Multi area generation system reliability calculation

The analysis involves two major steps for a given s e t of area reserve margin distributions and transmis- sion link capacity states.. The linear flow network model and the presented te

IEE T r a n s a c t i o n s on P o w e r A p p a r a t u s a n d S y s t e m s , vol PAS-94, no 2, MarchIApril 1975

MULTI-AREA GENERATION SYSTEM RELIABILITY CALCULATIONS

Power T e c h n o l o g i e s , I n c Schenectady, N Y

This paper presents the results of an investigation

of a technique for the evaluation of the reliability of

supplying power in a system with a nunber of intercon-

nected load-generation areas There is no restriction

as t o how the areas may be interconnected Most pre-

vious techniques using analytical methods (as opposed

t o Monte Carlo simulations) have been limited to sys-

tems w i t h a maximum of three interconnected areas Sys-

tems with more areas have been analyzed assuning that

the interconnecting e l e c t r i c a l network did not contain

e y loops The application of straightforward enunera-

tlve methods to systems w i t h more complex interconnec-

tians than these can result in an improbably large m-

ber of computatians The method of analysis described

in this paper is based upon the use of a linear flow

network t o model the transmission interconnections and

makes use of an e f f i c i e n t graph theory algorithm t o

segregate the failure states by finding critical mini-

mal am in the network The probabilities of failure

t o supply the various loads are ccmputed by evaluating

the various combined event probabilities associated with

these c r i t i c a l minimal cuts For the cases tested, the

technique reduces the number of probability evaluations

required by about one t o two orders of magnitude in com-

parison with complete state enmeration methods The

tested method provides r e l i a b i l i t y measures ( i e , the

probability of f a i l u r e t o meet the load) f o r each indi-

vidual area and the total system, and also allows the

canputation of the probability that each "link" (trans-

mission line or source) is a manber of a c r i t i c a l mini-

mal cut 'Ihe l a t t e r will f a c i l i t a t e the application of

the method t o the design of systems and specifically to

the problem of evaluating the reliability benefits of

increased transmission capacity versus added generation

The amputation of generation system r e l i a b i l i t y

indices, particularly the failure to supply the expected

load, has been done fo single areas on a regular basis

f o r a n h e r of years195 h e failure index, the loss-

of-load probability li.e , MLP) has been corrputed for

two or more interconnected system using analyticas

techniques as long as there are no loops in the system

Techniques to handle system with loops have been re-

ported but they i r e a transfoxmation into equivalent

radial systems For more complex interconnection

arrangements, sinulations using Monte Carlo methods have

been used The objective of this paper is to present a

method for calculating the generation system r e l i a b i l -

i t i e s f o r N intercannected load-generation areas using

linear flaw techniques

Engineering Committee of the IEEE Power Engineering Society for presentation at Paper

T 74 342-2, recommended and approved by the IEEE Power System

the IEEE PES Summer Meeting & Energy Resources Conf., Anaheim, Cal., July

14-19, 1974 Manuscript submitted February 1, 1974; made available for

printing April 16,1974

The straightforward extension of the analytical

methods used in the single-area, two-area and radially connected three or more area s i t u a t i o n t o a larger nun- ber of areas interc orme$gp in a complex fashion causes

coanputational problems which a r i s e from two basic sources These are the large d e r of events requiring

evaluation i n order t o make a complete r e l i a b i l i t y com-

putation and the necessity to be able to identify the

failure events : that is, the lack of sufficient genera- tion and/or transmission capability to meet the expected loads

The operation of individual areas in a single sys- tem, or pool, is aimed in part at the sharing of genera- tion reserves under emergency conditions Therefore , the number of possible events t o be cansidered (i.e., conbinations of independent occurrences of generation outage and loads) increases rapidly with the nmbr of pool mehers The presence of an interconnecting net- work which contains loops requires that the constraints imposed upon the sharing of reserves caused by the net- work configuration and transmission limits6 be recog- nized in the computation From a purely theoretical

point of view, this requires the inclusion of sone s o r t

of AC load f l o w model in the analysis A t the present time, t h i s is an appalling thought

Even the incorporation of a DC load f l o w model w i l l lead t o d i f f i c u l t i e s The use of a DC model of the transmission system requires the solution of a nunber

of linear programs t o separate the failure and success

s t a t e s i n events where it may be feasible to supply a deficient region of one or more areas fran one or more

sources and where the t i e l i n e s may impose limits on the

import and export capabilities These restrictions D ~ U S

the Dc load flow relationships form one s e t of linear constraints and lead to the use of linear p r o m

(U) methods t o maximize the flow into the deficient region The maximun number of these LP cases for N areas

is of the order of ZN which can, of course, become very large for a moderate value of N Therefore, we are led

t o consider the use of a simpler mcdel for the electri-

cal network; a linear f l o w network w i t h links which are characterized by several possible independent capacity

s t a t e s and their associated probabilities

FAILURE STA'IES

For the purposes of analyzing the r e l i a b i l i t y of a single generation system, a failure can be defined as

the existence of a load-generation s t a t e w i t h a negative reserve margin This failure probability, referred to

in t h i s paper as "LOLP," can also be canputed f o r two

or more independent areas which are connected by t i e lines With pooled operation of various areas, sane

thought nust be given t o t h e operating d e s of the pool

in defining failure states for the individual areas in the system

When shortages of generation occur in an area they can be made up by the transfer of reserves within the

capability and availability of the transmission system Suppose, hawever, that there is an overall negative reserve i n the total interconnected system which is pose further that these areas can be assisted t o cer-

caused by shortages in me or more of the areas Sup-

tain extent by the shifting of reserves from sane of the areas with excess generation The question is, w i l l

these reserves be shifted even thaugh t h i s act will

cause a curtailment i n the previously self-sufficient

area?

