Weibull PDF

The increasing wind power generation requires the updating of the computational tools that support decision making in the operation and planning of electric power systems. This work describes a methodology to model the spatial correlations among wind speeds probability distributions in probabilistic power flow. The proposed methodology is based on Nataf transformation and Monte Carlo Simulation. The proposed methodology is illustrated in case studies with the IEEE 24 bus test system, highlighting the importance of adequately considering such correlations.

Information about the diameter distribution of a forest stand is an essential component for deriving a variety of stand attributes, understanding stand dynamics under varying ecological conditions and planning future forest prescriptions. Diameter distributions from multiple even-aged stands of Pinus occidentalis from three different ecological zones within La Sierra region in the Dominican Republic were modeled using a three-parameter Weibull probability prediction method using either maximum likelihood or modified method of moments, and two bases on parameter recovery method using either percentile-based or cumulative distribution function regression. Stand characteristics such as age, total tree height, trees per hectare, site quality and basal area were employed as predictor variables. Methods were evaluated based on goodnessof-
fit of the distributions and RMSE of stand yield. In these data, the parameter recovery method based on percentile based distribution proved to be considerably more efficient in describing the diameter distribution.

The main objective of this research is to estimate the surface runoff which is
generated by rainfall for a residential area in Basrah Province, south of Iraq for
different recurrence intervals of rainfall. The depth of runoff is based on changing of
rainfall intensity when it exceeding the rainfall intensity of 17 mm / hr (the approved
value in the design of the stormwater network in the area under consideration). Two
common probability techniques (Weibull and Gamma distribution) were used for
developing the relationship between annual maximum rainfall intensity and
recurrence intervals. Weibull and Gamma probability distribution passed the Chisquared
test for degree of freedom equaled to three. EPA Storm Water Management
Model (SWMM) is used for simulating a rainfall –runoff routing process. Modified
Green-Ampt method is used to describe the infiltration of water into the soil.
Kinematic wave routing is used for routing flows through drainage system. The
recurrence intervals have been used for estimating of surface runoff are ranged from
5 to 50 years with increment is equaled to 5 years. The depth of runoff is ranged from
16.2 mm for recurrence interval (5 years) to 29.6 mm for recurrence interval (50
years); these values are related to storm with duration is equaled to one hour, while it
is ranged from 49.5 mm to 101.6mm for storm duration of three hours

The annual monthly and yearly mean wind speed variations in King Saud airport, Al-Aqiq, KSA are collected and studied to investigate the feasibility to generate electrical power. Average wind speeds and wind power have been determined daily, monthly and annual. The parameters of the shape and scale of the Weibull density distribution function and the scale of the Rayleigh distribution function have been calculated. Comparison of the Weibull model with Rayleigh's distribution of wind power densities. The comparison revealed that the Weibull distribution function well reflects wind data as compared to Rayleigh density distribution. The mean wind speed for the Al-Aqiq region is about 3.39 m/s over a 38-year time period with a main direction of southwest. The diurnal study showed that the wind speed remains above 3.0 m/s from 09:00 AM to 10:00 PM and below it during the rest of the day's hours. A seasonal analysis shows that in summer, the wind speed is highest (4.41 m/s with a mainly north- northwest direction). The generation of wind energy by Aeronautica Windpower, Dewind, Enercon, Soyut Wind, Nordex and SouthWest at a hub height of 50 m has been considered for 15 wind machines of various sizes. The highest capacity factors(35.31%) were calculated for Soyut Wind 250 (773.370 MWh/Year). These results indicate that it is feasible to generate power from wind energy in the Al-Aqiq region for providing energy demands.

Energy supply in Nigeria is a major problem for both large and small-scale purposes. The potential of wind power for generation and sustainable electricity supply is yet to be adequately harnessed. Therefore, this study empirically investigated wind characteristics and potentials over four meteorological stations (Calabar, Uyo, Warri, and Ikeja) within the coastal region of Nigeria using 4-year (2008-2011) wind speed data obtained from the archive of Nigerian Meteorological Agency (NiMET). The wind characteristics were evaluated using descriptive analysis. The Weibull and Rayleigh probability density function and cumulative distribution function were fitted with the actual observed wind frequencies at 10 m, and accuracy assessment of the goodness of fit was carried out using some selected error metrics. The Weibull shape factor (k) and scale parameter (c) were calculated and used to estimate Weibull wind speeds and wind power density at 10 m and hub height 70 m using power law. Results revealed that, the maximum mean monthly wind speeds of 3.88 m/s in December, 4.73 m/s in January, 3.98 m/s in April, and 8.37 m/s in August, were observed over Calabar, Uyo, Warri, and Ikeja respectively. The performance evaluation of the distribution fit revealed that RMSE (ranging from 0.01 to 0.06) and χ 2 (ranging from 0.001 to 0.23) are very low for Weibull and high RMSE (0.03-0.33) and χ 2 (0.003-13.91) for Rayleigh, while r (>0.9) and COE (>0.8) were both high over all the stations and for the periods of study. The study concludes that Weibull model could be adopted as the best over Rayleigh for representing wind speed distributions over the coastal region and also Ikeja area can be considered as very suitable for wind turbine applications thus, the wind power potential in this area should be adequately harnessed in order to increase the power supply and reduce carbon dioxide emissions.

