The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan ESTIMATION OF THE DESIGN WIND SPEED BASED ON UNCERTAIN PARAMETERS OF THE WIND CLIMATE Michael Kasperski 1 1 Head Research Team EKIB, Ruhr-University Bochum, 44780 Bochum, Germany michael.kasperski@rub.de ABSTRACT The analysis of the extreme wind climate suffers from such large statistical uncertainties that experts have not been able to agree on a general method to specify the design wind speed. The paper presents a new approach which accepts the fact that the 'true' parameters of the extreme wind climate can not be estimated from a few decades of observations. Consequently, it is understood that virtually every wind climate may hide behind the observed parameters, however, with different probability. These probabilities can be obtained from simulations and form the basis of constructing a non-exceedance probability of the uncertain design wind speed. The best estimate of the design wind speed then is obtained with a chosen confidence interval for avoiding an underestimation of the design wind speed. KEYWORDS: EXTREME WIND CLIMATE, DESIGN WIND SPEED, STATISTICAL UNCERTAINTIES Introduction Although the appropriate specification of the design wind speed is of vital importance for the wind resistance of a structure, experts have failed to agree on a method how to estimate this value based on meteorological observations of wind speeds. The discussions deal with almost any aspect of the problem, starting with the question on the basic ensemble (extremes over threshold or yearly extremes), the appropriate extreme value distribution (Gumbel or Reverse Weibull), the representative variable (v or v² or even v³), the fitting method (BLUE or least-square fit), the sampling period (calendar year or meteorological year) and so on [Kasperski, 2009]. Furthermore, many codes hide the design value by specifying a characteristic value and a partial factor. The large variety of code-concepts - with return periods from 5 to 1000 years and recommended partial factors from (at a first sight confusing) 0.93 to 1.6 - rather lects the basic uncertainty of experts than the different demands in regard to the reliability of the structures. The basic cause for never-ending discussions lies in the low statistical stability of an extreme value analysis which is based on only a few decades of meteorological observations. Today, this unfortunate feature can easily be identified based on simulations. The question arises, how a sufficient reliability in regard to wind load effects can be achieved. The paper introduces a new and consistent approach which considers the statistical uncertainties. Basic Parameters of the Wind Climate A general approach to the analysis of the extreme wind climate considers at least the following two basic variables: the number of events per year and the intensity of an individual event. The non-exceedance probability of a erence value per year is obtained as follows:
N (v v year) = p(n) p(v v ) N= 0 p (1) N - number of storms per year p(n) - probability of N storms per year p(v v ) - non-exceedance probability per event The non-exceedance probability of a erence value v over the anticipated (or design) working life of the structure is accumulated as follows: y s +L p(v v lifetime) = p (v v ) (2) i i=ys y s - starting year of exposure, L - design working life in years, p i - non-exceedance probability of v in year i For a stationary wind climate, i.e. for the non-exceedance probability of v being the same in each year, equation (2) becomes: p(v v lifetime) p(v v year) L = (3) A reasonable model for the number of events per year is obtained with the Poisson distribution: N λ p(n) = e N! λ λ - average number of events per year (4) For the intensity of the single events, the Generalised Pareto distribution may be used [Holmes & Moriarty, 1999]: s p(v v ) = 1-1+ k v v s 1/k (5) v s - threshold value s - scale parameter k - shape factor In case of a negative shape factor, the distribution has a finite upper tail which can not be exceeded. The corresponding largest value is given as: v = v s/k (6) max s
For k = 0, the exponential distribution is obtained: p(v v ) = 1 - exp ( v v ) s s (7) The two parameters s and k can easily be obtained from the mean value m and the standard deviation σ of the exceedances of the threshold value: 1 2 1 2 s = m 1+ ( m/ σ), k = 1-( m/ σ) 2 2 (8) Statistical Stability of Basic Parameters of the Wind Climate The statistical stability of the estimated parameters λ, s and k can be studied based on simulations. In the following, a wind climate is assumed with v s = 14.1 m/s, λ = 2, s = 2.0 m/s and k = -0.1. In figure 1, the probability densities of estimated λ, s and k values are shown based on each 10 6 independent runs, simulating 20, 30 or 50 years observation periods. As a matter of fact, none of the parameters can be identified with sufficient statistical stability. The deviations of the estimated values from the 'true' values are large enough to affect the estimation of the design wind speed. 0.12 0.20 Tobs = 20 yeras 0.08 0.06 0.04 0.02 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0-0.8-0.6-0.4-0.2 0.0 0.2 observed average number of events per year scale parameter s [m/s] slope parameter k 0.12 0.20 Tobs = 30 yeras 0.08 0.06 0.04 0.02 1.0 1.5 2.0 2.5 3.0 observed average number of events per year 1.0 1.5 2.0 2.5 3.0 3.5 4.0 scale parameter s [m/s] -0.8-0.6-0.4-0.2 0.0 0.2 slope parameter k 0.12 0.20 Tobs = 50 yeras 0.08 0.06 0.04 0.02 1.0 1.5 2.0 2.5 3.0 observed average number of events per year 1.0 1.5 2.0 2.5 3.0 3.5 4.0 scale parameter s [m/s] -0.8-0.6-0.4-0.2 0.0 0.2 slope parameter k number of events per year λ scale parameter s shape parameter k Figure 1: Statistical stability of estimated parameters of the extreme wind climate
