# Layout Selection for an Optimal Sewer Network Design Based on Land Topography, Streets Network Topology, and Inflows

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

## 3. Methodology

#### 3.1. Selection of an Initial Layout

#### 3.1.1. Criterion 1

#### 3.1.2. Criterion 2

#### 3.1.3. Criterion 3

#### 3.2. Iteration with Penalties in Excavation

#### 3.3. Case Studies

^{3}/s. The second sewer network was proposed by Moeini and Afshar [46]; it has 81 manholes, 144 pipes, a total flow rate of 0.593 m

^{3}/s, and its topography is completely flat since each manhole has the same elevation. The third sewer network is called Chicó and was proposed by Duque et al. [45]; it is part of a real sewer network located in Bogotá, Colombia. It has 109 manholes, 160 pipes, it is located in wavy topography terrain, and the total flow rate is 1.526 m

^{3}/s.

^{−2}), $h$ is the buried depth (m), β is the pipe cost per unit length (USD*m

^{−1}), ${m}_{\alpha}$, ${m}_{\beta}$, ${n}_{\alpha}$, and ${n}_{\beta}$ are constants and their values are presented in Table 2.

## 4. Results

#### 4.1. Benchmark Network Proposed by Li and Matthew

#### 4.2. Benchmark Network Proposed by Moeini and Afshar

#### 4.3. Benchmark Network Proposed by Duque et al.: Chicó

#### 4.4. Computational Effort

## 5. Discussion

## 6. Conclusions

- In the three case studies tested, the present methodology achieved the lowest construction cost reported in the literature. The cost reduction was more significant in the network with wavy topography, i.e., Chicó. While in the other networks, which are flat, the cost reduction was not so big, especially in the Moini and Afshar network, which is completely flat.
- The cost reduction was achieved in fewer iterations and in significantly less computational time when compared to the methodology of Duque et al. [1]. This shows that when selecting an optimal layout or one close to it, it is only required to perform the shortest path algorithm once to obtain a cost-effective sewer network design.
- Land topography turned out to be an important input in the layout selection model since whether the land topography is flat or not, a layout that follows the land slope and maximizes the number of inner-branch pipes allows a cost-effective layout to be obtained.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

From methodology proposed by Duque et al. [45]: | |

$\mathcal{M}$ | set of nodes representing manholes. |

$\mathcal{E}$ | set of undirected edges representing links between two nodes ${m}_{i}\in \mathcal{M};{m}_{j}\in \mathcal{M}.$ |

${\mathcal{A}}_{L}$ | set of directed links between two manholes, ${m}_{i}$ and ${m}_{j}$, so that (${m}_{i},{m}_{j})\in \mathcal{E}.$ |

$T$ | set of possible types of pipes, containing outer-branch pipes (${t}_{1}$) and inner-branch pipes (${t}_{2}$). |

${x}_{ijt}$ | binary decision variable that represents the flow direction and connection type in the network layout, for all (${m}_{i},{m}_{j})\in {\mathcal{A}}_{L}$ and $t\in T$. |

${q}_{ijt}$ | continuous decision variable that represents the flow through arc ${m}_{i},{m}_{j})$ of type $t$, for all for all (${m}_{i},{m}_{j})\in {\mathcal{A}}_{L}\mathrm{and}t\in T$. |

${a}_{ij}$ | fixed cost estimate for selecting the flow direction ${m}_{i}$ to ${m}_{j}.$ |

${c}_{ij}$ | estimation of cost per flow unit that traverses from ${m}_{i}$ to ${m}_{j}.$ |

From present methodology: | |

${b}_{ijt}$ | coefficient that depends on the land topography in the pipe from ${m}_{i}$ $\mathrm{to}{m}_{j}\in {\mathcal{A}}_{L}$ of type $t\in \mathcal{T}.$ |

${s}_{ijt}$ | land slope in the pipe from ${m}_{i}$ to ${m}_{j}\in {\mathcal{A}}_{L}$ of type $t\in \mathcal{T}.$ |

$\overline{{S}_{{t}_{1}}}$ | average installation slope of outer-branch pipes. |

$\overline{{S}_{{t}_{2}}}$ | average installation slope of inner-branch pipes. |

${L}_{ij}$ | length of the pipe from ${m}_{i}\mathrm{to}{m}_{j}\in \mathcal{M}$. |

${C}_{{t}_{1}}$ | average cost per unit length of outer-branch pipes. |

$\mu $ | penalty for outer-branch pipes in the selection of the initial layout. |

${\gamma}_{ij}$ | penalty for increments in excavation cost in pipe from ${m}_{i}$ to ${m}_{j}\in {\mathcal{A}}_{L}$. |

${\omega}_{ij}$ | bonus for reduction in excavation cost in pipe from ${m}_{i}$ to ${m}_{j}\in {\mathcal{A}}_{L}$. |

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Constraint | Value | Condition |
---|---|---|

Minimum diameter | 0.2 m | $\mathrm{Always}$ |

Maximum filling ratio | 0.6 | $d\le 0.3\mathrm{m}$ |

0.7 | $0.35\mathrm{m}\le d\le 0.45\mathrm{m}$ | |

0.75 | $0.5\mathrm{m}\le d\le 0.9\mathrm{m}$ | |

0.8 | $d\ge 1\mathrm{m}$ | |

Minimum velocity | 0.7 m/s | $d\le 0.5\mathrm{m}\mathrm{and}\mathrm{Flow}\mathrm{rate}0.015{\mathrm{m}}^{3}/\mathrm{s}$ |

