Even though the generalized modeling methodology apparentlyincreases the dimensionality of the problem, the computation of thedynamics is performed without incurring any penalty for the higherdimension. This is accomplished by using the invariance property ofthe first order linearization and the application of sparse matrixtechniques. By this procedure it is assured that the number ofequations to be solved by iterative process is consistent with theorder of the control system.Implementation of this generalized methodology concept in an

array processor should contribute in the long run to the de¬velopment of a power system training simulator.The potential future applications of the generalized modeling

methodology are:. Its application to modeling of power plants. Study of model order reduction. Design of power system training simulator. Exchanging of power system data among various organiza¬

tions.

January 1982, p. 147

Accurate Modeling of Frequency-Dependent Transmission Lines inElectromagnetic TransientSimulations

The network Zeq is a combination of resistance-capacitance (R-C)units and has a frequency response that matches that of the linecharacteristic impedance ZCM (Fig. 2(a)). The current sources lkhand lmh represent the weighted history of the voltage and currentwaves traveling along the line. These sources are derived from thesimulation of the system propagation function A fa) in the fre¬quency domain by means of a rational function approximation (Fig.2(b). In the time domain this rational function corresponds to a sumof exponential terms. This allows the evaluation of the convolutionintegrals that are necessary to determine lkh and lmh to be performedrecursively [2].Numerical TechniquesThe numerical techniques for the synthesis of Zfa) and the

simulation of A fa) are based on an asymptotic tracing of thecorresponding magnitude functions, from their initital to their finalasymptotic values. This results in a uniformly accurate represen¬tation over the entire frequency band, from the de condition to, forinstance, 106 Hz. The order of the approximations is automaticallydefined by the breaking points of the asymptotes. This processovercomes the limitations of previous rational function approxi¬mations where, due to the difficulty of pre-establishing the form ofthe approximating functions, the characteristic impedance has beenassumed constant and the propagation function has been simulatedwith only three exponentials [2] (resulting in loss of accuracy over anextended frequency range). The approximation of Zfa) in Fig. 2 (a)resulted in 8 R-C blocks, and the simulation of A fa) in Fig. 2 (b)resulted in 13 exponential terms.

J. R. Marti, Member IEEEUniversity of British Columbia, Department ofElectrical Engineering, Vancouver, B.C., Canada

Electromagnetic Transient SimulationsThe experience over the last ten years in the digital simulation of

electromagnetic transients in power systems has proven the ad¬vantages oftime domain formulations [1] for a large class of systemconditions. It has also been recognized, however, that not to accountfor the frequency dependence of the parameters of system com¬

ponents such as transmission lines with ground return can greatlyaffect the results of the simulations.

Different formulations have been suggested in order to developcircuit models that without restricting the generality of time domainformulations can incorporate the effect of the frequency depend¬ence of the parameters. The improvement obtained with thesemodels as compared to constant-parameter representations hasbeen quite encouraging. However, these models have encounteredin their application a series of numerical instability and inaccuracyproblems, and their use has required many particular considera¬tions.

New FormulationThe formulation presented in this paper avoids the numerical

problems encountered in previous formulations and leads to muchmore accurate models without the need for particular considera¬tions on the part of the users. The general form of the new models isshown in Fig. 1.

¡k(0 ¡m(t)

nVUÏ

Zeq lkhQj(T)lmhzeqjn m

Vm(t)

Ï

10 JO* NT*FREQUENCY IHZI

10" \(f Iff JO' IO*

(a)

20.4-f

02H

lunn imwii iirnw i nun iiimiiDmi hhmt rTmn nun limn nmnIO* IO1 IO"' I IO 10* IO* 10* IO* IO* IO' l(f

FREQUENCT IHZI

(b)

Fig. 1. New frequency-dependent line models at nodes kand m.Fig. 2. Simulation ofZfa) andA fa). Curves (I): Exact parameters.

Curves (II): New Modelparameters.

PER JAN 29

TestsThe validity and accuracy of the new method has been assessed

by comparisons with analytical results and with field tests. Theseresults are presented in the main paper. As an example of thecomputational efficiency of the routines, the simulation of aBonneville Power Administration field test required only about 20%extra computer time as compared to a constant-parameter rep¬resentation.

References[1] H. W. Dommel, "Computation of Electromagnetic Transients".

Proceedings IEEE, vol. 62(7), pp. 983-993, July 1974.[2] A. Semlyen and R. A. Roth, "Calculation of Exponential Step

Responses.Accurately for Three Base Frequencies". IEEETrans., PAS-96, pp. 667-672, March/April 1977.

