Flow-Induced Stress Analysis of a Large Francis Turbine Under Different Loads in a Wide Operation Range  
[PERSON]  
1 Department of Energy and Power Engineering, Tsinghua University, Beijing 100084, China; [PERSON]]  
2S.C.L. Energy (Swiss), Future Energy Research Institute, 8706 Zurich, Switzerland 2  
[PERSON]  
3 Xinhua HydroPower Co., Ltd., Beijing 100704, China; [EMAIL_ADDRESS] (H.O. ); [EMAIL_ADDRESS] (G.W.) [PERSON]  
4 Xinjiang Xinhua Zangao Hydropower Development Co., Ltd., Kashi 844700, China; [EMAIL_ADDRESS]  
4 Xinjiang Xinhua Zangao Hydropower Development Co., Ltd., Kashi 844700, China; huanghao@kthydropower. With the increase in grid demand and the rapid development of new energy sources such as photovoltaic and wind power in recent years, the operation mode of hydropower units has undergone significant changes. Francis turbines are the most widely used turbines in hydroelectric power plants because of their versatility and efficiency under various operating conditions. In order to adapt to the regulation needs of the power grid and to consume the highly volatile photovoltaic and wind power, Francis units are often operated within the load range from 0 to 100%, especially under low-load conditions [3]. This new operating mode goes beyond the recommended operating range of the Francis turbine unit when it was originally designed and poses new challenges to the Francis turbine units. When Francis turbine runners operate under off-designed conditions, they are susceptible to the impact of uneven water flow, which can cause large vibrations on the runner blades [4; 5]. As shown in Figure 1, many Francis turbines at different hydropower plants have suffered severe cracks and damage [6]. In order to avoid the damage problem happening in the Francis runners of the power station, and to guarantee the long-term safe and stable operation of the turbine units, this study focuses on analyzing the flow-induced dynamic characteristics of hydraulic turbine units under different loads. To better understand the complex flow characteristics in Francis turbines, researchers in various countries have contributed to the investigation numerically and experimentally. [PERSON]. [7] presented the hydraulic stability and pressure pulsations at the high part load for Francis turbines in the Three Gorges hydropower plant. [PERSON] et al. [8] carried out the flow characteristics analysis of the Francis turbine at the rated condition and overload operation condition and found significant differences in the pressure distribution and velocity distribution of the turbine flow domain. [PERSON] et al. [9] studied the flow characteristics in the vaneless space between the guide vanes and runner blades of the Francis unit with site measurement and numerical simulation. The dynamics of the rotating vortex in the draft tube of the Francis turbines under different operating conditions were investigated by both experimental flow survey and numerical simulations [10; 11]. The impact of the number of runner blades on the efficiency and flow pattern of the turbines under different operating conditions has also been investigated [12; 13; 14]. [PERSON], [PERSON] et al. [15] carried out experimental studies to analyze the effect of the splitter blades on the turbine performance characteristics, including efficiency, cavitation, and pressure fluctuation, and revealed that the splitter blades can increase the efficiency by approximately 2%, and they can reduce the pressure fluctuation in the vaneless space under high-head operating conditions. With the help of numerical simulation and experimental testing, [PERSON] et al. [16; 17] provided insight  
Figure 1: Damage of Francis turbine runners [6]. into flow characteristics including pressure and velocity distribution of Francis runners at different operating conditions. Under partial load or overload conditions, abnormal water flow distribution in the runner fluid channel can easily cause cavitation on the runner blades.