The Annual Rate of Independent Events for the analysis of the extreme wind speed

J. Wind Eng. Ind. Aerodyn. 156 (2016) 104–114

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Journal of Wind Engineering and Industrial Aerodynamics journal homepage: www.elsevier.com/locate/jweia

The Annual Rate of Independent Events for the analysis of the extreme wind speed Alessio Torrielli a, Maria Pia Repetto b,n, Giovanni Solari b a b

Siemens Wind Power A/S, WP OF EN ES MEC MMI, Fiskergade 1-9, 7100 Vejle, Denmark Department of Civil, Chemical and Environmental Engineering, University of Genoa, Via Montallegro, 1, 16145 Genoa, Italy

art ic l e i nf o

a b s t r a c t

Article history: Received 20 November 2015 Received in revised form 19 July 2016 Accepted 20 July 2016

The statistical modeling of the extreme mean wind speed is a very controversial subject. Several models are currently being applied in the literature, based on different assumptions and supplying quite different results. This paper takes a step back towards the common root of many extreme value models: the Annual Rate of Independent Events (ARIE). Starting from an analysis of large datasets of synthetic wind observations, the ARIE is inspected from a new point of view and some features so far hidden are brought to light. Based on that, a new effective approach to model the distribution of the extreme mean wind speed is proposed. The reliability of the proposed model is discussed processing both large datasets of synthetic wind observations, and small-sized datasets representative of real conditions. ARIE sensitivity to uncertainties in the parent distribution is also evaluated and critically discussed. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Annual Rate of Independent Events Extreme value distribution Large datasets Long-term Monte Carlo simulation Mean wind speed

1. Introduction In the last 50 years, a great effort has been made to establish efficient statistical models to fit the extreme mean wind speed. The relevance of extreme value (EV) analysis on the economic assessment of the design wind speed and reliability of structures subjected to wind actions justifies the ongoing debate about the most suitable distribution model (Kumar et al., 2015). So far, however, the wind engineering community has not yet reached a common viewpoint and the EV distributions typically applied to predict design wind speed lead to scattered results, with uncertainties that increase progressively as the duration of the wind record decreases or the length of the return period increases (Lagomarsino et al., 1992; Lombardo, 2012). Working in this field, the authors have carried out a research program aimed at creating synthetic case studies through extensive Monte Carlo (MC) simulations of long-term time series of the mean wind speed. Initially, a large dataset consisting of thousands of years of synthetic mean wind speed observations was generated (Torrielli et al., 2011) through a simulation algorithm (Masters and Gurley, 2003 ) that matches the target power spectral density function (PSDF) perfectly, while the probability distribution function (PDF) slightly deviates from the target one. Such observations were fitted n

Corresponding author. E-mail addresses: [email protected] (A. Torrielli), [email protected] (M.P. Repetto), [email protected] (G. Solari). http://dx.doi.org/10.1016/j.jweia.2016.07.010 0167-6105/& 2016 Elsevier Ltd. All rights reserved.

to some of the most common EV distributions (Torrielli et al., 2013), in order to evaluate and discuss their effectiveness by removing any dummy effect due to the limited number of observations. Later, refined studies were carried out on modeling and simulating the macro-meteorological component of the mean wind speed (Torrielli et al., 2014). Particular attention was paid to the main harmonics associated with the spikes of the spectrum. In addition, a simulation algorithm able to generate sample functions with exact PDF and very accurate PSDF was implemented (Nichols et al., 2013). In such a framework it was proved that the stochastic or deterministic modeling of the main harmonics has a limited impact on EV analysis; on the contrary, the adoption of a simulation algorithm that perfectly matches the target PDF substantially improves the quality of the simulation with special regard to the tail of the parent distribution. This paper takes a step back from the common EV analysis, towards the base that gave origin to many EV distributions, i.e. the Annual Rate of Independent Events (ARIE). The ARIE is a cloudy concept and some confusion exists about its estimation, basically depending on the features of the source data and the processing method (Davenport, 1967; Cook and Harris, 2004; Torrielli et al., 2013). So, inspired by a contribution from Harris (2014) and starting from the analysis of large datasets of synthetic wind observations, the ARIE is inspected from a new point of view and some features so far hidden are brought to light. Based on that, a new effective approach to model the distribution of the extreme mean wind speed is proposed.

A. Torrielli et al. / J. Wind Eng. Ind. Aerodyn. 156 (2016) 104–114

In such a framework, Section 2 presents the MC simulation of two large datasets, each consisting of 12,740 years of synthetic wind observations that reproduce two different reference wind climates. Section 3 focuses on the ARIE, discussing its physical meaning in the light of its variability as a function of the mean wind speed; based on an analysis of the simulated large datasets, a simple analytical model of the ARIE is also presented, which is the key point of a new technique for modeling the distribution of the extreme mean wind speed. The accuracy of this technique is measured in Section 4 through the evaluation of design mean wind speed in realistic cases, where only a few decades of wind observations are available. Section 5 investigates the sensitivity of this approach to the uncertainties in the parent distribution. Section 6 summarizes the main conclusions and draws some perspectives for future research.