The questitn has a real relevance to the evaluatitn

of generation systan reliability indices for intercon-

nected areas If the power pool i n question operates

an the basis of sharing reserves up t o the limit of the

t i e c a p a b i l i t i e s o r t o the point where the daating

areas individually have a zero reserve margin, then t h i s

type of event is not a failure for the self-sufficient

areas On the other hand, i f the pool's operating

policy is such that s n e r a t i m reserves are t o be shared

up t o the limits of the t i e l i n e s , even t o the point of

curtailing an individual area load, then this s o r t of

event mst be classified as a failure for a previously

self-sufficient area

There are many possible variations of the two lim-

iting operating policies described above In the analy-

sis and exanples that follow, only the two extreme pos-

s i b i l i t i e s a r e considered; that is,

1 A no load-loss sharing policy implies that re-

serves are transferred cmly up t o the limit of

the t i e lines or available reserves, whichever

is limiting

2 A load-loss sharing policy is defined as cm-

plete sharing of available generation up to the

limits of the interconnection

These two policies can be mst e a s i l y i l l u s t r a t e d by

reference t o a two-area situaticm Figure 1 shows a

V e n n diagram of the possible margin events in the two

areas for some specified condition Ihe tie is asnmred

t o be perfectly reliable and t o have transfer limits of

and T a in the two directions indicated ~n the

2 t c h 1Re various areas on this diagram represent

various possible events which are as f o l l m :

1 Both areas are deficient

2 A is deficient; B assists A, only a t

the expense of reduction i n its own

load and is not limited by the trans-

fer capability; also both areas com- bined are deficient

3 A ' s deficiency and B's surplus are

larger than the capacity of the t i e line

4 A is deficient; B can supply excess

r e s e m s up t o the t i e limit without any reduction of its own load

5,687 Identical to areas 2 , 3, and 4,

respectively, except that the roles

of A and B are interchanged

a Both A and B are independently s e l f -

sufficient

N o w , i f we l e t these areas represent the probabilities

of the various events described above, we can then de-

fine the system and area failure probabilities under the

two possible operating policies In either case, the

system L O is given by:

!System LOLP = 1+2+3+5+6

With no load-loss sharing the area LOLP values are:

LOWA = 1+2+3

LOUB = 1+5+6

Under a policy of fully sharing load losses:

LOPA= 1+2+3+5

LOWB = 1+5+6+2

Further, we can capUte the probability that the t i e is

loaded t o capacity because of reserve sharing This is

the swn of the probabilities represented by areas 3 and

6

Area A Reserve Margin

F i g u r e 1 Two-Area System and

Reserve Margin

P r o b a b i l i t y Diagram

The e x a p l e s discussed below were conplted for both policies W i t h load- loss sharing and perfectly reliable

t i e l i n e s , the area and system r e l i a b i l i t i e s will ap-

proach the s evalue as the t i e capacities are increased

t o i n f i n i t e values With no sharing of the load losses

t h i s is not true and the areas and system risk levels are generally different, Whi no load-loss sharing an increase i n intercannection capability will always sw

t o increase the area r e l i a b i l i t y ; w i t h load-loss sharing

an increase in transfer capability may actually reduce

t h e r e l i a b i l i t y of an area The method described will allow assessment of these effects within the limits of

the asd linear flaw transmission model

blE"Xl OF NULYSIS

Flow Networks

The objective of the analysis is to cosnpute the failure probabilities for interconnected generation-

load areas The transmission network w i l l be modelled

by a linear flow network so that the w e l l - h m minimdl cut- * flow algorithns may be used t o c o n p t e t h e

s y s t p - f e r m i l i t i e s 7 This s w l i f i e d model of

an e l e c t r i c power network would seem t o be a reasonable approximation for interconnected generation-load areas where it i s possible to control the flow of power be-

tween areas t o a reasonable degree This control capa-

b i l i t y does exist in a nunber of power pools

509

The various transmission links are described by

capacity states and the associated probability of exist-

ence That is, each interconnection may be assumed t o

be made up of a n d e r of p a r a l l e l c i r c u i t s , each of

which has a transfer capacity and an outage rate When

a linear flm network is used t o model the transmission

network, the power flow i n these interconnections s a t i s -

fies Kirchoff's current law, but not the vmltage law

The flow pattern for a given distribution of area re-

serve margins i s not unique and the network s t a t e is

characterized only by a maximum transfer capability

This maxirmrm flow capability may p" determined using

well-known and efficient algorithms

Reliability Analysis

W i t h this simplified t r m i s s i o n model, one of the

major computational problems mentioned above i s avoided

The other problan of the very large number of s t a t e s

that must be evaluated is also assisted by the use of

the flow model The technique used t o reduce the num-

ber of states requiring a probabilistic evaluation is a

modification of the method presented by Doulliez i n

Reference 8 The modified method is outlined below

The individual areas are described by the probabil-

ity distribution of available reserve margins These

may be generated by using well-known algorithms to cre-

ate tables of reserve margin availability from individ-

ual unit forced outage rates and expected loads The

use of area reserve margin distributions facilitates the

treatment of area loads which are completely random or

else perfectly correlated The examples presented below

assumed random area loads Correlated loads would re-

quire the computation of the r e l i a b i l i t i e s f o r individ-

ual sets of loads

The analysis involves two major steps for a given

s e t of area reserve margin distributions and transmis-

sion link capacity states These are:

1 Find the c r i t i c a l minimal cuts of the network

t o detennine a l l of the events k3lich a r e f a i l -

ures and,

2 Compute the failure.probabilities for each area

and the t o t a l system plus the probabilities

that individual transmission links are loaded

t o capacity

F i g u r e 2 shms a simplified flow chart of the re-

liability analysis technique All failure events for

each codination of self-sufficient and deficient areas

a r e f i r s t determined and then followed by the computa-

tion of the various failure probabilities Since a l l

combinations are mutually exclusive events, the various

over-all failure probabilities are obtained by sumning

over a l l combinations

The determination of all failure events is based

upon an expansion scheme which reduces greatly the num-

ber of states requiring evaluation over the ,number that

would be involved in a straightforward emmeration The

algorithm is based won the notion classifying the fail-

ure states by the network cut limiting reserve transfers

For a conbination of self-sufficient and deficient

areas, the expansion process begins by examining the

initial s t a t e formed by s e t t i n g a l l areas t o zero re-

verse margin and all t i e s t o mininun capacity A linear

flow would indicate if a given s t a t e is a success s t a t e

or a failure one A s t a t e is a success s t a t e i f t h e r e

is no load loss, otherwise it i s a f a i l u r e s t a t e New

states are eenerated and examined until the process t e r -

Find Firrt Gmbhaticn a t S?lf-%fficient

and Deficient Areas

1

F o nm initial state by setting all anzas to zero margin and a l l ties <

Perfom Linear Flar I , -

Success State

+

- of deficient areas

Gscnte new states by m k j q

I

F i g u r e 2 S i m p l i f i e d Flow C h a r t o f t h e

R e l i a b i l i t y A n a l y s i s A l g o r i t h m

'Ihese two ways are discussed below:

1

2

Given a success s t a t e , an increase in the mar-

g i n of self-sufficient area o r an increase in

any t i e capacity o r both would result in an-

other success s t a t e However, such subsequent success states can be ignored in the process

In order to find any failure states in exist- ence, margin of deficient areas must be re-

duced This may o r may not inmediately lead

t o a f a i l u r e s t a t e , but the search i s i n the right direction

Given a f a i l u r e s t a t e , one would want to find other failure states that may exist A f a i l - ure state is characterized by the minimal cut

in the system I t can be seen that a minimal

cut would s t i l l remain a minimal cut i f the

capacity of any of i t s members is reduced Such states would s t i l l be f a i l u r e st a t e s , their existence is clear and, therefore, these states need not be stored If the capacity of any member of a minimal cut i s increased or the capacity of any member of the system not

in the minimal c u t s i s reduced, a new minimal cut could occur "his new minimal cut may

correspond t o a different failure condition Thus, an increase in the margin of self-suf- ficient areas o r t i e capacity o r a reduction in the deficient areas whose loads are satisfied may change the minimal cut in the system This could lead to other failure states with d i f - ferent failure conditions

By systematically increasing the margin of s e l f - sufficient areas and the capacity of t i e s and decreasing margin of deficient areas in the manner described above,

a l l f a i l u r e s t a t e s will be obtained when the expansion process i s completed, the various probabilities are then

minates wit6 no more new states generated f i e expansion

scheme proceeds in one of two ways depending on whether ~

the s t a t e under examination is a success o r a failure one computed

The probability computation may be illustrated by

the five-area system with perfectly reliable, or firm,

t i e l i n e s shown in Figure 3

/

F i g u r e 3 A Five-Area System

Let a s t a t e X of the system be denoted by ( x 1 , x ~ x 3 , q ,

x ) where xi is the margin level of Area i SO, X i

takes on v a l ~ e s Y, 2, , ai, h e r e level 1 is the

lowest margin (generally margin 1s negative) and ai the

highest margin (positive)

Consider a cumbination with Areas 1, 3 and 5 s e l f -

sufficient and Areas 2 and 4 deficient Suppose a s t a t e

Y = ( ~ ~ ~ y ~ , y ,y ,y ) i s found such t h a t t i e l i n e s 2-4, 3-4,

and 5 a?e h ? y loaded and the deficiency in Area 2

i s met while that in Area 4 is not met even though there

may be surplus in Areas 1 and 3 Under such a situation,

a linear flow will identify the minimal cut C and the

direction of flow may be as shown in ' F i g u r e 3 In this

case a system failure is said to have ocmrred For a

no load- loss sharing policy, failure occurs only in

Area 4 However, f o r a policy with load-loss sharing,

failure occurs i n both areas 4 and 5

I t can be seen that the same failure condition

would cccur i f ( i ) margin of Area 1 and/or Area 3 is in-

creased, this increase w i l l not help Area 4 as a l l paths

leading fran Areas 1 and 3 t o Area 4 are already fully

loaded; (ii) margin of Area 4 is reduced, this only ag-

gravates the situation; (iii) both ( i ) and ( i i ) a r e oc-

curring Let E@) be defined as a s e t of a l l s t a t e s X

all A A t a ? e s x M e t i e 4 G G 2 '

I t is clear that llure conditim as

s t a t e Y Hence, instead of storing all the individual

states in E(Y) only s t a t e Y needs t o be stored The

failure probability contributed by E@) is given by:

with x >y x 7 2 9 x ,>y x

where

PGE(yi) = probability that margin of Area i

PE(yi) = probability that margin of Area i

PLEbi) = probability that margin of Area i

i s g r e a t e r or equal t o yi

is equal t o yi

i s less than o r equal t o yi

Bviously state Z i s not a menher of E(Y) Let E(Z) be defined in the same way as E(Y) , it can be seen that

s t a t e Y is not a member of E(Z) But, both E(Y) and E(Z) have some states i n c0"sl and any such s t a t e X is

given by the following:

x5 = ys = z5

A t t h i s juncture, a set E(YZ) is defined as the s e t of

a l l s t a t e s which are i n bot! E O and E(Z) In the set theory terminology, E m ) is the intersection of E(Y) and E(Z), or symbolically

The failure probability contribution fran all states in both EM and E(Z) is equal to the sum of FT{E(Y)I and FT{E(Z)I minus Pr{E(YZ)I The last term i s

necessary as the contribution by the states in E(YZ) has been added twice , once Q-I PrIECI) 1 and the other in Pr{E (Z) 1

In general, i f there are n sets of states , S

, , s , for the same f a i l u r e c o n ~ t i o n , thg 3;

t o t a l f a i l u b p r o b a b i l i t y Contribution is given as f o l -

laws:

Total failure probability =

(-l)n-l Pr(Slns2n ,

Pr{E(Y) I is a l s o a contribution to the probability that

cut C i s a minimal cut

For the above combination of self-sufficient and There may be other states which are not in E(Y) but deficient areas, suppose 4 minimal cuts, as shown in

also give r i s e t o the same failure condition as Y A n Figure 4, are encountered and the deficiency of areas t o example of such states is a s t a t e Z which is given as the right of each cut is not met

follows:

F i g u r e 4 Locations of Minimal Cuts

Let

Pr(Ci) = probability that cut Ci

h-(T.- .) = probability that tie i-j

is a minimal cut

Pr(Ai) = probability that failure

occurs in Area i Then,

4

i= 1 System failure probability = 1 Pr(Ci)

h-(T1-2) = Pr(C1)

Pr(T1-3) h-(C1)+Pr(C2)

Pr(T2-3) = Pr(C2)

fi(T2-4) = Pr(C2)+Pr(C3)+h-(Cq)

Pr(T3-4) = Pr(C3)+Pr(Cr)

Pr(T3-5) = WC,)

fi(T4-5) = Pr(C4)

The failure probabilities of individual areas f o ~

both load-loss and no load- loss sharing policies a n

given below:

For no load-loss sharing policy:

The above shows haw individual probabilities are computed for a given conhination of self-sufficient and deficient areas Total failure probabilities are ob- tained by sumring over a l l possible ccmbinations of the areas

RESULTS OF SAMPLE CASES

The algorithm described above has been implenented

in an experimental c o q u t e r program t o t e s t t h e e f f i - ciency of the method in reducing the nmber of computa- tions required and t o examine the computational e f f o r t involved i n determining the area and system r e l i a b i l i -

t i e s f o r a n h e r of interconnected areas The program

w a s written i n FORTRAN TV f o r a time-shared computer system involving a CDC Mod61 6400 ?he- relative central processor times obtained from these experiments give a measure of the algorithm's efficiency and the trend of the computational effort with increasing system size

Five different system configurations were studied

?he system designation and configuration for each is shown i n Table I

TABLE I

m SAMPLE

2 o o

4

Each system was studied for one s e t of area reserve

1,iargiLs ea& of w i l i c i ~ co~ltaineri from 6 t o 20 niargin levels The systems were studied both with multiple ca- pacity state transmissfon links and single state, per-

fectly reliable links A maxinnnn of three states was used for the interconnecting ties, though m r e s t a t e s are allowable The t o t a l number of possible states for the system is the product of the number of reserve mar- gin states in each area times the product of the number

of capacity states for each t i e l i n e This number var- ied between 63 and 128,000 for the five systems noted above

In addition, the 4-area, bridge-connected system

w a s used t o i l l u s t r a t e p o s s i b l e system design proce- dures Reliability indices were computed along with data showing the unreliability contribution of each area and each t i e l i n e as measured by the probability that each was a member of the set of c r i t i c a l c u t s of the system These data were then used t o decide which t i e

t o reinforce o r where t o add additional generating ca- pacity

Results Demonstrating Camputational Efforts The effect of t h i s algorithm i n reducing the number

of s t a t e s t h a t need to be analyzed can be illustrated

by considering the s t a t i s t i c s of a computation for sys- tem 3A, the delts-comectad 3-area system F i n t i e

51 2

lines were assumed and each area had 20 possible re-

serve margin s t a t e s thereby giving a t o t a l of 20 x 20 x

20 = 8000 possible system states There are six possi-

ble combinations of deficient systems that need t o be

analyzed since the conditions of a l l three being s e l f -

sufficient or a l l three being deficient are easily clas-

sified The procedure described here performed 643 l i n -

ear flows and evaluated the probabilities of only 243

s t a t e s ?his is only s l i g h t l y over 3% of the possible

s t a t e s (See Case 6 i n Table 11.)

The same tests were repeated with each t i e consid-

ered t o have multiple capacity states This increased

the total number of possible system s t a t e s by a factor

of 12 t o a level of 96,000 In t h i s c&e the algorithm

performed sane 1335 linear flows and evaluated the prob-

a b i l i t i e s of only 585 s t a t e s , 0.61% of the 96,000 pos-

sible These are also given by Case 7 i n Table 11 The

encoxaging characteristic exhibited here is that al-

though the number of possible states went up by a factor

of 1 2 , the actual number of state probabilities that had

t o be evaluated increased by a factor of about 2 4 and

the required computer time as measured by CPU seconds

increased by approximately a factor of 3 4

TABLE I 1

SLMKiRY OF COFlfUTATIONAL EFFORTS

FOR VARIOUS CASES STUDIED

CIse

1

2

3

4

5

6

7

8

9

10

11

2

2

2

3

3

3A

3A

4

4

4

5

Possible

No of States

6 3

126

189

600 3,600 8.000 96,000 4,200 33,600 128,000 37,800

Linear Flow No of Performed

10

12

17

98

133

643

1335

1853

780

6004

5543

*On a CDC 6400 time-shared computer

No of Probability

6

8

10

1 4

8 5

246

585

271

784

2160

1580

CPU

0.179 0.210 0.232 0.403 0.732 5.694 12.600

18.020 3.823

104.500 52.900

This s a m encouraging behavior w a s also exhibited

for the various different systems used Table I1 lists

the cases studied, number of possible states and number

of states requiring probability evaluations The r e l a -

t i v e computing times are also indicated Figure 5 shows

the variation of the number of state probability eval-

uations required in percent of the number of possible

s t a t e s as a function of the number of possible states

For a l l the cases tested, the number of required ccarpu-

tations w a s l e s s than 10% of the theoretical maximrm

possible The data also indicate an encouraging trend

i n that the ratio exhibited on this figure tended t o

decrease with system size

While i t i s not appropriate t o draw generalized

conclusions about the exact computational e f f o r t because

of the limited number of t e s t systems, we expect t h a t

the desirable features illustrated are independent of

the exact system configuration That is, t h i s technique

reduces the computational e f f o r t by about 1 t o 2 orders

of magnitude over t h a t required f o r a c q l e t e s t a t e

enumeration

System Design Example

The 4-area, bridge-connected system (sample system

number 4) can be used t o i l l u s t r a t e the use of this

method i n the design of r e l i a b l e power pools The sys-

tem set-up is shown again i n Figure 6 A simple example

I

!

