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1 INTRODUCTION
In recent decades, there has been a significant increase
in container transportation worldwide. This fact is
objective, since in addition to the growth of
production in the world industry, the number of
vessels is also rapidly increasing. A good
confirmation is statistical data on the growth of the
sea vessels quantity. According to [10] during last 15
years a number of sea vessels on the planet fleet
enlarged about 53% with growing of their gross by
47%.
By using a container the cargo can be stored in a
standard steel box during transport without opening.
Standardization leads to flexibility, low transport
costs and rapid transshipment, particularly when the
cargo is moved over long distances. Thanks to these
advantages, containers are widely used in a global
freight transport, which consists of an extremely large
and complex structure of distribution systems and
business activity. In these systems, a container is
typically intermodally transported from an origin to a
destination, where two or more transport units (e.g.
ships, barges, trains and trucks) are used in sequence.
Manufacturers, forwarders, shipping companies,
terminal operators and customers are involved in the
process of container cargo handling. All of them form
a large supply chain [10] and this technological chain
is constantly changing and mostly in almost all
seaports on the planet it is caused by varying
operational and safety standards.
Port terminals, as transport hubs, play an
important role in the container transport network.
They play main role in the vessel’s interaction with
various types of transport. Transshipment of
containers from one type of transport to another is
carried out at intermodal container terminals.
New Approach In Models for Managing the Vessel
Unloading Process
L.L. Nikolaieva, T.Y. Omelchenko & O.V. Haichenia
National University “Odesa Maritime Academy”, Odesa, Ukraine
ABSTRACT: New way of formalizing hybrid systems in models for managing the process of vessel unloading is
caused by the significant increase in container transportation around the world over the past decade. The
growth of container traffic is the main reason for the constant modernization of port container terminals and the
improvement of cargo unloading technology is most promising when using those methods, that allow obtaining
maximum results in terms of cargo processing speed. This problem is the subject of an article in which an
analytical review of container handling technologies existing in world ports is carried out, their key
performance indicators are formulated and it is shown how the use of a centralized hybrid control system based
on the dynamics of discrete events can lead to increased profitability of the port. Developed concept of a hybrid
control system makes possible to consider such features of the vessels unload process that have not been
considered until now.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 18
Number 4
December 2024
DOI: 10.12716/1001.18.04.11
848
An analysis of the current level of theoretical
statements and methodological principles related to
intermodal transportation, in particular, optimization
of the management of automated processes of
container ships cargo operations, shows that they
have remained understudied. The same applies to the
substantiation of trends in the use of automated
technical equipment, as well as the analysis of
conceptual approaches to improving the efficiency of
cargo handling of container ships.
It should be noted that many problems from the
list of port operation risks are still out of port's
control. Correct analysis of connection between these
risks and the reasons that caused them can lead to
creation of radically new models of managing the
process of ship unloading.
2 MATERIALS AND METHODS
2.1 Modern trends in the use of container ship processing
technologies
2.1.1 Trends in the development of automation of vessel’s
cargo handling processes
The limitation of the existing theoretical and
methodological developments on the above-
mentioned problems increases the relevance and
significance of conducting new research in the field of
forming methods for optimizing the management of
processes of container ships cargo operations.
A very important question is the generalization of
theoretical knowledge about container ships cargo
handling processes and the development of practical
recommendations for their application in the
management of automated technologies. Practically
all production problems in this case can be reduced to
the best level of solution. The subsequent
implementation of such a solution can be used in the
work of any port if there would be realized the
following operations: to justify the development
trends of the automation of cargo operations
processes; to analyze new concepts for increasing the
efficiency of cargo handling; to determine the
methodological basis for key performance indicators
evaluating; develop proposals for optimal
management of container ships cargo handling
processes; to evaluate general indicators of the
investing feasibility in the automation of cargo
terminals.
In fact, it can be stated that new theoretical and
methodological foundations of managing the
automated processes of container ships cargo
operations can give the highest indicators of the
profitability of port operations. During research
works, it was stated that the main and, at the same
time, the most promising trends in the development
of the work of port terminals consist in the realization
of the following directions:
− development of methodological bases for
evaluating the efficiency of processes of container
ships cargo operations at automated terminals
using analytical methods and considering the
dynamics of continuous time and discrete events;
− creation of a new principle of optimizing the
management of cargo handling processes of
container ships, expressed in the ratio between the
time of cargo operations and the energy efficiency
of automated equipment;
− the development of a technological system that
uses new criteria for evaluating indicators of
improving the efficiency of ship cargo handling in
the port;
− creation of new methods for assessing the
feasibility of investing in the automation of cargo
terminals.
2.1.2 Influence of automation on the quality of port
operations
During the research, there was used classification
of automatic equipment, which was proposed in [12].
In accordance with it, processing a vessel in the port
involves the use of only three components. The first
refers to those technological operations that occur on a
vessel in the port while it is berthed. The second refers
to those technological operations that are associated
with storing cargo at the port berth. The third relates
to the transport of cargo from the berth or to local
areas of the port or directly outside the port.
In a port, when carrying out cargo operations with
a ship that is berthed, three types of automated
equipment are usually used: remote quay cranes
(QC), automated cargo vehicles (AGV) and stacker
cranes (ASC). This equipment is energy-intensive and
an increase in overload abilities will always lead to a
significant increase in the energy consumed.
However, in accordance with the data from [13], the
fact of increasing energy consumption is an incentive
for the creation of new types of more autonomous
equipment.
Nowadays, more and more new technical
developments are appearing at port terminals. A good
example in this case is a GPS-based AGV. This type of
AGV ensures free behavior and significantly speeds
up travel along the standard path, which is fixed or
governed using various methods (wires, lasers, AI
optical vision, etc.), but it is necessary to note that
freedom in AGV’s behavior enlarges difficulties for
managing terminal operations. From one side,
preventing of two AGVs collision should be
considered from the safety point of view. On the other
hand, AGV interacts with other types of machines that
are used during shipboard operations (loading or
unloading).
Analysis of statistical data allows to conclude that
work in the port was and is extremely dangerous.
According to Pacific Maritime Association (PMA)
statistics, the injury rate by 1960 was three to four
injuries per full-time employee each year. This grim
picture has changed dramatically as continuity-
enhancing operations have been abandoned in favor
of containerization. By the end of 1970, the number of
injured people was approximately 15 per 100 full-time
workers, which is 95% lower than the level that
existed before containerization [5].
In the analytical article [14], the trends of
increasing the level of security after containerization
are revealed in detail and the possibilities of obtaining
additional advantages in the field of security by
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increasing automation and new algorithms for
managing ship cargo handling systems are studied.
Work [6] investigated the cases of injuries in the
ports of the West Coast of the United States (United
States West Coast, USWC) and predicted that the
frequency of injuries will steadily decrease over time.
Thus, since 1997, the frequency has decreased to 1
annual injury per 10 full-time workers (less than one-
thirtieth of the frequency in 1950). This was primarily
due to the improvement of the quality of safety
equipment and training, which allows consider past
accidents and change the behavior accordingly. In
subsequent years, the decrease in the number of
injuries is due to the use of more automated
equipment.
Statistics from the last two years continue to
indicate a very big importance of security issues
within the area of port or port terminals. In 2022, 50%
of maritime accidents took place in ports and port
terminals and as one can see in Figure 1 [3] main part
of these accidents answers to: berth, port using
facilities, during port and harbor transit. RightShip
data [3] shows that 818 accidents in 2022 has
happened in ports.
Figure 1. Number of maritime incidents by vessel location in
2022 [3]
From our point of view, the use of new cargo
handling management systems in seaports will lead to
an even greater reduction in injury rates. In general,
this will happen due to the use of non-standard
solutions when using control algorithms, since hybrid
systems allow to consider such features of the cargo
unloading process from a ship, which have not been
considered until now. One of these indicators can be
the velocity of movement of all system elements
depending on such external factors as weather
conditions, current time of the day, number of
workers in the dangerous zone, etc.
