# A Clustered, Decentralized Approach to Urban Water Management

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Proposed Water Supply System Clustering Method

#### 2.2. Source-Demand Distance

#### 2.3. Intra-Cluster Demand and Topographic Homogeneity

#### 2.4. Minimization of Source-Demand Distance

#### 2.5. Identification of Source Centers: Water Source Clustering

_{c}is source center, Q

_{i}and D

_{i}are the supply capacity of the source and the distance from reference water source.

#### 2.6. Assignment of Demand Parcel to the Nearest Source

_{1}, P

_{2},…,P

_{n}}, $P\in {\mathbb{R}}^{N}$ Euclidean norm defines, $\Vert P\Vert ={\left(P\ast P\right)}^{\frac{1}{2}},ifN=1then\Vert P\Vert =\left|P\right|,$ the absolute value of P. $\Vert P\Vert $ is the Euclidean norm of P that is used to measure the distance between points [28]. For example, suppose $P=\left(X,Y\right)\in {\mathbb{R}}^{2}$ and the source centers are defined by $C=\left({X}_{1},{Y}_{1}\right)\in {\mathbb{R}}^{2}$. Then the shortest distance from the source to the parcel is determined using Equation (2).

#### 2.7. Maximization Intra-Cluster Homogeneity and Connectivity

#### 2.8. K-means for Clustering WSS

_{1}, S

_{2},…,S

_{k}}. For the given cluster, assignment A that involves K groups, the total cluster variance is minimized through minimization of the sum of the squares of Euclidean norm for all clusters using Equation (4),

_{i}is the number dataset assigned to S

_{i}, µ

_{i}is mean of parcels in cluster S

_{i}and is calculated using Equation (5) [30].

#### 2.9. Intra-Cluster Parcels Connectivity

_{(m,n)}and neighbor parcels as P

_{(n±1,m±1)}, if parcel P

_{(m,n)}of one cluster neighbors two or more parcels from another cluster, and has only one neighbor from its own cluster, the evaluation of the minimum Euclidean norm of the parcel P

_{(m,n)}is performed with respect to the neighboring cluster centroid and is re-assigned to the closest one. In addition, the periphery parcels, which do not have many neighbors, are merged to the nearest cluster group in case they belong to another cluster. This connectivity analysis alone does not guarantee the existence of cluster members in another spatial location. One can use the smallest recommended size of cluster and/or the smallest demand that a cluster should supply to decide on merging isolated parcels to the neighboring cluster. An isolated parcel group will be kept as an independent cluster if the demand it supplies is greater than the required minimum size/demand within the cluster. However, a parcel group that does not satisfy the mentioned condition will be merged to the neighbor cluster. The decision of which cluster to combine is made by evaluating the minimum Euclidean norm value with respect to the centroid of neighboring clusters.

## 3. Results of Cluster Analysis Application to Arua, Uganda

#### 3.1. General Description of the Area

^{3}/d in the year 2032, which would worsen the water shortage. This predicted future demand takes into consideration the different population density and socio-economic status of each of the parish areas.

#### 3.2. Application of the Proposed Clustering Method and Results

#### 3.3. Source-Demand Distance minimization

`and`the sensitivity analysis of parcel resolution is not the focus of this paper. The available water-source abstraction locations of the area were aggregated into seven groups. The decision to propose a number of groups might depend on the size of the area, the size of clusters required, the numbers of water source locations available, etc. Different researchers have highlighted the need for case-by-case analysis to determine the population number that should be supplied by a single source to determine the smaller cluster size [8,25]. However, the determination of the number of groups required is not the focus of this paper. Thus, the minimum cluster size with a population of 10,000 was used in decentralizing the emerging area as suggested by Webster et al. [5] to determine the number of source centers for grouping. The evaluation of the distance between sources was preformed using Equation (2). The output of source-group identification process is shown in Figure 7a,b. Once the groups were identified the X, Y coordinate and supply capacity Qs were used to calculate source-centers. The source and source-center information is summarized in Table 1.

