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A new wind zone map of Germany
Journal of Wind Engineering and Industrial Aerodynamics 90 (2002) 1271–1287
A new wind zone map of Germany Michael Kasperski Department of Civil Engineering, Ruhr-University Bochum, D-44780 Bochum, Germany
Abstract The revised version of the wind zone map of Germany is based on a refined extreme value analysis for three physical phenomena which are leading to high gust wind speeds: the typical storms induced by strong frontal depressions, thunderstorms and additional gust fronts especially in weaker frontal depressions which are induced by down-drafts of cold air and/or by rain and which are not fully covered by the usual flow field models of frontal depressions. Individual ensembles for each of these storm phenomena have been sampled for a large number of stations. Furthermore, it has been tried to increase the statistical stability of the estimated representative values by summarizing stations with a similar wind climate. The fitting of the theoretical extreme value distribution is performed for only the right tail of the observed traces of non-exceedence probabilities. The proposed new map presents five wind zones with characteristic values from 22.5 to 32.5 m/s. The zoning follows mainly the boundaries of the administrative districts. Compared to the preceding wind zone map, the new map specifies for many sites smaller wind loads. Additionally, directional effects have been analysed, however, the results are too inhomogeneous to be included in the code. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Wind climate; Extreme value analysis; Design wind speed; Directional effects
1. Introduction In the scope of the Eurocode 1, part 2.4 wind loads, the respective national authorities are responsible for providing a wind zone map and other meteorological information which are required to specify the design wind load for a structure. In its Annex A, the actual draft of the Eurocode contains a wind map for Germany as shown in Fig. 1. There are a number of points which have suggested a new and refined analysis of the extreme wind climate in Germany. The proposed map in Fig. 1 is based on an analysis including data only up to 1980, i.e. today 20 further years of data are E-mail address: [email protected] (M. Kasperski). 0167-6105/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 0 2 ) 0 0 2 5 7 - X
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Fig. 1. The German wind map as proposed in the Eurocode 1.
available which could be included in a reviewing analysis. The then used method to specify the characteristic values has not been based on an extreme value analysis, but it was tried to extrapolate from the observed trace of the average number of hours per year above a certain threshold [1]. A single storm might then be included in this type of statistics with more than one value, i.e. the statistics contain dependent or correlated data. Furthermore, some doubts have occurred with regard to having cities with obviously different wind climates in one and the same wind zone, like e.g. Rostock and Bremen and Dusseldorf. . Finally, it might be noted that the characteristic values of the wind speeds jump at the boundaries to the neighbouring countries. This inconsistency is to one side only, i.e. the wind speeds are higher in Germany than in the neighbouring countries. This might be explained with the fact, that the method which has been used in Germany has not been used anywhere else, thus probably leading to a systematic bias in the analysis. If, however, a revised analysis of the German data is able to overcome this shortcoming, remains doubtful. Experts neither in Europe nor worldwide have been able to agree on one standardized method to analyse the wind climate properly, although the specification of the design wind speed is one of the central and important points in the codification of the wind loads. All these reasons convinced the Deutsche Institut fur . Bautechnik (DIBt) (German Institute of Structural Engineering) to sponsor a research project to develop a revised version of the German wind zone map. 2. Method of analysis The data basis is provided by the Deutsche Wetterdienst (DWD) (German Weather Service) in terms of 24 hourly mean wind speeds per day, the corresponding
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mean wind direction and the daily maximum gust wind speed (corresponding to a 2– 3 s gust) plus the respective time of occurrence. Altogether, 183 stations have been included in the study. On an average, the observation period is 40 years. The distribution of the stations over Germany is plotted in Fig. 2. The wind climate in Germany is mainly governed by two storm phenomena, frontal depressions and thunderstorms. Basically, for a proper extreme value analysis, the independent physical phenomena have to be separated and individual ensembles have to be sampled [2]. The sorting into several ensembles is accomplished by specifying respective thresholds, e.g. one threshold for the mean wind speed vm;lim and one threshold for the gust wind vg;lim : If the actual mean wind speed exceeds the threshold vm;lim ; obviously, a storm induced by a frontal depression is obtained. Usually, this threshold is exceeded for several hours which have not to be
Fig. 2. Meteorological stations used for the analysis of the extreme wind speeds in Germany.
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consecutive, i.e. some storms ‘take a break’ and slow down before they speed up again. The highest intensity may last for more than 1 h (Fig. 3). The task for the extreme value analysis now becomes to identify independent storm events for the ensemble. Clearly, it is not sufficient to sample only yearly extremes, since storms tend to occur in clusters or families, which means that the second or third strongest storm of 1 year might be considerably stronger than the strongest storm of another year (Fig. 4). Therefore, the analysis should be based on all extreme events. Each extreme event is sampled in the ensemble with its largest value, i.e. only the strongest storm hour of a frontal depression is used. The remaining storm hours are dependent data and should therefore not be included in the ensemble of independent events. Strictly speaking, they have to be included in the load model by introducing two further variables, which are the duration of a storm and the relative intensity of the 2–nth storm hour [3]. Since these variables are not influencing the specification of the design wind speed itself, but only the exceedence probability of the design wind load and thus the specification of the appropriate fractile of the aerodynamic coefficients, this aspect is not discussed in this paper. The identification of an independent storm event is based on an assumption about the longest possible break during a single storm. For the actual study, a value of 24 h has been chosen. Assembling the second ensemble is much easier. A thunderstorm is identified by a gust wind speed exceeding the threshold vg;lim while the corresponding hourly mean is well below the threshold vm;lim : Since per day only one value is sampled, the such identified thunderstorms are already independent events.
Fig. 3. Examples of time histories of hourly means in a frontal depression.
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mean wind speed [m/s] 25
20
15
yearly extremes
10
5
0 55
60
65
70
75
80
85
90
95
00
year of observation
25
mean wind speed [m/s]
20 independent extremes vm > 14.0 m/s
15
10
5
0 55
60
65
70
75
80
85
90
95
00
year of observation Fig. 4. Time distribution of the yearly extremes and the independent extremes with threshold 14.0 m/s (Dusseldorf . airport 1952–1999).
