^{1}

^{*}

^{2}

^{*}

As a result of social awareness of air emission due to the use of fossil fuels, the utilization of the natural wind power resources becomes an important option to avoid the dependence on fossil resources in industrial activities. For example, the maritime industry, which is responsible for more than 90% of the world trade transport, has already started to look for solutions to use wind power as auxiliary propulsion for ships. The practical installation of the wind facilities often requires large amount of investment, while uncertainties for the corresponding energy gains are large. Therefore a reliable model to describe the variability of wind speeds is needed to estimate the expected available wind power, coefficient of the variation of the power and other statistics of interest, e.g. expected length of the wind conditions favorable for the wind-energy harvesting. In this paper, wind speeds are modeled by means of a spatio-temporal transformed Gaussian field. Its dependence structure is localized by introduction of time and space dependent parameters in the field. The model has the advantage of having a relatively small number of parameters. These parameters have natural physical interpretation and are statistically fitted to represent variability of observed wind speeds in ERA Interim reanalysis data set.

In the literature typically cumulative distribution function (CDF) of wind speed W, say, is understood as the long-term CDF of the wind speeds at some location or region. The distribution can be interpreted as variability of W at a randomly taken time during a year. Weibull distribution gives often a good fit. Limiting time span to, for example, January month affects the W CDF simply because, as it is the case for many geophysical quantities, the variability of W depends on seasons. To avoid ambiguity when discussing the distribution of W, time span and region over which the observations of W are gathered need to be clearly specified. By shrinking the time span to a single moment t and geographical region to a location p, one obtains (in the limit) the distribution of

where S can be a month, a season or a year. Similarly the long-term CDF over a region A, say, is proportional to

In order to identify the distributions at all positions p and times t, vast amount of data are needed. Here the reconstruction of W from numerical ocean-atmosphere models based on large-scale meteorological data, called also reanalysis, is utilized to fit a model. The reanalysis does not represent actual measurements of quantities but extrapolations to the grid locations based on simulations from complex dynamical models. It is defined on regular grids in time and space, and hence convenient to use. In this paper, the ERA Interim data [

Modeling spatial and temporal dependence of wind speed is a very complex problem. Models proposed in the literature are reviewed in [

In Section 2, a general construction of non-stationary model for wind speed variability in time and space is presented. Section 3 presents probabilistic model for the velocity of storms movements. Then statistical properties of some storms characteristics are described in Section 4. The physical interpretations of the introduced parameters are also given in this section and in Appendix 1. In Section 5, on board measured wind speeds are used to validate the proposed model, where the long term CDFs of encountered wind speeds and persistence statistics are used. Total forty routes are used, see

In this section we shall introduce the transformed Gaussian model for the variability of wind speeds. In particular the transformation G making the transformed wind speed data

The wind speed

The parameter a is nonnegative with convention that the case

Mean and variance of

time. The temporal variability of mean and variance is approximated by seasonal components with trends defined as follows

Here t has units years. This type of model has been used in the literature, see e.g. classical paper [

Remark 1 For a fixed position p, the parameters a and m_{i} in Equation (3) are fitted simultaneously in such a way that the distance between yearly long-term empirical CDF of

More precisely for a wind data at fixed position p and parameter ^{*} that minimizes the distance is the estimate of a.

A table of a and m_{i} values as function of the location p is created. As additional parameters of the model will be estimated new columns with parameters estimates will be added to the table. For example, having estimated a and m_{i} the variance_{i} are saved in the table. More details of model estimations are given in Appendix 3.

Ten years of data _{i} and b_{i} in Equation (3), (4) in North Atlantic on a grid of 0.75 degree.

Usefulness of the proposed model relies on the accuracy of the approximation of

In the right plots of

instead. The values of the median for February and August are presented in two left plots of

Finally we check whether the regressions Equations (3) and (4) used to model seasonal variability of m and

where

A storm occurring at time t is a region where

Following [

where W_{t} is the time derivative of the wind speed,

The general assumption of this paper is that parameter a does not depend on time and changes much slower in space than the wind speed W varies, see

where

For a homogeneous Gaussian field the velocities have median values equal to

see [

Remark 2 The angle θ depends on properties of the covariance matrix Σ of the gradient_{x} and X_{y} become uncorrelated.

Let A_{θ} be the rotation by angle θ around the t-axis matrix making covariance between X_{x} and X_{y} zero. Then let denote by

where

Example 1 Let consider the following field

where

Obviously X_{x} is independent of X_{y} and hence θ = 90˚, see Remark 2. Further

while

In this simple example the median velocities agree with the velocity of the harmonic wave moving along the x-axis.

In

Main subject of the paper is development of a simple model describing variability of wind speeds time series encountered by a vessel or at a fixed location. In this section we will define the model and give means to estimate the long-term CDF of encountered winds; expected duration and strength of an encountered storm.

A ship route is a sequence of positions p_{i}, say, a ship intends to follow. We assume that a ship will follow straight lines between the positions having azimuth

A ship sailing along a route

where

The process

The long-term CDF of encountered wind speeds is defined by

The CDF given in Equation (16) could be be estimated by fitting an appropriate distribution to available data. (Weibull distribution is often used.) Alternative approach is to compute the theoretical CDF, viz.

