**3. Fault Simulation of the PEM Fuel Cell System**

The PEM fuel cell system can directly convert chemical energy into electricity through electrochemical reaction and produce water and heat at the same time. The PEM fuel cell simulator model uses controller strategies and nonlinear models presented by Pukrushpan et al. [1]. It is assumed that the system is in a constant temperature state, ignoring the influence of the double charge layer, and it is regarded as a rapid dynamic behavior near the electrode/electrolyte. Parameters commonly used in the PEM fuel cell simulator model are described in Table 1.


The PEM fuel cell simulator model was established by Pukrushpan, J.T. in [1] and some parameters of the PEM fuel cell simulator model are from [45–47] based on actual product parameters. The PEM fuel cell simulator model is widely used for the fault diagnosis of the PEM fuel cell system [7,8,12,14], and represents the75kW fuel cell system with 381 cells. The PEM fuel cell simulator model includes the fuel cell stack model, the compressor model, the supply manifold model, the return manifold model, the air cooler model, and the humidifier model. The PEM fuel cell system block diagram is shown in Figure 2. The five faults are partially quoted from the literature [1,14] and the amplitude of characteristic parameters is reduced to ±10%. The faults in the PEM fuel cell simulator model are described in Table 2.

**Figure 2.** The PEM fuel cell system block diagram [1].


The characteristic parameters remain unchanged and Fault0 is in normal state. Equations (15)–(21) [1] are used to simulate Fault1–Fault4. According to the thermodynamic formula, the compressor torque τ*cp* is expressed as:

$$\tau\_{cp} = \frac{\mathcal{C}\_p}{\omega\_{cp}} \frac{T\_{atm}}{\eta\_{cp}} \left[ \left( \frac{p\_{sm}}{p\_{atm}} \right)^{\frac{\gamma - 1}{\gamma}} - 1 \right] \mathcal{W}\_{cp} \tag{15}$$

where, τ*cp* is the torque needed to drive the compressor, *Cp* is the specific heat capacity of air, ω*cp* is the compressor speed, η*cp* is the compressor efficiency, *psm* is the supply manifold pressure, *patm* is the pressure of the air, *Tatm* is the temperature of the air, γ is the ratio of the specific heats of the air, and *Wcp* is the air mass flow of compressor.

A lumped rotational parameter model with inertia is used to represent the compressor speed:

$$J\_{cp}\frac{d\omega\_{cp}}{dt} = \left(\tau\_{cm} - \tau\_{cp}\right) \tag{16}$$

where *Jcp* is the combined inertia of the compressor and the motor, and τ*cm* is the compressor motor torque input.

The Fault1 state is simulated with the increment Δ*kv* in the compressor constant *kv*. The Fault2 state is simulated with the increment Δ*Rcm* in the compressor motor resistance *Rcm*:

$$
\pi\_{\rm cm} = \frac{\eta\_{\rm cm} k\_{\rm t}}{(R\_{\rm cm} + \Delta R\_{\rm cm})} \Big[ v\_{\rm cm} - (k\_{\rm \nu} + \Delta k\_{\rm \nu}) \omega\_{\rm cp} \Big] \tag{17}
$$

where, η*cm* is the motor mechanical efficiency, *kt* is the motor torque constant, *Rcm* is the compressor motor resistance, Δ*Rcm* is the increment in the compressor motor resistance, *kv* is the motor electric constant, and Δ*kv* is the increment in the motor electric constant.

The maximum mass of the vapor that the gas can hold is calculated from the vapor saturation pressure:

$$m\_{v, \text{max}, \text{ca}} = \frac{p\_{\text{sat}} V\_{\text{ca}}}{R\_v T\_{\text{st}}} \tag{18}$$

where, *mv*,*max*,*ca* is the maximum mass of the vapor, *psat* is the saturation pressure of the vapor, *Rv* is the gas constant of the vapor, and *Tst* is the temperature of the stack. If *mw*,*ca* ≤ *mv*,max,*ca*, so *mv*,*ca* = *mw*,*ca*, *ml*,*ca* = 0; if *mw*,*ca* > *mv*,max,*ca*,so *mv*,*ca* = *mv*,max,*ca*,*ml*,*ca* = *mw*,*ca* − *mv*,max,*ca*.

The total cathode pressure is the sum of oxygen, nitrogen, and vapor partial pressure:

$$P\_{\rm ca} = P\_{\rm O\_2,\rm az} + P\_{\rm N\_2,\rm az} + P\_{\rm v,\rm az} = \frac{m\_{\rm O\_2,\rm az}R\_{\rm O\_2}T\_{\rm st}}{V\_{\rm c2}} + \frac{m\_{\rm N\_2,\rm az}R\_{\rm N\_2}T\_{\rm st}}{V\_{\rm c2}} + \frac{m\_{\rm v,\rm az}R\_{\rm v}T\_{\rm st}}{V\_{\rm c2}} \tag{19}$$

where *Pca* is the cathode pressure; *Vca* is the cathode volume; *PO*2,*ca*, *PN*2,*ca* and *Pv*,*ca* are the partial pressure of oxygen, nitrogen, and vapor; *RO*<sup>2</sup> , *RN*<sup>2</sup> and *Rv* are the gas constants of oxygen, nitrogen, and vapor.

Fault3 is simulated with the increment Δ*kca*,*out* in the cathode outlet orifice constant *kca*,*out*:

$$\mathcal{W}\_{\text{ca},\text{out}} = (k\_{\text{ca},\text{out}} + \Delta k\_{\text{ca},\text{out}})(p\_{\text{ca}} - p\_{rm}) \tag{20}$$

where, Δ*kca*,*out* is the increment in the cathode outlet orifice constant, *kca*,*out* is the cathode outlet orifice constant, *Wca*,*out* is the air flow in the cathode outlet, *pca* is the cathode pressure, and *prm* is the return manifold pressure.

Fault 4 is simulated with the increment Δ*ksm*,*out* in the supply manifold outlet orifice constant *ksm*,*out*:

$$\mathcal{W}\_{\text{sm,out}} = (k\_{\text{sm,out}} + \Delta k\_{\text{sm,out}})(p\_{\text{sm}} - p\_{\text{cn}}) \tag{21}$$

where, *Wsm*,*out* is the outlet mass flow, Δ*ksm*,*out* is the increment in the supply manifold outlet orifice constant, and *ksm*,*out* is the supply manifold outlet orifice constant.
