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ELECTROMAGNETICTRANSIENTS

6.1 IntroductionA n electrical power system is subjected to many types of disturbances, all of whichresult in a transient. Accurate dynamic simulation is, therefore, essential in powersystem design to minimize the disruption and possible damage of equipment due tothe overvoltages and overcurrents caused by the disturbance. Unlike electronic systems,the size and cost of power system components makes it unrealistic to bread-boardalternative proposals. Each proposal m ust be checked thoroughly to ensure satisfactoryoperation and optimize com ponent parameters and controller settings, before alterationsare made. This may be done at the planning stage, where many scenarios for improvingthe transient performance must be investigated, or during operation to diagnose thecause of the disturbance.The term electromagnetic transient refers to transients that involve the interactionbetween the energy stored in the m agnetic fields of the inductances and electric field ofthe capacitances in the system. It does not include the slower electromechanical tran-sient response, which involve the interaction between the mechanical energy stored inthe rotating machines and the electrical energy stored in the electrical network; theseare covered in Chapters 7 and 8.However, no component model is appropriate for all types of transient analysis andmust be tailored to the scope of the study. The main criterion for the selection ofmodel components is the time range of the study. For instance, when analysing fasttransients, such as lightning phenomena, stray capacitance and inductances must berepresented and the solution step size needs to be at least h t h of the smallest timeconstant introduced by the stray parameters. For system disturbances, such as short-circuits, the dynamics of power electronic controllers, rather than stray components,will play the major role and the solution step size can be considerably larger.Regardless of the type of system equivalent and size of the integration steps, theEMTP concept proposed by Dommel [1-4) is now universally accepted for the simula-tion of complex power systems containing non-linearities, power electronic componentsand their controllers.

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162 6 ELECTROMAGNETIC TRANSIENTS

6.2 Background and DefinitionsMost of the programs developed for power system transient simulation are based onBergerons method [ 5 ] . This method uses linear relationships (characteristics) betweencurrent and voltage, which are invariant from the point of view of an observer travellingwith the wave. However, the time intervals or discrete steps required by the digitalsolution generate truncation errors which often leads to num erical instability. The useof the trapezoidal rule for the integration of the ordinary differential equations hasimproved the situation considerably.Dom mels EM TP method combines the method of characteristics and the trapezoidalrule into a generalized algorithm, which permits the accurate simulation of transientsin networks involving distributed as well as lumped parameters.

The method, generally referred to as Numerical Integration Substitution, makes useof difference equations for the digital simulation of the power system continuousdynamic behaviour.It is often referred to by alternative names. One of them, proposed by G.T. Heydt [ 6 ] ,is the Method Of Companion Circuits, as the difference equation can be viewed as aNorton equivalent (or companion circuit) for each element in the circuit. Another is theNodal Conductance Approach (NDA), to emphasize the use of the nodal formulation,each of the network com ponents being represented by its com panion circuit.6.3 Numerical Integrator Substitution

As the name implies, Numerical Integrator Substitution involves substituting a numer-ical integration formula into the differential equation and rearranging it to the appro-priate form.Dom mels method involves substituting the trapezoidal integrator into the differen-tial equation. While other integrators could have been used, the trapezoidal solutionwas preferred due to its simplicity as well as being A-stable and reasonably accuratein most circumstances. However, since it is based on a truncated Taylor series, numer-ical oscillations can still occur under certain conditions due to the neglected terms.Substituting an integrator is equivalent to substituting the appropriate finite differ-ence approximation into the differential equations. Another important contribution ofDom mels work is the replacement of inductors and capacitors by a resistor and currentsource in parallel (the latter representing previous history terms) and their integrationinto a nodal conductance matrix and injected currents vector, respectively, to find asolution for the complete system.These two com ponents are derived using the trapezoidal rule as shown in Figure 6.1.

1-At tFigure 6.1 Trapezoidal rule

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6.3 NUMERICAL INTEGRATOR SUBSTITUTION 163

'i" "r+Figure 6.2 Resistance6.3.1 Resistance

This case, shown in Figure 6 .2 , is straightforward, i.e.vk( t ) - m ( t > =R i k S r n ( t )

6.3.2 InductanceThe differential equation for the inductor shown in Figure 6 .3 is:

( 6 . 3 )dikmV L =vk - jn =L-,dtwhich must be integrated from a known state at t - A t to the unknown one at t , i.e.

and, applying the trapezoidal rule, Equation (6.4) can be replaced by:A t2LA t A t2L 2L

i k n d t ) =ikm(r-Ar) -k -((vk - m ) ( r ) +(uk - rn)(r-Af))= km(1-A') +-(vk(f-Af) - % ( f - A f ) ) +- ( v k ( t ) - m ( r ) ) * ( 6 . 5 )

This equation can be rewritten in the following form:1

R e f fik , n ( l ) = history(r-At) +- (vkt f ) - rn([)),which is recognized as a Norton equivalent, as shown in Figure 6.4.

Figure 6.3 Inductor

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Figure 6.4 Norton equivalent of inductorThe term l/R,ff =G,ff is the instantaneous term that relates the present time voltageto a present time current contribution, and involves a pure resistance. The term

Ihistory(t-At) is the history term as this current source value is a function of quantitiesat previous time steps whereA t2L2LA t

Ihistory(f-Af) =ikm(t-Af) +-(vk(t-Af) - m(f-At))and

Reff =-.Transforming to the z-domain gives:

6.3.3 CapacitanceThe d ifferential equation for the capacitor, shown in Figure 6.5, is:

(6.10)(vk -urn>ikm=c- =cdt dtRearranging it as an integral:

and applying trapezoidal integration:

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6.3 NUMERICAL INTEGRATOR SUBSTITUTION 165

iFigure 6.5 Capacitor

Hence the current in the capacitor is given by:

(6.13)which, again, is interpreted as the Norton equivalent shown in Figure 6.6.Transforming to the z-domain gives:

(6.16)Combinations of components can be replaced by a single Norton equivalent, therebyreducing the number of nodes and, hence, the computation at each time step. Forexample, each tuned RLC branch forming part of a harmonic filter bank can be

Figure 6.6 Norton equivalent of capacitor

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7Figure 6.7 Reduction of RLC branch to a Norton equivalentrepresented by a Norton equivalent, as shown in Figure 6.7. Equally, the completefilter bank can be represented by a single Norton equ ivalent that combines the equiv-alents of the individual filter branches. The reduction process involves a series ofThevenin to Norton transformations, the internal nodes being reduced by Gaussianelimination; the result is a greatly simplified nodal admittance matrix.

