IEEE Transactions on Power Apparatus and Systems, Vol. PAS-100, No. 3, March 1981

GENERALIZED GENERATION DISTRIBTlrION FACTOFRSFOR POWER SYSTEM SECURITY EVALUATIONS

Wai Y. Ng

Ontario Hydro, Canada

Abstract - A set of Generalized GenerationDistribution Factors (GGDF's) is developed to replacethe conventional Generation Shift Distribution Factors(GSDF's). This model relates the line flows withgenerations for a given network configuration. Beingin an integral form, new flows on lines' can beobtained directly without running load flows whentotal system generation changes. Conforming loadchange is also demonstrated. These new factors areespecially suitable for constraint formulation inmathematical programning, such as optimal generationdispatch with security contingencies 'considered.

different system generation levels. The basic conceptbehind this set of GGDF's will be given in Section 2.The formal derivation will be presented in Section 3.In Section 4, we shall show that the overallgeneration change is being absorbed by the system loadin a conforming manner, which is a very desirableresult. A simple numerical example, illustrating thecalculation and use of GGDF, is given in Section 5,followed by a conclusion in Section 6. An appendix isincluded with the derivation of the GSDF's for thepurpose of completeness.

2. The Concept of GGDF1. Introduction

The use of sensitivity methods in system securityand contingency analysis remained very popular [l, 2)

despite recent advancements in -fast load flowtechniques.' Their simplicity in derivations,linearity and physical comprehension remain as theirimportant appeals for wide acceptance in the utilityindustry.

Generation Shift Distribution Factors (GSDF's),together with the Outage Distribution Factors (ODF's),are perhaps the two most important sensitivity methodspresently used. We shall concern ourselves here onlywith GSDF's.

The number of GSDF's required for analysesdepends-on the number of lines to be monitored, whichusually is small even in a very large system. Hence,the use of GSDF's for calculating 'line flows aftershifts of generation is comparatively much simplerthan running a complete load flow. 'This is especiallytrue when second order multiple security contingencyconstraints have to be observed. In this case, a -hugenumber of load flow runs is required, which, up tonow, still poses unsurmuntable constraints on thesystem' operation computer.

The GSDF's, besides being linear factors, are notwithout limitations. As indicated bty the name,-GSDF'sare only useful for determining line flows whengenerations are shifted. 'Whenever the total systemgeneration changes, a new load flow has to be run toreestablish the initial line flows. This is ratherinconvenient because generation level is a forever-changing quantity in a system.

In order to overcome this limitation on GSDF's,we shall, in this paper, propose a set of GeneralizedGeneration Distribution Factors (GGDF's) which can beused independently to' establish line flows for

80 SM 591-8 A paper recommended and approved by theIEEE Power System Engineering Committee of the IEEEPower Engineering Society for presentation at theIEEE PES Summer Meeting, Minneapolis, Minnesota,July 13-18,1980. Manuscript submitted January 23,1980; made available for printing April 23, 1980.

The GSDFequations:

A Fi-kAG + AGR =

where: AGG

A Fj-k

AGR

can be defined by the following

(1)= AJk AGgA2kG

0 (2)

= change in generation of generatorg, excluding the referencegenerator R

= change in flow on line i-k (frombus to bus k), due to shiftingAGg amount of generation from a

predesignated reference generatorR to the generator g

Akg = a proportional constant, or GSDF,for line i-k, due to shift ofgeneration on generator g

= change in generationreference generator R

in the

The linearity of Equation (1) permits the use ofsuperpositions, ie, shifting generation from one

generator to any generator via the reference generatorR, provided that the total generation or load of thesystem remains unchanged. This can be formulated as a

constraint.

E G = E L. = constant (3)g i1

where g and i are suned through respectively all thegenerators and loads.

