A stochastic Markov chain model for simulating wind speed time series at Tangiers, Morocco

Time series of then years of Hourly Average Wind Speed, HAWS, data from Tangiers station are analysed on a statistical basis by applying the Markovian process. The limiting behaviour of the Markov chain is then examined and compared to the histogram of observed wind speed. It was found that a 12×12 transition probability matrix was necessary to generate an acceptable synthetic time series. The manner in which the Markovian model can be used to generate wind speed time series are also described.Using the transition probability matrix developed from the real wind data, the synthetic wind speed time series are generated. The comparison between the real wind speed and the synthetic one shows that the statistical characteristics of wind speed are faithfully reproduced. The synthetic HAWS may be utilised as input data for any wind energy system.     

Forecasting and simulating wind speed in Corsica by using an autoregressive model

Alternative approaches for generating wind speed time series are discussed. The method utilized involves the use of an autoregressive process model. The model has been applied to three Mediterranean sites in Corsica and has been used to generate 3-hourly synthetic time series for these considered sites. The synthetic time series have been examined to determine their ability to preserve the statistical properties of the Corsican wind speed time series. In this context, using the main statistical characteristics of the wind speed (mean, variance, probability distribution, autocorrelation function), the data simulated are compared to experimental ones in order to check whether the wind speed behavior was correctly reproduced over the studied periods. The purpose is to create a data generator in order to construct a reference year for wind systems simulation in Corsica.     

考虑扇区划分的自然风风向建模

以随时间变化的自然风为研究对象,考虑风向扇区划分的方法,计算指定时间段内的风向时间维变差,基于统计学方法马尔科夫链模型理论与实验基地实测自然风数据,构建呼和浩特郊区典型风况下自然风风向变化的转移概率矩阵模型,并根据模型生成随机时间序列进行反向验证,意在建立更完整的风模型。     

Stochastic modelling of daily beam irradiation

A dynamical statistical analysis of the daily sum of the beam irradiation measured, on a horizontal surface, in Genoa, Italy, has been done using a 9-year time series, with two substantially different methods: the Markov chain model and the first order autoregressive model. In the first case, the data range has been divided into five different equiprobable classes or “states”. The sequential characteristics of the obtained discrete time series have been described by four “seasonal” 5 × 5 transition matrices between the states of the process. Yearly series of daily beam irradiation have been simulated by associating suitable values of irradiation to every state of the chain. In the second case, data have been first modified in order to obtain a standard Normal frequency distribution; an autoregressive process of order 1 has been fitted to the transformed series. The autoregressive parameter has been estimated keeping it time invariant. Synthetic sequences of daily solar irradiations have been generated with the fitted model. The reliability both of the Markov chain model and of AR(1) model has been verified by comparing the artificial series to the empirical one. The autoregressive model has shown an appreciable superiority in reproducing the stochastic law of the daily sums of beam irradiation with respect to the Markov chain model.     

Quality of wind speed fitting distributions for the urban area of Palermo,Italy

This study investigates the wind speed characteristics recorded in the urban area of Palermo, in the south of Italy, by a monitoring network composed by four weather stations. This article has two main objectives: the first one, to describe with clarity and simplicity the numerical procedures adopted to perform a preliminary statistical analysis of wind speed data, providing at the same time, the necessary mathematical tools useful to perform this analysis also without special software. The second objective is to verify if there are more suitable probability distributions able to better represent the original data respect the traditional ones. After a preliminary statistical analysis, in which the wind speed time series are split and analysed for each month and season, seven probability density functions are employed to describe wind speed frequency distributions: Weibull, Rayleigh, Lognormal, Gamma, Inverse Gaussian, Pearson type V and Burr. Shape and scale parameters for each weather station, period and distribution are provided. Their estimation is performed using the maximum likelihood method and the maximum likelihood estimators for each probability density function are provided. The quality of the data-fit is assessed by the classic statistical test Kolmogorov–Smirnov. The statistical test is used to rank the selected distributions in order to identify the distribution better fitting with the wind speed data measured in the urban area of Palermo. The Burr probability density function seems to be the most reliable statistical distribution.     

