of the load flow voltage solution with respect to z near a Jacobian singularity. These theoretical results provide an analytic confirmation of several properties ...
IEEE Transactionson Power Systems, Vol. 8, No. 2, May 1993
646
A SIMPLIFIED APPROACH TO ESTIMATE MAXIMUM LOADING CONDITIONS IN THE LOAD FLOW PROBLEM
ZC. Zeng, F.D. Galiana, B.T.Ooi, N. Yorino Department of Electrical Engineering McGill University Montreal, Quebec, Canada ABSTRACT This paper describes a computationally efficient and simple approach to estimate maximum loading conditions in the load flow problem. These operating points are known to result in a number of undesirable phenomena such as the singularity of the Jacobian, solution bifurcations, and voltage collapse. The approach presented here generates precise estimates of the maximum possible amount of load increase that the system can tderate along a specified path, as well as the corresponding vdtage vector. The method described here Is simple and efficient since it is based primarily on conventional load flow solutions. Tests on several standard power networks confirm the accuracy and efficiency of the technique. INTRODUCTION The control of voltage and reactive power has become as critical to power system reliability as the control of real power. This was not always the case in earlier and less stressed power networks when var allocation and voltage control could be safely carried out in an ad hoc manner. Nevertheless, uncoordinated var and voltage control is still common today, and several recent major system failures have been directly attributed to inadequate voltage support [Fink, 19911. In particular, the problem of voltage collapse has received Considerable attention as a result of several events in Italy, Japan, France and Canada ascribed to this phenomenon. It soon became apparent that voltagereactive power phenomena are very complex and not well understood. Consequently, in the last ten years 92 WM 307-9 PWRS A paper recommended and approved by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society for presentation at the IEEE/PES 1992 Winter Meeting, New York, New York, January 26 - 30, 1992. Manuscript submitted August 28, 1991; made available for printing January 9, 1992.
or so, the power industry has devoted considerable research to deepen its understanding of these phenomena (as evidenced by the numerous presentations at the EPRl and NSF Workshops on Voltage Collapse and Stability of 1989 and 1991). The problems associated with voltage control are quite broad, and several causes for voltage instability have been identified [Yorino et a1,1991-a]. Some are due to static phenomena, others to dynamic characteristics, and still others to control mechanisms such as tap-changers [Yorino et al, 1991-b]. In this paper, attention is focused on the static load flow equations and in particular on those undesirable operating points akin to "black holes", where the load flow Jacobian is singular, system voltages collapse, voltage solutions bifurcate, and loads can no longer increase. These so-called maximum loading conditions (or load flow singularities) constitute hard limits which well designed and operated power networks must avoid. Evidently, in order to avoid such operating points, one has to be able to understand and identify them first. Methods to numerically find load flow singularities have been developed by several authors [Alvarado & Jung 1989, Dobson & Lu 1991, Ajjarapu 19911. They essentially determine the maximum possiblevariation of the real and reactive power injections before a Jacobian singularity condition is reached. The approach followed by these authors is to augment the load flow equations by a set of relations representing the null space of the singular Jacobian matrix (see equations (15)-(17)).This augmented set of equations is then solved numerically for the corresponding voltages and the null space vector. An alternative approach is presented in this paper to calculate maximum loading conditions (or load flow singularities). The method is simple since it relies primarily on conventional load flow simulations. The approach is based on recent results by [Galiana & Zeng, 19911, where the'behaviour of the voltages near
0885-8950/!33$03.00 0 1992 IEEE
647
a singularity is characterized in analytic closed form fashion as a function of the injections. The efficiency of the approach is demonstrated on the IEEE 30 and 57 bus networks. THEORETICAL BACKGROUND The theoretical background for this paper as well as numerical tests are detalled In [Galiana & Zeng,1991], and summarized in this section.
