Dependencia de La Frecuencia de Redes Electricas

8/13/2019 Dependencia de La Frecuencia de Redes Electricas

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1052 IEEE Transactions on Power Delivery, Vol. 14, NO. 3, July 1999RATIONAL APPROXIMATIONOF FREQUENCY DOMAIN RESPONSES BY VECTOR FllTlNG

B j ~ r n u sta vs en M)EFI, Trondheim, Norway

Email: [email protected]

Adam Semlyen LF)University of Toronto, Toron to, Canada

Email: semlyen @ecf.toronto.eduAbstract - The paper describes a general methodology for thefitting of measured or calculated frequency domain responses withrational function approximations. This is achieved by replacing a setof Starting poles with an improved set of poles via a scalingprocedure. A previous paper[5] has described the application of themethod to smooth functions using real starting poles. This paperextends the method to functions with a high number of resonancepeaks by allowing complex startingpoles. Fundamental propertiesofthe method are discussed and details of its practical imptementationare described. The method is demonstrated to be very suitable forfitting network equivalents and transformer responses. Thecomputer code is in the public domain, available from the firstauthor.

1 INTRODUCTIONOne of the problems encountered in power system transients

modeling is the accurate inclusion of frequency dependent effectsin a time domain simulation. Such effects arise from eddy currentsin conducting materials and sometime s from relaxation phenomenain dielectrics. These effects materialize as a frequency domainvariation in the resistance, inductance and capacitance matricesused in the formulation of the model. In practice, the frequencydependent responses are obtained via calculations or m easurementsas discre te functions of frequency.A linea r model of a power system compo nent can in general beincluded in a time domain simulation via convolutions betweenterminal quantities (e.g. node voltages) and impulse responsescharacterizing the dynamics of the model. A full numericalconvolution is always possible, but this becomes computationallyinefficient because of the many tim e steps in a simulation. A muchmore efficient implementation is achieved if the frequency domainresponses are replaced with low order rational functionapproximations, as the convolutions can then be given a recursiveformulation [l] The ability of finding a good rational functionapproximation is therefore important in power system modeling.In principle, an approximation of a given order can be found byfitting a ratio of two polynomials to the data [2]:

a , a , s a 2 s2+...+a N s N6 , 6,s 6,s2+.. .+6,sNs) =

Equation (1) is nonlinear in terms of the unknown coefficients butcan be rewritten as a linear problem of the type A x = b bymultiplying both sides with the denominator. However, theresulting problem is badly scaled and conditioned as t he column s inA are multiplied with different pow ers of s. This limits the methodto approximations of very low order, particularly if the fitting isover a wide frequenc y range.PE-I94-PWRDQll-1997 A paper recommended and approved bythe IEEE Transmission and DistributionCommittee of the IEEE PowerEngineeringSociety for publication n the IEEE Transactions on P rDelivery. Manuscript submitted July 2,1997; made available for printingDecember3,1997.

The difficulty in formulating a general fitting methodology hasresulted in many methods which are tailored for particularproblems. For inst ance , Bode type fitting restricted to real polesand zeros has been successfully applied to transmission linemodeling based on modal characteristics [3]. Transformer modelsand network equivalents need complex poles to representresonance peaks. Such responses have bee n approximated by fittingpartial fractions to the data in an optimization procedure, withprecalculated poles [4].An attempt at formulating a general fitting methodology wasintroduced in [ 5 ] . Th is method-vector fitting-was based ondoing the approximation in two stages, both with know n poles. Thefirst stage was carried out with real poles distributed over thefrequency range bf interest. In addition, an unknown frequencydependent scaling parameter was introduced which permitted the

scaled function to be accurately fitted with the prescribed poles.From the fitted function a new set of poles were obtained and thenused in the sec ond stage in the fitting of the unscaled function. Thisprocedure was very successful in fitting the smooth functionsoccurring in transmission line modeling [5-61. However, laterinvestigations by the authors have shown that the method failswhen there are many resonance peaks in the response to be fitted.This paper shows that the above mentioned limitations caneasily be overcome by using complex starting poles. This result isdemonstrated by numerical examples involving artificially createdfrequency responses, a measured transformer response, and anetwork equivalent. The paper also prov ides detail s on the practicalimplementation of vector fitting.

2 VECTOR FITTING BY POLE RELOCATIONCons ider the rational function approximation

Nf (s )= x L + d + s h (2) - a n

The residues c, and poles a , are eithe r real quantities or com e incomplex conjugate pairs, while d and h are real. The problem athand is to estimate all coefficients in (2) so that a least squaresapproximation of f ( s ) is obtained over a given frequency interval.We note that (2) is a nonlinear problem in terms of the unknowns,because the unknowns a , appear in the denominator.Vector fitting solves the problem (2) sequentially as a linearproblem in two stages, both times with known poles.Stage l:Dole identificationSpecify a set of starting poles Z, in 2), and multiply f ( s ) with anunknown function ~ ( s ) . n addition we introduce a rationalapproximation for ~ ( s )This gives the augmented problem :

Note that in (3) the rational approximation for ~ s ) as thesame poles as the approximation for ~ ( s ) s) . Also, note that the0885-8977/99/$10.00 1997 IEEE
http://ecf.toronto.edu/http://ecf.toronto.edu/http://ecf.toronto.edu/


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1053in (2) by a vector. This will result in all elements of the fittedvector sharing the same poles. The advantage of using the samepoles for all elements in a vector is that the time domainconvolutions for the vector become about twice as fast (see theclosure to [51).

