## 3 Results
### Demand Management of EDLC and the Hybrid Locomotive Through PBLM  
The instantaneous power p(t) and energy w(t) of EDLCs are strongly dependent on voltage and, consequently, the SoC of the overall unit proposed for the hybrid locomotive. It seems useful to remember the main equations for these magnitudes for an elemental EDLC device according to our model (see Figure 9 for the nomenclature of main magnitudes and parameters). The instantaneous power (generation criterium):  
\[p(t)=U(t)I_{g0}(t)=u_{c}(t)I_{g0}(t)-R_{S}I_{g0}^{2}(t) \tag{26}\]  
The available stored energy, w(t), in the EDLC at time t of a simulation starting at \(t_{0}\) (remember \(K_{P}=3K_{V}\)) is  
\[w(t)-w(t_{0}) =\int_{t_{0}}^{t}u_{c}(\tau)\frac{dq_{EDLC}(\tau)}{d\tau}d\tau= \int_{u_{c}(t_{0})}^{u_{c}(t)}u_{c}(\tau)\big{[}C_{0}+K_{P}u_{c}^{2}(\tau)\big{]} du_{c}=\] \[=\Big{(}\tfrac{1}{2}C_{0}u_{c}^{2}+\tfrac{1}{3}K_{P}u_{c}^{3} \Big{)}\Big{]}_{u_{c}(t_{0})}^{u_{c}(t)}=\Big{(}\tfrac{1}{2}C_{0}u_{c}^{2}+K_{ v}u_{c}^{3}\Big{)}\Big{]}_{u_{c}(t_{0})}^{u_{c}(t)} \tag{27}\]  
And SoC at the elemental (or overall) EDLC:  
\[SoC(t)=\frac{w(t)}{w_{max}}=\frac{\tfrac{1}{2}C_{0}u_{c}^{2}(t)+K_{v}u_{c}^{3} (t)}{\tfrac{1}{2}C_{0}u_{c-max}^{2}+K_{v}u_{c-max}^{3}} \tag{28}\]  
Previous equations clearly show that an accurate voltage must be forecasted to achieve the best performance of both EDLC (OESD) and locomotives. The model can help in the achievement of some further targets, such as optimization of energy or EDLC size according to train requirements (both targets are not objectives in this paper). The itinerary of freight trains is usually conditioned by passenger traffic and involves continuous stops. The energy management system of tandem locomotives should account for train energy requirements and the OESD state. Usually, several simulations must be applied according to changes in the itinerary. For this reason, commissioned and potential train stops define a partition of the route into several \"route sections\". An alternative is to consider signalization tracks. Moreover, the peak on the elevation profile also defines two sections between RS and ER stations. A simplified flow diagram is presented in Figure 19. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Property** & **Battery Pack** & **EDLC Pack** \\ \hline Module reference 1  & Victron 100 Ah & BMOD0500 P016 \\ Volume (module) & 0.0125 m3  & 0.0092 m3  \\ Modules in series per row & 62 & 56 \\ Rows in parallel & 10 & 40 \\ Capacity (Ah) & 500–1000 Ah & 4000–6000 Ah \\ Nominal energy capacity (kWh) & 740 kWh & 40 kWh \\ Operational limits & SoC (0.2–0.9) & Voltage (8.5–16.5 V) \\ Available energy & 520.8 kWh & 27.4 (max 29.2) kWh \\ Weight & 5.5 t & 12.5 t \\ Volume & 7.74 m3  & 10 m3  \\ Capital cost & 130 k€ & 1 M6  \\ Additional power electronics 2  & 500 kg & 300 kg \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Nominal OESD properties (common-sense criteria). Notice that values for battery and EDLC modules were presented in Tables 2 and 3. A simplified premise for initiating energy management is illustrated in Figure 20a. Essentially, the S269 locomotive compensates for the power surplus above 2230 kW (the rated power of the diesel S333) and absorbs the surplus braking energy (1650 kW for the S333). Naturally, it means that the train's initial acceleration is limited without the support of the hybrid unit. During the initial acceleration phase, both units contribute their forces, with the EDLC operating at a state of charge (SoC) near 100% at the CT station stop, particularly at the bifurcation point between the freight and passenger stations. Prior to departure, the proposed algorithm calculates the energy requirements for an upcoming segment (e.g., based on time from the starting point or target speed) and allocates a percentage of the total power demand to the EDLC (e.g., 50%) for the next section of track (e.g., Sec-1, from CT to an intermediate station located 12 km away, as shown in Figure 20b).