## 3 Numerical Simulations
### Path Planning with Ocean Currents
#### 3.4.2 Path Planning Considering Time-Varying Current Information  
The ocean current is assumed to be steady in Section 3.4.1. However, the ocean current varies with time due to dynamic phenomena, such as tides, eddies, etc. For example, Daya Bay is dominated by irregular semidiurnal tides, and the tidal current turns in direction every \(\sim\)3 h. In the experiments of Section 3.4.1, the total cruising times exceed 10 h. Therefore, it is necessary to consider the temporal variation of ocean currents, especially for long-term deployments.  
Short-period motions, such as sea surface waves, are excluded because they are a type of random process. Basin-scale motions, such as tides in a bay, usually vary in a slow period. Therefore, we can treat the time-varying current as a set of steady ocean currents in the simulation. In each time step, we calculate the optimal path of AUV at the current position with a steady ocean current at this moment, then continuously update the optimal path when the time and AUV's position change. Finally, we integrate all the AUV's positions to obtain the optimal path. Because the value iterative method updates the optimal path of all the points in the entire domain every time step, it is very convenient to implement the above method.  
Figure 9 shows the MDP algorithm path with consideration of the time-varying ocean current, and Table 7 is the cruising time. The ocean current is updated every hour. The cruising time in each case is over 14 h, which indicates that the tidal current turns direction more than 4 times during the AUV deployment.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Case 1** & **Case 2** & **Case 3** \\ \hline MDP + Current & 13.3308 & 14.4615 & 17.9979 \\ MDP No Current & 13.3308 & 14.7278 & 19.4622 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Same as Table 5, but the AUV moves along the ocean currents.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Case 1** & **Case 2** & **Case 3** \\ \hline MDP – Current & 40.4790 & 112.8243 & 108.2778 \\ MDP No Current & 40.4790 & \(\infty\) & \(\infty\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: The comparison of the travel time (h) along the optimal paths between with (MDP – Current) and without (MDP without Current) consideration of ocean currents. In this case, the AUV moves against the ocean current. \(\infty\) means that the AUV cannot finish the predicted path due to strong counter-currents.  
The MDP algorithm predicts two different paths between with and without consideration of the ocean current (Figure 10). With the ocean current, the optimal path requires that the AUV first floats up near the surface and then goes around the island 'A' from its east. Without the ocean current, the AUV first moves close to the bottom, floats up near the island 'A', and goes around it from its west. When the AUV travels inside the bay, the optimal paths predicted by these two scenarios are the same. If the AUV moves along the path predicted by the MDP algorithm without consideration to the tidal current, it will take 33.9324 h to arrive at the target, 3.6487 shorter than that with the tidal current, highlighting the necessity to incorporate the ocean currents in the AUV's path planning.