2.1.3 Analysis of modern cargo handling systems and
technical support of automated container terminals
The standard list of works, which is performed at the
berth during vessel’s cargo unloading, consists in the
use of quay cranes (QC) at the first stage. At the
second stage the cargo is moved to the cargo area or
warehouse with the help of vehicles. These types of
transport are mainly: trucks (YT), container loaders
(SC) and automated guided vehicles (AGV). During
inland operations, containers are delivered to the gate
by road trucks and all their documents and damage
are checked.
Container cargo handling systems in the port
terminal are recognized as standard and divided into
two types [3]. The first type of system uses indirect
transmission and involves two different types of
lifting equipment: trucks and conveyors. Appropriate
warehouse cranes (Yard Crane, YC) are used to
handle containers in the stacking area. The second
type of system uses a tractor or SC. When SCs are
used as conveyors, the stacking height is lower than in
the first type of system. Since in almost all ports on
the planet the need for storage is high and storage
space is very often not enough, and the first type of
systems is usually used.
A very high-quality example of how the
automation of cargo equipment can significantly
increase the operational speed of port container
terminals is Figure 2, which shows an ordinary quay
crane [7]. Its automation makes it possible to increase
the number of containers per lift when other things
being equal.
Figure 2. Automatization of QC [7].
Figure 2-a shows a standard QC design with one
cart. In fig. 2-b QC already uses two trolleys: the first
moves over the berth and the second moves over the
vessel on the same track. The whole process can be
divided into two cycles: transferring the container to
and from the platform. Figure 2-c shows the
SupertainerTM developed by Paceco Corporation and
in its design it has two carts with a platform for
moving containers from one cart to another. As a
result, the QC cycle is divided into three segments,
which further reduces the time of each individual
cycle. The conceptual project in which two lifts are
installed between the carts is shown in fig. 2-d. They
are responsible for the vertical movement of
containers. There are also two small conveyors that
move containers on different levels. It is expected that
this crane can increase the loading capacity to 94
containers per hour, which is a significant increase in
productivity compared to the productivity achieved
by traditional QCs, which is 38 containers per hour
[7].
A review of the transmission system shows that
the most popular transport equipment is a conveyor
consisting of a truck and a frame. The automated
version of trucks is the Automated Guided Vehicle
(AGV). One of the disadvantages of using trucks or
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AGVs is the possibility that a crane or transporter
may have to wait an arrival of another type of
equipment to transfer the container. Another type of
transporter is a container handler (SC). They can lift
containers directly from the berth, eliminating the
need for a transfer operation.
When using an AGV to transport containers, the
operation of the crane can be separated from the
operation of the conveyor. This can be done by using
buffer stations that eliminate the need to transfer
containers between cranes and vehicles. Such stations
are racks - steel platforms that are separated from the
AGV and on which containers are located. With such
a technological solution, QC or AGV can leave
containers on the rack even before other equipment
reaches the transfer position.
In addition to improving the equipment of
transporters in almost all modern ports, many port
authorities and operational teams are making
significant efforts to improve the commercial sense to
attract vessels in the field of operational efficiency
optimization. This is mainly implemented in practice
by using high operational standards, improving
algorithms and rules for the compatible operation of
equipment, reducing inherent risks. For automated
conveyors, methods of effective dispatching, routing,
planning and traffic control are constantly being
developed and applied in practice.
Simultaneously with the need for storage space,
the height of the stacks also increases. In the early
days of container terminals, the stack height was 1-3
tiers. When SC spread, the stack height was 2-3 tiers,
and after YS spread, the stack height was 4-6 tiers [17].
As the stack grows, more attention should be paid to
reprocessing operations. Various algorithms and rules
have been developed to minimize these operations.
An analysis of the transshipment capacities of all
ports on the planet allows us to draw an unequivocal
conclusion - the tendency to their growth will be
observed until the size of transport vessels increases.
2.1.4 Analysis of technologies for increasing the efficiency
of the process of container ships cargo handling
The efficiency of the process of container ships cargo
handling can be evaluated by using a set of indicators
that have specific numerical ranges. They should
change constantly to a greater extent and, depending
on the type of equipment, which is used at the
terminal, can be divided into three categories:
− for cranes: cycle time of processing one container;
ability for multi lifting; crane deployment density;
− for transporters: time necessary to transport one
container; carrying capacity;
− for warehousing equipment: maximal density of
containers storage with the maximum permissible
amount of overloading; number of cranes set to
increase throughput with low interference.
In addition to these basic criteria, it is possible to
use additional criteria to evaluate the efficiency of
container ship cargo handling process. They should
be considered when developing new concepts and the
main ones are: flexibility, cost, environmental
protection, technological feasibility, reliability.
Regarding flexibility, it should be considered that
cargo handling system at the terminal should be
applied with minor changes, even if the container
flow pattern is different or the logistics environment
changes. Once the control system is in place, it should
be easily adaptable to ever-changing situation at the
container terminal. Main characteristics of flexibility
are the probability of application to different
situations with the smallest modifications and the rate
of adaptation to a changing situation.
Regarding the cost, there should be considered
such numerical results as: a decrease in the amount of
investment and a decrease in the cost of operations. It
should be considered that the cost indicator depicts
not only the cost of updating the technology itself,
which is used at the terminal in the port, but also the
cost of its operation in the feature.
Estimating the cost of environmental protection
requires the use of volumes: total energy consumption
and CO2 emissions.
Reliability indicators are mainly based on the
assessment of the possibility of maintenance in the
future and the recovery time to the working state
during technological failures or accidents [15].
Many conceptual systems are used to improve the
efficiency of container ships cargo handling process.
The most effective include: Linear Motor Conveyance
System (LMCS), Automated Storage and Retrieval
Systems (AS/RS), Overhead Grid Rail (GRAIL),
SPEEDPORT, SuperDock, ZPMC automated system,
Teustack [7, 9, 11].
The unique features of LMCS are that the platform
on which containers are stacked is used as a conveyor
and can move along the track on a fixed trajectory
with high positioning accuracy and high reliability.
LMCS is an environmentally friendly system because
it uses electricity instead of organic diesel fuel, which
is the main energy source for trucks and most AGVs.
The main disadvantage of LMCS is the high
investment cost for initial construction. LMCS is
limited also in the number of routes for platforms and
this means that its conveyor routing flexibility is
relatively low in comparison to truck-based or AGV-
based systems.
Two main components of an AS/RS are Storage
and Retrieval Machine (SRM) and storage racks.
AS/RS has the advantages of providing high-density
storage capacity, high throughput, and random access
to the target container without overloading
operations. It possible to create this system on a small
spot of port territory and then easily add storage
capacity by increasing the number of tiers. This is
useful when space is limited and expensive.
Main disadvantage of SRM is high cost of
construction and possibility of blocking in local space
(at the entrance at the lowest level of each aisle there
is an AS/RS station, which is located at one end) of all
further operations in case of SRM failure.
GRAIL uses electric shuttles as main equipment
elements, which are used for storage on the territory
of the terminal and for containers delivery between
the storage place and the wharf. They can move
between suspended tracks when moving over stacks
and transporting containers. This movement is
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ensured by the operation of switches and can take
place directly between the pier and the railway
station. The connection points between QC and
shuttles are an advanced automated platform under
QC, where the QC operation and shuttle management
are separated. QCs pick up or place containers
without attending to the entrance to the shuttle, and
they in turn drop off or pick up containers without
waiting for QC arrival. This operational algorithm
reduces equipment waiting time.
Specificity of GRAIL's operation is that the plane of
movement for the shuttles is completely in the air
space. This makes it possible to save on redundant
aisles and avoid obstacles with container stacks on the
ground or with the trajectory of trucks with drivers.
Main disadvantage of GRAIL is a complex
management system and high investment costs. This
system is an attractive solution for locations where
high urgent productivity is required but land for
container storage is limited and expensive.