3 STARTING POLES3.1 Sianificance of staft ina DolesSuccessful application of vector fitting requires that the linearproblem (6) can be solved with sufficient accuracy. In ourexperience, difficulties may arise due to poor starting poles in thefollowing ways :1) The linear problem (6) becomes ill-conditioned if the startingpoles are real. This may result in an inaccurate solution.2) A large difference between the starting poles and the correctpoles may result in large variations in a(s) and o(s ) f (s ) .Because a least squares approach is used when solving (6), apoor fitting may result where these functions are sm all.The first problem is overcome by usage of complex starting poles.The second problem is overcome by sensible location of thestarting poles, and by using the new poles as starting poles in aniterative procedure.3.2 Reco mm ended DfOCedUre for selec tlon of star t inu DolesFunctions with distinct reson ance pea ksThe starting poles should be complex conjugate with imaginaryparts p linearly distributed over the frequency range of interest.Each pair is chosen as follows :

a , = - a + j p , a n + , = - a - j p 9 )

ambiguity in the solution for a@) has been removed by forcinga(s) to approach unity at very high frequencies.Multiplying the second row in (3) wth f ( s ) yields thefollowing relation :

or(of)&> = ol , s ) f s ) 6 ,

Equation (4) is linear in its unknowns c n , d , h , Z n .Writing (4) forseveral frequency p oi qs gives the overdetermined linear problemA x = b (6)

where the unknowns are in the solution vector x . Equation (6) ssolved as a least squares problem. Details about the formulation ofthe linear equa tions is shown in Appendix A.A rational function approximation for f ( s ) can now be readilyobtained from (4). This becomes evideht if each sum of partialfractions in (4) is written as a fraction :

n = l n=lFrom (7) we getN l

n=lEquation (8) shows that the poles of f ( s ) become equal to thezeros of a&) (Note that the starting poles cancel in the divisionprocess because we use the same starting poles for (afb, and forafi,(s). Thus, by calculating the zeros of of&) we get a good setof poles for fitting the original function f ( s ) The calculation ofzeros from the representation by partial fractions 4) is

straightforward, as shown in Appendix B.On occasion, some of the new poles may be unstable. Thisproblem is overcom e by inverting the sign of their real parts.Stage 2 : esidue identificationIn principle we could now calculate the residues for f ( s ) directlyfrom (8).However, a more accurate result is in general obtained bysolving the original problem in (2) with the zeros of a(s) as newpoles a,, for f ( s ) . This again gives an overdetermined linearproblem of form Ax = b where the solution vector x contains theunknowns c n , d and h. It is solved as (A.2) of Appendix A(without the ne gative term).General remarksNote that the numerator and denominator of a have beenspecified in (7) to be of the same order. This has the implicationthat if the starting poles are correct, then the new poles (zeros ofa,,) become equal to the starting poles (ofi,(s)=l).n practicalapplications, this has the consequence that the rationalapproximation will converge if the new poles are used as startingpoles in an iterative procedure.In (2) an unknown constant term and an unknown proportionalterm are included. It is straightforward to modify the method tohandle cases were these quantities are known.Vector fitting is equally well suited for fitting vectors as it isfor scalars (hence its name). It is done by replacing the scalar f ( s )

wherea=p /100

This simple procedure produces starting poles with sufficientlysmall real parts, thus avoiding the ill-conditioning problem as isexplained in section 6.Smooth functionsUse real poles, linearly or logarithmically spaced as function offrequency, (The ill-conditioning problem associated with real poleswill not lead to an inaccurate fitting when f ( s ) is smooth.)

4 FlmNG NON SMOOTH FUNCTIONS4.1 Freauencv resnons8In what follows we consider an artificially created frequencyresponse of order 18, defined by (2). The assumed poles, residues,constant term and proportional term are given below.Table 1 Coefficients of frequency response defined by (2)Poles [Hz] Residues [Hz]- 4500 -3000- 41000 -83000- 100 f j 5000 -5 f j 7 0 0 0-120 f j l 5000 20 f j 18000- 3000 f j 35000 6000 rt j 45000- 200 f j 45000 40 f j 60000- 1500 f j 45000 90 f j l O000- 500 f j 70000 50000 f j 80000- 1000 j 73000 1000 f j 45000- 2000 f j 90000 - 5000 f 92000d=O 2 , h=2E- 5


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1054This response contain s 2 real poles and 16 complex poles. Notethat two complex pairs have poles with identical imaginary parts

(45000 Hz) but different real parts. The magnitude of the resultingfunction is shown in figure 1.

