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

, H. Yan

, G. Zhou

& T. Liu

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)

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:

( ) ( )

t n n

h x t x t v



=−



(5)

The stopping position of n

:Safe distance

from ship's stop

N +1 stopping distance

The distance

traveled by the n+1

ship in T0 time

n-1

(t)

s(t)

(t)

: n stopping distance

The position where

the N begins to slow

down

n+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:

( )

i j r

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:

t i j i j

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

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

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

( ) / / /

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

h x t x t t M v



= − = +

(15)

The average bow time interval of the traffic flow is:

( ) ( )

1,1 2,2 1,2 2,1

(1 ) 1 1h p h p h p p h p p h=  + −  + −  + − 

(16)

( )

( ) ( )

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

, A=6.6×10

-3

m/s

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

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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