We consider the problem of wind energy production by using a second-order semi-Markov chain in state and duration as a model of wind speed. The model used in this paper is based on our previous work where we have shown the ability of second-order semi-Markov process in reproducing statistical features of wind speed. Here we briefly present the mathematical model and describe the data and technical characteristics of a commercial wind turbine (Aircon HAWT-10？kW). We show how, by using our model, it is possible to compute some of the main dependability measures such as reliability, availability, and maintainability functions. We compare, by means of Monte Carlo simulations, the results of the model with real energy production obtained from data available in the Lastem station (Italy) and sampled every 10 minutes. The computation of the dependability measures is a crucial point in the planning and development of a wind farm. Through our model, we show how the values of this quantity can be obtained both analytically and computationally. 1. Introduction Wind is one of the most important renewable energy sources. Wind energy is produced by converting the kinetic energy of wind into electrical energy by means of a generator. For this reason, it is important to dispose of an efficient stochastic model for wind speed changes. First- and second-order Markov chain models have been extensively used in wind speed modelling and synthetic time series generation; see, for example, Shamshad et al. [1], Nfaoui et al. [2], Youcef Ettoumi et al. [3], Sahin and Sen [4], and Torre et al. [5]. The Markovian assumption has, especially in the modeling of wind speed, several flaws. In discrete time Markov chain models, waiting times in a state before making a transition to another state are geometrically distributed. Therefore Markov chains impose artificial assumptions on the structure of the data that, very often, are inappropriate. This leads to a great simplification of the model which is unable to reproduce correctly the statistical properties of the real wind speed process. Semi-Markov chains do not have this constraint because the waiting time distribution functions in the states can be of any type, and this allows the data to speak for themselves without any restriction. For this reason, semi-Markov chains have been extensively applied to different fields [6–13]. D'Amico et al. [14] was the first paper where semi-Markov chains were applied in the modelling of wind speed. In that paper were proposed first- and second-order semi-Markov models with the aim of generate
References
[1]
A. Shamshad, M. A. Bawadi, W. M. A. Wan Hussin, T. A. Majid, and S. A. M. Sanusi, “First and second order Markov chain models for synthetic generation of wind speed time series,” Energy, vol. 30, no. 5, pp. 693–708, 2005.
[2]
H. Nfaoui, H. Essiarab, and A. A. M. Sayigh, “A stochastic Markov chain model for simulating wind speed time series at Tangiers, Morocco,” Renewable Energy, vol. 29, no. 8, pp. 1407–1418, 2004.
[3]
F. Youcef Ettoumi, H. Sauvageot, and A. E. H. Adane, “Statistical bivariate modelling of wind using first-order Markov chain and Weibull distribution,” Renewable Energy, vol. 28, no. 11, pp. 1787–1802, 2003.
[4]
A. D. Sahin and Z. Sen, “First-order Markov chain approach to wind speed modelling,” Journal of Wind Engineering and Industrial Aerodynamics, vol. 89, no. 3-4, pp. 263–269, 2001.
[5]
M. C. Torre, P. Poggi, and A. Louche, “Markovian model for studying wind speed time series in Corsica,” International Journal of Renewable Energy Engineering, vol. 3, no. 2, p. 3119, 2001.
[6]
G. D'Amico, J. Janssen, and R. Manca, “Semi-Markov reliability models with recurrence times and credit rating applications,” Journal of Applied Mathematics and Decision Sciences, vol. 2009, Article ID 625712, 17 pages, 2009.
[7]
G. D'Amico, J. Janssen, and R. Manca, “European and American options: the semi-Markov case,” Physica A, vol. 388, no. 15-16, pp. 3181–3194, 2009.
[8]
G. D'Amico, M. Guillen, and R. Manca, “Full backward non-homogeneous semi-Markov processes for disability insurance models: a Catalunya real data application,” Insurance, vol. 45, no. 2, pp. 173–179, 2009.
[9]
J. Janssen and R. Manca, Semi-Markov Risk Models for Finance, Insurance and Reliability, Springer, 2007.
[10]
N. Limnios and G. Opri？an, Semi-Markov Processes and Reliability, Statistics for Industry and Technology, Birkh？user, Boston, Mass, USA, 2001.
[11]
G. D'Amico and F. Petroni, “A semi-Markov model with memory for price changes,” Journal of Statistical Mechanics, no. P12009, 2011.
[12]
G. D'Amico and F. Petroni, “A semi-Markov model for price returns,” Physica A, vol. 391, no. 20, pp. 4867–4876, 2012.
[13]
G. D'Amico and F. Petroni, “Weighted-indexed semi-Markov models for modeling financial returns,” Journal of Statistical Mechanics, no. P07015, 2012.
[14]
G. D'Amico, F. Petroni, and F. Prattico, “First and second order semi-Markov chains for wind speed modeling,” Physica A. In press.
[15]
V. Barbu, M. Boussemart, and N. Limnios, “Discrete-time semi-Markov model for reliability and survival analysis,” Communications in Statistics, vol. 33, no. 11, pp. 2833–2868, 2004.
[16]
A. Blasi, J. Janssen, and R. Manca, “Numerical treatment of homogeneous and non-homogeneous semi-Markov reliability models,” Communications in Statistics, vol. 33, no. 3, pp. 697–714, 2004.
[17]
N. Limnios and G. Opri？an, “An introduction to semi-Markov processes with application to reliability,” in Handbook of Statistics, D. N. Shanbhag and C. R. Rao, Eds., vol. 21, pp. 515–556, North-Holland, Amsterdam, The Netherland, 2003.