It should also be noted that Equation (1) is inan incremental form. Given all the A's, only thechange in flow on a particular line due to generationshifts can be calculated. For security andcontingency analyses, however, the actual flowsinstead of their incremental values are required.Hence, the initial flows on all the lines must besuplied a priori by running a load flow. The valuesof GSDF's, depend only on the network configuration butare independent of the total generation on the systemand the distribution of generation or load. However,because the initial line flows must be known, to applythe GSDF's to generation shift analyses, any change in

1981 IEEE

1001

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1002total system generation will require that a new loadflow be run to reestablish the initial flows.Therefore, the GSDF's are useful only when the totalsystem generation (or load) remains unchanged.

In order to overcome this inconvenience in theuse of GSDF's, we shall propose a new set of GGDF'sdefined by the following equation:

Fk = D G (4)-k 9g f-k,g g

where the summation goes through all the generators,and

Fi-kG9

= actual power flow on line 2-k

= generation of generator g

D}ikg = GGDF for line 2-k, due to generator g

The advantage of using GGDF is apparent.Firstly, no reference generator has been necessaryhere, nor the constant total generation constraint asgiven by Equation (3). Furthermore, the physicalmeaning of Di-k,g is obvious: it represents theportion of generation supplied by generator g thatflows on line i-k.

It should be pointed out that, in fact, Equation(4) does not uniquely define the GGDF's. This meansEquation (4) cannot be used to determine the D's bymeasuring the F's and G's. However, given the D's andG's, Equation (4) defines the F's uniquely. We shallestablish that the D's are indeed unique and derivethe D's from the base case load flow, thus reducingthe GSDF method to a special case of GGDF.

3. Derivation of GGDF's

From Equation (4), the definition of GGDF, if aparticular generator g is increased in generation bysame amount AG , the flow on line J-k will be:

g

factors, we need, therefore, m umore equations. Fromthe definition of the GGDF, the flows and generationsin the base case load flow must also satisfy Equation(4), which now represents a group of m additionalequations for calculating all the D's.

Assuming all the A's are known (calculation ofA's given in the appendix), the solving of these (n xm) equations is, in fact, very simple. By shiftingall the generations from all the generators to thereference generator R, ie, AG = -Gp, from Equation(1), we have:

.,

Fk - F = - EAkA Gi-k j -k

p Jkppp # R (9)

where p sums over all the generators but R

F = final flow on line i-k after the shifts1Q-kFek = original flow on line Q-k before shifts

On the other hand, from Equation (4):

F" - ED G + D G2-k - 1-k,,p p T-k,R R

where p sums over all the generators but R and

to

G = final generation from generator p which isnow reduced to zero

GR = final generation from the referencegenerator R

Therefore:

.. D-kR CR (10)

of

Now GR, after the generation shifts, contains thetotal system generation, or:

F.2k = E D_kp Gp + D AGi- -,p I-k gg

where p sums through all the generators, including g.If we now choose arbitrarily a reference generator R(with R :' g) as in the case of the GSDF's, anddecrease its generation by the same amount AGg, theflow on line Q-k after this generation shift will be:

if

= E D G AG D AG-k -k,p p 2-k,,g g 1-k,R g

(6)

From Equation (4), we have:

ED G~ Fj-kp p i-k

where F,k is the original line flow.

Therefore, Equation (6) becomes:it

Fyk FQ-k (D2-k,g D,.kR) AG,

Fe k iF-k = Yv-k,g D>-k,R) 9F F (DI - D AGg (7)

Comparing Equation (7) with Equation (1), we obtain:

Dek,g Dj-k,R -k,g (8)

For a system of n generators and m transmissionlines, Equation (8) represents a family of mn(n - 1)equations. To fully determine all the (n x m) D

with q- surmming through all the generators including R.After substituting Equations (10) and (11) into

(9) and some rearrangements, we finally obtain:

Fe E A GDkqpR -k,p p(

If F -k and G are supplied by a base case load flowand all the A's are calculated, the calculation ofDJ-k,R for all the lines becomes trivial. Once all

the D2k,R are obtained, the rest of the D's can beeasily calculated using Equation (8).

