# Analysis of a Supercapacitor/Battery Hybrid Power System for a Bulk Carrier
## 3 Proposed Hybrid Power System
### Cargo Loading/Unloading Mode
If k\({}_{\rm a}\) was assumed to be 1.2, k\({}_{\rm s}\) was assumed to be 1.1, and k\({}_{\rm e}\) was assumed to be 0.9. The E\({}_{\rm sc\_min}\) is calculated as below:  
\[\rm E_{\rm sc\_min}=E_{\rm sc\_demand}\times(k_{\rm a}/k_{\rm e})\times k_{s} =1,005.92\ (Wh)\times(1.2/0.9)\times 1.1\cong 1.48\ (kWh)\cong 5,311\ (kJ). \tag{5}\]  
In most applications, the SC pack is assembled in modules, and these are connected in series and parallel to increase both the working voltage and overall capacitance. The total capacitance of the SC pack (C\({}_{\rm SC,t}\)) is then evaluated as:  
\[\rm C_{\rm SC,\it t}=C_{\rm SC,\it module}\times(P/S)\ (F), \tag{6}\]  
where P is the number of parallel strings and S is the number of series modules. The selected SC module specifications are as shown in Table 9. Then, the SC pack capacity was calculated using Equation (7). \[\rm E_{\rm SC}=(1/2)\times C_{\rm SC,\it t}\times V_{\rm SC,\it t}{}^{2}\ (J), \tag{7}\]  
where V\({}_{\rm SC,\it t}\) is the voltage of an SC pack, which is proportional to the number of SC modules. In general, the voltage variation of an SC pack is to be kept between 100% and 50% of its maximum voltage. As shown in Figure 6, the LIB offers a fairly constant discharge voltage performance throughout the spectrum of usable energy, whereas the SC voltage shows a linear and decreasing behavior from the maximum value up to 50% in general. Even if the stored energy is proportional to the product of the capacitance for the square of the voltage, the delivered power decreased in the discharging phase because the current is limited [31; 32]. Thus, the available energy of the designed SC pack (E\({}_{\rm SC\_design}\)) is calculated by the following equation [31; 33]:  
\[\rm E_{\rm SC\_design}=(1/2)\times C_{\rm SC,\it t}\times(V_{\rm SC,\it t, \max}{}^{2}-V_{\rm SC,\it t,\min}{}^{2})=(1/2)\times \tag{8}\] \[\rm C_{\rm SC,\it t}\times(V_{\rm SC,\it t,\max}{}^{2}-(1/2) \times V_{\rm SC,\it t,\max}){}^{2})=(3/8)\times C_{\rm SC,\it t}\times V_{ \rm SC,\it t,\max}{}^{2}\ (J),\]  
where V\({}_{\rm SC,\it t,\max}\) is the maximum terminal voltage, and V\({}_{\rm SC,\it t,\min}\) is the minimum terminal voltage of the SC pack. If the output voltage of the SC pack was determined to be DC 625 V, which was achieved by \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**No.** & **Step** & **Time (s)** & **Power (kW)** & **Energy (Wh)** \\ \hline \⃝ & Lowering with no load & 20 & \(-\)19.2 & \(-\)106.68 \\ \⃝ & Grab (close) & 10 & 20.0 & 55.56 \\ \⃝ & Hoisting with full load & 25 & 134.6 & 934.66 \\ \⃝ & Luffing in (up) & 5 & 86.5 & 120.15 \\ \⃝ & Slewing to port side & 15 & 34.6 & 144.18 \\ \⃝ & Lowering with full load & 15 & \(-\)67.3 & \(-\)280.44 \\ \⃝ & Grab (open) & 5 & 20.0 & 27.78 \\ \⃝ & Hoisting with no load & 10 & 38.4 & 106.68 \\ \⃝ & Slewing to ship side & 10 & 23.1 & 64.17 \\ \⃝ & Luffing out (down) & 5 & \(-\)43.3 & \(-\)60.14 \\ \hline \hline \end{tabular}
\begin{tabular}{c c c} \hline \hline  & About 4 min/cycle (including \\  & overhauling time) & 1005.92 Wh/cycle \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Expected power demands for deck crane operations. arranging five (5) SC modules in series, then, the SC pack needed have three (3) parallel strings to obtain the demand capacity according to the below equation:  
\[ \begin{array}{l}\text{E}_{\text{sc\_design}}=(3/8)\times(C_{\text{SC,module}} \times(\text{P/S}))\times\text{V}_{\text{SC,t,max}^{2}}=(3/8)\times\\ (63~{}(\text{F})\times(3/5))\times(625~{}(\text{V}))^{2}\cong 5,537~{}(\text{kJ}) \cong 1.54~{}(\text{kWh})>1.48~{}(\text{kWh}).\end{array}  \tag{9}\]  
The design specifications of the SC pack are as shown in Table 10. If the SC pack was discharged, it could be charged with a C-rate of 180 within about 20.1 seconds during the luffing (10) and lowering (1) steps.