105

Table 1 Parameters and first order moments of the HW distribution.

SIM1 SIM2

F0

k

c [m/s]

μV [m/s]

sV2 [m2/s2]

0.118 0

1.155 2

3.091 5.561

2.592 4.928

6.637 6.637

The second simulation (SIM2) disregards wind calms, F0 ¼0, and considers a shape parameter representative of Northern European countries, k¼ 2. The scale parameter c is determined by inverting the following equation under the assumption that the variance of SIM2 coincides with that of SIM1: 2 ⎧ ⎛ 2⎞ ⎡ ⎛ 1 ⎞⎤ ⎫ σV2 = c2 ⎨ Γ ⎜ 1 + ⎟ − ⎢ Γ ⎜ 1 + ⎟⎥ ⎬ ⎪ ⎪ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ k k ⎩ ⎭ ⎪

⎪

(2)

Γ{∙} being the Gamma function.

2. Mean wind speed simulations This section illustrates two extensive MC simulations of 10-min mean wind speeds, corresponding to two ideal wind climates characterized by the same macro-meteorological spectrum and by different parent distributions. The macro-meteorological spectrum originates from the wind climate of an area in the central part of Italy of approximately 200 km in diameter. A full description of this wind climate and of the spectral combination technique adopted to derive its PSDF (Fig. 1(a)) are provided in Torrielli et al. (2011, 2013, 2014). The Hybrid Weibull (HW) model (Takle and Brown, 1978) is used to describe the parent distribution of the two simulations, whose cumulative distribution function (CDF) is given by

⎧ ⎡ ⎛ v ⎞k ⎤ ⎫ FV (v) = F0 + (1 − F0 ) ⎨ 1 − exp ⎢ −⎜ ⎟ ⎥ ⎬ ⎣ ⎝ c ⎠ ⎦⎭ ⎩ ⎪

⎪

⎪

⎪

(1)

where F0 is the rate of the wind calms, c and k are the scale and shape parameters, respectively, estimated by processing only the non-null wind observations. Table 1 lists the parameters and the first order moments adopted for the two simulations. The first simulation (SIM1) uses the parameters estimated by real observations consistent with the macro-meteorological spectrum, i.e. F0 ¼0.118, c ¼3.091 m/s and k ¼1.155. This last value is quite low with respect to the values usually adopted in the U.K. and in other Northern European countries (Torrielli et al., 2011) whereas it matches previous estimates of the Italian wind climate (Castino et al., 2003; Freda and Solari, 2010; Solari et al., 2012; Burlando et al., 2013) and with other analyses carried out by Holmes (2015).

Two datasets consisting of 12,740 years of continuous 10-min mean wind speeds were generated through the MC algorithm described in Torrielli et al. (2014). This algorithm is based on the shuffling technique proposed by Nichols et al. (2010) for the simulation of spectrally colored non-Gaussian processes. Such a technique allows the generation of sample functions with a marginal PDF that perfectly matches the target one, while the PSDF approximates the target spectrum almost exactly. The original algorithm has been modified in order to suitably handle harmonics associated with annual and diurnal cycles. Fig. 1(b) shows the PDF of single samples of SIM1 and SIM2 on a probability paper that involves ln{v} on the abscissa and ln{  ln [(1  FV)/(1 F0)]} on the ordinate, ln{∙} being the natural logarithm. In this figure, lines are used for the target PDFs whereas symbols are used for the empirical distributions. The empirical distributions are modeled through the plotting positions of the order statistical median, as explained in Torrielli et al. (2011). The matching between the empirical and target distributions for both SIM1 and SIM2 is perfect over the entire speed range; only slight deviations from the target distribution occur on the upper-tail region. Such deviations are due to the unavoidable statistical uncertainties deriving from the generation of large but finite-size samples.

3. The Annual Rate of Independent Events Given the random variable V with parent distribution FV, for the Law of Compound Probability (Cramer, 1946) the distribution of

Fig. 1. Macro-meteorological spectrum of the mean wind speed (a) and Hybrid Weibull plot of the parent distribution (b).

