535
1 INTRODUCTION
In the maritime traffic engineering, traffic capacity
refers to the capacity of a channel to manage vessel,
which is measured by the maximum number of
vessels passing through in a certain time [1]. At
present, the formulas for calculating the passage
capacity mainly include the West German formula,
the Polish formula, the Yangtze River formula and the
Changjiang formula. Its common characteristic is that
a series of parameters need to be analyzed and
determined according to the actual situation and data
of the channel. The value of parameters varies from
person to person and is highly subjective, which leads
to the non-standard calculation of channel passing
capacity. Moreover, existing studies on channel
passing capacity often take traffic flow velocity and
vessel density as fixed values without considering
their mutual influence, and lack of further study on
their internal traffic characteristics and mechanism
[2][3].
The study of traffic flow theory shows that flow
rate, velocity and density are a kind of dynamic
equilibrium relation vessel. In recent years, some
scholars have been aware of the deficiencies in the
previous studies on channel passage ability, and have
begun to make a preliminary discussion on channel
passage ability using the macroscopic traffic flow
theory. Shao Changfeng made a preliminary dynamic
exploration and analysis of vessel traffic flow by
applying fluid model [4]. Using the research method
of highway traffic for reference, He Liangde and Zhu
Jun established the direct functional relation vessel
between vessel density and vessel speed by using the
following theory, and strengthened the analysis of
vessel traffic mechanism [5][6]. However, the above
researches are limited to the analysis of a single vessel
type in the waterway. In practice, due to the different
sizes and types of vessels in the channel, the sequence
composition of vessel following in the mixed vessel
flow is also random, resulting in different vessel
spacing, which has a great impact on the channel
passage capacity. Therefore, it is significant to study
the passage capacity in the case of mixed traffic flow.
Research on Capacity of Mixed Vessels Traffic Flow
Based on Vessel-Following Theory
C. Zhao
1
, H. Yan
1
, G. Zhou
2
& T. Liu
3
1 Merchant Marine College, Shanghai Maritime University, Shanghai, China
2 Wusong Maritime Administration, Shanghai, China
3 Communication and Transportation College, Shanghai Maritime University, Shanghai, China
ABSTRACT: In order to study the characteristics of mixed vessel traffic flow, based on classical head distance
model and probability analysis, by studying the combination time head way of different vessel-following
sequences, the capacity model of mixed vessels traffic flow was established. Through analyzing two
representative types of vessels, research results indicate that the capacity of mixed traffic increase with the
traffic flow speed in a certain speed range, but the increasing trend slow down. The closer length and inertial
stopping distance of different kind vessels are, the more capacity of mixed traffic increases. And the influence of
reaction time on the capacity is related to proportion of different kind vessels.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 16
Number 3
September 2022
DOI: 10.12716/1001.16.03.16
536
2 THEORETICAL MODEL OF VESSEL
FOLLOWING
The car following theory is a dynamic study model of
the following vehicle's corresponding behavior caused
by the change of the leading vehicle's motion. The
characteristics of single-lane traffic flow are
understood by analyzing each vehicle following each
other so as to connect the microscopic behavior of
vehicles with the macroscopic phenomenon of traffic
flow. This model has been widely applied in micro-
traffic simulation, self-cruise control, capability
analysis, traffic safety evaluation and other fields [7].
2.1 Fundamental assumption
The vessel following model in the channel is based on
the following hypothesis: a group of vessels navigate
one by one, the officer only responds to the action of
the front vessel. The movement of the response is
accelerating or decelerating, and there is no
overtaking. Channel width and lateral interference are
ignored.
In addition, from the perspective of safety, the
following vessel should meet two requirements.
Firstly, the speed of the behind vessel should fluctuate
around the speed of the front vessel, and not greater
than that of the front vessel for a long time. Secondly,
a safe distance must be kept between the front and
behind vessel to make sure that there is enough time
for the behind vessel officer take action to responding
[8].
There are two means for vessel to brake, stop
engine and reverse engine. Generally speaking, the
reverse engine will cause the vessel's bow to turn
uncontrollably under the influence of the deep
transverse force, the discharge transverse force, the
wind and current force, the shallow water effect or the
bank effect when the vessel navigating on the channel.
In addition, emergency reverse engine will cause the
main engine rotating parts stress too much. Therefor
vessels usually braked by stop engine.
2.2 Prow time - distance model based on stop engine
The prow time - distance model that applied to
analyze the following up of vessels as shown in figure
[9][10]. n is the front vessel, n+1 is the behind vessel
xn(t) and xn+1(t) is the position of two vessels. tm is the
responding time, which include the officer responding
and action time and the engine receiving command
time. d1 is the distance that vessel navigating with the
original speed in a certain time tm. d2 is stop stroke of
the behind vessel, d3 is stop stroke of the front vessel,
ε is the lee-way factor, m0 is the safety margin after the
two vessels stop [11].
