# Exploring Numerically the Benefits of Water Discharge Prediction for the Remote RTC of WDNs

^{1}

^{2}

^{3}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Control Valve Regulation

_{1}(s), let the water discharge in the pipe, temporally averaged over the control temporal step Δt (s), be equal to Q (m

^{3}/s). The valve closure setting is then assumed to be equal to α

_{1}(-), which can range from 0 (fully open valve) to 1 (fully closed valve). If ξ(α) is the valve curve, ξ

_{1}(-) is the local head loss coefficient associated with α

_{1}. As shown in [5], the corresponding local head loss Δh

_{1}(m) is then equal to:

^{2}) is the pipe inner cross section area and g = 9.81 m/s

^{2}is the gravity acceleration. In the downstream pipe end B, a temporally averaged pressure head value h

_{B}(m), far from the set point value h

_{sp}(m) by a quantity e (m), is observed. Incidentally, positive and negative values of e indicate larger and smaller values of h

_{B}(m) than h

_{sp}, respectively.

_{sp}at new time t

_{2}(s) (with t

_{2}= t

_{1}+ Δt), the valve has to be regulated in such a way that the new head loss Δh

_{2}(m) is equal to:

_{2}(-) at time t

_{2}has to be equal to:

_{2}corresponding to ξ

_{2}can be easily obtained. The setting variation from α

_{1}to α

_{2}has to be limited by the maximum correction allowed by the valve shutter velocity [9].

#### 2.2. Variable Speed Pump Regulation

_{1}, let the water discharge in the pipe, temporally averaged over the temporal step Δt, be equal to Q. If β

_{1}(-), which can range from 0 (pump speed equal to 0) to 1 (maximum pump speed), is the current speed setting for the pump, the head Δh

_{1}provided by the pump is obtained through the pump curve:

^{2}/m

^{5}), b (s/m

^{2}) and c (m) are the pump curve coefficients. Equation (6) was obtained by considering that a second order curve can fit effectively the curve data of head and water discharge from pump catalogues, and by applying the affinity laws for pumps [5].

_{B}in the downstream pipe end B be far from the set point value h

_{sp}by a quantity e.

_{sp}at new time t

_{2}, the pump has to be regulated in such a way that the head is equal to:

_{2}(-) can then be calculated starting from the following equation, derived from Equation (7) and from Equation (6):

_{2}in the previous equation is:

_{1}to β

_{2}has to be limited by the maximum correction allowed by the variable speed drive.

#### 2.3. Algorithm Refinement through State Prediction

_{1}. However, the algorithm can be improved if it is aimed at eliminating the expected pressure head deviation at new time t

_{2}, rather than at eliminating the pressure head deviation at current time t

_{1}. Therefore, reference should be made to the predicted water discharge Q

_{forecast}[m

^{3}/s] and deviation e

_{forecast}[m]. Q

_{forecast}is the predicted water discharge at the new time; e

_{forecast}, instead, is the deviation that would be expected at time t

_{2}if no setting variations were made from time t

_{1}.

_{forecast}, based on the value of the water discharge at time t

_{1}and at other previous times. At time t

_{1}, based on the series of temporally averaged values Q = Q(t

_{1}), Q(t

_{1}− Δt), Q(t

_{1}− 2Δt), …, Q(t

_{1}− (N − 1)Δt) derived from the available measurements, a regression is carried out to evaluate a smooth (fluctuation-free) trend of Q in the time interval from time t

_{1}− (N − 1)Δt to time t

_{1}, with the following second order polynomial:

_{1}(m

^{3}/s) r

_{2}(m

^{3}/s

^{2}) and r

_{3}(m

^{3}/s

^{3}) are derived through the application of linear regression [17]. This polynomial is used for calculating Q

_{forecast}, that is Q(t

_{2}= t

_{1}+ Δt).

_{1}, r

_{2}and r

_{3}, a value of N ≥ 3 must be considered. Incidentally, for r = 3, no best fit has to be searched for since only one parabola passes through three points. At each case study, an ad hoc analysis should be performed to understand the best window size N. As an explicative example of the prediction algorithm, let us consider a water discharge prediction at time t

_{1}, in the context of RTC with Δt = 180 s (see graphs in Figure 2, in which dots and lines represent the measured values of Q and the polynomials obtained through regression, respectively). As graph a) highlights, if the water discharge trend is smooth (regular trend of the dots), as is the case with the aggregation of stochastically reconstructed demands from very numerous users, N = 3 already gives excellent prediction results. When the trend features visible random fluctuations (irregular trend of the dots), as is the case with the aggregation of stochastically reconstructed demands from few users (e.g., see Figure 2b), a larger value of N has to be considered to obtain a polynomial that reflects reliably the global temporal trend of Q, though failing to catch its random fluctuations. A small value of N (e.g., N = 3), instead, makes the water discharge prediction too dependent on random fluctuations. At each case study, the optimal window size must be assessed with the objective to maximize the overall performance of water discharge prediction and, therefore, of RTC. In this context, the closeness of the controlled variable to the set point can be considered as an indicator of the RTC performance for the choice of N, as is shown in Appendix A.