6.4 Transmission Lines and CablesThe models used for overhead transmission lines and cables are identical except for thederivation of their electrical parameters (described in Chapter 2 ) . The work of Carson[7] forms the basis of overhead transmission line parameter calculations, where eithernumerical integration of Carson's integral equation, the use of Carson's series or acomplex depth approximation are used. Underground cable parameters are calculatedusing Pollaczek's equations [8].The three transmission line models commonly used in electromagnetic transientprograms are PI section, Bergeron and frequency-dependent line; the latter two beingclassed as travelling wave models.The PI section model is used for short lines, where the travel time is less than thetime step. This corresponds to approximately 15 km for a 50 ps time step. Typically

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6.4 TRANSMISSION LINES AND CABLES 167this occurs in distribution rather than transmission systems. This model is unsuit-able for longer lines as the number of nominal PI sections required for an adequaterepresentation makes the solution very inefficient.With the losses ignored, Bergerons method provides a simple and elegant travellingwave model, which is then completed by the addition of series resistance to representthe losses. In its original implementation, EMTP used the concept of natural modesto represent multi-conductor lines and the main effort went into the development offrequency-dependent modal transformations and fitting techniques to try and improveaccuracy and thus eliminate prospective instabilities. However, Gustavsen and Semlyen[9] have indicated that, although the phase domain problem is inherently stable, theassociated modal domain may be inherently unstable; hence, regardless of the fittingaccuracy, the modal line model will be unstable. This has spurned research effort intomodelling lines directly in the phase-domain despite the complication of representingcouplings between phases [lo- 151.

6.4.1 Bergeron line modelConsider a lossless distributed parameter line as depicted in Figure 6.8, with L induc-tance) and C (capacitance) per unit length. The wave propagation equations for thisline are:

(6.17a)(6.17b)

The general solutions of Equations (6.17a) and (6.17b) are:i(x, )=f l ( x - t ) +f 2 ( x +s t ) ,v(x , ) =Z f l ( x - t ) - f 2 ( x + t ) , (6.18a)(6.18b)

where f l ( x - t ) and f 2 ( x + t ) are arbitrary functions of ( x - t ) and ( x + r ) tobe determined from problem boundary and initial conditions, f ( x - t ) represents awave travelling at velocity s in a forward direction and f 2 ( x + t ) a wave travellingin a backward direction.

X = 0 x=dFigure 6.8 Propagation of a wave on a transmission line

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168 6 ELECTROMAGNETIC TRANSIENTSZ, the surge (o r characteristic) impedance and s, the propagation velocity for alossless line are given by:

1s =-m'Multiplying Equation (6.18a) by ZC and adding it to, and subtracting i t from, Equa-

(6.19a)(6.19b)

Note that v(x, )+Z c i ( x , t ) s constant when ( x - r ) is constant. This can be inter-preted by becom ing a fictitious observer travelling along the line as shown in Figure 6.8with the wave. Then (x - t ) and v ( x , )+Z c i ( x , ) will appear constant all along theline. If the travel time to get from terminal k to terminal m of a line of length d is

tion (6.18b) gives the required branch equations, i.e.v(x , )+Z c i ( x , t )=2 Z c f l ( x - t ) ,v (x , ) - Z&, r ) =- 2 Z ~ f 2 ( x +s t ) .

then the expression v(x , )+Z c i ( x , ) seen by the observer when leaving terminal k attime t - t,must be the same when the observer arrives at terminal m at time t , i.e.vk(t- t)+Z C i k m ( t - 7)=v m ( t ) +Z c ( - i m k ( t ) ) .

Rearranging the latter leads to the simple two-port equation for i m k , i.e.

where the current source from past history terms is:1ZCZ m ( t - t)=--vk(t - t)- k m ( t - t).

Similarly for the other end1

ZCi k m = -vk(t) + k ( t - t),

(6.20)

(6.21)

(6.22)where

II k ( t - t)=--vm(t - )- m k ( r - ). (6.23)ZCThe expressions ( x - t ) =constant and ( x + t ) =constant are called the character-istic equations of the differential equations.Figure 6.9 depicts the resulting two-port model. In this model, there is no directconnection between the two terminals and the conditions at one end are seen indirectly,and with time delay t (travelling time), at the other through the current sources. Thepast history terms are stored in a ring buffer and hence the maximum travelling time

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6.4 TRANSMISSION LINES AND CABLES 169

-

Figure 6.9 Equivalent two-port network for lossless linethat can be represented is the time step times the number of locations in the buffer.Since the time delay is not usually a multiple of the time step the past history termseither side of actual travelling time are extracted and interpolated to give the correcttravelling time.The distributed series resistance of the line is approximated by treating the line aslossless and adding lumped resistances at both ends. Although lumped resistances canbe inserted in many places along the line by dividing the total length into many linesections, this makes little difference and, hence, two sections are normally used, asdepicted in Figure 6.10. This lumped resistance model gives reasonable answers onlyif R/4

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170 6 ELECTROMAGNETIC TRAN SIENTS

512)

Figure 6.11 Half line section

Figure 6.12 Bergeron transmission line model(t - / 2 ) . For example, the expression for the current source at end k is:

The EM TDC line model separates the propagation into low and high frequency pathsso that the line can have a higher attenuation to higher frequencies.

