## 2 Modelling and Control Organization  
### Architecture and System Modelling  
The system architecture is described in Figure 1. Each source is connected to a DC-DC converter through a common DC bus voltage, such as suggested by [PERSON] et al [22]. It may be noted that other architectures have also been depicted in the literature [7; 23; 24]. The cells of the fuel cell system are based on proton exchange membrane (PEM) technology. Due to its high power density, high efficiency, and low temperature, the PEM technology is the most widely used in the fuel cell vehicle sector [25]. As the objective is to propose an optimal energy management of a multi-source vehicle, energetic models are sufficient. Hence, the modelling of the PEM-FC is achieved with a quasi-static model. A polarization curve is used to depict the main losses with the voltage drops [26] (Figure 2). The polarization curve has been characterized with a laboratory plant for a constant temperature of 346 K with reactant pressures of 250 kPa. The PEM-FC stack is composed of 80 cells and can develop a maximum power of 20 kW for a stack voltage range of 50-80 V for a stack maximum current of 360 A. More details on the tests can be found in the book of [PERSON] et al [27]. A time constant of 10 ms is furthermore added to represent the main FC dynamics. The battery is based on lithium iron phosphate (LFP) technology of the manufacturer A123. The LFP is adapted for high-power application, such as an secondary energy storage system, with a very good thermal stability [28]. The most dominating phenomena of LFP batteries are the ohmic effect  
Figure 1: System architecture. Figure 2: Polarization curve of the fuel cell. and the diffusion process [29]. Hence, the battery model is composed of an open circuit voltage (OCV), a series-resistance \(r_{b}\), and a constant phase element (CPE) impedance (Figure 3). The CPE, defined by  
\[Z_{CPE}=\frac{1}{Q(j\omega)^{k}}\;k\in[0,1] \tag{1}\]  
takes into account the voltage drop due to the diffusion process [30]. The battery model parameters have been identified by experimentation for an environmental temperature maintained at 25 \({}^{\circ}\)C into a thermal chamber. Each cell develops a nominal capacity of 20 Ah with a nominal voltage of 3.3 V, at 50% of the SoC, for a recommended maximum charge current of 100 A, and a maximum discharge continuous current of 200 A. The SC bank is modeled to take into account the fast dynamics with a two branch [PERSON] Zubieta and Bonert model [31] (Figure 4). The parameter \(r_{sc}\) represents the SC bank series resistance, \(C_{sc}\) is its main capacitance, and \(C_{il}\) is the coefficient of the variable capacitance evolution with its voltage. All parameters have been identified following the experimental protocols of [PERSON] et al [31]. For the majority of the traction applications, this kind of model is enough for energetic study. The SC bank is based on Maxwell BPAK0350-15 EA modules (San Diego, CA, USA). Each module develops a maximal voltage of 15 V for a nominal capacity of 58 F.  
The smoothing inductors are modeled by R-L series circuits. No saturation of the magnetic circuit is considered:  
\[L_{i}\frac{d}{dt}i_{Li}=u_{i}-u_{hi}-r_{Li}i_{Li} \tag{2}\]  
where \(r_{Li}\) and \(L_{i}\) are the resistance and the inductance of the inductor, respectively. The index \"\(i\)\" corresponds to each source (\"fc\" for fuel cell, \"sc\" for supercapacitors, \"b\" for battery). For the DC-DC converters, a response time of the current \(i_{Li}\) much greater than the switching frequency of the corresponding converter is considered [32]. In this way, an average model can be used, which is expressed by  
\[\left\{ \begin{array}{l}u_{dci}=m_{i}u_{i}\\ i_{dci}=m_{i}\eta_{i}^{\gamma_{i}}i_{Li}\end{array}