ADMITTANCE POLICY TIDAL BOUND SHIPS Design of a probabilistic computer model for determination of channel transit risks to a seaport

COVER PHOTO: BERGE STAHL

ADMITTANCE POLICY TIDAL BOUND SHIPS Design of a probabilistic computer model for determination of channel transit risks to a seaport

AVV Transport Research Centre

Section Navigation and Waterways

R. Bouw

November 2005

I

PREFACE This master thesis forms the completion of my education at Delft University of Technology, Faculty of Civil Engineering and Geosciences. The project has been preformed by order of AVV Transport Research Centre, section Navigation and Waterways. Readers who are especially interested in the physical and mathematical background of HARAP or the selected alternative are referred to chapter 2 and appendix 1. The project conclusions are given in chapter 6. The program language of the selected alternative can be found in appendix 2. Special thanks go to E. Bolt and N. Dofferhoff for the daily supervision, nice discussions and constructive criticism. I would like to thank my supervisors from the university for the personal help and the freedom I got in doing the project. E. Cusell and W. Duif, thanks for correcting my English. The graduation committee consists of: - Prof. drs. ir. J.K. Vrijling (Chief supervisor, DUT) - Dr. ir. P.H.A.J.M. van Gelder (Supervisor, DUT) - Prof. ir. A.C.W.M. Vrouwenvelder (Supervisor, DUT) - Ir. E.W.B. Bolt (Supervisor, AVV) Ruud Bouw Rotterdam, November 2005

II

III

SUMMARY An entrance channel ensures the accessibility of most seaports. These channels are created through natural processes or by dredging and form the only port access for deep draught vessels. The dimensions of entrance channels have to follow scale enlargement of ships. It is possible to make the channels deep and wide enough for even the largest ships to access the port at any time, however the dredging cost will be high. To avoid these high dredging costs, the waterway authorities apply an admittance policy based on the risk of touching the bottom. This policy allows deep draught vessels to use the entrance channel only under particular conditions. The risk of touching the bottom during a channel transit depends on conditions like: the wave climate, astronomical tide and meteorological water level variations in combination with the ship characteristics. The inaccessibility percentage appears to be a good measure for waterway authorities to characterise their admittance policy and for companies to support their location selection. To calculate the inaccessibility percentage and tidal windows, AVV Transport Research Centre developed the probabilistic computer model HARAP (HARbour APproach). This model takes the occurrence frequency of most normative conditions into account by numerical integration of the probability density functions. For runtime reduction, the functions are divided into relatively large classes and the channel in large parts. The consequences: the results depend on arbitrarily chosen class borders and tidal window issues are based on the classes to which the predicted conditions belong instead of actual predicted data. The project objective is to design a computer model that uses a continuous channel and continuous probability density functions to calculate the inaccessibility percentage and an entrance regime for a tidal bound ship. Important physical and mathematical relations are studied, three alternatives are generated and a case study is done to compare the selected alternative and HARAP. The first alternative uses the existing model HARAP. The alternative calculates the inaccessibility percentage with at least ten times more classes and channel parts. Alternative 2 determines the inaccessibility percentage using the Monte Carlo method. It simulates channel transits by drawing the conditions from their distribution functions and positions along the channel axis. The third alternative determines the inaccessibility percentage in approximately the same way as alternative 1, but now with a constant integration step size. In HARAP and alternative 1, the integration steps are defined by the users and are not necessary constant of size. Alternative 2 is selected because its runtime is approximately 1013 times faster than alternatives 1 and 3. Both the existing model HARAP and the alternative 2 calculate the expected number of bottom touches [ ]ξE during a channel transit. By substituting [ ]ξE in the Poison distribution, the probability of one or more bottom touches can be calculated. The probability has to be lower than a particular safety criterion. Besides a safety criterion, serviceability criteria are formulated to ensure for instance the manoeuvrability of the ship. During a transit, both the safety and serviceability criteria have to be checked.

IV

A tidal window issue is based on a set of predicted conditions. In reality the conditions vary around these predictions. Therefore alternative 2 draws ksimu times a set of “real” conditions. With each set of “real” conditions and Npksi drawn positions along the channel axis, the [ ]ξE is calculated. The mean of ksimu [ ]ξE is checked on the safety criterion. This is repeated for a number of starting moments in a tidal cycle. If a sequent series of starting moments fulfils the safety criterion, it forms a tidal window. Simulating lsimu times a tidal window issue by drawing the same number of times a set of predicted values from their distribution function, the inaccessibility percentage can be calculated (see figure). The Euro and Maas channel to Rotterdam are taken for the case study. HARAP calculated a percentage of 4.3% in a runtime of 5 minutes and the selected alternative 4% in a time of 80 hour. This difference between the inaccessibility percentages is acceptable. The runtime of 80 hour is relatively large. Optimising of the algorithm, limiting the number of draws and the use of a compiled computer language will reduce the runtime. With the results of the case study it can be concluded that it is possible to make a good model based on the Monte Carlo method. Alternative 2 calculates with: continuous probability density functions, a continuous course of the channel, real tidal data and uses actual predicted conditions for tidal window issues. The results seem to be promising, therefore further research is recommended.

predicted wave condition

predictedmeteorological

condition

predictedsailing speed

predicteddraught

real astronomical condition

real wave condition

real meteorological condition

real sailing speed

real draught

inputchannel

transit

bottom position

freq

uenc

y

ship motion

CHANNEL VIEW FROM ABOVE

CROSS-SECTION A - A

drawn position along channel axis

drawn condition

Draw predictedcondition

(lsimu times)

Draw realcondition

(ksimu times)

Draw channelposition

(Npksi times)

AA

freq

uenc

y

freq

uenc

y

freq

uenc

y

freq

uenc

yfr

eque

ncy

freq

uenc

y

freq

uenc

y

freq

uenc

y

wat

er le

vel

time

Repeat for Ntbi starting

moments

starting moment

Inaccessibilitypercentage

V

TABLE OF CONTENTS Preface I

Summary III

List of figures IX

List of tables XI

1 Introduction 11.1 Background 1

1.2 Objectives 2

1.3 Outline of the report 2

1.4 Shipping definitions with respect to accessibility 2

2 Project background 52.1 Maritime industry background 5

2.1.1 Scale of the maritime industry in the Netherlands 52.1.2 Procedure for incoming vessels 62.1.3 Actors around the admittance policy 72.1.4 A measure for the accessibility 8

2.2 Physical background of ship motions 82.2.1 Keel clearance 82.2.2 Ship movements due to horizontal velocities 92.2.3 Ship movements due to waves 112.2.4 Relations between the physical processes 12

2.3 Probabilistic background 132.3.1 From under keel clearence to probabilistic models 132.3.2 Chance of touching the bottom 132.3.3 From wave spectrum to the motion spectrum of ships 142.3.4 Probability of a condition 162.3.5 Admittance criteria 16

2.4 Calculation methods 182.4.1 Level-classification of calculation methods 182.4.2 Numerical Integration 182.4.3 Monte Carlo Simulations 20

2.5 The existing model HARAP 212.5.1 Mathematical background 212.5.2 Modules 222.5.3 Input variables 24

2.6 Chapter conclusions 27

VI

3 Problem analysis 293.1 Evaluation of the existing model HARAP 29

3.1.1 Degree of describing the physical processes 293.1.2 Advantages and disadvantages 30

3.2 Definition of the project 313.2.1 Problem definition and project objective 313.2.2 Project boundaries 31

3.2.3 Project assumptions 32

4 Alternatives 334.1 Alternative 1: Calculation with predefined classes 33

4.1.1 Description of the model 334.1.2 Advantages and disadvantages 34

4.2 Alternative 2: Monte Carlo method 344.2.1 Description of the model 354.2.2 Advantages and disadvantages 37

4.3 Alternative 3: Numerical integration 384.3.1 Description of the model 384.3.2 Advantages and disadvantages 41

4.4 Module 1: Runtime reduction 424.4.1 Description of the model 424.4.2 Advantages and disadvantages 43

4.5 Module 2: Optimisation for the conditions 434.5.1 Description of the model 444.5.2 Advantages and disadvantages 45

4.6 Module 3: Optimisation for the ships 464.6.1 Description of the model 464.6.2 Advantages and disadvantages 47

4.7 Degree of describing the physical processes 47

4.8 Chapter conclusion 48

5 Case study 515.1 Input data 51

5.1.1 Ship data 515.1.2 Channel data 525.1.3 Astronomical data 535.1.4 Meteorological data 555.1.5 Wave data 565.1.6 Criteria data 57

5.2 Number of draws in alternative 2 585.2.1 Number of draws: positions 595.2.2 Number of draws: real conditions 605.2.3 Number of draws: predicted conditions 67

VII

5.3 Number of draws in variant of alternative 2 685.3.1 Number of draws: positions 695.3.2 Number of draws: predicted conditions 71

5.4 Results case sudy 725.4.1 Calculated inaccessibility percentages 725.4.2 Calculated tidal window 72

5.5 Chapter conclusions 73

6 Conclusions and recommendations 756.1 Conclusions 75

6.1.1 Conclusions evaluation alternatives 756.1.2 Conclusions case study 76

6.2 Recommendations 776.2.1 Investigation of model faults 77

6.2.2 Investigation of model improvements 78

Literature 81

Appendix 1: Wiskundige achtergrond (in Dutch) a1

Appendix 2: Matlab program alternative 2 a11

Appendix 3: Random number generator a37

Appendix 4: Meandering due to external forces a41

Appendix 5: Explanation of the case study conclusions a43

Appendix 6: Type input data in the case study a47

VIII

IX

LIST OF FIGURES

fig. 1.1-1 Example causes of ship failure 1fig. 1.1-2 Example aspects that determine the probability of a bottom contact 1fig. 1.4-1 Tidal window 3fig. 1.4-2 Example calculation of the downtime and inaccessibility percentage 3fig. 1.4-3 Keel clearance 4fig. 2.1-1 Berge Stahl 5fig. 2.1-2 Freight import and export in the Netherlands in 2003 6fig. 2.1-3 Pilots transport 6fig. 2.1-4 Euro and Maas channel 7fig. 2.1-5 Actors around the admittance policy 7fig. 2.2-1 Keel clearance 8fig. 2.2-2 Trim of a ship 9fig. 2.2-3 Constant crosscurrent 9fig. 2.2-4 Large crosscurrent gradient 10fig. 2.2-5 Handling in a strong wind 10fig. 2.2-6 Six degrees of freedom of a ship 11fig. 2.2-7 Refraction 11fig. 2.2-8 Diagram of the theoretical framework of physical relations 12fig. 2.3-1 RAO function 15fig. 2.3-2 Process to find the linear relation between H E10 and Z s 15fig. 2.3-3 Run aground vessel 16fig. 2.3-4 Oil tanker 'Prestige' breaks due to a storm 17fig. 2.4-1 Flow chart Numerical Integration 19fig. 2.4-2 Flow chart Monte Carlo method 20fig. 2.5-1 Probability density distribution with discrete classes 22fig. 2.5-2 Differences sections and segments 22fig. 2.5-3 Principle description astro class 25fig. 2.5-4 Wave frequency distribution, H E10 26fig. 2.5-5 Wave frequency distribution, H res 26fig. 2.5-6 Joint probability density distribution wave height 26fig. 3.1-1 Arrangement of the classes 30fig. 4.1-1 Number of classes 33fig. 4.1-2 Small part of the structure program diagram of HARAP with the location 34

for the module refraction fig. 4.2-1 Drawing PmeanHE10 35fig. 4.2-2 Drawing astronomical tide 35fig. 4.2-3 Drawing H(k) 36fig. 4.2-4 Drawing positions 36fig. 4.2-5 Drawing d(p) 36fig. 4.2-6 Calculation of ksi(k) 36fig. 4.2-7 Converging speed calculation method 37fig. 4.3-1 Determining PmeanHE10(1) 38fig. 4.3-2 Determining the astronomical tide 38fig. 4.3-3 Determining PmeanHE10(Li3) 38fig. 4.3-4 Determining H(ki3) 39

X

fig. 4.3-5 Determining the position 39fig. 4.3-6 Determining d(kp) 39fig. 4.3-7 Calculation of ksi 40fig. 4.3-8 Stopping the algorithm 41fig. 4.4-1 Lower and upper boundary 42fig. 4.5-1 Principle sketch of the distribution of the single transit criterion over 44

the wave heightfig. 4.5-2 Construction of the distribution of ksi 45fig. 4.5-3 Distribution of the single transit criterion over the wave height and 45

the meteo-effectfig. 4.5-4 Difference between a distribution along one or two axes 45fig. 5.1-1 Traffic separation scheme 52fig. 5.1-2 Euro and Maas channel 53fig. 5.1-3 Channel input HARAP 53fig. 5.1-4 Channel input alternative 2 53fig. 5.1-5 Astronomical input alternative 2 54fig. 5.1-6 Discontinuous probability density function 55fig. 5.1-7 Discontinuous distribution function 55fig. 5.1-8 Continuous distribution function 55fig. 5.1-9 Discontinuous probability density function of H E10 56fig. 5.1-10 Definition of the HE10 value for a tidal window issue = maxHE10 56fig. 5.1-11 Discontinuous distribution function of H E10 56fig. 5.1-12 Continuous distribution function of H E10 56fig. 5.2-1 Influence of Npksi on ksi (set 1, 4 runs) 60fig. 5.2-2 Influence of Npksi on ksi (set 2, 4 runs) 60fig. 5.2-3 Moving mean of E[ ξ ] for ksimu (set 1, 3 runs) 61fig. 5.2-4 Moving mean of E[ ξ ] for ksimu (set 2, 2 runs) 61fig. 5.2-5 Expected number of bottom touches of a transit for various starting times,

values of Npksi and values of ksimu (set 1) 66fig. 5.2-6 Moving mean of the inaccessibility percentage for lsimu (set 1) 67fig. 5.2-7 Moving mean of the inaccessibility percentage for lsimu (set 2) 67fig. 5.3-1 Influence of Npksi on ksi (4 runs) 69fig. 5.3-2 Expected number of bottom touches of a transit for various starting times

and values of Npksi 70fig. 5.3-3 Moving mean of the inaccessibility percentage for lsimu 71fig. 5.4-1 Presentation examples of variable variation along the channel axis 73fig. 6.2-1 λ for different positions along the channel axis 79

XI

LIST OF TABLES

tab. 2.1-1 Trends in transport per harbour 6tab. 3.1-1 Relation matrix between wind and waves 29tab. 4.3-1 Comparison number of calculations HARAP and alternative 3 41tab. 4.8-1 Number of calculations of the models 48tab. 5.1-1 Astronomical input HARAP 54tab. 5.1-2 Calculation values meteo-classes with respect to Mean Sea Level [m] 55tab. 5.1-3 Prediction matrix in HARAP 57tab. 5.2-1 Sets of input values for the investigation of Npksi 59tab. 5.2-2 Sets of input values for the investigation of ksimu (one starting moment) 61tab. 5.2-3 Sets of input values for the investigation of ksimu (more starting moments) 62tab. 5.2-4 Tidal windows for various ksimu-values and input 62tab. 5.2-5 Expected number of bottom touches of a transit for various starting times and

values of ksimu (set 1) 63tab. 5.2-6 Expected number of bottom touches of a transit for various starting times and

values of ksimu (set 2) 64tab. 5.2-7 Expected number of bottom touches of a transit for various starting times and

values of ksimu (set 3) 65tab. 5.2-8 Expected number of bottom touches of a transit for various starting times,

values of Npksi and values of ksimu (set 1) 66tab. 5.2-9 Sets of input values for the investigation of lsimu 67tab. 5.3-1 Set input values for the investigation of Npksi 69tab. 5.3-2 Expected number of bottom touches of a transit for various starting times and

values of Npksi 70tab. 5.4-1 Tidal windows for astro class 2, meteo class 2 and wave classes 1 to 5 72

1

1 INTRODUCTION

1.1 BACKGROUND Ships can have internal problems like: fire, explosions and falling cargo. Extreme conditions like waves and wind can cause stability problems or failure of the construction. Also collisions occur with other ships, icebergs, obstructions or the sea bottom. This report is focussed on failure due to sea bottom contact (figure 1.1-1). To prevent contact with the sea bottom, entrance channels are used to sail to a seaport. These channels are dredged or created through natural processes and form the only access for relatively large ships. In general the largest ships are tidal bound ships. Despite the presence of a channel the distance between the keel and the bottom is not sufficient for these ships during the entire tidal cycle.

Whether the keel clearance is sufficient or not, depends on the risk of touching the bottom. The risk that a ship touches the bottom depends on various aspects like: its motions due to wind, waves and currents, draught of the ship and the water depth (fig.1.1-2). When the risk of a bottom touch is acceptable, the ship can sail to the port. Otherwise it has to wait for a better moment. Mostly these moments are available in a tidal cycle.

The period in a tide in which a ship is allowed to sail through the entrance channel is called a tidal window. The percentage of tides in which no tidal window is available, is the inaccessibility percentage. This percentage turns out to be a good measure for the applied admittance policy of a waterway authority. The smaller the inaccessibility percentage, the less delays ships will have.

The inaccessibility percentage depends on the occurrence probability of the conditions that determine the availability of a tidal window. Probability density functions describe the occurrence probability and are therefore important for the determination of the inaccessibility percentage. For tidal window issuing and the determination of the inaccessibility percentage the existing computer model HARAP (HARbour APproach) can be used. HARAP is a probabilistic model and is developed by AVV Transport Research Centre (part of the Dutch Ministry of Transport, Public Works and Water Management). The model uses a numerical integration method to take the probability density functions of the normative conditions (also astronomical data) into account.

EXPLOSION

FALLING CARGO

INTERNAL LEAKAGE

......

ICEBERG COLLISION

SHIP COLLISION

BOTTOM CONTACT

OBSTRUCTION COLLISION

WAVE FORCES

FAILURE OF THE SHIP:INSTABILITY

BREAKINGSINKING

MATERIAL DAMAGEHUMAN CASUALTIES

CAPSIZELEAKAGES

.....

WATER DEPTH

SHIP CHARACTERISTICS

SHIP MOTIONS

CURRENTS

HUMANINFLUENCES

BOTTOM CONTACT

......

fig. 1.1-1 Example causes of ship failure

fig. 1.1-2 Example aspects that determine the probability of a bottom contact

2

To reduce the calculation time, HARAP uses rather large integration steps. This way continuous probability density functions are transformed into discrete probability density functions. Therefore the results obtained with HARAP are partly influenced by arbitrarily chosen integration step sizes (HARAP calls these steps: classes). Consequences: the calculated safety level is doubtful, it is tempting (but unsafe) to play with the class boundaries to get the lowest inaccessibility percentage and tidal window issuing cannot be determined on actual predicted data.

1.2 OBJECTIVES The project objective is to design a better computer model than HARAP. The new model has to calculate the inaccessibility percentage and an entrance regime for a tidal bound ship using continuous input variables. First a literature study is done to the physical and mathematical background of the important relations and calculation methods. Then alternatives are generated and evaluated. The chosen alternative is compared with HARAP in a case study.

1.3 OUTLINE OF THE REPORT The results of the literature study are given in chapter 2 and will provide the project background information. Chapter 3 gives the problem analysis. To achieve the project objective three alternative computer models with three possible extension modules are generated. These alternatives and modules are described in chapter 4. The results of the case study with the selected alternative and HARAP are presented in chapter 5. An evaluation of the objectives and recommendations for further research to the topic and development of the program can be found in the last chapter, chapter 6.

1.4 SHIPPING DEFINITIONS WITH RESPECT TO ACCESSIBILITY Entrance channels The accessibility of many seaports has to be ensured by entrance channels. These channels are dredged, or created through natural processes. Channel-bound and tidal-bound ships Vessels that can only reach the port through the entrance channel are called channel-bound ships. Tidal-bound ships get also tidal windows because the water level is not always sufficient for those vessels. Only at high tide, they can reach the harbour.

3

Tidal window A tidal window is a time window in which a ship is allowed to enter the channel. During the channel transit, tidal windows indicate the different opening and closing times at different locations. Between those points in time it is safe to sail through a particular part of the channel. The length of the tidal window depends especially on the astronomical tide, ship characteristics (like draught), wave climate and meteorological conditions. The more unfavourable the situation gets (for example more wind or waves), the narrower the tidal window becomes. In the most extreme situation no tidal window is available and the vessel has to wait for a next tide and better circumstances (fig.1.4-1). Downtime and inaccessibility percentage The downtime equals the percentage of time during which a vessel cannot access the harbour because no tidal window is available. The inaccessibility percentage equals the percentage of tides during which no tidal window can be provided, due to unfavourable water level or weather conditions (fig.1.4-2). Gross keel clearance (according IMO1) The gross keel clearance is the distance between the keel of a motionless ship in motionless water and the intervention level for dredging (fig 1.4-3).

1 International Maritime Organisation

time

distancestart of channel end of channel

closing time

opening time

fig. 1.4-1 Tidal window

fig. 1.4-2 Example calculation of the downtime and inaccessibility percentage

Tidal window 1 (2:15 hour)

Tidal window 2(2:45 hour)

Tidal window 3(2:30 hour)

Tide 1(12:25 hour)

Tide 2(12:25 hour)

Tide 3(12:25 hour)

Tide 4(12:25 hour)

Downtime = x 100% = 85%length tide 1+2+3+4 - length tidal window 1+2+3

length tide 1+2+3+4

Inaccessibility percentage = x 100% = 25% number of tides without a tidal windownumber of tides

Time

Water level

M.S.L.

4

Net keel clearance (according IMO) The net keel clearance is the gross keel clearance diminished with the squat, vertical ship motions due to waves, a margin for unexpected sedimentation between two soundings and sounding inaccuracies (fig 1.4-3). ref. [15]

Draught

Squat

Safety margin for ship motions due to waves

Net keel clearance

Margin for sedimetation between two soundings

Sounding inaccuracies

Extra dredged depth

Dredging tolerance

Gross keel clearance

Intervention level

Nautical guaranteed depth

Actual water levelPredicted water level

fig. 1.4-3 Keel clearance

5

2 PROJECT BACKGROUND Chapter 2 gives the results of a literature study, preformed to research the following master thesis objective, formulated by Delft University of technology and the Transport Research Centre (AVV):

Design a new computer program that can be used to design an entrance regime and that calculates a measure for the accessibility for tidal bound ships to a harbour, given an accepted safety level.

On the basis of this objective the following seven research questions were formulated: 1. What is the procedure for tidal bound ships? 2. Who wants to know the measure for accessibility and why? 3. Which parameter gives a good measure for the accessibility? 4. What determines the accessibility? 5. What is an acceptable safety level? 6. Which calculation methods are available? 7. Which program is used at the moment? The answers to these questions will be used as background for the rest of the project and to define a more specific project objective in chapter 3. Paragraph 2.1 summarizes background information of the maritime industry. The physical background of ship motions is presented in paragraph 2.2. The safety level is discussed in paragraph 2.3. How the safety level can be determined is given in paragraph 2.4. An existing program that is used at the moment is discussed in paragraph 2.5. Subparagraph 2.6 gives a summary of all the answers to the questions.

2.1 MARITIME INDUSTRY BACKGROUND Some background information about the maritime industry is given to illustrate the project problem. Subparagraph 2.1.1 gives some figures about the scale of the Dutch maritime industry. The procedures around ships entering a harbour are presented in subparagraph 2.1.2. The actors who determine the admittance policy are given in 2.1.3. In subparagraph 2.1.4 is explaind which parameter gives the best measure for the accessibility.

2.1.1 SCALE OF THE MARITIME INDUSTRY IN THE NETHERLANDS The amount of transferred freight keeps growing in the Netherlands. From 1980 to 2003, the amount of transfer has increased 35%, while the amount of ships has stayed almost constant. This was possible due to scale enlargement of the ships. At the moment the biggest ship that enters Rotterdam is Berge Stahl. This ship is 343 m long (length overall), 63 m wide, has a draught of 22.6 m and can carry 365,000 ton of iron ore.

fig. 2.1-1 Berge Stahl

6

Import and export In the year 2003 sea-going vessels transported 76% of the import and 36% of the export:

Transport per harbour The port of Rotterdam is the largest harbour in the Netherlands and takes care of 78% of the total port traffic (table 2.1-1). ref. [11]

2.1.2 PROCEDURE FOR INCOMING VESSELS Only a small percentage of ships can enter the harbour without any pilot assistance and at any time. They are quite manoeuvrable, have no problems with the existing depths and know the map and regulations of the harbour. Ships who need help can ask for a pilot, and for most ships pilotage is compulsory. For instance in Rotterdam pilotage is compulsory in general when the length of the ship exceeds 70 m. A pilot helps the vessel to safely enter and leave the port. The pilot goes aboard for this, but in some weather conditions they keep in contact with a ship from ashore (fig. 2.1-3).

fig. 2.1-2 Freight import and export in the Netherlands in 2003 (millions of tons)

tab. 2.1-1 Trends in transport per harbour (millions of tons)

Year 1999 2000 2001 2002 2003Waddensea region 4 5 5 4 5Amsterdam region 56 64 68 65 70Rotterdam 314 334 327 339 333River Scheldt region 26 25 25 28 27Total 400 428 425 437 435

fig. 2.1-3 Pilots transport

Road haulage includes only Dutch road hauliers

Export

102

17110

52

Import

330

7

46

48

sea shipping

rail transport

inland shipping

road haulage

7

Channel bound ships that have to wait for a pilot or the opening of the tidal window can use the sea anchoring areas outside the channel. These areas are deep enough for the largest tidal-bound ships (fig 2.1-4). In general only channel-bound ships use the entrance channel to reach the harbour. In the port they can get assistance from one or more tugboats. Tugboats take a ship to the right location in the port. Once the ship has arrived, a boatman makes sure that the ship is securely moored alongside a quay, a jetty or buoys.

2.1.3 ACTORS AROUND THE ADMITTANCE POLICY In general the waterway authority decides when a ship is allowed to enter the harbour. They determine the tidal windows and pass these on to the pilots. The waterway authority is also responsible for the channel dimensions. A very simple representation of the actors and their relations may be as follows: Politicians determine the acceptable risk level. With this risk level the waterway authority determines its admittance policy. And this policy is one of the factors for companies to determine the location of their business. In reality there are interactions between the involved parties (fig. 2.1-5). For this report especially the interaction between companies and the waterway authority is important. To some extend the infrastructure capacity should follow the needs of the companies. For example, in Rotterdam the maximum possible draught was around 40 feet (12.2 m) till the late fifties. At the moment bulk carriers in Rotterdam can have draughts up to 74 feet (22.6 m).

fig. 2.1-5 Actors around the admittance policy

ParliamentSafetypolicy

Establishmentpolicy

Waterway authority

Companies

Admittancepolicy

fig. 2.1-4 Euro and Maas channel

VOORNE

GOEREE

SCHOUWEN

3°00' 3°30' 4°00'

52°00'0'

TURNING -AREA

TURNING -AREA

3°00' 3°30' 4°00'

EURO CHANNEL

MAAS CHANNEL

SHORT TERM ANCHORAGE FOR DEEP DRAUGHT SHIPS

EURO PLATFORM

LONG TERM ANCHORAGE FOR DEEP DRAUGHT SHIPS

H.v.H.

8

2.1.4 A MEASURE FOR THE ACCESSIBILITY Reliable infrastructure in shipping is infrastructure that can be used frequently and under most circumstances. For most companies it is acceptable that they cannot reach the harbour during extreme conditions when the delay does not last too long. Time costs money and companies are willing to pay this price as long as it is compensated by low costs for using the infrastructure. The costs for infrastructure depend partly on the building and maintenance costs of the entrance channel, which the waterway authority charges to the companies. To find an optimum between the costs and the accessibility of a harbour, it is important to find a measure for the accessibility. Two parameters are widely used as a measure for the accessibility: the downtime and the inaccessibility percentage. These two values are quite different. It is possible to have a downtime of 80% and an inaccessibility percentage of 0%. For tidal bound ships it is most important that there is an opportunity to reach the harbour each tidal cycle (low inaccessibility percentage). It is less important how long this opportunity exists during the cycle (low downtime). Therefore the inaccessibility percentage will be used as the most important measure for the accessibility.

2.2 PHYSICAL BACKGROUND OF SHIP MOTIONS To calculate the inaccessibility percentage it is important to know how a ship reacts to different circumstances. The motions depend on a lot of factors, like the wave climate, currents, bathymetry of the sea bottom, channel design, water levels and ship characteristics. A model is always a simplification of the reality. Some models describe the reality better than others. The best model takes all the physical processes into account in the right way. The important physical processes are mentioned in this paragraph. They are numbered {..} in the text. These numbered processes will be used to compare the different models with eachother in chapter 4. Subparagraph 2.2.1 considers vertical ship motions due to relatively slow processes, like the vertical astronomical tide. In subparagraph 2.2.2 the motions of a ship due to horizontal velocities are discussed. Vertical motions resulting from waves are described in 2.2.3. At the end of this paragraph a diagram is given with all the processes and their interrelations.