x. The higher order term Q(dx) being negligible for sufficiently small changes. However, if we let the expansion point x, = q,at which value the Jacobian matrix is singular, then equation (2) cannot be solved by the sensitivity relation (6) since the inverse of L(&) does not exist. Nevertheless, from numerical simulations, it is known that voltage solutions do occur for some injections z near z, = F ( 4 . In [GalianaUeng, 19911 it Is proven that the behaviour of x for z near a singular pdlnt, that is, for,
z=z.+~z Denote the load flow equations as,
is given by, z = F(x)
(1)
x=q+dx where z is the vector of bus injections, while x is the vector of bus voltage components In m a r or rectangular coordinates. If the Injections are functions of x ( e.g. voltage dependent loads), then this dependence is assumed to be imbedded in the fundon F(x), so that z represents only the constant component of the Injections.
where dx is a function of dz defined by,
+ Wdp
dx=vdo do =
?
J[2 aT dz / aT Q(v)]
L(4 v = 0 The behaviour of (1) around an arbitrary operating point, x,, is given by,
L(xJ~a =
o
dp = [WT L(xJ w]-' WT dz' dz'= dz - L(v)v [aTdz/aT Q(v)]
where the Jacobian matrix, L(q), is defined by,
(3) and where,
The term Q(dx) contains second order terms in dx (higher order terms are neglected), and is given by, Q,(dx) = $4 dxT {aFJ&)
dx
, k=l, ...,n
(5)
As long as the operating point, q,Is such that the Jacobian is non-singular, for small variations in z one can express,
Equation (6) represents the normal behaviour of the load flow problem where a 'small" change In the injections, z, results In a "small' change In the voltages,
Note that the vector v represents the null space of the Jacobian (equation (1l)),while the vector a remesents the null space of the transposed Jacobian (dquation (12)). Associated with L(xJT,there exists a range space spanned by n-1 independent vectors (wk} perpendicular to v. These vectors are represented by the n by n-1 matrix W. Equations (8)-(14) analytically describe the behaviour of the load flow voltage solution with respect to z near a Jacobian singularity. These theoretical results provide an analytic confirmation of several properties which have been experimentally observed by a number of authors, however to our knowledge, up to now, no one had derived explicit analytic relations predicting such a behaviour. One can summarize the above results by noting that the solution, dx, as a function of dz, has the following properties:
a) It is composed of the sum of two terms, v da and W d P.
648
b) The first term has a unique direction, v, but its magnitude is affected by do, a term dependent on the square root of aTdz
zs-zo+ 1 a z
c) The second term is orthogonal to the first, and it is linear in d z d) The square root term shows that zcannot vary along certain directions, ( aTdz 2 0 ), which explains the concept of maximum loadabllity. e) The square root term tends to dominate the behaviour near a singularity, yielding relatively large changes in x for small changes in z (e.g. a change in z inside the square root of 0.01 can yield a change of 0.1 in x, that is ten times bigger). Thls explains the phenomenon of voltage collapse near a singularity. f) The voltage collapse occurs along the vector v, that is, along the null space of the Jacobian, regardless of the direction of dz. The vector v therefore describes the sensitivity of the bus voltages to changes In injections near a singularity.
g) The square root component has two solutions, corresponding to the plus and minus signs. These predict and explain the bifurcation phenomenon observed by numerical simulatlons. h) These results are derived for an onedimensional singularity only. Higher dimehsional singularities are possible but very rare. MOTIVATION FOR THE PRQPOSED ESTIMATION APPROACH Although, from a theoretical point of view, these results unify and confirm observed phenomena, in their present form they are not directly useful in resolving some practical questions:
z-Zk=ZO+ 20
AZ
.