4. Conforming Load Change

In using GSDF, the total generation in the systemis kept constant. The local load of each bus remainsunchanged in the generation shift process. When usingGGDF, since the total system generation is no longer aconstraint, the net change in generation has to beabsorbed by the system load. We shall show that thisnet system generation change is being absorbedconformingly. This, indeed, is a very desirableresult as exactly the same is usually done in a change-

(5) G =1Gq

(11)

(12)

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case load flow to produce the new initial flows whensystem generation level changes. Specifically, ifonly shifts of generation occur, the change in eachindividual load will be zero, reducing to the case ofusing GSDF.

Busy1-y

1003

N

i l i-y,R y(17)

After substituting Equation (17) into Equation (16),we finally have:

AGR ALy

G Ly(18)

which represents conforming load change in bus y dueto change in generation on bus R. Since the selectionof bus y is arbitrary, all the loads in the system,including that of bus R, will change conformingly.Finally, since the choice of bus R is also arbitrary,the general result of conforming load change isestablished.

N-y

Ly

FIGURE 1

5. Illustrative Exanle

In this section, a simple five-bus system, takenfrom Reference [3] , is used to demonstrate thecalculation and usage of the generalized D factors.Conforming load change will also be illustrated.

Consider a particular bus y in the system with alocal load Ly and a generator injecting power Gy intothe bus. Connecting to this bus y are, say, N linesdenoted by i-y, where i runs from 1 to N as shown inFigure 1.

If we denote Fi.y to be the power flow on line i-y, from bus i into bus y, we can equate the net powerflow into and out of bus y by the following equation:

NE F. + G Li=l i-y y y

(13)

By considering in the system a bus R where R /d y, wecan differentiate Equation (13) with respect to GR toobtain:

N_

AF. AL

i=l v = A

Differentiating Equation (4) with respect to GR, wehave:

AF.Diy =RD

Ruto(1)ieby oe

Equation (14) then becomes:

N ALE D.i=l i-y,R AGR

(15)

and after multiplying both sides of Equation (15) byG, the total original system generation, we have:

N ALlyil Di-y,R G = GR (16)

Consider in the original system if all the generationswere shifted to the generator in bus R, according toEquation (4), the definition of GGDF: the flow on

line i-y will be (Di-y,R G), since the generationsfrom all other generators are reduced to zero. Itfollows that total net flow into bus y, will be:

(14)

BUS 1101.5 Zl11.11

110.668

144.0 86.'

N

0

BUS 598.1. 5.81

rt | 17AID

toN -

BUS 497.2Z 6.33 I

33.07

36.2

- j

Reactance on 200 MVA base

BUS 298.0Z. 8.00

BUS395.3Z/ 15.68

FIGURE 2

The system configuration is shown in Figure 2.Voltages and line flows from a base case load flow are

marked, together with the generations, loads and linereactances used.

Gy

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1004Using Equation (A.3) developed in the Appendix,

the GSDF or A factors were calculated with bus 2 asthe reference bus and are given in Table 1. Alsoincluded in Table 1 are the GGDF or D factorscalculated using Equations (12) and (8).

By shifting 10 MW of generation from bus 1 to bus3, the new flows on all the lines were calculatedseparately using both the A and D factors. Theresults recorded in separate columns in Table 2 areidentical as expected.

To illustrate the feature of conforming loadchange, the generation in bus 1 was increased by 10 MWto 154 MW, ie, an increase of 5.28 percent of thetotal system generation. The power flow on each linewas then calculated using Equation (4) with the newgeneration pattern employing the D factors. The newloads on each bus were obtained from Equation (13).In order to screen out the effects of losses in thecomparison, the original load on each bus wascalculated using the base case line flows and Equation(13). The change in each individual load was thencalculated. All these results were recorded in Table3. In column 7, we note that the change of load ineach bus is very close to 5.28 percent, thusdemonstrating conforming load change.