Similarly as in Section 3 we will say that a ship encounters stormy conditions at time t if wind speed

The region of stormy conditions consists of time intervals when the wind speed is constantly above threshold u. The intervals will be called storms. Then let N_{u} denote the number of encountered storms. For example,

Let the number of encountered storms for which event (statement) A is true be denoted by

Next the theoretical, based on model, probability of event A, e.g.

The proposed model Equation (15) will be validated by comparing the empirical distribution of storms strength A^{st} and the average durations of storms with theoretically computed

introduced in [

will be used for validation purposes. In Equation (20), T^{cl} denotes time period when wind speed is uninterruptedly below the threshold u, i.e. a time period between storms. The Equation (20) will be proved in Appendix 2.

In order to evaluate Equation (19) and Equation (20), the formula for ^{e} can be computed using the generalized Rice’s formula [

see also [

Remark 3 Consider a stationary Gaussian process X with mean m and variance_{w} be the number of upcrossings of level w by X in time interval of length S then classical result of Rice [

Consequently the average distance between upcrossing of the mean level m by X is

From definition of the encountered wind speed process W^{e} it follows that the number of upcrossings of the level w by

In the following we shall use an additional parameter

and write Equation (24) in an alternative form

Note that if X^{e} is stationary, then

The proposed model is validated by investigating the accuracy of the theoretically computed distributions with the empirical distributions estimated from data. Firstly at fixed positions p the theoretical statistics of the storm characteristics A^{st}, T^{st} and T^{cl} will be compared with estimates of the statistics derived using ten years of hind- cast data. Secondly, the long-term wind speed distributions encountered by vessels are compared with the theoretically computed distributions using the model and the estimates derived from the hind-cast. The expected number of encountered upcrossing will also be used in the validations. However statistics of encountered storm characteristics will not be used in the validation process. This is because the wind speeds measured on-board ships are biased by captains’ decisions to avoid sailing in heavy storms, reported also in [

Consider a buoy at position p then

In

The values

The probabilities

Measurements of the wind speed over ground, i.e. ten minutes averages, recorded each ten minutes on-board some ships, are used to validate the proposed model. Since the data are recorded much denser than the hind-cast we have removed high frequencies from the signals (periods above 1.5 hour were removed using FFT). The data used in this study is limited to the North Atlantic and western region of Mediterranean Sea. The accuracy of the theoretically computed long-term distribution of encountered wind speed will be investigated.

First a single voyage operated in late August, shown in the top left plot of

In the left bottom plot of

shown. Based on the results presented in

In _{w} (the solid irregular line) are compared with the theoretical _{w} for wind speeds above 12 m /s.

Common experience says that wind speeds vary in different time scales, e.g. diurnal patten due to different temperatures at day and night; frequency of depressions and anti-cyclones which usually occur with periods of about 4 days and annual pattern. To follow the claim the transformed observed wind speed field

Now for any voyage one can compute parameters

More precisely, for a ship route

Here

The process

Obviously the integrals in Equation (28) have to be computed numerically. This is carried out using the following approximation

where Z_{ij}, _{j} forms an equidistant grid covering the domain of the kernel

The proposed model gives means for efficient simulation of wind speeds along any ship routes. The parameters

Note that parameters

A statistical model for the wind speed field variability in time and over large geographical region has been proposed. The model was fitted to ERA Interim reanalyzed data. Validation tests show very good match between the distributions estimated from the data and the theoretical computed one from the model. The model was also used to estimate risk of encountering extreme winds and the theoretical estimates agree well with the empirical one. Realistic wind profiles can be simulated using the model.

Support of Chalmers Energy Area of Advance is acknowledged. Research was also supported by Swedish Research Council Grant 340-2012-6004 and by Knut and Alice Wallenberg Stiftelse . The authors also would like to thank Wallenius Lines AB for providing us with onboard wind measurement data.

The parameter

Obviously

where

The variance

and following off-diagonal elements

The ships velocity

Obviously

where the encountered velocity, e.g. the difference between the ship velocity and the wind field velocity is, in the rotated coordinates, given by

In order to interpret components in Equation (35), we need to introduce some additional parameters that describe average size of windy weather regions and some irregularity factors.

Recall that windy weather conditions region at time t is the region consists all p where wind speeds exceeds the median

see Equation (22) and Remark 3. Obviously the values of parameters are slowly changing functions of position and time and that why we call them local sizes of windy regions. However if the field

Now by multiplying both sides of the Equation (35) by

where

are useful irregularity factors. Roughly, smaller values of the factors higher risks of extreme storms, see [

If p has rotated coordinates then

For a homogeneous wind field

Let assume that

Since

and hence

The parameters of the model have been fitted for the North Atlantic. Here the ERA Interim data has been used, although in future work we plan to also use data from satellite based sensors. A moment’s method and regression fit were employed to estimate the parameters. In this section we give a short description of the applied estimation procedure. In the following the measured wind speeds at a location will be denoted by

Step 1: For a fixed geographical location and ^{*} minimizing the distance between the two distributions is selected as an estimate of a. The corresponding mean

residual

Step 2: Estimation of signals_{1} is estimated as follows; first one filters out from

Step 3: For a signal _{j}, assuming stationarity of _{j}, estimates of

tions

Step 4: Estimation of_{i}. The functions are changing slowly with season but spatial variability can be high, particularly at coastal and inland locations. Consequently we fit six seasonal components to the covariances for each of positions p on a grid with mesh 0.75 degree. The components are estimated in a similar way as discussed in Step 3.