## Abstract

Sewage systems are essential for the efficient functioning of cities. Wastewater contains solids and organic matter that pose health risks, making it necessary to optimize the sewage system design. In recent years, optimization tools have been introduced to minimize costs while still complying with regulations. Despite this, traditional designs still dominate, but the use of optimization methodologies can significantly reduce construction costs. For this reason, this research will compare the construction costs of a sewage system designed optimally and one designed using traditional methods, to determine the cost difference between the two. In a sector of Bogota, Colombia, a sewerage system was already built and designed according to Colombian laws using traditional methodologies. The information from this area was used to implement the UTOPIA program, created by the Universidad de los Andes. This program uses the Shortest Path Problem with the Bellman-Ford algorithm to design the network and minimize costs. The results show that the optimized system was about 15% cheaper than the traditional one, and it ensured that all pipelines met the design restrictions. Optimized sewage systems are a useful alternative for ensuring universal access to safe drinking water, increasing sewerage coverage, and reducing problems associated with inadequate design.

## HIGHLIGHTS

The paper makes a comparison between a traditional design methodology and an optimization one, for large sewers.

It proves that an optimized design not only obtains lower costs but for all pipes complies with the hydraulic restrictions.

The methodology introduced will help sewer coverage in developing countries.

It will help in reaching the SDG No. 6.

It is an interesting application of operational research.

## INTRODUCTION

A sewage system is characterized as a set of pipes whose objective is to collect and dispose of rainwater and/or wastewater, produced by human interaction with the hydrological cycle (Butler & Davies 2010). The proper functioning of a city depends on a properly designed sewage system, as wastewater carries solids and organic matter that can pose health and environmental hazards. For this reason, the proper design of a sewage system is a topic that in recent years has had great relevance (Butler & Davies 2010). However, it is considered that the sewerage infrastructure has two major problems: (1) The high costs of construction, operation, and maintenance and (2) the high difficulty of design since many variables must be considered.

The design of a sewage system is a complex process that involves two stages: the selection of the layout and the hydraulic design (Duque *et al.* 2016). The objective of the layout selection is to determine the direction of the pipes, design flows, and the type of the pipe. After the selection of the layout, the diameters, the slopes, and the excavation depths of each link must be determined; all these parameters belong to the hydraulic design stage. However, each of these stages has multiple variables to be considered, for this reason it is very difficult to have a single methodology that solves both processes together, which makes the design of sewage systems very demanding (Duque *et al.* 2016).

Obtaining a sewage system that complies with national legislation and minimizes cost is known as an optimized design (Duque *et al.* 2016). This optimized problem was proposed since the 1960s and, historically, the two design components (layout selections and hydraulic design) are covered separately. To solve the hydraulic design problem, different methodologies based on mathematical programming (MP) have been proposed. Some of the most common MP methods are linear programming (LP; Elimam *et al.* 1989; Swamee & Sharma 2013; Safavi & Geranmehr 2017), non-linear programming (NLP; Mansouri & Khanjani 1999; Swamee 2001), and dynamic programming (DP; Mays & Yen 1975; Gupta *et al.* 1983). In 2016, Duque *et al.* (2016) proposed a methodology based on DP and used the shortest path (SP) algorithm to find an optimal solution that meets the design constraints and minimizes the cost function.

However, the use of MP can have a high computational cost, which is why metaheuristic methods are frequently employed. Some examples of the methods are genetic algorithms (GAs; Afshar *et al.* 2006; Afshar 2012; Haghighi & Bakhshipour 2012), simulating annealing (SA; Yeh *et al.* 2011), ant colony optimization (ACO; Afshar 2010, 2012), and tabu search (TS; Haghighi & Bakhshipour 2015). In order to improve the optimal solution and decrease computational time, hybrid methodologies have been introduced. For example, Pan & Kao (2009) combined GAs and quadratic programming (QP) to obtain a network that was optimally designed with low computational times. Although the metaheuristic methodology requires low computational times, they do not ensure that a global optimum is reached (Pan & Kao 2009).

In 1990, Li and Matthew first proposed a methodology for the layout selection of a sewage system (Li & Matthew 1990). They used the search direction method for layout selection and discrete differential dynamic programming (DDDP) for hydraulic design. Their methodology was implemented on a network that would later become a benchmark for researchers. Their research showed that including both layout selection and hydraulic design in the design of a sewage system significantly reduced construction costs. They concluded that the flow rate, size, and gradient of pipes were important factors to consider in the optimal design of an urban drainage system (Li & Matthew 1990).

