Analysis of Effective Height of Green Water Loading for Tension Leg Platform Based on Statistical Process Control

WU Hao,LIN Yan

(Department of Naval Architecture,Dalian University of Technology,Dalian 116023,China)

0 Introduction

Wave flows onto the deck of marine structures,and it becomes green water.Green water can cause damage to the equipment and superstructure of marine vessels.It is a strong nonlinear problem,difficult to solve by mathematical theory,and it has very complex physical phenomenon.Water will inevitably flow onto the deck,when TLP is in service.It will cause deck wetness,affecting the normal operation of drilling equipment on the deck.Additionally,it will threaten the security of TLP if the wave height is too high.It is necessary to research on the effect of green water for TLP,and predict wave height and the probability of green water.

Based on model tests,researchers found that the green water occurrence and loading are strongly dependent on the wave period,wave height and current velocity and cannot be predicted with present prediction methods based on linear theory.The bow shape and the position of a structure with respect to the forward perpendicular have effect on the green water problem[4].Based upon the prediction of the vertical relative motion with respect to the wave surface at the bow calculated from the seakeeping analysis,a statistical model of green water occurrence is described,which can predict the level of wetness and water ingress for hatchcoverless containerships[11].Then a research about the probability of green water occurrence is investigated by taking into account the threshold of the vertical relative motion exceeding the freeboard.The number of wetting of the unit/vessel is predicted using probabilistic method[8].After four years,a new design tool has been developed for the calculation of water velocities on the deck of a ship as a result of green water incidents.This tool models the flow by simulating a shelf submerging into a pool of water.The flow of water is modelled by using a numerical method,which applies potential flow theory and uses a desingularised boundary integral equation method combined with an implicit time-stepping procedure[21].In the same year,OMAE’02,a study describes the influence of the water on deck on the dynamical behavior of an offshore supply vessel with a large open aft deck based on analysis and computational procedure.The deck under certain load and sea conditions can become partially or totally immersed.This study focuses on the roll motion that can be of large amplitude and therefore has implications and risks for the ship’s safety.The Glimm’s method is used to model the three dimensional flow of shallow water on the deck[12].A practical method for estimating the amount of shipping water into the hold over the hatch coaming of an open-top container ship was proposed by combining wave overtopping theory with flood wave theory[24].Based on ISSC wave spectrum with the velocity and heading angle of ship considered,the nonlinear dynamic response of ship is simulated by harmonic acceleration method.The safe probability Ps（H,T,X,U ）is obtained from a four parameterized function by applying statistic knowledge with wave height,wave period,heading angle,the velocity of ship and random phase angle of waves taken into account[7].Later,the measurement of velocity fields of a plunging wave impacting on a structure in a twodimensional wave tank was investigated experimentally.From both the PIV and BIV measurements,it was found that the maximum fluid particle velocity as well as the bubble velocity in front of the structure during the impinging process is about 1.5 times than the phase speed of the waves.While the maximum horizontal velocity above the deck is less than the phase speed.It was also found that the dam breaking solution does not work well in predicting the green water velocity.A prediction model is obtained for the green water velocity distribution[18].By taking advantage of the virtues of the potential flow theory and CFD,a technique of dynamic mesh is applied in a 3-D numerical wave tank to simulate the green water occurrence on an oscillating FPSO model in head waves[23].OMAE2009,an experimental research investigates the velocity fields of plunging breaking waves impinging on a three-dimensional simplified ship-shape structure.It was observed that in deck-impingement case,the maximum horizontal velocity is higher for the case with waves compacting on the deck and waves also passed the deck much quicker.The profiles of the green water velocity show a non-linear distribution with the maximum velocity occurring near the front of the water[2].The critical condition and wave load models were established and realized in large-scale ship maneuvering simulator environment by using probability theory and nonlinear dynamics method for ship sailing with slow speed[5].The rolling response of ship with water on deck in random beam wave was numerically simulated based on Bernoulli equation[13-14].A three-dimensional seakeeping numerical solver is developed to handle occurrence and effects of water-on-deck and bottom slamming.It couples the rigid-ship motion with 3-D weakly nonlinear potential flow solver based on the weak-scatterer hypothesis with(A)the water flowing along the deck and(B)bottom slamming events.Problem A,and so local and global induced green-water loads,are investigated by assuming shallow-water conditions onto the deck.Problem B is examined through a Wagner-type wedge-impact analysis.The resulting numerical solver can study efficiently the ship interaction with regular and irregular sea states and the forward motion with limited speed of the vessel[6].In statistical field,researchers propose and illustrate statistical modeling and fitting of time-series effects and the application of standard control-chart procedures to the residuals from these fits[1].Seven years later,SPC approaches based on multivariate statistical projection methods(PCA and PLS)have been developed.Multivariate control charts in the projection spaces provide powerful methods for both detecting out-of-control situations,and diagnosing assignable causes,and they are applicable to both continuous and batch processes[15].With statistical process control method development,it is applied in process of structural health monitoring.A research focuses on applying a Statistical Process Control(SPC)technique known as an ‘X-bar control chart’ to vibration-based damage diagnosis[22].

Many researchers have carried out extensive researches on the green water of marine structures.Researchers have meaningful attempt in the field of the probability of green water,green water loading,and green water field of ships,FPSO and other marine oil and gas production platforms with using experimental,probability theory,CFD,and many other methods.While model tests need lots of facilities,manpower,money and time,and CFD needs long time to programme and run,and probability method needs practicable model.Actually,there is not a powerful and quite less calculation time simplified methodology to solve green water problem now.Meanwhile,SPC method has been applied to engineering and industry,and solved many practical problems.So far,no scientists have researched on green water problem with SPC method.Green water problem combined with SPC method is an interesting topic,and has the potential to be a powerful and quite less calculation time simplified methodology to solve green water problem.

