Estimation of Electrical Power System Steady State Stability

IEEE Transactions on Power Apparatus and Systems, vol. PAS-94, no. 3 , May/June 1975

ESTIMATION OF ELECTRICAL R)WEB"SYSTDf STEADY-STATE STABILZTY

I N IDAD PIdJ CALCULATIONS

V. A. Venikov, V. A. Stroev, V. I. Idelchick, V. I. Tarasov

(U.S.S.R.)

ABSTBACT

The paper shows that in load f low calculat ions performed by Newton's method one can estimate, as a by-product of the solution, the s t eady- s t a t e s t ab i l i - ty of the operat ing condi t ion in quest ion. In order t o mke t h i s estimate -re r e l i a b l e and t o improve convergence of the i terative Solution it is expedient t o use a d i f i e d Newton's algorithm -- suppleaented wi th p roper a l te ra t ions o f the i t e ra t ive cor rec t ions . The p r a c t i c a l ways t o determine the steady-state sta- b i l i t y l imi t s a r e a l so desc r ibed i n t he pape r .

INTBODUCTION

S t e a d y - s t a t e s t a b i l i t y o f e l e c t r i c a l pover sys- tcms is frequently estimated by the methods based on the ve r i f i ca t ion o f t he a lgeb ra i c sign o f t h e l a s t term (ao) of the charac te r i s t ic equa t ion (which is equqk to the value of the h rac te r i s t ic de te rminant for opera tor p = d/d t = 0) '-'. These mthods are usu- a l l y c a l l e d as hose of s teady-s ta te aper iodic s ta - b i l i ty ana lys i s5 because they def ine opera t ing condi - t i o n s which can not loose s t a b i l i t y a p e r i o d i c a l l y , due t o p o s i t i v e real mot of t he cha rac t e r i s t i c equa- t ion. Such methods give pract ical ly exact s teady- s t a t e s t a b i l i t y e s t i m a t i o n i n t h e case of absence of s e l f - o s c i l l a t i o n s i n t h e system, i . e . t he o sc i l l a to ry process of loosing the Stabi l i ty (which corresponds to conjugate complex roots wi th pos i t ive real p a r t s i n cha rac t e r i s t i c equa t ion ) .

As it is known t h e s e l f - o s c i l l a t i o n s i n electri- c a l power systems are amstly caused by iqroper choice of autonretic control.' However, there exist thorough- l y developed methods fo r t he analysis of automatic c o n t r o l i n electrical power systems and for thechoice of such s t ructures and parameterr of theautomaticam- trol devices which practically exclude the influence o f s e l f - o s c i l l a t i o n s o n ' s t e a d y - s t a t e s t a b i l i t y l i m i t s o f t h e aystems. Thece is a la rge number o f t heo re t i - cal works and p r a c t i c a l means to ensure sa feopera t ion of automatic control, withoutcausingaelfoscillrtion i n t h e system and they provide reliable results.L2s a-B

Under these COUditiOM s t e a d y - s t a t e a p e r i o d i c B b b i l i t y a n a l y s i s becomes pr inc ipa l method for the ver - i f i ca t ion o f t he s t eady- s t a t e s t ab i l i t y o f e l ec t r i ca l power systems. Such ca lcu la t ion cons is t s in genera l

Engineering Committee of the IEEE Power Engineering Society for presentation at Paper T 74 320-8, recommended and approved by the IEEE Power System

the IEEE PES Summer Meeting & Energy Resources Conf., Anaheim, Cal., July 1419, 1974. Manuscript submitted August 7, 1973; made available for printing April 4, 1974.

