Maritime University of Szczecin
Akademia Morska w Szczecinie
2010, 21(93) pp. 62–66 2010, 21(93) s. 62–66
Investigation of vessel traffic processes at waterway
intersections
Badanie procesów ruchu na skrzyżowaniach torów wodnych
Piotr Majzner, Wojciech Piszczek
Maritime University of Szczecin, Faculty of Navigation, Institute of Marine Navigation Akademia Morska w Szczecinie, Wydział Nawigacyjny, Instytut Nawigacji Morskiej 70-500 Szczecin, ul. Wały Chrobrego 1-2, e-mail: p.majzner@am.szczecin.pl
Key words: fairways, intersection, traffic capacities, traffic flow Abstract
These authors present operational and safety aspects of research problems relating to waterway intersections and their safety and operational aspects. Attention is drawn to delay times that occur in the process of vessel traffic and the probability of close quarter situations. The results of simulations are compared with analytical research. Examples are given of real values of both system and traffic process measures.
Słowa kluczowe: tory wodne, skrzyżowanie, natężenie ruchu, proces ruchu Abstrakt
W artykule przedstawiono problemy badawcze występujące na skrzyżowaniu torów wodnych w aspekcie eksploatacyjnym i bezpieczeństwa. Zwrócono uwagę na czasy opóźnienia w procesie ruchu i prawdopodo-bieństwo nadmiernego zbliżenia. Wyniki badań symulacyjnych pokazano na tle badań analitycznych. Doko-nano egzemplifikacji w zakresie realnych wartości miar systemów i realnych wartości miar procesów ruchu.
Formulation of the problem
A waterway intersection where vessels move within water traffic engineering systems is one of the most dangerous elements of waterway infra-structure. Within an area of waterway intersection phenomena connected with dangerous encounters of vessels and traffic delays occur simultaneously.
Research problems of safety at an intersection are discussed in a number of publications [1, 2, 3, 4, 5], while delay time that occurs in the traffic process has been analyzed, inter alia, in [4].
While examining traffic processes at fairway in-tersections, the authors formulated the research problem and performed analytical research. Then a series of simulation tests was performed. Charac-teristics of traffic process were determined with a particular attention paid to the phenomena con-nected with:
times of delays that take place in the traffic process,
probabilities of avoiding potentially dangerous situations.
Figure 1 presents a graph of the traffic process at an intersection. The vessels proceeding along the fairway make up the longitudinal stream, while those sailing across the fairway are termed the cross stream. Vessels move in the longitudinal and cross streams at speeds of, respectively, vw and vp.
Analyzing traffic processes in real systems we can observe that usually some vessels have the right of way. It is assumed in the examined problem that vessels in the longitudinal stream have the right of way over those proceeding in the cross stream. The intensity of longitudinal traffic at the input of the intersection subsystem is denoted as λwwe(t), while its output denotation is λwwy(t). As longitudinal traf-fic vessels are privileged (stand-on vessels), λwwe(t) = λwwy(t). The cross vessel traffic entering the inter-section has the stream intensity λpwe(t), while the vessel stream leaving the intersection has the
intensity λpwy(t). Since vessels crossing the right-of- -way route cannot enter the intersection uncondi-tionally, having to give way to vessels that may be present in the longitudinal stream, the intensities
λpwe(t) and λpwy(t) are not always equal. As give-way vessels have to wait for a vessel or vessels in the longitudinal traffic to clear the intersection, there may be a number of waiting vessels No(t) in the cross stream.
Fig. 1. Graph of the traffic process at a waterway intersection (source: authors’ study)
Rys. 1. Schemat procesu ruchu na skrzyżowaniu dróg wodnych (źródło: badania własne)
Let us assume that a vessel in the cross traffic may enter the intersection, if the conjunction of the following conditions is satisfied:
the stern of vessel in the longitudinal stream is at least at a distance lr from the intersection point of the two routes;
a vessel in the cross traffic is able to leave the intersection before a vessel in the longitudinal stream reaches a distance lb before the route crossing;
a preceding vessel of the cross stream is at a minimum allowable distance lmin.
Vessels in the cross stream that may enter the in-tersection without delay form a stream with the intensity λbo(t). When a longitudinal stream vessel proceeding along the fairway is at a distance shorter than lb, or a cross stream vessel is already waiting, another cross stream vessel approaching the inter-section area stops and acquires a waiting status. Vessels changing their status to waiting make up a stream with intensity λdo(t). At the same time, vessels waiting start entering the intersection form-ing a stream with intensity λzo(t). There is still a number of vessels No(t) with the waiting status.