0 20 40 60 80 100Frequency [kHz]

Fig.1 18th order frequency response f ( s )4.2 Fitt lna w ith comDlex Doles10 pairs of complex starting poles were linearly distributed over thefrequency range, as explained in section 3.2. T he starting poles arelisted below :Table 2 Starting poles- 1. 00E-02f j l . OOE OO - 5. 553 02f j 5.55E 04- l . l l E 02 f j 1. 11E 04 - 6. 663 02f j 6. 66E 04- 2. 22E 02f j 2. 22E 04 - 7. 773 02 f j 7.77E 04- 3. 333 02 f j 3. 33E 04 - 8.883 02 f j 8.88E 04- 4. 443 02f j 4. 44E 04 - 1. 00E 03f j l . OOE OS

Using the starting poles in table 2, the first stage in vectorfitting was carried out. This produced rational approximationsof&) and (of)&). The magnitude of the responses is shown infigure 2. Also is shown the magnitude of the difference betweeno )f(s) and ( O ~ ) ~ , , ( S > hich corresponds to the error in (4). Thedifference is seen to be smaller than 1E-10.In the second stage in vector fitting, the zeros of csf,, s)werecalculated and used as new poles for the fitting of the originalfunction f(s) .Figure 3 shows that the resulting approximation fo rf(s) is extremely accurate The root-mean-square (RMS) errorwas found to be 3.8E-12. Table 3 lists the errors of the valuesobtained for the individual coefficients in table 1.They are all verysmall.Table3 Error in estimate of coefficients intable 1.Poles1E- 07

- 3E-081E- 11 f j 3E- 114E- 11 -I j 4E- 111E- 11 f j l E- 10

- 4E- 11f j 3E- 114E- 10f j 2E- 10- 4E-11 f j l E- 102E- 11 f j 5E- 117E- 12 f j l E- 10

Residues1E- 071E- 07

- 2E-09f j 5E-102E- 10 f j 5E- 09

- l E- 08 f j l E- 09SE- 09 f j 3E- 09- l E- 08 f j l E- 08

- l E- 08 f j l E- 08- 5E-09 f j l E- 08- 9E-09 f j l E- 09

d: - 2E- 12 h: - 5E- 18In this case we fitted an 18th order function with a 20th orderapproximation. The two "surplus" poles came out with very small

Poles Residues- 1. 03E-02 3. 78E-14- 1. 50E 04 - 4. 233-07The magnitude of the two corresponding partial fractions have botha maximum value less than 1E-11 . They do in practice notcontribute to the response and may therefore be removed.

l o5 r 1

120 40 60 80 100Frequency [kHz]

Fig.2 Rational approximations af,,(s) and (af)fi,(s)I I

i ccurate

C

deviation

C

deviation0 20 40 60 80 100

Frequency [kHz]Fig.3 Fitted function f ( s ) (20th order approximation)

A very attractive feature of the fitting methodology is that itwill not fail if one attempts to use a very high order approximation.For instance, if one increased the number of linearly spacedcomplex starting poles in table 2 from 20 to 40, the RMS-errorbecame 1.6E-12.4.3 Fltt lna w lth real startina DolesAs was pointed out in section 3.2, frequency responses withresonance peaks should be fitted using complex starting poles.However, we now attempt at fitting f ( s ) with 20 real poles,linearly spaced over the given frequency interval.The error of (4) became large, indicating that of&) isinaccurate. This resulted in the new poles to be incorrect and so thefitting of f ( s ) became poor, with an RMS-error of 7.1. However,several of the new poles were complex. After a few iterations withthe new poles as starting poles, a very accurate approximation wasachieved Table 4 shows how the error in the fitting of f(s)decreased dur ing the iterations.Table 4 Reduction in e rror by iterationIteration RMS-error

1 7. 12 1. OE- 11residues : 3 4. 2E- 13


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1055For starting poles we used 10 linearly spaced complex pairs.

6 shows the resulting fitting for f ( s ) after 4 iterations. It isseen that a fair approximation has been achieved.ccurate I

200

4.4 Reduced order f i t t lnqIn practical applications one often wants to find a low orderapproximation to a high order function. 7 peaks can readily beobserved in figure 1 so at least 14 poles should be used. Figure 4shows the resulting approximation after 3 iterations when using 7pairs bf complex starting poles. The deviation is seen to have itsmaximum around 45kHz, where f ( s ) has two complex pairs (seetable 1).