of two s tages: 1) load f low solut ion for the operating condition under consideration; 2) ca lcu la t ion a f the l as t t e rm of charac te r i s ic equation or one of the p rac t i ca l s t ab i l i t y c r i t e r i a ! . Recen t ly t he re ap - peared a number of publications where it is proposed to combine ( p a r t l y or completely) these two s tages i n order to s implify the analysis nd to accelerate the computations. For instance, in f c o e f f i c i e n t Prtrh of smell dis turbance equat ions for operator p li d/d t - 0 (the determinant of the matrix is equal to a )is c a l - culated by a amdification of a Jacobian matre, used i n Newton's load flow solution. In it is proposed t o es t imate s teady-s ta te s tab i l i ty judging by the convergence of specially developed load - f low i te ra t ive solution. Generally speaking the estimation oftheelec- t r ical power sys tem s teady-s ta te s tab i l i ty dur ing load flow calculations can be accomplished bydifferentme&- ods. Eowever i t seemed =st convenient to do t h i s d u r - ing Newton's load flow solutionlo l4 This method required computation of the coefficient matrix f o r l i n e - arized equations of steady-state condition (Jacobian matr ix) . Because a. i s equal to the determinant of the l i nea r i zed d i f f e ren t i a l equa t ions of the t rans ien t p r o - cesses in the system calculated for p i tmaycoinkide under certain conditions with the determinant of the Jacobian metrix.

STEADY-STATE APERIODIC STABILITY ANALYSIS DURING LOAD PLOW SOLUTION By

NEWTON'S MEIETBOD

Steady-state nodal power equat ions o f e lec t r ica l power system can be wr i t ten in genera l mat r ix form as

W(X, Y) = 0. (1)

where X -vector of dependent (uncontrolled) variables, Y -vector of independent (controlled) variables, W -vector-function of power res idua ls .

Newton's method employs the fo l lowing i t e ra t ive so lu- t ion of (1) .

x ( i+ l )= + A X (i) I x(i)

where X(i), values of the vector calculated a t s t e p (i) and (i + 11,

L J

Jacobian matrix of (1) computed a t t h e p o i n t X(i) , A X (i) = correction of X a t s t e p (i + 1) .

A t each s tep o f the i t e ra t ive process presented$ (2) one should invert Jacobian matrix or solve the fob lowing system of linear equations -

1034

In the so lu t ion of (3) the Jacobian of (1) is cal- cufated. This Jacobian coincides w i t h t h e l a s t term of the character is t ic equat ion providing (see &pen- dix) :

1) a c t i v e parer and voltage magnitude are specif ied for each of the generator buses;

2) load buses are specif ied with the s m s t e a d y - s t a t e c h a r a c t e r i s t i c s a s i n t h e s t eady- s t a t e s t ab i l i t y ana lys i s ;

3) i n f in i t e buses a r e t r ea t ed a s slack nodes.

Only under these conditions the sign of the Jaeo- b ian g ives in format ion on s teady-s ta te s tab i l i ty o f the operating condition being computed. For instance, i f is load f low calculations P and Q a re spec i f i ed as independent variables for generator buses then double rov of the Jacobian matrix for this node w i l l correspond t o t he gene ra to r con t ro l l ed i n such a way as to m a i n t a i n w o n s t . * As such a "computational" control does no t ex i s t i n r ea l i t y one can not estinute steady- s t a t e s t a b i l i t y o f t h e system by the latter Jacobian.

The choice of the slack nodes and independentvar- i a b l e s f o r t h e r e s t o f t h e d e s in load f low problems has usua l ly no connection with the physical features of actual operat ing condi t ions but is determined from the considerat ions of computat ional eff ic iency of the i te ra t ive p rocess w i t h due regard for the constraints imposed on the operating condition parameters. There- fore the Jacobian of the load f low equations w i l l not general ly coincide with the las t term of the charac- t e r i s t i c equa t ion . But whatever the form oftbcseequa- t ions is it always appears easy to determine the Jaco- b i an fo r so lu t ion po in t which s a t i s f i e s t h e above re- quirecaents i . e. coincides with the last term of the charac te r i s t ic equa t ion . It can be done by s l i g h t Podif icat ions of the exis t ing load f low programs o r even without the Podifications - by recoding some part o f t he i n i t i a l da t a .*

Thus in load f low solution by Newton's oethod one can e s t iPe te s t eady- s t a t e ape r iod ic s t ab i l i t y o f t he operating condition either without any additional calcu la t ions or (if the choice of independent variables in load f low ca lcu la t ions does no t sa t i s fy therequi re - mente fotmrlated above) with the addi t ionalcalculat ion of the Jacobian corresponding to the last term of the charac te r i s t ic equa t ion .