For the exemplification of research conducted in conditions as close to reality as possible, certain real values of traffic model and process parameters were assumed. These values were specified on the basis of the identification of actual traffic processes in restricted areas. In order to generalize the case examined, the number of vessels waiting to enter the intersection was not limited, although in prac-tice a quite significant limitation exists in ferry traffic.
The following assumptions were made for the investigation:
there is a rather uniform stream of vessels in the longitudinal fairway traffic;
longitudinal stream speed is constant and equals
vw = 8 knots (4.12 m/s);
vessels in the fairway do not exceed the mini-mum distance leb = 691 m (safe distance de-fined by experts);
length of vessel in the longitudinal stream is
Lsw = 159 m;
the period of time when vessels appear in the longitudinal stream is described with the expo-nential distribution with a mean value Twe and shift Tp = 129 s (complying with the required safety criterion);
minimum allowable distance at which a cross stream vessel may leave the intersection being ahead of a longitudinal stream vessel equals
lb = 1000 m;
minimum allowable distance at which a cross stream vessel may enter the intersection astern of a longitudinal stream vessel equals lr = 0.5Lsw = 79.5 m;
the cross traffic in the fairway consists of a uni-form stream of give-way vessels with a length
Lsp = 49 m;
cross traffic vessels proceed at a constant speed
vp = 5 knots (2.57 m/s);
cross traffic vessels have to move at a distance between each other of not less than lmin = 200 m;
period at which vessels join the cross traffic is described by the normal distribution with the standard deviation = 0 and mean value Twep equal to:
a) Twep = 3600 s (equivalent to pwe = 1 [1/h]), b) Twep = 1800 s (equivalent to pwe = 2 [1/h]), c) Twep = 1200 s (equivalent to pwe = 3 [1/h]), d) Twep = 360 s (equivalent to pwe = 10 [1/h]); breadth of the longitudinal fairway, i.e. length of
the intersection section that a crossing vessel has to go across is lw = 200 m. λ wwe(t) λ wwy(t) intersection λ bo(t) waiting λ pwe(t) λ zo(t) λ do(t) No(t) λ pwy(t)
Analytical research
In the work [4] the author built a model of wa-terway intersection traffic. Elements of probability calculus were used in the analysis and construction of some measures. It is a simplification to assume a uniform distribution only as the stochastic model, because the consequent results converge to reality only for rather deeply subcritical states and for such states exclusively the model can be applied. It should be noted that deeply subcritical states of system operation occur quite frequently. In Poland, for instance, these states occur in fairways of low traffic intensity.
One of the major assumptions of this method is the description of vessel traffic stream by means of kinematic equations and an observation that a ves-sel in the fairway occupies its segment (domain) determined by the vessel’s size and by the distance ahead and astern of it that should be clear of other vessels. This observation can also be expressed using the time dimension. Then it is considered that a vessel occupies the fairway (intersection) for some period of time (occupancy time) that can be referred to the mean time of vessels presence in the traffic stream. When a vessel enters the fairway (intersection) and occupies some time there, it faces a certain situation (i.e. period when vessels are present within the intersection). Should a vessel occupancy time overlap with another – which is not allowed – the entering vessel has to wait.
The range of model applicability is restricted by the value of privileged stream intensity for which the subordinate stream is still possible. Let this value be called the boundary value, expressed by this formula:
b r sw
w
w sp
p p w wg v l l L v l L v v [l/h] (1)The measure that matters to navigators conduct-ing cross traffic vessels is the mean delay time top of one vessel. The mean delay time of a vessel pro-ceeding in the subordinate stream has a formula developed from the said model:
wwe w p wwe z op T t t T T t 2 ) ( 2 2 2 [s] (2) where: Tz – occupancy time of vessels from the cross and longitudinal streams [s], tp – occupancy time of cross stream vessels [s], tw – occupancy time of longitudinal stream vessels [s], wwe – mean time [s] during which longitudinal stream vessels arrive at the intersection; note that this formula is applicable when Twwe > tw.
The mean delay time top according to the above model is the function of the period of time during which stand-on vessels arrive at the intersection, that is the function of longitudinal stream input intensity.
Another measure important for the description of phenomena at an waterway intersection is the probability of avoiding a close quarter situation –
pupnz. The premise for a close quarters situation is a situation when a crossing vessel enters the inter-section at the minimum allowable distance from a vessel proceeding along the privileged track. In the discussed model the relationship defining
pupnz is presented as a ratio of longitudinal traffic break time available for the cross traffic to the time of the subsequent appearance of the longitudinal stream vessels: p sp t w sw r b wwe wwe p w upnz v L l v L l l T t t p 1 1 (3) Simulation tests
Simulation experiments meeting the above con-ditions were performed. Figure 2 presents a graph of mean delay time top falling on a vessel of the cross stream as a function of longitudinal stream input intensity λwwe (for three values λpwe). The chart additionally shows the function top = f (λwwe) based on analytical relationships.
