*3.2. Mathematical Modeling of Wind Power*

The wind model consists of variation of wind velocity with gust and wind speed [30–36].

$$\mathcal{W}\_V = V\_w + \, V\_{\mathcal{g}} + V\_{wrr} \, \tag{14}$$

where, *V<sup>w</sup>* = Base wind velocity, *V<sup>g</sup>* = Gust wind velocity, and *Vwr* = ramp wind component. The gust speed is calculated by

$$V\_{\text{wg}} = \left\{ \begin{array}{l} 0 \ t < T\_1 \\ \text{C}\_2 \left\{ 1 - \cos \pi \left[ \frac{t - T\_1}{T\_2 - T\_1} \right] T\_1 \le t \le T\_2 \right\} \\\ 0 \ t \ge T\_2 \end{array} \right\} \tag{15}$$

$$V\_W = \left\{ \begin{array}{l} 0 \, s < T\_3\\ \, \text{C}\_3 \left\{ \begin{array}{l} \frac{s - T\_3}{T\_4 - T\_3} \, \middle| \, T\_3 \le s \le T\_4\\ 0 \, s \ge T\_4 \end{array} \right\} \end{array} \right\},\tag{16}$$

where, *C*<sup>2</sup> = maximum value of the gust component, *C*<sup>3</sup> = maximum wind speed caused by the ramp, and *T*<sup>3</sup> and *T*<sup>4</sup> are the cut-in and cut-out times of the ramp, respectively.

Wind power is calculated by

$$P\_W = \frac{d\mathcal{W}\_w}{dt} \tag{17}$$

Energy drawn by the wind turbine is

$$\begin{aligned} \mathcal{W}\_W &= V\_a \times \frac{1}{2} \rho (V\_1^2 - V\_3^2) \\ P\_W &= d \frac{V\_a \frac{1}{2} \rho (V\_1^2 - V\_3^2)}{dt} \end{aligned} \tag{18}$$

where, *W<sup>w</sup>* = energy drawn by wind turbine and *ρ* = Air density.

According to Betz, the maximum wind turbine power output is

$$P\_M = \frac{16}{27} A\_R \frac{3}{2} V^3 \tag{19}$$

Equation (19) is obtained by substituting the value for *V*1, and *V*3.

$$\begin{array}{l} V\_2 = \frac{2}{3} V\_1\\ V\_3 = \frac{1}{3} V\_1 \end{array} \tag{20}$$

The wind turbine model represents the output power captured by the turbine [33–36]. Figure 5 shows the characteristic curve for wind speed vs. power. The power in the wind (*Pw*) in an area is obtained by

$$P\_W = \frac{1}{2} \rho A \mathcal{W}\_V^{\;>\;} \tag{21}$$

$$P\_M = P\_W \mathbb{C}\_p \tag{22}$$

$$\mathcal{C}\_P = \frac{1}{2} [\delta \ - \ 0.22 \beta^2 \ - \ 5.6) e^{-0.17\delta} \tag{23}$$

*β* = Pitch angle of the blade in degrees, *δ* = the tip speed ratio of the turbine, and *C<sup>p</sup>* = Power coefficient.

**Figure 5.** Characteristic curve of wind system (wind speed vs. power).

Wind generated power is expressed as:

$$p\_G = V\_G I\_G \tag{24}$$

*3.3. Mathematical Modeling of Fuel Cell Power*

A different assumption [36] is made, which is described below:


$$\begin{aligned} PH\_2 &= \frac{-1}{lH\_2} \left[ PH\_2 + \frac{1}{KH\_2} \left( q\_{th}{}^{in} - 2K\_l i\_{\text{FC}} \right) \right] \\ PO\_2 &= \frac{-1}{lO\_2} \left[ PO\_2 + \frac{1}{KO\_2} \left( q\_{th}{}^{in} - 2K\_V i\_{\text{FC}} \right) \right] \\ PH\_2O &= \frac{-1}{lH\_2O} \left[ PH\_2O + \frac{1}{KH\_2O} K\_l i\_{\text{FC}} \right] \end{aligned} \tag{25}$$

Here, *KH*<sup>2</sup> = valve meter constant for hydrogen and *KO*<sup>2</sup> = valve meter constant for oxygen (2.52 <sup>×</sup> <sup>10</sup>−<sup>3</sup> kmol/s atm). *<sup>K</sup><sup>r</sup>* = constant defined by the rate of reactant hydrogen and fuel cell current. The reactant utilization factor *U* is defined as follows:

$$\mathrm{dL}\_{\mathrm{F}} = \frac{qH\_{2}^{\mathrm{in}} - qH\_{2}^{\mathrm{out}}}{qH\_{2}^{\mathrm{in}}} \tag{26}$$

$$\begin{aligned} V\_{act} &= \left[ E\_1 + E\_2 T + E\_3 T \ln(co\_2) + E\_4 \ln(i\_{FC}) \right] \\ co\_2 &= \frac{po\_2}{5.08 \times 10^6 \varepsilon \left( \frac{498}{T} \right)} \end{aligned} \tag{27}$$

where, *E*1, *E*2, *E*3, *E*<sup>4</sup> is the cell parameter coefficients, *CO*<sup>2</sup> = concentration of oxygen. Figure 6 represents the equivalent circuit of the fuel cell. It consists of cell voltage, actual resistance, concentration resistance, and ohmic resistance [37,38].

$$V\_{\rm ohm} = i\_{\rm F\mathbb{C}} \left( R\_{\rm m} + R\_{\rm ohm} \right) \tag{28}$$

$$R\_m = \frac{\rho\_m l}{A} \tag{29}$$

where, *ρ<sup>m</sup>* = specific resistivity of the membrane for electron flow (Ω cm). *A* = active cell area (cm<sup>2</sup> ) and *l* = thickness of the membrane.

**Figure 6.** Equivalent circuit of the fuel cell.

The concentration loss is due to the reactive excess concentration near the catalyst surface.

$$V\_{Con} = \frac{-R\_{con}T}{2F} \ln\left(1 - \frac{j}{j\_{\text{max}}}\right) \tag{30}$$

The fuel cell current can be determined as

$$\dot{\alpha} = \dot{\iota}\_o A \left( e^{\frac{\alpha \mathcal{H}}{P\_f \mathbb{C}}} V\_{act} - e^{\frac{(1-a)\kappa F\_{\text{Vac}}}{RTFC}} \right) \tag{31}$$

*I<sup>o</sup>* = exchange current density (A/m<sup>2</sup> ), *A* = catalyst layer surface (m<sup>2</sup> ), and *i* = fuel cell current. The Figure 7 shows the single fuel cell characteristics for stack current vs. cell voltage and power. The power of the fuel cell can be obtained from

$$P\_{fuel} = V\_{out} i\_{\mathbb{C}}.\tag{32}$$

**Figure 7.** Single fuel cell characteristics curve (stack current vs. cell voltage and power).