10accurate

Q) 0'I.-

cnf loo

10''

?I1 ,

0 20 40 60 ao 100Frequency [kHz]Fig. 4 14th order approximation of f s ) after 3 iterations

@ IIn som e applications f ( s ) will contain poles outside the frequencyinterval considered in the fitting process. To investigate the effectof this we attempt at fitting the response f ( s ) in the range 1Hz-6OkHz, so that several poles lie at higher frequencies than thefrequency interval considered in t he fitting.Starting poles were obtained by linearly distributing 8 complexpairs between 1Hz and 60kHz. Figure 5shows the resulting fittingfor f ( s ) in the range 1Hz-100kHz, after 3 iterations. The RMS-error for this interval was 3.3E-6, which is still very small.

I

0 20 40 60 80 100Frequency [kHz]Fig. 5 Fitting f ( s ) between 1Hz and 6OkHz (16th order approx.)

number of poles to 20.4.6 Effect of noiseWe have found that the ability of shifting the starting poles isreduced if a significant amount of noise is added to the response.Thus, convergence is slow and several iterations may be needed. Inthe following we add to f ( s ) noise which varies randomlybetween -10 and +IO.

The error further decreased to 3.2E-13 when increasing the

. i t teddeviationn

150?nY

0 20 40 60 80 100Frequency [kHz]

Fig.6 Adding random noise to f ( s ) (20th order approximation)The resulting RMS-error between the fitting function and f ( s )(including noise) was 5.0. For comparison, the RMS-value of thenoise alone was 5.3. Table 5 lists how the deviation decreased

during the iterations.Table 5 Reduction in error by iterationIteration RMS-error

1 18.22 9.53 5 . 34 5.0

4.7 Siani f lcance of start ina Dole locationIn general, the starting poles should be distributed so that theconsidered frequency range is covered. If, for instance, the startingpoles are confined to a small part of the frequency interval, thepoles will have to be shifted over a large distance during the fittingprocess.Figure 7 shows the resulting approximations af,,(s) and(o ,,(s) when using complex starting poles, linearly distributedbetween 1Hz and 20kHz. The functions are seen to have a verylarge variation, and (4) s not well satisfied at high frequencies.

1oo I0 20 40 60 80 ooFrequency [kHz]

Fig. 7 Approximation of ofjr s)nd (afifii,(s)sing starting 20complex starting poles in frequency interval 1Hz - 2OkHzThe resulting fitting for f ( s ) was poor at high frequencies.However, one more iteration with the new poles as starting polesreduced the RMS-error from 9.2 to 3.48E-10. (The variation inof&) and (ofifj,(s)as greatly reduced by each iteration as morepoles w ere shifted towards higher frequencies.)


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1056On the other hand, the iteration was much less successhi ifnoise was added to f ( s ) . When adding the same noise as infig ur e6 , no poles were produced for the fitting of the highestfrequency resonance peak, even after 50 iterations.

4.8 OPtimalitv of methodIn general, the method of vector fitting will not lead to an optimalfitting, as the resulting approximation may depend on the selectionof starting poles. To explain this, we attempt at fitting f ( s ) infigure 1with only one complex pair. This resulted in one of theresonance peaks being fitted. As to which of the peaks was fitteddepended on the location of the two starting poles.However, in the case of smooth functions, we have foundexperimentally that the fitting will (by iteration) converge to aresult which is independent of the starting pole locations.

5 FlrnNG SMOOTH FUNCTIONSIn some applications the frequency responses are very smoothfunctions, without resonance peaks. Exam ples of this are the modalresponses for characteristic admittance and propagationencountered in transmission line m odeling.In the following we c onsider an artificially created 18th orderfunctio n, defined by (2) with param eters as show n below :

Table 6 Coefficients of frequency responsePole Residue Pole Residue- 2000 1000 -34000 -12000- 4000 -1000 -44000 20000- 9000 7000 -48000 41000-15000 12000 -56000 8000-18000 5000 -64000 15600-21000 -12000 -72000 -10000-23000 -2000 -79000 -12000-29500 1500 -88000 50000-33000 31000 -93000 -2000d=O, h=O

Figure 8 shows the resulting fitting for f ( s ) when using 20real starting poles, linearly distri buted between 1Hz and 1OOkHz.The root-mean-square erro r was 5.9E-11, which is very small.

4, 1

00 20 40 60 80 100Frequency [kHz]

Fig. 8 Fitted function f(s)However, the parameters in table 6 were not accuratelyreproduced. Table 7 lists the calculated poles. It is seen that thepoles differ from those in table 6.The reason why an accuratefitting was achieved despite an inaccurate solution is related to thesolution not being well-defined, as will be explained in section 6.

Table 7 New poles produc ed by stage #1 in vector fitting.- 2 0 0 0 -36843 -58405 -91297-4000 -38736 -63158 -94737-9001 -47369 -68421 -99146-15128 -49859 -73684 -24393 f jl0589-30341 -52632 -84211

Smooth functio ns can often be fitted quite accurately with avery low order approximation. Table 8 shows the accuracy,dependent on how many real starting poles have been used (singleiteration).Table 8 Root-me an-square error in approximation of f ( s )

Order Error2 5.1E-24 7.1E-46 3.1E-58 6.23-620 5.9E-11

It should be noted that complex poles may also be used asstarting poles for smooth functions. The function f(s) was fittedusing 10 pairs of complex starting poles linearly distributedbetween 1Hz and 100kHz. This gave an approximation having anRMS-error of l .lE -7, which is somew hat less accurate than theresult obtained with real starting poles (5.9E-11). Also, several ofthe new poles (5 pairs) were complex.

