Small signal frequency domain model of an HVDC converter

C.M.Osauskas, D.J.Hume and A.R.Wood

Abstract: A small-signal analytic frequency domain model of a 6-pulse HVDC converter is presented. The model consists of a set of explicit algebraic equations which relate the transfer of distortion from AC voltage, DC current and firing angle modulation, to AC current and DC voltage. The equations represent the linearisation of the transfers around a base operating point, and are derived from a piecewise linear description of the AC current and DC voltage waveforms. The model provides an understanding of the transfer of distortion by the converter and is in excellent agreement with time domain simulations.

1 Introduction

The interaction of an HVDC converter with other power system components is of a complex nature, and has gained significant attention in the literature. Accurate modelling of these interactions requires a suitable converter model. The use of analytic small-signal converter models has provided useful insight and understanding into HVDC system behaviour [14].

The development of analytic small-signal converter models has generally been undertaken using a frequency domain transfer function based approach [5-71. Although the converter is a nonlinear time-variant device, it has the characteristics of an approximately linear modulator when represented in the frequency domain. Model accuracy is dependent on the approximations and simplifications made in their derivation. This relates, in particular, to the dynam- ics associated with the commutation period and switching instant variation, which have been identified as being of importance [7, 81 and present a significant modelling chal- lenge.

The analysis presented is based on a piecewise-linear state-variable representation of the converter, and is an extension of the previously used transfer function based analysis methods. A state-variable representation allows all dynamics associated with the operation of the converter to be described mathematically, the resulting model being exact in the small-signal sense. The modelling techniques developed have allowed for a structured analysis of the converter, and should improve the understanding of power electronic device modelling in general.

2 Formulation of the analysis

The converter circuit under consideration is a star-star connected 6-pulse Graetz bridge, where the AC terminal

0 IEE, 2001 ZEE Proceedings online no. 20010533 DOL lO.l049/ip-gtd:20010533 Paper first received 14th September 2000 and in revised form 17th May 2001 The authors are with the Department of Electrical and Electronic Engineering, University of Canterbury, Private Bag 4800, Christchiirch, New Zealand

voltage and commutation reactance are referred to the secondary side of the converter transformer, as shown in Fig. 1. It is assumed that the commutation impedance is purely inductive and balanced, the thyristor switches are ideal, and there are no parasitic components such as stray capacitance.

Fig. 1 Gruetz hridge converter

The selection of the converter input variables as the AC voltage, DC current and firing angle, allows the operation of the device to be described by eqn. 1, a form first proposed by Larson [9]: The quantities prefixed by A repre- sent the deviation of the converter waveforms from their base or nominal state. AC side variables are represented in terms of positive and negative sequence components, the zero sequence being omitted as it does not interact with the converter. In Sections 3 and 4 linear analytic expressions for the change in the outputs resulting from a change in the inputs are derived. The converter model can be linearly coupled to other components, such as ACiDC systems and controllers, for a full HVDC system analysis.

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2. I Specification of the input variables The input variables are defined as the combination of a base coniponent and a distortion component:

21, = V I $ + AV,

i d c = I d , + a t d c

a = a0 + Aa

(2)

( 3 )

(4) The base components are a positive sequence fundamental frequency voltage vlv,, and constant values of DC current 4,L and firing angle q, These components define the base operating point around which the transfer of distortion by the converter is linearised. The fundamental frequency volt- age is described as

~121, = VI sin(w0t + cp - $) ( 5 ) where q is the voltage phase angle, and I/J is 0, 120,240" for the three phases a, b, c. An ideal phase locked oscillator (PLO) firing-angle control system is assumed, which pro- vides thyristor firing ramp references locked to the base AC voltage. The difference between a practical and the ideal PLO system can later be incorporated using additional transfers.

The distortion components are described as sinusoidal with arbitrary magnitude, frequency and phase as follows

(6) AV+ = V, sin(nw0t - s$ + 6,) Aid, = Ih sin(kw0t + 6k) Acu = ~ \ r k sin(kw0t + 6 k )

(7)

( 8 ) where s indicates the sequence, + I for positive sequence, and -1 for negative sequence. AC and DC side distortions are assigned a harmonic multiple n and k respectively, and are, in general, noninteger numbers.

2.2 Piecewise linear analysis of the converter During each conduction period, the voltage and current waveforms are described as the combination of the steady- state and transient responses of a linear circuit [lo]. The steady-state response is at the same frequency as the applied distortion, and is the component of the waveform which would be established when the transient response has decayed to zero. Owing to the continually changing state of the converter this is never the case, and the responses are referred to as the partial steady state (PSS) and the partial transient (PT) [ 1 I].

The PSS response is determined by considering the circuit in the frequency domain, while the PT response is associated with the natural response and is used to satisfy the initial conditions at the switching instants. In the case of a commutation period circuit, the inductor current in one of the commutating phases is an independent state- variable. As it is assumed that there is no resistance in the commutation circuit, the associated PT response is a constant inductor current. The direct conduction circuit has no independent state variables and, consequently, no PT response.

