386
where S
v
= the shortest route length; T
v
= scheduled
voyage time; S(ℜ) = length depending on route ℜ
configuration.
Thus, the main voyage optimality criterion, with-
out risks consideration, is the minimum of addition-
ally performed work, appeared due to weather, time
and distance limitations. This work can be given as
voyage length integral of additional resistance R
W
arisen due to environmental disturbances:
(4)
From equations (2), (3) the additional work can
be obtained as:
(5)
Therefore, the objective function representing the
specified route optimality can be expressed as:
( )
min
,
v
TT
Z P t dt Z A
≤
= →
∫
(6)
For the full-valued solution of the problem, it is
also necessary to take into account corresponding
limitations. For this purpose the risk assessment
concept was used and next was formulated: the op-
timal route is found if the total work for the voyage
is closest to minimal, voyage time does not exceed
the scheduled one, and the risk level on each route
leg is less then specified limit. Thus, the objective
function will be given as:
( )
( )
( )
( )
( )
max
R,
min ,
R,
p
safe
TT
W safe
Ut
Z P dt
RU t
≤
=
+∆
∫
P
P
, (7)
where U
safe
= maximum safe speed, at which the
specified hazardous occurrence risk R is below the
critical limit; P
max
= maximum engine power; Р =
engine power needed to keep defined calm water
speed;
∆
Р(R
W
) = additional power needed to com-
pensate the resistance due to environmental disturb-
ances R
W
; R ∈ (0,1) = risk level on the specified
route leg.
3 RISK EVALUATION
3.1 Problem definition
According to the route optimality definition, given
above, the risk level conducted with ship activity in
prescribed weather conditions shall be determined
for each route leg. Therefore, we define the leg as
the part of the route on which ship control regime
(speed and heading) and weather conditions remain
constant. As opposed to classical definition two or
more different route legs may be situated on one line
between the waypoints, depending on weather grid
density.
Mathematically the risk level can be defined as
product of likelihood of hazardous occurrence and
its consequence. In our case we define likelihood as
probability of reaching defined dynamical motion
parameters that may lead to the series of negative
consequences, conducted with ship’s operation in
storm.
Assessing the risks of ship operation in heavy
weather conditions one can define the situations
connected with damages to hull structure, ship’s sys-
tems and machinery and the situations arising due to
violations of cargo handling technology.
For instance, the achievement of defined high
amplitudes of roll may lead to the series of situations
with different levels of consequences, such as shift-
ing or loss of cargo, flooding of ship’s compart-
ments, capsizing. Therefore, next risk levels can be
highlighted: insignificant, low, practically allowable
and not allowable. The risk management should
cover such measures which allow to vary the proba-
bility of definite event or to reduce the degree of its
consequence. When solving the problem of safe ship
control regime selection in heavy seas we assume
the degree of consequence as constant. From the
other hand by altering ship control settings operator
can affect the probability of reaching such ship mo-
tion parameters that lay beyond the limits of practi-
cally allowable risk. In this case the risk level can be
given as
, (8)
where p
1
, p
2
,…,p
n
= probabilities of reaching the
ship motion parameters, that may lead to definite
hazardous occurrence.
3.2 Seaworthiness criteria
To perform the risk assessment and to find a safe
control regime in given weather conditions it’s nec-
essary to define appropriate criteria, thereupon fol-
lowing factors should be taken into account:
− frequency and force of slamming;
− frequency of green water;
− motion amplitudes;
− hull stresses;
− propeller racing;
− accelerations in various ship points;
− forced and controlled speed redaction.