factorized load flow

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 4, NOVEMBER 2013 4607

Factorized Load FlowAntonio Gómez-Expósito, Fellow, IEEE, and Catalina Gómez-Quiles, Member, IEEE

Abstract—This paper extends to the load flow problem the fac-torized solution methodology recently developed for equality-con-strained state estimation. This is done by considering a redundantset of initial conditions in order to transform the undeterminedsystem arising in the first linear stage into an overdetermined one.The second nonlinear stage proceeds exactly like in the state esti-mation case. Experimental results are included showing that thefactorized load flow is more robust than the standard NR method,while remaining computationally competitive.

Index Terms—Equality constraints, factorized solution, loadflow, state estimation, weighted least squares (WLS).

I. INTRODUCTION

A LTHOUGH rudimentary load flow (LF) solution proce-dures were introduced late in the 1950s, it was not until

the seventies that efficient techniques, taking advantage of andpreserving the Jacobian sparsity, made the quadratically conver-gent Newton-Raphson (NR) scheme fully competitive [1], [2].Since then, a number of improvements (component mod-

eling, computational saving by parallel computation, solutionadjustments) and alternative solution techniques (constantJacobians, approximate DC solutions, second-order methods,globally convergent schemes, etc.) have been proposed, ofwhich very few have found widespread use in industry. Byfar, along with the basic NR algorithm in polar form, the mostsuccessful implementation is the so-called fast decoupled loadflow [3]. The reader is referred to the excellent survey [4] or themore recent compilation provided in [5, Ch. 3] for a detailedaccount of the power flow problem and associated issues.The state estimation (SE) notion for power systems was also

introduced early in the seventies [6], in an attempt to circum-vent the limitations arising in primitive SCADAs from the directuse of LF solvers for real-time network monitoring purposes. Inaddition to power injections and bus voltages, state estimatorscan readily use power flow measurements, relying on the avail-able redundancy to reduce the uncertainty of the computed statevector. Like in the LF case, the nonlinear weighted least squares(WLS) original formulation has been the subject of extensiveresearch. This includes numerical improvements to the iterativeprocedure based on the Normal equations (orthogonal factoriza-tion, equality-constrained and augmented schemes), robustnessagainst bad data (normalized residual test, non-quadratic SE) orthe use of increasingly higher redundancy levels to detect and

Manuscript received January 10, 2013; revised April 15, 2013; accepted May22, 2013. Date of publication June 13, 2013; date of current version October17, 2013. This work was supported by the Spanish Ministry of Science andInnovation, under grant ENE2010-18867. Paper no. TPWRS-00031-2013.The authors are with the Department of Electrical Engineering, University of

Seville, Seville, Spain (e-mail: [email protected]; [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TPWRS.2013.2265298

correct topology and parameter errors. The reader is referred to[7] and [8] for an overview of the SE problem (see also [9] fora more detailed treatment of this topic).As early noted by pioneers, the SE reduces to an LF solution

when there is no redundancy at all and the critical set of mea-surements does not contain power flows. In this case, the mea-surement Jacobian is nonsingular and there is no need to buildthe gain matrix.For the purposes of this paper, the equality-constrained SE

formulation, in which a subset of (real or virtual) measurementshas to be exactly enforced, is of most interest [10]. In this case,there is also a straightforward, yet not so well known, relation-ship between the equality-constrained SE and the LF, which isfully exploited in this work.Recently, a two-stage SE solution methodology has been pro-

posed [11], which is finding a number of applications in hierar-chical state estimation. Of particular interest is the case when thefirst stage reduces to a linear WLS problem, since this leads toa factorized formulation which shows better convergence ratesand has proved to be faster than the Gauss-Newton (GN) iter-ative scheme, based on the Normal equations [12], [13]. Sucha factorized scheme can be easily extended to the equality-con-strained case by simply augmenting the involved equation sys-tems [14].This paper is aimed at extending the factorized SE solution

methodology, developed so far, to the LF problem, which isstraightforward by adopting a unified vision that considers theLF as a particular case of the equality-constrained SE.The paper is organized as follows: Section II summarizes

the equality-constrained SE, which is formally related to theLF problem in Section III. Then, Section IV reviews the re-cently-introduced factorized SE models. This constitutes thebasis of the factorized LF models presented in Section V, whichare tested and compared to the NR method in Section VI.

