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Wind characterization analysis incorporating genetic algorithm: A case study in Taiwan Strait
Energy 36 (2011) 2611e2619
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Energy journal homepage: www.elsevier.com/locate/energy
Wind characterization analysis incorporating genetic algorithm: A case study in Taiwan Strait Feng-Jiao Liu a, Pai-Hsun Chen b, Shyi-Shiun Kuo b, De-Chuan Su b, Tian-Pau Chang b, *, Yu-Hua Yu b, Tsung-Chi Lin b a b
Department of Electrical Engineering, Nankai University of Technology, 568 ChungTseng Road, Caotun, Nantou 542, Taiwan Department of Computer Science and Information Engineering, Nankai University of Technology, 568 ChungTseng Road, Caotun, Nantou 542, Taiwan
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 August 2010 Received in revised form 29 January 2011 Accepted 1 February 2011 Available online 8 March 2011
In this paper, the genetic algorithm (GA) is originally applied to compute the Weibull parameters for wind characterization analysis, in which an objective function required in GA for searching optimization solution has been first defined as well. Wind data analyzed are observed at a wind farm in the Taiwan Strait from 2006 to 2008. To accurately describe wind speed distribution three kinds of probability density functions are compared, i.e. the Weibull, logistic and lognormal functions. Statistical parameters including the max error in the KolmogoroveSmirnov test, root mean square error, Chi-square error and relative error of wind power density are considered as judgment criterions. The results show that GA is a useful method, there is about 33% time saving when compared with conventional iteration method. Weibull function describes best the wind distribution, regardless of time periods. Accordingly, wind power density, availability factor and electrical energy output from an ideal turbine are assessed using the Weibull parameters; utilization rate of wind energy for the currently used turbine is discussed. Further the wind energy compensates very well with solar energy; when solar radiation is down in winter and spring, the wind power becomes greater; energy ratios for each month are calculated lastly. Ó 2011 Elsevier Ltd. All rights reserved.
Keywords: Genetic algorithm Wind speed Wind energy Probability density function Weibull function
1. Introduction Sufficient energy supply is very crucial for a country’s economic development. Due to the shortage of fossil fuels, seeking alternative energy source already becomes an urgent business [1e5]. Wind energy is inexhaustible natural resource that can be converted to electrical power through wind turbine. To effectively evaluate the wind power available for a particular site, studying the wind’s statistical characteristics is needed; meanwhile three indication quantities including the availability factor, capacity factor, and conversion efficiency of a wind turbine must be considered together while assessing wind resource [6]. Many kinds of probability density functions have been proposed in literature for wind distribution analysis such as the gamma function, beta function, logistical function, lognormal function, Rayleigh function and Weibull function. Among these, Weibull and Rayleigh distributions are the most frequently used ones for wind speed modeling due to their two flexible parameters [7e21]; Weibull shape parameter describes the width of data distribution, the larger the shape parameter the
* Corresponding author. Tel.: þ886 49 2563489; fax: þ886 49 2315030. E-mail address: [email protected] (T.-P. Chang). 0360-5442/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2011.02.001
narrower the distribution and the higher its peak value; scale parameter controls the abscissa scale of a plot of data distribution [22], Rayleigh function is a special case of Weibull function in which shape parameter is 2. However, it should be noted that the Weibull and Rayleigh distributions are not suitable for all the wind regimes encountered in nature. Akpinar and Akpinar [23] as well as Jaramillo and Borja [24] used the mixture distribution of Weibull function, while Li and Li [25] as well as Ramirez and Carta [26] used the probability function derived with maximum entropy principle to model wind characteristics. Luna and Church [27] proposed to use a universal distribution for wind speed, and discovered that the most applicable distribution is the lognormal function. However, they also mentioned that the universal distribution may not work well for every site due to the differences in climate and topography. Similarly, Garcia et al. [28] found that lognormal distribution is better than Weibull distribution for the wind speed data in the area of Navarre, Spain. Scerri and Farrugia [29] compared the ability of logistic function in wind speed modeling with that of Weibull function and found that the results obtained from the two functions follow the same trend. Taiwan locates between the world’s largest continent (Asia) and the largest ocean (Pacific). More than 95% of energy supply comes from imported fossil fuels. In winter and spring seasons the island is
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F.-J. Liu et al. / Energy 36 (2011) 2611e2619
Nomenclature A Af c erf() Eideal f() F() g() G() h() H() k n O() P Pr T T()
sweep area of turbine blade, m2 availability factor, dimensionless scale parameter of Weibull function, m/s error function output energy of ideal turbine, kWh Weibull pdf cumulative Weibull function logistic pdf cumulative logistic function lognormal pdf cumulative lognormal function shape parameter of Weibull function, dimensionless sample size, dimensionless cumulative frequency of observed data wind power density, W/m2 rated power, W time period, hr cumulative frequency of theoretical function
visited by strong northeastern monsoon. While in summer and autumn the southwestern monsoon prevails in this region and occasionally tropical cyclones come with extremely high wind. Overall the average wind speed for the whole year is high enough especially in the region of Taiwan Strait. As a result, among various renewable energy sources wind energy has much potential to compensate the lack of fossil fuels. However, relevant research focused on the Taiwan Strait is still few so far. With the fast growing of economic activities between Taiwan and Mainland China, investigating the wind characterization about this region is needed. In this paper the validity of the logistic, lognormal and Weibull functions in describing wind speed distribution will be compared considering various statistical errors such as the max error in the KolmogoroveSmirnov test, root mean square error, Chi-square error and relative error of wind power density; the solar irradiation observed in central Taiwan will be shown together with the wind power to see what the relationship is between them. Some numerical methods in literature can be employed to calculate the Weibull parameters such as the moment method [16e21], empirical method (a special case of moment method), graphical method (i.e. doing line regression for the cumulative distribution data based on least-squares method), maximum likelihood method, etc. Among these, as concluded by Seguro and Lambert [11], Genc et al. [16] as well as Chang [22], the graphical method performs worse than others when the number of data points used is smaller, although this method is widely applied in wind energy software. According to the tests of Monte Carlo simulations in [22], the maximum percent error in calculating the Weibull shape parameter is only 0.08% when random data number is 10,000, which is a little larger than the number of hours of a year 8760 (¼365 24); however, the same error reaches 1.3% and 5.2% if data number is 1000 and 100, respectively. Besides the relative errors between the wind potential energy calculated from actual time-series data and that from theoretical probability function by the graphical method reach even 23% in [11] and 45% in [22]. As mentioned by Seguro and Lambert [11] the basic reason for the inaccuracy of the graphical method is that the leastsquares regression is performed not on the actual wind speed data, but on its cumulative frequency distribution. Each point is equally weighted even if some data bins may represent more data points than others. Contrarily both the moment and maximum likelihood methods that require iterative procedures always get
v v vi vME vMP vo vr z
wind speed, m/s mean wind speed, m/s cut-in speed of turbine, m/s wind speed carrying maximum energy, m/s most probable wind speed, m/s cut-off speed of turbine, m/s rated speed of turbine, m/s anemometer height, m
Greek letters e objective function in GA s standard deviation of wind speed, m/s G() Gamma function r air density, kg/m3 a coefficient of power law, dimensionless d scale parameter of logistic function, m/s l mean of natural logarithm of wind speed, m/s f standard deviation of natural logarithm of wind speed, m/s
better performance, irrespective of data number. In the present study, the moment method is adopted for its accuracy. Determining the relevant parameters of Weibull probability function through numerical iteration way is a tedious problem. Another aim of this paper is to apply an optimization method, the genetic algorithm (GA), to calculate the Weibull parameters that would be the first presentation in wind energy field; its performance will be compared with conventional iteration method. Wind data analyzed is measured from 2006 to 2008 at a wind farm, Chungtun, located in the middle of Taiwan Strait (with longitude of 119.6 E and latitude of 23.6 N), which is handled by the Ministry of Economic Affairs. There are eight wind turbines installed in this wind farm ranging about 1500 m in length; they are designed with the same specifications as below: the rated power is 600 kW; the cut-in, rated, and cut-off speed is 2.5, 12.5, and 25 m/s, respectively. Rotor’s diameter is 43.7 m, which has blade sweep area of 1500 m2. Anemometer height is 46 m above ground level. Wind data recorded is not always complete for the eight turbines, for the consideration of study quality only three turbines are selected and analyzed in the present work. Although there is no wind farm other than Chungtun built in the Taiwan Strait; wind data from other turbines with different kinds of specifications is not available currently, but it would not influence the uniqueness of the present paper introducing originally the GA into the wind energy field. 2. GA GA is a kind of optimization searching techniques, which was first proposed by Holland [30] simulating the evolutionary idea of the ‘survival of the fittest’. Initially the possible solutions are randomly generated and are coded as strings, named chromosomes, a set of chromosomes is called the population of the generation; the chromosomes in a generation are forced to evolve toward better ones in the next generation using three basic operators: i.e. the reproduction, crossover and mutation, considering simultaneously a problem-dependent objective (fitness) function [31]. The computation flowchart of GA with 8 bits in gene coding is illustrated in Fig. 1. Reproduction is a process in which a number of selected chromosomes are exactly copied and become a part of the offspring according to the level of fitness values. The chromosomes with greater fitness value will have larger opportunity of contributing offspring to their next generation. However, a small part of
F.-J. Liu et al. / Energy 36 (2011) 2611e2619
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2 1=k vME ¼ c 1 þ k
(4)
The cumulative Weibull distribution function enables us to calculate the probability of the wind speed exceeding the value u:
k u Fðv uÞ ¼ exp c
(5)
For a wind turbine designed with a given cut-in speed (vi) and cut-off speed (vo), the operating probability of the wind turbine, named here availability factor, is equal to the probability of wind speed between vi and vo, i.e. Fðvi v < vo Þ calculated by:
k k v vo exp AF ¼ Fðvi v < vo Þ ¼ exp i c c
(6)
3.2. Wind power density Wind power density calculated from actual time-series data is expressed by:
P1 ¼
1 3 rv 2
(7)
where, r is the air density; v3 is the mean of wind speed cubes. Wind power density based on a theoretical probability density function can be obtained by the integration:
P2 ¼
Z 1 1 3 r v3 f ðvÞdv ¼ rc3 G 1 þ 2 2 k
(8)
