interstate container ship optimization - ode...
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Interstate Container Ship
Optimization
ME555 Final Report
Jason Strickland
Katharine Woods
Thomas Devine
Yan Liu
ABSTRACT Due to Jones Act legal requirements, aging containership vessels currently operate a
west coast run often referred to as the “Pineapple Run”. These vessels have reached
their intended service life and require replacement in the immediate future. Since
these vessels are required to be built, owned, operated, and maintained in the US costs
are high as compared to foreign competitors. To ease the financial burden, a design
optimization will be conducted with the goal of net reduction of cost through both
initial construction and lifecycle operation. Four individual sub-systems, hull form
resistance, vessel powering, mid-ship structural scantlings and regulatory and
operational decisions will be individually optimized each with a specific cost-related
objective function. Their resulting design solutions will then be incorporated into a single
functional architecture to produce a resulting optimized design.
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Table of Contents
ABSTRACT ....................................................................................................................................... 1
1 INTRODUCTION ........................................................................................................................... 4
1.1 THE PINEAPPLE RUN ............................................................................................................. 4
1.2 TRADITIONAL NAVAL DESIGN ............................................................................................ 5
1.3 US JONES ACT VESSELS ....................................................................................................... 6
2 RESISTANCE SUBSYSTEM ............................................................................................................. 8
2.1 PROBLEM STATEMENT .......................................................................................................... 8
2.2 NOMENCLATURE ................................................................................................................. 8
2.3 MATHEMATICAL MODEL ..................................................................................................... 9
2.3.1 OBJECTIVE FUNCTION .................................................................................................. 9
2.3.2 CONSTRAINTS .............................................................................................................. 12
2.3.3 DESIGN VARIABLES ..................................................................................................... 12
2.3.4 MODEL SUMMARY ...................................................................................................... 13
2.4 MODEL ANALYSIS .............................................................................................................. 13
2.5 OPTIMIZATION STUDY ........................................................................................................ 14
2.6 PARAMETRIC STUDY .......................................................................................................... 15
2.7 DISCUSSION OF RESULTS ................................................................................................... 15
2.8 SYSTEM INTEGRATION........................................................................................................ 16
3 PROPULSION SUBSYSTEM ......................................................................................................... 18
3.1 PROBLEM STATEMENT ........................................................................................................ 18
3.2 NOMENCLATURE ............................................................................................................... 18
3.3 MATHEMATICAL MODEL ................................................................................................... 19
3.3.1 OBJECTIVE FUNCTION ................................................................................................ 19
3.3.2 CONSTRAINTS .............................................................................................................. 20
3.4 MODEL ANALYSIS .............................................................................................................. 20
3.5 OPTIMIZATION STUDY ........................................................................................................ 21
3.6 PARAMETRIC STUDY .......................................................................................................... 24
3.7 RESULTS ............................................................................................................................... 26
4 SHIP MIDSHIP STRUCTURE SUBSYSTEM ..................................................................................... 29
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4.1 PROBLEM STATEMENT ........................................................................................................ 29
4.2 NOMENCLATURE ............................................................................................................... 29
4.3 MATHEMATICAL MODEL ................................................................................................... 31
4.4 MODEL ANALYSIS .............................................................................................................. 33
4.5 OPTIMIZATION REULTS ....................................................................................................... 37
4.6 PARAMETRIC STUDY .......................................................................................................... 38
4.7 SUBSYSTEM INTEGRATION ................................................................................................. 39
5 REGULATORY AND OPERATIONAL REQUIREMENTS SUBSYSTEM .......................................... 42
5.1 DESIGN PROBLEM STATEMENT.......................................................................................... 42
5.2 NOMENCLATURE ............................................................................................................... 43
5.3 MATHEMATICAL MODEL ................................................................................................... 43
5.3.1OBJECTIVE FUNCTION ................................................................................................. 43
5.3.2 CONSTRAINTS .............................................................................................................. 45
5.3.3 DESIGN VARIABLES AND PARAMETERS .................................................................... 46
5.3.4 ASSUMPTIONS .............................................................................................................. 46
5.3.5 MODEL SUMMARY ...................................................................................................... 47
5.4 MODEL ANALYSIS .............................................................................................................. 47
5.5 OPTIMIZATION STUDY ........................................................................................................ 49
5.6 PARAMETRIC STUDY .......................................................................................................... 50
5.7 DISCUSSION OF RESULTS ................................................................................................... 51
6 SYSTEM OVERVIEW CONCLUSION ......................................................................................... 52
REFERENCES ................................................................................................................................. 53
Appendix I: Project Variables ................................................................................................... 54
Appendix II: Weather Data ....................................................................................................... 55
Hawaii Weather Data ............................................................................................................ 55
Oakland Weather Data ......................................................................................................... 56
Tacoma Weather Data .......................................................................................................... 57
Appendix III: Fuel Oil Information.............................................................................................. 58
Appendix IV: The Beaufort Scale .............................................................................................. 59
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1 INTRODUCTION
1.1 THE PINEAPPLE RUN
The Pineapple Run is the nickname for the various ship routes that travel from the West
Coast of the United States to Hawaii. The route is either from Los Angeles, CA to
Honolulu, HI or from Tacoma, WA to Oakland, CA to Honolulu, HI and back.
Figure 1.1: Pineapple Run Visualization1
All routes are loops and the ships are on very strict schedules. Cargos include golf carts,
Christmas trees and various bulk foodstuffs on the route toward Hawaii and coffee,
pineapples and frozen fish on the return trip to the west coast. Ship cargo container
volume is measured in number of Twenty-foot Equivalent Units (TEUs). The Horizon Pacific,
one ship currently on the run, carries 2,361 TEUs when the ship is full. Table 1.1 provides
the principle dimensions for the existing ship. All ships travel the shortest distance
between ports. This route is known as the Great Circle Route. Ships operating on this
route are required to meet specific state, federal and international regulations as
published and enforced by the various regulatory bodies. These regulations are
discussed in depth in Section 5.1. This route was picked for this design problem because
the ships on the run are long past their expected lifespan and are due for replacement.
1 Source: Horizon Lines
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The regulations that apply are sufficiently complicated and restrictive that all together
this is a complicated and worthy design problem.
Figure 1.2: Horizon Pacific2
Table 1.1
Length Overall (L, meters) 236
Maximum Beam (B, meters) 27.43
Maximum Draft (T, meters) 11.14
Deadweight Tonnage (Δ, tons) 31,213
Power Output (PR, kW) 23,538
1.2 TRADITIONAL NAVAL DESIGN
Naval architecture has been, traditionally an iterative, and incremental design process.
To this day the design spiral is still heavily utilized. The ultimate hope is that with each
lap around the spiral as fidelity increases the design will converge. While this approach
has been successful in developing numerous outstanding vessels it succumbs to the
reality that the converged design is not likely the global optimum. Figure 1.3 below
outlines one possible incarnation of the design spiral as it pertains to ship design. It is
important to note that as the design matures and converges it is incumbent upon the
designer to ensure that all of the fundamental constraints have not been violated at
each major subject transit. If any of the constraints have been violated it will perturb
the convergence, retard the overall progress, and may require substantial rework in
previously closed areas.
2 Source: Containership-info.com
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For our project we have considered a subset of the major functional requirements
illustrated below on the design spiral. This subset includes the following four items:
regulatory and operational objectives, propulsion selection, hull resistance optimization,
and the configuration of a typical structural section. The overall vessel objective will
stem from a replacement-in-kind of an existing vessel on an existing route. However,
the goal will be to reduce a nominal cost objective locally with respect to each of the
discipline areas listed above and then globally once the different models have been
aggregated.
Figure 1.3: Design Spiral3
1.3 US JONES ACT VESSELS
The Jones Act is a US cabotage law originating in 1920 governing the ownership,
construction repair and operation of vessels conducting intrastate/territory trade. For
larger vessels, especially large containerships, this regulation has led to a rusting
outdated fleet without recapitalization due to the high cost of construction and
operation. Due to labor rates and cost of materials, a ship built in the US is
approximately triple the cost of a comparable ship built elsewhere.4 This is the
fundamental reason that the ships on this run have not been replaced yet. Currently,
this vessel class is almost exclusively produced in highly optimized yards in Korea Japan
and China. As a Jones Act vessel, designers are challenged to alter decisions due to
3 Source: Ship Construction, Sixth Ed, D. J. Eyres, 2007
4 Comparing the cost of a 3,100 TEU domestically-built ship for trade between the continental US and Puerto Rico ($350 million, from
http://www.aribbeanbusinesspr.com, a local Puerto Rican news source) and the cost of an 18,000 TEU container ship built in South Korea for trade in any ports
that are large enough to fit the ship ($190 million, from http://www.ship-technology.com, an industry magazine).Sources accessed 26JAN2013
Page | 7
the highly constrained nature of the problem. Trade-offs in materials, construction,
manning and operation need to be considered in order to allow the nominal vessel to
be properly classed as a Jones Act vessel.
