## 1 Introduction  
### Motivation  
Currently, the global community is transitioning into an era characterized by clean and efficient energy, with natural gas emerging as the predominant energy source. According to the Global Energy Outlook, worldwide consumption of natural gas is projected to increase continuously until 2050, with an average annual growth rate exceeding 1.4% ([PERSON], 2020). The significant rise in natural gas demand necessitates enhanced efforts toward the construction of surface gas-gathering pipeline networks. The investment required for gas-gathering pipeline networks is substantial, often comprising over 30% of the total development costs of a gas field [1]. Consequently, optimizing the design of these pipeline networks in a rational manner can yield considerable economic benefits. The system typically includes wells, gas-gathering stations, central treatment facilities, and pipelines (Figure 1). Initially, the natural gas extracted from the wells is conveyed to the gas-gathering station via production pipelines. Once collected at the gas-gathering station, the gas is then transported through gathering pipelines to the central treatment plant for purification, ensuring it meets the standards required for external distribution. Based on network topology, gas-gathering pipeline networks can be categorized into three fundamental types: star, tree, and cyclic topologies [2]. In a star topology, multiple wells are directly connected to a single station, as illustrated in Figure 1(a). In a tree topology, wells are linked to the station through adjacent wells, as depicted in Figure 1(b). A cyclic topology involves interconnected wells forming a loop, as shown in Figure 1(c). In practical large-scale gas-gathering pipeline networks, various topological combinations are often observed. At present, in the actual construction of gas fields, based on a variety of factors, including gas well distribution, construction cost, management convenience, operation efficiency, etc., star-star topology network structures and star-tree topology network structures have been widely used in gas field development. The star-star topology network structure of multiple gas wells directly connected to a central gas-gathering site makes the layout simple and easy to manage and control. Its advantages are that the total length of the pipeline can be reduced, the initial construction cost can be reduced, and it is convenient to centrally process and transport the natural gas produced by the gas field, which is suitable for a gas field with a small gas-bearing area, a large number of wellheads, and a low output of a single well. The star-tree topology network structure combines the characteristics of the star and dendritic structures. In this structure, each gas well and its gas-collecting stations are connected by a star shape, and the gas-collecting stations are connected by gas-collecting trunk lines to form a dendritic structure. This lay  
Figure 1: Typical gas-gathering pipeline network. Figure 2: Pipeline network structure: (a) star topology, (b) tree topology, (c) cyclic topology. out balances the cost and efficiency of the long-haul collection line, while providing some flexibility to adjust and expand according to the stage of development of the field. Figure 3 illustrates a typical star-tree gas-gathering pipeline network. This network structure is particularly suitable for scenarios where wells within local blocks are concentrated, but these blocks are widely dispersed with significant distances between them. In general, the optimization problem can be categorized into design optimization and operational optimization [3]. Design optimization involves selecting the most suitable layout configuration, station placements, pipe sizes, and equipment scales for a new pipeline network to achieve optimal construction investment costs.