3.3. Electrical energy generated by an ideal wind turbine Fig. 1. Computation flowchart of GA.
less fit chromosomes are still needed and included in optimization searching, this helps keep the diversity of the chromosomes preventing premature convergence on poorly local solutions. In crossover, stochastically selected subsections of two individual chromosomes are swapped each other to produce the offspring. In mutation, randomly selected genes in chromosomes are changed with a small mutation rate to maintain genetic diversity from one generation to the next. In binary coding, it just means that changes digit 1 to digit 0 and vice versa. The algorithm is terminated when either a maximum number of generations have been produced, or a satisfactory fitness level is reached in searching.
For an idealized wind turbine, it starts to run at cut-in speed then its output power (P(v)) increases with wind speed until the rated speed (vr). For wind speed between the rated speed and cutoff speed the output power remains a constant amount, i.e. the rated power (Pr). The turbine will be shut down while wind speed exceeds cut-off speed. The rated power is given by:
Pr ¼
where, A is the sweep area of the blade of wind turbine. Consequently, as illustrated in Chang et al. [6] and Mathew et al. [33], the output energy (Eideal) of an ideal turbine for a given time period T can be calculated by the following expression through numerical integration:
Eideal
3.1. Weibull function
¼ T
vMP
The wind speed that carries maximum energy is obtained by:
PðvÞf ðvÞdv þ
Pr f ðvÞdv vr
Zvr k kvk1 v dv v3 exp ¼ T 2rA c c c
ð10Þ
vi
k Zvo k1 k v v þv3r dv exp c c c vr
(2)
(3)
Zvo
vi
(1)
The most probable wind speed for a particular location with the given probability function can be calculated by [13,32]:
1 1=k ¼ c 1 k
PðvÞf ðvÞdv Zvr
where v is the wind speed, k is the shape parameter, and c is the scale parameter with the same unit as wind speed. Weibull cumulative distribution function (cdf) is expressed by:
k v FðvÞ ¼ 1 exp c
¼ T 0
Weibull probability density function (pdf) is given as:
k kvk1 v exp c c c
(9)
ZN
3. Wind and wind turbine characteristics
f ðvÞ ¼
1 rAv3r 2
3.4. Wind speed extrapolation To estimate wind speeds at different heights, empirical power law is considered:
v1 ¼ v2
a z1 z2
(11)
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F.-J. Liu et al. / Energy 36 (2011) 2611e2619
where v1 and v2 are the wind speeds at heights z1 and z2, respectively. Since the effect of height on air density for the elevations considered by the present work can be neglected, subsequently the power density of the wind at different height is mainly affected by the wind speed and can be estimated by:
3a z1 z2
wind speed observed at hub height of 46m
25
(12)
The parameter a is the power law coefficient representing the degree of roughness of ground surface. The typical value of 0.14 (one-seventh) for a wide-plain area is adopted in this study [34]. 4. Objective function
20
Wind speed (m/s)
P1 ¼ P2
30
Low wind period 15
10
Weibull shape (k) and scale (c) parameters can be estimated using iterative way based on the moments of wind distribution, i.e. the mean ðvÞ and standard deviation (s) of observed wind speeds, which is commonly adopted by other researches for its accuracy [16e22] and named conventional iteration method here.
1 v ¼ cG 1 þ k
s ¼ c G 1þ
2 1 1=2 G2 1 þ k k
0
(13)
2
v
Gð1 þ 2=kÞ G2 ð1 þ 1=kÞ G2 ð1 þ 1=kÞ
¼
(14)
s2 2
v
Gð1 þ 2=kÞ G2 ð1 þ 1=kÞ G2 ð1 þ 1=kÞ
(16)
5. Goodness of fit To test the goodness of fit between a theoretical probability function and observation data, four kinds of statistical errors are considered as the judgment criterions in this paper. The smaller the errors the better the fit is. The first one is the KolmogoroveSmirnov test (KS), which is defined as the max error in cumulative distribution functions [21].
(17)
where T(v) and O(v) are the cumulative distribution functions for wind speed not exceeding v in the theoretical and observed dataset, respectively. The second one is the root mean square error (RMSE) defined as:
1=2 X 1 n 2 RMSE ¼ ðy y Þ i ic n i¼1
2000
3000
4000
5000
6000
7000
8000
9000
where, yi is the actual value at time stage i, yic is the value computed from correlation expression for the same stage, n is the number of data. The third judgment criterion is the Chi-square error given as:
c2 ¼
In GA optimization searching, the residual value e is set to be the objective function. If the objective function converges to a predefined tolerance value then an acceptable shape parameter k is obtained; the scale parameter c can be thus determined using Eq. (13). To practically evaluate the performance of GA, the tolerance value of the objective function is defined to be 1 103 in the present study, similar research can be found in [35].
KS ¼ maxjTðvÞ OðvÞj
1000
Fig. 2. Hourly mean wind speed measured at a selected wind turbine throughout the year.
(15)
As known it usually exists a residual error e between the left and right side of Eq. (15) during iteration expressed as:
e¼
0
Hour of year
where G() is the Gamma function. To apply the GA to search the Weibull parameters, the moment equations above can be rewritten as:
s2
5
(18)
Xn 1 2 ðy y Þ i ic i¼1 y ic
(19)
Table 1 Wind and wind turbine characteristics for different time periods. Low wind period (Aprile September)
High wind period (Octobere March)
Yearly
Mean speed (m/s) Standard deviation (m/s) Skewness coefficient Kurtosis coefficient
6.75 3.49 0.76 3.77
12.61 4.96 0.22 2.41
9.67 5.19 0.41 2.32
Weibull shape parameter Weibull scale parameter (m/s) Most probable speed (m/s) Speed having maximum energy (m/s)
2.02
2.75
1.94
7.61
14.17
10.91
5.44
12.03
7.51
10.69
17.29
15.70
Capacity factor Availability factor Power density (kW/m2) Potential energy in specified area (kWh)
0.293 0.900 0.35 2338429.1
0.745 0.983 1.82 11903902.8
0.518 0.938 1.09 14326901.0
Output energy of ideal turbine (kWh) Output energy of actual turbine (kWh) Conversion efficiency (actual/specified) Conversion efficiency (actual/ideal)
2012999.4
5461497.7
7450772.5
797800.3
2018006.5
2815806.8
0.341
0.170
0.197
0.396
0.369
0.378
F.-J. Liu et al. / Energy 36 (2011) 2611e2619 2
0.26 wind speed histogram Weibull pdf logistic pdf lognormal pdf Weibull cumulative probability logistic cumulative probability lognormal cumulative probability observed cumulative probability
0.24
0.25
0.22 0.2
Probability density function
Objective function
0.2
0.15
0.1
0.05
0.18 0.16
1.8
1.6
1.4
January
1.2
0.14 1 0.12 0.8
0.1 0.08
Cumulative probability
0.3
2615
0.6
0.06 0.4 0.04
0
0.2
0.02 0
0 0
-0.05 0
20
40
60
80
100
120
140
160
180
2
4
6
8
10
14
16
18
20
22
24
26
28
30
Wind speed (m/s)
200
Generations in GA
12
Fig. 5. Comparison between the theoretical probability density functions and observed wind speed histogram for January.