There is a phase between the motion of wave and marine structure.The distance between wave height and deck can not describe the height of green water accurately.Considering phase,amplitude and statistical probability,using effective height of green water describe the actual relative location of wave and marine structures more accurately.This paper researches on the center line,upper control limit,lower control limits and exceeding probability of effective height of green water for TLP based on statistical process control theory to solve green water problem in a more powerful and quite less calculation time simplified methodology.

In this paper,we analyze effective height of green water loading for TLP based on statistical process control theory.The paper is organized as follows:In Chap.1,relative motion of TLP and numerical methods for effective height of green water are presented.In Chap.2,several wave spectra are given.In Chap.3,statistical process control theory is presented.In Chap.4,numerical results are presented to discuss effective height of green water under different probability and wave incidence angle with JONSWAP spectrum and Pierson-Moskowitz spectrum.Finally,the conclusions of this research are given in Chap.5.

1 Relative motion and effective height of green water of TLP

Wave height is a stochastic process,whose uncertainty leading that it is difficult to simulate mathematically.Statistical approach deals with stochastic process very well.Wave spectrum,as statistical result of wave,is a good tool to research on wave characteristics.We assume that wave is a stationary and ergodic stochastic process in time histories.Stochastic wave elevation is accumulated by infinite sine waves of random amplitude,period and initial phase as shown in Fig.1.

Fig.1 Random wave high superposition

Most of the wave energy is concentrated around a certain frequency.Wave spectral density function corresponds to a narrow band process.Wave spectrum is stationary and ergodic process,and follows Rayleigh distribution.

The mean or average of the envelope curve amplitude is

The mean or average of square the envelope curve amplitude is

Variance of wave spectrum with relative motion is

There are six degrees of freedom:surge ξ1,sway ξ2,heave ξ3,roll ξ4,pitch ξ5and yaw ξ6for the motion of TLP in ocean.Wave is function of time and space η0x,（）t.The relative motion of TLP deck center position and wave is

We assume that η3（t）and η0（x, t ）are harmonic function of time and space.The motion of TLP in ocean is

whereis a complex included amplitude and phase; ξ0is external excitation amplitude.

The relative motion of TLP and external excitation is

Response spectrum of relative motion of TLP and wave is[3]

Considering TLP RAO and wave phase,effective green water of TLP deck center position relative to wave is

whereis a probability of a certain value being exceeded;is the standard variance of response spectrum of relative motion of TLP deck center position and wave.

Fig.2 Liuhua 16-2 TLP model

Fig.3 Liuhua 16-2 TLP model for AQWA

So relative motion RAO is the key to obtain effective green water.This paper using TLP LH16-2 as example,the model information is shown in Fig.2.Using ANASYS AQWA to obtain heave RAO and phase with 0°,45°and 90°wave incident angles.The geometry used for AQWA is shown in Fig.3.Results of RAO and phase are shown in Fig.4 and Fig.5.

Fig.4 TLP heave motion RAO

Considering heave RAO and phase,obtain relative heaveof TLP and wave,shown in Fig.6.

Fig.5 TLP heave motion RAO phase

2 Wave spectrum

So far,there are many wave spectra,such as JONSWAP spectra,Pierson-Moskowitz spectra,Nuemann spectra and ISSC spectra.JONSWAP spectra are growing spectra.Pierson-Moskowitz spectra are fully developed spectra.

Fig.6 TLP complex relative heave motion|RAO|2

Fig.7 JONSWAP spectrum

The JONSWAP(Joint North Sea Wave Observation Project)spectra are an empirical relationship that defines the distribution of energy with frequency as parameter.It is a growing spectrum and never fully developed.It continues to develop through non-linear,wavewave interactions even for very long times and distances.Therefore,in the JONSWAP spectrum,waves continues to grow with distance(or time)as specified by the α(alpha)term,and the peak in the spectrum is more pronounced,as specified by the γ(gamma)term[10].When in 1966 Hasselmann found the latter to be particularly important as it led to enhanced non-linear interactions.Hence an extra and somewhat artificial factor was added to the Pierson-Moskowitz spectrum in order to improve the fit to their measurements.The JONSWAP spectrum is thus a Pierson-Moskowitz spectrum multiplied by an extra peak enhancement factor.JONSWAP spectra are very familiar in engineering project.The envelope curve of JONSWAP spectra is shown in Fig.7.

The Pierson-Moskowitz(PM)spectrum(Fig.8)is an empirical relationship that defines the distribution of energy with frequency as parameter.Developed in 1964,the PM spectrum is one of the simplest descriptions for the energy distribution.It assumes that if the wind blows steadily for a long time over a large area,then the waves will eventually reach a point of equilibrium with the wind.This is known as a fully developed sea.Pierson and Moskowitz developed their spectrum from measurements in the North Atlantic during 1964,and presented the relationship between energy distribution and wind.PM spectra have been fitted from data with wind speed of 20 to 40 knots.Most of the data come from the North Atlantic.It is a standard wave spectrum recognized by ITTC[17].

Fig.8 Pierson-Moskowitz spectrum

Another popular spectral formula is the ISSC(International Ship Structures Congress)two parameter formula.Both the significant wave height and the significant zero crossing frequency are required for ISSC spectra.The ISSC spectra shows how a time history is generated by using a finite number of sine waves.