!

0.1

1 0 l o 4 l o 3

No o f P o s s i b l e S t a t e s

5

F i g u r e 5 V a r i a t i o n o f t h e Number o f

S t a t e P r o b a b i l i t y E v a l u a t i o n s

i n P e r c e n t of t h e Number of

P o s s i b l e S t a t e s

F i g u r e 6 Four-Area Bridge-

Connected System

was studied in which each area w a s assumed t o have the discrete reserve margin states shown i n Table 111 As-

sociated ~ t each reserve margin s t a t e is an existence h probability value This table also shows the capacity states assumed for each transmission link or tie Each

of these s t a t e s had an assigned probability also

TABLE I11 INITIAL RESERVE MARGIN STATES AND TIE CAPACITIES OF THE 4-AREA BRIDGE-CONNECTED SYSTEM

MW Reserve Margin States Area 1 Area 2 Area 3 Area 4

2 5 0

150

150

5 0

200

- 5 0

- 2 5 0

-100 -200 - s o

- 1 0 0

-200 -300

-400

-500

MW Tie Capacity States

T A B L E IV LOLP OF SYSTEM AND INDIVIDUAL AREAS FOR T H E 4 - A R E A B R I D G E - C O N N E C T E D S Y S T E M

L- A r e a or

S stem

S y s tern

3

4

O r i g i n a l

No Ties

S y s t e m

B e t w e e n

A r e a s , 8 4 1 5 x

.1988 x

.3933 x

.2847 x lo-’

.1998 x

.7984 x

.2229 x l o m 1 .1696 x lo-’

.2337 x

.lo91 x

.1988 x

.7993 x

.2229 x 1 O - I .2186 x lo-’

.1998 x lo-‘

.4575 x lo-’

.2337 x lo-’

.1142 x

The area and system LOLP values were computed for

t5e two different policies of load-loss and no load-loss

sharing In addition, LOU values were ccgnputed assun-

ing both zero capacity and firm, i n f i n i t e capacity t i e

lines These r e l i a b i l i t y d a t a are shown i n t h e f i r s t

three columns o f Table IV

The advantages of interconnections could be seen

from t h e r e s u l t s i n columns 1 and 2 The r e l i a b i l i t y of

the system w a s increased by about four orders when the

t i e s given i n Table I11 were used t o interconnect the

four areas Column 3 shows the ultimate reliability

t h a t can be achieved through interconnection The sys-

tem LOLP will approach the value in this column asymp-

t o t i c a l l y as additional and m r e r e l i a b l e ties are used

An examination of the results i n column 1 of Table

IV indicates that the LOLP for Area 3 w a s the highest,

being m r e than two times larger than that f o r any other

area In f a c t , Area 3 f a i l u r e contributed t o a major

part of the total system f a i l u r e The r e s u l t s i n Table

V show that the mjor system “bottleneck” involved t i e -

lines connected d i r e c t l y t o Area 3 F u r t h e m r e , over

96% of the probabilities that tie lines 2-3 and 3-4

being loaded t o capacity occurred when power was flowing

i n t o Area 3 Therefore, it may be concluded that Area 3

was the most deficient and that other areas were capable

of providing further help had the capacities of the t i e

lines connected t o Area 3 been larger

To proceed with the design of the network, three

most effective alternatives are available to increase

t h e r e l i a b i l i t y of Area 3; namely, ( i ) add new units t o

Area 3 , ( i i ) add new lines to right-of-way 2-3 and ( i i i )

add new lines to right-of-way 3-4 Studies were per-

f o m d f o r t h e f i r s t two alternatives The f i r s t one

had a 100 MW unit added t o Area 3 while the other had a

100 MW p a r a l l e l c i r c u i t added between Areas 2 and 3

The results of these cases are shown i n columns 4 and 5

of Table IV For both cases, the reliability of Area 3

and the over-all system increased by about an order of

magnitude I n f a c t , r e l i a b i l i t i e s f o r a l l a r e a s i m -

proved The question as to which of the alternatives is

b e t t e r i s an economic area involving consideration of

both implemntation costs and r e l i a b i l i t y improvements

f Load Probability (LOLP) Firm

I n f i n i t e ’ A d d a 100 M W

T i e s Unit to Area 3 A r e a s 2 and 3

Circuit Between Add a 100 MW

.3905 x

.6055 x l o - ’

5 5 1 5 x l o m 5 , 1 8 2 1 x l o - ’

I O U x

.1181 x l o m 5 .8865 x

.3654 x 10.’

.3635 x lo-’

.lo42 x

.5554 x

1 9 ~ x

.1725 x

.1258 x

.NU x

.4646 x

.3803 x

, 1 9 1 1 x

.1102 x

.5159 x

I

.6148 x 10‘’

.1911 x lo-’ 2838 x 1434 x

TABLE V PROBABILITY TIE LINES ARE LOADED

TO CAPACITY FOR INITIAL 4-AREA BRIDGE-CONNECTED SYSTEM Tie Line Probability

1 - 2 .4012 x

CONCLUSIONS The conputation of multiple-area reliability in- dices for power pools using analytical techniques i s feasible and economic when the interconnecting network

i s modeled by a linear flow network Based on such a network model, an e f f i c i e n t algorithm t o evaluate the probabilities of f a i l i n g t o supply the load in individ- ual areas has been presented in zhis paper Also in- cluded i n the paper are the results of a study on a variety of systems using the proposed algorithm The

f i r s t of the two major steps in the technique used i s t o detennine a l l of the events which are failures by find- ing t h e c r i t i c a l m i n i m a l cuts, while the other step com- putes the various failure probabilities

