Topological analysis in bulk power system reliability evaluation

1, February 1997 Topological Analysis in Bulk Power System Reliabil Power Systems Research Group University of Saskatchewan Saskatoon, Saskatchewan Canada 456 Abstract-This paper presen

IEEE Transactions on Power Systems, Vol 12, No 1, February 1997 Topological Analysis in Bulk Power System Reliabil

Power Systems Research Group University of Saskatchewan Saskatoon, Saskatchewan

Canada

456

Abstract-This paper presents a new computationally efficient

methodology for calculating the reliability indices of a bulk

power system using the state enumeration approach The

proposed approach utilizes topological analysis to determine

the contribution of each system state to the frequency and

duration indices at both the system and the bus level In this

approach, there is no need to conduct additional adequacy

evaluations or utilize an optimization technique to identify the

boundary wall between success and failure states The

effectiveness of the proposed methodology is demonstrated by

application to the IEEE Reliability Test System (IEEE-RTS)

and a model of the existing network of SaskPower in Canada

The proposed approach can also be used for non-coherent

systems,

I INTRODUCTION

There are two fundamental approaches to composite

system reliability evaluation: analytical methods and Monte

Carlo simulation Both techniques have their merits and

have received considerable attention over the last two

decades[l-6] A wide range of reliability indices are used in

bulk power system planning There is a strong interest i n the

calculation of frequency and duration indices in composite

power systems [7-IO] Computation of accurate estimates of

actual frequency, duration and frequency related indices is a

difficult problem as it involves recognition of all the no-load

curtailment states that can be reached from a failure state in

one transition In the absence of an efficient procedure [8-

lo], there exists a requirement of n+l adequacy evaluations,

corrective actions etc, for a failure state with n components,

each of which is modeled by two states, to obtain the actual

frequency contribution of the state to the failure frequency

index Application of tree and subtree structures in the

reliability evaluation of bulk power systems were first used in

[6,7] An efficient calculation of upper and lower bounds of

system frequency and unavailability is presented in [6,7]

using a binary tree approach Reference 6 also extends the

application of binary trees to obtain the bounds of vanous

reliability indices[6] An approach is given in [8] to

calculate accurate estimates of frequency, duration and

related indices using a Monte Carlo approach

96 WM 313-7 PWRS A paper recommended and approved by the IEEE

Power System Engineering Committee of the IEEE Power Engineering

Society for presentation at the 1996 IEEEiPES Winter Meeting, January 21-

25, 1996, Baltlmore, MD Manuscript submitted December 16, 1994; made

available for printing December 5, 1995

Reference 9 presents an approach to duration indices using an optimization procedure to determine the boundary

states and performing Reference 10 provides an conditional probability approach which can be used for both analytic

methods

The problem of calculating acc duration indices can be

states, folIowed by an e ch for the boundary crossing neighboring states that can be reached from a given contingency state, and finally p

calculations to obtain the indices This inefficient as it

computer memory

to efficiently obtain all the acyclic sub graph This paper utilizes topologica system frequency, duratio

in composite system a methodology which can a given set of contingenc indices A basic assum evaluation of bulk power systems is coherency This paper also illustrates how to consider non-coherent systems using the topological approach The efficiency of the method is demonstrated with examples and application to the IEEE -

RTS [2,4], RBTS [15] and the SaskPower net This methodology is useful for practical bulk analysis

NCEPTS

A network can be ented by a graph branch corresponds to a generator or a tran The branches of the graph can be deleted by removing one branch at a time, which results in sub graphs This deletion procedure can be continued utilizing an efficient topological procedure 111,121 until the graph results in an empty graph, which is the graph with all branches deleted The total number of sub graphs will be equal to 2" -1, where n is the total number of branches in the graph These sub graphs correspond to the Markov states generated for the network

-;sing two state Markov models A simple power system is

shown in Figure 1 The two state Markov model for this system consists of 32 states The Markov states which correspond to the sub-graphs shown in Figure 1 can be

systematically obtained by deleting one branch at a time from the original graph GO in which all branches are present