The relationship between an input variable U and an output variable Y can be written as

y = Y P V ) Q P ( U ) (9) P

where 5) is the waveform associated with Y during the pth conduction period, and I/JP is a sampling function which has value 1 during the pth conduction period and is zero else- where. Differentiating eqn. 9 with respect to U allows the

574

transfer of distortion from U to Y to be described as

The terms on the right-hand side are referred to as the first and second distortion transfer mechanisms. The first distor- tion transfer mechanism is associated with the change of ql during the unmodified conduction periods, while the sec- ond is associated with the unchanged ql during the modifi- cation of the conduction periods.

An examination of the converter waveforms indicates that, in the case of the transfers to AC current, the change in the current-time area associated with the second distor- tion transfer mechanism is a nonlinear function of the applied distortion magnitude. This causes the second dis- tortion transfer mechanism to be zero, and, consequently, only the current-time areas associated with the first distor- tion transfer mechanism need to be considered. This is in contrast with the transfers to DC voltage, where the volt- age-time areas associated with both the first and second distortion transfer mechanisms are linear and therefore nonzero [8].

3 Transfers to AC current

3. I Sampled partial steady state response (SPSS) The analysis first determines the PSS responses, and then applies sampling functions to obtain the change in the AC waveform. The partial steady states are sampled during the unmodified conduction periods, and there are no PSS responses associated with firing angle distortion.

Recognising patterns in the form of the partial steady states of the commutation currents associated with AC voltage distortion, allows a general expression to be writ- ten. The PSS response when the phase is commutating on is given by

fiv, S7T AI&TS = -- cos(,n,wot - s$ - - + 6,) (11)

21aX 6 where X is the reactance in a phase. The change in the AC waveform in the W phase during the commutation on peri- ods of the q~ phase is obtained by sampling the partial steady state using Y,,:I(-cIL. The sampling function Wva'-oL is shown in Fig. 2 in relation to the base AC waveform. It has value 1 during the base commutation on periods, and value 0 elsewhere.

"0 '"0

--' -- In

Fig. 2 a Basc AC curlent waveform h Sainpliiig function V,,P-c'l

Don& from AC voltnge to AC curretii

The Fourier series of the sampling function Yc-f'c is

i ~ 7 - a ' = Lo + __ 4 sin ( Y$c) na 7r

7 T m

x cos mwot-nm (F+g+ao-cp+$)) 7T i I E E Pr.oc.-Cener. Ticinmi. Uivtrib., Vol. 148, No. 6, Noveriiher. 2001

where m = 2, 4, 6 .... The sampled PSS in the I) phase during its commutation off period is the negative of the SPSS in the phase which is commutating on. This observa- tion allows the SPSS waveform to be found by evaluating eqn. 13, which is a linear function of the AC voltage distortion:

AI;;spss = AI$;pssQ;-"'

- A I ~ ~ + 2 ~ / 3 ) , P S S ~ ; L ~ 1 ~ ~ / 3 , (13)

The PSS responses of the AC current associated with DC current distortion are half the value of the DC current dis- tortion during the commutation periods, and the value of the DC current distortion during the direct conduction periods, in either the positive or negative direction. The sampling fimction Y$+lC, which when multiplied by the DC current distortion results in the correct change in the AC current wavefoim, is shown in Fig. 3h.

The Fourier series of Y$+lC can be obtained by splitting the sampling function into two components, as shown in Fig. 3c. The first component, drawn with a solid line, assumes the commutation period is of zero length, while the second component, drawn with a dashed line, modifies the first to take account of the coinmutation period. The Fourier series of the sampling function Yv;ic-(lc is

( i+no-p+$) ) m

4 - sin ( y) sin ( y ) mT

+ m

7r nz (9 + - + w - 2

x sin mwot - ( where m = 1, 5, 7 .... Multiplying the DC current distortion eqn. 7 by the sampling functions gives the SPSS waveform

(15) Q&-rLC A 1 $ ~ L ~ ~ L ~ s = .$

which is a linear function of the DC current distortion.

3.2 Partial transient response (PT) The partial transients are such that the zero current initial conditions at the thyristor firing instants are satisfied. The PSS of the ith thyristor commutation on current associated with AC voltage distortion, referenced in the positive

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direction of the ith thyristor, is

where i = 1, ..., 6. The commutation current partial tran- sient response is the negative value of the PSS at the thyris- tor firing instant. Substitution of the firing instants of the ith thyristor motl into eqn. 16 allows the value of the partial transients to be written as

where woti = (i - 1)d3 + d6 + cr, The value of the partial transient response associated

with DC current distortion is the negative of the PSS response at the thyristor firing instants:

q.

The transfer from firing angle to AC current distortion dif- fers from the previous two cases as the thyristor firing instants deviate from cr,. Fig. 4u shows the change in the commutation current waveform when the thyristor firing instant is advanced by A@. The base current waveform is drawn as a dashed line, while the changed waveform is drawn as a solid line.

The PSS responses of both waveforms during the com- mutation period are the same, and can be obtained by sub- stituting the base AC voltage vlly parameters into eqn. 11. Examination of the lighter and darker shaded areas indi- cates that only the lighter shaded area is a linear function of the magnitude of the firing angle distortion. The differ- ence between the waveforms is modelled by the partial transient response shown in Fig. 4b, as described by Pers- son [2].