In figure 2, the 90%-confidence intervals are shown for the three parameters λ, s and k. Even for an observation period of 100 years, the possible deviations from the 'true' values remain large. The statistical uncertainties in the estimated values of λ, s and k lead to uncertain estimates for extrapolated values of the wind speed. The uncertainties will increase with decreasing exceedance probability. This is shown in figure 3 on the example of the characteristic wind speed with p(v > v k = 1/50 per year) and the design wind speed with p(v > v d = 1/1000 per year) considering a range from 1.5 to 3 for the scale parameter and a range from -0.45 to 0.1 for the shape parameter. Even for the characteristic wind speed, the uncertainties are large. The underestimations may reach values larger than 15%; the overestimations may exceed a factor of 1.4. For the design wind speed, the deviations become even larger. Underestimations up to 25% may occur; overestimations may reach a factor of 1.7. 2.6 3.0 0.1 identified number of storms per year 2.4 2.2 2.0 1.8 1.6 identified scale parameter 2.5 2.0 1.5 identified shape parameter 0.0-0.1-0.2-0.3-0.4 1.4 20 30 40 50 60 70 80 90 100 years of observation 1.0 20 30 40 50 60 70 80 90 100 years of observation -0.5 20 30 40 50 60 70 80 90 100 years of observation number of events per year λ scale parameter s shape parameter k Figure 2: Influence of the observation period on the 90%-confidence interval of the statistical parameters of the extreme wind climate shape parameter k - - - -0.20-0.25-0.30-0.35-0.40 0.85 0.90 0.95 1.00 1.05 1.30 1.25 1.20 1.15 1.10-0.45 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 scale parameter s shape parameter k - - - -0.20-0.25-0.30-0.35-0.40 0.75 0.80 1.50 1.60 1.40 1.30 1.20 0.90 0.95 1.00 1.05 1.10 1.15 0.85-0.45 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 scale parameter s v k Figure 3: Ratio of the estimated to the 'true' wind speed for different combinations of s and k v d Best Estimate of the Design Wind Speed Accepting that the 'true' parameters can not be identified leads to the understanding that virtually every wind climate may hide behind the observed parameters, however, with different probabilities. The probability p, that a specific triple (λ, s, k) randomly leads to the observed triple (λ obs, s obs, k obs ), can be obtained from simulations. In the next step, for each triple (λ, s, k), a design wind speed can be estimated. Sorting the design wind speeds in ascending order with the corresponding probabilities allows constructing a probability distribution of the uncertain design wind speed:
( ) i ( des des, i ) j des ( λj j j ) p v v = p v, s, k (9) j=1 The final design wind speed can be obtained based on a chosen confidence level. Basic demand of any estimation of a design value of actions is to avoid an underestimation of the design value with an appropriate one-sided confidence. The probability of underestimation can be understood as the error probability; the complementary probability is the confidence. The question arises, how small the error probability should be. An intuitive choice would lead to small values say in the range of 1% to 5%. It is however important to note that with decreasing error probability the probability of an overestimation increases. The question to the appropriate error probability has been answered in civil engineering already in the scope of estimating the design value of the resistance based on experiments. The Eurocode [CEN, 2002] for instance recommends as confidence interval 75%, i.e. the corresponding error probability is 25%. Consequently, the best estimate of the design wind speed is obtained as the value in the sorted list which has an exceedance probability of 75%. Design Target Values For the specification of design target values it is reasonable to distinguish structural classes in regard to their importance level. The recently published version of ISO 4354 [ISO 2009] distinguishes four classes: A - B - C - D - structures with a special post disaster function (hospitals, schools, transmission lines, bridges) buildings which as a whole contain people in crowds (high-rise buildings, stadia, concert halls) normal structures (office buildings, commercial buildings, factories, residential buildings) structures presenting a low degree of hazard to life and other properties (farm buildings, house chimneys, roofing tiles) The corresponding target values for the ultimate limit state are specified in ISO 4354 with erence to a single year and are summarized in table 1. A more general approach ers the target probabilities to the projected working life of the structure. The respective values are also given in table 1. Both demands are approximately the same for 50 years projected working life. Strictly speaking, the target values er to the design wind load. For a general solution it is reasonable to demand the same target exceedance probabilities for the design wind speed. Table 1: Target values of the exceedance probability of the design value w des of the wind load or wind load effect for the ultimate limit state structural class A B C D p target / year [ISO 4354, 2009] 1/2000 1/1000 1/500 1/200 p target /working life [Kasperski, 2009] 0.025 0.20