3 Statistical Process Control,SPC

Statistical Process Control(SPC)is a method applying mathematical statistical theory to monitor and control a process.If the data got from monitor devices in process is in the range of control limits,it indicates that the process is in control.Otherwise the process is out of control.The variation in the range of control limits is because of the natural attribute of the process and it is expected as part of the process.If the variation out of the range of control limits,it indicates that there is something changed or something wrong happened in the process,and should be fixed before out of control.If the process is in control,statistical characteristics of fluctuation about process have a stable stochastic distribution.If the process is out of control,statistical characteristics of fluctuation about process have an unstable stochastic distribution.SPC is an analytical decision making method based on the statistical characteristics of fluctuation about process to monitor and control a process,monitor common and assignable variation,warn unusual trends,eliminate assignable cause,recover the stability of process,and realize the purpose that control and improve a target finally.

SPC control charts are an essential graphical tool based on mathematical statistics theory for continuous process control to monitor common and assignable variation and judge whether the process is in control.Common variation is stochastic fluctuation in the range of control limits due to a common cause-the natural variation that is expected as part of the process,in other words,it is a normal fluctuation.Assignable variation is caused by assignable reasons in system,unusual to exist,and has a great influence on stability of process if it occurs.So it is necessary to monitor assignable variation,analyze factors of assignable variation,eliminate assignable cause,recover the stability of process,and realize the purpose that process is in control.SPC control charts use statistical ‘discovering unusual’ as a tool in the control process,which is a foundation for SPC control charts.Established control limits by valid data,it is upper control limit(UCL),lower control limit(LCL)and center line(CL).Generally,UCL is CL+3σ,and LCL is CL-3σ,where σ is standard variance.Data does not fall outside of control limits if there is not assignable cause in process.

Calculate effective green water distribution followed wave spectrum with relative motion of TLP and wave,denoted as effective green water spectrum.Effective green water spectrum represents changes of wave elevation distribution of wave spectrum for the reason of relative motion.Using Fourier transform to translate stochastic wave elevation accumulated by infinite sine waves of random amplitude,period and initial phase,we obtain wave spectrum.Wave spectrum includes most of wave information.Wave spectrum is stationary and ergodic process,so effective green water spectrum is stationary and ergodic process also.A process is in control,if it is affected by the common cause-natural variation that is expected as part of the process.We assume that effective green water distributes between upper control limit and lower control limit.Hypothesis test about stochastic wave process is not rejected.Combined with these precondition,we obtain center line and 3σ,and compare with center lines and 3σ with different probability.If effective height of green water falls out of the control limits,this indicates that it is a small probability event,and the probability about this is 0.27%.We assume that the effective height of green water with this spectrum parameter rarely occurs in actual marine environment.

There are variety of methods to obtain average center line of effective green water spectrum,such as statistical average method,weighted average method,modal method,mean area method,etc.Effective green water spectrum not only represents wave elevation characteristic,but also indicates wave elevation distribution under wave parameter.It is not the best tool to obtain the average of spectrum for statistical average method which is not well depicted in the form of distribution.Weighted average method is unsuitable for averaging effective green water spectrum because setting weight involves subjective factors which make wave statistical stochastic properties fail.Modal method is best for describing the wave elevation that appears most frequently,and does not show the distribution of wave elevation,so it is not suitable for seeking the mean.A mean area method for effective green water spectrum is presented to calculate average wave elevation of the spectrum,which fit well with wave elevation distribution form under the condition of wave parameter.We assume that average wave elevation of the spectrum is a certain value.The distribution area of the certain value under the condition of wave parameters is equal to the spectral area.This method is equivalent to transforming any form of spectrum area into rectangular spectrum,and the width of the rectangular is average wave elevation of spectrum.This method retains the distribution characters of effective green water spectrum under the condition of wave parameter.Effective green water spectrum is an energy distribution characterization under the condition of wave parameter and is a two-dimension distribution.In the form of two-dimensional distribution,the essence of distribution form is shape characteristics of variables in two-dimensional.The essence of shape characteristics is shape distribution features of function in two-dimension.For the mean value of distribution form,mean area method fits well with depict shape characteristics,and retains shape distribution features of function better in two-dimension.From the above,mean area method is the best tool to get average wave elevation under the condition of wave parameter combined with spectrum distribution characteristics.Variance of effective green water spectrum σ is calculated by R(0).

4 TLP effective height of green water

This paper mainly based on JONSWAP spectrum and Pierson-Moskowitz spectrum,takes a TLP LIUHUA16-2 as an example,and calculates average center line of effective height of green water.JONSWAP spectrum is a growing spectrum,so it can describe a growing wave height condition.Pierson-Moskowitz spectrum is a fully developed spectrum,so it can describe a stable wave height condition.

4.1 Wave incident angle is 0°,JONSWAP spectrum and Pierson-Moskowitz spectrum effective height of green water calculation

Wave incident angle is 0°,and calculate effective height of green water exceeding 1/1 000,1/100,1/10 and 1/3 probability followed JONSWAP spectrum and Pierson-Moskowitz spectrum by Eq.(8).Results are shown in Fig.9 and Fig.10.

Fig.9 JONSWAP spectrum effective wave height of green water

Fig.10 Pierson-Moskowitz spectrum effective wave height of green water

From Fig.9,wave frequency range varies from 0 to 2.3 rad/s.The effective green water spectrum curve is substantially closed.Other frequency wave has little impact on the effective height of green water.Compared with entire effective green water spectrum,the probability of effective height of green water is no more than 9.1%,if wave frequency greater than 1.0 rad/s.It is a small part of effective green water spectrum.We mainly focus on wave energy concentrated part,wave frequency range varies from 0 to 1.0 rad/s.From Fig.9,wave frequency range varies from 0.1 to 0.7 rad/s,effective height of green water has a big change under different probability.In this frequency range,where located in TLP rolling natural period range,effective green water spectrum and TLP RAO have great fluctuations.Drawing curves about effective height of green water in frequency dimension.It is an energy distribution,and indicates the distribution of effective height of green water under frequency.From Fig.9,as wave frequency decreases,effective height of green water increases,and the energy increases.As wave frequency increases,effective height of green water decreases,and the energy decreases.Effective height of green water changes significantly under wave frequency range from 0.1 to 0.7 rad/s.Effective height of green water changes slowly under wave frequency no less than 1.0 rad/s.

