Modelling a ship safety according to collision threat for ship routes

Scientific Journals
Zeszyty Naukowe
Maritime University of Szczecin
Akademia Morska w Szczecinie
2010, 20(92) pp. 120–127
2010, 20(92) s. 120–127
Modelling a ship safety according to collision threat
for ship routes crossing
Modelowanie bezpieczeństwa statku w aspekcie zagrożenia
kolizyjnego dla krzyżujących się szlaków wodnych
Zbigniew Smalko1, Leszek Smolarek2
1
Air Force Institute of Technology
Instytut Techniczny Wojsk Lotniczych, 01-494 Warszawa, ul. Księcia Bolesława 6
2
Gdynia Maritime University, Faculty of Navigation
Akademia Morska w Gdyni, Wydział Nawigacyjny, 81-225 Gdynia, ul. Morska 81–87
Key words: safety of maneuvering, Copula function, risk of navigation, queuing model
Abstract
Ship traffic on the Baltic Sea grows each year affecting the shipping safety and boosts chances of collision
with other vessel. In this article the modelling of hazard of collision is presented for ship routes crossing,
taking advantage of function Copula and methods of queuing theory.
Słowa kluczowe: bezpieczeństwo manewrowania, funkcje Copula, ryzyko nawigacyjne, model kolejkowy
Abstrakt
Gęstość ruchu statków na Bałtyku wzrasta każdego roku, powodując zwiększone ryzyko wystąpienia kolizji
statków. W pracy przedstawiono modelowanie zagrożeń dla kolizyjnych strumieni transportowych, wykorzystujące funkcje Copula i metody obsługi masowej.
Introduction
With respect to ship-to-ship collisions, the three
different collision scenarios should be examined
separately namely:
During the last few years the density of ship’s
traffic on Baltic Sea has increase importantly. Such
situation causes the growth of collision probability:
“Ship collision probabilities are higher than LNG
plant accidents, especially in approaches to harbours. They depend directly on the traffic and controls put in place. Without knowing the ship traffic
information (numbers, speeds, sizes) it is impossible to judge the probability. Ship collisions are
fairly common in and around port areas” [1].
The formal safety assessment process is made
up of three main steps such as risk analysis, risk
evaluation and risk assessment [2, 3]. A risk
assessment may be defined as an identification of
the hazards present in a task and an estimate of the
extent of the risks involved, taking into account
whatever precautions are already being taken [4, 5].
1. Head-on collision, in which two vessels collide
on a straight leg of a fairway as a result of twoway traffic on the fairway;
2. Collision, in which two vessels moving in an
opposite direction on the same fairway collide
on a turn of the fairway as a result of one of the
vessels neglecting or missing the turn (error of
omission) and thus coming into contact with the
other vessel;
3. Crossing collision, in which two vessels using
different fairways collide at the fairway
crossing.
In the paper a safety model according to third
type of collision is presented.
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Modelling a ship safety according to collision threat for ship routes crossing
No. of collisions
Fig. 3. Types of accidents, Baltic Sea, the period 2000–2008
[6]
Rys. 3. Typy wypadków, Morze Bałtyckie, lata 2000–2008 [6]
Fig. 1. Images of the traffic on the Baltic Sea in 2008 within
a time period of two days [6]
Rys. 1. Ruch statków na Bałtyku w 2008 roku wraz z kolizyjnymi (poprzecznymi) torami wodnymi, w okresie dwóch
dni [6]
Year
According to research carried by HELCOM,
number of case on Baltic Sea in 2007 year is almost
twice greatest in comparison to year 2003.
Total number of collisions
2000–2008: 288
Fig. 4. Collisions in the Baltic Sea during 2000–2008 [6]
Rys. 4. Liczby kolizji w latach 2000–2008 [6]
No. of accidents
Stochastic model of ship’s transport flow
Year
The lack of complete and certain information
about hydro meteorological parameters and ships
operational parameters makes necessary to use stochastic approach for modelling of ship transportation flow. The markov models such as Greenshield
or Greenberg are often used. The main assumption
of these models is that traffic flow is stationary and
can be characterized by average flow speed , flow
density  and flow intensity.
The Greenshield model is given by formula:
Total number of accidents
2000–2008: 910
Fig. 2. Number of reported accidents on the Baltic Sea, the
period 2000–2008 [6]
Rys. 2. Liczby wypadków w latach 2000–2008 [6]