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2 RESISTANCE SUBSYSTEM
2.1 PROBLEM STATEMENT
The moving container ship experiences resistance from water and therefore requires
power to overcome it. In the energy saving point of view, the reduction of resistance
will result in less power requirement and lead to a lower fuel consumption rate.
Therefore, ship resistance has an important effect on the profits a ship can earn. It is in
this consideration that the resistance performance is placed as a subsystem to be
optimized in the whole container ship design project. The objective of this subsystem
optimization process is to design the ship shape that has the minimum resistance while
satisfying all the design requirements.
Improving resistance performance is competing objective as there are requirements
regarding container ship design. In resistance’s consideration, a slim ship model would
be favorable, however that would make it hard to fulfill the containers arrangement
requirements. It is also necessary to take IMO (International Maritime Organization), ABS
(American Bureau of Shipping) and USCG (US Coastal Guard) rules into account if we
want to calculate a valid design.
2.2 NOMENCLATURE
Variables Definition
L Length of water line
B Breadth of the ship
D Depth of the ship
T Draft of the ship
CB Block coefficient
RT Total Resistance
RF Frictional Resistance
RR Residual Resistance
RA Additional Resistance
RAirdrag Air drag force
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2.3 MATHEMATICAL MODEL
Various methods for ship resistance calculation are reported in academic literature.
Traditional methods to estimate ship resistance are based on ship forms that may be
considered obsolete. Here we used the Hollenbach (1999) method to calculate the
resistance of the ship form. To evaluate the accuracy of the Hollenbach method, it was
compared in the test cases of Hamburg Ship Model Basin, which has 433 models with
protocols of 793 resistance tests and 1103 propulsion tests each for a set of different
speeds. The results showed that the Hollenbach method predicts much better the
resistance for single-screw ships. The method is introduced further in the following
paragraph.
2.3.1 OBJECTIVE FUNCTION
The objective of this subsystem is to minimize the resistance of a ship form.
Total resistance of a ship is expressed in Equation 2.1 below.
(2.1)
The Froude number in the following formula is based on the length LFn:
{
( )
(2.2)
Where the ‘Length over surface’ Los is defined in Figure 2.1:
Fig 2.1: Definition of length LWL, LOS, and LPP
In the Equation 2.1, RF is calculated by Equation 2.3:
(2.3)
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CF is calculated from ITTC (International Towering Tank Conference) (1957) line, which is
current international standard in resistance computation:
( ) (2.4)
Where
, .
In Equation 2.3 S means the wetted surface of ship hull. In Hollenbach’s method, the
following formula can be used for estimating the wetted surface of hull and
appendages for single-screw vessels:
( ) (2.5)
with
(
) (
) ( ) (
) (
) (
)
(
) (
)
(2.6)
TA is the draft at AP, TF the draft at FP, DP the propeller diameter.
The residual resistance RR in Equation 2.1 is given by:
(2.7)
The nondimensional coefficient CR is generally expressed as:
(
)
(
)
(
)
(
)
(
)
(
) (2.8)
NRud is the number of rudders [1 or 2].
(
) (
) (2.9)
( ) (2.10)
(2.11)
(2.12)
The maximum total resistance is
(2.13)
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All the coefficients needed are listed in Table 2.1.
Table 2.1: Resistance coefficients
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2.3.2 CONSTRAINTS
In ship design, there are certain rules to follow with regards to principal dimensions.
There are well-established dimension relationships studies. It is rational to use these rules
as guidance for our ship design. Watson and Gilfillan (1976) have noted that
containerships had Length-Beam ratio at around 6.25. This kind of guidance is very
useful to examine our optimal design. The Beam-Depth ratio is another important non-
dimensional ratio affecting a lot in the ship’s transverse stability. The third most important
dimensional ratio is the Beam-Draft ratio as it is influential on residuary resistance and
transverse stability. Besides, there are also restrictions of the Hollenbach’s resistance
model. All the constraints used in this subsystem are provided below:
T≤12.2m (maximum draft of Honolulu port)
2.25≤B/T≤3.75 (Watson suggestion)
4.71≤L/B≤7.11 (Hollenbach method requirement)
0.6≤CB≤0.83 (Hollenbach method requirement)
B/D≥1.65 (Stability requirement)
D-T≥4 (USCG freeboard requirement)
B≥12*2.463 (Container arrangement constraint)
L*B*T*CB-∇ =0 (Displacement requirement)
The last constraint is from the point view of containers arrangement (Taggart and 1980).
In order to maximize cargo storage, the Beam of a containership is normally a multiple
of the container spacing on deck. Therefore the ship breadth is typically a multiple of
2463mm with some additional margin. In our containership design, it is expected to
arrange 12 containers in the breadth wide.
2.3.3 DESIGN VARIABLES
In this container ship design, the ship resistance model is simplified and represented only
by principal dimensions of the ship hull. The design variables are listed in Table 2.2.
Table 2.2: Design variables for resistance subsystem
L Length on waterline
B Beam
D Depth
T Mean draft
CB Block coefficient
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2.3.4 MODEL SUMMARY
Objective function:
Min (L, B, D, T, CB)
Subject to:
∇
2.4 MODEL ANALYSIS
Before sending the problem into the optimizer, it is necessary to conduct some initial
study on the objective function and constraints. Monotonicity analysis could be
employed to help determine if the problem is well bounded and determine active
constraints. However, monotonicity analysis about the resistance objective function is
not straightforward. From a resistance point of view, the increase of length for a given
displacement will reduce the wave-making resistance but increase the frictional
resistance. The effects of other variables in resistance function are also complicated. As
can be seen in Equation 2.1, the total resistance is a combination of different
components of resistance. It is hard to determine if the function is monotonically
increasing or decreasing with regards to one variable.
Monotonicity analysis was conducted for all the constraints listed in the optimization
problem statement. Table 2.3 showed the analysis results for each constraint. In the
table, a plus sign means that the constraint is monotonically increasing with regard to
the variable, while a minus sign means the opposite. The dots mean that the variable is
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not included in the given constraint. It is noticed that each variable has at least one
upper and lower bound.
Table 2.3: Monotonicity table
L B D T CB
g1 . . . + .
g2 . - . + .
g3 . + . - .
g4 - + . . .
g5 + - . . .
g6 . . . . -
g7 . . . . +
g8 . - + . .
g9 . . - + .
g10 . - . . .
h1 + + . + +
2.5 OPTIMIZATION STUDY
As the resistance fitness function is not computationally expensive and can be
computed quickly. The fmincon function that is programmed in Matlab could efficiently
solve this optimization problem. The Hollenbach method script, the constraints script
and a main script that ran the optimization program are sent into Matlab to get the
results.
Table 2.4: Optimal Design
Variables L B D T CB
Optimal 210.17 29.56 15.8 11.8 0.66
To verify that the solution is the global optimal, different starting points were used to
rerun the optimizer. The same result in Table 2.4 was obtained at every run, indicating
that the result is indeed the global optimal in the function space. Also the optimal
design hit the boundary of constraint g5, g9, and g10, this means that these constraints
are active constraints.
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2.6 PARAMETRIC STUDY
The resistance function is high related with ship traveling speed. In this subsystem study,
a design speed of 10.5 knots was used based on operation study, which is another
subsystem in the project. In order to better understand the change of design related
with speed, another optimization run was conducted. In this case, the design speed
was increased to 20 knots to run the optimizer. A new set of optimal solution was
returned and listed in Table 2.5.
Table 2.5: Optimal solution at 20 knots
Variables L B D T CB
Optimal 200 29.56 15 7.88 0.6
The block coefficient hit the lower bound when the speed increases, which is
engineering logical. The curve presented in Figure 2.2 illustrates the trend in basic
container ship block coefficient as a function of speed.
Figure 2.2: Block coefficient versus speed
2.7 DISCUSSION OF RESULTS
Even though the resistance model is simplified, the optimal solution is not a biased ship
design. When studying successful container ships, the ship length in this design indicates
that this ship is large enough to carry the design containers. The curve in Figure 2.3
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presents the relationship between ship length and total TEU based on regression analysis
of real-world container ship designs.
The design load of this container ship project is 2400 TEUs, and in the optimal solution the
length is 210m, meaning that the optimal design fits well in the regression model. In this
respect, the optimal design is valid which could fully satisfy the loading requirement.
Figure 2.3: LBP versus TEUs
The optimization result could be improved by further considering the ship’s
maneuvering and seakeeping performance. These would be competing objectives
compared with resistance optimization. The multi-objective optimization results could
degrade the resistance performance. However, the optimal design would be more
developed and balanced ship design.