6.4.2 Multi-conductor transmission linesEquations (6.17a) and (6.17b) are also valid for multi-conductor lines if the scalarvoltages and currents are replaced by vectors and using inductance and capacitancematrices. Thus, in the frequency domain these equations are:

(6.27a)

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6 .4 TRANSMISSION LINES AND CABLES 171By differentiating a second time one vector either voltage or current may be elimi-nated giving:

Traditionally the complication of having off-diagonal elements in the matrices ofEquation (6.28) and (6.29) is overcome by using natural modes. Eigenvalue analysis isapplied to produce diagonal matrices thereby transforming from coupled equations inthe phase domain to decoupled equations in the modal domain. Each equation in themodal domain is then solved as for a single phase line by using modal travelling timeand modal surge impedance. The transformation matrices between phase and modalquantities are different for voltage and current, i.e.

Substituting Equation (6.30a) in (6.28) gives:

(6.30a)(6.30b)

(6.31)Hence

[%] [TYI-'[Zbhasel[Ybhasel[TVIIVmodel = [A][vmodel. (6.32)To derive the matrix IT,] that diagonalizes [ZLhase][ bhase], its eigenvalues and

eigenvectors must be found. However, eigenvectors are not unique, as when multi-plied by a nonzero complex constant they still are valid eigenvectors; therefore, somenormalization is desirable to allow [T,] from different programs to be compared.PSCADRMTDC uses the root squaring technique developed by W edepohl for eigen-value analysis [161. To generate frequency-dependent line models the eigenvectorsmust be consistent from one frequency to the next, such that the eigenvectors forma continuous function of frequency, so that curve fitting can be applied. A NewtonRaphson algorithm has been developed for this purpose [171.On completion of the eigenvalue analysis, the following relationships are used:[Zmodel =[Tvl-'[Zphasel[Til,[yrncdel=[Til-'[Yphasel[Twl~

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172 6 ELECTROMAGNETIC TRANSIENTSAs [z;ha,.[Ybha,] is different from [YbhaSe][Zbhase], he eigenvectors are also

and,ifferent, even though their eigenvalues are iden tical. How ever, [T i] =(therefore, only one of these matrices needs to be calculated.Looking at mode i , i.e. the ith equation of (6.32), gives:

and its general solution at point x in the line is:

where

(6.33)

(6.34)

yi =6,VF =forward travelling wave,VB =backward travelling wave.

Equation (6.34) contains two arbitrary integration constants. An n-conductor linehas n natural modes and thus requires 2n arbitrary constants. This is consistent withthe existence of 2n boundary conditions, one for each end of every conductor.In matrix form, Equation (6.34) becomes:Ymde(x) =[ e - Y X ~Y:,de(k) + * Y : d e (m ) (6.35)

Reintroducing phase quan tities by using Equation (6.32) gives:~ ( x ) [ e - r x l ~ F+[erx] yB, (6.36)

is the propagation matrix, and

Similarly the solution of (6.29) for cu rrent gives:r x F - rx B~ ( x )=re- 1~ [e IL =Y ~ ( [ ~ - ~ ~ I v ~erxIvB) ,

whereIF=forward travelling wave,IB=backward travelling wave.

Thus, the voltage and current at the k end of the line are:V(k)=(v+vB, ,

F BL(k)=(LF+lB ) Yc(V -V 1

(6.37)

(6.38)(6.39)

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6.4 TRANSMISSION LINES AND CABLES 173and, similarly, at the m end of the line:

- ( m ) [ e - r r ~ ~ F[ e r X ] ~ L ( m ) =Y ~ ( [ ~ - ~ I v ~er1yB).

(6.40)(6.41)

Hence, the forward and backward travelling voltage waves at terminal k are:VF =(V ) (k )+Z c L ( k ) / 2 ,-B =(Y) (k)- Z c I ( k ) / 2 .

[ Y ~ I V ~& =2 1 ~ 2 [ e - ] ~ ~

(6.42)(6.43)

And since

the forward and backward travelling current waves at terminal k are: (6.44)(6.45)

6.4.3 Frequency-dependent m odelAs the line parameters are functions of frequency, the relevant equations are firstexpressed in the frequency domain and extensive use m ade of curve fitting to incorpo-rate the frequency-dependent parameters [18-20]. Analysis proceeds by first consid-ering a frequency-domain solution for a single conductor line of length I.

V k ( 0 ) =c ~ s h [ y ( ~ ) l ] V m ( ~ )Z ; ( W ) inh[y(w)l]i,(w), (6.46)(6.47)

whereis the p ropagation constant, and

is the characteristic impedance.Y(w) =G +w C

is the per unit length shunt admittance obtained from conductor geometry, andZ ( w ) =R +wLis the per unit length series impedance, also obtained from conductor geometry. Let

Fk(U)=vk(w) +z C ( @ ) i & ( w ) ,Bk(w)=Vk(w)- C(w)ik(w),

Fm(w) =Vm(w>+z c ( ~ ) i m ( ~ ) ~(6.48)

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17 4 6 ELECTROMAGNETIC TRANSIENTSThe functions F and B correspond to forward and backw ard travelling waves, respec-tively in the frequency domain. From Equations (6.46), (6.47) and (6.48):

Bk w ) =A I ( w > F m w ) ,Bm(w)=A I ( W ) F k ( W ) , (6.49)where A l ( w )=e-v (wy =cosh(y(w)l) - inh(y(w)l).