The Speedport system is actually a modernized
analogue of the GRAIL system, as its track is extended
above the ship's hull for the possibility of its
operation. The main equipment is a spider, which
performs dual functions - a truck and a shuttle (which
is similar to the GRAIL shuttle). Each spider is self-
propelled and moves along a network of aerial beams
and ground tracks.
Speedport can reduce the time necessary to
transfer cargo from cranes to vehicles in a system that
uses traditional QC and YC with spiders. In this case,
they can perform the functions of both cranes and
trucks. The number of spiders that can work with one
ship is large and this contributes to a significant
increase in the overload capacity.
Disadvantage of Speedport is a very high cost of
building the structure and the spiders themselves.
Such a system has technical problems also. In most
cases, they are related to the lack of flexibility when
working with vessels of different sizes.
The SuperDock concept currently remains at the
stage of theoretical development, since the initial
investment costs for its implementation amount to
billions of US dollars. It was developed out of the
need for economically and environmentally beneficial
container terminals in North American ports.
Operation of SuperDock is based on the use of the
rail conveyor and universal stacking systems with a
long dock that uses many QCs. On the other hand,
SuperDock has a railway for trains. The use of
Superdock system should reduce air pollution and
noise.
In the ZPMC automated system, flat cars run on
rails on two levels: an overhead track installed
parallel to the QC track, and another laid on the
ground in a direction perpendicular to the
embankment. If necessary, more tiers can be added to
the upper track. After the QC places the container on
the flatbed car, it moves to a preset position to
transport the container to the Rail-Mounted Gantry
(RMG). At the beginning each RMG lifts the container,
changing its spatial orientation by 90 degrees. The
RMG container is then reloaded onto the lower AGV
and the container is transported to the next automated
RMG at the storage location.
Main advantages of ZPMC are: simplification of
control over vehicles compared to the AGV system;
increased reliability of the system and ease of support
compared to traditional automatic container
terminals; environmentally friendly system because it
uses only electrical energy sources.
Main problems of ZPMC are the great complexity
of planning the synchronous movement of all relevant
equipment; high construction costs; lack of high
operational flexibility in routing the movement of
flatbed cars.
The latest Teustack system is the most promising
and high-quality [4, 16]. This system is designed for
transportation and storage of all types of containers:
standard 20-foot and 40-foot containers, refrigerated
containers, containers of increased capacity. In
Teustack, as shown in Figure 3-a, after the moment
when containers are unloaded from the vessel, cranes
move them to specially designated receiving devices.
After that, platforms with containers are moved to the
first available storage space and come back to get new
containers. Inside the terminal, as shown in Figure 3-
b, there are shuttles. They are similar to rotary and
distribution platforms and provide horizontal
movement of containers on each floor.
Vertical movements in the system are provided by
lifting cranes distributed along the aisles. After
reaching the required level, containers are picked up
by shuttles and transported to the final destination.
All movements are performed simultaneously on
different levels. This allows the system to control
horizontal and vertical movements separately.
Containers stored in the terminal are freely accessible
and can be removed at any time without additional
movements. Vessel loading operations are carried out
similarly. Eight tiers allow storage of 6.4 thousand
TEU on 25000 sqr. m. At a normal terminal, 100,000
sqr. m is needed for storage of 6.4 thousand TEU,
which is 4 times more. Compared to standard ship
cargo handling systems, Teustack has 70% more
productivity, and also provides a sufficient level of
safety and reliability.
a) b)
Figure 3. Teustack. a - mooring crane transports container
to the Teustack platform; b - internal storage system
2.2 Technical support and equipment operational
characteristics of the biggest automated container
terminals
During the research, as automated terminals were
considered those terminals where at least one type of
equipment during its full operational cycle for
containers handling works without direct human
interaction. In most cases of container terminals
discussed below, operators are not physically
involved in the operation of the cranes, although
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sometimes they may be present in the cabins of the
equipment. Modern equipment of such container
terminals includes the following list: Automated
Stacking Crane; Rail Mounted Gantry; Rubber Tired
Gantry; Autostrad [8].
Automated Stacking Crane lift and transfer
containers one by one to their destination in the limits
of row. They are the current global standard for
automated container terminals and perform most of
the operating cycle autonomously without any
interaction with operators. If necessary, they can be
remotely controlled. Containers are delivered to them
by automatic carriers or other types of transport -
automated vehicles, container loaders, or human-
operated vehicles.
Rail Mounted Gantry can work parallel and
perpendicular to the wharf. The specific position of
RMG for containers processing is determined by the
density of their location. RMGs are used in many
terminals around the world, especially in Asia. They
are usually served by human-operated vehicles.
Rubber Tired Gantry are usually manually
operated by operators in cabs and serviced by vehicles
with drivers. The exception is Tobishima Container
Terminal, Nagoya, Japan. This terminal uses
unmanned RTGs and is serviced by automated
guided vehicles.
Autostrads require a lot of space to allow
movement and maneuverability. They do not have a
high stacking height because container loaders have
height restrictions. For this reason, the operation of
the Autostrad results in a very low density of
containers at the berth.
In the world most of the automated terminals, that
are under development, focus on the use of ASC. This
type of equipment is well compatible with various
types of automated transport [1].
Provision of automated equipment for 6 terminals
in Europe is given in Tables 1 and 2 [6]. Their analysis
shows that a stacking height of 6 tiers has become the
standard for automated cranes. Side-by-side ASCs
range in length from 36 to 59 total container spaces
(770 to 1,260 feet). Despite the fact that these
restrictions are not strict, in a short row of containers,
ASC is poorly implemented due to the high cost of the
equipment itself, and in a long row, the processing
time of containers increases significantly. The port
terminals, with the exception of the two in Hamburg,
use two identical ASCs on the same set of tracks.
The automated container handlers at both
Hamburg terminals (Altenwerder and Burchardkai)
have unique designs compared to other facilities
around the world. They use two different pairs of
tracks for the ASC, allowing the smaller loaders to
pass under the taller ones. Burchardkai Container
Terminal also has three ASCs, each with two smaller
ASCs, although this reduces the stacking height to 4
tiers, while all other terminals with ASCs can stack up
to 5 tiers.
Most modern terminals with ASC lay out
containers perpendicular to the berth. However, it is
increasingly possible to observe terminals where rows
of containers are arranged in parallel. The width of
the rows usually varies between 8 and 10 containers,
with the exception of 12 rows of containers at the CTA
and CTB terminals in Hamburg.
Main properties of automated equipment for
largest six terminals in Europe formulated in Tables 1
and 2.
2.3 Key Performance Indicators
In container terminals, performance can be evaluated
using a very large number of different indicators and
therefore it is always necessary to determine the main
ones - key performance indicators. The most
important indicator of the terminal efficiency is the
vessel’s service time. This indicator is connected with
other performance indicators that directly relate to the
terminal's transport processes. The main key
performance indicators are:
Table 1. Main technical characteristics of container terminal equipment
___________________________________________________________________________________________________
Main properties Name of ports and terminals
Hamburg Rotterdam Antwerp Algeciras Hamburg Norfolk
CTA Euromax DPW TTI CTB APMT
___________________________________________________________________________________________________
Height of ASC, TEU 4/5 5 5 5 4/5 5
Width ASC, TEU 10/12 10 9 8 10/12 8
Number of ASCs, pcs. 52 58 14 32 15 30
Type of motor vehicle AGV AGV SC Shuttle SC Shuttle
Length of the ASC, TGS row 37 36 41 45 45 59
General field ASC, TGS 9620 10440 7545 5760 17000 7080
Stack height, tiers 4 5 3 5 3 5
The highest use of the stack, % 75 75 75 75 75 75
Maximum stacking height 3.0 3.8 2.3 3.8 2.1 3.8
in tiers
Total capacity, TEU 28860 39150 17129 21600 35700 26550
Total terminal size, acres. 247 208 138 74 346 234
Expected annual throughput, 2,3 1,8 1,0 1,0 2,9 0,7
million TEU
Length of coastline, feet. 4590 4920 6100 3940 9350 3025
Number of port cranes, pcs. 15 16 9 8 25 6
Type of the port crane Double cart, Double cart, One cart One cart Double cart, One cart
automat. automat. automat.