2.8 SYSTEM INTEGRATION
The resistance subsystem interacts closely with all the other subsystems studied. The
optimal results calculated in this subsystem were sent to mid-ship structure strength
system. Also, with the design speed provided by the operation subsystem we can give
an estimated effective powering requirement to the propulsion system as a starting
point.
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This subsystem also takes feedbacks from the other subsystems. The integration of
propulsion and operation subsystems would generate an estimated fuel volume
needed to run this containership. To check that the optimal design hull has the
capacity to carry the containers and fuel volume required, a 3-D Rhino model was build
and a mid-ship section was plotted with design loaded containers inside. In Figure 2.5,
the space between the bottom and the inner bottom was design to filled with oil tanks.
After an approximation calculation, the design ship can successfully carry all the
containers and fuels. The design was considered to converge at this initial design level.
Figure 2.4: Rhino model for the ship hull
Figure 2.5: Mid-ship section with loaded containers
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3 PROPULSION SUBSYSTEM
3.1 PROBLEM STATEMENT
The prime mover selection for a vessel is a critical and essential portion of the total
vessel design. It marks a major milestone in the maturity of the design and is dependent
upon and influences several key performance parameters of the total system. This sub-
system will consist of several components that will have to be optimized and matched
for the desired performance of the overall system. The current effort will analyze a
variety of engine configurations in order to determine the optimal design configuration.
The original goal of optimizing the main engine, drive shaft, main reduction gear, and
the propeller, has been reduced in scope and limited to the main engine configuration.
The remainder of the proposed system can be handled with the aid of mark-downs for
efficiency losses.
Table 3.2 - Total Set of Engines Evaluated contains the 71 engines as a function of type
for the 714 viable architectural configurations. Each of these architectural
configurations was evaluated for each of the 71 engines. This yielded 50,964 scenarios
that need to be optimized to meet the prescribed constraints.
Table 3.2 - Total Set of Engines Evaluated
Main Engine
Type
Variants
Slow Speed
Diesel (SSDE)
41 engines, 7
families
Med Speed
Diesel
(MSDE)
24 engines, 6
families
Gas Turbine
(GT)
5 engines, 3
families
3.2 NOMENCLATURE
Variable Description
ni The configuration-vector a nine
position vector that allows each
position to vary from 0 to 4. However
the total component sum of the
vector is limited to 4.
A The coefficients that describe the
quadratic fuel loading curve for
each engine as a function of
percentage load
B
C
xi Percentage load applied to an
engine allowed to be continuous for
zero to one.
pi The maximum installed power for
Page | 19
each engine
wi The maximum installed weight for
each engine
PwrReq The total system required power in
order to make the prescribed speed.
3.3 MATHEMATICAL MODEL
3.3.1 OBJECTIVE FUNCTION
This is a multi-objective optimization problem. However system weight is a concern and
not a hard limitation. The prime mover system must deliver the required power in order
to maintain speed and minimize fuel efficiency across the required range. The two
functions that need to be minimized at the total fuel consumption and the total
installed weight. These functions are highlighted below.
The total fuel consumption is the summation of the product of the configuration vector,
the individual engine power at load, and the engine’s specific fuel consumption at
load. The engine power and the specific fuel consumption are both a function of the
applied load. This creates a summation of third order polynomials, represented in
Equation 3.1.
Equation 3.1
∑ (
)
The total weight installed is simply the summation of the individual engine weights
multiplied with the configuration vector. For this function the engine load is irrelevant.
Equation 3.2
∑
As the 50,964 scenarios are evaluated individually a Pareto front is developed for total
weight installed and total fuel consumption. The optimal points are a function of the
minimum required power from the system.
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3.3.2 CONSTRAINTS
The following initial constraints have been established. The installed power must be
greater than or equal to the total required power, Equation 3.3. This is required for the
vessel to make speed. The sum of the configuration vector will not exceed 4, Equation
3.4. In other words there will be no more than four installed engines. The engine
loading will be bounded between 0% and 100% of its maximum rating, Equation 3.5.
Equation 3.3
∑
Equation 3.4
∑
Equation 3.5
3.4 MODEL ANALYSIS
A monotonicity analysis for the generalized case illustrates that either constraint G1 or
G3 is active. Since constraint G3 would mean that the percentage load is zero this is
considered a trivial solution and removed from consideration. This implies that
constraint G1 is the active constraint. This result is consistent with modeling results for the
expanded case. Table 3.3 - Monotonicity Analysis for General Scenario highlights the
outcome of the generalized scenario analysis.
Table 3.3 - Monotonicity Analysis for General Scenario
F + ( ) Equation 3.1
G1 - Equation 3.3
G2 + Equation 3.5
G3 - Equation 3.5
Several simplifications have been incorporated into this model. The most notable is that
only one engine family is evaluated for optimality at a time. This mean there are no
multiple family solutions presented within any of this data. The second is that engines of
the same model within the family have been loaded identically. This means that if two
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or more of the same engine has been selected by a given configuration vector all of
these engines have the same applied loading. These simplifications are seldom true in
the marine market. Hybrid or multi-family solutions are employed extensively. This fact is
becoming more the standard as opposed to the norm as emission requirements
continue to become more stringent. Further if common engines are selected it is often
the case that the operator will choose to run one or more engine in a highly loaded
configuration with one on idle or standby in order to ensure fuel economy or
redundancy.
3.5 OPTIMIZATION STUDY
Input parameters to the optimization study include the required system power and the
fuel consumption curve coefficients. The following graphics were developed using a
required system power of 25,000 kW. This parameter was based on the vess that is to be
replaced by this proposed solution. Figure 3.1 - Engine Configuration Evaluations plots
the optimized solutions for every engine family for every possible configuration vector.
There are a few important items to note on this graphic. First the majority of the
solutions have converged to the minimum required power of 25,000 kW. This set of
converged solutions includes almost all of the diesel engine scenarios regardless of type.
The scattering of points above the 25,000 kW represent several of the GT scenarios. This
has occurred since the optimizer had minimized fuel consumption and the minimal
consumption for these combinations of engine family and configuration occur above
the minimum required power. Essentially these options consume more fuel and
produce more power than required for this application. There are some few points that
fail to meet the, 25,000 kW, requirement. These points are strictly infeasible for this
required power.
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Figure 3.1 - Engine Configuration Evaluations
Figure 3.2 - Best of Family plots the optimal solution for each family as a function of total
system weight and fuel consumption. This graphic represent the best, feasible, non-
dominated point for each family at the, 25,000 kW, power requirement. One can see
the development of a pseudo Pareto front. As expected the front follows an inverse
exponential pattern. However it does point out that if a closest to utopia approach
was adopted the MSDE group would be the preferred solution set for this power rating.
However if weight is not a constraining factor and fuel consumption is the only concern
then the SSDE group would be the preferred option. The only viable use for the GT
group would be in highly weight constrained applications, such as aircraft or other high
performance vessels.
0
2000
4000
6000
8000
10000
0
2
4
6
8
10
x 104
0
0.5
1
1.5
2
2.5
3
x 107
System Weight [mt]
Engine Evaluation
System Power [kW]
Fuel Load [
g/h
r]
V51/60DF
V48/60CR
V48/60B
L51/60DF
L48/60CR
L48/60B
S90ME-C8
G70ME-C9
S70ME-C8
G60ME-C9
K98ME-C7
K98ME7
S90ME-C9
LM1600
LM2500
LM2500+
LM2500+G4
MT30
Page | 23
Figure 3.2 - Best of Family
Figure 3.3 - Best of Type Fuel Consumption vs Load illustrates how the fuel consumption
of the three type optimal solutions will vary with respect to load. While at first glance
one could be tempted to approximate the MSDE and SSDE curves as linear, further
inspect reveals that there is an inflection point in each however the curvature is not
nearly as dramatic as the GT trend. Focusing on the diesel solutions one can also see
that the curves are essentially parallel and do not intersect. This would indicate that
there is not a portion of the loading profile that would change the optimal solution for
this power requirement. Additionally since there is not a large deviate between these
curves anywhere along the length the incremental advantage of one solution over the
other is not persuasive.
0 200 400 600 800 1000 12004
4.5
5
5.5
6
6.5x 10
6
System Weight [mt]
Fuel Load [
g/h
r]
Best of Family
V51/60DF
V48/60CR
V48/60B
L51/60DF
L48/60CR
L48/60B
S90ME-C8
G70ME-C9
S70ME-C8
G60ME-C9
K98ME-C7
K98ME7
S90ME-C9
LM1600
LM2500
LM2500+
LM2500+G4
MT30
GT
MSDE
SSD
E
Page | 24
Figure 3.3 - Best of Type Fuel Consumption vs Load
3.6 PARAMETRIC STUDY
Upon the completion of the modeling described above, it was desired to determine
the effect of the modeling input parameters on the solution set. To support this study
the input parameter of Required Power was varied from 75 MW to 5 MW in order to
observe the affect. These results are tabulated in Table 4.3 - Impact of Required Power.