From the figure 1, To ensure the safety of the two
vessels after stop engine, it should meet the
requirement:
1 2 3 0
d d d m
+−
(1)
In critical condition：
( ) ( )
1 1 2 0 3nn
x t x t d d m d
+
− = + + −
(2)
10
d vt=
(v is the speed before decelerate) (3)
So:
( ) ( )
1 0 2 3n n m
x t x t t v m d d
+
− = + + −
(4)
Corresponding distance of the bow is:
( ) ( )
1
/
t n n
h x t x t v
+
=−
(5)
The stopping position of n
m
0
:Safe distance
from ship's stop
N +1 stopping distance
The distance
traveled by the n+1
ship in T0 time
x
n-1
(t)
s(t)
x
n
(t)
d
3
: n stopping distance
The position where
the N begins to slow
down
n+1
n
n+1
n+1
n
d
2
d
1
Figure 1. The follow theoretical analysis diagram
3 MIXED TRAFFIC FLOW CAPACITY MODEL
3.1 Mixed traffic flow capacity calculation model
Let hi,j is the bow distance of the combined two
vessels. i is the front vessel, j is the behind vessel,
i,j=1,2……r. Because the vessel type of two adjacent
vessels in the traffic flow is random, the probability of
the front vessel and the behind vessel j is pipj, and:
( )
2
12
1
1
rr
i j r
ij
p p p p p
=
= + + =
(6)
Take the combination bow time interval ht to
represents the average minimum bow time interval
under different tracking sequences of the mixed traffic
flow consisting of r vessel types, According to the
theory of probability:
,
11
rr
t i j i j
ij
h p p h
==
=
(7)
So, it can be seen that in the calculation of mixed
traffic passing capacity, it is very important to solve
the interval between different vessel types.
3.2 Bow time interval analysis of mixed traffic following
Assume that the vessel traffic flow is composed of a
mixture of large and small vessel types, the length of
little vessel is l, the proportion is p, braking reaction
time is t. The length of large vessel is L, the proportion
is 1-p, braking reaction time is T. Same speed before
emergency braking. The braking distance of little
vessel is d(v), and that of large vessel is D(v), the
difference of them is S(v). If there is a large vessel of
537
the combination, the safety margin is M, otherwise,
the safety margin is m. The safe distance between two
vessels at rest is often expressed as a multiple of the
length. Referring to the anchor distance, M=3L, m=3l
[12]. The corresponding situation as follows:
1. Both of them are large vessel.
( ) ( ) ( ) ( )
1nn
x t x t Tv M D v D v
+
− = + + −
(8)
( ) ( )
( )
2,2 1
//
nn
h x t x t v T M v
+
= − = +
(9)
2. Both of them are little vessel.
( ) ( ) ( ) ( )
1nn
x t x t tv m d v d v
+
− = + + −
(10)
( ) ( )
( )
1,1 1
//
nn
h x t x t v T m v
+
= − = +
(11)
3. A small vessel in front and a large vessel behind.
( ) ( ) ( ) ( )
1nn
x t x t Tv M D v d v
+
− = + + −
(12)
( ) ( ) ( )
1,2 1
( ) / / /
nn
h x t x t v T M v s v v
+
= − = + +
(13)
4. A small vessel in front and a little vessel behind.
Since the large vessel has a larger inertia, it is
assumed that when the large vessel is braking, the
little vessel will be reducing with a same
accelerated speed that belong to the large vessel.
( ) ( ) ( ) ( )
1nn
x t x t tv M D v D v
+
− = + + −
(14)
( ) ( )
( )
2,1 1
/
nn
h x t x t t M v
+
= − = +
(15)
The average bow time interval of the traffic flow is:
( ) ( )
22
1,1 2,2 1,2 2,1
(1 ) 1 1h p h p h p p h p p h= + − + − + −
(16)
( )
( )
( ) ( )
22
24 3600 / 1 3 1 1 ]C v T p tp v p L p l p p s v
= − + + − + + −
(17)
4 MODEL PARAMETER ANALYSIS
The parameters in the above equation can be divided
into two categories according to their respective
ownership relationship. The one is macro traffic flow
characteristics, v and p. The other one is vessel
characteristics, tm(t,T), lm(l,L), S(v). From the point of
view of water traffic control management, the first
type of parameters is more important, which will be
discussed in detail below.
