Investigation of wind characteristics and assessment of wind energy potential for Waterloo region, Canada

Energy Conversion and Management 46 (2005) 3014–3033 www.elsevier.com/locate/enconman

Investigation of wind characteristics and assessment of wind energy potential for Waterloo region, Canada Meishen Li, Xianguo Li

*

Department of Mechanical Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ont., Canada N2L 3G1 Received 18 August 2004; received in revised form 18 August 2004; accepted 25 February 2005 Available online 20 April 2005

Abstract Wind energy becomes more and more attractive as one of the clean renewable energy resources. Knowledge of the wind characteristics is of great importance in the exploitation of wind energy resources for a site. It is essential in designing or selecting a wind energy conversion system for any application. This study examines the wind characteristics for the Waterloo region in Canada based on a data source measured at an elevation 10 m above the ground level over a 5-year period (1999–2003) with the emphasis on the suitability for wind energy technology applications. Characteristics such as annual, seasonal, monthly and diurnal wind speed variations and wind direction variations are examined. Wind speed data reveal that the windy months in Waterloo are from November to April, defined as the Cold Season in this study, with February being the windiest month. It is helpful that the high heating demand in the Cold Season coincides with the windy season. Analysis shows that the day time is the windy time, with 2 p.m. in the afternoon being the windiest moment. Moreover, a model derived from the maximum entropy principle (MEP) is applied to determine the diurnal, monthly, seasonal and yearly wind speed frequency distributions, and the corresponding Lagrangian parameters are determined. Based on these wind speed distributions, this study quantifies the available wind energy potential to provide practical information for the application of wind energy in this area. The yearly average wind power density is 105 W/m2. The day and night time wind power density in the Cold Season is 180 and 111 W/m2, respectively. Ó 2005 Elsevier Ltd. All rights reserved.

*

Corresponding author. Tel.: +1 519 888 4567x6843; fax: +1 519 888 6197. E-mail address: [email protected] (X. Li).

0196-8904/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2005.02.011

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Keywords: Maximum entropy principle (MEP); Wind characteristics; Wind energy potential; Wind speed distribution

1. Introduction Alternatives to conventional energy sources, and especially renewable energy, are becoming increasingly attractive because of the limited fossil fuel reserves and the adverse effects associated with the use of fossil fuels. The renewable energy resources include wind, wave, hydro, tidal, solar, geothermal and bio-energy. All these renewable energy resources are abundant, and the technologies, if well established, can provide complete security of energy supply. It is now evident that renewable energy technologies play a strategic role in achieving the goals of sustainable economical development and environmental protection. Wind energy, first used more than 3500 years ago in Egypt, has some key advantages such as cleanliness, abundance in most parts of the world, low cost, sustainability, safety, popularity etc. Ample attention has been directed toward the use of renewable wind energy, especially after the major energy crisis in the 1970s, and wind energy continues to be the fastest growing power generating technology in the world. The wind energy industry celebrated a near record breaking year in 2003, adding more than 8000 MW of wind energy capacity in more than two dozen nations. The total world capacity of 39,294 MW provides enough power to supply the equivalent of 9 million average American homes [1]. Canada lags behind other industrialized countries in the exploitation and implementation of wind energy. So far, the total installed wind power capacity is only 341 MW across Canada [2]. The goal of the Canadian Wind Energy Association (CWEA) is to encourage investment in wind energy for 10,000 MW by 2010, providing 5% of CanadaÕs electricity. The Canadian Atlas Level 0 shown in Fig. 1 is recently available from Environment Canada [3]. The wind energy potential

Fig. 1. Canadian Atlas Level 0.

