IEEE Transactions on Power Apparatus and Sys tem, Vol. PAS-96, no. 1, JanuaryjFebruary 1977

A SECOND ORDER LOAD FLOW TECHNIQUE

M. S. Sachdev T. K. P. Medicherla

Power Systems Research Group University of Saskatchewan

Saskatoon, Canada

ABSTRACT

This paper presents a secondorder load flow technique based on the Taylor series expansion of a multivariable function. An algorithm for obtainingdigital solutions by the proposed approach is described. Six simplified versions of this algorithm are also discussed. The proposed algorithm and its modifications have been tested by obtaining load flow solutions of a five bus system, the 26 bus Saskatchewan Power Corporation transmission network model and the IEEE 14 bus, 30 bus, 57 bus and 118 bus test systems. The effectiveness of these algorithms is compared with the performance of the Newton Raphson approach and some of the results from these studies are presented.

INTRODUCTION

Many load flow studies a r e conducted for power system planning and operation. Since the introduction of the f irst digital computer load flow algorithml, new methods have been suggested from time to time z. Convergence charact- eristics of different methods have been examined by Sasson and Jaimes 3. Two most commonly used methods are the Gauss-Seidel and Newton Raphson techniques 4. Both these have their relative advantages and disadvantages. However, the popularity of the Newton Raphson approach has been increasing since the introduction of optimally ordered tri- angular factorization by Tinney and Hart 5. Many variations of the Newton Raphson load flow have also been suggested and used 637,8.

A load flow problem consists of solving a set of non- linear equations. The Newton Raphson technique, which is one of the many load flow techniques available atpresent, uses the f irst two te rms of the Taylor series. This approach transforms the nonlinear load flow equations to a linear form before a solution is attempted. It is also possible to form- ulate the load flow problem by using the first three terms of the Taylor series. In other words, second order terms, whichare not insignificant, can be included in the algorithm and can be used during digital computation. This approach has been investigated and is presented in this paper. The significant second order terms are found to be minor vari- ations of the terms of the Jacobian matrix. It is shown in this paper that the coefficients of the second order terms are not required to be separately stored. In the proposed tech- nique, the state vector is first calculated by an iteration of the conventional Newton Raphson technique. Using the calculated state vector and elements of the Jacobian matrix,

Paper F 76 096-8, recommended and approved by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society for presentation at the IEEE PES Winter Meeting and Tesla Symposium, New York, N.Y., January

November 4,1975. 25-30, 1976. Manuscript submitted September 2, 1975;made available for printing

second order terms are estimated and subtracted from the residual vector. The modified residual vector obtained fn this manner is thenused to compute a new state vector. This procedure is repeated till the elements of the latest state vector are within permissible tolerance of those previously calculated. The magnitudes and phase angles of bus voltages a r e thenupdated. The total procedure is then repeated until a converged solution is obtained. Six variations of this al- gorithm were also investigated and are described in this paper. The proposed load flow technique and its variations were tested by computing load flows of a five bus system 1 , the 26-bus Saskatchewan Power Corporation transmission network model and the IEEE 14 bus, 30 bus, 57 bus and 118 bus test systems. For comparison, each study was repeated using the Newton Raphson technique. Some of the significant results of these studies are also presented.

TAYLOR SERIES

A function can be evaluated in a neighbourhood using theTaylor series expansion. The procedure is simple and well known for functions consisting of a single variable. Taylor series of multivariable functions can also be defined. For annvariable function, the series is expressed as follows:

f‘y + 9, ”2 + h”z .. ... xn + Axn) = f ( 3 ’ 3, . ...

. . . (1)

Because the summation in equation 1 consists of m terms, a residue, Rm, is introduced to take care of thesummation from (m+l)th term to infinity. This residue i s not known exactly because definite values cannot be assigned to ai’s. Neglecting the third and higher order terms in equation 1, equation 2 is obtained. Equation 2 can be rearranged to give equation 3 which can be expanded as equation 4.

+ - l a (- hx + - a Ax + .... -Axn) a 2 f (x l , “z ,... 2 axl 1 a% 2 axn . . . . (2)

1s9

......( 4)

SECOND ORDER LOAD FLOW MODEL

In a power system, real and reactive powers injected into a bus, say p, is equal to the net flow in the elements connected to this bus. Power flowing in an element connect- ing two buses, say p and q, can be defined in terms of the magnitudes and phase angles of the voltages at these buses and the parameters of the element. A load flow is, therefore, a problem of solving a set of nonlinear equations consisting of the magnitudes and phase angles of the system bus voltages as variables and the parameters of the system elements as constant coefficients. Let the power mismatch at a bus, p, of an n bus system be defined as the difference between the scheduled power injection into this bus and the sum of the calculated power flows in all the elements connected to this bus. Real power mismatch can be defined in te rms of the above mentioned variables by equation 5. This second order equationis similar to the Taylor series expansion excluding third and higher order terms given in equation 4.