In order to compare the predicted line flowsusing the D factors method, a full changed case loadflow was run with the new generation in bus 1 and allloads increased by 5.28 percent. The results,tabulated in Table 4, agree extremely well within theno-loss approximation used in the distribution factorsmethod.

Table 1

GSDF

Ai-k, l I-k, 31 0.00000 -0.47730 -0.47750 -0.52280 -1.0000

D-k, 10.54560.15480.06210.13780.0081

GGDF

D2-k, 2-0 .45440.15480.06210.13780.0081

'j -k, 3-0.4544-0 .3225-0.4154-0.3850-0.9919

Table 2

Line flows after 10 MW have been shifted froin bus 1 tobus 3.

Base Case58.0012.78-4.778.00

-33.07

Line Flows (MW)From GSDF48.008.009.552.77

-43.07

From GGDF48.008.009.552.77

-43.07

Table 3

change after generation in bus 1 increased by 10

Generation (MW)Old New A (MW)

144.000 154.000 1010.668 10.668 034.600 34.600 00.000 0.000 00.000 0.000 0

Load (MW)Old New86.00 90.5547.89 50.420.00 0.0036.30 38.2117.55 18.47

Al.5.295.28

5.265.24

Table 4

Comparison of power flow on each line using the Dfactors and a full load flow after generation in bus 1changed to 154 MW and all loads increased by 5.28percent.

Linesi-k1-22-55-42-44-3

Base CaseLoad Flow

58.0012.78-4.778.00

-33.07

Line Flows (M)Change-CaseLoad Flow

63.4614.30-4.229.36

-33.07

From GGDF

63.4514.33-4.159.37

-32.98

6. Conclusion

We have presented here a new set of generalizedsensitivity factors. In contrast to the conventionalGSDF's, the new GGDF's do not require a referencegenerator, nor the invariance of the total systemgeneration; the iimediate benefit being theelimination of running load flows whenever systemgeneration is changed for reestablishing initialflows. It has been shown that the GSDF's reduce to aspecial case of the more general GGDF. Conformingload change due to change in total system generationwas also demonstrated as a feature of these newsensitivity factors.

Besides replacing all the applications of GSDF's,the GGDF's, being in integral form, are especiallysuitable for formulating constraints in mathematicalprogranTring, such as to produce optimum generationschedules under security and contingency constraints[4]. Furthermore, fixed flows on certain lines caneasily be maintained.

The GGDF's have also been applied successfully tolarge interconnected systems where generation and loadchange in only one area is of interest. Fulldescriptions of the model and results will be reportedin another paper under preparations.

ACKNOWLEDGMENT

The author wishes to thank Mr. A.J. Harris andMr. P.T. Chan for their critical ccrurnts on thepresentation.

REFERENCES

H1H.D. Limner, "Techniques and Applications ofSecurity Calculation Applied to DispatchingComputers", Third Power System Computation

ri Conference, Rome, 1969.12 S.A. Anafeh, "Real-Time Security Assessment with

Fast Optimum Generation Shift Control", IEEEConference Proceedings PICA, May 1977.

E31 L.K. Kirchmeyer, "Economic Operation of PowerSystems", pp 118 to 120, Johns-Wiley and Sons,1967.

[4 D.M. Frances and P.T. Chan, "An LP Model forEconomic Power Generation Scheduling Subject toTransmission Limitations", TIMS/ORSA Conference,New Orleans, 1979.

Lines1-k1-22-55-42-44-3

Linesi-k1-22-55-42-44-3

LoadMW.

Busl2345

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APPENDIX

DERIVATION OF GSDF

From Equation (1), the definition of GSDF, wehave:

A AFj-kA-k,g AG9

= g-k&Ig

where AI t-k is the change in current in line J-k dueto shift of generation AGg from R to g, and AI is thechange in injection current into bus g; volUages atall busses being taken as 1 pu.

Using the definition of the reactance matrix X,and the dc approximations, we have:

AIik =A4 AVkXik

= Xig - Xk. AIg (A.2)

Xikwhere XI g, Xkg are elements of the reactance matrixand xik the line reactance.

