Published on


Published in: Technology, Business
  • Be the first to comment

  • Be the first to like this

No Downloads
Total Views
On Slideshare
From Embeds
Number of Embeds
Embeds 0
No embeds

No notes for slide


  1. 1. Electric Power Systems Research, 20 (1991) 225 - 235 225Studies of Load Flow Methods in Polar and R e c t a n g u l a r CoordinatesV. H. QUINTANADepartment of Electrical Engineering, University of Waterloo, Waterloo, Ont., N2L 3G1 (Canada)N. MULLEREnergy Management Systems Division, Control Data Corporation, Plymouth, MN 55441 (U.S.A.)(Received June 14, 1990)ABSTRACT some modified versions of the standard fast decoupled load flow method [5] (in polar coor- This paper presents a study on the perfor- dinates) have recently been published [6-8].mance of load flow methods (in polar and The objective of this paper is to brieflyrectangular coordinates) with regard to com- describe several load flow techniques and com-putation speed, convergence reliability and pare their performance regarding speed, con-memory requirements. The load flow techniques vergence and memory requirements underin polar coordinates that are studied are the various load conditions and R / X ratios. InNewton-Raphson and the fast decoupled load polar coordinates, the N ew t on-Raphson loadflow methods; in rectangular coordinates, the flow (NRLF) and fast decoupled load flowload flow methods considered are the second- (FDLF) are considered in this study; inorder load flow and a decoupled load flow. A rect angul ar coordinates, the second-ordermodification of the standard fast decoupled load flow (RSLF) and a decoupled load flowload flow for systems with high R / X ratio is (RDLF) are studied. Various versions of thesealso presented and compared. Numerical re- methods, suitable for systems with high R / Xsults were obtained on a 16 MHz, 80386-87per- ratios, are also presented and discussed.sonal computer. Simulations were performed on Load flow techniques in polar and rectan-the I E E E 24-bus reliability test system and the gular forms differ from each ot her in the wayI E E E 118-bus test system. These systems were the node complex voltages are represented:simulated with several different R / X ratios andunder various load conditions. Vi = Vi (cos 0i + j sin 0i) = ei + Jfi (1) 2. LOAD FLOWS IN POLAR COORDINATES1. INTRODUCTION In polar coordinates, the real and reactive Many numerical techniques have been de- power injections are given byveloped for load flow solutions. The Gauss-Seidel method was one of the first to be Pi = Re{ Viii*}implemented on a digital computer. It wasfollowed by Newtons iterative method [1] = Vi ~ Vj(G o cosO 0 + B o sin O 0) j=lwhich is numerically more efficient t han theGauss~Seidel method. Some load flow methods i = 1. . . . . n (2)are surveyed in refs. 2 and 3. However, ref. 2 Qi = Ira{ Viii*}does not include recent advances in load flowcomputation, and ref. 3 does not extend the = Vi ~ Vj(G o sin O - B o cos O ) o o j=lanalysis to various load conditions, does notinclude any special-purpose load flow methods i = 1. . . . . n (3)and does not evaluate the second-order load where Yo = Go + jB~j is the ijth element of theflow in r e c t a n g u l a r coordinates [4] properly. bus admittance matrix Yb,8, 0o = Oi -- Oj, and nJustifications and performance evaluation of is the total number of buses in the system.0378-7796/91/$3.50 © Elsevier Sequoia/Printed in The Netherlands
  2. 2. 226 Four load flow techniques in polar coordi- ory requirements, a few assumptions havenates are studied in this paper: been suggested for the NRLF [5]: (i) Newton-Raphson load flow (NRLF) [1]; (1) neglect the submatrices N and M of (ii) standard fast decoupled load flow eqn. (4)--thus eqn. (4) becomes decoupled into(FDLF), as presented in ref. 5; two systems, one relating AP with A0, and (iii) a modified FDLF for systems with high another relating AQ with AV;R / X ratio (FDLFR/x), as presented in ref. 8; (2) cos 0o ~ 1.0, G 0 sin 00 <B0, Qi < B~i V~2;and (3) omit the representation of shunt reac- (iv) a simplified version of the FDLF, de- tances and off-nominal in-phase transformernoted by FDLFs~mp, for which the assumptions taps from H, and do not consider the seriesmade in ref. 5 are modified in order to simplify resistances explicitly; andthe calculations. (4) omit the effects of phase shifter trans- Flat start is assumed throughout this pa- formers from L.per, that is, at iteration k = 0 all unspecified Based on these assumptions, the equationsvoltage magnitudes are set to 1.0 per unit and for the standard fast decoupled load flowthe voltage phase angles of all buses are set at (FDLF) [5] are0 °. Neglecting second- and higher-order terms [Apk/v k/ = B [A0 k/ (5)of the Taylor series expansion of eqns. (2) and(3), the following matrix equation can be writ- [AOk/V] = B" [ A V ] (6)ten for an iterative solution: where B and B" are simplified