## 4 Modeling of the Electrical Efficiency for the Rolling Piston Compressor  
The isentropic efficiency theoretical power and electrical power are used to evaluate the effective utilization of motor input power. This metric is influenced by several factors, including the indicated efficiency, heating efficiency, leakage efficiency, and mechanical efficiency [30]. The efficiency of the cycle is determined by comparing the input power of the actual cycle to that of the theoretical cycle. According to Equation (9), several parameters, including the temperature coefficient, leakage coefficient, compressor suction, discharge air pressure ratio, and compressor suction-specific volume c, are related to the indicated efficiency. The temperature coefficient is linked to the pressure ratio [30], while the suction-specific volume can be calculated using the evaporation temperature and the suction superheat. If there is a superheated compressor suction port, the leakage coefficient is primarily influenced by the pressure ratio. To sum up, the compressor's indicated efficiency is related to the pressure ratio, evaporation temperature, and suction superheat. The measurement of heating efficiency is a crucial factor that indicates the amount of heat loss during the suction process. To estimate the heating efficiency, a temperature coefficient is commonly utilized. It is important to note that the temperature coefficient serves as an approximation for the heating efficiency and should be carefully considered in any heating-related analysis or evaluation. Leakage efficiency is a measure of the energy lost due to cylinder leakage, which is estimated by the leakage coefficient. This parameter is influenced by the pressure ratio in the superheated compressor suction stage. Mechanical efficiency, on the other hand, accounts for the impact of frictional power, which is primarily affected by the viscosity of the lubricant and refrigerant and is thus dependent on the mixture's temperature and concentration. In the superheated suction stage, the lubricant shows low solubility in the gaseous refrigerant, making mechanical efficiency a constant value. Through the above analysis, it can be seen that under the same compressor frequency, when the compressor suction is superheated, the compressor electrical efficiency can be approximately equal to the function of the evaporation temperature, the suction superheat degree, and the suction and discharge air pressure ratio; that is,  
\[\eta_{el}=f(T_{e},T_{sh},\Pr) \tag{10}\]  
We can analyze the data shown in Figure 3 to show that the compressor electrical efficiency is linear with the suction superheat, i.e., when the evaporation temperature and the suction superheat are constant, Equation (10) can be simplified to Equation (11). \[\eta_{el}=a+b\cdot T_{sh} \tag{11}\]  
where \(a\) and \(b\) are constants. When the evaporation temperature and the suction superheat are not fixed, the constant \(b\) in Equation (11) is reduced to a quadratic polynomial related to the evaporation temperature and the suction superheat. \[\eta_{el}=a_{0}+\Big{(}a_{1}+a_{2}\cdot T_{e}+a_{3}\cdot\Pr+a_{4} \cdot T_{e}{}^{2}+a_{5}\cdot\Pr^{2} \tag{12}\] \[+a_{6}\cdot T_{e}\cdot\Pr)\cdot T_{sh}\]  
Equation (12) is the compressor electrical efficiency model for suction superheat at the same compressor frequency. From Figures 3 and 4, in the wet compression section, the compressor electrical efficiency and suction dryness are in a linear relationship, and in different frequencies, they are under basically the same slope; so, the wet compression according to the compressor electrical efficiency model is:  
\[\eta_{el,\ x}=\eta_{el,\ x}=1-a(1-x) \tag{13}\]According to the study in [21], a method was proposed for the construction of a performance model for variable-frequency compressors. This method involves selecting a specific frequency of the compressor as the standard frequency and standardizing both the electrical efficiency and the frequency of the compressor.