This paper presents a rational method for the selection of the most suitable directional sectors in the analysis of extreme wind loads on structures. It takes into consideration the main sources of uncertainty stemming from sector selection and leads to the definition of independent and statistically homogeneous directional sectors.

This method is applied to the selection of directional sectors for the calculation of the design wind speed of a structure located at the mouth of the Río de la Plata. The results in the estimated reliability and costs were compared to those obtained with conventional engineering methods, revealing significant differences. It was found that the proposed method is a simple and objective tool for the selection of directional sectors, which comply with the working hypothesis of the directional models, and offers better guarantees for dimensioning than the use of more traditional engineering approaches for sectorial division.

Wind directionality effects have a well recognized impact on the characteristics of the extreme wind loads of structures. The methods for dealing with it usually involve the division of available data into sectors (whose statistical behavior is assumed to be homogeneous) and the evaluation of the extreme behavior of the wind velocity in each of them. The implicit decisions involved in this procedure (and its uncertainty) include (i) the identification of extreme values, (ii) the selection of the optimal model for data fitting, (iii) the definition of the directional sectors for calculation, and (iv) the characterization of the dependence among directional extremes. From all of them, the selection of the sectors, which is the subject of this article, has received the least attention.

Wind tunnel laboratories and building codes have developed multiple methods
in order to consider the influence of directionality on the estimation of
extreme wind speeds and wind-induced quantities, such as the “up-crossing”
method, the “worst case” method, the “storm passage” method, etc. (see,
for example,

Recently,

Using a priori defined divisions to this end results, in general, in
correlated directional sectors. This complicates the use of these methods
since the dependence structure among the extreme directional values of wind
speeds must be modeled using any of the existing approaches (e.g.,

In this work an alternative methodology that defines the distribution of the extreme wind directional velocity in a nonarbitrary manner is proposed. Both intrasectorial homogeneity and intersectorial independence conditions are imposed, among others, to obtain the directional sectors. Unlike previous approaches, this methodology results in uncorrelated sectors, so it is not necessary to use dependence models (e.g., copulas) and allows the approximation of the multivariate (directional) extreme distribution simply as the product of the marginals. In addition, this method assures directional sectors that contain data in consonance with the working hypotheses of the directional model for the extremes.

This methodology was applied to the study of extreme values of wind speed at the study zone of the mouth of the Río de la Plata, where directional effects are of particular importance. The effect of directional sector selection on the design wind speed of a structure was also estimated. These calculations and their consequences for project design reliability were compared with the results obtained with traditional engineering methods based on the use of divisions with equal size sectors and a northern direction of origin.

The rest of this paper is organized as follows. After defining the research problem, Sect. 2 delimits the framework and specifies the main sources of uncertainty considered. The methodology for the selection of directional sectors is then explained. The case study in Sect. 3 describes the wind characteristics in the study zone, followed by the quantification of the impact of sector selection, based on indicators for each source of uncertainty. These results are compared to those obtained with conventional engineering methods. In Sect. 4, a simple example is used for illustrating the potential consequences of the selection of directional sectors for project design reliability. Finally, Sect. 5 presents the main conclusions that can be derived from this research.

The selection of calculation sectors affects the estimates of directional extreme values, which may impact the evaluation of project costs and structural reliability. The main factors that influence the result are the following: (1) the procedure followed to identify the extreme events of the sector samples, (2) the validity of the model used to characterize extreme behavior, (3) the goodness of parameter estimation, (4) the capacity of each model to represent extreme behavior in the total amplitude of the corresponding sector, and (5) the validity of the dependence model among extreme values in different sectors. All of these factors are in turn conditioned by the quantity of available data and their directional distribution.

For the selection of calculation sectors, this paper describes a procedure
that considers the main sources of uncertainty stemming from the choice of
sectors. Firstly, the candidate divisions are limited to those whose sectors
are compatible with the selected model of extreme values and which have a
given minimum quantity of information. Secondly, the consequences of this
selection are evaluated for each division by means of indicators that
characterize the intrasectorial homogeneity of the samples, the uncertainty
of the estimates of directional extreme values, and their intersectorial
independence. Finally, the division with the best overall behavior is
selected, based on the set of indicators. This methodology is outlined in the
flowchart in Fig.

Methodology for sector definition, based on the sources of uncertainty in the calculation of reliability in a system subjected to directional extreme values.

Extreme events are isolated in each sector for different possible angular
divisions by means of any appropriate technique, such as peaks over
threshold (POT), block maxima

Regarding the first requirement, the minimum acceptable quantity of data in
each sector (or semi-sector) should be such that the probability of a Type II
error (a false negative finding) in the statistical hypothesis tests that are
used in the proposed method is less than a given value

The next step involves the evaluation of the consequences of sector selection on (i) the intrasectorial homogeneity of the samples, (ii) the uncertainty of the estimates of directional values, and (iii) their intersectorial independence. To this end, the use of three indicators based on standard statistical analysis and which are measured on a 0–1 scale is proposed. This approach is also compatible with the use of other indicators specific to the problem under consideration.

