E J Obrien

X.-Y Zhou

C C Caprani

S Wilson

E Sheils

A Review of Probabilistic Methods of Assessment of Load Effects in Bridges

Keywords: Review, Bridge, Load, Traffic, Assessment, POT, Peaks-Over-Threshold, Extreme Value, GEV, Box-Cox, Rice, Predictive Likelihood

This paper reviews a range of methods used to infer characteristic values and probabilities of failure from a sample of measured or simulated load effect data. The popular methods of Peaks-Over-Threshold and Generalized Extreme Value (GEV) are considered but also other methods including the Box-Cox approach, fitting to a Normal distribution and the Rice formula. These five methods are fitted to the tail of the daily maximum data. Bayesian Updating and Predictive Likelihood are also considered, but are fitted to the entire data set. In general, the five tail fitting methods are reasonably accurate at inferring characteristic annual maximum values from 1000 days of data and the other methods less so. All methods are considerably less accurate at inferring probabilities of failure than characteristic values.

Introduction

A necessary part of bridge management is the assessment of the safety of bridge structures. In its simplest form, a bridge is safe when its capacity to resist load exceeds the load applied. More precisely, a bridge can be considered safe when there is an acceptably low probability that load exceeds capacity. A great deal of work has been carried out on methods of evaluating the loadcarrying capacity of bridges and the associated uncertainties. Load-carrying capacity can be reduced by different forms of deterioration, depending on factors such as the structural material, the quality of workmanship during construction, the age of the structure, the environment and the loading history. To carry out a more accurate assessment of the load-carrying capacity, non-destructive and/or destructive tests can be carried out to get more detailed site specific information on these deterioration mechanisms to reduce uncertainty and associated conservatism [START_REF] Al-Harthy | Concrete cover cracking caused by steel reinforcement corrosion[END_REF][START_REF] Frangopol | Maintenance and management of civil infrastructure based on condition, safety, optimization, and life-cycle cost *[END_REF][START_REF] Richard | A methodology for robust updating of nonlinear structural models[END_REF][START_REF] Rücker | Guideline for the assessment of existing structures[END_REF][START_REF] Suo | Corrosion cracking prediction updating of deteriorating rc structures using inspection information[END_REF]. These inspection results can be incorporated into time-dependent reliability-based assessments to give up-to-date structurespecific deterioration rates. These in turn can be used to accurately predict the capacity of the structure and to schedule maintenance and repairs [START_REF] Melchers | Structural reliability: Analysis and prediction[END_REF][START_REF] Orcesi | Optimization of management strategies applied to the national reinforced concrete bridge stock in france[END_REF][START_REF] Orcesi | A bridge network maintenance framework for pareto optimization of stakeholders/users costs[END_REF][START_REF] Sheils | Investigation of the effect of the quality of inspection techniques on the optimal inspection interval for structures[END_REF].

Traffic loading on bridges, perhaps the greatest source of uncertainty, has received less attention and is the focus of this paper. Historical developments in the field of traffic loading are reviewed. A wide range of statistical/probabilistic approaches have been applied to the problem with no clear 'winner' emerging.

Two Extreme Value examples are used here as benchmark tests, against which a range of approaches are compared. The first example is the problem of finding the maximum of numerous normally distributed random variables, a problem for which the exact theoretical solution is known. The many methods of estimating characteristic maxima are shown to give large errors, to varying degrees.

The second example is based on a carefully calibrated traffic load simulation model. The simulation is run for 5 000 years so that, while the exact solution is unknown, it can be estimated very well and there is a high degree of confidence in the lifetime maximum results. As for the first example, several methods of prediction, using modest quantities of data, are tested. Again, the errors in the predictions are found to be significant and to vary considerably, depending on the approach adopted.

Review of Literature

Load effects (LE's)bending moments, shear forces, etc.result from traffic passing over a bridge. The process varies in time with many periods of zero LE when there is no traffic on the bridge and peaks corresponding to heavy vehicle crossings or more complex vehicle meeting or overtaking scenarios. The majority of the local peaks in LE are due to cars which are relatively light and there have been many efforts to simplify the problem by excluding consideration of these data. The methods of statistical inference used in the literature to predict the extremes of traffic LE's are quite diverse.

Tail Fitting

In the context of this problem, many approaches fit a distribution to the tail of the Cumulative Distribution Function (CDF). This can be justified by the fact that the distribution is often made up of a mixture of load effect types -for example, LE's due to 2-axle trucks and those due to heavy lowloader vehicles. For bridge traffic loading, the heavier vehicles tend to dominate, with the lighter ones making very little contribution to the probability of exceedance at the extremes. The tail can be chosen by engineering judgement when the cumulative distribution is seen to change at a particular probability level. Alternatively, some authors have fitted to the top 2n of a distribution of n data, based on theoretical considerations [START_REF] Castillo | Extreme value theory in engineering (statistical modeling and decision science)[END_REF]. Others have fitted to the top 30% of data [START_REF] Enright | Simulation of traffic loading on highway bridges[END_REF] based on sensitivity analyses.

Two of the tail fitting approaches are particularly popular -Peaks-Over-Threshold (POT) and Block Maximum. POT considers the extent by which the peaks of LE exceed a specified threshold. The POT LE's are fitted to a probability distribution such as the Generalized Pareto distribution. In the Block Maximum approach, only the maximum LE's in given blocks of time (days, years, etc.), are considered. This has the advantage of time referencing the data which is necessary when calculating lifetime maximum probabilities of exceedance. Block maximum LE's can be fitted to one of a range of distribution types such as Generalised Extreme Value (GEV) (incorporating Gumbel, Weibull and Fréchet), or Normal. Fitting block maximum values to GEV and Normal distributions will be considered here.

The Block Maximum approach has the disadvantage that only one LE in each block of time is considered, even if several very large LE's are recorded. The POT approach addresses this issue but the selection of the threshold, below which LE's are discarded, is subjective. The Box-Cox approach is more general and aims to address the disadvantages of both POT and GEV. The Rice formula is also investigated as it was used for the extrapolations in the background study supporting the development of the Eurocode for traffic loading on bridges. However, while the Rice formula is a fitting to tail data, it is applied to a histogram of 'upcrossings' past a threshold, not to a CDF, and assumes a normally distributed process.

Full Distribution Fitting

Bayesian Updating is another approach that can be applied to bridge traffic loading. A probability distribution is assumed for the block maximum LE's and is updated using available LE data. While only tail data could be used, in this work, the Bayesian approach is used to update the entire distribution, not just the tail. Predictive Likelihood also seeks to develop a probability distribution for all LE's but uses a frequentist likelihood approach, assigning likelihoods on the basis of the quality of the fit to the measured data.

Peaks Over Threshold (POT)

Block Maximum approaches use only the maximum LE in each block of time. There is therefore a risk that some important data is discarded: if two unrelated extreme loading events occur in the same block of time, only one of the resulting LE's is retained. In such a case, the POT approach would retain both LE's as valid data.

To find characteristic maximum values of LE, data above the threshold must be fitted to a probability distribution. [START_REF] Coles | An introduction to statistical modeling of extreme values[END_REF] provides a brief outline proof that the Generalized Pareto (GP) distribution approximates the CDF of such POT data well. [START_REF] Crespo-Minguillón | A comprehensive traffic load model for bridge safety checking[END_REF] use the GP distribution to model the excesses of weekly maximum traffic LE's over a threshold. [START_REF] James | Analysis of traffic load effects on railway bridges[END_REF] applies the POT method to analyse load effects on railway bridges. [START_REF] Gindy | Comparison of traffic load models based on simulation and measured data[END_REF] analyse load effects caused by combined data from over 33 Weigh-in-Motion sites over an 11-year measurement period, and compare extreme values as predicted by both GP and GEV distributions.

A significant drawback of the POT approach is the issue of selecting the threshold. There are many different kinds of loading scenario on a typical bridge. For example, there are usually many singlevehicle crossings of standard 5-axle trucks. The probability distribution of LE's due to such an event type may be quite different from that due to large cranes or that due to 2-truck meeting events [START_REF] Caprani | The structural reliability of bridges subject to time-dependent deterioration[END_REF]. If the threshold is too low, there may be an excessive mixing of extreme event types with other less critical types which can result in convergence to an incorrect characteristic LE. On the other hand, if the threshold is too high, there will be too few peaks above the threshold, leading to high variance and unreliable results.

The basic principle in selecting a threshold is to adopt as low a threshold as possible, while maintaining a consistent trend. Two methods are available [START_REF] Coles | An introduction to statistical modeling of extreme values[END_REF]: one is an exploratory technique carried out prior to model estimation; the other is an assessment of the stability of parameter estimates, based on the fitting of models across a range of different thresholds. [START_REF] Crespo-Minguillón | A comprehensive traffic load model for bridge safety checking[END_REF] apply the latter method and select the optimal threshold based on the overall minimum least-squares value.

Having selected the threshold, the next step is to estimate the parameters of the GP (or other) distribution. [START_REF] Bermudez | Parameter estimation of the generalized pareto distribution-part ii[END_REF] consider several methods of estimating these parameters including the method of moments, the probability weighted method, the maximum likelihood method, and Bayesian updating. [START_REF] Crespo-Minguillón | A comprehensive traffic load model for bridge safety checking[END_REF] adopt a methodology that is based on the minimization of the weighted sum of squared errors. [START_REF] James | Analysis of traffic load effects on railway bridges[END_REF] and [START_REF] Gindy | Comparison of traffic load models based on simulation and measured data[END_REF] use maximum likelihood estimation.