Fig. 3. Variation of objective function with generation number in GA searching.
The fourth judgment is the relative error between the wind power density calculated from actual time-series data and that from theoretical probability function, which is most meaningful in wind energy assessment. 6. Results and discussion Fig. 2 shows the hourly mean wind speed measured at one of the selected wind turbines throughout the year in the Taiwan Strait. As shown, the data could be roughly classified into high wind period from October to March and low wind period from April to September, as marked by the vertical broken lines reflecting
different monsoons. Since the variations of wind speeds for other two turbines selected (not shown here) reveal similar trends with the Fig. 2; hereafter in the present study the subsequent analyses are based on the averaged wind data of the three turbines. Corresponding wind and turbine characteristics for different time periods are summarized in Table 1. Although the power density during high wind period is much greater than that during low wind period, the energy conversion efficiency for wind turbine currently used is lower, that will be analyzed later. Fig. 3 shows the variation of objective function with the generation number of GA in searching a Weibull shape parameter; the objective function converges to an acceptable level after only 43 generations. The convergent rate is dependent on GA operators and computer specifications. To objectively evaluate the superiority of GA in determining Weibull parameter, first we calculate the shape
100 2
0.4 0.38
wind speed histogram Weibull pdf logistic pdf lognormal pdf Weibull cumulative probability logistic cumulative probability lognormal cumulative probability observed cumulative probability
0.36 0.34 0.32
70
0.3
Probability density function
Time saved by using GA (%)
80
60 50 40 30
0.28 0.26
1.8
1.6
1.4
July
0.24
1.2
0.22 1
0.2 0.18
0.8
0.16 0.14
Cumulative probability
90
0.6
0.12 0.1
20
0.4
0.08 0.06 0.04
10
0.2
0.02 0
0
1
2
3
4
5
6
7
8
9
10
11
12
0 0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
Wind speed (m/s)
Month Fig. 4. Time saved by using GA compared with conventional iteration method.
Fig. 6. Comparison between the theoretical probability density functions and observed wind speed histogram for July.
F.-J. Liu et al. / Energy 36 (2011) 2611e2619 2
0.16 wind speed histogram Weibull pdf logistic pdf lognormal pdf Weibull cumulative probability logistic cumulative probability lognormal cumulative probability observed cumulative probability
mean of wind speed most probable speed speed having max. energy scale parameter shape parameter
1.8
26 1.6
24
1.4
0.1
1.2
Yearly
1
0.08
0.8
0.06
0.6 0.04
Wind speed & Scale parameter (m/s)
Probability density function
0.12
20
28
Cumulative probability
0.14
30
0.4
18 16
22 14 20 18
12
16 10 14 12
8
Shape parameter
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10 6 8 6
4
4
0.02
0.2
2 2
0
0 0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
0
0
30
1
2
3
4
5
6
7
8
9
10
11
12
Wind speed (m/s)
Month of year
Fig. 7. Comparison between the theoretical probability density functions and observed wind speed histogram for the entire year.
Fig. 8. Monthly Weibull scale and shape parameters and two meaningful speeds.
Table 2 Statistical errors for different distribution functions. Error
Distribution
Low wind period (AprileSeptember)
High wind period (OctobereMarch)
Yearly
Max error in cdf
Weibull Logistic Lognormal Weibull Logistic Lognormal Weibull Logistic Lognormal Weibull Logistic Lognormal
0.015 0.068 0.066 0.0312 0.0842 0.0974 0.1568 0.6449 0.5813 1.008 10.235 9.821
0.058 0.064 0.146 0.0645 0.1411 0.1229 0.1246 0.5643 0.6247 1.993 7.894 12.810
0.030 0.086 0.064 0.0428 0.1032 0.0827 0.1080 0.3167 0.4124 1.976 9.348 11.271
RMSE
c2 Potential energy error (%)
test, and so are the RMSE and Chi-square errors. Moreover the relative error of potential energy never exceeds 2%, whereas it is around 10% for other two functions. One can conclude that Weibull function outperforms both other functions. On the basis of the Weibull distribution, several important quantities concerning wind characteristic in Taiwan Strait are calculated using GA for each month. As shown in Fig. 8, shape parameter lies between 2 and 5; scale parameter is greater than the corresponding mean speed by about 10%. The smallest shape parameter appears in August corresponding to more dispersive or gusty winds. The speed carrying maximum energy is stronger than the mean speed and the most probable speed. Basically, different wind parameters reflect different wind regimes or energy potential, determining these parameters accurately for particular time period is necessary for the applications of wind energy.
0.2
0.18
Observed Weibull
0.16
Wind power density distribution
parameter for a given wind data set using conventional iteration method for 30 times in the range of [0.1e10] with the increment interval of 0.001 in accuracy, and count the total computer CPUtime used. Second we implement the same calculation using GA method for 30 times subject to the same degree of solution accuracy as in the iteration method; where the population size used is 20, crossover fraction is 0.5 and mutation rate is 0.2. Fig. 4 shows the time saved by using the GA method compared with the conventional one for different months. There is about 33% of time saving when GA method is employed. That means the GA is powerful in wind energy analysis. Figs. 5e7 compare the theoretical probability density functions with observed wind speed histogram in bin size of 1 m/s for January (representative of high wind month), July (low wind month) and for the entire year, respectively. Therein the corresponding cumulative probability distributions (cdf) are also plotted. The related parameters for logistic and lognormal distributions are obtained directly with the mean and standard deviation of wind data, without GA method, the details are provided in Appendix. Table 2 lists the corresponding statistical errors for the three functions. The Weibull function presents the smallest discrepancy with observation data, independent of time periods. For example, its max error in cdf is the smallest and is close to the critical value of 95% confident level in the KolmogoroveSmirnov
Yearly 0.14
0.12
0.1
0.08
0.06
0.04
0.02
0 0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
Wind speed (m/s) Fig. 9. Yearly wind power density distribution among different wind speeds.
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F.-J. Liu et al. / Energy 36 (2011) 2611e2619
2.8
wind power density availability factor capacity factor
2.6
1 energy in specified blade area energy from an ideal turbine energy actual output efficiency (actual / specified) efficiency (actual / ideal)
1.8
2.4
1.6 1.4
2 1.8
1.2
1.6 1 1.4 1.2
0.8
1 0.6
0.9 0.8 0.7
2E+006 Wind energy (kWh)
2.2
Availability & Capacity factor
Wind power density (kW/m2)
3E+006
2
0.6 0.5
Efficiency
3
2617
0.4 1E+006 0.3
0.8 0.6
0.2
0.4
0.4
0.1
0.2 0.2 0
0E+000
0 1
2
3
4
5
6
7
8
9
10
11
0 1
12
2
3
4
5
6
7
8
9
10
11
12
Month of year
Month of year Fig. 10. Monthly wind power density, availability factor and capacity factor.
Fig. 12. Monthly wind energy and turbine efficiencies obtained from different operations.