The spatial distribution of the reported accidents
in 2008 shows that groundings were the most
common type of accidents in the Baltic accounting
for almost a half of all reported cases (45%)
surpassing the number of collisions (32%).
The new ship traffic monitoring system (AIS),
which started at July 2005, makes situation a little
better but still the ship safety according to collision
threat is important problem.
Zeszyty Naukowe 20(92)
   0 1 

 

 0 
(1)
where: 0 – speed of free movement, 0 – maximum
density of flow.
Parameters of the model can be estimated using
large set of observation because it should take into
consideration navigation parameters for analysed
flow of traffic.
121
Zbigniew Smalko, Leszek Smolarek
While using the model for any traffic stream, it
is necessary to get the boundary values of free flow
speed and jam density. These values can be obtained from a number of speed and density observations and then fitting a linear equation between
them using linear regression method.
In Greenshielda and Greenberg models the dependence between an average speed of flow and
density of flow is not taken into consideration.
Studying the behavior of both indicators of
a random flow and how they affect the final result
needs using the two-dimension probability distribution.
It is possible to define one-dimension probability distributions F(), P(), means  ,  ,
it depends on regulations defining ship’s traffic,
hydrographic parameters and operational of water
route also current environmental conditions.
So, probability distribution of speed of a ship
should be modeled by limited probability distribution e.g. it is possible to use truncated gamma density as a marginal distribution of form [7]:
f ( ) 
(m,  , max ) m  m 1  e 


 ( m)
 ( m)
2
C (u, w)  u  w  1   1  u 1  w
Model of collision threat
(2)
The system
can be used to estimate two dimensional distribution of  and .
As the conformity criterion of modelled random
vectors [i, i] the equality of their means,
variances and correlation coefficient can be used
(in sense of correlation theory).
In this case joint probability density function,
which used Copula function and one dimensional
distributions, is described by:
h( ,  )  f ( )  p( ) 
 1   1  2 F ( ) 1  2 P( )
The system consists of two crossing waterways
(ship routes). The first route is called main route
and the ships traffic is described by parameters
such as density of a traffic flow and traffic flow
speed.
The second route with lower traffic density is
called collision flow and the time interval between
consecutive ships has exponential density function
with the parameter µ:
(3)
f ( x) 
where parameter  is given by formula:
  r 


0


  2  f ( ) F ( ) d 


0



1

x
e  , 0 x
(7)
where   0.

(4)
collision
area
0


   2   p ( ) P ( ) d  


1


where the domains of integration of parameters
 and  are   (0,0),   (1,0) respectively.
Speed Vs of a ship on water route fulfills following limitation:
Vs min  Vs  Vs max
(6)
Density of flow on water route, in optional moment of time (number of ships occupying individual
section of traffic route), in general case, is a discreet random variable.
variances ˆ , ˆ and correlation coefficient r
taking advantage of statistic methods.
The Copula function described by:
2
(m,  , min )

 ( m)
main route
collision
flow
(5)
where: Vs min – minimal manoeuvring ship’s speed,
it depends on construction of a ship and its
equipment like main engine, rudder, driving screw;
Vs max – maximum admissible speed of a ship;
Fig. 5. The system diagram
Rys. 5. Schemat systemu
The system state can be described by three
dimensional random vector (X(t), O(t), Y(t)) of
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Modelling a ship safety according to collision threat for ship routes crossing
FT (t )   pi pij Fij (t )
independent components where: X(t) – time
between ships on main route, O(t) – time of staying
of the ship on main route in collision area, Y(t) –
time between ships on collision flow.
For further analysis the following definitions are
given.
Definition 1. There is a collision situation if
a ship cannot continue to move in an unimpeded
manner and has to change the ship course or ship
speed.
Definition 2. There is a collision threat if the
span of time among ships on collision area, gauged
respect of point of potential meeting, is smallest
than set up safety (admissible) value i.e.:
X (t )  Y (t )  X (t )  O(t )
and threat probability:

p zag   e  t d FT (t )
Probability of appearance the event described by
(8) is:
P X (t )  Y (t )  X (t )  O(t )  
 P(Y (t )  X (t )  O(t ) / X (t )  Y (t )) 
 P( X (t )  Y (t )) 
t 
 i E[ i ]
  j E[ j ]
i = 1, 2, ...
To calculate the probability (13) we have to
count probability cumulative distribution function
of random variables X–Y and Y–X–O.
Simulation model
The system simulations were carried out using
different states of systems channel loads and
collision flow traffic intensity to assess collision
threat.
During the simulation following parameters
were assigned:  collision flow intensity,  service
intensity, n number of iterations (n = 1000),
probability distribution of random time between
ships on main route.
(9)
j 1
where i satisfy the system of equations:
[ i ]  [ i ][ pij ]


m
  j  1,

 j 1
(10)
and i is time of staying at state i.
Suppose that the embedded Markov chain is
ergodic.
Event of failure such as loss of request is
regarded as the collision e.g. request is lost if the
distance time between arriving requests is less than
residuary service time.
Moments of service starting are equals to the
moment of arriving a ship of collision flow at collision area [9].
Let Tα be the time distance between arriving
requests at main router then its probabilisty distribution is given by formula:
Zeszyty Naukowe 20(92)
(13)
 FY  X / O (0)1  FY  X (0)
(8)
We take into consideration the G/M/1 queuing
system with losses, a general arrival process, an
exponential service process (µ) and a single server.
The arrival process is a semi-Markov stationary
point process [8].
A semi-Markov process can be described by
transition matrix [pij] and matrix of conditional
transition times distributions markov kernel [Fij],
i, j = 1,2,..., i  j, where Fij is a cumulative
probability distribution of a holding time of a state
i, if the next state will be j.
The asymptotic probabilities pi(t) are given by
formulas:
m
(12)
0
Queuing system description
pi  lim pi (t ) 
(11)
i, j
Fig. 6. Graf of simulation algorithm
Rys. 6. Schemat algorytmu symulacyjnego
123
Zbigniew Smalko, Leszek Smolarek
Fig. 7. Simulation program results presentation
Rys. 7. Wyniki programu symulacyjnego, prezentacja
The set of collision threat events is divide into
three classes according to parameter TCPA (time to
closest point of approach) [10].
The following notations are used to describe the
model of collision threat (for the ship on main route
at collision area):
 B – no collision threat;
 MZK – small possibility of collision threat, no
special action is needed, just observation;
 SZK – medium possibility of collision threat,
stay ready for action;
 DZK – high possibility of collision threat, action
necessary to avoid collision.

otherwise
0

 t na 

 b a 


1

f X (t )  
(1)i 

k
(
n

1
)!
(
b

a
)
i

0


 k 
k 1
ka  t  kb
  i  t  na  i (b  a) 
 
(15)
where: [d] is the integer part of number d.
The random variable of cumulative time for ship
on main route at crossroad under the condition
where: O = O1 +  + On, there were n – ships on
main route, has the density function given by
formula:
Model 1 [11]
We assumed that the probability distribution of
random time Xk between ships (k and (k+1)) on
main route is uniform with parameters a and b.
The cumulative distribution function is:
 0
x a
FX k (t )  
b  a
 1
for
xa
for a  x  b
for
t0
0

 n n 1
f O (t )    t
e  t
 (n  1)!
t 0
(16)
The conditional distribution of waiting time for
the ship number m, on collision flow has the density function given by formula:
(14)
bx
0