0885-8950/97/$10.00 0 1996 IEEE

$j2 GO

61 G2

G4

6 3

Figure 1 A Simple Power System , the Graph and Sub graphs of the

System

111 TOPOLOGICAL ENUMERATION AND

CALCULATION OF RELIABILITY INDICES

This section illustrates the topological enumeration

application to a power system state space model and provides

the necessary definitions and properties [ 1 1,121

The enumeration of all the sub graphs of an original

graph GO can be obtained by successively deleting the

branches This deletion procedure can be represented in the

form of a tree The vertices generated in the tree represent

non-empty sub graphs of GO the root vertex being GO Any

vertex, say Gk, corresponds bne-to-one with a sub graph of

Gk in which one or more branches of GO are deleted Thus

this Gk represents a state in a state space model with one or

more components being out of service

A link directed from vertex Gi to Gj is weighted with a

factor W, deleted from Gi to obtain Gj, i.e., Gj = Gi - W In

the tree generated by the algorithm, a vertex Gi(Gj) is

referred to as futher(chi1d) of Gj(Gi) when there exists a link

from Gi to Gj Vertex Gi is the ancestor of Gj when Gi is

contained in the path from the root vertex GO to Gj (Gi

#Gj) The vertices having the same father are termed

brothers A vertex Gi is the younger (elder) brother of vertex

Gj, if the algorithm generates the children of Gi later

(earlier) than the children of Gj Thus starting from a root

vertex GO, the algorithm grows into a tree by progressively

generating children on all possible vertices of the tree

A vertex Gk is termed as a failed vertex if it corresponds

to a sub graph which causes system failure, i.e., load

curtailment or isolation of a bus in a power system

The procedure starts with GO, which is in level 0 and

generates all possible children The children are in level 1,

or in other words all the first order contingencies are

enumerated The following rule can be used for the efficient

enumeration of non-duplicate sub graphs or the generation of

children on a vertex Gk of the tree which is at a level one or

more

Rule a : Let {Wk+l, Wk+2, Wk+3 Wn) be the weights

of the links incident to the younger brothers of vertex Gk

Then Gki = Gk - Wi, (i = k+l, n) is a child of Gk

Consider the graph GO shown in Figure 1 Using the

stated methodology, the children G1 to G5 are obtained by

deleting links 1 to 5 GO is the father of the vertices GI to

G5 G1 is the elder brother of G2 as G1 is generated first G2

is the younger brother of G I These vertices are success

vertices as they do not cause a system failure G1 at level 1,

is considered to generate its children From the above rule a ,

457

the possible candidate for deletion of a link is Gl's younger brothers Le., links 2,3,4 and 5 Now consider the next vertex

G2 in level 1 Using rule a, it can be seen that there are three younger brothers to the vertex G2 which are links 3,4 and 5 Thus a repetition of deleting links 1 and 2 is avoided

as this combination is already considered, i.e., from G1

5

Figure 2 Tree for the Network Shown in Figure 1

Calculation of System Frequency, Duration and Availability Indices

An efficient procedure is given in [lo] to calculate frequency duration indices for bulk power systems using the failed states Alternately, these indices can also be obtained

by identifying all the success states and the boundary transitions This is computationally advantageous as the number of success states in bulk systems can be much less than the number of failure states It can be seen from Figure

2, that the success states above the boundary wall are necessary to compute the frequency, duration and availability indices Consider the RBTS [ 151 with nine transmission lines and eleven generators The total number

of success states is 6395, which is lower than 1% of the total number of state~(2~O) Similarly it can be seen that relatively fewer states must be enumerated for the IEEE-RTS and SaskPower networks to find accurate estimates of the frequency, duration and availability indices The important requirement, however, is the calculation of boundary crossing transitions An efficient calculation of upper and lower frequency bounds is presented in [6,7] using sorting of contingencies and using binary tree data structures Selection of a suitable tree and the derivation of data structures depend on the application [14] The approach described in this paper and the data structures are more suitable to obtain frequency and frequency related indices as the actual frequency contribution of a contingency involves the recognition of all neighboring states with less searching effort It should be recognized that the calculation of the actual contribution of a contingency to the system failure