II. EQUALITY-CONSTRAINED STATE ESTIMATION

For the sake of self-sufficiency, the equality-constrainedWLS SE formulation is reviewed below (the reader is referredto [9] and [10] for further details).Exact quantities, such as zero injections, can be easily accom-

modated into the WLS estimation model by explicitly addingthe necessary constraints to the regular measurement equations:

(1)

(2)

where

state vector to be estimated, composed of busvoltage magnitudes, , and phase angles, (size

, being the number of buses);measurement vector;

0885-8950 © 2013 IEEE

4608 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 4, NOVEMBER 2013

vector of functions relating error-free measurementswith state variables;vector of measurement errors, customarily assumed tobe and uncorrelated, with covariance matrix

;vector of exactly known quantities ( for nullinjections);vector of functions relating with state variables.

The associated optimization problem consists of minimizingthe Lagrangian:

(3)

which leads to the repeated solution, until convergence, of thefollowing system:

(4)

where , is the vector of Lagrangemultipliersand represents the iteration index.A simpler and more widespread scheme consists of con-

sidering exact quantities as pseudo-measurements with muchlarger weights than those associated with regular measure-ments, which leads to the following Normal equations, instandard form:

(5)

or, carrying out the products

(6)

where is a sufficiently small scalar.There is a risk of unacceptable ill-conditioning if is too

small, but values in the order of , or even smaller, are welltolerated by present computers, particularly if orthogonal de-composition of the Jacobian [15], rather than Cholesky factor-ization of the gain matrix, is adopted. For practical purposes, thesolutions provided by (4) and (6) are identical in most instances.

III. LOAD FLOW AS AN EQUALITY-CONSTRAINEDSTATE ESTIMATION

If the number of equality constraints (2) equals (state vectorsize), and the resulting Jacobian is of full rank, then the es-timate is fully determined, irrespective of the number, valuesand uncertainty of measurements in (1). Indeed, from the lowerequation in (4), or from (6) with , the standard Newton-Raphson (NR) load flow iterative process is easily obtained

(7)

The above statement is trivial and well-known, except forthe following key observation: whereas in the conventionalNR-based iterative scheme (7) the only degree of freedom isthe starting point , in the equality-constrained WLS approach

(4) we can additionally choose arbitrary values for and . Infact, no matter how many extra measurements or pseudomea-surements we include in , the solution will remain the same,so long as .For practical purposes, adding the redundant set of pseu-

domeasurements (1) to the conventional LF model (2), isuseless, since the solution of (4) does not provide any computa-tional advantage compared to the simpler system (7). However,as explained in the sequel, this pretty simple idea can be appliedto the factorized SE model recently proposed in [12], [13],yielding a new family of LF solution procedures which can beadvantageous over the conventional NR method.Note that the values of , which do not have to be actual

measurements in the SE sense, could even change on the flyduring the iterative process, without any consequence for thecomputed solution (provided convergence is finally achieved).This idea is the basis of the new iterative scheme specificallydeveloped for the LF problem in this paper (Section V-B).

IV. FACTORIZED STATE ESTIMATION

For the sake of completeness, both the unconstrained andequality-constrained factorized WLS SE models introduced re-spectively in [13] and [14] will be reviewed here.

A. Unconstrained Factorized Model

In this approach, the following state vectors are involved:• State vector . Unlike in the conventional formulation,voltage magnitudes are replaced for convenience in thefactorized model by their log counterparts:

(8)

where .• Intermediate vector :

(9)

where and, for each branch connecting busesand , the following pair of variables is adopted:

(10)

(11)

Vector then comprises variables ( being thenumber of branches).