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Table 1 Commercially international system of classification for wind by Elliott and Schwartz [4] from the Pacific Northwest Laboratory (PNL) Wind power class

10 m Wind power density (W/m2)

10 m Speed (m/s)

30 m Wind power density (W/m2)

30 m Speed (m/s)

50 m Wind power density (W/m2)

50 m Speed (m/s)

1 2 3 4 5 6 7

6100 6150 6200 6250 6300 6400 61000

64.4 65.1 65.6 66.0 66.4 67.0 69.4

6160 6240 6320 6400 6480 6640 61600

65.1 65.9 66.5 67.0 67.4 68.2 611.0

6200 6300 6400 6500 6600 6800 62000

65.6 66.4 67.0 67.5 68.0 68.8 611.9

Classes of mean wind speed and wind power density at 10, 30 and 50 m.

shown is the quantity at 50 m above ground and is based on five years (1996–2000) meteorological data. According to the commercially international system of classification by Elliott and Schwartz [4] from the Pacific Northwest Laboratory (PNL) shown in Table 1, southwestern Ontario, where the Waterloo region is located, falls into Class 3 and under. It is popularly accepted that Class 4 and above are suitable for large-scale electricity generation with modern wind turbine technology. It has already been proven that large-scale wind turbines with good enough wind sources are cost competitive and can become one of the least cost power sources. With the development of future generation wind turbine technology, Class 2 areas and under may be viable for large-scale applications in the near future. In his study, Celik [5] states that wind electricity by medium scale wind turbines is preferable in remote locations for being socially valuable and economically competitive. As for small-scale turbines, when sized properly and used at optimal working conditions, it also could be a reliable energy source and produce socioeconomically valuable energy. Other than electricity generation, wind with speeds ranging from 2.6 m/s to about 4 m/s at 10 m above ground level is utilizable for wind pumping and other mechanical conversion systems. In all cases, the wind resource has specific uses; lower to moderate wind speed for small-scale utilization and high wind speeds for community electricity supplies. To supply reliable electricity at a reasonable cost in a given location, detailed examination of the wind characteristics and an accurate assessment of the wind energy have to be conducted. The Waterloo region is also often referred to as the Canadian High-Tech Triangle, and the energy demands to power commercial and residential usages grow rapidly. Knowledge of the availability of wind energy for this area is helpful for development of the regional economy. However, the Canada Atlas is not fine enough to identify whether wind energy may be viable for this area. A few small-scale wind energy conversion systems do exist in the Waterloo region. However, there does not exist a detailed analysis of the wind characteristics and the wind energy potential. This study has been performed to investigate the wind characteristics and assess the wind energy resource for this area. A practical method of quantitative analysis of wind data can be performed by establishing the wind speed pattern, level and prevailing direction. To avoid the time and expense associated with processing multiple year data records of hourly wind speed data, it is very important to describe the variation of wind speeds with statistical functions for optimizing the design of the systems. Empirical distributions, such as the Weibull, the Rayleigh and the Lognormal

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have long been applied to fit the wind speed variations to create wind speed frequency distributions over a period of time. The two parameter Weibull function is accepted as the best one [6], and the application of this function for different sites can be found in many literatures [7–10]. However, one main limitation of the Weibull distribution is that it does not accurately represent the probabilities of observing zero and very low wind speeds. Recently, Li and Li [11] developed a theoretical approach to analytical determination of wind speed distributions based on the maximum entropy principle (MEP), and it was shown that this model can describe not only the actual data more accurately than the Weibull distribution but also a much wider range of data types. The objective of this study is to investigate the wind characteristics such as the existence of diurnal, monthly and annual trends and assess the wind energy potential for the Waterloo region, Canada, based on the newly developed wind speed frequency distribution model, the MEP based distribution. This can provide information for further developing wind energy conversion technologies (WECTs) and optimizing the design of WECTs, resulting in less energy generating costs.