458

frequency is different from the lower boundary frequency

calculation presented in [6,7] Each node in the tree

represents a contingency and all the nodes that can be

reached from a given state can be accessed without

exhaustive sorting of states and searching for the

neighboring stares in the entire list of contingent states

Another important feature with this method is that all the

lower order contingency states are encountered first before

higher order contingencies, Le., with increasing order

contingency depth It is, therefore possible to terminate the

tree progress in a particular path, if the contribution of

higher contingencies is found to be insignificant in order to

obtain accurate estimates

Figure 3 The Possible State Transitions

A state with n components, with x components out of

service needs n transitions to update the frequency as shown

in Figure 3 The approach described in this paper converts

the state space diagram into a tree in which each state is a

vertex and treated as one of the categories shown in Figure 4

and explained below

Parents

Parent

4 i

Figure 4 Possible Traversals to Obtain the Frequency Update

The case shown in 4.i represents a failure child vertex

generated from a success parent There will be a contribution

to the frequency related indices as there is a transition across

the boundary and there can also be contributions from

parents younger brothers if any of those are success vertices

Another case is shown in Figure 4 3 There can be a

frequency contribution even if there is a transition from

failure parent to failure child, if any of the parents younger

brothers are success vertices The other two cases represented

in Figures 4.iii and 4.iv are required for non-coherent cases

which will be discussed later

The failed vertices obtained from failed parents need not

be explicitly stored In coherent systems, frequency, duration

and availability can be obtained using only success vertices

and the transitions across the boundary The enumeration

progresses if a vertex is a failure and has any success vertex

among the parent and its younger brothers This can be

shown by the following analysis Let the total number of

branches in GO be n Let the present failed vertex be Gm in

which x components are out and Gm is i n level d of the tree

It corresponds to a sub graph GO in x branches are deleted There are x number of p

vertices from level "d-I" can reach vertex Gm in level "d" in one transition The possibilities are as follows

G(12 - 3 x - - 1, x n) G(i23 .=,? n) where Gm

- -

is represented as G ( i 2 3 x - l,x n), a sub graph in which x branches are deleted, the term on the left hand side represents the parent of

level, Le., level d-1

There remaining x- 1 pos

I

G ( l ? ? x -l,x n) -+G(iz3 x - 1,x n )

G ( 123 x-l,x n)-G(12 3 x-l,x n)

_ -

- The above x-1 terms on the the parent's younger

and the status of th will terminate if Gm is a failure and are failed subgrap

and availability indices can be obtai the success states and the transitions

update the frequency The number of

number of contingency le system shown in Figure 1

enumerated This can be which are needed while tr

all neighboring states can be re without any exhaustive search

require sorting of the contingencies and searching among all the sorted states to update the freq

vertices are automatically sorte the data structures This impr considerably

An Algorithm for Calculating th Availability Indices

The algorithm for calculating 1) Start with GO,

2) Delete Gk su

procedure describ 3) See if the status of vertex Gk in the tree is a failure

or success

3.1) If the vertex Gk is not a failed vertex, obtain the associated probability Probability of the vertex can

unavailabilities of the ancestors obtained by backtracking from the present vertex to

3.2)

4)

5 )

5 )

the root vertex GO, with the availabilities of the

remaining branches System probability of success

is obtained by accumulating the individual

availability of the success states

If the vertex Gk is a failed vertex, there could be a

contribution from this vertex, or the corresponding

state, to the system frequency index In order to

calculate the frequency of a vertex, determine the

status of its parent and the parent's younger brothers

which reach the present vertex by deletion of one

link from the previous level Check if there are any

success vertices among these parent's younger

brothers and the parent If there exists vertices that

correspond to the success states which can be

reached from the present failure state, the frequency

contribution of those vertices to the present vertex

can be computed by accumulating the product of the

associated probabilities and the failure rates of those

links which cause the failure

Terminate the enumeration of the tree from a vertex

if the status of a vertex is failed and also the status

of the parent and the its younger brothers

Consider the next vertex in the same level of the

tree Gk = Gk+l, go to Step 2

If all the vertices in the present level are

encountered, go to the next level's right most vertex

and go to Step 2

The above algorithm can be terminated if there is no

change in the reliability indices

Example

The system shown in Figure 1 may require the use of an

AC or DC load flow to test the condition of a contingency

state A DC load flow combined with suitable corrective

actions [8] was utilized in this case to determine if a vertex

in the tree is a failure or success The corresponding tree for

this network is shown in Figure 2

The failure probability, frequency and duration can be

obtained using the procedure explained earlier The failure

probability, frequency and duration indices are as follows

P = I - ( P g + P 1 + P 2 + P 3 + P 4 + P g + P 7 + P g + P g + P 1 1 +

f = ( P I 12 + P 2 11 ) + ( P 2 13 + P 3 1 2 )

