## 2 Methods
### Theoretical Modelling and Simulation Tools
In 3-3\({}^{\prime}\)-4\({}^{\prime}\) circuit, the refrigerant changes into super-cooled liquid after heat transfer in economizer and then expands through expansion valve A, the super-cooled liquid changes into low temperature and low-pressure gas-liquid mixture and is delivered into evaporator. In the evaporator, the refrigerant evaporates to low temperature and low-pressure gas phase refrigerant after heat exchanging with outdoor air. The refrigerant vapor from the evaporator is sucked into the compressor to start a new cycle. In the auxiliary circuit (3-4-1\({}^{\prime\prime}\)), refrigerant changes into low temperature and low-pressure gas-liquid mixture after expansion valve B, and transfers heat with high temperature refrigerant in circuit 3-3\({}^{\prime}\)-4\({}^{\prime}\) through the economizer. After evaporating in the economizer, the gas phase refrigerant is directly sucked into the auxiliary refrigerant inlet of the compressor, and the new cycle begins. The theoretical \(COP_{h}\) of a quasi-two-stage compression ASHP system for heating can be calculated with Equation (6)  
\[COP_{h-th}=\frac{h_{2}^{\prime}-h_{3}}{\alpha(h_{2}^{\prime}-h_{1}^{\prime})+( 1-\alpha)(h_{2}^{\prime}-h_{1}^{\prime\prime})} \tag{6}\]where \(h_{1}^{\prime}\) is the refrigerant enthalpy at the outlet point of the evaporator, kJ/kg. \(h_{1}^{\prime}\) is the refrigerant enthalpy at the outlet point of the economizer in 3-4-1\" circuit, kJ/kg. \(\alpha\) is the proportion of refrigerant going through 3-4\"-1\" circuit. The \(h_{2}^{\prime}\) in quasi-two-stage compression refrigeration cycle can also be calculated by Equation (4), and the \(h_{1}\) could be calculated with Equation (7)  
\[h_{1}=\alpha h_{1}^{\prime}+(1-\alpha)h_{1}^{\prime} \tag{7}\]  
Considering the energy loss through power-engine transfer in the compressor and heat loss during heat transfer between refrigerant and water in condenser, the actual \(COP_{h}\) of quasi-two-stage compression ASHP could be calculated with Equation (8). \[COP_{h}=\eta_{e}\eta_{d}\eta_{m}\eta_{t}\frac{h_{2}^{\prime}-h_{3}}{a(h_{2}^{ \prime}-h_{1}^{\prime})+(1-\alpha)(h_{2}^{\prime}-h_{1}^{\prime\prime})} \tag{8}\]  
To calculate the \(COP_{h}\) of ASHP with Equations (3)-(8), the physical parameters of different HFC refrigerants could be inquired with the software REFPROP. With the mixing HFC refrigerants and their proportions, the properties of RHR-1 could be calculated and obtained with REFPROP as well. The \(COP_{h}\) of different refrigerants could thus be obtained using the theoretical Equations (3)-(8) integrating REFPROP. Other parameters that affect the performance and practical utilization of ASHP using different refrigerants such as condensing pressure, evaporating pressure, compression ratio, and discharging temperature could also be calculated based on the theoretical model. During the simulation, an overall refrigeration efficiency \(\eta_{equil}\) was used to represent the effects of \(\eta_{e}\), \(\eta_{d}\), \(\eta_{m}\), \(\eta_{t}\) and \(\eta_{is}\). The \(\eta_{equil}\) was pre-set to be 0.65 in the original computation process. The isentropic efficiency was pre-set to be 0.76 [38]. A schematic is given as Figure 5 to show the input, output parameters and the computation process of the proposed model. The simulation equations were coded within MATLAB 9.0-R2016a (MathWorks, Natick, MA, USA, 2016) assisted by MS-Excel invoking REFPROP 9.1.