From Fig.10,with wind speed at 19.5 m above sea level increases,effective height of green water increases.It is an extremely dangerous situation that TLP is in survival conditions,because marine environment is extremely harsh,if wind speed exceeds 25 m/s.It is beyond the scope of research.With wind speed increases,effective height of green water grows approximation to second order with different probability,and maximum is 10.40 m.

Averaging effective height of green water in time dimension,space dimension and frequency dimension,indicates different wave energy distribution.For JONSWAP spectra,average effective green water spectrum in frequency dimension indicates average wave energy in frequency dimension.For Pierson-Moskowitz spectra,average effective green water spectrum in space dimension indicates average wave energy in space dimension.Wave spectrum is stationary and ergodic.There is much difference in frequency dimension and space dimension about effective height of green water.Considering distribution of wave energy and probability in frequency dimension and space dimension,mean area method is suitable for calculating center line,upper control limit and lower control limit of effective green water spectrum.

Fig.11 SPC chart with probability 1/1 000

Fig.12 SPC chart with probability 1/100

a.Wave incident angle is 0°,the probability is 1/1 000.JONSWAP spectrum,mean area method applied,center line of effective green water spectrum is 4.15 m,and corresponding to the wave frequency is 0.635 rad/s,also spectrum σ is 3.193.As we know from statistical process control theory,effective height of green water upper control limit is 13.72 m,lower control limit is 0 m.It is out of control if wave frequency is less than 0.5 rad/s.It is a small probability event for effective height of green water occurrence in this range of wave frequency.Pierson-Moskowitz spectrum,mean area method applied,center line of effective green water spectrum is 4.15 m,and corresponding to the wind speed is 14.35 m/s.Effective height of green water is pretty evenly distributed in entire effective green water spectrum,and it is in control.Effective green water spectrum center lines of JONSWAP spectrum and Pierson-Moskowitz spec-trum are approximately equal.As SPC control chart shown in Fig.11,most of effective heights of green water are between upper control limit and lower control limit,which are in control.Only effective heights of green water of small frequency wave are beyond upper control limit,which are out of control.It indicates that if the probability of spectrum is very little,the probability that the effective heights of green water of small frequency wave are out of control is small.SPC control chart controls the stochastic process of effective height of green water very well.

b.Wave incident angle is 0°,the probability is 1/100.JONSWAP spectrum,mean area method applied,center line of effective green water spectrum is 3.39 m,and corresponding to the wave frequency is 0.635 rad/s,also spectrum σ is 2.885.As we know from statistical process control theory,effective height of green water upper control limit is 12.04 m,lower control limit is 0 m.It is out of control if wave frequency is less than 0.4 rad/s.It is a small probability event for effective height of green water occurrence in this range of wave frequency.Pierson-Moskowitz spectrum,mean area method applied,center line of effective green water spectrum is 3.39 m,and corresponding to the wind speed is 14.35 m/s.Effective height of green water is pretty evenly distributed in entire effective green water spectrum,and it is in control.Effective green water spectrum center lines of JONSWAP spectrum and Pierson-Moskowitz spectrum are approximately equal.As SPC control chart shown in Fig.12,most of effective heights of green water are between upper control limit and lower control limit,which are in control.Only effective heights of green water of small frequency wave are beyond upper control limit,which are out of control.Additional,the level of out of upper control limit decreases.It indicates that if the probability of spectrum is very small,the probability that the effective heights of green water of small frequency wave are out of control is small.SPC control chart controls the stochastic process of effective height of green water very well.

c.Wave incident angle is 0°,the probability is 1/10.JONSWAP spectrum,mean area method applied,center line of effective green water spectrum is 2.39 m,and corresponding to the wave frequency is 0.635 rad/s,also spectrum σ is 2.426.As we know from statistical process control theory,effective height of green water upper control limit is 9.67 m,lower control limit is 0 m.It is out of control if wave frequency is less than 0.2 rad/s.It is a small probability event for effective height of green water occurrence in this range of wave frequency.Pierson-Moskowitz spectrum,mean area method applied,center line of effective green water spectrum is 2.39 m,and corresponding to the wind speed is 14.34 m/s.Effective height of green water is pretty evenly distributed in entire effective green water spectrum,and it is in control.Effective green water spectrum center lines of JONSWAP spectrum and Pierson-Moskowitz spectrum are approximately equal.As SPC control chart shown in Fig.13,most of effective heights of green water are between upper control limit and lower control limit,which are in control.Only effective heights of green water of small frequency wave are beyond upper control limit,which are out of control.Additionally,the level of out of upper control limit decreases further.The effective heights of green water that are out of upper control limit are approximately equal to upper con-trol limit.It indicates that if the probability of spectrum is small,the probability that the effective heights of green water of small frequency wave are out of control is small.SPC control chart control the stochastic process of effective height of green water very well.

d.Wave incident angle is 0°,the probability is 1/3.JONSWAP spectrum,mean area method applied,center line of effective green water spectrum is 1.65 m,and corresponding to the wave frequency is 0.635 rad/s,also spectrum σ is 2.016.As we know from statistical process control theory,effective height of green water upper control limit is 7.70 m,lower control limit is 0 m.All frequency waves are in control.It is a great probability event for effective height of green water occurrence in this range of wave frequency.So far as we know,with spectral probability increases,effective heights of green water center line decreases,the level of effective heights of green water out of control limit(upper control limit and lower control limit)decreases,until the condition that effective heights of green water are out of control is disappeared,where effective heights of green water of all frequency wave are in control.Pierson-Moskowitz spectrum,mean area method applied,center line of effective green water spectrum is 1.65 m,and corresponding to the wind speed is 14.34 m/s.Effective height of green water is pretty evenly distributed in entire effective green water spectrum,and it is in control.Effective green water spectrum center lines of JONSWAP spectrum and Pierson-Moskowitz spectrum are approximately equal.As SPC control chart shown in Fig.14,effective heights of green water of all frequency waves are between upper control limit and lower control limit,which are in control.It indicates that if the probability of spectrum is not big,effective heights of green water of all frequency waves are in control.SPC control chart controls the stochastic process of effective height of green water very well.