Figure 1: Direction of change of load flow injections.
for A, x,, and v. The value of )IIAzll is the "distance" to the singularity, q, (maximum loading condition) from z, along Az,x, is the singularity in voltage space, while v is a byproduct of the calculation, corresponding to the null space of the Jacobian matrix. These equations are solved through a Newton Raphson algorithm and provide exact solutions. This approach has some drawbacks however: (1)The dimension of the non-linear set of equations to be solved is twice that of the conventional load flow, (2)The set of equations to be solved, (15)-(17), is unconventional, (3)The approach requires a good initial estimate of v in order for the Newton method to converge, (4)Guessing a good initial value for v is not evident. What we demonstrate in this paper is how to exploit the new theoretical results summarized above in the section on theoretical background to numerically estimate A, x,, and v by using mainly conventional load flow solutions.
(i) How does one numerically find a singularity x,? (ii) More specifically, if we start at a given point ~0 and move along a given direction Az, how far can one move before hitting a singularity, z,? (Figure 1). Several authors [Alvarado & Jung 1989, Dobson & Lu 1991, Ajjarapu 19911 have dealt with this question by developing algorithmsto numerically solve the following problem,
Tl-lEPROPOSEDAPPROACH To estimate q, we first define a direction (and magnitude) of injection changes, A z , from a given initial operating point, q,.Then, define a sequence of injections { z , } by,
649
for k = 0,1,2 ,... . As seen In Figure 1, this sequence may eventually leave the load fluuv feasibility region for k greater than some limit wmsponding to maximum loading. Hwevw, near the boundary, the behaviour of x,, the corresponding voltage solution to 4, must satisfy the analytic equations (8)-(14). The proposed approach, therefore, obtains a number of pairs (x,,q) by &ng the load flow q=F(x,J for k=1,2, .... These pairs are then substituted into the analytic expressions (8)-(14), whose unknown parameters, I , x,, and v, are then fitted to the simulated data, as indicated below. Evidently, one objective of this estimation procedure is to obtain a good estimate of the unknown parameterswith as few load flow solutions as possible. The results section will give a number of numerical results to demonstrate that, indeed, relatively few load flow simulations are needed.
The basic Mea behind the proposed method is to estimate the unknown quantities [A, x,, v, 01 by adjusting their values to satisfy equation (23) as shown in the next section. For each given value of k and of the corresponding solution, x,, a new estimate is found. In contrast to most estimations which worsen as the singularity is approached, these actualiy improve as one approaches the singularity. Experience Indicates however that good estimates are found even though the simulations are far from maximum loading. THE ALGORITHM (1) For each value of k, solve a load flow for z given by equation (18) and obtain x,. (2) Define,
Now, by definition, the injection vector at maximum loading satisfies,
Thus, subtracting (19) from (18), we obtain,
The left hand side of equatiorl (20) represents the value of the differeritial dr with respect to the unknown maximum loading injection, 4, when z = q. The corresponding Value of dx must therefore be x, -x, =
d%If We now replace dx and dz in equations (8)-(14) respectively by dx, and dq, then after some manipulation one obtains,
dx,
vd
m ) + o ( I - k)
= v
[Jn-
+
As can be seen, the right side of equations (27) and (29), are dependent only on the unknowns v and A, while the left sides are known from load flow simulations. Although both (27) and (29) are linear in v, it was found by experimentation that (29) gave better results when the simulations were far from the maximum loading conditions.
(23)
(3) Now, one can obtain an equation only in A by defining,
where o is an unknown but constant (independent of k) vector. Similarly, the "distance" to maximum loading, 11 I A z I ) , the singular voltages, x,, and the null space vector, v, are unknown but constant and independent of k. The symbol indicates that the right hand side of the equations are approximations which are exact only near the expansion point, x,.
(4) Equation (31) is a scalar equation in one unknown,
xk
X,
+ V d(A - k) +
0
-
( A k)
650
A, which is readily solved by a Newton method for the Mh. estimate of I .
0.006
(5) Knowing k and the estimate of A, allows us to estimate v by a simple division in equation (27).
0.-
(6) Having estimates of v and I , allows us through equation (25) to readily estimate a.