matrices H and[Ap 1 [ p _ p k H k N,IFAo, l L, respectively. Both B and B" are functions (4) AQkJ=LQ-Qk]=[M k LkJLAVkJ of the network parameters only; therefore, they do not depend on the values of V and 0.where Hence, these matrices are constant through- = + - = +1 _ out the calculation and are factorized only once (at the beginning of the iterative pro-P~ and Q~ are calculated by eqns. (2) and (3), cess), unless P limit and/or Q limit violationsrespectively; H k, N k and M k, L k are Jacobian require a bus to change its status. Moreover,matrices of the first-order derivatives of P and B" is always symmetric and of order equal toQ, respectively, with respect to phase angles the number of buses whose reactive powerand voltage magnitudes; and superscript k injections are specified. Matrix B is of orderstands for the kth iteration number. equal to the number of system buses minus Equation (3) is used in computing the reac- one, and is symmetric for systems withouttive power of the P V and OV buses at every phase shifter transformers or for systems initeration. The Q limits of these buses are which these are accounted for by alternativechecked, and those buses with a violated limit means [5].change status to PQ or OQ, setting the reac- Equations (5) and (6) are solved alterna-tive power to the value of the violated limit. tively in an iterative process [5, 8], always using the most recent voltage values. Each2.1. The Newton-Raphson load flow method iteration comprises one solution for A0 (with The Newton-Raphson load flow (NRLF) which 0 is updated) and one solution for A Vmethod is a full-model method since it solves (with which V is updated). The iterative pro-eqn. (4) iteratively without any special as- cess is stopped when maxilAPil~<ep andsumptions or simplification [1]. The solution max/IAQII ~<eq are satisfied.of eqn. (4) is obtained when the absolute valueof every component of the mismatch vector 2.3. The simplified fast decoupled load flowlAP k, AQk] w is smaller than an acceptable tol- methoderance ~. In the simplified fast decoupled load flow (FDLF,imp) the matrices B and B" are deter-2.2. The standard fast decoupled load flow mined directly from --Bbu,. Thus both themethod shunt reactances and the series resistances of In order to simplify the calculations and the transmission branches are taken into con-obtain a much faster solution with less mem- sideration implicitly.
  3. 3. 2272.4. The high R / X ratio fast decoupled load equations are obtained for the real and reac-flow method tive power injections at node i: A fast decoupled load flow for systems withhigh R / X ratio of the transmission elements Pi = ~ [ei (ej Gij - fj Bij )(FDLFR/x) is presented in ref. 9. The method is j=lbased on the standard FDLF; now the series +f~(fjGo+ejBo)] i=1,2 ..... n (10)resistances are included in the calculations insuch a way that the range of convergence is Q~ = ~ [ f z ( e j V o - f s B o )enlarged. In order to include the effects of the j=lseries resistances, ref. 9 suggests the following -e~(fjG o + ejBo)] i = 1. . . . . n (11)procedure to modify matrix B of eqn. (5): The voltage equation is taken into accountBIj = - B i j - 0.4Gij - 0.3Gij2/Bo i #j (7) as follows:B~i = - ~ B~j V~2 = e~2 + f~2 (12) j¢:i Two basic load flow techniques, withwhere " . . . the coefficients of.0.4 and 0.3 were voltages in rectangular coordinates, are pre-found e x p e r i m e n t a l l y . . . " [9]. sented in this paper: a second-order load flow The effects of series resistances in eqn. (6) (RSLF) method based on ref. 4, and decoupledare taken into consideration by adding eqns. load flow (RDLF) methods [6, 11].(2) and (3):Pi -k- Qi = (Gii - - Bii) Vi 2 3.1. Second-order load flow method The second-order load flow method in + Vi ~ Vj [(Gij - B o) cos 0 o rectangular coordinates (RSLF) takes advan- j¢i tage of the fact that the functions of the +(G o + Bij) sin 0o] (8) right-hand side of eqns. (10), (11) and (12) are quadratic functions of the real and imaginaryBy assuming cos 0~j = 1 and components of the voltages (ei, fi); each ofIGij - Bol > I(Gij + Bo) sin 0ol these functions can be represented by a func- tion of the formeqn. (6) is replaced by y(x) = (1/2)xTAx[ A P k l V k + A Q h / V k] = B" [AV h] (9) where x is a vector comprising the real andwhere B" is given by B}j = G i s - Bis for all i imaginary components of the complexand j. voltages (except that of the 0V bus), A is a The power flow solution by the FDLFR/x is symmetric constant matrix (determined by thefound by first calculating B using eqn. (7) and bus admittance matrix), and superscript T de-then solving eqns. (5) and (9) by the same notes transpose. The Taylor series expansionprocedure employed for eqns. (5) and (6) of the of each of eqns. (10), (11) and (12) terminatesFDLF. at the second-order terms, with no truncation Appendix 5 of ref. 10 presents an alterna- error [4, 11], that is,tive compensation method for branches with ySpec = y~ (xest) + J~ A x + Yi ( A x ) esthigh R / X ratio; for each branch with prob-lems, the method artificially increases its sus- i = 1, 2 . . . . . 