Technical Paper Summaries - IEEE Xplore

Technical Paper Summaries

IEEE Transactions on Energy Conversion Energy Development and Power Generation capacities less than x#, i.e.,

89 WM 234-6 December 1989

Probabilistic Power System Production Cost and Reliability Calculation by the Z-Transform Method D. Sutanto, Senior Member, IEEE and H. R. Outhred, Member, IEEE University of New South Wales P.O. Box 1, Kensington 2033 Australia Y. B. Lee China Light and Power, Hong Kong

(LOLP)j = a,j

g , i=1

Ai

where xg < x,j c Xg+1. The total LOLP for the period is the sum of the LOLPs associated with each load impulse. T

(LOLP)period = E (LOLP),

where T is the total number of load impulses for the period. The EUE for the load a,j(x,j) is equal to the energy demanded less the energy supplied and is given by g

This paper describes a new method of probabilistic power system production cost and reliability calculation using the z-transform.

Proposed Method In the proposed method, a generating unit j is represented by an available capacity probability density function (pdf) pj: =

-

* *,

ajI(xj,)}

(1)

where aj; is the probability of unit j being available with capacity xj; (i 1, * *, n). xj; is an integer with xj; < xjk if i < k. The load for a study period is represented by a pdf

alT,(XIT)}

pl(x)={all(xl1), a12(X12),

(2)

where a,, is the probability of a load level x,; such that x,; < xlk if i < k and T is the number of separate load levels in the period. The equivalent available capacity pdf of the first k units in the loading order is formed by convolution of the unit pdfs,

Pk(X) =pl (X)*p2(X)*

.

*

pk(X)

(3)

The convolution is effected by the z-transform and the resulting pdf is represented by

Pk(x) ={A,(xi), *-

-

Ag(x,),

(7)

j=1

Introduction

pj(x) {aj1 (xj1),

(6)

*-

-

An(Xn)l

9

n

i=1

i=g+1

Ai

By calculating Pk,,(x) and Ek+1 in a similar manner, the expected energy contribution from unit (k + 1) is given by (Ek+1 - Ek). This approach is used to calculate the expected energy contribution of each unit towards supplying this particular load level. The whole process can be repeated for all load levels to obtain expected energy contributions by all units for the whole simulation period. The expected cost of production is calculated by costing the energy produced by each unit. For calculation of the loss of load probability (LOLP) and expected unserved energy (EUE), assume that the system has k units and the equivalent available capacity pdf of all the units is given by (4). For the load represented by a,j(x,j), the probability of it being not fully supplied is equal to the sum of the probabilities associated with available IEEE Power Engineering Review, December 1989

(8)

i= 1

The total EUE for the period is the sum of the EUEs associated with each load impulse. Algorithms for efficiently calculating production cost and reliability curves and incremental generation costs are also derived and are given in the paper. Multiple block representation of a generating unit is a special case in which the available capacity pdfs of the blocks are not independent. Two methods are proposed in the paper for evaluating the equivalent available capacity pdfs for units with multiblock representation.

Results of Simulation Trials A comparison was made of the proposed method with the cumulant method and traditional direct convolution procedure for the IEEE Reliability Test System. When compared with the cumulant method, results show that the unit energy contributions are in close agreement. The accuracy of the proposed method in calculating loss of load probability and expected unserved energy is however, superior. When compared to the direct convolution procedure, the results are identical. The computing time of the proposed method is about 3 times that of the cumulant method, one-sixth of that of the direct convolution method and about one-quarter of that of an hourby-hour simulation method. Discusser: J. K. Delson

(4)

where xi < xj for i < I. The expected energy contribution by the first k units to a Ikoad represented by alj(xlj) such that xg < xlj s xg+1 is given by

Ek-=alyA ixi+aljxj F

= al IAi(x1-xi) EUiE

89 WM 235-3 December 1989

Cumulant Based Probabilistic Power System Simulation using Laguerre Polynomials W. D. Tian EPRI, Beijing, China D. Sutanto, Senior Member, IEEE, Y. B. Lee, and H. R. Outhred, Member, IEEE The University of New South Wales, P.O. Box 1, Kensington 2033, Australia Introduction This paper describes a new cumulant method of probabilistic power system simulation using the Laguerre polynomial expansion, and compares its performance with the more commonly used Hermite

expansion.

33

Proposed Method In

the

proposed method of simulation,

outage probability density functions (pdf's)

the

following Laguerre polynomial expansion

f(x)=e-x

,

are

load

and

generator

represented by the

CrL,(x).

(1)

r=o

In equation (1), f(x) is the pdf of the random variable x and L,(X) is the

rth Laguerre polynomial. The Laguerre polynomial, L,(x), on the interval (0, m) with to the weight function e-x is defined as follows: ex

L, (x)

respect

d n (e -Xxn)

dx

nl

(2)

Xn =2

j=0

A (n, j)xi

(3)

where A (n, i)

(- 1 )

(n-j)!