Equations (6.49) yield the equivalent circuit of Figure 6.13 with the source termsBk and B , related to electrical quantities at the other end of the line.To convert the frequency domain circuit of Figure 6.13 to the t ime domain, it is onlynecessary to express the source terms Bk and Bm in the time domain. This requires aconvolution integral in place of the multiplication in Equations (6.49):

B k ( t ) =L 'A( u ) F , , ( t - )duB, , ( t ) =l A ( u ) F k ( r- )du . (6.50)

The lower integral limit of t s set equal to the shortest possible !ransmission delayof the line. Evaluating the convolution integrals (Equation (6.50)) at every time stepis very slow, and a recursive method is used instead [21]. The recursive convolutionmethod represents A ( u ) in Equation (6.50) by a sum of exponentials in u , and Fk bya low order polynomial. For example, if

n

/ = Iandthen

f m ( t - ) =f ( t )+au2+Bu,B k ( t ) =l e - " i u ( f ( t ) +au2 +L?u)du (6.51 )

=C [a rB t ( t- A t ) + l f m ( t ) + I . r f rn( t - A t ) +V / f m ( t - 2At)], (6.52)1

where 01 and ,L? are obtained by equating the interpolating quadratic to f ( t ) , ( t - At ) ,f ( t - A t ) and A, p , 21 are constant coefficients obtained by integrating Equation (6.5 )

A

vk ( w)

Figure 6.13 Single conductor line equivalent

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6.4 TRANSMISSION LINES AND CABLES 175

RC networkModal Modaltransform transformc--

from t - At to t . The convolution integral is, therefore, replaced by a past-history termB k ( t- A t ) , and a linear combination of past-history terms for electrical conditions atthe other end of the line.Curve fitting for Zc and A @ ) Both Zc and A are readily defined in the frequencydomain. Representation in the time domain proceeds by first finding a rational poly-nomial in s that matches Z or A along the j w axis. In EMTDC, the fitting takes placeat 100 requency po ints evenly distributed on a log scale between specified lower andupper frequencies. The choice of lower frequency affects the shunt conductance at d.c.,giving i t a maximum value. Specifying a very low start frequency affects the accuracyand efficiency of the fit at other frequencies.Rational polynomial fitting is an area of active research, with many applications inpower system analysis. The method employed in the EMTDC transmission line andcables program directly places poles and zeros in the s plane to obtain a match withthe interpolated function. The orders of the numerator and denominator polynomialsare incremented until a good match is obtained (up to a specified limit). Finding therational polynomial directly in terms of poles and zeros,

----.

-.

(6 .53)simplifies the implementation of the function in a form suitable for simulation. A partialfraction expansion yields:

(6.54)which can be readily represented by an RC network in the case of z , or transformingto the time domain for A @ ) ,

R ( r ) =kle- +k2e-P?+. . . (6 .55)k , =0 since i n this case the denominator is of higher order than the numerator.Implementation of the transmission line model is illustrated in Figure 6.14. The trans-mission line and CABLES program calculates the RC networks, exponential sum

k3+-+...( s )=k, +- -l k2S + P l S + P 2 S + P 3

Lc

,IIIIII

Coupled multi-conductor lineDecoupled m ulti-mode line// < I I

Figure 6.14 Implementation of the frequency-dependent transmission-line model in EMTDC

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176 6 ELECTROMAGNETIC TRANSIENTSapproximation to the propagation constant, and additionally an exponential sum forevery term in the modal transform matrices. This is because the modal transformmatrices are frequency dependent but can be treated in the same way as the propagationconstant using recursive convolution. In order to obtain continuity in calculated eigen-values and eigenvectors at the 100 sample points, the solution at frequency N is usedas the starting point for a Newton Raphson iterative refinement at frequency N +1.Numerical illustration The line propagation constant, a(@) + g ( w ) , is a functionof frequency, while the propagation function (e-(cr(w )+'p(w ))')s a function of frequencyand line length ( 1 )The variation of the amplitude term e-n(w)' of a single phase line is shown inFigure 6.15(a) for a line length 1 =100 km. These results illustrate the line low passcharacteristics. As the line length is in the exponent, the attenuation of travelling wavesincreases with it.The variation with frequency of the phase angle of the propagation function, e-jp(w)',is shown in Figure 6.15(b). A negative phase represents a phase lag in the travellingwaveform and its counterpart in the time domain is a time delay. The phase angle is acontinuous function, becoming more negative with frequency but, for display purposes,it is constrained to be within the range -180" to 180". It is a difficult function to fi tand requires a high order rational function to achieve sufficient accuracy. However,multiplication by e-Jsr, where r represents the time for a wave to travel from one endof the line to the other (in this case 0.33597 ms), results in the sm ooth function shownin Figure 6.15(b). With this procedure, referred to as backwinding, the attenuation(complex function) is easily fitted with a low order rational function. To obtain thecorrect response, the model must counter the phase advance applied in the frequencydomain fitting. This is performed in the time domain implementation by incorporating

Magnitude IAttenuationl =e-cr (w) '

Phase - (Degrees) Angle (Attenuation)=e-i@(w)'0

-1 00-200

With backwinding-3OOL I1 2 3 4 5 6 7(b ) Log ( 2 4

Figure 6.15 Propagation function

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6.4 TRANSMISSION LINES AND CABLESMagnitude Propagation constant

0.4 ' I(a ) 1 1.5 2 2.5 3 3.5 4 4.5 5Phase - (Degrees)

177

1 1.5 2 2.5 3 3.5 4 4.5 5(b ) Log(2nt)

Figure 6.16 Propagation constanta time delay T. With this purpose, a buffer of past voltages and currents at each endof the line is maintained and the values delayed by T are used. Because t n general isnot an integer multiple of the time step, interpolation between the values in the bufferis required to get the correct time delay.Figure 6.16 shows the match obtained when applying least squares fitting of arational function (numerator order 2, denominator order 3). The number of poles isnormally one more than the zeros for the attenuation function magnitude which mustgo to zero as frequency approaches infinity.Although the fitting is good, close inspection shows a slight error at fundamentalfrequency. Any slight discrepancy at fundamental frequency shows up as a steady-stateerror, which is undesirable. This occurs because the least squares fitting tends to smearthe error across the frequency range. To control this problem, a weighting factor canbe applied to specified frequency ranges (such as around fundamental frequency) whenapplying the fitting procedure. When the fitting has been completed any slight errorstill remaining is removed by multiplying the rational function by a constant k to givethe correct value at low frequency. This sets the d.c. gain (i.e. its value when s is setto zero) of the fitted rational function. The value of k controls the d.c. gain of thisrational function and is calculated from the d.c. resistance and the d.c. gain of the surgeimpedance, thereby ensuring that the correct d.c. resistance is exhibited by the model.At frequencies when the value of A ( w ) drops below a specified threshold, the trav-elling waves will have been attenuated by the time they reach the end of the line;therefore the associated terms in the fitted A ( w ) are neglected. Also frequencies wellabove the Nyquist rate (e.g. 10 x Nyquist frequency for selected time-step) will notinfluence the simulation results and the corresponding terms are removed.