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the maximum value VT̂ of N independent observations collected in the epoch T is N Pr VT̂ < v = ⎡⎣ FV (v) ⎤⎦

{

}

(3)

Galambos (1978) showed that Eq. (3) also holds in the case of serially correlated data, provided that the total number of observations N is replaced by the number of independent observations; it follows that: rT Pr VT̂ < v = ⎡⎣ FV (v) ⎤⎦

{

}

(4)

where r is the rate of independent events per epoch; since T is usually measured in years, r is the Annual Rate of Independent Events (ARIE). Eq. (4) is the base of several EV distributions that have been developed in the literature. Even though it provides an exact relationship for any finite value of r, Eq. (4) is rarely directly applied for two main reasons. First, there is some confusion about the estimation of the ARIE, depending on the features of the source data and the processing method: Davenport (1967) analyzed the up-crossings of records consisting of 15-min averaged wind speeds at Brookhaven (USA) and estimated 876 independent events over the total number of 15-min samples per year, i.e. N  35,000; in regard of independent storms from hourly averaged speeds, r  150 is accepted for temperate climates in North Europe (Cook and Harris, 2004); much larger values r 17,000 result from fitting EV distributions to the annual maxima of synthetic 10-min averaged speeds extracted from a known parent (Torrielli et al., 2013). Secondly, the wind community is not completely aligned with the model of the parent distribution FV (Harris, 2004). Under the assumption of r -1, and depending on the upper tail of FV, Eq. (4) converges to one of three possible asymptotic distributions (Gumbel, 1958), namely the Fisher and Tippet Type I (FT1), Type II (FT2) or Type III (FT3). Alternative approaches have also been developed, such as the peak over threshold method (POT) (Pickands, 1975) that relies upon the assumption r -1, the FT1 penultimate (Cook and Harris, 2004, 2008) and the XIMIS (Harris, 2009) that rely on the assumption that r is constant. A new point of view on the ARIE was proposed by Harris (2014) based on an analysis of long synthetic series of the mean wind speed. He pointed out that the ARIE for a correlated process is not constant but varies significantly with the velocity. In particular, Harris proved that the distribution of the maximum mean velocity in the epoch T, i.e. VT̂ , can still be derived from Eq. (4) provided that r is a function of the velocity, namely: r (v ) T Pr VT̂ < v = ⎡⎣ FV (v) ⎤⎦

{

}

(5)

The dependency of r on the velocity implies that the subset of rT independent events per year in Eq. (4) is not always the same but changes according to the velocity. Low wind velocities are frequent and correlated; on the contrary, high wind velocities are generally rare and independent events, thus it is reasonable to suppose that the ARIE increases as the wind velocity increases. However, the ARIE cannot exceed N, i.e. the number of observations collected in T. In this framework Galambos (1978) proved that r in Eq. (4) tends asymptotically to a fixed value for very large wind velocity; later Harris (2014) asserted the existence of such a limiting value by analyzing the synthetic data of Boscombe Down. Furthermore, it is worth noting that the Normal Comparison Lemma (NCL) (Slepian, 1962; Berman, 1964, 1971; Cramer and Leadbetter, 1967) shows that the extreme distribution of a stationary correlated Gaussian series is close to Eq. (3) as v increases. Even though the NCL holds for Gaussian processes (Leadbetter et al., 1983), it is

reasonable to expect that the same may also occur for nonGaussian processes. If this is the case, on increasing the wind velocity the ARIE approaches the total number of observations N. The following analysis is addressed to numerically verify such a qualitative guess. 3.1. Numerical estimate of the ARIE The two wind speed simulations described in Section 2 are used to numerically estimate the ARIE. Unlike usual real cases, the parent distribution FV is fully known here, and the large dataset of synthetic wind observations allows a reliable estimate of the EV distribution. The numerical estimate of the ARIE is obtained through Eq. (3) by assuming T ¼1 year:

r (v) =

⎡ ⎤ ln ⎣ Pr V1̂ < v ⎦ exp ( − y) = ln ⎣⎡ FV (v) ⎦⎤ − ln ⎡⎣ FV (v) ⎤⎦

{

}

(6)

where Pr{V1̂ ov} is the distribution of the annual maximum speed and y¼  ln[  ln(Pr{ V1̂ ov})] is the Standard Reduced Variate (SRV). Note that the dependence on r of the velocity prevents the linearization of Eq. (5) regardless of the actual shape of r(v); thus, an unbiased estimation of the plotting positions of the velocity is not available. Faced with this shortcoming, the simple and wellestablished plotting positions attributed to Weibull (1939) is used to model Pr{ V1̂ ov} in Eq. (6):