Block Maximum -Extreme Value Distributions

Extreme value theory is based around the extreme value theorem, proved by [START_REF] Gnedenko | Sur la distribution limite du terme maximum d'une serie aleatoire[END_REF] and based on initial work by [START_REF] Fisher | Limiting forms of the frequency distribution of the largest or smallest member of a sample[END_REF] and [START_REF] Gumbel | Les valeurs extrêmes des distributions statistiques[END_REF]. For a sequence of independent random variables X 1 , X 2 ,..., with distribution function F(x) = Prob(X ≤ x), the distribution of max(X 1 ,...,X n ) is F(x) n . As n gets large, this degenerates to 0 if F(x) < 1, as is usual. The Fisher-Tippett theorem shows that a non-degenerate distribution can be found using a linear function of x, say a n + b n x. Then, there is a non-trivial limit to F(a n + b n x) and this limit must be in the form of the Generalised Extreme Value distribution (GEV), also known as the Fisher-Tippett distribution [START_REF] Jenkinson | The frequency distribution of the annual maximum (or minimum) values of meteorological elements[END_REF][START_REF] Mises | La distribution de la plus grande de n valeurs[END_REF]: Equation 1defined in terms of parameters , and where is the location parameter, the scale parameter and the shape parameter, such that

  10 x       .
Hence, for an appropriately large n, the exact distribution, F(x) n , converges asymptotically to F GEV (x). For the Normal distribution, the theorem holds and it is well known that its limiting distribution is the Gumbel, the case of the GEV. However, convergence is slow [START_REF] Cramér | Mathematical methods of statistics[END_REF].

Each block maximum LE is the maximum of many traffic loading scenarios. As convergence may be slow, [START_REF] Caprani | Probabilistic analysis of highway bridge trafiic loading[END_REF] and OBrien et al. ( 2010) have fitted block maximum LE data with a 'Normal to the power of n', i.e., a Normal distribution raised to some power, n, whose value is found by fitting to the data. This has merit for smaller data samples. [START_REF] Ghosn | Design of highway bridges for extreme events Transportation Research Board National Research[END_REF] determine the distribution of lifetime maximum LE by raising the parent distribution of LE to an appropriate power. In this way they determine the mean and coefficient of variation of the maximum LE. [START_REF] Caprani | Probabilistic analysis of highway bridge trafiic loading[END_REF] describes a probabilistic convolution method to obtain bending moments for single truck loading events and obtains the distribution of lifetime maximum LE by raising the parent distribution to an appropriate power. Other authors attempt to calculate the exact distribution of extreme load effect, based on a fit to the parent distribution [START_REF] Bailey | Basic principles and load models for the structural safety evaluation of existing road bridges[END_REF][START_REF] Bailey | A parametric study of traffic load effects in medium span bridges[END_REF][START_REF] Cooper | The determination of highway bridge design loading in the united kingdom from traffic measurements[END_REF][START_REF] Getachew | Simplified site specific models for determination of characteristic traffic load effects for bridges[END_REF][START_REF] Ghosn | Markov renewal model for maximum bridge loading[END_REF][START_REF] Nowak | Bridge live-load models[END_REF], Nowak et al., 1993). This is done by raising the initial distribution to an appropriate power.

Most researchers fit block maximum LE data to one of the extreme value distributions described by the GEV equation: Gumbel, Fréchet or Weibull (also known as Types I, II or III). The three types of distribution have distinct forms of behaviour, corresponding to the different forms of the tail in the original distribution function [START_REF] Fisher | Limiting forms of the frequency distribution of the largest or smallest member of a sample[END_REF]. [START_REF] Coles | An introduction to statistical modeling of extreme values[END_REF] establishes the conditions under which the Gumbel, Fréchet and Weibull distributions are the limiting forms for various parent distributions [START_REF] Gumbel | Statistics of extremes[END_REF].

In early applications of Extreme Value theory, it was usual to adopt one of the three distributions, and then to estimate the relevant parameters. There are two weaknesses with this: first, a technique is required to choose which of the three distributions is most appropriate for the data at hand; second, once such a decision is made, subsequent inferences presume this choice to be correct, and do not allow for the uncertainty such a selection involves, even though this uncertainty may be substantial [START_REF] Coles | An introduction to statistical modeling of extreme values[END_REF]. Nevertheless, many studies [START_REF] Caprani | Statistical computation for extreme bridge traffic load effects[END_REF][START_REF] Caprani | Characteristic traffic load effects from a mixture of loading events on short to medium span bridges[END_REF][START_REF] Kanda | Formulation of load factors based on optimum reliability[END_REF][START_REF] O'connor | Traffic load modelling and factors influencing the accuracy of predicted extremes[END_REF] indicate that LE data is either Weibull or Gumbel and, given that Gumbel is a special case of Weibull (with shape parameter,  = 0), an assumption that LE is always of the form of Equation 1, with   0, seems reasonable. (2001). [START_REF] González | An assessment of the influence of dynamic interaction modelling on predicted characteristics load effects in bridges[END_REF] also use the Gumbel and Weibull distributions to extrapolate bridge load effect. [START_REF] Getachew | Simplified site specific models for determination of characteristic traffic load effects for bridges[END_REF] fits the Generalized Extreme Value distribution to the LE's from simulated 2-truck meeting events representing two weeks of traffic. [START_REF] Bailey | Basic principles and load models for the structural safety evaluation of existing road bridges[END_REF] describes the use of plots of the mean and standard deviation of load effects, to estimate the appropriate extreme value distribution. [START_REF] Bailey | Basic principles and load models for the structural safety evaluation of existing road bridges[END_REF], [START_REF] Bailey | A parametric study of traffic load effects in medium span bridges[END_REF] and [START_REF] Bailey | Site specific probability distribution of extreme traffic action effects[END_REF] describe a qualitative analysis of 500 simulated upper tails of mean maximum load effects plotted against the number of events that contribute. They determine that the Weibull distribution is most appropriate to model these tails and use maximum likelihood estimation. [START_REF] Cooper | Development of short span bridge-specific assessment live loading[END_REF] presents a traffic model of about 81 000 measured truck events, and uses it to determine the distribution of LE's due to a 'single event'. He raises this distribution to powers to determine the distribution of LE for 1, 4, 16, 256 and 1024 such events. A Gumbel distribution is then fitted to this 1024-event distribution and used to extrapolate to a 2400-year return period. [START_REF] Cooper | Development of short span bridge-specific assessment live loading[END_REF] converts histograms of two-week traffic LE's into CDF's, which he then raises to a power equal to the number of trucks per day, to give the distribution of daily block maxima. [START_REF] Moyo | Highway bridge live loading assessment and load carrying estimation using a health monitoring system[END_REF] plot daily maximum strain values on Gumbel probability paper and use a leastsquares fit to determine the parameters of the distribution. [START_REF] Buckland | Proposed vehicle loading of long-span bridges[END_REF] use a Gumbel distribution to fit the 3-monthly maximum LE's and extrapolate to find characteristic values. [START_REF] Getachew | Simplified site specific models for determination of characteristic traffic load effects for bridges[END_REF] uses the GEV distribution to model the parent distribution of load effect, but not as an asymptotic approximation to the distribution of extreme values. [START_REF] Sivakumar | Protocols for collecting and using traffic data in bridge design[END_REF] adopt the Gumbel distribution to project the statistics of the maximum LE's for different return periods.

Box-Cox Approach

Researchers commonly debate the merits of the POT method relative to the Block Maximum approach. The Box-Cox transform [START_REF] Box | An analysis of transformations[END_REF] is used by [START_REF] Bali | The generalized extreme value distribution[END_REF] to introduce a more general extreme value distribution that encompasses the Generalised Pareto and Generalised Extreme Value distributions [START_REF] Caprani | Estimating extreme highway bridge traffic load effects[END_REF][START_REF] Rocco | Extreme value theory for finance: A survey[END_REF]. This transformation offers the possibility of improving the rate of convergence to the limiting extreme value form, since different distributions converge at different rates. This approach restricts the methodology to cases where the extreme data are strictly positive [START_REF] Wadsworth | Accounting for choice of measurement scale in extreme value modeling[END_REF] but still encompasses a wide variety of practical problems including traffic loading on bridges. The use of the Box-Cox transformation in extreme value analysis was considered before in an entirely different context by [START_REF] Eastoe | Modelling non stationary extremes with application to surface level ozone[END_REF].

The Box-Cox-GEV extreme value distributions are given by [START_REF] Bali | The generalized extreme value distribution[END_REF] as: Equation 2in which Equation 3The parameters of this distribution are those of the GEV (μ, σ, ξ) plus a 'model parameter', λ. As λ →1, Box-Cox converges to the GEV distribution. Conversely, as λ →0, by L'Hôpital's Rule, it converges to the GP distribution. To apply this model, a high threshold is set on the parent distribution [START_REF] Caprani | Estimating extreme highway bridge traffic load effects[END_REF][START_REF] Rocco | Extreme value theory for finance: A survey[END_REF]. [START_REF] Bali | The generalized extreme value distribution[END_REF] uses a threshold of two standard deviations about the sample mean. [START_REF] Caprani | Estimating extreme highway bridge traffic load effects[END_REF] thresholds are taken in steps of 0.5 standard deviations in the range from -2.5 to +2.5 standard deviations about the sample mean. [START_REF] Tötterman | Applying extreme value theory and tail risk measures to reduce portfolio losses[END_REF] suggests that the additional parameter should increase the accuracy for Box-Cox, compared with GEV and GP. [START_REF] Bali | Risk measurement performance of alternative distribution functions[END_REF] evaluate the performance of three extreme value distributions including the GP, GEV and Box-Cox. The empirical results show that the asymptotic distribution of the maximal and minimal returns fits the Box-Cox-GEV distribution in this case. A likelihood ratio test between the GEV and Box-Cox results in a rejection of the former [START_REF] Bali | Risk measurement performance of alternative distribution functions[END_REF][START_REF] Caprani | Estimating extreme highway bridge traffic load effects[END_REF].

Block Maximum -Normal Distribution

Block maximum data is often fitted with extreme value distributions as each data point represents the maximum of a number of parent values. However, block maximum data is also sometimes fitted to a Normal distribution. [START_REF] Nowak | Calibration of lrfd bridge design code[END_REF] uses a form of Normal (Gaussian) probability paper, i.e., he fits the block maximum data to a Normal distribution and extrapolates to find the characteristic maximum. In an earlier study, [START_REF] Nowak | Live load model for highway bridges[END_REF] uses 2.4 hours as the block size and fits the maximumper-block data to a Normal distribution. This distribution is then raised to an appropriate power to obtain the 75-year maximum LE distribution.

