## 3 Calculation Model
### Operating Condition Settings  
Two operating conditions were set for the large and small units (Figure 3) based on the actual layout of the turbines in the Xinlongkou Hydropower Station. In this figure, D1 and D2 represent large units, whereas X1 and X2 represent small units. Single- and two-stage linear valve closure methods were modeled to calculate the water hammer pressure.  
The formula for calculating the water hammer wave speed in pipelines is as follows:  
\[a=\sqrt{\frac{K}{\rho(1+\frac{KD}{Ee})}} \tag{3}\]  
The meanings of the symbols are as follows: \(a\) represents the water hammer wave speed, measured in m/s; \(D\) is the pipeline diameter, in m; \(K\) denotes the bulk modulus of elasticity of the fluid, in Pa; \(\rho\) is the fluid density, in kg/m\({}^{3}\); \(E\) refers to the elastic modulus of the pipeline material, in Pa; and \(e\) is the pipe wall thickness, in m.  
In this study, the fluid in the pipeline is water, with a density of 1000 kg/m\({}^{3}\) and a bulk modulus of elasticity of 2.2 \(\times\) 10\({}^{9}\) Pa. The steel pipe has a wall thickness of 0.04 m and an elastic modulus of 2.1 \(\times\) 10\({}^{11}\) Pa, with a pipeline diameter of 4.1 m. Based on these parameters, the calculated wave speed in the pipeline is approximately 1000 m/s.  
In practical engineering applications, to ensure that the initial wave reaches the opposite end of the pipeline and begins to reflect before the valve is fully closed, it is possible to reduce premature wave overlap and excessive reflection peaks. Therefore, some design standards or empirical guidelines often use half of the water hammer propagation time  
Figure 3: Schematic Diagram of Branch Pipe Arrangement and Monitoring Point Locations.  
(i.e., \(2\,L/a\)) as a conservative reference for the \"minimum\" valve closing duration. Given that the pipeline in this study is relatively short and the water hammer event occurs rapidly, we adopt the half-cycle of the water hammer as the reference for the minimum valve closing time.  
\[\mu=2\,L/a \tag{4}\]  
In the equation, \(\mu\) represents the wave phase; \(L\) is the length of the pipe segment, in meters (m); and \(a\) is the average wave speed in the pipe segment, in meters per second (m/s).  
Based on the findings presented in Equation (4), the complete water hammer period for the pipeline is determined to be 2.9 s; consequently, the minimum valve closure time established for this study is set at 3 s, resulting in a total computational duration of 120 s. Building upon the research conducted by [PERSON] et al. [38], who explored the effects of continuous valve closure on water hammer pressure increases in multi-branch pipelines, as well as empirical evidence gathered from actual engineering practices, it was established that the maximum water hammer pressure occurs when all turbines simultaneously reject their loads; consequently, this study investigates the maximum and minimum pressure variation patterns of water hammer from the terminal valve to the main pipeline segment by implementing single- and two-stage linear valve closures. The single-stage linear closure times were set to 3, 6, 9, and 12 s for the large unit and 3, 4, 7, and 10 s for the small unit. For the two-stage linear valve closure, both the large and small units shared the same fast and slow closure times, specifically characterized by fast closure occurring at 80% in 3, 4.5, and 6 s for the large unit (with slow closure at 20%) and at 80% in 2, 3.5, and 5 s for the small unit (with slow closure at 20%), along with an additional setting for the large unit of fast closure at 60% in 3, 4.5, and 6 s (with slow closure at 40%) and for the small unit of fast closure at 60% in 2, 3.5, and 5 s (with slow closure at 40%). Thus, based on the differing closure times for the large and small units, combined operating conditions are established to study the maximum and minimum pressure variations of water hammer from the terminal valve to the main pipeline segment, with the specific operating conditions detailed in Table 1.