Fig.13 SPC chart with probability 1/10

Fig.14 SPC chart with probability 1/3

From cases a,b,c and d with different spectral probability,as probability increases,the level of effective heights of green water out of control limit(upper control limit and lower control limit)decreases.If the probability is 1/10,the condition that effective heights of green water are out of control is close to disappearing.If the probability is 1/3,the condition that effective heights of green water are out of control disappears completely.Effective heights of green water are all in control if spectral probability is no less than 1/3.Effective heights of green water are more controllable,and effective heights of greenwater center line are lower,and the probability of effective green water spectrum increases.

4.2 Wave incident angle is 45°,JONSWAP spectrum and Pierson-Moskowitz spectrum effective height of green water calculation

Wave incident angle is 45°,and calculate effective height of green water exceeding 1/1 000,1/100,1/10 and 1/3 probability followed JONSWAP spectrum and Pierson-Moskowitz spectrum by Eq.(8).Results are shown in Fig.15 and Fig.16.

Fig.15 JONSWAP spectrum effective wave height of green water

Fig.16 Pierson-Moskowitz spectrum effective wave height of green water

From Fig.15,wave frequency range varies from 0 to 2.3 rad/s.The effective green water spectrum curve is substantially closed.Other frequency wave has little impact on the effective height of green water.Compared with entire effective green water spectrum,the probability of effective height of green water is no more than 9.1%,if wave frequency is greater than 1.0 rad/s.It is a small part of effective green water spectrum.We mainly focus on wave energy concentrated part,wave frequency range from 0 to 1.0 rad/s.From Fig.15,wave frequency range from 0.1 to 0.7 rad/s,effective height of green water has a big change under different probability.In this frequency range,where located in TLP rolling natural period range,effective green water spectrum and TLP RAO have great fluctuations.Drawing curves about effective height of green water in frequency dimension.It is an energy distribution,and indicates the distribution of effective height of green water under frequency.From Fig.15,as wave frequency decreases,effective height of green water increases,and the energy increases.As wave frequency increases,effective height of green water decreases,and the energy decreases.Effective height of green water changes significantly under wave frequency range from 0.1 to 0.7 rad/s.Effective height of green water changes slowly under wave frequency no less than 1.0 rad/s.

From Fig.16,with wind speed at 19.5 m above sea level increases,effective height of green water increases.It is an extremely dangerous situation that TLP is in survival conditions,because marine environment is extremely harsh,if wind speed exceeds 25 m/s.It is beyond the scope of research.With wind speed increases,effective height of green water grows approximation to second order with different probability,and the maximum is 10.52 m.

Averaging effective height of green water in time dimension,space dimension and frequency dimension,indicates different wave energy distribution.For JONSWAP spectra,average effective green water spectrum in frequency dimension indicates average wave energy in frequency dimension.For Pierson-Moskowitz spectra,average effective green water spectrum in space dimension indicates average wave energy in space dimension.Wave spectrum is stationary and ergodic.There is much difference in frequency dimension and space dimension about effective height of green water.Considering distribution of wave energy and probability in frequency dimension and space dimension,mean area method is suitable for calculating center line,upper control limit and lower control limit of effective green water spectrum.

a.Wave incident angle is 45°,the probability is 1/1 000.JONSWAP spectrum,mean area method applied,center line of effective green water spectrum is 4.21 m,and corresponding to the wave frequency is 0.635 rad/s,also spectrum σ is 3.209.As we know from statistical process control theory,effective height of green water upper control limit is 13.84 m,lower control limit is 0 m.It is out of control if wave frequency is less than 0.5 rad/s.It is a small probability event for effective height of green water occurrence in this range of wave frequency.Pierson-Moskowitz spectrum,mean area method applied,center line of effective green water spectrum is 4.21 m,and corresponding to the wind speed is 14.35 m/s.The effective height of green water is pretty evenly distributed in entire effective green water spectrum,and it is in control.Effective green water spectrum center lines of JONSWAP spectrum and Pierson-Moskowitz spectrum are approximately equal.As SPC control chart shown in Fig.17,most of effective heights of green water are between upper control limit and lower control limit,which are in control.Only effective heights of green water of small frequency wave are beyond upper control limit,which are out of control.It indicates that if the probability of spectrum is very little,the probability that the effective heights of green water of small frequency wave are out of control is small.SPC control chart can control the stochastic process of effective height of green water very well.

b.Wave incident angle is 45°,the probability is 1/100.JONSWAP spectrum,mean area method applied,center line of effective green water spectrum is 3.44 m,and corresponding to the wave frequency is 0.635 rad/s,also spectrum σ is 2.900.As we know from statistical process control theory,effective height of green water upper control limit is 12.14 m,lower control limit is 0 m.It is out of control if wave frequency is less than 0.4 rad/s.It is a small probability event for effective height of green water occurrence in this range of wave frequency.Pierson-Moskowitz spectrum,mean area method applied,center line of effective green water spectrum is 3.44 m,and corresponding to the wind speed is 14.35 m/s.The effective height of green water is pretty evenly distributed in entire effective green water spectrum,and it is in control.Effective green water spectrum center lines of JONSWAP spectrum and Pierson-Moskowitz spectrum are approximately equal.As SPC control chart shown in Fig.18,most of effective heights of green water are between upper control limit and lower control limit,which are in control.Only effective heights of green water of small frequency wave are beyond upper control limit,which are out of control.Additional,the level of out of upper control limit decreases.It indicates that if the probability of spectrum is very small,the probability that the effective heights of green water of small frequency wave are out of control is small.SPC control chart can control the stochastic process of effective height of green water very well.