6 ANALYSIS BY SINGULAR VALUE DECOMPOSITIONIn the first stage of vector fitting we have to solve the linear

A x = b (1 1)where each row k in A is built from the starting poles a as follows:problem

The accuracy to which (11) can be solved is best analyzed usingsingula r value decomposition. This allows A to be actorized :A = USVT 13)

S is a diago nal matrix con taining the singular values of A , and U sa matrix with orthogonal columns. Th us, the colum ns of U may beconsidered as basis functions in the representation of A. There areas many si ngular values as there are columns in A.Figure 9 show s the singular values for some of the previousexamples :1) Fitting a response with resonance peaks using 20 complexpoles (section 4.2).2) Fitting a response with resonance peaks using 20 real poles(section 4.3).3) Fitting a smooth response using 20 complex poles (section 5).4) Fitting a sm ooth response using 20 real poles (section 5).

From numerical mathematics it is known that the contributionto A associated with a small singular value is inaccurately handledin the solution process of ( ll ) , if its ratio to the largest singularvalue approaches the m achine precision, e.g. below 1E-12.The pattern of the singular values in figure 9 implies thataccurate results would be obtained for problem 1). This is exactlywhat we found in section 4.2 : all poles were estimated with veryhigh accuracy.In problem 2) se veral of the singular values are very small. Thisimplies inaccurate solution of (1 1). This is de trimental because the


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1057a resistive network connected to the high voltage terminals. Thre ecomplex pairs of starting poles were linearly distributed over thefrequency range in figure 11. The resulting fitting is seen to bequite good (5 iterations were used).

new poles (which are calculated from the solution) must beaccurately identified for a func tion with resonanc e peaks. This is inaccordanc e with the results obtaine d in section 4.3.In both problems 3) and 4 ) there are many small singularvalues which implies inaccurate solution to (12). It was found insection 5 that the poles w ere inaccurately estimated in both cases.However, a very acc urate fitting was still obtained because smoothfunctions can be approximated quite accurately using slightlyincorrect poles.

.... .b ; *b b..

.....0 10 20 30 40Count

Fig.9 Singular values of A7 APPLICATIONS

In the following we demonstrate vector fitting for a fewapplications involving frequency responses with resonance peaks.

We first consider the positive sequence admittance for a highlycomplex distribution network. In this case, all lines were modeledas distributed parameter lines evaluated at 50Hz. Thus, thefrequency dependent effects are not fully represented.Figure 10shows the resulting fitting after 20 iterations, whenusing 60 pairs of complex starting poles, linearly spaced over thegiven frequency interval. It is seen that a very good approximationhas been achieved. The RMS-error was 3.8E-3. The error wasfurther reduced to 6.OE-4 when increasing the order to 240.7.2 Transformer resno nseFigure 11 shows the fitted zero sequence admittance of a1 kV/230V transformer, seen from the low voltage terminals with

i o o r 3~ easured

...... itted

0 1 2 3 4Frequency [MHz]

Fig. 11 Zero sequence transformer admittance (6th order approx.)However, the phase angle was not fitted very accurately at lowfrequencies. Therefore, the number of complex starting poles wasincreased from 6 to 30, and 15 iterations were carried out. Figure12 shows that a significant improvement has been achieved for thephase angle at low frequencies. (The RMS-error for the entirefrequency interval decreased from 1OE-3 to 5.4E-5.)

measureda 6th order approx.I 30th order approx. \:60}

--0 0.2 0.4 0.6 0.8 IFrequency [MHz]

Fig. 12 Fitted phase angle at low frequencies

IO

1 occurate-

nY9 IOrC0

1 -2

I o - ~0 50 100 150 250 250 300 350 400 450 500Frequency [kHz]

Fig. 10 Positive sequ ence admittance (120th orde r approximation)


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=l 1mm

deviation

p = 10000QrnFig. 13 Transmission line system

The open circuit line admittance Y ( w ) = I (w)IV(w) wascalculated for the overhead line in figure 13,taking frequencydependent effects in condu ctors and earth into account.Figures 14 and 15 show th e fitted positive- and zero sequenceadmittances, using 36 and 44 omplex starting poles, respectively.(Two iterations were carried out). The resulting fitting is seen to bevery accurate. The d eviation was further reduced when increasingthe order.

