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Energy and reliability benefits of wind energy conversion systems
Renewable Energy 36 (2011) 1983e1988
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Renewable Energy journal homepage: www.elsevier.com/locate/renene
Energy and reliability benefits of wind energy conversion systems Kaigui Xie a, *, Roy Billinton b a
State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University, Chongqing 400044, PR China b Power System Research Group, University of Saskatchewan, Saskatoon, Canada S7N5A9
a r t i c l e i n f o
a b s t r a c t
Article history: Received 3 August 2010 Accepted 4 December 2010 Available online 8 January 2011
The electrical energy production and reliability benefits of a wind energy conversion system (WECS) at a specific site depend on many factors, including the statistical characteristics of the site wind speed and the design characteristics of the wind turbine generator (WTG) itself, particularly the cut-in, rated and cut-out wind speed parameters. In general, the higher the degree of the wind site matching with a WECS is, the more are the energy and reliability benefits. An electrical energy production and reliability benefit index designated as the Equivalent Capacity Ratio (ECR) is introduced in this paper. This index can be used to indicate the electrical energy production, the annual equivalent utilization time and the credit of a WECS, and quantify the degree of wind site matching with a WECS. The equivalent capacity of a WECS is modeled as the expected value of the power output random variable with the probability density function of the site wind speed. The analytical formulation of the ECR is based on a mathematical derivation with high accuracy. Twelve WTG types and two test systems are used to demonstrate the effectiveness of the proposed model. The results show that the ECR provides a useful index for a WTG to evaluate the energy production and the relative reliability performance in a power system, and can be used to assist in the determination of the optimal WTG type for a specific wind site. 2011 Elsevier Ltd. All rights reserved.
Keywords: Wind energy conversion system Equivalent capacity ratio Electrical energy production Reliability benefit Analytical model
1. Introduction Wind power is considered to be a very promising and encouraging alternative for power generation due to its tremendous environmental and social benefits, together with public support and government incentives. The utilization of a large amount of energy from the wind is being considered by many utilities. In North America, many US states and Canadian provinces have agreed to generate between 5% and 25% of their electrical power by 2010e2015 from renewable energy resources, most of which will come from wind [1]. Wind power additions to electric power systems provide a range of benefits including electrical energy production and reliability improvements. The electrical power output from a wind energy conversion system (WECS) at a specific site depends on many factors [2e7], including the wind regime at the site, and the characteristics of the wind turbine generator (WTG), such as the cut-in, rated and cut-out wind speed parameters. It is important to select a matching WTG for a specific wind site from the candidate WTG in order to obtain maximum energy and reliability benefits. The power
* Corresponding author. Tel./fax: þ86 23 65112729. E-mail address: [email protected] (K. Xie). 0960-1481/$ e see front matter 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2010.12.011
generation of a WECS does not vary linearly with the wind speed due to the nonlinear relationship between the WTG parameters and the wind speeds. It is therefore important to consider a site-matching WTG from the candidate WTG. References [2e7] illustrate some of the impacts of the above factors on the power system reliability. Case studies in references [2e5] show that the WTG cut-in and rated wind speeds can have a significant effect on the reliability of a power system and the cutout wind speed has almost no effect. The power system reliability indices of Loss of Load Expectation (LOLE) and Loss of Energy Expectation (LOEE) decreases somewhat exponentially with the number of WTG units added to the system, but tend to saturate when wind speeds continue to increase. The forced outage rate of the WTG has relatively little impact on the reliability performance of a power system [5,6]. Reference [7] indicates that wind speed correlation between two wind regimes can have a considerable impact on the system reliability. There has, however, been relatively little research on the impacts of the above factors on the electrical energy production of a WECS. Reference [3] introduces two risk-based indices designated as the load carrying capacity benefit ratio (LCCBR) and the equivalent capacity ratio (ECR) to determine the optimum site-matching WTG for a specific wind site. The optimum WTG can be determined through a comparison of the LCCBR and ECR indices. The process,
K. Xie, R. Billinton / Renewable Energy 36 (2011) 1983e1988
however, does not include electrical energy production. The derated adjusted forced outage rate (DAFOR) [8] of a WECS can be evaluated from the multi-state generating unit output model [9]. This paper presents an index of ECR to determine the electrical energy production, reliability benefit of a WECS and the degree of wind site matching with a WTG. 2. Wind speed model Analysis of electrical energy production and reliability benefits require an accurate wind speed simulating technique for each specific wind site. Many methods, such as auto-regressive and moving average (ARMA) models [10], require historical wind speed data collected over a number years to determine the necessary parameters of the wind speed model for a specific site. The wind speed time series can usually be modeled by many distributions, including Weibull distribution and normal distribution (ND). A simplified normal wind speed model is proposed in reference [1] to evaluate the reliability of a power system containing WECS with a reasonable accuracy. The results show that the normal wind speed model can provide very close results compared to the statistical wind speeds. This indicates that the ND can be used to simulate a wind speed time series with a higher accuracy. The biggest differences are likely to be in the tails of the normal and the Weibull distributions, but these contribute little to the WECS energy production. In other words, the wind speed less than Vci (cut-in wind speed) has no contribution to the energy production. The wind speeds can, therefore, be considered as a ND from an energy production point of view. In many cases, the wind speed data can be reasonably represented by a ND provided that the wind speed data at a specific site are collected over a long period of time. The actual hourly wind speeds from a 20-year database (from 1 January 1984 to 31 December 2003) [1,9] for four sites: Swift Current, Saskatoon, Regina and North Battleford, were obtained from the Environment, Canada, and show that each site can be reasonably represented in this manner. The analysis above shows that the ND can be used to describe the wind speed at a specific site. The required data in the ND model are the mean wind speed and its standard deviation at the specific wind site. Assuming that a ND is used to describe the wind speeds at a specific site, the density function of the ND is given by:
# " 1 ðs mÞ2 ðN s NÞ f ðsÞ ¼ pffiffiffiffiffiffiffiffiffiexp 2s2 2ps
(1)
where s is the wind speed, and m and s are the mean wind speed and standard deviation, respectively. Negative values in (1) are converted to zeros as recommended in reference [1]. 3. Generation power model There is a nonlinear relationship between the power output of a WTG and the wind speed [6,11]. A WTG is designed to start generating at the cut-in speed Vci and is shut down for safety reasons if the wind velocity is higher than the cut-out speed Vco. In both cases, the power output is zero. The power output of a WTG unit increases with the wind speed between the cut-in speed and the rated speed Vr; after that the power output remains constant at the rated power Pr. Fig. 1 shows a typical power output curve of a WTG. The relationship between the power output and the wind speed at a wind site can be determined from the operational parameters
Pr
Power Output (kW)
1984
0
Vci
Vr
Vco
Wind Speed (km/hour) Fig. 1. Power curve of a WTG.
of the WTG. Wind speed s can be approximately transformed into a series of power terms P(s) as follows [6,11]:
8 < Pr A þ B s þ C s2 PðsÞ ¼ Pr : 0
Vci s < Vr Vr s < Vco otherwise
(2)
The constants A, B, and C are presented in references [6,11].