2. Wind energy assessment 2.1. MEP based wind speed distribution The maximum entropy principle (MEP) has been successfully applied to many problems arising in a wide variety of fields. Shannon [12] discovered a measure of uncertainty for any probability distribution and proposed the concept of information entropy, as defined by the following expression: X ð1Þ S ¼ k P i ln P i where Pi is the probability of occurrence of state i and k is the Boltzmann constant. Jaynes [13] extended this concept into a now well known method of maximum entropy formalism in the direction of statistical mechanics, which can be applied to problems that involve probability. As a statistical tool, the MEP allows one to determine the least biased probability distribution function when the information available is limited by some macroscopic constraints. It is recognized that many physical systems can be described by averages that may be known for the particular system. These observations can be expressed mathematically by the following constraints: n X

P i gr;i ¼ hgr i;

r ¼ 1; 2; 3; . . . ; m

ð2Þ

i¼1

where m is the number of physical constraints for the particular system, gr,i is some function evaluated at state i and hgri is the expectation of the average value of the function g over the entire system. The additional constraint is the definition of probability. n X Pi ¼ 1 ð3Þ i¼1

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Jaynes suggested that the most likely probability distribution should be the one that maximizes ShannonÕs entropy subject to the given information. Based on the LagrangeÕs method, the most likely distribution maximizing the entropy function under the constraints Eqs. (2) and (3) can be derived as [13] ! m X ð4Þ ar gr;i P i ¼ exp a0  r¼1

where the aiÕs are the Lagrangian multipliers. When the MEP is applied to the wind energy field to determine the wind speed distribution, the constraint equations imposed must be based on physical principles, i.e. the conservation of mass, momentum and energy for the air stream flowing with the wind, that can be expressed as follows: X qP i V i A ¼ qAV 10 ð5Þ Mass conservation: m_ ¼ i

Momentum conservation:

_ ¼ M

X

ðqP i V i AÞ  V i ¼ qAV 220

ð6Þ

i

Energy conservation: E_ ¼

X i

  1 2 1 ðqP i V i AÞ  V i ¼ qAV 330 2 2

ð7Þ

where q is the air density, V represents the speed at which the air stream flows and A means the sweep area of a turbine blade. V10, V20 and V30 are the mean velocities of the wind, which are defined mathematically through P P i V ri rq ð8Þ V rq ¼ Pi q iP iV i where r equals 1, 2 and 3, respectively, corresponding to V10, V20 and V30 when q is set to 0. Here, V10 is the same as the average velocity commonly used; V20 and V30 represent the wind speed at which the wind flowing through the rotor produces the same force or energy, respectively, as the wind flowing at variable speeds. These mean velocities can be obtained from the measured data. They also may be determined if the measurement of the wind mass flow rate, momentum flow rate (force) and the energy flow rate (or power) is available. It is evident that it is much easier to measure these three mean wind velocities than to measure the instantaneous wind speed over a long period of time followed by a tedious data analysis. Under the above constraints, Eqs. (5)–(7), and the definition of probability, maximizing ShannonÕs entropy gives the discrete wind speed probability, similar to Eq. (4), as   ð9Þ P i ¼ exp a0  a1 V i  a2 V 2i  a3 V 3i For continuous variables, such as the wind speed, the subscripts can be dropped and the summation can be replaced by integrals with the corresponding limits from minimum to maximum. The continuous probability density function can be obtained as [11]   ð10Þ f ðV Þ ¼ exp a0  a1 V  a2 V 2  a3 V 3

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The Lagrangian multipliers, aiÕs, can be determined with the following set of equations, which are the continuous form of Eqs. (3) and (5)–(7): Z V max

 exp a0  a1 V  a2 V 2  a3 V 3 dV ¼ 1 ð11Þ V min

Z

V max

 V exp a0  a1 V  a2 V 2  a3 V 3 dV ¼ V 10

ð12Þ

 V 2 exp a0  a1 V  a2 V 2  a3 V 3 dV ¼ V 220

ð13Þ

 V 3 exp a0  a1 V  a2 V 2  a3 V 3 dV ¼ V 330

ð14Þ

V min

Z

V max

V min

Z

V max

V min

This model can fit a wider collection of wind speed data accurately and much better than the empirical Weibull distribution. 2.2. Average power density in wind The power of the wind that flows at speed V through a blade sweep area A increases with the cube of the wind speed and the area, that is, 1 P ðV Þ ¼ qAV 3 2

ð15Þ

The wind power density of a site based on a probability density function can be expressed as follows: Z 1 P 1 3 E¼ ¼ ð16Þ qV f ðV Þ dV A 2 0 This equation can be used to calculate the available wind energy for any defined period of time when the wind speed frequency distributions are for a different period of time. The Betz limit, which has been commonly used now for decades, gives that a wind turbine would not extract more than 59% of the available wind power. Therefore, the maximum extractable power from the wind will be the product of the factor 0.59 and the calculated result from Eq. (15).