n ap n aP a% hp = E 2 AAq + E 2 AVq + $ -$ (A6q)2

q=l q q=l aVq q=l a 5 9

n a% n-1 n a% + 3 X --$ (AVq)' + C E A6qA6r

q=l 9 q=l r=q+l q r

n n a% n-1 n a% q=l ~l q r q=l avqavr

+ X X & A L ~ ~ A V ~ + X Z AvqAVr

........( 5) where Pp is the real power injection into bus p,

B q is the phase angle of the voltage at bus q,

V is the magnitude of the voltage at bus q. q A defines small changes in the variables.

It isinteresting to note that the terms of the first two ser ies in equation 5 are similar to the terms of the Jacobian matrix used in the Newton Raphson load flow approach. The remain- ingflve series constitute the second order terms. In a simi-

lar manner, reactive power mismatch can be defined a s follows:

n-1 n 1 X 1 A6 A6 +

I- q=l a tl q=l q r

. . . . (6) Expressing equation 5 for all system buses. except the swing bus and equation 6 for all load buses, a set of equations is obtained in the following form:

[:] = [$I [-] +

The submatrices Si through s6 include series of second order terms of all buses similar to those given in equatione 5 and 6. Suitablevalues areassigned to subscripts i, j, k, 1, Sand t. Many elements of the second order coefficient matrix, [SI , arevery small and canbeneglected. Also, it is not essential to compute the elements of matrix IS] and store it in computer memo-ry. These two aspects will now be discussed.

Real and reactive power injection into a bus p of ann bus system can be mathematically expressed by equations 8 and 9 respectively. Second order coefficients can be derived from these equations and can be grouped into twenty cata- gories. These coefficients are given in equations 10.1 through 10.20. A comparison with the elements of the Jacobian matrix indicatesthatthe second order terms given in equations 10.1 through 10.10 are minor modifications of the Jacobian ele- ments. This aspect is also indicatedin these equations. The te rms defined by equations 10.11 through 10.20 can be neg- lected aa will be discussed later in this section.

n P = '7 I V V Y I cos(6 - 6 4 ) P q=l P 9 w P 9 W

. . . . ( 8 )

n 0 = E IV V Y I S h ( 6 -6 -8 P q=l P 9 w P 9 w> . . . . (9)

where Ypqis the admittance of the element connecting buses p and q and is given by I Ypq &

190

a% n &- = - z I V Y I sin(s -6 -e 1 = - J1(P,P)

P P q=l P 9 W vP ....( 10.1)

n

a'% J1(P,9) A= - I V Y 1 sin(s -6 -e ) = - - a 6 aV P 9 p W P 9 W vq . . . . (10.2)

a% J1(PA) = lvpyml cos(& -6 -e &j-= Iv Y 1 Sin(6 -6 -e ) = - P 9 P 9 W

4 P 9 w P 9 W V P . . . . (10.3) . . . (10.18)

a% J1(P,9) & = I V Y I sin(& -6 -e =v

4 9 P W P 9 W 9 . . . . (10.4)

n

3 = -1V V Y I Sin(& -6 -8 ) = -J1(p,q) , . . .(10.6) a s2 P q W P 9 W

9

2 a$ = IVVY I Sin(6 -6 -e ) = J1(p,q) .... (10.8) as as

P 9 P 9 W P 9 W

% J1(P,q) P 9 p q W P 9

= Iym1 sin(6 -6 -e ) = vv . . . . (10.9)

a% --$ = mpp coSepp = ;;z 2 IJ2(P,p) - PPI ....( 10.10) P P

3 = o 9

....( 10.11)

3= 0 ....( 10.12) a$

q n

... (10.20)

The elements defined by equations 10.11 and 10.12 a r e equal to zero. Equations 10.13 through 10.20 define the second order elements which include Coe($-6q-9pq) a s a multi- plier. Since epqis close to 9d and (JP-6q) isusuallysmall, Cos(Gp-6q-epq) is quite small and therefore, thesecoeffi- cients can be neglected. This assumption and the simple relationship of the second order coefficients with the elements of the Jacobian matrix makes the application of second order load flow model straight-forward and with minimal additional computing effort. The double summations in equations 5 and 6 have I Ypq 1 as multipliers. For any bus p, I Ym I has non- zerovalues only for the buses q connected to it; for all other values of q, I Ypq I is zero. Therefore, the double summations for abus can be reduced to single summations which include only a few te rms depending on the number of buses connected to that particular bus. Deleting the insignificant terms from equations 5 and 6 and converting the double summations to single summations, equations 11 and 12 a r e obtained.

