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1394 Transactions on Power Systems, Vol. 7, No . 3. August 1992A LEAST SQUARES SOLUTION FOR OPTIMAL POWER FLOW SENSITIVITY CALCULATION

S.V. Venkatesh Wen-Hsiung E. LiuEmpros Systems InternationalMinneapolis, Minnesota

Abstract: Sensitivities of an Optimal Power Flow (OPF)solution to small changes in bus loads, flow limits, bus voltagelimits and other OPF constraints are becoming increasinglyimportant to the utility industry. Many of these sensitivitiesare not produced by most of existing OPF algorithms. In thepaper, a least squares based algorithm is proposed that issuitable for post-OPF sensitivity calculations. The algorithmprovides acceptable sensitivities of optimal solutions under theconsideration of all important operational and securityconstraints. The optimal solutions need not be exact. This isespecially important for real-time applications where onlyapproximate, but close to the true optimum, solutions areavailable. The proposed algori thm has been implemented in aproduction grade computer software and is sufficiently generalto be used by other electric power utilities. It is intended to beused as part of the OPF calculations in the new EMS of thePacific Gas and Electric Company. Test results on a 1700 bussystem are presented.Keywords:Incremental Costs, Least Squares Estimation.

Optimal Power Flow, OPF sensitivities, B us

I. INTRODUCTIONOptimal Power Flow (OPF) functions constitute a set ofmathematical tools that help utilities optimize variousoperating conditions of their power generation-transmissionsystem. [1.2] The objectives of optimization may beminimization of the total production cost, minimization of thetransmission line losses, or maintaining secure operation withminimum movement of control variables. The settings of theavailable controls. such as active power generation output,generator bus voltages, transformer tap positions etc., areadjusted, subject to some constraints of the system, to achievethe specific objective. Mathematically, there are two kinds ofconstraints -- equality constraints such as power flowconstraints. and inequality constraints such as limits on thecontrol variables or operating limits on the equipment andtransmission lines. Several OPF solution techniques such assuccessive linear programming approach [3,4]. successivequadratic programming approach [ 5 ,6 ] , and Newton approach[7]have been proposed and implemented. These techniqueshave attained a fair level of maturity in terms of flexibility,reliability and performance requirements for real lifeapplications.

Papers presented at the Seventeenth PICAConference at the Hyatt Regency BaltimoreHotel, Baltimore, Maryland, M a y 7 - 10, 1991Sponsored by the IEEEPower Engineering Society

Alex D. PapalexopoulosPacificGas md Electric CompanySan Frmcisco, California

The effects of changes in the system conditions on the optimaloperating states are usually very important for applications inthe real time environment. where OPF solutions track thesuccessive power system changes and continually steer thesystem in the direction of optimal and secure operation.Information regarding the sensitivities of the optimal stateswith respect to system changes, such as load variations,operating limit changes, or constraint parameter changes. canbe directly used in many practical applications. For example,the sensitivities of the production cost of generation withrespect to changes in the bus active power injections are calledBus Incremental Costs @ICs). A quantity such as BIC can beused to price co-generation. Other similar sensitivities can beused to establish proper pricing methodologies fortransmission services such as wheeling. These problems arebecoming increasingly important, as the utility industry's focusis shifted from building generation capacity to the transmissionpart of the business. Other OPF sensitivities such assensitivity of production cost of generation to change inbranch flow limits, can be useful in the planning environment.These sensitivities can help utilities identify expensivebottlenecks in the transmission system and take the necessaryactions.BICs and other sensitivities will be automatic by-products ofthe OPF solution if a full OPF using Newton approach [7 ] isperformed. However, most of the OPF techniques used in theEMS nvironment, rely on decoupling of the active and reactiveformulation [8.9]. Specifically, the OPF problem is decoupledinto an active power subproblem and a reactive powersubproblem. The active power subproblem determines thevalues of the active power controls that minimize an objectivewhich is only a function of active power variables whilesatisfying the active power constraints. The sequential linearprogramming is usually used to solve this subproblem.Similarly. the reactive power subproblem determines the valuesof the reactive power controls that minimize an objectivefunction which is a function of reactive power variables whilesatisfying the reactive power constraints. Usually, a Newtonbased approach is used to solve this subproblem. During theoptimization of the active power (or reactive power)subproblem, the reactive power (or active power) controlvariables are kept constant. The decoupled approach providesacceptable accuracy and meets the necessary computationalperformance requirements for real time applications.Unfortunately. it does not provide the necessary BICs and othersensitivities as part of the OPF solutions. Hence, a separateeffort is required to obtain the sensitivities.In the paper, a post-OPF sensitivity calculation algorithm isproposed. It computesLagrange multipliers. which are indeedthe required sensitivities [10.11]. after an OPF solution isobtained. The proposed algorithm involves the formulation ofa Lagrangian which includes the appropriate objective function(usually production cost of generation) and all the enforcedconstraints ( equality and binding inequality constraints). Byapplying the Kuhn-Tucker conditions. the derivatives of theLagrangian at the solution with respect to free variables (statevariables and control variables) provide a set of over-determined equations for the Lagrange multipliers. This set ofequations, ' however, may not be consistent since the OPF