0.003
0.004
0.002
om1
(7) Finally, from equation (23), the estimate of the singularity, x,, Is determined.
0 . 4.001 4
It must be noted that the computational effort associated with the above mentioned estimation calculations is minor compared with the load flow calculations needed to find %. TEST RESULTS The proposed method was tested on several standard IEEE networks (57 and 30-bus) as well as on a 5-bus network. The programs were written in Matlab and were run on a 33 MHz PC386 computer. Larger networks can be handled by using a commercial load flow algorithm whose precision is sufficient to implement our method. The methodology used for testing was as follows: (A) A sequence of load flows was run for {q,k= 1,2,...) starting from a standard base case. The injections were varied along a given direction and magnitude Az according to equation (18). A full Newton polar load flow was used, but a fast decoupled load flow could be substituted Instead. The direction of Az typically involved increasing loads and generations uniformly in proportion to their base values. This will generally result in a voltage collapse situation. The magnitude of Az was of the order of 0.05 pu for each power component. Enough load flows were run until they no longer converged. This indicated that either we were close to the Singularity or that we had gone outside the load flow feasibility region. It is Important to note, however, that the method does M require running load flows until divergence occurs. As is shown below, good estimates of the singularity are obtained after relatively few load flow runs.
(B) The algorithm was run on a variety of cases to estimate the problem unknowns, [A, x,, v, 4. The results of the algorithm were tested against the true singularity values obtained by using an independent method proposed in [Alvarado & Jung, 19891. The
QO10 4014 dol2
dol
4 . -
4.006 4.W -0.002
Real Part of AsX k 'igure 2 Straight line behaviour of
b,Xk
vs k
convergence of this independent method is however very sensitive to a good initial estimate of v which is not easily found. In addition, this independent method involves the solution of a set of non-linear equations of dimension twice that of the load flow. Since our method depends primarily on conventional load flow solutions, it was considerably faster and more robust. The results of the tests summarized in Figures 2-6. are from The IEEE 57-bus network.
id Q
'"1
10
20
3 0 4 0 5 0
Bus Number I
Figure 3 Exact and estimated voltage profile at maximum loading Figure 2 illustrates the exact simulated behaviour of the real and imaginary components at buses 10,30,31, 32 of the vector d,Xk parameterized by k starting at k=4 at the right hand bottom of the figure till k=21 at the top left. Note that the curves are nearly straight lines. This validates equation (29) which says that A,x, is parallel to the constant null space vector v. This is true
651
even for low values of k which correspond to injections far from maximum loading. 2.5
I
-$
s 8
Figure 6 represents the behaviour vs k of 11 h Az 11, the estimated "distance" from a given operating point to maximum loading along a given direction. This is an important measure when monitoring and controlling the proximity to a singularity. As can be seen, the estimate reaches a level very close to its exact value after 5 load flows.
-
Ec .-
E
5, 4
W
s
3 0
5
10
15
20
25
Number of Load Flow Solutions, k Ygure 4 % error in phase angles vs k Figure 3 shows both the exact and the estimated voltage magnitude profile of all 57 buses at maximum loading, clearly showing the voltage collapse at a number of buses. Note also that the deviation between the estimated voltage profile curves and the exact one is practically not noticeable (dark portions above bars). Figures 4 and 5 illustrate the behaviour of the estimated voltages and phase angles at maximum loading as a function of the number of load flow simulations, k. The %error of these vectors with respect to the exact value at maximum loading is shown versus k. After 5 load flows, the voltage magnitude errors and phase angles are less than 1%, decreasing rapidly and monotonically for higher values of k.
Q , 1
9 .e
0.8
E Q)
-B >" .-c
E
0.6 0.4
0.2
W
$
0
Number of Load Flow Solutions, k Figure 5 % error in voltage magnitude vs k
2 1 0,
0
5
10
15
x)
Number of Load Flow Solutions, k igure 6 Estimated distance to maximum loading vs k.