• Intermediate vector , composed also of variables

(12)

where

When those vectors are introduced, the conventional non-linear measurement model (1) can be decomposed into twolinear models:

(13)

(14)

GÓMEZ-EXPÓSITO AND GÓMEZ-QUILES: FACTORIZED LOAD FLOW 4609

which become coupled through a trivial nonlinear transform(change of variables)

(15)

The nonlinear function , as well as the constant matricesand , are easily obtained from the above definitions (see [13]

for the details). Of particular interest is theJacobian of , composed of diagonal scalars plus ,2 2 diagonal blocks, as follows:

(16)

(17)

the inverse of which are

(18)

(19)

Therefore, unlike the Jacobian of the conventional nonlinearmodel, obtaining involves virtually no computations.For the sake of compactness, the two-stage solution proce-

dure in the unconstrained case will not be detailed herein, sinceit is a particular case of the equality-constrained version de-scribed next (simply remove equality constraints).

B. Equality-Constrained Factorized Model

In the factorized formulation the nonlinear constraints (2)become also linear in terms of the auxiliary vector . There-fore, adopting the same notation as in the unconstrained caseabove, the nonlinear equality-constrained model (1)–(2) can berewritten as

(20)

(21)

(22)

where the only nonlinear term is .The best estimate of the pair is provided by solving the

following equality-constrained optimization problem:

(23)

which reduces to the sequential solution of two WLS simplerproblems, as follows:Step 1) Obtain a preliminary estimate by solving the

augmented system

(24)

where . The symmetric coefficient matrix istermed the “augmented gain matrix” (in the sequel, the subindex“a” will be used to denote augmented matrices).

Step 2.1: Use the estimate provided by Step 1 to ini-tialize by solving the following augmented system:

(25)

where

Step 2.2: Starting with , repeatedly solve the followingsystem until is sufficiently small:

(26)

where and the Jacobian is computed at.

In summary, this formulation proceeds by first computingfrom the linear system (24) and then iterating with (26) untilnecessary to obtain . With high redundancy levels and ratheraccurate measurements, it was found that the estimate pro-vided by (25) can be sufficiently accurate in practice.It is worth stressing again that both and are triv-

ially obtained, which is the main advantage of the factorizedformulation (the nonlinear functions arising in the con-ventional formulation cannot be inverted in this fashion).Both the unconstrained and equality-constrained factorized

solution schemes have proved to be computationally more ef-ficient than their GN-based counterparts. Lower solution timesstem from both lower costs per iteration and reduced iterationcount. Owing to the improved estimate provided by the firstWLS linear problem, the second WLS stage provides satisfac-tory solutions typically in just one or at most two iterations. Bycontrast, the conventional nonlinear formulation has to resort tothe flat start in absence of previous information.

V. FACTORIZED LOAD FLOW FORMULATIONS

With the above notation, the factorized load flow model con-sists of finding the pair simultaneously satisfying the fol-lowing two equations:

(27)

(28)

which reduce to the standard model when is eliminated:

(29)

In accordance with the state vector introduced above, whichcomprises for convenience all bus voltage magnitudes in logform, the vector of specified quantities must also contain allPV-bus voltage magnitudes:


Even though PV-bus voltage magnitudes can be removed fromboth and , like in the standard NR-based polar formula-tion, retaining those quantities will be very helpful in the fac-torized formulation to deal with PV-bus reactive power limits(see Section V-D).As shown in Section III, the load flow can be regarded as a

critically-constrained state estimation, to which arbitrary pseu-domeasurements of the form (20) can be addedwithout affectingthe “estimate”, which becomes in fact the load flow solution.While this is useless in the compact LF formulation, in the fac-torized model those pseudomeasurements are helpful, in com-bination with the underdetermined system (27), to render theintermediate vector fully observable, which is a requirementof the two-step procedure reviewed in the previous section.The factorized load flow problem reduces therefore to solving

the constrained WLS problem (23), to be compared with the so-lution of the more compact conventional system (7). The fol-lowing remarks are in order:• The number and choice of “pseudomeasurements” in(20) should guarantee that, when used in combination with(21), is observable. The simplest choice for , assuringnot only observability of but also certain redundancy, is

, where is a set of assumed values for , in whichcase reduces to the identity matrix.

