# Implementation of DMAs in Intermittent Water Supply Networks Based on Equity Criteria

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Water Supply Equity

#### 2.2. Supply Time

#### 2.3. Theoretical Maximum Flow

_{maxt}, or network capacity defines the maximum flow that a network can supply with at least a minimum pressure, P

_{min}, at every node. The lowest pressure node must have the predefined minimum pressure [30]. The theoretical maximum flow value is determined through a demand-driven-analysis (DDA) hydraulic modeling of the network, in which nodes are associated with a given average demand. For this determination, several working conditions are evaluated and the peak factor is modified until the minimum pressure at the most unfavorable network node is guaranteed.

#### 2.4. Sector Development

- Calculation of weights in pipes and nodes
- Calculation of criteria weights
- Critical node selection
- Shortest path between critical node and source
- Node clustering
- Hydraulic calculation and verification of water supply equity

#### 2.4.1. Calculation of Weights in Pipes and Nodes

_{maxt}, as described above (see [30] for specific details). Pipes are subjected to their maximum to fulfill the minimum pressure requirements. The calculated pressure, ${P}_{n}^{Qmaxt}$, at each node n; and the obtained flow, ${Q}_{p}^{Qmaxt}$, and head loss, ${h}_{p}^{Qmaxt}$ for each pipe p, are used in weight calculation as follows.

_{n}, are directly related to the pressure at each node n:

_{p}, is determined by the inverse of the power dissipation [42], a function of the water specific weight, γ, and the calculated flow and head loss on pipe p, as in (3).

_{n}, which is later used to determine the degree weight, w

_{gn}, and the similarity distance (see below).

#### 2.4.2. Calculation of Criteria Weights

_{m}, (m = 1 for east coordinate, m = 2 for north coordinate, m = 3 for elevation, and m = 4 for service pressure) are derived from the opinion of the water company experts, since they are fully acquainted with the network characteristics and performance. To derive those weights we use pairwise comparison matrices and their Perron eigenvectors to transform opinions into weights or priorities, as in the analytic hierarchy process (AHP) [43,44]. A different treatment is given to the connection degree, as explained below.

#### 2.4.3. Critical Node Selection

_{crit,i}, in the set of remaining nodes, V

_{i}(initially, all the nodes of the entire network belong to this set). This critical node is selected to be the least-supply-pressure node during the maximal theoretical flow working condition, according to (2):

_{i}under development. Thus, it is the first element in cluster C

_{i}, and must be included in this set: n

_{crit,i}ϵ C

_{i}.

_{i+}

_{1}, it must be removed from the previous set, V

_{i}:

#### 2.4.4. Shortest Path between Critical Node and Source

_{p}, of every pipe, the critical node as a start, and the supply source as a destination, we determine the shortest path between both using the Dijkstra algorithm [45]. If there is more than one supply source, the shortest path must be determined for all sources. This step is essential to identify sectors, since each sector will have its own starting shortest path. Due to pipe weight characteristics, the shortest path will usually be made up of larger diameter pipes.

_{i}, which groups shortest path nodes, is also defined, as well as a pipe subset, F

_{i}, which groups the shortest path pipes.

#### 2.4.5. Node Clustering

_{c}, (see (17) below for an exemption) and the next node is selected from subset V

_{i}. This selection is determined by using the similarity distance,

_{cm}and normalized value x

_{nm}for each node n, depending on the m criteria, and on the cluster connection through an edge (pipe). Before stating the selection mechanism, we first explain (6) further.

_{gn}is described below. Note that normalization for each criterion is performed by dividing each value by the sum of the criterion values.

_{gn}, which depends on the degree, g

_{n}, of node n in the network. Nodes with a low connection degree are prioritized in the selection by means of

_{gn}= 1 may be adopted. Prioritizing low connection degree nodes may increase pressure differences between the highest and lowest pressure nodes in the cluster. Consequently, smaller sectors are created.

_{sel}, is the graph node minimizing (6), that is to say, the graph node with the smallest similarity distance:

_{sel}, from the set of current available pipes, E

_{i}, is the pipe with lowest weight w

_{p}:

_{sel}, and pipe, p

_{sel}, must be included in the developing cluster, C

_{i}(n

_{sel}ϵ C

_{i}) and in the shortest path pipe subset, F

_{i}, (p

_{sel}ϵ F

_{i}), respectively. Moreover, the subset of the critical path nodes, S

_{i}, must also join the cluster node subset, C

_{i}, to obtain node subset B

_{i}, as in (10), which is the base for the new graph, H

_{i}, as specified in (11). This graph is used for hydraulic calculations.

_{i+}

_{1}, and edge set, E

_{i}

_{+1}, used in the next iteration:

_{i}.

#### 2.4.6. Hydraulic Calculation and Verification of Water Supply Equity

_{i}are considered open, while the remaining pipes are considered closed until a node that connects them to the developing sector is selected. This situation may have a huge influence in the sector capacity calculation and, consequently, in equity and supply times.

