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TABLE II
COMPARISON OF FUNCTIONS BETWEEN LSM AND HUBER
increased level of confidence in each of the estimates, in spite of structural imperfections in H .
V. CONCLUSIONS
TABLE III
STATISTICAL TEST RESULTS FOR TWO CASE STUDIES
The Huber function technique is demonstrated for a robust estimation application in power engineering. A novel statistical test (see
Table I) is presented to examine structural imperfections in the process
matrix. The results offer an improvement over least-squares estimation in cases where the process matrix is characterized by structural
imperfections.
REFERENCES
TABLE IV
LSM AND HUBER ESTIMATES FOR SYNCHRONOUS GENERATOR PARAMETERS
synchronous generator parameters using actual data. Usually, available
data for synchronous generators are the stator phase currents and voltages at the terminals of the machine as well as the field voltage and
current. Real data may be contaminated due to meter and communication errors, incomplete metering, or inaccuracy of metering equipment.
Further, model selection and specifications may introduce structural defects in the process matrix.
Actual data from a synchronous generator are considered. The
generator is located in the southwest USA and is rated 213.7 MVA and
18 kV. The linearized generator model derived in [4] is used in this
example. The least-squares method has provided satisfactory estimates
of the mutual inductances LAD and LAQ and the field resistance
rF . However, when the above parameters and the stator resistance
r are estimated simultaneously, unrealistic values of r are obtained
due to multicollinearity in H . This is evidenced by the results of the
statistical test described in Section II and shown in Table III.
The results of the statistical test show a “not high” statistical confidence for both cases. An examination of the flags raised by the test
indicates that there are problems associated with multicollinearity in H
(VIF flag), the condition number of H (eigenanalysis flag), and change
in the confidence interval (Willan–Watts test flag). Table IV depicts the
least-squares and Huber estimates of the synchronous generator parameters for both cases considered in Table III. For the Huber method, a
bisectional search was used to find the optimum value of t. In this case,
t = 2:1 yields optimum results.
Table IV shows that even though statistical confidence is “not high”
in case 1, the estimated parameters are satisfactory. The Huber technique does not produce estimates that are significantly different from
the least-squares technique. This result is coincidental; it is generally
advised to use robust estimation methods when the statistical confidence in the estimates is not high. This becomes evident in case 2,
where four parameters are estimated simultaneously; a significant decrease in the percent deviation from manufacturer data is observed
when robust estimation is used. Perhaps the most important benefit
of this method is that the estimated parameter r is no longer negative
(unrealistic for this application). The robust estimation results offer an
[1] D. C. Montgomery, E. A. Peck, and G. G. Vining, Introduction to Linear
Regression Analysis. New York: Wiley, 2001.
[2] L. Mili, G. Steeno, F. Dobraca, and D. French, “A robust estimation
method for topology error identification,” IEEE Trans. Power Syst., vol.
14, no. 4, pp. 1469–1476, Nov. 1999.
[3] S. Suryanarayanan, “Accommodation of loop flows in competitive electric power systems,” Ph.D. dissertation, Dept. Elect. Eng., Arizona State
Univ., Tempe, 2004.
[4] E. Kyriakides, G. T. Heydt, and V. Vittal, “On-line estimation of synchronous Generator parameters using a damper current observer and a
graphic user interface,” IEEE Trans. Energy Convers., vol. 19, no. 3, pp.
499–507, Sep. 2004.
Introduction to Cumulant-Based Probabilistic Optimal
Power Flow (P-OPF)
Antony Schellenberg, William Rosehart, and José Aguado
Abstract—This letter introduces the cumulant method for the probabilistic optimal power flow (P-OPF) problem. The proposed adaptation of
the cumulant method to the P-OPF problem is given.
Index Terms—Cumulants,
optimization.
optimal
power
flow,
probabilistic
I. INTRODUCTION
Optimal power flow (OPF), which is a mathematical programming
extension of the power flow problem, is a tool that has been commonly
used within the power systems industry for many years [1]. In addition,
many papers have been published on the subjects of probabilistic
power flow (P-PF) [2] and probabilistic optimal power flow (P-OPF)
[3], [4].