• Also, the values of are theoretically irrelevant. In prac-tice, however, the more accurate they are the better estima-tions are provided by the first WLS problem arising in thefactorized approach.

• For the same reason, the weighting matrix , reflecting inthe SE case the relative accuracy of components, can bearbitrarily chosen in the LF case. The default choice in thiswork will be .

In what follows, the two-stage WLS solution approachreviewed above, initially devised for the SE problem, will beadapted to the simpler LF case.

A. Equality-Constrained Formulation

The two-step scheme described in Section IV-B is of directapplication to the LF case simply by assuming that representsa set of equally-uncertain initial conditions, rather than actualmeasurements or pseudomeasurements. The simplest choice is

and , leading to

For the sake of clarity, the two steps arising in this simplifiedcase will be repeated herein:Step 1: Obtain a preliminary estimate by solving the

augmented system

(30)

which, in turn, can be easily solved in two stages as follows:Step 1.1: Compute from

Step 1.2: Obtain

In this particular case the estimate can be interpreted as thevector that, satisfying the network bus constraints (27), mini-mizes the Euclidean distance to . In other words, the WLSproblem arising in Step 1 reduces to a constrained least-distanceproblem.Step 2.1: Use the estimate provided by Step 1 to

initialize by solving (25). In this case, with , theweighting matrix reduces to

(31)

where can be shown to be a diagonal matrix.Step 2.2: Starting with , repeatedly solve (26) until

is sufficiently small.In the LF case the iterative process is customarily stopped

when rather than is sufficiently small. Sincehas to be computed anyway to solve (26), the mismatch vector

is readily obtained as a byproduct during theiterative process. Note however that the choice of the stoppingcriterion may affect the number of iterations.

B. Sequential Iterative Scheme

The factorized solution scheme described above is directlyadapted from the SE problem, in which vector represents ac-tual measurements with given uncertainty. As this informationremains constant throughout the solution process it makes nosense to repeat Step 1. In the factorized load flow case, however,vector is a vector of pseudomeasurements ( ) intendedto render observable. The more accurate is the more appro-priate is to start iterating in Step 2.Based on this observation, an alternative iterative scheme is

possible in the LF case by updating the values of in Step 1after each iteration of Step 2. In this scheme, steps 1 and 2 aresequentially performed until convergence, as follows:Step 1: Obtain by solving the augmented system (30).Step 2.1: Obtain by solving (25).Step 2.2: Update . If is small

enough, then stop. Otherwise go back to Step 1.From the computational point of view, it is worth noting that

successive runs of Step 1 are very fast, provided the solutionadjustments discussed in Section V-D do not modify the co-efficients of the gain matrix. In such cases, Step 1 reduces toupdating the right-hand side vector in (30), followed by for-ward/backward elimination (this is called “warm” start, to bedistinguished from the initial “cold” start).

C. Formulations With Two Weight Scales

The preceding formulations can be advantageously replacedin practice by unconstrained ones if “pseudomeasurements”(20) are considered several orders of magnitude less accuratethan equality constraints (21). As discussed below, these for-mulations are very well tailored to easily incorporate reactivepower limits in PV buses, provided includes, in addition to, an estimation of the injected reactive power for each PV


bus. For this reason, the general relationship , ratherthan , will be used in this section, with .The two steps of the equality-constrained formulation

(Section V-A) can then be restated like in the unconstrainedcase of Section IV-A, as follows:Step 1: Obtain by solving

(32)

where the gain matrix is in this case

(33)

A more compact expression is obtained if all products are car-ried out

(34)

Step 2.1: Initialize by solving

(35)

where

Step 2.2: Starting with , repeatedly solve the followingsystem until convergence:

(36)

where the matrix is evaluated atOn the other hand, the sequential iterative scheme of

Section V-B can be replaced by the following procedure, inwhich Lagrange multipliers are not explicitly computed:Step 1: Obtain by solving (34).Step 2.1: Obtain by solving (35).Step 2.2: Update . If is small

enough, then stop. Otherwise go back to Step 1, with .