When specifying a design wind load, extreme wind speeds are combined with extreme aerodynamic coefficients which have been obtained from a properly scaled boundary layer wind tunnel experiment. In the respective modelled flow, a certain level of gustiness is included and is – of course – influencing the extreme aerodynamic coefficients. As a matter of fact, only turbulent wind fields as they are expected in strong frontal depressions are modelled. As typical value of the gustiness in open country at 10 m height, an average gust factor of 1.6 is obtained. It should be noted that this gust factor is a random variable, i.e. it has a certain scatter in the wind
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tunnel flow as well as in nature (Fig. 5). If the turbulence intensities and the spectral densities modelled in the wind tunnel flow meet the corresponding parameters specified in the theoretical model, the shown random gust factors represent wind tunnel results as well. Extreme loads in a wind tunnel experiment therefore contain a certain contribution from the gustiness of the oncoming flow. Specifying the design wind load for a frontal depression based on an extreme value distribution for the observed gust wind speeds would therefore include this scatter twice. A consistent wind load model for wind loads induced by a strong frontal depression has therefore to be based on the maximum mean wind speeds in frontal depressions (either hourly means or 10 min means). For the induced wind loads in thunderstorms a consistent load model – i.e. an analysis of the gust wind speeds in thunderstorms plus an appropriate aerodynamic coefficient obtained from a wind tunnel test simulating wind fields in thunderstorms – does not exist. Therefore, as an interim solution, the gust wind speeds of thunderstorms are translated to equivalent mean wind speeds of frontal depressions. Thunderstorms and strong frontal depression differ considerably in their duration. As already mentioned, this difference in duration is affecting only the specification of the aerodynamic coefficient and not that of the design wind speed. A closer look at the recorded wind speed data reveals the fact that with the abovedescribed procedure some high gust wind speeds are not included and covered with the two sampled ensembles. This can be seen in a plot of the observed gust factors. The scatter in the observed gust factors is much larger than is predicted from the theoretical model. Typically, these high gusts occur in weaker frontal depressions before the strongest storm hour. They are induced by flows similar to flows in thunderstorms, i.e. cold air eventually together with rain is falling down and leads to a gust front which is riding in front of the storm. A third ensemble has to be introduced to cover these additional gusts. It is obtained by specifying a third threshold variable which is the gust factor. In this study, a threshold value of 1.8 is
Fig. 5. Comparison of observed gust factors to the scatter of gust factors as obtained from a numerical simulation based on the theoretical model of a flow field corresponding to open country. (a) Observed gust factors at Dusseldorf . airport from 1960–1995 with expected mean gust factors for a turbulent flow field corresponding to open country. (b) Random gust factors obtained from a numerical simulation of a turbulent flow field corresponding to open country with a mean wind speed of 20 m/s.
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introduced, i.e. if during a weaker frontal depression a gust wind speed has a higher gust factor than 1.8, the corresponding gust is sampled in the third ensemble of ‘gust fronts’. The next step is to perform order statistics to estimate the non-exceedence probabilities of the maximum hourly mean wind speeds in frontal depressions and of the gust wind speeds in thunderstorms and ‘gust fronts’, respectively. For consistency of the analysis, the gust wind speeds are ‘translated’ to equivalent hourly mean wind speeds by dividing the observed gust wind speeds with a gust factor of 1.6. In case of yearly extremes, the non-exceedence probability of an observed wind speed in a specific ensemble is obtained as follows: pðvpvi Þ ¼
ia ; N þb
ð1Þ
where i is the rank in list of ascending order, highest value rank N; lowest value rank 1, N is the ensemble size and a; b the parameters to define the ‘optimum’ plotting position [4]. Jensen and Franck [5] proposed for the ensemble with more than one storm per year to estimate the non-exceedence probability by introducing additionally an exponent which is obtained as the average number of storms per year, i.e. the nonexceedence probability becomes: i a N=K pðvpvi Þ ¼ ; ð2Þ N þb where K is the number of observation years. The respective traces are plotted in Gumbel probability paper. As a matter of fact, the choice of the threshold affects the number of sampled storms and therefore the trace of the non-exceedence probability. Basically, only the extremes of the sampled events should follow an extreme value distribution, i.e. the fitting of the observed trace to a theoretical expression should be performed only for the right tail (the largest wind speeds). This region is less affected by the actual sample size. An ‘optimum’ sample size can be specified by demanding that all sampled events should be displayed in the probability plot which should have a ‘symmetric’ vertical axis, i.e. the smallest probability value plow and the highest probability value phigh should add up to unity. Then, the ‘optimum’ number of events Nopt in K years is obtained from: 1 a Nopt =K ! plow ¼ pðvpv1 Þ ¼ : ð3Þ Nopt þ b For a range of the probability paper from non-exceedence probability 0.001–0.999 per year, these optimum ensemble sizes are given in Table 1. It is worth mentioning that the optimum sample size depends on the actual length of the observation period.
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Table 1 Optimum ensemble size in dependence on the length of the observation period Length of observation period (years) Optimum ensemble size Nopt
10 20
20 35
30 48
40 61
50 74
60 86
70 97
80 109
90 120
100 131
As basic model for the extreme value distribution, the reverse Weibull distribution (extreme value distribution type III) is introduced: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v m1=t F ðvÞ ¼ exp f1 f2 ; f1 ¼ Gð1 þ tÞ; f2 ¼ Gð1 þ 2tÞ f12 ; s ð4Þ where G is the gamma function, m is the mean value of the extreme wind speeds, s is the rms value of the extreme wind speeds and t the curvature parameter. This distribution has a finite upper tail which seems from geophysical aspects especially appropriate for extreme wind speeds [6]. The extreme value is given as: xmax ¼ m þ s f1 =f2
ð5Þ
The type III distribution has been applied already successfully to some stations in Germany [3,7], to a large number of stations in the USA [8] and to some stations in Australia [9]. If t becomes 0, the Gumbel distribution (extreme value distribution type I) is obtained: " " #!# p xm F ðxÞ ¼ exp exp g þ pffiffiffi ; ð6Þ 6 s where g is the Euler constant=0.5772. This distribution has an infinite upper tail. The fitting of the data to the theoretical distribution is based on a least linear error sum for the wind speed deviations between observation and prediction from the theoretical model. As mentioned above, the fitting procedure is using only the right tail of the observed non-exceedence probabilities. Typically, the fitting range starts from non-exceedence probabilities larger 0.3, which means that in 2 years on average three extreme storms induced by the respective storm phenomenon are obtained. However, it should be noted, that the lower bound for the fitting procedure has to be checked in the actual probability plot of the observed trace of non-exceedence probabilities (Fig. 6). The exceedence probability of a specific wind speed level is finally obtained from the non-exceedence probabilities of the contributing storm phenomena as follows: pðv > vref Þ ¼ 1 pfd ðvpvref Þ pth ðvpvref Þ pgf ðvpvref Þ;
ð7Þ
where pfd ðvpvref Þ is the non-exceedence probability of vref in frontal depressions, pth ðvpvref Þ is the non-exceedence probability of vref in thunderstorms and pfg ðvpvref Þ is the non-exceedence probability of vref in gust fronts. An example of a local strong wind climate is shown in Fig. 7. The three contributions have different importance. In this example, thunderstorms have almost no
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Fig. 6. Example of fitting an observed trace to the theoretical distribution (fitting range p > 0:3; m ¼ 16:2 m/s, s ¼ 1:89 m/s, t ¼ 0:125).
influence on the strong wind climate. For wind speeds from 15 m/s to about 20 m/s, the strong wind climate is governed by the ‘gust fronts’. Beyond that level, storms induced by strong frontal depressions govern the wind climate. The single contributions for different wind speed levels are summarized in Table 2. The storm phenomenon with the largest contribution for a specific wind speed level can be identified by the largest value in the respective line in Table 2. It should be noted that the identified theoretical models are valid only above the respective threshold introduced in the fitting procedure. Below the respective threshold, a different approach is required. The values in Table 2 as well as the plot in Fig. 7 include this effect. The aim of the extreme value analysis is to estimate the characteristic value of the extreme wind speeds which is the 98%-fractile, and the design value of the extreme wind speeds which is the 99.9%-fractile. For the example, 20.7 m/s are obtained for the characteristic value and 23.8 m/s for the design value. While the first value is reasonably stable from a statistical point of view, the second value requires larger extrapolations and it should be noted that, then the question of an appropriate confidence interval has to be answered. For developing a wind zone map in accordance to the Eurocode, however, only the 98%-fractile is required.
