Modeling Uncertainties in Power System by Generalized Lambda Distribution

Qing Xiao

doi:10.1515/ijeeps-2013-0152

Article

Modeling Uncertainties in Power System by Generalized Lambda Distribution

Qing Xiao

Published/Copyright: March 22, 2014

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From the journal International Journal of Emerging Electric Power Systems Volume 15 Issue 3

Abstract

This paper employs the generalized lambda distribution (GLD) to model random variables with various probability distributions in power system. In the context of the probability weighted moment (PWM), an optimization-free method is developed to assess the parameters of GLD. By equating the first four PWMs of GLD with those of the target random variable, a polynomial equation with one unknown is derived to solve for the parameters of GLD. When employing GLD to model correlated multivariate random variables, a method of accommodating the dependency is put forward. Finally, three examples are worked to demonstrate the proposed method.

Keywords: uncertainty; generalized lambda distribution; probability weighted moment; correlated random variables

Appendix A

The parameters ci (c=0,…,11) are as follows:

(34)c11=(B−C)(3B2−BC−2AC)(A−3B+2C)c10=4A4−28A3B+12A3C+71A2B2−90A2BC+47A2C2−29AB3127AB2C−213ABC2+91AC3−132B4+311B3C−212B2C2+41BC3c9=36A4−312A3B+156A3C+921A2B2−1014A2BC+401A2C2−864AB3+1,723AB2C−1,974ABC2+831AC3−627B4+1,979B3C−1589B2C2+333BC3c8=56A4−896A3B+612A3C+3634A2B2−4034A2BC+1,168A2C2−5,458AB3+9,254AB2C−8,112ABC2+3,528AC3+108B4+3,828B3C−4,888B2C2+1,200BC3c7=−672A4+3,576A3B−696A3C−4,830A2B2+3,726A2BC−3,968A2C2−7,305AB3+11,952AB2C−2,757ABC2+2,958AC3+12,213B4−18,864B3C+4,535B2C2+132BC3c6=−4,236A4+33,276A3B−13,284A3C−90,969A2B2+91,218A2BC−43,565A2C2+65,883AB3−101,553AB2C+121,815ABC2−40,977AC3+48,780B4−142,041B3C+92,516B2C2−16,863BC3c5=−10,524A4+100,800A3B−44,988A3C−337,455A2B2+350,990A2BC−155,035A2C2+368,022AB3−573,617AB2C+581,208ABC2−217,705AC3+78,723B4−424,845B3C+360,205B2C2−75,779BC3c4=−13,456A4+151,744A3B−70,764A3C−606,848A2B2+651,402A2BC−290,770A2C2+838,004AB3−1,341,124AB2C+1,341,342ABC2−543,922AC3+30,012B4−745,154B3C+776,776B2C2−177,242BC3c3=(−4)(2,98A4−29,616A3B+13,962A3C+138,963A2B2−154,935A2BC+75,310A2C2−234,204AB3+395,191AB2C−429,195ABC2+193,482AC3+8,427B4+221,852B3C−264,859B2C2+63,324BC3)c2=(−16)(194A4−2,750A3B+1,266A3C+14,614A2B2−17,420A2BC+10,392A2C2−27,982AB3+53,159AB2C−73,125ABC2+38,442AC3−2,772B4+50,271B3C−58,209B2C2+13,920BC3)c1=(−96)(4A4−58A3B+24A3C+320A2B2−480A2BC+460A2C2−479AB3+1,551AB2C−3,862ABC2+2,460AC3−1,152B4+4,938B3C−4,830B2C2+1,104BC3)c0=1,152(B−2C)(BC−3B2+2AC)(A−6B+6C)

Appendix B

Below are the procedures of evaluating u for a given value of x of GLD:

Determine the search interval u∈[a,b], set a=0, b=1. Give the precision δ.
Denote u∗=(a+b)/2, if (b−a)<δ, stop and output u=u∗, otherwise evaluate x∗: x∗=λ1+u∗λ3−(1−u∗)λ4λ2.
If u∗=u, stop and output u=u∗, otherwise set a=a, b=u∗ for x∗>x, set a=u∗, b=b for x∗<x, and go to Step 2.

Appendix C

Suppose x∗ is a random variable with zero mean and unit variance, it can be represented by Cornish–Fisher expansion as follows:

(35)x∗=z+16(z2−1)κ3+124(z3+3z)κ4−136(2z3−5z)κ32+1120(z4−6z2+3)κ5−124(z4−5z2)κ2κ3+132412z4−53z2κ33

where z is a standard normal random variable with CDF Φ(z), κi denotes the ith cumulant of x (i=1,…,5).

Rewrite the Cornish–Fisher expansion in the form of a fourth polynomial of z:

(36)x∗=b4z4+b3z3+b2z2+b1z+b0

where bi (i=0,…,4) are:

(37)b4=1120κ5+127κ33−124κ3κ4b3=124κ3−118κ32b2=−120κ5+16κ3+524κ3κ4−53324κ33b1=536κ32−18κ4+1b0=140κ5−16κ3−112κ3κ4+17324κ33

Then, for a random variable x with mean μ and standard deviation σ, it can be represented by:

(38)x=μ+σx∗=μ+σ∑k=04bkzk=∑k=04akzk

where ak=σbk (k=1,2,3,4), a0=μ+σb0.

For a given percentage, the associated percentile xp can be calculated as follows:

(39)xp=∑k=04akzpk zp=Φ−1(p)

where Φ−1(⋅) is the inverse CDF of z.

Then, the PDF of x can be constructed with the aid of the PDF of z:

(40)f(xp)=φ(zp)∑k=14kakzpk−1

where φ(⋅) is PDF of the standard normal variable.

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Published Online: 2014-3-22

Published in Print: 2014-6-1

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Articles in the same Issue

https://doi.org/10.1515/ijeeps-2013-0152

Keywords for this article

uncertainty; generalized lambda distribution; probability weighted moment; correlated random variables