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 Smalko
1, Leszek Smolarek
21 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, wykorzy-stujące funkcje Copula i metody obsługi masowej.
Introduction
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 har-bours. They depend directly on the traffic and con-trols put in place. Without knowing the ship traffic information (numbers, speeds, sizes) it is impos-sible 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].
With respect to ship-to-ship collisions, the three different collision scenarios should be examined separately namely:
1. Head-on collision, in which two vessels collide on a straight leg of a fairway as a result of two-way traffic on the fairtwo-way;
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.
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 koli-zyjnymi (poprzecznymi) torami wodnymi, w okresie dwóch dni [6]
According to research carried by HELCOM, number of case on Baltic Sea in 2007 year is almost twice greatest in comparison to year 2003.
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.
Fig. 3. Types of accidents, Baltic Sea, the period 2000–2008 [6]
Rys. 3. Typy wypadków, Morze Bałtyckie, lata 2000–2008 [6]
Fig. 4. Collisions in the Baltic Sea during 2000–2008 [6] Rys. 4. Liczby kolizji w latach 2000–2008 [6]
Stochastic model of ship’s transport flow
The lack of complete and certain information about hydro meteorological parameters and ships operational parameters makes necessary to use sto-chastic approach for modelling of ship transporta-tion 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: 0 0 1 (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. N o. o f ac ci de nt s
Total number of accidents 2000–2008: 910 Year N o. o f co ll isi on s
Total number of collisions 2000–2008: 288 Year
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 ob-tained from a number of speed and density observa-tions and then fitting a linear equation between them using linear regression method.
In Greenshielda and Greenberg models the de-pendence 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 distribu-tion.
It is possible to define one-dimension proba-bility distributions F(), P(), means , , variances
ˆ
2,
ˆ
2 and correlation coefficient rtaking advantage of statistic methods. The Copula function described by:
u w
w u w u C( , ) 11 1 (2) can be used to estimate two dimensional distribu-tion 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:
1 1 2 ( ) 1 2 ( )
) ( ) ( ) , ( P F p f h (3) where parameter is given by formula:
0 0 1 0 d ) ( ) ( 2 d ) ( ) ( 2 P p F f r (4)where the domains of integration of parameters and are (0,0), (1,0) respectively.
Speed Vs of a ship on water route fulfills
follow-ing limitation:
max
min s s
s V V
V (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;
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 distribu-tion e.g. it is possible to use truncated gamma den-sity as a marginal distribution of form [7]:
) ( ) ( ) , , ( ) ( ) , , ( ) ( 1 max min m e m m m m f m m (6)
Density of flow on water route, in optional mo-ment of time (number of ships occupying individual section of traffic route), in general case, is a dis-creet random variable.
Model of collision threat The system
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:
x e x f x 0 , 1 ) ( (7) where 0.
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
main route colli-sion area colli-sion flow
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.:
) ( ) ( ) ( ) (t Y t X t Ot X (8)
Queuing system description
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 j j j i i i t i E E t p p 1 ] [ ] [ ) ( lim i = 1, 2, ... (9)where isatisfy the system of equations:
m j j ij i i p 1 , 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 colli-sion area [9].
Let Tα be the time distance between arriving
requests at main router then its probabilisty distri-bution is given by formula:
j i i ij ij T t p p F t F , ) ( ) ( (11)and threat probability:
0 ) ( dF t e pzag t T (12)Probability of appearance the event described by (8) is:
1 (0)
) 0 ( )) ( ) ( ( )) ( ) ( / ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( /O Y X X Y F F t Y t X P t Y t X t O t X t Y P t O t X t Y t X P (13)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.
Fig. 6. Graf of simulation algorithm Rys. 6. Schemat algorytmu symulacyjnego
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.
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:
x b b x a a b a x a x t F k X for 1 for for 0 ) ( (14)
If n ships have appeared on main route then the random variable X = X1 + · · · + Xn has the density
function given by formula:
kb t ka a b i na t i k a b n t f k a b na t i i k X 1 0 ) ( ) 1 ( ) ( )! 1 ( 1 otherwise 0 ) ( (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: 0 )! 1 ( 0 0 ) ( 1 e t n t t t fO n n t (16)
The conditional distribution of waiting time for the ship number m, on collision flow has the densi-ty function given by formula:
0 )! 1 ( 0 0 ) ( 1 t e m t t t fY m m t (17)
Fig. 7. Simulation program results presentation Rys. 7. Wyniki programu symulacyjnego, prezentacja
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 odpowie-dnio w proporcjach 4:1 i 4:3 (model 1)
Fig. 9. Probability of collision threat as a function of utili-zation parameter and mean of random variable 0
Rys. 9. Prawdopodobieństwo zagrożenia kolizją jako funkcja parametrów użyteczności i średniej zmiennej losowej 0 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: ] , 0 [ , ) ( ) , 1 ( 1 ) ( 1 m p m x t t t t p B t f (18)
where: tm is the maximal time distance between
ships,
1
0 1 11 d ,b x x x a B a bis 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:
1 1( ) ) , ( 1 ) ( p m n x x B n p x t x f (19)
Beta curve numbers a = 50 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. 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: c x g x f ) ( ) ( (20) for all x, we can use g to get simulations from beta.
MZK; 1,01 SZK; 3,03 DZK; 3,03 B; 92,93 MZK; 10,1 SZK; 7,1 DZK; 11,1 B; 71,7 n = 5, p = 1 n = 1, p = 3 n = 2, p = 5 utilization P ro ba bil ity [% ] o = 1/8 o = 1/32
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.
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]
Conclusions
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 mo-del 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
12 ) ( , 2 ) ( 2 a b X Var a b X E (21) model 2
( 1) ) ( ) ( 2 p n p n np X Var p n n X E (22)and moment method [15], to estimate parameters a and b.
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 mini-mum and maximini-mum value.
Immediate work in simulation follows better evaluation measures and improvement of duration modelling, model system and model confidence levels.
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)
Pr ob ab ili ty X Pr ob ab ili ty Pr ob ab ili ty Pr ob ab ili ty Beta CDF (0.5, 0.5) Beta CDF (0.5, 2) Beta CDF (2, 0.5) X Beta CDF (2, 2) X X
Presented approach has to be further developed by more comprehensive experimental evaluations, examples of applications, analytical models relating selected simulation responses with model para-meters.
Discrete event models allow inclusion of indi-vidual 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.
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