The first indicator characterizes the variability in the statistical behavior of extreme events along the arc of each sector. Significant discrepancies among the subsamples of a sector can indicate the presence of different populations, which is incompatible with the hypotheses of the model used. The second indicator reflects the uncertainty of the fit of extreme values by analyzing their asymptotic distribution. Finally, the third indicator evaluates the incompatibility of the sectors with the independence among sectorial extreme values. This independence occurs when each storm event is restricted to one sector and does not move to neighboring ones.

The division selected is the one that shows the best overall performance as reflected in the set of characteristics evaluated by the indicators. This leads to the creation of a new global indicator, which is a function of the other three and which allows for sorting the candidate divisions according to the selected criterion.

In order to compare the statistical homogeneity of extreme values in
different regions of the same sector, the sector is divided into two
subsectors of equal amplitude (

As an indicator of this characteristic in a sector, the

This indicator gives a measure of the uncertainty stemming from the choice of
sectors, in the estimations of extreme values. This uncertainty is
characterized by analyzing the asymptotic distribution of the extreme values
within the context of delta method hypotheses (e.g.,

In the case of completely independent sectors, the following relation is verified:

The

In order to consider the previous indicators as a whole, the expression in
Eq. (

The procedure involves the following steps:

identification of extreme events per sector;

definition of requirements and conditioning factors (Sect.

requirements regarding data quantity,

requirements for the validity of the models by sector;

preselection of the sets of sectors that meet these requirements;

selection of the calculation sectors
(Sect.

evaluation of the indicators in each candidate division
(Sect.

selection of the set of sectors with the best overall indicator value.

The study site is located in front of the mouth of the Río de la Plata
(36

Location of the study site at the mouth of the Río de la Plata.

Atmospheric circulation in the area is controlled by the South Atlantic
high-pressure system, which brings hot, humid air to the estuary. In
addition, cold-air systems from this anticyclone bring masses of cold air to
the zone approximately every 4 days. This means that wind direction
frequently varies since northeasterly winds alternate with southeasterly
winds every few days

Furthermore, intense storms, known as

The research data used in this study come from reanalysis time series of the
ERA-Interim program

Independent events were identified by applying the POT
method with a time window of 5 days between storms to the omnidirectional
data. In this way, a total of 270 storms were isolated. Each storm was
characterized by its maximum wind speed

There were significant variations in direction with regard to the values
where the peaks occurred. More specifically, there were displacements greater
than 45

To take into account this directional dissipation in the extreme-value
modeling

The generalized extreme value (GEV) model is often chosen to describe the extremes of natural agents in
wind engineering

To consider the displacement of the storms among sectors, the distribution
of the annual maxima in a given sector

Storms were identified as the clusters of sequential values of the omnidirectional wind speed exceeding a given threshold, with a time lag of at least 5 days between their peaks to ensure their independence. In this way, the sequences of wind speeds and directions of the 270 omnidirectional storms were isolated.

From the set of 270 storms, the subset of those whose direction belongs at
some point to the sector under consideration was selected. The number of
these storms was

For each one of the

A GPD with parameters (

The Poisson–Pareto model from Eq. (

Directional sectors resulting from applying the different selection criteria.

The power curves of the Anderson–Darling test

Furthermore, the study did not consider any division containing sectors that
rejected the null hypothesis of the Anderson–Darling test

Finally, some practical limitations to the size of the sectors were
considered for this case study in order to reduce the range of divisions
considered and to limit computational costs. Specifically, the sector
amplitude was restricted to the range from 30 to 300

The effect of different criteria for sector selection on the extreme-value
models used to fit the available data was characterized. From now on, the
following nomenclature is used to present the results: C0 is the criterion
proposed in this work (whose definition is summarized in
Sect.

Sectors defined according to criteria T90, T45, and C0.

When applying criteria C0–C3, the selection requirements
(Sect.

Figures

Sectors defined according to criteria 1–3.

Also, for each criterion there is a scatter plot showing the data that were
used for the fit of the directional extreme values. During any given storm,
wind direction may vary in more than one sector and, in these cases, every
storm produces more than one extreme value (one for each sector in which it
has data). The number of points in each sector is indicated with the letter

A comparison of the accuracy of the extreme-value models that were used to
fit the directional data of each criterion is shown in Fig.

Figure

Values of indicators for each criterion considered.

Criteria C0 to C3 lead to divisions with three sectors in all cases: a larger one, which roughly covers the W–SE region, and two more in the range where larger and more frequent storms occur. These divisions are consistent with the analysis of regional wind characteristics. Furthermore, the divisions of criteria T90 and T45 show worse results in all indicators with striking differences in intrasectorial homogeneity and independence of the sectorial extremes.

