# Construction of an Integral Model by the Example of Wind Turbine Dynamics

S.V. Solodusha, D.O. Gerasimov, K.V. SuslovThis study addresses the application of Volterra integral-power series to describe the nonlinear dynamic 'input-output' systems. The universality of this mathematical tool makes it possible to create a software for computer experiments. This study is a continuation of the research on the identification of the Volterra kernels, which was started at the Energy Systems Institute, Siberian Branch of the Russian Academy of Sciences. The first part of the paper presents a new algorithm for the identification of the second-degree Volterra polynomials for the systems which can be used for an experiment based on the test sets of disturbances. In the second part the numerical calculation results for a 'reference' dynamic system are given. The reference system is represented by a model of a horizontal-axis wind turbine. Quadratic Volterra polynomials are constructed. They describe the nonlinear dynamics of the angular velocity of the wind turbine components, depending on a blade lean angle and wind speed. The Volterra kernels were practically identified with respect to some chosen stationary state of the simulated system.Full text

- Keywords
- nonlinear dynamic system; quadratic Volterra polynomial; horizontal - axis wind turbine.
- References
- 1. Venikov V.A., Sukhanov O.A. Kiberneticheskie modeli elektricheskikh sistem [Cybernetic Models of Electric Power Systems]. Moscow, Energoizdat, 1982. 328 p. (in Russian)

2. Pupkov К.А., Kapalin V.I., Yushenko A.S. Funktsional'nye ryady v teorii nelineynykh sistem [Functional Series in the Theory of Non-Linear Systems]. Moscow, Nauka, 1976. 448 p. (in Russian)

3. Stegmayer G. Comparison of Volterra Models Extracted from a Neural Network for Nonlinear Systems Modelling. Lecture Notes in Computer Science, 2005, vol. 3697, pp. 457-463. DOI: 10.1007/11550907_72

4. Tong Zhou G., Giannakis G.B. Nonlinear Channel Identification and Performance Analysis with PSK Inputs. First IEEE Signal Processing Workshop on Signal Processing Advances in Wireless Communications, New York, 1997, pp. 337-340.

5. Chen C.H., Powers E.J. Fifth-Order Volterra Kernel Estimation for a Nonlinear Communication Channel with PSK and QAM Inputs. Ninth IEEE Signal Processing Workshop on Statistical Signal and Array Processing, New York, 1998, pp. 435-438.

6. Lin J.N., Unbehauen R. 2-D Adaptive Nonlinear Equalizers. Proceedings of EUSIPCO-92, Sixth European Signal Processing Conference, vol. 1, Brussels, 1992, pp. 135-138.

7. Minu K.K., Jessy John C. Volterra Kernel Identification by Wavelet Networks and Its Applications to Nonlinear Nonstationary Time Series. Journal of Information and Data Management, 2012, vol. 1, no. 1, pp. 4-9.

8. Volterra V. Teoriya funktsionalov, integral'nykh i integro-differentsial'nykh uravneni [A Theory of Functionals, Integral and Integro-Differential Equations]. Мoscow, Nauka, 1982. (in Russian)

9. Frechet M. Sur les Funktionnoles Continues. Ann. de l'Ecole Normale Sup, 1910, vol. 27, pp. 193-216.

10. Apartsyn A.S. [Nonclassical Volterra Equations of the First Kind in Integral Models of Dynamic Systems: Theory, Numerical Methods, Application. Dissertation of the Doctor of Physical and Mathematical Sciences]. Irkutsk, Irkutsk State University, 2000. (in Russian)

11. Danilov L.V., Matkhanov L.N., Filippov V.S. Teoriya nelineynykh dinamicheskikh tsepey [A Theory of Non-Linear Dynamic Circuits]. Мoscow, Energoizdat, 1990. (in Russian)

12. Ljung L. Identifikatsiya sistem. Teoriya pol'zovatelya [System Identification. Theory for the User]. Moscow, Nauka, 1991. (in Russian)

13. Apartsyn A.S., Solodusha S.V., Spiryaev V.A. Modelling of Nonlinear Dynamic Systems with Volterra Polynomials: Elements of Theory and Applications. International Journal of Energy Optimization and Engineering, 2013, vol. 2, no 4, pp. 16-43. DOI: 10.4018/ijeoe.2013100102

14. Apartsyn A.S. Nonclassical Linear Volterra Equations of the First Kind. Utrecht, Boston, VSP, 2003.

15. Apartsyn A.S., Solodusha S.V. Test Signal Amplitude Optimization for Identification of the Volterra Kernels. Automation and Remote Control, 2004, vol. 65, no 3, pp. 464-471. DOI: 10.1023/B:AURC.0000019379.43119.d0

16. Solodusha S.V. [Numerical Modelling of Heat Exchange Dynamics by Modified Quadratic Volterra Polynomial]. Vychislitel'nye tekhnologii [Computational Technologies], 2013, vol. 18, no 2, pp. 83-94. (in Russian)

17. Apartsyn A.S. [By the Identification of Nonlinear Nonstationary Dynamic Systems]. Kraevye zadachi [Boundary Value Problems], Irkutsk, Irkutsk State University, 1997, pp. 91-99. (in Russian)

18. Sidorov D.N. Medody analiza integralnykh dinamicheskikh modeley: teoriya i prilozheniya [Methods of Analysis of Integrated Dynamic Models: Theory and Applications]. Irkutsk, Irkutsk State University, 2013. (in Russian)

19. Pronin N.V., Martyanov А.S. Model of Wind Turbine VEU-3 in the Package MATLAB. Bulletin of the South Ural State University. Series: Power Engeneering, 2012, no. 37 (296), pp. 143-145. (in Russian)

20. Perdana A., Carlson O., Persson J. Dynamic Response of Grid-Connected Wind Turbine with Doubly Fed Induction Generator During Disturbances. Proc. of IEEE Nordic Workshop on Power and Industrial Electronics, Trondheim, 2004.

21. Sedaghat A., Mirhosseini M. Aerodynamic Design of a 300 kW Horizontal Axis Wind Turbine for Province of Semnan. Energy Conversion and Management, 2012, vol. 63, pp. 87-94.

22. Suslov K.V., Gerasimov D.O., Solodusha S.V. [Increasing Power Quality During Control of Smart Grid Elements]. Upravlenie kachestvom elektricheskoy energii [Proceedings of the International Conference Power Quality Management], Moscow, National Research University 'MPEI', 2014, pp. 191-198. (in Russian)