To calibrate the traffic load model for the AASHTO load and resistance factor design (LRFD) approach, Nowak and others use Normal probability paper to extrapolate the maximum LE's for time periods from 1 day to 75 years, based on a set of 9250 heavy vehicles representing about two weeks of heavy traffic measured on a highway in Ontario [START_REF] Kulicki | Updating the calibration report for aashto lrfd code[END_REF][START_REF] Moses | Calibration of load factors for lrfr bridge evaluation[END_REF][START_REF] Nowak | Load model for bridge design code[END_REF][START_REF] Nowak | Reply: Load model for bridge design code[END_REF][START_REF] Nowak | Calibration of lrfd bridge design code[END_REF][START_REF] Nowak | Bridge live-load models[END_REF], Nowak et al., 1993[START_REF] Sivakumar | Protocols for collecting and using traffic data in bridge design[END_REF]. The expected values of the lifetime maximum LE's are found by fitting a straight line to the tails of the data on Normal probability paper. [START_REF] Kulicki | Updating the calibration report for aashto lrfd code[END_REF] identify the fact that block maximum LE's due to measured trucks are not Normal but fits tail data to the Normal distribution. In the background studies for Eurocode 1, [START_REF] Flint | Extreme traffic loads on road bridges and target values of their effects for code calibration[END_REF] fit half-normal curves to the ends of the histograms of LE. They adopt a least-squares best fit method to estimate the distribution parameters. Multimodal (bimodal or trimodal) Gumbel and Normal distributions are also used.

Rice Formula

The Rice formula, introduced by Rice (1945) and described more recently by [START_REF] Leadbetter | Extremes and related properties of random sequences and processes[END_REF], can be used to find a parametric fit to statistical data. [START_REF] Ditlevsen | Traffic loads on large bridges modeled as white-noise fields[END_REF] suggests that a load effect created by the traffic on a long span bridge can be modelled as a Gaussian random process. Under that hypothesis, the mean rate of up-crossings for a threshold level, during a reference period , can be expressed by the Rice formula: Equation 4where, x is the threshold value of LE, is the mean value, is the standard deviation and is the first derivative of with respect to time.

The CDF can be found from the definition of return period which is the mean period between two occurrences, or the value with an expectation of being crossed one time during the return period, R [START_REF] Cremona | Optimal extrapolation of traffic load effects[END_REF]: Equation 5where, v 0 is /2π . [START_REF] Cremona | Optimal extrapolation of traffic load effects[END_REF] suggests the Kolmogorov test [START_REF] Degroot | Probability and statistics[END_REF] to select the optimal number of class intervals and starting points.

Fitting Distributions to Extreme Data & Bayesian Inference

The concept of Bayesian Updating stems from Bayes' Theorem and is a major pillar of modern statistics. Bayesian Updating involves the adoption of an initial (prior) probability distribution, perhaps based on past experience, and updating it on the basis of measured data to give a posterior distribution [START_REF] Basu | Estimates of reliability for some distributions useful in life testing[END_REF][START_REF] Bhattacharya | Bayesian approach to life testing and reliability estimation[END_REF][START_REF] Holla | Bayesian estimates of the reliability function[END_REF]. [START_REF] Sinha | Bayes estimation of the parameters and reliability function of the 3-parameter weibull distribution[END_REF] use Bayesian Inference to find the full 3-parameter Weibull distribution from measured data. They propose the use of Bayes Linear Estimate to approximate the posterior expectations and formulate the corresponding calculations for the Weibull parameters. [START_REF] Smith | A comparison of maximum likelihood and bayesian estimators for the threeparameter weibull distribution[END_REF] work with the 3-parameter Weibull distribution, comparing Maximum Likelihood with Bayesian estimators, using specially adapted versions of numerical quadrature to perform the posterior calculations. Although the priors they work with are arbitrary, they are chosen to reflect a range of potential scientific hypotheses. They report that the Bayesian inferential framework as a whole proves more satisfactory for their data analysis than the corresponding likelihood-based analysis. The issue of prior elicitation is pursued by [START_REF] Singpurewalla | Reliability analysis using weibull lifetime data and expert opinion. Reliability[END_REF], who restrict attention to the 2-parameter Weibull model. The predictive density function [START_REF] Aitchison | Statistical prediction analysis[END_REF]) is defined as: Equation 6where x represents historical data, y a future observation,  the vector of parameters describing the distribution, the likelihood and the posterior distribution of given x. Thus, the predictive distribution averages the distribution across the uncertainty in as measured by the posterior distribution. [START_REF] Lingappaiah | Bayesian prediction regions for the extreme order statistics[END_REF] develops bounds for the predictive probabilities of extreme order statistics under a sequential sampling scheme, when sampling is carried out from either an exponential or Pareto population. From a practical viewpoint, the most important issues arising from the Bayesian literature are the elicitation and formulation of genuine prior information in extreme value problems, and the consequent impact such a specification has on subsequent inferences. [START_REF] Coles | Modelling extremes: A bayesian approach[END_REF] consider a case study in which expert knowledge is sought and formulated into prior information as the basis for Bayesian analysis of extreme rainfall.

Predictive Likelihood

The relatively new theory of frequentist Predictive Likelihood can be used to estimate the variability of the predicted value, or predictand. [START_REF] Fisher | Statistical methods and scientific inference[END_REF] is the first clear reference to the use of likelihood as a basis for prediction in a frequentist setting. A value of the predictand (z) is postulated and the maximized joint likelihood of the observed data (y) and the predictand is determined, based on a probability distribution with given parameters. The graph of the likelihoods thus obtained for a range of values of the predictand, yields a predictive distribution. Such a predictive likelihood is known as the profile predictive likelihood. Denoting a normed likelihood by this is given by: Equation 7This formulation states that the likelihood of the predictand, z, given the data, y, is proportional to the likelihood of both the data (L y ) and the predictand (L z ) for a maximized parameter vector, [START_REF] Caprani | The use of predictive likelihood to estimate the distribution of extreme bridge traffic load effect[END_REF]. [START_REF] Mathiasen | Prediction functions[END_REF] appears to be the first to study Fisher's Predictive Likelihood and notes some of its problems. Foremost in this work is the problem that it does not take into account the parameter variability for each of the maximizations of the joint likelihood function required [START_REF] Bjornstad | Predictive likelihood: A review[END_REF][START_REF] Lindsey | Parametric statistical inference[END_REF]. [START_REF] Lejeune | A simple predictive density function[END_REF] propose a similar predictive likelihood, but include a normalizing function.

Predictive Likelihood is a general concept and in the literature many versions have been proposed.

Cooley and Parke have a number of papers dealing with the prediction issue [START_REF] Cooley | Likelihood and other approaches to prediction in dynamic models[END_REF][START_REF] Cooley | Asymptotic likelihood-based prediction functions[END_REF][START_REF] Cooley | Predictive efficiency for simple non-linear models[END_REF]. However, their method relies on the assumption that the parameters are normally distributed. [START_REF] Leonard | Comment on "a simple predictive density function[END_REF] suggests a similar approach while [START_REF] Davison | Bootstrap methods and their application[END_REF] use a different form of Predictive Likelihood.

Caprani & OBrien (2010) use the Predictive Likelihood method proposed by [START_REF] Butler | Predictive likelihood inference with applications[END_REF], based on that of [START_REF] Fisher | Statistical methods and scientific inference[END_REF] and [START_REF] Mathiasen | Prediction functions[END_REF] and also considered by [START_REF] Bjornstad | Predictive likelihood: A review[END_REF]. [START_REF] Lindsey | Parametric statistical inference[END_REF] describes the reasoning behind its development. This Predictive Likelihood is the Fisherian approach, modified so that the variability of the parameter vector resulting from each maximisation is taken into account.

Simple Extreme Value Problem

To assess the safety of a bridge, a limited quantity of data is generally used to infer a probability of failure, a characteristic maximum or a statistical distribution of maximum load effects. Probability of failure is clearly the most definitive measure of bridge safety. However, it is strongly influenced by resistance which varies greatly from one example to the next. In order to retain the focus on load effect, the resistance distribution is here assumed to be a mirrored version of the exact LE distribution, shifted sufficiently to the right to give an annual probability of failure of 10 -6see Figure 1. Three thousand values of Z are considered in a given block, say per day, with maximum:

  max 1,2, ,3000 i X Z i  Equation 9
Typically, a finite number of days of data is available and extreme value distributions are inferred from a dataset of daily maximum values. Hence, a finite number of daily maxima (X values) may be used to infer, for example, annual maximum distributions. In all cases, the days are considered to be working days and a year is taken to consist of 250 such days.

The exact solution to this problem is readily calculated. The annual maximum can be expressed as: Equation 10where n is the number of values in a year, equal to (250×3000 =) 750 000.

Methods of Inference

Three alternative quantities of daily maximum data are considered: 200, 500 and 1000 working days. A wide range of statistical extrapolation methods are tested in each case to estimate the distribution for annual maximum LE:

 Peaks Over Threshold (POT) data, fitted to the Generalized Pareto distribution;  Generalized Extreme Value (GEV) fit to tail of daily maximum data;  Box-Cox fit to tail of daily maximum data;  Normal distribution fit to tail of daily maximum data;  Fit of Upcrossing frequency data tail to Rice formula;  Bayesian fit to all daily maximum data;  Predictive Likelihood (PL) fit to all daily maximum data.

In each case, the probability distribution of LE is inferred and the theorem of total probability is used with the exact resistance distribution to determine the probability of failure (defined as LE exceeding resistance).