2.2.1 KEEL CLEARANCE Consider a motionless ship in motionless water. The water column under the keel of the ship is called the gross keel clearance. It depends on the water level wl , draught Td and bottom level d (fig. 2.2-1).

fig. 2.2-1 Keel clearance

Draught

Gross keel clearance

Intervention level

Water level

9

hg

vnumberFroudeFC

LF

FZ s

nhsquatppnh

nhv ⋅

=−=→⋅∇−

=22

2

1

Variations of the water level wl over a longer period, like an hour, occur due to the tide {1} and meteorological conditions {2}. The tide is the vertical rise and fall of the sea level surface, caused primarily by the change in gravitational attraction of the moon, and to a lesser extent the sun. Due to differences in atmospheric pressure and strong winds from a constant direction the sea surface can be pushed up or drawn off. The effect mainly depends on the wind intensity, wind direction, fetch length and water depth. The draught Td of a particular ship depends mainly on the degree of loading, the shape of the hull and the trim of a ship {3}. The trim defines the longitudinal inclination of the ship. Trim may be expressed as the angle between the baseline of the ship and the water plane, but it is usually defined as the difference in draughts at the bow and the stern. Morphological factors like: sedimentation, sand waves and the dredging cycle determine the position of the channel bottom d {4}. ref. [15]

2.2.2 SHIP MOVEMENTS DUE TO HORIZONTAL VELOCITIES Now consider a sailing ship on a sea with currents and a flat surface. A ship moving through water modifies the pattern of flow lines and thus changes the dynamic pressure distribution around the hull. Due to a small distance between the keel and the bottom of the channel, a low-pressure area under the ship can occur, which increases the ship draught. The effect depends on the speed of the ship {5} with respect to the water. The resulting change in draught and trim is known as squat {6}. An estimation of the maximum squat is given by the method of Tuck-Taylor (adopted by ICORELS2):

(2.2-1)

vZ maximum squat [m] ∇ water displacement [m3]

sv sailing speed [m/s] ship length between g acceleration of gravity [m/s2] perpendiculars [m] h water column [m] squat coefficient [-] The crosscurrent {7} influences the orientation of the ship with respect to its sailing course. In a flow field a ship has to correct the crosscurrent to stay on its course (fig 2.2-3). In this way the crosscurrent affects the squat and the angle of the incoming waves.

2 International Commission for the Reception of Large Ships

fig. 2.2-2 Trim of a ship (exaggerated for clarity)

ppL

fig. 2.2-3 Constant crosscurrent (exaggerated for clarity)

Sailing course

Cross current

vground

vtrue

squatC

10

A large cross current gradient in a harbour entrance affects the handling of the ship. The falling away of the cross current on the bow, while the stern is still under influence of the current, causes the stern to be pushed away. When the ship-handler reacts to late, the ship will hit a breakwater (fig 2.2-4). If the ship-handler interferes too early, the ship may leave its sailing course. The current is a vector summation of the following physical phenomena: - horizontal astronomical tide {8}

In the North Sea the tide enters the basin along the North of Great Britain to the South. Before the Strait of Dover it bends towards the Netherlands and goes northwards. It leaves the North Sea along Norway.

- local wind driven currents {9} Local meteorological conditions can cause wind driven currents, resulting from transfer of wind energy to the water. The main wind direction along the coast of the Netherlands is South-West. This main direction causes a net current along the coast to the North.

- river discharges {10} The discharge of the river leads to currents and a difference in the density of the seawater. When a ship is sailing from the sea into a port it sails from salt to more fresh water. Because of the density differences the draught of the ship changes. Also density waves can cause ship motions.

- ocean currents The dominant forces in physical oceanography are gravity, buoyancy due to difference in density of seawater and wind stress. These forces create horizontal pressure gradients that are balanced almost exactly with the Coriolis force resulting from horizontal currents. This balance is known as the geostrophic balance. The balance is for a model for Dutch harbours of less importance.

Crosswind will cause the ship to drift sideways or to take up an angle of leeway (fig 2.2-5). It affects the squat and influences the approach angle of the waves with respect to the ship {11}. The crosswind effects depend on the wind speed and direction relative to the ship, the windage area of the ship and the depth/draught ratio. The meandering of the ship around its intended course due to response time of the ship-handler on the influence of wind, waves, currents, etc. does not lead to significant effects (see appendix 4). ref. [12]

fig. 2.2-5 Handling in a strong wind (exaggerated for clarity)

Wind

Channel boundary

Channel boundary

fig. 2.2-4 Large crosscurrent gradient (exaggerated for clarity)

Sailing course

Cross current

Breakwater

Breakwater

11

2.2.3 SHIP MOVEMENTS DUE TO WAVES The motion of a ship can be separated in six components. The three translational components are: surging along the longitudinal X-axis, swaying along the lateral Y-axis and heaving along the vertical Z-axis. The three rotational components around these axes are called: rolling, pitching and yawing, respectively (fig. 2.2-6). The translation and rotation of a ship depend on the characteristics of the waves, ship, water, bottom material and the bathymetry. It is a rather complex interaction between these factors. The ships motions can be modelled with coupled differential equations of motion for six degrees of freedom. External force In these equations the waves are the external forces on the ship. The wave height, period and the approach angle of the waves with respect to the ship are the most important characteristics of this force. The approach angle of the waves with respect to the vessel has a large influence on the ship movements. Waves approaching perpendicular to the sailing direction have less influence compared to waves oblique of behind. The characteristics of a wave spectrum depend on the characteristics of the wind fields {12} and the bathymetry. Waves generated by local wind fields {13} are steep waves: relatively high wave height/length ratio. The opposite is true for swell waves {14}, which are generated by wind fields from elsewhere. Incoming waves generated by passing ships have no influence3 due to the relative short period of the waves. When waves interact with the sea bottom, their shape and direction will change. Sometimes this can lead to breaking {15} before the channel and refraction of the waves {16} in the channel (fig.2.2-7). Inertia Ships with a large water displacement react slowly to incoming waves. Therefore they can only react to the waves with a long period. Damping Not all wave energy is translated into motions of the ship. Energy dissipation occurs due to the interaction of the ship between water and bottom. The water displacements due to ship motions cause damping. Proximity of the bottom causes an increase of damping forces. ref. [13]

3 Waves from high-speed boats, like fast ferries excepted.

fig. 2.2-6 Six degrees of freedom of a ship

Yaw

Heave Roll

Surge

Pitch

Sway

x

y

z

fig. 2.2-7 Refraction

Channel boundary

Channel boundary

Waves

Current

12

2.2.4 RELATIONS BETWEEN THE PHYSICAL PROCESSES The following processes were mentioned above: The relations between processes that lead to the net keel clearance are presented in figure 2.2-8.

fig. 2.2-8 Diagram of the theoretical framework of physical relations

{1} vertical tide {9} meteorological currents{2} meteorological water level {10} river discharges{3} draught variation {11} crosswind effects{4} morphological factors {12} relation wind and waves{5} sailing speed variations {13} local wind field waves{6} squat {14} swell{7} crosscurrent effects {15} breaking of waves{8} horizontal tide {16} refraction of waves

ASTRONOMICALTIDE

CURRENTS

WATER LEVEL

WIND SET-UP

WINDFIELDS

LOCALWINDFIELD

WINDFIELDELSEWHERE OR

EARLIERWINDFIELD

LOCALLY GENERATEDWAVES

SWELL

BREAKING ANDREFRACTION

SIGNIFICANTWAVE HEIGHT

PEAKPERIOD

WAVEDIRECTION

DREDGINGCYCLUS

MORPHOLOGICALPROCESSES

BATHYMETRY

WATER DEPTH

SQUAT

VERTICAL SHIPMOTIONS

LONG-TERM WAVE CHARACTERISTICS

RIVERDISCHARGE

NET KEEL CLEARANCE

SHIP

DRAUGHTTRIMHULL

SAILING SPEED

13

2.3 PROBABILISTIC BACKGROUND Paragraph 2.2 described how a ship reacts to different circumstances. This paragraph presents how the frequency of occurrence of the circumstances can be taken into account and how the safety of the channel can be quantified. Most harbours define their admittance policy based on under keel clearance models. The advantages and disadvantages of these models compared to probabilistic models are presented first. The probability of touching the bottom is described in subparagraph 2.3.2. Subparagraph 2.3.3 gives the relation between a wave spectrum and the motion spectrum of a ship. The probability density functions of the variables are mentioned in subparagraph 2.3.4. The admittance policy is based on admittance criteria. Subparagraph 2.3.5 gives the criteria that are used in the Netherlands. The derivations of the equations in subparagraph 2.3.2 can be found in appendix 1.

2.3.1 FROM UNDER KEEL CLEARENCE TO PROBABILISTIC MODELS Most under keel clearance (UKC) models examine if the gross keel clearance is large enough during the transit. The required keel clearance depends on safety factors, for example the vertical ship motions, depth variations and draught inaccuracies. These safety factors are based mostly on experience. In more advanced models the safety factors can vary for different combinations of conditions. For instance during a period in which low waves are predicted, the factors are smaller. Also exclusions during particular conditions can be applied, like exclusions due to strong crosscurrents near the port entrance. UKC models have the advantages that they are easy to construct and there is no need for long-term statistics. The disadvantages of these models are: - On beforehand, it is impossible to quantify the safety level of the admittance policy. - It is only possible to calculate the downtime and inaccessibility percentage when the safety

factors do not vary for different conditions and there are no conditional exclusions. - To find the economical optimum of the entrance channel, it is necessary to know the risks.

These are not possible to determine with UKC methods. - Due to the accumulation of the safety factors, the policy is very conservative.

2.3.2 CHANCE OF TOUCHING THE BOTTOM The probability that a ship touches the channel bottom is equal to the probability that the vertical downward movement due to waves of the most critical point of the ship exceeds the keel clearance KC . The keel clearance is here defined as:

(2.3-1) depth water level

draught ship maximum squat

vZ

dwl

dT

vd ZTwldKC −−+=

14

[ ] pasTE ⋅= λξ

( ) ( )…,2,1,0

!=⎯→⎯== −

kek

TkP forT

kpas pasλλ

ξ

216

10 sZm =

Consider a ship that sails from points A to B through a channel. In advance it is not known when or where a bottom touch will occur, but imagine that the following is true: - The mean frequency of bottom touches λ is constant in time. In a time interval with length t

the expected value of bottom touches is t⋅λ . - The bottom touches per transit are independent random variables: one bottom touch tells

nothing about the others.

(2.3-2) expected number of bottom touches during a transit total number of bottom touches during a transit passage time The discrete random variable ξ proves to be Poisson distributed:

(2.3-3) The mean frequency of bottom touches λ is equal to the mean frequency of vertical ship motions that are bigger than the keel clearance. Based on the formula of Rice this is given by:

(2.3-4)

mT mean period of a ship motion

0m 0th spectral moment of ship motions Like the distribution of the incoming waves it is assumed that also the vertical motions of a ship are Raleigh distributed. Therefore the following relation is true:

(2.3-5)

sZ significant vertical ship motion

2.3.3 FROM WAVE SPECTRUM TO THE MOTION SPECTRUM OF SHIPS The only unknown parameters in (2.3-4) and (2.3-5) are the mean period of the ship motions mT and the significant vertical ship motion sZ ; both follow from the ship motion spectrum. In paragraph 2.2.3 it is already mentioned that the motions of a ship can be described by coupled differential equations. The computer model that will be used for the case study in this report to determine the response of a ship to waves in an entrance channel is SEAWAY. SEAWAY is based on the strip theory and calculates ship motions for six degrees of freedom. The strip theory divides the ship into a number of slices. The thickness of a slice is small enough to allow a two-dimensional treatment of the flow in its plane. Per slice it determines the two-dimensional hydrodynamic coefficients: added mass, damping, wave excitation and the external forces. By integrating these values over the length of the ship it is possible to solve the coupled equations of motions.

pasT

[ ]ξEξ

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

0

2

2exp

1

m

KC

Tm

λ

15

The output of SEAWAY consists of Response Amplitude Operators (RAO’s) for a particular sailing speed, depth/draught ratio and approach angle of the waves with respect to the ship. The RAO function gives the relation between a wave spectrum and a motion spectrum of a ship. Figure 2.3-1 presents two examples of a RAO function, for a small and large ship. When the wave frequencies are very low, the ship response will be hydrostatic, this means that the RAO’s for these frequencies are one. When the wave frequencies are very high the ship will not react at all (RAO’s are zero). In between the RAO varies for different frequencies, with a peak response at the natural frequency of the ship. Large ships, like tidal bound ships to Rotterdam, only react to relatively long waves, like swell. If the shapes of the long wave spectra do not fluctuate much, they could be described by one parameter. This is the case for the port of Rotterdam. As parameter a characteristic wave height of the low frequent wave energy 10EH is used. The shape of the ship motion spectrum depends on the wave spectrum and the RAO function. In the case of Rotterdam it is therefore possible to describe the motion spectrum also with one parameter. Here the significant vertical ship motion sZ is used. A linear relation is assumed between the 10EH -value and the sZ -value. To find the linear relation, AVV Transport Research Centre developed a program additional to SEAWAY. This program uses thousands of measured wave spectra and the calculated RAO function. For each wave spectrum a motion spectrum is calculated with the RAO function. The program determines from the wave spectrum the 10EH -value and from the motion spectrum the sZ -value (fig. 2.3-2).. fig. 2.3-2 Process to find the linear relation between HE10 and Zs

WAVE SPECTRUM iS

frequency

SHIP MOTION SPECTRUM iS

frequency

RAO function

HE10 i S iZ

E10H

ZS

Repeat n times

= a H + bE10SZ

σ S E10= a H + b H + cE101 1 12

RA

O

frequencysmall shiplarge ship

fig. 2.3-1 RAO function

1

0

16

Through the coordinates ( )sE ZH ,10 the program calculates a linear regression line and the standard deviations sσ of the scatter around this line. Next the program determines a polynomial relation between the standard deviations and the 10EH -values:

(2.3-6) (2.3-7)

significant vertical ship motion standard deviation of the normal distributed scatter around the regression line motion characteristics significant wave height from the low frequent wave energy

2.3.4 PROBABILITY OF A CONDITION Conditions may change in time, in space, during a transit and/or between transits. The predictions are based on deterministic models or statistical data. The astronomical water level is an example of a deterministic value. It can be predicted quite accurately for the coming years for different locations. The wave height, meteorological conditions and the depth are examples of statistics. It is impossible to predict these values exactly in time and space. Only based on historical data something can be said about the expected frequency of occurrence of a particular condition in the future. The condition and its frequency of occurrence are described by probability density functions. To calculate the inaccessibility percentage that can be expected in the future, these functions are important. Statistical variables can be divided into four groups: 1 channel data, like: depth variation, width variations and material variations 2 meteorological data, like: water level and current variations 3 hydro dynamical data, like: wave conditions from local wind fields, swell and density waves 4 ship data, like: draught variations, sailing speed variations and transit frequencies A certain correlation can exist between data of the same group and between groups. For instance the meteorological current and water level are strongly correlated in most cases. A correlation between groups can be found between 2 and 3. For instance during Eastern wind in the Netherlands the water level will be lower and the waves from local wind fields will be relatively small due to absence of fetch length.

2.3.5 ADMITTANCE CRITERIA The admittance policy is based on admittance criteria. These criteria are related to safety and serviceability. In most situations the safety criteria are normative. Serviceability criterions become an issue in the case of manoeuvrability. For instance when the keel clearance is small, the turning ability is affected. Dutch Policy The Dutch government applies the following safety standard on the North Sea:

fig. 2.3-3 Run aground vessel

sZ

111 ,,,, cbaba

10EH

bHaZ Es +⋅= 10

11012

101 cHbHa EEs +⋅+⋅=σ

sσ

17

105.0)9.0ln(%101)0( 2525 =−=→=−=> −−

− yearsyears eP ξξ ξ

004214.0)004214.01ln(

004206.01.011)0())0(1(11.0)0( 252525

=−−=

→=−−=>→>−−==> −

year

yearyearyears PPP

ξξξξ

- The probability of a ship-bottom contact resulting in the loss of the ship or in large-scale contamination of the marine environment or the beaches of the North Sea, should be virtually zero.

- The probability of a ship touching the seabed resulting in damage to the ship, personal injuries, minor cargo spills and temporary blockage of the approach channel should be small comparable to other possible risks related to navigation.

Risk is defined here as: For Dutch channels this leads to the following safety criterion:

During a 25-year period the probability of touching the channel bottom with maximum minor damage must not be more than 10%.

Probability of touching the bottom With use of the Poisson distribution (2.3-2) the probability is equal to one bottom touch in 237 years: Number of bottom touches ξ per year: This leads to one bottom touch per T year on average: The economic life of a channel is assumed to be 25 years. After this time the channel is likely to be redesigned. Consequence of a bottom touch Research to the effects of touching the channel bottom has been carried out. TNO-IWECO4 concluded that a bottom touch does not always result in major damage. When a vessel penetrates 0.25 m into the bottom, only minor damage will occur. By a penetration of 0.40 m plastic deformation will occur, but still the failure of single plates is unlikely. The probability of leaks is therefore minor. Some conservative assumptions were made in these calculations, like: - The calculations were made with a solid channel bottom. In most channels in the Netherlands a

layer of mud is present. - The channel bottom is assumed to be flat, without any ridges. When a sailing vessel touches a

ridge, the top of the ridge will be flattened and the reaction force will be less. ref. [16]

4 Toegepast Natuurwetenschappelijk Onderzoek - Instituut voor Werktuigkundige Constructies

econsequencyprobabilitrisk ⋅=

fig. 2.3-4 Oil tanker ‘Prestige’ breaks due to a storm

(Spain 2002)

yearsT year 237004214.0/1/1 ≈== ξ

18

2.4 CALCULATION METHODS There are different methods to calculate the probability of a bottom touch during a channel transit. The most accurate and intensive method is a level III method. Two essentially different level III methods will be described in this paragraph. Subparagraph 2.4.2 describes the Numerical Integration method and subparagraph 2.4.3 the Monte Carlo Method.

2.4.1 LEVEL-CLASSIFICATION OF CALCULATION METHODS Structural engineers investigate the strength R (from Resistance) and load S (from Solicitation) of a mechanism. When the load exceeds the strength, the mechanism fails. The state just before failure occurs, is the limit state. The reliability of the mechanism is the probability that this limit state is not exceeded. Using limit states, it is often possible to define so-called reliability functions. The general form of a reliability function is: The probability of failure is: The reliability is the probability P(Z>0) and therefore it is the complement of the probability of failure, in formula: A parallel can be made between this reasoning and the probability of touching the bottom. The gross keel clearance can be seen as the strength in the model and the currents and waves as the loads. The probability of failure is defined as the probability of one or more bottom touches (2.3-2): In the structural domain, the Joint Committee on Structural Safety proposed a level-classification of the calculation methods. This classification includes the following three levels: Level III The level III methods calculate the probability of failure, by considering the

probability density functions of all strength and load variables. The reliability is directly linked to the probability of failure.

Level II This level assumes the reliability function to be linear in a carefully selected point. It approximates the probability functions of each variable by a standard normal distribution.

Level I At this level no failure probabilities are calculated. The level I calculation considers a mechanism sufficiently reliable if a certain margin is present between the representative values of the strength and the loads. ref. [2]

2.4.2 NUMERICAL INTEGRATION Before a channel transit the conditions are predicted, like the expected wave height, water level variation, sailing speed and currents. In reality the actual values vary around the expected values. For each condition the variations around the expected value are described by their probability density functions ( )iiX Xf . The number of bottom touches can be described as a function of the conditions X :

(2.4-1) The probability of failure can then be calculated with the integral:

(2.4-2)

SRZ −=

( ) ( )RSPZPPf ≥=≤= 0

( ) fPZP −=> 10

ξξ −−=>= ePPf 1)0(

( ) ( ) nnXXXff dXdXdXXXXfXPPn

……… … 2121,,,

0

,,,21

⋅⋅⋅= ∫∫ ∫>

ξξ

( )nXXXg ,,, 21 …=ξ

19

Equation (2.4-2) can be written as:

(2.4-3) in which: If the base variables iX of the statistical vector X are independent, then:

(2.4-4) To solve equations (2.4-3) and (2.4-4) various numerical integration methods are available. A simple method of numerical integration is the Riemann-procedure. A flowchart of how the Riemann procedure could be applied is given in figure 2.4-1. According to this procedure the probability of failure is approximated by:

(2.4-4) The number of integration steps is given by m . ref. [2]

( ) nXXdXdXdXXfdF …21 ⋅⋅=

( ) ( ) ( ) ( )nXXXff XdFXdFXdFXPPn

…… 21

1

0

1

0

1

021

⋅⋅= ∫ ∫ ∫ ξ

fig. 2.4-1 Flow chart Numerical Integration

Lower boundary of the integral:

yes

yes

yes

nXXX 000 ,,,

21…

1101 1XiXX ∆⋅+=

2202 2XiXX ∆⋅+=

( )Xξ

nnn XiXXn

∆⋅+= 0

fff PPP ∆+=

nn mi <

22 mi <

11 mi <

fP

( ) ( ) nXff XXXXfPP ∆∆⋅∆⋅⋅=∆ …21ξ

( ) ( ) n

m

i

m

i

m

innXff XXXXiXiXiXfXPP

n

n

∆∆⋅∆⋅∆++∆+∆+⋅≈∑∑ ∑= = =

……… 210 0 0

22110

1

1

2

2

ξ

( ) ( ) ( ) ( )11,,121

1

0

1

0

1

0

1

00

,,11121

XXXdFXXdFXdFXPdFdFP nnXXXXXXfXXf nn……… … −

>−

⋅⋅=== ∫ ∫ ∫∫∫ ξξ

20

2.4.3 MONTE CARLO SIMULATIONS A Monte Carlo simulation is a simulation of a transit with one statistical vector X for the conditions drawn out of the joint probability density function ( )Xf . By repeating these simulations a number of times, the probability of failure fP can be calculated. To draw a value out of a distribution the following statement is used:

The non-exceedence probability of an arbitrary random variable is uniformly distributed between zero and one, regardless of the distribution of the variable.

In formula:

(2.4-5) uniformly distributed variable between zero and one non-exceedence probability P(X<X) Equation (2.4-5) rewritten:

(2.4-6) inverse of the probability distribution function of X Examples of derivations of random number generators are given in appendix 3. To draw a value out of a joint probability density function, the function must be formulated as the product of the conditional probability distributions of the base variables. In formula:

(2.4-7) By taking n realizations of the uniform distribution between zero and one, a value can be determined for every iX (fig. 2.4-2):

(2.4-8)

ref. [2], [17]

( ) uX XXF =

uX

( )XFX

( )uX XFX 1−=

( )uX XF 1−

( ) ( ) ( ) ( )121,,,121 ,,,121121 −−

⋅= nnXXXXXXXXXXXXFXXFXFXF

nn…… …

( )( )

( )1211

,,,

11

2

11

,,,121

212

11

−−

−

−

−=

⋅⋅

=

=

nuXXXXn

uXX

uX

XXXXFX

XXFX

XFX

nnn……

Number of basic variables = n

Number of simulations = m

Draw n times out of the

uniform distribution

yes

uX

( )( )

( )1211

,,,

11

2

11

,,,121

212

11

−−

−

−

−=

⋅⋅

=

=

nuXXXXn

uXX

uX

XXXXFX

XXFX

XFX

nnn……

( )Xiξ

( )Xsumsum iξξξ +=

mi <

m

sumξξ =

( )ξfP

fig. 2.4-2 Flow chart Monte Carlo method

21

2.5 THE EXISTING MODEL HARAP HARAP (HARbour APproach) is a probabilistic computer model that calculates the inaccessibility percentage, the downtime and the permitted tidal windows for entrance channels. The model was developed by the Dutch Ministry of Transport, Public Works and Water Management. At the moment, only the port of Rotterdam uses the program to support their admittance policy, in the near future probably the port of IJmuiden will follow. First the mathematical background of HARAP is given in subparagraph 2.5.1. Subparagraph 2.5.2 describes the modules of HARAP and subparagraph 2.5.3 the input variables.

2.5.1 MATHEMATICAL BACKGROUND HARAP is a mix of a level I and a level III method. It uses numerical integration as calculation method and some allowances. The core program of HARAP calculates the probability of touching the channel bottom during a transit for a given combination of conditions. This is repeated for all possible combinations over a certain period of time T. The weighted sum of all the small chances of touching the bottom during the simulated transits should be smaller than the long-term safety criterion as given in subparagraph 2.3.5. This is in short the working of HARAP. The mathematical description is as follows: Number of bottom touches ξ during a channel transit at wave condition j (eq. 2.3-1):

(2.5-1)

With the expected number of channel transits B during wave condition j :

(2.5-2) intensity of channel transits frequency of occurrence of the wave condition period of the calculation number of channel transits it is possible to calculate the number of bottom touches in period T during wave condition j :

(2.5-3) By summing all the bottom touches in the transits the total number of bottom touches during period T is known:

(2.5-4) The intensity of bottom touches λ is not (as suggested) constant during a transit, because it depends on the keel clearance and the wave spectrum. The keel clearance is given by equation (2.3-5) and depends partly on the actual water level and the depth of the channel. Over a part of the channel the water level and depth are assumed to be constant. With nT the total amount of time in period T in which a certain water level and depth occurs, equation (2.5-4) can be extended:

(2.5-5)

pasjj T⋅= λξ

fδ

TN

[ ]T

TNTfBE j

Tj

withjjj

)(lim

∞→=⎯⎯→⎯⋅⋅= δδ

∑ ⋅⋅⋅⋅=+⋅⋅⋅⋅++⋅⋅⋅⋅=j

jjjpasjjjpaspas fTTfTTfTT δλδλδλξ ……111

jjjpasj fTT ⋅⋅⋅⋅= δλξ

nn

jnjnj

jpas TfT ∑∑ ⋅⋅⋅= δλξ

22

( ) ( ) ξξξ −== ek

kPk

!

Also the wave conditions and the sailing speed can change during transit. Therefore the channel is split in different parts with different conditions. In each section different speeds and wave climates are possible. With m the number of sections, ξ will be:

(2.5-6) The probability of k bottom touches, is known by the Poisson distribution, equation (2.3-2):

(2.5-7) In the most accurate model, the number of summations in (2.5-6) goes to infinity. This will also happen with the calculation time. Therefore HARAP calculates with classes for the conditions j (fig. 2.5-1) and divides the channel in finite different parts m and n . HARAP distinguishes the following j conditions: - wave climate - meteorological conditions - astronomical conditions - sailing speed - draught In HARAP it is assumed that the wave climate does not change during a transit through channel section m . The water level, depth and sailing speed can fluctuate more. Therefore the sections are divided into smaller segments n . During a transit through a segment, the water level, sailing speed and depth distribution are assumed to be constant. The draught variations are neglected during the transit of the channel, so density variations of the water are not taken into account (fig. 2.5-2). ref. [6] and [15]

2.5.2 MODULES HARAP has four modules. Each module contains different programs. The following modules are distinguished: - preparation - keel clearance: checks the nautical criterions - ksi: checks the safety criterions - window: calculates the tidal windows

∑ ∑∑ ⋅⋅⋅⋅=j

nn

jnjmnmpasm

jm TTf δλξ _

freq

uenc

y

condition

fig. 2.5-1 Probability density distribution with discrete classes

segment 5 segment 4 segment 3 segment 2 segment 1

section 2section 1

constant wave climate

constant water level, sailing speed and constant distribution of depth

fig. 2.5-2 Differences sections and segments

23

Module 1 In the module “preparation” the programs verify the correctness of the input data and transform these data into calculation values for the next modules. Module 2 HARAP checks two safety and two nautical criteria: safety criterions: - long-term criterion - single transit criterion nautical criterions: - manoeuvrability criterion - cross current criterion The module “keel clearance” checks the nautical criterions for different starting moments of a transit. If during that transit the net keel clearance is less then 1 m (manoeuvrability criterion) or the cross current exceeds a certain value, that starting moment is excluded. This module makes a first selection of the starting times. Module 3 Module “ksi” calculates the probability of touching the bottom for different conditions and for the remaining starting moments from the module keel clearance. With equation (2.3-4) it is determined how many bottom touches ξ can be expected during a transit, given a predicted combination of conditions. The calculated ξ has to satisfy the single transit criterion. The single transit criterion is a conditional criterion. Given a combination of circumstances the probability of a bottom touch during the transit should be less than 1%:

(2.5-8) When a transit under particular conditions satisfies the single transit criterion, ξ is multiplied with the probability that these conditions occur. This is repeated for all transit starting moments. The two successive starting moments that have the lowest probability of a bottom touch, form the optimal tidal window: ksitopt . All the ksitopt , according to equation (2.5-6) are summed: ksitot . This value has to satisfy the long-term criterion that is given in subparagraph 2.3-5. If not, HARAP keeps expelling optimal tidal windows with the largest contribution to the criterion, until it does. Module 3: subprogram CALCULATE KSI The subprogram CALCULATE KSI takes prediction inaccuracies into account. A ship gets a tidal window on the basis of predicted conditions. In reality, a probability exists that a different value occurs. This real value determines the probability that a ship touches the bottom.