___________________________________________________________________________________________________
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Table 2. Estimated operational parameters of container terminals
___________________________________________________________________________________________________
Operational parameters Name of ports and terminals
Hamburg Rotterdam Antwerp Algeciras Hamburg Norfolk
CTA Euromax DPW TTI CTB APMT
___________________________________________________________________________________________________
TGS on 1 acre 39 50 55 78 49 30
Static capacity of 1 acre 117 189 124 291 103 113
Berth length for 1 crane, m 306 308 678 493 374 504
Annual number of containers per 1 acre 9300 8700 7300 13500 8400 3000
Annual number of lifts per 1 crane 87000 63000 67000 75000 68000 67000
Annual number of lifts per foot of pier 280 200 100 150 180 130
Waiting time, days 4,6 7,9 6,3 7,9 4,5 13,8
___________________________________________________________________________________________________
1. Service time, hours It is defined as the time during
which the vessel is at the berth for the purpose of
loading or unloading. This indicator is defined as
the most important factor in the total transport cost
of containers because it directly reflects the
productivity of terminal operators.
2. Time of works completion, hours. This indicator
corresponds to the time of cargo operations
completion using that part of the terminal
equipment that is directly related to the time of
ship service.
3. Energy consumption, kWh. This indicator
corresponds to the total electricity that was used to
transport containers between ship and storage
location or vice versa.
4. Time for calculations, sec. This indicator
corresponds to the time period that was spent on
solving a specific optimization problem related to
container processing.
5. Average distance of AGV movement, m. This
indicator corresponds to the average distance AGV
moves between the point of transfer at the wharf
and the point of stacking in the terminal.
6. Relative distance of AGV, m. This indicator
corresponds to the distance between the two AGVs
used for transporting containers.
7. Operation of QC, %;
8. Operation of AGV, %;
9. Operation of ASC, %.
The last three indicators mean the average value of
the percentage of time during which the respective
equipment, i.e. QC, AGV and ASC, was used during
the vessel’s unloading or loading.
3 FORMALIZATION OF HYBRID SYSTEMS IN
MANAGEMENT MODELS WITH CONSIDERING
OF KEY PERFORMANCE INDICATORS
3.1 Dynamics of discrete events
In quayside operations of automated container
terminals, QC, AGV and ASC work together to load
or unload a vessel. When formalizing hybrid systems
in terminal management models, the simplest case is a
small container terminal with one QC, one AGV and
one ASC. During its operation, a distributed method
is always used to control the equipment.
The structure of distributed control is shown in
Figure 4, where one can see, that the interaction of
different parts of the equipment follows the dynamics
of a discrete event and the controller of each
equipment for loading and unloading containers. The
continuous dynamics of the object is controlled
locally.
Figure 4. Dynamics of discrete events in a distributed
control system
Energy efficiency is consistent with both load
capacity and energy consumption. Power output
depends on the dynamics of a discrete event, while
energy consumption is determined by continuous
time dynamics in which position and speed change
over time. During formalization, the following rule
should always be followed: in order to increase the
energy efficiency of operational control at container
terminals, the dynamics of discrete events and the
dynamics of continuous time should be considered
together.
It should be expected that at the operational level
energy efficiency will be achieved for real-time
operation. Unexpected operations (delays in work,
imprecise arrival time of new containers, etc.) can
change logistics processes of container transportation
in real time and ultimately affect the energy efficiency
of the container handling system.
For energy efficiency, a combination of discrete-
event dynamics and continuous-time dynamics, called
hybrid systems, can be smoothly modeled using
interconnected hybrid models.
Since the studied system, presented in fig. 4,
includes a combination of discrete event dynamics
and continuous time dynamics, it is possible to
represent the dynamics using the theory of hybrid
automata [18]. The general model was formulated as
( )
, , , , , , , , =H f S X U f Init Inv E G R
(1)
S ‒ final set of discrete operational modes; X ‒ final set
of continuous state variables; U ‒ final set of control
variables; f: S×X×U ‒ describes evolution of
continuous variables in a certain discrete mode of
operation; Init ‒ set of possible initial states; Inv: S
→
P(X) describes an invariant set that defines possible
regions of continuous variables in a certain discrete
854
mode of operation, where P(X) denotes the power set
(set of all subsets) of X; E: S×S ‒ set of boundaries
representing possible switches between discrete
modes of operation; G = G(sα, sβ): S
→
P(X, U)
limiter, which provides conditions for transition of the
operation discrete mode from s
α
to s
β
; R: E × X
→
P(X) ‒ limiter that resets continuous variables
between discrete mode switches.
In this case, sets of interconnected hybrid automata
are considered. Automata interact through
constraints: transitions between certain discrete
modes are possible only when delays containing
variables from several automata are fulfilled. For this,
it is necessary to expand the description of the general
hybrid automaton. A hybrid interconnected
automaton was described as
( )
, , , , , , , , , , =
inter inter
H f S X U f Init Inv E G R V G
(2)
V ‒ final set of variables of other hybrid automata;
G
inter
= G
inter
(s
α
, s
β
): S
→
P (X, U) ‒ connecting function
that includes variables from X, U and V.
In an interconnected hybrid automaton, the
discrete mode of operation S, the state variables X,
and the state variables V can cause the relationship of
G
inter
functions. G
inter
indicates a function in which
another interacting device is involved. After G
inter
is
activated, the discrete mode can be switched between
each other. By formulating the values of V and G
inter
,
the interaction between the two machines can be more
clearly represented. For example, an interoperable
function may represent the point at which a single
container can be transferred from an AGV to an ASC.
There is a difference between controlled and
uncontrolled components when a container is
transported from a ship's berth to a stack in a storage
area. QC, AGV and ASC are controlled components as
the actions of these equipment elements should be
determined by the control system. The vessel and
storage location are uncontrolled components because
they do not move when container is moved through
the terminal.
3.2 Modeling of controlled components.
QCs, AGVs, and ASCs can be considered controlled
components that transport a container between two
points: the location where the component collects or
accepts the container and the location where it loads
or offers the container. This is shown in Figure 5
where controlled object picks up one container at
position A and transports it from A to B where it
would be then unloaded. The dynamics of one
controlled object can be described as an
interconnected hybrid automaton, which is shown in
Fig. 6. The dashed line in Fig. 6 means that the
interaction between system elements depends on the
presence of another object.
Figure 5. General model of controlled component
The details of the controlled type hybrid
automaton were formulated as
( )
, , , , , , , , , , =
inter inter
с с с с с с с с с
H f S X U f Init Inv E G R V G
(3)
1 2 3 4 5
,
c c c c c c
S s s s s s
‒ discrete states of the system;
( ) ( )
( ) ( )
( )
,,
pos vel pos vel
c c c c c
X x k x k x k x k
‒ set
of continuous states: position
( )
pos
c
xk
,
( )
pos
c
xk
, m and
velocity
( )
vel
c
xk
, m/sec of the system component;
( )
cc
U u k
‒ set of control variables representing
the acceleration of the component, m/sec2; fc ‒
function which describes continuous time dynamics
in every discrete mode.
Figure 6. Hybrid automaton of the controlled object
The controlled component can be in one of five
discrete states Sc. In state
1
c
s
(waiting) ‒ the
controlled object is waiting for another interacting
component to drag the container. In state
2
c
s
(pickup) ‒ picks up the container at point A. In state
3
c
s
(transportation) ‒ moves the container from point
A to B. In state
4
c
s
(unload) ‒ unloads the container
at point B if another interacting component is
available for container unloading. In
5
c
s
(return)
mode, the object moves from B to A to take the
container back to A.
In the following we define ∆t as the sampling time
and
( ) ( ) ( )
=
T
pos vel
c c c
x k x k x k
. Then in state 1
(waiting), state 2 (pickup), and state 4 (unloading): the
component's position and velocity are unchanged.