4. Analytical formulation of the equivalent capacity ratio 4.1. Equivalent capacity model for a WTG The power output P(s) of a WTG is related to the wind speed s, which is described by a ND. There is only one power output P(s) corresponding to a specified wind speed s. Hence, the probability density function f(s) of the wind speed can be directly linked to the density function of the power output of a WTG at a specific wind site due to the same wind speed parameter s. The equivalent capacity of a WTG denoted as EC can therefore be calculated using the definition of expectation for a random variable:
ZþN EC ¼
f ðsÞPðsÞds
(3)
N
Eq. (3) is a general formulation of the EC for a WTG and its analytical formulation is presented in the following sections.
4.2. Standard normal distribution The standard normal distribution (SND) is a powerful and useful probability distribution in engineering applications. It is used to derive the analytical formulation of the EC of a WTG. The density function of the SND is defined as
2 1 z ðN z NÞ f ðzÞ ¼ pffiffiffiffiffiffi exp 2 2p
(4)
The probability z˛½z0 ; þNÞ under the standard normal density function curve shown in Fig. 2 is given by
K. Xie, R. Billinton / Renewable Energy 36 (2011) 1983e1988
Zb
f(z)
a
Zb a
1985
2 1 z pffiffiffiffiffiffi z exp dz ¼ RðbÞ RðaÞ 2 2p
(11)
2 1 z pffiffiffiffiffiffi z2 exp dz ¼ WðbÞ WðaÞ 2 2p
(12)
Q(z0) Eqs. (8), (11) and (12) can be analytically calculated. 4.4. Analytical formulation of the equivalent capacity ratio
0
z0
The analytical formulation of the EC for a WTG in (3) is derived as follows. Based on (1)e(3),
z
Fig. 2. Probability of z˛½z0 ; þNÞ under the standard normal density function.
ZþN Q ðz0 Þ ¼ z0
2 1 z pffiffiffiffiffiffi exp dz 2 2p
ZþN (5)
i h Q ðz0 Þ ¼ f ðz0 Þ b1 t þ b2 t 2 þ b3 t 3 þ b4 t 4 þ b5 t 5
(6)
where t ¼ 1=ð1 þ rz0 Þ, r ¼ 0.2316419, b1 ¼ 0.31938153, b2 ¼ b4 ¼ 1.821255978 and 0.356563782, b3 ¼ 1.781477937, b5 ¼ 1.330274429. The maximum error in (6) is smaller than 7.5 108 [8,12]. If z0 < 0, the probability of Q(z0) can be calculated using the symmetry of the function f(z). Using (5),
Zz
f ðsÞPðsÞds N
ZVr
Although there is no explicit analytical expression for the cumulative distribution function of the SND [8, 12], Eq. (5) can be calculated using the following polynomial approximation [8,12] for z0 0:
GðzÞ ¼
EC ¼
2 1 t pffiffiffiffiffiffi exp dt ¼ 1 Q ðzÞ 2 2p
¼
f ðsÞPr AþBsþCs
a
z ¼
(15)
s
Vci m Vr m
(17)
s
0 Vco ¼
Vco m
(18)
s
þðBs þ2C msÞzþC s2 z2 aA1 þB1 zþC1 z2
ZVr EC1 ¼ Vci0
2 1 z pffiffiffiffiffiffi exp Pr A1 þB1 zþC1 z2 dz 2 2p 0
ZVr ¼ A1 Pr Vci0
0
2
2
ZVr 1 z 1 z pffiffiffiffiffiffi exp pffiffiffiffiffiffiexp dzþB1 Pr zdz 2 2 p 2p 2 0 Vci
0
(10)
where C1 and C are the two arbitrary constants. Based on (9) and (10), the following definite integral can be obtained
ð19Þ
Substituting (19) into EC1 in (13), 0
(9)
2 2 1 z 1 z pffiffiffiffiffiffi z2 exp dz ¼ zpffiffiffiffiffiffiexp 2 2 2p 2p 2 Z 1 z dz þ C1 þ pffiffiffiffiffiffiexp 2 2p ¼ zRðzÞ þ GðzÞ þ CaWðzÞ þ C
(16)
s
AþBsþCs2 ¼AþBðzs þ mÞþCðzs þ mÞ2 ¼ AþBm þC m2
The following equations can be obtained using the indefinite integral principle and (7):
Z
(14)
sm
4.3. Analytical formulation of integral functions
2 2 1 z 1 z pffiffiffiffiffiffiz exp dz ¼ pffiffiffiffiffiffiexp þ CaRðzÞ þ C 2 2 2p 2p
(13)
Vcr
Based on Eqs. (15)e(18), we have
where a and b are the two arbitrary real numbers.
Z
f ðsÞPr dsaEC1 þEC2
dsþ
Consider the following substitutions:
Vr0 ¼
(8)
ZVco
EC ¼ ECðm; s; Vci ; Vr ; Vco Þ
In other words, G(z) in (7) is an integral primitive function of f(z) in (4). G(z) can be analytically evaluated based on (6) and (7). The definite integral is given by (8)
2 1 t pffiffiffiffiffiffi exp dt ¼ GðbÞ GðaÞ 2 2p
It can be seen from (13) that the EC of a WECS is a function of the five variables m, s, Vci, Vco and Vr, i.e.
Vci0 ¼
Zb
2
Vci
(7)
N
ZVr þC1 Pr Vci0
2
1 z pffiffiffiffiffiffi exp z2 dz 2 2p
Using (8), (10) and (12),
(20)
1986
K. Xie, R. Billinton / Renewable Energy 36 (2011) 1983e1988
EC1 ¼ A1 Pr G Vci0 G Vr0 þ B1 Pr R Vr0 R Vci0 þ C1 Pr W Vr0 W Vci0
Table 2 Wind turbine parameters.