To ensure e f fec t iveness o f such s tab i l i ty ana ly- sis i t is necessary to secure the convergence of New- t on ' s i t e r a t ive a lgo r i thm a t least fo r s t ab le ope ra t ing condi t ions.

Fast convergence of Newton's method is one of i t s Pein advantages which ensured wide application of the metho f o r l e c t r i c a l power systems load flow problems p3 , '' However, t h i s method is not absolutely *It can be shown with the help of transformations sim- i l a r t o t h o s e used i n Appendix f o r t r a n s i t i o n from (X) t o (XII).

W i t h the exception of the cases when s teady-state cha rac t e r i s t i c s o f t he l oads used i n t h e s t a b i l i t y a m l - ysis d i f f e r from those used in load flow solution as well as when there is no i n f h ' t e bus i n t h e system. In these cases the computation of a. m y r e q u i r e e i t h e r s u b s t a n t i a l d i f i c a t i o n o f J a c o b i a n m t r i x ( i n c l u d i n g the effect of f requency var ia t ions in the system) or spec ia l program f o r t h e s t a b i l i t y c a l c u l a t i o n s .

convergent as there ply be cases when i ts convergence is considerably upset or even violated. Bperience from load flow calculations by Newton's r?thod f o r s a era1 electrical pawar systems showed that the convergence of this nethod depends not only on c i rcu i t d ia - gram, paraaeters and operating condition but also on solae f ac to r s s t i pu la t ed by the algorithm of calcula- tions15-17. As a r e s u l t Newton's i t e ra t ive p rocess may converge to unstable operat ing condi t ion or d i - verge for a s t a b l e one*-

Besides, even i f Newton's i t e r a t i o n 8 vould con- ve rge t he s t eady- s t a t e ape r iod ic s t ab i l i t y mrg in by corresponsing Jacobian m y not be single-valued. De- t a i led cons idera t ion of these cases is outlined below

If for cont inuously changing var iables during- i terat ive process the Jacobian of the load-flow equa- t i o n s (1) does not change the s ign (i. e. does not cross zero) and i f operat ing condi t ion corresponding t o t h e i n i t i a l p o i n t o f the process in aper iodica l ly stable then operating condition corresponding to the f i n a l p o i n t of the process w i l l a l s o be sper iodica l ly stable. That means tha t cons t an t sign of the Jacobian for continuous change of variables during convergent i t e r a t i v e Newton's process for s t a b l e i n i t i a l p o i n t is s u f f i c i e n t f o r the s t ab i l i t y o f ope ra t ingcond i t ion being calculated.

A s i t e ra t ive p rocess i s executed ind iscre t steps i t is possible to check the sign of the Jacobian only a t t h e s e steps. Eere one m y not notice double change of the sign of the Jacobian during one i teration. In th i s sense cons tan t sign of the Jacobian dur ingi te re t ive process does-not mean tha t opera t ing condi t ion under consideration is aper iodica l ly s tab le .

I f t he s igns o f t he J acob ian fo r i n i t i a l and5nd poin ts o f the i t e ra t ive p rocess a re d i f fe ren t then the operating condition is unstable . It is a s u f f i c i e n t condi t ion of aper iodic ins tab i l i ty . This condi t ion is not a necessary one as loadflow may correspond to in- s t ab i l i t y a l so i n t he ca ses when s i g n s o f t h e J r c o b i a n i n t h e f i n a l and i n i t i a l s t a b l e p o i n t s a r e t h e same. The double change of the sign of the Jacobian which may occur during one i teration (as mentioned above)or a t d i f f e r e n t i t e r a t i o n s may serve as an example.

Thus t h e s t e a d y - s t a t e a p e r i o d i c s t a b i l i t y e s t i - mation for operating condition calculated by Newton's method may not genera l ly l ead to cor rec t resu l t as : 1) Newton's i t e ra t ive p rocess may d iverge for s tab le operat ing condi t ion, 2) the double change of the sign of the Jacobian m y take place during the i t e r a t i o n s which would not permi t to es t imate the s tab i l i ty o f the operating condition being calculated. ** I t should be noted that the above s i tua t ions a r e ve ry r a r e i n p r a c t i c e . For instance the resul ts of load f low calculat ions for a l a rge number of various systems have not shown even a single case of divergence f o r Newton's method provided the Jacobian coincided with the las t term of the character is t ic equat ion Double change of the sign of the JacobianhssaPpeard f o r one system when a c t i v e and reac t ive m e r f o r a l l the independent nodes were taken as independentvaria- b les which does not correspond to thecoincidencebet- ween the Jacobian and 17

a. *

m+Comection between s t a b i l i t y and convergence is ddsplayed i n the de te r io ra t ion o f convergence for ill- conditioned Jacobian matrix which m y i n d i c a t e t h e prox-ty of the ca lcu la ted opera t ing condi t ion to the s t a b i l i t y l i m i t .