The network of Li and Matthew was used in 2013 by Haghighi (2013), who proposed an algorithm based on graph theory to solve the layout selection, which he called the loop-by-loop cutting algorithm. In this methodology, the network was represented as a graph with an undirected loop to form a feasible tree layout, and a genetic algorithm was used to find the optimal layout. This methodology was further improved by Haghighi and Bakhshipour in 2015, who combined the loop-by-loop algorithm with TS to find the optimal solution for both the layout selection and hydraulic design of the sewage system design (Haghighi & Bakhshipour 2015). Other methodologies used for the layout include GA (Walters & Lohbeck 1993; Walters & Smith 1995), heuristic approaches (Diogo & Graveto 2006; Bakhshipour *et al.* 2017), and ACO (Afshar & Mariño 2006).

Nevertheless, the optimized design of sewage systems cannot only be based on the minimization of the cost function, since there are other important variables for the optimal design solution, such as network resilience. For this reason, Bakhshipour *et al.* (2019) proposed a methodology that integrates structural resilience in the design of urban drainage networks.

The objective of this research is to use the UTOPIA (Underground Topology for Optimal Pipeline Infrastructure Assessment) research program, which is based on the methodology proposed by Duque *et al.* (2016), to determine the cost of an optimized sewage system and compare it to the costs of a sewage system designed with traditional methodologies. The use of an optimized sewage system is believed to have multiple economic benefits.

## BACKGROUND

The present study will employ the UTOPIA program developed by the Universidad de los Andes. This section will briefly summarize what this methodology consists of.

Duque *et al.* (2016) divided the methodology into two components: layout selection and hydraulic design. The layout selection defines flow rates, flow direction, and connection type of every edge through mixed-integer programming, resulting in a tree-like network.

The input for UTOPIA (layout selection) consists of an undirected graph composed of a set of edges (undirected connection between two nodes) and a set of nodes (manholes of the sewer network). In addition, the program requires the *x*, *y*, and *z* coordinates, and inflow rate for each manhole (Saldarriaga *et al.* 2021).

For a layout to be feasible there can be no recirculation of water through the pipes, all manholes must be connected to a single outfall and there must be a pipe in each edge. This means that the network is modeled as a tree-like structure, with two types of pipes: outer-branch and inner-branch. An outer-branch pipe is the first pipeline in a series that receives the flow only from the upstream manhole (Duque *et al.* 2020). On the other hand, the inner-branch pipes are the rest of the pipes in the system. In the model, the type of pipe is represented by the set , where represents the outer-branch pipe and represents the inner-branch pipe (Saldarriaga *et al.* 2021).

The layout selection component has two decision variables: a binary variable that takes the value of one (1) if the flow direction between manholes is from to and the connection is of type . The second variable is a non-negative real variable ( that represents the amount of flow from to if the connection type is (Duque *et al.* 2020). To consider all possible feasible layouts, the model incorporates a set of arcs (directed links between two nodes) (.

*et al.*2021).

After making the layout selection, the hydraulic design is carried out, which defines the diameter and elevation of each pipe. This problem has input data such as the tree-like network layout, the flow of each pipe, the cost function, the roughness, the discrete set of commercially available diameters, and some hydraulic constants. Some hydraulic constraints considered in this problem are minimum pipe diameter, minimum wall shear stress, minimum and maximum velocities, maximum filling ratio, and minimum and maximum slopes. Additionally, the upstream and downstream invert elevations of each pipe are chosen to ensure gravity-driven flow.

To solve the hydraulic design problem, Duque *et al.* (2016) proposed a single series of pipes methodology, in other words, this problem can be solved as an SP problem. In the SP problem, each graph is assigned a cost equation that assigns a weight to every arc, and the objective is to find a path between a pair of nodes that has the lowest possible cost. A suitable design of a graph according to Duque *et al.*'s (2016) methodology corresponds to a path in the graph that cipher diameter and invert elevation for every pipe of the network. The nodes cover all possible combinations between the available diameters and the inverse elevation within the established design constraints. An arc in this graph is the diameter and the invert elevation for a specific pipe, ‘To obtain an optimal design, costs are assigned to the arcs of the graph and an SP algorithm retrieves the minimum-cost path’ (Duque *et al.* 2020).

Constraint . | Value . |
---|---|

Minimum depth | 1.20 m |

Minimum diameter | 0.17 m |

Minimum shear strength | 1.2 Pa |

Maximum velocity | 5 m/s for concrete |

10 m/s for thermoplastic | |

Maximum filling ratio | 85% |

Constraint . | Value . |
---|---|

Minimum depth | 1.20 m |

Minimum diameter | 0.17 m |

Minimum shear strength | 1.2 Pa |

Maximum velocity | 5 m/s for concrete |

10 m/s for thermoplastic | |

Maximum filling ratio | 85% |

The methodology implemented in UTOPIA has been widely used in numerous investigations. For instance, in Duque *et al.*'s (2020) research, this methodology was applied to two benchmark networks: the first proposed by Li & Matthew (1990), and the second proposed by Moeini & Afshar (2012). The proposed methodology achieved better results in both benchmark networks because the cost reduction was up to 42% compared to the best-performing methodologies reported in the literature for the same case studies. As a result, one of the conclusions drawn from this research is that this new methodology could identify near-optimal solutions in terms of construction costs, satisfying the hydraulic and pipe connectivity constraints of the system.