Pm : M =

m M+1

(7)

where 〈Pm:M〉 is the best unbiased estimate of the mean probability of the mth smallest value in a ranked sample of size M. The wind speed simulations provide datasets consisting of M¼12,740 annual maxima. The sample size guarantees a proper modeling of Pr{ V1̂ ov}. In order to increase the accuracy of the lower tail, the distribution of the annual maxima is also derived from the 8-day maxima (Simiu and Heckert, 1996; Harris, 2009). A minimum separation time of 48 h is assumed to ensure the independence of 8-day maxima (Cook, 1986; Gusella, 1991). Fig. 2 shows the ARIE related to SIM1 (a) and SIM2 (b) (abscissa) versus the mean wind speed (left ordinate) and the associated SRV y (right ordinate), estimated from the annual (ryear) and the 8-day (r8-day) maxima. The vertical dash-dotted line corresponds to Ny ¼ 52,596, i.e. the number of wind observations per year, the speed values being separated by 10-min intervals. Both the estimates of r(v) show a 2-step pattern: an initial monotone increasing trend in the speed range up to 30 m/s (SIM1) and 21 m/ s (SIM2), followed by fluctuations around the vertical dash-dotted at Ny ¼52,596. Focusing on the initial increasing trend, slight detachments between the two estimates of r(v) related to ryear and r8-day occur for speeds less than 25 m/s (SIM1) and 19 m/s (SIM2); r8-day can be considered more reliable, since 8-day maxima are more numerous than the annual maxima in the middle-low wind speed range. Comparing the curves of r(v) related to SIM1 and SIM2, it is noteworthy that the greater shape parameter (k¼ 1.155 for SIM1 and k¼ 2 for SIM2) results in a stronger downward curvature of r(v) in the middle-low wind speed range. The fluctuations of r(v) at high wind speeds are due to the irregular upper tail of the empirical distributions of the annual maxima that was estimated by a large but finite-size sample. Furthermore, it is stressed that the uneven trend of the upper tail of Pr{ V1̂ ov} is amplified by the denominator of Eq. (6), which tends to 0 as the velocity increases. Assuming that these fluctuations are a numerical effect, and bearing in mind that Ny is a barrier that physically cannot be

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107

Fig. 2. Variability of the ARIE with the velocity from the annual and 8-day maxima of SIM1 (a) and SIM2 (b).

exceeded, it seems reasonable to conclude that the ARIE approaches the number of wind observations Ny for sufficiently high wind speeds. It is also noted that the NCL may raise the doubt that the limit trend pointed out by Fig. 2 is an artefact of the MC simulation; this does not seem to be the case because Nichol's method, unlike the MC method initially adopted by Torrielli et al. (2011, 2013), does not involve any static translation of a Gaussian process (Grigoriu, 1984). This remark further supports the applied MC simulation algorithm. Finally, comparing the numerical estimates of r(v) in Fig. 2 with the ARIE provided by Harris (2014), they show the same qualitative behavior: an initial increasing trend, followed by a segment characterized by fluctuations whose magnitude grows with the velocity (or y). Nevertheless, Harris's estimate of r(v) is lower than the limit value Ny ¼8766 related to the hourly speed observations at Boscombe Down. Observing that the extreme wind values obtained from Harris's simulation are unrealistically low, it is argued that this misalignment is due to Harris's questionable choice of disregarding the daily and annual harmonic components in the simulation of the mean wind velocity. 3.2. Analytical model of the ARIE A simple analytical model of the ARIE is proposed, taking into account the features highlighted in the previous numerical analysis. This model consists of a 2-piece function involving a monotonically increasing segment followed by a plateau:

⎧ r (v) = f (v) for v ≤ ur ⎨ ⎩ r (v) = Ny for v > ur ⎪

⎪

(8)

where f(v) may assume different shapes according to the type of maxima and parent distribution; Ny is the total number of the observations per year; ur is the wind speed above which the plateau starts. The model of r(v) in Eq. (8) is applied to the two previous numerical estimates of the ARIE, considering both annual and 8-day maxima. Table 2 summarizes the resulting parameters, while Fig. 3 superimposes the analytical models (AM) of r(v) to the empirical estimates in Fig. 2. The horizontal straight lines identify the value of the threshold ur estimated from annual (dashed line) and 8-day maxima (solid line). For SIM1 two different AMs are used to fit f(v) to the annual and 8-day maxima, respectively a linear and a quadratic function.