0885-8950/92%03.00 0 1992 IEEE

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solution in the real-time environment is usually not exactConsequently, a least squares estimation of the Lagrangemultipliers with the overdete rmind set of equations is used.Mathematical formulation of the least squares problem for thesensitivity calculation is presented. In addition. acomputationally efficient method of solving the normalequation is derived. Special care has been taken to accuratelymodel all controls rind all operating and security constraints.The algorithm has been tested on a reasonably large systemThe results indica te the validity of the approach. Thisaccomplishment fulfils an important need of the utilities,especially in an EMS environment.The paper is organized as follows: In section II, themethodology for the sensitivity calculation is presented. Insection III. the modeling of various OPF constraints andvariables in the sensitivity calculation is discussed. Testresults on a 1700-buses system are reported in section W.Finally, conclusions are drawn in section V.

II. METHODOLOGY FOR SENSITIVITYCALCULATION

The optimal power flow problem can be formulated as (21:Minimize f(x)subject to g(x)=0and h(x)s

where:xf(x)g(x)h(x)

is the set of free variables consisting of controlvariables and state variablesis a scalar objective functionrepresents the power flow constraintsconsists of the limits on the control variables andthe operating limits on the power system.

This is a general constrained optimization problem. Thesensitivities of the objective function to the bindingconstraints are equal to the corresponding Lagrange multipliers'at the solution point [10,11]. In other words, the desiredsensitivities, such as bus incremental costs and bindingconstraint sensitivities. can be obtained from the Lagrangianwhich explicitly includes the active constraint set at thesolution. Consequently, the sensitivity calculation algorithmis a post-calculation procedure for computing the Lagrangemultipliers.The Lagrangian, with the knowledge of the active constraintset, can be formulated by including the objective of interest(such as production cost of generation or transmission linelosses) and all binding constraints.

where:AC(x)

is a vector of Lagrange multipliersis the active Constraints which include all thequality and binding inequality constraints at thesolution point.At the optimum, xs, the derivatives of the Lagrangian withrespect to the free variables. are zero, i.e.:

x=xs

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(3 )

(4)x=xs x=xs

Since an OPF has already been performed, he only u n l m om ineq. (4) is A For conciseness, eq. (4) can be rewritten as:

AA=b ( 5 )where:

A is an nx l vectorb = d 'x is.nmx1vect.mmn

is the number of free variablesis the number of binding constraints.Usually, m is greater than n. Therefore, the set of equations in(5) is overdetermined. If an exud OPF solution were available.it would be possible to choose any n independent equations outof the m equations and obtain the solution for the Lagrangemultipliers. A, since all m equations are consistent. However,in a real-time enviroment, improvements in the solution speedare more important than the determination of exact optimalsolutions. Therefore, at best, only approximate solutions areavailable. Decoupled formulations. for example, are usuallyadopted for real-time applications because of theircomputational performance. Such formulations CM onlyprovide approximate solutions which are reasonably close tothe true optimum. Hence, the m equations in (5) are notguaranteed to be consistent. The solution obtained from achoice of one set of n independent equations may be differentfrom the solution obtained from another set of n independentequations. It is very difficult to determine which set ofequations would produce an acceptable solution for A .Furthermore, from the numerical point of view, the selection ofa set of independent equations is not trivial, either. A betteralternative is to use lemst squares estimation where the errors inthe estimated A are"Ill]. The residuals of the leastsquares estimation can then be used as an index of the quality ofoptimization as well as the sensitivity estimation. This is verysimilar to the philosophy of bad data detection in power systemsta te estimation where the residuals are used to detect andidentify bad measurements.The least squares estimation of the sensitivities is formulated asfollows:

The solution which minimizes J(X). A*, can be determined bythe normal equation approach