n AS" C 3 A6 + Z 3 av AV + $3 (AVp) 2 -I- q=l a6q q q=l q 9

P

a% & = IVpVqYml -(6p-6q-em) ....( 10.15) P 9

191

. . . . . ( 1 2 )

ALGORITHMS FOR SOLVING THE SECOND ORDER MODEL

The secondorder model defined by equations 7, 11 and 12 cannot be solvedin a straightforward manner. If the second order terms are neglected, this model reduces to the first order form which is usually solved by the well known Newton Raphson technique. The proposed model is nonlinear be- cause of the presence of the products of unkaowns (66&%63, AVk hVe and A C ~ ~ AVt) in equation 7. For obtaining a solution of this set of nonlinear equations without excessive computing effort, these are converted to a set of linear equations and solved using the algorithm consisting of the following dis- crete steps.

1. Compute the real and reactive power mismatches and elements of the Jacobian matrix using the spec- ified loads, generator outputs, system parameters and estimated magnitudes and phase anglesof bus voltages.

2. Neglect the second order terms; nowtheproblem is the same as in Newton Raphson load flows. Using one of the many available matrix a l g e b p techni- ques evaluate the state vector IA16 d v / V ] .

3. For each AJ? and AQ of the residualvector, compute the second order terms (defined in equations 11 and 12) using A6's and dV's evaluated in step 2.

At this stage the residual vector is known and the sum of second order series for each residual term has been estima- ted. Transferring the second order terms to the left hand side, equation 7 is modified as follows:

A 6 AVt S

............. (13)

3 a secondorder series is evaluated, it is subtracted from the corresponding residual term which procedure provides a modified residual vector defined by the left hand side of equation 13. It is important to realize that the elements of the matrix IS] a re not computed and therefore, no additional storage is used for this matrix.

4. Using the modified residual vector and equation 13, recalculatethestatevector I A S N/VJT. The tri- angularized Jacobian (or an inverse of the Jacobian) used in step 2 is reused at this stage.

5 . If the solution is within permissible tolerances of that obtained previously, proceed to step 7, other- wise to step 6.

6. Repeat step 3 using the state vector computed in step 4 and proceed to step 4.

7. Update the magnitudes and phase angles of the bus

voltages using the state vector of step 4. 8. Calculatethe real andreactivepowermismatches.

If the solution is within permissible tolerances of that obtained in the previous iteration proceed to step 10; otherwise to step 9.

9. Compute the elements of the Jacobian matrix and proceed to step 2.

10. Calculate power flows and print out the required load flow information.

Attempts were made to simplify the above procedure and to eliminate ineffective steps. To identi@ different al- gorithms, the one descrlbed above will be referred to a s version I. In this version steps 4, 5 and 6 form a subiter- ation. A reduced algorithm (version was obtained by deleting steps 5 and 6 which eliminated the subiteration. Flve variations of version IIwere also tried. In these variations, modifications due to the second order terms were applied to:

(a) AP's and AQ's of load buses only. (b) EQ's only. (c) AP's and AQ's in the first iteration only. (d) & P I S and A Q ' s of load buses in the first

(e) AQ's in the first iteration only. iteration only.

Digital computer programs based on the above algorithm were prepared at the University of Saskatchewan. These programs were used on an IBM 370/158 digital computer in order to determine the suitability of the second order load flow models described above.

SYSTEM STUDIES

The approach described in the previous section was tested by computing load flows of the following six systems:

(i) A five bus system9. (ii) IEEE 14 bus test system. (iii) Twenty six bus model of the Saskatchewan

(iv) IEEE 30 bus test system. (v) IEEE 57 bus test system.

(vi) IEEE 118 bus test system.

Power Corporation transmission systedo.

These systems consist of lines and transformers ranging from6, inthe five bus model, to 186 in the IEEE 118 bus teat system. System (i) above has only two generator buses com- pared to 54 in the 118 bus system. Digital computer programs based on the seven versions of the algorithm, described in theprevious section, were usedto obtain load flow solutions of the six systems listed above. For comparison, all load flow studies were repeated using the Newton Raphson techni- que. The results of studies relating to the IEEE 57 bus and 118 bus test systems are presented and discussed in detail. The studies of all the systems are then briefly compared.

IEEE 57 Bus Test System

This system consists of 78 lines andkansformers. and seven voltage controlled buses. Load flow solutions of this system were obtained using the digital computer programs based on versions I, 11, II. a, II. b, II. c, U.d, and II. e des- cribed earlier and the Newton Raphson approach. Version II of the algorithmis similar to Version I except that the inner iteration, consisting of steps 4, 5 and 6, is not executed in full. Only step4isretainedwhereassteps5and6aredeleted.