Statistical bivariate modelling of wind using first-order Markov chain and Weibull distribution

This paper studies the statistical features of the wind at Oran (Algeria). The data used are the wind speed and wind direction measurements collected every 3 h at the meteorological station of Es Senia (Oran), during the 1982/92 period. The eight directions of the compass card have been considered to build the frequency distribution of the wind speed for each month of the year and each direction. The three-hourly wind data have been modelled by means of Markov chains. First-order nine-state Markov chains are found to fit well the wind direction data, whereas the related wind speed data are well fitted by first-order three-state Markov chains. The Weibull probability distribution function has also been considered and found to fit the monthly frequency distributions of wind speed measurements. Two methods of wind data retrieval are thus made available. In fact, two models of chronological bi-series are obtained describing wind speed and wind direction.     

基于逐步回归分析—马尔可夫链模型的大坝变形预测

针对预测大坝变形准确性难度较大的问题,综合逐步回归分析和马尔可夫链的优点,采用逐步回归分析法对大坝原型观测资料进行分析,得到回归模型,并判别回归方程的有效性和精度,同时利用马尔可夫链确定位移时序的状态转移概率矩阵,通过划分残差状态、修正实测值与逐步回归模型拟合值的绝对误差与相对误差,建立了大坝变形预测的逐步回归分析—马尔可夫链预测模型(SRA-MC)。实例应用结果表明,模型的拟合值与实测值吻合良好,预测效果好,可见逐步回归分析—马尔可夫链模型在进行大坝变形预测时具有有效性,可应用于大坝变形预测分析及大坝安全监控预警中。     

基于EMD与加权马尔可夫链QR法的风电功率区间预测

提出一种基于经验模式分解(EMD)、加权马尔可夫链与分位数回归(quantile regression,QR)的风电功率概率区间预测方法。由于风功率数据与风速显著相关,首先对历史风速进行经验模式分解,得到不同频率段的风速,再以不同频率段的风速为样本,分别对其进行加权马尔可夫链预测,相加得到最终预测风速。最后将所得的预测风速代入QR预测模型,得到一定置信水平下的风电功率概率区间的上下限。以区间覆盖率和区间平均带宽为评价指标,与马尔可夫链下的QR法和加权马尔可夫链下的QR法的对比仿真表明,提出的基于经验模式分解与加权马尔可夫链下的QR法具有风电功率概率预区间预测的覆盖率更高,平均带宽更窄,精度更好的预测效果。     

Simulation of aggregate wind farm short-term production variations

The variability of wind power production poses the greatest challenge in the integration of large-scale wind power in power systems. Furthermore, larger-scale penetration implies a wider geographical spreading of installed wind power, resulting in reduced variability and the smoothing effect of total power generation. Therefore, analysis of the impact of wind power variations on power system operation requires adequate modeling of aggregate power output from geographically dispersed wind farms. This paper analyzes different aspects of Markov chain Monte Carlo simulation methods for the synthetic generation of dependent wind power time series. However, testing indicates that these approaches do not adequately model the stochastic dependence between wind power time series in conjunction with individual persistence which is necessary to obtain realistic distributions of aggregate power output and total power variations. Consequently, a novel approach based on a modified second-order Markov chain Monte Carlo simulation is proposed. Simulation results show that this method obtains synthetic time series of aggregate wind power which very closely fit the original data, with respect to both the cumulative density function of total output power and the probability density function of power variations.     

A Markov model of solar energy space and hot water heating systems

This paper presents a Markov model approach to the generalized solar energy space heating performance analysis problem. Specifically, Markov chain models are developed to represent ambient temperature, insolation, hot water load and system performance. From the Markov transition probability matrices for these variables, long-term expected performance is calculated. The theoretical development is implemented in FORTRAN IV on a Control Data 6400 Computer System. Computational experience gained, using STOLAR 3.1 (STOchastic soLAR energy systems model), indicates the stochastic approach requires approximately five per cent of the time necessary for standard dynamic simulation approaches with comparable performance results. The method also compared favorably with FCHART, a simplified design procedure.     