Figure 2 uses Gumbel probability paper to illustrate the first four methods of tail fitting to the CDF's: POT, GEV, Box-Cox and Normal. For all four cases, a least squares fit is found for the top 30% of values from 1000 daily maximum LE's. The exact distribution is shown for comparison. All distributions give good fits, with the Normal being more 'bounded' than the others in this example, i.e., tending more towards an asymptote at extremely low probabilities. The Rice formula fit is illustrated in Figure 3 which gives the histogram of upcrossings above each threshold, for the same 1000 daily maxima. While [START_REF] Cremona | Optimal extrapolation of traffic load effects[END_REF] has considered a variable quantity of data, the top 30% is used here to provide a direct comparison with the other tail fitting methods. Bayesian Updating is the sixth method considered. In this case, unlike the tail fitting methods, all 1000 daily maximum LE's are used. The method is therefore a Bayesian approach applied to block maximum data. The data is assumed to be GEV except that, in this case, a family of GEV distributions is considered. The GEV parameter values are initially assumed to be equally probable within specified ranges (uniform prior distributions). The daily maximum data is then used to update their probabilities.

The final method applied to this problem, Predictive Likelihood, is also based on the entire dataset of 1000 block maximum values and an assumed GEV distribution. The method is based on the concept of calculating the joint likelihood of a range of possible values at a given level of probability (predictands), given the value of that predictand and the available daily maxima. For example, Figure 4 shows the joint fit to Point A, given the daily maximum data and the joint fit to Point B, given that same set of daily maxima. The likelihood of actually observing Point A is less than that of Point B, given the measurements available. In this way, the joint likelihoods of a wide range of possible predictands are calculated and used to infer a probability distribution for a given time period, such as a year.

Inference of Annual Maximum Results from Daily Maximum Data

For the first four tail fitting methods -POT, GEV, Box-Cox and Normalthe parameters of the daily maximum distributions are inferred from the best fits to the top 30% of the daily maximum data, i.e., the block size is one day. Allowing for public holidays and weekends, 250 days are assumed per year. The annual maximum distribution can then be found by raising the CDF for daily maximum to the power of 250.

The Rice formula approach is also a tail fitting method but, in this case, the CDF for annual maximum is found directly from Equation 5. Bayesian Updating and Predictive Likelihood both infer the annual maximum distribution directly as described above.

Figure 5 illustrates the annual maximum CDF's inferred from all 7 approaches, together with the corresponding exact distribution. For this example, most of the tail fitting methods and Predictive Likelihood are more bounded than the exact solution, while Bayesian Updating is less so. The horizontal line corresponds to a return period of 75 years and it can be seen that all methods except Bayesian Updating, Box-Cox and GEV are slightly non-conservative. Characteristic values are calculated for a 75-year return period. The process is repeated for three different quantities of daily maximum data: 200, 500 and 1000 days. For each of the three quantities, the characteristic values are calculated 20 times so that a measure of the variability in the results can be found. Figure 6 shows the mean of the 20 runs in each case, ± one standard deviation. For 1000 days of data, the results are moderately accurate in most cases, generally falling in the 66 to 71 range. For POT, GEV and Box-Cox, the exact value falls within the error bars and the mean error is less than 1 from that value. Errors in individual results are less good, being as high as 6.1 in one case for GEV. There is no significant difference between these three methods. The Rice formula is relatively good. Results from tail fitting to a Normal distribution do not include the exact value in the error bars. However, the mean error is only 0.94 from the exact value and all the results are reasonably close. Predictive Likelihood is goodthe mean is very close to the exact value and the error bars are small. For Bayesian Updating, the error bars are very smallresults are highly repeatablebut it is consistently a little conservative for these 20 examples.

Not surprisingly, results are considerably less accurate when fewer days of data are available for inference. With 500 days of data, Normal includes the exact result within its error bars. Bayesian Updating looks better than before with the error bars coming close to the exact solution for both 500 and 200 days of data. Rice is again better than POT, GEV and Box-Cox with a mean very close to the exact and reasonably small error bars. For 200 days of data, PL looks less good than before, with the error bars becoming greater than Normal and Rice.

In order to compare inferred probabilities of failure, the exact annual maximum probability density function is mirrored to give a resistance distribution that implies a failure probability of 10 -6 . This resistance distribution is then used with each of the inferred distributions to determine the apparent probability of LE exceeding resistance. The calculated probabilities are illustrated in Figure 7. Even when plotted on a Normal distribution scale, the probabilities for this example are quite inaccurate. This exercise is analogous to an extrapolation from 200 -1000 days of data to 1 million years (i.e., annual probability of failure of 10 -6 ). While the variability in the results is hardly surprising, it has significant implications for any Reliability Theory calculation.

As before, for inference using POT, GEV and Box-Cox, the exact value falls within the error bars. On a scale, the mean error from 1000 days of data is less than about 0.5 from the exact value. Errors in individual results are considerably worse, being as high as 2.1 in the case of one outlier for GEV. The Rice formula is again relatively good, perhaps benefiting from not having an inferred daily maximum distribution raised to the power of 250.

Predictive Likelihood is relatively good and, while results from the Normal distribution do not include the exact value in the error bars, all results are reasonably close to the exact. Bayesian Updating is similar to the results for characteristic value. The error bars are again small and the mean is not near the exact value.

Traffic Load Effect Problem

As part of the European 7 th Framework ARCHES project [1], extensive WIM measurements were collected at five European sites: in the Netherlands, Slovakia, the Czech Republic, Slovenia and

Poland. The ARCHES site in Slovakia is used as the basis for the simulation model presented here.

Measurements were collected at this site for 750 000 trucks over 19 months in 2005 and 2006. The traffic is bidirectional, with average daily truck traffic (ADTT) of 1100 in each direction. Very heavy trucks were recorded at all sites, with a maximum gross vehicle weight (GVW) of 117 t being recorded in Slovakia.

A detailed description of the methodology adopted is given by Enright & OBrien (2012), and is summarised here. For Monte Carlo simulation, it is necessary to use a set of statistical distributions based on observed data for each of the random variables being modelled. For gross vehicle weight and vehicle class (defined here simply by the number of axles), a semi-parametric approach is used as described by [START_REF] Obrien | Importance of the tail in truck weight modeling for bridge assessment[END_REF]. This involves using a bivariate empirical frequency distribution in the regions where there are sufficient data points. Above a certain GVW threshold value, the tail of a bivariate Normal distribution is fitted to the observed frequencies which allows vehicles to be simulated that may be heavier than, and have more axles than, any measured vehicle. Results for lifetime maximum loading vary to some degree based on decisions made about extrapolation of GVW, and about axle configurations for these extremely heavy vehicles, and these decisions are, of necessity, based on relatively sparse observed data.

Bridge load effects for the spans considered here (Table 1) are very sensitive to wheelbase and axle layout. Within each vehicle class, empirical distributions are used for the maximum axle spacing for each GVW range. Axle spacings other than the maximum are less critical and trimodal Normal distributions are used to select representative values. The proportion of the GVW carried by each individual axle is also simulated in this work using bimodal Normal distributions fitted to the observed data for each axle in each vehicle class. The correlation matrix is calculated for the proportions of the load carried by adjacent and non-adjacent axles for each vehicle class, and this matrix is used in the simulation using the technique described by [START_REF] Iman | A distribution-free approach to inducing rank correlation among input variables[END_REF].

Traffic flows measured at the site are reproduced in the simulation by fitting Weibull distributions to the daily truck traffic volumes in each direction, and by using hourly flow variations based on the average weekday traffic patterns in each direction. A year's traffic is assumed to consist of 250 weekdays, with the very much lighter weekend and holiday traffic being ignored. This is similar to the approach used by [START_REF] Caprani | Characteristic traffic load effects from a mixture of loading events on short to medium span bridges[END_REF] and [START_REF] Cooper | The determination of highway bridge design loading in the united kingdom from traffic measurements[END_REF]. For same-lane multi-truck bridge loading events, it is important to accurately model the gaps between trucks, and the method used here is based on that presented by [START_REF] Obrien | Headway modelling for traffic load assessment of short to medium span bridges[END_REF]. The observed gap distributions up to 4 seconds are modelled using quadratic curves for different flow rates, and a negative exponential distribution is used for larger gaps.

The modelled traffic is bidirectional, with one lane in each direction, and independent streams of traffic are generated for each direction. In simulation, many millions of loading events are analysed, and for efficiency of computation, it is necessary to use a reasonably simple model for transverse load distribution on two-lane bridges. For bending moment the maximum LE is assumed to occur at the centre of the bridge, with equal contribution laterally from each lane. In the case of shear force at the supports of a simply supported bridge, the maximum occurs when each truck is close to the support, and the lateral distribution is very much less than for mid-span bending moment. In this case a reduction factor of 0.45 is applied to the axle weights in the second lane. This factor is based on finite element analyses performed for different types of bridge (OBrien & Enright, 2012). The load effects and bridge lengths examined in the simulation runs are summarized in Table 1. Two series of simulation runs are performedone to represent possible measurements over 1000 days, repeated 20 times, and another to represent the benchmark ('exact') results, consisting of 5000 years of traffic. For the benchmark run, the outputs consist of annual maximum LE's, and these can be used to calculate the characteristic values and annual maximum distributions to a high degree of accuracy.

Sample results are plotted on Gumbel probability paper in Figure 8 for the 5000-year simulation run. Two load effects are shownshear force (LE2) on a simply supported 15 m bridge, and hogging moment (LE3) over the central support of a two-span bridge of total length 35 m. Due to the randomness inherent in the process, there is some variability in the results, particularly in the upper tail region (top 1% of data approximately). Weibull fits to the upper 30% tail are used to smooth this variability (as shown in figure ), and these are used to calculate the characteristic values. This long-run simulation process is considered to be highly accurate, subject to the assumptions inherent in the model and is used as the benchmark against which the accuracy of all other methods is measured. The assumed measurements consisting of 1000 simulated daily maxima are used as the basis for extrapolation using each method to estimate the 'true' results calculated from the long-run simulation.

