## 1 Introduction
In reality, VS-PSUs can be divided into two types: doubly fed units and full-power variable frequency units. The doubly fed units can regulate speed from 80% to 100% of their rated power, while the full-power variable frequency units have a wider range, operating from 60% to 100% of their rated power [20]. Furthermore, the PSUs in the pumped storage of the aforementioned references all belong to VS-PSUs. In fact, there is no full variable-speed pumped storage (VS-PS). Consequently, research on the optimal scheduling for pure VS-PS is deemed impractical. As for T-PSUs, their disadvantage lies in that the cost per kilowatt is relatively high; however, it has unique technical advantages in many aspects [10; 11]. One of the main advantages of T-PSUs is that their capability to operate not only in modes where the turbine and pump operate independently (i.e., the conventional mode), but also simultaneously. Therefore, when the pump operates at its maximum pumping power, the turbine can still adjust its output simultaneously. By adjusting the output of the turbine, the overall pumping power of the unit is changed, which can be dynamically varied from zero to its maximum [21]. Since T-PSUs have a wider range of regulation in the pump mode, they can pump at a more precise power when the wind power is high, without pumping more or less. This advantage can be applied to the combined operation of a PS and wind farm, thus achieving the effect of mitigating wind power fluctuations and enhancing the integration ability of wind power. However, the current academic research on the ternary pumped storage (T-PS) remain primarily in the technical demonstration stage [10; 22], which mainly focuses on the transition performance between the pump and turbine modes, as well as the steady-state and dynamic regulation characteristics of the power regulation of units in the pump mode, with few systematic studies analyzing and demonstrating the effect of T-PSUs on improving the integration effect of new energy. Therefore, to explore the impact of T-PSUs in solving new energy integration, it is necessary to study the optimization scheduling of the power system including T-PSUs. Furthermore, the comparative analysis and comprehensive benefit assessment of optimal scheduling among FS-PSUs, VS-PSUs, and T-PSUs are notably absent. At present, most pumped storage plants in the world are installed with only FS-PSUs [21]. However, with the continuous development of variable-speed technology, the pure fixed-speed pumped storage (FS-PS) will be gradually upgraded in the future, leading to the coexistence of FS-PSUs and VS-PSUs [23]. Refs. [20; 24] forms a hybrid pumped storage with FS-PSUs and VS-PSUs by upgrading and transforming some units of a pure FS-PS into VS-PSUs. The results show that after the transformation, the wind and solar power curtailment rate of the system decreases, and the peak regulation pressure of the thermal power can be reduced to a certain extent. The above reference studies the participation of PS with both FS-PSUs and VS-PSUs in power system optimization scheduling. Compared to pure FS-PS, a hybrid PS with both FS-PSUs and VS-PSUs has stronger regulation capabilities. Given the flexibility advantage of T-PSUs, it is also very meaningful to upgrade the FS-PSUs to T-PSUs. Nevertheless, there remains a scarcity of references studying a hybrid PS with FS-PSUs and T-PSUs. In conclusion, VS-PSUs offer distinct advantages over FS-PSUs, including the enhanced utilization of new energy sources, decreased reliance on peak regulation by thermal power plants, and lowered carbon emissions. Furthermore, compared to VS-PSUs, T-PSUs possess superior flexibility and full-range power modulation capabilities in the hydraulic short-circuit modes. Given these benefits, the harmonious integration of both FS-PSUs and VS-PSUs is anticipated to lead the evolution of PS.