2(n -- 1) (13)ceptance by, say, bd, and puts a new line inparallel with the branch to cancel it out; this where x = col[f e] is the voltage column vec-new line is composed of a dummy additional tor; Yi = (1/2)xWAi x stands for the real power,bus and two dummy branches, each with sus- reactive power or voltage magnitude at nodeceptance - 2bd. i; superscripts spec and est denote specified and initial estimate values, respectively; jiest = ((~Yi/~x)TI = (xest)TAi3. LOAD FLOWS IN RECTANGULAR COORDINATES is the ith row of the Jacobian matrix; and Representing the complex voltages by their Ax = x - x ost. Notice that the second-orderreal and imaginary components the following terms of the Taylor series are simply expressed
  4. 4. 228as a function of Ax, that is, y,(Ax). The Equation (16) is used in this paper for thevoltage V of the 0V bus, also specified compo- rectangular coordinate second-order load flownent-wise, corresponds to the nth bus. (RSLF) method. Since the Jacobian matrix is Equation (13) leads to the following vector- constant throughout the calculations, it is im-matrix equation for all the specified Ps, Qs, portant to choose an initial estimate properlyand Vs (except at the OV bus): in order to ensure good convergence charac-jest Ax = y s p e c __ y ( x e s t ) __ y(Ax) (14) teristics. In this paper we chose to set the initial voltage of each bus equal to the valueEquation (14) is exact; there is no approxima- of the voltage of the 0 V bus which, in turn, istion in its derivation. Several schemes can be set equal to Vov =esw+j0, where e~w is thedevised for its numerical solution. In ref. 4, swing bus voltage magnitude. With thiseqn. (14) is solved numerically for the Ax of choice, the elements of the Jacobian matrixthe left-hand side by an iterative process in evaluated at the initial estimate are constantwhich the Ax of the right-hand side takes the and very much simplified [11].most recently calculated value at the previous Assuming that e = col[e L eg], where e Literation; that is, and eg are the voltage real parts of the P Q andjest Ax k + 1 _ y s p e c __ y ( x e s t ) __ y(Ax k) (15) P V buses, respectively, two observations can be made at this point [11].where Ax = 0 at iteration k = 0. (1) The equations for the voltage magni- It is important to note that the Jacobian tudes of the P V buses are independent of thej e s t is constant and calculated only once, at equations for the Ps and Qs; hence, eqn. (16)the initial estimate x est. Also, notice that Ax can be split into two systems of equations sois always measured with respect to the initial that eg can be computed separately.estimate x est. Thus the updated value of x is (2) For most practical power systems, thegiven by shunt conductances are negligible, thereforeXk+ 1 = xeSt _~_ Axk+ 1 6,, = - %The iterative solution of (15) is stopped when j~imax, IAx + - < Splitting eqn. (16) into two separate sys- tems, we haveIt can easily be shown (discussion by Duran inref. 4, and ref. 11) that the convergence char-acteristics of eqn. (15) are the same as those [B, G, B JLAe l:r J APle"-G Aeil(17a ) L-AQ Ie,.- e,jusing the Newton-Raphson method with theconstant Jacobian jest. That is, eqn. (15) can andbe written as [Ae~] = (1/2)[(AV2)k/es,] (17b)jest(xk + 1 __ xk) = y S p e e __ y ( x k) (16) The Jacobian matrix in eqn. (17a) is symmetricEquation (16) is the Newton-Raphson itera- a n d constant. Only its lower triangular part istive equation with constant Jacobian (evalu- stored and its factorization needs to be com-ated at the initial estimate). Since the puted only once. The order of the JacobianJacobian is constant, the solution of (16) is matrix for this method is the same as that ofequivalent to the so-called linear iteration the Jacobian matrix in the NRLF method.[12], whose general recursion formula is of the Additional storage is required for matrices G2form x k ÷ 1 = g(x~). Therefore, the rate of con- and B3, which are sparse, and do not requirevergence of the solution obtained by (16) is factorization.linear (or geometric), not quadratic as in the The iterative solution of the RSLF is per-standard Newton-Raphson method. However, formed by calculating [Ae~] from eqn. (17b)the rate of convergence can be improved by and then solving eqn. (17a). Values of f andrestarting the solution of (16) every, say, three e are updated and used in calculating AP,iterations, and using an updated Jacobian. A Q and (Air)2 for the n e w iteration. The itera-Obviously, there is a price to pay for updating tive solution is stopped w h e n the mismatchthe Jacobian: it may have different structure vectors AP, A Q and A V 2 are within acceptableand need to be triangular-factorized again. limits.