Table 2 Analytical modeling of the ARIE in Eq. (8). Simulation Maxima Equation for f(v)

Parameters a [s2/m2] b [s/m]

c [–]

ur [m/s]

SIM1

Annual 8-day

f(v) ¼b  v þ c – f(v) ¼a  v2 þb  v þ c 65.51

33,770.3  46,432.2 29.324  171.07 1,289.62 29.321

SIM2

Annual 8-day

f(v) ¼exp(b  v þ c) f(v) ¼exp(b  v þ c)

0.24277 5.71325 0.21724 6.29111

– –

21.080 21.243

Fig. 3(a) points out that the two AMs tend to converge as the wind speed increases, providing basically coincident estimates of ur, with the exception of the middle-low speed values. In such a range the analysis of the 8-day maxima should be preferred since they are more numerous than the annual maxima. For SIM2 the same model is used to fit f(v) to the annual and 8-day maxima. The resulting parameters of the two AMs in Table 2 are in good agreement with one another, as well as the estimates of ur. The coherence between the two AMs is also proved graphically by Fig. 3(b), especially for high wind velocities. 3.3. Extreme wind speed from the ARIE The distribution of the annual maximum speed Pr{ V1̂ ov} is derived from Eq. (5) by combining the analytical model of the ARIE with the parent distribution used in SIM1 and SIM2. The resulting distribution is hereafter referred to as the ARIE approach. Fig. 4 plots, on a Gumbel probability paper, the annual maximum distributions from the ARIE approach and from Eq. (1) under the assumption of Ny, Independent Observations (IO distribution). The IO distribution can be viewed as a limit case of the ARIE method, where r(v) ¼Ny for any wind velocity. The ARIE distributions show a perfect matching with the data over the entire speed range and coincide almost exactly with the IO distribution for v 4ur. In order to quantify the performance of the ARIE approach, the relative distributions are used to predict the 1:10-year, 1:50-year, 1:200-year and 1:2000-year design speeds. Table 3 lists the results (ARIE) and the reference values (data) derived from the empirical distribution of the annual and 8-day maxima. The results from the IO distribution are also given in this case. The accuracy of the

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Fig. 3. Analytical models of the ARIE from the annual and 8-day maxima of SIM1 (a) and SIM2 (b).

results provided by the ARIE approach is apparent: the deviations of the predicted design wind speeds from the reference values are always below 1.3% (absolute values). Furthermore, Fig. 4 and Table 3 reveal that the prediction of design speeds with return periods in the range 50–2000 years can be carried out without modeling the initial increasing trend of r(v), since such estimates fall within the plateau of r(v), where the ARIE distribution coincides with the IO distribution. Actually, the IO distribution might be used to derive conservative estimates of design wind speeds with low return periods. In fact, as shown in Appendix A, this method always leads to estimates on the safe side.

Table 3 Design speed from the distribution of the annual maximum speeds. Simulation

Maxima

Method

V10 [m/s]

V50 [m/s]

V200[m/s]

V2000[m/s]

SIM1

Annual

data ARIE data ARIE IO

28.310 28.350 28.297 28.340 28.472

31.457 31.580 31.449 31.580 31.579

34.118 34.165 34.075 34.165 34.165

38.452 38.377 38.453 38.377 38.377

data ARIE data ARIE IO

19.953 19.890 19.945 19.955 20.144

21.260 21.374 21.251 21.374 21.374

22.283 22.360 22.280 22.360 22.360

23.870 23.901 23.871 23.900 23.901

8-day

SIM2

Annual 8-day

4. Realistic case studies This section investigates the modeling of the ARIE and its applicability in realistic case studies, where only a few decades of wind observations are available. Different scenarios are explored, characterized by M¼ 10, 30, 50 years of synthetic wind observations. The parameters of the parent distribution in Eq. (1) are

a

8-day max - data

8

2000

Annl max - data

estimated case by case, in order to take into account the influence of the sample size. The results are listed in Table 4, together with the analytical law used to model the initial trend of the ARIE and the parameters estimated by analyzing the maximum values. Datasets consisting of 10–50 items are too small to provide a reliable modeling of the ARIE, therefore the dataset of the annual maxima

b

8

2000

500 200

6

500 200

50

4

50

2

10

2

10

0

2

0

2

8-day max - AM

6

Annl max - AM

IO Cramer

4

Design speed

-2

-2

-4

-4

-6

-6 5

10

15

20 25 30 velocity [m/s]

35

40

9

12

15 18 velocity [m/s]

21

Fig. 4. Gumbel plot of the distribution of the annual maximum speed relative to SIM1 (a) and SIM2 (b).

24

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109

Table 4 Parent distribution and modeling of the ARIE for sample of different size M. Simulation

M [yr]

Parent

ARIE equation

F0

k

c [m/s]

Parameters a [s2/m2]

b [s2/m2]

c [–]

ur [m/s]

SIM1

10 30 50 12,740

0.112 0.107 0.110 0.118

1.152 1.151 1.156 1.155

3.072 3.061 3.075 3.091

f(v) ¼ a  v2 þ b  v þc

73.77 65.95 66.61 65.51

 141.18  290.80  287.26  171.07

202.74 2854.34 2658.58 1289.62

27.624 29.757 29.622 29.321

SIM2

10 30 50 12,740

0.000 0.000 0.000 0.000

2.004 2.003 2.003 2.000

5.555 5.559 5.559 5.561

f(v) ¼ exp(b  v þc)