(21)
The term EC2 in (13) can be calculated using (8):
0 EC2 ¼ Pr G Vr0 G Vco
(22)
The EC of a WTG can be calculated using (13), (21) and (22) and the analytical formulation of G(z), R(z) and W(z) in (8), (11) and (12). The equivalent capacity ratio index (ECR) of a WTG associated with a specific wind site is defined as
ECR ¼
EC Pr
(23)
Types
Cut-in speed (km/h)
Rated speed (km/h)
Cut-out speed (km/h)
A B C D E F G H I J K L
10 10 11 11 12 12 14 14 16 16 18 18
35 38 35 38 38 41 35 38 41 35 38 41
48 60 70 60 70 60 70 80 60 52 70 65
The annual equivalent utilization time index (AEUT) of a WTG associated with a specific wind site can be obtained using the ECR
5. Case studies and applications
AEUT ¼ ECR 8760
5.1. Accuracy analysis of the proposed model
(24)
The AEUT indicates the annual equivalent utilization hours of a WTG and the wind energy produced by a WTG with a per unit rated capacity. The ECR measures the electrical energy production performance or energy benefit without linking it to the rated capacity of the WTG. A high value of this index indicates good utilization of wind power. It can also be used to reflect the reliability benefits and the degree of wind site matching with the WTG. Energy production and reliability benefits of a WTG are directly related to the site ECR. Based on the previous discuss and (14), (23) and (24), the EC, ECR and AEUT indices respectively can be analytically evaluated using the m, s for a wind site and the three parameters Vci, Vco and Vr for a WTG.
4.5. Analysis of Weibull wind speed model When the wind speeds are represented by a Weibull distribution, the m and s of the Weibull distribution can be evaluated from the scale (a) and shape (b) parameters [12] as follows:
m ¼ aG 1 þ
1
(25)
b
s2 ¼ a2 G 1 þ
2
b
G2 1 þ
1
(26)
b
where G() is the gamma function which is defined by
GðxÞ ¼
ZN
t x1 et dt
(27)
The wind speed time series at Lauwersoog wind site in 2001e2007 [13] were used to verify the calculation accuracy of the proposed model. The mean and standard deviation of the statistical wind speeds are 6.23 and 3.05 m/s, respectively. A WTG with a rated power output of 1 MW is assumed to be located at a site with the Lauwersoog wind regime. The cut-in, rated and cut-out wind speeds for the WTG are 3, 15 and 22 m/s, respectively. Annual energy production using the proposed ND, Weibull distribution and statistical wind speeds are 999.75, 999.37 and 1001.81 MWh, respectively, which indicates that the normal and Weibull distributions have evaluation errors of 0.21% and 0.24%, respectively. If only the data in 2007 were used, the mean and standard deviation of statistical wind speeds are 6.67 and 3.30 m/s, respectively. The evaluated annual energy productions are 1268.44, 1248.08 and 1257.43 MWh, which indicates that the normal and Weibull distributions have evaluation errors of 0.88% and 0.74%, respectively. Although the Weibull distribution provides a more accurate evaluation of energy production, the normal distribution also provides a considerable accuracy. These results show that the proposed analytical ND model has a considerable high accuracy. 5.2. Analysis of the equivalent capacity ratio The four wind farm locations noted in Section 2 were used in the following case studies. The basic wind speed statistics for the four specific sites are shown in Table 1. Table 2 shows the 12 WTG types utilized in the case studies. The cut-in wind speed parameters range from 10 to 18 km/h, the rated speeds from 35 to 41 km/h, and the cut-out speeds from 48 to 80 km/h. Table 3 ECR for the four wind sites with different WTG.
0
Types
Similarly, the EC, ECR and AEUT indices can be analytically evaluated using the m and s for a wind site and the three parameters Vci, Vco and Vr for a WTG.
Table 1 Site wind speed statistics. Sites
Swift Current
Saskatoon
Regina
North Battleford
Mean wind speed (km/h), m Standard deviation (km/h), s
19.46 9.70
16.78 9.23
19.52 10.99
14.62 9.59
A B C D E F G H I J K L
Wind sites Swift Current
Saskatoon
North Battleford
Regina
0.2684 0.2174 0.2682 0.2159 0.2141 0.1718 0.2593 0.2088 0.1614 0.2495 0.1908 0.1530
0.1922 0.1513 0.1907 0.1499 0.1480 0.1165 0.1813 0.1426 0.1063 0.1715 0.1249 0.0983
0.2766 0.2267 0.2774 0.2253 0.2235 0.1807 0.2689 0.2184 0.1707 0.2592 0.2013 0.1627
0.1512 0.1181 0.1497 0.1169 0.1152 0.0902 0.1409 0.1103 0.0812 0.1321 0.0947 0.0743
K. Xie, R. Billinton / Renewable Energy 36 (2011) 1983e1988 Table 4 AEUT for the four wind sites with different WTG (h/year). Types
Table 6 Reliability comparison of the IEEE-RTS with different WTG types.