For the five tail fitting methods, the distributions are fitted to the top 30% of data. For some load effects and spans, the distribution of the data is multi-modal (see Figure 9), i.e., there is a change in slopearound 400 kN in this caseimplying data from a different parent distribution. In the case illustrated, there is a change around this point from (i) daily maxima arising from regular trucks and cranes to (ii) maxima arising from extremely heavy and rare low-loader vehicles. The 75-year characteristic maximum LE's are inferred from the assumed measurements. This process is carried out for the 5 load effects and repeated 20 times to determine the variability in results. The results are illustrated in Figure 10 which shows, in each case (i) the median value, (ii) the 25% to 75% range (boxed), (iii) the 0.7% to 99.3% range (median ± 2.7 standard for normally distributed data) (dashed lines) and (iv) individual outliers beyond that range.

Figure 10 shows that the first three tail fitting methods are reasonably good, with modest range and median value close to the benchmark result from the 5000 year run. As for the simple example, fitting to a Normal distribution gives a lesser range of results which, in this case, are all reasonably close to the benchmark. The Rice method is generally better than all the others. Predictive Likelihood gives poor results for these traffic loading problems. Characteristic values are sometimes under-estimated and other times over-estimated, with no clear trend. Sensitivity studies of these results show that there is significant influence of the Fréchet (unlimited) tail of the GEV distribution that is used to jointly maximize the likelihood of observing the data and the predictand. If the fit is limited so that Fréchet tails are not permitted (admitting only Weibull or Gumbel tails), as could be argued from the physical bounds of the traffic loading phenomenon, then the fits improve.

Bayesian Updating is surprisingly poor for this class of problem. As for the simpler example, the 20 results are consistent but deviate significantly from the benchmark results. Several variations were tested in attempts to find a Bayesian approach that is consistently good. The GEV distribution was fitted to the top 30% of data, as an alternative to fitting it to all the data. Different numbers of parameters of the GEV distribution were updated: two ( and ) and three (,  and ). Different prior distributions were assumed for these parameters -Normal and Uniform. For the latter, different ranges were tested for the parameter values. None of these variations produced consistently better results for the five LE's and spans. The results shown are based on use of all the data; updating just two parameters ( and ) with a Uniform prior distribution and a limit on the range of  to be nonpositive.

Exact

Annual probabilities of failure are also inferred for the five combinations of load effect and span. As before, the probability of failure for the benchmark example is set at 10 -6 in each case and the resistance distribution is taken to be a mirrored version of the benchmark LE distribution.

The results are illustrated in Figure 11. As for the simple example, the errors in the probabilities, even when plotted on a Normal scale, are much higher than for characteristic values. Most of the tail fitting methods -POT, GEV, Box-Cox and Rice formulagive relatively good results, with the Rice formula generally beating the others. As before, when fitting to a Normal distribution, the benchmark result is sometimes outside the 25%-75% range, but not by a great deal. As for the characteristic values, Bayesian Updating and Predictive Likelihood are less accurate than the other methods.

Conclusions

In this paper, seven methods of statistical inference are critically reviewed. Each method is also tested using two examples. The first example is derived from a Normal distribution and the exact solution is known. A total of 3000 normally distributed values (e.g., vehicle weights) are considered per day and the daily maxima are used to infer the characteristic maximum and the probability of failure in a year.

In the second example, a sophisticated algorithm is used to generate a train of vehicles with weights and axle configurations consistent with measured Weigh-in-Motion data. Five different combinations of load effect and span are considered and, in each case, characteristic values and probabilities of failure are again calculated. In these cases, the exact solutions are not known but the simulation is run for 5000 years to obtain accurate benchmark references against which inferences based on 1000 days of data can be compared.

Of the seven methods considered, five are tail-fitting approaches, i.e., a distribution is fitted to the tail of the data. Peaks-Over-Threshold (POT) is popular in some sectors but is not time-referenced and selecting the threshold is a subjective process. Fitting the tail of block-maximum data to a Generalized Extreme Value (GEV) is perhaps the most popular used for bridge traffic loading, with a typical block size of a day. Box-Cox could be considered to be a hybrid between POT and GEV. These three methods are generally good for inferring the characteristic values, both for the simple and the more complex examples. There is no theoretical justification for fitting block maximum data to the tail of a Normal distribution but it is sometimes done. It is found here to give reasonably accurate results, with a small standard deviation. Finally, the Rice formula is an indirect approach as it is the upcrossing frequencies that are fitted to the formula, rather than the data itself. Nevertheless, it performs well in these tests, generally better than POT and GEV.

Bayesian Updating is used here to fit the block maximum data to a family of GEV distributions. The parameters of the GEV are allowed to vary, their associated probabilities being updated as the data is considered. The results from the Bayesian approach are found to be generally poor and several variations in the approach did not produce consistently good results. Finally, Predictive Likelihood is considered, a method where the likelihood of each inferred characteristic value is considered, given the available data. This is also found to give poor results.

All seven methods are used to infer the annual probabilities of failure as well as the characteristic values. To avoid the need for any assumption on the distributions for resistance, the benchmark load effect distribution is mirrored and this mirrored version is used in the calculation of probability of failure.

The inferred failure probabilities are considerably less accurate than the inferred characteristic values, perhaps not surprising given that such a small failure probability was being considered (10 -6 in a year). As for characteristic values, the tail fitting methods are better than the others but none of the methods gives an accurate inference with 1000 days of data.

  Grave (2001) uses a weighted least-squares approach to fit Weibull distributions to critical LE's. O'Connor (2001) fits Gumbel and Weibull distributions to a population of 'extreme' LE's. OBrien et al. (2003) plot hourly maximum strain values on Gumbel probability paper. A least-squares, straightline fit is made to the upper data points in a similar manner to Grave (2001) and O'Connor

  [START_REF] Getachew | Traffic load effects on bridges[END_REF] adopts Cremona's approach for the analysis of traffic LE's on bridges induced by measured and Monte Carlo simulated vehicle data. O'Connor & OBrien (2005) compare the predicted extremes of simply supported moment for a range of span lengths by the Rice formula, Gumbel and Weibull Extreme Value distributions: they find about 10% difference between Rice and the others. Finally, Jacob (1991) uses Rice's formula to predict characteristic LE's for the cases of free and congested traffic in background studies for the development of the Eurocode.

Figure 1 -

 1 Figure 1-Load Effect and Mirror Resistance A simple example is used here to compare the alternative methods of extrapolation. A Normally distributed random variable (such as vehicle weight in tonnes) is first considered:

Figure 2 -

 2 Figure 2 -Best fit distributions using four tail-fitting methods inferred from 1000 days of data

Figure 3 -

 3 Figure 3 -Rice Formula Fit to Tail for 1000 days of data

  Figure 4-Predictive Likelihood

Figure 5 -

 5 Figure 5 -Inferred Annual Maximum CDF's

Figure 6 -

 6 Figure 6 -Inferred 75-year Characteristic Values (mean ± one standard deviation)

Figure 7 -

 7 Figure 7 -Mean ± One Standard Deviation of Inferred Probabilities of Failure, P f plotted to a Normal scale (ϕ = cumulative distribution function for Normal distribution)

Figure 8 -

 8 Figure 8 -Annual maximum LE's from 5000 years of simulation

Figure 9 -

 9 Figure 9 -Daily Maximum LE2 -15 m span Data and Inferred Distributions

  Figure 10 -Range of Inferred 75-year Characteristic Values from 1000 Days of Data

Figure 11 -

 11 Figure 11 -Probabilities of Failure Inferred from Sets of 1000 Daily Maximum LE's

Table 1 -

 1 Load effects and bridge lengths

	Load Effect

Acknowledgment

This work is supported under the TEAM project. TEAM is a Marie Curie Initial Training Network and is funded by the European Commission 7 th Framework Programme (PITN-GA-209-238648).


Student Member, IEEE Zhibin Zhou
email: zhibin.zhou26@gmail.com

Member, IEEE Franck Scuiller
email: franck.scuiller@ecole-navale.fr

Member, IEEE Jean Frédéric Charpentier
email: frederic.charpentier@ecole-navale.fr

Senior Member, IEEE Mohamed E H Benbouzid
email: mohamed.benbouzid@univ-brest.fr

Senior Member, IEEE Tianhao Tang
email: thtang@shmtu.edu.cn

Power Control of a Non-Pitchable PMSG-Based Marine Current Turbine at Over-rated Current Speed with Flux-Weakening Strategy

Keywords: current turbine, fixed-pitch, PMSG, flux-weakening, high marine current speed

This paper deals with power control strategies for a fixed-pitch direct drive marine current turbine (MCT) when the marine current velocity exceeds the rated value corresponding to the MCT nominal power. At over-rated marine current speed, the MCT control strategy is supposed to be changed from maximum power point tracking (MPPT) stage to constant power stage. In this paper, flux-weakening strategy is investigated to realize appropriate power control strategies at high marine current speeds. During flux-weakening operations, the generator can be controlled to produce nominal or over-nominal power for a specific speed range (constant power range). These two power control modes are compared and the constant power range is calculated in this paper. The relationship between the expected constant power range and generator parameters requirement (stator inductance, permanent magnet flux, nominal power coefficient) is analyzed in this paper. A Torque-based control with a robust feedback flux-weakening strategy is then carried out in the simulation. The proposed control strategies are tested in both high tidal speed and swell wave cases; the results validate the analysis and show the feasibility of the proposed control method.

I. INTRODUCTION

Due to high predictability of the marine tides and high power potential of marine tidal currents, various marine current turbine (MCT) technologies have been developed to capture tidal current energy in recent years [START_REF] Bahaj | Generating electricity from the oceans[END_REF][START_REF] Benelghali | Marine tidal current electric power generation technology: State of the art and current status[END_REF]. However, difficulties of underwater accessibility make compact structure and low maintenance highly expected for MCTs. Several industrial MCT projects such as the OpenHydro (tested by French utility company EDF), Hydro Beluga 9 (Alstom) and Voith Hydro turbine system adopt nonpitchable turbine blades and use permanent magnet synchronous generators (PMSG). Two advantages of these projects can be noticed: fixed blades enable simplifying the turbine system (avoiding pitchvariable mechanism); and PMSGs enable realizing direct-drive systems (eliminating gearbox).