The annual monthly and yearly mean wind speed variations in King Saud airport, Al-Aqiq, KSA are collected and studied to investigate the feasibility to generate electrical power. Average wind speeds and wind power have been determined daily, monthly and annual. The parameters of the shape and scale of the Weibull density distribution function and the scale of the Rayleigh distribution function have been calculated. Comparison of the Weibull model with Rayleigh's distribution of wind power densities. The comparison revealed that the Weibull distribution function well reflects wind data as compared to Rayleigh density distribution. The mean wind speed for the Al-Aqiq region is about 3.39 m/s over a 38-year time period with a main direction of southwest. The diurnal study showed that the wind speed remains above 3.0 m/s from 09:00 AM to 10:00 PM and below it during the rest of the day's hours. A seasonal analysis shows that in summer, the wind speed is highest (4.41 m/s with a mainly northnorthwest direction). The generation of wind energy by Aeronautica Windpower, Dewind, Enercon, Soyut Wind, Nordex and SouthWest at a hub height of 50 m has been considered for 15 wind machines of various sizes. The highest capacity factors (35.31%) were calculated for Soyut Wind 250 (773.370 MWh/Year). These results indicate that it is feasible to generate power from wind energy in the Al-Aqiq region for providing energy demands.

In order to analyse failures of an ageing water pipe, some methods such as the loss-of-section require remaining wall thickness (RWT) along the pipe to be fully known, which can be measured by the magnetism based non-destructive evaluation sensors though they are practically slow due to the magnetic penetrating process. That is, fully measuring RWT at every location in a water pipe is not really practical if RWT inspection causes disruption of water supply to customers. Thus, this paper proposes a new data prediction approach that can increase amount of RWT data of a corroded water pipe collected in a given period of time by only measuring RWT on a part (e.g. 20%) of the total pipe surface area and then employing the measurements to predict RWT at unmeasured area. It is proposed to utilize a marginal distribution to convert the non-Gaussian RWT measurements to the standard normally distributed data, which can then be input into a 3-dimensional Gaussian process model for efficiently predicting RWT at unmeasured locations on the pipe. The proposed approach was implemented in two real-life in-service pipes, and the obtained results demonstrate its practicality.

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.

Analisi statistica dell’angolo d’inclinazione di frane superficiali mediante l’utilizzo della distribuzione di Weibull a 3 parametri.Nel presente studio e stata analizzata l'influenza dell’angolo di pendio quale fattore predisponente nello sviluppo di frane superficiali coinvolgenti sequenze prevalentemente argillose.Un approccio statistico basato sulla distribuzione di frequenza dell’angolo d’inclinazione di 2.520 frane superficiali inventariate nella provincia di Benevento ha permesso di osservare un ottimo adattamento alla distribuzione di Weibull con un valore dell’esponente b pari a circa 3.Gli indicatori statistici derivati dalla funzione di densita di Weibull hanno mostrato che la frequenza statisticamente significativa dei corpi di frana si registra su pendii con inclinazione da 6-17 gradi. Inoltre, il 63% del campione di frane analizzato ha una tendenza alla riattivazione su superfici inclinate di circa 14 gradi.

Among the various renewable energy
growing six times fast in recent years. This paper designates the systems of
the wind speed for the evaluation
Burr, Logistic and Weibull
wind speed density functions. The wind power density and cumulative density
functions are derived using the
distributions are estimated using the maximum likelihood method. The
these distributions are analyzed and compared using the
Darling test. An application of the mathematical model is demonstrated by a case
study that involves wind speed data from three different stations. Also, the descriptive
statistics of the wind speed
better awareness about the characteristics and properties of power density.
discussed distribution functions, the Burr probability density function appears to be
the most reliable statistical distribution for the stations taken for the analysis

Oil, gas, coal and uranium reserves of Kazakhstan are among the biggest in the world. Therefore, energy production is based on relatively cheap fossil fuels. However, it is little known about enormous renewable energy potential of Kazakhstan. In particular, as for wind power, nine sites are recognized by UNDP within Wind Power market development initiative as perspective. The average annual wind speed of all sites at a height of 80 meters is around 7.5 m/s and the estimated output can satisfy 2% of electricity demand of Kazakhstan by 2014. In this paper wind energy potential is assessed and power outputs for all locations are estimated using technical data of Vestas90-3MW turbine. Then, the obstacles preventing introduction of wind farms in Kazakhstan are discussed: a) problems related to the structure of energy sector; b) legislative issues and c) possible organizational and operational risks specific to each location. The paper describes existing barriers and suggests the possible changes to be made in order to introduce wind power in Kazakhstan.