D. Handling Reactive Power Limits in PV Buses

Any load flow procedure should consider in practice anumber of adjustments so that the solution obtained takes intoaccount the action of control variables and associated limits(see [5, Section 3.7], for a thorough discussion on this issue).Providing a detailed account of how any conceivable solutionadjustment can be handled by the factorized formulation pre-sented in this work is clearly beyond a single paper. However,the most basic one, namely the enforcement of reactive powerlimits in PV buses, will be discussed herein, for which only thesequential solution approach of Section V-B will be considered.As explained above, the state vector comprises all bus

voltage magnitudes while the vector of specified quantitiescontains all PV-bus voltages. Keeping this in mind, let us

assume that after the th execution of Step 2 the net reactivepower injection at PV bus exceeds one of its limits

. Then, the next time Step 1 is performed the specifiedlogarithmic voltage magnitude is replaced in vector by

. The opposite is just done when the resulting PQ busmust be reverted to the PV type. Therefore, reactive powerlimits are checked after Step 2 but enforced in Step 1. Noticethat, even though the size of matrix remains constant after abus-type switching, its nonzero pattern changes when the rowcorresponding to is replaced by that of (or vice versa).The factorized version based on two weight scales

(Section V-C) offers a simpler strategy to accommodatereactive power limits, allowing the structure of matricesand to remain constant throughout the iterative process,as follows: assume that, in addition to the trivial components( ), we also include in vector an estimate of foreach PV bus (initially, in absence of a better value, ,but this is updated after each execution of Step 2). This way,the combined “measurement” vector in (34), composed ofboth subvectors and , contains all injected powers P&Q(except for the slack bus) as well as all voltage magnitudes, andchanging the bus type (from PV to PQ, or vice versa) reduces toexchanging the corresponding element of by the appropriateelement of . This can be easily done, without modifying thestructure of matrices and , by properly modifying just twocoefficients of the weighting matrix . More specifically, acomponent of is converted into a component of by adding

to the respective diagonal of , whereas a must besimultaneously added to the component of being shifted to .In summary, the new matrix has the form

VI. TEST RESULTS

In this section test results corresponding to the following net-works are reported and discussed:• IEEE 14-, 118- and 298-bus benchmark systems.• 30- and 85-bus radial systems, introduced, respectively, in[16] and [17]. Those systems have been regarded in theliterature as ill-conditioned, owing to large R/X ratios andvery short and long sections connected to the same node.

• Large Polish systems (over 2000 buses) corresponding topeak loading conditions, profusely used for benchmarkingpurposes since they became available in Matpower release[18].

In addition to the conventional NR method in polar form, theproposed factorized schemes are coded in Matlab, taking fulladvantage of available sparse matrix capabilities.The following acronyms will be used hereinafter for the

tested algorithms:• 1LkNL: Single execution of Step 1 (linear) followed byiterations of Step 2 (nonlinear), as required until full con-vergence (Section V-A). This iterative scheme is directly


TABLE INUMBER OF ITERATIONS FOR BASE CASES

adapted from the SE factorized solution proposed else-where.

• kLkNL: Concatenated execution of Steps 1 and 2 ( itera-tions) until full convergence (Section V-B). This is the newiterative scheme specifically developed for the LF problemin this work.

