# Modelling and Incorporating the Variable Demand Patterns to the Calibration of Water Distribution System Hydraulic Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emissions and consumption of non-renewable resources. While a water column is moving through the pipe network, it involves energy expenditure and minimum energy is sought for sustainability and efficiency aspects. A WDS subjected to low flows with high frequency can potentially reduce the energy footprint and CO

_{2}emissions [26]. Controlling the pressure throughout the network also helps to improve the efficiency of the system [26]. The calibration objective function of a WDS hydraulic model should constitute the KPI variables to ensure sustainability and efficiency. While developing a hydraulic model, the operating rules in the model should be defined to reflect the KPIs. For instance, pump operating rules should be aligned with the minimum energy price of the day in cases where the energy price changes throughout the day to ensure cost and energy efficiency. Similarly, tank operating rules should be defined such that they correspond to a minimum water age to ensure the quality of water. The parameters representing the variable demand patterns in a WDS hydraulic model can be used with other available KPIs to formulate the objective function and optimise the model parameters.

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Hydraulic Modelling Tool

#### 2.3. Optimisation Tool

^{th}measurement, ${x}_{i}^{obs}$ and ${x}_{i}^{sim}$ are the observed and the simulated values, respectively, and n is the total number of data points. The minimum value of $\varnothing $ is zero which means a perfect fit. However, it is very unlikely to achieve a zero $\varnothing $ and, hence, minimum value is sought.

#### 2.4. Optimisation Algorithm

^{2}, C) where m is the mean vector, σ is the step-size and C is the covariance matrix, which is a positive definite convex quadratic function, similar to the inverse Hessian matrix. The parameters m and σ determine the centre and spread of the distribution, while parameter C is mainly related to the shape of the distribution [36]. For each iteration, these candidate solutions and the corresponding objective functions are evaluated and ranked. Based on the rank, the parameters of the multivariate normal distribution are updated. The CMAES algorithm adaptively increases or decreases the search space in the next iteration based on the result of the previous iteration. The number of candidate solutions, referred to as population size, is the most critical strategy parameter. The default population size is given by 4 + 3 ln(n) where n is the number of variables to optimise. The population size needs to be increased in case of multi-modal and/or noisy objective function. The global search performance is improved by increasing the population size [37]. However, a large population size significantly increases the convergence time, while a relatively small population size allows faster convergence. The initial step-size, σ, should be carefully defined such that it can be reasonably applied to all variables.

#### 2.5. Hydraulic and Calibration Setup

#### 2.6. Goodness-of-Fit (GOF)

^{2}), root mean square error (RMSE) and NSE were used to assess the fit between the observed and the simulated time-series. These are given by the following equations

^{2}lies between 0 and 1, where 1 indicates perfect fit and 0 means no fit at all. On the other hand, a better calibrated model will encounter a relatively lower RMSE value.

## 3. Results and Discussion

^{−1}and 5.06 L S

^{−1}, respectively, with a mean value of 7.94 L S

^{−1}and standard deviation of 1.64 L S

^{−1}. Several reasons, including seasonal variation of temperature, the day in the week (weekdays or weekend) and holidays can influence people’s water consumption pattern; hence, the demand multipliers on different days can also change.

^{−1}, 136.77 m and 42.53 m, respectively, while the same obtained through model simulation were 54.71 L S

^{−1}, 136.78 m and 42.54 m, respectively. These statistics indicate that the observed data were reasonably reproduced.

^{2}and RMSE for the first criterion at PS2 were 0.73, 0.77 and 27.63, respectively, while for the second criterion they were 0.77, 0.80 and 26.80, respectively. Similarly, at Binnies tank, NSE, R

^{2}and RMSE between the observed and simulated time-series were 0.86, 0.89 and 0.26, respectively, under criterion 1, while for criterion 2, they were 0.91, 0.92 and 0.24, respectively. On the other hand, at Meningie tank, the values of NSE, R

^{2}and RMSE for criterion 1 were 0.72, 0.76 and 0.20, respectively, while for criterion 2 they were 0.74, 0.77 and 0.19, respectively. These statistics indicate that the proposed method to model the variable demand patterns can improve the fit between the observed and simulated time-series. Although the calibration performances under both criteria during the studied period do not differ much, incorporating the linear modelling parameters into the calibration process allows modelling of the demand patterns on days with a relatively flatter peak.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**(

**a**) Flow pattern on different days during the simulation period and (

**b**) estimated average demand multipliers from flow patterns.

**Figure 6.**Plot of observed vs model simulated values (

**a**) flow at PS2 (

**b**) head at Binnies tank and (

**c**) head at Meningie tank.

**Figure 7.**Plot of observed vs model-simulated values (

**a**) flow at PS2 (

**b**) head at Binnies tank and (

**c**) head at Meningie tank.

**Figure 8.**Comparison of calibration performance while modelling the variable demand patterns using daily multiplication factors (criterion 1) and linear models (criterion 2) (

**a**) GOF for PS2 (

**b**) GOF for Binnies tank and (

**c**) GOF for Meningie tank.

Parameter Group | No. of Parameters | Group Range |
---|---|---|

Pipe roughness | 9 | 87–165 |

Pump settings/rotor speed | 42 | 0.80–1.33 |

Time-based controls for pump operation | 83 | 2–644 |

Morning and evening peak shifting | 56 | ±3 |

Linear model slope parameter | 28 | 0.70–1.40 |

Linear model Intercept parameter | 28 | 0.00–0.30 |

Optimisation Algorithm | Algorithm Type | Number of Model Runs | Elapsed Time | % Reduction of Objective Function |
---|---|---|---|---|

CMAES | Global search | 14,338 | 16 h using 28 processors | 29 |

SCE | Global search | 90,000 | 9 days and 16 h using 5 processors | 22 |

GLMA | Local search | 3196 | 1 day and 15 h using single processer | 19 |

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**MDPI and ACS Style**

Hossain, S.; Hewa, G.A.; Chow, C.W.K.; Cook, D.
Modelling and Incorporating the Variable Demand Patterns to the Calibration of Water Distribution System Hydraulic Model. *Water* **2021**, *13*, 2890.
https://doi.org/10.3390/w13202890

**AMA Style**

Hossain S, Hewa GA, Chow CWK, Cook D.
Modelling and Incorporating the Variable Demand Patterns to the Calibration of Water Distribution System Hydraulic Model. *Water*. 2021; 13(20):2890.
https://doi.org/10.3390/w13202890

**Chicago/Turabian Style**

Hossain, Sharif, Guna A. Hewa, Christopher W. K. Chow, and David Cook.
2021. "Modelling and Incorporating the Variable Demand Patterns to the Calibration of Water Distribution System Hydraulic Model" *Water* 13, no. 20: 2890.
https://doi.org/10.3390/w13202890