As evidenced the family solution for the SSDE and the MSDE is remarkably stable. This
would indicate that these family solutions are superior and non-dominated with regards
to fuel efficiency characteristics. It is important to note that the configuration vector
however is highly volatile. The optimizer has chosen a solution that effectively ‘rides’ the
bottom of the fuel consumption curve and loads the engines accordingly. This
observation holds for both the SSDE and the MSDE types. The GT are much more
volatile and finely tuned as expected. While the diesel engines provide a broader
operational availability the GTs, are niche solutions for a very narrow band of
operational considerations. It would be interesting to determine if a bi-modal
operational constraint would produce similarly consistent results.
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Table 4.3 - Impact of Required Power
Power
Required
(kW)
SSDE ni pi MSDE ni pi GT ni pi
75,000 G60ME-C9 2
0
0
2
13,400
16,080
18,760
21,440
V48/60CR 0
0
0
4
14,400
16,800
19,200 21,600
LM2500+ 3 30,201
65,000 G60ME-C9 0
0
2
1
13,400
16,080
18,760
21,440
V48/60CR 0
2
0
2
14,400
16,800
19,200
21,600
LM2500+G4 2 35,324
55,000 G60ME-C9 0
3
1
0
13,400
16,080
18,760
21,440
V48/60CR 0
0
0
3
14,400
16,800
19,200
21,600
LM2500+ 2 30,201
45,000 G60ME-C9 0
0
3
0
13,400
16,080
18,760
21,440
V48/60CR 0
2
1
0
14,400
16,800
19,200
21,600
LM2500 2 25,056
35,000 G60ME-C9 0
0
0
2
13,400
16,080
18,760
21,440
V48/60CR 0
0
1
1
14,400
16,800
19,200
21,600
MT30 1 36,000
25,000 G70ME-C9 0
0
0
1
18,200
21,840
25,480
29,120
V48/60CR 2
0
0
0
14,400
16,800
19,200
21,600
LM2500+ 1 30,201
15,000 G60ME-C9 0
0
1
0
13,400
16,080
18,760
21,440
V48/60CR 0
1
0
0
14,400
16,800
19,200
21,600
LM1600 1 14,914
5,000 G60ME-C9 1
0
0
0
13,400
16,080
18,760
21,440
L48/60CR 1
0
0
0
7,200
8,400
9,600
10,800
MT7 1 5,000
Page | 26
3.7 RESULTS
The Required Power parameter ultimately was 2,831 kW. This value was rounded to
3,000 kW to allow for some modest margin for unaccounted for shafting and tranmission
losses. The following three figures reanalyize the base data set in a similar manner that
was previously presented with the finalized parameters. Figure 3.4 - Best of Family Final
illustrates the same pareto front of Fuel Consumption versus System Weight analogous
to Figure 3.2. One can immediatle see that the required powere reduction as
subsequently reduced the ‘gaps’ between engine types and flattened the overall
curve. This is expected since fuel consumed and power required are directly
porportional. Since these ‘gaps’ have closed so significantly it is necessary to expand
the lower corner near the utopian point, see Figure 3.5.
Figure 3.4 - Best of Family Final
0 200 400 600 800 1000 12000.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2x 10
6
System Weight [mt]
Fuel Load [
g/h
r]
Best of Family
V51/60DF
V48/60CR
V48/60B
L51/60DF
L48/60CR
L48/60B
S90ME-C8
G70ME-C9
S70ME-C8
G60ME-C9
K98ME-C7
K98ME7
S90ME-C9
LM500
LM1600
LM2500
LM2500+
LM2500+G4
MT30
MT7
TF40
ETF40B
TF50A
GT
MSDE SSDE
Expanded
below
Page | 27
Figure 3.5 - Best of Family Final Expanded
From Figure 3.5 - Best of Family Final Expanded one can determine that the G60ME-C9,
L48/60CR, and TF40 are the preferred solutions. These results continue to illustrate the
trend stability in the family solution of the parametric study. Also in this case the
configuration vector is the same as presented for the 5 MW case within the parametric
study. Once again the GT is more sensitive to the input parameters; as such the
preferred solution is again different. Since this is an input to the other areas of the
project consumption versus fuel loading curve was developed for the preferred engine
selections. Figure 3.6 - Best of Type Fuel Consumption vs Load Final depicts the trends of
fuel consumption versus engine loading. The chart is similar to Figure 3.3, only in this
case there are very mixed results. There is no single dominant engine choice; therefore
the selection will largely depend on other operational considerations. Also it is worth
noting that the 50% load for all three engine types does not equate to a common
power level the minimum required power of 3,000 kW has been marked with a star on
Figure 3.6. At this point it may be prudent to consider additional smaller diesel engines
in order to further develop the design space and evaluate all options.
GT
MSDE
SSDE
Page | 28
Figure 3.6 - Best of Type Fuel Consumption vs Load Final
In conclusion Table 3.5 – Cost Comparison of Final Selections documents the most
preferred solution of each type, its associated variables, and an estimated cost. The
estimated cost is based on total installed horse power and was developed from a
regression fit of diesel engines. Its applicability to aero derivative gas turbines is suspect
and should be reevaluated when time allows.
Table 3.5 – Cost Comparison of Final Selections
Costs CYL KW n A B C hp $M
L48/60CR 6 7200 1 129.52 -186.29 239.76 9655.359 60.9E+6
G60ME-C9 5 13400 1 84 -115 202 17969.7 110.8E+6
TF40 2983 1 1480.8 -2340.7 1218.8 4000.269 27.0E+6
Page | 29
4 SHIP MIDSHIP STRUCTURE SUBSYSTEM
4.1 PROBLEM STATEMENT
The structure of the vessel must be adequately design such that it can withstand the
environment it is exposed to in a safe and reliable manner. At the same time, designers
seek to remove parasitic weight, often adding little strengthening effects while reducing
the efficiency of the transportation system. In order to make a significant reduction of
cost in the overall operation and lifespan of the newly designed container vessel, a
proper optimization of the ships primary hull structure will be conducted. For vessels like
a containership, a large percentage of the body is referred to as the parallel midbody,
where little to no significant curvature at the design waterline occurs. The structure in
this region is the main load bearing portion of the vessel, responding to the critical
bending and pressure loads. The proposed system takes a basic model of the midship
section and optimizes it for cost while constraining it to a set of published classification
standards.
Figure 4.7: Example Midship Containership Section5
Modern classification rules have been developed to incorporate traditional design
wisdom and tradeoffs as well as more modern state of the art analysis to quickly and
efficiently evaluate the design.
4.2 NOMENCLATURE
To model this structure, a series of structural elements are arranged in such a manner to
effectively recreate a midship section. The most basic element is a plate/stiffener
combination, often referred to as a stiffened T-panel. The T panel, Figure 4.8, is defined
by 5 primary characteristics, plate thickness, web thickness, web height, flange
thickness, and flange breadth.
5 Source: http://upload.wikimedia.org/wikipedia/commons/e/e9/General_cargo_ship_midship_section_english.png
Page | 30
Figure 4.8: Stiffened T Panel Geometry6
These stiffened T panels are arranged from span end to span end to form grillages,
often incorporating transverse members. For the purpose of this model, only
longitudinal effects are modeled. The grillages may then be arranged in location with
orientations to form a basic structure. Additional plates referred to as structural details
are often added to add structural integrity where needed, whether it be to decrease
the span of the overall grillage or to add moment of inertia to prevent deformation.
The problem presents designers with a decision to make; Either attempt to model each
dimension independently, thus producing 5 variables for every stiffened T panel,
multiplied by the number of panels in a grillage multiplied by the number of grillages in
a section, or seek to reduce the dimensionality of the problem. Choosing the second
option, stiffener libraries are often employed, this making the selection of stiffener
dimensions a single variable. For this problem, the following libraries were used:
Table 4.6: Stiffener Library (mm)
Dimension Stiffener
1
Stiffener
2
Stiffener
3
Stiffener
4
Stiffener
5
Stiffener
6
Stiffener
7
Stiffener
8
Web
Thickness
2 4 4 6 6 10 10 12
Flange
Thickness
2 3 4 5 6 8 10 10
Web
Height
100 200 200 300 300 400 400 500
Flange
Breadth
40 40 80 50 90 60 100 120
Table 4.7: Plate Library (mm)
Dimension Plate 1 Plate 2 Plate 3 Plate 4 Plate 5 Plate 6 Plate 7 Plate 8
Plate
Thickness
2.5 5 10 15 20 25 30 35
6 http://ars.els-cdn.com/content/image/1-s2.0-S0167473012000409-gr1.jpg
Page | 31
Furthermore, a working model of the ship is required. For structural purposes, the ship
will be defined by a series of parameters to be used in strength definitions. The
parameters Length(L), Beam(B), Draft(T), Depth(D), Block Coefficient ( ), and velocity
will be used for calculation of the strength constraints.