$0.8\mathrm{m}/\mathrm{s}$ | $d>0.5\mathrm{m}\mathrm{and}\mathrm{Flow}\mathrm{rate}0.015{\mathrm{m}}^{3}/\mathrm{s}$ | |

Maximum velocity | $5\mathrm{m}/\mathrm{s}$ | $\mathrm{Always}$ |

Minimum gradient | 0.003 | $\mathrm{Flow}\mathrm{rate}0.015{\mathrm{m}}^{3}/\mathrm{s}$ |

Minimum depth | 1 m | $\mathrm{Always}$ |

**Table 2.**Constants of the cost function proposed by Maurer et al. [47].

Constant | Value | Units |
---|---|---|

${m}_{\alpha}$ | $110$ | ${\mathrm{USD}\ast \mathrm{m}}^{-3}$ |

${m}_{\beta}$ | $1200$ | ${\mathrm{USD}\ast \mathrm{m}}^{-2}$ |

${n}_{\alpha}$ | $127$ | ${\mathrm{USD}\ast \mathrm{m}}^{-2}$ |

${n}_{\beta}$ | $-35$ | ${\mathrm{USD}\ast \mathrm{m}}^{-1}$ |

**Table 3.**Construction cost for each criterion in the benchmark proposed by Li and Matthew [34].

Scenario | Construction Cost × 10^{6} (CNY)Function of Li and Matthew [34] | Construction Cost × 10^{6} (USD)Function of Maurer et al. [47] |
---|---|---|

Criterion 1 | 1.36 | 20.06 |

Criterion 2 | 1.33 | 19.91 |

Criterion 3 | 1.42 | 19.58 |

**Table 4.**Construction cost with different methods for the benchmark proposed by Li and Matthew [34].

Method | Researchers | Construction Cost × 10^{6} (CNY)Function of Li and Matthew [34] |
---|---|---|

MGA | Pan and Kao [32] | 1.91 |

Adaptative GA | Haghighi and Bakhshipour [20] | 1.84 |

Loop-by-loop cutting algorithm and GA-DDDP | Haghighi and Bakhshipour [35] | 1.59 |

SDE-GOBL | Liu, Han, Wang, and Qiao [48] | 1.53 |

Loop-by-loop cutting algorithm and TS | Haghighi and Bakhshipour [28] | 1.43 |

Reliability-DDDP | Haghighi and Bakhshipour [1] | 2.41 |

MILP | Safavi and Geranmehr [7] | 1.57 |

ACOA-TGA-NLP | Moeini and Afshar [46] | 1.39 |

MIP and DP | Duque et al. [45] | 1.29 |

MIP and DP Extension | Present work | 1.12 |

**Table 5.**Construction cost for each criterion in the benchmark proposed by Moeini and Afshar [46].

Scenario | Construction Cost × 10^{4} (CNY)Function of Li and Matthew [34] | Construction Cost × 10^{4} (USD)Function of Maurer et al. [47] |
---|---|---|

Criterion 1 | 36.86 | 817.83 |

Criterion 2 | 35.99 | 813.46 |

Criterion 3 | 45.55 | 862.07 |

**Table 6.**Construction cost with different methods for the benchmark proposed by Moeini and Afshar [46].

Method | Researchers | Construction Cost × 10^{4} (CNY)Function of Li and Matthew [34] |
---|---|---|

ACOA-TGA-NLP | Moeini and Afshar [46] | 64.08 |

MIP and DP | Duque et al. [45] | 36.95 |

MIP and DP Extension | Present work | 35.99 |

**Table 7.**Construction cost for each criterion in the benchmark proposed by Duque et al. [45].

Scenario | Construction Cost × 10^{4} (CNY)Function of Li and Matthew [34] | Construction Cost × 10^{4} (USD)Function of Maurer et al. [47] |
---|---|---|

Criterion 1 | 38.22 | 843.38 |

Criterion 2 | 38.12 | 856.89 |

Criterion 3 | 60.01 | 1093.93 |

**Table 8.**Construction cost with different methods for the benchmark proposed by Duque et al. [45].

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**MDPI and ACS Style**

Saldarriaga, J.; Zambrano, J.; Herrán, J.; Iglesias-Rey, P.L. Layout Selection for an Optimal Sewer Network Design Based on Land Topography, Streets Network Topology, and Inflows. *Water* **2021**, *13*, 2491.
https://doi.org/10.3390/w13182491

**AMA Style**

Saldarriaga J, Zambrano J, Herrán J, Iglesias-Rey PL. Layout Selection for an Optimal Sewer Network Design Based on Land Topography, Streets Network Topology, and Inflows. *Water*. 2021; 13(18):2491.
https://doi.org/10.3390/w13182491

**Chicago/Turabian Style**

Saldarriaga, Juan, Jesús Zambrano, Juana Herrán, and Pedro L. Iglesias-Rey. 2021. "Layout Selection for an Optimal Sewer Network Design Based on Land Topography, Streets Network Topology, and Inflows" *Water* 13, no. 18: 2491.
https://doi.org/10.3390/w13182491