I

ccurate

.-CFnr lo8.-deviation

0 20 40 60 80 100Frequency [ktiz]Fig. 14 Positive sequence admittance(36th order approximation)

J

MethodologyAlthough the method of vector fitting is very easy to implement ina computer program, it may at first glance be difficult to see howand why it works so well. To see this we go back to equation S).which is repeated below for convenience :0 >fi , 3,f = 0 (14)

f is a given rational function with unknown poles a, , while(of)f l , nd ofl,-are unknown rational functions with given poles. The poles a,, are specified (starting poles). In order to satisfy(14), of,,ust have zeros which equal the poles of f , and poleswhich equal the zeros of f .Thus, in the product of, he originalpoles a, of f become replaced by the starting poles Z .Consequently, both and ori get the same poles E m , ndan accurate solution of (14) exists After solvin g (14), the original

a, of f are calculated as the zeros of ofi t .f course, accurate solution of (14) requires that the number ofsfarting poles E,, equals or exceeds the number of poles a,. If atoo low order is used, the replacement of poles in f will beincomplete, and the solution of (14) will be only an approximation.

AccuracyIn principle, a perfectIy accurate approximation should beobtained for the response of a rational function when a sufficientlyhigh order is used in the fitting process. T he results in section 4 oran artificially created response showed that extremely accurateresults were achieved, provided that the starting poles weresensibly selected.Starting poles should in general be complex with weakattenuation, and should cover the frequency range of interest. Itwas shown that even if starting poles were poorly selected, a veryaccurate result was still achieved by reusing the new poles asstarting poles in an iterative procedure. However, the speed ofconvergence was reduced if the order of the approximation wasreduced, or noise was added to the response.If the considered response f ( s ) is rational, then vector fittingwill (with a sufficiently high order) give a rational approximationwhose frequency response almost perfectly matches the originalone. If in addition the poles of the considered frequency responseare complex with fairly weak attenuation, then vector fitting iscapable of recovering the unknown poles and residues with a veryhigh accuracy (section 4.2). This in not possible if f ( s ) has densereal poles (section 5) as the solution is ill-defined. Nevertheless, thepoles and residues produced by vector fitting will still give a veryaccurate approximation for the given frequency response.

It was further shown that the accuracy will not be reduced ifone tries to use an excessive number of poles.In section 7 it was shown that vector fitting gave a goodapproximation for a measured transformer response. Increasing theorder of the fitting gave a substantial increase in the accuracy forthe phase angle. This accuracy property may b e very useful becausethe passive circuit behavior of an admittance representation couldbe lost in some frequency interval if the admittance is not wellfitted. Vector fitting was also shown to give very good results forthe admittance of a highly complex distribution network, and forthe open circuit admittance of an overhead line.EfficiencyConsider the fitting of a scalar f(s) using N , frequency pointsand N poles. This leads to the following operations :1) Calculation of new poles : Solving linear system A x = b ,where A has dimension 2 N , x ( 2 N +2 ) .2 ) Calculation of zeros : Calculate eigenvalues of matrix ofdimension N x N .3 ) Calculation of residues : Solving linear system Ax = b whereA has dimension 2 N , x ( N + 2 ) .For the example in section 4 we had N , = 100 , N = 20. UsingMatlab running on Windows NT on a Pentium PC, the aboveproblems were solved in 0 . 0 4 O . O l and 0.01 seconds, respectively.


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105912 APPENDICES

A- ole IdentificationTo simplify notation we will in the following use a instead of Efor starting poles. Equation (4) an be rewritten as

9 CONCLUSIONSThe paper has extended the method of vector fitting toapplications involving frequenc y domain responses with resonancepeaks. This has been achieved by introduction of complex startingpoles. Important features of the improved vector fitting are :The method is very accurate. In the case of an artificiallycreated response with known poles and residues, theseparameters were estimated with extreme accuracy. The methodwas demonstrated to give very accurate results for a measuredtransformer response and a network equivalent with a litrgenumber of resonance peaks.Sensible selection of starting poles will in many cases obviatethe need for iteration. Starting poles should be complex withweak attenuation, distributed over the considered frequencyinterval. A recommended procedure for the selection of startingpoles is given in section 3.2.The method is very robust. It will not fail if attempting to use afitting of very high order, or poorly selected starting poles.The method is very easy to implement in a computer program.It essentially consists of building matrices from simplefractions. The resulting matrix problems are solved usingstandard software packages.The method is very efficient. Application involves the solvingof two overdetermined linear matrix equations A x = b ofmoderate size.The method permits matrix elements to be fitted simultaneouslywith identical poles. This permits increased efficiency of thetime domain convolutions [ 5 ] .

