Determination of design wind speeds and the associated uncertainty is the most essential step in wind-resistant structural designs and in quantatative wind risk assessments. Design wind speeds are usually obtained by analyzing the recorded field wind speeds using various statistical methods.
However, recorded field wind data are often quite limited and are not sufficient for reliable statistical analyses, especially when one is predicting long-return-period design wind speeds or directional design wind speeds (Jain, etc., 2001; Tamura, 2007;
Vickery, etc., 1995). As a result, generation of arti-ficial wind speed time series becomes a critical re-search topic; the developed models are expected to produce synthetic wind speed time series whose
sta-tistical properties are compatible to those of field re-corded data.
Wind speed time series are generally non-zero-mean, non-Gaussian as well as non-stationary. Traditional time series models, assuming that the data are Gaus-sian and stationary, cannot be applied directly. The researchers (Brown, etc., 1984; Daniel, etc., 1991;
Nfaoui, etc., 1996; Poggi, etc., 2003) adopted Dubey method or power transformation method to trans-form data into Gaussian distributions; applied a sim-ple normalization to remove the diurnal non-stationarity; and then fitted the transformed data with traditional AR (Auto-Regressive) models or ARMA (Auto-Regressive Moving-Average) models with constant coefficients.
This paper proposes a non-stationary auto-regressive model in conjunction with a hidden Markov chain to
Simulation of Wind Speeds Using a Hidden Markov Chain Model
Rwey-Hua Cherng & Shih-Che Kao.
Department of Construction Engineering,
National Taiwan University of Science and Technology, Taipei, Taiwan
Jian-Ye Ching
Department of Civil Engineering,
National Taiwan University, Taipei, Taiwan
ABSTRACT: Determination of design wind speeds and the associated uncertainty is the most essential step in wind-resistant structural designs and in quantatative wind risk assessments. Design wind speeds are usually obtained by analyzing the recorded field wind speeds using various statistical methods. However, recorded field wind data are often quite limited and are not sufficient for reliable statistical analyses, especially when one is predicting long-return-period design wind speeds or directional design wind speeds. Accordingly, the generation of artificial wind speeds whose statistical properties are compatible to those from observed wind speeds is a critical research topic.
This paper proposes a non-stationary auto-regressive model with a hidden Markov chain to analyze field wind speeds and then to simulate synthetic wind speed time series. Compared to the models adopted by previous re-searches, the chosen model is different in the way that it is a non-stationary model. Moreover, new algo-rithms are developed to generate synthetic wind speeds based on the model and field wind speeds.
The daily wind speeds at a weather station are collected to demonstrate the implementation of the proposed approach. The long-term trend is first modeled and filtered out of the collected data. The empirical cumu-lative distribution function derived from the filtered data is modified by a generalized Pareto distribution; the resulting compound distribution function is used for transforming the filtered data into 'standard' data with a Gaussian distribution. Next, a non-stationary auto-regresive model with random coefficients modeled by a hidden Markov chain is developed for analyzing the 'standard' data. The maximum-likelihood estimates for the variances of daily winds are computed by an Expectation-Maximization algorithm; the random coefficients are sampled based on a Kalman filter algorithm. It is shown in the current study that the statisti-cal properties (e.g., means, standard deviations, auto-correlation functions as well as exceedance probabilities) of the simulated wind speeds are quite close to those of the field wind speeds. In addition, the simulated wind speeds exhibit the similar non-stationarity inherent in the field wind speeds; it reveals that the proposed model is capable of simulating non-stationary wind speeds.
.
analyze field wind speeds and then to simulate syn-thetic wind speeds. Compared to the models adopted by previous researches, the chosen model is different in the way that it is a non-stationary model with random time-varying model parameters.
Moreover, new algorithms are developed to estimate model parameters.
2 STANDARDIZATION OF WIND SPEED TIME SERIES
2.1 Probability distributions for wind speeds in each month
The daily maximum 10-minute wind speeds from 1961 to 1999 at a weather station are collected as a database to demonstrate the implementation of the proposed approach. The raw wind speeds are first converted to 10-meter, open-terrain wind speeds
{
Oki :i=1,...,39;k =1,...,365}
orO11::36539consider-ing the changes of anemometer heights and sur-rounding terrain roughness with time.