However, the MCT rated power would not be designed for the peak marine current speed due to the fact that the peak current speed may appear only at large spring tides and corresponds only to a small part of the statistical resource [START_REF] Benelghali | A simulation model for the evaluation of the electrical power potential harnessed by a marine current turbine[END_REF]. When the marine current speed is higher than the rated value, a fixedpitch MCT is unable to limit the extracted power via pitch control as in large wind turbines [START_REF] Polinder | Overview of and trends in wind turbine generator systems[END_REF] and some pitchable MCT turbines [START_REF] Thiringer | Flicker evaluation of the SeaGen tidal power plant[END_REF]. Therefore, an appropriate generator-side control strategy should be applied to control and limit the MCT output power.

Based on the turbine power characteristic, one possible solution is to operate the turbine at overnominal speed during high marine current speed periods for reducing turbine power coefficient and the harnessed power. However, over-nominal speed operation leads to high electromotive force (EMF) in PM machines and may cause saturations in the generator regulators. Although various flux-weakening control strategies have been studied to realize over-base speed operations for PM machines [START_REF] Morimoto | Effects and compensation of magnetic saturation in flux-weakening controlled permanent magnet synchronous motor drives[END_REF][START_REF] Lu | A review of flux-weakening control in permanent magnet synchronous machines[END_REF][START_REF] Tursini | Feedforward flux-weakening control of surface-mounted permanentmagnet synchronous motors accounting for resistive voltage drop[END_REF][START_REF] Song | A new robust SPMSM control to parameter variations in flux weakening region[END_REF][START_REF] Maric | Two flux weakening schemes for surface-mounted permanent-magnet synchronous drives -Design and transient response considerations[END_REF][START_REF] Liu | Comparison of drive performance of PM synchronous machine fed by inverters with different PWM strategies in constant torque and constant power regions[END_REF][START_REF] Lin | Voltage control technique for the extension of dc-link voltage utilization of finite-speed SPMSM drives[END_REF][START_REF] Kwon | Novel antiwindup of a current regulator of a surface-mounted permanent-magnet motor for flux-weakening control[END_REF][START_REF] Kwon | A novel flux weakening algorithm for surface mounted permanent magnet synchronous machines with infinite constant power speed ratio[END_REF][START_REF] Liu | Flux-weakening control of nonsalient pole PMSM having large winding inductance, accounting for resistive voltage drop and inverter nonlinearities[END_REF], the flux-weakening operation mode for PMSG in renewable energy systems is still a new topic. The joint operating characteristics of the MCT and the PMSG are focused in this paper. In a previous work [START_REF] Zhou | Power limitation control for a PMSGbased marine current turbine at high tidal speed and strong sea state[END_REF],

the authors proposed to apply flux-weakening strategy for limiting the MCT power to its nominal power at high current speeds. However, the maximal generator output power during the flux-weakening operation and the impacts of generator parameters on the MCT operation range are not discussed. In this paper, these issues will be focused. Figure 1 shows the general system scheme, and the generator-side control will be focused in this paper.

In Section II, the turbine power characteristic and the generator model are presented. In Section III, the flux-weakening operation and the generator parameters impacts on the MCT operational zone are discussed. In Section IV, the proposed robust power control scheme is presented and the simulation results of the two power control modes at high marine current speed are compared. And the conclusion is then given in Section V.

II. MARINE CURRENT TURBINE AND GENERATOR MODELS

A. Marine Current Turbine Model

The power harnessed by a horizontal-axis MCT can be calculated as

2 3 1 ρ π 2 p P C RV = (1)
The turbine power coefficient C p depends on the turbine blade structure and hydrodynamics. For typical MCTs, the optimal C p value for normal operation is estimated to be in the range of 0.35-0.5 [START_REF] Benelghali | A simulation model for the evaluation of the electrical power potential harnessed by a marine current turbine[END_REF]. For a given turbine and based on the experimental results, the C p curve can be approximated as a function of the tip speed ratio (

m λ= / R V ω
) and the pitch angle [START_REF] Slootweg | General model for representing variable speed wind turbines in power system dynamics simulations[END_REF]. In this paper the considered system is a nonpitchable MCT, therefore the C p is supposed to depend only on λ.

Figure 2 shows the C p curve used in this paper, which is referenced from [START_REF] Myers | Power output performance characteristics of a horizontal axis marine current turbine[END_REF][START_REF] Batten | The prediction of the hydrodynamic performance of marine current turbines[END_REF]. The maximum C p value is 0.45 which corresponds to a tip speed ratio of 6.3. This value is considered as the optimal tip speed ratio for realizing maximum power point tracking (MPPT) under the rated marine current speed. One 1.52 MW turbine is studied in this paper. The turbine maximum speed to follow MPPT is 24 rpm (2.52 rad/s) corresponding to a marine current of 3.2 m/s. When the marine current exceeds 3.2 m/s, the extracted power will be limited by operating the rotor speed over 2.52 rad/s as illustrated in Fig. 3.

B. Marine Current Generator Model

Surface-mounted PMSG (non-salient machine) is considered in this paper. Although salient PM machines can have wider flux-weakening range, non-salient PM machines are usually easier to manufacture and are capable of achieve a smoother electromagnetic torque [START_REF] Chaithongsuk | Optimal design of permanent magnet motors to improve field-weakening performances in variable speed drives[END_REF]. The PMSG model in the d-q frame is described by the equations (2).

i v Ri L L i i v Ri L L i T n i J T T f ⎧ = + - ⎪ ⎪ ⎪ = + + + ⎪ ⎨ ⎪ = ⎪ ⎪ ⎪ = -- ⎩ (2)

III. FLUX-WEAKENING FOR THE PMSG IN A MCT SYSTEM

A. Generator Operating Characteristics

In the synchronous rotation frame, the stator current is decomposed into a flux component i d and a torque component i q ; and the following voltage and current limits should be guaranteed at any time.

2 2 2 d q m a x v v V + ≤ (3) 2 2 2 d q m a x i i I + ≤ ( 4
)
V max is limited by the available DC-bus voltage V dc and the converter modulation method.

1/ 3 max dc V V =
for the sinusoidal wave model (linear space vector modulation) and

/

max dc V V = π for the full six-step operation [START_REF] Maric | Two flux weakening schemes for surface-mounted permanent-magnet synchronous drives -Design and transient response considerations[END_REF]. I max is determined by machine torque and converter power ratings. For a surface-mounted PM machine, the inductances in d-q axes are almost equal (L d = L q = L s ). By neglecting stator resistance, the voltage limitation (3) in steady-state can then be expressed by From [START_REF] Thiringer | Flicker evaluation of the SeaGen tidal power plant[END_REF], the voltage and current constrains can be drawn as circles shown in Fig. 4. The solid-line circle indicates the current limitation and the dotted-lines circles indicate voltage constraints which have a centre of (0, -ψ m /L s ) and shrink with increased operating speeds.

The generator parameters are set to match the turbine characteristics: the generator nominal speed is set at the maximum turbine speed to realize the MPPT stage (24 rpm, 2.52 rad/s); the generator nominal active power is designed as the same value of the turbine nominal one (1.52 MW); the generator maximum torque is required at the base speed to obtain the nominal power as given by

ω ω eb N e N m N T m a x p P T K I n = = (6)
where

(3 / 2) ψ T p m K n =
It should be noted that if the generator maximum torque is set to a smaller value that will be required by the MPPT strategy below the nominal speed (this also means a smaller current rating of I max ), the flux-weakening operation range will be reduced; while a larger torque rating can result in a wider fluxweakening operation range but the generator weight and cost will be increased [START_REF] Moncada | Axial flux permanent-magnet machine under optimum control strategy for wind power generation[END_REF][START_REF] Morandin | Torque and power rating of a wind-power PM generator drive for maximum profit-to-cost ratio[END_REF]. The typical control strategy for a non-salient PM machine under the base speed is to set i d = 0 for maximizing the torque per ampere ratio. When rotor speed is over the machine base speed, the voltage limitation circle will then shrink to the left side of the q-axis (as shown in Fig. 4). It indicates that negative d-axis current should be injected to decrease the total flux in the machine windings and to keep the operating points within the voltage and current limitation circles at flux-weakening operations.

Finite constant power speed ratio machine (ψ m > L s I max ) is considered in this paper. The maximum torque in the flux-weakening operation is obtained when the machine operates at the intersection points of the voltage and current limitation circles. The intersection point (point A in Fig. 4) can be calculated when the equality sign is used in ( 4) and [START_REF] Thiringer | Flicker evaluation of the SeaGen tidal power plant[END_REF]; and the corresponding i d and the i q can be obtained as follows.

2 2 2 s s s ψ 2ψ ω max m d m a x m e L V i I L L * ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ = - - ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ (7) 2 2 qmax max d i I i * = ± - (8)
If the generator output power is supposed to be limited to the nominal power at over-base speed operation (ω e > ω eb ), the q-axis current reference should be calculated by

N ω (ω / ) ω eb q m a x q m a x T T e p e T P i I i K K n * * ⎛ ⎞ = = = ≤ ⎜ ⎟ ⎝ ⎠ (9)
Figure 4 shows that for a given i q reference under i qmax , there exists a range for setting i d reference values (shown as the C-D line segment). It is not obvious to determine the optimal i d reference due to various optimization criteria. The copper loss is proportional to the square value of stator current, while the iron loss is approximately proportional to the square value of local flux density for a given speed [START_REF] Aubry | Sizing optimization methodology of a surface permanent magnet machineconverter system over a torque-speed operating profile: application to a wave energy converter[END_REF][START_REF] Li | Optimization of multibrid permanent-magnet wind generator systems[END_REF]. So the point D corresponds to the lowest copper losses while the point C corresponds to the lowest iron losses. If the maximum torque per ampere (MTPA) strategy is applied, then the point D will be chosen as the operating point for the given i q reference. Figure 5 illustrates the generator power characteristics in this paper. Under the base speed, the generator is supposed to operate in MPPT mode enabling the turbine to capture maximum energy from marine currents. Over the base speed, a demagnetizing current (negative i d current) is injected and the generator will operate in the flux weakening range. At the flux weakening stage, there exists a constant power range during which the generator power can be either over the nominal power with i qmax or limited to the nominal power with i q given by ( 9). The i d current curve in Fig. 5 is calculated by [START_REF] Lu | A review of flux-weakening control in permanent magnet synchronous machines[END_REF]; the two different i q current trajectories correspond to the coordinates described by points A and B in Fig. 4 when the rotor speed increases. However, when the rotor speed exceeds the constant power range, i q will be limited to i qmax and the operating points of the two power control modes are the same. Over the constant power range, the generator power will decrease rapidly below the nominal value with the increase of the rotor speed.