4.3 MATHEMATICAL MODEL
Based on the aforementioned scenario, a mathematical cost model will be developed
and optimized. For the structural scantlings, this project seeks to minimize total structural
cost, .
Equation 4.1
( )
A primary reference, Rahman and Caldwell, advocated a simplistic model in 1995
considering the plate and stiffener combination. Resultant cost is calculated based on
the volume of material, ( ) used in the design and a weighting factor on
the weld length, ( ). Specifically, 4 partial costs contribute:
Equation 4.2
The optimizer must balance the desire to utilize many smaller stiffeners versus single
larger stiffeners. The weighting factors will be critically important in determining the
design space. Experimental results have suggested that the cost is most dependent on
the material cost assumed for the vessel. At presents the cost used in the modern cost
per weight value for mild steel. This model as described in the 1995 publication has
been replicated and implemented. All parameter inputs are listed below.
Table 4.8: Cost Model Coefficients
Cost Coefficients Material Cost ($/ton) 800.00
Material Density (ton/m^3) 7.85
Stiffener cost Coefficient 1.05
Labor Rate ($/hr) 27.00
Weld rate (m/hr) 1.2
Fabrication rate (m/hr) 1.5
Electric Utilization rate (m/hr) 0.9
Page | 32
The objective will be constrained by multiple structural rules (ABS, 2013). The most basic
and design driving constraint is one on Section moment of inertia, .
Equation 4.3
( )
This minimum sectional moment of inertia is responsible for resisting the bending force
applied in the maximum design sea state (Hughes & Paik, 2010). Further granularity
indicates:
Equation 4.4
( )
This further reduces to:
Equation 4.5
( ) ( )
Where:
Equation 4.6
(
)
Or
And:
For the initial pass design. Thus we expect for the initial design, the require longitudinal
vertical moment of inertia in units of m^4 to be
At the individual plate level, the structural requirement is often that of pressures.
Specifically, the different regions as defined by the previously mentioned regions affect
the maximum design pressure hull plating must withstand.
Equation 4.7
( )
Page | 33
This ensures that regions of plating will remain elastic during previously calculated
slamming loads. Specifically, the following values were applied:
Table 9.4: Minimum Plate Thickness Constraints by Location
Location: Minimum Thickness Equation Minimum thickness resultant dependent
on spacing
Side Shell
(
) √( ) (
) ( )
( )
Bottom Shell
(
) √( ) (
) ( )
( )
Strength
Deck
( ) ( )
The final structural consideration is for stiffener buckling or tripping a failure mode in
which excess displacements of the stiffener web cause the structure to become
unstable and collapse. We require that:
Equation 4.8
( )
Where (248 MPa) is the yield stress of mild steel and E (217 GPA) is the elastic modulus
of mild steel and l (2 m) is the transverse frame spacing determining the un-bisected
length of the stiffener member. Again this ensures through empirical results that the
stiffener will remain in a load bearing capacity and thus structurally sound. We apply a
yield stress and elastic modulus of steel. Numerically, this constraint is defined to be a
value of .
This gives us the optimization expression:
( )
( )
( )
4.4 MODEL ANALYSIS
For the structural subsystem a notional hull was created to test the optimizer on.
Ideally this hull would closely mirror the future integrated ship so that changes within
Page | 34
parameters were kept to a minimum Model geometry was formulated using the
following ship hull characteristics (All Values in meters):
Table 4.10: Ship Characteristics
Length Beam Draft Depth Block
Coefficient
Inner bottom
Height
Bilge
Radius
Overhang
320 32 12 20 0.6 1.5 2 1
Due to the discrete nature of the libraries, gradient based optimization is somewhat less
useful and patches would be required to work around the discrete nature of the
problem. Instead, the problem was optimized using evolutionary algorithms. Initial
attempts were made to analyze the structure using Matlab and its Optimization toolbox.
However after multiple attempts, python was adopted as a base computing language.
The optimizer selected was an open source variant of the Inspyred repository Single
Objective Genetic Algorithm (SOGA). A 90 bit chromosome with 30 genes, interprets
integer inputs to create 8 stiffened panels requiring 9 bit places and 6 (7 shown, though
the bilge radius plate thickness is slaved to its neighboring bottom shell thickness value)
plate values requiring 3 bits.
Figure 4.9: Representative 9 bit Grillage Gene with Binary input on top and Discrete
Library Values on Bottom Each gene corresponds to an independent variable, selecting 1 through 8 from the
associated libraries. For a given grillage, the first 3 bits correspond to the plate thickness,
the second 3 bits correspond to the stiffener type and the third 3 bits correspond to the
number of stiffeners on the given grillage. Due to the indexing standard in python, the
binary conversion has a factor of +1 as python relies of 0 indices. A post processing
script produces the following image, where stiffened panel plates are shown in black,
the details are shown in blue and the stiffeners in red.
Page | 35
Figure 4.10: Representative Test Geometry (Dimensions in millimeters)
This geometry balances the tradeoff between granularity and producibility. The
problem becomes more simplified and manageable as the plate sections become
longer and longer. At the same time in the most complex case each individual stiffener
would be appropriately sized for its specific ship location. The decision was made to
limit maximum plate sizes to 10 m for producibility reasons. Ten meters is a rough
standard maximum size of industrial panel lines, though the exact length will vary from
manufacturer to manufacturer. This resulted in the largest plate being 9500 mm in
length.
The costing model applied normalization with external penalty terms for each violation
of the penalty:
Equation 4.9
∑
The model produced 17 net penalty terms, 2 terms per each of the 6 grillages and 1
term for the global moment of inertia. The external penalty term method was selected
for its ability to contain the infeasible region and keep the optimizer searching feasible
space and its edge. In this manner, the genetic algorithm was shown to remain most
Page | 36
stable, and the normalizing factor of 50,000 was determined empirically to scale the
results. The following figure shows a basic pseudo-code for the determination of
structural cost:
START
DECODE BINARY GENES INTO
DISCRETE LIBRARY ENTRIES
ASSEMBLE SECTION GEOMETRY FROM
INPUT LIBRARY PARAMETERS
CALCULATE VERTICAL CENTROID
FOR EVERY GRILLAGE WITHIN
THE SECTION:
DETERMINE REQUIRED AND EFFECTIVE PLATE THICKNESS
DETERIME EFFECTIVE STIFFENER MOMENT OF INERTIA TO AREA RATIO
CALCULATE GLOBAL EFFECTIVE MOMENT
OF INERTIA
DETERMINE UNPENALIZED SECTION COST
IS EFFECTIVE THICKNESS > REQUIRED
THICKNESS
ADD PENALTY TERM
IS EFFECTIVE INERTIA TO AREA RATIO < MAXIMUM
RATIO
NO
YES
YES
NO
ADD PENALTY TERM
IS GLOBAL MOMENT OF INERTIA > REQUIRED MOMENT OF INERTIA
NO
ADD PENALTY TERM
YESSUM PENALTY
TERMS AND ADD TO UNPENALIZED COST
RETURN RESULTANT COST
STOP
Figure 4.11: Structural Optimization Pseudo code Flow Chart
The model was created and run in python to test the bounded-ness and output of the
costing function. Four Random seeds were taken and used to run the SOGA. The
following algorithm parameters were used:
Table 4.11: Single Objective Genetic Algorithm Parameters
SOGA Parameters
Chromosome size (bits) 90
Crossover operator Single point per gene
Crossover rate 0.95
Mutation rate 0.01
Mutation Exponent 4
Population size 1000
Max Generation Number 100
Random Seed 1021,1022,1023,1024
Elitism Percentage 1
Selector Mechanism Two-pass Tourney
Page | 37
Initial results demonstrated that without a more complex model of the structure and at
the given test dimensions, the global moment of inertia constraint would not be met.
As only the primary longitudinal structure without tween deck is modeled, constraint
relaxation was necessary for this model. Therefore, the constraint was reduced to a
value of 50% of the nominal result or 220 m^4. The analysis was then repeated.