3. Wind characteristics 3.1. Measurement site The Waterloo region extends from latitude 43°16 0 N to 43°41 0 N and from longitude 80°11 0 W to 80°52 0 W. Wind data have been measured at the Waterloo Wellington meteorological station from 1974 to the present. The data can be found in the National Climate Data and Information

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Table 2 Yearly cumulative times the wind blows according to wind speeds in an hour Wind speed (m/s)

1999

2000

2001

2002

2003

5-Year

0–1 1–2 2–3 3–4 4–5 5–6 6–7 7–8 8–9 9–10 10–11 11–12 12–13 13–14 14–15 15–16 16–17 17–18 18–19 19–20 20–21

1299 2091 129 944 902 1104 580 453 206 99 120 35 20 12 3 2 1 0 0 0 0

1239 1748 502 1025 888 857 574 441 182 122 134 29 36 15 7 5 0 0 0 0 0

1315 1441 573 1253 1001 1085 718 571 288 125 133 50 62 19 11 16 4 3 1 0 0

954 1500 624 1185 1038 992 730 512 289 167 135 46 48 13 7 17 1 3 1 0 2

782 2053 735 1366 1043 857 644 450 323 180 113 49 33 19 14 14 2 0 0 0 0

5589 8833 2563 5773 4872 4895 3246 2427 1288 693 635 209 199 78 42 54 8 6 2 0 2

Total

6701

7804

8669

8264

8677

41,414

Archive on line on the Environment Canada web site [14]. The coordinates of this site are latitude 43°27 0 N and longitude 80°22 0 W, and the elevation is 317.00 m. The wind speed is observed at 10 m above the ground in km/h. The wind direction is expressed in tens of degrees. In this study, the recent data from 1999 to 2003 (5-year period) has been processed and analyzed. The original wind speed data are in time series format. Since it is accepted that hourly time resolution provides satisfactory accuracy in wind resource estimation [5], the diurnal, monthly, seasonal and yearly cumulative times the wind blows are arranged in hours according to the wind speeds. An example in Table 2 shows the yearly counts based on the original data, and then the frequency distribution can be obtained based on this format of data. 3.2. Wind pattern 3.2.1. Annual and overall mean wind speeds The yearly mean wind speeds can be obtained by averaging all the available wind speed in the year. For the Waterloo region, Canada, the average values for each year from 1999 to 2003 and the overall five year average are listed in Table 3. It can be found that all the mean wind speeds are lower than 4.4 m/s. The highest mean wind speed appears in 2002 and is only 4 m/s. According to the PNL classification system, the Waterloo region almost falls out of the category if based on the yearly mean wind speed. This means that, under the current wind turbine technique, this area may not be suitable for year round large-scale electricity generation due to the cost factor. However,

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Table 3 Yearly mean wind speed variations for overall and individual five years (1999–2003) in Waterloo region, Canada Year

Mean wind speed (m/s)

1999 2000 2001 2002 2003 5-Year

3.4 3.7 3.9 4.0 3.9 3.8

for small-scale applications, and in the long run with the development of wind turbine technology, the utilization of wind energy is still promising. 3.2.2. Monthly and seasonal wind speed variations Monthly mean wind speed variations for the overall and individual five years are presented in Fig. 2. The trends of the monthly means for the different years are similar. For the overall five years, the mean wind speeds are above 4 m/s from November to April. The maximum monthly mean wind speed of 4.8 m/s arises in February, while the minimum of 2.5 m/s occurs in August. In the Waterloo region, the higher heating demand also occurs from November to April, which can be grouped as the Cold Season. The wind energy may be applied as a supplement to the current gas or electricity heating. In this study, May to October is defined as the Warm Season. For the Cold Season, the 5-year overall mean wind speed is 4.5 m/s. Table 4 shows that the mean wind speeds in the Cold Season are much higher than those in the Warm Season, and the higher wind speeds coupled with the colder dense air combine to deliver a big energy yield when the power demand is higher in these months. 3.2.3. Diurnal wind speed variations The diurnal wind speed variations are illustrated in Fig. 3 for the yearly overall and the two seasons defined above based on the 5-year data. The bell shaped trend can be found for all three