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650

500

Som e fitting techniques force the poles and zeros to be real and stable (in the left handhalf of the s-plane) wh ile others allow complex poles and use other m ethods to ensurestable fits (such as vector-fitting). A comm on approach is to assume a minimum-phasefunction and use real half-plane poles. Fitting can be performed either in s-domain orz-domain, each alternative having advantages and disadvantages.The same fitting algorithm can be used for fitting the characteristic impedance (oradmittance if using the Norton form). The number of poles and zeros is the same inboth cases. Hence the partial expansion of the fitted rational function is:

.

t-- tart frequency

This can be implemented by using a series of RC parallel blocks (the Foster Irealization), which gives Ro =k o , R; =k i / p i . and C; = I / k ; . Either the trapezoidalrule can be applied to the RC network or better still recursive convolution.The shunt conductance G ( w ) is not normally calculated. At low frequencies thesurge impedance becomes larger as the frequency approaches zero, i.e.

7 7 ;-- tart frequencyThis trend can be seen in Figure 6.17 which shows the characteristic (o r surge)impedance calculated by the transmission line parameter program down to 5 Hz.

1

Magnitude IZsurgel650

600

550500

Phase - (Degrees) Angle (Zsurg e)

-151 2 3 4 5 6 7(b) Log(2 n f )

Figure 6.17 Characteristic impedance

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6.5 FORMULATION AND SOLUTION OF THE SYSTEM NODAL EQUATIONS 179In practice, the characteristic impedance does not tend to infinity as the frequencygoes to zero; instead r ( w )+ wL ( w )

G ( w )+wC ( w )This shortcoming is mitigated by entering a starting frequency, which flattens theimpedance curve at low frequencies and thus makes i t mor e realistic. Entering a startingfrequency is equivalent to introducing a shu nt conductance G. The higher the startingfrequency the greater the shunt conductance and, hence, the shunt loss. On the otherhand, choosing a very low start frequency will result in poles and zeros at these lowfrequencies and the associated large time constants cause long settling times to reach

the steady state.

6.5 Form ulation and Solution of the System NodalEquationsOnce all the network components are represented by an equivalent current source anda resistance in parallel, a nodal formulation is used to solve for the complete system.The nodal equation is:

IGlv( t ) =i ( t ) -t Hi s t o ry , (6.56)where[GI =conductance matrix,v(r) =vector of nodal voltages.i ( t ) =vector of external current sources,

I H , ~ ~ ~ , ~ ~vector current sources representing past history terms.Note that [G ] is real and sy mm etric when incorporating network c om pone nts. If controlequations are incorporated into the same [GI matrix, the symmetry is lost but theseare normally solved separately. As the elements of [GI are dependent on the timestep, by keeping the time step constant [GI is constant and triangular factorization canbe performed before entering the time step loop. Moreover, each node of the powersystem is connected to only a few other nodes an d, therefore, the conductance matricesare sparse. This can be exploited by only storing non-zero elemen ts and using optimalordering elimination schemes.Some of the node voltages will be known due to the presence of voltage sources inthe system but the majority are unknown. When there is an impedance in series withthe voltage source, the combination can be converted to a Norton equivalent and thealgorithm remains unchanged. However, a more general approach is to partition thenodal equation as follows:

where the subscripts K and U indicate connections to nodes with known and unknownvoltages, respectively.

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180 6 ELECTROMAGNETIC TRANSIENTSUsing triangular factorization (arrow l), Equation (6.57) reduces to:(1)I

(6.58)Forward reduction (arrow 2), followed by back substitution (arrows 3) is then usedto get V , ( t ) , i.e.(3) (2)

(6.59)where

Once V u ( t )has been found, the history terms for the next time step; are calculated.6.5.1 Modification for switching and varying param eters

To represent switching operations, or time varying parameters, matrices [ G u u ] and[ G u K ]need to be altered and retriangularized. The solution is simplified by placingthe nodes with switches last, such that the initial triangular factorization is reduced tothe nodes without switches, i.e.

Switching components(6.60)

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6.5 FORMULATION AND SOLUTION O F THE SYSTEM NODAL EQUATIONS 181followed by complete triangulation (arrow 2)

(6.61)and then forward reduction (arrow 3) and back substitution (arrow 4). i.e.

(4 ) (3)T - - - - - I (6 .62)Transmission lines do not introduce off-diagonal elements. This produces a blockdiagonal structure for [ G u u ] . Each block represents a subsystem that can be solvedfor independently, as any influence from the rest of the system is implemented in theform of history terms. This allows parallel computation and is used extensively inreal time digital simulators. PSCADEMTDC performs the triangular factorization ona subsystem basis rather than on the entire matrix.

6.5.2 Non-linear or time varying parametersTypical non-linearities requiring representation are the saturated inductances oftransformers and reactors and the resistances of surge arresters. Non-linear effects

in synchronous machines are incorporated directly in the machine equations. As thenumber of non-linear elements are limited, i t is more efficient to modify the linearsolution than using a non-linear solution method for the entire network. The threealternative approaches [22] used for this purpose are:00 Compensation method.0 Piecewise linear representation.Current-source representation This method uses a current source to model thecurrent drawn by the non-linear component, its value calculated from informationat previous time-steps. Therefore it lacks an instantaneous term and appears as anopen circuir to the voltages at the present time-step. Thus, to avoid possible relatedinstabilities, a correction source is used in the form of a large fictitious Nortonresistance. However there is a one time-step delay in the correction source.Compensation method If there is only one non-linear branch a compensationmethod can be applied, whereby the non-linear branch is excluded from the networkand replaced by a current source. The total network solution v(t) is then equal to the

Current-source representation (w ith one time step delay).