Finally, most sewer network optimization problems are typically addressed using a predetermined tree structure. However, the layout selection significantly impacts the determination of the minimum-cost network. In this regard, UTOPIA performs hydraulic calculations and determines the most efficient tree layout for the network. For this reason, the methodology implemented in UTOPIA gives results that are very close to optimal.

## METHODS

The implementation of the methodology outlined by Duque *et al.* (2016) was conducted to optimize the design of the sewer system, utilizing the UTOPIA program offered by the Universidad de los Andes. To effectively use this program, specific input information is required, including the flow rate, *x*, *y*, and *z* coordinates, and the type of the pipe (outer-branch or inner-branch) for each manhole. The optimized design will be compared against a traditional sewer system design to determine the most cost-efficient option. In order to perform the economic comparison between the optimized and traditional designs, the same hydraulic conditions were implemented for both designs. Additionally, the layout selection, the flow rate, and pipe characteristics (material, diameter, and wall thickness) used in the traditional design were also used in the optimized design.

The methodology was divided into four stages, including the selection of a case study, layout selection, determination of hydraulic conditions, and calculation of cost equations for comparison. Each stage will be thoroughly explained in the following sections.

### Case study

For the design of the optimized sewer system, a sector of Bogotá, Colombia, was selected as a case study. This sector was previously built and designed following traditional design methodologies and was undertaken by one of the largest construction companies in Colombia.

The case study is a macro-project carried out by one of the largest construction companies in Colombia. The area of the sector is 328 hectares and includes single-family and multi-family homes, commercial and institutional buildings, inhabited by 240,000 people. The sewer network of this sector consists of 414 manholes, 336 inner-branch pipes, and 77 outer-branch pipes, with a minimum diameter of 227 mm and a maximum diameter of 1,300 mm. The sewer network is designed to have a single discharge point.

### Layout selection

### Hydraulic constraints

In order to carry out the optimized design of the sewer system, it is imperative to have the hydraulic restriction for the design, which vary based on the country in which the system will be implemented. In Colombia, the design restrictions for the sewer systems are established by the Ministry of Housing and Cities (Resolución 0330 del 2017). The hydraulic constraints used in the optimized sewer system are presented in Table 1 and are the same as those applied in the traditional sewer design.

Manning's equation was used for the velocity calculation with a coefficient *n* = 0.010. The set of diameters, in meters, used are = {0.227, 0.284, 0.327, 0.362, 0.407, 0.452, 0.595, 0.671, 0.747, 0.823, 0.975, 1.051, 1.127, 1.3}. All these values were used in the design of the sewage system with traditional methodologies.

### Cost equation

The cost of a sewerage network is a crucial aspect in its design. Over time, numerous cost equations have been developed that consider not only the costs associated with the diameter of pipes but also excavation costs. These cost equations are often used as the objective function in the optimization of sewer network design, as the objective is usually to minimize them. In this study, both construction costs and manhole costs are considered in the cost equations. A detailed description of each equation will be provided below.

#### Construction costs

*ij*($ COP), is the length of the section

*ij*(m), is the diameter of the pipe

*ij*(m), is the excavation volume of the section

*ij*(m

^{3}), and

*k*is the pesos conversion factor from December 2007 to July 2018, equal to 1.53 (adimensional). In regards to the excavation volume, Equation (3) shows the parameters used, where is the excavation volume of the section

*ij*(m

^{3}),

*H*is the excavation depth of the upstream pipe (m), is the excavation depth of the downstream pipe (m),

*d*is the diameter of the pipe (m),

*e*is the thickness of the pipe wall (m),

*h*is the low tubing filling (m),

*B*is the side space next to the pipe (m),

*s*is the slope of the pipe (–), and

*l*is the length of the pipe (m):

#### Manhole costs

## RESULTS AND DISCUSSION

### Sewage system with traditional methodologies

In the design carried out with traditional methodologies, not all of the pipes comply with the restrictions specified in the Colombian regulations, since a shear stress of less than 1.2 Pa is used, and the minimum depth of excavation is not the one recommended in the law either. Another parameter that does not meet the standard is the maximum excavation depth; however, the Colombian law specifies that if it is necessary to design the system with depths greater than 5 m, a justification must be provided and the pipe must be protected, which was done in the constructed system.