Wind Speed (m/s)

6.0 5.0 4.0 3.0 2.0 1.0 0.0 Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Month 1999

2000

2001

2002

2003

5-year

Fig. 2. Monthly mean wind speed variations for overall and individual five years (1999–2003) in Waterloo region, Canada.

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Table 4 Yearly mean wind speed variations for Cold Season (November–April) and Warm Season (May–October) based on 5-year data (1999–2003) in Waterloo region, Canada 1999 2000 2001 2002 2003 5-year

Cold Season

Warm Season

4.3 4.4 4.3 4.9 4.5 4.5

2.7 3.0 3.6 3.1 3.4 3.1

7

Wind Speed (m/s)

6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Hour of Day Cold Season

Warm Season

Year

Fig. 3. Diurnal mean wind speed variations for Cold Season (November–April), Warm Season (May–October) and whole year based on 5-year data (1999–2003), in Waterloo region, Canada.

curves. It can be found that the day time, from 8 a.m. to 8 p.m., is windy for all seasons, while the night time is relatively quiet. The hourly means increase at around 6 a.m. and the peaks at around 2 p.m. After that, the afternoons are characterized by decreasing wind speeds, which tend to settle to lows after 9 p.m. There is a good coincidence between the energy demands and the characteristics of the Waterloo wind speeds since normally the energy demand is higher in the day time. Fig. 4 shows the diurnal wind speed variations for the individual months. It can be found that for January, February and December, the wind speeds are almost greater than 4 m/s all the time and the highest wind speed reaches 6 m/s in the daytime. For the other months in the Cold Season, November, December and April, the wind speeds are nearly up to or greater than 3 m/s almost all the day, and the wind flowing with speed above 5 m/s lasts at least one quarter of the day. The calmest month is August, and in that month, the wind speed is lower than 4 m/s for all the day.

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7 6 5 4 3 2 1

00 8: 00 9: 0 10 0 :0 11 0 :0 12 0 :0 13 0 :0 14 0 :0 15 0 :0 16 0 :0 17 0 :0 18 0 :0 19 0 :0 20 0 :0 21 0 :0 22 0 :0 23 0 :0 0

00

7:

00

6:

00

5:

00

4:

00

3:

00

2:

1:

0:

00

0

Jan Jul

Feb Aug

Mar Sep

Apr Oct

May Nov

Jun Dec

Fig. 4. Diurnal mean wind speed variations for individual month based on 5-year data (1999–2003), in Waterloo region, Canada.

3.2.4. Wind direction Analysis for wind direction is of importance for planning wind turbine installations, multi-wind or single wind turbine. Wind direction frequency distributions for the individual and overall five years are shown in Fig. 5. It is noted that for all the years, the wind directions demonstrate a similar pattern. The predominant distribution of the wind direction falls in the sector between 0° North counterclockwise to almost 180° South, and the range of 30° around 90° East also has a high frequency. The wind direction frequency distributions for the Cold and Warm Seasons are also investigated and shown in Fig. 6(a) and (b), respectively. For both seasons, the dominant

NNW NW WNW W

0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

N NNE NE ENE E

WSW

ESE SE

SW SSW S 1999

2000

2001

SSE

2002

2003

5-year

Fig. 5. Wind direction frequency distributions for overall and individual five years (1999–2003) in Waterloo region, Canada.

