5. The Proposed Optimal Navigation Search Method for Emission and Energy
Emissions and energy consumption during the navigation of an oceangoing vessel depend on the vessel’s speed. The navigation route for a voyage is prespecified on the basis of the regulations and environmental conditions. The speed of an oceangoing vessel is affected by external forces such as tidal currents, waves, and wind. Tidal currents show some patterns according to the lunar calendar time and date, and wind shows seasonal patterns.
Section 3 shows how to extract the patterns of externally forced speed changes for days of the year in regions from big marine environmental data. The effective speed of a navigating vessel is computed by combining the SOW and the externally forced speed change. The effective speed allows us to compute the real emissions and energy use over the voyage. The vessels usually navigate an ocean route at a fixed speed; it is, however, possible to save emissions and energy by adjusting the speed. An optimal speed adjustment plan should be computed by considering the SOW, externally forced speed changes, and the arrival time.
Here, we propose a DP-based method that efficiently computes an optimal speed adjustment plan along with total emissions. DP is a means for solving a complex problem by breaking it into smaller subproblems, solving each of them once and storing their solutions, and using them to construct the solution to the original problem [
29].
In DP, we use the following settings and notations: A navigation route
is expressed in a sequence of positions,
, where
is the location of index
and
N is the number of position indices;
indicates the time at which a vessel is at position
, and it has a value from the set
,
, …,
;
is the earliest time at which the vessel arrives at location
starting from time
, and
, where
is the time interval of considered arrival times at each position;
indicates the minimum time taken for the vessel to travel from
to
with the fastest speed allowed [
30]. Therefore, the following relationship holds:
In Equation (4),
denotes the index of the most recent time at which the vessel can depart from
to reach
within the considered time, that is,
where
is the minimum time taken for the vessel to travel from
to
;
denotes the externally forced speed at time
at position
indicates the energy consumption to travel from
to
with SOW
and externally forced speed
; and
is the time for the vessel to travel from
to
with speed
and externally forced speed
at time
. Therefore,
is computed as follows:
The accumulated energy consumption (
Figure 4) for a vessel to arrive at position
at time
can be recursively calculated as follows:
To determine an optimal speed adjustment plan, the proposed DP method uses Equation (6) for computing the accumulated energy consumption. It uses two-dimensional arrays
and
, where
stores the value of
and
stores index
that satisfies the following relationship for Equation (6):
The following procedure, DP-for-Energy-Consumption-Computation, is a DP algorithm to compute the accumulated energy consumption:
procedure DP-for-Energy-Consumption-Computation (, maxS, minS, Pmcr, )
input: for externally forced speed changes, where
maxS for the maximum SOW
minS for the minimum SOW
Pmcr for maximum continuous rating power
for the time interval of considered arrival time
for positions
output: C[1‥N, 1‥M] for accumulated energy consumption, where
B[1‥N, 1‥M] for preceding position’s time index of value used for
U[1‥N, 1‥M] for speech adjustment for
for the earliest arrival time at each position
for the latest arrival time at each position
begin
1. initialize C[1‥N, 1‥M] to be zero
2. initialize B[1‥N, 1‥M] to be zero
3. initialize U[1‥N, 1‥M] to be zero
4.
5.
6. for to
7. travelDist distance between and
8. highest-speed maxS
9. elapsed-time1 travelDist / highest-speed
10. + elapsed-time1
11. LF (highest-speed/maxS)3
12. consumed-energy Pmcr LF elapsed-time1
13. consumed-energy
14. slowest-speed minS
15. elapsed-time2 travelDist / slowest-speed
16. num-intervals floor(elapsed-time2 elapsed-time1)/
17. for = 2 to num-intervals
18. bmax max-prior-index()
19. maxValue
20. for j = 1 to bmax
21. elapsed-time ()
22. u travelDist / elapsed-time
23. LF (U/maxS)3
24. consumed-energy Pmcr LF elapsed-time
25. candidate-energy consumed-energy
26. if < candidate-energy then
27. candidate-energy
28. u
29. j
30. end if
31. end for
32. end for
33. (num-intervals 1)
34. end for
end
The following procedure, max-prior-index, determines the last index to be considered at the preceding position when the accumulated energy consumption is computed:
procedure max-prior-index()
input: for the latest arrival time at each position
for positions
for position index
for time index
for the time
output: for the last index to be considered at the preceding position
begin
1. distance between and
2. elapsed-time /maxS
3. floor(t elapsed-time [])/
end
Once DP-for-Energy-Consumption-Computation is executed, we can extract an optimal speed adjustment plan using the following procedure, Find-Optimal-Plan:
procedure Find-Optimal-Plan()
input: C[1‥N, 1‥M] for accumulated energy consumption, where
B[1‥N, 1‥M] for preceding position’s time index of value used for
U[1‥N, 1‥M] for speech adjustment for
for the time index for the arrival time
output: for the recommended speed adjustment plan
consumed-energy for the estimated energy consumption for the plan
begin
1. for to 1
2. prev
3.
4. end
5. consumed-energy
end
The time complexity of DP-for-Energy-Consumption-Computation is and the time complexity of Find-Optimal-Plan is .