The graphs allow to assess the degree of con-vergence of the results obtained by computer-based simulations and analytical method. Assuming the criterion of 10% technical tolerance, we can state that the results of the two methods are convergent when the stream intensity wwe does not exceed approx. 30% of the limit value wg, i.e. the value at which the cross traffic is still possible according to the analytical model. The graph can be used for the optimization of relations between the longitudinal and cross traffic intensities. Starting from the con-dition of the acceptable mean delay time of a cross traffic vessel one can read out the value of the allowable longitudinal stream intensity at which that condition will still be satisfied. In practice many water engineering systems (including the Szczecin – Świnoujście fairway) operate at traffic intensity level far below its maximum capacity. One can prove the thesis that sometimes such inten-sity will be justified from the economical point of view.
Fig. 3. Graph of the probability of avoiding a premise for a close quarters situation pupnz as the function of longitudinal stream input
intensity λwwe [vessel/h] [7]
Rys. 3. Wykres prawdopodobieństwa ominięcia miejsca zbliżenia pupnz jako funkcja zmian w intensywności strumienia wzdłużnego
λwwe [statek/h] [7]
0 1 2 4 6 8 10 12 14
Longitudinal stream input intensitywwe [1/h]
0,0 0,2 0,4 0,6 0,8 0,9 1,0 p ro b a b ili ty o f a vo id in g a p re m ise f o r a cl o se q u a rt e rs si tu a tio n pupnz a s th e fu n ct io n o f lo n g itu d in a l st re a m in p u t in te n si ty pwe2jdn/h] pwe 10[jdn/h] confidence level 0.95 Piszczek model
Fig. 2. Graph of the mean delay time top [s] of a vessel in the subordinate stream as the function of longitudinal stream input
intensityλwwe [unit/h] [6]
Rys. 2. Wykres średniego opóźnienia statku top [s] w podrzędnym strumieniu jako funkcja zmian w instensywności strumienia podłużnego λwwe [jdn/h] [6]
0 1 2 3 4 5 6 7 8 9
Longitudinal stream input intensitywwe [1/h]
0 100 200 300 400 500 600 700 800 900 M e a n d e la y tim e o f a ve sse l i n th e su b o rd in a te st re a m top [s /jd n] pwe1[jdn/h] pwe2[jdn/h] pwe 3[jdn/h]
S. Olszamow ski model W. Piszczek model
Longitudinal stream input intensity wwe [1/h]
M ea n de lay time o f a ve ss el in th e su bo rd in ate strea m top [s/ un it ] pwe = 1 [unit/h] pwe = 2 [unit/h] pwe = 3 [unit/h] S. Olszewski model W. Piszczek model 0 1 2 3 4 5 6 7 8 9 900 800 700 600 500 400 300 200 100 0 P ro ba bil ity o f av oid in g a pre m ise fo r a clo se q ua rters situ ati on pupnz as th e fu nc tio n of lo ng itu din al str ea m in pu t in ten si ty
Longitudinal stream input intensity wwe [1/h]
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 pwe = 2 [unit/h] pwe = 10 [unit/h] confidence level 0.95 W. Piszczek model 1.0 0.8 0.6 0.4 0.2 0.0
Figure 3 illustrates the probability of avoiding a premise for a close quarters situation pupnz as the function of the longitudinal stream input intensity
wwe.
The relationship pupnz = f (λwwe) of simulation models shows a visible correlation with the analyti-cal model. One can read from the graph, inter alia, the value of the intensity wwe, for a required level of safety. For instance, to assure a 90% safety level, the longitudinal traffic intensity should be restricted to slightly above 10% of the defined limit intensity. An increase of wwe approaching wg will cause a number of undesired phenomena, such as:
significant drop of the safety level (pupnz); stronger dependence of the safety level (pupnz)
on pwe;
difficulties in the longitudinal traffic when indi-vidual vessels move at various speeds. Solutions to this problem should engage traffic control areas (e.g. VTS).
Conclusions
The results of analytical and simulation research lead to the following conclusions:
convergence of the results obtained from the two models occurs only for deeply subcritical states of operation of the WTE (water traffic engineer-ing) systems (wwe wg);
taking into account economical factors connected with the construction and operational use of the two models, the analytical model has advantages over the other in deeply subcritical conditions of the WTE system operation (to 30% wg);
for wwe > 30% wg the simulation model should be used;
for near critical conditions it seems necessary to extend the simulation model with a module connected with phenomena of strong autocorre-lation of longitudinal stream vessels and with a module of traffic control (e.g. VTS);
the models herein discussed may be used auto-nomously or as components of complex optimi-zation models for seeking solutions to problems relating to water traffic engineering.
References
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