The sewage system for the case study using traditional methodologies had already been constructed and the necessary information was available to determine the costs of construction and of each manhole with Equations (3) and (4), respectively. The costs of this system are shown in Table 2.

Construction costs (USD$) | $ 30,900,000 |

Manholes costs (USD$) | $ 31,480,000 |

Total cost (USD$) | $ 62,380,000 |

Construction costs (USD$) | $ 30,900,000 |

Manholes costs (USD$) | $ 31,480,000 |

Total cost (USD$) | $ 62,380,000 |

The total cost of the sewer system using traditional methodologies was USD$ 62,380,000.

### Optimized system

Having the layout of the system and the information provided by the construction company, it was possible to enter all the inputs in the UTOPIA program to perform the optimized design of the system. By implementing the cost Equations (3) and (4), the costs shown in Table 3 were obtained.

Construction costs (USD$) | $ 23,720,000 |

Manholes costs (USD$) | $ 30,430,000 |

Total cost (USD$) | $ 54,145,000 |

Construction costs (USD$) | $ 23,720,000 |

Manholes costs (USD$) | $ 30,430,000 |

Total cost (USD$) | $ 54,145,000 |

The total cost of the optimized sewage system is USD$ 54,145,000. By using a program for the design of the system, it is being ensured that all the restrictions entered into the program are met, which were obtained from the Colombian standard. The results of the optimized design can be found in the supplementary material.

### Comparison of the two design methodologies

Comparing the design produced through both methodologies, it is evident that the optimized design adheres to all the design restrictions found in the regulations; the same does not occur in the system designed with traditional methodologies since it does not comply with the values of shear stress and minimum depth of excavation, which within the design period can generate operational issues.

*et al.*(2016), the cost difference between the optimized and traditional designs could be even greater. However, this step was not taken to facilitate a comparison between the systems produced using traditional and optimized methodologies.

Comparing the costs of each of the manholes, it can be analyzed that the optimized design is 3% more cost-effective compared to the system designed using traditional methodologies. This difference is due to the optimized design utilizing, on average, less excavation volume than the traditional design. Although the difference in costs between the two methodologies for this parameter is not significant, it highlights the ability of the optimized design to reduce excavation costs in the system.

Finally, analyzing the total costs of the system (construction costs and manhole costs), the cost difference between the optimized system and the system designed with traditional methodologies is nearly 15%. The biggest difference is in the construction costs due to the difference in the size of the pipes that are used because the optimized design uses smaller diameter pipes than the traditional design.

## CONCLUSIONS

One of the great priorities, worldwide, is to be able to provide drinking water and sewage service to the entire population since this would help reduce health and environmental problems. The greatest challenge is found in developing countries because there is low coverage of the system and there are multiple economic difficulties. As a solution in recent years, research has been carried out on optimized sewer systems, which aim to reduce economic costs and provide the service to as many people as possible.

Since the 1960s, different methodologies have been carried out whose objective is to solve the problem of layout selection and hydraulic design, with the objective function of minimizing construction costs. Historically, the two problems (layout selection and hydraulic design) have been resolved separately due to the large number of variables that must be analyzed. For the solution of the hydraulic design, some methodologies used are based on MP or metaheuristic algorithm. On the other hand, for the layout selection, each author has proposed different methodologies, for example, Li and Mathew used the searching direction method.

To compare the sewage system carried out with traditional methodologies and the optimized sewage system, a sector of Bogotá, Colombia, in which a sewage system had already been constructed with traditional methodologies, was chosen as the study area. For the optimized sewer, the UTOPIA program was used, which was created by the Universidad de los Andes. Subsequently, the costs of construction and of each manhole were determined for each of the designs, in order to compare and analyze which would have greater economic benefits.

To economically compare the sewage system designed with traditional methodologies and the optimized one, the same design restrictions found in the Colombian regulations were implemented and, in addition, the layout selection was the same. Therefore, in the optimized sewer design, it was only necessary to perform the hydraulic design of the sewer since the layout selection was given as initial information.

The design with the two methodologies showed that the optimized design complies with all the parameters of the regulations, while the design with traditional methodologies does not comply with shear stress and minimum excavation depth values. This unfulfillment of the regulations could be detrimental to the performance of the system within the design period.

The total cost of the optimized design using traditional methodologies was $62,380,000, while using optimization methodologies, a cost of $ 54,145,000 was obtained. Therefore, the optimized design is 15% cheaper than the design using traditional methodologies. The main reason why the optimized design is more economical is because it implements smaller diameter pipelines than the traditional design, which helps to significantly reduce pipeline costs. The second reason is that the optimized design requires shallower excavation depths than the other design. These two changes in the optimized design allow for considerable cost reduction. In conclusion, the design using optimization methodologies is the best alternative as it reduces construction costs and ensures compliance with the standard.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.