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182 6 ELECTROMAGNETIC TRANSIENTSvalue vo(t) found with the non-linear branch omitted, plus the contribution producedby the non-linear branch, i.e.

v(l) =vO(t) - Theveninikm(t)t (6.63)whereRmevenin=Thevenin resistance of the network without non-linear branch

vg(t) =open circuit voltage of network, i.e. voltage between nodes k and mThe T hevenin resistance of the linear network is calculated by taking the differencebetween the rn th and k th columns of matrix [Guul- ' . This is achieved by solving

[ G u ~ ] v ( t )=1 with 1 set to zero except for the rn and k elements, which are -1and 1, respectively. It is equivalent to finding the terminal voltage when connecting acurrent source (of magnitude 1) between nodes k and m. he Thevenin resistance is pre-computed only once, before entering the time step loop, and only needs re-computingwhenever switches open or close.Two scalar equations are then solved simultaneously, as shown in the diagram ofFigure 6.18, i.e.Vkrn(t) =VkmO(t) - RTheveninikmT (6.64)Vkrn(l) = ( i k m r dikmldt, t , * * *), (6 .65)

If Equation (6.65) is given as an analytic expression, a Newton Raphson solutionis used. When E quation (6.65) is defined point-by-point as a piecewise linear curve, asearch procedure is used instead.Even though there may be more that one non-linear branch, using the subsystemconcept described in Section 6.6, the compensation approach can still be used, as longas there is only one non-linear branch per subsystem.If the non-linear branch is defined by Vkm = ( i k m ) or Vkm =R(t)iknl the solution isstraightforward. For a non-linear inductor the flux (A = ( i k m ) ) is the integral of thevoltage with time, i.e.

connected between nodes k and m,without non-linear branch connected.

A ( t ) =A ( l - A t ) + (6.66)

Figure 6.18 Pictorial view of simultaneous solution of two equations

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6.6 USE OF SUBSYSTEMS 183

Figure 6.19 Artificial negative dampingEquation (6 .66) is solved using the trapezoidal rule, giving:

Af2( f > = -v( f> AHistory(f -

where(6.67)

However, numerical problems can occur with non-linear elements when Af is toolarge because the regions of the non-linear characteristic between samples will bemissed. This, as shown in Figure 6.19, may produce fictitious negative damping orhysteresis.Piecewise linear representation The piecewise linear inductor characteristic, asdepicted in Figure 6.20, can be represented as a linear inductor in parallel with a currentsource representing saturation. This model is very accurate when the compensation issmall.

A t2h ~ i ~ t ~ ~ ~A(f - A l ) +-v(f - A f) .

6.6 Use of SubsystemsThe presence of transmission lines and cables in the system being sim ulated introducesdecoupling into the conductance matrix. This is because the transmission line modelinjects current at one terminal as a function of the voltage at the other at previous timesteps. In the present time step, there is no dependence on electrical conditions at distantterminals of the line, frequently leading to a block diagonal conductance matrix, i.e.

Y = (6.68)

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A

'Compensation

i' - j icompensation 6Linear inductorFigure 6.20 Piecewise linear inductor

(b )Figure 6.21 Separation of two coupled subsystems by means of linearized equivalent sources

Each decoupled block in this matrix is a subsystem, and can be solved at each timestep independently of all other subsystems.Figure 6.21(a) illustrates coupled systems that are to be separated into subsystems.Each subsystem in Figure 6.21(b) is represented in the other by a linear equivalent.The Norton equivalent is constructed using information from the previous time step,looking into subsystem (2) from bus ( A ) . The shunt connected at ( A ) is considered

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6.6 USE OF SUBSYSTEMS 185

part of (1). The Norton admittance is

The Norton current is

The Thevenin impedance is

and the voltage source is

If Y A is a capacitor bank, Z is a series inductor, and Y B s small, thenY N % Y A and Y N= B A ( t - A?) (the inductor current),

VTH =v A ( t - A t) (the capacitor voltage).TH% 2 andWhen simulating HV DC systems, it can frequently be arranged that the subsystemscontaining each end of the link are small, so that only a sm all conductance matrix needbe refactored after every switching. Even if the link is not terminated at transmissionlines or cables, a subsystem boundary can still be created by introducing a one-timestep delay at the commutating bus.A d.c. link subdivided into subsystems is illustrated in Figure 6.22.The method of interfacing subsystems by controlled sources is also used to interfacesubsystems with component models solved by another algorithm, e.g. componentsusing Numerical Integration Substitution on a state variable formulation. Synchronousmachine and SVC models are framed in state variables in PSCA DE M TD C and appear

IIIIIIIIIIIIIII

y

Figure 6.22 A typical subsystem equivalent for a d.c. link

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186 6 ELECTROMAGNETIC TRANSIENTSto their parent subsystems as controlled sources. As for interfacing subsystems, bestresults are obtained if the voltage and current at the point of connection are stabilized,and if each componendmodel is represented in the other as a linearized equivalentaround the solution at the previous time step. In the case of synchronous machines, asuitable linearizing equivalent is the sub transient reactance, which should be connectedin shunt with the machine current injection.

6.7 Switching DiscontinuitiesThe basic EMTP-type algorithm requires modification in order to m odel accurately andefficiently the switching actions associated with HVDC, thyristors, FACTS devices,or any other piecewise linear circuit. The simplest approach is to simulate normallyuntil a switching is detected and then update the system topology andor conductancematrix.A switch can be represented, as shown in F igure 6.23, by either an ordoff resistanceor an off resistance only. The former method cannot represent an ideal switch, since Ronmust be large enough not to de-condition the system conductance matrix. In practice,this represents well real switching devices, which are themselves not ideal. The secondmethod avoids the use of small resistances, but requires a more severe change insystem topology since there is one fewer node in the conduction state than in theoff-state.For either representation, the system conductance matrix must be reformed andfactorized after each change in conduction state. This considerably increases the compu-tational requirements of the simulation in p roportion to the number of switching actions(recall that the efficiency of the EMTP technique lies in the fact that the conduc-tance matrix is held constant to avoid refactorization). Nevertheless, for HVDC andmost FACTS applications, the switching rate is only several kHz, so that the overallsimulation is still fast.The efficiency and elegance of the EMTP method relies on the step length beingconstant. This, however, causes firing errors when modelling switching elements asthe switching instants will not normally coincide with the time steps, as shown in

