# Optimal Node Grouping for Water Distribution System Demand Estimation

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Node Groups and Demand Patterns

#### 2.2. Field Measurements

#### 2.3. KF-Based Demand Estimation

**A**

_{k}relates the state at the previous time step k − 1 to the state at the current step k. This matrix is updated at each time step and calculated from the historical mean node group demand values;

**w**

_{k}is a random variable representing the process noise; and

**Q**

_{k}is the process noise covariance. Note that node grouping is provided from the optimization algorithm submodule (Figure 2) based on the methodology described in Section 2.1.

**u**

_{k}, operational information) to the measurements (pipe flows) as follows:

**z**

_{k}denotes the measurement variables;

**v**

_{k}is a random variable representing the measurement noise; and

**R**

_{k}is the measurement noise covariance.

**K**

_{k}is the Kalman gain matrix expressed as follows:

**H**

_{uk}is the Jacobian matrix of the partial derivatives of h with respect to x and u, and a unique

**H**

_{uk}is computed for each system operational state (

**u**

_{k}) around the a priori state estimate ${x}_{k}^{-}$; and ${P}_{k}^{-}$ is the a priori estimate error covariance calculated as follows:

**z**

_{k}and predicted $h\left({x}_{k}^{-},{u}_{k}\right)$ measurements. For example, a large measurement error covariance

**R**

_{k}results in a small update correction to the forecast state vector ${x}_{k}^{-}$.

**H**in Equations 2 (${z}_{k}=H{x}_{k}^{-}+{v}_{k}$) and 3 (${x}_{k}^{+}={x}_{k}^{-}+{K}_{k}\left({z}_{k}-H{x}_{k}^{-}\right)$). The LKF can be used for non-linear systems h(x) with weak non-linearity, but may perform poorly as the non-linearities increase. Both the LKF and NKF use first-order approximations for the error covariance propagation (${H}_{uk}{P}_{k}^{-}{H}_{uk}^{T}$ in Equation 4 and ${A}_{k}{P}_{k-1}^{+}{A}_{k}^{T}$ in Equation 5). A perturbation method, wherein the derivatives are approximated by the numerical forward finite differences, is used to calculate

**H**

_{uk}.

#### 2.4. Demand Estimation Accuracy Indicator: RMSE

_{k}is the estimated group demand at time step k (obtained from the KF-based demand estimation method described in Section 2.3); and O

_{k}is the true group demand at the k-th time step (synthetic measurements are obtained by the methodology described in Section 2.2).

#### 2.5. Elitism-Based GA

#### 2.6. Optimal Node Grouping Model

_{i}is the RMSE of the i-th node group; ng is the total number of node groups (i = 1, 2, ..., ng); and m is the total number of node groups predefined by the user generally equal to the total number of pipe flowmeters in the WDS.

## 3. Application

^{53}(= 14

^{47}). Note that the well-known Hanoi network design problem has 2.865 × 10

^{26}(6

^{34}) possible solutions [37]. The crossover and mutation processes are conducted with probabilities of 85% and 5%, respectively. The genetic traits of the chromosomes are shared at multiple scattered points, while the standard mutation is employed. The node grouping determined by the engineering decision of Jung and Lansey [8] (i.e., a node group comprising nodes of the same user type) is seeded as an initial solution, whereas the other initial solutions, where the population is 100 (ni = 100), are randomly generated by uniform sampling among 1–14 integer values. The eGA returns the best solution when the solution is not changed over 300 iterations.

## 4. Application Results

#### 4.1. Optimal Node Grouping Results

#### 4.2. Accuracy of Individual Nodal Group Demand Estimation

## 5. Summary and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Residential and commercial demands represented as nodes in a hydraulic WDS model. The dashed ellipse shows a potential node group.

**Figure 2.**Structure of the proposed optimal node grouping (ONG) model. OF indicates objective function.

**Figure 4.**Diurnal demand curves for five user types: Residential 1 (apartments), Residential 2 (houses with 1/2-acre lots), Residential 3 (houses with large home lots), commercial and industrial.

**Figure 5.**Synthetic field measurement generation steps: (

**a**) an identical demand pattern (Residential 1) is assigned to each node (i.e., time-varying demand factors are multiplied by the base demand); (

**b**) random variability is added to the nodal demands; (

**c**) the true node group demand (i.e., the sum of the three nodes' demand generated in the previous step); and (

**d**) the true pipe flow and the addition of measurement error.

**Figure 6.**Study network layout. A thicker pipe represents a larger diameter, while a larger node represents a larger demand.

**Figure 8.**Node groups of a representative initial random solution and meter locations. Each node group is delineated by either a different shape outline or filled by different colors/styles.

**Figure 9.**Optimal node groups identified by the proposed model. A node that is not grouped (boxed) has no external demand. Therefore, demand estimation is not required.

**Figure 10.**Actual (circle) and estimated (line) group demands at 5-min time steps for the first 24 h: (

**a**) NG1; (

**b**) NG4; (

**c**) NG7; (

**d**) NG8; (

**e**) NG12; and (

**f**) NG13 (NG = node group).

Node Group (NG) | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

RMSE (L/s) | 7.7 | 4.9 | 2.4 | 1.8 | 4.4 | 2.0 | 1.2 |

Node group (NG) | 8 | 9 | 10 | 11 | 12 | 13 | 14 |

RMSE (L/s) | 4.3 | 1.9 | 3.8 | 4.2 | 1.6 | 15.0 | 1.0 |

© 2016 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/).

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**MDPI and ACS Style**

Jung, D.; Choi, Y.H.; Kim, J.H.
Optimal Node Grouping for Water Distribution System Demand Estimation. *Water* **2016**, *8*, 160.
https://doi.org/10.3390/w8040160

**AMA Style**

Jung D, Choi YH, Kim JH.
Optimal Node Grouping for Water Distribution System Demand Estimation. *Water*. 2016; 8(4):160.
https://doi.org/10.3390/w8040160

**Chicago/Turabian Style**

Jung, Donghwi, Young Hwan Choi, and Joong Hoon Kim.
2016. "Optimal Node Grouping for Water Distribution System Demand Estimation" *Water* 8, no. 4: 160.
https://doi.org/10.3390/w8040160