192

This modification slightly reduces the computation effort. Figure 1 shows the magnitude of the maximum real and re- active power mismatches, max I d p I and maxl AQI , a$ the end of each iteration when versions I and 11 of the second order load flow algorithm and the Newton Raphson approach are used. It is noticedfrom this figure that the mismatches experienced during the load flow based on versions I and 11 a re l e s s than those observed in the Newton Raphson case. A comparison of mismatches noticed during load flows by versions1 and II indicates that there is no significant deter- ioration in the convergence pattern if the subiteration is deleted. Max 1 AP 1 and max I A Q I experienced at the end of each iteration of the Newton Raphson and second order tech- niques are given in Table I.A. A persual of this table indi- cates that the maximum mismatches noticed in all the second order load flows a re l e s s than those in the Newton Raphson approach; except the real power mismatches at the end of first iteration of versions II. b and II. e. Even in these cases, the differences in mismatches are very small. Fig. 2 shows typical convergnece characteristics of the voltage magnitude at a bus as calculated at the end of each iteration of the NewtonRaphson andversionII of thesecond order technique. It is notioed that in the second order technique the voltage descends into the final solution at a faster rate than in the Newton Raphson approach. A review of the algorithm will indicate that the computational effort in versions II. c, II. d and II. e is only slightly greater than that required by the NewtonRaphsontechnique. This is because the second order terms are evaluatedandusedinthe first iteration only. Com- putational effort in all subsequent iterations is equal to that of a NewtonRaphson iteration. From the studies using ver- sions II, II. a and II. b, it was noticed that the contributions of the second order terms after the second iteration are insignificant.

IEEE 118 Bus Test System

This system consists of 186 lines and transformers, and54 voltage controlled buses. Load flow solutions of this system were also obtained using the digital computer pro- grams based on the Newton Raphson technique and versions 11, II. a, II. b, II. c, II. d and 11. e described earlier. Figure 3 shows the maximum real and reactive power mismatches at the endof eachiteration when versions I and II of the pro- posed algorithm and the Newton Raphson technique a r e used. Like the 57 bus system, the mismatches experienced in ver- BiOnS I and II are less than those in the Newton Raphson case.

I 1 0 1 2 3 1

ITERATION NUMBER

Fig. 1. Convergence patterns of load flows of the IEEE 57 bus test system by (a) the Newton Raphson approachand, (b) version I and (c) version 11 of the proposed technique.

I 0 1 2 3 4

ITERATION NUMBER

Fig. 2. Typical voltage convergence charactersitics of the IEEE 57 bus test system using (a) version 11 of the proposed technique and (b) the Newton Raphson approach.

Table I. Misnatches observed during t h e load flow s t u d i e s of t h e I= 57 and 118 bus test system.

second order load flow version I

5.764 0.409 0.007 0.002

30.957 0.364 0.003

2.513 2.618 0.016 0.023 0.001 0.001

9.185 8.773 0.144 0.147 0 . m 0 . m

2.616 0.023 0.001

15.112 0.120 0.003

1 I . b 1I.c 1 I .d 1I.e I 1 I

A. IEEE 57 BUS TEST EXSIB!

6.686 2.618 2.616 6.686 17.347 3.354 0.154 0.085 0.085 0.145 0.616 0.024 0.001 0.001 0.001 0.002 0.006 0.002

0.003

B. IEEE 118 BUS TEST SY’SlXN

The t h e

maxirmn values of 57 bus systan and

I @I 589. 2Mw

and at the wing of i t e r a t i o n 1 263.0 WAr respect ively for t h e

193

101 -

100-

10-1 I-

1 0 1 2 3 4

ITERATION NUMBER

Fig. 3. Convergence patterns of load flows of the IEEE 118 bus test system by (a) the Newton Raphson approach and, (b) version I and (c) version I1 of the proposed technique.

Also, the convergence pattern of version 11 is quite similar to that of version I. Max I B P I and maxlb.Q&l experiencedat the end of each iteration of the Newton Raphson and second order load flowtechniques are giveninTable I. B. A Perusal of this table indicates that the maximum mismatches in all the second order load flows are less than in the Newton Raphson approach; except the reactivepower mismatches at the end of second iteration of versions II. c and II. e. Even in these cases, the mismatches are only slightly higher.

Comparative Performance

Load flow solutions of all the systems listed earlier in this section were obtained by the Newton Raphson approach and by each variation of the second order technique. Table II lists the number of iterations required for convergence of these load flows. (Magnitude and phase angle corrections calculated in each iteration being less than 0.0001 tolerance indicated convergence of the solution. ) A perusal of this table indicates that in most cases, the second order load flow technique requires less iterations for convergence than the Newton Raphson approach. Max I AP1 and max I AQ I observed at the end of first iteration of the load flow solutions a r e given in Table III. A study of this table indicates that in all cases, except one, the maximum mismatches are smaller when second order algorithms are used. This indicates that the solution descends into the final state at a faster rate in the second order load flows than in the Newton Raphson cases. This was also observed to be the case at the end of subsequent iterations. In all cases the contributions of the second order terms were noticed to be negligible after the first iteration. Because of this reason, the performance of versions II. c, II. d and II. e is very similar to that of versions 11, II. a and II. b respectively.