Fig.17 SPC chart with probability 1/1 000

Fig.18 SPC chart with probability 1/100

c.Wave incident angle is 45°,the probability is 1/10.JONSWAP spectrum,mean area method applied,center line of effective green water spectrum is 2.43 m,and corresponding to the wave frequency is 0.635 rad/s,also spectrum σ is 2.438.As we know from statistical process control theory,effective height of green water upper control limit is 9.74 m,lower control limit is 0 m.It is out of control if wave frequency is less than 0.2 rad/s.It is a small probability event for effective height of green water occurrence in this range of wave frequency.Pierson-Moskowitz spectrum,mean area method applied,center line of effective green water spectrum is 2.43 m,and corresponding to the wind speed is 14.35 m/s.Effective height of green water is pretty evenly distributed in entire effective green water spectrum,and it is in control.Effective green water spectrum center lines of JONSWAP spectrum and Pierson-Moskowitz spectrum are approximately equal.As SPC control chart shown in Fig.19,most of effective heights of green water are between upper control limit and lower control limit,which are in control.Only effective heights of green water of small frequency wave are beyond upper control limit,which are out of control.Additional,the level of out of upper control limit decreases further.The effective heights of green water that are out of upper control limit are approximately equal to upper control limit.It indicates that if the probability of spectrum is small,the probability that the effective heights of green water of small frequency wave are out of control is small.SPC control chart can control the stochastic process of effective height of green water very well.

Fig.19 SPC chart with probability 1/10

Fig.20 SPC chart with probability 1/3

d.Wave incident angle is 45°,the probability is 1/3.JONSWAP spectrum,mean area method applied,center line of effective green water spectrum is 1.68 m,and corresponding to the wave frequency is 0.635 rad/s,also spectrum σ is 2.026.As we know from statistical pro-cess control theory,effective height of green water upper control limit is 7.76 m,lower control limit is 0 m.All frequency waves are in control.It is a great probability event for effective height of green water occurrence in this range of wave frequency.So far as we know,with spectral probability increases,effective heights of green water center line decreases,the level of effective heights of green water out of control limit(upper control limit and lower control limit)decreases,until the condition that effective heights of green water are out of control is disappeared,where effective heights of green water of all frequency waves are in control.Pierson-Moskowitz spectrum,mean area method applied,center line of effective green water spectrum is 1.68 m,and corresponding to the wind speed is 14.35 m/s.The effective height of green water is pretty evenly distributed in entire effective green water spectrum,and it is in control.Effective green water spectrum center lines of JONSWAP spectrum and Pierson-Moskowitz spectrum are approximately equal.As SPC control chart shown in Fig.20,effective heights of green water of all frequency waves are between upper control limit and lower control limit,which are in control.It indicates that if the probability of spectrum is not big,effective heights of green water of all frequency waves are in control.SPC control chart can control the stochastic process of effective height of green water very well.

From cases a,b,c and d with different spectral probability,as probability increases,the level of effective heights of green water out of control limit(upper control limit and lower control limit)decreases.If the probability is 1/10,the condition that effective heights of green water are out of control is close to disappearing.If the probability is 1/3,the condition that effective heights of green water are out of control disappears completely.Effective heights of green water are all in control if spectral probability is no less than 1/3.Effective heights of green water are more controllable,and effective heights of greenwater center line are lower,and the probability of effective green water spectrum increases.

4.3 Wave incident angle is 90°,JONSWAP spectrum and Pierson-Moskowitz spectrum effective height of green water calculation

Wave incident angle is 90°,and calculate effective height of green water exceeding 1/1 000,1/100,1/10 and 1/3 probability followed JONSWAP spectrum and Pierson-Moskowitz spectrum by Eq.(8).Results are shown in Fig.21 and Fig.22.

From Fig.21,wave frequency range varies from 0 to 2.3 rad/s.The effective green water spectrum curve is substantially closed.Other frequency wave has little impact on the effective height of green water.Compared with entire effective green water spectrum,the probability of effective height of green water is no more than 9.1%,if wave frequency is greater than 1.0 rad/s.It is a small part of effective green water spectrum.We mainly focus on wave energy concentrated part,wave frequency range from 0 to 1.0 rad/s.From Fig.21,wave frequency range from 0.1 to 0.7 rad/s,effective height of green water has a big change under different probability.In this frequency range,where located in TLP rolling natural period range,effective green water spectrum and TLP RAO have great fluctuations.Drawing curves about effective height of green water in frequency dimension.It is an energy distribution,and indicates the distribution of ef-fective height of green water under frequency.From Fig.21,as wave frequency decreases,effective height of green water increases,and the energy increases.As wave frequency increases,effective height of green water decreases,and the energy decreases.The effective height of green water changes significantly under wave frequency range from 0.1 to 0.7 rad/s.The effective height of green water changes slowly under wave frequency no less than 1.0 rad/s.