Quantile plots for the fitted model in each sector (empirical quantile
on the

This section evaluates the effect of directional sector selection on design
values and structure reliability. For this purpose, we used the simple
example of a structure with three straight sections whose design wind speeds
should be adapted to the directional variability in the extreme values of the
agent. The normal directions of the sections form angles of 60,
180, and 300

Failure of the whole structure occurs when at least one of the sections fails.

The failure mode does not depend on the direction of the agent's incidence
but rather on the section type. In this case, the failure in each section
occurs when wind action in the normal direction

The response coefficients of the structure are all equal to 1.

Each directional sector isolates a population of the agent's extreme values with homogeneous characteristics.

Indicators

Given the requirement that the overall failure probability in the useful life
of the structure is lower than a given value

Failure regions in each section and their relation with the sectors
of criteria T90

We compared the design obtained with the sectors defined according to
criterion C0 with the result of applying (a) the omnidirectional analysis
(in which the design value of the wind speed is the same for the three sections)
and (b) the sectorial divisions obtained from criteria T90 and T45. For
each section and criterion, Fig.

Parameters of the optimization problem for criteria T90, T45, and C0.

Assuming a section

For the omnidirectional analysis we fit a GPD to the omnidirectional POT regimen. Next, we used a Poisson–Pareto model to estimate the distribution for the annual maxima in the range of directions that can cause failure of each sector. To take into account the width of the resulting directional ranges, the Poisson parameter was evaluated into them. For the directional results we followed the same scheme but the GPD was fit to the directional POT regimes according to the divisions of each criterion.

Table

The optimization problem was solved with an interior point algorithm

Optimization results for criteria T90, T45, C0, and omnidirectional
analysis for

All the comparison criteria (T90, T45, and omnidirectional) show design
wind speeds for each section that differs from those of criterion C0.
Consequently, directional sector selection can be decisive in the project
design if cost is a relevant factor. The greatest discrepancies occurred in
section 1. With criterion T90, there are variations of 6.54 % with

By definition, the design wind speeds corresponding to each criterion fulfill
the requirements of the optimization problem, in accordance with their
respective probability models. However, only the sectors of criterion C0 have
been selected objectively in consonance with the working hypothesis of the
directional model for the extremes and, therefore, they offer better
guarantees for dimensioning. Thus, in order to compare the impact of design
wind speeds on both the failure probability during the useful life of the
structure and the cost function (Eq.

Table

Solving the optimization problem for criterion C0 led to solutions far from
the edge of the validity region. The design wind speeds obtained with the
other criteria increase the probability of failure but fulfill the design
requirements, with the exception of the T90 criterion. Particularly noteworthy
is the result with T90 for

On a final note, the selection of the threshold is an additional source of
uncertainty that can affect the results. Different thresholds imply different
definitions of what is an extreme value and hence result in different
directional sectors. If there is no strong criterion for an a priori
selection of the threshold, a sensitivity analysis of the results is
recommended. In this situation,

Failure probabilities and cost function
(Eq.

This paper has described a procedure for the selection of directional sectors in a nonarbitrary manner, considering the following sources of uncertainty: (1) the validity of the model used to characterize the extreme behavior of the sector samples, (2) the goodness of parameter estimation, (3) the capacity of each model to represent extreme behavior in the total amplitude of the corresponding sector, and (4) the validity of the working hypothesis of the independence among extreme values in different sectors.

This research led to the following conclusions. Firstly, the results of modeling the directional extreme behavior of natural agents can be affected by the choice of directional sectors used for calculation. Secondly, the selection of sectors without considering the extreme properties of the data negatively affects the confidence in the estimates on which the project design is based. This makes the use of sectors of equal amplitude not recommended, without sufficient justification. In this sense, the method presented in this research is an objective tool for the selection of directional sectors, which also facilitates the application of standard calculation procedures since it leads to homogeneous and uncorrelated sectors. The results obtained show that it offers better guarantees for dimensioning than the use of more conventional engineering approaches based on divisions arbitrarily chosen because it reduces the sources of uncertainty in the estimation of design values. Furthermore, this method also assures that sector division by direction is in consonance with the working hypotheses of the directional model. This means that quantification of probabilities is applied within the validity range of this model.

The method was applied to the selection of directional sectors for the calculation of the design wind speed of a structure located at the mouth of the Río de la Plata. The impact that choice of method would have on the failure probability during the useful life of a structure was analyzed, and the results with the proposed method were compared to those based on divisions with equal size sectors and a northern direction of origin. It was found that the procedure followed can have significant repercussions on the cost estimate and reliability, and thus condition the viability of an investment project. Consequently, decisions regarding sector selection should be an integral part of the project design process.

The data used in this study are available in

The doctoral studies that led to this work were partially funded by Abengoa Research within the framework of a cooperation agreement between the company and the Environmental Fluid Dynamics Research Group of the University of Granada. Edited by: Mauricio Gonzalez Reviewed by: two anonymous referees