Sewer Network Layout Selection and Hydraulic Design Using a Mathematical Optimization Framework
Abstract
:1. Introduction
- we propose an MIP to model the LS problem over a network with general topography,
- we extend the methodology in Duque et al. [9] to generate a hydraulic design, and
- we propose a novel iterative scheme in which the objective function in the LS model approximates the true hydraulic-based cost, and this approximation is refined as the method progresses.
2. Problem Statement and Definitions
- The network must transport water from a discrete number of sources (manholes, sinks, etc.) to an outfall at a specific location [8].
- The flow moves by gravity over a tree-like structured network.
- There is a manhole between adjacent pipes due to changes in slope, diameter and/or flow direction [40].
- A uniform flow is assumed for the hydraulic design of each pipe [40].
- The HD methodology can use the Manning or Darcy–Weisbach and Colebrook–White resistance equations.
2.1. Layout Selection
2.2. Hydraulic Design
3. Methodology
3.1. Layout Selection Model
- is a binary variable that takes the value of one if the flow direction between manholes and is from to and the connection is of type .
- is a nonnegative real-valued variable that represents the amount of flow from to if the connection type is , in the same units as the inflow in each manhole .
3.2. Layout Representation as a Tree-Like Directed Graph
- ,
- , and
3.3. Hydraulic Design Model and Solution Approach
- is the set of nodes in the HD graph , where is the th design node associated to the tree node and is the last design node in the outfall node . Therefore, is the union of disjoint subsets for every tree node , i.e., .
- is the set of arcs of the HD graph .
- Add a dummy design node connecting every design node in the outfall (see Figure 1);
- Assign a cost value for every arc in
- Reverse all arcs in and execute the Bellman–Ford algorithm [45] to obtain the one-to-all shortest paths from the outfall node to every other node in ;
- Retrieve the best paths from to any design node in ;
- Select the deepest and largest diameter pipe when multiple design arcs that are part of different shortest paths belong to the same tree arc , since only one pipe should exist in each section.
3.4. Iterative Scheme
3.5. Software and Data
3.6. Case Studies
4. Results
4.1. Benchmark Li and Matthew
4.2. Benchmark Moeini and Afshar
4.3. Real Network: Chicó
4.4. Convergence Curves
5. Discussion
6. Conclusions
Supplementary Materials
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Nomenclature
Layout Selection | |
is the graph that represents the layout selection problem. | |
is the set of nodes representing manholes. | |
is the set of undirected edges representing links between two nodes and . | |
is the inflow at manhole . | |
is the total flow in the system reaching the outfall manhole . | |
is the ground elevation at manhole . | |
is the set of possible types of pipes, containing outer-branch pipes () and inner-branch pipes (). | |
is the set of directed links between two manholes, and , so that . is the binary decision variable that represents the flow direction and connection type in the network layout, for all and . | |
is the continuous decision variable that represents the flow through arc of type , for all and . | |
is a large positive number. | |
Tree-structured Layout | |
is the graph that represents the selected layout as a tree-structured network. | |
is the set of nodes in the tree-structured graph | |
is a label that represents the index of the manhole associated to tree node | |
is a subset of that contains tree nodes associated with manhole . | |
is the set of arcs in the tree-structured graph. | |
Hydraulic Design | |
is the auxiliary graph used to represent the hydraulic design problem. | |
is the set of nodes in the hydraulic design graph , which is divided in subsets of nodes related with the tree node . | |
is the set of arcs of the hydraulic design graph . | |
is the discrete set of commercially available pipe diameters. | |
is the discrete decision variable that represents a possible diameter for an upstream pipe of tree node and manhole . | |
is the continuous decision variable that represents the elevation above a reference level of a node of the graph . | |
is the slope for each pipe, fully determined by the invert elevations and at the extremes of the arc which represents a pipe. | |
is the absolute roughness of the pipes. |
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Constraint | Value | Condition | |
---|---|---|---|
1 | Minimum diameter | 0.2 m | Always |
2 | Maximum filling ratio | 0.6 | |
0.7 | |||
0.75 | |||
0.8 | |||
3 | Minimum velocity | 0.7 | |
0.8 | |||
4 | Maximum velocity | 5.0 | |
5 | Minimum gradient | 0.003 | Flow rate < 0.015 |
Method | Researchers | Optimized Layout | Construction Cost (Yuan) × 106 |
---|---|---|---|
MGA | Pan and Kao [46] | No | 1.91 |
Adaptive GA | Haghighi and Bakhshipour [12] | No | 1.84 |
MILP | Safavi and Geranmehr [4] | No | 1.57 |
SDE-GOBL | Liu et al. [47] | No | 1.53 |
Loop-by-loop cutting algorithm and GA-DDDP | Haghighi and Bakhshipour [28] | Yes | 1.59 |
Loop-by-loop cutting algorithm and TS | Haghighi and Bakhshipour [20] | Yes | 1.43 |
Reliability - DDDP | Haghighi and Bakhshipour [48] | Yes | 2.41 |
ACOA-TGA-NLP | Moeini and Afshar [31] | Yes | 1.39 |
MIP and DP | (Present work) | Yes | 1.29 |
ACOA-TGA-NLP [31] | DP (Present work) with Fixed Layout [31] | MIP and DP (Present Work) | |
---|---|---|---|
Costs | |||
Construction Cost | ¥ 640,845 * | ¥ 400,940 | ¥ 369,498 |
Maintenance Cost | ¥ 269,155 * | ¥ 168,395 | ¥ 155,189 |
Total Cost | ¥ 910,000 * | ¥ 569,335 | ¥ 524,687 |
Computational effort | |||
Iterations [–] | 10 | 1 | 30 |
Time [min] | 198 | 4 | 115 |
Physical characteristics | |||
Average diameter [m] | 0.331 ** | 0.262 | 0.267 |
Average depth [m] | Not reported | 2.07 | 1.95 |
Outfall diameter [m] | 1.05 ** | 0.7 | 0.8 |
Outfall depth [m] | Not reported | 6.6 | 5.4 |
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Duque, N.; Duque, D.; Aguilar, A.; Saldarriaga, J. Sewer Network Layout Selection and Hydraulic Design Using a Mathematical Optimization Framework. Water 2020, 12, 3337. https://doi.org/10.3390/w12123337
Duque N, Duque D, Aguilar A, Saldarriaga J. Sewer Network Layout Selection and Hydraulic Design Using a Mathematical Optimization Framework. Water. 2020; 12(12):3337. https://doi.org/10.3390/w12123337
Chicago/Turabian StyleDuque, Natalia, Daniel Duque, Andrés Aguilar, and Juan Saldarriaga. 2020. "Sewer Network Layout Selection and Hydraulic Design Using a Mathematical Optimization Framework" Water 12, no. 12: 3337. https://doi.org/10.3390/w12123337