Therefore, the continuous time dynamics with respect
to these three states
( ) ( )
( )
1
,
c c c
f x k u k
,
( ) ( )
( )
2
,
c c c
f x k u k
and
( ) ( )
( )
4
,
c c c
f x k u k
was
described as
( ) ( )
1+=
cc
x k x k
(4)
In state 3 (transition) and state 5 (return) a double
integrator can be considered for continuous time
dynamics. This was done without considering
resistance to air resistance and rolling resistance and
therefore discretized the continuous-time dynamics in
states 3 and 5, namely
( ) ( )
( )
3
,
c c c
f x k u k
and
( ) ( )
( )
5
,
c c c
f x k u k
, was written as
855
( ) ( ) ( )
2
1
0,5
1
01
+ = +
c c c
T
T
x k x k u k
T
(5)
For a given controlled component, the value of Invc
was formulated as
( )
( )
1
==
pos unload
c c c
Inv s x k x
, (6)
( )
( )
2
==
pos unload
c c c
Inv s x k x
, (7)
( )
( )
3
=
load pos unload
c c c c
Inv s x x k x
, (8)
( )
( )
4
==
pos load
c c c
Inv s x k x
, (9)
( )
( )
5
=
load pos unload
c c c c
Inv s x x k x
, (10)
load
c
x
та
unload
c
x
‒ positions for loading and unloading
containers; Ec ‒ defining as the set
( ) ( ) ( ) ( ) ( )
1 2 2 3 3 4 4 5 5 1
, , , , , , , , ,
c c c c c c c c c c
s s s s s s s s s s
; Gc ‒
interaction function of the controlled component;
( )
c
sk
‒ discrete state of the component at time k.
Interaction
( )
( ) ( )
1 2 1
, , = = =
load
c c c c c c
G s s s k s x k x
depends on the presence of another component to
pick up the container. This dependence is represented
by the dashed line in Fig. 6.
When
( )
( )
2 3 2
, ==
c c c c
G s s s k s
‒ the component
finishes pick up.
When
( )
( ) ( )
3 4 3
, , = = =
unload
c c c c c c
G s s s k s x k x
‒
component reaches the loading position and waits for
unloading.
When
( )
( )
4 5 4
, ==
c c c c
G s s s k s
‒ component
completes unloading.
When
( )
( )
51
, ==
load
c c c c
G s s x k x
‒ component
reaches the loading position.
A continuous state does not change as a result of
switching discrete states. That is why,
( )
22
, | ,
− + − − − +
= =
c c c c c c c
R x x x x та x x
.
The final set of variables Vc is associated with the
variables of other hybrid automata interacting with it.
The interaction state of variables of other hybrid
automata is used to launch interconnected functions.
The
inter
c
G
function describes the interaction of
controlled components with different hybrid systems
simultaneously. In fact, this indicates that two
inter
c
G
of every interconnected hybrid automaton are
connected.
QC, AGV and ASC are the controlled components.
That is why
( )
G ,
inter
cc
ss
can be represented as
( )
G ,
inter
qc qc
ss
,
( )
G ,
inter
agv agv
ss
and
( )
G ,
inter
asc asc
ss
.
In particular, container is transferred from the QC to
the AGV, in which the initiators
( )
12
G ,
inter
qc qc
ss
and
( )
34
G,
inter
agv agv
ss
are triggered simultaneously.
Similarly,
( )
12
G,
inter
agv agv
ss
and
( )
34
G,
inter
asc asc
ss
are
triggered simultaneously when a container is
transported from AGV to ASC. Interrelated functions
of controlled components are shown in table. 3.
Table 3.
inter
c
G
function with related functions
________________________________________________
G
inter
Related
G
inter
________________________________________________
( )
12
G ,
inter
qc qc
ss
( )
34
G,
inter
agv agv
ss
( )
12
G,
inter
agv agv
ss
( )
34
G,
inter
asc asc
ss
________________________________________________
The dynamics of uncontrolled component can be
described as a hybrid automaton with
( )
, , , , , , , , =
uc uc uc uc uc uc uc uc uc uc
H S X U f Init Inv E G R
(11)
12
,=
uc uc uc
S s s
‒ two discrete states in which
uncontrolled component can be;
( )
( )
( )
=
uc uc uc
X N k N k
;
( )
uc
Nk
‒ limited
number of containers in this component; fuc ‒
represents dynamics of this uncontrolled component;
Euc ‒ is defined as the set
( ) ( )
1 2 2 1
, , ,
c c c c
s s s s
.
Let
( ) ( )
=
uc uc
x k N k
. In discrete state
1
uc
s
(action),
one container is loaded or unloaded from this
uncontrolled component. In discrete mode
2
uc
s
(waiting), this component waits for a container. The
dynamics of the uncontrolled component is shown in
Fig. 7
Figure 7. Hybrid automaton for uncontrolled component
The continuous dynamics of two discrete states is
modeled as follows:
1. In state 1 (action), the number of containers
changes and
( )
1
uc uc
f N k
can be written as
( ) ( )
1+ = +
uc uc uc
x k x k a
(12)
1=−
uc
a
, if the container is on the ship;
1=
uc
a
, if the
container is stacked.
2. In state 2 the number of containers does not change
and
( )
2
uc uc
f N k
can be written as
( ) ( )
1+=
uc uc
x k x k
(13)
Invuc is then defined for this unmanaged
component in the form
( )
( )
1
0=
uc uc
Inv s x k N
(14)
( )
( )
1
0=
uc uc
Inv s x k N
(15)
N ‒ capacity of this component.
856
When
( )
( )
12
, ==
pos act
uc uc uc c
G s s x k x
‒ the
interaction depends on the arrival of the controlled
component.
When
( )
( )
2 1 1
, ==
uc uc uc uc
G s s s k s
‒ the processing
of the container ends.
( )
uc
sk
is used to describe the
discrete state of the uncontrolled component at time k.
A continuous state does not change as a result of
switching discrete states. That is why
( ) ( )
1 2 2 1
, ,
−+
= = =
uc uc uc uc uc uc
R s s R s s N N
.
The
inter
uc
G
function describes interaction of
uncontrolled components with controlled
interconnected hybrid automata. The term
uncontrolled can be replaced by vessel and stack to
denote a vessel and a place of storage. In the
simulation,
( )
21
,
inter v v
G s s
and
( )
34
,
inter qc qc
G s s
are
connecting when the QC collects the container from
the vessel. Also,
( )
21
,
inter v v
G s s
and
( )
34
,
inter qc qc
G s s
are synchronized when the ASC unloads the container
onto the stack. These related conjugate functions are
shown in Table 4.
Table 4.
inter
c
G
u
with related functions
________________________________________________
G
inter
Related
G
inter
________________________________________________
( )
12
G ,
inter
vv
ss
( )
34
G,
inter
qc qc
ss
( )
21
G,
inter
ss
ss
( )
12
G,
inter
asc asc
ss
________________________________________________
Figure 8. Simplified representation of the complete hybrid
system
When a container is transported from a vessel’s
berth to a stack storage location, the QC and AGV
interact in the berth area, while the AGV interacts with
the ASC in the storage area. This interaction is shown
in Figure 8. Five components are connected by
interaction functions labeled A, B, C, and D.
The interaction between two different components,
marked with letters A, B, C and D in fig. 8 allows us to
conclude that two functions of two interacting
components are realized and take place in time
simultaneously.
For interaction A, when
( )
21
G,
inter
ss
ss
is triggered,
the vessel make transition from the discrete state
2
v
s
to
1
v
s
to ship the container. At the same time
( )
21
G,
inter
ss
ss
and
( )
14
G,
inter
qc qc
ss
coincide when
discrete state of the vessel would be changing from
2
v
s
to
1
v
s
to pick up the container. Similarly, the
synchronization of two interacting components can be
specified for B, C, and D.
The integration of the five components mentioned
above forms a hybrid system, including continuous
time linear dynamics and discrete event dynamics.
