Enhancing Urban Energy Infrastructure by Optimizing Underground Transmission Line Routing in Phnom Penh ^†

Abstract

Swift urbanization and technical progress in Cambodia, specifically in Phnom Penh, require underground transmission lines (UGTL) as a viable substitute for overhead transmission lines (OHTL). However, the substantial cost of UGTL has prevented its extensive integration. In this respect, we identified the most cost-effective technological route for an underground transmission line between substations. Using geographic information system (GIS) data, we generated algorithms to define the optimal route for the installation of a UGTL and minimize the costs of the material and labor required. The research results presented an automated tool for route optimization which simplifies the planning of energy projects and partially relieves the financial burden of UGTL integration. The proposed method radically changes the planning of urban energy infrastructure, as it provides a technology-based, cost-efficient, and environmentally favorable decision for UGTL routing. It also fosters the development of sustainable and resilient urban energy systems in similar urban locations.

1. Introduction

Underground transmission lines (UGTLs) are widely used for electrical power systems in urban regions. These lines are more expensive than overhead transmission lines (OHTL) due to the special physical, geographical, and construction requirements [1,2]. However, UGTLs are chosen at times due to technical reasons, but there are challenges in defining the optimal route. While looking for the optimal route of UGTL, a complex problem involving the most optimal route measured according to different geographical data and assigned by the cost of the creation of the line must be solved. Therefore, the shortest path for a UGTL must be determined using different algorithms from graph theory, such as Bellman–Ford, Dijkstra, or A*. These algorithms are used to identify the shortest path between two points, which is important in transport management, electricity grid location, and the design of infrastructure [3,4].

The research on the shortest paths is a key research area, given its numerous applications. The ‘ideal’ shortest route is the least distance between the starting point and the endpoint. The increase in research on the shortest route comes from the challenging problems appearing in several fields. Various methods, notably exact methods and meta-heuristic algorithms, provide exact and approximate solutions for the shortest paths. Heuristic methods are employed to minimize the time required for solving the problems. Several studies present the results of path-finding in various instances [5].

The shortest-route problem is a significant issue in computer systems due to its widespread application, including communication networks, transportation, and traffic management. According to ref. [6], the three most successful algorithms are A*, Dijkstra, and ant colony optimization (ACO). Specifically, Dijkstra’s algorithm assesses possible solutions to indicate the optimal path, whereas A* improves performance efficiency by assuming that some paths are more relevant. ACO is inspired by the behavior of the insects and uses pheromone paths that function as heuristics to probabilistically identify the best path at a specific point [7,8]. Moreover, researchers have developed algorithms to improve cycling, time management, and route efficiency [9]. The algorithms are crucial in the use of UGTL networks to solve the rapid urban electricity demand and congestion. UGTLs with high voltage require maximum energy transmission, especially in OHTL. While the shortest route needs to be appropriately analyzed to decrease the construction cost of the cables and system, a longer route might be more feasible when it encounters fewer obstacles or less traffic [10].

This study aims to efficiently fill the research gap on UGTL routing in an urban setting using an analytical method. Route choice concerns the geographical patterns related to the construction environment. Eight shortest path algorithms including Dijkstra, Floyd–Warshall, Bellman–Ford, A*, Improved Dijkstra, Breadth-First Search, Minimum Spanning Tree, and Highway Hierarchy (HH) were reviewed in this study. These algorithms are instrumental in estimating the best UGTL routes connecting different start and destination pairs. Four case studies in Phnom Penh, Cambodia, were analyzed in terms of traffic volume during rush hours. For each case study, R-squared coefficients, mean absolute percentage error (MAPE), root mean square error (RMSE), mean absolute error (MAE), and mean square error (MSE) were used to validate algorithm performance. The eight algorithms were applied to find the best UGTL routing, and their performances were compared using the metrics to select the best one.

2. Material and Methodology

2.1. Study Area

The research area covers two substations located in the Chamkar Mon district, the heart of Phnom Penh, Cambodia. The coordinates of the area are 11°32′36.246″ N and 104°55′26.793″ E, with an area of 10.56 km². Approximately, 70,772 people are living in the area as of 2019. The country’s capital and largest city, Phnom Penh, is Cambodia’s political, commercial, cultural, religious, and economic center. It is also the country’s main transport hub. Its area is 289 km² with a population of 2,129,371. The city experiences significant population growth due to urbanization and other factors [11].

2.2. Study Procedure

This study was carried out to determine an appropriate route for a UGTL to meet the particular requirements. User preferences were surveyed for decision-making, and edge weights were estimated for traffic volumes, closure of roads, and route lengths. To predict congestion, 3–172, 7–159, and 8–181 nodes were considered. The case studies were conducted during the peak hours in the morning, midday, and evening. In each case study, data on the specific site were collected. The GIS data is presented in Figure 1a, and the flowchart of this study is shown in Figure 1b.