1035

Nevertheless, in o r d e r t o g e t r e l i a b l e r e s u l t s i t i s expedient to d i f y Newton's method in such a wayas to avoid the drawbacks mentioned above.

The i terat ive process considered is a general iza- fion of Newton's method: ,-

L

where A ( i )

i n t h i s manner one can de te rmine the s tab i l i ty l imi t with suff ic ient accuracy.

This d e is most e f f e c t i v e when the change i n i n i t i a l o p e r a t i n g c o n d i t i o n is achieved by the increase of mer generation in one or severa l nodes or by the r e d i s t r i b u t i o n of the generation between thenodes.Its advantage is t h a t s p e c i f i c a t i o n o f t h e v a r i a t i o n s o f the angles instead of act ive power does no t l ead tod i - vergence of Newton's method whi l e app roach ings t ab i l i - t y limit.

The second d e suggests that the unstablecondi- t i o n is achieved by the changes in in i t ia l s tab le con- d i t i o n up to the point when Newton's method diverges= modified Newton's method "suspends". After that one should specffy a new in i t ia l es t imate wi th an oppos i te sign of the Jacobian and with the values of P f o r gen- erators corresponding to the last i t e r a t ion be fo re t he mentioned "suspending". In t h i s ca se t he i t e r a t ions w i l l converge to the solut ion with the opposi te s ignof the Jacobian corresponding to unstable operatingcondi- t i on . In o rde r t o ob ta in t he i n i t i a l e s t ima tewi th the sign of the Jacobian opposite to that of the stable- d i t i o n it i s s u f f i c i e n t t o s p e c i f y f o r t h e nodes where power is varied during the burdening of the operating condi t ion the values of the vol tages angles s l ig 'ht ly larger than those for the previous s tep. The values of a l l the o ther dependent var iab les may be l e f t unchang- ed. It should be noted tha t w i th t h i s =de of calcula t ion o f the l imi t ing opera t ing condi t ion Newton's method may not converge from the i n i t i a l e s t i m a t e chosen a s t he l a t t e r may be very close to the space IdW /

---

ax1 = 0.

This mode of calculation of the unstable condi- t ions is most e f f ec t ive i n t he ca se when the changes in operating, condition of the system are achieved by increasing the loads in the specif ied nodes of thesys- tem and/or by red is t r ibu t ing the genera t ion between the nodes.

The following feature of the modified Newton's method which allows to implement the second mdein.the most simple way i s worth n o t i n g. While approaching t h e s t e a d y - s t a t e a p e r i o d i c s t a b i l i t y limit the value of K in (5) increases with each step of change in opera t - ing condition. This value is defined by the component of vector A having maxinum modulus. Select ing several components with large uodul i and learning the numbers of these components and consequently the numbersofcor responding nodes one can determine most "strained" nodes. Knowing the mode of variation of angles and vol tages in these nodes for several operat ing condi- t i o n s one may choose the i n i t i a l e s t i m a t e c l o s e t o t h e s t a b i l i t y l i m i t and change only values of 6 and U i n the above nodes in such a way so a s t h e i t e r a t i o n s w i l l converge to uns tab le so lu t ion . In p rac t ica l use o f the d i f i e d Newton's method i t appeared suf f ic ien t to determine the number of the node corresponding to the determination of K. The experience gained by numerous ca lcu la t ions have shown tha t the most "strained" node determined in this way coincides with (or i s e l e c t r i - cally connected to) one of the controlled nodes the parameter of which was varied according to the burdening.