01.0%1)0( ≤⎯⎯ →⎯≤> ξξ PoissonconditionsP

24

HARAP distinguishes the following inaccuracies and in most cases they are assumed to be normal distributed around the predicted value: - draught prediction - sounding inaccuracy - water level prediction - inaccuracy in the linear relation between the significant wave height and motions of the ship - wave prediction A program structure diagram of module 3 is given below:

Module 4 The main program of module 4 is the program window. It calculates the final tidal windows. The calculated optimal tidal windows are stretched until the long-term criterion is reached and a minimal inaccessibility percentage is found. The final tidal windows are saved and plotted in different tables. ref. [3], [4] and [5]

2.5.3 INPUT VARIABLES The model HARAP has the following input values: - Ship data - Channel data - Astronomical data - Meteorological data - Wave data Ship data HARAP divides ships into classes by type, draught and dimensions. The representative ship of a class is a ship with the most common dimensions. The movements of this ship at different sailing speeds and depth/draught ratio are also representative for all ships in that class.

ksitot=0for all astro classes

for all meteo classesfor all wave climates

for all saling speed regimesfor all allowed starting moments (tb = begin time)

ksi0(tb)=0for all channel sections m

for all channel segments nfor all depths

for all draughts

CALCULATE KSI

tb = allowed tb = not allowedif tb = allowed: ksi0t(tb) = ksi0t(tb) + ksi(tb)*f(astro)*f(meteo,wave)*f(speed)

ksitopt = sum of the lowest two successive ksi0t(tb) in a tideksitot = ksitot + ksitopt

throw away the biggest ksitopt

ksi(tb) < single transit criterion

ksitot < long term criteriontrue false

true false

25

The sailing speeds of a ship class are established in speed regimes. A regime prescribes the speed for each channel segment. Channel data The probability of touching the bottom is calculated in HARAP in a semi-stationary situation: a situation with a constant water level and channel depth. To vary these and other values the channel is divided in a number of parts. A channel exists of sections and the sections are divided into segments. It is possible to define the depth in a segment at a constant value or with depth classes. To compensate the influence of sand waves on the ship motions an allowance is possible. The sounding inaccuracies are compensated by a standard deviation. Astronomical data The astronomical water level is defined in HARAP by astro classes; each has a certain frequency of occurrence. An astro class exists of a water level curve with the length of a tidal period. The spreading of the real water level around the schematized curve is given by a standard deviation. An astro class is classified by the tide difference. This is the difference between the astronomical high water level and the astronomical low water level of the preceding low water. With an astronomical tidal series over a couple of years, it is possible to define the distribution of the tidal difference (fig. 2.5-3). Currents are coupled to the astronomical water level and defined by a current velocity and direction. The inaccuracies are taken into account by the definition of the crosscurrent criterion. For the current along the ship an allowance is taken into account. Meteorological data The meteorological water level is defined as the difference between the real water level and the astronomical water level. The deviation of the astronomical water level is called the meteo-effect and is caused by wind and air pressure effects. The input is again in the form of classes. Similar to the position of the stroke middle of the astro curve, the meteo-effect causes a shift along the vertical axis. The independence of these two variables makes it possible to combine them in one distribution. The accuracy of the meteo-effect depends on the following factors: - uncertainties in the frequency distribution of the meteo-effect - uncertainties as a result of the schematization of the astro curve - interpolation errors between the water level stations: HARAP interpolates linear between

locations - assumption error: it is assumed that the meteo-effect is constant during the tidal period.

fig. 2.5-3 Principle description astro class

position stroke middle

time

wat

er le

vel

tide

diff

eren

ce

stroke middle

M.S.L.

26

Wave data HARAP offers three ways to describe the wave climate, by: 1. low-frequent wave energy lfe 2. a HE10- Hres relation 3. a Hs- Tp relation (significant wave height and peak period) The first description is a rather simple one-parameter schematisation of the wave climate. This is useful because for most applications more detailed descriptions are not available. The lfe parameter gives the amount of low-frequency energy in the range between 0.03 and 0.1 s-1. This parameter is translated to a characteristic wave height of the low frequent wave energy, the HE10:

(2.4-9) 0.03 [s-1] 0.1 [s-1] wave frequency wave density spectrum The Hres parameter equals the amount of energy in the range between 0.1 and 0.5 s-1. Depending on the water level it is possible to reduce the wave height linear to the depth. This can be desirable when the water depth is relatively low to the wave height (shallow waves). In two different ways it is possible to take the wave prediction accuracy into account: by a measure for the spreading or by a prediction matrix. A prediction matrix is a joint probability density function of the predicted wave and the wave that was measured in reality (fig 2.5.6). ref. [7] and [15]

∑ ∆⋅=→=b

aE ffSmmH )(4 0010

ba

f)( fS

occu

renc

e

frequency0.03 0.10

occu

renc

e

frequency0.500.10

fig. 2.4-4 Wave frequency distribution, HE10

fig. 2.4-5 Wave frequency distribution, Hres

10

9

8

7

6

5

4

3

2

1

1 2 3 4 5 6 7 8 9 10

freq

uenc

y

frequency

1 2 3 4 5 6 7 8 9 10

10

9

8

7

6

5

4

3

2

1

real

wav

e he

ight

predicted wave height

fig. 2.5-6 Joint probability density distribution wave height

[s-1]

[s-1]

27

2.6 CHAPTER CONCLUSIONS Based on the previous paragraphs the following answers can be given: 1. What is the procedure for tidal bound ships?

Tidal bound ships wait in anchoring areas for a pilot and the opening of a tidal window. The pilot helps the ship to sail safely into the harbour. In the harbour the ship can get assistance of one or more tugboats. Tugboats take a ship to the right place in the harbour. Once the ship has arrived, a boatman makes sure that the ship is securely moored alongside a quay, a jetty or buoys.

2. Who wants to know the measure for accessibility and why?

The waterway authority wants to classify their admittance policy and the companies need information to support their establishment policy.

3. Which parameter gives a good measure for the accessibility?

The inaccessibility percentage: the percentage of tides during which no tidal window can be provided, due to unfavourable water level or weather conditions.

4. What determines the accessibility?

The following variables are needed to make a reliable probabilistic model to calculate the accessibility: - channel dimensions and morphological processes around and in the channel - ship characteristics that define the draught and sailing speed and describe the behaviour of

the ship in waves and currents in a given environment - wave climate and its variation in time and space - influence of the variation in time and space of meteorological conditions on the water

level, wind and currents. 5. What is an acceptable safety level?

The following safety level is acceptable for the Dutch government: During a 25 year period the probability of touching the channel bottom with maximum minor damage must not be more than 10%.

6. Which calculation methods are available?

The most accurate are the level III methods. Two examples are numerical integration and Monte Carlo simulation.

7. Which program is used at the moment?

HARAP (HARbour APproach) is used by the port of Rotterdam.

28

29

3 PROBLEM ANALYSIS In the beginning of the previous chapter, seven questions were formulated. Although the answers give a lot of information, it is still not clear why a new program should be made. Therefore the following question: Why is the existing program insufficient? An evaluation of HARAP is given in paragraph 3.1. Paragraph 3.2 gives the project definitions based on this evaluation and on the project background, presented in chapter 2.

3.1 EVALUATION OF THE EXISTING MODEL HARAP The evaluation how HARAP describes the processes is given in subparagraph 3.1.1. Subparagraph 3.1.2 gives the advantages and disadvantages.

3.1.1 DEGREE OF DESCRIBING THE PHYSICAL PROCESSES Paragraph 2.2 gives the background of the physical processes that are important to calculate the probability of touching the bottom. This subparagraph 3.1.1 evaluates how well HARAP describes these processes by placing a score from 1 to 5 behind the processes. A one means that HARAP does not take the process into account. If it takes the process into account, but not very accurate, it gets a two, three or four, depending on the accuracy. A five is given for a good translation of the processes. Explanation - Due to the classification of astronomical data, HARAP converts deterministic variables into

statistical uncertainties. {1, 8} - Because of the transformation of continuous probability density functions into functions with

discrete classes, HARAP never gets the full score of five points. {2, 5, 6} - The difference between the predicted draught and the real

draught can be taken into account. The draught variations due to density variations are not taken into account. {3}

- The influence of sand waves is taken into account by an allowance. {4}

- The cross current criterion, that excludes transits when a certain cross current is exceeded, is not a proper description of the cross current effects. {7}

- Meteorological current and river discharge variations are taken into account as an allowance in the crosscurrent criterion. {9,10}

- It is possible to take the dependence between the meteo-effect and waves into account, by a matrix (table 3.1-1). {12}

{1} vertical tide 3 {9} meteorological currents 2{2} meteorological water level 4 {10} river discharges 2{3} draught variation 3 {11} cross wind effects 1{4} morphological factors 2 {12} relation wind and waves 3{5} sailing speed variations 4 {13} local wind field waves 3{6} squat 4 {14} swell 3{7} cross current effects 2 {15} breaking of waves 2{8} horizontal tide 3 {16} refraction of waves 1

table 3.1-1 Relation matrix between wind and waves

meteo classes1 2 3

1 0.20 0.10 0.042 0.15 0.08 0.033 0.05 0.08 0.024 0.03 0.06 0.025 0.02 0.03 0.026 0.01 0.01 0.017 0 0 0.018 0 0 0.019 0 0 0.01

10 0 0 0.01tot. 0.46 0.36 0.18 1.00

wav

e cl

asse

s

30

- No distinction is made between swell and waves from local wind fields, but the wave climate can be described rather accurately with a wave direction, period and height. {13, 14}

- HARAP does not use a model to calculate the interaction between the bottom and the waves. Only a rough limitation of the wave height is possible. {15}

3.1.2 ADVANTAGES AND DISADVANTAGES Advantages - Since the introduction of HARAP in Rotterdam in 1985, no bottom touches of tidal bound ships

have been recorded. - The inaccessibility percentage of all ships is lower since the introduction. - It is easy to determine the tidal window for an incoming ship. Tables are used, so calculations

are not needed. - Using the short version without much optimisation, the calculation time is about 5 minutes. Disadvantages - The choice of the number of classes and their arrangement influence the results. Figure 3.1-1

shows two different arrangements: one with a constant class width and one with varying class widths. The study group “Evaluation 74-feet channel” researched a possible lowering of the inaccessibility percentage of 74-feet ships to Rotterdam. They concluded that the inaccessibility percentage could be lowered from 37% to 23.5% by shifting only one border of a class. Besides shifting the class borders, it is possible to shift the calculation value of the class. It is for instance not always logical to take the class middle. Due to the exponential function (2.3-3) the adverse class border has the most influence.

- A tidal window issue is not based on actual predictions of the astronomical tide, currents, meteorological water level variations and wave spectrum.

- HARAP uses ship classes for its calculation. In the future, it is desired to go to an admittance policy for individual ships.

- Some allowances in HARAP can no longer be retrieved. - The semi-stationary calculation method due to the division of the channel in sections and

segments has the same kind of variation possibilities as the class arrangements. - HARAP allows ships to sail under heavy circumstances without satisfying the long-term

criterion, if they satisfy to a stricter criterion during gentle conditions. This implies that the allowed ξ may vary for different tidal windows as long as the expected value over all the circumstances is smaller than the long-term criterion. The long optimisation algorithm that divides the starting moments of transits over all circumstances has a long calculation time. Sometimes the calculation times are in the order of days or even without any results.

- It is not easy to expand HARAP with an extra module, because there is no clear distinction between the physical and probabilistic calculations.

freq

uenc

y

condition

freq

uenc

y

condition1 2 9876543 1 2 76543 8 9

constant class width various class width

fig. 3.1-1 Arrangement of the classes

31

- The astro-classes in HARAP make it impossible to sail from one tidal cycle to another. Mirroring the tidal window at the end of the cycle solves this, but this is not a very good solution.

ref. [8], [18], [19], [20] and [21]

3.2 DEFINITION OF THE PROJECT The problem definition and project objective are defined in subparagraph 3.2.1. Subparagraphs 3.2.2 and 3.2.3 give respectively the project boundaries and assumptions.

3.2.1 PROBLEM DEFINITION AND PROJECT OBJECTIVE From the previous paragraph the following can be concluded:

Why is the existing program insufficient? The distribution of input variables into discrete classes is the biggest disadvantage of HARAP. Due to this results of HARAP are partly determined by arbitrarily chosen class bounds. Given the possibilities of modern technology, like the calculation speed of a computer, these disadvantages are not acceptable anymore. Only by making large adjustments to HARAP, it could be improved. With these adjustments, one could speak of a new program.

Problem definition Due to the schematisation of input variables into discrete classes in the most extensive program in the Netherlands (HARAP), its results are partly determined by arbitrarily chosen class bounds. Project objective Design a better computer model than HARAP that calculates the inaccessibility percentage and an entrance regime for a tidal bound ship with a given draught, using continuous input variables. More specific, the program has to calculate with: - continuous probability density functions - continuous course of the channel - actual predicted conditions Project steps - Generate alternatives and evaluate each of them in the same way as HARAP. (chapter 4) - Compare the selected alternative with HARAP in a case study. (chapter 5) - Draw conclusions and give recommendations. (chapter 6)

3.2.2 PROJECT BOUNDARIES - The third dimension, the width of the channel, is left out of the calculations. - The exceptional situation that a ship comes outside the channel is not taken into account. - Only bulk carriers are taken into account. - The alternatives only take channels in relatively deep areas into account. - The programming language will only be applicable to a specific case study.

32

3.2.3 PROJECT ASSUMPTIONS - Results from the program SEAWAY and the additional program of AVV are correct. - The drift due to wind on fully loaded bulk carriers can be neglected. - Refraction and breaking of the waves can be neglected in the case study. - The input data for the case study of the alternatives will be the same as for HARAP. To

translate discrete classes into continuous probability density functions, it is allowed to interpolate linearly between the class borders.

- To determine the squat, the method of Tuck-Taylor may be used. - To make a comparison between the alternatives and HARAP, it is allowed to interpolate

linearly between different input data in the same way as HARAP does (for example the interpolation between two measurement stations along the channel).

33

4 ALTERNATIVES In this chapter, three possible computer models are described to achieve the project objective. The alternatives use the level III calculation methods: Numerical Integration or Monte Carlo Simulations. A description of both methods can be found in paragraph 2.4. Alternatives 2 and 3 may be extended with different modules: a module to reduce the run time and two modules to reach a lower inaccessibility percentage. A lower inaccessibility percentage for a given channel design is possible with a more optimized distribution of the tidal windows over conditions and ships. As long as the expected number of bottom touches [ ]ξE over the years is lower than the long-term criterion, the safety level for a single transit is allowed to vary for different conditions and ships. For instance during heavy conditions, some ships can only get a tidal window during the tidal cycle, if the waterway authority uses a relatively low safety level. This is allowed when the tidal windows are issued on the basis of a stricter long-term safety criterion during relatively gentle conditions. The same holds for a distribution of the tidal windows over the ships. Ships with a relatively large draught can satisfy to a more tolerant safety level as long as the ships with relatively small draughts compensate this. The following alternatives and modules are described in this chapter Alternative 1: Calculation with predefined classes Alternative 2: Monte Carlo method Alternative 3: Numerical integration Module 1: Runtime reduction Module 2: Optimisation for the conditions Module 3: Optimisation for the ships Each alternative or module is described in a different paragraph. A paragraph contains a model description and a list of the advantages and disadvantages. Chapter 4.7 gives an evaluation of the way the alternatives describe the physical processes (which were mentioned in paragraph 2.2). The chapter ends with some conclusions in paragraph 4.8.

4.1 ALTERNATIVE 1: CALCULATION WITH PREDIFEND CLASSES The first alternative considers adjustments of the existing computer model HARAP. In paragraph 2.5 a description of HARAP is given and its evaluation can be found in paragraph 3.1. HARAP uses numerical integration as calculation method, but with predefined integration steps. The step size is defined by the users of HARAP and is not necessary constant.

4.1.1 DESCRIPTION OF THE MODEL - HARAP schematises the probability density functions into discrete classes. By taking more

classes, for example 100 classes instead of the now used 10, the functions are almost continuous (fig. 4.1-1).

-4 -3 -2 -1 0 1 2 3 40

1000

2000

3000

4000

5000

6000

7000

-4 -3 -2 -1 0 1 2 3 40

1000

2000

3000

4000

5000

6000

7000

11 classes 100 classes

fig. 4.1-1 Number of classes

34

- The model calculates with more channel parts, for example 100 instead of 10. - Module 4: Venster has to be adjusted. It is not efficient anymore to work with printed tables if

HARAP distinguishes 100 classes for every condition. In that case already for one ship class with a certain draught, 10,000 tables are needed. Each table contains 100 x 100 tidal windows. In total, with all the ship classes, 1010 tidal windows will be needed. These tidal windows have to be saved on a computer.

- Still a solution for the problems with the optimisation algorithm has to be found, see subparagraph 3.1.2.

4.1.2 ADVANTAGES AND DISADVANTAGES Advantages - Relatively easy to realise. - The alternative is based on a tested model. - There are possibilities to work with an individual admittance policy. - The alternative works with ‘continuous’ probability density functions. - The choice of the class calculation value is less important (subparagraph 3.1.2). - The inaccessibility percentage is relatively low due to the optimisation algorithm. Disadvantages - The calculation time will be long. - The model will still calculate with astro-classes instead of the real tide: to issue a tidal window,

one of the 100 tides has to be selected. - A database of at least 10 gigabytes is needed. - For the calculation speed, physical and probabilistic relations are not wise to keep separated. In

some cases it is useful to calculate physical relations between probabilistic integration steps. This can considerably reduce the calculation time. When expanding a model with for instance a module for refraction, it is recommended to put this module after the bottom position loop and before the ship draught loop (see figure 4.1-2). Due to the independency of the draught and the wave refraction, it is not necessary to calculate the effects of the bottom position on waves for every draught, because it will not vary. By placing physical modules between probabilistic calculation steps, the model gets less transparent.

- The problem with transits to other tides remains (see subparagraph 3.1.2).

4.2 ALTERNATIVE 2: MONTE CARLO METHOD Alternative 2 uses the Monte Carlo method to calculate the probability of a bottom touch during a transit for a given predicted set of conditions. This probability has to be smaller than a constant long-term criterion.

for all astro classesfor all meteo classes

for all wave climatesfor all saling speed regimes

for all allowed starting moments for all channel sections

for all channel segmentsfor all depths

Module wave refractionfor all draughts

calculate ksi

fig. 4.1-2 Small part of the structure program diagram of HARAP with the location for the module refraction

35

4.2.1 DESCRIPTION OF THE MODEL - The following three steps form the calculation sequence of the model for issuing a tidal

window: 1. Predict the following conditions:

- mean wave height - mean wave direction - meteorological conditions (water level and current) - mean sailing speed

2. Determine for different starting moments of a transit (with the above conditions) whether the nautical criteria are fulfilled during the transit. The nautical criteria are defined as follows: 1. The keel clearance should never be less than ‘x’ metre during a transit. 2. The cross current should never exceed a certain value at a certain position.

3. Determine the probability of touching the bottom for the remaining starting moments, by taking the variations around the predicted conditions into account. This probability should be less than the long-term criterion that is defined in paragraph 2.3.

- By simulating a tidal window issue a number of times, the inaccessibility percentage can be calculated. A simulation is done with a drawn set of predicted conditions and their probability density functions.

- The calculation uses: - real tidal data: it draws tides from a succession of tides. - continuous probability density distributions of the conditions - a channel that is schematised by a continuous line: the model draws different positions

along this line to calculate the probability of touching the bottom - a period of 15 minutes between the transit starting moments: this is copied from HARAP

- Issuing tidal windows is no longer based on tables, but on running step 1 once and then steps 2 and 3.

The program structure diagram of the “Monte Carlo” alternative is given below:

H prediction

freq

uenc

y

E10

PmeanHE10(Li3)

fig. 4.2-1 Drawing PmeanHE10

timewat

er le

vel

time

wat

er le

vel

fig. 4.2-2 Drawing astronomical tide

1 read input2 PmeanT = draught = constant3 lsimu = number of draws from the distributions of the predicted conditions4 ksimu = number of draws from the distributions around the predicted conditions5 NpKC = number of positions along the channel axis to determine the keel clearance6 Npksi = number of positions, drawn along the channel axis to determine ksi7 IP_sum = 0 (IP = Inaccessible Percentage)8 L = 19 step 110 draw a set of predicted conditions from their distributions:11 PmeanHE10(L) = wave height (fig. 4.2-1)12 PmeanRHE10(L) = wave direction13 PmeanM(L) = meteo-effect14 Pmeanvs(L) = sailing speed15 draw a tidal cycle from a sequence of tides (uniform distribution ) (fig. 4.2-2)

16 tb(1) = begin time of tidal cycle17 Ntbi = the number of starting moments of a transit during the tide18 delta_t = (channel lengh / (NpKC - 1)) / Pmeanvs(L)

36

freq

uenc

y

PmeanHE10(Li3)

wave height

H(k)

fig. 4.2-3 Drawing H(k)

Drawing

Channel boundary

boundaryChannel

fig. 4.2-4 Drawing positions

d(p)Pmeand(position)

MSLpositon

dept

h

fig. 4.2-6 Calculation of ksi(k)

labd

a

time(time, labda)area = ksi

fig. 4.2-5 Drawing d(p)

19 step 220 i = 121 position (of the ship) = 0 (at the beginning of the channel)22 time = tb(i) = starting moment of the channel transit23 wl_a = astronomical water level for given time and position24 c_a = astronomical current for given time and position25 wl_m = meteo water level rise for given position26 c_m = meteo current for given position27 Pmeand = mean bottom level for given position28 c = current = c_a + c_m (vector summation)29 Zv = squat (with Tuck-Taylor formula and current influence)30 KC = keel clearance = wl_a + wl_m + d - PmeanT - Zv31 check nautical criteria32 OK not OK33 time = time + delta_t tb(i) = not allowed34 position = position + Pmeanvs * delta_t35 tb(i) = allowed36 repeat until position = end channel or tb(i) = not allowed37 tb(i) = tb(1) + 15 minutes * i38 i = i + 139 repeat until i = Ntbi40 step 341 i = 142 tb(i) = allowed43 yes44 k = 1 i = i + 145 draw a set of conditions from distributions around 46 Pmean (normal distribution ):47 H(k) = wave height (fig. 4.2-3)48 Hr(k) = wave direction49 M(k) = meteo-effect50 vs(k) = sailing speed51 T(k) = draught52 p = 153 draw a position along the channel axis54 (uniform distribution ) (fig. 4.2-4)55 determine the time at this position56 time = tb(i) + position(p) / vs(k)57 determine for position and time:58 wl_a, c_a, wl_m and c_m59 determine for position: Pmeand60 draw d(p) (= dept) from distribution 61 around Pmeand (fig. 2.4-5)62 determine for position: KC and Zv63 determine PmeanZs = mean significant 64 ship motion65 draw Zs (= significant ship motion) from 66 distribution around PmeanZs67 determine Tm = mean period of ship motion68 m0 = 1/16 * Zs^26970 labda(time) =7172 p = p + 173 repeat until p = Npksi74 ksi(k) = area under labda(time) (fig. 4.2-6)75 k = k + 176 repeat until k = ksimu

no

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

0

2

2exp

1m

KC

Tm

λ

37

If for example lsimu = 10,000, ksimu = 1,000, Ntbi = 100 and Npksi = 100 the number of calculations is approximately 1011. Tidal window issue: lsimu = 1 and the number of calculations is approximately 107.

4.2.2 ADVANTAGES AND DISADVANTAGES Advantages - It calculates with a continuous channel, continuous distributions and real tidal data. - Tidal window adapted to each individual ship. - Relatively convenient program. - It is relatively easy to expand the model without losing the clear distinction between physical

and probabilistic relations. For every drawn set of conditions the physical relations have to be recalculated once. The core program of the model consists of a set of physical calculations and the input for this program is the set of draws from the probability density functions. The output of the physical calculations is used to calculate the probability of touching the bottom.

- Entering another tidal cycle during the transit causes no problems. - Relatively few calculations are needed. - Tidal window issues are based on calculations with actual tidal data instead of tables. Disadvantages - The inaccessibility percentage will probably be higher than in alternative 1, due to the absence

of an optimisation algorithm. Instead module 2 could be used. - The accuracy of the results depends on the number of draws from the distributions: the larger

lsimu, ksimu and Npksi, the more accurate the results will be. The project objective is to make a model that is not influenced by classes. If this influence is replaced by the influence of the number of draws, the objective is not achieved. However compared to numerical integration, less calculation steps are needed to remove the influence of the calculation method (fig. 4.2-7).

77 ksi_total(i) = mean of all the ksi(k)78 ksi_total < long criterion79 yes no80 tb(i) = allowed tb(i) = not allowed81 i = i + 1 i = i + 182 repeat until i = Ntbi83 If during the tidal cycle two sequent tb(i) are allowed IPL = 0 else IPL = 184 IPsum = IPsum + IPL85 L = L + 186 repeat until L = lsimu87 IP = IPsum / lsimu * 100%

fig. 4.2-7 Converging speed calculation method (principle sketch)

inac

cess

ibil

ity p

erce

ntag

e

number of calculationsNumerical integration

Monte Carlo simulations

38

4.3 ALTERNATIVE 3: NUMERICAL INTEGRATION The third alternative has the same basic principles as alternative two and uses the same calculation method as HARAP: Numerical integration, but now the step sizes are constant and internal defined.

4.3.1 DESCRIPTION OF THE MODEL - The same three steps as in alternative 2 can be distinguished: 1 predict the conditions, 2 check

the nautical criteria and 3 check the long-term criterion. - The calculation uses:

- tidal data that is schematised into many classes - continuous probability density distributions of the conditions - a channel that is schematised with a continuous line: the model calculates the probability

of touching the bottom and the keel clearance for many fixed positions on this line - a period of 15 minutes between transit starting moments: this is copied from HARAP

- The issue of tidal windows is no longer based on tables, but on running once step 1 and then steps 2 and 3.

A program structure diagram of alternative 3 is given below:

fig. 4.3-1 Determining PmeanHE10(1)

H prediction

freq

uenc

y

E10PmeanHE10(1)

H prediction

freq

uenc

y

E10PmeanHE10(Li3)

delta_HE10

fig. 4.3-2 Determining the astronomical tide

fig. 4.3-3 Determining PmeanHE10(Li3)

(PmeanA(Li1))

wat

er le

vel

frequency

astr

onom

ical

tide

time

time

time

time

(PmeanA(L1))

(PmeanA(1))

(spring tide)

(neap tide)

1 read input2 Li1 = Li2 = Li3 = Li4 = Li5 = 13 determine the lower boundary:4 PmeanA(1) = astronomical tidal cycle5 PmeanM(1) = meteo-effect6 PmeanHE10(1) = wave height (fig. 4.3-1)7 PmeanRHE10(1) = wave direction 8 Pmeanvs(1) = sailing speed9 determine the number of integration steps:10 L1 = number of integration steps of PmeanA11 L2 = number of integration steps of PmeanM12 L3 = number of integration steps of PmeanHE1013 L4 = number of integration steps of PmeanRHE1014 L5 = number of integration steps of Pmeanvs15 k2 = number of integration steps of M (= meteo-effect arround PmeanM)16 k3 = number of integration steps of H (= wave height around PmeanHE1017 k4 = number of integration steps of RH (= wave direction around PmeanRHE10)18 k5 = number of integration steps of vs (= sailing speed around Pmeanvs)19 p = number of integration steps along the channel20 pd = number of integration steps of d = depth21 pZs = number of integration steps of Zs = significant ship motion22 determine the integration step size:23 delta_A = step size of PmeanA (size depends on L1)24 delta_M = step size of PmeanM (size depends on L2)25 delta_HE10 = step size of PmeanHE10(size depends on L3)26 delta_RHE10 = step size of PmeanRHE10 (size depends on L4)27 delta_vs = step size of Pmeanvs (size depends on L5)28 IPsum = 0 (IP = Inaccessible Percentage)29 step 130 PmeanA(Li1) = PmeanA(1) + (Li1 - 1) * delta_A (fig. 4.3-2)31 PmeanM(Li2) = PmeanM(1) + (Li2 - 1) * delta_M32 PmeanHE10(Li3) = PmeanHE10(1) + (Li3 - 1) * delta_HE10 (fig. 4.3-3)33 PmeanRHE10(Li4) = PmeanRHE10(1) + (Li4 - 1) * delta_RHE1034 Pmeanvs(Li5) = Pmeanvs(1) + (Li5 - 1) * delta_vs35 tb(1) = begin time of transit = 036 Ntbi = the number of starting moments of a transit during the tide

39

wave height

freq

uenc

y

PmeanHE10(Li3)

delta_Hk

Hk(ki3)

fig. 4.3-4 Determining H(ki3)

fig. 4.3-5 Determining the position

fig. 4.3-6 Determining d(kp)

dept

h

positonMSL

Pmeand(position)d(kp) delta_d

Channel boundary

Channel boundary

Position

delta_p

37 step 238 i = 139 delta_t = (channel length / p) / Pmeanvs(Li5)40 delta_p = delta_t / Pmeanvs(Li5)41 position (of the ship) = delta_p / 2 (= almost at beginning of channel)42 time = tb(i) + delta_t / 243 wl_a = astronomical water level for given time and position44 c_a = astronomical current for given time and position45 wl_m = meteo water level rise for given position46 c_m = meteo current for given position47 d = bottom level for given position48 c = current = c_a + c_m (vector summation)49 Zv = squat (with Tuck-Taylor formula and current influence)50 KC = keel clearance = wl_a + wl_m + d - PmeanT - Zv51 check nautical criteria52 OK not OK53 time = time + delta_t tb(i) = not allowed54 position = position + delta_p55 tb(i) = allowed56 repeat until position = end channel or tb(i) = not allowed57 tb(i) = tb(1) + 15 minutes * i58 i = i + 159 repeat until i = Ntbi60 step 361 i = 162 tbi(i) = allowed63 yes no64 k2i = k3i = k4i = k5i = 1 i = i + 165 ksi_total = 066 determine M(1) = meteo-effect67 delta_Mk = step size of M (size depends on k2)68 M(ki2) = M(1) + (ki2 - 1) * delta_Mk69 determine H(1) = wave height70 delta_Hk = step size of H (size depends on k3)71 H(ki3) = H(1) + (ki3 - 1) * delta_Hk (fig. 4.3-4)72 determine RH(1) = wave dierction73 delta_RHk = step size of RH (size depends on k4)74 RH(ki4) = RH(1) + (ki4 - 1) * delta_RHk75 determine vs(1) = sailing speed76 delta_vsk = step size of vs (size depends on k5)77 vs(ki5) = vs(1) + (ki5 - 1) * delta_vsk78 delta_t = (channel length / p) / vs(ki5)79 delta_p = delta_t / vs(ki5)80 position = delta_p / 2 (fig. 4.3-5)81 time = tb(i) + delta_t / 282 ksi = 083 determine for position and time:84 wl_a, c_a, wl_m and c_m85 determine for position: Pmeand 86 kp = 187 determine d(1) = depth88 determine delta_d = step size of d89 labda(time) = 090 d(kp) = d(1) + (kp-1)*delta_d (fig. 4.3-6)91 determine for position: Zv and KC 92 determine PmeanZs = mean 93 significant ship motion94 kZ = 195 determine Zs(1) = significant ship motion96 determine delta_Zs = step size of Zs

40

fig. 4.3-7 Calculation of ksi

area = ksi(time, labda)

time

labd

a

97 Zs(kZ) = Zs(1) + (kZ-1) * delta_Zs98 determine Tm = mean period of ship99 motion100 m0 = 1/16 * Zs(kZ)^2101102 delta_labda =103104 Prl = f(d) *delta_d * f(Zs) * delta_Zs105 labda = labda + delta_labda * Prl106 kZ = kZ + 1107 repeat until kZ = pZs108 kp = kp + 1109 repeat until kp = pd110 delta_ksi = labda(time) * delta_t111 time = time + delta_t112 position = position + delta_p113 ksi = ksi + delta_ksi (fig. 4.3-7)114 repeat until position = end channel115 delta_k = delta_Mk * delta_Hk * delta_RHk * 116 delta_vsk117 Prk = f(M, H, RH, vs) * delta_k118 ksi_total = ksi_total + ksi * Prk119 ki5 = ki5 + 1120 repeat until ki5 = k5121 ki4 = ki4 + 1122 repeat until ki4 = k4123 ki3 = ki3 + 1124 repeat until ki3 = k3125 ki2 = ki2 + 1126 repeat until ki2 = k2127 ksi_total < long criterion128 yes no129 tb(i) = allowed tb(i) = not allowed130 i = i + 1 i = i + 1131 repeat until i = Ntbi132 delta_L = delta_A * delta_M * delta_HE10 * delta_RHE10 * delta_vs133 PrL = f(PmeanA, PmeanM, PmeanHE10, PmeanRHE10, Pmeanvs) *134 delta_L135 If during the tidal cycle two sequent tb(i) are allowed IPL = 0136 else IPL = 1137 sumIP = sumIP + IPL * PrL138 Li5 = Li5 + 1139 repeat until Li5 = L5140 Li4 = Li4 + 1141 repeat until Li4 = L4142 Li3 = Li3 + 1143 repeat until Li3 = L3144 Li5 = Li5 + 1145 repeat until Li2 = L2146 Li4 = Li1 + 1147 repeat until Li1 = L1148 IP = IPsum * 100%

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

0

2

2exp

1

m

KC

Tm

λ

41

In table 4.3-1 a comparison is made between the number of calculations using alternative 3 and the existing model HARAP. The calculation time with alternative 3 will be roughly 1015 times longer. Stopping the integration algorithm when a tidal window no longer is available could shorten the calculation time (fig 4.3-8). If for instance for a predicted wave height no tidal window is available, this will also be the case for unchanged other conditions and larger wave heights. With this procedure the number of calculations could be reduced by roughly a factor 100. This would lead to 1022 calculations, which is still a lot compared to HARAP.