CONCLUSIONS This paper has presented a new efficient and simple approach to estimate the characteristics of the load flow solution when a given set of injections is uniformly varied along a given direction until maximum loading is reached. The method is simple since it is based primarily on conventional load flow simulations. Its efficiency is attributed to the use of newly discovered analytical relations between the voltages and the injections near maximum loading. The technique finds the distance from the initial operating point to the maximum loading point, as well as the corresponding vector of voltages and injections. These quantities are estimated by fitting exact load flow solutions to the analytic relations. Experiments show that good estimates are obtained after only five load flow simulations, and that these simulations need not be near maximurn loading to achieve good accuracy. The method has been evaluated on a number of standard networks and has proven to be both efficient and accurate. Although the results show cases with constant P and Q loads, the method is also valid for voltage dependent loads.
652
ACKNOWLEDGEMENTS
BIOGRAPHIES
The authors are grateful for the support of the Fonds pour la Formation de Chercheurs et d'AMe B la Rechercheand to the Natural Sclencesand Engineering Research Council.
Francisco D. Galiana obtained his B.Eng. (Hon) at McGill University followed by the S.M. and Ph.D from M.I.T. He spent several years at Brown Boveri Research Center and at the University of Michigan. He is presently a professor of electrical engineering at McGill University where he teaches and conducts research in power system analysis and control for planning and operation. Recently, his interests have included the development of expert systems for transmission and operation planning.
REFERENCES V. Ajjarapu, "IDENTIFICATION OF STEADY-STATE VOLTAGE STABILITY IN POWER SYSTEMS', InternationalJournal of Enerav Svsterns, Vd. 11, No. 1, 1991, PP.43-46. F.L. Alvarado, T.H. Jung, "DIRECT DETECTION OF VOLTAGE COLLAPSE CONDITIONS", Proceedinas: Bulk Power Svstem Voltaae Phenomena - Voltaae Stabilitv and Security, Report EPRl El-6138, Jan. 1989, pp. 5-23 5-38.
-
1. Dobson, L Lu, "USING AN ITERATIVE METHOD TO COMPUTE A CLOSEST SADDLE NODE BIFURCATION IN THE LOAD POWER PARAMETER SPACE OF AN ELECTRICAL POWER SYSTEM", Proceedinas of the NSF InternationalWorkshoD on Bulk Power Svstem Voltaae Phenomena. Voltaae Stability and Security, Maryland, August 1991. L.H. Fink, "INTRODUCTORY OVERVIEW OF VOLTAGE PROBLEMS", Proceedinas of the NSF InternationalWorkshoe on Bulk Power Svstem V Phenomena. Voltaae Stabilitv and Secur@, Maryland, August 1991. F.D. Galiana, Z.C. Zeng, "ANALYSIS OF THE LOAD FLOW BEHAVIOUR NEAR A JACOBIAN SINGULARITY", Proceedinas of the IEEE Power lndustrv Comwter ADDlications Conference, May 1991, pp. 149-155. N. Yorino, H. Sasaki, Y. Masuda, Y. Tamura, M. Kitagawa, A. Ohshimo, "AN INVESTIGATION OF VOLTAGE INSTABILITY PROBLEMS", Paper 91WM202-2PWRS, IEEE Winter Power Meeting, N.Y., 1991.
N. Yorino, H. Sasaki, A. Funahashi, F.D. Galiana, M. Kitagawa, "ON THE CONDITION FOR INVERSE CONTROL ACTION OF TAP CHANGERS", Proceedinas of the NSF InternationalWorkshoD on Bulk Power Svstem Voltaae Phenomena. V d t w Stability and Security, Maryland, August 1991.