There is no r e s t r i c t i o n on the configuration of R

system whose r e l i a b i l i t y is t o be evaluated Also, t i e

l i n e outages are recognized and taken into cansidera- tion when computing the failure probabilities In ad- dition, the method is capable of handling power pool policies which involve load-loss and no load-loss shar- fng The results will yield numerical data which f a c i l -

l t a t e the decisions t o be made whenever r e l i a b i l i t y levels are to be increased by adding t o t i e c a p a c i t i e s and/or area reserve margins

514

Results of study show that the presented technique

significantly reduced the computational effort when com-

pared with ccnnplete state enumeration methods An en-

couraging feature revealed in the results is F a t the

reduction in conputational effort tended to LLBcrease

with system size (number of possible states)

The linear flow network model and the presented

technique permit the development of practical multi-

area power system reliability calculations

REFERENCES

1

2

3

4

5

6

7

8

AIEE Subcomaittee Report, "Application of F-robabil-

ity Methods to Generating Capacity Problems," AIEE

Transaction (Powsr Apparatus and Systems) , Vol 79,

pp 1165-1182, 1960 (February 1961 issue)

R Billinton, R J Ringlee and A J Wood, "Power

System Reliability Calculations," The MIT Press,

1973

V M Cook, C D Galloway, hi J Steinberg and

of ' h o Interconnected Systems," IEEE Transactions,

Vol PAS-82, pp 18-33, 1963

G S Vassell and N Tibbets, "Analysis of Generat-

ing Capacity Reserve Requirements for Interconnected

Power Systems , I t IEEE Transactions , Vol PAS-91,

pp 638-649, 1972

H T Spears, K L Hicks and S.T.Y Lee, "Probabil-

ity of Loss of Load for Three Areas," IEEE Trans-

actions, Vol PAS-89, pp 521-526, 1970

E Jamoulle, "The Use of Models in the Network In-

vestment Planning," Paper 1.1/8, Fourth Power System

Computation Conference, Grenoble , France , September

11-16, 1972

L R Ford and D R Fulkerson, "Flows in Networks,''

Princeton University Press, 1962

P Doulliez, "Optimal Capacity Planning of Multi-

Terminal Networks , I 7 F%.D Thesis, Universite

Catholique de Lowain, Belgium, 1970

Discussion

A P Bonaert (Interactive Systems, Brussels, Belgium): The authors

have written an extremely interesting and well presented paper They

have succeeded in solving a problem that has frustrated many active

people in power system reliability and the contribution offered by their

paper is quite impressive:

i) clarification between the two extreme interconnection policies

ii) consideration of complex (i.e with links) interconnection net-

iii) an elegant algorithm that uses quite efficiently a "truncated"

work

search

Two points had awakened my curiosity:

loads However, if i) The examples presented in the paper deal with independent loads are fully correlated (and hence not inde-

pendent), the evaluation of Pr [ E(y)] below figure 3 is likely to be more

difficult than a product of probabilities It would be helpful if the

authors could in their closure outline the difference in the computa-

tions given below figure 3 for this case

ii) The numerical examples of Tables I to V refer to relatively

simple interconnection networks

complex interconnection network that is given only by its topology and

Is the programming fairly evolved to handle the general use of a

the distribution of its tie-lines capacities and nodes reserve margins?

Let me reiterate my congratulations to the authors and hope for

reading more of their valuable contributions

Manuscript received July 17, 1974

R L Sullivan (University of Florida, Gainesville, Fla.): The authors are

to be congratulated for presenting a somewhat new approach for network flow algorithms and basic probability theory the authors have solving a most difficult analysis problem Through the use of standard

produced a rather simple technique for evaluating the reliability of interconnected areas and simultaneously identifying bottlenecks in the interconnected network By judiciously segregating the failure states by finding the critical minimal cuts, the authors have managed to avoid, quite successfully, the problems associated with complete enumeration The authors point out quite clearly that for the basic technique to

be viable, the use of d s or a s load flows, as opposed to network flows, must be avoided This approach unfortunately ignores the more general

concept of service quality in which voltage profiles and the like become important It would be interesting to hear the authors comments on how their basic approach could be extended to include the more general notion of service quality

does not include the outage characteristics of a given areas transmission

In the paper, the authors appear to have used an area model which system It seems that to include in detail the probability distributions for the interconnections while omitting individual area transmission outage distributions is questionable Perhaps the authors could comment and also discuss how it would influence the results of a particular study

on how each areas transmission network could be taken into account, The minimal cut approach to identifying fully loaded interconnec- reasonable approach However, it is not clear how the author's program tions as well as bottlenecks is quite interesting and appears to be a handles redundant minimal cuts, i.e., it seems that it is quite possible mal cut value, and if the minimal cut algorithm always chooses a that more than one combination of tie-lines could have the same mini- particular combination of tie-lines a pessimistic reliability level would

be calculated for that particular combination If the authors could clarify this point, it would be very helpful

but the discusser would like the authors to indicate how the generator The paper does not address the automatic design problem directly, and/or interconnection capacities could be increased to achieve a desired reliability level at minimum cost Unfortunately, as the paper points out, changing the element capacities tends to alter the minimal cut structure and hence the indices of reliability, which appears to compli- cate automatic design procedures

In conclusion, the discusser found this paper very interesting, and

is certain that more work along these lines will be stimulated as a result

of the work recorded

Manuscript received July 29, 1974

P B Shortley (Westinghouse Electric Corp., East Pittsburgh, Pa.): Cal- culation of the loss of load probability index of system reliability for a general system of any number of individual areas tied together in a general configuration, with limited transfer capability between them, has received very little attention in the technical literature T h i s is true despite the wide spread use of the L O U index for generation planning

by individual companies and a number of large power pools throughout the industry This is due in part to technical and computational prob-

lems that the authors have pointed out The authors are to be congratu- lated for their extensive research in this area and their contributions to the solution of some of these problems