  5. 5. 2293.2. Decoupled load flow in rectangular 4. COMPUTER SIMULATIONScoordinates The decoupled load flow method in rectan- To solve Ax = b for x, Cholesky factoriza-gular coordinates (RDLFb), presented as tion can be applied for all symmetric positive-method (iii) in ref. 6, is based on assumptions definite cases, such as in the FDLF, RSLF andsimilar to those of the FDLF; that is, DLFR methods. In these situations, A is fac- torized into L D L T, where L is a lower triangu-[e,Bijl}>lfiGij I and ei ~ej lar matrix and D is a diagonal matrix. Since Aif bus i is connected directly to bus j. Al- is not symmetric in the NRLF, A may only bethough the author of ref. 6 does not make it factorized to LU, where U is an upper triangu-clear, it is assumed that Ifi[<le, I (i.e lar matrix. The solution x is obtained by back-cos Oij "~ 1.0 and sin 0u ~ Ou); otherwise, the as- ward and forward substitutions [12, 13].sumptions made in ref. 6 would not hold true. In order to keep A sparse during the factor-The following system of equations is formed: ization and reduce the number of multiplica- tive operations, and consequently reduce G IrAfl (18) round-off errors, the minimum-degree row and (AQIe)~I -Ol B~ JLAe~J column ordering scheme [13, 14] to reduce thewhere G~ is formed by elements of the Gbus number of fill-ins in A has been used.matrix, and B~ and B~ are formed by elements A Fortran-77 program has been written forof the Bbus matrix. Reference 6 simply neglects the purpose of testing the several load flowG~ in eqn. (18), creating two decoupled sys- methods and versions. It is a user-friendlytems of equations that are solved separately: program developed for an 80386-87 personal computer. It allows the user to choose the[(AP/e) k] = B~ [Afh] (19a) load flow method and its various versions, and[(AQ/e) k] = B~ [Ae k] (195) set data and output files. Computation of the condition number of the Jacobian matrices is In ref. 6, the P V buses are considered by done according to the users wish.properly adding eqn. (12) to the systems of Simulations were performed on the IEEEequations (19a) and (19b). The Taylor series 24-bus reliability test system [15], and theexpansion of eqn. (12), neglecting the second- IEEE ll8-bus test system [16]. These systemsorder terms, is described below, with notation were simulated with several different R / X ra-appropriate to an iterative process: tios and under various load conditions.V~2= (e~)2+(f~)2+2e~ Aek + 2f~ Af k (20) 4.1. Memory requirement Large errors may be introduced if any of The matrix storage scheme suggested in ref.the terms involving Ae h or Afk is neglected. 17 was used in the program. The exact amountTaking both terms into consideration would of storage requirement depends on the fill-inscouple eqns. (19a) and (19b) in such a way necessary in the Jacobian matrices during thethat separate solutions would not be possible factorization process. Although the number ofany more. fill-ins depends on the structure of the matrix, In order to overcome this situation in the which is a function of the number of buses andRDLFb, an alternative decoupled load flow in transmission elements of the system as well asrectangular coordinates (RDLF)--based on on the configuration of the network, it is pos-the RSLF method--is presented in this paper. sible to-estimate the total memory require-As already mentioned, the Jacobian matrix ment from the network data.does not require updating in the RSLF The order of the Jacobian matrices in themethod, and Aeg is determined separately by NRLF and RSLF is the same. However, theeqn. (17b). The equations corresponding to the Jacobian associated with the RSLF is symmet-voltage magnitude of the P V buses are not ric and therefore only its lower triangularincluded in eqn. (17a). By neglecting subma- part is stored. The Jacobians related to thetrices G1 and G1 w from the Jacobian matrix, FDLF and RDLF are of the same order and ineqn. (17a) may be decoupled into two systems both methods the matrices are symmetric, sug-of equations: one relating P with f, the other gesting similar memory requirements. Al-relating Q with e. though the calculations were made in single