Zhao-Chang Zeng has an M.Eng. degree from the Nanjing Automation Research Institute (NARI), China. He spent several years at the NARI and CAE Electroncs working on power system automation and control centers. He is presently a graduate student at McGill University working on the problem of voltage collapse. Boon-Teck Ooi (S'69-M'71-SM'85) was born in Kuala Lumpur, Malaysia. He received the B.Eng. (honours) degree from the university of Adelaide, Australia, the S.M. degree from the Massachusetts Institute of Technology, Cambridge, and the Ph.D. degree from McGill University, Montreal PQ,Canada, all in electrical engineering. He is currently a Professor in the Department of Electrical Engineering, McGill University. His research interests are in the areas of linear motors, repulsive magnetic leviation for high-speed ground transport, HVDC, static var controllers, power electronics, subsynchronous resonance instability in turbogenerators and voltage instability. Dr. Ooi is registered Engineer in the Province of Quebec. Naoto Yorino was born in Tokyo, Japan on January 24, 1958. He received the B.S., M.S., and Ph.D degrees in electrical engineering in 1981, 1983, and 1987, respectively, all from Waseda University, Tokyo, Japan. He joined Fuji Electric Co., Ltd, Japan from 1983 to 1984, then worked as Research Assistant of Waseda University from 1985 to 1987, and Research Associate of the Department of Electrical Engineering, Hiroshima University, Japan, from 1987 to 1990. He is presently an Associate Professor of Hiroshima University. His research interestsare mainly power system stability and control problems. Dr. Yorino is a member of IEE of Japan. He is now a visiting professor at McGill University, Montreal, PQ, Canada.
653
DISCUSSION
In order to obtain a good estimation of the maximum loading
F. ALVARADO and Y. HU (The University of Wisconsin,
point, does one have to consider the singularity assumption on the Jacobian via the selection of small AZ and large k when solving
Madison, WI):
The authors are to be commended for their
equation F ( X k ) = Z , + k A Z
for x k , Xk-1, xk.2 and x k - 3 ? If
interesting paper on a simplified method for computing the power
one has to, how exactly should Az and k be selected?
system maximum loading conditions. The proposed method uses several ordinary power flow computations to estimate the maximum
Note that equation (30) is dependent on AZ while equation (31) is
loading conditions. Existing methods for computing these conditions include the direct method (PoC method) [Alvarado &
equations (30) and (31).
Jung 19891 and the continuation method [Dl]. It should be noted that both methods have been successfully applied to large power systems [D2]. The direct method solves a large set of equations simultaneously, and it is very effective but requires a reasonable initial guess for convergence. The continuation method uses the ordinary power flow equations together with one more equation to solve the problem iteratively, and the convergence is usually assumed. It also avoids solving larger sets of equations. We observe that the proposed method, in addition to being useful on its own, may also be useful as another way to obtain an initial guess for the direct method. We would like the authors’ comments. An important aspect in determining the maximum loading conditions for large practical systems is handling a variety of limits. Both direct and continuation methods can use standard optimization techniques to deal with limits during computations. Our question about the proposed method is: how do the authors estimate the maximum loading conditions when limits such as reactive power limits are imposed on the system? [Dl]
K. Iba, et al, “Calculation of Critical Loading Conditions with Nose Curve Using Homotopy Continuation Method,”
[D2]
IEEE Trans. on Power Systems, Vol. 6, No. 2, May 1991. C. Canizares and F. L. Alvarado, “Computational Experience with the Point of Collapse Method on Very Large AC/DC Systems,” presented at IEEEPES Winter Meeting, New York, New York, January 26-30, 1992.
Manuscript r e c e i v e d February 19. 1992.
T. WU and R. FISCHL (Drexel University, Philadelphia, PA): This paper addresses a very important issue for finding the maximum loading point. A good initial guess is required for all the existing methods to find the maximum loading point efficiently. The authors are to be commended for developing a method for providing a good initial guess of the maximurn loading point. The authors response to the following comments and questions would be appreciated. 1 Equations (9)-(14) are derived in [Galiana & Zeng 19911 under the assumption that the Jacobian is singular with nullity one. However Equations (9)-(14) are used in this paper when the operating point is far from maximum loading, i.e. when the Jacobian is not singular. Is there any theoretical explanation for the validity of this transition?
independent of
&. Does the selection of AZ affect the validity of
The consideration of on-load tap changer and the generator reactive control makes it more difficult to find the maximum loading efficiently. Can this method be used when on-load tap changer and/or generator is considered? M a n u s c r i p t r e c e i v e d F e h r u a r y 2 4 , 1992.
