# Comparison of Flow-Dependent Controllers for Remote Real-Time Pressure Control in a Water Distribution System with Stochastic Consumption

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## Abstract

**:**

**l**ess proportional controller with known

**c**onstant pressure control valve

**f**low); and LVF (parameter-

**l**ess controller with known

**v**ariable pressure control valve

**f**low), which uses a flow rate forecast. Indeed, this study considered an upgrade of LVF, in which the flow rate forecast was tailored to the conditions of stochastic demand. The application in a specific example network proved the performance of these controllers to be quite similar, although LCF was preferable due to its simple structure. For LCF, the average pressure at the critical node had a clear relationship to the consumption pattern. LVF outperformed when the hourly variation dominates the fluctuations in the flow. The conditions under which this out-performance occurred are comprehensively discussed.

## 1. Introduction

## 2. Head-Loss Controller

## 3. Controllers Based on Known PCV Flow Rate

**g**eneral parameter-less controller with known

**v**ariable PCV

**f**low” (GVF). It is parameter-less, because it contains no tunable parameter. Specifically, ${S}_{i}$ is not tunable.

**l**ess proportional controller with known

**c**onstant PCV

**f**low” (LCF) [21]. With ${S}_{i}=-1$, it is first derived in [19]; and is also called “valve resistance” (RES) control [11,20].

## 4. Modification of Controllers: Stochastic Consumption

## 5. Numerical Study in the Example WDS

## 6. Results

## 7. Discussion

## 8. Conclusions

- The magnitude of the average stochastic fluctuation in consumption decreases.
- There are many hours with a sizeable magnitude of the rate of change of the flow rate through the valve $|{Q}_{trend}^{\prime}|$.
- There are hours with a large $|{Q}_{trend}^{\prime}|$.

## Author Contributions

## Funding

## Conflicts of Interest

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

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**Figure 2.**(

**a**) Flow-rate Q every second and its value ${Q}_{av}$ averaged over 3 min; (

**b**) valve setting $\alpha $ (evaluated every 3 min); and (

**c**) ressure head H every second.

**Figure 3.**Results for LVFn with ${T}_{c}=3$ min over one day: (

**a**) ${\left|e\right|}_{mean}$; and (

**b**) $\mathsf{\Sigma}\left|\mathsf{\Delta}\alpha \right|$. For $n=1,2$, the values are out of range at $11.1$ and $7.4$, respectively.

**Figure 4.**${\left|e\right|}_{mean}$ evaluated during an hour period preceding the time of the datum shown ${T}_{c}=3$ min.

**Figure 5.**e

_{mean}evaluated during an hour period preceding the time of the datum shown (T

_{c}= 3 min): (

**a**) e

_{mean}; and (

**b**) out-performance of LVF7 over LCF, defined using e

_{mean}. Out-performance is positive if the e

_{mean}of LVF7 is nearer to 0 than the e

_{mean}of LCF.

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**MDPI and ACS Style**

Page, P.R.; Creaco, E. Comparison of Flow-Dependent Controllers for Remote Real-Time Pressure Control in a Water Distribution System with Stochastic Consumption. *Water* **2019**, *11*, 422.
https://doi.org/10.3390/w11030422

**AMA Style**

Page PR, Creaco E. Comparison of Flow-Dependent Controllers for Remote Real-Time Pressure Control in a Water Distribution System with Stochastic Consumption. *Water*. 2019; 11(3):422.
https://doi.org/10.3390/w11030422

**Chicago/Turabian Style**

Page, Philip R., and Enrico Creaco. 2019. "Comparison of Flow-Dependent Controllers for Remote Real-Time Pressure Control in a Water Distribution System with Stochastic Consumption" *Water* 11, no. 3: 422.
https://doi.org/10.3390/w11030422