## 1 Introduction  
The water supply and distribution network plays a pivotal role in urban water management, agricultural irrigation, and various other sectors. In recent years, the development of irrigation pipelines has substantially enhanced water distribution efficiency and facilitated more precise water metering in irrigation areas [1,2], thereby contributing to the establishment of large-scale pipeline networks [3]. In Xinjiang, China, inland irrigation areas have capitalized on terrain elevation differences to implement self-pressurized irrigation systems, constructing multiple large-scale automated self-pressurized pipeline networks for water transmission and distribution. These networks offer a range of advantages, including improved water use efficiency, conservation of water resources, reduced energy consumption, and simplified operational management [4]. The reliability of the water transmission and distribution pipeline network is essential for ensuring the network's integrity and maintaining a consistent water supply [5,6]. In addition, the hydraulic reliability of irrigation networks differs from that of urban water networks. It is influenced by topographic elevation, crop water demand patterns, supply flow rates during the irrigation season, and the alternating cycle of pipe filling and draining during non-irrigation periods [7]. Current research methods for evaluating pipeline network reliability include analytical methods, simulation approaches, and proxy index techniques [8,9]. [PERSON] et al. [10,11] developed a reliability evaluation system for mine water supply pipeline networks using the Analytic Hierarchy Process (AHP), while [PERSON] et al. [12] applied the AHP-fuzzy comprehensive evaluation method to assess the drainage pipeline network in Huai'an City. [PERSON] et al. [13] evaluated the hydraulic reliability of urban water supply networks using indicators such as the ratio of node pressure head to minimum pressure head, the ratio of node flow to total flow, and flow velocity. [PERSON] et al. [14] identified key hydraulic performance indicators for urban water supply networks, including node pressure head, node pressure head qualification rate, pressure head uniformity, and pipe section flow velocity. [PERSON] et al. [15] proposed five indicators to evaluate the hydraulic reliability of urban water supply networks: water supply guarantee rate, node water pressure head ratio, water pressure head distribution uniformity, node water volume reliability, and remaining water supply capacity. [PERSON] et al. [16] suggested that the hydraulic reliability of urban drainage networks can be assessed using indicators such as pipeline water depth, pipeline flow velocity, pipeline filling degree, and pipeline bearing capacity. [PERSON] [17] emphasized that key hydraulic reliability indicators for urban water supply networks include flow rate, node hydraulic availability, and the number of pipe sections. These evaluation indicators are primarily derived from the reliability assessment of urban water supply systems, which are typically large-scale, structurally complex, and operate under high water supply pressure head [18]. Such systems generally provide 24-h uninterrupted water supply, often include regulatory structures, and require high operational and maintenance standards, with evaluation metrics that are easily quantifiable [19]. The reliability assessment standards for irrigation networks differ from those established for urban water supply systems. Research on the hydraulic reliability of irrigation networks remains limited. With the advancement of large-scale agriculture, these irrigation systems are expanding in size and increasing in complexity, incorporating various water supply modes such as segmented, uniform pressure head reduction, and random water supply [20,21]. However, a comprehensive methodology for evaluating the reliability of large-scale irrigation pipeline networks has yet to be established. Field research on large-scale irrigation pipelines presents significant challenges.