Fig. 9 illustrates how the yearly wind power density distributes among different speeds. The observed speed carrying maximum energy, 17 m/s, is slightly greater than the one calculated from the theoretical Weibull function, 15.7 m/s. The highest wind power, occupying near 11% of the annual power generation, comes from the speed of 17 m/s, which contributes only 4% of the total wind speed probabilities (Fig. 7). So choosing a probability function having good agreement with observation data in high wind ranges is more practical than that in low wind ranges, since wind power is proportional to the cube of wind speed. It has a maximum power density in November, 2.4 kW/m2, as shown in Fig. 10. The turbine-dependent availability factors lie above 0.9 implying that a great percentage of operation time would be expected except in August. Capacity factor is an essential indicator in estimating turbine’s profit which is defined as the ratio of the actual energy generated by turbine for a particular time period
to its rated energy for the same time period; it presents greater value for high wind months. The hourly variations for mean wind speed and power density are shown in Fig. 11. Their peaks appear at 4 pm, but there is no significant discrepancy from day to night. This is because the wind farm studied is located in a small island of open ocean area and therefore the wind speed is affected only a little by the changes of temperature within a day. Fig. 12 demonstrates the wind energy obtained from different operating situations and relevant turbine efficiencies. During the low wind season from April to September the wind energy calculated from ideal turbine using Eq. (10) is close to the potential energy within a specified blade area of 1500 m2, suggesting that the operating speeds (cut-in, rated and cut-off speed) of the turbine presently studied are applicable for this season. However, in high wind season, both the energy output from ideal turbine and that
12
1 wind speed at hub height wind power density
11.5
0.9 0.8
Wind speed (m/s)
0.7 10.5
0.6
10
0.5 0.4
9.5
0.3
Wind power density (kW/m2)
11
9 0.2 8.5
0.1
8
0 0
2
4
6
8
10
12
14
16
18
20
22
24
Hour of day Fig. 11. Hourly variations of wind speed and power density.
Fig. 13. Daily wind speed, wind power density and solar irradiation throughout the year.
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F.-J. Liu et al. / Energy 36 (2011) 2611e2619
applied for the first time to calculate the Weibull parameters for wind energy assessment. The applicability of three probability functions in describing wind distribution was compared considering various statistical errors. The conclusions can be summarized as follows. (a) GA could be a useful alternative in the calculation of Weibull parameters, there is about 33% time saving when compared with conventional iteration method. (b) Weibull probability function fits the observed wind speed distribution better than both logistic and lognormal functions, irrespective of time periods. (c) Based on the analyses aforementioned, the designed speeds of the wind turbines studied are very suitable for low wind season. There is much room to enhance in the utilization rate of wind potential energy during high wind season. (d) Wind and solar energy potential in Taiwan present a great complementary relationship and balance near April and September that would be a useful finding in the applications of natural resources. Acknowledgment Fig. 14. Monthly energy ratio for wind and solar data.
from actual turbine are significantly less than the potential energy available in the specified blade area. As compared with the energy output from ideal turbine, the conversion efficiency of the actual turbine studied is between 0.35 and 0.46. While compared with the potential energy available the efficiency becomes lower and lies between 0.13 and 0.42, meanwhile it reveals an opposite trend with that of wind speed, i.e. 0.341, 0.170 and 0.197 for the low wind, high wind and the whole year period, respectively, as listed in Table 1. It implies that there is a lot of room to enhance the utilization rate of wind energy during high wind season or for a place where the appearance probability of wind speed between the rated and the cut-off speed is high; e.g. it might be done through choosing a wind turbine with higher rated power, even this will make its capacity factor down. Similar analyses can be found in Refs. [6,33]. Fig. 13 compares the daily wind speed, wind power density and solar irradiation throughout the year. Herein wind speed has been transferred by using one-seventh power law to the standard height of 10 m above the ground level adopted by the World Meteorological Organization. Solar irradiation is the averaged global radiation from 1990 to 1999 observed at the ground surface of Taichung conducted by the Central Weather Bureau [36]. Since Taichung situates in central Taiwan (with geographic latitude similar to the wind farm); the radiation incident on it could properly represent the general property of Taiwan. The extraterrestrial radiation under air mass zero (AM0, free any shadings) is plotted too [36]. Fig. 14 shows the energy ratio for wind and solar energy. Herein the energy ratio is defined as the ratio of wind or solar energy to the sum of the both energy for a specific month. The data is fitted with a cosine curve; related constants are available in the figure; the R-square coefficient of determination reaches 0.92084 (same for both curves). Wind and solar energy present complementary characteristic and balance near April and September, they could support each other while utilizing the natural resources. 7. Conclusions In this study, wind characteristics in the Taiwan Strait had been investigated systematically for different time periods. The GA was
The authors would deeply thank the Central Weather Bureau and Taipower Company for providing observation data and appreciate Dr. Wu C.F. and Dr. Huang M.W., researchers of the Institute of Earth Sciences, Academia Sinica, Taiwan, and international reviewers for their useful suggestions. This study is supported partly by the National Science Council under the contract of NSC99-2221-E-252-011. Appendix Logistic probability density function [29,37]
gðvÞ ¼
exp½ ðv vÞ=d df1 þ exp½ ðv vÞ=dg2
(A1)
The cumulative logistic function is given by:
GðvÞ ¼
1 1 þ exp½ ðv vÞ=d
(A2)
The curve of the logistic probability density function is symmetric to the mean wind speed, v, where the frequency g(v) is a maximum. d is the scale parameter of logistic distribution. It determines the variation rate of the logistic probability density function and can be calculated by:
d¼
pffiffiffi 3s=p
(A3)
where s is the standard deviation of the observed frequency distribution. Lognormal probability density function [4]
(
1 pffiffiffiffiffiffi exp hðvÞ ¼ vf 2p
½lnðvÞ l 2f2
2
) (A4)
Lognormal distribution is a kind of probability density function for any random variable whose logarithm is normally distributed. Parameters l and f are the mean and the standard deviation of the variable’s natural logarithm, respectively.
F.-J. Liu et al. / Energy 36 (2011) 2611e2619
The cumulative lognormal distribution function is expressed as:
HðvÞ ¼
# " 1 1 lnðvÞ l pffiffiffi þ erf 2 2 f 2
(A5)
where erf () is error function given by:
2 erfðxÞ ¼ pffiffiffi
p
Zx
exp t 2 dt
(A6)
0
References [1] Ackermann T, Soder L. An overview of wind energy-status 2002. Renew Sustain Energy Rev 2002;6:67e128. [2] Raichle BW, Carson WR. Wind resource assessment of the Southern Appalachian Ridges in the Southeastern United States. Renew Sustain Energy Rev 2009;13:1104e10. [3] Carta JA, Ramirez P, Velazquez S. A review of wind speed probability distributions used in wind energy analysis: case studies in the Canary Islands. Renew Sustain Energy Rev 2009;13:933e55. [4] Zhou J, Erdem E, Li G, Shi J. Comprehensive evaluation of wind speed distribution models: a case study for North Dakota sites. Energy Convers Manage 2010;51:1449e58. [5] Morgan EC, Lackner M, Vogel RM, Baise LG. Probability distributions for offshore wind speeds. Energy Convers Manage 2011;52:15e26. [6] Chang TJ, Wu YT, Hsu HY, Chu CR, Liao CM. Assessment of wind characteristics and wind turbine characteristics in Taiwan. Renew Energy 2003;28:851e71. [7] The world of wind atlases, http://www.windatlas.dk. [8] The wind atlas analysis and application program, http://www.wasp.dk/. [9] Jowder FAL. Wind power analysis and site matching of wind turbine generators in Kingdom of Bahrain. Appl Energy 2009;86:538e45. [10] Ucar A, Balo F. Evaluation of wind energy potential and electricity generation at six locations in Turkey. Appl Energy 2009;86:1864e72. [11] Seguro JV, Lambert TW. Modern estimation of the parameters of the Weibull wind speed distribution for wind energy analysis. J Wind Eng Ind Aerodynam 2000;85:75e84. [12] Zhou W, Yang HX, Fang ZH. Wind power potential and characteristic analysis of the Pearl River Delta region, China. Renew Energy 2006;31:739e53. [13] Jamil M, Parsa S, Majidi M. Wind power statistics and an evaluation of wind energy density. Renew Energy 1995;6(5):623e8. [14] Yang HX, Lu L, Burnett J. Weather data and probability analysis of hybrid photovoltaic-wind power generation systems in Hong Kong. Renew Energy 2003;28:1813e24. [15] Dorvlo ASS. Estimating wind speed distribution. Energy Convers Manage 2002;43:2311e8.