Wind sites
A B C D E F G H I J K L
Types ECR
Swift Current
Saskatoon
North Battleford
Regina
2352 1905 2349 1892 1875 1505 2271 1829 1414 2185 1671 1341
1684 1325 1670 1313 1296 1020 1588 1249 932 1502 1094 861
2423 1986 2430 1973 1958 1583 2356 1913 1496 2271 1763 1425
1325 1035 1311 1024 1009 790 1234 966 711 1157 829 651
The ECR and AEUT for the 12 WTG types applied to the four wind sites are shown in Tables 3 and 4. The values given in bold in Tables 3 and 4 are the highest ECR and AEUT for each wind site. It can be seen from these tables that the most suitable WTG types for the Swift Current, Saskatoon, North Battleford and Regina wind sites are A, A, C and A, respectively. The highest ECR for the four wind sites are 26.84%, 19.22%, 27.74% and 15.12%, respectively, and the highest AEUT are 2352, 1684, 2430 and 1325 h/year, respectively. It can also be seen from these tables that there is a large difference between the highest and lowest ECR or AEUT for different WTG at a wind site. Consider a wind farm at Swift Current as an example. In these cases, the maximum ECR and AEUT are 26.84% and 2352, respectively, whereas the minimum ECR and AEUT are 15.30% and 1341 respectively. These show that the energy production at a wind farm strongly depends on the WTG type. The ECR and AEUT can be used to measure the degree of site wind speed matching with the WTG. Selecting a suitable site-matching WTG for a specific wind site is therefore important in order to achieve maximum energy benefits from a WECS. 5.3. Analysis of adequacy evaluation The reliability benefit of a WECS at a specific site depends on many factors, including the wind speed distributions, the ECR of the WTG, the capacity distributions and reliability performance of traditional generating units and the load distributions. The ECR impacts of a WTG on the system reliability of power systems were investigated using the RBTS [14] and IEEE-RTS [15]. The RBTS consists of 11 traditional generating units with a total capacity of 240 MW and a peak load of 185 MW. The IEEE-RTS consists of 32
Table 5 Reliability comparison of the RBTS with different WTG types. Types ECR
Ranked by LOEE Ranked by AEUT Ranked by LOLE (h/year) the ECRa (h/year) the LOEEa (MWh/year) The LOEEa
A B C D E F G H I J K L
2352 1905 2349 1892 1875 1505 2271 1829 1414 2185 1671 1341
0.2684 0.2174 0.2682 0.2159 0.2141 0.1718 0.2593 0.2088 0.1614 0.2495 0.1908 0.153
1 5 2 6 7 10 3 8 11 4 9 12
0.612 0.674 0.615 0.677 0.682 0.739 0.637 0.695 0.765 0.662 0.741 0.787
1 5 2 6 7 9 3 8 11 4 10 12
5.465 6.023 5.489 6.053 6.094 6.610 5.697 6.218 6.857 5.928 6.653 7.063
1987
1 4 2 5 7 9 3 8 11 6 10 12
a The ECR indices are ranked in using decreasing values order; the LOLE and LOEE indices are ranked in using increasing values order.
A B C D E F G H I J K L
Ranked by the ECR
0.2684 1 0.2174 5 0.2682 2 0.2159 6 0.2141 7 0.1718 10 0.2593 3 0.2088 8 0.1614 11 0.2495 4 0.1908 9 0.153 12
LOLE (h/ Ranked by year) the LOLE
LOEE (MWh/ year)
Ranked by the LOEE
4.72 5.19 4.74 5.22 5.25 5.69 4.91 5.35 5.89 5.09 5.71 6.06
541.15 596.99 543.47 599.95 604.08 655.67 564.17 616.35 680.21 587.00 659.45 700.65
1 5 2 6 7 9 3 8 11 4 10 12
1 5 2 6 7 9 3 8 11 4 10 12
traditional generating units with a total capacity of 3405 MW and a peak load of 2850 MW. The generating unit ratings and reliability parameters are shown in [14,15]. The hourly per unit chronological load demand of the IEEE-RTS [15] was used in both test systems. A wind farm with 20 WTG units is assumed to be located at a site with the Swift Current wind regime and added to the RBTS. Each WTG unit has a rated power output of 2 MW. The WTG types in Table 2 are used in these studies. Algorithms for modeling the wind farm multi-state capability distributions given in references [6,9] were also used. Each traditional generating unit is represented using a two-state model (up and down states), and each WTG is represented using a five-state model. System states were enumerated by considering up to the fifth contingency level, i.e., up to five units in an outage or derated event. The RBTS indices obtained using the analytical enumeration approach are shown in Table 5. The WTG types are ranked according to the ECR or AEUT of the WTG associated with the Swift Current wind site. It is obvious that the ECR and AEUT have the same order due to having a constant multiplier between the two indices. It can be seen from Table 5 that the ECR and the reliability indices, such as LOLE or LOEE, mainly have the same order for the WTG. In general, the higher the ECR of a WTG is, the higher is the reliability benefit. Some orders of the ECR in Table 5, such as those for types F and K, are not consistent with the reliability index order due to the different wind speed distributions. One conclusion that can be drawn from Tables 3 and 5 is that the ECR provides a useful index for evaluating the energy production of a WTG and the potential power system reliability benefit. The ECR can also be used to judge the degree of a WTG matching with a wind site. In other words, the ECR can be used to assist in the determination of the optimal WTG type for a specific wind site. A wind farm with 250 WTG units and the Swift Current wind regime was added to the IEEE-RTS. The rated capacity and the WTG types are the same as those used in the previous section. It can be seen from Table 6 that the ECR and the reliability indices generally exhibit the same order for the WTG. Table 6 indicates that similar conclusions to those obtained from the RBTS study can be drawn for the IEEE-RTS. 6. Conclusion The contributions in terms of energy production and reliability benefits of including wind energy conversion systems (WECS) in power systems are dependent on a wide range of factors including the wind speed characteristics and the WTG design parameters, such as the cut-in, rated and cut-out wind speed. The degree of the
1988
K. Xie, R. Billinton / Renewable Energy 36 (2011) 1983e1988
wind site matching with the WTG has a significant impact on the electrical energy produced by a WECS and the power system reliability benefits. The equivalent capacity ratio (ECR) of a WTG is introduced to quantify the electrical energy production performance or energy benefit, the reliability benefit, and the degree of wind site matching with a WTG. The ECR is formulated as the expected value of the power output random variable and the probability density function of the site wind speed. This index can assist system planners and utility managers to assess the energy and reliability worth of a WECS and add useful information to the managerial decision process. The ECR index can be extended to indicate the amount of annual equivalent utilization time (AEUT) of a WTG. The major advantage of the proposed method is that probability and integral theories are used to mathematically derive the analytical formulation of the ECR and AEUT. Once the parameters of the wind speed at a site (mean and standard deviation), the WTG parameters (the cut-in, rated and cut-out wind speed) are available, the ECR and AEUT can be directly and analytically evaluated. The impacts of different WTG design parameters on the energy and reliability benefits associated with a specific wind site are illustrated in this paper. The case studies show that the ECR provides a direct and physical indication of the energy benefit for a WECS, and an estimation of its contribution to the reliability of a power system. The ECR index can be used to judge the degree of a WTG matching with a wind site and assist in the determination of the optimal WTG type for the potential site. Significant energy and reliability benefits can be obtained using the proposed technique to select appropriate site-matching WECS. Acknowledgements This work was supported in part by the National Natural Science Foundation of China (No. 51077135), Natural Science Foundation Project of CQ CSTC (No. CSTC2010BA3006) and Scientific Research
Foundation of Stage Key Lab. of Power Transmission Equipment and System Security (No. 2007DA10512709103).