4.3.2 ADVANTAGES AND DISADVANTAGES Advantages: - It calculates with one continuous channel and with continuous distributions. - When the number of integration steps of the astronomical tide (L1) is large enough, the tidal

window issuing can be based on real tidal data. - Tidal window adapted to each individual ship. - Entering another tidal cycle during the transit causes no problem. Disadvantages - The inaccessibility percentage will probably be higher than alternative 1 due to the absence of

an optimisation algorithm. - The accuracy of the results depends on the number of integration steps: the larger L1-5, k2-4,

dp and pZs, the more accurate the results will be. - The number of calculations is relatively large compared to HARAP and alternative 2. The

calculation time depends on the number of stochastic variables. Therefore the calculation time will increase quickly with each addition of a stochastic variable.

- It is relatively hard to expand the model with physical modules without loosing the clear distinction between probabilistic and physical calculations for the same reason as for HARAP.

fig. 4.3-8 Stopping the algorithm

tab. 4.3-1 Comparison number of calculations HARAP and alternative 3

Never a tidal windowavailable during a tidal cycle

H prediction

freq

uenc

y

E10

stop

Steps Alternative 3 HARAPL1 100 3L2 100 5L3 100 10L4 100 1L5 100 3k2 50 10k3 50 5k4 50 5k5 50 5p 100 5pd 50 5pZs 50 5Ntbi 100 100Number of

calculations: 1024 109

42

4.4 MODULE 1: RUNTIME REDUCTION The runtime can be reduced with a preparation calculation before a run with alternative 2 or 3.

4.4.1 DESCRIPTION OF THE MODEL The preparation calculation determines a lower and upper boundary for a condition. For a predicted condition below the lower boundary the ships always get a tidal window and above the upper boundary never (fig. 4.4-1). If a drawn value (alternative 2) or integration step value (alternative 3) lays between the boundaries the model has to run steps 2 and 3, otherwise the model can skip these steps. In some cases this can reduce the calculation time by 80%. Determination of the upper and lower boundary: - Determine the most unfavourable set of conditions. Calculate if a tidal window is available by

varying one condition from the most favourable value until a value at which no tidal window is available. By repeating this for all the stochastic conditions the lower boundaries for all the conditions can be calculated. For example, in Rotterdam a set of the most unfavourable conditions will be: Eastern wind, neap tide, swell from the North and low sailing speed. Now vary the wave height from zero with steps until the value at which no tidal window is available and keep the other conditions constant. The lower boundary is the value at which no tidal window is available.

- Determine for the upper boundary the most favourable set of conditions. Calculate the availability of a tidal window by varying one condition from the most unfavourable value until the value at which a tidal window is available. Repeat this algorithm for all stochastic conditions, by starting each calculation with the most favourable set and determine the upper boundaries of all the conditions.

- It is easier to assume that the conditions are independent from each other during the determination of the boundaries. If a dependency is taken into account, it is harder to determine the most unfavourable set of conditions. Assuming independency leads to wider boundaries.

- The distance between the boundaries depends on the condition influence on the availability of a tidal window. When one condition is dominant (for instance the wave height), the boundaries of the other conditions remain almost unchanged. In this example the time benefit depends only on waves.

fig. 4.4-1 Lower and upper boundary

E10

freq

uenc

y

H prediction

Always at least one tidal windowavailable during a tidal cycle

Never a tidal windowavailable during a tidal cycle

Upperboundary

Lowerboundary

43

252525 1.011)0())0(1(11.0)0( −−=>→>−−==> − yearyearyears PPP ξξξ

A program structure diagram of the module for alternative 2 is given below:

The same principle holds for alternative 3 and the program structure diagram will not differ much.

4.4.2 ADVANTAGES AND DISADVANTAGES This module has the following advantage and disadvantages, additional to the alternatives: Advantage: - A shorter runtime than alternative 2 or 3. Disadvantages - The preparation calculation has to be done before every model run and it therefore must

become a standard procedure. It is easy to forget this procedure when a run has to be done for a channel redesign.

- Some knowledge and experience with respect to the influences of the conditions is needed to determine the most (un)favourable set of conditions.

4.5 MODULE 2: OPTIMISATION FOR THE CONDITIONS As long as the criteria are fulfilled, it is allowed to distribute the tidal windows over the conditions. This can be explained using a different formulation of the long-term criterion. The probability of more than one bottom touch in a year ( )0>ξP year:

1 read input2 PmeanT = draught = constant3 lsimu = number of draws from the distributions of the predicted conditions4 ksimu = number of draws from the distributions around the predicted conditions5 NpKC = number of positions along the channel axis to determine the keel clearance6 Npksi = number of positions, drawn along the channel axis to determine ksi7 IP_sum = 0 (IP = Inaccessible Percentage)8 determine the lower boundaries by numerical integration9 determine the upper boundaries by numerical integration10 L = 111 step 112 draw a set of predicted conditions from their distributions:13 PmeanHE10(L) = wave height (fig. 4.2-1)14 PmeanRHE10(L) = wave direction15 PmeanM(L) = meteo-effect16 Pmeanvs(L) = sailing speed17 if PmeanHE10(L) < lower boundary wave height => IPL = 018 if PmeanHE10(L) > upper boundary wave height => IPL= 119 if PmeanRHE10(L) < lower boundary wave direction => IPL = 020 if PmeanRHE10(L) > upper boundary wave direction => IPL = 121 if PmeanM(L) < lower boundary meteo => IPL = 022 if PmeanM(L) > upper boundary meteo => IPL = 123 if Pmeanvs(L) < lower boundary sailing speed => IPL = 024 if PmeanRHE10(L) > upper boundary sailing speed => IPL = 125 else: do line 15 up to and including line 83 of alternative 294 IPsum = IPsum + IPL95 L = L + 196 repeat until L = lsimu97 IP = IPsum / lsimu * 100%

44

When n ships sail to a port in a year, every ship may have ξ ship-allowed number of bottom touches:

(4.5-1) The long-term criterion can also be defined as:

The expected number of bottom touches during one transit of a ship over all the conditions [ ]ξE has to be equal or lower than the allowed number of bottom touches ξ ship-allowed:

(4.5-2) This means that it is allowed to have more bottom touches ξ than ξ ship-allowed during relatively rough conditions as long as ξ is smaller during gentle conditions. It can be seen as a distribution of the number of bottom touches over the conditions with an expected number of bottom touches smaller than or equal to ξ ship-allowed:

(4.5-3)

4.5.1 DESCRIPTION OF THE MODEL - The module calculates for each condition how the single transit criterion ξ can vary for

different values of the condition. An example for the predicted wave height is given in figure 4.5-1, where the factor q determines the position of ξ with respect to the long-term criterion ξ ship-allowed. The value max_q is equal to the maximum single transit criterion, which has the same definition as the single transit criterion in HARAP: equation (2.5-8).

A program structure diagram of the module for the wave condition is given below, but can be extended with other conditions:

( ) ( ) ( )( )( ) ( ) ( )

( )( ) ( )( )( )n

allowedship

yearwithn

yearshipallowedship

withnyearship

nshipyearyear

shipsn

PPP

ePPP

PPP

25

25

9.0ln

9.01)0(01ln01ln

100110

01100

−=

→−=>⎯⎯→⎯>−−=>−−=

→−=>⎯⎯→⎯>−−=>

→>−−=>=>

−

−

−

−

ξ

ξξξξ

ξξξ

ξξξξ

[ ] allowedshipE −≤ ξξ

fig. 4.5-1 Principle sketch of the distribution of the single transit criterion over the wave height

[ ] ( ) ( )∫ ⋅⋅=condition

dconditionconditionfconditionE ξξ

HE10

ξ

allowedship−ξmax_qq

1 run alternative 2 or 3 with a constant single transit criterion equal to the long-term 2 criterion, the calculated inaccessible percentage = IPmax3 maxHE10(1) = max wave height 4 M = number of integration steps to calculate the distribution over the wave height5 delta_Mi = step size of PmeanM (size depends on M1)6 delta_q = step size of q7 q_max = maximum single transit criterion = 0.018 E[ksi] = long-term criterion9 Mi = 110 minHE10 = 0

45

- The same steps are possible for the meteorological water level variations. The distribution of

the single transit criterion over both wave height H E10 and the meteo-effect wl m is shown in figure 4.5-3.

- The program structure diagram could be extended, to achieve better results. The largest ships for instance can never get a tidal window during the most extreme wave heights, despite a wider single transit criterion. Therefore it is not useful to allow more bottom touches during these transits. Here it is wise to distribute the single transit criterion also along the horizontal axis (fig 4.5-4).

4.5.2 ADVANTAGES AND DISADVANTAGES Advantage: - The inaccessibility percentage will be lower. Disadvantages - The downtime will be higher. - If the probability density distributions of the predicted conditions are not correct, the long-term

criterion can be exceeded in reality. The tidal windows are issued on the basis of variable single

H

freq

uenc

y

E10maxHminH

delta_H

delta_Mi

H

E10

E10

E10

qmax

delta

_q

E[ksi]

ksi

fig. 4.5-2 Construction of the distribution of ksi

10EH

[ ]ξE

ξ

mwl

fig. 4.5-3 Distribution of the single transit criterion over the wave height and the meteo-effect (principle sketch)

fig. 4.5-4 Difference between a distribution along one or two axes (principle sketch)

E10Hoptimisation along the vertical axis

optimisation along both axes

ξ

allowedship−ξ

11 q(Mi) = E[ksi] + delta_q12 PrMi = f(PmeanHE10(Mi)) * delta_Mi13 delta_H = PrMi / f(minHE10(Mi))14 ksi(maxHE10(Mi) - delta_Mi / 2) = q(Mi)15 ksi(minHE10 + delta_H / 2)) = q(Mi) - 2 * (q(Mi) - E[ksi]) (fig. 4.5-2)16 run alternative 2 or 3 with the adjusted criterions17 IP < IPmax18 yes no19 IP = IPmax q(Mi) = q_max20 minHE10 = delta_H + minHE1021 q(Mi) = q(Mi) + delta_q22 repeat until q = q_max23 Mi = Mi + 124 repeat until Mi = M

46

transit criteria. These criteria are calculated with probability density functions of predicted conditions. If for instance more frequently a relatively high wave is predicted than was assumed beforehand, the probability of a bottom touch will be larger than the long-term criterion. In other words: the probability of a bottom touch in the 25 years will be more than 10%.

- If the conditions can change during the transit it becomes a rather complex calculation to understand. If for instance two wave fields are defined for two different channel parts, the single transit criterion would change during the transit. Therefore the single transit criterion should be renamed into a single channel part transit criterion. Another possibility is to calculate the expected single transit criterion during the transit time. The same can be done for other conditions when they vary along the transit. The expected value of the single transit criteria over the conditions at the end should be lower than the long-term criterion. However mistakes are easily made in this process.

- The calculation time will be longer. The time depends on the number of integration steps M , the step size delta_q and the number of conditions. If for instance M = 100, the number of steps to max_q = 10 and the number of conditions = 2 the calculation time will be 2,000 times longer than the initial alternative.

4.6 MODULE 3: OPTIMISATION FOR THE SHIPS Besides a distribution of the allowed number of bottom touches over the conditions, the number of bottom touches can be distributed over the ships. Ships with large draught, during equal conditions, have in general a larger probability of touching the bottom compared to smaller ships. By varying ξ ship for different ships the inaccessibility percentage for all ships together can be lower. This is only allowed when the expected number of bottom touches of the ships E[ξ ship] is equal to or smaller than the allowed number of bottom touches ξ ship-allowed (see equation 4.5-1):

(4.6-1) with n the number of ships.

4.6.1 DESCRIPTION OF THE MODEL Any combination of alternative 2 or 3 with extension modules 1 and 2 can be used in this module as core program. A program structure diagram of the module is given below:

[ ] ( ) ( )nallowedship

n

ishipship i

nE 25

1

9.0ln1 −=≤⋅= −

=∑ ξξξ

1 number of ships = n2 Q = 03 long-term criterion is equally distributed over the ships4 IPsum_max = 0 (IP = Inaccessibility Percentage)5 ni = 16 calculate IP(ship(ni)) for ship(ni) with alternative 2 or 37 IPsum_max = IPsum_max + IP(ship(ni))8 ni = ni + 19 repeat until ni > n10 sort the ships on IP and renumber the ships: largest IP: ni = 1 and lowest IP: ni = n11 IPsum = IPsum_max12 IPsum = IPsum - IP(ship(n)) - IP(ship(1))13 add by the long-term criterion of ship(1) a factor q and subtract this factor from 14 the criterion of ship(n)15 calculate IP for ship(1) and ship(n) with alternative 2 or 316 IPsum = IPsum + IP(ship(n)) + IP(ship(1))

47

The presented program structure diagram is a simple algorithm. With a more advanced diagram a lower inaccessibility percentage could be achieved.

4.6.2 ADVANTAGES AND DISADVANTAGES Advantage - The inaccessibility percentage for all ships together is lower. Disadvantages: - The downtime of all ships together will be higher. - No fair treatment of ships: this module causes a higher inaccessibility percentage for small

ships. - Large ships have a larger probability of a bottom touch. With the assumption that the

consequences of a bottom touch for large ships are bigger compared to small ships, the risk of all ships together becomes higher in spite of the constant probability of a bottom touch E[ξ ship]. Therefore the consequences have to be taken into account, which for each ship separately are hard to determine.

4.7 DEGREE OF DESCRIBING THE PHYSICAL PROCESSES The 3 alternatives describe the physical processes in the same way; they only use different calculation methods. Therefore they are evaluated here together. The extension modules do not change the translation of the physical processes and are therefore left out of the evaluation. Like the evaluation of HARAP behind every physical process of paragraph 2.2 a score from 1 to 5 is placed. A one means that the alternative does not take it into account. If it takes the process into account, but not very accurate, it gets a two, three or four depending on the accuracy. A five is given for a good translation of the processes.

HA

RA

P

alte

rnat

ives

HA

RA

P

alte

rnat

ives

{1} vertical tide 3 5 {9} meteorological currents 2 4{2} meteorological water level 4 5 {10} river discharges 2 4{3} draught variation 3 4 {11} cross wind effects x x{4} morphological factors 2 2 {12} relation wind and waves 3 4{5} sailing speed variations 4 5 {13} local wind field waves 3 4{6} squat 4 5 {14} swell 3 4{7} cross current effects 2 2 {15} breaking of waves x x{8} horizontal tide 3 5 {16} refraction of waves x x

17 IPsum < IPsum_max18 yes no19 IPsum_max = IPsum Q=120 sort the ships on IP and renumber the ships: 21 largest IP: ni = 1 and lowest IP: ni = n22 save the long-term criteria for the ships23 repeat until Q = 124 IPsum = IPsum_max

48

Explanation - The alternatives score 5 points for processes {1} and {8}, because the alternatives calculate

with the real tide to determine the tidal windows. - Because the occurrence of different processes is schematised by continuous probability density

functions, the tidal windows are issued on actual predicted values. Therefore these processes get one point more in this evaluation than the evaluation of HARAP. {2, 3, 5, 6, 12, 13 and 14}

- Morphological factors, like sand waves, could easily be implemented into the alternatives 2 or 3 if the physical relations and the probability of occurrence are known. For the physical relations it is necessary to investigate how a ship reacts on sand waves. {4}

- An implementation of the meteorological currents and currents due to river discharges is possible. Therefore the physical relations and the probabilistic dependency between the astronomical tide, river discharge and meteorological conditions have to be investigated for the project area. If this is done properly the real predicted current curve along the channel could be used to issue a tidal window. At the moment these relations are not implemented in HARAP for the port of Rotterdam. {9 and 10}

- A better description of the ship movements is possible, by using the real predicted wave spectrum and the RAO-function of the ship instead of a representation by the H E10-value and a linear relation between H E10 and Z s. Therefore the maximum score of 5 is not given. {12, 13 and 14}

- Processes {11}, {15} and {16} are not taken into account in the evaluation, because they lay outside the project boundaries (see paragraph 3.2).

4.8 CHAPTER CONCLUSION The alternatives have the following advantages compared to HARAP: - They give a better description of the physical processes of subparagraph 2.2. - They calculate with a continuous channel, continuous distributions and real tidal data and

therefore take away the influences of the number of classes and channel parts on the results and makes it possible to issue tidal windows on actual predicted data instead of tables.

- Tidal window adapted to each individual ship. The difference between the 3 alternatives can be found in the used calculation method. Alternative 1 and 3 use numerical integration and alternative 2 uses the Monte Carlo method. The advantage of numerical integration compared to the Monte Carlo method is the higher accuracy when enough integrations steps are used. Monte Carlo analysis on the other hand has: a more conveniently arranged program structure and a relatively short calculation time. A rough prediction of the number of calculations is given in table 4.8-1.

model number of calculations

HARAP 109

Numerical integration 1024

Monte Carlo simulation 1011

tab. 4.8-1 Number of calculations of the models

49

From the table can be concluded that alternative 2 roughly 100 times slower is than HARAP, but 1013 times faster than alternative 1 and 3. Because of the relatively fast calculation method: the Monte Carlo method compared to the Numerical integration method, alternative 2 is selected to work out further in the case study. The three modules for runtime reduction and a lower inaccessibility percentage can only be used when some experience with the model have been gathered. For proper use, first the following points should be known: - the influence of the conditions on the availability of tidal windows (for module 1) - the accuracy of the probability density functions of predicted conditions (for module 2) - the consequences of a bottom touch per ship (for module 3) When these points are not known, the calculated inaccessibility percentage can be inaccurate, or worse, the calculated risks could be structural lower than the real risks.

50

51

5 CASE STUDY The selected alternative of chapter 4: alternative 2: Monte Carlo Method will be compared to HARAP in a case study. Both the inaccessibility percentage and the presentation of the tidal windows will be compared. For this case study, the port of Rotterdam is taken, because: - the waterway authorities in Rotterdam are the only users of HARAP at the moment - a lot of reliable statistical data of the entrance channel is available - most of the physical processes of subparagraph 2.2 are present in this area. Paragraph 5.1 gives the input of the case study. How many draws alternative 2 and a variant on this alternative need to converge to the theoretical inaccessibility percentage is described in respectively paragraph 5.2 and 5.3. The variant draws, compared to alternative 2, on a different moment during the calculations from the distribution functions. The results of the case study for HARAP and alternative 2 are presented in paragraph 5.4. The chapter ends with conclusions in paragraph 5.5. Alternative 2 and its variant are programmed in the mathematical computer programme Matlab, version 7 release 14. The program languages can be found in appendix 2.

5.1 INPUT DATA Alternative 2 uses the same input data as HARAP in this case study. The transformation of discrete input data into continuous data will be done by linear interpolation. When the waterway authorities decide to take the model from the design phase into the operational phase, proper continuous input data has to be used. The following input data is distinguished: - Ship data - Channel data - Astronomical data - Meteorological data - Wave data - Criteria data In separate subparagraphs each data sort is described for both HARAP and alternative 2. An explanation about the input variables of HARAP can be found in subparagraph 2.5.3. A complete list of the input data with its type (stochastic or deterministic) is given in appendix 6.

5.1.1 SHIP DATA The Berge Stahl is taken as ship for the case study. Berge Stahl transports iron ore and has the following characteristics: = 316 [m] length between perpendiculars = 1.75 [-] squat coefficient = 22.63 [m] mean draught = 0.1 [m] standard deviation predicted draught (normal distribution) = 372750 [ton] water displacement = 4 [m/s] mean sailing speed with respect to the bottom = 0.5 [m/s] standard deviation sailing speed (normal distribution) = 0.1 [m/s] standard deviation predicted sailing speed (normal distribution) The motion coefficients are calculated with SEAWAY (see subparagraph 2.3.3) for two different depth/draught ratios h/Td, three different sailing speeds sv and one wave direction. For Rotterdam

ppL

squatC

dT

dPTσ∇

sv

svσsPvσ

52

HARAP calculates with the most unfavourable wave direction, because the wave direction cannot be measured accurate enough for swell waves. In the future this will be possible. The following motion coefficients are determined ( sZ [m] and mT [s]):

Combination: sv = 3.1 m/s, h/Td = 1.125 Combination: sv = 5.1 m/s, h/Td = 1.125

Combination: sv = 3.1 m/s, h/Td = 1.175 Combination: sv = 5.1 m/s, h/Td = 1.175

Combination: sv = 3.1 m/s, h/Td = 1.250 Combination: sv = 5.1 m/s, h/Td = 1.125

5.1.2 CHANNEL DATA The port of Rotterdam is accessible via the Euro and Maas channel. In accordance with the traffic separation scheme, ships can reach the Euro channel from the Dover Strait (fig. 5.1-1). The Euro channel is 45.65 km ( ≈ 25 mile) long, 600 m wide and the nautical guaranteed depth (with respect to Mean Sea Level) varies from 25.4 to 24.8 m. The direction is 82 . On both sides of the channel lays a 300 m wide bank with a depth of LLWS 5 -22.00 until -22.10 m. From 0 km until 34.5 km sand waves occur.

5 LLWS = mean Low LowWaterSprings

93.12

198.0

060.0488.1

=

=+⋅=

m

Z

s

T

HZ

sσ

25.13

123.0

039.0492.1

=

=+⋅=

m

Z

s

T

HZ

sσ

45.12

143.0

019.0114.2

=

=+⋅=

m

Z

s

T

HZ

sσ

99.12

117.0

050.0838.1

=

=+⋅=

m

Z

s

T

HZ

sσ

17.12

130.0

105.0069.2

=

=+⋅=

m

Z

s

T

HZ

sσ

46.12

143.0

154.0114.2

=

=+⋅=

m

Z

s

T

HZ

sσ

fig. 5.1-1 Traffic separation scheme

TO EURO

CHANNEL

RYE

BRIGHTON

BOULOGNE-SUR-MER

THAMES ESTUARYLONDON

DOVER

LE TRÉPORT

DUNKERQUECALAIS

MEDWAY

53

The Maas channel is 11.35 km ( ≈ 6 mile) long, the width of the channel varies from 600 m at Maas Centre until 500 m at the port entrance and the nautical guarantied depth is 24.3 m. The direction is 112 (fig. 5.1-2). The channel has no banks. For the case study the project area ends at the port entrance. Normally HARAP also takes the area in the port into account. HARAP distinguishes two sections: the Euro and Maas Channel. The Euro channel contains four segments and the Maas channel one. Each segment has a constant depth (fig. 5.1-3). Alternative 2 translates the segments into one continuous line (fig. 5.1-4). This is a rough translation and a bit at the unsafe site. For the case study is assumed that the depths does not fluctuate in time. Also the occurrence of sand waves is ignored.

5.1.3 ASTRONOMICAL DATA At three locations along the channel astronomical data is available. The data consists of the predicted astronomical: water level, current and current direction. The locations are: the Euro Platform (2.83 km), Maas Centre (45.65 km) and the harbour entrance (57.00 km). HARAP distinguishes three classes for each location with the following probability of occurrence: ASTRO 1 = 25% ASTRO 2 = 50% ASTRO 3 = 25% An overall water level spreading of 0.12 m is assumed in HARAP.

fig. 5.1-2 Euro and Maas channel

fig. 5.1-3 Channel input HARAP

fig. 5.1-4 Channel input alternative 2

52°00'

VOORNE

H.v.H.

EURO PLATFORMMAAS CHANNEL

EURO CHANNEL

4°00'3°30'3°00'

MAASCENTRE

57 km

34.5 km45.65 km

23 km11.5 km

0 km

N

S

EW

82°112°

botto

m p

ositi

on

(with

res

pect

to M

.S.L

.)

position0 km 11.5 km 23 km 34.5 km 45.65 km 57 km

-25.20 m-25.10 m

-25.00 m

-24.80 m

-24.30 m

Euro channel Maaschannel

Maas centre Maas centre

Maaschannel

Euro channel

-25.00 m

position

botto

m p

ositi

on

(wit

h re

spec

t to

M.S

.L.)

