Probability Modelling of Wind Velocity for Assessment of Wind Energy at Alexandria Coast

For the wind energy assessment, it is important to know the probability density distribution of the wind speed in order to calculate the mean power of a wind. In this study wind data, for the period from January 2000 to December 2002, hourly records of speed and direction at Alexandria coast are analyzed as random monthly observations. Statistical methods are used for determination of best fit random distributions of wind speed and direction using statistical standard software that helps in evaluation of wind speed energy potential in a study area. In the present paper, different theoretical Probability Density Functions (PDF) have been tried to determine the appropriate models, including Weibull distribution and their parameters. The minimum wind power density at Alexandria found in July and August (2000-2002) were 20.41 and 21.34 w/m and the maximum was in December and January (110.81 and 123.34 w/m). The most important distributions of the wind speed were Weibull and Dagum and the direction followed mostly the Johnson SB model.


INTRODUCTION
The world energy need increases every year by 4-5% whereas fossil fuel reserves decrease much faster than the need.In addition, with increasing negative effects of fossil fuels on environment, mainly developed countries and others have begun using renewable energy sources.Wind energy is a form of solar energy; it is an air current created by the balance between pressure and temperature differences due to the different distribution of solar heat coming to Earth.Nowadays, the fastest developing and most common used energy source is the wind energy.It is a clean and renewable alternative source of energy potential to fossils based energy sources polluting the lower layer atmosphere.Therefore, wind energy systems transforming wind power to electrical energy has been developing quite fast (Aras et al., 2003).Alexandria is a rapidly growing energy consumer and the use of renewable energy, such as wind energy will be of great importance as friendly source to environment.
Alexandria city is located in the south of the eastern Mediterranean at the Egyptian coast.It lies at (lat.31°12' N, long.29°57' E), shown in Fig. 1 is in the southern part of the Levantine Sea and it is one of the most important Egyptian cities that is affected by prevailing winds from north west with exceptional strong wind in winter during storms.

LITERATURE REVIEW
The minimum wind speed values occurred during night from 9.00 p.m. to 6.00 a.m.The monthly mean wind speed has the highest values during winter when it reached its maximum in January 1978 (9.2 knots) and in March 1978 (11.6 knots) for the meteorological station under study, Ras El-tin (Sabra, 1979).
According to Shalaby (1999), the wind speed along the Egyptian coast of the Mediterranean Sea, during the period from January 1961 to January 1990, the daily speed records showed higher values in winter season and lower ones in the autumn.The monthly means of the wind speed reached 11-16 knots at Alexandria during January and has minimum values during October, 7-10 knots.The high wind speed ranges, 17-21 and >22 knots, had higher percentage ratio during the winter season at Alexandria.During spring season at Alexandria, the wind speed ranges of 11-16 and 17-21 knots, had high percentage ratio and values>22 knots showed less frequent percentage.During summer season, the wind speed range, 7-10 knots, represented the high percentage at Alexandria.During autumn season, the wind speed ranges 7-10 and 11-16 knots had the high percentage with less frequent high values in the ranges 17-21 and greater than 22 knots.The maximum wind speed occurred during July 1977 to June 1978 round afternoon from 12.00 to 3.00 p.m.
During winter season, the prevailing winds were between south west, west and North West.They were During the autumn season, the prevailing winds were mainly between north west, north and north east with same occasional blow from south west and west directions.The most prevailing direction of wind was north west and north directions (Sabra, 1979;Shalaby, 1999).
Some authors have shown that the wind speed is fitting to the Weibull and other distributions for producing wind energy such as Lun and Lan (2000), Quine (2000), Toure (2005) and Zhou and Yang (2006).Quine (2000) estimated the mean wind climate and the probability of strong winds for wind risk assessment.He found that the relationship between wind strength and probability is commonly derived from several years of measurements at the site of interest.This relationship can be derived from the mean wind climate and represented by parameters of the parent Weibull distribution.Mortensen et al. (2006) provided a coherent and consistent overview of the wind energy resource over the land and sea of Egypt.The wind resource assessments and the site of wind turbines and wind farms may be employed directly by the mesoscale modelling results of the numerical wind atlas database.Yilmaz and Çelik (2008) estimated the wind speed probabilities using the probability distributions of Beta, Erlang, Exponential, Gamma, Log-Logistic, Lognormal, Pearson V, Pearson VI, Uniform and Weibull.Baran et al. (2013) described two possible Bayesian Averaging (BMA) models for wind speed data of the Hungarian Meteorological Service and showed that BMA post-processing significantly improves the calibration and precision of forecasts.
This study deals with the statistical analysis of hourly wind speed and direction data using probability models and the descriptive statistical measures at Alexandria Egypt.