2.3. Shortest-Path Algorithm

The shortest route algorithm minimizes the sum of edge weights to determine the shortest path between two nodes. Such a problem is found in computer science to determine network routing, logistics, or transportation. Vertices or nodes are linked with edges or links as connections in a graph where every edge has a certain weight or cost denoting the time or distance to move from one vertex to another. The path with the lowest overall cost is considered to be the shortest route. In this study, we modified the shortest path algorithm to contain a global restriction on the number of edges to obtain the closest destination vertex in a graph and make the algorithm feasible in applications.

• Dijkstra’s algorithm is a method for finding the shortest path in a weighted graph $G=(V,E)$ from a source vertex $s$ to any other vertex, where edges have non-negative weights. The algorithm uses labels to represent the shortest path estimates to each vertex. A label $l\left(v\right)$ for a vertex $v\in V$ indicates the shortest distance from $s$ to $V$, which can be either temporary or permanent. Once a label is marked as definitive, the shortest path to that vertex is known.
• The Bellman–Ford algorithm is used to find the shortest path from a source vertex $s$ to all other vertices in a weighted graph $G=(V,E)$, even when edge weights can be negative. It also identifies negative cycles that reduce the total path weight indefinitely indicating that no solution exists in such cases.
• The Floyd–Warshall algorithm is a dynamic programming method that allows negative edge weights but not negative cycles for determining the shortest pathways between any pair of vertices in a weighted graph.
• The HH algorithm optimizes the shortest path computations by transforming the graph structure by finding and prioritizing the most relevant edges. The algorithm transforms the input graph G obtained from the common denial to both graph H. The computation of the graph G is based on the set of parameters, such as weight function lengths d, “modulating constants” g, width w, and the same numbers of reports in n. Such parameters are correlated to the definitions of relevant and non-relevant edges. The resulting narrow graph fits in the memory resources based on a reliable ratio estimation of the actual length d. The precondition for the correctness of this transformation method is the inclusion of a unary node, so the modulating constant must be a non-decreasing function. We defined the four sub-problems and related them to the overall problem for maintaining the functionality, ensuring isolation, balancing weight functions between the problems, and reorientation using the same set sizes. The total memory size is dependent on two parameters, w and n, similar to the reporting problem solution.
• The A* algorithm is one of the most excellent pathfinding and graph traversal algorithms that find the shortest path between two nodes. It combines the use of the cost and the heuristic estimate to simplify the search process. The A* algorithm is widely used in GPS navigation, robotics, and game development, as it can find optimal paths.
• The BFS and DFS essential graph traversal algorithms explore the vertices and edges of a graph. BFS and DFS are fundamental techniques in graph theory, each with its unique advantages depending on the context of the problem being solved.
• The combination of Dijkstra’s algorithm with the A* algorithm enhances pathfinding efficiency by leveraging the strengths of both methods. The integration of Dijkstra’s and A* algorithms enables solving complex pathfinding problems in various applications, particularly in urban environments with diverse routing conditions.
• Prim’s algorithm is a greedy method used to find the MST of a connected, weighted graph. Prim’s algorithm is widely used in network design, including telecommunications and computer networks, where minimizing connection costs is essential.

3. Results and Discussion

3.1. Experimental Design

A personal computer with Windows 10 Pro-64-bit, Intel Core i5-7200U CPU, 8 GB RAM, and PyCharm Community Edition 2023.3.5 was used in this study. The shortest-path algorithm simulated navigation to identify the shortest path between an origin and a destination. The algorithm was tested in Phnom Penh, Cambodia. The prediction error between the algorithm’s output and the real shortest path length was measured using four evaluation metrics to evaluate the performance of the algorithm.

• MAE

  $$\mathrm{{\rm M}}\mathrm{A}\mathrm{{\rm E}}={\displaystyle \frac{1}{\mathcal{N}}}\sum _{\mathcal{i}=1}^{\mathcal{N}}\left|{\mathcal{Y}}_{\mathcal{i}}-\widehat{{\mathcal{Y}}_{\mathcal{i}}}\right|$$

  (1)
• MAPE

  $$\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}={\displaystyle \frac{1}{\mathcal{N}}}\sum _{\mathcal{i}=1}^{\mathcal{N}}\left|{\displaystyle \frac{{\mathcal{Y}}_{\mathcal{i}}-\widehat{{\mathcal{Y}}_{\mathcal{i}}}}{{\mathcal{Y}}_{\mathcal{i}}}}\right|$$

  (2)
• RMSE

  $$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{\mathcal{N}}}\sum _{\mathcal{i}=1}^{\mathcal{N}}{\left({\mathcal{Y}}_{\mathcal{i}}-\widehat{{\mathcal{Y}}_{\mathcal{i}}}\right)}^2}$$

  (3)
• R-squared

  $${\mathrm{R}}^2=1-{\displaystyle \frac{{\sum }_{\mathcal{i}=1}^{\mathcal{N}}{\left({\mathcal{Y}}_{\mathcal{i}}-\widehat{{\mathcal{Y}}_{\mathcal{i}}}\right)}^2}{\sum _{\mathcal{i}=1}^{\mathcal{N}}{\left({\mathcal{Y}}_{\mathcal{i}}-\overline{{\mathcal{Y}}_{\mathcal{i}}}\right)}^2}}$$