Load f l w s o l u t i o n s f o r s e v e r a l e l e c t r i c power systems by the modified Newton's method have ahownthat the rate of convergence does not dependconsiderablyon the proximity of the solut ion to the s tabi l i ty limit. When the burdening was car r ied on with the constant s t e p and ca1cu:ations for each subsequent step began

from the solution of the previous one the number of i t e r a t ions fo r t he l a s t s t eps i nc reased on the average by 1 o r 2. I f t h e s o l u t i o n f o r t h e i n i t i a l step was chosen a8 an in i t ia l po in t for each of the fo l lowing s teps the number o f i t e r a t ions fo r f i na l a t epa i nc rease by 2 o r 4. As an example t h e t o t a l numbers of in te r - a t ions for the load f law solut ions by the d i f i e d New- ton 's method f o r t h r e e e l e c t r i c a l pover systems are given in Table (2). For each of the step changes of opera- t ing condi t ion for these systems the ini t is lapproxima- t i o n was a normal f l a t s t a r t . It ia seen from Table(2) t ha t t he number of i terations does not increase consid- e rab ly while approaching the s tab i l i ty limit. It should be noted tha t t he i nc rease i n t he number of i t e r a t i o n s is a l s o caused by ' b v i n g o f f " t h e i n i t i a l p o i n t from the solution.

Table (11

Character is t ics of Test Systems

System Branches Nodes

Number of Number of

1 7 8 ~

2

52 3

47 25

63

Table (2)

Number of I t e r a t ions fo r D i f f e ren t

Operating Conditions

System Operating Condition 1 5 S.L. 4 3 2

1

7 7 7 5 4 3

8 7 6 5 4 2

7 5 5 4 3

S. L. ( S t a b i l i t y L i m i t ) CONCLUSION

1. It is expedient to combine the analysis of steady- s t a t e a p e r i o d i c s t a b i l i t y o f e l e c t r i c a l power systems with load f low calculations for these systems which may considerably reduce the volume of compu- ta t ion .

2. It is most convenient to carry out such analysis with Newton's load f low program. In order to improve convergence of the iterations and o b t a i n r e l i - ab l e r e su l t s i t is exped ien t t o md i fy Newton'r method by introducing addi t ional correct ions of the i t e ra t ive s teps .

3. It is proposed t o modify the existing Newton's load flow programs SO as t o ob ta in s t eady- s t a t e aperiod- i c s t a b i l i t y e s t i m a t i o n .

103 7

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1038

f APPENDIX

Relationship between the Jacobian of load flow equa- t ions and the last t e rm o f t hecha rac t e r i s t i cequa t ion .

Load f low ca lcu la t ion for an e l e c t r i c power system cons i s t so f the so lu t ion o f t henon l i aea requa t ions ,

w(xsY) 0 , (1)

where W - m-order vec tor func t ion of paver res idua lsa t system nodes (m

n, - operat ing condi t ion parameter used for exci- ta t ion control of gcneratok "i",

W,(p) - t ransfer func t ion of the exci ta t ion regu- l a to r o f the 1- th genera tor wi th respec t to n,. The other designat ions are the same a s i n (11) - ( I V ) .

Let the system has a t l e a s t one node where vol t - age is constant i n magnitude and angle ( i .e. there is a n i n t i n i t e bua i n the system). Then the last term of the charac te r i s t ic equa t ion for (v) - (VIII) is equal to the determinant of the system of the equations c a l c u l a t e d a t p - 0. For this case the equations(VI1- (VIII) w i l l not change and "equations (V) - (VI) will be as follows:

i - l , . . . ,k. Taking into account that i n general 1

where Ui,II - terminal generator voltage and s t a t o r cur ren t ;

khi, koIi- regulator vol tage deviat ion and cur ren t gains ,

one gets equat ions (X) i n the form

i - * l,..., k.