P12);

+ ( P 3 14 + P4 1 3 1 + ( P 3 1 5 + p 5 1 3 1 + ( P 4 1 5 + p5 1 4 1

+P7 h 2 + P g 12 + P i 1 h l + P g h 2 + P 1 2 h l

( p 7 1 4 + P g 1 3 )

( p8 15+P9 1 4 )+ p1113

( p 7 1 5 + p 9 I 3 )

PI2 1 3 1 l 5 + p12 l 4 )

d = p/f

In order to illustrate the frequency calculation, consider

vertex, G19 It is a failed vertex corresponding to the state in

which branches 1, 3 and 4 are out of service In order to

calculate the frequency contribution, the status of its parent

which is G7 and the parents younger brothers, G8 and G13

(generated later, right to left in Figure 2) are required It can

be seen from the tree shown in Figure 6 that G7 and G8 are

459

success vertices and G13 is a failed vertex The frequency contribution due to vertex G19 is (P7 A4 + Pg 13)

Calculation of Reliability Indices

Reliability indices such as expected energy not supplied, expected load curtailed etc., can also be obtained using this approach These indices will be based on the failure states and should be considered after boundary crossing Steps 3 and 4 of the algorithm presented are modified The probability of each failure state should be calculated in Step

3 and the indices such as expected energy not served, demand not served are accumulated for each failed vertex Step 4 is not required for calculation of these indices The main advantage with this approach is, however, that more probable vertices are encountered first and the algorithm can terminate after there is no significant change in the reliability indices In order to compute reliability indices, the only information to be stored is the weights of the success links

SASKPOWER NETWORKS

Reliability indices were obtained for the IEEE -RTS[2,3], RBTS[15], and SaskPower networks utilizing the topological methodology described earlier The SaskPower network model has 45 buses, 71 lines/transformers and 29 generating units The annualized indices for selected buses of the IEEE- RTS are presented in Table 1 and the values for the

SaskF'ower network in Table 2 Due to space limitations,

only a few indices at selected buses are presented in Tables 1 and 2 The system indices for these networks are shown in Table 3

The indices given in Tables 1 and 2 are as follows

PLC - Probability of load curtailed EDLC - Expected duration of load curtailed (hrs) EFLC - Expected frequency of load curtailed (f/yr) ADLC - Average duration of load curtailed (hrs)

EDNS - Expected demand not served (MW) EENS - Expected energy not supplied (MWhr) BPECI - Bulk power energy curtailment index

Table 1 Selected Annualized Bus Indices for the IEEE - RTS

0.043885 7.985004 48.0124 5919.22 0.034958 4.561291 66.9532 7981.04 0.039016 4.706072 72.4264 6230.93

15 0 057319 9.142908 54.7680 21145.92

18 0 060672 9.333662 56.7870 26321.30

Table 2 Selected Annualized Bus Indices for the SaskPower Model

2 0.000213 ~ ~~~ 0.494235 3.7650 31.24309

I 0.000152 0.331478 4.0059 8.856329

11 0.000055 0.128981 3.7252 28.57590

22 0.000120 0.084225 1.0458 16.55165

38 0.000002 0.004796 3.6430 0.1 19957

V COMPARISON WITH OTHER METHODS

The results obtained utilizing the topological method described in this paper have been' compared with those obtained using a Monte Carlo technique [8] Table 4 shows the system indices for the IEEE-RTS using the topological

460

method and the Monte Carlo approach [SI It can be seen

from Table 4 that the results are very close despite the

fundamental differences in the two approaches Table 4

shows the indices for a single load level and for a 70 step

load representation

Table 3 Annualized System Indices for the IEEE - RTS, SaskPower Model

and RBTS

Index Annualized Annualized Annualized

System Indices System System (IEEE - RTS ) Indces Inhces

Present State

Method Transition [8]

(SaskPower) RBTS

Annual (70 Step [SI) Present State Method Transition [8]