The flux-weakening range and constant power range depend on machine parameters. The maximum speed for the flux-weakening range can be obtained from Fig. 4 as The flux-weakening range and constant power range depend on machine parameters. The maximum speed for the flux-weakening range can be obtained from Fig. 4 as

ω ψ max emax m s m a x V L I = - (10)
ω ψ max emax m s m a x V L I = - (10)
The maximum speed for the constant power range can be calculated based on (7)~( 9) with the equality sign in (9), ( )

2 ω ω 1 2 eb econ pu x = - (11)
where

ω / pu eb s max max x LI V =
, and the parameter relationship at the base speed conforms to ( ) ( )

2 2 ω ψ ω max eb m eb s max V L I = + (12)
Then, the flux-weakening range and the constant power range can be described by the ratio of their maximum operational speeds with the base speed as follows (

)

ω _ = ω ω ψ emax max eb eb m s max V Ratio fx L I = - (13)
( )

2 ω 1 _ = ω 1 2 econ eb pu Ratio con x = - (14)
In this paper, the generator (parameters illustrated in the Appendix) is able to have a flux-weakening range of Ratio_fx = 3.3 and constant power range of Ratio_con = 2.4 as shown in Fig. 5. This constant power range is sufficient for the generator to cope with the MCT and achieve both over-nominal power and nominal power modes during high marine current speeds. In the following, we will use the maximum active power (MAP) mode and the constant active power (CAP) mode to distinguish the two generator power control modes at the flux-weakening stage.

B. Impacts of Generator Parameters on the MCT Operation

It can be seen from Fig. 5 that both MAP and CAP modes are effective in the constant power range, while for a rotor speed over this range the generator output power will decrease rapidly below the nominal value. Therefore, it is important to study the requirements of the generator parameters for an expected constant power range. In this section, generator base speed ω eb , nominal torque T eN and the maximum stator voltage V max are fixed and we therefore focus on the parameter requirements of stator inductance and rotor permanent magnet flux. Figure 6 shows that the generator nominal power angle φ, defined at the base speed, is decided by the ratio between L s I max and ψ m with the MTPA strategy.

From ( 13) and ( 14), it can be seen that the flux-weakening and constant power ranges strongly depend on L s I max and ψ m. For an expected constant power speed ratio (CPSR), the required minimum values for L s I max and L s can be calculated as follows.

Fig. 6. Generator parameters relationship at base speed.

_ 1 ( ) ω 2 _ max s max eb V Ratio con L I Ratio con * - = (15)
( ) different CPSRs. From ( 16) and Fig. 7, it can be seen that a wider CPSR requires smaller ψ m and higher L s and I max ; these results are consistent with the studies in [START_REF] Bianchi | Parameters and volt-ampere ratings of a synchronous motor drive for fluxweakening applications[END_REF]. Although interior PM machines are easier to realize wider flux-weakening range with their saliency, surface-mounted PM machines can also have wider flux-weakening range via proper machine design. For example, additional leakage inductance can be obtained by introducing additional cores enclosing end-windings, thus realizing a wider CPSR [START_REF] Kwon | Design and control of an axial-flux machine for a wide fluxweakening operation region[END_REF]. The generator parameters used in this paper enable the machine to have a CPSR of 2.4 with a nominal power factor of 0.84. A higher CPSR requires large stator inductance and L s I max values thus reducing the nominal power factor. For example, stator inductance value should increase by 8.3% to obtain a CPSR of 5 but the resulted nominal power factor will be reduced to 0.77. Smaller stator inductance can improve the generator nominal power factor and then reduce the V-I rating of the generator-side converter, however the CPSR will also be decreased and the system operable area will be limited.

2 2 * 3 ( ) ω 3 ψ ( ) 2
For example if the inductance is designed with 1.0 mH (16.7 % smaller than L s used in this paper), the PMSG will have a nominal power factor of 0.9, but with a small CPSR of 1.6. Figure 8 shows the generator operating points with CPSR=1.6. The dotted-lines illustrate the turbine powers characteristics.

In this case, both MAP and CAP modes can be applied when the marine current speed is between 3.2 m/s and 3.6 m/s. However, the MCT system can not produce power at a current speed of 4.0 m/s due to small CPSR and low flux-weakening capability of the generator.

Figure 9 shows the case with generator parameters chosen in this paper. It can be seen that with CPSR = 2.4, the generator enables the MCT system to produce power for a wide range of over-rated marine current speeds. At a high current speed of 4.0 m/s, the MCT system is still able to produce 1.77 MW under the MAP mode or 1.52 MW under the CAP mode.

It should be noticed that the flux-weakening operation is on the decreasing slope of the C p (λ > λ opt ) curve to reduce the turbine power coefficient during over-rated current speeds. Therefore, the turbine C p characteristic can also influence the expected generator CPSR for non-pitchable MCTs. The C p curve used in this paper has a common shape for three-blade turbines [START_REF] Myers | Power output performance characteristics of a horizontal axis marine current turbine[END_REF][START_REF] Batten | The prediction of the hydrodynamic performance of marine current turbines[END_REF], [START_REF] Eltamaly | Modeling of wind turbine driving permanent magnet generator with maximum power point tracking system[END_REF]. Increasing turbine blade number could have sharper C p and better stall performance at high current speeds. However, flatter C p can be found in two-blade turbines or when the blade chord and twist distribution is designed to keep high C p values for a wide TSR range [START_REF] Coiro | Experimental test campaign on an innovative device to harness clean energy from tidal and river current[END_REF][START_REF] Scherillo | Numerical and experimental analysis of a shrouded hydroturbine[END_REF]. In this case, it would require large CPSR for the generator to cope with the turbine at over-rated marine current speed.

IV. SYSTEM SIMULATION AT HIGH MARINE CURRENT SPEED

A. Robust MCT Power Control Strategy

From [START_REF] Lu | A review of flux-weakening control in permanent magnet synchronous machines[END_REF] it can be seen that the model-based d-axis current reference requires accurate information about machine parameters and operating speed. Actually, robust flux-weakening methods are more favorable in real applications to overcome the parameter indeterminacy and variations. The fluxweakening method proposed in [START_REF] Kwon | Novel antiwindup of a current regulator of a surface-mounted permanent-magnet motor for flux-weakening control[END_REF][START_REF] Kwon | A novel flux weakening algorithm for surface mounted permanent magnet synchronous machines with infinite constant power speed ratio[END_REF] is chosen in this paper for its robustness and high utilization of the DC-bus voltage. Figure 10 The torque-based MPPT strategy is applied when the generator speed is under the nominal value.

This torque-based MPPT calculates the reference generator torque as a function of C pmax , λ opt and ω m .

When the rotor speed is gradually driven by the high marine current speed to be over the nominal speed, the generator will be accelerated to over-base speed operation points with the flux-weakening algorithm.

The reference torque can be calculated by the constant power requirement. Equation ( 17) illustrates the torque control reference used in this paper. Fig. 10. Generator-side control scheme with flux-weakening strategy.

5 2 3 if ω ω if ω > ω ω 1 ρπ ω 2 λ m m pmax m op N m mN m t N C R T P * ⎧ ⎪ ≤ ⎪ = ⎨ ⎪ ⎪ ⎩ (17)
The MTPA strategy is embodied in this control scheme. Indeed, i d * is generated to maintain the operating points on the voltage constraints. Therefore, this control scheme is able to achieve the maximum active power (MAP) mode as at point A (in Fig. 4) and the constant active power (CAP) mode with minimum copper losses at point D (in Fig. 4). For the CAP mode, i q * is calculated from the torque reference (second equation in ( 17)) to keep the generator output power at its nominal value; and for the MAP mode, i q * is set to

2 2 max d I i * -
for maximizing the generator torque and output power.

B. Under High Tidal Speed

The MCT is supposed to be installed at a proper depth under the sea-surface so that normal sea waves generated by local winds will have negligible influence on the MCT system. High speed marine currents would occur at large spring tides or under strong sea states (caused by swell waves or storms).

In this section, the sea is supposed to be calm (without significant swell waves) and the high marine current speed driven by spring tides are considered. m/s for this site at spring tides. Without power limitation control, the maximum extractable power at the peak tidal speed would be over 2 MW for the MCT.

In the simulation of this section, the marine current speed is set initially at 2.8 m/s and then rises to 1. It can be seen that when the tidal current speed rises to the rated value of 3.2 m/s at about 47 s (Fig. 12), the MCT rotor speed will reach the nominal value of 24 rpm (Fig. 13) and the generator torque will reach its maximum value about 600 kNm (Fig. 14). When the current speed rises over the rated value, the flux-weakening strategy will be triggered and the rotor speed will rise over the nominal speed. Both MAP and CAP modes are simulated. Figure 13 and 14 show that at over-rated marine current speeds, the CAP mode features lower generator torque to accelerate the turbine to a higher speed than the MAP mode. The reason is that the CAP mode requires reducing the turbine C p value more than the MAP mode in order to keep the generator output power at the nominal value for a given over-rated current speed. While MAP mode aims to generate maximum torque and then maximize the generator output power as shown in Fig. 14 and Fig. 15.

P i i R = + (18)
The specific iron losses (W/kg) can be calculated based on the following common relationship.