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Renewable Energy journal homepage: www.elsevier.com/locate/renene
Energy and reliability benefits of wind energy conversion systems Kaigui Xie a, *, Roy Billinton b a
State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University, Chongqing 400044, PR China b Power System Research Group, University of Saskatchewan, Saskatoon, Canada S7N5A9
a r t i c l e i n f o
a b s t r a c t
Article history: Received 3 August 2010 Accepted 4 December 2010 Available online 8 January 2011
The electrical energy production and reliability benefits of a wind energy conversion system (WECS) at a specific site depend on many factors, including the statistical characteristics of the site wind speed and the design characteristics of the wind turbine generator (WTG) itself, particularly the cut-in, rated and cut-out wind speed parameters. In general, the higher the degree of the wind site matching with a WECS is, the more are the energy and reliability benefits. An electrical energy production and reliability benefit index designated as the Equivalent Capacity Ratio (ECR) is introduced in this paper. This index can be used to indicate the electrical energy production, the annual equivalent utilization time and the credit of a WECS, and quantify the degree of wind site matching with a WECS. The equivalent capacity of a WECS is modeled as the expected value of the power output random variable with the probability density function of the site wind speed. The analytical formulation of the ECR is based on a mathematical derivation with high accuracy. Twelve WTG types and two test systems are used to demonstrate the effectiveness of the proposed model. The results show that the ECR provides a useful index for a WTG to evaluate the energy production and the relative reliability performance in a power system, and can be used to assist in the determination of the optimal WTG type for a specific wind site. 2011 Elsevier Ltd. All rights reserved.
Keywords: Wind energy conversion system Equivalent capacity ratio Electrical energy production Reliability benefit Analytical model
1. Introduction Wind power is considered to be a very promising and encouraging alternative for power generation due to its tremendous environmental and social benefits, together with public support and government incentives. The utilization of a large amount of energy from the wind is being considered by many utilities. In North America, many US states and Canadian provinces have agreed to generate between 5% and 25% of their electrical power by 2010e2015 from renewable energy resources, most of which will come from wind [1]. Wind power additions to electric power systems provide a range of benefits including electrical energy production and reliability improvements. The electrical power output from a wind energy conversion system (WECS) at a specific site depends on many factors [2e7], including the wind regime at the site, and the characteristics of the wind turbine generator (WTG), such as the cut-in, rated and cut-out wind speed parameters. It is important to select a matching WTG for a specific wind site from the candidate WTG in order to obtain maximum energy and reliability benefits. The power
* Corresponding author. Tel./fax: þ86 23 65112729. E-mail address: [email protected] (K. Xie). 0960-1481/$ e see front matter 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2010.12.011
generation of a WECS does not vary linearly with the wind speed due to the nonlinear relationship between the WTG parameters and the wind speeds. It is therefore important to consider a site-matching WTG from the candidate WTG. References [2e7] illustrate some of the impacts of the above factors on the power system reliability. Case studies in references [2e5] show that the WTG cut-in and rated wind speeds can have a significant effect on the reliability of a power system and the cutout wind speed has almost no effect. The power system reliability indices of Loss of Load Expectation (LOLE) and Loss of Energy Expectation (LOEE) decreases somewhat exponentially with the number of WTG units added to the system, but tend to saturate when wind speeds continue to increase. The forced outage rate of the WTG has relatively little impact on the reliability performance of a power system [5,6]. Reference [7] indicates that wind speed correlation between two wind regimes can have a considerable impact on the system reliability. There has, however, been relatively little research on the impacts of the above factors on the electrical energy production of a WECS. Reference [3] introduces two risk-based indices designated as the load carrying capacity benefit ratio (LCCBR) and the equivalent capacity ratio (ECR) to determine the optimum site-matching WTG for a specific wind site. The optimum WTG can be determined through a comparison of the LCCBR and ECR indices. The process,
K. Xie, R. Billinton / Renewable Energy 36 (2011) 1983e1988
however, does not include electrical energy production. The derated adjusted forced outage rate (DAFOR) [8] of a WECS can be evaluated from the multi-state generating unit output model [9]. This paper presents an index of ECR to determine the electrical energy production, reliability benefit of a WECS and the degree of wind site matching with a WTG. 2. Wind speed model Analysis of electrical energy production and reliability benefits require an accurate wind speed simulating technique for each specific wind site. Many methods, such as auto-regressive and moving average (ARMA) models [10], require historical wind speed data collected over a number years to determine the necessary parameters of the wind speed model for a specific site. The wind speed time series can usually be modeled by many distributions, including Weibull distribution and normal distribution (ND). A simplified normal wind speed model is proposed in reference [1] to evaluate the reliability of a power system containing WECS with a reasonable accuracy. The results show that the normal wind speed model can provide very close results compared to the statistical wind speeds. This indicates that the ND can be used to simulate a wind speed time series with a higher accuracy. The biggest differences are likely to be in the tails of the normal and the Weibull distributions, but these contribute little to the WECS energy production. In other words, the wind speed less than Vci (cut-in wind speed) has no contribution to the energy production. The wind speeds can, therefore, be considered as a ND from an energy production point of view. In many cases, the wind speed data can be reasonably represented by a ND provided that the wind speed data at a specific site are collected over a long period of time. The actual hourly wind speeds from a 20-year database (from 1 January 1984 to 31 December 2003) [1,9] for four sites: Swift Current, Saskatoon, Regina and North Battleford, were obtained from the Environment, Canada, and show that each site can be reasonably represented in this manner. The analysis above shows that the ND can be used to describe the wind speed at a specific site. The required data in the ND model are the mean wind speed and its standard deviation at the specific wind site. Assuming that a ND is used to describe the wind speeds at a specific site, the density function of the ND is given by:
# " 1 ðs mÞ2 ðN s NÞ f ðsÞ ¼ pffiffiffiffiffiffiffiffiffiexp 2s2 2ps
(1)
where s is the wind speed, and m and s are the mean wind speed and standard deviation, respectively. Negative values in (1) are converted to zeros as recommended in reference [1]. 3. Generation power model There is a nonlinear relationship between the power output of a WTG and the wind speed [6,11]. A WTG is designed to start generating at the cut-in speed Vci and is shut down for safety reasons if the wind velocity is higher than the cut-out speed Vco. In both cases, the power output is zero. The power output of a WTG unit increases with the wind speed between the cut-in speed and the rated speed Vr; after that the power output remains constant at the rated power Pr. Fig. 1 shows a typical power output curve of a WTG. The relationship between the power output and the wind speed at a wind site can be determined from the operational parameters
Pr
Power Output (kW)
1984
0
Vci
Vr
Vco
Wind Speed (km/hour) Fig. 1. Power curve of a WTG.
of the WTG. Wind speed s can be approximately transformed into a series of power terms P(s) as follows [6,11]:
8 < Pr A þ B s þ C s2 PðsÞ ¼ Pr : 0
Vci s < Vr Vr s < Vco otherwise
(2)
The constants A, B, and C are presented in references [6,11].