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13%

In applications such as state estimation, the dimension of thegain matrix, ATA, s equal to the number of buses. The matrixgraph used lo form ATA CM be obtained directly from the powersystem network without going through actual matrix,multiplication [121. However, in the sensitivity estimation,the "nodes" of the matrix graph for ATA in (7) are theconstraints which are enforced at the solution obtained fromOPF. Hence, in this case. there is no physical network graphCorresponding to the gain matrix. In order to have an efficientapproach to build both symbolic and numerical ATA for sparsematrix computations. the Hachtel augmented matrix, H. s used.Consider the matrix H f dimension m+n:

m nl w k - 4(9 )

L d

It is shown in the Appendix that if the fust m rowslcolumns ofH are ordered and factorized first, ATA will be produced in thelower portion of H. The sparsity structure of the resultingsubmatrix ATA will not be affected by the ordering of the first mrows/columns as long as they do not mix with the rest of nrows/columns. The remaining n rows/columns are then orderedby minimum degree criterion to provide a sparsity structure forthe submatrix ATA. The techniques implemented here do notrequire major modifications of the traditional sparsity routines.On the other hand, they do eliminate the burden of matrixmultiplication when creating the gain matrix in (7).

III. MATHEMASE ?CAL'ICAL MODELLING INSITIVITYCULATION

All binding constraints at the optimal solution, have to bemodeled explicitly in the Lagrangian to obtain accurateconstraint sensitivities. Variables have to be modeled as freevariables or parameters based on their ability to move at theoptimal point obtained by an OPF. These three importantissues - binding constraints. free variables and parameterizedvariables - are discussed in the following subsections.The modeling information provided below is important becauseconstraints and variables may be modeled differently in themain OPF algorithm. For instance, the successive linearprogramming approach [3,4] will linearize the constraintsabout the operating point: this modeling is different from whatis needed for the sensitivity calculation. On the other hand, adecoupled approach [8,9] will model active variables and activeconstraints in the active power sub-problem and reactivevariables and reactive constraints in the reactive power sub-problem. Furthermore, the main OPF algorithm may notexplicitly model power flow constraints such as areainterchange constraints and transformer remote voltageconstraints. These constraints may be handled in between OPFiterations. The modeling provided below clearly illustrates howall of the constraints and variables can be modeledsimultaneously, to produce accurate estimates of the OPFsensitivities. The inclusion of all important operationalconstraints and their treatment in a unified way in thesensitivity calculation algorithm is one of the contributions ofthis work.

1. Biadlng ConstralntsIn general, there are three ypes of binding constraints:1) Equality constraints, such IS activeheactive powerinjection constraints;2) b c r l control constraints, such as area interchangecontrols, phase shifter controls, and generator andtransformer remote voltage controls;

Functional binding constraints, such as MW reserveconstraint. net area generation constraints, branch flowconstraints and transmission corridor constraints.3)

A brief description of some of the above constraints is givenbelow.a .Active Power ConsqgiDt (PIC)

An active power injection is present at every bus in thenetwork. The k corresponding to these constraints are thedesired Bus ncremental Costs. The constraint equation for eachbus is written as follows:

Pg - P1+a *E - Z Pflow=0where:

Pg = Total MW njection of generators connectedto the busP1= TotalMW oad of the busa= Participation factor of the generators

connected to the bus for area interchangeand/or dismiuted slackE = Slack error for area interchange and/or

distributed slackx flow = Sum of MW lows of branches connected tothe bus.A reactive power injection constraint is present at every bus inthe network. If the bus is a generator bus and it is participatingin voltage control. then the reactive injection constraint ismodified to include terms indicative of the generator'sparticipation in voltage control. The equation for each bus iswritten as follows:

Q~ -Q, +6 *a *E+Zy * G - Qflow= owhere:

Qg=Ql=

a =B =

E =

Z.r=G=

Q k w =

Sum of fixed MVAR injection of generatorsconnected to the busTotal MVAR load of the busFactor to change MVAR corresponding toMW lack adjustmentsparticipation factor of slack as in eq. (10)Slack enor for area interchange and/ordistributed slackTotal participation factor of generators forvoltage controlTotalMVAR equired for voltage controlSum of MVAR flows of branches connectedto the bus.