– – – –

0.17 0.20 0.21 0.22

6.90 6.55 6.41 6.29

22.817 21.646 21.255 21.080

is disregarded and the ARIE model is fitted only to the 8-day maxima. A right censorship of 20 and 16 m/s is applied, respectively, for SIM1 and SIM2. Table 4 points out that the convergence of the parent parameters towards the exact values cannot always be found on increasing M. Nevertheless, the parameters are always accurately estimated, because of the great number of observations; also in the case M¼ 10 years, the amount of data (  525,960) guarantees a proper modeling of the parent distribution. Even though slight deviations from the right parameters occur, which are likely ascribable to the uncertainty of the fitting procedure, these are taken into account in the modeling of r(v) through Eq. (6). The first column of Figs. 5 and 6 plots the numerical estimates of r(v) and the relative analytical models described in Table 4. The horizontal dash-dotted lines identify the position of the benchmark threshold ur corresponding to 29.3 m/s (SIM1) and 20.08 m/s (SIM2).The numerical estimates of the ARIE show a regular increasing trend at low velocity while fluctuations appear as the speed increases. These fluctuations are due to the finite-size dataset, therefore a right-censorship of 20 m/s (SIM1) and 16 m/s (SIM2) is applied in order to fit the analytical model of r(v) only to the regular section. Both figures point out that the AMs of r(v) approach the benchmarks as M increases. The ARIE approach is used to predict the distribution of the annual maxima by processing only M years of observations through the combination of the analytical models of r(v) with the parent distribution in Table 4. The second column of Figs. 5 and 6 compares the empirical, ARIE and IO distributions on Gumbel plots. For each M-case, two empirical distributions are provided, one related to the full dataset of 12,740 years of observations, the other related to the subset of M years of observations. The ARIE distribution is in perfect agreement with both the empirical distributions even though only 10 years of observations are analyzed. It is also noted that the ARIE and IO distributions gradually diverge as the speed decreases below ur, i.e. for design speeds with return period less than 10 years. The accuracy of the ARIE approach is tested in prediction of the 1:10-, 1:50-, 1:200- and 1:2000-year design wind speeds from the analysis of M years of observations. Table 5 compares these results with the reference values derived from the empirical distribution of the annual maxima associated with the full dataset. The estimates provided by the ARIE approach very closest match the reference values, with a deviation always below 0.7% (absolute value). Excellent estimates are derived even considering only 10 years of observations. 5. Sensitivity of the ARIE model The ability of the ARIE model to describe the distribution of the annual maxima by processing large datasets of synthetic wind

observations as well as small-sized datasets representative of realistic conditions was proved in the previous section. However, the previous applications are based on the perfect knowledge of the marginal distributions. In the case of real wind observations, instead, both the model and the parameters of the parent distribution are approximated. This paragraph investigates the sensitivity of the ARIE approach to the parameters of the HW-parent distribution. The high velocity range, v4 ur, corresponding to the plateau of the ARIE, is the most sensitive, because possible distortions in the parent are amplified by rising FV to the power of Ny. Since the velocity threshold ur assumes values with return period typically in the order of Rp ¼10–50 years (see Tables 3 and 5), the plateau of the ARIE has a key role in the estimation of the design wind speed Vd. In order to evaluate the error resulting from uncertain estimates of the parent parameters, Vd is expressed in terms of the return period Rp. This is done by replacing Eq. (1) in Eq. (5), in the case r(v)¼ Ny. It follows that:

⎡ 1 ⎞ ⎤1/ k ⎛ ⎢ ⎜ 1 − ( 1 − 1/Rp ) Ny ⎟ ⎥ Vd = c ⎢ − ln ⎜ ⎟⎥ 1 − F0 ⎜ ⎟⎥ ⎢ ⎝ ⎠⎦ ⎣

(9)

where F0, k and c are the parent parameters. Considering a generic parameter x, the relative error is defined as εx ¼ (x ̂ − x ) /x , where the overhead symbol represent the ‘estimate’. Thus, εVd|F0, εVd|k and εVd|c are the relative errors in the estimate of Vd resulting from the uncertain estimate of F0, k and c, respectively. They are given by

Vd̂ εV

d | F0

=

=

εV

=

k

− Vd

Vd

Vd̂ d |c

− Vd

Vd

Vd̂ εV

d |k

F0

c

⎡ εF log 1 − 1 / F 0− 1 ⎢ 0 = ⎢1 + Vd k ⎢ ⎣ c

(

( )

) ⎤⎥⎥

1/ k

⎥ ⎦

−1 (10)

−εk

⎛ V ⎞1 + ε = ⎜ d⎟ k − 1 ⎝ c ⎠

(11)

− Vd

Vd

= εc

(12)

where εF0, εk, εc are the relative errors involved by the estimation of the single parameter F0, k and c. Fig. 7 plots the relative errors in (Eqs. (10)–12) associated with the return periods Rp ¼50 years (a, c) and Rp ¼2000 years (b, d). It is worth noting that an uncertain estimate of F0 has a negligible influence on the prediction of Vd, irrespective of the return period.