 m m1
fY (t )    t
e t
 (m  1)!
If n ships have appeared on main route then the
random variable X = X1 + · · · + Xn has the density
function given by formula:
124
t0
t 0
(17)
Scientific Journals 20(92)
Modelling a ship safety according to collision threat for ship routes crossing
where: tm is the maximal time distance between
ships,
SZK;
3,03
MZK;
1,01
1
DZK;
3,03
Ba, b    x a 1 1  x 
b 1
dx
0
is called a Beta function.
If n ships have appeared on main route then the
random variable X = X1 + · · · + Xn has the density
function given by formula:
f x ( x) 
B; 92,93
(19)
Beta curve numbers
MZK;
10,1
a = 50
SZK; 7,1
DZK;
11,1
B; 71,7
b = 41
Mean
0.5510512971422693
SD
0.051116482746800834
Count
0
0
0
0
0
0
0
2
26
129
Lo
0 to
.05
.10
.15
.20
.25
.30
.35
.40
.45
Hi
.05
.10
.15
.20
.25
.30
.35
.40
.45
.50
Count
333
343
149
17
1
0
0
0
0
0
Lo
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
Hi
.55
.60
.65
.70
.75
.80
.85
.90
.95
1
Fig. 10. Betas simulation for n = 50 [12]
Rys.10. Wyniki symulacji rozkładu beta dla n = 50 [12]
Fig. 8. The distribution of ship collision for tree classes
and proportion between intensities 4:1 and 4:3 respectively
(model 1)
Rys. 8. Rozkład dla kolizji statków i intensywności odpowiednio w proporcjach 4:1 i 4:3 (model 1)
n = 5, p = 1
n = 1, p = 3
n = 2, p = 5
o = 1/8
Probability [%]
1
x n 1 (t m  x) p 1
B(n, p)
o = 1/32
 utilization
Fig. 9. Probability of collision threat as a function of utilization parameter and mean of random variable 0
Rys. 9. Prawdopodobieństwo zagrożenia kolizją jako funkcja
parametrów użyteczności i średniej zmiennej losowej 0
Fig. 11. Examples of the beta probability density function [13]
Rys. 11. Wykresy gęstości rozkładu beta [13]
The rejection method can be used to simulate
beta distribution. Suppose there is a density g(x)
which is “close” to the density beta that we wish to
simulate from but it is much easier to simulate from
g than beta (g might be Weibull). Then provided c
such that:
f ( x)
(20)
c
g ( x)
Model 2
We assumed that the probability distribution of
random time Xk between ships (k and (k+1)) on
main route is beta B(1, p).
The density distribution function is:
f x (t ) 
1
(t m  t ) p 1 , t [0, t m ]
B(1, p)
Zeszyty Naukowe 20(92)
(18)
for all x, we can use g to get simulations from beta.
125
Zbigniew Smalko, Leszek Smolarek
Thus it is important to choose g so that c is small
because the number of iterations until an
acceptance will be geometric with mean c and
choosing c very conservatively large resulting in
high computational costs.
Beta CDF (0.5, 2)
Probability
Probability
Beta CDF (0.5, 0.5)
Fig. 13. Frequency histogram, 1 – collision threat, and 0 –
lack collision threat (model 1)
Rys. 13. Histogram częstości dla 1 – zagrożenia kolizyjnego
i 0 – braku zagrożenia kolizyjnego (model 1)
X
Beta CDF (2, 2)
Probability
Probability
X
Beta CDF (2, 0.5)
model 2
E( X ) 
X
X
Fig. 12. The beta cumulative distribution function with the
same values of the shape parameters [5]
Rys. 12. Wykres dystrybuanty rozkładu beta dla ustalonego
parametru kształtu [5]
Var ( X ) 
The modelling of hazard of collision for ship
routes crossing, taking advantage of function
Copula and methods of queuing theory is presented.
The semi-Markov model is described by
transition matrix and semi-Markov process kernel.
The semi-Markov process kernel counting is based
on probability distribution of random vector
(X(t), O(t), Y(t)) and multilayer structure of traffic
flow at main route. Using formulas (3–7) we can
find the probability distribution of random variables
X and O [14, 15].
The model of collision threat allows estimating
stationary probabilities for each of three classes
MZK, SZK, DZK.
