## 1 Introduction  
When confronted with the increasingly serious energy crisis, wind energy, as a renewable and clean energy, has been extensively developed and utilized in power systems [1, 2]. However, the significant volatility [3] and the inverse peak regulation characteristics [4] of such wind power impose considerable pressure on both the smooth and economic operation of the power system and the integration of wind power [5]. Therefore, it is necessary to provide a flexible power supply for wind power transmission. Pumped storage (PS) is a kind of low-carbon and clean flexible power supply, with the characteristics of pumping water during high wind conditions and releasing it to generate electricity during low wind conditions [6]. In the future, thermal power will gradually transition from supplying base loads to serving as adjustable power sources [7]. For large-scale wind power delivery systems, studies on the output strategies of pumped storage units (PSUs) and thermal power units can scientifically guide the daily operational arrangement of the system, which plays a crucial role in improving the smoothness and economy of system operations [8]. Therefore, it is necessary to carry out research on the optimization scheduling of the power system equipped with wind power, pumped storage, and thermal power. According to different speed regulation types, pumped storage units can be divided into three types: fixed-speed pumped storage units (FS-PSUs),variable-speed pumped storage units (VS-PSUs), and ternary pumped storage units (T-PSUs) [9; 10]. Given the influence of the different speed regulation characteristics of the PSUs participating in power system optimization scheduling [11], the academic community has conducted some studies. As for FS-PSUs, their characteristic is that the power cannot be adjusted during operation in the pump mode [12]. Ref. [13] conducted the short-term optimization scheduling of a wind-pumped storage complementary system, intending to maximize the total revenue of the whole system. The results show that PS effectively compensates for the volatility of the wind power output and significantly enhances the total power generation revenue of the complementary system. However, when the wind power output is high, and if the excess wind power does not reach the integer multiple of the maximum pumped power of the FS-PSUs, pumping at this time will lead to \"over-peak regulation\". Conversely, refraining from pumping to store the excess wind power may lead to wind curtailment. Ref. [14] compared the cost-effectiveness of the combined operation of FS-PSUs and VS-PSUs with renewable energy sources. The results show that compared with VS-PSUs, FS-PSUs exhibit weaker load adaptability, leading to the frequent start-up and shutdown of thermal power units, which increases the fuel consumption and CO\({}_{2}\) emissions. Therefore, it can be found that when the FS-PSUs are involved in the optimal scheduling of the power system, the pumping power remains inflexible and cannot be adjusted. This will increase the pressure of peak regulation operations for thermal power units, which does not satisfy the requirements for the low-carbon operations of the new power system. As for VS-PSUs, their characteristic is that the power can be adjusted within a certain range in the pump mode [15]. Currently, the research on VS-PSUs primarily focus on unit dynamics modeling, control strategies, and stability evaluation [16; 17], with quite limited investigation in power system optimization scheduling. In Refs. [18; 19], a joint dispatching model including VS-PSUs and other power units is established, and it can be found that VS-PSUs cannot only improve the levels of new energy consumption but also reduce the pressure of thermal power peak regulation. However, the aforementioned references suffer from the assumption of a minimum pumping power of zero for VS-PSUs during their pumping phase, which does not match well with practical scenarios.