-25.10 m

-24.80 m

-24.30 m

-25.20 m

57 km45.65 km34.5 km23 km11.5 km0 km

54

The calculation values of the ASTRO-classes for the Euro Platform are given in table 5.1-1. Alternative 2 calculates with a continuous tidal sequence in time. For the input, the period between 1 February 2005 and 1 August 2005 is used. The month April is presented in figure 5.1-5.

tab. 5.1-1 Astronomical input HARAP (Euro Platform)

0 5 10 15 20 25 30-1.5

-1

-0.5

0

0.5

1

1.5Water level Euro Platform

Time [days]

Wat

erle

vel

[m]

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1Current Euro Platform

Time [days]

Cur

rent

[m

/s]

0 5 10 15 20 25 300

50

100

150

200

250

300

350

400Direction current Euro Platform

Time [days]

Dir

ecti

on [

degr

ee]

0 5 10 15 20 25 30-1

-0.5

0

0.5

1

1.5Water level Maas Centre

Time [days]

Wat

erle

vel

[m]

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9Current Maas Centre

Time [days]

Cur

rent

[m

/s]

0 5 10 15 20 25 300

50

100

150

200

250

300

350

400Direction current Maas Centre

Time [days]

Dir

ecti

on [

degr

ee]

0 5 10 15 20 25 30-1

-0.5

0

0.5

1

1.5Water level port entrance

Time [days]

Wat

erle

vel

[m]

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1Current port entrance

Time [days]

Cur

rent

[m

/s]

0 5 10 15 20 25 300

50

100

150

200

250

300

350

400Direction current port entrance

Time [days]

Dir

ecti

on [

degr

ee]

fig. 5.1-5 Astronomical input alternative 2 (April)

ASTRO 1 ASTRO 2 ASTRO 3Water Current Direction Water Current Direction Water Current Direction

level [m] [m/s] current [degree] level [m] [m/s] current [degree] level [m] [m/s] current [degree]-0.62 0.2500 215 -0.78 0.4500 215 -0.95 0.6500 215-0.57 0.3500 220 -0.72 0.500 220 -0.87 0.6500 220-0.51 0.3375 220 -0.64 0.4937 220 -0.79 0.6500 220-0.44 0.3250 220 -0.57 0.4875 220 -0.70 0.6500 220-0.37 0.3125 220 -0.49 0.4812 220 -0.61 0.6500 220-0.28 0.3000 218 -0.41 0.4750 218 -0.52 0.6500 218-0.17 0.2250 218 -0.32 0.3562 218 -0.42 0.4875 218-0.01 0.1500 218 -0.19 0.2375 218 -0.27 0.3250 2180.19 0.0750 218 0.03 0.1187 218 -0.02 0.1625 2180.40 0.0000 0 0.35 0.0000 0 0.35 0.0000 00.56 0.1125 25 0.63 0.1562 25 0.72 0.2000 250.63 0.2250 25 0.79 0.3125 25 0.94 0.4000 250.63 0.3375 25 0.81 0.4687 25 0.99 0.6000 250.55 0.4500 25 0.74 0.6250 25 0.91 0.8000 250.44 0.4375 25 0.60 0.6000 25 0.77 0.7625 250.29 0.4250 25 0.44 0.5750 25 0.60 0.7250 250.13 0.4125 25 0.25 0.5500 25 0.42 0.6875 25

-0.06 0.4000 25 0.05 0.5250 25 0.22 0.6500 25-0.25 0.3000 25 -0.19 0.3937 25 0.00 0.4875 25-0.47 0.2000 25 -0.46 0.2625 25 -0.25 0.3250 25-0.67 0.1000 25 -0.75 0.1312 25 -0.50 0.1625 25-0.85 0.0000 0 -0.98 0.0000 0 -0.69 0.0000 0-0.95 0.0625 215 -1.11 0.1125 215 -0.80 0.1625 215-0.99 0.1250 215 -1.13 0.2250 215 -0.83 0.3250 215-0.97 0.1875 215 -1.09 0.3375 215 -0.79 0.4875 215

55

5.1.4 METEOROLOGICAL DATA For the three locations that were mentioned in subparagraph 5.1.3, the distributions of the meteorological water level are known. HARAP distinguishes five classes for each location with the following probability of occurrence: METEO 1 = 15.0% METEO 2 = 52.0% METEO 3 = 30.7% METEO 4 = 2.2% METEO 5 = 0.1% The calculation values are the class middles and are given in table 5.1-2. Alternative 2 calculates with continuous distribution functions. Therefore the discontinuous probability density function ( )Mf of HARAP (fig. 5.1-6) is first transformed into a distribution function ( )MF (fig. 5.1-7). This distribution function is made continuous by linear interpolation between the class boundaries (fig. 5.1-8). For the three different locations these distribution functions are made. Alternative 2 uses these distribution functions as input.

Alternative 2 calculates with a spreading of 0.1 m around the predicted mean. HARAP has put this spreading in the 0.12 m around the predicted astronomical water level, which is not taken into account in alternative 2. For the case study it is assumed that the wave climate is independent of the meteorological conditions. By ignoring this physical relation, the calculated inaccessibility percentage will be higher. Also the influence of the meteorological conditions on the currents is left out of the calculations. This has influence when exclusions on the basis of cross currents are done in the model. Otherwise the influence will be minor.

Location Meteo 1 Meteo 2 Meteo 3 Meteo 4 Meteo 5Euro Platform 0.39 0.11 -0.20 -0.58 -0.87Maas Centre 0.47 0.24 0.02 -0.28 -.053port entrance 0.50 0.22 -0.09 -0.38 -0.62

tab. 5.1-2 Calculation values meteo-classes with respect to Mean Sea Level [m]

f (M

) [%

]

Meteo prediction [m]

-1.0

15

0.1

-0.7

25

-0.3

90

-0.0

45

0.25

0

0.53

0

2.2

30.7

52.0

15.0

100.

0

85.0

33.0

2.3

0.53

0

0.25

0

-0.0

45

-0.3

90

-0.7

25

0.1

-1.0

15

Meteo prediction [m]

F (M

) [%

]

Meteo prediction [m]

-1.0

15

0.1

-0.7

25

-0.3

90

-0.0

45

0.25

0

0.53

0

2.3

33.0

85.0

100.

0

F (M

) [%

]

fig. 5.1-6 Discontinuous probability density function (Euro Platform)

fig. 5.1-7 Discontinuous distribution function (Euro Platform)

fig. 5.1-8 Continuous distribution function (Euro Platform)

56

5.1.5 WAVE DATA HARAP distinguishes ten classes for modelling the probability density function of HE10 (fig 5.1-8). The function is based on a set of predicted wave heights in the past. A tidal window is issued on the basis of the maximum HE10 value that is predicted in the period six to twelve hours after the prediction moment (fig 5.1-9). The tidal window is issued just after the prediction moment: almost six hours before the opening of the tidal window. The transformation of the discontinuous probability density function into a continuous distribution function for alternative 2 is done in the same way as in the previous subparagraph (fig. 5.1-10 and 5.1-11). The HMR6 issues tidal windows for Rotterdam. They also predict the HE10 –value. The relation between the predicted and the actual measured HE10 –value is described by a conditional probability density function. HARAP schematises this function by a prediction matrix.

6 Hydro Meteo centrum Rijnmond

f (H

)

[%

]

E10H prediction [m]

0.00

0.40

0.55

0.70

0.85

1.00

1.15

1.30

1.45

2.95

0.25

81.9

1

9.63

2.65

1.60

1.09

0.61

0.49

0.36

0.22 1.44

E10

100.

00

98.5

698

.34

97.9

897

.49

96.8

895

.79

94.1

991

.54

81.9

10.

25

2.95

1.45

1.301.15

1.00

0.85

0.70

0.55

0.40

0.00

H prediction [m]E10

E10

F (H

)

[%

]

E10H prediction [m]

0.00

0.40

0.55

0.70

0.85

1.00

1.15

1.30

1.45

2.95

0.25

81.9

1 91.5

494

.19

95.7

996

.88

97.4

997

.98

98.3

498

.56

100.

00

F (H

)

[%

]E

10fig. 5.1-9 Discontinuous probability density

function of HE10

fig. 5.1-11 Discontinuous distribution function of HE10

fig. 5.1-12 Continuous distribution function of HE10

time [hour]

H

pre

dict

ion

[m]

E10

126Prediction moment: time =0

max H E10

fig. 5.1-10 Definition of the HE10 value for a tidal window issue = maxHE10

57

( )( ) ( ) 004206.01.01100111.0)0( 252525 =−−=>→>−−==> − jaarjaaryear PPP ξξξ

( )( ) ( ) 5250250 10686.1)0(110011)0( −⋅=>−−=>→>−−=> yearshipshipyear PPPP ξξξξ

The real wave height is assumed to be normal distributed around the predicted wave height. The mean and standard deviation are given by the following equations:

(5.1-1) This means that the HMR in general overestimates the HE10 –value. The prediction matrix for HARAP is given in table 5.1-3. Alternative 2 can calculate with the equations (5.1-1). For the case study it is assumed that the probability density function is constant along the channel. In reality it can be expected that the wave spectrum in the Maas channel will be different compared to the Euro channel.

5.1.6 CRITERIA DATA The following criteria are taken into account: - long-term criterion - single transit criterion (only HARAP) - manoeuvrability criterion At the moment the tidal window issue is based on 250 tidal bound ships per year. This number turns out to be overestimated, but for now 250 ships is assumed to be true. The following definition of the long-term criterion was given in subparagraph 2.3.5:

During a 25-year period the probability of touching the channel bottom with maximum minor damage must not be more than 10%.

The probability of a bottom touch in a year is given by: The allowed probability of a bottom touch for a ship is given by:

0047929.03406.0

0019204.0*8914.0

10

10

10

10

+⋅=

−=

EH

EH

PmeanH

PmeanH

E

E

σµ

tab. 5.1-3 Prediction matrix in HARAP

Predicted wave class1 2 3 4 5 6 7 8 9 10

1 0.9985 0.3718 0.1516 0.0804 0.0512 0.0367 0.0284 0.0233 0.0198 0.01172 0.0015 0.4626 0.2970 0.1575 0.0900 0.0565 0.0384 0.0277 0.0210 0.00763 0 0.1540 0.3311 0.2525 0.1614 0.1038 0.0698 0.0493 0.0364 0.01154 0 0.0114 0.1730 0.2566 0.2138 0.1537 0.1081 0.0774 0.0569 0.01675 0 0.0002 0.0422 0.1651 0.2091 0.1834 0.1424 0.1071 0.0806 0.02326 0 0 0.0048 0.0673 0.1509 0.1763 0.1597 0.1308 0.1034 0.03107 0 0 0.0003 0.0174 0.0804 0.1366 0.1523 0.1410 0.1199 0.03998 0 0 0 0.0028 0.0316 0.0852 0.1237 0.1341 0.1260 0.04949 0 0 0 0.0003 0.0092 0.0429 0.0855 0.1126 0.1197 0.0587

10 0 0 0 0 0.0023 0.0249 0.0916 0.1969 0.3163 0.7502

Rea

l wav

e cl

ass

58

( ) 510686.1)0(1ln1)0( −− ⋅=>−−=→−=> shipshipship PeP ship ξξξ ξ

410686.1 −⋅=shipξ

With the Poisson distribution (see subparagraph 2.3.2) the allowed number of bottom touches for a transit is equal to:

In this case study the recommendation from the TNO-IWECO investigation (see paragraph 2.3.5) to multiply the long-term criterion by a factor 10 is adopted. A more constructive method would be to allow a bottom penetration of 25 cm. This is not done in the case study; otherwise a comparison with HARAP would not be possible. With the factor 10 the long-term criterion is:

(5.1-2) HARAP uses a single transit criterion of 1%. The manoeuvrability criterion is fixed on 1 m keel clearance. The cross current criterion is left out of the calculations. This criterion, like the manoeuvrability criterion, does not influence the probability of a bottom touch. It only excludes starting moments of a transit on the basis of a deterministic value. By leaving the cross current criterion out of the model it is easier to investigate the influence of various conditions. The manoeuvrability criterion disturbs the evaluation of the influence of various conditions much less, because there is always a starting moment in the tidal cycle at which this criterion is fulfilled. It is likely that this moment also has a larger probability of satisfying the long-term criterion compared to a moment without a satisfied manoeuvrability criterion. By first excluding starting moments based on the manoeuvrability criterion, the calculation time can be reduced.

5.2 NUMBER OF DRAWS IN ALTERNATIVE 2 Statistics enable to predict an inaccessibility percentage, but there will always be a difference between the calculated inaccessibility percentage and the measured inaccessibility percentage in practice. The difference has multiple causes. The following causes can be distinguished: - Incorrectness of the statistics

The probability density functions of the conditions determine the magnitude of the calculated inaccessibility percentage. The correctness of these functions has influence on the accuracy of the percentage. If for example the tidal windows are issued with more large waves than was assumed beforehand, the measured inaccessibility percentage will be higher than the calculated. Besides the correctness of the probability density functions, the dependency between various conditions has influence.

- Inaccuracy of the calculation method The Monte Carlo method consists of drawing a number of values out of their distribution functions. The number of draws determines the accuracy of the calculated inaccessibility percentage. The accuracy of the calculation method of HARAP has no influence on the difference between the calculated and measured inaccessibility percentage. If more classes are taken into account both the calculated and measured inaccessibility percentage will change. This is the advantage of working with tables for issuing tidal windows instead of a calculation based on actual data.

- Experience of the ship handler Nowadays it is easier to gather weather predictions. When a ship handler knows from experience that he will not get a tidal window with the predicted conditions, he may change his

59

sailing schedule or decide to sail with less draught. Both will lead to a lower measured inaccessibility percentage.

This paragraph gives the results of an investigation to the calculation method accuracy of alternative 2. The following research question is formulated:

What is the influence of the number of draws on the calculated inaccessibility percentage? Alternative 2 draws: - values from the distribution functions of the predicted conditions (lsimu times) - values from the distribution functions around the predicted condition (ksimu times) - positions along the channel axis (Npksi times) The parameters lsimu, ksimu and Npksi are the number of draws. For a tidal window issue, lsimu is always one. Therefore the number of draws from the distribution functions of the conditions has no influence on the safety level7. Because ksimu and Npksi are not one for the calculation of a tidal window, they influence both the accuracy of the inaccessibility percentage and the safety level. Investigation results to the influence of each parameter: lsimu, ksimu and Npksi are given in different paragraphs.

5.2.1 NUMBER OF DRAWS: POSITIONS Alternative 2 calculates for Npksi positions in the channel, the frequency of a bottom touch λ . The frequency is defined by equation (2.3-4). The number of bottom touches ξ for a transit is linearly dependent of the frequency. By running the model for different values of the parameter Npksi, its influence on the number of bottom touches ξ can be investigated. For 15 values of Npksi, runs were done. This was repeated 4 times for 2 sets of input values. At the end 1202415 =⋅⋅ runs were done. The 2 sets of input values are given in table 5.2-1 and the following values of Npksi were used: Npksi = [100; 200; 300; 400; 500; 1000; 2000; 3000; 4000; 5000; 10000; 50000; 100000; 500000;

1000000]

7 This is not true when module 2: optimisation for the conditions is used (see subparagraph 4.5.2).

tab. 5.2-1 Sets of input values for the investigation of Npksi

Variable Set 1 Set 2 Unit Descriptionvs 4.000 4.00 [m/s] sailing speedMe 0.250 0.250 [m] meteo-effect Euro PlatformMm 0.355 0.355 [m] meteo-effect Maas CentreMh 0.360 0.360 [m] meteo-effect harbour entranceH 0.600 0.800 [m] wave heighttb 3850800 3853500 [s] begin timeTd 22.630 22.630 [m] draught

60

Figure 5.2-1 gives the results for set 1 as input values and figure 5.2-2 for set 2. Conclusion When Npksi approaches a value of 1,000,000, the number of bottom touches ξ goes to a constant value. If a variation of ξ of about a factor 4 is accepted, a value of Npksi of 1000 will do. For a factor of 2, 10,000 draws are needed.

5.2.2 NUMBER OF DRAWS: REAL CONDITIONS A tidal window issue is based on a set of predicted conditions. In reality a different set of conditions will occur. Alternative 2 takes the prediction inaccuracies into account by drawing ksimu times a value around the predicted value. Simulations are done to investigate the effect of ksimu on: the expected number of bottom touches for one starting moment and the length of a tidal window.

fig. 5.2-1 Influence of Npksi on ksi (set 1, 4 runs)

fig. 5.2-2 Influence of Npksi on ksi (set 2, 4 runs)

102

103

104

105

106

0

1

2

3

4

5

6

7

8x 10

-4

Number of drawn positions (Npksi)

Num

ber

of b

otto

m t

ouch

es (

ksi)

102

103

104

105

106

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8x 10

-3

Number of drawn positions (Npksi)

Num

ber

of b

otto

m t

ouch

es (

ksi)

61

One starting moment The expected number of bottom touches [ ]ξE during a transit for a given starting moment is the mean ξ of ksimu simulated transits. For two sets of input values the moving mean for ksimu up to 100,000 is calculated; three times for set 1 and two times for set 2. Table 5.2-2 shows the sets of input values: The results of the runs with set 1 are given in figure 5.2-3 and the results with set 2 in figure 5.2-4.

fig. 5.2-3 Moving mean of [ ]ξE for ksimu (set 1, 3 runs)

fig. 5.2-4 Moving mean of [ ]ξE for ksimu (set 2, 2 runs)

(inset: entire ksi-domain)

tab. 5.2-2 Sets of input values for the investigation of ksimu (one starting moment)

Variable Set 1 Set 2 Unit DescriptionPmeanvs 4.000 4.000 [m/s] sailing speedPmeanMe 0.100 0.100 [m] meteo-effect Euro PlatformPmeanMm 0.100 0.100 [m] meteo-effect Maas CentrePmeanMh 0.100 0.100 [m] meteo-effect harbour entrancePmeanHE10 0.300 0.300 [m] wave heighttb 8903700 8903700 [s] begin timeTd 22.630 22.630 [m] draughtNpksi 100 1000 [-] number of position

100

101

102

103

104

105

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

-4

Number of simulated channel transits (ksimu)

Num

ber

of b

otto

m t

ouch

es (

ksi)

100

105

0

0.5

1

x 10-3

100

101

102

103

104

105

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

-4

Number of simulated channel transits (ksimu)

Num

ber

of b

otto

m t

ouch

es (

ksi)

62

Conclusions one starting moment - For ksimu 1000, [ ]ξE varies with a factor 2.5, ksimu = 10,000 the factor is 1.5 and for ksimu =

100,000 a factor of 1.25 can be expected. - It looks like that there is some relation between [ ]ξE and ksimu when Npksi = 100. In reality

this relation does not exists although it is very likely that the graph has this shape. An explanation can be found in appendix 5.

More starting moments For a tidal window issue it is necessary to calculate the expected number of bottom touches for different starting moments. The starting moments of channel transits are assumed to be uniform distributed over the tidal window. This means that the ship handler has no preferential starting moment. In reality the most ship handlers want to sail directly after the opening of the tidal window. The uniform distribution is chosen to be similar to HARAP. For each quarter of an hour in a tidal cycle the expected number of bottom touches is calculated. If the mean number of bottom touches of a series quarters is smaller than the long-term criterion, the length of the tidal window is known. For 4 different values of ksimu and 3 input sets, the lengths of the tidal windows are calculated. This leads to 1234 =⋅ runs. After these runs the influence of Npksi on the length of tidal windows is tested. The 3 sets of input values are given in table 5.2-3 and the following values of ksimu are used: ksimu = [1; 50; 100; 1000] Table 5.2-4 gives the lengths of the tidal windows for each set of conditions. The [ ]ξE -values for each starting moment is given in tables 5.2-5 until 5.2-7.

tab. 5.2-3 Sets of input values for the investigation of ksimu (more starting moments)

tab. 5.2-4 Tidal windows for various ksimu-values and input (time relative to first low water)

Variable Set 1 Set 2 Set 3 Unit DescriptionPmeanvs 4.000 3.500 3.500 [m/s] sailing speedPmeanMe 0.100 0.250 0.250 [m] meteo-effect Euro PlatformPmeanMm 0.100 0.355 0.355 [m] meteo-effect Maas CentrePmeanMh 0.100 0.360 0.360 [m] meteo-effect harbour entrancePmeanHE10 0.300 0.600 0.600 [m] wave heighttb 22.630 22.630 22.630 [s] begin timeTd 8897400 3845400 3845400 [m] draughtNpksi 50 50 100 [-] number of position

ksimu1 0:45 - 8:15 1:15 - 6:30 1:45 - 6:00

50 1:15 - 8:00 4:00 - 4:30100 1:15 - 8:15 3:15 - 4:45

1000 1:30 - 8:00 3:15 - 4:15 no tidal window

Set 2 Set 3

no tidal windowno tidal window

Set 1

63

tab. 5.2-5 Expected number of bottom touches of a transit for various starting times and values of ksimu (set 1)

ksimu = 1 ksimu = 50 ksimu = 100 ksimu = 1000tidal window : 0:45 - 8:15 tidal window 1: 1:15 - 8:00 tidal window 1: 1:15 - 8:15 tidal window 1: 1:30 - 8:00

time ksi ksi ksi ksi0:00 KC < 1m KC < 1m KC < 1m KC < 1m0:15 KC < 1m KC < 1m KC < 1m KC < 1m0:30 KC < 1m KC < 1m KC < 1m KC < 1m0:45 0.00000624323591 true 0.04328122528001 false 0.01822134808894 false 0.05707608869780 false1:00 0.00000128071752 true 0.00420680482770 false 0.02332598585342 false 0.01977108948036 false1:15 0.00000000024992 true 0.00388342376982 false 0.00057006758702 false 0.00415060695310 false1:30 0.00000000000024 true 0.00009276550364 true 0.00009321841403 true 0.00098453897810 false1:45 0.00000000005042 true 0.00003636278688 true 0.00018480619261 false 0.00018330230817 false2:00 0.00000000000754 true 0.00000673828518 true 0.00000130079981 true 0.00001729960594 true2:15 0.00000000000004 true 0.00000998916860 true 0.00000009491401 true 0.00000326346050 true2:30 0.00000004886653 true 0.00000007197234 true 0.00000079144008 true 0.00000126129319 true2:45 0.00000000000001 true 0.00000040453324 true 0.00000000761985 true 0.00000014481177 true3:00 0.00000000000000 true 0.00000002752230 true 0.00000003579483 true 0.00000004130447 true3:15 0.00000000000000 true 0.00000000102143 true 0.00000000625776 true 0.00000002455593 true3:30 0.00000000000000 true 0.00000000044543 true 0.00000000000207 true 0.00000000504302 true3:45 0.00000000000000 true 0.00000000003294 true 0.00000000007745 true 0.00000001061594 true4:00 0.00000000000000 true 0.00000000000576 true 0.00000000000052 true 0.00000000000278 true4:15 0.00000000000000 true 0.00000000000041 true 0.00000000000001 true 0.00000000000308 true4:30 0.00000000000000 true 0.00000000000004 true 0.00000000000014 true 0.00000000000816 true4:45 0.00000000000000 true 0.00000000000033 true 0.00000000000002 true 0.00000000015581 true5:00 0.00000000000000 true 0.00000000000080 true 0.00000000000596 true 0.00000000200653 true5:15 0.00000000000000 true 0.00000000002272 true 0.00000000003809 true 0.00000000008061 true5:30 0.00000000000000 true 0.00000000000020 true 0.00000000000873 true 0.00000000076286 true5:45 0.00000000000000 true 0.00000000000063 true 0.00000000003272 true 0.00000001349292 true6:00 0.00000000000000 true 0.00000000000195 true 0.00000000059756 true 0.00000004753420 true6:15 0.00000000000000 true 0.00000000319208 true 0.00000000007518 true 0.00000001462879 true6:30 0.00000000000000 true 0.00000005416719 true 0.00000000172766 true 0.00000002713203 true6:45 0.00000000000000 true 0.00000005680898 true 0.00000006098938 true 0.00000281038149 true7:00 0.00000000027643 true 0.00000050333943 true 0.00000121840309 true 0.00000361650213 true7:15 0.00000000000063 true 0.00000874227814 true 0.00000114645486 true 0.00002337887263 true7:30 0.00000000002433 true 0.00005048647520 true 0.00000301137034 true 0.00001355401142 true7:45 0.00000000040522 true 0.00001305807123 true 0.00001013444174 true 0.00021721543124 false8:00 0.00000000000165 true 0.00024203185677 flase 0.00053371506025 false 0.00100268206977 false8:15 0.00004526469937 true 0.00108699926615 flase 0.00097367651201 false 0.00306856171056 false8:30 KC < 1m KC < 1m KC < 1m8:45 KC < 1m KC < 1m KC < 1m9:00 KC < 1m KC < 1m KC < 1m9:15 KC < 1m KC < 1m KC < 1m9:30 KC < 1m KC < 1m KC < 1m9:45 KC < 1m KC < 1m KC < 1m

10:00 KC < 1m KC < 1m KC < 1m10:15 KC < 1m KC < 1m KC < 1m10:30 KC < 1m KC < 1m KC < 1m10:45 KC < 1m KC < 1m KC < 1m11:00 KC < 1m KC < 1m KC < 1m11:15 KC < 1m KC < 1m KC < 1m11:30 KC < 1m KC < 1m KC < 1m11:45 KC < 1m KC < 1m KC < 1m12:00 KC < 1m KC < 1m KC < 1m12:15 0.00000562513050 true 0.00557715690007 flase 0.00636833324029 false 0.01696443883310 false

TID

AL

WIN

DO

W

TID

AL

WIN

DO

W

TID

AL

WIN

DO

W

TID

AL

WIN

DO

W

ksi<1.686e-04 ksi<1.686e-04ksi<1.686e-04ksi<1.686e-04

64

tab. 5.2-6 Expected number of bottom touches of a transit for various starting times and values of ksimu (set 2)

ksimu = 1 ksimu = 50 ksimu = 100 ksimu = 1000tidal window 1: 1:15 - 6:30 tidal window 1: 4:00 - 4:30 tidal window 1: 3:15 - 4:45 tidal window 1: 3:15 - 4:15

time ksi ksi ksi ksi0:00 KC < 1m KC < 1m KC < 1m KC < 1m0:15 KC < 1m KC < 1m KC < 1m KC < 1m0:30 0.01945689833760 false 1.48637220135825 false 0.95437483981030 false 0.81698488092413 false0:45 0.02003946394143 false 0.83809983333149 false 0.46680982728262 false 0.43433470064568 false1:00 0.00457138838034 false 0.33091232331331 false 0.22800212098368 false 0.20035280739158 false1:15 0.00021122289500 false 0.17189456643775 false 0.07590819466052 false 0.09978159924700 false1:30 0.00024404358419 false 0.05929093374975 false 0.04111148677291 false 0.04525484181039 false1:45 0.00002827580515 true 0.03458840117698 false 0.01698092858204 false 0.01690634035327 false2:00 0.00006747303468 true 0.01853963426318 false 0.00986079560094 false 0.00753672276340 false2:15 0.00000710056316 true 0.00415208932881 false 0.00577373741589 false 0.00478625499888 false2:30 0.00000027549816 true 0.00285932318485 false 0.00219981299074 false 0.00204549293054 false2:45 0.00000083111460 true 0.00196693665717 false 0.00105722403456 false 0.00114282088309 false3:00 0.00000762431679 true 0.00078965976260 false 0.00048576669112 false 0.00056245244014 false3:15 0.00000001154279 true 0.00014891448617 true 0.00010778855355 true 0.00024403590626 false3:30 0.00000000620294 true 0.00035556879057 false 0.00022785911461 false 0.00022782358406 false3:45 0.00000000014768 true 0.00021044683297 false 0.00007257365587 true 0.00014953182250 true4:00 0.00000018264179 true 0.00006604179326 true 0.00008794905920 true 0.00009915204810 true4:15 0.00000000255180 true 0.00023613564463 false 0.00012110515301 true 0.00010605305007 true4:30 0.00000002893334 true 0.00018360895748 false 0.00013789391513 true 0.00025548968312 false4:45 0.00000001357445 true 0.00129727606579 false 0.00029243682744 false 0.00032496112834 false5:00 0.00000000411662 true 0.00038480057804 false 0.00027079742317 false 0.00054136309227 false5:15 0.00000001719468 true 0.00089617565011 false 0.00134048289914 false 0.00120503242631 false5:30 0.00000058634828 true 0.00134540944507 false 0.00373870580540 false 0.00208817031695 false5:45 0.00003001397554 true 0.00773810113571 false 0.00417403854462 false 0.00537848097400 false6:00 0.00000000136240 true 0.02991082152547 false 0.03019347187426 false 0.01694097972063 false6:15 0.00130190590262 false 0.05139418216507 false 0.05376405036661 false 0.02976970076310 false6:30 0.00118827224792 false 0.20211303750250 false 0.11527996761627 false 0.08342539283189 false6:45 KC < 1m KC < 1m KC < 1m KC < 1m7:00 KC < 1m KC < 1m KC < 1m KC < 1m7:15 KC < 1m KC < 1m KC < 1m KC < 1m7:30 KC < 1m KC < 1m KC < 1m KC < 1m7:45 KC < 1m KC < 1m KC < 1m KC < 1m8:00 KC < 1m KC < 1m KC < 1m KC < 1m8:15 KC < 1m KC < 1m KC < 1m KC < 1m8:30 KC < 1m KC < 1m KC < 1m KC < 1m8:45 KC < 1m KC < 1m KC < 1m KC < 1m9:00 KC < 1m KC < 1m KC < 1m KC < 1m9:15 KC < 1m KC < 1m KC < 1m KC < 1m9:30 KC < 1m KC < 1m KC < 1m KC < 1m9:45 KC < 1m KC < 1m KC < 1m KC < 1m

10:00 KC < 1m KC < 1m KC < 1m KC < 1m10:15 KC < 1m KC < 1m KC < 1m KC < 1m10:30 KC < 1m KC < 1m KC < 1m KC < 1m10:45 KC < 1m KC < 1m KC < 1m KC < 1m11:00 KC < 1m KC < 1m KC < 1m KC < 1m11:15 KC < 1m KC < 1m KC < 1m KC < 1m11:30 KC < 1m KC < 1m KC < 1m KC < 1m11:45 KC < 1m KC < 1m KC < 1m KC < 1m12:00 KC < 1m KC < 1m KC < 1m KC < 1m

TID

AL

WIN

DO

W

TW

TW

TW

ksi<1.686e-04 ksi<1.686e-04ksi<1.686e-04ksi<1.686e-04

65

To investigate the influence of Npksi on the length of a tidal window 4 runs are done. The input consists of set 1 and the following combinations of ksimu and Npksi : ksimu, Npksi = [100, 1000; 1000, 100; 1000, 1000; 1000, 10000] The results are presented in table 5.2-8 a graphical representation of a part of the table is given in figure 5.2-5.