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IO

Fig. 5. Modeling of the ARIE (column 1) and Gumbel plot of the annual maxima (column 2) from M-year datasets relative to SIM1.

Instead, the uncertainty on k and c significantly affects the prediction of Vd,: errors on c give rise to errors on Vd that are exactly the same involved by the parameter c, while errors on k propagate

in a more complex way, following the law in Eq. (11). The combined effect of εk and εc on the estimation of the design wind speed Vd is expressed by

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111

IO

Fig. 6. Modeling of the ARIE (column 1) and Gumbel plot of the annual maxima (column 2) from M-year datasets relative to SIM2.

Vd̂ εV

d | k, c

=

k, c

− Vd

Vd

−εk

⎛ V ⎞1 + ε = (1 + εc )⋅⎜ d ⎟ k − 1 ⎝ c ⎠

(13)

Fig. 8 plots the relative error in the estimate of Vd associated with Rp ¼50 years (a, c) and Rp ¼2000 years (b, d), resulting from uncertain estimates of the k and c parameters of the parent

112

A. Torrielli et al. / J. Wind Eng. Ind. Aerodyn. 156 (2016) 104–114

Table 5 Design speed from the ARIE approach by analyzing M years of synthetic wind observations. Simulation

M [yr]

V10 [m/s]

V50 [m/s]

V200 [m/s]

V2000 [m/s]

SIM1

10 30 50 12,740

28.472 28.273 28.293 28.297

31.580 31.580 31.580 31.449

34.165 34.165 34.165 34.087

38.3768 38.3768 38.3768 38.4793

SIM2

10 30 50 12,740

19.726 19.870 19.929 19.945

21.165 21.328 21.374 21.251

22.297 22.360 22.360 22.280

23.9005 23.9005 23.9005 23.8705

distribution of SIM1 and SIM2. It is worth noting that particular combinations of the errors εk and εc can generate very large errors in the prediction of the design speeds, e.g. εVd|k,c  0.46 for εk   0.1 and εc  0.1 (SIM1, Rp ¼2000 years). It is also apparent that the magnitude of the error does not change significantly when moving from Rp ¼ 50 to Rp ¼ 2000 years, thus revealing a slight variability with the return period. Instead, moving from SIM1 to

SIM2, εVd|k,c almost halves its magnitude, suggesting a strict dependence on the parent distribution; in particular, based upon the available data, this analysis suggests that the lower k is the larger the error on Vd is. Some recently published papers (Harris and Cook, 2014; Chiodi and Ricciardelli, 2014; Drobinski et al., 2015) critically discuss the use of the Weibull model, pointing out its many shortcomings in representing the parent distribution. Alternative distributions have also been proposed as parent, whose implementation in the ARIE approach in principle leads to different results. The extreme sensitivity of the design wind speed to the modeling of the parent distribution and the uncertainties of its parameters is a major shortcoming for the application of the ARIE approach in real cases. In spite of this, the authors believe that the ARIE approach retains at least two main advantages. Firstly, it provides a theoretical key for interpreting some inconsistencies involved in previous studies concerning EV distributions, these including some issues that remained pending in the results shown in Torrielli et al. (2013). Secondly, it provides a base and a framework for the development of consistent and robust advanced EV distributions.

Fig. 7. Error on the design speed relative to SIM1 (a, b) and SIM2 (c, d) induced by erroneous estimate of a single parameter of the parent.

A. Torrielli et al. / J. Wind Eng. Ind. Aerodyn. 156 (2016) 104–114

0.1

0.1

-0.1 -0.1

-0.05

0

-0 .2 -0 .14

28 0. 4 3 0.

0.05

-0.1 -0.1

0.1

0. 16

0. 1

0. 0 1 .0 -0 4 .0 0 .0 8 2

k

0. 1

Rp=50 yr

0

-0.05

0. 22

-0.05

0. 16 0.1 0.04 -0 . 02

0

0.05

-0 0. .02 04

-0 .2 -0 .08

-0 .14

0.05

b

-0 .08

a

22 0. 8 2 0. 4 Rp=2000 yr 0.3 4 0.