_{forecast}, which is associated with Q

_{forecast}, the two following expressions can be used in the case of control valve or variable speed pump, respectively:

_{forecast}, in the absence of device setting variations. Potentially, two additional terms could be inserted in the right side of the expressions (12) and (13): the former associated with the predicted pressure-head variation upstream from the device (i.e., at the source) from time t

_{1}to time t

_{2}; the latter associated with the predicted head loss variation in the WDN from time t

_{1}to time t

_{2}. However, these terms are neglected in the present work because they are usually smaller than the contribution related to the variation in head loss (Equation (12)) or in head gain (Equation (13)) across the device. Furthermore, the assessment of these terms would require other variables to be monitored in the WDN. Ultimately, the presence of the corrective term K in Equation (5) and in Equation (10) enables taking account of all the effects not explicitly considered in the analysis.

## 3. Results

_{1}= 0.75 and c

_{2}= 3.25 are the coefficients calculated to best fit the data provided by a valve manufacturer. In Equation (14), α was allowed to range from 0 (fully open) to 0.95 (almost fully closed).

_{sp}= 25 m.

_{sp}as the pressure head for full demand satisfaction at all nodes. Furthermore, leakage q

_{leak}(m

^{3}/s) was related to the nodal pressure head h through the Tucciarelli et al. [21] formula:

**∑**L (m) is the total length of the pipes connected to the node. Furthermore, α

_{leak}(m

^{2−γ}/s) and γ (-) are the leakage coefficient and exponent, respectively. While γ depends on pipe material and leak opening shape [22], α

_{leak}depends on the number of leak openings along the pipe and then grows with pipe age. In this work, these parameters were assumed uniform over the network. γ was set to 1 (typical value for plastic pipes) while α

_{leak}was set to 9.4 × 10

^{−9}m/s in order to have in the first scenario a leakage percentage rate close to 20% of the total outflow from the source, in the case of fully open control valve.

_{i}at the control node, to the set point value h

_{sp}. It is calculated as:

_{tot}= 2400 is the number of temporal steps Δt in the five days of simulation.

_{i}and β

_{i}are the values of valve setting α and pump setting β, respectively, at the generic i-th time. Generally speaking, a good controller is expected to keep the controlled variable close to the set point with small device setting variations, to avoid the wear of the control device. Therefore, it will feature a low value of both indices.

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 1.**Schematic for the description of the operation of logic controller for the control valve (left) and for the variable speed pump (right): situation at the current (above) and new (below) times. HGL stands for head grade line.

**Figure 2.**Example of application of the polynomial regression method for water discharge prediction in smooth (

**a**) and irregular (

**b**) demand trends.

**Figure 4.**Real time control (RTC) of control valve through LCb. In the five days of network operation, trends of total network demand (

**a**), device setting (

**b**) and controlled pressure head (

**c**).

**Figure 5.**RTC of variable speed pump through LCb. In the five days of network operation, trends of total network demand (

**a**), device setting (

**b**) and controlled pressure head (

**c**).

**Figure 6.**RTC of control valve through LCb under conditions of modified demand. In the five days of network operation, trends of total network demand (

**a**), device setting (

**b**) and controlled pressure head (

**c**).

**Table 1.**Results of simulations 1a and 1b related to scenario 1 (control valve), in terms of mean|e| and ∑|Δα| for the optimal value of K. LCa (logic controller a); LCb (logic controller b).

Simulation | Logic Controller | K (-) | Mean|e| (m) | ∑|Δα| (-) |
---|---|---|---|---|

1a | LCa | 0.5 | 0.79 | 26.59 |

1b | LCb | 0.5 | 0.70 | 25.11 |

**Table 2.**Results of simulations 2a and 2b related to scenario 2 (variable speed pump), in terms of mean|e| and ∑|Δβ| for the optimal value of K. LCa (logic controller a); LCb (logic controller b).

Simulations | Logic Controller | K (-) | Mean|e| (m) | ∑|Δβ| (-) |
---|---|---|---|---|

2a | LCa | 1.4 | 0.25 | 10.66 |

2b | LCb | 1.4 | 0.22 | 9.64 |

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**MDPI and ACS Style**

Creaco, E. Exploring Numerically the Benefits of Water Discharge Prediction for the Remote RTC of WDNs. *Water* **2017**, *9*, 961.
https://doi.org/10.3390/w9120961

**AMA Style**

Creaco E. Exploring Numerically the Benefits of Water Discharge Prediction for the Remote RTC of WDNs. *Water*. 2017; 9(12):961.
https://doi.org/10.3390/w9120961

**Chicago/Turabian Style**

Creaco, Enrico. 2017. "Exploring Numerically the Benefits of Water Discharge Prediction for the Remote RTC of WDNs" *Water* 9, no. 12: 961.
https://doi.org/10.3390/w9120961