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[16] Genc A, Erisoglu M, Pekgor A, Oturanc G, Hepbasli A, Ulgen K. Estimation of wind power potential using Weibull distribution. Energy Sources 2005;27:809e22. [17] Kantar YM, Senoglu B. A comparative study for the location and scale parameters of the Weibull distribution with given shape parameter. Comput Geosci 2008;34:1900e9. [18] Bechrakis DA, Deane JP, McKeogh EJ. Wind resource assessment of an area using short term data correlated to a long term data set. Solar Energy 2004;76:725e32. [19] Lai CM, Lin TH. Technical assessment of the use of a small-scale wind power system to meet the demand for electricity in a land aquafarm in Taiwan. Renew Energy 2006;31:877e92. [20] Akpinar EK, Akpinar S. Determination of the wind energy potential for Maden-Elazig, Turkey. Energy Convers Manage 2004;45:2901e14. [21] Sulaiman MY, Akaak AM, Wahab MA, Zakaria A, Sulaiman ZA, Suradi J. Wind characteristics of Oman. Energy 2002;27:35e46. [22] Chang TP. Performance comparison of six numerical methods in estimating Weibull parameters for wind energy application. Appl Energy 2011;88:272e82. [23] Akpinar S, Akpinar EK. Estimation of wind energy potential using finite mixture distribution models. Energy Convers Manage 2009;50:877e84. [24] Jaramillo OA, Borja MA. Wind speed analysis in La Ventosa, Mexico: a bimodal probability distribution case. Renew Energy 2004;29:1613e30. [25] Li M, Li X. MEP-type distribution function: a better alternative to Weibull function for wind speed distributions. Renew Energy 2005;30:1221e40. [26] Ramirez P, Carta JA. The use of wind probability distributions derived from the maximum entropy principle in the analysis of wind energy. A case study. Energy Convers Manage 2006;47:2564e77. [27] Luna RE, Church HW. Estimation of long term concentrations using a universal wind speed distribution. J Appl Meteorol 1974;13:910e6. [28] Garcia A, Torres JL, Prieto E, De Francisco A. Fitting wind speed distributions: a case study. Solar Energy 1998;62:139e44. [29] Scerri E, Farrugia R. Wind data evaluation in the Maltese Islands. Renew Energy 1996;7:109e14. [30] Holland J. Adaptation in natural and artificial systems. Ann Arbor: The University of Michigan; 1975. [31] Nara K, Shiose A, Kitagawa M, Ishihara T. Implementation of genetic algorithm for distribution systems loss minimum re-configuration. IEEE Trans Power Syst 1992;7:1044e51. [32] Akpinar EK, Akpinar S. An assessment on seasonal analysis of wind energy characteristics and wind turbine characteristics. Energy Convers Manage 2005;46:1848e67. [33] Mathew S, Pandey KP, Kumar A. Analysis of wind regimes for energy estimation. Renew Energy 2002;25:381e99. [34] Tar K. Some statistical characteristics of monthly average wind speed at various heights. Renew Sust Energy Rev 2008;12:1712e24. [35] Ghosh AA. FORTRAN program for fitting Weibull distribution and generating samples. Comput Geosci 1999;25:729e38. [36] Chang TP. Performance evaluation for solar collectors in Taiwan. Energy 2009;34(1):32e40. [37] Assuncao HF, Escobedo JF, Oliveira AP. A new algorithm to estimate sky condition based on 5 minutes-averaged values of clearness index and relative optical air mass. Theor Appl Climatol 2007;90:235e48.
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Energy journal homepage: www.elsevier.com/locate/energy
Wind characterization analysis incorporating genetic algorithm: A case study in Taiwan Strait Feng-Jiao Liu a, Pai-Hsun Chen b, Shyi-Shiun Kuo b, De-Chuan Su b, Tian-Pau Chang b, *, Yu-Hua Yu b, Tsung-Chi Lin b a b
Department of Electrical Engineering, Nankai University of Technology, 568 ChungTseng Road, Caotun, Nantou 542, Taiwan Department of Computer Science and Information Engineering, Nankai University of Technology, 568 ChungTseng Road, Caotun, Nantou 542, Taiwan
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 August 2010 Received in revised form 29 January 2011 Accepted 1 February 2011 Available online 8 March 2011
In this paper, the genetic algorithm (GA) is originally applied to compute the Weibull parameters for wind characterization analysis, in which an objective function required in GA for searching optimization solution has been first defined as well. Wind data analyzed are observed at a wind farm in the Taiwan Strait from 2006 to 2008. To accurately describe wind speed distribution three kinds of probability density functions are compared, i.e. the Weibull, logistic and lognormal functions. Statistical parameters including the max error in the KolmogoroveSmirnov test, root mean square error, Chi-square error and relative error of wind power density are considered as judgment criterions. The results show that GA is a useful method, there is about 33% time saving when compared with conventional iteration method. Weibull function describes best the wind distribution, regardless of time periods. Accordingly, wind power density, availability factor and electrical energy output from an ideal turbine are assessed using the Weibull parameters; utilization rate of wind energy for the currently used turbine is discussed. Further the wind energy compensates very well with solar energy; when solar radiation is down in winter and spring, the wind power becomes greater; energy ratios for each month are calculated lastly. Ó 2011 Elsevier Ltd. All rights reserved.
Keywords: Genetic algorithm Wind speed Wind energy Probability density function Weibull function
1. Introduction Sufficient energy supply is very crucial for a country’s economic development. Due to the shortage of fossil fuels, seeking alternative energy source already becomes an urgent business [1e5]. Wind energy is inexhaustible natural resource that can be converted to electrical power through wind turbine. To effectively evaluate the wind power available for a particular site, studying the wind’s statistical characteristics is needed; meanwhile three indication quantities including the availability factor, capacity factor, and conversion efficiency of a wind turbine must be considered together while assessing wind resource [6]. Many kinds of probability density functions have been proposed in literature for wind distribution analysis such as the gamma function, beta function, logistical function, lognormal function, Rayleigh function and Weibull function. Among these, Weibull and Rayleigh distributions are the most frequently used ones for wind speed modeling due to their two flexible parameters [7e21]; Weibull shape parameter describes the width of data distribution, the larger the shape parameter the
* Corresponding author. Tel.: þ886 49 2563489; fax: þ886 49 2315030. E-mail address: [email protected] (T.-P. Chang). 0360-5442/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.energy.2011.02.001
narrower the distribution and the higher its peak value; scale parameter controls the abscissa scale of a plot of data distribution [22], Rayleigh function is a special case of Weibull function in which shape parameter is 2. However, it should be noted that the Weibull and Rayleigh distributions are not suitable for all the wind regimes encountered in nature. Akpinar and Akpinar [23] as well as Jaramillo and Borja [24] used the mixture distribution of Weibull function, while Li and Li [25] as well as Ramirez and Carta [26] used the probability function derived with maximum entropy principle to model wind characteristics. Luna and Church [27] proposed to use a universal distribution for wind speed, and discovered that the most applicable distribution is the lognormal function. However, they also mentioned that the universal distribution may not work well for every site due to the differences in climate and topography. Similarly, Garcia et al. [28] found that lognormal distribution is better than Weibull distribution for the wind speed data in the area of Navarre, Spain. Scerri and Farrugia [29] compared the ability of logistic function in wind speed modeling with that of Weibull function and found that the results obtained from the two functions follow the same trend. Taiwan locates between the world’s largest continent (Asia) and the largest ocean (Pacific). More than 95% of energy supply comes from imported fossil fuels. In winter and spring seasons the island is
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F.-J. Liu et al. / Energy 36 (2011) 2611e2619
Nomenclature A Af c erf() Eideal f() F() g() G() h() H() k n O() P Pr T T()
sweep area of turbine blade, m2 availability factor, dimensionless scale parameter of Weibull function, m/s error function output energy of ideal turbine, kWh Weibull pdf cumulative Weibull function logistic pdf cumulative logistic function lognormal pdf cumulative lognormal function shape parameter of Weibull function, dimensionless sample size, dimensionless cumulative frequency of observed data wind power density, W/m2 rated power, W time period, hr cumulative frequency of theoretical function