3. Further processing towards a wind zone map Usually, further processing from the ‘raw’ data of the characteristic values (or design values) to a wind zone map means summarizing similar stations to one wind
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probability of non-exceedence
probability of non-exceedence
0.999
0.999
0.995 0.990
0.995 0.990
0.950 0.900
0.950 0.900
0.500
0.500
0.100 0.010 0.001
0.100 0.010 0.001
0
5
10
15
20
25
0
5
10
15
20
25
mean wind speed [m/s]
mean wind speed [m/s]
non-exceedence probability in strong frontal depressions
non-exceedence probability in thunderstorms
probability of non-exceedence
probability of non-exceedence
0.999
0.999
0.995 0.990
0.995 0.990
0.950 0.900
0.950 0.900
0.500
0.500
0.100 0.010 0.001
0.100 0.010 0.001
0
5
10
15
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25
0
5
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mean wind speed [m/s]
mean wind speed [m/s]
non-exceedence probability in gust fronts
overall non-exceedence probability
25
Fig. 7. Example of three contributing non-exceedence probabilities to the overall non-exceedence probability.
zone. For the actual study, a different approach is used alternatively: the traces of similar stations are summarized thus trying to increase the statistical stability of the extrapolated fractiles. Typically, a storm occurs with high mean wind speeds at several stations. A further step is therefore required which sorts out the maximum storm hour of a single storm at the respective stations. In Fig. 8, an example of this kind of summarizing is shown for five stations in the northern part of Germany. Basically, stations can be summarized if they are in the same region and show a similar individual trace of the observed non-exceedence probabilities. In the example, altogether, 244 samples have been found for the five stations (threshold value 14 m/s). Removing multiple samples for each individual storm leads to remaining 128 independent storms. The new trace has been fitted to an extreme value distribution for non-exceedence probabilities larger than 0.2, leading to the following parameters: m ¼ 17:0 m/s, s ¼ 1; 95 m/s, t ¼ 0; 10: The two fractile values are 21.7 m/s for the 98%-fractile and 24.7 m/s for the 99.9%-fractile. It
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Table 2 Exceedence probabilities of different wind speed levels for the three contributing storm phenomena and the composed wind climate vref (m/s)
pfd (v > vref )
pgf (v>vref )
pth (v>vref )
p (v>vref )
14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.5 25.0
1.0000 0.9723 0.7000 0.5949 0.4612 0.3440 0.2491 0.1762 0.1225 0.0840 0.0570 0.0383 0.0256 0.0170 0.0112 0.0073 0.0048 0.0031 0.0020 0.0013 0.0008 0.0005 0.0003
1.0000 1.0000 0.9990 0.9789 0.8841 0.7000 0.5298 0.3655 0.2327 0.1379 0.0763 0.0395 0.0190 0.0085 0.0034 0.0012 0.0004 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000
0.9801 0.8757 0.6667 0.4362 0.2559 0.1400 0.0733 0.0372 0.0185 0.0090 0.0043 0.0021 0.0010 0.0004 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
1.0000 1.0000 0.9999 0.9952 0.9536 0.8308 0.6728 0.4967 0.3392 0.2175 0.1328 0.0782 0.0451 0.0257 0.0148 0.0087 0.0052 0.0032 0.0020 0.0013 0.0008 0.0005 0.0003
Fig. 8. Non-exceedence probability of the maximum mean wind speed in frontal depressions for the region around Celle, Luchow, . Fassberg, Hannover and Wunstorf.
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should be noted that in this approach the problem of the confidence interval is not removed completely, which means that the estimated design wind speed (99.9%fractile) still contains a certain amount of randomness which cannot be neglected. For the further zoning, it is useful to plot the typical ‘foot-prints’ of single storms. A special software has been developed to extract the wind direction and the mean wind speed for specific storm hours. Two typical examples of storm situations over Germany are shown in Fig. 9. The length of the arrows represent the observed hourly mean wind speed, the direction of the arrows corresponds to the observed wind direction. The final wind zone map for Germany (Fig. 10) introduces five wind zones specifying the characteristic value of the mean wind speeds in steps of 2.5 m/s starting with a value of 22.5 m/s. It is worth mentioning that the Eurocode and consequently the DIN 1055 present 10 min means. Therefore, all results from the statistical analysis have been multiplied by a factor of 1.06 to convert the hourly means to 10 min means. Generally, the zoning follows the administrative boundaries. Compared to the previous map, the new map reflects in a similar manner the decrease of the characteristic wind speeds with increasing distance from the coast. In the southern part between the Alps and the Swiss Jura and the Swabian Alb, respectively, form a corridor for the storms leading to higher wind speeds in this
Capella 3. January 1976,
Daria 25. January 1990
Fig. 9. Two typical foot-prints of storms over Germany (direction of the arrows corresponds to the observed wind direction, length of the arrows corresponds to the observed hourly mean wind speed).
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Fig. 10. Proposal for a new wind zone map for Germany.
region. While the bow of the low-mountain range of the Teutoburger Forest leads to a favourite wind climate for the westerly lying Munsterland . (zone 1), the Harz- and Ore-mountains seem to induce an increase of the wind speeds for Saxony. This increase of wind speeds is reflected in a wind zone 3. Strictly speaking, for some of
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the southern parts of Saxony, a wind zone 2 should occur, since the wind speeds decrease in these regions with further approach to the low-mountain range. For Dresden, e.g. the characteristic wind speed is considerably lower than is suggested with zone 3. An appropriate zoning, however, was difficult due to an insufficient density of meteorological stations. The eastern parts of Saxony, the Oberlausitz, clearly have a stronger wind climate than Dresden and again require a wind zone 3. Therefore, the actual proposal chose a conservative approach, i.e. Dresden is included in the higher wind zone 3. However, compared to the previous proposal, for many regions smaller wind loads are obtained. For the densely populated RuhrArea, e.g. the reduction for the characteristic wind speed is 22.5/27.6=0.815, which means a reduction in terms of wind loads of 33%.