I would appreciate the authors' comments on the following ques-

tions and observations from the paper

The authors have introduce! the concept of "the probability that the tie is loaded to capacity However, it seems to me that there is the possibility of the ties being loaded to capacity under many other conditions than those stated in the paper Would not the regions of transfers equal to the tie capacity also be a function of the operating Fig 1 which represent the margin conditions which result in reserve policy of the pool, just as the regions which define loss of load as pointed out in the paper? If the policy of the pool is not to share load

loss, then regions 3 and 6 are the conditions of margins which result in

transfer of reserves equal to the capacity of the ties in order to minimize load loss However, if the policy of the pool is to share equally the load loss when the net margin of the pool is negative, then would not the regions shaded in Fig 7 represent the conditions which possibly result in ties loaded to capacity? Under this policy, the pool may try t o equalize margins in the two areas by transfer of capacity

over the ties in order to minimize the magnitude of load loss in either of the areas

I feel that the tzrminology of "the probability that the tie is loaded to capacity may be misleading Many times the tie capacity

used in reliability analysis of interconnected systems does not represent

Manuscript received July 30, 1974

the capacity of the physical ties at all It represents the maximum

transfer capability from one area to the other which may be limited by

any one of a number of possible factors including internal transnission

networks and operating constraints of each of the areas This frequently

leads to the concept of a two directional tie with different transfer

limits in the two directions as pointed out in Fig 1 of the paper

calculated These regions are the conditions of system margin which

The probability associated with the shaded regions of Fig 8 can be

result in loss of load to one or the other of the areas only because of the

transfer capability limitations That is, the probability associated with

these regions is part of the system L O U , but the size of the regions and

thus their contribution to system L O U may be reduced simply by

Fig I Regions of Margin Representing Possible Loading of Ties to

Full Capacity

Fig 8 Regions of Margin Contributing to System LOLP Because Of

Transfer Limitations

increasing the transfer capability between the areas This probability can

thus serve as an aid to system design decisions particularly regarding possible increases in transfer capability However, care must be exercised since the reliability of an individual area can actually be reduced by in- creasing the transfer capability, as pointed out in the paper for the case

increases the sizes of regions 2 and 5 in Fig 1 even though the shaded

of load loss sharing This is because an increase in transfer capability

regions of Fig 8 are reduced

The probability which the authors refer to as the probability of the ties being loaded to capacity was calculated for this purpose, I believe,

since it is applied in this manner in the results section of the paper In this respect the concept is useful However, I do not agree with the ter- minology and offer Fig 8 as possibly better serving the purpose I would greatly appreciate the author’s comments regarding these obser- vations

It is not clear to me from the “flow networks” section of the paper whether the various systems maximum transfer capabilities are input or

whether the program actually calculates these values If the program calculates them, is the internal transmission of each area represented?

I also have some questions regarding load models used by the authors in their multi area analysis I get the impression that the authors use the term random to imply independence of loads between areas If

this is so, what procedures were used in developing the margin distribu-

probabilistic model of loads used to represent the fact that they are tions to achieve the assumption of load statistical independence? Was a indeed random or was a deterministic load model used and a procedure implemented to simulate the assumption of independence?

As pointed out in the paper, area loads may in fact be statistically

dependent or correlated The various area loads, and consequently margins, may be neither perfectly correlated or perfectly independent Load correlation can have a significant affect on the system and area reliability calculations, particularly in multi area analysis Procedures are available for representation of the degree of load correlation in two area models and hopefully these can be extended to the authors a p proach of multi area analysis as intimated in the paper

A related but different phenomenon present in area loads is that of daily diversity The authors make no mention of this important char- acteristic Is load diversity taken into account in the model? I have found this consideration like correlation to greatly complicate the calculation of the joint margin probabilities associated with the various regions of Fig 1 In fact it alters the regions of Fig 1 which constitute loss of load since the system margin is not the sum of the margins of each area For a two area system, the system and area loads have the following relationships

% = L A + LB - d where Lp = pool daily peak load

LA = area A daily peak load

LB = area B daily peak load

d = daily diversity

Also L p , LA, and LB are random variables and, as pointed out, may

a random variable, although I have obtained good results by treatingit

not be independent The diversity should theoretically also be treated as

as a deterministic quantity %is results in system margin with the fol- lowing relationship:

M ~ = M A + MB + d where Mp = pool margin of available capacity

MA = area A margin of available capacity

MB area B margin of available capacity

that area In the system design example of the paper, the authors point out 3 of the system is the least reliable Consequently de*

alterations to improve the system reliability are concentrated in efforts

to improve the reliability of area 3 However, it is very interesting to note that with each area operating by itself (no ties), area 3 is the most reliable Are the authors aware of any particular characteristics of the 4

a r e a which might account for these results? I have Seen similar phe- nomena in studies i n v o h g one system tied to a much larger system volving the comparison of relative reliabilities of the areas, often lead to Unfortunately, unusual but perfectly possible results of this type, in- questionable conchsions regarding who benefits from the ties and to what degree

I wish to commend the authors for a very interesting paper I look

forward to additional work in this very important area resulting from the efforts and contributions of the authors in overcoming some very

6

difficult technical problems I will appreciate any comments the authors

may have regarding the observations and questions I have raised

L L Carver and R W Moisan (General Electric Company, Schenectady,

N Y.): This paper documents the signifcant fact that computation

times increase rapidly as the number of areas, margin states and tie

sizes are increased, Table 11 The authors have introduced a linear flow

model to keep the computations to a minimum They indicate that by

keeping the number of margin states small, ten or less in their Table 111

example, four-area and fwe-area systems can be solved Our experience

with actual utility systems indicates that several hundred to over one

thousand margin states are common in capacity outage tables and that

reducing these to approximately ten will sisnificantly affect the results

The issue of accuracy will be an important consideration in applying

this technique to system reliability studies

How did the authors classify the system state when the sufficient

areas can serve some but not all of the deficient areas? For example, in

Table 111, if Area 1 has a 50-MW margin, Area 2 a 5GMW margin, while

Area 3 is deficient by 100 MW and Area 4 is deficient by 50 M W , then

the system is deficient 50 M W , a system failure However, with a no

load-loss sharing policy the state is a success for 1 and for 2 Assuming

the non-zero ties as indicated in Table III how should 3 and 4 be

classified? Is the state a success for 3 with 1 and 2 providing their

reserves to cover the deficiency, and a failure for 4, or shall 4 be

classed as a success and 3 the failure?

