Automated Pump Placement Algorithms for Optimal Sewer Network Design in Areas with Complex Terrain ^†

Abstract

We present a set of algorithms for automatically determining the best locations for lift stations and pressurized pipes in sewer networks. These algorithms are integrated into an optimization framework for automatic sewer network planning. The algorithms are developed based on graph theory and metaheuristic optimization to optimize the allocation of lift stations and pressure pipes. The proposed algorithms are applied to a real large-scale test case in Paranatinga, Brazil, and the results are compared with an existing design. This comparison highlights the algorithms’ effectiveness in designing cost-efficient sewer networks in areas with complex terrain.

1. Introduction

The automatic planning of pumps in sewer networks is increasingly recognized as a crucial aspect of urban infrastructure development, especially in flat or complex terrains where gravitational flow is insufficient for sewage transport. Traditionally, the manual design of these networks has led to configurations with an extensive number of lift stations and pressurized pipes. This approach incurs significant disadvantages, including high initial investment costs, elevated operational expenses, and substantial energy consumption [1,2]. By integrating automated planning techniques, cities and municipalities can optimize the layout and operation of sewer networks. This streamlines the design process and potentially reduces the infrastructure’s financial and environmental footprint, ensuring a more sustainable and cost-efficient management of wastewater systems [3].

2. Mathematical Optimization of UDS

The optimal design of Urban Drainage Systems (UDSs) requires tackling two interconnected sub-problems: network layout and component sizing. Both are mathematically complex: layout is a combinatorial optimization problem in graph theory, and component sizing is a mixed-integer, non-linear program, making global optimization methods (metaheuristics) essential. Layout decisions directly influence component sizing, such as sewer sizing and pump station placement, and thus determine construction and operational costs [1]. In this study, we conducted a mathematical optimization of UDSs using following the formula:

d_opt = arg min_d∈D [f_d]

(1)

where d_opt represents the optimal decision variable for defining the sewer system’s layout, including parameters such as diameters, slopes, and pump station locations. The feasible space D ensures compliance with layout and hydraulic constraints with ${\mathit{f}}_{\mathit{d}}$ denoting the construction cost function.

UDN layouts are represented as graphs derived from a base graph encompassing all drainage possibilities, with manholes as vertices and sewers as edges. A layout is chosen from this base graph, forming a directed, tree-like structure leading to an outfall. This study adopts the loop-by-loop cutting algorithm (LBLCA) [4] to generate layout alternatives satisfying the case-specific constraints. The component sizing sub-problem addresses sewer diameters, slopes, and pump station placement, ensuring adherence to hydraulic and technical constraints. The adaptive sizing algorithm [5] is adopted to optimize the component sizes for each layout.

A Genetic Algorithm (GA) is employed as the optimization engine, utilizing the LBLCA for layout generation and the adaptive sizing algorithm for component dimension optimization. Pump station locations are determined subsequently using methods detailed in the next section. The cost function (${\mathit{f}}_{\mathit{d}}$) is evaluated iteratively within the GA until convergence.

3. Algorithm Design and Operations: Key Concepts

Pump placement within predefined networks can effectively reduce sewer depth, leading to cost savings and preventing depth thresholds from being surpassed. Two primary algorithms are identified for pump placement, with the potential for a hybrid approach.

Concept A: The first variant focuses on lifting stations and monitoring and adjusting depths to avoid exceeding maximum thresholds. Figure 1 illustrates this adaptive procedure, which calculates buried depths at each manhole, assigning lifting stations where necessary to maintain depths within limits. During optimization, the LBLCA generates layouts using layout decision variables derived from the GA. The adaptive algorithm then designs these layouts. A lift pump station is placed if the maximum depth exceeds the allowable limit. Therefore, the optimization engine endeavors to discover layouts that minimize the total construction costs, encompassing the expenses associated with lift stations. However, this method is solely applicable to lifting stations and does not determine the maximum number of pumps required. It is advisable to utilize lifting stations strategically to manage average insertion depths before reaching maximum thresholds.

Concept B: The following describes the second variant of the pump placement algorithm utilizing decision variables. The algorithm commences with predefined nodes, treating all nodes as potential pump locations (refer to Figure 2a). It employs binary decision variables to determine the role of each pump node, either as a lifting station or as part of a pressure pipeline. When a node is designated as the start of a pressure pipe, it initiates a subnetwork using the Hanging Garden Algorithm, potentially dividing it into subnetworks (refer to Figure 2b). Subsequently, the algorithm specifies the pump-to-pump connections within each sub-network, ensuring all subnetworks ultimately connect to the outlet.

To mitigate the risk of cyclic networks (as depicted in Figure 2c), the networks are transformed into trees, optimizing connections based on distance. This transformation yields a simplified network structure (as illustrated in Figure 2d), with specific boundary nodes identified at the edges of subnetworks serving as potential pump end nodes. The metaheuristic engine also optimizes these nodes’ selection, factoring in the distance from start nodes to boundary nodes (as depicted in Figure 3). This approach enhances flexibility and thoroughness in pump placement, aiming for optimal network performance. However, it entails significant computational effort due to the extensive consideration of potential locations and the intricate decision-making process involved.

4. Evaluating Algorithm Performance: A Case Study

The proposed algorithms were applied to a real large-scale test case in Paranatinga, Brazil, and the results were compared with an existing design. The network consists of two outlets and 1123 pipes with a total length of 7.3 km. The results showed that the applied algorithms could remarkably reduce the number of pressure pipes in the test case from eight to five. We also explored an alternative design using only seven lift stations, which turned out to be more cost-effective than the existing design with several pressurized pipes (refer to Figure 4). For this large test case, the optimization was completed in under 8 h using an eight-core processor clocked at 3.8 GHz, compared to the weeks typically required by planning offices for similar designs.

5. Conclusions

In conclusion, this study highlights the pivotal role of automated planning and optimization algorithms in the design of urban sewer networks, particularly in complex terrains. By employing an optimization integrated with novel pump allocating algorithms, we have demonstrated a significant enhancement in the efficiency and cost-effectiveness of sewer network designs. The case study of Paranatinga, Brazil, serves as a testament to the potential of these methodologies, showcasing a reduction in the number of required pumps and overall system cost.