A 2 Gen, 2 Line, 1 Gen, 1 Line, 1 Gen + 1 Line

B Contingencies of A + 3 Gen

C Contingencies of B + 4 Gen

D Contingencies of C + 3 Line, 2 Gen + I Line, 2 Line + 1 Gen

E Contingencies of D + 5 Gen, 4 Line, 3 Gen + 1 Line

2 5 TI - - O - T o p o l o p i c d -Approach [lo] 1

- E 2 0 +

t

Figure 5 Relative Convergence of the System Frequency Index of the IEEE-

RTS using the Topological Method and the Approach in Reference I O

VI NON-COHERENCE

Non-coherence has both practical and theoretical importance in power system networks [3,10,13], There are many possibilities that can make a system non-coherent The flows are distributed in a network based on the network impedances and other electrical paramet

impedance and the rating of a branch are low, there can be a

an overload problem in that branch[l3] It is a difficult task

to obtain reliability indices for a non-coherent network as the failed state can transit to a success state by further failures Step 3.1 of the proposed algorithm

consider non-coherent systems Gk is co

a success vertex If the vertex is a success state, then the status of its parent's younger brothers are determined If any

of those is a failed vertex, backtrack to the vertex and perform the coherency correction, i.e., subtract the frequency contribution of that state This corresponds to Figure 4.iii Similarly Step 3.2 is modified to include the transitions from the failure state to the success state which corresponds to Figure 4.iv It should be noted that calculation of the frequency contribution from a vertex's parent or its younger brother does not require any additional adequacy evaluations The only information required is the probabilities of the states and the status of the vertices in the previous level which can be obtained from the tree Hence to calculate system frequency in a non-coherence network, there is no necessity to perform additional adequacy evaluations Step 4

of the algorithm is not valid for non-coherent networks This approach can also be used to calculate various reliability indices such as expected energy not served, and frequency related indices More storage is required if the reliability indices for all buses have to be calculated

Consider the example shown in Figure 1 This system can become non-coherent if the rating of line 2 is changed The tree is shown in Figure 6 If thc vertices G11 and 6 1 2 are visited and found to be success vertices then the statuses of their parent and younger brothers are det utilizing the modified Step 3.1 of the topological procedure

It can be seen from Figure 6 that 6 2 , which is the parent of

611 and G12 is a failure vertex This indicates that failed

46 1

Failure (Parent or

Parent's Younger

vertex G2 can transit into successful vertices G11 and GI2

by deleting links 4 or 5 The non-coherent contribution to

frequency and probability, which in this example are P0h2

and P2 are obtained by back tracking Contribution of non-

coherence to frequency, unavailability and energy not served

in this example is more than 90% as non-coherent failure

occurres with a single order contingency

Success with an Addition outage of (child)

Zt -1

Figure 6 Tree of the Network Shown in Figure 1 with Non-Coherence

A test system with 23 generators, 24 lines and 15 buses is

considered and shown in Figure 7 The system becomes non-

coherent if the rating of the transmission line between buses

4 and 8 is changed For an example, if any of the generators

G16,G17,G18 connected to bus 4, is out of service, there

would be overload in line 4 (flow is 0.7954 p.u and the

rating of line 4 is 0.7) This overload would be eliminated by

curtailing load at certain buses if no non-coherence detection

procedure is utilized It can be seen that this overload can be

alleviated by switching out any one of the lines 4, 10, 15 or

17 A sample of non-coherent failures are presented in Table

8

-

B U

c"

1

3

3

4

7

7

9

6

Brother )

G16 or Gl7or G I 8 I LA,L15,L17

L4, L17

LA, L10, L17

L4, L15

The contribution of noncoherence to system frequency and energy not served is approximately 30% for this system The approach can be extended to multistate components and

it can be applied to other applications of power systems

VII CONCLUSIONS

This paper presents an efficient topological procedure to determine frequency, duration and other reliability indices utilizing the state enumeration approach The proposed approach provides the required indices directly without any additional analysis and provides accurate estimates of the indices for both coherent and non-coherent networks It can

be argued that non-coherence can be eliminated by proper system design The procedure proposed in this paper can be used to examine the existence of non-coherence in a given system The effectiveness of the proposed approach is demonstrated in this paper with examples and by application

to the IEEE-RTS and a model of the SaskPower network

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

VIII REFERENCES

Billinton, R and Allan, R N., Reliability Evaluation of Power Systems, Pitman Books, New York and London, 1984

Billinton, R and Allan, R N., "Reliability Assessment Of Large Electric Power Systems", Kluwer Academic Publishers, Boston, USA, 1988 Singh, C., and Billinton, R., "System Reliability Modeling and Evaluation", Hutchinson, London, 1977

