## 4 Results
### Reward Shaping
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\boldsymbol{\gamma}\)} & \multirow{2}{*}{**DM**} & \multicolumn{8}{c}{_SoC\({}_{f}\) (-)_} & \multicolumn{8}{c}{_FC (g)_} \\ \cline{2-13}  & & _DP_ & _R1_ & _R2a_ & _R2b_ & _R2c_ & _R3_ & _DP_ & _R1_ & _R2a_ & _R2b_ & _R2c_ & _R3_ \\ \hline \multirow{8}{*}{0.99} & CLUST1 & 0.6000 & 0.5939 & - & - & - & 0.6018 & 246.6 & 263.20 & - & - & - & 280.80 \\  & CLUST2 & 0.6000 & 0.5962 & - & - & - & 0.6031 & 197.2 & 255.38 & - & - & - & 254.28 \\  & CLUST3 & 0.6000 & 0.5754 & - & - & - & - & 152.6 & 169.26 & - & - & - & - \\  & CLUST4 & 0.6000 & 0.6019 & - & - & - & 0.5984 & 53.9 & 106.56 & - & - & - & 88.48 \\ \cline{2-13}  & \multicolumn{8}{c}{rFC (g)} & \multicolumn{8}{c}{_\(\Delta\)FC (\%)_} \\ \cline{2-13}  & CLUST1 & 246.6 & 272.02 & - & - & - & 280.83 & - & 10.31 & - & - & - & 13.88 \\ \cline{2-13}  & CLUST2 & 197.2 & 260.9 & - & - & - & 254.28 & - & 32.30 & - & - & - & 28.95 \\ \cline{2-13}  & CLUST3 & 152.6 & 204.7 & - & - & - & - & 34.14 & - & - & - & - \\ \cline{2-13}  & CLUST4 & 53.9 & 106.56 & - & - & - & 90.86 & - & 97.70 & - & - & - & 68.57 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Reward shaping analysis considering \(\gamma\) = 0.99. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\boldsymbol{\gamma}\)} & \multirow{2}{*}{**DM**} & \multicolumn{8}{c}{_SoC\({}_{f}\) (-)_} & \multicolumn{8}{c}{_FC (g)_} \\ \cline{2-13}  & & _DP_ & _R1_ & _R2a_ & _R2b_ & _R2c_ & _R3_ & _DP_ & _R1_ & _R2a_ & _R2b_ & _R2c_ & _R3_ \\ \hline \multirow{8}{*}{0.9} & CLUST1 & 0.6000 & 0.5838 & - & - & - & 0.5977 & 246.6 & 229.91 & - & - & - & 259.89 \\  & CLUST2 & 0.6000 & 0.5826 & - & - & - & 0.6000 & 197.2 & 227.19 & - & - & - & 282.63 \\  & CLUST3 & 0.6000 & 0.5817 & - & - & - & - & 152.6 & 173.02 & - & - & - & - \\  & CLUST4 & 0.6000 & 0.5819 & - & - & - & 0.5884 & 53.9 & 52.07 & - & - & - & 66.52 \\ \cline{2-13}  & \multicolumn{8}{c}{rFC (g)} & \multicolumn{8}{c}{_\(\Delta\)FC (\%)_} \\ \cline{2-13}  & CLUST1 & 246.6 & 253.21 & - & - & - & 263.13 & - & 2.68 & - & - & - & 6.70 \\ \cline{2-13}  & CLUST2 & 197.2 & 252.28 & - & - & - & 282.63 & - & 27.93 & - & - & - & 43.32 \\ \cline{2-13}  & CLUST3 & 152.6 & 199.33 & - & - & - & - & - & 30.62 & - & - & - \\ \cline{2-13}  & CLUST4 & 53.9 & 78.21 & - & - & - & 83.29 & - & 45.10 & - & - & - & 54.53 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Reward shaping analysis considering \(\gamma\) = 0.9. Recalling Section 2.4, the reward function R1 is _SoC_ oriented while R2 and R3 are FC oriented. Such a consideration is fundamental to justify the difficulty of the agent to solve the problem in case of R2 and R3. In fact, a more complex problem has to be faced when R2 and R3 are considered since a lower priority is given to the battery _SoC_ trajectory. In this case, unfeasibility conditions related to the battery _SoC_ exceeding the limits imposed by the battery _SoC_ window are more likely to be exploited. Therefore, higher discount factors (i.e., larger time horizons for the discounted return) could be considered as they have proved to allow for stronger agents in cases of very complex problems. Consistent with the results of the reward shaping analysis, a strong interaction between discounted return and reward function has been demonstrated. A non-trivial agent response can be obtained when reward functions with different levels of complexity are coupled with the discount factors and tested on different driving conditions.