MATERIALS AND METHODS
The data for speed and direction are obtained at Alexandria western harbour (Ras el Tin station), for the period of three years from January 2000 to December 2002, as hourly records.The western harbour lies at latitude 31°10' N and longitude 29°52.5 E with 7500000 m 2 surface area, 7 km length, 2 km maximum width and average 5.5 m depth (MRCC, 1993).It consists of two main basins, the inner basin and the outer basin (Fig. 1).
The data of wind speed and direction are modelled mathematically by the Probability distributions.This analysis was done on monthly bases from January to December.Foreach month, the data of the 3 years of analysis in the years 2000-2002 are used.In addition, the seasonal data for each year separately: winter (January, February and March), spring (April, May and June), summer (July, August and September) and autumn (October, November and December) are analyzed.For wind velocity, the proper distributions are determined by comparing 14 PDF (Weibull, Burr, Dagum, Gamma, Generalized (Gen.)Gamma, Gumbel Max, Maximum Extreme Value Type 1, Normal, Pearson Type 6, Johnson SB, Generalized Extreme Value, Nakagami, Beta and Cauchy) in the study area.The proper PDF was chosen by comparing with the most matching theoretical distributions with the observations.The goodness of fit plots (the Probability-Probability (P-P) and the Quantile-Quantile (Q-Q)) was used to find the convenient theoretical PDF distribution.The Easy Fit 5.5 Professional software (www.qweas.com)was used for these calculations.
Theory/calculation: The equations of the used distributions of the wind velocity probability.

Nakagami distribution:
The parameters: m-continuous parameter (m≥0.5),Ωcontinuous parameter (Ω>0), where 0≤x≤∞.Probability Density Function can be expressed by Eq. ( 25): And the Cumulative Distribution Function Eq. ( 26): Wind power density: Wind power density is proportional to the mean of wind speed cu be.It can be calculated by: (27) where, (kg/m 3 ) is the mean air density (1.069 kg/m 3 ) is used in this study.This depends on altitude, air pressure and temperature (Chang, 2010).

Testing of fitting of the probability distributions with observations has been done by:
 The Probability-Probability (P-P) plot which is a graph of the empirical CDF values plotted against the theoretical CDF values.It is used to determine how well a specific distribution fits to the observed data.This plot will be approximately linear if the specified theoretical distribution is the correct model. The Quantile-Quantile (Q-Q) plot which is produced by plotting the observed data values x i (i = 1, ... , n) along the X-axis, against: . as Y-axis where, F -1 (x) is Inverse Cumulative Distribution Function (ICDF), Fn (x) is empirical CDF and n is sample size.The Q-Q plot will be approximately linear if the specified theoretical distribution is the correct model.

RESULTS AND DISCUSSION
The above methods of analysis have been applied at Alexandria, Egypt, meteorological station in the study period on monthly and seasonal bases and the main results are shown below.

Probability distributions of wind velocity:
Probability distributions of wind speed: Monthly wind speed for the period of study: The Weibull probability distribution was good for all months of the study period.In addition to this model.Other models were also found to be in good fitting with data.During January 2000, the Dagum model was good fitting, during January, July and October 2001, the Burr model had good fitting, during January 2002, December    Seasonal wind speed: During all seasons of the study period, the Weibull distribution was the best fitting one.However, during winter 2000, summer 2000 and autumn 2001, Dagum, Johnson SB and Gamma models were comparable in their fitting.During winter (2001, 2002), spring (2000, 2001, 2002), summer (2001, 2002) and autumn (2000,2002), the Dagum model was better fitting than Weibull.Table 2 shows the parameters of the good models.
Descriptive statistics of wind speed and wind generation possibility at Alexandria, Egypt: The amount of energy harvestable from a wind turbine in a particular location depends on the characteristics of the wind turbine and wind conditions.It is based on the output power curve of a wind turbine and wind speed statistics.
It is important to know the minimum and the maximum mean wind speed for the generation of wind energy by using turbines.For Alexandria coast, during the years months 2000, 2001 and 2002 and the period from 2000 to 2002 of all months of the year are illustrated in Table 3. From this table, the minimum wind power density was found in July and August (2000)(2001)(2002) were 20.41 and 21.34 w/m 2 respectively that the corresponding mean wind speed were 2.70±1.45 and 2.74±1.49m/sec.respectively, the maximum power in the months December and January (110.81 and 123.34 w/m 2 ) and the corresponding mean wind speed were (4.05±3.10 and 4.11±3.28m/sec).