Example of Application The proposed method is applied to the extreme wind climate at Düsseldorf, Germany, which is governed by strong frontal depressions. As representative value, the largest hourly mean wind speed of independent events is used. In figure 4, the identified probability densities and distributions are shown. These results can be used directly for the estimation of the design wind speed. As target value, an exceedance probability of 5% in the projected working life is chosen. The resulting design values and the corresponding yearly exceedance probabilities are summarized in table 2 for different values of the working life L for class B buildings. In table 3, the influence of the building class is shown for the lifetime L = 50 years. Changing the design working life from 10 years to 50 years leads to 14% increase of the wind loads. The wind loads for a class A building and a class C building differ by 10% assuming 50 years design working life. 0.999 relative frequency 0.40 0.35 0.30 0.25 0.20 observed fitted non-exceedance probabilty 0.99 0.9 0.5 observed fitted 0.1 0 1 2 3 4 5 6 7 8 number of storms per year 0.01 1 10 15 20 25 velocity [m/s] λ obs = 92/55 v s, obs = 14.1 m/s s obs = 1.803 m/s k obs = -63 Figure 4: Observed parameters of the extreme wind climate at Düsseldorf airport (1952-2007) Table 2: Design wind speeds based on the identified parameters, class B L [years] 10 20 50 80 p yearly (v > v d ) 1/195 1/390 1/975 1/1560 v d [m/s] 21.90 22.55 23.34 23.71 Table 3: Design wind speeds based on the identified parameters, L = 50 years class A B C D p yearly (v > v d ) 1/1975 1/975 1/475 1/225 v d [m/s] 23.89 23.34 22.72 22.03
Figure 5 presents the probability densities of the triples (λ, s, k) randomly leading to the observed values, assuming that the number of storms per year and the intensity of the storms are statistically independent. The full simulation involves 10 6 independent runs each corresponding to the sub-period of 55 years. The resulting possible range for λ is from 1.18 to 2.27. For the scale parameter, the simulation identifies a range from about 1.0 to 3.5 m/s, while the shape factor probably lies between -0.5 and +0.25. The cumulative probability distributions for the design wind speeds are shown in figure 6 for different values of the design working life and for different classes. The best estimates of the design wind speed are obtained as the 75%-fractile. The respective values are summarized in table 4 and 5. Additionally, the resulting wind loads are compared to values from the 'classical' approach. 0.04 10 of getting 92 events 0.03 0.02 0.01 60 70 80 90 100 110 120 130 true number of events in 55 years relative frequency 08 06 04 02 00 1.0 1.5 2.0 scale 2.5 3.0-0.6-0.2-0.4 0.2 0.0 shape 3.5 4.0-0.8 probability density for observing 92 events in 55 years for different λ-values joint-probability density for getting the observed parameters s obs and k obs from an ensemble with 92 events Figure 5: Randomness in the identified parameters 1.0 1.0 0.9 0.9 non-exceedance probability 0.8 0.7 0.6 0.5 0.4 0.3 10 20 50 80 non-exceedance probability 0.8 0.7 0.6 0.5 0.4 0.3 A B C D 0.2 0.2 0.1 0.1 0.0 20 21 22 23 24 25 26 27 28 29 30 design wind speed [m/s] variation of lifetime L, class B 0.0 20 21 22 23 24 25 26 27 28 29 30 design wind speed [m/s] variation of building class, L = 50 years Figure 6: Cumulative probability distributions for the design wind speed
There are large differences between the direct estimations based on the observed wind climate parameters and the new approach. This is mainly due to the large uncertainties in the shape factor. For a class B building, the best estimate of the design wind load is about 18% to 26% larger than the wind load estimated with the observed parameters. The influence of the building class leads to a similar range of differences. Table 4: Best estimates of the design wind speed and comparison to the wind loads obtained with the 'classical' approach, class B L [years] 10 20 50 80 v d [m/s] 23.74 24.71 25.98 26.62 wind load increase +17.5% +20.1% +23.9% +26.1% Table 5: Best estimates of the design wind speed and comparison to the wind loads obtained with the 'classical' approach, L = 50 years class A B C D v d [m/s] 26.95 25.98 24.98 23.93 wind load increase +27.3% +23.9% +20.9% +18.0% Acknowledgement Part of this work has been sponsored by the Federal Ministry of Research and Education under the scope of the joint-research project RegioExAKT. This support is gratefully acknowledged. Conclusions The low statistical stability of the extreme value analysis of wind speeds has led in recent decades to completely diverging concepts for the wind resistant design. A new approach is presented which considers the statistical uncertainties in the observed parameters for the extreme wind climate. Based on simulations, the non-exceedance probability of the uncertain design wind speed can be obtained. The best estimate of the design wind speed is given with a corresponding confidence level. The new method allows overcoming the discrepancies between the actually used concepts to specify the design wind speed. References C.E.N. (2002) EN 1990: Basis of structural design Holmes, J.D. & Moriarty, W.W. (1999) "Application of the generalised Pareto distribution to wind engineering", Journal of Wind Engineering and Industrial Aerodynamics 83, pp 1-10 ISO (2009) "ISO 4354 - Wind actions on structures" Kasperski, M. (2009) Specification of the design wind load - a critical review of code concepts", Journal of Wind Engineering and Industrial Aerodynamics, available online July 2009