0 0 0 a b Fe Fe Fe Fe B f P P B f ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (19)
where B Fe and f are respectively the flux density and the field frequency in the stator iron core, and P Fe0 represents the iron losses per mass at the given frequency f 0 and the flux density B Fe0 . The typical value of a and b can be set as 2.2 and 1.5, respectively [START_REF] Drouen | Study of an innovative electrical machine fitted to marine current turbine[END_REF]. In a simplified approach, the fundamental harmonic of the flux density B Fe can be considered to be proportional to V s /f and ( 19) can then be rewritten as

2.2 0.7 0 0 0 s Fe Fe V f P P V f - ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (20)
In this paper, the typical point for calculating the iron losses is chosen at the base point (V 0 = V max and f 0 = ω eb /2π ) and the corresponding P Fe0 is 2.5 W/kg. The total iron mass in the stator core of the 1.5 MW generator is estimated to be about 4 tons [START_REF] Li | Optimization of multibrid permanent-magnet wind generator systems[END_REF]. The iron losses calculation results under both CAP and MAP modes are shown in Fig. 17.

As analyzed in the previous section, the MAP mode operates on the intersection points of voltagecurrent limitation circles and maintains the stator current at its maximum magnitude I max , while the CAP mode operates at a smaller stator current. The results from Fig. 16 confirm that the generator copper losses are kept at the maximum value by the MAP mode, and the CAP mode enables to greatly reduce the copper losses at over-rated marine currents. The iron losses difference is not very big because both modes operate at V max with the MTPA strategy during the flux weakening operation. The iron losses under MAP mode are a little higher than under CAP mode. This indicates that CAP mode reduces more flux density in the stator core.

From the simulation results and Table 1, it can be seen that at steady-state with a marine current speed of 3.6 m/s, the MAP mode will accelerate the turbine to 1.45 times of the nominal speed and produce 19.7% power more than the nominal value; while the CAP mode will accelerate the turbine to 1.58 times of the nominal speed and limit the generator power to the nominal power of 1.52 MW.

Compared to the MAP mode, the generator copper losses are reduced by 29.5% and the iron losses are reduced by 6.4% with the CAP mode at steady-state under a marine current speed of 3.6m/s.

C. Under Swell Effect

Swell refers to long-length ocean waves (usually over 150 m) generated by distant storms [START_REF] Goda | Random Seas and Design of Maritime Structures[END_REF]. Long distance dispersion makes the swell energy more accumulated than local wind-generated waves. Swells can propagate very deep below the sea surface and therefore have a non-negligible effect on the MCT system. Indeed, swell effect can lead to marine current speed fluctuations on a period of about 10 to 20 s. In this paper, a medium-strong sea state with significant wave height H s = 3 m and typical wave period T p = 13.2 s is considered (it corresponds to typical sea state in the winter off the western coast of Europe). Details of the swell modeling can be found in [START_REF] Zhou | Power smoothing control in a gridconnected marine current turbine system for compensating swell effect[END_REF]. The total marine current speed is calculated as the sum of tidal current speed and swell-induced current speed variations. It should be noticed that the tidal current speed can be considered as a constant during a short-time period due to tidal phenomenon. Figure 18 shows the marine current speed under swell effect: it can be seen that even the tidal current speed is fixed at 2.8 m/s, the total marine current speed will exceed the rated value of 3.2 m/s during some short periods caused by the swell-induced current variations.

Figures 19 and20 show the rotor speed and generator power responses in MAP and CAP modes respectively. It should be noticed that under swell effect, the marine current speed is fluctuating and there is no steady-state for the turbine. The turbine will not be accelerated to the same operating points as in steady-state cases, but the CAP mode will still accelerate the rotor speed a little higher than in the MAP mode to limit the generator power to its nominal value. In this case, the MAP can produce about 4% power more than the CAP mode during the main flux-weakening stages (40-45 s, 78-83 s and 93-98 s) as shown in Fig. 20. Figure 21 compares the d-q currents with the MAP and CAP modes under swell effect. Lower i q (and generator torque) can be noticed in the CAP mode than in the MAP mode. The difference of d-axis currents is very small because the CAP mode operates at a little higher speed than the MAP mode. In this case, the CAP mode has about 8% less copper losses than the MAP during the main flux-weakening stages. Figure 22 shows the turbine and generator torques under the CAP mode and this points out that for a large MCT system (with large total inertia), a big difference between the turbine and generator torques is required to accelerate or decelerate the system in dynamic stages.

V. CONCLUSION

This paper has investigated power control strategies for a PMSG-based MCT with non-pitchable blades at over-rated current speed. Flux-weakening control strategy has been proposed to accelerate the turbine and generator system over the nominal speed for power limitation control at high marine current speed. Two power control modes (the maximum active power mode and the constant active power mode) on the generator constant power range have been investigated. In this context, a torque control scheme with a robust flux-weakening algorithm has been applied. High speed marine currents caused by high tidal current speed and swell effect have been both considered.

The achieved simulation results have confirmed that at over-rated current speed, the constant active power mode enables to limit the generator output power to the nominal value with lower losses while the maximum active power mode enables to produce over-nominal power (better use the generator-side converter V-I rating) but with high copper losses. The parameter requirements of the non-salient PM machine for expected constant power speed ratio are provided in this paper. The compromise between high power factor and large constant power range should be noticed for appropriate generator parameter design.

APPENDIX

  power speed ratio; MAP = Maximum active power; MCT = Marine current turbine; MPPT = Maximum power point tracking; MTPA = Maximum torque per ampere; PMSG = Permanent-magnet synchronous generator; TSR = Tip speed ratio;H S , T p = Swell significant height and peak period;C p = Power coefficient; f B = Friction coefficient;i d , i q = Generator d-and q-axis currents;I max = Maximum phase current magnitude; J = Total system inertia;K T = Generator torque constant;L d , L q = Generator d-and q-axis inductances; resistance;T e , T m = Generator and turbine torque; V = Marine current speed;V s = Generator phase voltage magnitude;V dc = DC-bus voltage;V max = Maximum phase voltage magnitude; v d , v q = Generator d-and q-axis voltages; λ = Turbine tip speed ratio; ρ = Sea water density; φ = Generator power factor angle; ψ m = Permanent magnet flux; ω e , ω eb = Electrical angle and base speeds; ω m = Mechanical angle speed; ω mN = Rotor nominal angle speed.

Fig. 1 .

 1 Fig. 1. General scheme of a PMSG-based direct drive MCT system.

Fig. 2 .

 2 Fig. 2. MCT C p curve.

Fig. 3 .

 3 Fig. 3. The MCT power characteristic under different current speeds.

Fig. 4 .

 4 Fig. 4. Voltage and current limitation circles.

Fig. 5 .

 5 Fig. 5. Generator operating characteristics (corresponding to the Appendix data).

Figure 7 Fig. 7 .

 77 Figure 7 shows the calculation results; the (L s I max )* and L s * values in this figure are normalized by L s I max and L s (in the Appendix) respectively to illustrate the parameter change requirements according to

Fig. 8 .

 8 Fig. 8. Generator power characteristics with CPSR = 1.6.

Fig. 9 .

 9 Fig. 9. Generator power characteristics with CPSR = 2.4.

  illustrates the generator-side control scheme with the flux-weakening algorithm for the MCT. The difference between the current controller output (voltage reference) and the converter output voltage is used to generate the i d reference. The low-pass filter (LPF) is used to reject voltage signals high-frequency components.

Fig. 11 .

 11 Fig. 11. Tidal speed in the Raz de Sein for one month [3].

Figure 11

 11 Figure 11 gives an example of tidal current speed in the Raz de Sein (potential site for MCT project off the coast of Brittany, France) during one month based on tidal current data from SHOM (French Navy Hydrographic and Oceanographic Service). It can be seen that the peak tidal speed can reach 3.6

3. 6 m

 6 Figs.[START_REF] Kwon | Novel antiwindup of a current regulator of a surface-mounted permanent-magnet motor for flux-weakening control[END_REF] to 17. The steady-state results at 3.6 m/s are then summarized in Table1.

Fig. 12 .

 12 Fig. 12. High tidal speed.

Fig. 13 .

 13 Fig. 13. Rotor speed responses at high tidal speed.

Fig. 14 .

 14 Fig. 14. Generator torque responses at high tidal speed.

Fig. 15 .

 15 Fig. 15. Generator produced power at high tidal speed.

Fig. 16 .

 16 Fig. 16. Copper losses of the generator.

Fig. 17 .

 17 Fig. 17. Iron losses of the generator.

Fig. 18 .

 18 Fig. 18. Marine current speed under swell effect.

Fig. 19 .

 19 Fig. 19. Rotor speed responses under swell effect.

Fig. 20 .

 20 Fig. 20. Generator produced power under swell effect.

Fig. 21 .Fig. 22 .

 2122 Fig. 21. Generator stator currents in the d-q axis.

Table 1 .

 1 Steady-State Performance at Current Speed of 3.6 m/s.

					Control mode	Rotor speed (rpm)	Electromagnetic torque (kNm)	Electromagnetic power (MW)	Copper losses (kW)	Iron losses (kW)
					MAP	34.5	505	1.82	21	7.8
					CAP		38	381	1.52	14.8	7.3
	The generator copper losses (Joule losses) are calculated by
	copper	3 2	(	2 d	2 q	)	s

TABLE 2 . SYSTEM PARAMETER LIST

 2

	Sea water density	1027 kg/m 3
	Turbine blade radius	8 m
	System total inertia	1.3131×10 6 kg . m 2
	Maximum C p value	0.45
	Optimal TSR for MPPT	6.3
	Rated marine current speed	3.2 m/s
	MCT nominal power	1.52 MW
	Generator nominal phase voltage	649 V (RMS)
	Generator nominal phase current	928 A (RMS)
	DC-bus voltage	1500 V
	Rotor nominal speed	24 rpm
	Pole pair number	125
	Permanent magnet flux	2.458 Wb
	Generator stator resistance	0.0081Ω
	Generator d-q axis inductance	1.2 mH