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(b)

N NNW NW WNW W

0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

N NNE

NNW

0.12

NNE

0.1

NE

NW ENE

NE

0.08 0.06

WNW

ENE

0.04 0.02

E

WSW

ESE

SW

SE SSW

SSE S

W

E

0

WSW

ESE

SW

SE SSE

SSW

Cold Season

S Warm Season

Fig. 6. Wind direction frequency distributions for Cold Season (November–April), Warm Season (May–October) based on 5-year data (1999–2003), in Waterloo region, Canada ((a) Cold Season; (b) Warm Season).

wind direction is similar to the yearly distribution. However, for the Warm Season, the northwest wind seems weighted more. Since the utilization of wind power in the Cold Season is quite promising, the monthly wind direction variations for each month in this season, from November to April, are shown in Fig. 7(a)–(f), respectively. The dominant wind direction still falls in 0° North counterclockwise to 180° South except for April, for which the dominant wind direction is in 90° East counterclockwise to 270° West.

4. Wind speed frequency distribution and energy assessment The MEP based distribution is used here to model the wind speed frequency variations. All the input parameters, such as the mean velocities, can be obtained from the measured data for this site. A modified Newton–Raphson method is used to calculate the Lagrangian parameters. The details about the numerical method can be found in Refs. [15,16]. Weisser [8] pointed out that if the diurnal wind patterns are neglected, the wind energy assessment for a site can result in significant under/overestimation of the wind energy potential. To examine if this is the case for the Waterloo region, in this section, not only the monthly, seasonal and yearly distributions but the diurnal ones as well are all calculated based on the 5-year data. Based on these wind speed distributions, the wind energy potential is calculated. 4.1. MEP based wind speed distribution: comparison with the data All the Lagrangian parameters for the monthly distributions are listed in Table 5, and the distributions for each month are also shown in Fig. 8 along with the measured data. It can be found that the Waterloo Region does have a relatively high frequency of calm spells, especially for the Warm Season. The MEP model can represent this type of data much better than the empirical Weibull

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(a)

NNW

0.14

N

NW

N

(b)

NNE

0.12

NNW

0.06

ENE

0.04

WNW

WSW SW

E

W

ESE

WSW

NW WNW W

ESE

WSW

0

NW

WSW SW

ENE

WNW

E

W

ESE

WSW

0.16

ESE SE SSE

SSW S January

(f)

SSE

NE

WNW

ENE

ENE 0.05

E

W

ESE

WSW

SE SSE

SSW

February

NNE

0.1

0.06 0.04 0.02 0

S

0.2 0.15

NW

NE

SW

SE

N NNW

NNE

0.14 0.12 0.1 0.08

E

SW

N NNW

NE

SSW

W

SSE

(e) NNE

0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

E

December

N 0.16

NE ENE

S

November

NNE

WNW

SE SSW

S

N

0.05

SW

SSE

SSW

NNW

ENE

0.02 0

SE

0.2 0.15

NW

0.04

0

(d)

NNW

0.1

0.06

0.02

W

NE

0.1 0.08

0.08

WNW

(c)

NNE

0.12

NW

NE

0.1

0.16 0.14

3025

E

0

ESE SE

SW SSE

SSW

S

S

March

April

Fig. 7. Wind direction frequency distributions for individual months in Cold Season (November–April) based on 5-year data (1999–2003), in Waterloo region, Canada ((a) November; (b) December; (c) January; (d) February; (e) March; (f) April).

Table 5 The computed Lagrangian multipliers for individual months based on 5-year data (1999–2003) in Waterloo region, Canada January February March April May June July August September October November December

a0

a1

a2

a3

1.073454424 1.002964971 0.876949691 0.757186496 0.629153038 0.431994116 0.453965384 0.363907231 0.301536069 0.480286867 1.007062170 0.675255528

1.272019103 1.382797991 1.037660138 0.893662735 0.200414529 0.270983814 0.239087533 0.290283200 0.778694959 0.156198974 1.416612789 0.443583538

0.565085118 0.954589136 0.758626011 0.851942168 0.141500579 0.027518670 0.040564301 0.125327994 0.430180682 0.467299937 0.996692564 0.390897827

0.120420072 0.038404796 0.015031385 0.076855040 0.103529397 0.094857035 0.105371368 0.032219174 0.177991535 0.033731012 0.050768095 0.039055337

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Fig. 8. Wind speed frequency distributions for individual month based on 5-year data (1999–2003), in Waterloo region, Canada ((a) January; (b) February; (c) March; (d) April; (e) May; (f) June; (g) July; (h) August; (i) September; (j) October; (k) November; (l) December).

