## 2 Air Pressure Compensation Based on IE-FF Coupling Model
### Air Pressure Compensation System
Assume that the curve of the pressure dependent on the flow rate of a single fan changes quadratically based on experience, the curves of the intake and exhaust fans are, respectively, expressed as:  
\[\left\{ \begin{array}{c}P_{S1}=a_{1}Q_{1}{}^{2}+b_{1}Q_{1}+c_{1}\\ P_{S2}=a_{2}Q_{2}{}^{2}+b_{2}Q_{2}+c_{2}\end{array} \right., \tag{16}\]  
where \(P_{S1}\) and \(P_{S2}\) are, respectively, the static pressure of the intake and exhaust fans; \(Q_{1}\) and \(Q_{2}\) are, respectively, the volume flow rate of the intake and exhaust fans; \(a_{1}\), \(b_{1}\), \(c_{1}\), \(a_{2}\), \(b_{2}\), and \(c_{2}\) are the fitting coefficients. According to the fan similarity theorem, when the fan speed is changed, the characteristic curve of the fan will also change. If the fan speed changes from the nominal speed \(n_{0}\) to \(n_{1}\), the Equation (16) becomes:  
\[\left\{ \begin{array}{c}P_{S1}=a_{1}Q_{1}{}^{2}+b_{1}Q_{1}+\eta_{1}d_{1}\\ P_{S2}=a_{2}Q_{2}{}^{2}+b_{2}Q_{2}+\eta_{2}d_{2}\end{array} \right., \tag{17}\]  
where \(\eta_{1}\) and \(\eta_{2}\) are, respectively, the speed ratio of the intake and exhaust fans, and the speed ratio of the fan can be expressed as \(\eta=\left(\frac{n_{1}}{n_{0}}\right)^{2}\). The fan speed can be controlled by the input frequency:  
\[n=60\cdot\frac{f}{p}\, \tag{18}\]  
where \(n\) is the rotational speed of the fan; \(f\) is the input frequency of the fan; and \(p\) is the number of pole pairs of the fan motor. The actual working parameters of the fan are determined by the intersection of the ventilation duct characteristic curve and the fan flow characteristic curve, as shown in Figure 5. It can be seen that the fan can change the input frequency by changing the air volume when the air pressure changes. Intersection A is the working point of the fan with input frequency \(f\). Figure 5: The general flow characteristic curve of the fan. Therefore, in order to determine the specific parameters of the fan during its operation, it is also necessary to obtain the characteristic curve of the ventilation duct connected to the fan. The characteristic curve of ventilation duct is:  
\[\left\{ \begin{array}{l}\quad H_{1}=P_{i}(t)-P_{o}(t)+S_{1}\cdot Q_{1}{}^{2}\\ \quad H_{2}=P_{o}(t)-P_{i}(t)+S_{2}\cdot Q_{2}{}^{2}\end{array} \right., \tag{19}\]  
where \(H_{1}\) and \(H_{2}\) are the total pressure loss of the ventilation duct connected to the intake fan and exhaust fan, respectively; \(S_{1}\) and \(S_{2}\) are the resistance coefficient of the ventilation duct connected to the intake fan and exhaust fan, respectively; and \(P_{o}(t)\) and \(P_{i}(t)\) are, respectively, the inlet and outlet static pressures of the ventilation duct at time \(t\). That is, the actual static pressure of the intake and exhaust fans are:  
\[\left\{ \begin{array}{l}\quad P_{i}(t)-P_{o}(t)+S_{1}\cdot Q_{1}{}^{2}=a_{1} Q_{1}{}^{2}+b_{1}Q_{1}+\eta_{1}c_{1}\\ \quad P_{o}(t)-P_{i}(t)+S_{2}\cdot Q_{2}{}^{2}=a_{2}Q_{2}{}^{2}+b_{2}Q_{2}+ \eta_{2}c_{2}\end{array} \right. \tag{20}\]  
The mass flow equation of the fan has a nonlinear and time-varying relationship with the air pressure difference between the inside and outside of the car and the frequency of the inverter. Therefore, a simplified expression of the air pressure change in the car can be expressed as:  
\[\left\{ \begin{array}{l}\quad\dot{P}_{i}(t)=[q_{mi}(t)-q_{mo}(t)\pm q_{ml}(t) ]\cdot\frac{RT_{i}}{MV_{i}}\\ \quad q_{mi}(t)=\alpha_{1}(P_{i}(t)-P_{o}(t))+\beta_{1}\cdot f^{2}(t)\\ q_{mo}(t)=\alpha_{2}(P_{o}(t)-P_{i}(t))+\beta_{2}\cdot f^{2}(t)\end{array} \right., \tag{21}\]  
where \(f(t)\) is the frequency of the fan at time \(t\); and \(\alpha_{1}\), \(\alpha_{2}\), \(\beta_{1}\) and \(\beta_{2}\) are constants, which can be calculated by linearizing the nonlinear expression Equation (20) in the neighborhood of working point A, as shown in Figure 5.