## 1 Introduction  
As the demand for electricity and the requirements for developing a low-carbon economy continues, the driving force for energy development will gradually shift to renewable and clean energy such as photovoltaic power and wind power [1; 2; 3]. Due to its fast start-stop speed and flexible working conditions, the pumped storage units (PSUs) can quickly respond to the requirements of the power system for frequency and phase modulation, peak load shifting, rotation, and accident reserve, which can enhance the grid's ability to absorb wind power and photoelectricity [4; 5]. The pump turbine governing system (PTGS) is a complex hydraulic-mechanical-electrical-magnetic coupling system,which is time-varying, stochastic, and nonlinear [6; 7]. The precise modeling of the PTGS is of great significance in analyzing the dynamic response of PSUs under complex operational conditions [8; 9]. As a crucial part of PSU, an accurate pump turbine model is the key to the accurate modeling and simulation of PTGS [10]. In the current research work, the modeling of a pump turbine is mainly based on complete characteristic curves where different operational conditions of this model can be described [11; 12; 13]. The complete curves of pump turbine include the flow characteristic curve and the moment characteristic curve. These curves contain the inherent nonlinear characteristics of the pump turbine under various normal operational conditions or transient working conditions [14; 15]. However, due to the existence of the \"S\" characteristic and hump area, the estimation of complete characteristic curves is highly complex [16; 17]. It is easy for the complete characteristic curves to gather, cross, and twist in the pumps and anti-pump regions, which cause a huge interpolation error and non-convergence problem to the model when calculating the transient parameters of the runner boundary of the pump turbine. An in-depth and extensive research study has been conducted to extend the complete characteristic curves in recent years. The common methods to estimate the complete curves of pumped turbine mainly include the assistant mesh processing [18], the Suter transformation, its improved versions [19; 20], and the 3D surface fitting [14]. The assistant mesh processing method divides the flow characteristic curve or moment characteristic curve into multiple sections by introducing the idea of piecewise linearization, and then the original curve is approximated using small line segments. An assistant mesh line, which is approximately orthogonal to the opening line, is drawn to facilitate calculation. The characteristic curves of the pump turbine have been obtained and the modeling accuracy has been improved through the assistant mesh processing technique [8]. However, the above method requires prior artificial meshing of which the workload is heavy and the piecewise approximation also causes errors in the orthogonal curvilinear grid. The Suter transformation proposed by [PERSON] [19] transformed the original complete characteristic curves into \(WH(x,y)\) and \(WM(x,y)\) curves using dimensionless parameters. The abscissa of the curves is determined according to the relative flow and the relative speed. Because the Suter transformation stretches the flow and moment characteristic curves to both sides of the coordinate axis, the difficulties of the interpolation calculation caused by curve bending, torsion, anti-S, and multi-value characteristics can be alleviated effectively. However, scholars have found that the complete curves using Suter transformation or improved Suter transformation still have the following drawbacks: the length of the opening lines is different, some opening lines have singularities and are not conductive, and the density of data points on the opening lines is different. These non-uniform curves will affect the interpolation accuracy and computational efficiency of the model.