probability studies The authors have provided several stimulating ideas for multi-area

Manuscript received August 6 , 1974

C K Pang and A J W o o d : We sincerely appreciate the contributions,

Garver and Moisan

discussions and questions offered by M e m Bonaert, Sullivan, Shortley,

The treatment of correlated loads (or “load diversity”) in the in-

dividual areas was raised The illustrations in the paper assumed statis-

tical independent and uncorrelated loads in each area The treatment of

perfectly correlated loads (i.e., the computation of R[E(Y)J )is identical

proceed level by level We have not considered the treatment of partial

to the computations illustrated in the paper except that they must now

correlation for more than two areas One would suspect that, as in many

other cases, the theoretical solution to the question of partial correlation

in general will be easily formulated but expenhely computed

Mr Bonaert asks if the method is applicable to more than a few

areas The algorithm described in the paper was recently applied to a 19

node system with 22 links between nodes We feel that the method

described in the paper is perfectly general, but with more complex

systems it might be appropriate to invest the effort to improve the

coding effiiency in order to reduce the computational costs The 19

node case required between 75 and 100 CPU seconds, depending upon

the total load level

to extend Mr Sullivan points out several key this approach to the treatment of voltage profiles The flow areas We have not attempted

networks of the paper are approximations to the d c power flow models

Resumably this same type of approximation could be made to the d c

var flow model Several questions arise in this sort of extension For

instance;

(1 ) What link capacity should be used with both real and reactive

power flowing simultaneously on the same link?

flow model? (2) How should one model the voltage regulated nodes in a linear

It has been our experience that voltage problems tend t o be cured

by the application of reactive generation (capacitors, condensers, etc.)

and tap changing transformers rather than by increasing circuit capabil-

ity

Both Messs Sullivan and Shortley asked about transmission limits

The method presented in the paper is applicable to generation-transmis-

sion systems generally so that, if it is important, individual area trans-

mission may be represented explicitly in the model In fact, this was the

Manuscript received October 25,1974

case for the 19 node example cited above In our experience the in- clusion of area transmission limitations can be signifkant and should be considered The penalty paid for including this is the increased cost of computations In all of these cases the various transmission link states must be described by data input

Mr Sullivan surmises correctly that the treatment of multiple minimum cuts is a problem We recognize that whenever cuts occur which have identical capacities, the probabilities should be allocated properly Our present practice is to assign the probability to whichever minimum cut is encountered fust This would not affect the assessment

of the area reliability levels, but would tend to give a pessimistic ap- praisal to some link values

The use of a network linear flow model in automatic system design is indicated in reference 9 In this application outages are neglected and a branch and bound optimization technique is used to direct the automated design The reported method Seems efficient for

moderately sized systems We feel that this method could be expanded

to include the reliability technique described in our paper, but such a step might require substantial computational effort for large systems

Mr Shortley’s discussion contains several worthwhile additions to

the paper We agree with the analysis he has indicated on Figure 7 under

the conditions he has assumed We should like to take the opportunity

to reemphasize the point that the analysis presented in this paper hopefully, stated in an explicit fashion in the paper The number of applies only for the two extreme conditions which we assumed and, possibilities appears to be quite large and one should attempt, by all means, to clarify the pool operating policies when analyzing multi-area reliability measures

quite in order We also feel Mr Shortley’s comments concerning Our choice of terminology was primarily intended to Figure 8 are facilitate the explanation presented in the paper

The system design example was designed to illustrate some of the major points in the method rather than to simulate an actual case

While it is true that in this example area 3 has the “best” reliability level

when isolated and the “worst” when interconnected, the calculated

reliability level of area 3 was improved by an order of magnitude by the

interconnection

As an extreme example to further illustrate Mr Shortley’s point

concerning the possible confusion that may arise in this type of study, consider an isolated generation area with practically no load Its com- puted loss-of-load probability is very small Now connect it to an area with almost zero installed reserve margin and a fairly high load level If the tie capacity is large and the two area pool is operating under a

area with the high reserve margin will undoubtedly increab The policy of load loss sharing, the calculated loss-of-load probability of the question of the tie “benefit” in this case is somewhat subjective since

the overall pool and other area’s loss-of-load probability have undoubt- edly been reduced greatly by the tie

This question gives us an opportunity to comment on one of the

statements concerning tie benefits in the paragraph immediately pre- ceeding the section Method of Analysis in the paper Under the assump- tion of no load loss sharing, increasing the tie capability will, of course, not decrease the lowof-load probability indefinitely as the tie size in-

creases The increased in one area’s reliability is always limited by the maximum amount of reserve available in the other area, that is by the maximum positive margin state

Messrs Garver and Moisan may be overly concerned with accuracy are felt to be required may be represented in the implementation of this and the limited number of margin states represented As many states as technique at the cost of increased computational expenses

current logic of this program would classify both areas 3 and 4 as

For the situations postulated by Messs Garver and Moisan, the failures It is, of course, a relatively simple matter to incorporate priority rules in the logic to allocate the failure probabilities in some prescribed order under these situations Such rules would be another form of pool operating policy

Again we wish to thank the discussers for their questions and contributions We hope that this paper will stimulate further activity in this area

REFERENCES [9] H Baleriaux, E Jamoulle, P Doulliez and J VanKelecom, “Opti-

mal Investment Policy for a Growing Electrical Network by a

Sequential Decision Method”, CIGRE Paper 3248,1970

5 1 7