  6. 6. 230TABLE 1Storage r e q u i r e m e n t s (bytes)Precision Method NRLP FDLF RSLF RDLFSingle 128n + 128b 92n + 52b 120n + 80b 92n + 52bDouble 208n + 196b 152n + 84b 192n + 124b 152n + 84bn = n u m b e r of buses; b = n u m b e r of t r a n s m i s s i o n branches.precision arithmetic, Table 1 shows an esti- concern and numerically stable methods, suchmate of the storage requirements of each as orthogonal transformations which limit themethod presented in this paper using both error growth, would not substantially improvesingle and double precision arithmetic. The the numerical solution.program was compiled by Watcoms Fortran- In certain network configurations a decou-77 and it creates an executable module of pled load flow, either in polar or rectangularapproximately 280kbytes (the program in- coordinates, may encounter difficulties in get-cludes all versions of the four load flow meth- ting convergence. Suppose that a bus, say busods, ordering and factorization routines, and i, is connected only to one other bus, say buscondition number routines). The executable j, and the net real or reactive power injectionmodule of any of the load flow methods alone, at bus i is large. In such a case the voltage phasewithout the condition number routines, re- angle difference between buses i and j is alsoquires approximately 100 kbytes of memory. large; consequently, the matrices associated with the decoupled load flows do not represent4.2. Power system conditioning the nonlinear load flow equations correctly. The condition number of a matrix is defined The RDLF failed to converge for the IEEEas the ratio between the largest and smallest ll8-bus system. The reason for failure, how-(in magnitude) eigenvalues of the matrix [12]. ever, may be due to a wrong assumption. It isThe smaller the condition number, the less assumed in the RDLF that the relationshipsensitive to round-off errors the linear system between P and e is negligible, and thereforeof equations, that is, a small condition number submatrix JPe is set equal to zero. However, asuggests a numerically stable system of equa- more careful observation of the elements of Jvetions. shows that the assumption should not be made. However, a good condition number does not These elements, prior to any simplification, areguarantee that a power system is well-condi- given by the following expressions:tioned, since conditioning of a power system isassociated with the ability to solve a set of ~2eiVii + ~_, (ejVij -fjBij ) for i =j ~Pi j~nonlinear equations that describes the physi- ~ej = ~eiGij +fiBij for i #j (21)cal behavior of the system mathematically.Although in some cases the matrices related At the first iteration, when fi = 0, eqn. (21) canto the linear models are well-conditioned, indi- be replaced by the real part of the Ybu~ matrixcating that solutions of the linear models are and may be neglected for networks with lownot subject to round-off errors, a load flow R / X ratios. However, at a point closer tomay be very difficult to converge. In such convergence, the term fi may not be assumedcases, the difficulty is more the result of poor zero. The voltage magnitude of bus 41 of thelinear models, which may not even approxi- IEEE 118 system is 0.967 p.u. and its voltagemately represent the nonlinear systems, than phase angle is -22.875 °. In rectangular coordi-the conditioning of the matrices associated nates this voltage is 0.8910-j0.3759. Clearly,with the linear models. the imaginary part of V41 cannot be neglected; In all systems and load flow methods used if it is neglected, large errors may be intro-in this paper the condition number of the duced (as is the case for the RDLF).Jacobian matrices is smaller than 20, suggest- It must be noted that the concepts behinding that round-off errors are not a problem of the decoupling performed in the load flow in
  7. 7. 231TABLE 2 (2), (3)) require 2n(t +2) additions andN u m b e r of iterations to achieve convergence: RTS 24 and ( 4 t - 1)n ÷ 1 multiplications for each compo-I E E E 118 nent of the mismatch vector, where t is the number of terms in the power series represen-Method P o w e r system tation of the trigonometric functions, while the RTS 24 I E E E 118 load flow equations in rectangular coordinates (eqns. (10), (11)) require 4n additions and 6nNRLF 3 4 multiplications for each component of the mis-FDLF 3 5FDLFsimp 4 6 match vector. Therefore, eqns. (2) and (3) re-FDLFR/x 4 5 quire 2(t - 1)/3 more multiplicative operationsRSLF 5 6 than eqns. (10) and (11) for each component ofRDLF 4 the mismatch vector. The NRLF method also requires update andpolar coordinates are very different and more factorization of the Jacobian matrix in everyacceptable than those applied in the load flow iteration, and every nonzero component of thein rectangular coordinates. For example, in Jacobian requires several multiplicative andpolar coordinates, the term Gij sin Oij is negligi- trigonometric operations. In addition, the Ja-ble compared with B~j because, in general, cobian matrix related to the NRLF method issin •ij ~ Oij ~ O. In rectangular