The coefficients C, in equation (1) Cr=

0

are

(j!)2

Conclusion

given by

f (x) L,(x) dx

=, A (r, j)tij

(4) (5)

j=0

where iz is the jth order moment of f (x) about zero. The basic procedure of the cumulant method is to calculate the cumulants of the load and generator outage pdf's, convolve them by addition and obtain the equivalent load pdf from the convolved cumulants. For the calculation of moments and cumulants, a recursive relationship between the moments and cumulants of any order of a pdf is used. Given the equivalent load pdf, f(x), the loss of load probability (LOLP) is given by LOLP = e-x {1

The expected unserved

-

energy

EUE = He-X {1 -Ci( +x)+

z Cr[Lr,_ (x) - Lr(X)]}

(6)

(EUE) is calculated by

Cr[L, 2(x)-2Lr 1(x)+L,(x)]}

A new cumulant method has been developed based on the representation of pdfs by the Laguerre polynomial expansion and a recursive relationship between cumulants and moments. The proposed method has been implemented and tested using the data of an existing medium-sized power system. Test results indicate that the proposed method is more accurate than the existing method in which pdfs are represented by the Hermite polynomial expansion, especially in the calculation of LOLP and EUE. The CPU time required by the proposed method is comparable to that of the Hermite expansion when the same number of cumulants are used. The Laguerre method, however, requires less CPU time to achieve the same accuracy of simulation because of its better convergence property. Discusser: E. Breitenberger

88 WM 242-0 December 1989

Fire Protection Wrapped Cable Tray Ampacity (7)

where H is the load duration in hours. It is noted that evaluation of the LOLP and EUE by (6) and (7) respectively does not require numerical integration. The energy contribution of a generating unit (or block capacity) is calculated by taking the difference between the EUE before and after the unit (or block capacity) is loaded.

Test Results The cumulant method based on the Laguerre expansion has been implemented and tested on a VAX-1 1/780 computer using data for the IEEE Reliability Test System. Tests were carried out to evaluate the accuracy of the Laguerre expansion to fit the load pdf of the test system. The results were compared with those of the Hermite expansion. Results show that the accuracy of the two expansions is comparable when the same number of cumulants are used and that the accuracy is improved by using more cumulants. Tests were also carried out to compare the accuracy of the Laguerre and Hermite expansions to fit the equivalent load pdf. Results indicate that whilst they afford comparable accu-

34

racy in fitting the overall equivalent load pdf, the Laguerre expansion achieves significantly higher accuracy in fitting the tail of the pdf. Studies were made of the accuracy in the calculation of LOLP and EUE using the two methods. In order to study the convergence properties of the two methods, up to 50 cumulants were used. Results reveal that the Laguerre expansion exhibits faster convergence than the Hermite expansion with respect to the number of cumulants employed. Significantly higher accuracy is achieved by the Laguerre expansion using the same number of cumulants. The accuracy of the two methods in the calculation of expected generating unit energies has also been assessed. Test results indicate that the accuracy of the two methods is comparable for the same number of cumulants, and that accuracy increases with the number of cumulants used. The computation time of the two methods has been investigated for calculating the generating unit energies, LOLP and EUE when different number of cumulants are used. Results show that the CPU time increases approximately linearly with the number of cumulants employed. It is also observed that the CPU time required by the Laguerre and Hermite expansions are comparable for the same number of cumulants. As the convergence of the Laguerre expansion is faster than that of the Hermite expansion, especially in the calculation of LOLP and EUE, less CPU time is required with the Laguerre expansion to achieve the same level of accuracy.

Phil Save, Member, IEEE and Gary Engmann, Senior Member, IEEE Southern California Edison Company This paper presents a method of calculation of ampacity of cables in fire protection wrapped cable tray. The method, based on accepted heat transfer analysis, is an extension of a method presented in References (1) and (2). Using a computerized iterative solution, the method was applied to typical configurations. The results closely verified several manufacturers tests (3M and TSI). More generally, it was found that the Cable derating: 1. Is mostly

a function of the fire wrap material and thickness (the emissivity e and the factor z/k, where z is the total wrap thickness (top and bottom) and k the thermal conductivity). 2. Is not significantly affected by the cable mass thickness. 3. Is independent of the loading depth of the cable tray.

Based on these findings Fig. 1 is proposed to provide a practical and simple determination of the cable derating due to fire wrap. Fig. 1 consists of three curves representing the percent correction factors to be applied to the ICEA P-54-440 ampacity tables, as a function of the factor z/k (ranging from 0.05 to 1.40 C x m2/w), respectively for outside emissivity eo of 0.9, 0.5, and 0.1.

IEEE Power Engineering Review, December 1989