Table 11. Nunber of i t e r a t i o n s * to convergene for the six test svstars.

wises I I1 1I.a 1 I . b 1I.c 1I .d 1I.e

30

118 3

* ’Ihe nunber of i t e r a t i o n s do not include the last iter- a t ion requid for checking t h e oonveqence.

‘hble 111. Misnatches a t the end of first i t e r a t i o n for t h e test systars.

(m) bfa.xl~QI (mh) NO. of N . R . ~ proposed version* N.R.~ proposed version*

I I I1 1II.a 1 I I . b I I I1 1II.a 1 I I . b 5

6.7 7.1 7.1 3.4 17.4 6.7 2.6 2.6 2.5 5.8 57 1.8 2.2 2.1 3.3 10.8 10.8 10.3 6.6 4.7 11.2 30 48.0 48.8 48.5 3.3 60.0 15.7 13.7 13.9 4.7 18.3 26 3.0 3.7 3.7 2.8 16.3 10.8 10.4 6.0 5.9 12.1 14 0.6 0.4 0.4 1.1 3.8 0.0 1.6 1.6 1.3 3.5

118 21.0 9.2 8.8 15.1 15.1 3.9 3.9 9.2 2.7 75.2 * The mietches at the end of f i r s t i t e r a t i o n in t h e load f l aw s tud ie s by the second order versions II.c, 1 I . d and 1I.e are t h e same as in versions I1,II.a and I1 .b respec t ive ly .

CONCLUSIONS

The second order load flow model presented in this paper has some advantages over the conventional Newton Raphson approach. From the studies presented in this paper, i t is obvious that the proposed technique and all its variations have better convergence characteristics in almost all the load flows attempted by the authors. The magnitudes and phase angles of bus voltages descendinto the final solution at a faster rate than that observed in the Newton Raphson load flows. Therefore, in many cases, the secondorder approach requires lesser iterations than the Newton Raphson technique. Moreover, theadditional computing effort, especially in ver- sions II. c, II. d and II. e, is only slightly greater comparedto aNewton Raphson load flow. It has also been shown that the elements of the second order coefficient matrix need not be stored separately. The proposed technique can be conven- iently incorporated in the existing Newton Raphson load flows by adding a subroutine for computing the second order terms and calculating the modified residual vector.

ACKNOWLEDGEMENTS

The authors express their appreciation to the Saskat- chewan Power Corporation and the University of Saskatchewan for providing financial support for this work.

194

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

REFERENCES

J. B. Ward and H. W. Hale, ''Digital computer solution ofpower flow problems", Trans. AIEE (Power Appara- tus and Systems), Vol-75, pp. 398-404, June 1956.

B. Stott, ''Review of load flow calculation methodst7, Proc. IEEE, Vol-62, pp. 916-929, July 1974.

A. M. Sasson and F. J. Jaimes, "Digital methods ap- plied to power flow studies", IEEE Transactions on Power Apparatus and Systems, Vol. PAS-86, pp. 860-867, July 1967.

G. W. Stagg and A . H. El-Abiad, Computer Methods in Power System Analysis. New York: McGraw-Hill Book Company, 1968.

W. F. Tinney and C. E. Hart, "Power flow solution by Newton's method", IEEE Transactions on Power Appa- ratus and Systems, Vol. PAS-86, pp. 1449-1460, Novem- ber 1967.

A. M. Sasson, C. Trevino and F. Aboyles, ''Improved Newton's load flow through a minimization technique", Ibid, Vol. PAS-90, pp. 1974-1981, Sept./Oct. 1971.

B. Stott and 0. Alsac, "Fastdecoupled load flow", Ibid, Vol. PAS-93, pp. 859-869, May/June 1974.

A. K. Laha, K. E. Bollinger, R. Billinton and S. B. Dhar, "Modified form of Newton's method for faster load flowsolutions", Proc. IEE, Vol-121, pp. 849-853, Aug. 1974.

W. D. Stevenson, J r . , Elements of Power System Ana- lysis. New York: McGraw-Hill Book Company, 1962, p. 219.

M. S. Sachdev and S. A. Ibrahim, "A fast approximate technique for outage studies in power system planning and operation", IEEE Transactions on Power Apparatus and Systems, Vol. PAS-93, pp. 1133-1142, July/Aug. 1974.