Such a class of hybrid systems can be described as
mixed logic dynamic systems. In such systems, part of
continuous time is described by linear dynamics, and
part of a discrete event is modeled as a set of linear
constraints on dual variables and continuous
variables. This type of model is very good for
formulating control prediction model problems for
hybrid systems.
3.3 A model of mixed logic dynamic system.
The general model of the mixed logic dynamic system
was described by the following equations
( ) ( ) ( ) ( ) ( )
1 2 3
x 1 Ax B u B δ B z+ = + + +k k k k k
, (16)
( ) ( ) ( ) ( ) ( )
1 2 3
y Cx D u D δ D z = + + +k k k k k
, (17)
( ) ( ) ( ) ( )
2 3 1 4 5
E δ E z E u E x E+ + +k k k k
, (18)
To use it, input signals should have the following
structure
( ) ( ) ( )
with
=
T
TT
rb
x k x k x k
( )
n
r
xk
‒
continuous part of the state vector.
( )
0,1
b
n
b
xk
‒
part of the state vector corresponding to the discrete
part.
Output signals should have analogical structure
( ) ( ) ( )
=
T
TT
rb
y k y k y k
with
( )
m
r
yk
‒
continuous part of the output and
( )
0,1
b
m
b
yk
‒
discrete part of the output. y(k) ‒ output vector.
The input vector
( ) ( ) ( )
=
T
TT
rb
u k u k u k
consists
of continuous part
( )
r
l
r
uk
and discrete part
( )
0,1
b
l
b
uk
.
z(k) ‒ auxiliary integer; matrices A, B1 ~ B3, C, D1 ~
D3 and E1 ~ E4 denote real constant matrices; E5 ‒
real vector.
Written as the set of equations (16)-(18), the form
of a mixed logic dynamic system during the
simulation of terminal operation actually allows to
solve the problem of the development of continuous
variable functions using linear dynamic equations and
discrete variables. It is possible to work with the help
of described functions and their interaction with each
other.
Boundary conditions for the model consist of x(k)
values. Individual geometry described using
load
c
x
and
unload
c
x
can be mapped to a mixed logic dynamic
system model. Uncertainties, such as delay of
operations and exact arrival time of new containers,
can be incorporated based on the mixed logic
dynamic system model by measuring states and
adding new variables.
857
4 RESULTS
4.1 Centralized hybrid model of predictive management
The term predictive control model refers to a control
methodology that makes explicit use of dynamic
model to derive control actions. In the predictive
control model, a dynamic model was developed to
predict the future state of the terminal system based
on the current state and proposed future actions.
Predictive control models can be applied to hybrid
systems that simultaneously considers the dynamics
of a discrete event and the dynamics of a continuous
event. The general structure of the centralized hybrid
model of predictive control is shown in Figure 9.
Figure 9. The structure of centralized hybrid model of
predictive management
The aim of management is to transport containers
from the vessel’s berth and one stack using
components, that presented in Fig. 9. The purpose of
management is to balance the load capacity and
energy consumption of the controlled components.
The problem of predictive management model was
formulated in the following way
( ) ( )
( )
( ) ( )
( )
1
12
0
1 , 1 ,
−
=
+ + + + + + +
p
N
l
min J x k l u k l J x k l u k l
(19)
( ) ( ) ( ) ( ) ( )
1 2 3
1
+ + = + + + + + + +x k l Ax k l B u k l B k l B z k l
(20)
( ) ( ) ( ) ( ) ( )
1 2 3
+ = + + + + + + +y k l Cx k l D u k l D k l D z k l
(21)
( ) ( ) ( ) ( )
2 3 1 4 5
+ + + + + + +E k l E z k l E u k l E x k l E
(22)
( )
+
min max
u u k l u
(23)
( )
1 + +
min max
x u k l x
(24)
( )
+
min max
y u k l y
(25)
( ) ( )
( )
( ) ( )
1
1,+ = −
vs
J x k u k N k N k
;
( ) ( )
( )
( ) ( ) ( )
2 1 2 3
1,
+ = + +
qc agv asc
J x k u k u k u k u k
;
Nv(k) – describes containers on the ship; Ns(k) –
describes containers in the stack; uqc(k), uagv(k) and
uasc(k) ‒ accelerations of QC, AGV and ASC,
respectively; Np - forecast horizon;
( )
1++x k l
‒
predicted state at time
1++kl
based on input
( )
+u k l
.
It should be noted that weights λ1, λ2 and λ3 are
used to balance the loading power and energy
consumption. At the same time, umin, umax, xmin, xmax and
ymin, ymax are the boundaries on inputs, states and
outputs, respectively.
The function
( ) ( )
( )
1
1,+J x k u k
is dedicated to
consider the problem of overload power management.
The vessel is emptied as quickly as possible,
minimizing Nv(k), but this parameter cannot
guarantee the arrival of the last container in the stack
after it was removed from the vessel. The value Ns(k)
is added to J1 to ensure that last container arrives in
the stack.
The function
( ) ( )
( )
2
1,+J x k u k
is dedicated to
consider the process of simplifying the consumption
of kinetic energy of all controlled components.
Continuous time dynamics is a double integrator that
ignores air resistance and rolling resistance. For this
reason, the absolute value of acceleration is
considered as a cost criterion arising from the
problem of optimal fuel management. This fuel-
optimal criterion makes it easier to solve the
optimization problem.
In the objective function of the proposed hybrid
predictive control model, the processing part J1 and
the energy-efficient part J2 can be balanced by
changing λ1, λ2 and λ3.
Considering that
( ) ( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
, 1 , , 1 , , 1 , , 1 , , 1 , , 1
= + + − + + − ++ + −
T
T T T
T T T T T T
p p p
u k u k u k u k N k k k N z k z k z k N
time step k can be formulated as
( )
( ) ( ) ( ) ( )
0 1 2 3
min
+++
TTTT
uk
f u k f u k f u k f u k
, (26)
(26) was written under the condition that
( )
min max
b Au k b
, (27)
( )
min max
u u k u
(28)
f0 ‒ relates loading capacities to the objective function;
f1, f2 and f3 ‒ spending related to energy consumption
by QQ, AGV and ASC, respectively; bmin and bmax ‒
lower and upper boundaries of the corresponding
inequality;
min
u
and
max
u
‒ lower and upper limits
of the controlled variables.
In the objective function (26), the scale
( )
0
T
f u k
will be much larger than
( )
1
T
f u k
,
( )
2
T
f u k
and
( )
3
T
f u k
.
To reduce the share of loading power and keep the
cost of operation in a relatively constant range, it is
necessary to use the adaptive weight
( )
0
1−
T
f u k
. In
this case, the effect of λ on power consumption and
throughput can be seen more clearly. The new
objective function in this case can be written as
( )
( )
( )
( ) ( ) ( )
0
1 2 3
0
min ( )
1
+ + +
−+
T
TTT
T
uk
f u k
f u k f u k f u k
f u k
(29)
ε is a very small number in the case
( )
0
10−=
T
f u k
.
858
In the formulated objective function (29), the
variable λ can affect the energy consumption and
throughput of a piece of equipment.
The function of controller is to obtain the
minimum time for each operation and assign specific
equipment to perform them. Considering one QC, we
denote the time limits for two operations at stage 1 as
11 11
,,
,
start i end i
tt
.
In the second stage, there is no difference between
selection of certain equipment because all AGVs are
identical. We define
1,2, , =
agv agv
n
, which
represent a set of AGVs.
: →
agv agv
f Ф
(30)
fagv ‒ function that depicts a set of tasks Ф for AGV by
the number Ψagv.
Function fagv(i) describes the specific AGV assigned
to work i. The time limits for work i were written as
( ) ( )
21 21
, , , ,
,
agv i agv i
start i f end i f
tt
and
( ) ( )
22 22
, , , ,
,
agv i agv i
start i f end i f
tt
.