coordinates, the nonsymmetric. While the Jacobian matricesterm f~ G~j is assumed negligible compared with associated with the FDLFs and RSLFs do notB~j. However, the magnitude of f~ is, in general, need to be updated, they are symmetric andmuch larger than that of sin O~j. As for the case their components are obtained directly fromof bus 41 of the IEEE 118 system, f41 = -0.3759 the network parameters. These matrices can bewhile sin 841.40 = -0-0078 and sin 841_42 = factorized only once at the first iteration. -0.0279, where buses 40 and 42 are the only Moreover, the factorization of the matricestwo buses directly connected to bus 41. For the related to the FDLFs require fewer multiplica-reasons mentioned above, further theoretical tive and additive operations than the matricesinvestigation of the RDLF is necessary. associated with the nondecoupled load flows. Table 2 shows the convergence of the vari- The FDLF is the fastest method, followed byous algorithms for the two networks. As seen the RSLF and NRLF methods, as long as allin the Table, no decoupled load flows in rectan- methods converge in the same number of itera-gular coordinates converge for the IEEE 118 tions. However, in general, the decoupled loadsystem. flows do not have the same convergence char- acteristics as the nondecoupled ones since the4.3. Speed of convergence Jacobian matrices associated with the FDLFs Every load flow method requires an update represent an approximation to the tangent ofof the mismatch after every iteration. Load the right-hand-side functions of eqns. (2) andflow equations in polar coordinates (eqns. (3).TABLE 3Average CPU time (s) per iteration: RTS 24 and I E E E 118System Method NRLF FDLF RSLFRTS 24 Mismatch 0.610 1.000 0.200 Jacobian N.A. 0.150 0.700 Factorization 14.204 2.801 7.303 Solve A x = b 0.400 0.400 1.050I E E E 118 Mismatch 3.778 1.6902 1.000 Jacobian N.A. 0.200 1.530 Factorization 522.794 93.715 243.080 Solve A x = b 0.611 0.941 2.460N.A. = not applicable since the m i s m a t c h vector and J a c o b i a n are calculated s i m u l t a n e o u s l y .
  8. 8. 232 Table 3 summarizes the computation time in the nominal case; the Q limits of the gener-per iteration for the NRLF, FDLF and RSLF. ators were not modified. Tables 4 and 5 showAll CPU times are in seconds and were ob- the number of iterations required to convergetained on an 80386-87 microcomputer. Some the RTS 24 and IEEE 118, respectively.CPU time has been saved in the NRLF method Tables 4 and 5 show t hat the convergenceby computing the mismatch vector and Jaco- characteristic of the NRLF is better than t hatbian matrix simultaneously; consequently, of the other load flow methods for the systemsonly the total CPU time to calculate the mis- studied. Moreover, the results with thematch and Jacobian is available for this FDLFR/X were better for the IEEE 118 than formethod. the RTS 24, probably due to the higher R / X ratio of the IEEE 118 than t hat of the RTS 24.4.4. Convergence reliability The largest R / X of the IEEE 118 is 0.4735, Load conditions and R / X ratios were while the largest R / X of the RTS 24 is onlymodified artificially on the IEEE 24-bus reli- 0.2590.ability test system (RTS 24) and IEEE 118 inorder to simulate systems that might operate 4.4.2. Changes in real and reactive powerunder these adverse conditions. load Results of changes in both the real and 4.4.1. Changes in reactive power load reactive power loads have also been obtained. Simulations on the RTS 24 and IEEE 118 The load demands and real power generationfor different reactive power loads were per- of all generating buses were modified by theformed and are presented. The objective is to same factor; the 0 V bus is the only exception,evaluate the performance of the load flows since it must compensate for the losses of theunder conditions similar to those observed for system. Similar simulations to those presentedlight and heavy reactive power loads. The in the previous section were performed. Loadsreactive power loads were modified such that and generations were varied from 20% toloads as low as half and as high as 2.5 times 300% of the nominal values, for both the RTSthe nominal values are obtained. The voltages 24 and the IEEE 118 the P V buses were set to the same values as No load flow method converged for loadsTABLE 4N u m b e r of i t e r a t i o n s for c o n v e r g e n c e : RTS 24 ( v a r i o u s r e a c t i v e power loads)Method L o a d c o n d i t i o n (%) 50 80 100 120 150 180 200 250NRLF 3 3 3 3 4 6 5 6FDLF 3 3 3 4 4 6 6 9FDLFsimp 4 4 4 4 4 6 6 9FDLFR/x 4 5 4 5 6 6RSLF 6 5 5 5 5 6 6 8TABLE 5N u m b e r of i t e r a t i o n s for c o n v e r g e n c e : I E E E 118 ( v a r i o u s r e a c t i v e p o w e r loads)Method L o a d c o n d i t i o n (%) 50 80 100 120 150 180 200 250NRLF 5 5 4 5 6 5 5 5FDLF 6 6 5 6 6 6 6 8FDLFsImp 6 6 6 6 6 7 6 7FDLFR/x 5 5 5 5 5 6 6 7RSLF 7 6 6 6 6 7 8 7