Computational complexity of mathematical model gives rise to usage of simulation approach.
The simulation model presented in the paper is
rather not complicated but reflects the changes of
main route traffic intensity on collision threat.
Where the traffic intensity on main route was count
using formulas:
model 1
ba
,
2
b  a 2
Var ( X ) 
np
(22)
n  p 2 (n  p  1)
and moment method [15], to estimate parameters
a and b.
Conclusions
E( X ) 
n
n p
Fig. 14. Influence of proportion between traffic intensity and
probability of collision
Rys. 14. Wpływ intensywności ruchu na prawdopodobieństwo
kolizji
The beta density function has the form of
different shapes depending on the values of the two
parameters, for example if n = p = 1 it is the
uniform [0, tm] distribution. The beta distribution
can be used to model events which are constrained
to take place within an interval defined by a minimum and maximum value.
Immediate work in simulation follows better
evaluation measures and improvement of duration
modelling, model system and model confidence
levels.
(21)
12
126
Scientific Journals 20(92)
Modelling a ship safety according to collision threat for ship routes crossing
Presented approach has to be further developed
by more comprehensive experimental evaluations,
examples of applications, analytical models relating
selected simulation responses with model parameters.
Discrete event models allow inclusion of individual variables without creating compound states,
which could improve the model precision
Some interesting questions are still open. For
example possible questions can relate to correlation
between random variables.
5. MERKISZ J., NOWAKOWSKI T., SMALKO Z.: Bezpieczeństwo
w transporcie – wybrane zagadnienia. Uwarunkowania
rozwoju systemu transportowego Polski, pod red.
Bogusława Liberadzkiego i Leszka Mindura. Wyd.
Instytutu Technologii Eksploatacji – Państwowy Instytut
Badawczy, Warszawa–Radom 2007, 499–561.
6. www.helcom.fi
7. CHAPMAN D.G.: Estimating the Parameters of a Truncated
Gamma Distribution. Ann. Math. Statist. 1956, Vol. 27,
No. 2, 498–506.
8. SMOLAREK L.: Finite Discrete Markov Model of Ship
Safety. TransNav Procedings, Gdynia 2009, 589–592.
9. KONIG D., STOYAN D.: Metody obsługi masowej. Wyd.
Naukowo-Techniczne, Warszawa 1979.
10. SMOLAREK L.: Human reliability at ship safety
consideration. Journal of KONBiN 2008, 2(5), 191–206.
11. SMALKO Z., SMOLAREK L.: Estimate of Collision Threat for
Ship Routes Crossing. Proc. Marine Traffic Engineering
Conference MTE09, Malmo, Sweden, 19–22 October
2009, 195–199.
12. http://bayes.bgsu.edu/nsf_web/jscript/betasim/betas.htm
13. http://isometricland.com/geogebra/geogebra_beta_distributions.php
14. SHEMYAKIN A., YOUN H.: Statistical aspects of joint-life
insurance pricing. 1999 Proceedings of Amer. Stat. Assoc.,
2000, 34–38.
15. SHEMYAKIN A., YOUN H.: Bayesian estimation of joint
survival functions in life insurance. [in:] Bayesian Methods
with Applications to Science, Policy and Official Statistics,
European Communities, 2001, 489–496.
References
1. RONALD P., KOOPMAN PH.D.: P.E review of the “QRA done
for the proposed Shannon LNG Terminal. Land Use
Planning QRA Studies of the Proposed Shannon LNG
Terminal, September 2007” – December 2007.
2. SMALKO Z.: Modelowanie eksploatacyjnych systemów
transportowych. Instytut Technologii Eksploatacji, Radom
1996.
3. SMALKO Z.: Application of expert methods for risk
assessment of transport systems. Technology, Law and
Insurance, 1999, Vol. 4.
4. BUKOWSKI L., FELIKS J., SMALKO Z.: Analiza niezawodności systemów podwyższonego ryzyka. 35 Zimowa Szkoła Niezawodności: Szczyrk 2007. PAN, Warszawa–Radom
2007, 103–116.
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