Final Report, "Composite -System Reliability Evaluation: Phase-1 - Scoping Study", Tech Report EPRI EL - 5290, Dec 1987

IEEE Tutorial Course, "Reliability Assessment of Composite Generation and Transmission Systems", Course Text, 90EH031 I-I-PWR

EPRI, "Transmission System Reliability Methods", Report No EL-2526, Vol 1, July 1982

Clements, K A, Lam, B P., Lawrence, D J., and Reppen, N D., "

Computation of Upper and Lower Bounds on Reliability Indices for Bulk Power Systems", IEEE Trans on PAS, Vol PAS-103, No 8, Aug 1984, Billinton, R., and Li, W., " A System State Transition Sampling Method for Composite System Reliability Evaluation", IEEE Trans on Power Systems, Vol 8 No 3, August 1993

Melo, A C G., Pereira, M V F., k i t e da Silva, A M.," Frequency and

Duration Calculations in Composite Generation and Transmission Reliability Evaluation", IEEE Trans on Power Systems, Vol 7, No 2, May 1992

Melo, A C G., Pereira, M V F., k i t e da Silva, A M.," A Conditional Probability Approach to the Caculation of Frequency and Duration Indices

in Composite Reliability Evaluation", IEEE Trans on Power Systems, Vol

8, No 3, August 1993

Satyanarayana, A,, and Hagstrom, J N., " A New Algorithm for the

Reliability Analysis of Multi-Terminal Networks", IEEE Transactions on Reliability, Vol R-30, No 4, October 1981, pp 325-333

Satyanarayana, A., and Prabhakar, A., "New Topological Formula and

Rapid Algorithm for Reliability Analysis of Complex Networks", IEEE Transactions on Reliability, Vol R-27, No 2, June 1978, pp 82-100

Li, W., et al, "Application of Transmission Reliability Assessment in Probabilistic Planning of BC Hydro Vancouver South Metro System", IEEE Trans on Power Systems, Vol 10, No 2, May 1995, pp 964-970 Horowitz, E., and Sahni, S., "Fundamentals of Data Structures", Computer

Science Press

Billinton R., et al, "A Reliability Test System for Educational Purposes - Basic Data", IEEE Transactions on Power Apparatus and System, Vo1.4,

pp 2318-2325

NO 3, AUg 1989, pp 1238-1244

BIOGRAPHIES Roy Billinton is presently C.J MacKenzie, Professor of Engineering and Associate Dean, Graduate Studies, Research and Extension of the College of Engineering at the University of Saskatchewan

Satish Jonnavithula is in a Ph.D program at the University of

Saskatchewan

462

Discussion A.C.G Melo (CEPEL, Rio de Janeiro, Brazil), A.M Leite

da Silva (EFEI, Itajuba, Brazil), M.V.F Pereira (PSRI, Rio

de Janeiro, Brazil), J.C.O Mello (CEPEL, Rio de Janeiro,

Brazil): The authors are congratulated for their paper

exploring the use of topological analysis to determine the

contribution of each system state to F&D indices in

generation and transmission reliability evaluation based on an

enumeration approach

We would like to make a clarification related with the

comparison between the authors' method and the Conditional

Probability Approach described in Reference 10 of the paper

Reference 10 provides a simple and efficient formula to

implicitly and recursively identify the boundary wall between

the failure and success states of the system (or vice-versa),

and thus calculating the frequency of system failure (LOLF

index)

The formula can deal equally well with both enumeration and

Monte Carlo simulation (MCS) techniques However, due to

the dimensionality of bulk power systems used in case

studies, our paper was more oriented to MCS

As described in [lo], the system frequency of failure can be

calculated using the concept of incremental transition rate:

X € X f

where:

&

zn

h (x) incremental transition rate associated with the failure

set of system failure states

state x This rate is given by:

j = 1 u = s + 1 v = 1 ' J

where:

m number of system components

mj

s

h j

number of states of component j

state of componentj in state vector x

transition rate of component j fiom state s to u

su

As we can see, expressions D1 and D2 can easily consider

multi-state Markov models, for example, to represerit derated

states of generation and multiple load levels In case of two-

state model representation, the incremental transition rate is

simplified to:

in

a (x) =

where LY and U x correspond respectively to the down and up

components in the system failure state x

As we can see, this formula (D2 or D3) has two terms The positive part is related with those transiti

may cross the boundary wall, i.e., those

to the equipment repair rates The negative part is just a

"trick" to cancel out the internal transition rates, which do not contribute to the frequency ind For ease of explanation, assume that all equipment is resented by a two-state model Suppose that a specific equipment is up in a system failure state Xa If this equipment fails, the new system state xb will also be a failure state and its failure rate will receive a

minus signal However, when the system state x b is visited,

this equipment will be in the down state and its repair rate will now receive aplus signal, which will cancel out the negative

value assigned to the equipment transition in state 9 Observe that the basic assumption in this formula is that when we assign a negative value to a failure rate in a specific state, the new resulting state will be visited In other words, the state space should be consistently analyzed

Now a question that arises is what would happen if we decide

to stop the analysis in a given contingency level Clearly, the previous assumption is no longer valid As a consequence, the frequency index is underestimated, as shown in Fig 5 of the

paper

This limitation can be alleviated by in modification in the application of the previous formula: if we are analyzing a system state that belongs to the deepest allowed contingency level, we only have to neglect the negative part of the formula (D3), since we will not visit the corresponding deeper sates In this case, expression (D3) can

be rewritten as:

pk - ~ hk i f x E to the deepest level

R E L Y ~ E I P :

034)

R E L Y

For example, consider the three-component system illustrated

in Fig D, where u, and a, denote the unavailability (xz = 0) and availability (x, = 1) of component i

Applying expressions D1 and D3 to this system, we have:

LOLF = u1 a,a, x (pl - h, - a,) + al u2a

Rearranging and canceling terms results in the same expression obtained by inspection:

which is the the true index obtained by inspection

I

X I = o x1 = 1 X 1 = l

x 2 = 1 x 2 = 0 , x2 = 1

x 3 = 1 x 3 = 1 x 3 = o

!

I

A P I A P3

h3 h2

X I = o X I = 1 X I = o

x 2 = 0 x 2 = 0 x2 = 1

x 3 = 1 x3 = 0 x 3 = 0

x1 = o

x2 = 0

Fig D - Three- Component System

Now the frequency index will be calculated by limiting the

analysis to single and double contingencies By inspection, as

the transitions related to the failure of the three components

are internal ones, the frequency is the same as calculated in

(D6) Applying (D4), i.e., neglecting the negative terms

associated with the deepest allowed contingency (double) we

have:

Clearly, taking into account only single contingencies, the

frequency index is given by:

463 Applying (D4) by neglecting the negative terms associated with all single contingencies (deepest level) which result in system failure, we also obtain expression D9:

As we have seen, with this modification, the LOLF index will never assume negative values, as presented in the paper In fact, equation D4 had been used in the implementation of the state enumeration procedure in our production grade software

We apologize for not clarifying this issue in our original paper

As shown in Ref 9, also cited in the authors' paper, one related concern has been with the major contribution to the frequency of failure index which comes from load transitions None of the examples of the paper deals with more complex load models, for instance, considering hourly load cycles along the whole year Is there any difficulty to deal with these load models? And to represent derated states of generating units? How would be the increase in the associated CPU time?

Finally, it would be interesting to know about the possibility

of extending topological concepts to Monte Carlo simulation approach

We would again like to compliment the authors for their interesting paper

Manuscript received February 20, 1996

would like to thank the discussers for their interest in our paper Their comments extend the material provided in Reference 10 We do not see any difficulty in including derated state representation in the topological approach It will, however, clearly increase the required CPU time due to the creation of additional branches The question of load

transitions is more complicated as the individual bus loads in

a composite system do not transit at the same time We have approached this problem using a sequential Monte Carlo approach in which each bus load has the customer

characteristics associated with that load point [A] We do not feel that this form of individual load bus representation can

be included in the method of Reference 10 or in the topological approach

In conclusion, we would again like to thank the discussers for their comments

[A] A Sankarakrishnan and R Billinton, "Sequential Monte Carlo Simulation for Composite Power System Reliability Analysis With Time Varying Loads", IEEE Transactions on Power Systems,

Vol 10, No 3, Aug 1995, pp 1540-1545

Manuscrbt received March 22 1996