January 1982, p. 158

An Efficient Constrained Power FlowTechnique Based on Active-ReactiveDecoupling and the Use of LinearProgrammingP. A. Chamorel and A. J. Germond, Member IEEEFederal Institute of Technology, Lausanne,Switzerland

Active and reactive power optimization is becoming an

increasingly important dispatching center function, in order toimprove security in general and to judiciously operate existingresources, by minimizing generation costs and system losses.The general optimization problem has to be treated globally, and

includes in its nature a great amount of variables and constraints.Several optimization methods have been developed for solving thisproblem with respect to planning. On the other hand, these methodsare unsuitable for on-line analysis, due to their large computerstorage and time requirements.

Linear programming is certainly a promising tool for solving thistype of problem, but at the price of a light inaccuracy due to thelinearization of nonlinear functions. Its application for solvingoptimal active power generation is well known, and gives satis¬factory results. However, the same technique is more difficult toapply to reactive power optimization, due to the nonlinearity ofrelations between reactive power injections and voltages. As mosthigh voltage power systems encounter problems in reactive powerand voltage management, linear programming extension to reactivepower is of great interest. An iterative and decoupled method, basedon the optimization of injected active and reactive current, isproposed.The objective functions consist in minimizing the overall pro¬

duction costs or priorities for a defined period, as follows:

active power: min [ ^FPj * \pj . U¡ . IP¡ 4- ^ppf . A<xf j

reactive power: min (Y¿FQ} . \q¡ . U¡ . IQ¡ + J)ws . ¿\Tg Jwhere

IP, IQ are the active and reactive generated currentsAa is the angular displacement of phase shifting

transformersAT is the difference oftransformer ratio oftap changing

transformersU is the bus voltage

Xp, \q, ¡ip, i±q are the linear costs or priorities of decision variables

The penalty factors FP and FQ are introduced if one of theobjectives is to minimize the system active losses.

Production, bus voltage and branch current constraints are ex¬

pressed in terms of generated currents and of variable transformers,which can be directly controlled, as follows:

2 C/ . IP-^U . /PL- Pioss = 0

^U . IQ 2 U . IQL Qloss = 0

VPBmin)<lAi)VP) + IBi) + iA2)(Aa) <(/PBmax)UQBm in) < (A3)(/Q) + (B2) + <C2) 4- {A4)IAT) < (IQBmax){Um]n) < (As)(IQ) + (B3) + (C3) + {A6)(AT) 4- Uref < (Umax)

with

where

IP, IPLIQ, IQLPloss, QlossIPB, IQB"ref(A,)f(A2)AA3)t

(B,),(B2),(B3)

(C2), (C3)

UPninXUPXVPmax)UQmin)<UQXUQmBX)(Acxmin)<(Aa)<(Ac*max)(Armin)<(Arx(Armax)

are the active generator and load currentsare the reactive generator and load currentsare the active and reactive system lossesare the active and reactive branch currentsis the reference bus voltage

[A4) are the matrices of decision variable con¬tributions to branch currents and bus volt¬agesare the vectors of fixed load contributions tobranch currents and bus voltagesare the vectors of fixed transformer contri¬butions to reactive branch currents and busvoltages

This set of linear equations is solved by the dual simplex method,with upper bound technique. The mentioned contribution matricesand vectors are computed by means of a dc-flow program, based on

the solution of the general complex equation, with active-reactivedecoupling:

where

(/net)m(U)

Unet) = (y)(U)

is the complex impressed current vectoris the complex nodal admittance matrixis the complex bus voltage vector

Optimization is performed by an iterative process. At the be¬ginning, only equality constraints are present before introduction ofbranch flow and bus voltage constraints, and the optimization is firstachieved by a simple sort of costs or priorités. If one or severalinequality constraints are violated, the latter are introduced insuccession of their occurrence, and optimization is thereafterperformed by linear programming. Generally, convergence is ob¬tained quickly especially in case of severe constraints. Contrary toactive problem, reactive power optimization is more difficult, be¬cause each voltage modification leads to a general modification ofloads, reactive losses, penalty factors and of several contributionmatrices and vectors.The proposed example deals with an actual 220/125 kV system

with 47 busses and 16 generating units, where the objective is tominimize active losses due to reactive productions. Nine iterationswere necessary for reaching convergence after introducing 7 busvoltage and 1 branch flow violated constraints. This optimizationtook8 seconds ofCPU on VAX-11/780. Checking the results has beenperforrried with a full Newton-Raphson load-flow, and shows in thiscase a very good accuracy of the linearized model.

30 PER JAN