4.5 OPTIMIZATION REULTS
Using the modified constraint, the structure was developed and tested with the 4
random seed inputs. The results are shown below:
Table 4.12: Random Seed Optimization Results
Initial System Results
Random Seed 1021 $379,150 per m $140,740,000
Random Seed 1022 $379,150 per m $140,740,000
Random Seed 1023 $379,150 per m $140,740,000
Random Seed 1024 $379,150 per m $140,740,000
The four random seeds selected all converged to near identical values (within $100
which is beyond the accuracy of the model). The minimum cost of the structure per
unit length was determined to be $379,150 per meter. Applying a 1.4 complexity factor
for the high curvature in the bow and stern regions and assuming a standard
containership with 60% parallel midbody, this equates to an initial ship cost of
$140,740,000. This value is slightly high as compared to modern containerships, but the
labor rate within the United States, mandated by the Jones act is roughly 2x to 2.5x that
of competing ship yards in China. The resulting structure appears below:
Page | 38
Figure 4.12: Initial Optimized Midship Half Section
4.6 PARAMETRIC STUDY
As previously mentioned the cost model is well studied and it is understood that
the cost of the material will drive the ultimate price of the structure, with labor rate
playing a secondary role. The critical sensitivity observed in this optimization is the input
ship dimensions and their effect on moment of inertia globally. The 320m length and 32
m beam cause the global required moment of inertia to be far too high for a structure
prototyped in this manner. In fact, this is part of the reason tween decks are often used
in ships of this scale, as they contribute to the moment of inertia. The results show that
due to the requirement to meet that strict constraint, the bottom shell plating thickness,
and the overhangs, which have the greatest effect on the moment as these
components are farthest from the vertical centroid are increased in thickness until the
constraint is met. This causes the hull bottom to have a high moment of inertia to area
ratio locally. This in turn causes the number of stiffeners per grillage to increase and the
span of the hull plate to decrease, keeping the constraint unviolated. Indeed we see
numerous very small stiffeners in the strength deck for the same reason. On the side
shell, where the thickness has less effect on the global moment of inertia, we see a
decrease plate thickness. This decreased thickness allows the stiffeners to increase in
size. Therefore, we expect the global moment of inertia to have the greatest sensitivity.
Decreases in magnitude will have a drastic effect on the resulting output. And initially,
Page | 39
the hullform subsystem communicated that there would be large changes to the
dimensions causing the global moment to decrease.
4.7 SUBSYSTEM INTEGRATION
The structural subsystem integrates mainly with the Resistance and Hullform
Subsystem. Again using the design spiral approach, the following parameters were
used for a final optimization:
Table 4.13: Final Optimization Hull Parameters
Length Beam Draft Depth Block Coefficient Inner bottom Height
210 29.56 11.8 15.8 0.66 1.5
This critically changes the constraint parameters. The new required global moment of
inertia becomes 110.54 m^4. The minimum thicknesses are affected in the following
way:
Table 4.14: Updated Minimum Thickness Constraints
Location: Minimum Thickness Equation Minimum thickness resultant dependent
on spacing
Side Shell
(
) √( ) (
) ( )
( )
Bottom Shell
(
) √( ) (
) ( )
( )
Strength
Deck
( ) ( )
In general we see a decrease or relaxation of constraints as compared to the initial
study and development. Again from the parametric study we expect this to drastically
change the optimized structure. To attempt to model a better notional geometry, the
outputs of the resistance calculation were mocked up in the ORCA3D suite of
Rhinoceros. The resulting final midship section as considered by the structural subsystem
is shown below.
Page | 40
Figure 4.13: Representative Half Section for final Resistance Outputs
This section was best modeled by a total of 10 grillages and 5 structural details. Note,
stiffener intersections were assumed to be insignificant in this updated geometry. This
added to the length of the chromosome making it 105 bits long, with 35 3 bit genes.
This geometry may be viewed below in the finalized output, Figure 4.14.
Again, identical parameters were used for the SOGA. The same random seeds were
run to completion. The results are shown below, this time with some variability in the
optimized solution.
Table 4.15: Random Seed Optimization Results
Initial System Results
Random Seed 1021 $296,700 per m $72,276,000
Random Seed 1022 $296,820 per m $72,305,000
Random Seed 1023 $296,750 per m $72,880,000
Random Seed 1024 $296,660 per m $72,266,000
The most exciting thing to see is that the global moment of inertia constraint and the
finalized ship cost scale linearly. In the most efficient structures, there is no parasitic
material, so a direct change to a driving constraint will have the same effect on the
Page | 41
structure. The global moment of inertia was halved, and in turn, the cost was roughly
halved, from $140 MM to $ 72 MM. This would seem to indicate that the program is
performing very well with the new design. Additionally viewing the outputs, one can
deduce where the relaxation has an effect.
Figure 4.14: Final Structurally Optimized Midship Section
First, we can see that the optimizer seeks to move steel away from the centroid of the
section, located roughly 5 meters above the baseline. The stiffeners in the inner bottom
have grown in web height because the relaxation in thickness has allowed more
contribution to the stiffener moment of inertia. The new division of geometry has also
shifted some of the location of stiffeners around and as grillages away from the
centroid grow their plate thicknesses, the other constraint respond accordingly. We see
that in Figure 4.15 the last two genes are both 7 corresponding to the thickest detail
plate thickness in the overhangs at the top of the structure. Again this corresponds to
the desire to raise the moment of inertia of the structure as efficiently as possible and
will cost less than a stiffener as the stiffener has 105% of the cost of plate steel.
5 7 5 5 4 5 4 6 5 4 0 5 4 3 0 6 0 5 3 5 0 4 4 5 4 1 3 7 7 0 4 7 4 7 7
Figure 4.15: Real Coded Gene Values for Optimized Midship Section
The Value of $72 MM is a very good real world estimate for early stage design. One
would expect the structural cost to be about $30 MM for a highly optimized
containership of this size from China and with the mark up for US labor prices, this seems
to fit accordingly.
Page | 42
5 REGULATORY AND OPERATIONAL REQUIREMENTS SUBSYSTEM
5.1 DESIGN PROBLEM STATEMENT
Since all ships must meet certain regulatory and operational requirements, optimization
in this arena can be a bit of a challenge. For the purposes of this project, this subsystem
is defined in two parts. The first is optimizing the volume of fuel that must be carried
onboard from the beginning of the trip to make the entire loop. This portion takes into
account that the ship may encounter storms and must have enough fuel to avoid the
storms and still deliver the cargo. The second part is defining regulatory and operational
constraints for the entire project. Since fuel is the second largest cost of operating a ship
(personnel costs being the first), fuel volume optimization is an area where significant
savings can be realized. This system has been simplified by disregarding all regulatory
safety factors, ballast needs, and the effect of fuel consumption on the trim of the
vessel. These assumptions definitely impact the ability to model reality properly, but this
model would still be useful as a starting point for a more sophisticated optimization
project.
The relevant regulations for this project are those published by the class society, the
American Bureau of Shipping, federal regulations as enforced by the US Coast Guard
and international regulations as published by the International Maritime Organization.
For the purposes of this optimization, the only regulations that will be discussed are those
that pertain to the environment. This decision was made because environmental
regulations will drive the cruising speed (fuel consumption, route, scheduling, etc.),
propulsion prime mover selection (steam/diesel/diesel electric) and other equipment
decisions. All regulations regarding ship structure, minimum engineering design
standards, and other aspects of ship design will be considered constraints of the design
space and uninteresting as a topic for discussion. If this project were to result in an
actual ship, state and port regulations might also have to considered depending on the
route the ship was taking.
The regulations that are primarily of interest are MARPOL 73/76 (international) and 40
CFR 94 and 1042 (domestic). MARPOL 73/76 regulates all pollution from ships; everything
from oil or hazardous waste to various types of air emissions. Annex VI of MARPOL 73/76
contains the international regulations for all air emissions, primarily sulfur oxides (SOx)
and nitrous oxides (NOx) in exhaust gases and chlorofluorocarbon (CFC) refrigerants.
Since this project will only be driven by exhaust gas emissions, this report will disregard all
other types of pollution since those regulations pertain to treatment equipment, record
keeping and proper operation of the ship. These regulations (both domestic and
international) dictate what type of engines new ships are allowed to use. The engines
must meet a certain pollution standard for NOx and SOx by a certain year. The provision
that applies to the ship that results from this design project is Tier 3, which comes into
force in 2016. This means that any engine the ship uses must not emit more than 1.96-3.4
Page | 43
grams of NOx per kilowatt-hour the engine produces, 2 grams of hydrocarbons per
kilowatt-hour and 5 grams of carbon monoxide per kilowatt-hour. Currently, SOx
emissions are controlled by legislation regarding the sulfur content of the fuels used in
the engines. For context, current limits for NOx emissions are 9.8-17 grams per kilowatt-
hour. The domestic regulations contain the exact same standards as MARPOL 73/76.
The ships are permitted to use various types of emissions-reducing technology including
exhaust gas scrubbers to meet these standards. This aspect of this project will address
the propulsion plant holistically, rather than trying to optimize the combination of
emissions-reducing components. However, this project will optimize the volume of fuel
carried based on number of nautical miles travelled and the probability of needing to
divert from the Great Circle Route due to weather or other mishap. The safety factor will
be disregarded to allow the problem to be optimized.