visited by strong northeastern monsoon. While in summer and autumn the southwestern monsoon prevails in this region and occasionally tropical cyclones come with extremely high wind. Overall the average wind speed for the whole year is high enough especially in the region of Taiwan Strait. As a result, among various renewable energy sources wind energy has much potential to compensate the lack of fossil fuels. However, relevant research focused on the Taiwan Strait is still few so far. With the fast growing of economic activities between Taiwan and Mainland China, investigating the wind characterization about this region is needed. In this paper the validity of the logistic, lognormal and Weibull functions in describing wind speed distribution will be compared considering various statistical errors such as the max error in the KolmogoroveSmirnov test, root mean square error, Chi-square error and relative error of wind power density; the solar irradiation observed in central Taiwan will be shown together with the wind power to see what the relationship is between them. Some numerical methods in literature can be employed to calculate the Weibull parameters such as the moment method [16e21], empirical method (a special case of moment method), graphical method (i.e. doing line regression for the cumulative distribution data based on least-squares method), maximum likelihood method, etc. Among these, as concluded by Seguro and Lambert [11], Genc et al. [16] as well as Chang [22], the graphical method performs worse than others when the number of data points used is smaller, although this method is widely applied in wind energy software. According to the tests of Monte Carlo simulations in [22], the maximum percent error in calculating the Weibull shape parameter is only 0.08% when random data number is 10,000, which is a little larger than the number of hours of a year 8760 (¼365 24); however, the same error reaches 1.3% and 5.2% if data number is 1000 and 100, respectively. Besides the relative errors between the wind potential energy calculated from actual time-series data and that from theoretical probability function by the graphical method reach even 23% in [11] and 45% in [22]. As mentioned by Seguro and Lambert [11] the basic reason for the inaccuracy of the graphical method is that the leastsquares regression is performed not on the actual wind speed data, but on its cumulative frequency distribution. Each point is equally weighted even if some data bins may represent more data points than others. Contrarily both the moment and maximum likelihood methods that require iterative procedures always get
v v vi vME vMP vo vr z
wind speed, m/s mean wind speed, m/s cut-in speed of turbine, m/s wind speed carrying maximum energy, m/s most probable wind speed, m/s cut-off speed of turbine, m/s rated speed of turbine, m/s anemometer height, m
Greek letters e objective function in GA s standard deviation of wind speed, m/s G() Gamma function r air density, kg/m3 a coefficient of power law, dimensionless d scale parameter of logistic function, m/s l mean of natural logarithm of wind speed, m/s f standard deviation of natural logarithm of wind speed, m/s
better performance, irrespective of data number. In the present study, the moment method is adopted for its accuracy. Determining the relevant parameters of Weibull probability function through numerical iteration way is a tedious problem. Another aim of this paper is to apply an optimization method, the genetic algorithm (GA), to calculate the Weibull parameters that would be the first presentation in wind energy field; its performance will be compared with conventional iteration method. Wind data analyzed is measured from 2006 to 2008 at a wind farm, Chungtun, located in the middle of Taiwan Strait (with longitude of 119.6 E and latitude of 23.6 N), which is handled by the Ministry of Economic Affairs. There are eight wind turbines installed in this wind farm ranging about 1500 m in length; they are designed with the same specifications as below: the rated power is 600 kW; the cut-in, rated, and cut-off speed is 2.5, 12.5, and 25 m/s, respectively. Rotor’s diameter is 43.7 m, which has blade sweep area of 1500 m2. Anemometer height is 46 m above ground level. Wind data recorded is not always complete for the eight turbines, for the consideration of study quality only three turbines are selected and analyzed in the present work. Although there is no wind farm other than Chungtun built in the Taiwan Strait; wind data from other turbines with different kinds of specifications is not available currently, but it would not influence the uniqueness of the present paper introducing originally the GA into the wind energy field. 2. GA GA is a kind of optimization searching techniques, which was first proposed by Holland [30] simulating the evolutionary idea of the ‘survival of the fittest’. Initially the possible solutions are randomly generated and are coded as strings, named chromosomes, a set of chromosomes is called the population of the generation; the chromosomes in a generation are forced to evolve toward better ones in the next generation using three basic operators: i.e. the reproduction, crossover and mutation, considering simultaneously a problem-dependent objective (fitness) function [31]. The computation flowchart of GA with 8 bits in gene coding is illustrated in Fig. 1. Reproduction is a process in which a number of selected chromosomes are exactly copied and become a part of the offspring according to the level of fitness values. The chromosomes with greater fitness value will have larger opportunity of contributing offspring to their next generation. However, a small part of
F.-J. Liu et al. / Energy 36 (2011) 2611e2619
2613
2 1=k vME ¼ c 1 þ k
(4)
The cumulative Weibull distribution function enables us to calculate the probability of the wind speed exceeding the value u:
k u Fðv uÞ ¼ exp c
(5)
For a wind turbine designed with a given cut-in speed (vi) and cut-off speed (vo), the operating probability of the wind turbine, named here availability factor, is equal to the probability of wind speed between vi and vo, i.e. Fðvi v < vo Þ calculated by:
k k v vo exp AF ¼ Fðvi v < vo Þ ¼ exp i c c
(6)
3.2. Wind power density Wind power density calculated from actual time-series data is expressed by:
P1 ¼
1 3 rv 2
(7)
where, r is the air density; v3 is the mean of wind speed cubes. Wind power density based on a theoretical probability density function can be obtained by the integration:
P2 ¼
Z 1 1 3 r v3 f ðvÞdv ¼ rc3 G 1 þ 2 2 k
(8)
3.3. Electrical energy generated by an ideal wind turbine Fig. 1. Computation flowchart of GA.
less fit chromosomes are still needed and included in optimization searching, this helps keep the diversity of the chromosomes preventing premature convergence on poorly local solutions. In crossover, stochastically selected subsections of two individual chromosomes are swapped each other to produce the offspring. In mutation, randomly selected genes in chromosomes are changed with a small mutation rate to maintain genetic diversity from one generation to the next. In binary coding, it just means that changes digit 1 to digit 0 and vice versa. The algorithm is terminated when either a maximum number of generations have been produced, or a satisfactory fitness level is reached in searching.
For an idealized wind turbine, it starts to run at cut-in speed then its output power (P(v)) increases with wind speed until the rated speed (vr). For wind speed between the rated speed and cutoff speed the output power remains a constant amount, i.e. the rated power (Pr). The turbine will be shut down while wind speed exceeds cut-off speed. The rated power is given by:
Pr ¼
where, A is the sweep area of the blade of wind turbine. Consequently, as illustrated in Chang et al. [6] and Mathew et al. [33], the output energy (Eideal) of an ideal turbine for a given time period T can be calculated by the following expression through numerical integration:
Eideal
3.1. Weibull function
¼ T
vMP
The wind speed that carries maximum energy is obtained by:
PðvÞf ðvÞdv þ
Pr f ðvÞdv vr
Zvr k kvk1 v dv v3 exp ¼ T 2rA c c c
ð10Þ
vi
k Zvo k1 k v v þv3r dv exp c c c vr
(2)
(3)
Zvo
vi
(1)
The most probable wind speed for a particular location with the given probability function can be calculated by [13,32]:
1 1=k ¼ c 1 k
PðvÞf ðvÞdv Zvr
where v is the wind speed, k is the shape parameter, and c is the scale parameter with the same unit as wind speed. Weibull cumulative distribution function (cdf) is expressed by:
k v FðvÞ ¼ 1 exp c
¼ T 0
Weibull probability density function (pdf) is given as:
k kvk1 v exp c c c
(9)
ZN
3. Wind and wind turbine characteristics
f ðvÞ ¼
1 rAv3r 2
3.4. Wind speed extrapolation To estimate wind speeds at different heights, empirical power law is considered:
v1 ¼ v2
a z1 z2
(11)
2614
F.-J. Liu et al. / Energy 36 (2011) 2611e2619
where v1 and v2 are the wind speeds at heights z1 and z2, respectively. Since the effect of height on air density for the elevations considered by the present work can be neglected, subsequently the power density of the wind at different height is mainly affected by the wind speed and can be estimated by:
3a z1 z2
wind speed observed at hub height of 46m
25
(12)
The parameter a is the power law coefficient representing the degree of roughness of ground surface. The typical value of 0.14 (one-seventh) for a wide-plain area is adopted in this study [34]. 4. Objective function
20
Wind speed (m/s)
P1 ¼ P2
30
Low wind period 15
10
Weibull shape (k) and scale (c) parameters can be estimated using iterative way based on the moments of wind distribution, i.e. the mean ðvÞ and standard deviation (s) of observed wind speeds, which is commonly adopted by other researches for its accuracy [16e22] and named conventional iteration method here.
1 v ¼ cG 1 þ k
s ¼ c G 1þ
2 1 1=2 G2 1 þ k k
0
(13)
2
v
Gð1 þ 2=kÞ G2 ð1 þ 1=kÞ G2 ð1 þ 1=kÞ
¼
(14)
s2 2
v
Gð1 þ 2=kÞ G2 ð1 þ 1=kÞ G2 ð1 þ 1=kÞ
(16)
5. Goodness of fit To test the goodness of fit between a theoretical probability function and observation data, four kinds of statistical errors are considered as the judgment criterions in this paper. The smaller the errors the better the fit is. The first one is the KolmogoroveSmirnov test (KS), which is defined as the max error in cumulative distribution functions [21].