10 ACKNOWLEDGMENTSThe authors would in particular thank Dr. T. Henriksen (Em)for suggesting to use the compact formulation (4)nstead of using(3) directly. Financial support by EFI and the Natural Sciences andEngineering Research Council of Canada is gratefullyacknowledged. The authors thank Mr. H.K. Hoidalen (NTH) and

Dr. T. Henriksen (E n ) for providing the frequency responses forthe transformer and the complex network equivalent, respectively.11 REFERENCES

A. Semlyen and A. Dabuleanu, "Fast and Accurate Switching TransientCalculations on Transmission Lines With Ground Return UsingRecursive Convolutions", IEEE Trans. PAS, vol. 94, MarcWAprilA.O. Soysal and A. Semlyen, "Practical Transfer Function Estimationand its Application to Transformers", JEEE Trans. PWRD, vol. 8, no.J.R. Marti, "Accurate Modelling of Frequency-DependentTransmission Lines in Electromagnetic Transient Simulations", IEEETrans. PAS, vol. 101, no. 1, January 1982, pp. 147-157.A. Morched, L. Marti and J. Ottevangers, "A High FrequencyTransformer Model for the EMTP", IEEE Trans. PWRD, vol. 8, no. 3,B. Gustavsen and A. Semlyen, "Simulation of Transmission LineTransients Using Vector Fitting and Modal Decomposition", paper PE-347-PWRD-0-01-1997, presented at the 1997 IEEWPES WinterMeeting, New York.B. Gustavsen and A. Semlyen, "Combined Phase and Modal DomainCalculation of Transmission Line Transients Based on Vector Fitting",paper PE-346-PW RD-0-01-1997, presented at the 1997 E W E SWinter Meeting, New York.

1975, pp. 561-571.

3, July 1993, pp. 1627-1637.

July 1993, pp. 1615-1626.

For a given frequency point st we get

where

X = [Cl e .. CN d h E ... E N I T bk =f(s,) (A.4)Note that Ak and x are row and column vectors, respectively.In the case of complex poles, a modification is introduced toensure that the residues come in perfect conjugate pairs. Assumethat the partial fractions i and i+l constitute a complex pair, i.e.,

U =a + j a ' . , ~ , + ~a - j a , c i = c '+ jc',ci+l =c' - j c " ( A . 5 )and AkI+, are modified as

j j (A.6)This has the effect that the corresponding residues in the solutionvector x become equal to c' and C , espectively.Writing (A.2) for several frequency points gives anoverdetermined linear m atrix equation :

A x = b ( A . 7 )In the fitting process we use only positive frequencies. In order topreserve the conjugacy property we have to formulate (A.7) interms of real quantities :

The two corresponding elementsfollows :1 1Ak,i=-+-, Ak,i l=---sk -ai sk -ai St -ai Sk -ai

[;]x=[JB- alculation of zerosAfter solving (A.@, the zeros are calculated as the eigenvalues ofthe matrix

H = A - b Z T (B.1)where A is a diagonal matrix containing the starting poles and b isa column vector of ones. Z is a row-vector containing theresidues for CJ . In the case of a complex pair of poles, thecorresponding submatrices in (B.l) are modified (via a similaritytransformation) as follows :

LThis modification has the effect that H becomes a real matrix andso its complex eigenvalues come out as perfect complex conjugatepairs.13 BIOGRAPHIES

For authors' biographies, see [5].


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1060Discussion

N. R. Watson (Department of Electrical ElectronicEngineering, University of Canterbury, Private Bag 4800,Christchurch, New Zealand): The Authors have m ade a veryvaluable contribution to rational approximation of frequencyresponses. Stability of the rational approximation is animpor tant issue. Our experience has been that unstable rationalapproximation results when a higher order than necessary isused. Would the au thors comment on their method of ensuringstability by inverting the sign of the real paft of unstable poles,and the effect this has on the accuracy of the fit. How doesthis com pare to simply removing the unstable poles?Maria Sabrina Sar to (Univ. of Rome "La Sapienza", Rome,Italy). The au thors should be commended for their interestingpaper, which gives an important contribution to the long-lasting issue concerning the rational approximation oftranscendent functions. The vector fitting method proposed bythe autho rs is particularly useful not only in the modeling ofpower system transients, but in general in the tim edom ainanalysis of electromagnetic problems. It c n be alsoconsidered as a genera l approach to compute numerically theinverse Fourier transform of functions describingelectromagnetic systems in the frequency domain. In fact,these functions are often characterized by very broadfrequency spectra; they can be either smooth functions or canhave many resonance peaks, so that the calculation of theirdiscrete inverse Fourier transform can be expensive andinefficient.For these reasons, some extra-information concerning thepractical use of the vector fitting procedure and a fewtheoretical aspects is requested.A key point i n the iterative procedure seems to be the choiceof the starting poles. In particular, it is said that "thecomputed approximation may depend on the selection of thestarting poles". In other words, this means that the iterativeprocedure described in the paper can stop in local minima.Have the author investigated about this point? Have theyexperienced the efficiency of a different approach to solve t h i sproblem?An other point concerns the approximation of smoothfunctions having very bro ad frequency spectra. It is observed,as it is said in the paper, that such functions, withoutresonance peaks can be fitted rather accu rately with low orderapproximations, by using real poles preferably. However, ifthe approximation is required in a wide frequency range andthe order of the rational fitting function is increased, high-frequency complex poles appear. Comments of the authorsabove this p oint would be greatly appreciated.The last point that is addressed to the authors is thefollowing. It seems that often the o rder of the rational fittingis merely related to the desired accuracy of theapproximation: the higher is the order of the rational

approximation, the h igher is the accuracy. It would be usefulif the authors can give any guideline concerning themaximum number of poles to consider in the fitting and thebest choice of the number of the ex act function samples to usein the iterative procedure.Finally, the authors are asked to provide more specificinformation about the form ulation of the method for vectors.Manuscript received April 9, 1998.