4. Analytical formulation of the equivalent capacity ratio 4.1. Equivalent capacity model for a WTG The power output P(s) of a WTG is related to the wind speed s, which is described by a ND. There is only one power output P(s) corresponding to a specified wind speed s. Hence, the probability density function f(s) of the wind speed can be directly linked to the density function of the power output of a WTG at a specific wind site due to the same wind speed parameter s. The equivalent capacity of a WTG denoted as EC can therefore be calculated using the definition of expectation for a random variable:
ZþN EC ¼
f ðsÞPðsÞds
(3)
N
Eq. (3) is a general formulation of the EC for a WTG and its analytical formulation is presented in the following sections.
4.2. Standard normal distribution The standard normal distribution (SND) is a powerful and useful probability distribution in engineering applications. It is used to derive the analytical formulation of the EC of a WTG. The density function of the SND is defined as
2 1 z ðN z NÞ f ðzÞ ¼ pffiffiffiffiffiffi exp 2 2p
(4)
The probability z˛½z0 ; þNÞ under the standard normal density function curve shown in Fig. 2 is given by
K. Xie, R. Billinton / Renewable Energy 36 (2011) 1983e1988
Zb
f(z)
a
Zb a
1985
2 1 z pffiffiffiffiffiffi z exp dz ¼ RðbÞ RðaÞ 2 2p
(11)
2 1 z pffiffiffiffiffiffi z2 exp dz ¼ WðbÞ WðaÞ 2 2p
(12)
Q(z0) Eqs. (8), (11) and (12) can be analytically calculated. 4.4. Analytical formulation of the equivalent capacity ratio
0
z0
The analytical formulation of the EC for a WTG in (3) is derived as follows. Based on (1)e(3),
z
Fig. 2. Probability of z˛½z0 ; þNÞ under the standard normal density function.
ZþN Q ðz0 Þ ¼ z0
2 1 z pffiffiffiffiffiffi exp dz 2 2p
ZþN (5)
i h Q ðz0 Þ ¼ f ðz0 Þ b1 t þ b2 t 2 þ b3 t 3 þ b4 t 4 þ b5 t 5
(6)
where t ¼ 1=ð1 þ rz0 Þ, r ¼ 0.2316419, b1 ¼ 0.31938153, b2 ¼ b4 ¼ 1.821255978 and 0.356563782, b3 ¼ 1.781477937, b5 ¼ 1.330274429. The maximum error in (6) is smaller than 7.5 108 [8,12]. If z0 < 0, the probability of Q(z0) can be calculated using the symmetry of the function f(z). Using (5),
Zz
f ðsÞPðsÞds N
ZVr
Although there is no explicit analytical expression for the cumulative distribution function of the SND [8, 12], Eq. (5) can be calculated using the following polynomial approximation [8,12] for z0 0:
GðzÞ ¼
EC ¼
2 1 t pffiffiffiffiffiffi exp dt ¼ 1 Q ðzÞ 2 2p
¼
f ðsÞPr AþBsþCs
a
z ¼
(15)
s
Vci m Vr m
(17)
s
0 Vco ¼
Vco m
(18)
s
þðBs þ2C msÞzþC s2 z2 aA1 þB1 zþC1 z2
ZVr EC1 ¼ Vci0
2 1 z pffiffiffiffiffiffi exp Pr A1 þB1 zþC1 z2 dz 2 2p 0
ZVr ¼ A1 Pr Vci0
0
2
2
ZVr 1 z 1 z pffiffiffiffiffiffi exp pffiffiffiffiffiffiexp dzþB1 Pr zdz 2 2 p 2p 2 0 Vci
0
(10)
where C1 and C are the two arbitrary constants. Based on (9) and (10), the following definite integral can be obtained
ð19Þ
Substituting (19) into EC1 in (13), 0
(9)
2 2 1 z 1 z pffiffiffiffiffiffi z2 exp dz ¼ zpffiffiffiffiffiffiexp 2 2 2p 2p 2 Z 1 z dz þ C1 þ pffiffiffiffiffiffiexp 2 2p ¼ zRðzÞ þ GðzÞ þ CaWðzÞ þ C
(16)
s
AþBsþCs2 ¼AþBðzs þ mÞþCðzs þ mÞ2 ¼ AþBm þC m2
The following equations can be obtained using the indefinite integral principle and (7):
Z
(14)
sm
4.3. Analytical formulation of integral functions
2 2 1 z 1 z pffiffiffiffiffiffiz exp dz ¼ pffiffiffiffiffiffiexp þ CaRðzÞ þ C 2 2 2p 2p
(13)
Vcr
Based on Eqs. (15)e(18), we have
where a and b are the two arbitrary real numbers.
Z
f ðsÞPr dsaEC1 þEC2
dsþ
Consider the following substitutions:
Vr0 ¼
(8)
ZVco
EC ¼ ECðm; s; Vci ; Vr ; Vco Þ
In other words, G(z) in (7) is an integral primitive function of f(z) in (4). G(z) can be analytically evaluated based on (6) and (7). The definite integral is given by (8)
2 1 t pffiffiffiffiffiffi exp dt ¼ GðbÞ GðaÞ 2 2p
It can be seen from (13) that the EC of a WECS is a function of the five variables m, s, Vci, Vco and Vr, i.e.
Vci0 ¼
Zb
2
Vci
(7)
N
ZVr þC1 Pr Vci0
2
1 z pffiffiffiffiffiffi exp z2 dz 2 2p
Using (8), (10) and (12),
(20)
1986
K. Xie, R. Billinton / Renewable Energy 36 (2011) 1983e1988
EC1 ¼ A1 Pr G Vci0 G Vr0 þ B1 Pr R Vr0 R Vci0 þ C1 Pr W Vr0 W Vci0
Table 2 Wind turbine parameters.