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Fig. 8 (continued )

distribution [11]. To evaluate the performance of the theoretical MEP based distributions, the coefficient of determination (COD) is used here. This coefficient, expressed as a percentage, indicates

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how much of the total variation in the dependent variable can be accounted for by the theoretical or empirical distribution. A higher COD represents a better fit using the theoretical or empirical function. Normally, a value higher than 70% of COD is acceptable. The definition of the COD is COD ¼ R2 ¼ 1 

r2y;x r2y

ð17Þ

where R is the correlation coefficient [17], and ry is the standard deviation of the measured data y from its own mean value ym, which is conventionally defined as "P # N 2 1=2 i¼1 ðy i  y m Þ ry ¼ ð18Þ N 1 where N is the total number of measurements. Similarly, "P # N 2 1=2 ðy  y Þ ic i¼1 i ry;x ¼ N 2

ð19Þ

where the yi are the actual values of y, and yic are the values computed from the correlation equation for the same value of x. Two other goodness-of-fit parameters in statistical analysis, Chi-square error and root mean square error (RMSE), are also introduced for additional evaluation of the performance of the MEP distributions. The expressions for these parameters are 2 N X ðy i  y ic Þ v ¼ yi i¼1 2

"

N 1 X ðy  y ic Þ2 RMSE ¼ N i¼1 i

ð20Þ #1=2 ð21Þ

The smaller the values of these two parameters are, the better the proposed distribution function approximates the measured data (or the better the curve fits). In the ideal case, the values should be zero for these two parameters. The parameters for the statistical analysis: the COD, RMSE and Chi-square error, are given in Table 6 for all the monthly MEP based wind speed distributions. It can be seen that the CODs vary from 78% to 85% for the individual months. A similar procedure is also performed for the Cold and Warm Seasons. The Lagrangian parameters for both seasons are shown in Table 7, and the comparison between the MEP based distribution and the measured data are shown in Fig. 9(a) and (b). The parameters of the statistical analysis for the two seasons can be found in Table 8, and both the CODs are greater than 83%. Fig. 10 shows the wind speed distributions for the day time, night time and the whole day based on the 5-year data along with the correspondingly measured data. The distributions for the day and night times show a significant deviation from the yearly whole day distribution shown in the same figure. The corresponding Lagrangian multipliers for the day time, night time and whole day

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Table 6 The statistical analysis parameters for monthly wind speed distributions in the Waterloo region, Canada COD RSME v2

COD RSME v2

January

February

March

April

May

June

0.8165 0.02279 0.000550

0.8399 0.01980 0.000414

0.7790 0.02558 0.000693

0.7792 0.02731 0.000787

0.825 0.02516 0.000670

0.8413 0.02906 0.000905

July

August

September

October

November

December

0.8357 0.03097 0.001027

0.8158 0.03810 0.001583

0.8501 0.02794 0.000837

0.7967 0.03014 0.000962

0.8055 0.02537 0.000684

0.8298 0.02183 0.000506

Table 7 The computed Lagrangian multipliers for Cold Season (November–April) and Warm Season (May–October) based on 5-year data (1999–2003) in Waterloo region, Canada Cold Season Warm Season