Z. C. Zeng, F. D. Galiana, B. T. Ooi and N. Yorino: We very much appreciate the discussers’ comments and the time that they took to read the paper. In response to the comments of Profs. Wu and Fischl: (1) It is true that the particular behavior of the load flow voltages described by equations (9) to (14) is valid for voltages “near” a singularity only. This is equivalcnt to stating that a sensitivity relation is valid only “near” the expansion point. We view equations (9) through (14) as a sensitivity relation but a more general one also valid at or near a singularity. We have no precise measure of what the term “near” implies and that is why we experimentally tested the validity of the relation “far” from the expamion point with surprisingly good results. We should point out, however. that although the operating points, zk,are “far” from the expansion point, the quantities estimated by the method ( x \ , v and A) are thow charactct-iTing the singular point that one would find if the load inlcctions z were increased along the direction A z until the boundary of the load flow feasibility region. We believe that it is because of thc particular structure of the sensitivity relation (square root term and a linear term) that the projected or estimated singular values are so closc to their true values even though the load flow injections, zk, arc far from the singularity. ( 2 ) Actually, Az essentially primarily defines a direction of change of the load flow injections. Its dimension should be selected so that it represents a small change relative to the starting point, z,,, e.g. a 1% change in loads and generation along some chosen direction. Then, k is selected as indicated in the paper, as k = 0. 1,2,3,. _ _ where k = 0 is the starting point and subsequent k’s define injections closer to a singularity. Typically, in a load flow, if you increase the loads and the generation, you will eventually rcach a feasibility boundary. If A z is rcasonably sized (as indicated above), the Jacobian is not singular at k = 0 nor is it for k = 1,2,3,4. In fact, it is very well behaved and a fast decoupled load flow will converge normally. Experimentally, we found out that k need not be large (k < 6 ) to be able to estimate the singularity well. Again, this is probably due to the fact that the sensitivity relation predicts the valuc and location of the singularity quite accurately even from far away. ( 3 ) The discussers are correct in their statement. Equation (31) assumes that the sensitivity relation is cxact while equation (30) is the cxperimcntally obtained result. The selection of Az has an effect on (30), particularly its direction. However. once its direction is picked, therc should be only one value of A (distance to the singularity). To ensure that (31) is accurate, we believe that it is enough to make the magnitude of A z not be very large compared to z,,. at least small enough that one can solve several load flows without numerical instability. (4)The incorporation of other typcs ot states (taps, phase-shifters) or of controls which depend on the states can also be treated by this theory as long as the equations can tie written in the form of equation (1) and if a second order expansion of F(x) is possiblc. We have not, however, cxpcrimentally tested this yet. In response to the comments fi-om Dr. Alvarado: We agree that the estimate provided here for the null space vector could he used as a good initial estimate for the direct method. Moreover. the second voltage wlutioii estimate can also be used as an
654 initial estimate in a load flow algorithm to find a second close-by solution. In fact, by choosing different Az, one finds different singularity points, each of which could yield different load flow solutions to the starting point zo. This could be used to systematically find a number of solutions to a given load flow problem. The treatment of load flow limits has not yet been implemented but we believe that the following approach may be useful: Find an esti-
mate of the singularity along a given direction, Az. If there are var violations at the estimated singularity, try to find the value ,of k where these violations first take place. At this point, the var source should be fixed at its limit and the voltage released. The process of estimating a singularity is repeated until no more var limits are violated. Manuscript received April 13, 1992.
