## 3 DC-DC Boost Converter  
DC-DC boost converter is a high step up power converter, which is used to boost and regulate an input DC voltage. In this work, the main objective of using the boost converter is to provide an efficient power conversion from the cell stack to the load and offer a regulated output voltage. As shown in Figure 4, an ideal boost converter consists of linear (filtering capacitor \(C\), load resistor \(R\), and inductor \(L\)) and nonlinear (diode \(D\), switching transistor \(S\)) elements.  
According to [65; 66], the relationship between input and output voltage in a boost converter is presented by Equation (15), where \(V_{in}\) is the input voltage source, \(V_{out}\) is the output voltage, and \(d\) is the control signal which represents the switch position:  
\[V_{out}=\left(\frac{1}{1-s}\right)\times V_{in}. \tag{15}\]  
Under continuous conduction mode (CCM), the operation of a DC-DC boost converter becomes fairly simple. Thus, using the inductor \(L\) and the two switches \(D\) and \(s\), the circuit alternates between two states (ON and OFF) for each complete switching cycle \(T_{s}\). Each state of ON and OFF has a varying duration. The ON time \(T_{on}\) can be calculated by multiplying the switching cycle \(T_{s}\) with the duty cycle \(d\). The OFF time \(T_{off}\) can be found by subtracting the ON time \(T_{on}\) from the complete switching cycle \(T_{s}\). The waveforms of inductor current \(i_{L}\), inductor voltage \(V_{L}\), switching current \(i_{s}\), switching voltage \(V_{s}\), and diode current \(i_{d}\) are shown in Figure 5.  
* The ON state: When the switch \(S\) turns ON, the inductor \(L\) connects to the DC source voltage. Therefore, the current \(i_{L}\) moves across the inductor \(L\) and the transistor switch \(s\), which results in an increase in the magnitude of \(i_{L}\) and \(i_{s}\), while \(V_{L}\) is approximately equal to the input voltage \(V_{in}\). On the other hand, during this state, the capacitor \(C\) discharges through the load  
Figure 4: Ideal boost converter circuit.  
Figure 3: PEMFC 10 stack V-I and proportional–integral (PI) characteristics.  
\(R\). The obtained differential equations of the inductor current \(i_{L}\) and the output voltage \(V_{out}\) are expressed as follows: \[\left\{\begin{array}{l}\frac{d{i_{L}}}{dt}=\frac{1}{L}(V_{in})\\ \\ \frac{dV_{out}}{dt}=\frac{1}{C}(-i_{out})\end{array}\right..\] (16)
* The OFF state: When the switch \(S\) turns OFF, the inductor \(L\) connects to the capacitor \(C\) and the load \(R\). Therefore, the current \(i_{L}\) moves across the inductor \(L\), the diode \(D\), the capacitor \(C\), and the load \(R\), which results in a decrease in the magnitude of \(i_{L}\) and \(i_{d}\) (discharging of the inductor \(L\) into the load \(R\) and the capacitor \(C\)). During this state, \(V_{L}\) is approximately equal to \(V_{in}-V_{out}\). The obtained differential equations of the inductor current \(i_{L}\) and the output voltage \(V_{out}\) are expressed as follows: \[\left\{\begin{array}{l}\frac{d{i_{L}}}{dt}=\frac{1}{L}(V_{in}-V_{out})\\ \\ \frac{dV_{out}}{dt}=\frac{1}{C}(i_{L}-i_{out})\end{array}\right..\] (17)  
The global state-space representation of the high step-up DC-DC boost converter can be expressed in Equation (18), where \(x=\left[\begin{array}{c}x_{1}\\ x_{2}\end{array}\right]\), \(x_{1}\) represents the inductor current, and \(x_{2}\) represents the output voltage.  
\[\left\{\begin{array}{l}\dot{x}=\left[\begin{array}{cc}0&\frac{d-1}{L}\\ \frac{1-d}{C}&-\frac{1}{RC}\end{array}\right]x+\left[\begin{array}{c}\frac{1 }{L}\\ 0\end{array}\right]u\\ \\ y=\left[\begin{array}{cc}0&1\end{array}\right]x\end{array}\right.. \tag{18}\]  
Figure 5: Waveforms of different currents and voltages under continuous conduction mode (CCM) operation.