(17)
where T(v) and O(v) are the cumulative distribution functions for wind speed not exceeding v in the theoretical and observed dataset, respectively. The second one is the root mean square error (RMSE) defined as:
1=2 X 1 n 2 RMSE ¼ ðy y Þ i ic n i¼1
2000
3000
4000
5000
6000
7000
8000
9000
where, yi is the actual value at time stage i, yic is the value computed from correlation expression for the same stage, n is the number of data. The third judgment criterion is the Chi-square error given as:
c2 ¼
In GA optimization searching, the residual value e is set to be the objective function. If the objective function converges to a predefined tolerance value then an acceptable shape parameter k is obtained; the scale parameter c can be thus determined using Eq. (13). To practically evaluate the performance of GA, the tolerance value of the objective function is defined to be 1 103 in the present study, similar research can be found in [35].
KS ¼ maxjTðvÞ OðvÞj
1000
Fig. 2. Hourly mean wind speed measured at a selected wind turbine throughout the year.
(15)
As known it usually exists a residual error e between the left and right side of Eq. (15) during iteration expressed as:
e¼
0
Hour of year
where G() is the Gamma function. To apply the GA to search the Weibull parameters, the moment equations above can be rewritten as:
s2
5
(18)
Xn 1 2 ðy y Þ i ic i¼1 y ic
(19)
Table 1 Wind and wind turbine characteristics for different time periods. Low wind period (Aprile September)
High wind period (Octobere March)
Yearly
Mean speed (m/s) Standard deviation (m/s) Skewness coefficient Kurtosis coefficient
6.75 3.49 0.76 3.77
12.61 4.96 0.22 2.41
9.67 5.19 0.41 2.32
Weibull shape parameter Weibull scale parameter (m/s) Most probable speed (m/s) Speed having maximum energy (m/s)
2.02
2.75
1.94
7.61
14.17
10.91
5.44
12.03
7.51
10.69
17.29
15.70
Capacity factor Availability factor Power density (kW/m2) Potential energy in specified area (kWh)
0.293 0.900 0.35 2338429.1
0.745 0.983 1.82 11903902.8
0.518 0.938 1.09 14326901.0
Output energy of ideal turbine (kWh) Output energy of actual turbine (kWh) Conversion efficiency (actual/specified) Conversion efficiency (actual/ideal)
2012999.4
5461497.7
7450772.5
797800.3
2018006.5
2815806.8
0.341
0.170
0.197
0.396
0.369
0.378
F.-J. Liu et al. / Energy 36 (2011) 2611e2619 2
0.26 wind speed histogram Weibull pdf logistic pdf lognormal pdf Weibull cumulative probability logistic cumulative probability lognormal cumulative probability observed cumulative probability
0.24
0.25
0.22 0.2
Probability density function
Objective function
0.2
0.15
0.1
0.05
0.18 0.16
1.8
1.6
1.4
January
1.2
0.14 1 0.12 0.8
0.1 0.08
Cumulative probability
0.3
2615
0.6
0.06 0.4 0.04
0
0.2
0.02 0
0 0
-0.05 0
20
40
60
80
100
120
140
160
180
2
4
6
8
10
14
16
18
20
22
24
26
28
30
Wind speed (m/s)
200
Generations in GA
12
Fig. 5. Comparison between the theoretical probability density functions and observed wind speed histogram for January.
Fig. 3. Variation of objective function with generation number in GA searching.
The fourth judgment is the relative error between the wind power density calculated from actual time-series data and that from theoretical probability function, which is most meaningful in wind energy assessment. 6. Results and discussion Fig. 2 shows the hourly mean wind speed measured at one of the selected wind turbines throughout the year in the Taiwan Strait. As shown, the data could be roughly classified into high wind period from October to March and low wind period from April to September, as marked by the vertical broken lines reflecting
different monsoons. Since the variations of wind speeds for other two turbines selected (not shown here) reveal similar trends with the Fig. 2; hereafter in the present study the subsequent analyses are based on the averaged wind data of the three turbines. Corresponding wind and turbine characteristics for different time periods are summarized in Table 1. Although the power density during high wind period is much greater than that during low wind period, the energy conversion efficiency for wind turbine currently used is lower, that will be analyzed later. Fig. 3 shows the variation of objective function with the generation number of GA in searching a Weibull shape parameter; the objective function converges to an acceptable level after only 43 generations. The convergent rate is dependent on GA operators and computer specifications. To objectively evaluate the superiority of GA in determining Weibull parameter, first we calculate the shape
100 2
0.4 0.38
wind speed histogram Weibull pdf logistic pdf lognormal pdf Weibull cumulative probability logistic cumulative probability lognormal cumulative probability observed cumulative probability
0.36 0.34 0.32
70
0.3
Probability density function
Time saved by using GA (%)
80
60 50 40 30
0.28 0.26
1.8
1.6
1.4
July
0.24
1.2
0.22 1
0.2 0.18
0.8
0.16 0.14
Cumulative probability
90
0.6
0.12 0.1
20
0.4
0.08 0.06 0.04
10
0.2
0.02 0
0
1
2
3
4
5
6
7
8
9
10
11
12
0 0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
Wind speed (m/s)
Month Fig. 4. Time saved by using GA compared with conventional iteration method.
Fig. 6. Comparison between the theoretical probability density functions and observed wind speed histogram for July.
F.-J. Liu et al. / Energy 36 (2011) 2611e2619 2
0.16 wind speed histogram Weibull pdf logistic pdf lognormal pdf Weibull cumulative probability logistic cumulative probability lognormal cumulative probability observed cumulative probability
mean of wind speed most probable speed speed having max. energy scale parameter shape parameter
1.8
26 1.6
24
1.4
0.1
1.2
Yearly
1
0.08
0.8
0.06
0.6 0.04
Wind speed & Scale parameter (m/s)
Probability density function
0.12
20
28
Cumulative probability
0.14
30
0.4
18 16
22 14 20 18
12
16 10 14 12
8
Shape parameter
2616
10 6 8 6
4
4
0.02
0.2
2 2
0
0 0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
0
0
30
1
2
3
4
5
6
7
8
9
10
11
12
Wind speed (m/s)
Month of year
Fig. 7. Comparison between the theoretical probability density functions and observed wind speed histogram for the entire year.
Fig. 8. Monthly Weibull scale and shape parameters and two meaningful speeds.
Table 2 Statistical errors for different distribution functions. Error
Distribution
Low wind period (AprileSeptember)
High wind period (OctobereMarch)
Yearly
Max error in cdf
Weibull Logistic Lognormal Weibull Logistic Lognormal Weibull Logistic Lognormal Weibull Logistic Lognormal
0.015 0.068 0.066 0.0312 0.0842 0.0974 0.1568 0.6449 0.5813 1.008 10.235 9.821
0.058 0.064 0.146 0.0645 0.1411 0.1229 0.1246 0.5643 0.6247 1.993 7.894 12.810
0.030 0.086 0.064 0.0428 0.1032 0.0827 0.1080 0.3167 0.4124 1.976 9.348 11.271
RMSE
c2 Potential energy error (%)
test, and so are the RMSE and Chi-square errors. Moreover the relative error of potential energy never exceeds 2%, whereas it is around 10% for other two functions. One can conclude that Weibull function outperforms both other functions. On the basis of the Weibull distribution, several important quantities concerning wind characteristic in Taiwan Strait are calculated using GA for each month. As shown in Fig. 8, shape parameter lies between 2 and 5; scale parameter is greater than the corresponding mean speed by about 10%. The smallest shape parameter appears in August corresponding to more dispersive or gusty winds. The speed carrying maximum energy is stronger than the mean speed and the most probable speed. Basically, different wind parameters reflect different wind regimes or energy potential, determining these parameters accurately for particular time period is necessary for the applications of wind energy.