I IFigure 6.23 Switch representation

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6.7 SWITCHING DISCONTINUITIES 187

Figure 6.24 Apparent reverse current in a diode due to placement of simulation time stepsFigure 6.24 for the case of a diode-ending conduction. At (A) the diode should beswitched off, but the negative current is only detected at (B) and the conductancematrix is reformed at (C).An effective solution of this problem is to apply a linear interpolation at (B) tofind all the nodal voltages at a point very close to (A). An exact solution for the zerocrossing is a non-linear problem; however, given the close proximity of points t andt +A t, a linear interpolation between them introduces no significant error [22,23].In the above example, the diode model must include logic to detect the switchingevent and then estimate the instant t +T I of its occurrence between t and t +A t . Fora d iode, a linear interpolation on the forward current yields:

A t i f ( t )if(?) - f ( f +A t )r = (6.73)

The nodal voltages at t +T are then approximately:t~ ( t A t) 2 ~ ( t ) - [ [ v ( t +A t ) - ~ ( t ) ] ,At (6.74)

and similarly for branch currents in the system. If simulation proceeds normally fromt +T with the new conductance matrix, subsequent solution points will be shifted by t.For convenience, an additional interpolation can be made after the first time stepwith the new conductance matrix to bring the simulation back onto the original timesequence, yielding the sequence of steps illustrated in Figure 6.25. Another option is

JI >New conductance matrixFigure 6.25 A double interpolation scheme to find the switching instant and re-synchroniset ime steps

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L9UaC- ,

to accept unequally spaced points in the solution (which complicates Fourier anal-ysis of the resulting waveforms), or to fit a cubic spline to the solution and re-indexto any desired time base. In the PSCADEMTDC package, the interpolation schemeof Figure 6.25 is used but with two additional interpolations introduced to eliminatevoltage and current chatter, to be discussed in the next section.

T-2Af T-At T T+At T+2Af \

6.7.1 Voltage and current chatter due to discontinuitiesThe trapezoidal integration has the effect of increasing the apparent impedance ofinductors at high frequencies approaching the Nyquist rate. A corollary is that largevoltages will be developed across inductors if high frequency currents are injected intothem. This effect is manifested most notably with respect to switching actions or pointdiscontinuities, which necessarily contain high frequency components [25-271.Figure 6.26 shows an inductor in series with a diode ending conduction. Assumingthat the inductor current falls steadily to zero at T, the voltage will be constant at V L .At T he diode switches off and the inductor voltage will fall instantaneously to zero,since dil/dr =0. The inductor voltage is shown in Figure 6.27(a).

Pf

T-2Af T-Af T T+Af T+2A tFigure 6.26 ~ - circuit M ..ich will display voltage chatter after the diode switches off

-"1 I

Figure 6.27 (a) Correct solution for voltage across the inductor. (b) Chatter voltage at thediode anode

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6.7 SWITCHING DISCONTINUITIES 189The trapezoidal integration equation for the inductor is:

2LAtV L ( t )=- ( ( i ~ ( t ) - L(r- A t ) )- V L ( t- Ar).

From ( t +A t ) onwardsi L ( t ) = L ( t - A t ) =0,

so V L ( t )=- VL ( t - A t ) .

(6.75)

(6.76)The inductor voltage consequently oscillates between ~ V Lnstead of falling to zero

(Figure 6.27(b)). Note, however, that the average inductor voltage is correct at zero,and that the chatter does not grow larger.PSCADEMTDC uses a double half-time step interpolation method to remove chatterwhich relies only on trapezoidal integration [28]. With reference to Figure 6.28, thefollowing steps are involved:trapezoidal integration,interpolation back to discontinuity, between (1) and (2),trapezoidal integration with new conductance m atrix,interpolation between (3) and (4 ) of half time step to removechatter,trapezoidal integration to get past t +A t ,interpolation between (6) and (5 ) to re-synchronise time steps,normal integration resumes.

Figure 6.28 Chatter removal

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190 6 ELECTROMAGNETIC TRANSIENTSThe chatter problem discussed above is associated with the numerical error of thetrapezoidal (or any other integration) rule, i.e. it is inherent in the numerical integratorsubstitution method. With the use of root matching techniques instead of numericalintegrator substitution [29], a chatter removal p rocess is not required. Root matching isalways num erically stable and more efficient numerically than trapezoidal integration.Root matching is only formulated for the branches containing at least two or moreelements (i.e. RL, RC.. RLC, LC.. ) and these can be intermixed in the same solutionwith branches solved with other integration techniques.

6.8 Root-matching TechniqueSince digital simulation requires the transformation from the s- to the z - plane, thepoles and zeros of the continuous process must be correctly represented in the digitalsolution. However, when using the num erical integrator substitution method to derive adifference equation, the poles and zeros are not inspected. They will, therefore, matchpoorly those of the continuous system that is being simulated.Ensuring the correct match for the poles and zeros is the basic purpose of the root-matching technique [29 ,30] .The difference equations generated by this m ethod involveexponential functions, as the transform equation 2- =e-SAs used rather than someapproximation to it. This concept is explained next.

6.8.1 Exponential form of difference equationThe application of Dommels method to a series RL branch produces the followingdifference equation for the branch:

AtR At(6.77)

Careful inspection of this equation shows that the first term is a first order approxi-mation of e- where x =A t R / L . The second term is a first order approximation of( 1 - -X)/2.This suggests that using the exponential expressions in the d ifference equation willeliminate the truncation error and give accurate and stable simulations regardless ofthe time step.Equation (6.77) is thus expressed as:

(6.78)Although the exponential form can be deduced from the difference equation devel-oped by numerical integrator substitution, such an approach is unsuitable for mosttransfer functions or electrical circuits, due to the difficulty in identifying the formof the exponential that has been truncated. The root-matching technique provides arigorous method.