De‐trending of wind speed variance based on first‐order and second‐order statistical moments only

The lack of efficient methods for de‐trending of wind speed resource data may lead to erroneous wind turbine fatigue and ultimate load predictions. The present paper presents two models, which quantify the effect of an assumed linear trend on wind speed standard deviations as based on available statistical data only. The first model is a pure time series analysis approach, which quantifies the effect of non‐stationary characteristics of ensemble mean wind speeds on the estimated wind speed standard deviations as based on mean wind speed statistics only. This model is applicable to statistics of arbitrary types of time series. The second model uses the full set of information and includes thus additionally observed wind speed standard deviations to estimate the effect of ensemble mean non‐stationarities on wind speed standard deviations. This model takes advantage of a simple physical relationship between first‐order and second‐order statistical moments of wind speeds in the atmospheric boundary layer and is therefore dedicated to wind speed time series but is not applicable to time series in general. The capabilities of the proposed models are discussed by comparing model predictions with conventionally de‐trended characteristics of measured wind speeds using data where high sampled time series are available, and a traditional de‐trending procedure therefore can be applied. This analysis shows that the second model performs significantly better than the first model, and thus in turn that the model constraint, introduced by the physical link between the first and second statistical moments, proves very efficient in the present context. © 2013 The Authors. Wind Energy Published by John Wiley & Sons Ltd.     

Simple procedure for generating sequences of daily radiation values using a library of Markov transition matrices

This article describes how sequences of daily global radiation can be generated for any location, using as input only the average monthly radiation for that location (or the average monthly number of sunshine hours—insolation). The generated sequences are statistically indistinguishable from real ones and the method derived here is, therefore, a way of obtaining radiation sequences for locations were such sequences have not been measured, and for which many types of long-term performance calculations could not be made until now. The method is based on the observation that (1) there is a significant correlation only between radiation values for consecutive days and (2) that the probability of occurrence of radiation values is the same for months with the same K_t (clearness index). The method employs a library of Markov transition matrices, each corresponding to a specific interval in K_t. This article explains the derivation of the matrices, how they are to be used to generate radiation sequences, and compares synthetized and measured sequences.     

Using temperature and state transitions to forecast wind speed

A major issue in forecasting wind speed is non‐linear variability. The probability distribution of wind speed series shows heavy tails, while there are frequent state transitions, in which wind speed changes by large magnitudes, over relatively short time periods. These so‐called large ramp events are one of the critical issues currently facing the wind energy community. Two forecasting algorithms are analyzed here. The first is a regression on lags, including temperature as a causal factor, with time‐varying parameters. The second augments the first using state transition terms. The main innovation in state transition models is that the cumulative density function from regressions on the states is used as a right‐hand side variable in the regressions for wind speed. These two methods are tested against a persistence forecast and several non‐linear models, using eight hourly wind speed series. On average, these two models produce the best results. The state transition model improves slightly over the regression. However, the improvement achieved by both models relative to the persistence forecast is fairly small. These results argue that there are limits to the accuracy that can be achieved in forecasting wind speed data. Copyright © 2007 John Wiley & Sons, Ltd.     

Modelling of turbulent wind flow using the embedded Markov chain method

Small wind turbines are usually installed to provide off-grid power and as such can be situated close to the load in a less-than-ideal wind resource. These wind regimes are often governed by low mean speeds and high wind turbulence. This can result in energy production less than that specified by the manufacturer's power curve. Wind turbulence is detrimental to the fatigue life of key components and overall turbine reliability and therefore must be considered in the design stage of small wind turbines. Consequently it is important to accurately simulate wind speed data at highly turbulent sites to quantify loading on turbine components. Here we simulate wind speed data using the Markov chain Monte Carlo process and incorporate long term effects using an embedded Markov chain. First, second and third order Markov chain predictions were found to be in good agreement with measured wind data acquired at 1 Hz. The embedded Markov chain was able to predict site turbulent intensity with a reasonable degree of accuracy. The site exhibited distinctive peaks in wind speed possibly caused by diurnal heating and cooling of the earth's surface. The embedded Markov chain method was able to simulate these peaks albeit with a time offset.     

