## 1 Introduction  
New energy policies have remarkably promoted the rapid increase in the proportion of renewable clean energy, such as wind power, solar energy, and hydropower [1, 2, 3, 4, 5]. As a result, ultra-supercritical thermal power plants (USC-TPPs) are now frequently expected to participate in deep peak shaving. However, this action will produce constant start-up and shut-down for devices, causing a wider range of temperature fluctuation for structural components [6, 7, 8, 9, 10]. As such, higher requirements have been put forward for the structural integrity of key components, particularly for the flue-side water-cooled wall. This component plays a crucial role in cooling flue gas and recovering heat, which directly relates to the safety and economy of USC-TPPs. Unfortunately, in recent years, the large temperature gradient and load changes triggered by deep peak shaving have led to frequent cracking failures of structural components, for example, the weld seam and connecting fins on the flue-side wall. Figure 1 shows a typical failure diagram of a flue-side wall. Overall, this kind of accident seriously hinders the safe operation and economic benefit of USC-TPPs [11, 12, 13, 14]. Additionally, due to the practical constraints on the installation of the flue-side wall during welding, the size of the connecting fins varies accordingly. To reduce the cracking probability of welds and connecting fins and ensure the safe operation of thermal power plants, it is necessary to clarify the impact of connecting fin dimensions on residual welding stress. Due to the large structural dimensions and demanding working conditions associated with flue-side walls or membrane-type water-cooled walls (structures sharing similar designs, but differing in nomenclature), researchers have typically relied on numerical simulations to study temperature and stress distribution. [PERSON] et al. [15] investigated the critical conditions in a sub-critical drum boiler and found that water wall tubes of all lengths had a serious risk of combustion and overheating if the fluid flow in the tubes was disturbed. [PERSON] et al. [16] analyzed the failure locations in the water-cooled wall tubes of a specific power plant. Following extensive experimental analysis, numerical simulation methods were employed to calculate the residual stresses and equivalent stresses of water-cooled wall tubes under bending conditions. [PERSON] et al. [17] investigated the impact of combustion condition symmetry on temperature, heat flux density, strain, and stress distribution. This study provided, for the first time, a comprehensive overview of the entire operational state of a boiler's water-cooled wall, particularly focusing on strain and stress distribution. [PERSON] et al. [18] explored the heat transfer characteristics between the membrane wall and syngas under different working conditions, and found that when the membrane water wall was not protected, the inner surface part of the fin would overheat and the maximum stress would exceed the allowable stress. [PERSON] et al. [19] established a numerical framework that coupled fluid dynamics and solid mechanics to obtain temperature profiles within a furnace, thermal stress, and strain fields on a furnace wall. This framework was employed to predict critical regions essential for mechanical reliability. [PERSON] et al. [20] conducted experimental research and numerical simulations for 17 typical flexible operating conditions to investigate the safety of water-cooled walls for combustion boilers under flexible load conditions. Furthermore, research has revealed that the failure and cracking of furnace side walls or membrane-type water-cooled wall structures result from the synergy of many factors, including high-temperature creep, corrosion, and scaling [21; 22; 23; 24; 25; 26]. However, no previous studies have considered the impact of welding structure dimensions on the residual stress and deformation of water-cooled walls.