Seasonal probability distributions of wind direction:
The Weibull distribution does not fit the direction observations for all seasons during the period (2000)(2001)(2002).The Johnson SB model was good for all seasons except in summer, 2000, the Cauchy model was more suitable than Johnson SB in summer, 2000.In winter,

CONCLUSION
The off shore Wind speed and proximity of coasts is interest for producing energy.The wind energy resource assessments are important for sitting the wind turbines.The probability density distribution of the wind speed is used to calculate the mean power from a wind turbine over a range of mean wind speeds.In this study, the wind data for the period from January 2000 to December 2002, hourly records for speed and direction at Alexandria coast are analyzed as random monthly observations.The proper PDF's for both speed and direction were chosen by comparing the most matching theoretical distributions with the observations.The results of monthly and seasonal probability distributions of wind speed indicated that the Weibull probability distribution was good for all months and seasons of the study period, in the study area.In addition, Dagum model was similar to Weibull in winter 2000.It is more fitting than Weibull in February (2001,2000), March 2000, April 2002, May (2000, 2001, 2002), June (2001, 2002), July 2000, September 2000, October 2000, November 2000, December 2002, the months (February, March, April, May, June, July, August, October), winter (2001, 2002), spring (2000, 2001, 2002), summer (2001, 2002) (2000)(2001)(2002).In this period, Johnson SB, was good for all months and seasons except in summer, 2000, Cauchy model was the best.In addition to Johnson SB, Beta model was good in December of the study period and winter, 2001.The general descriptive monthly statistics of wind speed and direction are also presented.
The wind power density average of the three years was maximum in January and December (about 123 and 110 w/m 2 ) and minimum in July and august (about 20 to 22 w/m 2 ).
These results can be used for the calculation of the probability of occurrence of a certain speed that helps in the choice of the best wind generating device in the specific area to get maximum wind energy at the site.

Fig. 1 :
Fig. 1: Alexandria map (admiralty chart), black triangle shows the meteorological stationbetween north west, north and north east during spring season.Sometimes, strong winds blow between from south east and south west.The prevailing winds were mainly from the North West direction and frequently from west and north directions during summer season.During the autumn season, the prevailing winds were mainly between north west, north and north east with same occasional blow from south west and west directions.The most prevailing direction of wind was north west and north directions(Sabra, 1979;Shalaby, 1999).Some authors have shown that the wind speed is fitting to the Weibull and other distributions for producing wind energy such asLun and Lan (2000),Quine (2000),Toure (2005) andZhou and Yang (2006).Quine (2000) estimated the mean wind climate and the probability of strong winds for wind risk assessment.He found that the relationship between wind strength and probability is commonly derived from several years of measurements at the site of interest.This relationship can be derived from the mean wind climate and represented by parameters of the parent Weibull distribution.Mortensen et al. (2006) provided a coherent and consistent overview of the wind energy resource over the land and sea of Egypt.The wind resource assessments and the site of wind turbines and wind

Fig. 2 :
Fig. 2: April 2002 wind speed probabilities, at Alexandria, Egypt.This Figure shows Dagum model is more proper than Weibull

Fig. 3 :
Fig. 3: December wind direction probabilities for the period 2000-2002, at Alexandria, Egypt.It shows that beta model is more proper than Johnson SB

Fig. 4 :
Fig. 4: Summer, 2000, wind direction probability distributions at Alexandria, Egypt.It shows that Cauchy model is more convenient than Johnson SB 2001, the Beta model was similar to Johnson SB.The parameters of the good fitting models are mentioned in Table 5 and examples of the best fit distributions are shown in Fig. 4 that indicates that Cauchy model is more convenient than Johnson SB during summer, 2000.
, autumn 2002.Burr model was similar to Weibull in January 2001, July 2001 and October 2001.Pearson 6 was similar to Weibull in January 2002 and December 2001.Nakagami model was similar to Weibull in March 2001, August 2001 and November 2001.The months November and December were the best in autumn 2000.Gen Extreme Value was similar to Weibull in March 2002.Gen. Gamma was similar to Weibull in April 2000, July 2002 and September 2002 and it was the best in October 2002, November 2002, the months January and august.Normal model was similar to Weibull in august 2000 and the best in August 2002.Gamma was similar to Weibull in autumn 2001 and the best in December 2000.Johnson SB was similar to Weibull in summer 2000.For wind direction, the Weibull model does not fit for all months and seasons of the period

Table 2 :
Parameters of wind speed probability distributions at Alexandria, Egypt, in the period 2000-2002 seasons

Table 3 :
The mean and the standard deviation of the wind speed data in m/sec and the wind power in w/m 2 at Alexandria, Egypt, during different months of the years2000, 2001 and 2002 individually and in the total period (2000-2002)

Table 4 :
Parameters of wind direction probability distributions at Alexandria, Egypt, in the period