It has been concluded (Kao, 2004) that the wind
2.3 Transformation to Gaussian distribution
Observations stated in Section 2.1 and additional goodness-of-fit test results reflect that V11:365:39 is non-Gaussian. The tail of its empirical cumulative dis-tribution function is first modified by a generalized Pareto distribution; the resulting compound distribu-tion funcdistribu-tion is subsequently used for transforming
39
3 DEVELOPMENT OF A HIDDEN MARKOV CHAIN MODEL
3.1 Governing equations for the proposed model The standard deviations of standardized daily wind speeds in a year can be computed based onX11::36539 ; the results, depicted in Figure 2, suggest that the wind speed time series are indeed non-stationary. To analyze X11::36539 , a non-stationary auto-regresive model with random time-varying model parameters realized by a hidden Markov chain is developed as
where m (=8 according to Cherng, etc. (2008))is the model order; a (j=1,…, m; k=1,…365) are model kj parameters; e (k=1,…365) follow zero-mean Gaus-k sian distributions with variances of
2
σk (k=1,…365); εkj (j=1,…, m; k=1,…365) also follow zero-mean Gaussian distributions with vari-ances of δ2.
The auto-regressive model parameters a (j=1,…, kj m; k=1,…,365) vary with days according to the re-alization of the hidden Markov chain governed by Eq. (3).
Figure 2 Plot of variation of standard deviations in a year
0 5000 10000
3.2 Estimation of model parameters
The maximum-likelihood estimates 21ML:365
σˆ for the variances σk2(k=1,…365) can be derived by an Ex-pectation-Maximization algorithm (Shumway and Stoffer, 1982). A set of realization for e (k=1,…, k Kalman filter algorithm (Cherng, etc., 2008).
3.3 Generation of wind speed time series
A year of standardized daily wind speedsZ1:365can be generated by substituting a set of aˆ11::365m and ˆe1:365, sampled in Section 3.2, into Eq. (2). Subsequently, thirty-nine years of standardized daily wind speeds
39
4 COMPARISON OF FIELD AND ARTIFICIAL WIND SPEEDS
4.1 Comparison between X11::36539 and Z11::36539
To verify if the simulated wind speed time series
39 intervals of their statistical properties are compared with corresponding statistical properties ofX11::36539 . It
X1 fall within the 95% confidence intervals of the associated standard deviations ofZ11::36539 . Figures 3 and 4 show the comparison of auto-correlation func-tions in March and September respectively; Figure 5 shows the comparison of exceedance probability curves. All three figures indicate that the statistical properties of Z11::36539 are quite close to those of
Figure 3. Comparison of auto-correlation functions of
39
Figure 4. Comparison of auto-correlation functions of
39
Figure 5. Comparison of exceedance probability curves of
39 com-puted and the 95% confidence intervals of their sta-tistical properties are compared with corresponding statistical properties of V11:365:39. It is observed that monthly standard deviations of V11:365:39 locate within the 95% confidence intervals of the associated stan-dard deviations ofW11:365:39. Figures 6 and 7 display the comparison of cumulative distribution functions and exceedance probability curves respectively.
They also demonstrate that the statistical properties of V11:365:39 fall within the corresponding 95% confi-dence intervals for those ofW11::36539.
Figure 6. Comparison of cumulative distribution functions of
39
Figure 7. Comparison of exceedance probability curves of curves, shown in Figure 8, again reflects that the ex-ceedance probability curve ofO11::36539 locates within
Figure 8 Comparison of exceedance probability curves of
39
A non-stationary auto-regressive model in conjunc-tion with a hidden Markov chain is proposed to model field wind speed time series and to simulate artificial wind speed time series. It is shown in the current study that the statistical properties (e.g., means, standard deviations, auto-correlation func-tions as well as exceedance probabilities) of the simulated wind speeds are quite close to those of the
field wind speeds. In addition, the simulated wind speeds exhibit the similar non-stationarity inherent in the field wind speeds. In fact, the proposed model can be extended to deal with wind speeds and associated wind directions simultaneously; the results will appear in another paper.
6 ACKNOWLEDGEMENT
The financial support from National Science Council (Taiwan) under grants NSC 96-2221-E-011-047 and NSC 97-2221-E-011-067 is greatly appreciated.
7 REFERENCES
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