  9. 9. 233a n d g e n e r a t i o n s l a r g e r t h a n 150% o f t h e n o m - compare the reliability of the FDLF methodi n a l v a l u e s o n t h e R T S 24, a n d 200% o f t h e designed specifically for this purpose w i t hb a s e c a s e o n t h e I E E E 118 s y s t e m . T a b l e s 6 t h e o t h e r l o a d flow m e t h o d s . T h e s e r i e sand 7 show the n u m b e r of i t e r a t i o n s for e a c h r e s i s t a n c e s of all t r a n s m i s s i o n lines andsimulation. t r a n f o r m e r s o f t h e R T S 24 a n d I E E E 118 systems were, for each simulation, multiplied 4.4.3. Changes in the R / X ratio of the system s i m u l t a n e o u s l y by a f a c t o r t h a t i n c r e a s e s Some distribution systems are character- or reduces the R/X ratio of all transmissioni z e d by a h i g h R / X t r a n s m i s s i o n l i n e r a t i o . elements. Tables 8 and 9 summarize theseT h e o b j e c t i v e is t o s i m u l a t e t h i s c o n d i t i o n a n d results.TABLE 6 TABLE 7Number of iterations for convergence: RTS 24 (various Number of iterations for convergence: IEEE 118 (variousreal and reactive power loads) real and reactive power loads)Method Load condition (%) Method Load condition (%) 20 50 80 100 120 150 20 ~ 80 1~ 1~ 1~ 200NRLF 5 3 3 3 4 5 N-RLF 4 5 4 4 5 6FDLF 5 4 3 3 4 8 FDLF 5 6 5 5 6, 5 4 3 4 4 8 FDLF,~, 6 6 5 6 6 7FDLFR/x 4 5 FDLFR/x 6 5 4 5 6 6 10RSLF 5 7 5 5 6 11 RSLF 5 5 6 6 7 12TABLE 8Number of iterations for convergence: RTS 24 (various R[X ratios)Multiplication Largest Methodfactor R/X NRLF FDLF FDLF.~p FDLFRIx RSLF0.0 0.0 3 3 30.5 0.1295 3 3 31.0 0.2590 3 3 41.5 0.3885 3 4 42.0 0.5180 4 5 52.5 0.6475 4 7 73.0 0.7770 5 8 114.0 1.0360 4 115.0 1.2950 6TABLE 9Number of iterations for convergence: IEEE 118 (various R/X ratios)Multiplication Largest Methodfactor R /X NRLF FDLF FDLF.~p FDLFR/x RSLF0.0 0.0 4 5 6O.5 0.2368 4 5 61.0 0.4735 4 5 61.5 0.7103 5 6 62.0 0.9470 5 8 72.5 1.1838 5 83.0 1.4205 5 94.0 1.8940 125.0 2.3675
  10. 10. 2345. SOME COMMENTS AND CONCLUSIONS the other methods. High reactive power loads increased the number of iterations of the Memory requirements are highly dependent FDLF and FDLFsimp, suggesting t hat for ex- on the programming style; however, the de- tremely high reactive power loads these two coupled load flows in polar and r ect ang ul ar methods would fail to converge. coordinates are the most efficient in terms of Changes in both the real and reactivememory usage. The Newton-Raphson is the power loads affect the convergence of allmethod that requires the most memory to methods. The FDLFR/x was the method t hatstore its variables. performed best on a system with high R/X Ill-conditioned systems are more likely to ratios, but it was also t hat which had thebe related to poor linear models that may not worst performance on a system with low R/Xeven closely represent the behavior of the ratios. No substantial difference was observednonlinear equations than to the condition in the convergence of the other load flownumber of the matrices associated with the techniques. However, all methods failed tolinear models. The network configuration and converge on heavily loaded systems.operating point are two of the factors t hat For systems with high R/X ratios themost influence the convergence of a load flow. FDLFR/X, as expected, performed best, fol- The performance of the RDLF is severely lowed by the RSLF, NRLF, FDLF andcompromised by the assumptions and simplifi- FDLFsimp.cations; it did not converge for any of the The following can be concluded from thesimulations with the IEEE 118 system. Al- simulations performed.though it is a fast technique, its use is limited (1) The decoupled load flows are muchto a very narrow range of possible networks faster t han the nondecoupled ones.with some specific characteristics. (2) The convergence of the decoupled load The FDLFs are the fastest methods avail- flow in rect angul ar coordinates is very unreli-able, followed by the RSLFs and NRLF. The able.Jacobian matrix associated with the NRLF is (3) Within the range of the simulationsnonsymmetric and needs updating and factor- performed, the decoupled load flow proved toization at every iteration. The speed of the be the most reliable method for systems withload flow methods in polar coordinates may be high R/X ratios.improved by disregarding the higher order (4) The second-order load flow in rectangu-terms of the power series representation of the lar coordinates appears to be more reliabletrigonometric functions that have a negligible and is computationally much more efficienteffect on the value of the function. t han the Newton-Raphson method in polar The computation speed of all load flows can coordinates with regard to speed and memorybe improved by using sparse vector methods requirements.