4. Directional effects In the scope of the revised analysis, directional effects have been investigated, too. Especially, extreme wind speeds induced by frontal depressions show a considerable dependency on the wind direction. The ‘conditional’ probability that a wind speed level vref is exceeded in a sector F can be obtained as follows: pk ðv > vref jFÞ ¼ 1 pk ðvpvref Þhrel ðFÞ ;
ð8Þ
where hrel ðFÞ is the relative frequency of the storm phenomenon k in sector F and pk ðvpvref Þ is the non-exceedence probability of the storm phenomenon k: The conditional exceedence probability for a specific sector is always smaller or equal than the total exceedence probability, i.e. the conditional 98%-fractile is smaller or equal the total 98%-fractile. The following example should illustrate this more clearly. For a specific station, the frontal depressions are distributed only over the following four 301-sectors: 1801, 2101, 2401 and 2701. The corresponding observed relative frequencies are as follows: 0.03, 0.42, 0.41 and 0.14. The probability distribution is given with the parameters m ¼ 16:1 m/s, s ¼ 1:65 m/s and t ¼ 0:025: The characteristic wind speed, i.e. the 98%-fractile is 20.3 m/s. This wind speed level is exceeded in the sector 1801 with a probability of 10.980.03=0.0006, in the sector 2101 with a probability of 10.980.42=0.0084, in the sector 2401 with a probability of 10.980.41=0.0082 and in the sector 2701 with a probability of 10.980.14=0.0028. The conditional 98%-fractiles for the individual sectors are the 98%1=hrel ðFÞ -fractiles of the original probability distribution. For 1801, a characteristic wind speed of 15.8 m/s is obtained, for 2101 and 2401 the characteristic wind speed is 19.3 m/s and for 2701 the characteristic wind speed is 17.9 m/s. It is important not to confuse the ratios of the ‘conditional’ characteristic wind speeds to the total characteristic wind speed and the directional factors. If one would introduce the first set of values in the design, for each direction an exceedence probability of 0.02 is obtained, i.e. the accumulated exceedence probability for the analysed four sectors ends up with 0.08. Obviously, a lower exceedence probability per sector is required to end up with the target value of 0.02. A strict solution of this problem has to be based on the convolution of probability distributions of the wind
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speeds and the aerodynamic coefficients [3]. An enveloping solution can be obtained by demanding that the target value for each direction should be 0.02/N; where N is the number of sectors. For the above example N is 4 and the new target value for a directional analysis becomes 0.02/4=0.005, i.e. the new target fractile is 99.5%, where 0.9954 leads to the required 0.98 target value. The respective wind speeds for each direction are obtained as the corresponding values to 99:51=hrel ðFÞ and are 17.6, 21.0, 20.9 19.6 m/s. The directional factors then become: 17.6/20.3=0.87 for 1801, 21.0/20.3=1.04 for 2101, 20.9/20.3=1.03 for 2401 and 19.6/20.3=0.97. It is worth mentioning that the directional factor usually becomes larger than 1.0 for the strongest wind direction. Directional effects especially can be observed for frontal depressions. In Fig. 11, the relative frequency of strong storms induced by frontal depressions are shown for six stations which are supposed to be in the same wind zone. For each station, only a few sectors are effected, however, the respective relative frequencies of the sectors differ considerably. An enveloping solution based on the largest value of each affected sector would lead to an overdesign and is therefore less appropriate. The alternative would be a very detailed zoning for the directional effects. This solution was rejected for sake of simplicity of the new map.
5. Summary and conclusions On behalf of the Deutsche Institut fur . Bautechnik (DIBt) (German Institute of Structural Engineering), the old wind zone map of Germany, developed in the 1980s, was reviewed. The revised version of the wind zone map of Germany is based on a refined extreme value analysis for three physical phenomena which are leading to high gust wind speeds: the typical storms induced by strong frontal depressions, thunderstorms and additional gust fronts especially in weaker frontal depressions which are induced by down-drafts of cold air and/or by rain and which are not fully covered by the usual flow field models of frontal depressions. Individual ensembles for each of these storm phenomena have been sampled for a large number of stations. The ensembles are not based on yearly extremes but on all extreme events. Furthermore, it has been tried to increase the statistical stability of the estimated representative values by summarizing stations with a similar wind climate. As basic theoretical expression, an extreme value distribution type III is used. This distribution has a finite upper tail. The fitting of the theoretical extreme value distribution is performed for only the right tail of the observed traces of nonexceedence probabilities. The proposed new map presents five wind zones with characteristic values from 22.5 to 32.5 m/s. The zoning follows mainly the boundaries of the administrative districts. Compared to the preceding wind zone map, the new map specifies for many sites of smaller wind loads. Additionally, directional effects have been analysed. A consistent method is proposed which allows to develop the directional factors based on the relative frequency of a storm phenomenon in the respective sector. However, the results obtained for stations in Germany are too inhomogeneous to be included in the code.
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Düsseldorf
Essen
0 0.6 0.5 0.4 300 0.3 0.2 0.1 0.0 270 0.0 0.1 0.2 0.3 240 0.4 0.5 0.6
0 30
330
60
90
120
150
210
0.6 0.5 0.4 300 0.3 0.2 0.1 0.0 270 0.0 0.1 0.2 0.3 240 0.4 0.5 0.6
330
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Haltern
Hameln 0
0 30
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0.6 0.5 0.4 300 0.3 0.2 0.1 0.0 270 0.0 0.1 0.2 0.3 240 0.4 0.5 0.6
60
90
120
180
Werl 0
0 30
60
90
120
150
210 180
150
210
Münster
330
30
330
180
0.6 0.5 0.4 300 0.3 0.2 0.1 0.0 270 0.0 0.1 0.2 0.3 240 0.4 0.5 0.6
150
210
180
0.6 0.5 0.4 300 0.3 0.2 0.1 0.0 270 0.0 0.1 0.2 0.3 240 0.4 0.5 0.6
30
0.6 0.5 0.4 300 0.3 0.2 0.1 0.0 270 0.0 0.1 0.2 0.3 240 0.4 0.5 0.6
30
330
60
90
120
150
210 180
Fig. 11. Sector-wise relative frequency of strong wind conditions induced by frontal depressions.
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Acknowledgements The financial support of the Deutsche Institut fur . Bautechnik (DIBt) and the technical support of the Deutsche Wetterdienst (DWD) is gratefully acknowledged. . The author acknowledges the support of Dr.-Ing. N. Holscher for organizing the handling of the huge amount of data and the support of Dipl.-Ing. U. Versen who carried out most of the calculations.
References . . [1] G. Bollmann, G. Jurksch, Ein Beitrag zur Festlegung der Grundwind und Nennboengeschwindigkeit im Binnenland der Bundesrepublik Deutschland fur . die DIN-Norm 1055, Teil 4, Meteorologische Rundschau 37, 1–10, February. 1984. [2] L. Gomes, B.J. Vickery, Extreme wind speeds in mixed climates, J. Ind. Aerodyn. 2 (1978) 341–344. [3] M. Kasperski, Specification and codification of design wind loads, Habilitation Thesis, RuhrUniversity Bochum, Department of Civil Engineering, 2000. [4] C. Cunnane, Unbiased plotting positions—a review, J. Hydrol. 37 (1978) 205–222. [5] M. Jensen, N. Franck, The Climate of Strong Winds in Denmark, Danish Technical Press, Copenhagen, 1970. [6] E. Simiu, N.A. Heckert, Extreme wind distribution tails: a ‘peaks over threshold’ approach, J. Struct. Eng. 122 (5) (1996) 539–547. [7] M. Kasperski, Climate change and design wind load concepts, Wind Struct. 1 (2) (1998) 145–160. [8] E. Simiu, Estimation of extreme wind speeds, A state of the art in wind engineering, Proceedings of the ninth ICWE, New Delhi, India, 1995. [9] J.D. Holmes, W.M. Moriarty, Application of the generalized Pareto distribution to wind engineering, J. Wind Eng. Ind. Aerodyn. 83 (1999) 1–10.