Discussion

RP. Sod and, T.S.M. Rao (University of Roorkee, Roorkee, India): The authors are to be complimented for presenting an excellent paper in the area of digital load flow studies. The discussers would offer the following comments and seek some clarification:-

Newton Raphson algorithm for load flow gained popularity after the development by Tinney and others of very efficient sparsity programmed ordered elimination. This eliminated the drawback of large computer storage required for this algorithm. The main advantage of N.R. method is that the number of iterations required to obtain a solution is practically independent of system size but the algorithm is susceptible to failure if starting values of voltage profile are not chosen comctly.

The important contribution of this paper is the extension of the present Newton Raphson Technique to include the second order terms without increasing the storage requirements to any great extent. The

the principle of superposition. The authors have demonstrated the authors have linearized the second order load flow model by utilising

better convergence characteristics and accuracy compared to con- ventional Newton Raphson Method through extensive tests on a number

Manuscript received January 19, 1976

of small and large systems. This is an expected result since by including

Method is improved. The merit of the proposed algorithm lies in the the higher order terms of the Taylor's series, the accuracy of Newton's

fact that by adding a subroutine, the existing programmes can be ex- tended to include second order terms without any major modifications.

Since load flow is the most frequently performed routine digital network calculation, the discussers would like to know what are the computation times per iteration of the conventional Newton Raphson Method and the proposed second order load flow for the various systems studies so that a comparison of computation cost could be made.

It appears that all the system studied are well conditioned i.e. having diagonal dominance. Have any ill conditioned systems (systems containing cables etc.) been studied by the proposed algorithm?

We once again commend the authors for their very fine effort.

Balbu S. Sandhu (Manitoba Hydro, Winnipeg, Manitoba): Dr. Sachdev and Mr. Medicherla have prepared an interesting and valuable paper describing the second order load flow technique. The authors deserve to be congratulated for presenting a load flow method based upon Taylor series expansion of multi-variable functions. The proposed technique seems to have better convergence characteristics from the load flow cases carried out and the final solution is achieved at a faster rate than

of the paper, the following points are not dealt with and the writer that obtained by other load flow solutions. However due to the brevity

would appreciate comments with regard to the following questions. 1. Has the second order load flow technique been applied to

systems for which it is difficult to obtain a converged load flow solution by iterative and Newton Raphson methods?

2. For all the cases reported in the paper, load flow solutions have been obtained by both (i) second order load flow technique (ii) Newton Raphson method. What are the respective computation times required by these methods?

The authors work is a logical innovation which uses the significant second order terms of the Taylor series for the solution of load flow equations.

Manuscript received January 26, 1976

J.C. Tiburcio and A. Brameller (U.M.I.S.T. Power Systems Laboratory, Manchester, England): The authors ought to be congratulated for presenting an interesting paper which describes an algorithm to take account of the second order terms in the Taylor series expansion of the load flow equations, with the advantage of requiring no extra storage for the second order coefficients.

compared with the ordinary Newton Raphson algorithm, in the major- The tests performed seem to indicate a faster rate of convergence,

ity of the cases studied. Since solutions obtained with all versions of the proposed method,

as well as with Newton Raphson's, converged to the same accuracy, and since there is no difference in storage requirements, the best way to as- sess the performance of the new scheme for the systems investigated, would be by comparison of solution times. Could the authors supply some comparative figures?

Also, we would be interested to know if the method has been used in systems where the ordinary Newton Raphson algorithm fails

Could the proposed technique cope satisfactorily with such cases? We or presents difficulties in convergence. If s o , what were the results?

have recently tried a network which does not converge with Newton Raphson's method. If the authors are interested, we could supply them with the pertinent data.

method with algorithms other than Newton Raphson's, for instance, Finally, it would also be interesting to compare the proposed

that of reference 7 which is reported to be superior to Newton Raphson in terms of time and storage.

Manuscript received February 9, 1976.

Y. Robichaud and G.T. Vuong (HydroQuebec Institute of Research Varennes, Quebec, Canada): The authors should be commended for their refinements on Newton-Raphson technique.

Particularly, it is felt that this second order load flow model is much less sensitive to divergence problems which are sometimes ex- perienced with difficult networks.

Manuscript received February 11,1976.

195

Considering that many efficient load flow techniques have been reported in the last few years, it would be interesting to compare t e performance of t h i s method with a decoupled load flow technique 9 .