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constraint (mArea interchange control ensures that tie l i e flows match thescheduled net interchange for each company. When the areainterchange option is exercised, an area interchange equation iswritten for all but the primary company of the network. Thereasons are: 1) Specifying interchange for all but onecompany is sufficient- the interchange for the unspecifiedcompany is automatically satisfied; and 2) T he primarycompany is th e appropriate choice for not ,specifyinginterchange. since there is no slack variable corresponding toit- all slack is taken care of by the generator banks on MWcontrol.The area interchange equation for each external company ismitten as follows:

PM +PEIf+PEIw -CTie flows=0where:

PM= Desired area interchange for an externalcompanyFirm MW transaction of an externalcompany with primary companyWithdrawable MW transaction of anexternal company with primary companySum of tie line MW flows leaving theexternal company.

PEIf=

P a w=

ZTi e flows=

vo frvC)When several transformers control I remote bus, they share theMVARs required to maintain the voltage at its scheduled values.An equation is written for every participating transformer asfollows:

B *F Qflow = Owhere:B =

F =Qflow=

Participation factor of transformer in remotevoltage controlTotal MVAR required for remote voltagecontrolMVAR flow through the transformer.

Flow MWThe branch flow MW constraint is mitten as follows:

where:Pm= MW limit on branch

Pf= MW flow through the branch.m i s s i o n w r ons& (CRC)A transmission conidor constraint is a constraint on the s um ofthe M W flows through a set of lines included in the "corridor."The constraint is written as follows:

1397

(15)

where:Pa=

f P f i=

M W limit on the transmission comdorSum of MW flows through lines in thetransmission corridor.

2. Free VarlablcsDerivatives of the Lagrangian are taken with respect to all freevariables. These variables are free to move during theoptimization. There are three types of free variables:1 . State variables2 . Active control variables3 . Reactive control variables.State variables are :1 .2.3.4.5 .6 .7.

Voltage anglesVoltage magnitudesTransformer taps used for remote voltage controlPhase-shifters on MW flow controlTotal MVAR required for remote voltage control bytransformers (See handling of remote voltage control bytransformers)Total MVAR required for voltage control by generators(Similar to 5 above)Slack variables for each company for the interchangemodeling.

Active control variables are :1. Generatorbanks2. Phase-shifters3. Sheddableloads4.5.6.Reactive control variables are :

Economy interchange - Finn transactionsEconomy interchange - Withdrawable transactionsEconomy interchange - Capacity transactions.

1. Generator voluge magnitudes2. Static VAR compensators3. Voltage transformers4. Capacitor/Reactor shunts.3. Parameterlzed VariablesWhen a state variable such as voltage magnitude at a bus is at itslimit in the OPF solution, it can be handled explicitly byincluding it as a constraint in the Lagrangian. Altematively. itmay be handled by treating the limited variable as a parameter.The former results in a straightforward implementation; thelatter will significantly reduce the dimension of matrix, used tosolve for sensitivities, since at the solution point manyvariables could be at their limits. If a limited variable is treatedas a parameter, its Lagrange multiplier, and hence itssensitivity, can be determined as a function of other Lagrangemultipliers as follows:

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1398

Number of busesNumber of BranchesNumber ofGenerators

where:Lagrange multiplier of parameter -sensitivity of objective with respect toparameter-r. = Partial derivative of Lagrangian with respecta p to parameter

'=

17002130330

aLi=aCiap =

Partial derivative of objective with respecta p to parameterLagrange multiplier of constraint iPartial derivative of constraint i with respectto parameter.

-

Number of KV ControlsNumber of LTC ControlsNumber of Shunt ConmlsNumber of External Comnanies

IV. TEST RESULTS

48033011012

The proposed least squares based sensitivity calculationalgorithm has been implemented and tested on an actual 1700-bus system. Table 1 shows the main characteristics of thetested network.. .

Total Generation I49700MWTotal Load 147600M W

The algorithm has been inco rated with a successive linearprogramming based OPF, a Te wto n based OPF and also acombination of both. In the test results, the successive linearprogramming based OPF that minimiis the production cost ofgeneration, was used to provide the basic exact OPF solution.Subsequently, the sensitivity algorithm was used to calculatethe sensitivit ies of the production cost of generation in thesystem with respect to changes in the bus active powerinjections (BICs). The fmt column n Table 2shows the resultsof this calculation for 10 sample buses. In the second column,the estimated BICs for an approximate (but close to the trueoptimum) OPF solution are illustrated. This solution isobtained by the same OPF algorithm after appropriatelyrelaxing the convergence tolerances and keeping the penaltyfactors copstant throughout the optimization. The residual(sum of squares of errors in the estimation) is 0.019 for theexact OPF solution and 0.244 for the approximate solution. Inaddition to the results from the sensitivity method. the valuesof the BICs at these buses calculated by the differencing method(two OPF executions with and without 1 MW load increase ateach bus in the system) are presented in the third column ofTable 2. The binding sets (including power flow constraints)

should stay the same for both executions in order to getmeaningful results.Table2. BICs for Selected Buses