A new wind zone map of Germany Michael Kasperski Department of Civil Engineering, Ruhr-University Bochum, D-44780 Bochum, Germany
Abstract The revised version of the wind zone map of Germany is based on a refined extreme value analysis for three physical phenomena which are leading to high gust wind speeds: the typical storms induced by strong frontal depressions, thunderstorms and additional gust fronts especially in weaker frontal depressions which are induced by down-drafts of cold air and/or by rain and which are not fully covered by the usual flow field models of frontal depressions. Individual ensembles for each of these storm phenomena have been sampled for a large number of stations. Furthermore, it has been tried to increase the statistical stability of the estimated representative values by summarizing stations with a similar wind climate. The fitting of the theoretical extreme value distribution is performed for only the right tail of the observed traces of non-exceedence probabilities. The proposed new map presents five wind zones with characteristic values from 22.5 to 32.5 m/s. The zoning follows mainly the boundaries of the administrative districts. Compared to the preceding wind zone map, the new map specifies for many sites smaller wind loads. Additionally, directional effects have been analysed, however, the results are too inhomogeneous to be included in the code. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Wind climate; Extreme value analysis; Design wind speed; Directional effects
1. Introduction In the scope of the Eurocode 1, part 2.4 wind loads, the respective national authorities are responsible for providing a wind zone map and other meteorological information which are required to specify the design wind load for a structure. In its Annex A, the actual draft of the Eurocode contains a wind map for Germany as shown in Fig. 1. There are a number of points which have suggested a new and refined analysis of the extreme wind climate in Germany. The proposed map in Fig. 1 is based on an analysis including data only up to 1980, i.e. today 20 further years of data are E-mail address: [email protected] (M. Kasperski). 0167-6105/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 0 2 ) 0 0 2 5 7 - X
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Fig. 1. The German wind map as proposed in the Eurocode 1.
available which could be included in a reviewing analysis. The then used method to specify the characteristic values has not been based on an extreme value analysis, but it was tried to extrapolate from the observed trace of the average number of hours per year above a certain threshold [1]. A single storm might then be included in this type of statistics with more than one value, i.e. the statistics contain dependent or correlated data. Furthermore, some doubts have occurred with regard to having cities with obviously different wind climates in one and the same wind zone, like e.g. Rostock and Bremen and Dusseldorf. . Finally, it might be noted that the characteristic values of the wind speeds jump at the boundaries to the neighbouring countries. This inconsistency is to one side only, i.e. the wind speeds are higher in Germany than in the neighbouring countries. This might be explained with the fact, that the method which has been used in Germany has not been used anywhere else, thus probably leading to a systematic bias in the analysis. If, however, a revised analysis of the German data is able to overcome this shortcoming, remains doubtful. Experts neither in Europe nor worldwide have been able to agree on one standardized method to analyse the wind climate properly, although the specification of the design wind speed is one of the central and important points in the codification of the wind loads. All these reasons convinced the Deutsche Institut fur . Bautechnik (DIBt) (German Institute of Structural Engineering) to sponsor a research project to develop a revised version of the German wind zone map. 2. Method of analysis The data basis is provided by the Deutsche Wetterdienst (DWD) (German Weather Service) in terms of 24 hourly mean wind speeds per day, the corresponding
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mean wind direction and the daily maximum gust wind speed (corresponding to a 2– 3 s gust) plus the respective time of occurrence. Altogether, 183 stations have been included in the study. On an average, the observation period is 40 years. The distribution of the stations over Germany is plotted in Fig. 2. The wind climate in Germany is mainly governed by two storm phenomena, frontal depressions and thunderstorms. Basically, for a proper extreme value analysis, the independent physical phenomena have to be separated and individual ensembles have to be sampled [2]. The sorting into several ensembles is accomplished by specifying respective thresholds, e.g. one threshold for the mean wind speed vm;lim and one threshold for the gust wind vg;lim : If the actual mean wind speed exceeds the threshold vm;lim ; obviously, a storm induced by a frontal depression is obtained. Usually, this threshold is exceeded for several hours which have not to be
Fig. 2. Meteorological stations used for the analysis of the extreme wind speeds in Germany.
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consecutive, i.e. some storms ‘take a break’ and slow down before they speed up again. The highest intensity may last for more than 1 h (Fig. 3). The task for the extreme value analysis now becomes to identify independent storm events for the ensemble. Clearly, it is not sufficient to sample only yearly extremes, since storms tend to occur in clusters or families, which means that the second or third strongest storm of 1 year might be considerably stronger than the strongest storm of another year (Fig. 4). Therefore, the analysis should be based on all extreme events. Each extreme event is sampled in the ensemble with its largest value, i.e. only the strongest storm hour of a frontal depression is used. The remaining storm hours are dependent data and should therefore not be included in the ensemble of independent events. Strictly speaking, they have to be included in the load model by introducing two further variables, which are the duration of a storm and the relative intensity of the 2–nth storm hour [3]. Since these variables are not influencing the specification of the design wind speed itself, but only the exceedence probability of the design wind load and thus the specification of the appropriate fractile of the aerodynamic coefficients, this aspect is not discussed in this paper. The identification of an independent storm event is based on an assumption about the longest possible break during a single storm. For the actual study, a value of 24 h has been chosen. Assembling the second ensemble is much easier. A thunderstorm is identified by a gust wind speed exceeding the threshold vg;lim while the corresponding hourly mean is well below the threshold vm;lim : Since per day only one value is sampled, the such identified thunderstorms are already independent events.
Fig. 3. Examples of time histories of hourly means in a frontal depression.
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mean wind speed [m/s] 25
20
15
yearly extremes
10
5
0 55
60
65
70
75
80
85
90
95
00
year of observation
25
mean wind speed [m/s]
20 independent extremes vm > 14.0 m/s
15
10
5
0 55
60
65
70
75
80
85
90
95
00
year of observation Fig. 4. Time distribution of the yearly extremes and the independent extremes with threshold 14.0 m/s (Dusseldorf . airport 1952–1999).