A. Convergence Rates for Base Cases

The most important check relates to the convergence rate,since any new proposal should compete with the well-knownquadratic convergence rate of the NR method. Table I gathersthe required number of iterations for two convergence thresh-olds (0.001 and 0.00001). As can be seen, both factorizedschemes converge in the same or fewer iterations than thoseneeded by the NR method, the differences being more notice-able for more exigent thresholds. It should be noted that, in1LkNL method iterations of Step 2 are preceded by a singleexecution of Step 1, while in kLkNL scheme iterations ofboth steps are concatenated. This is surely the reason whykLkNL performs better than 1LkNL.The whole iteration numbers in Table I may give the reader

the impression that the convergence rate of factorized schemesis very close to that of the NR, which is not actually the case.This is more clearly seen in Fig. 1, where the maximum mis-match vector component is shown for the 298-bus system. Notefor instance that, after three iterations, the largest power mis-match for kLkNL scheme is nearly four orders of magnitudesmaller than that of the conventional NR method. This more ro-bust behavior is characteristic of both factorized schemes, par-ticularly kLkNL, for all test cases.It is also worth noting that the two radial systems, that were

deemed as ill-conditioned in the old times of single-precisionarithmetic, are solved without any problem by all methods withtoday’s processors and mathematical packages.

B. Convergence Rates for Limit Cases

In addition to the base cases reported in Table I, more stressfulscenarios are tested in order to confirm the enhanced behaviorof the factorized schemes in such cases. For this purpose, allspecified powers (i.e., active and reactive power at load busesand active power at generation buses) are multiplied by the max-imum scalar for which the NR method converges from flat start,

Fig. 1. Convergence profile for 298-bus base (convergence threshold 0.00001).

TABLE IINUMBER OF ITERATIONS FOR LIMIT LOAD CASES

when reactive power limits are ignored. For each test system,this value is obtained, up to two decimal digits, through a trial-and-error procedure. The results are collected in Table II. Forinstance, for the 298-bus network, this factor is 1.32 (1.33 givesrise to divergence).In those scenarios, 1LkNL typically saves an iteration

whereas kLkNL tends to save two. Fig. 2 is the counterpart ofFig. 1 for this experiment, where it is clearly seen that eventhough the convergence rate slows down for all methods, thefactorized schemes achieve better performance.One more experiment is performed with the 118-bus system,

by converting all PV buses (except the slack bus) into equiva-lent PQ buses, with computed from the base-case solution.In this case, both 1LkNL and kLkNL take 4 iterations (i.e., justone more than in the base case) whereas the conventional NRmethod behaves erratically and fails to converge from flat startin 35 iterations (the specified limit), as shown in Fig. 3. Whenvoltage magnitudes at PV buses are initialized to their specifiedvalue, convergence is not achieved either. However, if the re-sults obtained after the first iteration by any of the factorizedschemes are used to initialize the NR process, then it convergesin three iterations. This proves that the NR method is more sen-sitive to the starting point than factorized methods, probably


Fig. 2. Convergence profile for the modified 298-bus case with bus powersmultiplied by 1.32.

Fig. 3. Convergence profile for the modified 118-bus case with PV buses con-verted to PQ buses.

owing to the fact that the latter use a redundant set of initialvalues, including branch variables.

C. Reactive Power Limits Enforcement

Next, the influence of many PV buses simultaneouslyreaching their reactive power limits is assessed. For this pur-pose, the 118-bus system is well tailored, as nearly one halfof its buses are voltage regulated ones. Different scenarios arecreated by replacing the original reactive power limits withsymmetrical limits given by , fordecreasing values of . This way, as approaches zeroall PV buses tend to PQ buses with , eventually leadingto infeasible cases. Table III shows the iterations required byboth the NR method and the kLkNL factorized scheme, for

in the interval 0.8–0.4 (for no limits are reachedwhereas leads to an infeasible case), as well as thenumber of PV buses which are switched to PQ buses in eachscenario. As can be seen, the convergence of the NR method is

TABLE IIINUMBER OF ITERATIONS WITH REACTIVE POWER LIMITS (118-BUS CASE)

TABLE IVRUN TIMES (ms) FOR BASE CASES

TABLE VRUN TIMES (ms) FOR LIMIT LOAD CASES

TABLE VIRUN TIMES (ms) WITH REACTIVE POWER LIMITS (118-BUS CASE)

more affected by the bus switching logic, particularly for veryexigent convergence thresholds.

