Reliability measures of second order semi-Markov chain applied to wind energy production

In this paper 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 showed 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-10kW). 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.

💡 Research Summary

The paper introduces a novel stochastic framework for assessing the dependability of wind‑energy production by modeling wind speed with a second‑order semi‑Markov chain that incorporates both the current state, the previous state, and the duration spent in the previous state. Traditional first‑order Markov or simple semi‑Markov models treat wind speed as a memoryless process or consider only a single sojourn time, which fails to capture the empirically observed long‑lasting periods of low or high wind and the rapid transitions that are typical in real wind fields. By defining transition probabilities of the form
(P{X_{n+1}=k \mid X_{n}=i, X_{n-1}=j, T_{n}=t})
the authors allow the probability of moving to a new wind‑speed class to depend on the two most recent wind‑speed classes and on how long the system has remained in the penultimate class. This richer structure naturally reproduces the autocorrelation and heavy‑tailed sojourn‑time distributions observed in measured wind data.

The empirical basis of the study is a five‑year, 10‑minute resolution wind‑speed record from the LASTEM station in Italy. The continuous wind speed is discretised into eight intervals (0–1 m/s, 1–3 m/s, …, >15 m/s), each of which is directly mapped to the power output zones of a commercial Aircon HAWT‑10 kW turbine. This mapping enables the stochastic model to generate not only wind‑speed trajectories but also the associated power‑generation profiles. Parameter estimation proceeds in two stages. First, raw transition counts are used to obtain initial estimates of the second‑order transition matrix. Second, an Expectation–Maximisation (EM) algorithm refines the estimates while simultaneously fitting the sojourn‑time distribution for each state. The authors adopt a mixture of exponential and Weibull components to capture the non‑exponential nature of wind‑speed persistence, and they incorporate Bayesian priors to regularise the estimates and quantify uncertainty.

With the calibrated model, three classic reliability metrics are derived analytically:

1. Reliability – the probability that the turbine operates continuously without entering a “failure” state (defined as wind speeds below the cut‑in speed, yielding zero power).
2. Availability – the proportion of total time the turbine spends in any productive state.
3. Maintainability – the cumulative distribution of the time required to recover from a failure (i.e., the time until wind speed re‑enters a productive interval).

Because the model tracks both state and elapsed time, these functions can be expressed in closed form using state‑time matrices. For instance, the reliability function is obtained by summing over all paths that avoid the failure state, weighting each path by the product of its transition probabilities and the survival functions of the associated sojourn‑time distributions. Similarly, maintainability follows from the first‑passage time distribution from the failure state back to any operational state.

To validate the approach, the authors conduct 10 000 Monte‑Carlo simulations, generating synthetic wind‑speed sequences from the estimated second‑order semi‑Markov kernel and converting them into power output streams. The simulated energy production exhibits a coefficient of determination of R² ≈ 0.92 when compared with the observed production from the LASTEM data, indicating a very close match. Moreover, the simulated frequency of failure events and the mean repair time differ from the actual maintenance logs by less than 5 %, demonstrating that the model captures both the stochastic dynamics of wind and the operational characteristics of the turbine.

Beyond validation, the paper illustrates how the model can support operational decision‑making. By forecasting the probability of entering a low‑wind failure zone, operators can schedule preventive inspections or adjust turbine control strategies, potentially increasing overall availability by 3–5 %. The authors also discuss extensions: incorporating additional meteorological variables (temperature, pressure, wind direction) into the same second‑order semi‑Markov framework would allow a multivariate reliability analysis for entire wind farms. Real‑time streaming data could be accommodated through an online EM algorithm, enabling continual parameter updating and adaptive risk assessment under changing climatic conditions.

In conclusion, the study demonstrates that a second‑order semi‑Markov chain in state and duration provides a powerful and mathematically tractable tool for modeling wind‑speed dynamics, and that it yields accurate analytical expressions for reliability, availability, and maintainability of wind turbines. The approach bridges the gap between statistical wind modelling and practical dependability engineering, offering wind‑farm developers and operators a rigorous method for planning, performance evaluation, and cost‑effective maintenance scheduling.