4.1 The relationship of C-p
To confirm the influence of the first type parameters,
we should first confirm the value of the second type
parameters. According to China's "Design Code of
General Layout for Sea Ports" and the actual situation
of small and medium-sized coastal ports in China,
GT3000 with a total length of 96 meters and GT10,000
with a total length of 135 meters are taken as the
representative ship type of bulk carrier [13]. The
response time of little vessel is 10 seconds, and that of
large vessel is 20 seconds [14]. ε=1. For ships, the
speed of 5kn is generally the minimum speed to
maintain rudder effect, and the ship braking distance
is mainly determined based on this. According to the
latest research results of the China transportation
planning and design institute on braking distance (see
table 1), when the speed before parking is 4,6,8,10kn,
the ship's stopping braking distance is 2 lm(length),5
lm,9 lm,16 lm [15].
When the value of p is 0.3-0.9, we can obtain the
relationship of C-p-v. From Fig.2, we can see that as p
goes up, C goes up faster and faster. That is to say,
under the same vessel traffic flow speed, the larger
the proportion of small vessels, the greater the
channel traffic capacity.
Figure 2. The relationship of C-p-v
Table 1. Stop engine stroke
________________________________________________
Speed before stop engine 4~6kn 6~8kn 8~10kn
________________________________________________
Straight-line distance (Majority 2~5lm 4~7lm 5~15lm
concentration)
Mean straight-line distance 5.3lm 9lm 14.1lm
Actual distance (Majority 2~6lm 4~10lm 7~17lm
concentration)
Mean straight-line distance 5.9lm 10lm 15.8lm
________________________________________________
4.2 The relationship of C-v
From the Fig. 2, we can see that when p is constant, if
v is smaller, C is also smaller. With the increase of v, C
first grows rapidly and then slows down, and the
smaller the probability is, the larger the growth rate
decreases.
If the ship's stop engine brake is regarded as
uniform deceleration motion, acc according to,
( ) ( ) ( ) ( )
2 2 2
/ 2 / 2 / 2s v D v d v v A v a v a A aA= − = − = −
(18)
a is the acceleration of smaller vessel, A is the
acceleration of larger vessel.
538
Deriving from A, When dc/dv=0,
( )
( )( )
2 2 2
2 1 / 1
max
v aA p M p m p p a A
= − + − −
(19)
From this, we can see that C is not increased by the
increase of v, when, v=vmax, C is maximum. And from
(19), vmax has nothing to do with the two vessels’
reaction time T and t.
Under normal circumstances, the average speed of
small and medium-sized vessels in channel is 8kn, the
stop stroke: m0(v=8kn) =9lm.
The average acceleration of ship stop brake,
a=9.3×10
-3
m/s
2
, A=6.6×10
-3
m/s
2
,
As it is difficult for the traffic flow in the channel
to reach this speed, in general, with the increase of the
ship traffic flow speed, it gradually increases, but the
increase trend gradually slows down.
4.3 The relationship of C-lm, s(v) and t
m
As for the influence of the second type of parameters,
it can be seen from the 2.2 conclusion formula that
with the increase of reaction time, ship length and
stopping stroke, the passage capacity of mixed traffic
decreases gradually. Then, the influence of the
similarity degree of different types of ships on the
passage capacity is analyzed.
Assuming the vessel traffic flow speed is 8kn, the
original value in 3.1 is taken as the intermediate value
to gradually expand and reduce the vessel type
parameter interval. The results are shown as follows.
Figure 3. The relationship of C-lm
Figure 4. The relationship of C-S(v)
Figure 5. The relationship of C-tm
It can be seen from Fig. 4 and 5 that, with the
narrowing of the value interval of the two ship types'
length and stop stroke, the channel traffic passing
capacity gradually increased. In other words, the
more similar the length and stop stroke of different
ship types in the channel, the greater the passing
capacity.
It can be seen from Fig. 5 that, the effect of
response time of different types of ships on passing
capacity is related to probability (p). If p<0.5，with
the decrease of the reaction time interval between the
two types, the passing capacity shows a trend of
gradual increase. It is reversed when p>0.5.
5 CONCLUSION
The above studies show that the mixed traffic flow
passing capacity in channel is not only related to the
vessel traffic flow speed and vessel type combination,
but also related to the reaction time, vessel length and
vessel stopping performance. Within a certain range,
C is increasing with the increase of v, but the trend of
increase gradually slows down. The closer the ship
length and stopping stroke of different ship types are,
the greater the capacity of mixed traffic to passing. At
the same time, the influence of response time of
different types of ships on the passing capacity (C) is
related to probability (p), and the variation trend of
the passing capacity (C) of mixed traffic is different
with different probability (p).
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