5.2 NOMENCLATURE
All variables for this project are outlined in Annex I. The variables needed for this system
are the speed of the ship on each leg of the trip (velocity, V, meters per second),
distance travelled (Dt, meters), specific fuel consumption of the engine(s) at the
different speeds (SFC, cubic meters per kilowatt of engine output) and the sea state (Ss,
Beaufort Scale). The outputs of this optimization will be the final fuel cost for a full tank
(Fc, $) and the final fuel volume (Fv, cubic meters) that will be given to the hull resistance
subsystem and the propulsion subsystem at various stages in the project to aid in the
project’s progress through the design spiral. The final cost for fuel will be part of the final
cost of the ship.
5.3 MATHEMATICAL MODEL
5.3.1OBJECTIVE FUNCTION
The objective function will be a function of specific fuel consumption, speed, distance
travelled, and weather encountered in the ship’s path. Specific fuel consumption and
speed will be constraints determined analytically given the demands of the schedule
and the specifications of the propulsion engine(s). Instead of modeling the weather as
a continuous function of some sort, the model calculates the individual fuel cost and
volume values given a specific scenario. For each engine set, there will be sixteen
scenarios; eight to model the combinations of storm/no storm on each leg and two sets
of eight to model the ship going in a loop versus returning to the previous port. Please
see the map diagram on the following page. The extra distance is captured in the Dt
variable.
Page | 44
Equation 5.1
( ) ( ) ( )
Equation 5.2
( ) [
] [
]
[
]( [
]
)
The equation above is modified to calculate the loop trip costs by replacing the matrix
after the storm matrix with the values for the loops. The values in the matrices are
number of days since specific fuel consumption is units of cubic meters per kilowatt
hour.
Table 5.1 – Storm Matrix
TacomaOakland OaklandHonolulu HonoluluTacoma
1 1 1
1 1 0
1 0 1
1 0 0
0 1 1
0 1 0
0 0 1
0 0 0
In the storm matrix, a value of one indicates that the ship encounters a storm, a value of
zero indicates that the ship does not encounter a storm.
Page | 45
Figure 5.1 – Leg and Loop Duration
The percentages indicate the percentage of sea speed. For example, 100% of sea
speed = 10.5 knots. 75% sea speed = 8 knots.
The cost of fuel is given as $650 per metric ton since that is approximately the current
market price of marine diesel fuel. This type of fuel was selected since higher grade
fuels are not necessary for proper operation of the vessel and lower grade fuels do not
comply with environmental restrictions as outlined previously.
5.3.2 CONSTRAINTS
Since the regulatory constraints and the realistic smaller tank volumes are being ignored
in favor of optimizing one volume, there are few physical constraints. This problem is a
minimization problem that will be bounded by a minimum volume calculated from the
shortest distance the ship must travel (once around the loop, no diversions). However,
this bound should not be active because the realistically optimal volume of fuel is more
than that volume. This constraint is a practical constraint. The practical constraints for
this problem come in upper and lower bounds (or even given specific values) for the
specific fuel consumption and ship’s velocity. The distance travelled and sea state will
be outputs of the probability function with the maximum sea state the ship can
withstand set at 9 on the Beaufort Scale.
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5.3.3 DESIGN VARIABLES AND PARAMETERS
Table 5.2 –Variable Nomenclature
Symbol Definition Bound
Dt Distance Travelled Minimum = entire loop
Fc Final Fuel Cost Minimum = cost of fuel to travel one loop,
maximum = cost if there are storms on each leg
Fv Final Fuel Volume Minimum = 138.75 m3, maximum = 672.79 m3
SFc Specific Fuel Cost $650 per metric ton
SFC Specific Fuel
Consumption
Specified for chosen engine(s)
All variables used in the model are described in the table above. However, it should be
noted, that as the model is developed, other variables and parameters might be
added. At this point, the problem has two degrees of freedom. One feasible solution is
the minimum volume calculated based on perfect weather.
5.3.4 ASSUMPTIONS
This model has been simplified through use of several assumptions:
If the ship encounters a storm, the storm is assumed to be located such that the ship
cannot complete delivery of the cargo on that leg. This choice was made to ensure
that the optimization only analyzed the worst case scenarios.
The ship travels at a constant speed (sea speed) from sea buoy to sea buoy. This means
that the model does not have to take maneuvering in and out of port or in-port fuel
consumption into account. This assumption simplifies the model so that the optimization
can use readily available data rather than making potentially unreasonable
assumptions about the behavior of the machinery at low speeds.
The number of days of each storm avoidance loop was calculated using the number of
days that storm events lasted as indicated by the data. For example, if the storm event
lasted three days, the storm avoidance loop duration was set to three days. This ensures
that the optimization only takes into account the worst case scenario.
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5.3.5 MODEL SUMMARY
( ) ( ) ( )
Equation 5.3
( )
( )
The number of days for the 75% leg and the loop are assumed based on data analysis.
The data indicated that at the various locations, unacceptable weather would occur
for a certain number of days. The cumulative probability for each trip was calculated
by multiplying the probabilities for each respective leg of the trip.
5.4 MODEL ANALYSIS
The only constraint that is currently active is where the risk cutoff is placed. At this point,
the design is proceeding by only considering the three most likely storm profiles for both
the loop and multi-trip scenarios.
As could be predicted, the lowest storm probability is the most expensive, and the
highest probability is the least expensive. However, the hump in the middle of the curve
is interesting and was unexpected. The volumes and costs were calculated using an
assumed specific fuel consumption value. The chart below was developed using the
data for the medium speed diesel propulsion plant. However, the shape of the curve is
the same for each propulsion plant.
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Figure 5.2 – Cost vs Probability Chart
The final calculations for the cost of each scenario for each type of propulsion plant are
included in the table below. At the bottom of the table, the minimum, mean and
maximum values are calculated to allow the designer to make inferences about the
data. The colors in each column capture the relationship between the values in that
column. The red values are the most expensive scenarios, the green values are the least
expensive and the yellow and orange values fall somewhere in between. The row of
the storm matrix and the ship navigation strategy that correspond to each scenario are
listed in the same row as the various cost values. The table is organized from lowest to
highest and then highest to lowest probability.
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Table 5.3 – Final Data Table
Storm Matrix
Scenario Probability Med. Spd. Slow Spd Gas Turbine TO OH HT L/MT
1 0.36% $155,197.63 $74,022.64 $300,162.72 1 1 1 Loop
2 0.98% $137,375.90 $66,355.89 $262,474.13 0 1 1 Loop
3 1.42% $119,554.16 $58,689.13 $224,785.54 1 0 1 Loop
4 3.91% $101,732.42 $51,022.38 $187,096.94 1 1 0 Loop
5 4.98% $146,286.76 $70,189.27 $281,318.43 0 0 1 Loop
6 13.69% $128,465.03 $62,522.51 $243,629.83 0 1 0 Loop
7 19.91% $110,643.29 $54,855.75 $205,941.24 1 0 0 Loop
8 54.76% $92,821.55 $47,189.00 $168,252.65 0 0 0 Loop
9 54.76% $92,821.55 $47,189.00 $168,252.65 0 0 0 Multi-Trip
10 19.91% $141,088.76 $69,404.72 $264,717.50 1 0 0 Multi-Trip
11 13.69% $199,009.41 $96,063.60 $380,475.32 0 1 0 Multi-Trip
12 4.98% $247,276.61 $118,279.32 $476,940.17 0 0 1 Multi-Trip
13 3.91% $179,702.52 $87,177.31 $341,889.38 1 1 0 Multi-Trip
14 1.42% $227,969.73 $109,393.03 $438,354.23 1 0 1 Multi-Trip
15 0.98% $285,890.38 $136,051.90 $554,112.05 0 1 1 Multi-Trip
16 0.36% $334,157.58 $158,267.63 $650,576.90 1 1 1 Multi-Trip
Min 0.36% $92,821.55 $47,189.00 $168,252.65
Mean -- $168,749.58 $81,667.07 $321,811.23
Max 54.76%
$334,157.58 $158,267.63 $650,576.90
5.5 OPTIMIZATION STUDY
The output of this optimization is the data in the table above, but such a table is hardly
useful. When rearranged such that a Pareto front can be plotted, the data is much
more useful. In this case, the Pareto front is defined as the percentage of cases
covered (X-axis) at what cost ($, Y-axis). This plot, for all three propulsion plants, is
included on the following page. Since this is a stochastic model and the optimization
function produces sixteen discrete values, none of the usual optimization validation
tools apply (such as KKT conditions and local vs. global solutions) and each solution is
feasible. In this case, each iteration of the model calculations led to a higher degree of
reality. The first few rounds used assumed values for the speed, fuel consumption and
loop distances.
