b indicates that LVF7 substantially outperformed LCF during Hours 5–8. This coincided with the hours when *Qtrend* changed the most quickly. In addition, during the second fastest flow change during 20–24 h, LVF7 outperformed LCF.

The sum of the out-performance amounts for each hour in Figure 5b was 0.028 m. Hence, with the performance measure *emean*, LVF7 outperformed LCF insignificantly over the day. Taking into account the results from all three performance measures, it is fair to say that the performance of LVF7 was the same as LCF over the entire day.

#### **7. Discussion**

Water consumption shows stochastic fluctuation in a real WDS. There are also unsteady flow processes that can cause sudden variations in flow and pressure [4,5,28]. For RRTC, the effect on a proportional-integral controller of adding random consumption fluctuation at each time-step *Tc* to smooth water consumption is initially studied in [18]. The bottom-up approach used in this work incorporates both fully stochastic consumption fluctuation and unsteady flow processes.

From the viewpoint of the derivation of the controllers, LVF should at first glance be an improvement on LCF. However, LVF depends on a future change Δ*vi*, which can only be modelled by estimating it from the past [11]. The way to decrease the effect of stochastic fluctuation in consumption is to estimate Δ*vi* by looking far into the past (Figure 3). However, relying on the far past is undesirable, as shown by the case of no fluctuation. (Assuming *Q* is a smooth function of time, estimating Δ*vi* from the most recent past velocities should be the most accurate). Hence, the larger is the fluctuation, the greater is the performance of LVF weakened.

The performance of LVF relative to LCF depends on a large number of factors. For a given WDS, these were argued to include |*Q trend*| at a certain time. It is postulated that this is to be compared to the magnitude of the average fluctuation at a certain time. The following study indicates what happens if |*Q trend*| dominates the fluctuations. For non-stochastic consumption, LVF1 was found to strongly outperform LCF at almost all times in two WDSs (Figures 3, 5 and 6 of [27]). In another study, a controller that uses future flow forecasting (LCb [23]) shows a clear advantage above the case when the future forecasting is neglected (LCa), when the fluctuations in consumption are small compared to its hourly variation. The advantage is obtained for a flow rate that has smaller fluctuations and much larger hourly variation than in Figure 2a.

In the current study for *Tc* = 3 min, it was found that LVF7 outperformed LCF when |*Q trend*| was large at a certain time. On the other hand, LVF7 and LCF performed the same during the entire day, for the assumed consumption pattern. However, it is expected that LVF*n* can outperform over the entire day (for some *n*) when there are more hours when |*Q trend*| dominates the fluctuations, or there are hours when |*Q trend*| strongly dominates the fluctuations.
