## 1 Introduction  
Vibrations within the vicinity of both the plant and the diversion pipeline are generated throughout the running phase of a pumped-storage power station due to its inherent features of high head, high speed, and high capacity [1]. Previously, some power stations have caused severe accidents because of these vibrations [2; 3; 4; 5; 6], and the incidents resulted in casualties. The influences of vibrations can also give rise to issues for ground architectures and residents. The vibration sources have been researched in extensive studies. Hydraulic, mechanical, and electrical factors are the main reasons for vibrations while running units [7]. The hydraulic factor is the most crucial of these [8], which mainly results from uneven flow between the guide blades and the runner. The region without blades between the movable guide blades and the runner is referred to as the vaneless area [9]. During the operation of the units, the vaneless area generates multiple overlapping pulsating pressures with a main frequency approximately equal to the passing frequency of the runner blades (referred to as the blade frequency, calculated by multiplying the number of runner blades by the speed frequency) [10]. This high-frequency pressure pulsation is transmitted upstream via the water hammer effect, resulting in vibrations along the pipeline [11]. Researching the propagation of vibration, [PERSON] analyzed the response of the concrete structure in the steel pipe pressure compensation section of the Degorsk PSPS. The findings demonstrated that incorporating a compensator into the pipe section significantly enhanced its stability, which reduced the vibrations of the diversion pipeline [12]. [PERSON] utilized the finite element method to analyze the natural frequencies of functionally graded graphene sheets (GPL) in different distribution modes while considering different material parameters and boundary conditions. The findings revealed that FG-O types exhibited the minimum natural frequencies [13]. [PERSON] investigated the secondary vibration and noise of the surrounding ground induced by high-speed trains and analyzed the law of vibration propagation in different soil layers [14]. [PERSON] proposed a comprehensive set of calculation methods and an evaluation standard for assessing railway ground vibrations and noise, which can also be utilized for the ground vibration of PSPSs [15]. The problem of vibration necessitates the implementation of vibration reduction measures. [PERSON] utilized finite element software to simulate the cushion pipeline thickness and elastic modulus of a diversion pipeline. The results demonstrated that increasing the cushion pipeline thickness while decreasing the elastic modulus achieved better vibration reduction effects [16]. [PERSON] researched the global collapse resistance capacity of a seismic-damaged steel-reinforced concrete (SRC) frame structure strengthened with an enveloped steel jacket. The results demonstrated the ability of the enveloped steel jacket reinforcement to effectively enhance the SRC frame structure's resistance to global collapse. It is also valuable for the vibrations of diversion pipelines [17]. [PERSON] used the characteristic line method and software to research the impact of the presence of a surge chamber on hydraulic transients under diverse working conditions, concluding that implementing a surge chamber can effectively mitigate the occurrence of cavitation [18]. [PERSON] utilized MATLAB to numerically simulate the control equation of transient flow in a pipeline with or without a surge chamber. The results demonstrated the effectiveness of the surge chamber in regulating the pressure along the pipeline, and the vibration level was significantly diminished [19]. Although there are studies on the law of vibration propagation, research regarding ground analysis and vibration in the diversion pipeline areas of PSPSs is lacking. Most studies are limited to numerical simulations and analyzing the vibration frequency.