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6.8 ROOT-MATCHING TECH NIQUE 1916.8.2 Root-matching implementation

The steps followed in the application of the root-matching technique are:1.2.

3.4.5 .

6.

Determine the transfer function in the s-plane, H s) and the position of its polesand zeros.Write the transfer function H ( s ) in the z-plane using the mapping z =eSA, husensuring the poles and zeros are in the correct place. Also, add a constant to allowadjustment of the final value.Use the final value theorem to compute final value of H ( s ) for a unit step input.Determine the final value of H ( z ) for unit step input and adjust the constant to bethe correct value.Add additional zeros depending on the assumed input variation between solutionpoints.Write the resulting z-domain equation in the form of a difference equation.

Until recently, i t has not been appreciated that the exponential form of the differ-ence equation can be applied to the main electrical components as well as to controlequations, in time domain simulation. Both can be formed into Norton equivalents,entered in the conductance matrix and solved simultaneously with no time step delayin the implementation.Structurally, the new algorithm is the same as Dommels, the only difference beingin the formula used for the derivation of the conductance and past history terms.Moreover, although root-matching can also be applied to single L or C elements, thereis no need for that, as in such cases the response is no longer of an exponential form.Hence, Dommels algorithm is still used for converting individual L and C elementsto a Norton equivalent. This allows difference equations, hence Norton equivalents,based on root-matching methods to be easily used in ex isting electromagnetic transientprograms, yet giving unparalleled improvement in stability and accuracy, particularlyfor large time steps.In the new algorithm, Z H ~ ~ ~ ~ ~ncludes the history terms of both Domm els and root-matching method. Similarly, the conductance matrix, which contains the conductanceterms of the Norton equivalents, includes some terms from Dommels technique andothers of the exponential form developed from the roo t-matching technique.The main characteristics of the exponential form that permit an efficient implemen-tation are:0 The exponential term is calculated and stored prior to entering the time step loop.0 During the time step loop, only two multiplications and one addition are required

to calculate the Z H ~ ~ ~ ~erm.Only the previous time step current is required, while Dommels method requiresboth current and voltage at the previous time step.0

6.8.3 Numerical illustrationThe effectiveness of the root-matching technique can be illustrated by the step responseto switching a series RL branch. Figure 6.29 shows the current response derived from

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192 6 ELECTROMAGNETIC TRANSIENTSExponential orm - - Dommels method -- Theoretical Curve

100.0

80.0

60.0hv)Q-w

40.0

20.0

0.0

50

0.0008 0.0011 0.0014 0.0017 0.002Time (seconds)

Figure 6.29 Step response of switching test system for A t =T- xponential orm -.---Dommels method- - - - .Theoretical curve

Figure 6.30 Step response of switching test system for A t =5 rDom mels method, the exponential method and continuous analysis for a case where thetime step A t is equal to the time constant of the circuit r . For this time step, Dom melsmethod shows no numerical oscillations, although it introduces considerable error.However, when the time step is increased to 5r , Figure 6.30 shows that Dommelsmethod experiences truncation errors, whereas the exponential form gives the correctanswer. A further increase of the time step, i.e. A = lo r , shown in Figure 6.31, resultsin much greater numerical oscillation for Domm els method, while the exponential formcontinues to provide the exact answer.

6.9 a.cJd.c. ConvertersPSCADEMTDC provides as a single component a six-pulse valve group, shown inFigure 6.32(a) with associated PLU firing control and sequencing drop and parallel

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6.9 A.C./D.C. CONVERTERS

abC - w -

f7

193

- xponential fo rm ........ ' Dommel's method- - - - - Theoretical curve

Figure 6.31 Step response of switching test system fo r Ar =10sI/

III

III Ro n

, 7, I

I

,, I,,\\\\\

Figure 6.32 (a ) PSCAD six pulse group model (b) Thyristor and snubber equivalent circuitmodelsnubber, as shown in Figure 6.32(b). The combination of on resistance and forwardvoltage drop can be viewed as a two-piece linear approximation to the conductioncharacteristic. The interpolated switching scheme, described in Figure 6.28, is used foreach valve.The factorization scheme used in EMTDC is optimized for the type of conduc-tance matrix found in power systems, and for the presence of frequently switchedelements. The block diagonal structure of the conductance matrix caused by travellingwave transmission line and cable models is exploited by processing each associ-ated subsystem separately and sequentially. Within each subsystem, nodes to whichfrequently switched elements are attached are ordered last, so that the matrix refactor-ization after switching need only proceed from the switched node to the end. Breakersand faults are not ordered last, however, since they switch only once or twice in thecourse of a simulation. This means that the matrix refactorization time is affectedmainly by the total number of switched elements in a subsystem, and not by the totalsize of the subsystem. Only non-zero elements in each subsystem are processed by

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194 6 ELECTROMAGNETIC TRANSIENTSemploying sparse matrix indexing methods. A further speed improvement, and reduc-tion in algorithm complexity, is to store the conductance m atrix for each subsystem infull form, including the zero elements. This avoids the need for indirect indexing ofconductance matrix elements by pointers.Although the user has the option of building up a valve group from individualthyristor components, the complete valve group model includes sequencing and firingcontrol logic.The firing controller implemented is of the phase-vector type, shown in Figure 6.33,which em ploys the trigonometric identities to operate on an error signal following thephase of the positive sequence component of the commutating voltage. The outputof the PLO is a ramp, phase shifted to account for the transformer phase shift. Afiring occurs for valve 1 when the ramp intersects the instantaneous value of the alphaorder from the link controller. Ramps for the other five valves are obtained by addingincrements of 60" to the valve 1 ramp. This process is illustrated in Figure 6.34.As for the six-pulse valve group, where the user has the option of constructing it fromdiscrete component models, HVDC-link controls can be modelled by synthesis from

--cVA

VB--.-)vc-

Figure 6.33 Phase-vector type phase-locked oscillator

Interpolated firing Interpolated firingof valve 1 of valve 2

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