Bjern Gustavsen and Adam Semlyen: We wish to thank Drs.Watson and Sarto for their valuable comments and usefulcontributions. We offer the following clarifications.1In Vector Fitting, unstable poles may occur incidentally during thefirst iteration(s) because the new set of poles is generally verydifferent from the previous one. The unstable poles vanish as themethod converges. Thus, if unstable poles were to be deleted in eachiteration, the order of the fitting could become lower than intended.This problem can be overcome in at least two different ways :1) In each iteration, flip unstable poles into the left half plane (usedin this paper)2) Accept unstable poles in the iterations.Then make a final iterationin which any unstable poles are deleted.The two procedures work equally well, and our implementation ofVector Fitting allows both. In the normal situation (where the finalresult does not contain unstable poles) the two approaches will arriveat practically the same result.Unstable poles in the final result may indeed occur when using avery high order fitting. Regarding the accuracy, we find there is littledifferencebetween the two approaches. We have previously used thepoledeleting technique in [ 5 ] .In reply to Dr. Sarto :Choice of starting d e sIt is correct that the computed approximation may depend on theselected startingpoles. However, this is of concern only for functionswith resonance peaks when we use a too low order. For instance, ifthere are n dominant resonance peaks in the considered frequencyinterval, then at least 2n poles should be used for the approximation.If we use a lower order, it will not be possible to fit all of the peaks.In such situationswe have found that the choiceof starting poles mayhave an influence on which of the resonance peaks will be fitted. Inpractice this will not be a problem as one will require a reasonablygood approximationand thus a sufficiently high order.The main reason why we talk about the selection of starting polesis that a good choice reduces the number of iterations that are neededfor the method to converge. At some time we tried to scan thefrequency response, followed by an assignment of starting poles toeach resonance peak. In addition, a few extra poles were distributedover the considered frequency range. This procedure gave a very fastconvergence. However, we found that by simply distributing the polesas described in this paper, we achieved an acceptable speed ofconvergence. We chose he latter method for its simplicity.Smooth functionswith broad frauencvSDectrumEven though real poles are eminently suited for the fitting of smoothfunctions, they are sometimes complemented with complex poleshaving strong attenuation. We often encounter such poles when fittingmodal propagation functions for transmission lines. The complexpoles then occur at high frequencies near the toe portion of theresponse. It appears that the complex poles are more suited than realpoles in producing the strongattenuation required at high frequencies.Auulication to vectorsThe paper shows the formulation of Vector Fitting for scalar


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Order RMSI Order RMSZ

functions. However, Vector Fitting can also be applied directly tovector functions, with the assumption that all elements in the vectorhave identical poles. The vector formulation is shown below for thecase that the vector consists of two elements. (The generalization tovectors with more elements is straightforward.)

RMSZ/RMSI

-=Using the same starting poles for both vector elements, and acommon scaling function a quation (A.l) now becomes :

where superscripts 1 and 2 for the residues refer to element 1 and 2,respectively. (The d and h terms in (A.l) have been neglected forsimplicity.)For a given frequency point Sk we get :

where

Selection of aDDroximation order and f ra ue nc v samDlesThe samples should be chosen so densely that the frequency responseis fully resolved. In addition, one should always use at least as manyfrequency samples as there are poles, in order to get anoverdetermined problem. Otherwise, spurious poles may arise whichleads to highly inaccurate behaviour betwe en frequency samples.Sometimes one may want to achieve high accuracy at certainfrequency points or frequency intervals. This can easily be achievedby weighting the rows of the least squares problem (At and bk in (1 8)and (19)). For instance, in figure 17one may consider to increase theaccuracy at low frequencies at the expense of the accuracy at highfrequencies. Figure 18shows the same result as in figure 17,when therows have been m uliplied with the inverse of the m agnitu e ofthe off-diagonal elements.

. . . .. . . . . . .1 2 3 4 5 6 7 8

010 20 40 60 80 100Horizontal distance [m]Fig. 16 500 kV DC line in parallel with tw o 300kV lines.

1oo

al.eC24 o4I

I 1 o 1o2 1o4 1osFrequency [Hz]

Fig. 17 First column of H itted using 1 0 poles. = IOOnm )

loo , 724 o4QUC.-H

ine., 1 o 1 2 1o4 1O6Frequency [Hz]Fig. 18 Effect of weighting

Dependencia de La Frecuencia de Redes Electricas

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