G.T. Heydt (Purdue University, West La Fayette, Indiana): Sachdev

useful improvement to the Newton-Raphson power flow method. The and Medicherla have presented a most interesting and apparently quite

improvement is an update of the [ A P AQ] vector using an approxima- tion of the second order terms of the Taylor series. They have clearly demonstrated that the second order terms are easily obtainable from the Jacobian matrix thus making the proposed technique computer

does not result in a radical improvement in computation, it is a signif- efficient. Although the second order power flow method presented

icant improvement to the conventional Newton-Raphson method.

whether the decoupled power flow method of Stott and Alsac may The discussor would like the authors to briefly comment on

entries of matrix S, Eq. ( 1 3) of the paper, will have special properties. be applied to the proposed second order technique. In this case, the

The update of [ A P AQ] should improve the decoupled power flow method.

number of iterations required for convergence. What are the correspond- Table I1 of the paper shows noteworthy improvements in the

ing fwres for central processing time? If the update method proposed

then the second order correction should be used in the fmt iteration by the authors has a significant effect on computing time per iteration,

only (method 1 l.C).

found a system which will not converge using the conventional Newton- Finally, would the authors please comment on whether they have

Raphson solution but will converge after applying the second order corrections.

count of their work. I t is clear that there will be many potential users Sachdev and Medicherla have presented a very well written ac-

for this improvement.

Manuscript received February 17,1976.

S. Vemuri (University of Alaska, Fairbanks, Alaska): The authors are

material of sohing power system load flow problems. The matrix to be commended for providing an interesting addition to the present

representation of second order terms of equations (7) and (13) is not consistant. Instead, equation (7) should have been

where the Jacobian terms J1, J2, J3 and J4 are matrices, the second order terms SI, 52, . . . , s6 are vectors, Avk'= Avk/vk and T is the transpose of the matrix.

I would appreciate the authors' comments on the following points.

less number of iterations for convergence than the Newton-Raphson 1. The conclusion that the second order load flow techmques take

(NR) method is misleading. On the contrary, the results of Table 1 and Table I1 suggest that there is not significant difference in the

This saving in the number of iterations, in general, is obtained at the number of iterations of the proposed method and the NR approach.

cost of computation time. How do the computation times of the proposed techniques compare with the NR method?

2. To reduce the computation time of the proposed method, one can explore the weak coupling between P-V and Q-6 terms. This sug- gests an alternate method of solving the nonlinear second order equa- tions.

(D7), the simplified model takes tkie form o? (D8). Neglecting the weak coupling terms J , J3, s2 and s4 of equation

hp_ = Jp A$

AQ = J AV' Q - where

This problem can be solved by the following iterative procedure. (i) Assume initial state vector 6 and y. (ii) Calculate J1, J4, SI, S3, s5 and s6 about the known

values of _S and v. Calculate the power misnatches hp and AQ.

(iii) Calculate Jp and JQ from the calculated values of As and

For the fmt iteration obtain Ah and AY 'from (D7) without

and AY' = J4- AQ. For the subsequent iterations A4 and considering th second order terms or from M = J1-1 A I

Ay' are the values obtained in the last iteration (step iv).

Ay :

P

(iv) Calculate the modified corrections from (D9).

(v) At this stage, update _S and y 'and go to step (ii). The iterative procedure is terminated when the modified values of

A4 and AY ' of step (iv) are within permissible tolerances. In each iteration, the Jp-l(J -1) of (D9) can be directly obtained from J1'I (J4-l) by the matrix gmma of (D 10).

where

Jpl = J1 + slAST

and

Here the vector quantities of SI, s3, MT and AYT result in a simple scalar multipfcation .and division of (D10). Similar results can be obtained for JQ- . In thls form the sparse matrix techniques can still be used to obtam the inverses of the matrices J1 and J

3. In general, the second order terms should make &e proposed algorithm to converge over a wide range of initial guesses of the state vector. In the authors' experience, how does the range of initial guesses of the proposed and the NR methods compare?

4. The results of Table I1 suggest that the six test systems of the paper are "well-behaved" systems and for such systems one would not expect any convergence problems with the NR method. In general, the reliability of the NR method is comparatively good. But failures do occur on some ill-conditioned problems [2] . Did the authors test the proposed method on ill-conditioned problems?

the version IIc, IId and IIe suggest the proposed method is a powerful 5. The results of Table 111 and the conclusions of the authors for

tool to obtain the starting values for the NR method.

ME. El-Hawary and WJ. Vetter (Faculty of Engineering & Applied Science, Memorial University of Newfoundland, St. John's, New- foundland, Canada): We would like to commend the authors for an interesting paper. The presentation of the method by illustrating the Taylor expansion for a single function of many variables helps in conveying the basic idea of the method. This is in contrast to ambi- guity that may result if the vector function Taylor expansion formula- tion is attempted.

The basic idea of what we refer to as Taylor-based iteration schemes is to seek an incremental change A in the current guess value (x). This hopefully will result in a new value that satisfies the relation f (x + A) = 0. For quadratic functions the derivatives of order higher

in one iteration in the single variable problem. For multi-variable than two are zero. Thus a secondader Taylor-based iteration converges

vector-function problems such as our Load Flow problem the problem is that of solving a set of simultaneous second order equations in a

Manuscript received February 17,1976, Manuscript received February 17,1976.