As can be seen, the largest difference between the results of thesensitivity calculation algorithm for the exact solution and thedifferencing method did not exceed 1.0percent. And also it didnot exceed 4.0 percent between the results of the sensitivityanalysis for the approximate OPF solution and the differencingmethod. This illustrates the fact that the least squares approachfor sensitivity calculation can be successfully used after anapproximate (but close to the true optimum) OPF solution hasbeenobtained.The largest differences between the exact and theapproximate OPF solutions are presented in Table 3.

Table 3. LXffqgces of the Exact and Approximate Solution

BICs provide an insight into the economic dispatchingmechanism and the ability to estimate the influence of activepower injection variations on the optimal solution, while alloperating and security constraints are satisfied. They can beused to establish proper pricing methodologies forcogeneration, wheeling and other transmission services. Intoday's competi tive environment, BICs can enhance a utility'sability to maximize services that involve economicallybeneficial transactions, to provide fair and equitable rates and tomitigate, to a large extent, the level of subsidization of oneclass of customers by another. For example, in the above test,a customer connected to bus 1 should compensate the utility bya larger amount than a customer connected to bus 10 for thesame type of service.Estimation of BICs for the approximate solution for the 1700bus transmission system using a CYBER 860 required 8.0 CPUseconds that can be further improved. The dimension of theHachtel matrix in eq. (9) is 6736 x 6736.Table 4compares the CPU time required to estimate the BIC atone of the system buses using the proposed algorithm and thedifferencing method. The time for the BIC estimate consists ofthe CPU time of a simple backward and forward substitution.(The time reported for the differencing method does not include

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SampleBindingConstraints

the overhead needed to start the OPF for the second execution --this overhead time can be substantial. ) Naturally. the time forthe differencing method depends on the OPF algorithm used toproduce the optimal solution. This does not affect thesensitivity calculation because the sensitivity algorithmoperates "outside" an OPF environment. As can be seen, theefficiency gained from the use of the sensitivity calculation issubstantial and this conclusion is expected to be applicableregardless of the OPF algorithm used.

BCS by the BCS by theSensitivity DifferencingMethod Method

. .Table 4. CPU Time for BIC E s mSensitivity Differencing

A Sample Bus from 0.584 16.00the 1700-Bus Network

Similar sensitivities can be computed for all controls andoperating and security constraints that are binding at theoptimal solution. These sensitivities can provide usefulinformation in identifying expensive bottlenecks in the systemwhose elimination, in the form of incremental generationadditions and transmission upgrades. could produce substantialsavings. Table 5 shows the binding constraint sensitivities(BCS) of an exact solution for one control (a generator bank)and four constraints (an interchange constraint between twocompanies, a branch flow. a transmission corridor and a net areageneration constraint). The sensitivities for the bindingcontrols/constraints calculated by the differencing method arealso included in Table 5. The largest difference between theresults of the proposed OPF sensitivity calculation algorithmand the differencing method did not exceed 1.0 percent.

T I'v't'e ect

Generatorh4WBranch Flow M W 16.14 16.28Transmission CorridorFlowMw

12.55

The sensitivity algorithm can also be used to evaluate theimpact of reasonable changes in system parameters andconstraint limits on the optimal solutions (by multiplying theestimated Sensitivities with the magnitude of the change inparameters). Multiple OPF studies (differencing method) canalso be used to produce the same results with higher accuracy,however, at a substantially higher computational cost. Toensure acceptable results, for these applications with theproposed algorithm, it is essential to verify that the active set(i.e.. the set of binding constr8jnts)does no t change because ofthe system parameter and constraint limit changes.