tab. 5.2-7 Expected number of bottom touches of a transit for various starting times and values of ksimu (set 3)

ksimu = 1 ksimu = 50 ksimu = 100 ksimu = 1000tidal window 1: 1:45 - 6:00 tidal window 1: - tidal window 1: - tidal window 1: -

time ksi ksi ksi ksi0:00 KC < 1m KC < 1m KC < 1m KC < 1m0:15 KC < 1m KC < 1m KC < 1m KC < 1m0:30 0.78115996183728 false 1.99347723159853 false 1.71694342084785 false 1.28675809967124 false0:45 0.18152776371007 false 1.05622800441405 false 0.97285677584754 false 0.77405726515730 false1:00 0.08279939664725 false 0.56992849025418 false 0.42870055660112 false 0.43607770819052 false1:15 0.01790490265449 false 0.25648120476896 false 0.19856548451485 false 0.22155206697971 false1:30 0.00392228997625 false 0.11559373337596 false 0.10411580487802 false 0.10149956797284 false1:45 0.00040433132185 false 0.05100048749559 false 0.04362867979684 false 0.05203037062063 false2:00 0.00020535745113 false 0.03569877287211 false 0.01974205767681 false 0.03268584191754 false2:15 0.00004065885081 true 0.01218062360102 false 0.01038902784145 false 0.01840226095811 false2:30 0.00002659306993 true 0.01367297943128 false 0.00469724775675 false 0.01104955985053 false2:45 0.00003138218682 true 0.00435460548720 false 0.00379715222796 false 0.00619117974123 false3:00 0.00000232786747 true 0.00345180944039 false 0.00172943551440 false 0.00418922504319 false3:15 0.00000004546544 true 0.00166117971469 false 0.00079797510323 false 0.00263084409497 false3:30 0.00000039150049 true 0.00104843071655 false 0.00065396045099 false 0.00136870529154 false3:45 0.00000236823736 true 0.00059120795485 false 0.00040056394669 false 0.00144373117858 false4:00 0.00000000850554 true 0.00064141300901 false 0.00026994767157 false 0.00098977821660 false4:15 0.00000010117996 true 0.00050083283937 false 0.00029186187574 false 0.00115132078253 false4:30 0.00000027453919 true 0.00045779348447 false 0.00148262412103 false 0.00147236657169 false4:45 0.00000001059504 true 0.00115584950291 false 0.00120315207220 false 0.00171148886765 false5:00 0.00001596682118 true 0.00436611481633 false 0.00155618747358 false 0.00327107965028 false5:15 0.00000214008955 true 0.00549889467203 false 0.00291574064357 false 0.00455687374989 false5:30 0.00019223874270 false 0.00504130330544 false 0.00586113777578 false 0.01147407467816 false5:45 0.00027878628035 false 0.02083219790176 false 0.02539421978917 false 0.02277303028641 false6:00 0.00027420161512 false 0.05867579181952 false 0.02822555425832 false 0.03779302682023 false6:15 0.00276323165316 false 0.10354423416324 false 0.08156006854764 false 0.08268969681061 false6:30 0.00718993741549 false 0.27093969499208 false 0.21261814115996 false 0.18431438408491 false6:45 KC < 1m KC < 1m KC < 1m KC < 1m7:00 KC < 1m KC < 1m KC < 1m KC < 1m7:15 KC < 1m KC < 1m KC < 1m KC < 1m7:30 KC < 1m KC < 1m KC < 1m KC < 1m7:45 KC < 1m KC < 1m KC < 1m KC < 1m8:00 KC < 1m KC < 1m KC < 1m KC < 1m8:15 KC < 1m KC < 1m KC < 1m KC < 1m8:30 KC < 1m KC < 1m KC < 1m KC < 1m8:45 KC < 1m KC < 1m KC < 1m KC < 1m9:00 KC < 1m KC < 1m KC < 1m KC < 1m9:15 KC < 1m KC < 1m KC < 1m KC < 1m9:30 KC < 1m KC < 1m KC < 1m KC < 1m9:45 KC < 1m KC < 1m KC < 1m KC < 1m

10:00 KC < 1m KC < 1m KC < 1m KC < 1m10:15 KC < 1m KC < 1m KC < 1m KC < 1m10:30 KC < 1m KC < 1m KC < 1m KC < 1m10:45 KC < 1m KC < 1m KC < 1m KC < 1m11:00 KC < 1m KC < 1m KC < 1m KC < 1m11:15 KC < 1m KC < 1m KC < 1m KC < 1m11:30 KC < 1m KC < 1m KC < 1m KC < 1m11:45 KC < 1m KC < 1m KC < 1m KC < 1m12:00 KC < 1m KC < 1m KC < 1m KC < 1m

TID

AL

WIN

DO

W

ksi<1.686e-04 ksi<1.686e-04ksi<1.686e-04ksi<1.686e-04

66

tab. 5.2-8 Expected number of bottom touches of a transit for various starting times, values of Npksi and values of ksimu (set 1)

1.0E-12

1.0E-11

1.0E-10

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0:45

1:15

1:45

2:15

2:45

3:15

3:45

4:15

4:45

5:15

5:45

6:15

6:45

7:15

7:45

8:15

ksimu = 100 Npksi = 1000

ksimu = 1000 Npksi = 100

ksimu = 1000 Npksi = 1000

ksimu = 1000 Npksi = 10000

long-term criterion

fig. 5.2-5 Expected number of bottom touches of a transit for various starting times, values of Npksi and values of ksimu (set 1) (inset: Water level for various channel positions and various starting times)

0 2 4 6 8 10 12 14-1

-0.5

0

0.5

1

1.5

Starting times [hour]

Wat

erle

vel [

m]

Water level Euro PlatformWater level Maas CentreWater level port entrance

ksimu = 100 ksimu = 1000 ksimu = 1000 ksimu = 1000Npksi = 1000 Npksi = 100 Npksi = 1000 Npksi = 10000tidal window 1: 1:30 - 8:00 tidal window 1: 1:30 - 8:00 tidal window 1: 1:30 - 8:15 tidal window 1: 1:30 - 8:15

time ksi ksi ksi ksi0:00 KC < 1m KC < 1m KC < 1m KC < 1m0:15 KC < 1m KC < 1m KC < 1m KC < 1m0:30 KC < 1m KC < 1m KC < 1m KC < 1m0:45 0.05483500157204 false 0.07095708666818 false 0.07760430558033 false 0.07037673659620 false1:00 0.02134817323932 false 0.02498553233327 false 0.02270464253422 false 0.02227692241517 false1:15 0.00638867139139 false 0.00543120820186 false 0.00651649814813 false 0.00556159543585 false1:30 0.00137185889062 false 0.00173193748054 false 0.00144806430935 false 0.00114555780576 false1:45 0.00012219106220 true 0.00020049507258 false 0.00029187274049 false 0.00016014190111 true2:00 0.00001873367601 true 0.00002952179690 true 0.00003117078197 true 0.00002220779516 true2:15 0.00000376885262 true 0.00000279347660 true 0.00000632513671 true 0.00000323315367 true2:30 0.00000033776967 true 0.00000098630849 true 0.00000115484392 true 0.00000104220259 true2:45 0.00000018268527 true 0.00000024244742 true 0.00000030224214 true 0.00000025808391 true3:00 0.00000004497405 true 0.00000006915143 true 0.00000021511026 true 0.00000009604204 true3:15 0.00000000611688 true 0.00000002960016 true 0.00000001389857 true 0.00000001990398 true3:30 0.00000000035532 true 0.00000000380552 true 0.00000000388926 true 0.00000000684528 true3:45 0.00000000004135 true 0.00000001314757 true 0.00000000057206 true 0.00000000086575 true4:00 0.00000000004717 true 0.00000000009452 true 0.00000000016863 true 0.00000000010287 true4:15 0.00000000002484 true 0.00000000002463 true 0.00000000009306 true 0.00000000007375 true4:30 0.00000000000667 true 0.00000000004080 true 0.00000000032393 true 0.00000000005675 true4:45 0.00000000000914 true 0.00000000008391 true 0.00000000030270 true 0.00000000015849 true5:00 0.00000000106840 true 0.00000000037784 true 0.00000000035706 true 0.00000000027437 true5:15 0.00000000039607 true 0.00000000400888 true 0.00000000061922 true 0.00000000129581 true5:30 0.00000000118652 true 0.00000000064904 true 0.00000000153969 true 0.00000000200795 true5:45 0.00000000066202 true 0.00000000068952 true 0.00000000982637 true 0.00000000434843 true6:00 0.00000001939611 true 0.00000000524023 true 0.00000002380692 true 0.00000002413211 true6:15 0.00000000465534 true 0.00000014226829 true 0.00000003667661 true 0.00000004314691 true6:30 0.00000016762132 true 0.00000029137654 true 0.00000031978901 true 0.00000018076396 true6:45 0.00000004452239 true 0.00000110374572 true 0.00000058630053 true 0.00000058206992 true7:00 0.00000088460276 true 0.00000341860037 true 0.00000213886831 true 0.00000346119448 true7:15 0.00000538265956 true 0.00006240657122 true 0.00000827681669 true 0.00000892178564 true7:30 0.00002139247808 true 0.00003912581473 true 0.00003086382361 true 0.00003876540935 true7:45 0.00008256385531 true 0.00037134528773 false 0.00015498684228 true 0.00012046242015 true8:00 0.00028886145422 false 0.00072840553810 false 0.00054261411636 false 0.00047919259450 false8:15 0.00108611927963 false 0.00222171829330 false 0.00160719066560 false 0.00154571248033 false8:30 KC < 1m KC < 1m KC < 1m KC < 1m

12:00 KC < 1m KC < 1m KC < 1m KC < 1m12:15 0.00979260100921 false 0.01376787299430 false 0.01533530580631 false 0.01382901826210 false

ksi<1.686e-04 ksi<1.686e-04ksi<1.686e-04ksi<1.686e-04

TID

AL

WIN

DO

W

TID

AL

WIN

DO

W

TID

AL

WIN

DO

W

TID

AL

WIN

DO

W

Exp

ecte

d nu

mbe

r of

bot

tom

touc

hes

Starting times [hour]

67

Conclusions more starting moments - Tables 5.2-5, 5.2-6 and 5.2-7 show that the tidal window length does not vary much for a ksimu

of 50 to 1000. - Small values of the expected number of bottom touches are calculated less accurately than

larger values (see tables 5.2-5, 5.2-6, 5.2-7 and 5.2-8). This means that more draws for a stricter long-term criterion are needed.

- Table 5.2-8 and figure 5.2-5 show that with both a ksimu and Npksi of 1,000, the tidal window is calculated accurately enough. With these values, a tidal window of 10 starting moments is calculated with 10 million draws. For the calculation of the inaccessibility percentage, a ksimu of 100 and a Npksi of 1,000 should be enough.

5.2.3 NUMBER OF DRAWS: PREDICTED CONDITIONS Alternative 2 calculates the inaccessibility percentage by drawing lsimu times a set of conditions from their distribution functions. To investigate the influence of lsimu on the inaccessibility percentage, again a number of runs are made. For two sets of input values the moving mean for lsimu up to 10,000 is calculated. Table 5.2-9 shows the sets of input values. The results of the runs with the two sets are given in figures 5.2-6 and 5.2-7.

tab. 5.2-9 Sets of input values for the investigation of lsimu

fig. 5.2-6 Moving mean of the inaccessibility percentage for lsimu (set 1)

fig. 5.2-7 Moving mean of the inaccessibility percentage for lsimu (set 2)

100

101

102

103

104

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Number of simulated tidal window issues (lsimu)

Inac

cess

ibil

ity

rati

o

100

101

102

103

104

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Number of simulated tidal window issues (lsimu)

Inac

cess

ibil

ity

rati

o

Variable Set 1 Set 2 Unit Descriptionksimu 50 100 [-] number of transit simulationsNpksi 1000 1000 [-] number of positions

68

Conclusions Set 1 gives an inaccessibility percentage of approximately 4.3% at lsimu = 10,000, but is not stable yet. Set 2 gives approximately 4.0% at lsimu = 5,000 and is stable.

5.3 NUMBER OF DRAWS IN VARIANT OF ALTERNATIVE 2 Alternative 2 simulates ksimu times a transit for each starting moment. Therefore ksimu draws of the conditions wave height, sailing speed and meteo-effect are done and ksimuNpksi ⋅ draws around the bottom profile and the significant ship motion. Instead of drawing the conditions prior to the channel transit, it is also possible to draw these during the transit. The results should be the same because of the averaging of the number of bottom touches over ksimuNpksi ⋅ calculations. In the variant ksimu is one and Npksi has to be approximately 1,000 times larger. The influence of ksimu then disappears. The program structure diagram will be different from line 44 compared to alternative 2.

44 time = tb(i) i = i + 145 position(1) = 046 p = 247 draw Npksi positions along the channel axis

48 (uniform distribution )

49 arrange the positions50 draw a set of conditions from distributions around 51 Pmean (normal distribution ):

52 H(p) = wave height 53 Hr(p) = wave direction54 M(p) = meteo-effect55 vs(p) = sailing speed56 T(p) = draught57 determine the time at this position58 time = time + (position(p)-position(p-1)) / vs(p)59 determine for position and time:60 wl_a, c_a, wl_m and c_m61 determine for position: Pmeand62 draw d(p) (= dept) from distribution 63 around Pmeand 64 determine for position: KC and Zv65 determine PmeanZs = mean significant 66 ship motion67 draw Zs (= significant ship motion) from 68 distribution around PmeanZs69 determine Tm = mean period of ship motion70 m0 = 1/16 * Zs^27172 labda(time) =7374 p = p + 175 repeat until p = Npksi + 176 ksi = area under labda(time) 77 ksi < long criterion78 yes no79 tb(i) = allowed tb(i) = not allowed80 i = i + 1 i = i + 181 repeat until i = Ntbi82 If during the tidal cycle two sequent tb(i) are allowed IPL = 0 else IPL = 183 IPsum = IPsum + IPL84 L = L + 185 repeat until L = lsimu86 IP = IPsum / lsimu * 100%

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

0

2

2exp

1m

KC

Tm

λ

69

5.3.1 NUMBER OF DRAWS: POSITIONS First the influence of Npksi on the expected number of bottom touches for one starting moment is checked, then the influence of Npksi on the length of a tidal window. One starting moment For 13 values of Npksi, runs were done. This was repeated 4 times for a set of input values. At the end 52413 =⋅ runs were done. The set of input values is given in table 5.3-1 and the following values of Npksi were used: Npksi = [100; 200; 300; 400; 500; 1000; 2000; 3000; 4000; 5000; 10000; 50000; 100000] The Results are given in figure 5.3-1. Conclusions one starting moment - From Npksi 1,000, [ ]ξE varies with a factor 10. - It is not clear where [ ]ξE converges to. Therefore more draws are needed. More starting moments For 2 different values of Npksi, the lengths of the tidal windows are calculated. The set of input values is the same as above (tab. 5.3-1), but now with more starting moments tb. The results are presented in table 5.3-2 and figure 5.3-2 for the following values of Npksi: Npksi = [1000; 10000]

Variable Set 1 Unit DescriptionPmeanvs 4.000 [m/s] sailing speedPmeanMe 0.100 [m] meteo-effect Euro PlatformPmeanMm 0.100 [m] meteo-effect Maas CentrePmeanMh 0.100 [m] meteo-effect harbour entrancePmeanHE10 0.300 [m] wave heighttb 8903700 [s] begin timeTd 22.630 [m] draught

tab. 5.3-1 Set input values for the investigation of Npksi

fig. 5.3-1 Influence of Npksi on ksi (4 runs)

102

103

104

105

0

0.5

1

1.5

2

2.5

3x 10

-3

Number of drawn positions (Npksi)

Num

ber

of b

otto

m t

ouch

es (

ksi)

70

Npksi = 1000 Npksi = 10000tidal window 1: 1:30 - 8:00 tidal window 1: 1:30 - 8:00

time ksi ksi0:00 KC < 1m KC < 1m0:15 KC < 1m KC < 1m0:30 KC < 1m KC < 1m0:45 0.08527070753681 false 0.05824198029346 false1:00 0.01326645521343 false 0.01937027423491 false1:15 0.00402044881231 false 0.00380233069791 false1:30 0.00024960126524 false 0.00105068906404 false1:45 0.00004445650537 true 0.00017044772480 false2:00 0.00003017175421 true 0.00015102093220 true2:15 0.00001924649325 true 0.00001428687619 true2:30 0.00000013691248 true 0.00000329345713 true2:45 0.00000003219093 true 0.00000059632235 true3:00 0.00000024313121 true 0.00000015012171 true3:15 0.00000000076651 true 0.00000000506952 true3:30 0.00000000000037 true 0.00000000636751 true3:45 0.00000000000035 true 0.00000000056129 true4:00 0.00000000000008 true 0.00000000004010 true4:15 0.00000000000000 true 0.00000000000016 true4:30 0.00000000000000 true 0.00000000000000 true4:45 0.00000000000627 true 0.00000000006629 true5:00 0.00000000000000 true 0.00000000001163 true5:15 0.00000000000000 true 0.00000000000035 true5:30 0.00000000000000 true 0.00000000000348 true5:45 0.00000000000079 true 0.00000000032314 true6:00 0.00000000000017 true 0.00000001400805 true6:15 0.00000000030560 true 0.00000000233747 true6:30 0.00000000011221 true 0.00000003378840 true6:45 0.00000000000278 true 0.00000006185413 true7:00 0.00000000048335 true 0.00006595679610 true7:15 0.00000209926485 true 0.00002406535928 true7:30 0.00005017633032 true 0.00006809231640 true7:45 0.00053103505444 false 0.00093117923247 false8:00 0.00172331863997 false 0.00249558822447 false8:15 0.00303017168124 false 0.00901581503933 false8:30 KC < 1m KC < 1m

12:00 KC < 1m KC < 1m12:15 0.00393162054517 false 0.03738167415157 false

TID

AL

WIN

DO

W

TID

AL

WIN

DO

W

ksi<1.686e-04ksi<1.686e-04

1.0E-17

1.0E-16

1.0E-15

1.0E-14

1.0E-13

1.0E-12

1.0E-11

1.0E-10

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0:45

1:15

1:45

2:15

2:45

3:15

3:45

4:15

4:45

5:15

5:45

6:15

6:45

7:15

7:45

8:15

Npksi = 1000

Npksi = 10000

ksimu = 1000 Npksi = 10000

long-term criterion

tab. 5.3-2 Expected number of bottom touches of a transit for various starting times and values of Npksi

fig. 5.3-2 Expected number of bottom touches of a transit for various starting times and values of Npks (For comparison, the line ksimu 1000 and Npksi 10000 from subparagraph 5.2.2 is added.)

Starting times [hour]

Exp

ecte

d nu

mbe

r of

bot

tom

touc

hes

71

Conclusions more starting moments The expected number of bottom touches fluctuates a lot for both values of Npksi, but converges to the line that was found with alternative 2.

5.3.2 NUMBER OF DRAWS: PREDICTED CONDITIONS With a Npksi of 1,000, the moving mean of the inaccessibility percentage for lsimu up to 10,000 is calculated. The results are presented in figure 5.3-3. Conclusions - The variant of alternative 2 with Npksi = 1,000 gives an inaccessibility percentage of 4.3% at

lsimu = 10,000, but is still unstable. - More simulations have to be done, to find the number of draws (lsimu and Npksi) at which the

inaccessibility percentage and the length of the tidal windows are stable. - Alternative 2 and its variant give more or less the same results as expected (see appendix 5).

The advantages of the variant are: there is no influence anymore of ksimu and with fewer calculations more draws from the distributions around the predictions are possible. However it is relatively easy to program alternative 2, because some conditions, like the wave height, are constant during the transit. At the moment calculates alternative 2 faster, but this will probably change when the models are written in a compiled language.

fig. 5.3-3 Moving mean of the inaccessibility percentage for lsimu

100

101

102

103

104

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Number of simulated tidal window issues (lsimu)

Inac

cess

ibil

ity

rati

o

72

5.4 RESULTS CASE SUDY Both HARAP and alternative 2 calculated the inaccessibility percentage and tidal windows for a given set of input variables. The calculated inaccessibility percentage is given in subparagraph 5.4.1 and some examples of tidal windows are given in paragraph 5.4.2.

5.4.1 CALCULATED INACCESSIBILITY PERCENTAGES HARAP calculated an inaccessibility percentage of 4.28% and alternative 2 calculated 3.94%. The difference is acceptable. The lower percentage of alternative 2 can partly be explained by the different input and chosen class boundaries in HARAP. The chosen class boundaries are unfavourably wide for the Berge Stahl in the critical situations and uselessly narrow for situations without the possibility of a tidal window. The run with HARAP took 5 minutes and alternative 2 needed 80 hours. For the run of alternative 2, extension module 1: Runtime reduction was used, which made the program a factor 8 faster. Both models were run on the same computer. Until 1985 Rotterdam used an under keel clearance (UKC) model to calculate its entrance regime. For the advantages and disadvantages of such a model, one is referred to subparagraph 2.3.1. Rotterdam used an under keel clearance percentage of 20%. Running step 1 and 2, with a manoeuvrability criterion of 5.42.063.22 =⋅ m, an inaccessibility percentage of 100% is found for the Berge Stahl.

5.4.2 CALCULATED TIDAL WINDOW A tidal window issue with HARAP is based on tables. These tables are made in module 4: window after the calculation of the inaccessibility percentage. Alternative 2 calculates the tidal windows by running steps 2 and 3 for a set of predictions. With HARAP a tidal window is determined instantly and it takes approximately an hour with alternative 2. For astro class 2, meteo class 2 and wave classes 1 to 5 the tidal windows for both HARAP and alternative 2 are given in table 5.4-1. Alternative 2 gives wider tidal windows for wave classes 1 to 3 and a smaller tidal window at the highest wave class, compared to HARAP. This is partly due to the optimisation algorithm of HARAP. HARAP distributes the number of bottom touches over the wave height, so the length will be more constant. By using extension module 2: Optimisation for the conditions the same is possible with alternative 2.

tab. 5.4-1 Tidal windows for astro class 2, meteo class 2 and wave classes 1 to 5 (starting times with respect to high water at port entrance)

wave classopen close open close

1 -4:45 -0:45 -6:45 -0:302 -4:30 -0:45 -6:15 -0:303 -4:15 -0:45 -5:30 -1:004 -3:15 -0:45 -3:15 -2:005 no tidal window no tidal window

HARAP alternative 2

73

For each starting moment, it is possible to present the variation of every variable along the transit. Figure 5.4-1 gives some examples of the possibilities for starting moment –2:45 and wave class 4.

5.5 CHAPTER CONCLUSIONS - The more accurate the inaccessibility percentage and the tidal window issue have to be, the

more draws are needed and the longer the calculation time will be. - A stricter long-term criterion leads to more draws, if a constant accuracy is required. - The difference between the calculated inaccessibility percentage with HARAP and alternative 2

is acceptably small. The calculation time of HARAP is about 5 minutes and alternative 2 needs 80 hours.

- The optimisation algorithm in HARAP partly causes the difference between the calculated tidal window lengths.

- Alternative 2 needs approximately an hour to determine a tidal window. - For a tidal window issue there are a lot of possibilities to present different variables in graphs. - Although not proved, it is very likely that the variant on alternative 2 will converge to a

constant value. It is wise to use the variant instead of alternative 2 if it is possible to rewrite the variant in a faster algorithm. The dependency of the inaccessibility percentage accuracy on two values (Npksi and lsimu) instead of three (Npksi, ksimu and lsimu) is the biggest advantage of the variant compared to alternative 2.

fig. 5.4-1 Presentation examples of variable variation along the channel axis (starting moment –2:45, wave class 4)

0 10 20 30 40 50 600.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3Water level

Position [km]

Wat

er le

vel

[m]

0 10 20 30 40 50 60-0.5

0

0.5

1

calo

ng (

+ =

tail

cur

rent

) [m

/s]

Position [km]

Along and corss current

0 10 20 30 40 50 60-0.5

0

0.5

1

-. c

cros

s (+

= t

o po

rt s

ide)

[m

/s]

0 10 20 30 40 50 60

0.3

0.32

0.34

0.36

0.38

0.4

0.42

0.44

0.46Squat

Position [km]

Squa

t [m

]

0 10 20 30 40 50 602.3

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2Keel clearance

Position [km]

Kee

l cle

aran

ce [

m]

0 10 20 30 40 50 601

1.1

1.2

1.3

1.4

1.5

1.6

1.7Ship motions +/- sigma

Position [km]

Ver

tica

l shi

p m

otio

ns [

m]

0 10 20 30 40 50 60-26

-25

-24

dept

h [m

]

Position [km]

Depth and direction channel

0 10 20 30 40 50 6080

100

120

-. D

irec

tion

[de

gree

]

74

75

6 CONCLUSIONS AND RECOMMENDATIONS This last chapter summarizes the conclusions, made in the previous chapters, and gives recommendations for developing the program and further research to the topic.

6.1 CONCLUSIONS In paragraph 3.2 the following project objective is formulated:

Design a better computer model than HARAP that calculates the inaccessibility percentage and an entrance regime for a tidal bound ship with a given draught, using continuous input variables.

More specific, the program has to calculate with: - continuous probability density functions - continuous channel - actual predicted conditions Project conclusion: The objective is reached because: - The designed computer model calculates with a continuous channel, continuous probability

density functions and real tidal data for the determination of the inaccessibility percentage. - For the entrance regime the model uses actual predicted conditions. - With a case study it is proved that the used calculation method, in spite of the longer

calculation time, is suitable for the practice. More detailed conclusions will be given in the following subparagraphs. Subparagraph 6.1.1 gives the selection background of the alternatives. Conclusions from the case study are summed up in subparagraph 6.1.2.

6.1.1 CONCLUSIONS EVALUATION ALTERNATIVES Three alternatives and three extension modules are formulated in chapter 4 to achieve the objective: Alternative 1: Calculation with predefined classes Alternative 2: Monte Carlo method Alternative 3: Numerical integration Module 1: Runtime reduction Module 2: Optimisation for the conditions Module 3: Optimisation for the ships Alternative 1 and alternative 3 use numerical integration as calculation method, like HARAP. Alternative 2 uses the Monte Carlo method. Alternatives versus HARAP - The alternatives use continuous probability density functions, a continuous channel and real

tidal data. - The alternatives describe occurring physical processes like tidal currents and wave fields better. - Alternative 2 has the fastest runtime of the alternatives, but is still 100 times slower than

HARAP.

76

Selected alternative versus the other alternatives - Alternative 2 is selected, because it calculates the inaccessibility percentage about 1013 times

faster than alternatives 1 and 3. If for instance alternative 2 has a runtime of 1 hour, the other two alternatives will calculate approximately 1 billion years.

- Alternative 2 is easier to expand with physical modules without loosing the distinction between physical and probabilistic calculations.

Advantages and disadvantages modules - The modules have the advantage that they reduce the calculation time (module 1) and lower the

inaccessibility percentage (modules 2 and 3). - Modules 2 and 3 lead to a larger downtime. - When not enough experience with the model is gathered, use of the modules can lead to an

inaccurate calculated inaccessibility percentage, or worse, the calculated risks could be structural lower than the real risks.

6.1.2 CONCLUSIONS CASE STUDY In chapter 5 it is investigated how long alternative 2 needs to give a stable result. Then a variant on alternative 2 is investigated in the same way. The variant draws the conditions at a different moment during the calculations compared to alternative 2. After investigation of the needed number of draws, the results of runs with HARAP and alternative 2 are compared. The following is concluded: - The more accurate the inaccessibility percentage and the tidal window issue have to be, the

more draws are needed and the longer the calculation time will be. - A stricter long-term criterion leads to more draws, if a constant accuracy is required. - The difference between the calculated inaccessibility percentage with HARAP and alternative 2

is acceptably small. The calculation time of HARAP (compiled computer language) is about 5 minutes and alternative 2 (interpreted computer language) needs 80 hours.

- The optimisation algorithm in HARAP partly causes the small differences between the calculated tidal window lengths.

- Alternative 2 needs approximately one hour to determine a tidal window. - Alternative 2 has many possibilities to present different variables in graphs at a tidal window

issue. - It is very likely that the variant on alternative 2 will converge to the same constant value as

alternative 2. It is wise to use the variant instead of alternative 2 if it is possible to rewrite the variant in a faster algorithm, because the variant accuracy depends on two values instead of three. Two values makes it easier to find the optimal number of draws.

The users of HARAP in Rotterdam are probably especially interested in the practical use of alternative 2 compared to HARAP. Assumed that the calculation time can be reduced by a factor 10 when alternative 2 is rewritten in a compiled computer language, they may question:

What is the practical use of alternative 2 instead of HARAP for Rotterdam if the calculation time is one hour instead of 5 minutes, the tidal window issue takes 5 minutes instead of instantly and the results do not differ much?

77

Answer: - With alternative 2 there are no problems with existence of a temporary shallow spot in the

entrance channel. In HARAP the depth of an entire segment has to be adjusted, in alternative 2 just the spot.

- There are no problems anymore in choosing the meteo class or astro class for a tidal window issue. At the moment the most unfavourable class of the three stations: Euro Platform, Maas Centre and port entrance is normative Alternative 2 can use the actual predicted values of the separated stations.

- With alternative the accuracy of the calculated risks no longer depends on a class schematization.

- HARAP has a maximum wave class. For predicted waves, larger than the highest wave class (in Rotterdam 2.2 m), no tidal windows are issued. Alternative 2 has no restriction.

- Alternative 2 gives the users, like the pilots, more confidence when they know that the tidal window issue is based on actual tidal data.

- Compared to HARAP alternative 2 has more possibilities to extend the model with physical modules.