_{i}for a working condition u. We also determine the maximum, P

_{max}, and the minimum, P

_{min}, pressures for the selected set of nodes, B

_{i}. Thus, we are able to determine the peak factor, k

_{u}, and, using the average demand, Q

_{j}, for any selected node j, j = 1, …, ns, we obtain for this working condition

_{s}, we assume that the consumed water volume in continuous supply equals that of intermittent supply, ${V}_{s}={\displaystyle \sum _{j=1}^{ns}{Q}_{j}\cdot 24}={Q}_{\mathit{max}t}^{h}\cdot {t}_{s}$. Furthermore, we consider that the average flow is distributed 24 h a day, and the network capacity [30] is high enough to supply a high flow, ${Q}_{maxt}^{u}$, in a short supply time. Thus,

_{u}value, so supply periods tend to 24 h. If the number of nodes is low, the peak factor increases, and thus supply time is shorter. In this case, having fewer supply hours is useful for avoiding supply schedule overlap.

_{eq}, which assures water supply equity:

_{i}is not empty) a new centroid, µ

_{cm}, is determined for each criterion m. We use the normalized values x

_{qm}for each node q, which makes up the developing sector B

_{i}, N

_{c}being its total number of nodes:

_{eq}is effectively surpassed and there are still unassigned nodes (V

_{i}is not empty), we start the iteration process in Section 2.4.3 again, and use the next critical network node, which is selected from all the excluded nodes (already grouped in previous clusters). From this new critical node, a new sector is built. Each network sector is built this way until all the nodes are assigned to a sector. This ends the sectorization process.

## 3. Case Study Description

#### Preliminary Evaluation of Water Supply Equity

_{min}= 5.30 m, which rearranges the setting curve to a fulfillment of the demanded flow (Figure 6).

_{int}= 12.64 L/s) to a value that equals the current network capacity (Q

_{maxt}= 10.04 L/s), using

_{s}= 4 h, to a minimum supply time, t

_{min}= 5.04 h, the demand is satisfied by the network capacity, 10.04 L/s, and the pressure at each node is over 10 m. Nevertheless, we must also evaluate pressure differences. We determined the pressure difference ΔP = 7.88 m between the maximum pressure 17.88 m and the minimum pressure 10.00 m, which clearly exceeds 5 m and thus equity is not guaranteed.

## 4. Results and Discussion

_{g}= 1 for each node.

_{s}, is reduced. Moreover, we increase the pressure difference value to analyze the sector configuration behavior at high pressures (Figure 9).

_{s}= 3732 m (Figure 10). Under these conditions, we create the second sector (Figure 11) and guarantee the desired equity.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of interest

## References

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**Figure 1.**Theoretical maximum flow for a tank fed network [30].

**Figure 2.**Flowchart, including high-level pseudocode and conceptual description, for implementation of sectors in intermittent water supply networks.

**Figure 9.**Variation of the pressure difference as a function of the selected nodes and the head in the water supply source.

**Figure 10.**Variation of the water supply time as a function of the selected nodes and the head in the water supply source.

Variable | Definition | Unit |
---|---|---|

R | Graph of whole network, used as hydraulic model | - |

V(R) | Set of whole network nodes | - |

E(R) | Set of whole network pipes | - |

${I}_{R}$ | Incidence relation of graph | - |

Q_{maxt} | Theoretical maximum flow or network capacity | L/s |

n | Node | - |

p | Pipe | - |

${P}_{n}^{Qmaxt}$ | Pressure at nodes at theoretical maximum flow working condition | m |

${Q}_{p}^{Qmaxt}$ | Flow at pipes at theoretical maximum flow working condition | L/s |

${h}_{p}^{Qmaxt}$ | Head loss at pipes at theoretical maximum flow working condition | m |

w_{n} | Weight in node n | m |

w_{p} | Weight in pipe p | m^{-1} |

z_{1} | Weight for east coordinate criterion | - |

z_{2} | Weight for north coordinate criterion | - |

z_{3} | Weight for elevation criterion | - |

z_{4} | Weight for service pressure criterion | - |

n_{crit,i} | Critical node at the developing sector i | - |

i | Developing sector | - |

C_{i} | Subset of selected nodes or developing sector | - |

V_{i} | Set of remaining nodes | - |

E_{i} | Set of remaining pipes | - |

S_{i} | Subset of shortest path nodes | - |

F_{i} | Subset of shortest path pipes | - |

d(µ_{c},x_{j}) | Similarity distance | - |

m | Number of criteria weight, m = 1 for east coordinate, m = 2 for north coordinate, m = 3 for elevation, and m = 4 for service pressure | - |