The goal of the P-OPF problem is to determine the probability density functions (PDFs) for all unknown variables present in the problem.
These PDFs are the distributions of the optimal solutions. This letter
proposes the adaptation of the cumulant method used in P-PF studies
[5]–[7] to the P-OPF problem based on a logarithmic barrier interior
Manuscript received March 17, 2004; revised June 18, 2004. Paper no. PESL00032-2004.
A. Schellenberg and W. Rosehart are with the University of Calgary, Calgary,
AB T2N 1N4 Canada (e-mail: schellen@enel.ucalgary.ca; rosehart@enel.ucalgary.ca).
J. Aguado is with the University of Malaga, 29071 Malaga, Spain (e-mail:
jaguado@uma.es).
Digital Object Identifier 10.1109/TPWRS.2005.846188
0885-8950/$20.00 © 2005 IEEE
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 2, MAY 2005
1185
point method (LBIPM) [8]-type solution when random bus loading is
considered.
II. PROBABILITY AND STATISTICS BACKGROUND
The cumulant method is based on a statistical measure known as
cumulants. Therefore, some background information in probability and
statistics is presented to provide a basis for the cumulant method.
The expected value of a random variable x is defined as
E [x ] =
1
01
xfx (x)dx
(1)
where fx (x) is the PDF of x.
The nth-order raw moment mn is defined in the following manner:
m n = E [x n ] :
the moment generating function, 8z (s), for the random variable z can
be written as follows:
8z (s) = E [esz ]
= E [es(a x
= E [es(a x
+a
)
e
x
+
s(a x
111+a
)
x
...e
)
]
s(a x
(7a)
)
]:
Since x1 ; x2 ; . . . ; xn are independent
8z (s) = E [es(a x ) ]E [es(a x ) ] . . . E [es(a x ) ]
= 8x (a1 s)8x (a2 s) . . . 8x (an s):
(3)
The nth raw moment is computed from the moment generating function by taking the nth derivative with respect to s and evaluating at
s = 0. For example, the third raw moment can be computed as follows:
d 3 8 x (s )
ds3
3
sx
= d Eds[e3 ]
3 sx
=E ddse3
m3 =
9z (s) = ln(8z (s))
= ln(8x (a1 s)8x (a2 s) . . . 8x (an s))
= ln(8x (a1 s)) + ln(8x (a2 s))
+ + ln(8x (an s))
= 9 x (a 1 s ) + 9 x (a 2 s ) + + 9 x (a n s ):
111
s=0
s=0
The cumulant generating function [9], denoted by 9x (s), can be
written in terms of the moment generating function 8x (s) as follows:
(5)
The cumulant generating function is employed in the same manner as
the moment generating function; successive derivatives are taken with
respect to s and evaluated at s = 0. The nth cumulant is denoted as
n .
(9b)
(9c)
(9d)
(10)
111
(11)
111
(12)
Evaluating (12) at s
(4)
(9a)
The zero-, first- and second-order cumulants for a random variable
9 z (s ) = 9 x (a 1 s ) + 9 x (a 2 s ) + + 9 x (a n s )
9z0 (s) = a1 90x (a1 s) + a2 90x (a2 s)
+ + an 9x0 (an s)
00
9z (s) = a21 9x00 (a1 s) + a22 9x00 (a2 s)
+ + a2n 9x00 (an s):
s=0
9x (s) = ln 8x (s):
(8b)
z are computed as
111
= E x3 esx s=0
= E [x 3 ]:
(8a)
The cumulants for the variable z can be computed using the cumulant generating function (5) in terms of the component variables as
follows:
111
8x (s) = E [e ]:
(7c)
(2)
It is possible to compute the raw moments through the use of the
moment generating function 8x (s) [9]. Mathematically, this function
is stated as
sx
(7b)
= 0 gives
9z00 (0) = a21 9x00 (0) + a22 9x00 (0) +
111
+ a2n 9x00 (0):
(13)
It is noted that there is no limitation to second-order cumulants. Therefore, third and higher order cumulants can be computed following the
same procedure. In general, the nth-order cumulant for z, which is
a linear combination of independent random variables, can be determined with the following equation:
n = 9z(n) (0)
= an1 9x(n) (0) + an2 9x(n) (0) + 1 1 1 + ann 9x(n) (0)
(14a)
(14b)
where the exponent (n) denotes the nth derivative with respect to s.