- It is not investigated what will happen with the inaccessibility percentage when smooth continuous probability density functions will be used, but the results will be more accurate.

6.2 RECOMMENDATIONS There are some faults in both HARAP and alternative 2 that can easily be corrected in alternative 2, but are left for the moment for two reasons: to make a good comparison possible between the two models and the availability of other statistical data in a short period of time. The faults can be found in subparagraph 6.2.1. Recommendations for possible improvements of alternative 2 are described in subparagraph 6.2.2.

6.2.1 INVESTIGATION OF MODEL FAULTS - It is assumed that the real significant vertical ship motion is normally distributed around the

regression line (see subparagraph 2.3.3). Therefore it is possible (just like in HARAP) that for small waves a sZ -value is drawn smaller than zero metre. This is of course very wrong from a physical point of view. A different type of distribution function (for instance a Rayleigh-distribution), especially for the smaller wave heights, is therefore recommended. The results for the case study are minimally influenced by this fault, because the inaccessibility percentage depends mainly on the larger waves. Besides that, the fault is on the safe side: by squaring the negative sZ -value to determine the 0th spectral moment of ship motions 0m , a higher probability of a bottom touch during a transit is calculated.

- In the case study a correlation coefficient of 1 for the meteo-effect is assumed between the three stations: Euro Platform, Maas Centre and the harbour entrance. Both HARAP and alternative 2 calculate with coupled values for the meteo-effect. If for instance the meteo-effect for the Euro Platform is predicted, the other two values are also known. It is recommended to investigate the dependency between the stations. Then a tidal window issue can be based on “uncoupled” values for the three stations.

- It is recommended to investigate the starting moment distribution over a tidal window. In the case study the starting moments of a channel transit are assumed to be uniform distributed. For very long and short tidal windows this is a rather good assumption, but on average ship

78

handlers will start their transit more at the opening than just before the closing of a tidal window.

- The position variation of the stroke-middle of an astro-class is integrated in the meteo-effect (see subparagraph 2.5.3). Due to the absence of astro-classes this position variation is not relevant anymore in alternative 2.

- For the long-term criterion of the port of Rotterdam it is recommended to use 510686.1 −⋅ instead of 410686.1 −⋅ and an allowed bottom penetration of 0.25 m instead of none. From a probabilistic point of view this is a better description of the findings of TNO-IWECO (see subparagraph 5.1.6). A variable allowed bottom penetration dependent on the ship type is even better.

- For the squat calculation in step 3 of alternative 2 and in module KSI of HARAP, the relation between the draught of the ship and its water displacement has to be implemented. The influence is negligible, but better from a physical point of view.

- The dependency between the following conditions for the port of Rotterdam have to be investigated: - meteorological circumstances and wave climate - tidal current, river discharge and meteorological circumstances - currents and wave climate - wave height and wave direction (therefore wave direction data has to be gathered) The dependency will lower the calculated inaccessibility percentage.

- It is recommended to use the mean position of the bottom over the width of the channel instead of the nautical granted depth.

- For alternative 2 it is recommended to use smooth continuous probability density functions as input instead of a translation of the discrete HARAP functions.

- In HARAP it is possible to vary some statistics along the channel, like the sailing speed and wave height. Alternative 2 can easily be adjusted to take this variation also into account, but first the dependency between the values along the channel axis has to be determined.

6.2.2 INVESTIGATION OF MODEL IMPROVEMENTS - Instead of using a characteristic wave height for the low frequent wave energy 10EH and the

significant vertical ship motion sZ it is better to use the real spectra of the waves and ship motions. With the wave spectrum and the response amplitude operator function of SEAWAY the ship motion spectrum can be determined. By taking the 0th and 2nd spectral moment of this spectrum the following equation to determine the frequency of bottom touches can be used:

(6.2-1)

Finding the statistical relation between the predicted wave spectrum and the wave spectrum that the ship will encounter during the transit is the biggest obstacle of this method. It is also rather difficult to determine the distribution of the wave spectra in time for the calculation of the inaccessibility percentage. But it is recommended to research the possibilities because of the better ship motions description. A relatively easy improvement, to prevent a negative sZ and to couple the ship spectrum and its mean ship motion period Tm, is a linear relation between both 0m - 10EH and 2m - 10EH , similar to the sZ - 10EH relation. Also the standard deviation of both 0m and 2m and their

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

0

2

0

2

2exp

m

KC

m

mλ

79

correlation have to be determined. By drawing 0m and 2m for a given 10EH from their joint probability density function, labda can be determined with equation (6.2-1).

- It is recommended to investigate the implementation possibilities of modules that can describe the conditions like water level, currents and waves more accurate. These conditions are now determined by linear interpolation between measuring stations.

- The Maas channel is the most critical part of the channel transit. Therefore it is recommended to draw relatively more positions in this channel compared to the Euro channel. An example of the calculated mean frequencies of bottom touches λ along the channel axis is given in figure 6.2-1. The relatively small depth in the Maas channel is the main reason for the relatively large λ ’s.

- The third dimension, width of the channel, can be integrated into the calculations, by drawing positions of the ship along the width of the channel. By doing this, the probability that a ship sails into the channel bank can also be calculated. To determine the distribution of the ship positions over the width of the channel, human influences are important.

- Tidal window issues for container ships should take the wind drift into account. - In the case study the transit ended at the port entrance. HARAP normally takes also the area

inside the port into account. For this area it is recommended to extend the model with a module that calculates the seiches.

- When a faster algorithm for the variant on alternative 2 is possible, it is recommended to use the variant instead of alternative 2. Before implementation it has to be checked when the results go to a constant value.

- It is recommended to use alternative 2 or the variant on alternative 2 instead of HARAP. For the implementation the following should be done: - program the model in a compiled computer language - test it with more ships - work out at least the recommendations of subparagraph 6.2.1 - run alternative 2 at the background for a while to gather confidence

- Program and test the modules. Use the modules only when enough experience is gained.

fig. 6.2-1 λ for different positions along the channel axis

0 1 2 3 4 5 6

x 104

0

0.5

1

1.5

2

2.5

3

3.5x 10

-3

Position [m]

Lab

da [

s-1]

80

81

LITERATURE

[1] Blaauw H.G., T. Schilperoort en J. Strating, Optimalization of depths of channels. Delft: Delft Hydraulics Laboratory, 1982.

[2] CUR190, Probability in Civil Engineering, Part 1: The theory of probabilistic design. Gouda:

Stichting CUR, 1997.

[3] Doorn, J.T.M. van, Probabilistische berekeningen voor de Selected Route. Delft: Waterloopkundig laboratorium, 1990.

[4] Doorn, J.T.M. van, Harap, specificatie rapport. Delft: Waterloopkundig laboratorium, 1992.

[5] Harap, Functioneel Ontwerp. Rotterdam: Ministerie van Verkeer en Waterstaat, 1994a.

[6] Harap, Ontwerphandleiding. Rotterdam: Ministerie van Verkeer en Waterstaat, 1994b.

[7] Harap, Gebruikershandleiding. Rotterdam: Ministerie van Verkeer en Waterstaat, 1994c.

[8] Hoek, A. van der, Pilot Berge Stahl Fase 1. Rotterdam: Adviesdienst Verkeer en Vervoer, 1999.

[9] Holthuijsen, L.H., Wind waves, collegedictaat ct 5316. Delft: Technische Universiteit Delft, 2005.

[10] Informatie voor de vaart met geulgebonden schepen naar de haven van Rotterdam. Rotterdam: Rijkswaterstaat, e.a., 1995.

[11] Kerncijfers Goederenvervoer. Internetsite www.rws-avv.nl: Adviesdienst Verkeer en Vervoer,

2004.

[12] PIANC-IAPH Working Group II-30 in cooperation with IMPA and IALA, Approach Channels, A guide for Design. Brussels: PIANC, 1997.

[13] Principles of Naval Architecture. United States of America: The Society of Navel Architects and Marine Engineers, 1974.

[14] Rice, S.O., ‘Mathematical analysis of random noise’. In: N. Wax, e.a., Selected Papers on Noise and Stochastic Processes. New York: Dover Pub. Inc., 1954, p.184-195.

[15] Savenije, A.C., ‘Probabilistic admittance policy deep draught vessels’. PIANC bulletin, 91 (1996), p.25-36.

[16] Savenije, A.C., Safety Criteria for Approach Channels. Rotterdam: AVV Transport Research Centre, 1998.

[17] Vrijling, J.K., and P.H.A.J.M. van Gelder, Probabilistic design in hydraulic engineering, college dictaat ct5310. Delft: Technische Universiteit Delft, 2002.

82

[18] Vrijling, J.K., Audit HARAP. Delft: Technische Universiteit Delft, 2004.

[19] Wijnstra, R., Meerjarig veiligheidscriterium Euro-maas geul, Notitie. Rotterdam: Rijkswaterstaat,

1993.

[20] Wust, J.C., Invoer randvoorwaarden, Evaluatie 74-voets geul. Rijswijk: 1994. [21] Wust, J.C., Doorvaart naar volgend getij, Notitie. Rijswijk: 1995.

Websites: [22] http://mathworld.wolfram.com [23] http://www.shipmotions.nl

ADMITTANCE POLICY TIDAL BOUND SHIPS Design of a probabilistic computer model for determination of channel transit risks to a seaport.

APPENDICES

aI

TABLE OF CONTENTS a1 Wiskundige achtegrond a1

a1.1 Poissonverdeling a1

a1.2 Kielspeling a2

a1.3 Verticale scheepsbewegingen a31.1.1 Golfspectrum a31.1.2 Random-fase/amplitude model a41.1.3 Variantie-dichtheidspectrum a51.1.4 Momenten van het golfspectrum a61.1.5 Waterstanduitwijking op een willekeurig tijdstip a61.1.6 Verwachte aantal golven per seconde a61.1.7 Verwachte aantal golven per seconde groter dan bepaald niveau a81.1.8 Golftoppen a81.1.9 Golfhoogte a91.1.10 Het verticale scheepsbewegingsspectrum a10

a2 Matlab program alternative 2 a11a2.1 Remarks about the program a11

a2.2 Programming language alternative 2 a14

a2.3 Programming language variant of alternative 2 a32

a3 Examples random number generators a37a3.1 Weibull distribution a37

a3.2 Normal distribution a37

a4 Meandering due external forces a41

a5 Explanation of the case study conclusions a43a5.1 Estimation of the expected ksi for small Npksi a43

a5.2 Example to illustrate the conclusions a45

a6 Type inpute data in the case study a47

aII

aIII

LIST OF FIGURES

LIST OF TABLES

fig. 1.1-1 Tijdintervallen met random bodemberoeringen a1fig. 1.3-1 Golfrecord model a3fig. 1.3-2 Observatie van de oppervlakte-uitwijking in één golfrecord en het bijbehorende

amplitude en fase spectrum a4fig. 1.3-3 Random-fase/amplitude model a4fig. 1.3-4 Transformatie naar de continue verdeling a5fig. 1.3-5 Cosinuskwadraat a5fig. 1.3-6 Verschil tussen de harmonische en de Stokes-type golf a6fig. 1.3-7 Opwaartse kruisingen door een bepaald niveau a8fig. 1.3-8 Verschil breed en smal spectrum a8fig. 1.3-9 Significante golfhoogte a9fig. 2.1-1 Two examples of the determination of calong a13fig. 4.1-1 Real sailing course a41fig. 4.1-2 Difference in length between intended and real course a42fig. 5.1-1 Fluctuation of ksi a44fig. 5.2-1 Results example a45

tab. 2.1-1 Used Matlab functions a11tab. 4.1-1 Maximum angle between intended and real course a41

a1

( ) pRjP −== 10

( ) ( )npP −== 10ξ

( ) ( ) nppP n ⋅−⋅== −111ξ

( ) ( ) knk ppknk

nkP −−⋅⋅

−== 1

)!(!

!ξ

a1 WISKUNDIGE ACHTEGROND Deze bijlage geeft de wiskundige achtergrond van HARAP en de alternatieven. Daarbij staat de volgende stelling centraal:

De kans dat een schip met inkomende golven de geulbodem raakt, is gelijk aan de kans dat de verticaal neergaande beweging, van het meest kritieke punt van het schip, de kielspeling overschrijdt.

Paragraaf a1.1 beschrijft hoe de kans op een bodemberoering tijdens een vaart kan worden bepaald, gegeven de statistische gegevens van de verticale scheepsbewegingen en de kielspeling. Op welke wijze de kielspeling en de verticale scheepsbewegingen berekend kunnen worden, staat omschreven in respectievelijk paragraaf a1.2 en a1.3.

a1.1 POISSONVERDELING Een schip vaart van A naar B door een geul. Vooraf is niet bekend wanneer of waar een bodemberoering zal plaatsvinden, maar stel dat het volgende geldt: - De gemiddelde frequentie van bodemberoeringen λ is constant over de tijd. In een subinterval

van de lengte u is de te verwachte aantal bodemberoeringen u⋅λ . - De bodemberoeringen per vaart zijn onafhankelijke random variabelen. Dat wil zeggen dat de

ene bodemberoering niets zegt over de andere. Hieruit volgt: verwachte aantal bodemberoeringen tijdens een vaart ξ totaal aantal bodemberoeringen tijdens een vaart tijdsduur van één vaart Om de verdeling van ξ te bepalen wordt eerst het interval [0, Tpas] opgedeeld in n intervallen met een lengte Tpas/n. Als n groot genoeg gekozen wordt, zal elk interval 0 of 1 bodemberoering bevatten. Als iR het aantal bodemberoeringen in het interval iI is, dan kan iR de waarde 0 of 1 aannemen (fig. 1.1-1). Hierdoor is de waarde iR Bernoulli verdeeld: De kans dat op geen enkel interval een bodemberoering plaats vindt: De kans op één bodemberoering op een willekeurig tijdsinterval: De kans op k bodemberoeringen gedurende één vaart: Deze verdeling wordt ook wel de binomiaal-verdeling genoemd.

[ ] pasTE ⋅= λξ

pasT

[ ]ξE

( ) [ ]n

Tn

EpRjP pasλξ ====1

time

R = 01

2R = 1

iR = 0

n-1R = 0

nR = 1

time = 0

time = Tpas

pasTn

fig. 1.1-1 Tijdintervallen met

random bodemberoeringen

a2

nT

p pasλ=

( )kn

pas

k

pas

nT

nT

knk

nkP

−

⎟⎠⎞

⎜⎝⎛ −⋅⎟

⎠⎞

⎜⎝⎛⋅

−== λλξ 1

)!(!

!

!

1

!

111lim

1

)!(!

!lim

kkn

kn

n

n

n

n

nknk

nnkn

=⋅+−−⋅=⋅− ∞→∞→

11lim =⎟⎟⎠

⎞⎜⎜⎝

⎛−

−

∞→

k

pas

n n

Tλ

( ) ( )pasT

kpas ek

TkP

λλξ −==

!

( ) pasTePP

λξξ −−==−=> 101)0(

Met: is deze verdeling ook te schrijven als: Deze vergelijking geldt alleen als n voldoende groot is om de Bernoulli verdeling te kunnen toepassen. Als n naar oneindig gaat: De kans op k keer raken van de bodem gedurende een vaart is dan te schrijven als: Deze verdeling staat bekend als de Poisson-verdeling. De kans op één of meerdere bodemberoeringen is dan gelijk aan:

(1.1-1) De enige onbekende parameter in deze verdeling is de frequentie van bodemberoeringen λ . De frequentie wordt bepaald door de kielspeling KC en de verticale scheepsbeweging ( )tZs . [1]

a1.2 KIELSPELING De kielspeling KC (Keel Clearence) is de afstand tussen de geulbodem en laagste punt van het schip. De volgende variabelen bepalen de kielspeling: - waterstand ws - diepgang en squat van het schip vd ZT + - bodemligging d De waterstand wordt bepaald door het getij en de weersomstandigheden. Een aanhoudende wind uit een bepaalde richting kan water opstuwen of afwaaiing veroorzaken. Daarnaast kunnen hoge en lage drukgebieden de waterstand beïnvloeden. De diepgang van het schip is afhankelijk van de scheepsvorm, de stroomsnelheid, de vaarsnelheid, beladingsgraad en hoe goed het schip getrimd is. Samen met de scheepsvorm en de stroomsnelheid bepaalt de vaarsnelheid de squat, ofwel de inzinking van het schip. Een niet gelijklastig getrimd schip ligt schuin over de lengte richting. De maximum squat is te schatten met de methode van Tuck-Taylor:

( )

( ) pasTpas

paspasn

reeksTaylor

n

pas

n

eTn

nnn

Tn

nnT

n

n

n

T

λλ

λλλ

−

∞→

−

∞→

=+−−

−−+−⎯⎯⎯⎯ →⎯⎟⎟⎠

⎞⎜⎜⎝

⎛−

3

3

2

2

)2)(1(

)1(1lim1lim

a3

( ) ∑=

+⋅=n

iiii tfat

1

)2cos( απη

(1.2-1)

waarbij:

vZ maximum squat [m]

sv scheepssnelheid [m s-1] g gravitatieversnelling [m s-2] h waterdiepte [m] ∇ waterverplaatsing [m3]

ppL scheepslengte [m] c squatcoëfficiënt [-] Morfologische factoren bepalen de ligging van de geulbodem. Sedimentatie, zandgolven en de baggercyclus zijn hierin bepalend. De kielspeling KC is als volgt te bepalen:

(1.2-2) ref. [4]

a1.3 VERTICALE SCHEEPSBEWEGINGEN Verticale bewegingen van schepen blijken voornamelijk afhankelijk te zijn van de laagfrequente golven (swell). Mede hierom is het bewegingsspectrum goed te beschrijven met dezelfde vergelijkingen waarmee het golfspectrum wordt beschreven. Eerst wordt in deze paragraaf het golfspectrum wiskundig behandeld en later zullen deze vergelijkingen omgeschreven worden naar de vergelijkingen voor het bewegingsspectrum van schepen.

1.1.1 GOLFSPECTRUM Golven op zee veroorzaken een onregelmatig bewegende wateroppervlak. De bewegingen van dit oppervlak worden meestal gemeten in een vast punt, gedurende een tijdsinterval, en vastgelegd in een record. Het record kan wiskundig beschreven worden door een groot aantal sinusgolven met verschillende amplitude, frequentie en fase te sommeren, Fourier series (fig 1.3-1). (1.3-1)

( )tη wateroppervlakte-uitwijking [m]

ia amplitude [m]

if frequentie [s-1]

iα fase [rad] n een groot getal [-] Met deze vergelijking kan één record wiskundig beschreven worden. Echter door vele invloedsfactoren zijn de records nooit aan elkaar gelijk. Om tot een beschrijving te komen van alle mogelijke observaties (records), die kunnen voorkomen onder bepaalde omstandigheden, wordt hieronder het zeeoppervlak beschreven als een stationair stochastisch proces. Stochastisch omdat de variabelen willekeurig (random) zijn en stationair omdat statistische eigenschappen van de variabelen, bepaald door middeling over alle records, niet variëren in de tijd. ref. [9]

-4

-2

0

2

4

6

8

10

12

14

+

hg

vgetalFroudeFc

LF

FZ s

nhppnh

nhv ⋅

=−=→⋅∇−

=22

2

1

fig. 1.3-1 Golfrecord model

vd ZTwsdKC −−+=

a4

∑=

=m

jii a

ma

1

1

1.1.2 RANDOM-FASE/AMPLITUDE MODEL Om statistische gegevens te verkrijgen uit de records worden de frequenties opgedeeld in intervallen

Df /1=∆ , met D de tijdsduur van de records. Per interval en per record wordt de amplitude en fase bepaald (fig. 1.3-2). Uit een groot aantal records zijn vervolgens per frequentie de gemiddelde waarden van de amplitude en fase te berekenen. Voor de meeste golfrecords blijkt dat de fase elke waarde tussen de 0° en 360° kan aannemen zonder een voorkeur voor een bepaalde waarde. De gemiddelde amplitude is te bepalen met:

voor alle frequenties if en m het aantal golfrecords. (1.3-2) De eigenschappen van de fase en amplitude liggen vast in kansdichtheidsverdelingen. De fase is uniform verdeeld en de amplitude heeft een Rayleigh-verdeling. In het random-fase/amplitude model hoort zo bij elke discrete frequentie Difi /= één uniforme verdeling voor de fase en één Rayleigh-verdeling voor de amplitude, met een verwachtingswaarde als eigenschap. Voor een realisatie van ( )tη wordt een waarde voor de amplitude ia en fase iα getrokken uit hun kans-dichtheidsverdelingen voor elke frequentie apart en onafhankelijk (fig. 1.3-3). ref. [9]

fig. 1.3-2 Observatie van de oppervlakte-uitwijking in één golfrecord en het bijbehorende amplitude

en fase spectrum

fig.1.3-3 Random-fase/amplitude model

( )tη ia

t

iα

if

if

Df /1=∆ Df /1=∆

amplitude spectrum fase spectrum

1f 2f

[ ]11 aE=µ [ ]22 aE=µ

)( 1ap )( 2ap

1a 2a

[ ]iaE

1f 2f if

)( 1αp )( 2αp

π− π0 π− 0 π

[ ]1aE [ ]iaE[ ]2aE

amplitude spectrum

Rayleigh

Uniform

Rayleigh

Uniform

et cetera 1α 2α

π2/1 π2/1

a5

( ) [ ]22

1

0

1lim i

if

aEf

fSi ∆

=→∆

[ ] [ ] [ ] [ ] faEf

aEEEn

ii

i

n

ii

n

ii ∆

∆==== ∑∑∑

=== 1

22

1

1

22

1

1

222 1ηηη

[ ] [ ] ( )dffSfaEf

E i

n

ii

ifi

∫∑∞

=→∆→∆

∆=

01

22

1

0

2 1limη

1.1.3 VARIANTIE-DICHTHEIDSPECTRUM Het amplitude spectrum [ ]iaE als functie van de frequentie if geeft genoeg informatie om de wateroppervlakte-uitwijkingen te beschrijven. Er blijken echter voordelen te zijn om niet [ ]iaE , maar de variantie [ ]2

21

iaE 1 als functie van de frequentie if te nemen. Twee voordelen zijn: - de variantie is statistisch gezien relevanter dan de amplitude: Bijvoorbeeld de som van

varianties van de golfcomponenten is gelijk aan de variantie van de som van de golfcomponenten. Dit geldt niet voor de amplitudes.

- de energie van golven is proportioneel aan de variantie Om de discrete waarde van de frequentie if om te zetten naar een continue verdeling wordt de variantie verdeeld over de frequentie-intervallen en de breedte van de intervallen wordt oneindig klein gemaakt (fig. 1.3-4):

(1.3-3)

De functie ( )fS beschrijft het variantie-dichtheids-spectrum. De variatie van de oppervlakte-uitwijking ( )tη blijkt gelijk te zijn aan het gemiddelde van het kwadraat van de oppervlakte uitwijking 2η . Dit wordt hieronder afgeleid. Kwadrateer de oppervlakte-uitwijking per discrete frequentie: middel deze over de tijd en gebruik de regel2

[ ] ][XEaaXE ⋅= waarin a een constante is (fig. 1.3-5):

Sommeer de varianties: Als 0→∆ if dan: (1.3-4) ref. [9]

1 de factor ½ is een hulpwaarde die later duidelijk zal worden.

2

][)()(][ XEadxXfXadxXfXaXaE ⋅=⋅⋅=⋅⋅=⋅ ∫∫

∞

∞−

∞

∞−

( )( ) ( )22 )2cos( iiii tfat απη +=

[ ] [ ]

[ ] [ ]22

1

0

22

0

2222

)2(cos1

)2(cos1

i

T

iii

T

iiiii

aEdttfT

aE

dttfaET

E

=+⋅

=+==

∫

∫

απ

απηη

fig. 1.3-4 Transformatie naar de

continue verdeling

fig. 1.3-5 Cosinuskwadraat

-1,5

-1

-0,5

0

0,5

1

1,5

( )xcos

( )x2cos

x

[ ]iaE

[ ]22

1iaE

[ ]22

11

ii

aEf∆

( )fS

if

f

if

if

f∆

f∆

f∆

Amplitude spectrum

Variantie-dichtheids-spectrum (discontinu)

Variantie-spectrum

Variantie-dichtheids-spectrum (continu)

a6

( )( ) ( )22 )2sin(2 iiiii tfaft αππϕ +⋅−=

[ ] [ ]

[ ] [ ]22

122

0

2222

0

222222

4)2(sin1

4

)2(sin41

ii

T

iiii

T

iiiiii

aEfdttfT

aEf

dttfaEfT

E

⋅=+⋅⋅

=+⋅==

∫

∫

παππ

αππϕϕ

1.1.4 MOMENTEN VAN HET GOLFSPECTRUM De hierboven berekende variantie is het 0e-moment van het variantie-dichtheidspectrum. Daarnaast zijn er nog n andere momenten als volgt gedefinieerd:

(1.3-5) ref. [9]

1.1.5 WATERSTANDUITWIJKING OP EEN WILLEKEURIG TIJDSTIP De lineaire benadering van de oceaangolven (het random-fase/amplitude model) veronderstelt dat op een willekeurig tijdstip de waterstanduitwijking normaalverdeeld is. De kansdichtheidsverdeling met verwachtingswaarde 0 en variantie 0m :

(1.3-6)

Dit is een benadering omdat golven in werkelijkheid niet sinusvormig zijn maar meer een Stokes-type vorm hebben (fig. 1.3-6). ref. [9]

1.1.6 VERWACHTE AANTAL GOLVEN PER SECONDE Een golf bestaat uit twee as-doorkruisingen, een opwaartse en een neergaande. Om de gemiddelde golffrequentie te bepalen worden hier de opwaartse geteld. Anders gezegd; de afgeleide van )(tη is positief. In een tijdsinterval dt net voorafgaand van een opwaartse as-door-kruising, kan de uitwijking η alleen een waarde aannemen tussen:

0<<− ηϕdt ϕ de afgeleide van η naar de tijd Het interval dt is hier zo klein genomen dat de lijn η in het interval lineair mag worden verondersteld. Aangezien η normaal verdeeld is geldt dat ook voor ϕ . Voor de afleiding van de variantie van ϕ gelden dezelfde stappen als eerder. Kwadrateer ϕ per discrete frequentie: middel deze over de tijd:

( )dffSfm nn ∫

∞

=0

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅=

0

2

0 2exp

2

1)(

mmp

ηπ

η

fig. 1.3-6 Verschil tussen de harmonische en de Stokes-type golf

t

Stokes-type golf

sinus golf

normale verdeling

geobserveerde verdeling

)(ηp

)(tη η

0

a7

[ ] [ ] [ ] [ ] faEff

aEfEEn

iii

i

n

iii

n

ii ∆⋅

∆=⋅=== ∑∑∑

=== 1

22

122

1

22

122

1

222 144 ππϕϕϕ

[ ] [ ] ( ) 22

0

22

1

22

122

0

2 441

4lim mdffSffaEff

E i

n

iii

ifi

πππϕ =⋅→∆⋅∆

= ∫∑∞

=→∆

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅=

22

2

22 8

exp)4(2

1)(

mmp

πϕ

ππϕ

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

⋅=

22

2

0

2

22

082

exp42

1),(

mmmmp

πϕη

ππϕη

∫ ∫∞

−

⋅0

0

),(dt

ddpϕ

ϕηϕη

),0(),0(),(00

0ϕϕηϕηϕη

ϕϕ

pdtdpdpdtdt

t⋅⋅== ∫∫

−−→∆

∫ ∫∞ ∞

⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅⋅=⋅⋅

0 0 22

2

22

08

exp42

),0( ϕπϕ

ππϕϕϕϕ d

mmmdtdpdt

0

2

022

2

22

0

22

22

22

08

exp44

8

8exp

44 m

mdt

mmm

dtmdu

m

u

mm

dt

u

⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅

⋅

⋅−=⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅

⋅

∞

=∫

ϕπϕ

πππ

πππ

0

20 m

mf =

Sommeer de varianties: Als 0→∆ if dan:

(1.3-7) De kansdichtheidsverdeling van ϕ : De gezamenlijke kansdichtheidsverdeling van η enϕ , waarbij η en ϕ onafhankelijk van elkaar zijn:

(1.3-8)

Het aantal te verwachten nullen waarbij ϕ positief is gedurende een tijdsinterval dt :

(1.3-9)

De binnenste integraal herschreven3: geeft: Met behulp van de substitutieregel 2ϕ=u : De gemiddelde frequentie van nulkruisingen is hiermee bekend:

(1.3-10)

ref. [14]

3 Als t∆ naar nul gaat is )(ηp over het interval t∆ constant.

a8

∫ ∫∞

−

⋅0

** ),(η

ϕη

ϕηϕηdt

ddp

∫ ∫∞ ∞

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

⋅⋅=⋅⋅

0 0 22

2

0

2

22

082

exp42

),( ϕπϕη

ππϕϕϕηϕ d

mmmmdtdpdt

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

00

2

2exp

mm

mf

ηη

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

0

2

0 2exp

mmp

top

ηηηη

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−−=>−=≤

0

2

2exp11

mPP toptop

ηηηηη

),(),(),( **

0

* ϕηϕηϕηηϕηη

ϕη

η

ϕη

pdtdpdpdtdt

t⋅⋅=⋅= ∫∫

−−→∆

1.1.7 VERWACHTE AANTAL GOLVEN PER SECONDE GROTER DAN BEPAALD NIVEAU Interessant is om te weten wat de frequentie is van waterstandsuitwijkingen groter dan een bepaald niveau η . Analoog kan dan met dezelfde soort vergelijking bepaald worden hoe vaak de scheepsuitwijkingen groter zijn dan de kielspeling. Voor de gemiddelde frequentie van η -kruisingen moeten de grenzen van de binnenste integraal van vergelijking (1.3-9) aangepast worden in ηηϕη <<− *dt (fig. 1.3-7): De binnenste integraal herschreven: geeft: De gemiddelde frequentie van η -kruisingen is als volgt:

(1.3-11) ref. [14]

1.1.8 GOLFTOPPEN Een golftop is het hoogste punt van een golf. Deze kan in een breed spectrum een positieve en negatieve waarde aannemen. Golven die echter een rol spelen in een toegangsgeul zijn gegenereerd in diepwater. Deze golven hebben een relatief smal spectrum, waardoor de waarden van alle golftoppen positief zijn (fig.1.3-8). Het aantal golftoppen is gelijk aan het aantal opwaartse nulkruisingen. Het aantal golftoppen met een waarde groter dan een bepaald niveau η is dan gelijk aan het aantal opwaartse kruisingen door dat niveau. De fractie van golftoppen door een bepaald niveau ηη >top gedurende een bepaalde periode D : De kansverdeling ( )ηη ≤topP : De kansdichtheidsverdeling: (1.3-12) Deze verdeling staat ook wel bekend als de Rayleigh-verdeling. ref. [9]

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=⋅⋅=>

0

2

0

2

0

2

0

2

0

2exp

2exp

m

m

m

mm

m

Df

DfP n

top

η

η

ηη

fig. 1.3-7 Opwaartse kruisingen door

een bepaald niveau

fig. 1.3-8 Verschil breed en smal spectrum

)(tη

)(tη

t

t

topη

breed spectrum

smal spectrum

negatieve toppen

nT nT nTη

)(tη

t

a9

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−==

0

2

0 8exp

4 m

H

m

H

dH

dpHp topη

η

[ ]∫

∫∞

∞

≥

⋅

⋅⋅==

*

*

*

)(

)(

H

HHHs

dHHp

dHHpH

HEH

∫

∫∞

∞

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

*

*

0

2

0

0

2

0

2

8exp

4

8exp

4

H

Hs

dHm

H

m

H

dHm

H

m

H

H

1.1.9 GOLFHOOGTE Uitgegaan van sinusvormige golven geldt dat de golfhoogte twee keer de amplitude is: topH η2= . De kansdichtheidsverdeling van de golfhoogte kan dan bepaald worden met de Jacobiaan

21=dHd topη :

(1.3-13)

De significante golfhoogte sH is gedefinieerd als de gemiddelde waarde van de 3

1 -hoogste golfhoogten. De significante golfhoogte is daarmee de ligging van het zwaartepunt ten opzichte van de lijn 0=H van het gearceerde oppervlak in figuur 1.3-9. Het zwaartepunt is gelijk aan het oppervlaktemoment van het gearceerde vlak ten opzichte van de lijn 0=H gedeeld door zijn oppervlak. (1.3-14) Invullen van (1.3-13): De integraal in de noemer moet gelijk zijn aan 3

1 en daarmee kan *H bepaald worden: Met de ondergrens *H en de noemer gelijk aan 3

1 is vergelijking 1.3-14 te bepalen door partieel te integreren:

fig. 1.3-9 Significante golfhoogte

)3ln(8

8exp

8exp

8exp

8exp

8

1

2,8

exp4

0*

31

0

2*

*0

2

000

23

1

0

2

0

2

*

⋅=

→=⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−−⎯⎯ →⎯⎟⎟

⎠

⎞⎜⎜⎝

⎛−−=⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛−

⎯⎯ →⎯⋅==⎯⎯⎯⎯ →⎯=⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−

∞

=

−∞

∫

∫

mH

m

H

m

H

m

udu

m

u

m

dHHduHudHm

H

m

H

H

Hu

uu

invullenregelesubstituti

H

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−⋅−⋅=⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛−⋅

⎯⎯ →⎯

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−−=⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

==⎯→⎯⋅−⋅=⋅

∫∫

∫ ∫

∞∞∞

* 0

2

*0

2

* 0

2

0

2

0

2

0

2

0

8exp

8exp3

8exp

43

8exp,

8exp

4

,

HHH

invullenmet

dHm

H

m

HHdH

m

H

m

H

m

HvdH

m

H

m

Hdv

dHduHu

duvvudvu

)( Hp

sH*H

31

*

)( =⋅∫∞

dHHpH

H

a10

mm Tm

m 1

0

2 ==λ

*H Vervangen door )3ln(8 0 ⋅m :

(1.3-15) Hierin is )(xerf de error functie met )3ln(=x : De significante golfhoogte sH is met de standaard error functie in 1.3-15 te schrijven als functie van het 0e-moment van het variantie-dichtheidsspectrum:

(1.3-16) ref. [9]

1.1.10 HET VERTICALE SCHEEPSBEWEGINGSSPECTRUM Het bewegingsspectrum van schepen kan wiskundig met dezelfde vergelijkingen beschreven worden als het golfspectrum. Het aantal scheepsbewegingen per seconde is dan analoog aan vergelijking (1.3-10):

(1.3-17)

met

mλ gemiddelde bewegingsfrequentie van een schip

20 ,mm 0e en 2e spectrale moment van de scheepsbewegingen

mT gemiddelde bewegingsperiode van een schip Het gemiddeld aantal scheepsbewegingen per seconde die groter zijn dan de kielspeling KC is analoog aan (1.3-11):

(1.3-18)

De significante scheepsbeweging sZ is op dezelfde manier gedefinieerd als de significante golfhoogte sH : de gemiddelde waarde van de 3

1 -grootste scheepsbewegingen (1.3-16):

(1.3-19) ref. [4]

⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

0

2

0

2

0

2

2exp

1

2exp

m

KC

Tm

KC

m

m

m

λ

216

1004 ss ZmmZ =→=

00 40043,4 mmH s ≈=

( )

( )( ))3ln(123)3ln(22

exp2

23)3ln(22)3ln(8

,8

,88

exp3)3ln(22

00

)3ln(

200

0

**

00* 0

2

0

erfmm

dUUmmm

HU

m

dHdU

m

HUdH

m

Hm

invullen

regelesubstituti

H

−⋅⋅+⋅

=⋅−⋅+⋅⎯⎯ →⎯==

==⎯⎯⎯⎯ →⎯⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−+⋅

∫

∫∞

−∞

π

ππ

( ) dUUxerfx

⋅−⋅= ∫ 2

0

exp2

)(π

a11

a2 MATLAB PROGRAM ALTERNATIVE 2 This appendix gives the programming language of alternative 2 and its variant, but first some remarks are summed in paragraph a2.1.

a2.1 REMARKS ABOUT THE PROGRAM - For the case study, alternative 2 is programmed in the mathematical computer-programming

environment Matlab, version 7 release 14. Matlab is an interpretor and is therefore slower than the compiled programming languages such as: C, C++ and Fortran. However, programming with Matlab goes faster and graphs are easier to make. When alternative 2 will be used in practice, it is recommended to program the model in a faster language.

- Alternative 2 consists of three steps (see paragraph 4.2 for the program structure diagram): 1 draw values from the long-term statistics 2 check on the manoeuvrability criterion 3 check on the long-term criterion The checking on the manoeuvrability criterion is done in the programme ‘keel clearance’. This programme does not use the possibility in Matlab to calculate with matrices. Instead of checking the criterion for all the positions in the channel at once, it determines separately for each position in the channel and moment in time the keel clearance. Therefore relatively much repetitions loops are needed and at the end the calculation time will be longer. The advantage of this way of programming is the clear programme structure. Especially when the reader is not familiar with the Matlab language. Steps 1 and 3 use the possibility to calculate with matrices. This reduces the calculation time with Matlab a lot4. To show how steps 1 and 3 could be programmed in the same way as step 2 the following adjustments are made: - draws from the wave height and meteorological distribution functions are programmed in

the same way as step 2 - step 2 uses more or less the same symbols as step 3, so it is easier to see how step 2 has to

be extended to program step 3 - The programme calculates in seconds, metres and degrees. - The input of astronomical data is given in centimetres and the values are predicted every ten

minutes. - The used Matlab functions are given in table 2.1-1:

4 Using third or higher dimensions of matrix spaces could make the programme again much faster. However

the programme would be less clear to oversee.

Function Descriptionrand(a,b) draws a*b values of a uniform rand(2,3)= 0.4057 0.2785 0.5565

distribution 0.5389 0.8784 0.5497randn(a,b) draws a*b values of a standard randn(2,3)= 2.2523 0.7399 -0.3911

normal distribution -1.6148 1.1263 -0.1539interp1(A,n) one dimensional data interpolation A= 6.2000

in a column matrix 5.40009.70008.3000

n= 3.5000interp1(A,n)= 9.0000

Example

tab. 2.1-1 Used Matlab functions

a12

Function Descriptioninterp1(b,A,bi) one dimensional data interpolation a= 1.0000

in a column matrix 2.00003.00004.0000

ai= 1.30002.00002.8000

interp1(a,A,ai)= 5.96005.40008.8400

interp2(a,b,B,ai,bi) two dimensional data interpolation a= 1.0000(table lookup) 2.0000

3.0000b= 10.0000

20.000030.000040.000050.0000

B= 3.4000 4.1000 4.60003.5000 4.4000 5.10004.2000 5.3000 5.90005.4000 5.8000 6.20004.7000 5.5000 6.1000

ai= 1.30002.00002.8000

bi 13.000040.000028.0000

interp2(a,b,B,ai,bi)= 3.65805.80005.6160

sort(A) sort the matrix A A= 6.2000 5.4000 9.7000 8.3000sort(A)= 5.4000 6.2000 8.3000 9.7000

length(A) length of matrix A length(A)= 4.0000round(A) rounds the matrix A round(A)= 6.0000 5.0000 10.0000 8.0000min(A) minimum of matrix A min(A)= 5.4000max(A) maximum of matrix A max(A) 9.7000find(A==c) finds the matrix position find(A==5.4) 2.0000sum(A) sum of matrix elements sum(A) 29.6000cumsum(A) cumulative sum cumsum(A) 6.2000 11.6000 21.3000 29.6000B = k:l:m B is a row from k with step size l until m B=2:1:5= 2.0000 3.0000 4.0000 5.0000cosd(alpha) Cosine of an argument in degrees cosd(90)= 0.0000sind(alpha) Sine of an argument in degrees sind(90)= 1.0000+ - * / ^ Arithmetic operations B= 1.0000 2.0000

3.0000 4.0000B^2= 7.0000 10.0000

15.0000 22.0000.+ .- .* ./ .^ Arithmetic operations for matrix entries B.^2= 1.0000 4.0000

9.0000 16.0000% The percent symbol denotes a comment

Example

a13

- For the squat calculation, the sailing speed with respect to the ground is corrected with the astronomical current speed along the ship: calong. To avoid problems by interpolating between

a−360 and b+0 , the program interpolates between the (co)sine of the angle. In step 2, calong in the Euro channel is calculated with the following equation (fig. 2.1-1):

( )α−⋅= 82coscurrentcalong and the Maas Channel:

( )α−⋅= 112coscurrentcalong Because step 3 calculates for all positions at once the mean frequency of bottom touches λ , it is easier to work with the following equation5:

5

fig. 2.1-1 Two examples of the determination of calong

c

d

aβ−αβ

α

α

r = 1

b

( ) ( ) ( ) ( ) ( )αβαβαβ sinsincoscoscos ⋅+⋅=−

( )( )

( )( )

( ) ( )( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )αβαβαβ

αββααβα

βα

βαβ

sinsincoscoscos

tansincoscoscos

cos

sin

tan

cos

cos

⋅+⋅=−⋅+⋅=−

⋅+==

⋅==

−=

cba

d

dc

b

a

360°, 0°

90°270°

180°

112°

82°

α

α2

1

curre

nt 1

current 2 MAASCHANNEL

EURO

CHANNELcalong 1 EC

calong 2 EC

calong 2 MC

calong 1 MC

a14

a2.2 PROGRAMMING LANGUAGE ALTERNATIVE 2

a15

a16

a17

a18

a19

a20

a21

a22

a23

a24

a25

a26

a27

a28

a29

a30

a31

a32

a2.3 PROGRAMMING LANGUAGE VARIANT OF ALTERNATIVE 2

a33

a34

a35

a36

a37

a3 EXAMPLES RANDOM NUMBER GENERATORS The Monte Carlo method draws random conditions of the distribution functions. The transformation derivations of random numbers from a uniform distribution to random numbers from two various distributions are given in this appendix. The derivations use equation (2.4-5) of paragraph 2.4: The following distribution functions will be used as example: - Weibull distribution - Normal distribution Each transformation of uniformly distributed random variable uX will be derived in a different paragraph.

a3.1 WEIBULL DISTRIBUTION The probability density function of the Weibull distribution: The distribution function is equal to a uniform distributed variable between zero and one uX : Rewritten:

a3.2 NORMAL DISTRIBUTION An analytical formulation of the distribution function of the normal distribution is not possible to give. Therefore it is relatively hard to derive a random number generator for the normal distribution compared to the Weibull distribution. The Box Muller transformation is a rather efficient method. It uses two randomly uniform distributed values 1,uX and 2,uX to draw two values 1z and 2z from a standard normal distribution:

(3.2-1) (3.2-2)

This can be verified by solving these equation for 1,uX and 2,uX . Square both equations for solving 1,uX ,

( )uX XFX 1−=

( )β

αβ

ααβ ⎟

⎠⎞

⎜⎝⎛−−

⋅⎟⎠⎞

⎜⎝⎛⋅=

X

X eX

Xf1

( ) ( ) u

X

XX XedxXfXF =−=⋅=⎟⎠⎞

⎜⎝⎛−

∫β

α1

( )( )βα1

1ln uXX −−⋅=

( )( )2,1,2

2,1,1

2sin)ln(2

2cos)ln(2

uu

uu

XXz

XXz

⋅⋅⋅−=

⋅⋅⋅−=

π

π

( ) ( )( ) ( )2,

21,

22

2,2

1,21

2sinln2

2cosln2

uu

uu

XXz

XXz

⋅⋅⋅−=

⋅⋅⋅−=

π

π

a38

Sum the equations:

(3.2-3) For solving 2,uX rewrite equation (3.2-1): Fill this equation in equation (3.2-2):

(3.2-4) By using the following rule6: The derivatives of 1,uX and 2,uX to 1z and 2z , are:

6

The derivative of ( )xy 1tan −= with respect to x is:

If u is a differentiable function of x , the Chain rule can be applied to find:

( ) ( ) ( )( ) ( )( )

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=

→⋅−=+=⋅+⋅⋅⋅−=+

2exp

ln22sin2cosln222

21

1,

1,22

212,

22,

21,

22

21

zzX

XzzXXXzz

u

uuuu ππ

( ) ( )2,

11,2,1,1 2cos)ln(22cos)ln(2

uuuu X

zXXXz

⋅=⋅−→⋅⋅⋅−=

ππ

( ) ( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛=→⋅=→⋅⋅

⋅= −

1

212,2,

1

22,

2,

12 tan

2

12tan2sin

2cos z

zXX

z

zX

X

zz uuu

u πππ

π

( )

( )

2

1

212

2,

2

1

2

21

2

1

2,

22

21

22

1,

22

21

11

1,

1

1

2

1

1

1

2

2exp

2exp

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⋅⋅

=∂

∂

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⋅⋅

−=∂

∂

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−⋅−=∂

∂

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−⋅−=∂

∂

z

zzz

X

z

zz

z

z

X

zzz

z

X

zzz

z

X

u

u

u

u

π

π

( )2

1

1

/tan

u

dxdu

dx

ud

+=

−

( ) ( )( ) ( )( ) ( ) ( )( )

( )( ) ( )2

tan2

2

22

1

111tan

1cos

cossintan1

tantan

xdx

dy

dx

dyy

dx

dy

y

yy

dx

dy

dy

yd

dx

ydxy

xy

ruleChain

+=⎯⎯⎯ →⎯=⋅+

→=⋅+=⋅⎯⎯⎯ →⎯=→=

=

−

( )( )2

1

1

1tan

xdx

xd

+=

−

( )2

1

1

/tan

u

dxdu

dx

ud

+=

−

a39

The Jacobian gives the final prove: To draw a value from a normal distribution with mean xµ and standard deviation xσ the following equation can be used: Matlab has a pre-programmed routine (randn) that is used in the case study. ref. [22]

( )( )

( )

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅⋅⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅−

=⎟⎟⎠

⎞⎜⎜⎝

⎛ +−⋅−=

∂∂

∂∂

∂∂

∂∂

=∂

∂

2exp

2

1

2exp

2

1

2exp

2

1

,

,

22

21

22

21

2

2,

1

2,

2

1,

1

1,

21

2,1,

zz

zz

z

X

z

Xz

X

z

X

zz

XX

uu

uu

uu

ππ

π

( )2,1, 2cos)ln(2 uuxx XXX ⋅⋅⋅−⋅+= πσµ

a40

a41

a4 MEANDERING DUE EXTERNAL FORCES Currents, waves, wind etc. causes perturbations of the intended course (fig. 4.1-1). The width of the swept path depends on a number of factors, the key elements are: - the magnitude of the external forces; - the inherent manoeuvrability of the ship (which will vary with water depth/draught ratio); - the ability of the ship-handler; - the visual cues (e.g. buoys) available to the ship-handler; - the overall visibility. The meandering has an influence on the following aspects: - the determination of the dimensions in the design phase; - the distribution of the ships over the width of the channel; - the variation of the angle of incoming waves; - the length of the sailing course. The first two are not taken into account because this project deals only with the depth and length of the channel (see paragraph 3.2 Project definitions). To find out the influence of the third and fourth point, a calculation is done. For sine shaped paths with different amplitude (0 - 50 m) and a constant length (1000 m), the following is calculated: the maximum angle and the percentage of length difference between the real course and the intended course (see tab. 4.1-1 and fig. 4.1-2).

CurrentWaves

Channel boundary

Channel boundary

Theoretical course

Real course

fig. 4.1-1 Real sailing course

(exaggerated for clarity)

Amplitude Max. angle (in degree)5 0.0005

10 0.001115 0.001620 0.002225 0.002730 0.003335 0.003840 0.004445 0.004950 0.0055

tab. 4.1-1 Maximum angle between intended and real course

a42

Maximum angle The maximum angle of 0.0022 degree can be neglected. Length difference An amplitude of 50 m (this is a lot, an amplitude of 20 m is more likely) leads to more or less 2.5% length difference. This is 6 minutes on a sailing time of 4 hour, which can be neglected. ref. [12]

0 100 200 300 400 500 600 700 800 900 1000-50

-40

-30

-20

-10

0

10

20

30

40

50Different sailing courses

Distance [m]

Am

plit

ude

[m]

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5Difference real and theoretical course

Amplitude [m]

Per

cent

age

of l

engt

h di

ffer

ence

[%

]

fig. 4.1-2 Difference in length between intended and real course

a43

a5 EXPLANATION OF THE CASES SYUDY CONCLUSIONS The following conclusions of the case study will be explained: - why there is no relation between the number of bottom touches and ksimu in spite of the trend

in figure 5.2-3 (subparagraph 5.2.2 of the report) - why the results of alternative 2 and the variant of alternative 2 will converge to the same

constant value With a simple example these two conclusions can be explained. The example needs however some introduction. First will be explained how the expected number of bottom touches for small Npksi-values can be estimated.

a5.1 ESTIMATION OF THE EXPECTED KSI FOR SMALL NPKSI On page a31 of the appendices (paragraph a2.2) nine figures are given. From figure Tidal window availability by predicted HE10 (figure 7 of 9) can be concluded that for low wave heights (smaller than 0.3m) almost always a tidal window will be available and for HE10-values larger than 1m probably never. Such a conclusion cannot be found in the figures 8 and 9. Therefore the availability of a tidal window for the port of Rotterdam depends mainly on the wave conditions and in less extent on the sailing speed and meteorological conditions. The significant ship motions sZ in alternative 2 are linearly related to the wave conditions (HE10). With the significant ship motions the 0th-spectral moment of the ship motions 0m is calculated by: The expected number of bottom touches [ ]ξE is given by: with: Tpas transit time Tm mean period of a ship motion KC keel clearance Figures 5.2-3 and 5.2-4 are calculated with a predicted wave height of 0.3m. The wave height that can be expected to occur in reality and its variation is given by the formulas in subparagraph 5.1.5 of the report: The significant ship motion depends linearly on the wave height and is with the smallest depth/draught ratio (DDR) approximately given by (see subparagraph 5.1.1 of the report):

[ ] ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

0

2

2exp

m

KC

T

TE

m

pasξ

216

10 sZm =

11.00047929.03406.0

27.00019204.0*8914.0

10

10

10

10

=+⋅=

=−=

EH

EH

PmeanH

PmeanH

E

E

σµ

15.0

5.1

=

⋅=

s

z

Z

Z H

σµ

a44

Due to the dependency the standard deviation of the ship motions will be larger. With the following Matlab lines the standard deviation can be calculated: drawings=1000000 HE10=0.27; sigmaHE10=0.11; H=HE10+randn(drawings,1)*sigmaHE10; meanZs=1.5*H; sigmaZs=0.16; Zs=meanZs+randn(drawings,1)*sigmaZs; realmeanZs=mean(Zs) realsigmaZs=std(Zs) The aswers: realmeanZs = 0.41 realsigma = 0.23 Because the standard deviation is half the mean, the significant ship motions fluctuate strongly. The number of bottom touches ksi fluctuates even stronger, because of the exponential function. Figure 5.1-1 presents the fluctuation of ksi for Tpas/Tm = 1000 and KC = 2. The following Matlab lines are used: drawings=100 HE10=0.27; sigmaHE10=0.11; H=HE10+randn(drawings,1)*sigmaHE10; meanZs=1.5*H; sigmaZs=0.16; Zs=meanZs+randn(drawings,1)*sigmaZs; m0=1/16*Zs.^2; ksi=1000*exp(-2^2./(2.*m0)); plot(ksi,'k*') Npksi can be compared to the variable drawings in preceding Matlab lines. The expected number of bottom touches of a transit is then equal to the mean of ksi. Because of the exponential fluctuation, the mean ksi (when Npksi is small) can be estimated by: For instance the mean of 10-250, 10-50, 10-125, 10-10 is approximately 10-10/4. A set of input conditions determines the ksi value. A more unfavourable set of conditions gives a higher ksi. The probability that an unfavourable set is drawn during a transit depends on Npksi. For alternative 2 Npksi values of 100 or 1000 are relatively small. Therefore the preceding estimation rule counts.

[ ] ( )Npksi

ksiE

max=ξ

0 10 20 30 40 50 60 70 80 90 10010

-300

10-250

10-200

10-150

10-100

10-50

100

Drawing number

ksi

fig. 5.1-1 Fluctuation of ksi

a45

a5.2 EXAMPLE TO ILLUSTRATE THE CONCLUSIONS Assume that the following values for ksi are calculated for 10 sets of conditions: ksi = [0 0 0 0 0 0 0 0 0 100] The value 100 is obviously the maximum of the set. In this example a transit consists of drawing uniformly Npksi values from this set. The probability that the mean ksi is larger than 0 when Npksi = 1, is equal to 0.10. When Npksi = 10 this probability is ( ) 65.01.011 10 =−− . It is therefore very likely that the number of bottom touches for one transit with Npksi = 10 lager is than 0 and for Npksi = 1 not. However the probability that ksi is 100 is very small for Npksi = 10 and relatively large for Npksi = 1. When these transits are repeated ksimu times the mean ksi will be equal to the mean of

ksimuNpksi ⋅ ksi’s. The mean ksi will converge to a constant value by increasing ksimu values. With the following program this becomes clear: A=[0 0 0 0 0 0 0 0 0 100] Npksi=10 %or 1 ksimu=1000 d=0; for i=1:ksimu a=round(10*rand(Npksi,1)+0.5); b=A(a); c=mean(b); d=d+c; ksi(i)=d/i; end plot(i,ksi) The shape of figure 5.2-1 looks like the shapes of figures 5.2-3 and 5.2-4 in subparagraph 5.2.2. From figure 5.2-1 could be assumed, like in figure 5.2-3, that for small Npksi and ksimu values, ksi will be relatively small. However now it is clear that this assumed relation is wrong. With a probability of 10% (for Npksi = 1) it is for instance possible that the first time the value 100 is drawn. Then this relation will be gone. With this example it also becomes clear that the results of alternative 2 and its variant will converge to the same value. In the example the variant can be seen as the lines Npksi = 1, because in the variant it does not matter if Npksi = 1 or ksimu = 1.

fig. 5.2-1 Results example

100

101

102

103

0

5

10

15

20

25

Number of transits i

ksi

Npksi=10Npksi=10Npksi=1Npksi=1

a46

a47

a6 TYPE INPUT DATA IN THE CASE STUDY Variable Type (HARAP) Type (alternative 2) Unit Description

Ship data

deterministic deterministic [m] length between perpendiculars

deterministic deterministic [-] squat coefficient

deterministic deterministic [m] draught

normal distributed normal distributed [m] real draught given predicted draught

deterministic deterministic [m3] water displacement

normal distributed normal distributed [m/s] sailing speed

normal distributed normal distributed [m/s] real sailing speed given predicted sailing speed

normal distributed normal distributed [m] significant ship motion given real saling speed and h/Td*1

deterministic deterministic [s] mean period ship motions given real saling speed and h/Td

Channel data

deterministic deterministic [m] length of channel

deterministic deterministic [m] depth of channel

deterministic deterministic [m] direction of channel axis

Astronomical data

tidal cycle emperical distributed uniform distributed [-] tidal cycle

normal distributed deterministic [m] water level given tidal cycle

deterministic deterministic [m/s] current given tidal cycle

deterministic deterministic [degree] current direction given tidal cycle

Meteorological data

emperical distributed emperical distributed [m] meteo effect

normal distributed normal distributed [m] real meteo effect given predicted meteo effect

Wave data

emperical distributed emperical distributed [m] wave height

normal distributed normal distributed [m] real wave height given predicted wave height

*1 h/Td = depth/draught ratio

ppL

squatC

∇

sv

realsv _

mT

mwl

realmwl _

10EH

realEH _10

realwl

realc

realDc

d

l

r

dT

realdT _

sZ

a48

a49

a50

Literatuurlijst [1] Blaauw H.G., T. Schilperoort en J. Strating, Optimalization of depths of channels. Delft: Delft

Hydraulics Laboratory, 1982.

[2] CUR190, Probability in Civil Engineering, Part 1: The theory of probabilistic design. Gouda: Stichting CUR, 1997.

[3] Doorn, J.T.M. van, Probabilistische berekeningen voor de Selected Route. Delft: Waterloopkundig laboratorium, 1990.

[4] Doorn, J.T.M. van, Harap, specificatie rapport. Delft: Waterloopkundig laboratorium, 1992.

[5] Harap, Functioneel Ontwerp. Rotterdam: Ministerie van Verkeer en Waterstaat, 1994a.

[6] Harap, Ontwerphandleiding. Rotterdam: Ministerie van Verkeer en Waterstaat, 1994b.

[7] Harap, Gebruikershandleiding. Rotterdam: Ministerie van Verkeer en Waterstaat, 1994c.

[8] Hoek, A. van der, Pilot Berge Stahl Fase 1. Rotterdam: Adviesdienst Verkeer en Vervoer, 1999.

[9] Holthuijsen, L.H., Wind waves, collegedictaat ct 5316. Delft: Technische Universiteit Delft, 2005.

[10] Informatie voor de vaart met geulgebonden schepen naar de haven van Rotterdam. Rotterdam: Rijkswaterstaat, e.a., 1995.

[11] Kerncijfers Goederenvervoer. Internetsite www.rws-avv.nl: Adviesdienst Verkeer en Vervoer, 2004.

[12] PIANC-IAPH Working Group II-30 in cooperation with IMPA and IALA, Approach Channels, A guide for Design. Brussels: PIANC, 1997.

[13] Principles of Naval Architecture. United States of America: The Society of Navel Architects and Marine Engineers, 1974.

[14] Rice, S.O., ‘Mathematical analysis of random noise’. In: N. Wax, e.a., Selected Papers on Noise and Stochastic Processes. New York: Dover Pub. Inc., 1954, p.184-195.

[15] Savenije, A.C., ‘Probabilistic admittance policy deep draught vessels’. PIANC bulletin, 91 (1996), p.25-36.

[16] Savenije, A.C., Safety Criteria for Approach Channels. Rotterdam: AVV Transport Research Centre, 1998.

[17] Vrijling, J.K., and P.H.A.J.M. van Gelder, Probabilistic design in hydraulic engineering, college dictaat ct5310. Delft: Technische Universiteit Delft, 2002.

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[18] Vrijling, J.K., Audit HARAP. Delft: Technische Universiteit Delft, 2004.

[19] Wijnstra, R., Meerjarig veiligheidscriterium Euro-maas geul, Notitie. Rotterdam: Rijkswaterstaat, 1993.

[20] Wust, J.C., Invoer randvoorwaarden, Evaluatie 74-voets geul. Rijswijk: 1994.

[21] Wust, J.C., Doorvaart naar volgend getij, Notitie. Rijswijk: 1995.

Websites: [22] http://mathworld.wolfram.com [23] http://www.shipmotions.nl