µ_{cm} | Centroid depending on the m criteria | - |

x_{nm} | Normalized value for each n node, depending on the m criteria | - |

g_{n} | Node degree | - |

w_{gn} | Weight for node degree | - |

M | Constant depending on the node degree importance | - |

n_{sel} | Selected node | - |

p_{sel} | Selected pipe | - |

q | Node of subset C_{i} | - |

x_{qm} | Normalized value for each q node, depending on the m criteria | - |

B_{i} | Node subset used for hydraulic calculations | - |

N_{c} | Total node number of a sector | - |

H_{i} | Graph of developing sector i, used as hydraulic model | - |

u | Working condition for developing sector | - |

${Q}_{maxt}^{u}$ | Theoretical maximum flow for working condition u | L/s |

k_{u} | Peak factor for working condition u | - |

P_{min} | Minimum pressure in subset B_{i} | m |

P_{max} | Maximum pressure in subset B_{i} | m |

j | Node of subset B_{i} | - |

Q_{j} | Average demand for each j node in subset B_{i} | L/s |

ns | Total node number of subset B_{i} | - |

t_{s} | Supply time | h |

V_{s} | Total supplied water volume in continuous and intermittent supply | m^{3} |

ΔP | Pressure difference | m |

P_{eq} | Limit value of pressure difference that assures water supply equity | m |

t_{min} | Minimum supply time, depending on the network capacity | h |

Q_{int} | Average flow in intermittent water supply | L/s |

H_{s} | Water level in tank or supply source | m |

Description | Value |
---|---|

Number of network nodes | 56 nodes |

Number of network pipes | 61 pipes |

Average demand flow in intermittent water supply | 12.64 L/s |

Current supply time | 4 h |

Minimum pressure | 5.30 m |

Maximum pressure | 17.20 m |

Node | East Coordinate (m) | North Coordinate (m) | Elevation (m) | Pressure (m) | Degree |
---|---|---|---|---|---|

J-2 | 698,074.22 | 8,010,604.23 | 3719.00 | 17.87 | 4 |

J-3 | 697,855.66 | 8,010,454.61 | 3719.00 | 16.05 | 3 |

J-4 | 697,853.41 | 8,010,448.53 | 3718.98 | 16.06 | 3 |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

J-57 | 697,801.55 | 8,010,310.70 | 3718.60 | 13.14 | 2 |

Sum | - | 448,583,649.73 | 208,217.36 | 808.56 | - |

Node | x_{n}_{1} | x_{n}_{2} | x_{n}_{3} | x_{n}_{4} |
---|---|---|---|---|

J-2 | 0.00155617 | 0.01785755 | 0.01786114 | 0.02210271 |

J-3 | 0.00155569 | 0.01785721 | 0.01786115 | 0.01984983 |

J-4 | 0.00155568 | 0.01785720 | 0.01786103 | 0.01985699 |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

J-57 | 0.00155557 | 0.01785689 | 0.01785921 | 0.01625678 |

Criterion | East and North Coordinates | Elevation | Pressure | Eigenvector |
---|---|---|---|---|

East and north coordinates | 1 | 3 | 1/2 | 0.333 |

Elevation | 1/3 | 1 | 1/3 | 0.140 |

Pressure | 2 | 3 | 1 | 0.528 |

Criterion | Expert 1 | Expert 2 | Expert 3 | Geometric Mean | Normalized Weight |
---|---|---|---|---|---|

East and north coordinates | 0.333 | 0.333 | 0.200 | 0.281 | z_{1} = z_{2} = 0.291 |

Elevation | 0.140 | 0.333 | 0.200 | 0.210 | z_{3} = 0.218 |

Pressure | 0.528 | 0.333 | 0.600 | 0.473 | z_{4} = 0.490 |

Total | 1 | 1 | 1 | 0.964 | 1 |

Sector | P_{max} (m) | P_{min} (m) | ΔP (m) | Q_{maxt} (L/s) | Supply Time t_{s} (h) | H_{s} (mca) | Clustering Nodes C_{i} |
---|---|---|---|---|---|---|---|

Sector 1 | 14.81 (J-30) | 10.00 (J-57) | 4.81 < 5 | 2.76 | 8.46 | 3737 | 26 |

Sector 2 | 13.54 (J-10) | 10.00 (J-34) | 3.54 < 5 | 6.18 | 4.41 | 3732 | 30 |

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## Share and Cite

**MDPI and ACS Style**

Ilaya-Ayza, A.E.; Martins, C.; Campbell, E.; Izquierdo, J.
Implementation of DMAs in Intermittent Water Supply Networks Based on Equity Criteria. *Water* **2017**, *9*, 851.
https://doi.org/10.3390/w9110851

**AMA Style**

Ilaya-Ayza AE, Martins C, Campbell E, Izquierdo J.
Implementation of DMAs in Intermittent Water Supply Networks Based on Equity Criteria. *Water*. 2017; 9(11):851.
https://doi.org/10.3390/w9110851

**Chicago/Turabian Style**

Ilaya-Ayza, Amilkar E., Carlos Martins, Enrique Campbell, and Joaquín Izquierdo.
2017. "Implementation of DMAs in Intermittent Water Supply Networks Based on Equity Criteria" *Water* 9, no. 11: 851.
https://doi.org/10.3390/w9110851