a0

a1

a2

a3

0.9338269099 0.4593753338

1.190979052 0.04345354839

0.8422791429 0.2368777867

0.02391028773 0.02463423533

distributions are listed in Table 9. The statistical analysis in Table 10 shows reasonable agreement between the MEP based distributions with the actual measured data. 4.2. Wind energy potential Wind power density can be calculated by integrating the corresponding MEP based wind speed distribution function according to Eq. (15), and the resulting monthly wind power density can be found in Table 11. It can be seen that the wind power density for each month in the Cold Season exceeds 100 W/m2, falling into the classification of Class 2 of the PNL classification system. Class 2 areas are considered marginal for wind power development. Moreover, the wind power density for three months, January, February and March, even exceeds 150 W/m2 with the highest power density, 188 W/m2, arising in February. This falls into the Classification of Class 3, which is suitable for wind energy development using taller wind turbine towers. The lowest wind potential appears in August, in accordance with the lowest monthly mean wind speed in this month. The seasonal and yearly wind power potential is shown in Table 12. The estimations of wind power potential of the day time and night time are also listed in the table. In accordance with the trends of wind speed, the wind power density in the day time and night time is higher and lower than the whole day estimation, respectively. Just as Weisser stated, estimating only the whole day distribution of winds does lead to an assessment that underestimates the wind power potential during the day light hours and overestimates the wind power potential at night time. According to the PNL classification system, the yearly wind energy potential estimations for the whole day and day time fall in Class 2. All the estimations, day time, night time and whole day, of the wind energy potential of the Cold Season exceeds 100 W/m2. The day light hours have the highest wind power potential, 180 W/m2.

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Fig. 9. Wind speed frequency distributions for Cold Season (November–April) and Warm Season (May–October) based on 5-year data (1999–2003), in Waterloo region, Canada ((a) Cold Season; (b) Warm Season).

Table 8 The statistical analysis parameters for seasonal (Cold Season: November–April; Warm Season: May–October) wind speed distributions in Waterloo region, Canada COD RSME v2

Cold Season

Warm Season

0.8333 0.02192 0.000505

0.8448 0.02760 0.000806

M. Li, X. Li / Energy Conversion and Management 46 (2005) 3014–3033

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Fig. 10. Wind speed frequency distributions for overall five years (1999–2003) in Waterloo region, Canada.

Table 9 The computed Lagrangian multipliers for day time (from 8 a.m. to 8 p.m.), night time (from 9 p.m. to 7 a.m.) and whole day wind speed distribution based on 5-year data (1999–2003) in Waterloo region, Canada Whole day Day time Night time

a0

a1

a2

a3

0.6148072548 1.363354351 0.2207082868

0.3659797102 2.461481140 0.5528055670

0.4360674768 1.687198340 0.1055049466

0.007723161330 0.1638990393 0.007436289014

Table 10 The statistical analysis parameters for diurnal (day time: from 8 a.m. to 8 p.m.; night time: from 9 p.m. to 7 a.m.) wind speed distributions in Waterloo region, Canada COD RSME v2

Day time

Night time

Whole day

0.8243 0.02381 0.000599

0.8817 0.02519 0.000666

0.8400 0.02380 0.000595

Table 11 Monthly wind power density (unit: W/m2) based on 5-year data (1999–2003) in Waterloo region, Canada January

February

March

April

May

June

Wind power density

157

188

152

136

98

56

July

August

September

October

November

December

Wind power density

47

36

59

95

122

146

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Table 12 Diurnal (day time: from 8 a.m. to 8 p.m.; night time: from 9 p.m. to 7 a.m.) wind power density (unit: W/m2) based on 5-year data (1999–2003) in Waterloo region, Canada Whole day Day time Night time

Cold Season

Warm Season

Year

149 180 111

66 99 31

106 135 69

5. Conclusion Wind characteristics have been analyzed for the Waterloo Region, Canada, in this study based on a measured data source over a 5-year period (1999–2003). Characteristics such as annual, monthly and diurnal wind speed variations are examined. The annual and monthly wind direction variations are also investigated. The wind speed data reveal that the windy months in Waterloo are from November to May, with February being the windiest month. The diurnal trend shows that day time is windier than night time in this region. The model derived from the maximum entropy principle is applied to determine the wind speed distributions, and the available wind power potential of the area is assessed based on the resulting distributions. The wind power density in the day time for the Cold Season is 180 W/m2. It is helpful that the higher power demands in the day time of the Cold Season coincide with the windy period of time.

Acknowledgment The financial support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.

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