III. BASIC CUMULANT METHOD DERIVATION
IV. ADAPTATION TO P-OPF
Given a new random variable z, which is the linear combination of
independent random variables, x1 ; x2 ; . . . ; xn
The cumulant method is adapted from the basic derivation above
to accommodate the P-OPF problem when an LBIPM-type solution is
used. The Hessian of the Lagrangian is necessary for the computation
of the Newton step in the LBIPM. The inverse of the Hessian, however,
can be used as the coefficients for the linear combination of random
z = a 1 x1 + a 2 x2 + 1 1 1 + an xn
(6)
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 2, MAY 2005
variables around the current operating point. The pure Newton step is
computed at iteration k of the LBIPM using the following equation:
H (yk )1yk
= 0F (y )
(15)
k
( )
1
where y is the vector of system variables, H yk is the Hessian of the
Lagrangian function evaluated at the point yk , yk is the pure Newton
step, and F yk is the mismatch vector evaluated at yk . Replacing yk
with yk+1 0 yk in (15) and rearranging yields
( )
1
yk+1 = 0H 01 (yk )F (yk ) + yk :
(16)
The similarity between (16) and a general linear equation is noted.
Therefore, the matrix H 01 yk , which is the inverse Hessian, in (16)
contains the multipliers for a linear combination of PDFs. This can be
used to map the PDFs from known random variables into unknown
random variables. In the case of random bus loading, a change in the
bus load maps directly into a change in the mismatch vector. Therefore,
the linear mapping information contained in the inverse Hessian can be
used to determine cumulants for other variables when bus loading is
treated as a random variable. If the multipliers in row i of the inverse
; ai;n , then the v th cumulant for the ith variHessian are ai;1 ; ai;2 ;
able in y is computed using the following equation:
( )
...
yi;v
=a
1 x
v
i;
;v
+a
2 x
v
i;
;v
+ +a
111
v
i;n
x
;v
REFERENCES
[1] M. Huneault and F. Galiana, “A survey of the optimal power flow literature,” IEEE Trans. Power Syst., vol. 6, no. 2, pp. 762–770, May 1991.
[2] M. T. Schilling, A. L. da Silva, R. Billinton, and M. El-Kady, “Bibliography on power system probabilistic analysis (1962–98),” IEEE Trans.
Power Syst., vol. 5, no. 1, pp. 1–11, Feb. 1990.
[3] M. Madrigal, K. Ponnambalam, and V. H. Quintana, “Probabilistic optimal power flow,” in IEEE Canadian Conf. Elect. Comput. Eng., vol. 1,
May 1998, pp. 385–388.
[4] G. Viviani and G. Heydt, “Stochastic optimal energy dispatch,” IEEE
Trans. Power App. Syst., vol. PAS-100, no. 7, pp. 3221–3228, Jul. 1981.
[5] W. Tian, D. Sutanto, Y. Lee, and H. Outhred, “Cumulant based probabilistic power system simulation using Laguerre polynomials,” IEEE
Trans. Energy Convers., vol. 4, no. 4, pp. 567–574, Dec. 1989.
[6] J. Stremel, R. Jenkins, R. Babb, and W. Bayless, “Production costing
using the cumulant method of representing the equivalent load curve,”
IEEE Trans. Power App. Syst., vol. PAS-99, no. 5, pp. 1947–1956,
Sep./Oct. 1980.