196

number of variables. The authors have chosen the polar form of Load Flow formulation which introduces trignometric functions. We note that the rectangular form offers only terms up to second order in the components of bus voltages. Thus the second order scheme will be exact. This suggests that the rectangular form will be superior to the polar form when using an approximate second order scheme such as any of the six algorithms offered by the authors. We would like to invite the authors comments on this.

ing time necessary for, say three, or four iterations using Newton’s The discussors would be interested in a comparison of the comput-

method and the author’s proposed method. This will enable a better feel for the success of the method with larger,size systems.

We once again compliment the authors for a well organized paper.

W.F. Thiele (Saskatchewan Power Corp., Regina, Saskatchewan, Can- ada): The authors have presented some interesting results of load flow studies under taken to compare the second order technique to the conventional Newton Raphson approach. Although the mismatches at the end of the first iteration for each of the test systems (Table 111) are significantly lower when using the new technique, it is noted that after the fmal iteration (Table I), there is little difference in mismatch between the two methods. From a user’s point of view, the advantage of the new technique with regard to mismatch would not appear to be signifxant.

Since the new technique involves additional terms, the load flow computation time per iteration will be longer. However, the discusser assumes that total computation time for each of the 57 bus test system

corresponding load flows based on the Newton-Raphson method load flows employing the new technique was less than that for the

because of the fact that one less iteration is required in each case. With regard to the 118 bus test system where the same number of iterations is required, it would be interesting to know how much additional computation time was necessary using the new second order technique.

The authors have shown that computer time can be shortened for some systems by utilizing one or another version of the second order

new version or one already developed by the authors - for application technique. Further work would determine the best version - be it a

to a specific system.

Manuscript received March 22, 1976.

M.S. Sachdev and T.K.P. Medicherla: The authors thank all the discus- sors for their interest in this paper and for their encouraging comments.

Messrs. Brameller, Hawary, Heydt, Rao, Sandhu, S o o d , Tiburcio, Vemuri and Vetter have enquired about the comparative computation

The CPU time for obtaining load flow solution of the IEEE 57-bus test times of the Newton-Raphson and Second Order Load Flow techniques.

system by Versions I, 11, II.a, Kb , Kc , II.d, and 1I.e of the Second

Manuscript received July 21,1976.

Order Load Flow technique are 105,89,88,84,84,82 and 82 percent of the time taken by the Newton-Raphson load flow. Version I of Second Order Load Flow technique requires more CPU time than the Newton-Raphson load flow. Version 1I.e takes minimum time for obtaining a load flow of this IEEE 57-bus test system.

Messrs. Brameller, Heydt, Rao, Sandhu, Sood, Tiburcio and Vemuri have enquired about the effectiveness of applying the Second Order Load Flow technique to nonconvergent systems. One such system, whlch 1s given in Reference 7, was chosen. This system is composed of one generator bus and 42 load buses. The load flow of this system does not converge when Newton-Raphson technique is used. However, a converged solution of this system was obtained in 8 itera- tions when second order corrections were applied in the second and subsequent iterations.

The decoupled load flow technique7 requires less CPU time than

authors have applied the principle of decoupling to the second order the second order approach described in this aper. However, the

load flow model. The results of this investigation have been reported in the paper “A Second Order Decoupled Load Flow Technique” (A76 491-1) presented at the 1976 IEEE PES Summer meeting. These results indicate that the benefits of including second order terms in the decoupled load flow model are not substantial. This is probably

icant and cannot be substantially offset by the use of second order because the approximations used in the decoupled approach are signif-

terms.

ple to the second order load flow model. This a proach when applied Dr. Vemuri suggests a direct application of the decoupling princi-

to the conventional first order load flow modell P was proved to be less effective than the decoupled model developed in Reference 7. In view of the results given in References 7 and 11, Dr. Vemuri’s approach is not expected to be superior to that of the technique suggested in our paper A76 491-1 mentioned above. Also, S 1 through S6 are sub- matrices and therefore equation D7 suggested by Dr. Vemuri is in- correct.

Dr. Vemuri has also enquired about the effect of initial guess

ed this aspect. However, in our opinion, including the second order on the convergence of second order load flows. We have not investigat-

terms causes the load flow to converge faster only if the estimate at any stage is close to the solution. The second order load flow technique is not likely to be effective when the initial estimates of the system voltages are poor.

Using AV/V’s in frst order terms and AV’s in second order terms is computationally more convenient than using AV/V’s in the fmt and second order terms.

We agree with Messrs. Hawary and Vetter that the rectangular form of the second order load flow model is in some ways superior to the polar form. However, considerable additional storage is required when rectangular form of the Second Order Load Flow model is used.

3

REFERENCE

[ 1 1 ] B. Scott, “Decoupled Newton Load Flows”, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-91, pp. 1955-57, September/October 1972.

1 9 7