# Adaptive Feedback Control for Four-Phase Interleaved Boost Converter Used with PEM Fuel Cell +
## 3 Controller Design
### PI Design  
To design the PI controller, we implemented two loops to ensure the stability of the output voltage and output current. Next, transfer functions from the duty cycle to the inductor current \(F_{iu}\) and from the duty cycle to the output voltage \(F_{eu}\) were determined based on the small-signal model of the system's mathematical model. The PI parameters were synthesized using the Particle Swarm Optimization (PSO) algorithm, as indicated in reference [18]. The PSO algorithm was chosen due to its efficiency in handling multi-objective optimization problems, its ability to minimize tracking error, and its demonstrated success in optimizing control parameters in power electronic systems.  
Figure 2d illustrates the PI control architecture for the four-phase IBC, with two loops incorporated into the structure. To generate the transfer functions of the inner and outer loops, a small-signal model of the four-phase IBC can be validated by expanding Equation (10) and using the corresponding DC and AC components of the voltage, current, and duty cycle, as follows:  
\[\left\{\begin{array}{c}i_{i}(t)=I_{i}+\hat{i}_{i}(t)\\ v_{out}(t)=V_{out}+\phi_{out}(t)\\ u(t)=U+\hat{u}(t)\end{array}\right. \tag{11}\]  
where \(U\), \(I_{i}\), and \(V_{out}\) are the DC components of the duty cycle, current, and voltage, respectively. \(\hat{i}\), \(\phi_{out}\), and \(\hat{u}(t)\) represent the AC components of the current, voltage, and duty cycle. Thus, the small-signal model is given by the following equation:  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Item** & **Components** & **Simulation Value** \\ \hline \multirow{4}{*}{Fuel cell stack} & Number of cells in stack & 46 \\  & Voltage range & 22 to 50 V \\  & Nominal fuel cell stack voltage & 26 V \\  & Nominal current & 0 to 46 A \\  & Auxiliary power & 24 V (60 W) \\ \hline \multirow{4}{*}{Four\(-\)phase IBC} & \(V_{out}\): Output voltage & 40\(\sim\)70 V \\  & \(C\): Capacitor & 600 uF \\  & \(f\): Working frequency & 5 kHz \\  & \(U\): Duty cycle & 0.458 \\  & \(i_{out}\): Current load & 3.75\(\sim\)8.75 A \\  & \(L_{i}\): Inductors & 1 mH \\ \hline \multirow{2}{*}{BS + STSM} & Outer loop & \(\alpha_{1}=0.0027\), \(\alpha_{2}=47.53\) \\  & Inner loop & \(c_{1}=2970\), \(c_{2}=6019\) \\ \hline \multirow{2}{*}{DL-STSM} & Outer loop & \(\alpha_{o}=0.0471\), \(\lambda_{o}=95.64\) \\  & Inner loop & \(\alpha_{k}=0.1044\), \(\lambda_{k}=213.2\) \\ \hline \multirow{2}{*}{PI} & Outer loop & \(K_{pv}=0.5\), \(K_{lv}=25\) \\  & Inner loop & \(K_{Pi}=0.3\), \(K_{li}=65\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Values of physical parameters.  
\[\left\{\begin{array}{c}\frac{d_{i}^{2}}{dt}=-\frac{1-U}{L}\phi_{out}+\frac{1}{L} \phi_{in}+\frac{V_{out}}{L}\hat{a}(t)\\ \frac{dt_{out}}{dt}=\frac{1-U}{C}\sum_{i=1}^{4}\hat{i_{i}}-\frac{1}{RC}\phi_{ out}-\frac{\hat{B}(t)}{C}\sum_{i=1}^{4}I_{i}\end{array}\right. \tag{12}\]  
assuming \(L_{j}=L\) and \(I_{i}=I\). The two transfer functions between current and duty cycle, and voltage and duty cycle, \(F_{iu}\) and \(F_{vu}\), respectively, can be found by applying the Laplace transform to Equation (12). So, the transfer functions are given by  
\[F_{iu}=\frac{v_{in}}{1-U}\frac{CRs+2}{LCRs^{2}+Ls+4R(1-U)^{2}} \tag{13}\]  
\[F_{vu}=\frac{v_{in}}{(1-U)^{2}}\frac{-ks+1}{RCks^{2}+ks+1} \tag{14}\]  
where \(k=\frac{L}{4R(1-U)^{2}}\).  
The system control design process involves determining the \(PI_{i}\) controller, which enables the phase currents to track a reference current generated by the transfer function of the \(PI_{v}\) controller, allowing the desired voltage to be followed as well. The transfer functions for the voltage loop control and the current loop control can be expressed as follows: \(F_{cv}=K_{Pv}+K_{Iv}s\) and \(F_{ci}=K_{Pi}+K_{Ii}s\). The PI controller parameters are optimized using the Particle Swarm Optimization (PSO) algorithm. Thus, the parameter values are as follows: \(K_{Pv}=0.5\) and \(K_{Iv}=25\) for the voltage loop, and \(K_{Pi}=0.3\) and \(K_{Ii}=65\) for the current loop.