V. CONCLUSION

A least squares based algorithm, which computes thesensitivities of optimal operating states with respect to systemchanges, has been proposed. These changes include loadvariations. operating and security limit changes and constraintparameter changes. The optimal solutions. that can be providedby any OPF algorithm. do not need tobe exact.The OPF sensitivities of the proposed algorithm can be veryuseful to utilities in today's competitive energy markets. Theycan be used to establish proper pricing methodologies forcogeneration, wheeling and other transmission services.Furthermore, they can provide important information inidentifying expensive bottlenecks in the system, whoseelimination could produce substantial savings.The major contributions of t h i s work include:

The suitability of the proposed algorithm for computingsensitivities for approximate (but close to the trueoptimum) OPF solutions - an important need of utilities,especially m a real-time environment.The accurate modeling of all activeheactive. local/globalcontrols and all operating and security constraints in theOPF algorithm. L s k of modeling of the constraints. suchas those for area interchange control, in the OPFformulation, leads to inaccurate results. The OPFalgorithm used may not explicitly model these constraints,even if they use a Lagrangian formulation.The implementation of a new approach to form andfactorize the gain matrix, which considerably eases theprogramming effoxt

The least squares algorithm has been implemented and issufficiently general to be used by other utilities. It has beentested under a wide variety of conditions on a large scale powersystem. Test results prove the validity of the methodology.Low residuals obtained for all test cases strongly indicate thatthe produced OPF sensitivities are accurate when the OPFsolutions are reasonably close to the true optimum. Theproposed algorithm operates in conjunction with a successivelinear programming-based OPF. a Newton-based OPF and acombination of both. All four components are intended to bepart of the optimization capability of a major EMS. Thesensitivity calculation methodology developed in this paperprovides the utilities a well tested approach for OPF sensitivitycalculation.

VI. ACKNOWLEDGEMENTS

The authors acknowledge the many useful technical discussionsthey had with Prof. B. Wollenberg. Mr. W. Tmey . and Mr.C. Impamtoduring the development of this work.VII. REFERENCES

[ l ] J. Carpentier, "Contribution a. 1'etude du DispatchingEconomique," Bulletin de la Societe Francaise desElectriciens. Vol. 3. pp. 431447 , Aug. 1962.

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DiscussionA. P. Sakis Meliopoulos and X. Y. Chao: The authors should becommented for introducing Least Square estimation methods to esti-mate the sensitivities associated with constraints in an Optimal PowerFlow. The concept is extremely practical since many times one doesnot have a converged optimal power flow for a variety of reasons, suchas load changes since last OPF solution etc.In a real time application it is also desirable to know a measure ofthe accuracy of the estimated sensitivities.A direct way to compute anaccuracy measure of the sensitivities is

Define W-/ =diag(abs( A i - ) )Where i s the estimated sensitivities

Compute z =(A T W A ) - The diagonals of the matrix Z provide an estimate of the square errorof the computed sensitivities (see reference 1).On another matter, it appears to use that there may be the caseswhere the inverse of the gain matrix A%, [ A% ] - may be ill-condi-tioned. Have you encountered these conditions and if yes, whatsafeguards are you using to avoid the problem.

111Reference

A. P. Meliopoulos, A. D. Papalexopoulos, R. P. Webb, and C.Blattner, Estimation of Soil Parameters from Driven Rod Mea-surements, IEEE Transactions of Power Apparatus and Systems,Vol. PAS-103, No . 9, pp. 2579-2587, September 1984.

S. V. Venkatesh, Wen-Hsiung E. Liu, and Alex D. Papalexopoulos: Wethank Professor Meliopoulos and Dr. Chao for their valuable discus-sion and comments.The sensitivity calculation algorithm presented in the paper is apost-calculation procedure for computing the Lagrange multipliers.These Lagrange multipliers are solutions of the set of overdeterminedlinear equations in ( 5 ) . These equations are not guaranteed to beconsistent. Hence, a least squares solution was used in our algorithm.The value of the minimized objective function (6), i.e. summation ofthe squares of estimation residuals, will be a good index for theconsistency of these linear equations. Consequently, one may use it asan index for the quality of the optimization solutions produced byOPF algorithms. The method suggested by the discussers will be usefuladditional information regarding the estimation variance for eachsensitivity calculated by the least squares algorithm.The possibility of numerical ill-conditioning for the normal equationhas been well investigated in the power system state estimationproblem. Many different alternative solution algorithms, such asHachtel method, orthogonal method, etc., are available to mitigate theeffects of ill-conditioning. Any one of these numerically robust algo-rithms can be applied to ou r sensitivity calculation, if necessary. As amatter of fact, as a fall-back position, we have implemented anorthogonal based method to solve this problem. However, during ourdevelopment and testing, we have never encountered any numericalinstability. This may be due to the nature of the matrix A that doesnot contain either small or large numbers and its singular values arenot small. In other words, the condition number of matrix A is smalland the condition number of AA is still acceptable. Another problemthat we watched very carefully was the closeness of the optimalsolution to other non-binding constraints which may render the sensi-tivities invalid. The results of testing strongly indicated that this wasnot a problem, either.