Substituting (A.2) into (A.1), we obtain finally:

AP = -P + Q1 D P ( ) 01 g1 d1 1ilD 1()

AP =P0 +9i p+p Vm+l dm+l m+l D + Pm+l (

APm Qdm+l + tm+lQ D + Qm+ (V) = 0

(A.1)or E7(PgIv PD) = 0 for short

where the l's are the participation factors of the frequency-dependent portion of loads and/or generators. The v vector is:

7t= [62 (63p. . m+lIVm+l1 6m+2I Vm+2IF ...]

Thus, there are n + (n-m) equality constraints and n + (n-m)unknowns (all the v's plus pD, the total system frequency-sensitive de-mand). The distribution factors can then be found from:

drPt_*|'~~~~~~~~~~~o!FJZF-k d;Y +FtX dPD

0L-k,g P h(PgV PD)- V- g 7 ;

The term is very simple in structure: it is a row vector of dimen-sion (n-i) + (n-m) with exactly four nonzeros. Through a reasoningsimilar to the one in [1] it can be concluded that:

J Qk,gD + D= aF-k)tDk 1 t -

Wai Y. Ng (M'80) received hisB.A.S . degree in engineeringphysics from the University ofToronto in 1967 and Ph.D. intheoretical physics in 1973.

In 1977 he joined OntarioHydro Power System OperationsDivision, where he develops new

techniques for resource

utilization.Dr. Ng is also a menber of

the American Physical Society anda registered Professional Engineerin the Province of Ontario inCanada.

Discussion

F. L. Alvarado, (University of Wisconsin, Madison, WI): The author isto be complimented for an interesting paper in which he has correctlyperceived the limitations of Generation Shift distribution factors. Weoffer two comments, aimed at extending the usefulness of the paper.

(1) We feel that rather than defining new non-incremental distribu-tion factors, as in (4), it is better to stick to the traditional ones as

in (1), but to allow for a dependence of these on the total genera-

tion of the system. We thus suggest defining:

0 + D .PFi k =F-k + gi

(2) The distribution factors can be calculated from the transposedJacobian Equation (15) of [1] expressed the system equations in auseful manner. Assume n buses and m generators (no "slack"generation is needed, but the angle d, is fixed):

.t pQ1 DQZk,g + Q1I DQZk,l 1

From this expression the exact distribution factors are found. J is the or-

dinary power flow Jacobian matrix. Although D,,, is a vector of dimension(n-I) + (n-m), we are only interested in the values of D,,, that correspond togenerators. Thus, J should be ordered to keep all generators in the lower right-hand corner. This way only a partal back substitutions is required to solve foreach D,k,,,, vector. Note that further simplifications are possible, as in 11].

REFERENCE

[1] F. L. Alvarado, "Penalty Factors from Newton's Method", IEEETransactions on PowerApparatus and Systems, NovemberDecember1978, pp. 2031-2039.

Manuscript received August 11, 1980.

Wal Y. Ng: The author wishes to thank the discusser for his interest in

this paper.For checking system security, it is the total flow on a line, rather than

its incremental value that is often required. However, the definition of

GGDF given in equation (4) is very general. It can be used to calculate

either the total or incremental flow on a line. When the latter is of in-terest, the LHS will be the incremental flow and the RHS generationswill be the incremental generations. It should perhaps be noted that the

symbol P,, used in the first equation in the discussion should really beAP,1 = P, - P°,,.The concept of GGDF and its application are independent of its

derivations. One method of deriving GGDFs from the conventionalGSDFs was given in the paper. The alternate approach pointed out bythe discussor for calculating the distribution factors directly from thetransposed Jacobian is enlightening, and probably would be more ac-

curate as well, although the physical meaning of generation distributionfactors may be lost within the mathematics.

Manuscript received September 22, 1980.

A1-k, gX - x=q kg

XSk(A.3)

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