Fig.21 JONSWAP spectrum effective wave height of green water

Fig.22 Pierson-Moskowitz spectrum effective wave height of green water

From Fig.22,with wind speed at 19.5 m above sea level increases,effective height of green water increases.It is an extremely dangerous situation that TLP is in survival conditions,because marine environment is extremely harsh,if wind speed exceeds 25 m/s.It is beyond the scope of research.With wind speed increases,effective height of green water grows approximation to second order with different probability,and the maximum is 10.40 m.

Averaging effective height of green water in time dimension,space dimension and frequency dimension,indicates different wave energy distribution.For JONSWAP spectra,average effective green water spectrum in frequency dimension indicates average wave energy in frequency dimension.For Pierson-Moskowitz spectra,average effective green water spectrum in space dimension indicates average wave energy in space dimension.Wave spectrum is stationary and ergodic.There is much difference in frequency dimension and space dimension about effective height of green water.Considering distribution of wave energy and probability in frequency dimension and space dimension,mean area method is suitable for calculating center line,upper control limit and lower control limit of effective green water spectrum.

a.Wave incident angle is 90°,the probability is 1/1 000.JONSWAP spectrum,mean area method applied,center line of effective green water spectrum is 4.15 m,and corresponding to the wave frequency is 0.635 rad/s,also spectrum σ is 3.193.As we know from statistical process control theory,effective height of green water upper control limit is 13.73 m,lower control limit is 0 m.It is out of control if wave frequency is less than 0.5 rad/s.It is a small probability event for effective height of green water occurrence in this range of wave frequency.Pierson-Moskowitz spectrum,mean area method applied,center line of effective green water spectrum is 4.15 m,and corresponding to the wind speed is 14.35 m/s.The effective height of green water is pretty evenly distributed in entire effective green water spectrum,and it is in control.Effective green water spectrum center lines of JONSWAP spectrum and Pierson-Moskowitz spectrum are approximately equal.As SPC control chart shown in Fig.23,most of effective heights of green water are between upper control limit and lower control limit,which are in control.Only effective heights of green water of small frequency wave are beyond upper control limit,which are out of control.It indicates that if the probability of spectrum is very little,the probability that the effective heights of green water of small frequency wave are out of control is small.SPC control chart can control the stochastic process of effective height of green water very well.

Fig.23 SPC chart with probability 1/1 000

Fig.24 SPC chart with probability 1/100

b.Wave incident angle is 90°,the probability is 1/100.JONSWAP spectrum,mean area method applied,center line of effective green water spectrum is 3.39 m,and corresponding to the wave frequency is 0.635 rad/s,also spectrum σ is 2.885.As we know from statistical process control theory,the effective height of green water upper control limit is 12.04 m,lower control limit is 0 m.It is out of control if wave frequency is less than 0.4 rad/s.It is a small probability event for effective height of green water occurrence in this range of wave frequency.Pierson-Moskowitz spectrum,mean area method applied,center line of effective green water spectrum is 3.39 m,and corresponding to the wind speed is 14.35 m/s.The effective height of green water is pretty evenly distributed in entire effective green water spectrum,and it is in control.Effective green water spectrum center lines of JONSWAP spectrum and Pierson-Moskowitz spectrum are approximately equal.As SPC control chart shown in Fig.24,most of effective heights of green water are between upper control limit and lower control limit,which are in control.Only effective heights of green water of small frequency wave are beyond upper control limit,which are out of control.Additionally,the level of out of upper control limit decreases.It indicates that if the probability of spectrum is very small,the probability that the effective heights of green water of small frequency wave are out of control is small.SPC control chart can control the stochastic process of effective height of green water very well.

c.Wave incident angle is 90°,the probability is 1/10.JONSWAP spectrum,mean area method applied,center line of effective green water spectrum is 2.40 m,and corresponding to the wave frequency is 0.635 rad/s,also spectrum σ is 2.426.As we know from statistical process control theory,the effective height of green water upper control limit is 9.67 m,lower control limit is 0 m.It is out of control if wave frequency is less than 0.2 rad/s.It is a small probability event for effective height of green water occurrence in this range of wave frequency.Pierson-Moskowitz spectrum,mean area method applied,center line of effective green water spectrum is 2.40 m,and corresponding to the wind speed is 14.35 m/s.The effective height of green water is pretty evenly distributed in entire effective green water spectrum,and it is in control.Effective green water spectrum center lines of JONSWAP spectrum and Pierson-Moskowitz spectrum are approximately equal.As SPC control chart shown in Fig.25,most of effective heights of green water are between upper control limit and lower control limit,which are in control.Only effective heights of green water of small frequency wave are beyond upper control limit,which are out of control.Additionally,the level of out of upper control limit decreases further.The effective heights of green water that are out of upper control limit are approximately equal to upper control limit.It indicates that if the probability of spectrum is small,the probability that the effective heights of green water of small frequency wave are out of control is small.SPC control chart can control the stochastic process of effective height of green water very well.

Fig.25 SPC chart with probability 1/10

Fig.26 SPC chart with probability 1/3

d.Wave incident angle is 90°,the probability is 1/3.JONSWAP spectrum,mean area method applied,center line of effective green water spectrum is 1.65 m,and corresponding to the wave frequency is 0.635 rad/s,also spectrum σ is 2.016.As we know from statistical process control theory,the effective height of green water upper control limit is 7.70 m,lower control limit is 0 m.All frequency waves are in control.It is a great probability event for effective height of green water occurrence in this range of wave frequency.So far as we know,with spectral probability increases,effective heights of green water center line decreases,the level of effective heights of green water out of control limit(upper control limit and lower control limit)decreases,until the condition that effective heights of green water are out of control is disappeared,where effective heights of green water of all frequency waves are in control.Pierson-Moskowitz spectrum,mean area method applied,center line of effective green water spectrum is 1.65 m,and corresponding to the wind speed is 14.35 m/s.The effective height of green water is pretty evenly distributed in entire effective green water spectrum,and it is in control.Effective green water spectrum center lines of JONSWAP spectrum and Pierson-Moskowitz spectrum are approximately equal.As SPC control chart shown in Fig.26,effective heights of green water of all frequency wave are between upper control limit and lower control limit,which are in control.It indicates that if the probability of spectrum is not big,effective heights of green water of all frequency wave are in control.SPC control chart can control the stochastic process of effective height of green water very well.