In the case of ASC, scheduling of work i is
determined in advance, since each container has a
specific position on the ship and a specific destination
in the stack. For this reason, the function fasc(i) was
used to describe the assigned ASC for job i. The time
limits of operation i for a certain ASC were written as
( ) ( )
31 31
, , , ,
,
asc i asc i
start i f end i f
tt
and
( ) ( )
32 32
, , , ,
,
asc i asc i
start i f end i f
tt
.
The time of cargo operations
12
hh
i
t
depends on the
initial time values of ai, bi and ci. The time intervals of
operation
12
O
hh
i
in three stages are shown in table 5.
We enter:
12
,
hh
start i
t
‒ start time of the work and
12
,
hh
end i
t
‒
time of the end of the corresponding cargo operation
12
O
hh
i
(h1 ∈ {1,2,3}, h2 ∈ {1,2}).
Table 5. Time intervals of operations
12
hh
i
O
at three stages
________________________________________________
Operation Equipment Time Time Operation
of start of end execution time
________________________________________________
11
i
O
QC
i
a
11
+
ii
at
11
i
t
12
i
O
QC
11
+
ii
at
11 12
++
i i i
a t t
12
i
t
21
i
O
AGV
i
b
21
i
t+
i
b
21
i
t
22
i
O
AGV
i
c
22
i
t+
i
c
22
i
t
31
i
O
ASC
i
c
31
i
t+
i
c
31
i
t
32
i
O
ASC
31
+
ii
ct
31 32
i
t++
ii
ct
32
i
t
________________________________________________
At the lower level, the controller provides
management of each device and makes decisions
about the continuous trajectory of a piece of
equipment. In each controller at the lower level, the
task of optimal control should be formulated in a form
which gives an ability to finish the operation within
the set working time. A lower-level controller may
consider additional objectives, such as minimizing
energy consumption. The specific task of equipment
management depends on the operation it should
perform.
Let r0 = [r0 0]
T
rf = [rf 0]
T
describes initial position and
destination of the equipment. In cargo operations, the
corresponding piece of equipment will move from r0
to rf within a given time. This management
optimization problem can be presented in the form
( )
( ) ( )
( )
min ,
ut
J r t u t
, (31)
with condition of fulfillment of the following
equations
( ) ( ) ( )
( )
g,=r t r t u t
, (32)
( )
( )
0 0 0
, , ,
= =
f f f
r t r r t r t t t
, (33)
( ) ( )
( )
( )
0
2
2
( , 0,5=
f
t
t
J r t u t mr t
‒ function that quantifies
the energy consumption of equipment with mass m
and speed r2.
The terminal equipment starts its work at t0 and
should finish it before tf. The initial and final states in
equation (33) ensure that operation is complete.
To solve the problem of optimal control (31), an
analytical approach was used, where optimal solution
is obtained by reducing to the corresponding
extremum the quadratic function of energy
consumption, considering, that the general system
model is linear.
The aim of control optimization is to minimize the
mechanical energy of a piece of equipment when
moving from initial position r0 to final tf during
corresponding time interval (from t0 to tf) for the
operation
12
O
hh
i
. For discretization, time step was
written as ΔT, and then
12
t
1
ΔT
+
hh
i
‒ number of time
steps for the time interval from t0 to tf. The discrete
dynamic model based on (32) and (33) for the part of
port terminal equipment was written as
( ) ( ) ( ) ( ) ( )
2
1 ΔT
0,5ΔT
1,
01
ΔT
+ = + = +
r k r k u k Ar k Bu k
(34)
r(k) = [r1(k) r2(k)]
T
describes position r1(k) and the
velocity r2(k) of the piece of equipment,
( )
uk
‒
acceleration of the piece of equipment.
To minimize the mechanical energy of the
equipment from k = 0 to k = Ns considering its
dynamics and limitations, after calculating u=[u(0),
u(1), … , u(Ns−1)]
T
, the optimization problem was
formulated as follows:
( )
( )
2
2
1
min 0,5
=
s
N
u
k
m r k
, (35)
with condition that k = 0,1, ..., Ns−1, and also provided
that are realized equalities
( ) ( ) ( )
1+ = +r k Ar k Bu k
(36)
( )
min max
r r k r
(37)
( )
min max
u u k u
(38)
( )
( )
0f
0 , ==
s
r r r N r
(39)
859
( )
( )
2
2
0,5m r k
describes the kinetic energy at the
time moment k; rmin and rmax ‒ restrictions on r(k)
states; umin, аnd umax ‒ limits of control variable u(k).
At the moment when problem of minimizing the
mechanical energy of equipment is solved, the lower-
level controller will set the calculated trajectories as a
reference for the part of equipment.
To plan operations with a hierarchical
management system it is necessary to determine the
minimum time required for a processing operation
with one piece of equipment. This time depends on
condition and continuous dynamics of equipment. Its
numerical value can be obtained using the theory of
optimal control due to the use of Pontryagin's
maximum principle [2]. Applying the principle gives
a control action u(t) that minimizes time to complete
the task. This control action u(t) was written as
( )
2
12
1
for t , ,
0 for t , ,
for t 0 , ,
+
+−
+−
− =
= =
=
max b
max
u t t
u t t t
ut
, (40)
where t1 and t2 are the variables which determine
acceleration.
At the same time,
11
+
+tt
,
22
+
+tt
,
11
−
−tt
and
22
−
−tt
(where ε is a small positive
number), and t1 and tb are calculated as
2
1
2
if
if
=
max max
t
max max
t max
t
max max
vv
d
uu
t
dv
d
uu
, (41)
2
2
2
2 if
2 if
−
+
=
max
t
max max max
t
max max max
b
t max
t
max max
v
d
v u v
d
u v u
t
dv
d
uu
, (42)
The minimum time depends on the ratio between
dt and
2
max
max
v
u
. Thus, the minimum time required for
treatment operation with one piece of equipment can
be obtained from the Pontryagin maximum
conditions.
Figure 10. The minimum working time for the distance dt
Different plots of the minimum time to complete
the work depending on the distance dt are shown in
Figure 10. When
2
max
t
max
v
d
u
(Figure 10-a) and when
2
max
t
max
v
d
u
(Figure 10-b )
The hierarchical system emphasizes the
interdependence of the planning problem regarding
the dynamics of discrete events of all port terminal
equipment units and task of optimal control regarding
individual equipment elements considering dynamics
of continuous time. When using it, it is important to
display correctly dynamics of discrete events with
dynamics of continuous operation of a particular type
of equipment. In this case, the use of a constant
movement velocity when calculating equipment’s
operational time can lead to difficulties in controlling
the equipment. Examples of such error are when
dynamics and hardware limitations (such as velocity
and acceleration) are considered.
4.2 Simulation results
During simulation of the hybrid system in the port
terminal management model, as a container terminal
was taken variant which contains three QCs, four
AGVs, and four ASCs. It was accepted that a
distributed method is used to control the operation of
all equipment during terminal operation. Throughout
the modeling process, the operational management of
container terminal was focused on the dynamics of
discrete events during planning. Optimal control of
equipment based on the centralized hybrid model of
predictive control was considered as an option for
obtaining the minimum period of time for the all
works completion. This parameter was considered as
the final result of numerical calculations. The time of
work completion was determined in accordance with
the second main indicator of the port terminal
operation - total energy consumption.
Simulation was carried out with consideration of
different options for containers position on the ship.
Systematization of results was carried out in
comparison with energy efficiency indicators for two
options for container transportation.
The first transportation option corresponded to the
case of closest choice, when the sequence of N jobs
1
ij
at stage 1 ranged from the nearest to the farthest
place on the container ship.
The second variant of transportation corresponded
to the case of random selection, when the sequence of
N workplaces
2
ij
at stage 1 was determined
randomly.
