Stochastic Subspace Modelling of Turbulence

Authors: M. T. Sichani, B. J. Pedersen, S. R. K. Nielsen

Abstract:

Turbulence of the incoming wind field is of paramount importance to the dynamic response of civil engineering structures. Hence reliable stochastic models of the turbulence should be available from which time series can be generated for dynamic response and structural safety analysis. In the paper an empirical cross spectral density function for the along-wind turbulence component over the wind field area is taken as the starting point. The spectrum is spatially discretized in terms of a Hermitian cross-spectral density matrix for the turbulence state vector which turns out not to be positive definite. Since the succeeding state space and ARMA modelling of the turbulence rely on the positive definiteness of the cross-spectral density matrix, the problem with the non-positive definiteness of such matrices is at first addressed and suitable treatments regarding it are proposed. From the adjusted positive definite cross-spectral density matrix a frequency response matrix is constructed which determines the turbulence vector as a linear filtration of Gaussian white noise. Finally, an accurate state space modelling method is proposed which allows selection of an appropriate model order, and estimation of a state space model for the vector turbulence process incorporating its phase spectrum in one stage, and its results are compared with a conventional ARMA modelling method.

Keywords: Turbulence, wind turbine, complex coherence, state space modelling, ARMA modelling.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1085395

References:

[1] M. Shinozuka and C.M. Jan, Digital Simulation of Random Processes and its Applications. Journal of Sound and Vibration, 25(1), 111-128,1972.
[2] G. Solari and F.Tubino, A turbulence Model based on Principal Components. Probabilistic Engineering Mechanics, 17, 327-335, 2002.
[3] A. Kareem, Numerical simulation of wind effects: A probabilistic perspective. Journal of Wind Engineering and Industrial Aerodynamics, 96, 1472-1497, 2008.
[4] X. Chen,A. Kareem, Aeroelastic analysis of bridges under multicorrelated winds: integrated state-space approach. Journal of Engineering Mechanics ASCE, 127 (11), 1124-1134, 2001.
[5] J.C. Kaimal, J.C. Wyngaard and Y. Izumi, O.R. Cote, Spectral Characteristics of Surface-Layer Turbulence. Quarterly Journal of the Royal Meteorological Society, 98, 1972.
[6] M. Shiotani and Y. Iwayani, Correlation of Wind Velocities in Relation to the Gust Loadings. Proceedings of the 3rd Conference on Wind Effects on Buildings and Structures, Tokyo, 1971.
[7] E. Samaras, M. Shinozuka and A. Tsurui, ARMA representation of random processes. Journal of Engineering Mechanics ASCE, 111(3), 449461, 1985.
[8] A. Papoulis, Probability, Random Variables and Stochastic Processes, 2nd Ed. Mc Graw-Hill, 1984.
[9] W. Gersch and J. Yonemoto, Synthesis of multivariate random vibration systems: A two-stage least squares AR-MA model approach. Journal of Sound and Vibration, 52(4), 553-565, 1977.
[10] Y. Li and A. Kareem, ARMA systems in wind engineering. Probabilistic Engineering Mechanics, 5(2), 50-59, 1990.
[11] P. Van Overschee and B. De Moor, Subspace Identification for Linear Systems: Theory-Implementation-Applications, Dordrecht, Netherlands: Kluwer Academic Publishers, 1996.
[12] H. Akaik, Stochastic theory of minimal realization, IEEE Transactions on Automatic Control 19, 667-674, 1974.
[13] T. Katayama, Subspace Methods for System Identification, first ed., Springer, 2005.
[14] H. Akaik, Markovian representation of stochastic processes and its application to the analysis of autoregressive moving-average processes, Annals of the Institute of Statistical Mathematics 26(1), 363-387, 1974.