As data was gathered and analyzed, speed and loop distances became more
accurate. After the most recent round of optimization, the model was expanded to
Page | 50
include three types of propulsion and utilized the most accurate specific fuel
consumption and power values.
Figure 5.3 – Cost vs. Risk Pareto Fronts
5.6 PARAMETRIC STUDY
The only parameter that can be changed to change results is the specific fuel
consumption. This implies that the two project design variables that have the most
effect on the optimization of the fuel volume are the prime mover’s power (slow speed
diesel, medium speed diesel or gas turbine) and that prime mover’s specific fuel
consumption. Changing these values has a significant effect on the output of the
objective function. This can easily be seen in the chart above (Cost vs. Risk, Three
Propulsion Types), since each type of propulsion has a different power and specific fuel
consumption rating. The data for these propulsion types are outputs of the optimization
performed by Jason Strickland and thus this part of the design has no control over those
parameters. The only other parameters that could conceivably affect this optimization
are the cost of fuel and the speed of the ship. Since the speed of the ship is fixed due to
the schedule, this parameter cannot be changed. Fuel cost will go up over the life of
the ship, but it will simply shift the entire curve up, it will not change the shape of the
curve.
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5.7 DISCUSSION OF RESULTS
The curves that resulted from this optimization lead to an interesting optimization
problem. Essentially, the decision must be made to cut off the curve at some point and
that is where the optimum solution exists. For each mode of propulsion, this cut off
occurs at 62.5% of cases. On each curve, the slope increases after that point which
indicates that any increase in cases covered will cost more than before. Put a different
way, covering one more case before 62.5% of cases is cheaper than covering one
more case after 62.5% of cases. Since this is true of each propulsion plant, the optimum
values are in the following table:
Table 5.4 – Engine Costs
Med. Speed Diesel Slow Speed Diesel Gas Turbine
$155,197.63 $74,022.64 $300,162.72
Clearly, the optimal propulsion plant is the slow speed diesel since the cost at the
optimal coverage is so much lower than the other two options. This analysis is born out
in reality in that container ships are almost always powered by slow speed diesels.
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6 SYSTEM OVERVIEW CONCLUSION
True to the spiral nature of the design process, to converge the total system, Hullform
and Resistance was selected to control the critical parameters used by the other
systems. The subsystems in structural and propulsion required geometric and resistance
outputs to conduct final optimizations of their respective architectures. Geometrically,
the optimized result showed a 26 meter decrease in length and two meter increase in
beam. This is likely due to previous limiting constraints on crane arm reach when the
legacy Horizon Pacific was designed. Draft increased by 0.7 meters and this may be
due to the specific nature of the route considered for the new replacement ship.
Volumetric constraints on both fuel and TEU capacity were shown to be inactive.
The parameters of the other systems were then optimized using the existing framework
to be combined in a linear manner to determine total cost as described in Equation 6.1
below.
Equation 6.1
( )
The structural and engine costs are grouped together because they are initial costs
associated with building the ship. The fuel cost was optimized on a per-trip basis,
making it an operational cost. The final structural cost is anticipated to be slightly higher
than initial estimates based on additional curvature of geometry. The structural cost is
estimated to be $72.3 million. The Propulsion and Operational interaction lead to the
selection of the slow speed diesel engine at a cost of $110.8 million. This projects to an
initial build cost of $183.1 million. This compares favorably to existing modern costs,
matching order of magnitude for domestic construction of containerships. The final fuel
cost per trip for this engine is approximately $74,000. When this per-trip fuel cost is
compared to the existing ship, there are two critical improvements. The first is a gross
savings of approximately $30,000 per trip. The second, more interestingly, is that there is
a volumetric fuel savings. This allows the owner to build a smaller, more cost-effective
interstate shipping option. Additionally, it coincides with the general trend determined
by the resistance and hullform subsystem. The optimization succeeds in joining four
complex and dissimilar naval architecture disciplines to produce a working model of a
modern containership. The net result of this optimization was a rapid early stage design
demonstrating the extent of the design space and an existing optimal solution within it.
Page | 53
REFERENCES
1. American Bureau of Shipping. “ABS Rules for Building and Classing Steel Vessel
Rules (2013)”. Houston, Texas, 2013.
2. Ford, William. Third Mate, M/V Otto Candies, Interview on February 3, 2013.
3. Gillmer and Johnson. Introduction to Naval Architecture, 1987
4. Horizon Pacific Machinery Operation Manuals (Various)
5. Hollenbach, Uwe. "Estimating Resistance and Propulsion for Single-Screw and
Twin-Screw Ships in Preliminary Design”, Proceedings of the 10th ICCAS
Conference, June 7-11, 1999
6. Hughes, Owen F. and Paik, Jeom Kee. Ship Structural Analysis and Design. Jersey
City, New Jersey 2010.
7. International Convention on Tonnage Measurement of Ships, 1982 (ITC)
8. International Towing Tank Convention (Various Publications)
9. International Convention for the Prevention of Pollution from Ships, 1973 as
modified by the Protocol of 1978 (MARPOL 73/78)
10. Merchant Marine Act of 1920 (Jones Act)
11. Rahman, M. K. and Caldwell, J. B., “Rule-Based Optimization of Midship
Structures”, Marine Structures, 1992
12. Skerlos, Steve. Air Transport and Econ Example F10 Power Point. Fall 2012. ME589,
University of Michigan.
13. Society of Naval Architects and Marine Engineers, Principles of Naval
Architecture (SNAME PNA Vol. 1-3)
14. Society of Naval Architects and Marine Engineers, Ship Design and Construction
(SNAME SDC)
15. Society of Naval Architects and Marine Engineers, Ship Structural Analysis and
Construction
16. Watson and Gilfillan. Some Ship Design Methods. 1977
17. Taggart, Robert, and Society of Naval Architecture and Marine Engineers (U.S.).
Ship Design and Construction, 1980.
Page | 54
Appendix I: Project Variables
Symbol Variable Units
B Beam Meters
b_f Flange Breadth Meters
BP Bottom Plate --
BS Bottom Shell --
C_b Block coefficient --
C_p Prismatic Coefficient --
C_s Structural Cost $
C_x Sectional Area Coefficient --
C_t Total resistance coefficient --
D Depth Meters
D_t Distance Travelled Meters
E Elastic Modulus Pascals
F_c Final Fuel Cost $ per Full Tank
f_t Flange Thickness Meters
F_v Fuel Volume Cubic meters
H_f Specific energy of the fuel Joules/gram
h_w Height of web Meters
I_x Cross Sectional Area Moment of Inertia Meters4
L Length Meters
L_W Weld Length Meters
n Rotation Rate RPM
p Pressure Pascals
p_t Plate Thickness Meters
Q_s Shaft Torque Newton meters
R_t Total Resistance Newtons
rho Fluid Density Kilogram per meter3
SA Surface Area Meters2
s Stiffener Spacing Meters
S_s Sea State Beaufort Scale
SD Strength Deck --
SF_c Specific Fuel Cost $ per cubic meter
SFC Specific Fuel Consumption Meters3/KW Output
sigma_ult Ultimate Stress Pascals
sigma_y Yield Stress Pascals
SS Side Shell --
t Thrust Deduction --
Page | 55
T Draft Meters
Tr Thrust Newton
V Velocity Meters per second
V_m Structural Material Volume Meters3
w_t Web Thickness Meters
WD Wet Deck --
Appendix II: Weather Data
All weather data was gathered from the National Buoy Data Center. Data queries were
performed for the most recent five years. If such data was not available, data queries
for five consecutive years were gathered. In a few cases, only three consecutive years
were available. All data gathered is listed in the tables below.
Hawaii Weather Data
The worst year of weather was 2009 with two weather events. The probability of a storm
in Hawaii was calculated from this data as 2 days out of a 30 day month = 6.7%.
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Oakland Weather Data
Page | 57
The worst year in the San Francisco Bay was 2002, with eight events in the month of
December. The probability for Oakland was calculated as 8 days out of a 30 day
month = 26.67%.
Tacoma Weather Data
The weather data for the Puget
Sound indicates that the worst year
was 2004 and the worst month was
November. The probability of
encountering a storm in the Puget
Sound was calculated as 6 days out
of a 30 day month = 20%.
Page | 58
Appendix III: Fuel Oil Information
The information about the properties of the fuel was taken from the Material Safety
Data Sheet for Marine Diesel Fuel Oil as published by Environment Canada,
Emergencies Science and Technology Division. The pertinent portion of the MSDS is
included below:
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Appendix IV: The Beaufort Scale
The Beaufort Scale is a way of measuring the force of a storm based on indicators that
are easily observable to the mariner on a boat or ship. The chart below also includes
directions for sailing vessels to minimize mishap under the various conditions. This project
assumes that sea state 9 is the state at which ships would turn around or divert to avoid
a storm. The data point used to query the weather databases was waves higher than 7
meters.