## 3 Residential Electric Heating System Model
### Performance Maps of IAC  
#### 3.2.1 Performance Calculation of IAC  
Steady-state IAC models can be used to study the cycling characteristics of IAC under different environment and control conditions. Specifically, the performance of IAC includes power consumption \(P_{AC}\) and coefficient of performance \(COP\), which are affected by compressor speed \(N_{comp}\), outdoor temperature \(T_{o}\) and indoor temperature \(T_{a}\). Figure 3 shows the main components of the air conditioning heating system and the cycle process, which meets the conservation of energy and mass. Starting from point \(A\), the cycle is analyzed. The specific enthalpy \(h_{A}\) of refrigerant steam is calculated from the evaporation temperature \(T_{evap}\) and the degree of superheat \(SH\), and the steam changes into high-temperature and high-pressure steam through the compressor module (including the accumulator, suction pipe, compressor and exhaust pipe, etc.). Then the compressor power \(W_{comp}\) and the mass flow rate \(m_{comp}\) can be obtained. The high-temperature and high-pressure steam releases heat to the indoor air through the condenser and cools to subcooled liquid. The degree of subcooling \(SC\), is calculated by the condenser model. The refrigerant liquid with medium temperature and high pressure is depressurized by the thermal expansion valve, and the mass flow rate \(m_{exp}\) is calculated according to the expansion valve model. The low-temperature and low-pressure liquid absorbs heat and turns into a gas through the evaporator, and the output specific enthalpy \(h_{evap}\) is calculated according to the evaporator  
Figure 2: ETP model of residences.  
model. There are three independent variables in the above cycle, namely [\(D_{camp}\), \(D_{cand}\), \(SH\)], where \(D_{camp}\) is the difference between the inlet temperature at the air side of the evaporator and the evaporation temperature, and \(D_{cand}\) is the difference between the condensation temperature and the inlet temperature at the air side of the condenser. Accordingly, the cycle should also meet three balance constraints, namely, specific enthalpy constraints at the start and end of the cycle, condenser subcooling constraints and mass flow balance constraints. The constraints are represented by residuals [\(\Delta\)1, \(\Delta\)2, \(\Delta\)3] respectively. When the steady state model converges, the residual values are all zero.  
\[P_{AC}=W_{comp}+W_{cond,fan}+W_{evap,fan} \tag{16}\]  
\[COP=\frac{Q_{cond}+W_{cond,fan}}{P_{AC}} \tag{17}\]  
Figure 4 shows the numerical calculation flow of the system, which is mainly divided into four parts: initial value calculation, independent variable update, pressure drop correction and performance parameter calculation. The operating condition parameters include the indoor air temperature \(T_{a}\), the outdoor temperature \(T_{out}\) and the rotor speed \(N_{comp}\) of the inverter compressor. The model first solves the independent variables in the minimal model as the initial value of the iteration. Secondly, the Broyden algorithm is used to solve the independent variable iteratively to make the residual close to zero, and the pressure drop at the high pressure side \(\Delta P_{h+}\), and low pressure side \(\Delta p_{l.}\) are calculated when converged. The pressure drop is introduced into the compressor model, and the pressure drop and independent variables are updated iteratively under this condition. The iteration stops when the pressure drop difference is less than the threshold. Finally, the performance parameters are calculated. As shown in Equations (16) and (17), the power consumption of IAC is mainly related to the compressor, condenser fan and evaporator fan. At the same time, the electric energy consumed by the condenser fan is eventually converted into the internal energy of the indoor air. Therefore, in addition to the heat release of the condenser, the numerator of \(COP\) also contains the internal energy.