As it i a laown the value of k f o r modern regu- l a t o r s is of the order of 50-200 u%!tr of excitation per uni t o f vo l tage and fur ther increase i n k h 8 S practically very #light inf luence on at.&- s t a te c h a r a c t e r i s t i c s o f t h e s y r t e l . Thus without a n o t i c e able error i n t h e r e s u l t one may ass-

koui + ? (i - l j * - - , k ) s and (XI) i r reduced t o

AUi 0, i l , . . . ,k (XI1 1

Subs t i tu t ing (HI) into (U), the f o l r n o g equations f o r the geoe r r to ra t o ca l cu la t e the last term of the c h a r a c t e r i s t i c equation of the system considered is obtained

i-l,......, k

From comparison of (11) - (IV) with (XIII),(VII)and (V 111) it follows that the Jacobian of load flow equa- t i o n s c o i d i d e s w i t h t h e last term of cha rac t e r i s t i c equation (with the error caused by the subs t i tu t ionof

f o r f i n i t e v a l v e s of k )only i f the sys- h d s - a n i n f i n i t e bus and i f i n %!id f low calcula- t ions : 1) P and U are specified as independent variables for generator buses; 2) loads buses are taken into account by t h e same s t eady- s t a t e cha rac t e r i s t i c s as i n s t a b i l i t y c a l c u l a t i o n s ; 3) i n f in i t e buses a r e taken as s lack nodes.

These conclusions are also valid when U' and U" are chosen as dependent variables.

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Discusoion

R H. hpp (General Electric Company, Schenectady, New York): I compliment the authors on their work, There is indeed a close connec-

work in the steady state, hnd steady state stability. This paper focuses tion between a load flow, which itself represents the solution of the net-

upon that connection in detail, and some questions came to mind to which the authors may wish to addres themselves.

We have had similar experiences as the authors, that Newtons Method may fail to converge to a stable operating condition and may converge at an unsuitable operating condition. The authors have intro- duced a modified Newtons Method by the introduction of a X parameter in the correction equations of Newtons Method (equ. 4). The parameter appears to be a deceleration factor. In ordinary load flow execution, do the authors only use a nonunity X if they sense that Newtons Method is not properly converging? It was also interesting to observe from Table 2 that the number of iterations required to converge did not increase as the stability limit was approached for the three

question arises whether this conclusion holds true for systems of say systems given. Since the three systems are relatively small in size, the

an explanation of (5) and (6) for the readers convenience. Finally, per- two thousand buses? It may be helpful to include a brief derivation and

haps the authors could discuss their work in the Appendix from a purely physical or system standpoint, since it appears to represent the justifica- tion of the procedure.

Manuscript received August 5,1974.

V. A. Venikov, V. A. Stroev, V. 1. Idelchick, and V. I. Tarasov: The authors wish to thank Mr. H. H. Happ for his comments.

Manuscript received January 7,1975.

would like to note that algorithm (4)-(6) with automatical setting of X Regarding the question concerning the choice of the parameter we

(either X = 1 or X < 1) has been used. For the convergence estimation of this algorithm operating conditions close to stability limit and unstable ones were computed for 14 different electric power systems (with the number of modes up to 102). The majority of the systems considered had 500 KV transmission lines, series capacitors, three winding trans-

very low impedance. Newtons method (x = 1) diverged for some of the formers and autotransformers having medium voltage winding with

conditions of these systems in oscillatory manner. Calculations according to (4)-(6) had always ensured solution. Besides, for monotonic iterative process X = 1, that is the algorithm coincided with that by Newton. For oscillatory process X was set less than 1 and in 2-3 iterations, as a rule, the process changed to a monotonic one with X in- creased to 1.

Large systems (of more than 200 nodes) were not investigated. However, the ability of the modified Newtons method to converge to zero Jacobian solution (19) Ieads to the conclusion that a small increase in the number of iterations while approaching stability limit will be characteristic of the solution of larger systems as well.

process by setting X automatically. The mathematical derivation of The purpose of (5), (6) is to account for any kind of the iterative

these expressions is rather cumbrous for the discussion. The necessary details are outlined in (1 9).

The steady-state stability criterion used in the paper is in essence the synchronizing power criterion generalization for complex power systems, Intuitively it is clear that this criterion is of the same nature as the Jacobian of the converged load flow, both describing incremental behaviour of the power system in the small vicinity of the steady-state operating condition considered. As it is shown in the paper (in Appen- d i x , in particular) for the criterion and Jacobian to coincide the load flow problem formulation should reflect the actual behaviour of the system after a small disturbance.

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Estimation of Electrical Power System Steady State Stability