Table 6. The time of all works completion in relation to
different approaches to terminal operation, sec
________________________________________________
Test No. Optimal Nearest Random
________________________________________________
1. 477 477 496
2. 476 542 552
3. 496 573 540
4. 478 502 478
5. 476 516 504
6. 496 546 520
7. 478 551 488
8. 476 570 543
9. 481 476 532
10. 467 490 539
________________________________________________
Average 480.1 524.3 519.2
________________________________________________
Ten independent tests were conducted, as a result
of which sufficiently characteristic data were
obtained. The generalized simulation results are
860
shown in Table 6 and from the main efficiency
indicator point of view they are displayed in the form
of the most important result - time of works
completion in relation to different approaches of
terminal operation.
Figure 11 shows how the variance of the work
execution time D is distributed in all ten cases of the
test simulation. Each vertical section corresponds to a
separate modeling case. Red bars answer to the first
transportation option - the nearest container, black
bars answer to the second transportation option -
random container selection, blue bars answer to the
developed transportation option - a hybrid model of
managing the process of container selection and
transportation.
Figure 11. Distribution of time variance D for three variants
of container selection and transportation. 1 – selection of the
nearest container, 2 – random selection of container, 3 –
hybrid model of container selection and transportation.
The analysis of the displayed results allows us to
formulate an unequivocal conclusion that the same
result was obtained in nine out of ten simulations - the
hybrid control model was always characterized by the
shortest time to achieve the final result. The variance
was minimal in all nine cases of terminal operation
only in that case when the centralized hybrid
predictive control model developed during the
research was used to control its operation.
Table 7 shows the results generated using an
energy-saving schedule with optimal functioning of
the terminal due to the operation of the equipment
based on the centralized hybrid model of predictive
control. The indicator of energy costs, as it possible to
see, is very low.
Table 7. Performance indicators of the hybrid management
model
________________________________________________
Test No. Work completion time, s Energy consumption, kW
________________________________________________
1. 477 4.82
2. 476 7.05
3. 496 8.82
4. 478 5.15
5. 476 5.13
6. 496 6.54
7. 478 4.97
8. 476 7.61
9. 481 6.76
10. 467 6.29
________________________________________________
Average 480.1 6.314
________________________________________________
Figure 12. Time of works completion. 1 – selection of the
nearest container, 2 – random selection of container, 3 –
hybrid model of container selection and transportation.
Comparative analysis of results that describing
work completion time is shown in Figure 12. The
graph shows that the deviation of the time for
carrying out a set of container handling operations on
a ship from the optimal value invariably in all
modeling cases corresponded to the hybrid control
model (Figure 12, curve 3)
This result is quite non-standard since it is logical
to assume that the best result should correspond to
the closest time for every operation completion. The
graph clearly shows that the smallest time oscillations
correspond to the hybrid control model. The
amplitude of the deviation is minimal and, in general,
the transition from one point to another is smooth,
unlike the first and second options for controlling the
process of operating a port container terminal. The
graph shows that the curves that correspond to the
first and second control options gave an almost
negative result in terms of the quality of logistics for
container movement through the transport terminal.
Sudden jumps in time indicate obvious operational
problems in management. The third operating option,
based on a hybrid control model, is the most effective
because it is characterized by an almost complete
absence of pulsations and has minimal numerical
indicators.
The analysis of received simulation data allows us
to draw an unequivocal conclusion - the centralized
hybrid model of predictive control under all constant
conditions allows to obtain the best performance of
the terminal in comparison with standard methods of
managing the process of unloading containers from a
ship. The time to complete the work in this case is the
shortest, and the energy consumption is minimal.
5 DISCUSSION
The whole investigation described in the article was
devoted to solution of one problem – to arise
operational quality of the container terminal. The
centralized hybrid model of predictive management
showed very high results, but the main issue from the
point of view of port's operation profitability remains
the issue of automation of its operation management.
Terminal automation involves a high investment cost,
but is offset by a reduction in operational costs. By
developing its infrastructure, terminal capacity
increases and operating costs decrease due to
861
increased supply. Usually, ports with more than one
terminal operate at more competitive prices and
therefore have more lines of service for ships and
users in general.
The operating costs of a container terminal can be
reduced by investing in automation. Fixed costs (port
tariffs, state taxes, salaries, etc.) in ports, although
they increase over a long period, are practically
constant and do not depend on how many ships
docked each month or how many movements were
made at the berth. Variable costs are related to the
level of terminal service and are very flexible and
easily changed to provide different levels of service.
These costs can also vary in the short term and thus
constitute a large-scale scenario for further terminal
automation.
The main direction of further research in this way
can be described as: if variable costs of container
terminal can be reduced, then in all ports they can be
improved in the direction of the costs of cargo
operations.
If one considers all operations related to
maneuvering, mooring, wiring and security of vessels,
the total cost of arrival of one vessel can reach 14,000
USD, which is partly explained by the high risks
associated with mooring operations. Improvement
due to the automation of berth equipment operating
on the basis of the centralized hybrid model of
predictive management proposed in the article can
reduce risks and associated costs. In this case, the
services related to shipping of vessels may be reduced
up to 50% of the total cost of vessel’s arrival.
Terminal automation, which contributes to the
reduction of operating costs, also includes the amount
of fuel, electricity, materials and human resources
used by machines. This direction of spending can
make up to 60% of the total cost of cargo handling of
containers and it is very promising from the point of
view of terminal automation. Due to the introduction
of new automatic systems, it is possible to reduce the
total number of terminal operators and minimize the
cost of handling containers. The main obstacle in this
case can be the legislation on unemployment.
For the case of investment research, which requires
only initial spending, and then immediately begins to
provide income, it is recommended to use the
following equation
( )
0
0
1
=
=−
+
T
t
t
t
S
NPV i
i
(43)
T – quantity of years; St – net monetary flow in the
period; i – discount rate for one year; i0 – amount of
initial investment.
Return on Investment in the case of high values of
the obtained return on investment means a high
return that will be received by the investors who
financed the project. Evaluation of return on
investment can be done as
1
=
=
T
t
t
t
t
ROI
P
(44)
πt – profit received in each year; Pt – initial
investments.
In the optimal case of using new terminal
automation systems which are operating using the
proposed centralized hybrid model of predictive
control, equations (43) and (44) should give identical
results, but the first indicator (NPV) is more
qualitative and safer from the financial point of view.
As the main factors when assessing the level of
necessary investments in terminal automation, it is
possible to recommend an assessment of three
directions: the terminal operating system; if there is a
need for new equipment; changes in the terminal
infrastructure (coverage of the terminal territory
depending on the pressure of new cranes, alignment
of paths, increase of territory in the area of berths,
etc.).
Secondary factors that should be investigated
during terminal automation include: industrial
development of the country; labor cost; type of
existing road surface; dimensions of terminal;
technical specification and quality of necessary
equipment; production costs compared to other
countries; order sizes (scale effect).
During automation of terminal, every relevant
detail of the port should be considered. It will give an
ability in feature to estimate the necessary reductions
in operating costs to balance the capital investment in
automation.
6 CONCLUSIONS
The research described in the article is of significant
practical importance as it gives an ability in feature to
develop new concepts of automation of cargo
operations due to the improvement of: modern port
terminal management systems, terminal operating
systems, decision support systems, technological
planning of logistics schemes for handling container
ships.
The use of the proposed centralized hybrid model
of the predicted management of cargo handling
process in seaports will lead to an even greater
reduction in injury rates. This will happen due to the
fact that the developed concept of the hybrid
management system makes it possible to consider
such features of the process of unloading cargo from
the ship, which until now have not been considered
yet. An example of one of the indicators is the
movement velocity of all system elements depending
on various external factors - weather conditions, the
current time of day, the number of workers in the
dangerous zone.
Centralized hybrid model of predictive control
under all constant conditions allows to obtain the best
performance of the terminal in comparison with
standard methods of managing the process of
unloading containers from a ship. The time to
862
complete the work in this case is the shortest, and the
energy consumption is minimal.
Terminal automation involves a high investment
cost, but is compensated by a reduction in operating
costs. By developing its infrastructure, terminal
capacity increases and operating costs decrease due to
increased supply. The choice of the terminal operating
system, the need for new equipment, changes in the
terminal infrastructure are the main factors on which
investments depend.
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