#### 3.4. Maximizing Homogeneity and Connectivity Analysis

## 4. Discussion

#### 4.1. Performance Analysis

#### 4.2. Method for Power Usage Analysis

^{3}, g is the gravitational acceleration in m/s

^{2}, Q is the flow rate in m

^{3}/s, and h

_{p}is the total head in meters. The total head is the summation of the velocity head (${h}_{v}$), friction head (${h}_{f}$), and elevation head (${h}_{E}$) are found in Equation (7). The velocity head was calculated using $\frac{{V}^{2}}{2g}$. Over large distances, the friction loss of the fluid flowing through the pipes must be calculated. In 2015, Hunter Industries published the article Friction Loss Tables [34], which includes standard PSI loss for various types and sizes of piping. The article also contains Equation (8), the Hazen-Williams formula used for pipe head loss calculations,

#### 4.3. Power Usage-Scenario Analysis

#### 4.4. Water Security and Resiliency

#### 4.5. Future Research

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Assignment of parcels to the source center. (X, and Y are location parameters, Z is elevation above sea level (asl), Qd is parcel demand, Qs and Qg are capacity of local water sources and group source respectively).

**Figure 7.**(

**a**) Available water sources and their groups; (

**b**) Water-source centers (based on minimized Euclidean distance).

**Figure 8.**Maps showing parcel assignment (Minimized Euclidean norm) (

**a**) and parcel membership-M based on source-demand distance (

**b**).

**Figure 9.**Maps showing elevation in m (asl) in the study area (

**a**) and parcel water demand in m

^{3}/d (

**b**).

**Figure 10.**Maps showing (

**a**) initial K-means clusters and (

**b**) and clusters after merging isolated parcels.

**Figure 11.**Maps showing clusters after re-distributing small groups (

**a**) and WSS cluster boundaries with source centers (green circles) (

**b**).

**Figure 12.**The bar graph represents the total power requirements per cluster. The line graph represents the excess power of the WSS scenarios compared to the proposed WSS.

Source Group | Source no. | Water Source-Center Location | |
---|---|---|---|

X (m) | Y (m) | ||

1 | 1 | 1992 | 6772 |

2 | |||

16 | |||

2 | 3 | 5461 | 4650 |

4 | |||

5 | |||

3 | 6 | 6181 | 3600 |

7 | |||

4 | 9 | 6150 | 600 |

5 | 10 | 3110 | 510 |

13 | |||

6 | 11 | 2062 | 3108 |

12 | |||

14 | |||

7 | 15 | 1650 | 4950 |

Forest (8) | 8 | 5400 | 2850 |

Municipality (9) | 2400 | 3150 |

**Table 2.**Power requirements of each cluster, total power and operational cost per year for every WSS scenario.

Proposed Cluster WSS | Scenario 1 Cluster | Scenario 2 Cluster | Scenario 3 Cluster | Scenario 4 Cluster | Centralized WSS | |
---|---|---|---|---|---|---|

Cluster Number | Power Per Cluster (W) | Power Per Cluster (W) | Power Per Cluster (W) | Power Per Cluster (W) | Power Per Cluster (W) | Power for Centralized (W) |

1 | 852.14 | 804.53 | 739.50 | 934.16 | 956.58 | 18,463.78 |

2 | 2967.08 | 3497.88 | 3990.10 | 4420.44 | 4742.18 | |

3 | 1154.50 | 971.45 | 820.32 | 868.80 | 941.99 | |

4 | 4597.13 | 4296.83 | 4435.66 | 4472.43 | 2986.95 | |

5 | 460.32 | 398.16 | 366.06 | 526.45 | 587.94 | |

6 | 4545.88 | 5074.51 | 5183.64 | 5124.28 | 5089.55 | |

7 | 2.46 | 2.38 | 101.09 | 256.46 | 636.46 | |

Total Power | 14,579.51 | 15,045.74 | 15,636.38 | 16,603.01 | 15,941.65 | 18,463.78 |

Additional Power Usage | 0.00 | 466.23 | 1056.87 | 2023.50 | 1362.14 | 3884.27 |

US$ Per Year | $23,659 | $24,416 | $25,374 | $26,943 | $25,870 | $29,962 |

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**MDPI and ACS Style**

Tsegaye, S.; Missimer, T.M.; Kim, J.-Y.; Hock, J. A Clustered, Decentralized Approach to Urban Water Management. *Water* **2020**, *12*, 185.
https://doi.org/10.3390/w12010185

**AMA Style**

Tsegaye S, Missimer TM, Kim J-Y, Hock J. A Clustered, Decentralized Approach to Urban Water Management. *Water*. 2020; 12(1):185.
https://doi.org/10.3390/w12010185

**Chicago/Turabian Style**

Tsegaye, Seneshaw, Thomas M. Missimer, Jong-Yeop Kim, and Jason Hock. 2020. "A Clustered, Decentralized Approach to Urban Water Management" *Water* 12, no. 1: 185.
https://doi.org/10.3390/w12010185