  (4)
• MSE

  $$\mathrm{M}\mathrm{S}\mathrm{E}={\displaystyle \frac{1}{\mathcal{N}}}\sum _{\mathcal{i}=1}^{\mathcal{N}}{\left({\mathcal{Y}}_{\mathcal{i}}-\widehat{{\mathcal{Y}}_{\mathcal{i}}}\right)}^2$$

  (5)

  where ${\mathcal{Y}}_{\mathcal{i}}$ is the actual length, $\widehat{{\mathcal{Y}}_{\mathcal{i}}}$ is the resulting length, $\overline{{\mathcal{Y}}_{\mathcal{i}}}$ is the average of the actual length, and $\mathcal{N}$ is the number of forecasting samples.

3.2. Comparison with Conventional Methods

We determined how well the eight shortest-path algorithms solved routing problems of underground transmission lines. To evaluate the performance of the shortest path algorithm, we used four traditional assessment metrics: MAE, RMSE, MAPE, and R². Tests were conducted by providing the current direction and speed to each edge. Nodes in restricted zones and traffic congestion zones were discarded.

Experiment 1: Traffic Volume in the Morning

Experiment 1 was conducted to find the shortest path during traffic congestion in the morning, from 06:30 to 08:30.

Table 1. The result of the comparative case study 1 from node 3 to 172.

Nº	Shortest-Path Algorithms	Route [m]	MAE	MAPE	RMSE	R2
1	A* Star	4241	9.7	29.65	9.7	0.41
2	Bellman–Ford	4581	13.1	40.05	13.1	0.49
3	Breadth First Search	3524	2.53	7.73	2.53	0.14
4	Dijkstra	4581	13.1	40.05	13.1	0.49
5	Floyd–Warshall	3440	1.69	5.17	1.69	0.10
6	HH	4241	9.7	29.65	9.7	0.41
7	Improved Dijkstra	3502	2.31	7.06	2.31	0.13
8	Minimum spanning tree	6883	36.12	110.42	36.12	0.77

shows the result of the efficiency of all the shortest paths based on MAE, MAPE, RMSE, and R². Two algorithms found the shortest path, which was 4241 m long. MAE was 9.7%, MAPE was 29.65%, RMSE was 9.7%, and R² was 0.41%. The A* and HH algorithms and Dijkstra’s and Bellman–Ford’s algorithm showed a length of 4581 m with MAE of 13.1%, MAPE of 40.05%, RMSE of 13.1%, and R² of 0.49% (Figure 2). The BFS algorithm showed a length of 3524 m. Its MAE was 2.53%, MAPE was 7.73%, RMSE was 2.53%, and R² was 0.14% (Figure 3). Floyd–Warshall showed a length of 3440 m, with an MAE of 1.69%, MAPE of 5.17%, RMSE of 1.69%, and R² of 0.10% (Figure 3). Improved Dijkstra’s algorithm showed a length of 3502 m. Its MAE was 2.31%, MAPE was 7.06%, RMSE was 2.31%, and R² was 0.13% (Figure 4). The minimum spanning tree algorithm showed a length of 6883 m with an MAE of 36.12%, MAPE of 110.42%, RMSE of 36.12%, and R² of 0.77% (Figure 4).

The results of this investigation enabled the determination of the best options for UGTL routing. First, we examined multiple algorithms to find the shortest path. The route corresponded to the most direct way determined by the shortest-route algorithm. However, an alternative path was slightly longer but more feasible due to fewer obstacles on the path.

The previous path might show a serious traffic jam. Therefore, the algorithms might not determine the most rapid route in all instances. However, a path was selected when it had a considerably reduced number and intensity of obstacles. A peculiar pattern was observed on the longer path. The best algorithms for routing in the context of UGTL included Dijkstar’s, Bellman, Improved Dijkstra’s, A*, and HH algorithms. These algorithms modeled the number of cars and their capacity to avoid obstacles. The algorithms provided effective path selections. The results of experiments 2 and 3 were similar to that of experiment 1.

4. Conclusions

The shortest-path algorithms were used for UGTL planning in highly populated areas. Using graph-based methodology associated path network, the connected highway system provides the capability to design pathways. Eight algorithms were compared to determine the shortest distance between the two nodes: Dijkstra, Floyd–Warshall, Bellman–Ford, A*, BFS, improved Dijkstra, MST, and HH. Road limitations, the connected network’s hierarchy, precision, speed, and real-time changes were subsequently taken into account in determining the appropriate algorithm. Considering precision, performance, and adaptability, the A* algorithm was selected as the appropriate algorithm for UGTL routing. The highway hierarchy, improved Dijkstra, and A* algorithms were exceptionally fast in real-time assessment, enabling users to offer paths with the lowest number of constraints.