D. Solution Times

Both the polar-coordinates conventional NR and the pro-posed factorized solution schemes have been coded in Matlab(version R2008a) and run under Windows 7 on a 64-bit i5 IntelCore laptop (2.27 GHz, 4 GB of RAM).Tables IV–VI provide execution times (in ms) for the sce-

narios considered in Tables I–III, respectively. Each value isthe average of 100 runs, as individual recorded times are some-what affected by other computer activities. Note that, for the NRmethod, solution times are of the same order as those obtainedwith Matpower [18].


Even though solution times significantly depend on thecomputer platform and software environment (C, Java, Matlab,Python, etc.), they can be useful to assess the relative compu-tational cost if similar efforts are made to optimize the codeof the procedures involved in the comparison. In view of datashown in Tables IV–VI , it can be concluded that the factorizedschemes are also computationally competitive compared withthe NR method. For the two largest systems, speedups in therange 1.67–1.85 are obtained with kLkNL scheme for thebase-case solutions. Higher speedups (up to 2.3) are achievedin the limit load cases, owing to the additional iteration saving.Considering that the factorized methods involve two steps,

the fact that lower solution times are obtained may be counter-intuitive. In addition to the enhanced convergence rate (i.e., lessiterations), the following remarks can help explaining those re-sults:• Unlike in the NR case, the coefficient matrices involvedin the factorized schemes are symmetric (and also positivedefinite in the reduced formulation based on two weights).Therefore, their symmetry can be fully exploited by usingCholesky factorization (which deals only with upper tri-angular matrices). In fact, when computing gain matricesby carrying out matrix products of the form , al-most half of the computational cost can be saved if this iskept in mind.

• In the factorized formulations, costly expressions in-volving trigonometric functions do not appear. Exceptfor the trivial Jacobian of the nonlinear functions ,all involved matrices are constant. Moreover, the firstlinear stage is computationally less expensive for repeatedsolutions, in which only the right-hand side vector getsmodified (this applies to the kLkNL scheme).

VII. CONCLUSION

This paper extends the factorized solution notion, developedelsewhere for WLS SE, to those cases in which the number ofequations exactly matches the number of unknowns, like in theLF problem. This provides a new factorized scheme to solveload flows as a sequence of two WLS problems (the first onelinear and the second nonlinear), fully departing from the well-known NR-based iterative approach.Two iterative schemes are possible in this context, one of

them specifically designed for the LF case, both of which can bebased in practice either on strictly-enforced equality constraintsor by means of two weight scales. The latter one is particularlywell suited to allow reactive power limits to be easily accom-modated in the iterative process.Several benchmark systems of different sizes have been

tested, both in base-case and modified stressful scenarios, inorder to assess the convergence pattern and robustness of theproposed schemes. In all tested cases, the two-stage approachreduces either the number of iterations (one or two iterationsare frequently saved), or the maximum mismatch vectorcomponent after every iteration. A prototype implementation

shows that the proposed factorized LF is also computationallycompetitive.Future efforts will be devoted to investigating the possibility

of applying the factorized approach to other power system re-lated problems.

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[18] [Online]. Available: http://www.pserc.cornell.edu/matpower/.

Antonio Gómez-Expósito (F’05) received the electrical engineering and doctordegrees from the University of Seville, Spain.He is currently the Endesa Red Industrial Chair Professor at the University

of Seville. His primary areas of interest are optimal power system operation,state estimation, digital signal processing and control of flexible ac transmissionsystem devices.

Catalina Gómez-Quiles (M’13) received the engineering degree from the Uni-versity of Seville, Spain, in 2006 and the M.Eng. Degree from McGill Univer-sity, Montreal, QC, Canada, in 2008, both in electrical engineering. In 2012 shereceived the Ph.D. degree from the University of Seville.Her research interests include mathematical and computer models for power

system analysis and risk assessment in competitive electricity markets.

factorized load flow

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