When specifying a design wind load, extreme wind speeds are combined with extreme aerodynamic coefficients which have been obtained from a properly scaled boundary layer wind tunnel experiment. In the respective modelled flow, a certain level of gustiness is included and is – of course – influencing the extreme aerodynamic coefficients. As a matter of fact, only turbulent wind fields as they are expected in strong frontal depressions are modelled. As typical value of the gustiness in open country at 10 m height, an average gust factor of 1.6 is obtained. It should be noted that this gust factor is a random variable, i.e. it has a certain scatter in the wind
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tunnel flow as well as in nature (Fig. 5). If the turbulence intensities and the spectral densities modelled in the wind tunnel flow meet the corresponding parameters specified in the theoretical model, the shown random gust factors represent wind tunnel results as well. Extreme loads in a wind tunnel experiment therefore contain a certain contribution from the gustiness of the oncoming flow. Specifying the design wind load for a frontal depression based on an extreme value distribution for the observed gust wind speeds would therefore include this scatter twice. A consistent wind load model for wind loads induced by a strong frontal depression has therefore to be based on the maximum mean wind speeds in frontal depressions (either hourly means or 10 min means). For the induced wind loads in thunderstorms a consistent load model – i.e. an analysis of the gust wind speeds in thunderstorms plus an appropriate aerodynamic coefficient obtained from a wind tunnel test simulating wind fields in thunderstorms – does not exist. Therefore, as an interim solution, the gust wind speeds of thunderstorms are translated to equivalent mean wind speeds of frontal depressions. Thunderstorms and strong frontal depression differ considerably in their duration. As already mentioned, this difference in duration is affecting only the specification of the aerodynamic coefficient and not that of the design wind speed. A closer look at the recorded wind speed data reveals the fact that with the abovedescribed procedure some high gust wind speeds are not included and covered with the two sampled ensembles. This can be seen in a plot of the observed gust factors. The scatter in the observed gust factors is much larger than is predicted from the theoretical model. Typically, these high gusts occur in weaker frontal depressions before the strongest storm hour. They are induced by flows similar to flows in thunderstorms, i.e. cold air eventually together with rain is falling down and leads to a gust front which is riding in front of the storm. A third ensemble has to be introduced to cover these additional gusts. It is obtained by specifying a third threshold variable which is the gust factor. In this study, a threshold value of 1.8 is
Fig. 5. Comparison of observed gust factors to the scatter of gust factors as obtained from a numerical simulation based on the theoretical model of a flow field corresponding to open country. (a) Observed gust factors at Dusseldorf . airport from 1960–1995 with expected mean gust factors for a turbulent flow field corresponding to open country. (b) Random gust factors obtained from a numerical simulation of a turbulent flow field corresponding to open country with a mean wind speed of 20 m/s.
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introduced, i.e. if during a weaker frontal depression a gust wind speed has a higher gust factor than 1.8, the corresponding gust is sampled in the third ensemble of ‘gust fronts’. The next step is to perform order statistics to estimate the non-exceedence probabilities of the maximum hourly mean wind speeds in frontal depressions and of the gust wind speeds in thunderstorms and ‘gust fronts’, respectively. For consistency of the analysis, the gust wind speeds are ‘translated’ to equivalent hourly mean wind speeds by dividing the observed gust wind speeds with a gust factor of 1.6. In case of yearly extremes, the non-exceedence probability of an observed wind speed in a specific ensemble is obtained as follows: pðvpvi Þ ¼
ia ; N þb
ð1Þ
where i is the rank in list of ascending order, highest value rank N; lowest value rank 1, N is the ensemble size and a; b the parameters to define the ‘optimum’ plotting position [4]. Jensen and Franck [5] proposed for the ensemble with more than one storm per year to estimate the non-exceedence probability by introducing additionally an exponent which is obtained as the average number of storms per year, i.e. the nonexceedence probability becomes: i a N=K pðvpvi Þ ¼ ; ð2Þ N þb where K is the number of observation years. The respective traces are plotted in Gumbel probability paper. As a matter of fact, the choice of the threshold affects the number of sampled storms and therefore the trace of the non-exceedence probability. Basically, only the extremes of the sampled events should follow an extreme value distribution, i.e. the fitting of the observed trace to a theoretical expression should be performed only for the right tail (the largest wind speeds). This region is less affected by the actual sample size. An ‘optimum’ sample size can be specified by demanding that all sampled events should be displayed in the probability plot which should have a ‘symmetric’ vertical axis, i.e. the smallest probability value plow and the highest probability value phigh should add up to unity. Then, the ‘optimum’ number of events Nopt in K years is obtained from: 1 a Nopt =K ! plow ¼ pðvpv1 Þ ¼ : ð3Þ Nopt þ b For a range of the probability paper from non-exceedence probability 0.001–0.999 per year, these optimum ensemble sizes are given in Table 1. It is worth mentioning that the optimum sample size depends on the actual length of the observation period.
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Table 1 Optimum ensemble size in dependence on the length of the observation period Length of observation period (years) Optimum ensemble size Nopt
10 20
20 35
30 48
40 61
50 74
60 86
70 97
80 109
90 120
100 131
As basic model for the extreme value distribution, the reverse Weibull distribution (extreme value distribution type III) is introduced: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v m1=t F ðvÞ ¼ exp f1 f2 ; f1 ¼ Gð1 þ tÞ; f2 ¼ Gð1 þ 2tÞ f12 ; s ð4Þ where G is the gamma function, m is the mean value of the extreme wind speeds, s is the rms value of the extreme wind speeds and t the curvature parameter. This distribution has a finite upper tail which seems from geophysical aspects especially appropriate for extreme wind speeds [6]. The extreme value is given as: xmax ¼ m þ s f1 =f2
ð5Þ
The type III distribution has been applied already successfully to some stations in Germany [3,7], to a large number of stations in the USA [8] and to some stations in Australia [9]. If t becomes 0, the Gumbel distribution (extreme value distribution type I) is obtained: " " #!# p xm F ðxÞ ¼ exp exp g þ pffiffiffi ; ð6Þ 6 s where g is the Euler constant=0.5772. This distribution has an infinite upper tail. The fitting of the data to the theoretical distribution is based on a least linear error sum for the wind speed deviations between observation and prediction from the theoretical model. As mentioned above, the fitting procedure is using only the right tail of the observed non-exceedence probabilities. Typically, the fitting range starts from non-exceedence probabilities larger 0.3, which means that in 2 years on average three extreme storms induced by the respective storm phenomenon are obtained. However, it should be noted, that the lower bound for the fitting procedure has to be checked in the actual probability plot of the observed trace of non-exceedence probabilities (Fig. 6). The exceedence probability of a specific wind speed level is finally obtained from the non-exceedence probabilities of the contributing storm phenomena as follows: pðv > vref Þ ¼ 1 pfd ðvpvref Þ pth ðvpvref Þ pgf ðvpvref Þ;
ð7Þ
where pfd ðvpvref Þ is the non-exceedence probability of vref in frontal depressions, pth ðvpvref Þ is the non-exceedence probability of vref in thunderstorms and pfg ðvpvref Þ is the non-exceedence probability of vref in gust fronts. An example of a local strong wind climate is shown in Fig. 7. The three contributions have different importance. In this example, thunderstorms have almost no
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Fig. 6. Example of fitting an observed trace to the theoretical distribution (fitting range p > 0:3; m ¼ 16:2 m/s, s ¼ 1:89 m/s, t ¼ 0:125).