From cases a,b,c and d with different spectral probability,as probability increases,the level of effective heights of green water out of control limit(upper control limit and lower control limit)decreases.If the probability is 1/10,the condition that effective heights of green water are out of control is close to disappearing.If the probability is 1/3,the condition that effective heights of green water are out of control disappears completely.Effective heights of green water are all in control if spectral probability is no less than 1/3.Effective heights of green water are more controllable,and effective heights of green water center line are lower,and the probability of effective green water spectrum increases.

4.4 The effect of wave spectrum and wave incident angle on effective heights of green water

Effective green water spectrum center lines by mean area method about JONSWAP spectrum and Pierson-Moskowitz spectrum with 0°,45°and 90°wave incident angles are approximately equal.It indicates that the distribution of wave energy with different wave spectra is the same.Effective heights of green water center lines do not change with wave spectra form.Averaging effective height of green water in time dimension,space dimension and frequency dimension is approximately equal to the same value,what indicates that the distribution of wave energy in time dimension,space dimension and frequency dimension is uniform.

Tab.1 Information of green water with JONSWAP spectrum and Pierson-Moskowitz spectrum

Drawing curves about center lines of effective green water spectrum with different probability,analyzes the effect of wave incident angle and probability on effective heights of green water.As shown in Fig.27,there is little effect of wave incident angle on effective heights of green water.Effective heights of green water are identical with 0°and 90°wave incident angles.It indicates that effective heights of green water are the same with four orthogonal symmetric direction wave,because TLP is symmetrical.Conversely,there is a sizeable effect of probability on effective heights of green water.As probability increases,the effective height of green water decreases,and decreased slope decreases as well.As shown in Fig.28,there is much effect of probability on standard variance of effective green water spectrum.As probability increases,standard variance decreases.There is almost no influence of wave incident angle on standard variance of effective green water spectrum.

Fig.27 Influence of wave incident angle and probability on wave height of the green water

Fig.28 Influence of wave incident angle and probability on standard variance

Applied mean area method,average effective height of green water is 2.34 m,and corresponding to the probability is 1/8.5.In order to make more effective heights in control,let upper control limit is maximum of results,σ=3.209.The range of effective heights of green water which is in control is 0～11.967 m,and corresponding to the probability is 99.73%.Lower control limit is 0 m,upper control limit is 11.967 m,corresponding to the probability is 99.73%,actual probability of exceeding upper control limit is 1/8.5×(1-99.73%)=0.03%.

5 Conclusions

In this paper,a calculation model is established to research on effective height of green water loading of TLP based on statistical process control theory.This paper analyzes relative motion and effective height of green water with wave spectra as parameter for TLP,and calculates center lines,upper control limits and exceeding probability about effective height of green water.For the reason that the wave height is a random process whose uncertainty leading that there is no certain solution about the height of green water,presents mean area method to calculate center line of effective height of green water,which fit well with wave height distribution form under the condition of wave parameters.Using mean area method to calculate center lines and upper control limits of effective height of green water,we find that wave energy distribution between different wave spectra are the same,because the center lines of effective height of green water of different wave spectra with the same wave parameters are the same.It is evident that the wave energy distribution is accordant in time dimension,spatial dimension and frequency dimension.Based on a TLP that is in design stage for the case,for the condition that occurrence exceeding a certain probability in wave spectra,center line,lower control limit and upper control limit of effective height of green water are obtained to estimate whether effective height of green water is in control.We find that it is much easier to arrive at in control state for the condition that effective height of green water with great exceeding probability in wave spectrum.A rule is put forward that the influence of wave incident angle on effective height of green water is little by compared with the center lines of effective height of green water with different wave incident angle.Based on SPC,we conclude that:

(1)Statistical process control and SPC control chart control the stochastic process of effective height of green water very well,if probability of effective green water spectrum is not very small.

(2)Average effective heights of green water in time dimension,space dimension and frequency dimension are approximately equal,what indicates that the distribution of wave energy in time dimension,space dimension and frequency dimension is uniform.

(3)Mean area method is a good method to calculate center line and control limits of effective height of green water,which fit well with wave height distribution form under wave parameters.

(4)Average effective height of green water is 2.34 m.The range of effective heights of green water which is in control is 0～11.967 m,and corresponding to the probability is 99.73%.The rest 0.27%probability effective heights of green water may be in the range of 0～11.967 m,or may be out of the range.It needs additional structural reinforcement and corrosion protection treatment to against the waves wash over between lower control limit 0 m and upper control limit 11.967 m,because wave reciprocate frequently in this range with a great probability.TLP is much easier suffered from fatigue and corrosion damage with alternating load of wave and atmosphere between lower control limit and upper control limit.

(5)Distance from main deck to sea level is 31.85 m.Effective heights of green water is much less than the distance.Therefore,there is little effect of green water on deck machinery and operation.In practice,wave in ocean is not as the same as wave spectra,so actual heights of green water do not equal to the results as calculated absolutely.Effective heights of green water may be in the range with a great probability,and may be out of the range with small probability.It also may be greater than 31.85 m,even if it is a small probability event,in which people should adopt measures to protect from damage.On the whole,effective heights of green water are much less than the distance from main deck to sea level,and have little effect on deck machinery and operation.

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