0.2
0.18
Observed Weibull
0.16
Wind power density distribution
parameter for a given wind data set using conventional iteration method for 30 times in the range of [0.1e10] with the increment interval of 0.001 in accuracy, and count the total computer CPUtime used. Second we implement the same calculation using GA method for 30 times subject to the same degree of solution accuracy as in the iteration method; where the population size used is 20, crossover fraction is 0.5 and mutation rate is 0.2. Fig. 4 shows the time saved by using the GA method compared with the conventional one for different months. There is about 33% of time saving when GA method is employed. That means the GA is powerful in wind energy analysis. Figs. 5e7 compare the theoretical probability density functions with observed wind speed histogram in bin size of 1 m/s for January (representative of high wind month), July (low wind month) and for the entire year, respectively. Therein the corresponding cumulative probability distributions (cdf) are also plotted. The related parameters for logistic and lognormal distributions are obtained directly with the mean and standard deviation of wind data, without GA method, the details are provided in Appendix. Table 2 lists the corresponding statistical errors for the three functions. The Weibull function presents the smallest discrepancy with observation data, independent of time periods. For example, its max error in cdf is the smallest and is close to the critical value of 95% confident level in the KolmogoroveSmirnov
Yearly 0.14
0.12
0.1
0.08
0.06
0.04
0.02
0 0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
Wind speed (m/s) Fig. 9. Yearly wind power density distribution among different wind speeds.
30
F.-J. Liu et al. / Energy 36 (2011) 2611e2619
2.8
wind power density availability factor capacity factor
2.6
1 energy in specified blade area energy from an ideal turbine energy actual output efficiency (actual / specified) efficiency (actual / ideal)
1.8
2.4
1.6 1.4
2 1.8
1.2
1.6 1 1.4 1.2
0.8
1 0.6
0.9 0.8 0.7
2E+006 Wind energy (kWh)
2.2
Availability & Capacity factor
Wind power density (kW/m2)
3E+006
2
0.6 0.5
Efficiency
3
2617
0.4 1E+006 0.3
0.8 0.6
0.2
0.4
0.4
0.1
0.2 0.2 0
0E+000
0 1
2
3
4
5
6
7
8
9
10
11
0 1
12
2
3
4
5
6
7
8
9
10
11
12
Month of year
Month of year Fig. 10. Monthly wind power density, availability factor and capacity factor.
Fig. 12. Monthly wind energy and turbine efficiencies obtained from different operations.
Fig. 9 illustrates how the yearly wind power density distributes among different speeds. The observed speed carrying maximum energy, 17 m/s, is slightly greater than the one calculated from the theoretical Weibull function, 15.7 m/s. The highest wind power, occupying near 11% of the annual power generation, comes from the speed of 17 m/s, which contributes only 4% of the total wind speed probabilities (Fig. 7). So choosing a probability function having good agreement with observation data in high wind ranges is more practical than that in low wind ranges, since wind power is proportional to the cube of wind speed. It has a maximum power density in November, 2.4 kW/m2, as shown in Fig. 10. The turbine-dependent availability factors lie above 0.9 implying that a great percentage of operation time would be expected except in August. Capacity factor is an essential indicator in estimating turbine’s profit which is defined as the ratio of the actual energy generated by turbine for a particular time period
to its rated energy for the same time period; it presents greater value for high wind months. The hourly variations for mean wind speed and power density are shown in Fig. 11. Their peaks appear at 4 pm, but there is no significant discrepancy from day to night. This is because the wind farm studied is located in a small island of open ocean area and therefore the wind speed is affected only a little by the changes of temperature within a day. Fig. 12 demonstrates the wind energy obtained from different operating situations and relevant turbine efficiencies. During the low wind season from April to September the wind energy calculated from ideal turbine using Eq. (10) is close to the potential energy within a specified blade area of 1500 m2, suggesting that the operating speeds (cut-in, rated and cut-off speed) of the turbine presently studied are applicable for this season. However, in high wind season, both the energy output from ideal turbine and that
12
1 wind speed at hub height wind power density
11.5
0.9 0.8
Wind speed (m/s)
0.7 10.5
0.6
10
0.5 0.4
9.5
0.3
Wind power density (kW/m2)
11
9 0.2 8.5
0.1
8
0 0
2
4
6
8
10
12
14
16
18
20
22
24
Hour of day Fig. 11. Hourly variations of wind speed and power density.
Fig. 13. Daily wind speed, wind power density and solar irradiation throughout the year.
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F.-J. Liu et al. / Energy 36 (2011) 2611e2619
applied for the first time to calculate the Weibull parameters for wind energy assessment. The applicability of three probability functions in describing wind distribution was compared considering various statistical errors. The conclusions can be summarized as follows. (a) GA could be a useful alternative in the calculation of Weibull parameters, there is about 33% time saving when compared with conventional iteration method. (b) Weibull probability function fits the observed wind speed distribution better than both logistic and lognormal functions, irrespective of time periods. (c) Based on the analyses aforementioned, the designed speeds of the wind turbines studied are very suitable for low wind season. There is much room to enhance in the utilization rate of wind potential energy during high wind season. (d) Wind and solar energy potential in Taiwan present a great complementary relationship and balance near April and September that would be a useful finding in the applications of natural resources. Acknowledgment Fig. 14. Monthly energy ratio for wind and solar data.
from actual turbine are significantly less than the potential energy available in the specified blade area. As compared with the energy output from ideal turbine, the conversion efficiency of the actual turbine studied is between 0.35 and 0.46. While compared with the potential energy available the efficiency becomes lower and lies between 0.13 and 0.42, meanwhile it reveals an opposite trend with that of wind speed, i.e. 0.341, 0.170 and 0.197 for the low wind, high wind and the whole year period, respectively, as listed in Table 1. It implies that there is a lot of room to enhance the utilization rate of wind energy during high wind season or for a place where the appearance probability of wind speed between the rated and the cut-off speed is high; e.g. it might be done through choosing a wind turbine with higher rated power, even this will make its capacity factor down. Similar analyses can be found in Refs. [6,33]. Fig. 13 compares the daily wind speed, wind power density and solar irradiation throughout the year. Herein wind speed has been transferred by using one-seventh power law to the standard height of 10 m above the ground level adopted by the World Meteorological Organization. Solar irradiation is the averaged global radiation from 1990 to 1999 observed at the ground surface of Taichung conducted by the Central Weather Bureau [36]. Since Taichung situates in central Taiwan (with geographic latitude similar to the wind farm); the radiation incident on it could properly represent the general property of Taiwan. The extraterrestrial radiation under air mass zero (AM0, free any shadings) is plotted too [36]. Fig. 14 shows the energy ratio for wind and solar energy. Herein the energy ratio is defined as the ratio of wind or solar energy to the sum of the both energy for a specific month. The data is fitted with a cosine curve; related constants are available in the figure; the R-square coefficient of determination reaches 0.92084 (same for both curves). Wind and solar energy present complementary characteristic and balance near April and September, they could support each other while utilizing the natural resources. 7. Conclusions In this study, wind characteristics in the Taiwan Strait had been investigated systematically for different time periods. The GA was
The authors would deeply thank the Central Weather Bureau and Taipower Company for providing observation data and appreciate Dr. Wu C.F. and Dr. Huang M.W., researchers of the Institute of Earth Sciences, Academia Sinica, Taiwan, and international reviewers for their useful suggestions. This study is supported partly by the National Science Council under the contract of NSC99-2221-E-252-011. Appendix Logistic probability density function [29,37]
gðvÞ ¼
exp½ ðv vÞ=d df1 þ exp½ ðv vÞ=dg2
(A1)
The cumulative logistic function is given by:
GðvÞ ¼
1 1 þ exp½ ðv vÞ=d
(A2)
The curve of the logistic probability density function is symmetric to the mean wind speed, v, where the frequency g(v) is a maximum. d is the scale parameter of logistic distribution. It determines the variation rate of the logistic probability density function and can be calculated by:
d¼
pffiffiffi 3s=p
(A3)
where s is the standard deviation of the observed frequency distribution. Lognormal probability density function [4]
(
1 pffiffiffiffiffiffi exp hðvÞ ¼ vf 2p
½lnðvÞ l 2f2
2
) (A4)
Lognormal distribution is a kind of probability density function for any random variable whose logarithm is normally distributed. Parameters l and f are the mean and the standard deviation of the variable’s natural logarithm, respectively.
F.-J. Liu et al. / Energy 36 (2011) 2611e2619
The cumulative lognormal distribution function is expressed as:
HðvÞ ¼
# " 1 1 lnðvÞ l pffiffiffi þ erf 2 2 f 2
(A5)
where erf () is error function given by:
2 erfðxÞ ¼ pffiffiffi
p
Zx
exp t 2 dt
(A6)
0
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