influence on the strong wind climate. For wind speeds from 15 m/s to about 20 m/s, the strong wind climate is governed by the ‘gust fronts’. Beyond that level, storms induced by strong frontal depressions govern the wind climate. The single contributions for different wind speed levels are summarized in Table 2. The storm phenomenon with the largest contribution for a specific wind speed level can be identified by the largest value in the respective line in Table 2. It should be noted that the identified theoretical models are valid only above the respective threshold introduced in the fitting procedure. Below the respective threshold, a different approach is required. The values in Table 2 as well as the plot in Fig. 7 include this effect. The aim of the extreme value analysis is to estimate the characteristic value of the extreme wind speeds which is the 98%-fractile, and the design value of the extreme wind speeds which is the 99.9%-fractile. For the example, 20.7 m/s are obtained for the characteristic value and 23.8 m/s for the design value. While the first value is reasonably stable from a statistical point of view, the second value requires larger extrapolations and it should be noted that, then the question of an appropriate confidence interval has to be answered. For developing a wind zone map in accordance to the Eurocode, however, only the 98%-fractile is required.
3. Further processing towards a wind zone map Usually, further processing from the ‘raw’ data of the characteristic values (or design values) to a wind zone map means summarizing similar stations to one wind
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probability of non-exceedence
probability of non-exceedence
0.999
0.999
0.995 0.990
0.995 0.990
0.950 0.900
0.950 0.900
0.500
0.500
0.100 0.010 0.001
0.100 0.010 0.001
0
5
10
15
20
25
0
5
10
15
20
25
mean wind speed [m/s]
mean wind speed [m/s]
non-exceedence probability in strong frontal depressions
non-exceedence probability in thunderstorms
probability of non-exceedence
probability of non-exceedence
0.999
0.999
0.995 0.990
0.995 0.990
0.950 0.900
0.950 0.900
0.500
0.500
0.100 0.010 0.001
0.100 0.010 0.001
0
5
10
15
20
25
0
5
10
15
20
mean wind speed [m/s]
mean wind speed [m/s]
non-exceedence probability in gust fronts
overall non-exceedence probability
25
Fig. 7. Example of three contributing non-exceedence probabilities to the overall non-exceedence probability.
zone. For the actual study, a different approach is used alternatively: the traces of similar stations are summarized thus trying to increase the statistical stability of the extrapolated fractiles. Typically, a storm occurs with high mean wind speeds at several stations. A further step is therefore required which sorts out the maximum storm hour of a single storm at the respective stations. In Fig. 8, an example of this kind of summarizing is shown for five stations in the northern part of Germany. Basically, stations can be summarized if they are in the same region and show a similar individual trace of the observed non-exceedence probabilities. In the example, altogether, 244 samples have been found for the five stations (threshold value 14 m/s). Removing multiple samples for each individual storm leads to remaining 128 independent storms. The new trace has been fitted to an extreme value distribution for non-exceedence probabilities larger than 0.2, leading to the following parameters: m ¼ 17:0 m/s, s ¼ 1; 95 m/s, t ¼ 0; 10: The two fractile values are 21.7 m/s for the 98%-fractile and 24.7 m/s for the 99.9%-fractile. It
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Table 2 Exceedence probabilities of different wind speed levels for the three contributing storm phenomena and the composed wind climate vref (m/s)
pfd (v > vref )
pgf (v>vref )
pth (v>vref )
p (v>vref )
14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0 20.5 21.0 21.5 22.0 22.5 23.0 23.5 24.0 24.5 25.0
1.0000 0.9723 0.7000 0.5949 0.4612 0.3440 0.2491 0.1762 0.1225 0.0840 0.0570 0.0383 0.0256 0.0170 0.0112 0.0073 0.0048 0.0031 0.0020 0.0013 0.0008 0.0005 0.0003
1.0000 1.0000 0.9990 0.9789 0.8841 0.7000 0.5298 0.3655 0.2327 0.1379 0.0763 0.0395 0.0190 0.0085 0.0034 0.0012 0.0004 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000
0.9801 0.8757 0.6667 0.4362 0.2559 0.1400 0.0733 0.0372 0.0185 0.0090 0.0043 0.0021 0.0010 0.0004 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
1.0000 1.0000 0.9999 0.9952 0.9536 0.8308 0.6728 0.4967 0.3392 0.2175 0.1328 0.0782 0.0451 0.0257 0.0148 0.0087 0.0052 0.0032 0.0020 0.0013 0.0008 0.0005 0.0003
Fig. 8. Non-exceedence probability of the maximum mean wind speed in frontal depressions for the region around Celle, Luchow, . Fassberg, Hannover and Wunstorf.
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should be noted that in this approach the problem of the confidence interval is not removed completely, which means that the estimated design wind speed (99.9%fractile) still contains a certain amount of randomness which cannot be neglected. For the further zoning, it is useful to plot the typical ‘foot-prints’ of single storms. A special software has been developed to extract the wind direction and the mean wind speed for specific storm hours. Two typical examples of storm situations over Germany are shown in Fig. 9. The length of the arrows represent the observed hourly mean wind speed, the direction of the arrows corresponds to the observed wind direction. The final wind zone map for Germany (Fig. 10) introduces five wind zones specifying the characteristic value of the mean wind speeds in steps of 2.5 m/s starting with a value of 22.5 m/s. It is worth mentioning that the Eurocode and consequently the DIN 1055 present 10 min means. Therefore, all results from the statistical analysis have been multiplied by a factor of 1.06 to convert the hourly means to 10 min means. Generally, the zoning follows the administrative boundaries. Compared to the previous map, the new map reflects in a similar manner the decrease of the characteristic wind speeds with increasing distance from the coast. In the southern part between the Alps and the Swiss Jura and the Swabian Alb, respectively, form a corridor for the storms leading to higher wind speeds in this
Capella 3. January 1976,
Daria 25. January 1990
Fig. 9. Two typical foot-prints of storms over Germany (direction of the arrows corresponds to the observed wind direction, length of the arrows corresponds to the observed hourly mean wind speed).
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Fig. 10. Proposal for a new wind zone map for Germany.
region. While the bow of the low-mountain range of the Teutoburger Forest leads to a favourite wind climate for the westerly lying Munsterland . (zone 1), the Harz- and Ore-mountains seem to induce an increase of the wind speeds for Saxony. This increase of wind speeds is reflected in a wind zone 3. Strictly speaking, for some of
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the southern parts of Saxony, a wind zone 2 should occur, since the wind speeds decrease in these regions with further approach to the low-mountain range. For Dresden, e.g. the characteristic wind speed is considerably lower than is suggested with zone 3. An appropriate zoning, however, was difficult due to an insufficient density of meteorological stations. The eastern parts of Saxony, the Oberlausitz, clearly have a stronger wind climate than Dresden and again require a wind zone 3. Therefore, the actual proposal chose a conservative approach, i.e. Dresden is included in the higher wind zone 3. However, compared to the previous proposal, for many regions smaller wind loads are obtained. For the densely populated RuhrArea, e.g. the reduction for the characteristic wind speed is 22.5/27.6=0.815, which means a reduction in terms of wind loads of 33%.