[7] P. Zhang and S. T. Lee, “Probabilistic load flow computation using the
method of combined cumulants and Gram–Charlier expansion,” IEEE
Trans. Power Syst., vol. 19, no. 1, pp. 676–682, Feb. 2004.
[8] G. Torres and V. Quintana, “An interior-point method for nonlinear optimal power flow using voltage rectangular coordinates,” IEEE Trans.
Power Syst., vol. 13, no. 4, pp. 1211–1218, Nov. 1998.
[9] A. Papoulis and S. Pillai, Probability, Random Variables, and Stochastic
Processes, 4th ed. New York: McGraw-Hill, 2002.
(17)
where yi is the ith element in y , and x ;v is the v th cumulant for the
j th component variable. During an iteration of the cumulant method
for P-OPF, the cumulants for unknown random variables are computed
from known random variables. Next, PDFs are reconstructed using the
Gram–Charlier/Edgeworth Expansion theory [7]. A statistical step is
computed and combined with the pure Newton step in the LBIPM. The
statistical step is computed by examining the system variable’s PDFs
to increase the probability of occurrence.
V. CONCLUSIONS
This letter introduces the cumulant method for the P-OPF problem.
By noting that the inverse of the Hessian used in the logarithmic barrier interior point can be used to perform linear mapping, cumulants
can be computed for unknown system variables. The distributions are
reconstructed from the PDFs, and a statistical step is computed. The
statistical step is combined with the pure Newton step to determine a
new step that attempts to increase the likelihood of occurrence of a solution while retaining the convergence properties associated with the
pure Newton step.
Although results are not included in a tabular form due to space limitations, numerical results using this approach support its validity. Results from the proposed approach are compared against Monte Carlo
simulations in a nine-bus system. The mean values computed are well
within 1% of the Monte Carlo result, and the variances are within 6%.
As proposed here, the method makes the assumption that bus loads
are independent. However, results can be modified to include covariances between two random variables.
The method proposed in this letter is different from existing methods
since it uses cumulants rather than moments for random variable information in a P-OPF setting. In addition, it makes use of the Hessian
of the Lagrangian directly without any additional sensitivity/derivative techniques. Finally, the proposed method makes additional use of
statistical information in step determination without including random
variable information directly in the system equations.
A Novel Approach to Estimate Load Factor of
Variable-Speed Wind Turbines
D. D. Li and C. Chen, Senior Member, IEEE
Abstract—This paper proposes an approach to estimate the load factor
for variable-speed wind turbines, which is an index of high interest for
economic evaluation of the wind power generation systems. The proposed
method is based on the statistical discipline of the wind resource and shorttime performance of wind turbines. Calculation results are given for a case
study.
Index Terms—Load factor, spectral analysis, statistics, wind energy.
I. INTRODUCTION
Due to changeability of the wind energy, a wind power generation
system does not always operate at its rated condition, leading to inconvenience for calculation of actual output of the system. As the energy product is an important index for economic evaluation of the wind
power generation systems, efforts on this topic have been reported in
the literature [1]–[4]. Efficiencies of the electrical part of different wind
power generation systems are examined with a statistical method in [1].
The statistical method is also used to study the annual production of
wind energy systems [2]. Energy capture between various types of systems is compared based on a particular short-time wind speed series in
[3]. Calculation of capacity credit of wind energy conversion systems
is performed with chronological and probability methods in [4]. This
paper will focus on the load factor of variable-speed wind turbines in
Manuscript received April 7, 2004. This work was supported in part by the
Alliance for Global Sustainability. Paper no. PESL-00041-2004.
The authors are with the Department of Electrical Engineering, Shanghai
Jiao Tong University, Shanghai, China (e-mail: sjtuldd@yahoo.com.cn;
chchen@online.sh.cn).
Digital Object Identifier 10.1109/TPWRS.2004.841154
0885-8950/$20.00 © 2005 IEEE