*2.5. Battery Model*

The battery model represents a pack of Model 21700 lithium polymer battery cells. The battery pack is assembled in such a way that the cells are connected in series. According to [37], the open-circuit voltage of the battery can be estimated as:

$$
\mathcal{U}\_{\rm B,OC} = \text{SOC}(\mathcal{U}\_{\rm B,max} - \mathcal{U}\_{\rm B,min}) + \mathcal{U}\_{\rm B,min}.\tag{21}
$$

The battery power and the fuel cell load demand power sum up to provide the electrical power to the motor such that:

$$P\_M = P\_B + P\_{FC,load}.\tag{22}$$

Further, the current drawn from the battery set is obtained by solving:

$$P\_B = n\_B(lI\_{B,OC}I\_B - I\_B^2 R\_{B,int})\_\prime \tag{2.3}$$

which should not exceed its maximum discharge current *IB*,*max*.

The battery Coulombic efficiency in the battery model is assumed to be 100%. Thus, the *SOC* is satisfied as:

$$\text{SOC}(t) = \text{SOC}\_0 - \int\_{t\_0}^{t} \frac{I\_B(t)}{C\_B} dt. \tag{24}$$

where *t*, *t*0, and *SOC*<sup>0</sup> are the current time, initial time, and initial *SOC*, respectively.

#### **3. Hybrid System Model and Problem Formulation**

*3.1. Hybrid System*

The fuel cell load demand power, which will be indicated as *PFC* in the following section, is the only variable under control. Due to the output characteristic of the PEMFC, the change of *PFC* is chosen to be 5% of its maximum power, which depends on *γ* according to (18). The fuel cell load demand power dynamics are then:

$$P\_{\rm FC}(t\_{n+1}) - P\_{\rm FC}(t\_n) = u \cdot \Delta P\_{\rm FC} \tag{25}$$

where *u* ∈ {−1, 0, 1}, Δ*PFC* = 5%*PFC*,*max*, and *PFC*(*tn*) is the fuel cell load demand power at *t* = *tn*. Here, three different values of *u* correspond to decreasing, sustaining, or increasing *PFC*. According to Equation (19), the maximum load cell power can be calculated as:

$$P\_{\rm FC,max} = \frac{4n\_{\rm FC}R\_{\rm FC}'P\_0 - \left(n\_{\rm FC}L\_{\rm OC} - n\_{\rm FC}\kappa\_{\rm aux}\right)^2}{-4n\_{\rm FC}R\_{\rm FC}'}.\tag{26}$$

Using Equations (25) and (26), the final expression representing the fuel cell load demand power dynamics is given by:

$$\frac{dP\_{\rm FC}}{dt} = \mu \cdot 5\% \frac{4n\_{\rm FC}R\_{\rm FC}'P\_0 - (n\_{\rm FC}lL\_{\rm CC} - n\_{\rm FC}\kappa\_{\rm max})^2}{-4n\_{\rm FC}R\_{\rm FC}'}.\tag{27}$$

The *SOC* dynamics are obtained by differentiating both sides of (24) with respect to time,

$$\frac{dSOC}{dt} = -\frac{I\_B}{\mathbb{C}\_B}.\tag{28}$$

Combing (28) with (21) and (23) yields,

$$\frac{dSOC}{dt} = \frac{-n\_B l I\_{B,OC} + \sqrt{(n\_B l I\_{B,OC})^2 - 4n\_B R\_{B,int} P\_B}}{2n\_B R\_{B,int} C\_B} \tag{29}$$

where *UB*,*OC* = *SOC*(*UB*,*max* − *UB*,*min*) + *UB*,*min*.

The motor power and battery power are related by:

$$P\_B = S\_f P\_{M\nu} \tag{30}$$

where *Sf* is referred to as the split fraction, which could be calculated from (22) as:

$$S\_f = \frac{P\_M - P\_{FC}}{P\_M}\_{-}$$

Using Equations (29) and (30), the final expression representing the *SOC* dynamics is given by:

$$\frac{dSOC}{dt} = \frac{-\mathcal{U}\_{B,OC} + \sqrt{(\mathcal{U}\_{B,OC})^2 - \frac{4\mathcal{R}\_{B,int}S\_fP\_M}{n\_B}}}{2\mathcal{R}\_{B,int}C\_B},\tag{31}$$

where the internal resistance *RB*,*int* and the battery capacity *CB* are assumed to be constant [38].

The mass of remaining fuel dynamics is obtained from Faraday's law as:

$$\frac{dm\_{FR}}{dt} = -\frac{n\_{FC}I\_{FC}}{n\_{c}F}M\_{h\nu} \tag{32}$$

where *IFC* is calculated from *PFC* as shown in (20).

Equations (25), (29), and (32) are the final form of the state equations used in this study, where the states of the system are *<sup>x</sup>* = [*SOC*, *MFR*, *PFC*], the control is *<sup>u</sup>* ∈ {−1, 0, 1}, the outputs of the system are *y* = [*Sf* , *PB*], and the operating variables are *w* = [*PM*, *γ*]. These operating variables are treated as measured disturbances in the model.

Based on the above modeling assumptions and parameters in Table A1, the maximum fuel cell output power is 795 W at *γ* = 0 deg, 496.14 W at *γ* = ±10 deg, and 335.71 W at *γ* = ±20 deg. The theoretical maximum power for the battery series (of eight batteries) is 2940 W, due to the limitation of the discharge current (35 A); the maximum power of the battery is 1176 W at any climb angle.