[18, 19]. The NRLF method can be made more (5) The memory requirements and speed ofcompetitive by using partial matrix refactor- convergence of the N ew t on-Raphson methodization [20] after the first few iterations. are a major drawback on the use of this From all simulations performed, it can be method for large power systems on todaysconcluded th at the FDLF has a slightly better microcomputers.convergence characteristic than FDLFsim,. (6) The fast decoupled load flow is almostThe R/X ratio of the transmission branches of as reliable as the Newton-Raphson and sec-the system appears to have much more influ- ond-order load flow, except for systems withence on the convergence of the FDLFR/x than high R/X ratios. Simplifications on the stan-the loading of the system. The FDLFR/x con- dard fast decoupled load flow worsen the con-verged for most loading conditions on systems vergence characteristics and do not improvewith high R/X, but did not converge under the speed of calculations significantly.light or heavy load conditions on a systemwith low R/X ratios. High and low reactive power loads did not ACKNOWLEDGEMENTaffect the convergence of the NRLF, FDLF,FDLFsim, and RSLF significantly, although The authors acknowledge the N at ural Sci-the NRLF converged in fewer iterations than ence and Engineering Research Council of
  11. 11. 235Canada, under grant A8835, for financial sup- 10 S. Deckmann, A. Pizzolante, A. Montecelli, B. Stottport. and O. Alsac, Numerical testing of power system load flow equivalents, IEEE Trans., PAS.99 (1980) 2292- 2300. 11 N. Mfiller and V. H. Quintana, Performance evalua-REFERENCES tion of load flow techniques in polar and rectangular coordinates, Tech. Rep. No. UWEE 89.01, Univ. Water- 1 W. F. Tinney and C. E. Hart, Power flow solution by loo, Canada, 1989. Newtons method, IEEE Trans., PAS-86 (1967) 1449- 12 G. Strang, Linear Algebra and Its Applications, Aca- 1460 demic Press, New York, 1976. 2 B. Stott, Review of load flow calculation methods, 13 A. George and J. W. H. Liu, Computer Solutions of Proc. IEEE, 62 (1974) 916- 929. Large Sparse Positive Definite Systems, Prentice-Hall, 3 A. Keyhani, A. Abur and S. Hao, Evaluation of power Englewood Cliffs, NJ, 1981. flow techniques for personal computers, IEEE PES 14 A. Gomez and L. G. Franquelo, An efficient ordering Summer Meeting, Portland, OR, U.S.A., 1988, Paper algorithm to improve sparse vector methods, IEEE No. 88 SM 539-9. Trans., PWRS-3 (1988) 1538- 1544. 4 S. Iwamoto and Y. Tamura, A fast load flow method 15 Reliability Test System Task Force of the Application retaining nonlinearity, IEEE Trans., PAS-97 (1978) of Probability Methods Subcommittee, IEEE reliabil- 1586 - 1599. ity test system, IEEE Trans., PAS-98 (1979) 2047- 5 B. Stott and O. Alsac, Fast decoupled load flow, IEEE 2054. Trans., PAS-93 (1974) 859- 869. 16 J. Zaborszky, K. W. Whang and K. V. Prasad, Monitor- 6 B. S. Babic, Decoupled load flow with variables in ing, evaluation and control of power system emergen- rectangular form, Proc. Inst. Electr. Eng., Part C, 130 cies, Systems Engineering for Power: Emergency (1983) 98- 102. Operating State Control, U.S. Dep. Energy Rep. No. 7 R. A. M. Amerongen, A general-purpose version of Conf. 790904 - P1, 1979. the fast decoupled loadflow, IEEE PES Summer 17 G. W. Stewart, Introduction to Matrix Computations, Meeting, Portland, OR, U.S.A., 1988, Paper No. 88 SM Academic Press, New York, 1973. 735-3. 18 W. F. Tinney, V. Brandwajn and S. M. Chan, Sparse 8 A. Montecelli, A. Garcia and O. R. Saavedra, Fast vector methods, IEEE Trans., PAS.104 (1985) 295- decoupled load flow: hypothesis, derivations, and test- 301. ing, IEEE PES Summer Meeting, Portland, OR, 19 A. Gomez and L. G. Franquelo, Node ordering al- U.S.A., 1988, Paper No. 88 SM 172-8. gorithms for sparse vector method improvement, IEEE 9 D. Rajicic and A. Bose, A modification to the fast Trans., PWRS-3 (1988) 73- 79. decoupled power flow for networks with high R]X 20 R. Betancourt, An efficient heuristic ordering al- ratios, Proc. 1987 PICA Conf., Montreal, Canada, gorithm for partial matrix refactorization, IEEE IEEE, New York, pp. 360-369. Trans., PWRS-3 (1988) 1181 - 1187.