4. Directional effects In the scope of the revised analysis, directional effects have been investigated, too. Especially, extreme wind speeds induced by frontal depressions show a considerable dependency on the wind direction. The ‘conditional’ probability that a wind speed level vref is exceeded in a sector F can be obtained as follows: pk ðv > vref jFÞ ¼ 1 pk ðvpvref Þhrel ðFÞ ;
ð8Þ
where hrel ðFÞ is the relative frequency of the storm phenomenon k in sector F and pk ðvpvref Þ is the non-exceedence probability of the storm phenomenon k: The conditional exceedence probability for a specific sector is always smaller or equal than the total exceedence probability, i.e. the conditional 98%-fractile is smaller or equal the total 98%-fractile. The following example should illustrate this more clearly. For a specific station, the frontal depressions are distributed only over the following four 301-sectors: 1801, 2101, 2401 and 2701. The corresponding observed relative frequencies are as follows: 0.03, 0.42, 0.41 and 0.14. The probability distribution is given with the parameters m ¼ 16:1 m/s, s ¼ 1:65 m/s and t ¼ 0:025: The characteristic wind speed, i.e. the 98%-fractile is 20.3 m/s. This wind speed level is exceeded in the sector 1801 with a probability of 10.980.03=0.0006, in the sector 2101 with a probability of 10.980.42=0.0084, in the sector 2401 with a probability of 10.980.41=0.0082 and in the sector 2701 with a probability of 10.980.14=0.0028. The conditional 98%-fractiles for the individual sectors are the 98%1=hrel ðFÞ -fractiles of the original probability distribution. For 1801, a characteristic wind speed of 15.8 m/s is obtained, for 2101 and 2401 the characteristic wind speed is 19.3 m/s and for 2701 the characteristic wind speed is 17.9 m/s. It is important not to confuse the ratios of the ‘conditional’ characteristic wind speeds to the total characteristic wind speed and the directional factors. If one would introduce the first set of values in the design, for each direction an exceedence probability of 0.02 is obtained, i.e. the accumulated exceedence probability for the analysed four sectors ends up with 0.08. Obviously, a lower exceedence probability per sector is required to end up with the target value of 0.02. A strict solution of this problem has to be based on the convolution of probability distributions of the wind
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speeds and the aerodynamic coefficients [3]. An enveloping solution can be obtained by demanding that the target value for each direction should be 0.02/N; where N is the number of sectors. For the above example N is 4 and the new target value for a directional analysis becomes 0.02/4=0.005, i.e. the new target fractile is 99.5%, where 0.9954 leads to the required 0.98 target value. The respective wind speeds for each direction are obtained as the corresponding values to 99:51=hrel ðFÞ and are 17.6, 21.0, 20.9 19.6 m/s. The directional factors then become: 17.6/20.3=0.87 for 1801, 21.0/20.3=1.04 for 2101, 20.9/20.3=1.03 for 2401 and 19.6/20.3=0.97. It is worth mentioning that the directional factor usually becomes larger than 1.0 for the strongest wind direction. Directional effects especially can be observed for frontal depressions. In Fig. 11, the relative frequency of strong storms induced by frontal depressions are shown for six stations which are supposed to be in the same wind zone. For each station, only a few sectors are effected, however, the respective relative frequencies of the sectors differ considerably. An enveloping solution based on the largest value of each affected sector would lead to an overdesign and is therefore less appropriate. The alternative would be a very detailed zoning for the directional effects. This solution was rejected for sake of simplicity of the new map.
5. Summary and conclusions On behalf of the Deutsche Institut fur . Bautechnik (DIBt) (German Institute of Structural Engineering), the old wind zone map of Germany, developed in the 1980s, was reviewed. The revised version of the wind zone map of Germany is based on a refined extreme value analysis for three physical phenomena which are leading to high gust wind speeds: the typical storms induced by strong frontal depressions, thunderstorms and additional gust fronts especially in weaker frontal depressions which are induced by down-drafts of cold air and/or by rain and which are not fully covered by the usual flow field models of frontal depressions. Individual ensembles for each of these storm phenomena have been sampled for a large number of stations. The ensembles are not based on yearly extremes but on all extreme events. Furthermore, it has been tried to increase the statistical stability of the estimated representative values by summarizing stations with a similar wind climate. As basic theoretical expression, an extreme value distribution type III is used. This distribution has a finite upper tail. The fitting of the theoretical extreme value distribution is performed for only the right tail of the observed traces of nonexceedence probabilities. The proposed new map presents five wind zones with characteristic values from 22.5 to 32.5 m/s. The zoning follows mainly the boundaries of the administrative districts. Compared to the preceding wind zone map, the new map specifies for many sites of smaller wind loads. Additionally, directional effects have been analysed. A consistent method is proposed which allows to develop the directional factors based on the relative frequency of a storm phenomenon in the respective sector. However, the results obtained for stations in Germany are too inhomogeneous to be included in the code.
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Düsseldorf
Essen
0 0.6 0.5 0.4 300 0.3 0.2 0.1 0.0 270 0.0 0.1 0.2 0.3 240 0.4 0.5 0.6
0 30
330
60
90
120
150
210
0.6 0.5 0.4 300 0.3 0.2 0.1 0.0 270 0.0 0.1 0.2 0.3 240 0.4 0.5 0.6
330
60
90
120
180
Haltern
Hameln 0
0 30
330
60
90
120
150
210
0.6 0.5 0.4 300 0.3 0.2 0.1 0.0 270 0.0 0.1 0.2 0.3 240 0.4 0.5 0.6
60
90
120
180
Werl 0
0 30
60
90
120
150
210 180
150
210
Münster
330
30
330
180
0.6 0.5 0.4 300 0.3 0.2 0.1 0.0 270 0.0 0.1 0.2 0.3 240 0.4 0.5 0.6
150
210
180
0.6 0.5 0.4 300 0.3 0.2 0.1 0.0 270 0.0 0.1 0.2 0.3 240 0.4 0.5 0.6
30
0.6 0.5 0.4 300 0.3 0.2 0.1 0.0 270 0.0 0.1 0.2 0.3 240 0.4 0.5 0.6
30
330
60
90
120
150
210 180
Fig. 11. Sector-wise relative frequency of strong wind conditions induced by frontal depressions.
M. Kasperski / J. Wind Eng. Ind. Aerodyn. 90 (2002) 1271–1287
1287
Acknowledgements The financial support of the Deutsche Institut fur . Bautechnik (DIBt) and the technical support of the Deutsche Wetterdienst (DWD) is gratefully acknowledged. . The author acknowledges the support of Dr.-Ing. N. Holscher for organizing the handling of the huge amount of data and the support of Dipl.-Ing. U. Versen who carried out most of the calculations.
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