-0.05

0

c

0.05

c

0.1

-0.05

0

0.05

-0 .1 4

0.0 4

0.1

c

-0.1 -0.1

0.1

-0 . 02

-0.05

0.1 6

0.0 4 0.1

-0 . 08

0.1

-0.05

0.1 6

Rp=50 yr

0

0.2 2

0.1

-0.05

-0.1 -0.1

k

-0 .0 2 0.0 4

0

-0 .0 8

0.0 4

0.05

-0 .0 8

d

-0 .0 2

-0 .1 4

0.05

-0 .0 8

c

-0 . 02

0.1

k

0.1

Rp=2000 yr 0

0.05

0.2 2

k

113

0.1

c

Fig. 8. Error on the design speed relative to SIM1 (a, b) and SIM2 (c, d) induced by erroneous estimate of the parent parameters k and c.

6. Conclusions and perspectives Inspired by a contribution provided by Harris (2014) on the variability of the ARIE with the wind velocity of a correlated process, this paper digs into its physical meaning and sheds new light on some key features ignored until now, at least in the wind engineering field. This study leads to the proposal of a simple analytical model of the ARIE, consisting of a monotonically increasing function of the velocity followed by a plateau. The two sections of the ARIE modeling are separated by a velocity threshold. Combining this analytical model with the knowledge of the parent, a new the distribution of the annual maximum speed is obtained alternative to the commonly applied asymptotic analysis. The initial segment of r(v) is modeled by fitting a set of observed maximum values, suitably scaled through the parent distribution, to an appropriate mathematical function. The applications in this study highlight that the modeling of the initial segment of r(v) affects only he estimate of low return period design wind speeds. Differently, the shape of r(v) for high design wind speeds is fixed a-priori by postulating r(v) ¼Ny; accordingly, the upper tail of the EV distribution depends on the tail of the parent but not on observed maximum speeds. On the whole the ARIE approach provides extremely accurate

estimates of the distribution of the annual maximum by analyzing large datasets of synthetic wind observations as well as smallsized datasets representative of real conditions. This is true as long as the parent distribution is known with sufficient confidence. Of course, the exact knowledge of the parent distribution cannot be achieved by analyzing finite-size datasets. However, parametric analysis shows that the ARIE approach is very sensitive to the uncertainties in the parent distribution, since moderate errors in the estimate of the parent parameters may generate extremely large errors in the prediction of high design wind speeds. Such sensitivity clearly represents a major shortcoming for the practical application of this method. In spite of this, the ARIE approach retains at least two main advantages. On the one hand, it provides a key for interpreting some inconsistencies involved in previous studies concerning EV distributions, including some pending issues in Torrielli et al. (2013). On the other hand, the ARIE approach may be interpreted as a potential basis aimed at pursuing a refined modeling of the EV distribution, provided that a robust and consistent representation of the parent distribution is preliminarily developed. The discussion is open and the authors are working on both these aspects.

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Acknowledgements This research has been carried out first in the framework of the Project ”Wind, Ports, and Sea”, funded by the European Crossborder Program ”Italy-France Maritime 2007–2013”, then in the framework of the Project “Wind monitoring, simulation and forecasting for the smart management and safety of port, urban and territorial systems”, funded by Compagnia di San Paolo in the period 2016–2018.

Appendix A. Comparison between ARIE and IO distributions To demonstrate that the IO distribution provides an upper bound of the ARIE method, the definition of the SRV is applied to Eq. (5) in the case T ¼1 yr:

y = − ln

{ − ln⎡⎣ Pr { V̂ < v}⎤⎦} T

= − ln ⎡⎣ r (v) ⎤⎦ − ln − ln ⎡⎣ FV (v) ⎤⎦

{

}

(14)

For a given return period (or y), the corresponding speed value can be derived from the ARIE by using directly Eq. (14), otherwise from the IO distribution by replacing r(v) with Ny:

y = − ln ⎡⎣ r (vARIE ) ⎤⎦ − ln − ln ⎡⎣ FV (vARIE ) ⎤⎦

{

= − ln ⎡⎣ Ny ⎤⎦ − ln − ln ⎡⎣ FV (vCRMR ) ⎤⎦

{

}

}

(15)

Since r(v)rNy by definition, and ln{  } is a monotonic function, it follows that:

− ln ⎡⎣ r (vARIE ) ⎤⎦ ≥ − ln ⎡⎣ Ny ⎤⎦

(16)

Therefore, the equality in Eq. (16) holds only if:

− ln − ln ⎡⎣ FV (vARIE ) ⎤⎦ ≤ − ln − ln ⎡⎣ FV (vCRMR ) ⎤⎦

{

}

{

⇒ vARIE ≤ vCRMR

} (17)

Eq. (17) proves that for a given return period (or y) vARIE is necessary lower than vCRMR; so, the IO distribution always provides conservative estimates of the design wind speed.

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