(21)
The term EC2 in (13) can be calculated using (8):
0 EC2 ¼ Pr G Vr0 G Vco
(22)
The EC of a WTG can be calculated using (13), (21) and (22) and the analytical formulation of G(z), R(z) and W(z) in (8), (11) and (12). The equivalent capacity ratio index (ECR) of a WTG associated with a specific wind site is defined as
ECR ¼
EC Pr
(23)
Types
Cut-in speed (km/h)
Rated speed (km/h)
Cut-out speed (km/h)
A B C D E F G H I J K L
10 10 11 11 12 12 14 14 16 16 18 18
35 38 35 38 38 41 35 38 41 35 38 41
48 60 70 60 70 60 70 80 60 52 70 65
The annual equivalent utilization time index (AEUT) of a WTG associated with a specific wind site can be obtained using the ECR
5. Case studies and applications
AEUT ¼ ECR 8760
5.1. Accuracy analysis of the proposed model
(24)
The AEUT indicates the annual equivalent utilization hours of a WTG and the wind energy produced by a WTG with a per unit rated capacity. The ECR measures the electrical energy production performance or energy benefit without linking it to the rated capacity of the WTG. A high value of this index indicates good utilization of wind power. It can also be used to reflect the reliability benefits and the degree of wind site matching with the WTG. Energy production and reliability benefits of a WTG are directly related to the site ECR. Based on the previous discuss and (14), (23) and (24), the EC, ECR and AEUT indices respectively can be analytically evaluated using the m, s for a wind site and the three parameters Vci, Vco and Vr for a WTG.
4.5. Analysis of Weibull wind speed model When the wind speeds are represented by a Weibull distribution, the m and s of the Weibull distribution can be evaluated from the scale (a) and shape (b) parameters [12] as follows:
m ¼ aG 1 þ
1
(25)
b
s2 ¼ a2 G 1 þ
2
b
G2 1 þ
1
(26)
b
where G() is the gamma function which is defined by
GðxÞ ¼
ZN
t x1 et dt
(27)
The wind speed time series at Lauwersoog wind site in 2001e2007 [13] were used to verify the calculation accuracy of the proposed model. The mean and standard deviation of the statistical wind speeds are 6.23 and 3.05 m/s, respectively. A WTG with a rated power output of 1 MW is assumed to be located at a site with the Lauwersoog wind regime. The cut-in, rated and cut-out wind speeds for the WTG are 3, 15 and 22 m/s, respectively. Annual energy production using the proposed ND, Weibull distribution and statistical wind speeds are 999.75, 999.37 and 1001.81 MWh, respectively, which indicates that the normal and Weibull distributions have evaluation errors of 0.21% and 0.24%, respectively. If only the data in 2007 were used, the mean and standard deviation of statistical wind speeds are 6.67 and 3.30 m/s, respectively. The evaluated annual energy productions are 1268.44, 1248.08 and 1257.43 MWh, which indicates that the normal and Weibull distributions have evaluation errors of 0.88% and 0.74%, respectively. Although the Weibull distribution provides a more accurate evaluation of energy production, the normal distribution also provides a considerable accuracy. These results show that the proposed analytical ND model has a considerable high accuracy. 5.2. Analysis of the equivalent capacity ratio The four wind farm locations noted in Section 2 were used in the following case studies. The basic wind speed statistics for the four specific sites are shown in Table 1. Table 2 shows the 12 WTG types utilized in the case studies. The cut-in wind speed parameters range from 10 to 18 km/h, the rated speeds from 35 to 41 km/h, and the cut-out speeds from 48 to 80 km/h. Table 3 ECR for the four wind sites with different WTG.
0
Types
Similarly, the EC, ECR and AEUT indices can be analytically evaluated using the m and s for a wind site and the three parameters Vci, Vco and Vr for a WTG.
Table 1 Site wind speed statistics. Sites
Swift Current
Saskatoon
Regina
North Battleford
Mean wind speed (km/h), m Standard deviation (km/h), s
19.46 9.70
16.78 9.23
19.52 10.99
14.62 9.59
A B C D E F G H I J K L
Wind sites Swift Current
Saskatoon
North Battleford
Regina
0.2684 0.2174 0.2682 0.2159 0.2141 0.1718 0.2593 0.2088 0.1614 0.2495 0.1908 0.1530
0.1922 0.1513 0.1907 0.1499 0.1480 0.1165 0.1813 0.1426 0.1063 0.1715 0.1249 0.0983
0.2766 0.2267 0.2774 0.2253 0.2235 0.1807 0.2689 0.2184 0.1707 0.2592 0.2013 0.1627
0.1512 0.1181 0.1497 0.1169 0.1152 0.0902 0.1409 0.1103 0.0812 0.1321 0.0947 0.0743
K. Xie, R. Billinton / Renewable Energy 36 (2011) 1983e1988 Table 4 AEUT for the four wind sites with different WTG (h/year). Types
Table 6 Reliability comparison of the IEEE-RTS with different WTG types.