Application of Bayesian model averaging in modeling long-term wind speed distributions

Accurate estimation of wind speed distribution is critical to the assessment of wind energy potential, the site selection of wind farms, and the operations management of wind power conversion systems. This paper proposes a new approach for deriving more reliable and robust wind speed distributions than conventional statistical modeling approach. This approach combines Bayesian model averaging (BMA) and Markov Chain Monte Carlo (MCMC) sampling methods. The derived BMA probability density function (PDF) of the wind speed is an average of the model PDFs included in the model space weighted by their posterior probabilities over the sample data. MCMC method provides an effective way for numerically computing marginal likelihoods, which are essential for obtaining the posterior model probabilities. The approach is applied to multiple sites with high wind power potential in North Dakota. The wind speed data at these sites are the mean hourly wind speeds collected over two years. It is demonstrated that indeed none of the conventional statistical models such as Weibull distribution are universally plausible for all the sites. However, the BMA approach can provide comparative reliability and robustness in describing the long-term wind speed distributions for all sites, while making the traditional model comparison based on goodness-of-fit statistics unnecessary.     

Assessing the suitability of wind speed probabilty distribution functions based on wind power density

Three functions have so far predominantly been used for fitting the measured wind speed probability distribution in a given location over a certain period of time, typically monthly or yearly. In the literature, it is common to fit these functions to compare which one fits the measured distribution best in a particular location. During this comparison process, parameters on which the suitability of the fit is judged are required. The parameters that are mostly used are the mean wind speed or the total wind energy output (primary parameters). It is, however, shown in the present study that one cannot judge the suitability of the functions based on the primary parameters alone. Additional parameters (secondary parameters) that complete the primary parameters are required to have a complete assessment of the fit, such as the discrepancy between the measured and fitted distributions, both for the wind speed and wind energy (that is the standard deviation of wind speed and wind energy distributions). Therefore, the secondary statistical parameters have to be known as well as the primary ones to make a judgement about the suitability of the distribution functions analysed. The primary and secondary parameters are calculated from the 12-month of measured hourly wind speed data and detailed analyses of wind speed distributions are undertaken in the present article.     

Improved synthetic wind speed generation using modified Mycielski approach

In this paper, novel approaches for wind speed data generation using Mycielski algorithm are developed and presented. To show the accuracy of developed approaches, we used three‐year collected wind speed data belonging to deliberately selected two different regions of Turkey (Izmir and Kayseri) to generate artificial wind speed data. The data belonging to the first two years are used for training, whereas the remaining one‐year data are used for testing and accuracy comparison purposes. The concept of distinct synthetic data production with correlation‐wise and distribution‐wise similar statistical properties constitutes the main idea of the proposed methods for a successful artificial wind speed generation. Generated data are compared with test data for both regions in the sense of basic statistics, Weibull distribution parameters, transition probabilities, spectral densities, and autocorrelation functions; and are also compared with the data generated by the classical first‐order Markov chains method. Results indicate that the accuracy and realistic behavior of the proposed method is superior to the classical method in the literature. Comparisons and results are discussed in detail. Copyright © 2011 John Wiley & Sons, Ltd.     

Cox–Ingersoll–Ross model for wind speed modeling and forecasting

We propose a dynamic model for the squared norm of the wind speed which is a Markov diffusion process. It presents several advantages. Since the transition probability densities are in closed form, it can be calibrated with the maximum likelihood method. It presents nice modeling features both in terms of marginal probability density function and temporal correlation. We have tested the model with real wind speed data set provided by the National Renewable Energy Laboratory. The model fits very well with the data. Besides, we obtained a very good performance in forecasting wind speed at short term. This is an interesting perspective for operational use in industry. Copyright © 2015 John Wiley & Sons, Ltd.