Wind sites
A B C D E F G H I J K L
Types ECR
Swift Current
Saskatoon
North Battleford
Regina
2352 1905 2349 1892 1875 1505 2271 1829 1414 2185 1671 1341
1684 1325 1670 1313 1296 1020 1588 1249 932 1502 1094 861
2423 1986 2430 1973 1958 1583 2356 1913 1496 2271 1763 1425
1325 1035 1311 1024 1009 790 1234 966 711 1157 829 651
The ECR and AEUT for the 12 WTG types applied to the four wind sites are shown in Tables 3 and 4. The values given in bold in Tables 3 and 4 are the highest ECR and AEUT for each wind site. It can be seen from these tables that the most suitable WTG types for the Swift Current, Saskatoon, North Battleford and Regina wind sites are A, A, C and A, respectively. The highest ECR for the four wind sites are 26.84%, 19.22%, 27.74% and 15.12%, respectively, and the highest AEUT are 2352, 1684, 2430 and 1325 h/year, respectively. It can also be seen from these tables that there is a large difference between the highest and lowest ECR or AEUT for different WTG at a wind site. Consider a wind farm at Swift Current as an example. In these cases, the maximum ECR and AEUT are 26.84% and 2352, respectively, whereas the minimum ECR and AEUT are 15.30% and 1341 respectively. These show that the energy production at a wind farm strongly depends on the WTG type. The ECR and AEUT can be used to measure the degree of site wind speed matching with the WTG. Selecting a suitable site-matching WTG for a specific wind site is therefore important in order to achieve maximum energy benefits from a WECS. 5.3. Analysis of adequacy evaluation The reliability benefit of a WECS at a specific site depends on many factors, including the wind speed distributions, the ECR of the WTG, the capacity distributions and reliability performance of traditional generating units and the load distributions. The ECR impacts of a WTG on the system reliability of power systems were investigated using the RBTS [14] and IEEE-RTS [15]. The RBTS consists of 11 traditional generating units with a total capacity of 240 MW and a peak load of 185 MW. The IEEE-RTS consists of 32
Table 5 Reliability comparison of the RBTS with different WTG types. Types ECR
Ranked by LOEE Ranked by AEUT Ranked by LOLE (h/year) the ECRa (h/year) the LOEEa (MWh/year) The LOEEa
A B C D E F G H I J K L
2352 1905 2349 1892 1875 1505 2271 1829 1414 2185 1671 1341
0.2684 0.2174 0.2682 0.2159 0.2141 0.1718 0.2593 0.2088 0.1614 0.2495 0.1908 0.153
1 5 2 6 7 10 3 8 11 4 9 12
0.612 0.674 0.615 0.677 0.682 0.739 0.637 0.695 0.765 0.662 0.741 0.787
1 5 2 6 7 9 3 8 11 4 10 12
5.465 6.023 5.489 6.053 6.094 6.610 5.697 6.218 6.857 5.928 6.653 7.063
1987
1 4 2 5 7 9 3 8 11 6 10 12
a The ECR indices are ranked in using decreasing values order; the LOLE and LOEE indices are ranked in using increasing values order.
A B C D E F G H I J K L
Ranked by the ECR
0.2684 1 0.2174 5 0.2682 2 0.2159 6 0.2141 7 0.1718 10 0.2593 3 0.2088 8 0.1614 11 0.2495 4 0.1908 9 0.153 12
LOLE (h/ Ranked by year) the LOLE
LOEE (MWh/ year)
Ranked by the LOEE
4.72 5.19 4.74 5.22 5.25 5.69 4.91 5.35 5.89 5.09 5.71 6.06
541.15 596.99 543.47 599.95 604.08 655.67 564.17 616.35 680.21 587.00 659.45 700.65
1 5 2 6 7 9 3 8 11 4 10 12
1 5 2 6 7 9 3 8 11 4 10 12
traditional generating units with a total capacity of 3405 MW and a peak load of 2850 MW. The generating unit ratings and reliability parameters are shown in [14,15]. The hourly per unit chronological load demand of the IEEE-RTS [15] was used in both test systems. A wind farm with 20 WTG units is assumed to be located at a site with the Swift Current wind regime and added to the RBTS. Each WTG unit has a rated power output of 2 MW. The WTG types in Table 2 are used in these studies. Algorithms for modeling the wind farm multi-state capability distributions given in references [6,9] were also used. Each traditional generating unit is represented using a two-state model (up and down states), and each WTG is represented using a five-state model. System states were enumerated by considering up to the fifth contingency level, i.e., up to five units in an outage or derated event. The RBTS indices obtained using the analytical enumeration approach are shown in Table 5. The WTG types are ranked according to the ECR or AEUT of the WTG associated with the Swift Current wind site. It is obvious that the ECR and AEUT have the same order due to having a constant multiplier between the two indices. It can be seen from Table 5 that the ECR and the reliability indices, such as LOLE or LOEE, mainly have the same order for the WTG. In general, the higher the ECR of a WTG is, the higher is the reliability benefit. Some orders of the ECR in Table 5, such as those for types F and K, are not consistent with the reliability index order due to the different wind speed distributions. One conclusion that can be drawn from Tables 3 and 5 is that the ECR provides a useful index for evaluating the energy production of a WTG and the potential power system reliability benefit. The ECR can also be used to judge the degree of a WTG matching with a wind site. In other words, the ECR can be used to assist in the determination of the optimal WTG type for a specific wind site. A wind farm with 250 WTG units and the Swift Current wind regime was added to the IEEE-RTS. The rated capacity and the WTG types are the same as those used in the previous section. It can be seen from Table 6 that the ECR and the reliability indices generally exhibit the same order for the WTG. Table 6 indicates that similar conclusions to those obtained from the RBTS study can be drawn for the IEEE-RTS. 6. Conclusion The contributions in terms of energy production and reliability benefits of including wind energy conversion systems (WECS) in power systems are dependent on a wide range of factors including the wind speed characteristics and the WTG design parameters, such as the cut-in, rated and cut-out wind speed. The degree of the
1988
K. Xie, R. Billinton / Renewable Energy 36 (2011) 1983e1988
wind site matching with the WTG has a significant impact on the electrical energy produced by a WECS and the power system reliability benefits. The equivalent capacity ratio (ECR) of a WTG is introduced to quantify the electrical energy production performance or energy benefit, the reliability benefit, and the degree of wind site matching with a WTG. The ECR is formulated as the expected value of the power output random variable and the probability density function of the site wind speed. This index can assist system planners and utility managers to assess the energy and reliability worth of a WECS and add useful information to the managerial decision process. The ECR index can be extended to indicate the amount of annual equivalent utilization time (AEUT) of a WTG. The major advantage of the proposed method is that probability and integral theories are used to mathematically derive the analytical formulation of the ECR and AEUT. Once the parameters of the wind speed at a site (mean and standard deviation), the WTG parameters (the cut-in, rated and cut-out wind speed) are available, the ECR and AEUT can be directly and analytically evaluated. The impacts of different WTG design parameters on the energy and reliability benefits associated with a specific wind site are illustrated in this paper. The case studies show that the ECR provides a direct and physical indication of the energy benefit for a WECS, and an estimation of its contribution to the reliability of a power system. The ECR index can be used to judge the degree of a WTG matching with a wind site and assist in the determination of the optimal WTG type for the potential site. Significant energy and reliability benefits can be obtained using the proposed technique to select appropriate site-matching WECS. Acknowledgements This work was supported in part by the National Natural Science Foundation of China (No. 51077135), Natural Science Foundation Project of CQ CSTC (No. CSTC2010BA3006) and Scientific Research
Foundation of Stage Key Lab. of Power Transmission Equipment and System Security (No. 2007DA10512709103).
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