U0

- < A0 i- +A&+ . . . . . ,

I 1 L b d . .

+ 110 Fig. 4 ( I Base (daqhed) and distorted (solid) waveforms h Partial transient waveform

~ u t i ~ ~ e ~ ~ ~ t n ~ i r j t i g - ~ i i i g l ~ to AC wrreiii

Two small-signal approximations are required to relate the magnitude of the partial transient to the firing-angle distortion. The thyristor firing instants occur when the instantaneous value of the thyristor PLO ramp reference and firing-angle control signal become equal. These instants are approximated by the value of the firing-angle control signal at the unmodified firing instant q. The second approximation relates to the commutation current, which is represented by a tangent to the commutation current at q during the modification of the commutation period. The sides of a triangle are formed by the modification of the fir- ing instant, commutation current tangent and the value of the partial transient, which allows the value of the partial transients to be evaluated as

SI5

Table 1: Applied distortion and commutation current partial transient modulation signals

Transfer Applied distortion Partial transient magnitude

The values of the PT responses associated with the distor- tion inputs are summarised in Table 1, where the q t i used previously has been replaced with q t . In each case, the value of the partial transient is given by the value of a continuous sinusoidal signal, of the general form bcos(kqt + a), at the unmodified thyristor firing instants *ti. The partial transients can be viewed as being modulated by the applied distortion. The relationship between the distortion signal and modulation signal is linear, the modulation signal being a linear scaled, frequency shfted, and phase shifted version of the original distortion signal.

Fig. 5 shows a partial transient modulation signal and the sampled PT waveform associated with a phase current. The sampled PT waveform is modelled using four pulse amplitude-modulated waveforms having appropriate signs and time shifts.

"P

Fig .5 N Base AC current b Partial transient modulation signal and sampled PT waveform

l'run@rx to AC current

4 Transfers to DC voltage

4.7 Sampled partial steady-state response (SPSS) The commutation current partial transient is a constant and does not induce voltage in the inductive commutation impedance. As a result, the voltage-time area associated with the first distortion transfer mechanism of eqn. 10 only has a PSS component.

The sampling function Y7F-'/', which when multiplied by the AC voltage distortion results in the correct change in the DC voltage, is the same as eqn. 14 used for the analysis of the transfer from DC current to AC current. The contri- bution of the sampled partial steady-state to the change in the DC voltage is

AV::SPSS = ! € J Y - ~ " A V , (20) i

The DC voltage PSS responses associated with a DC current distortion is given by

(21) - -kXIncos(kwot + b h )

-2kX1, cos(kwot + &) (22)

3 2

during a commutation period and

during a direct conduction period. The DC voltage

516

sampled PSS can be written as

where YdcJc is the sampling function shown in Fig. 6, and has the Fourier series:

7T x cos mwot - m ( + a0 - ( 9)) (24)

Fig.6 n Base AC current waveform b Sampling function Yk dc

Transjievfi.om DC current to DC voltuge

4.2 Switching instant variation (SIV) This Section is concerned with the evaluation of the second term of eqn. 10. The first step in the analysis is the determi- nation of the effect that the applied distortion has on the switching instants of the converter. While the exact value of the switching instants cannot be found analytically, accu- rate linearisations can be made. In the case of firing-angle distortion, the variations of the thyristor firing instants are approximated by the value of the firing-angle control signal at the unmodified firing instants coot;

AO, = a k sin(kwoti + 6h) (25) Fig. 7 shows the effect of an AC voltage distortion on the AC current in the vicinity of an end of commutation period instant. The dashed line represents the base AC current waveform, while the thicker solid lines represent the modi- fied waveform. To obtain a linear analytic relationship between the input distortion and the switching-instant vari- ation, the effect of the distortion is only considered during the unmodified conduction periods. The mismatch between the commutation current and the DC current at (q, f h) is given by I,, which in this case has a negative value. Dur- ing the modification of the commutation period, the com- mutation current is approximated by a tangent to the base commutation current at (q, + h). The lines whch are tan- gent are indicated by the double perpendicular lines. The current mismatch, the tangent line and the variation of the switching instant form the three sides of the triangle ABC, which allows the variation of the switching instant to be evaluated as:

(26) IA4 --

t an u

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For the case shown in Fig. 7, the linearisation predicts the end of commutation instant to be at C, while the actual end of conmutation period instant is at D. The accuracy of the linearisation requires that the slope of the base commu- tation current is much greater than the slope of the distor- tion component of the commutation current at (CQ + ,I& and that the slope of the base commutation current does not change significantly during the modification of the con- duction periods.

uo++o

Fig. 7 hy AC voltage dktortion

Anulyyis of inodijkation of end of conwiutution period instant caused

The slope of the base commutation current tangent at (q + h) is the same for all commutation periods and is obtained by differentiating the partial steady state of the commutation current associated with the base AC voltage V1U

&VI tan D = - sin(ao + p g ) 2x The current mismatch at the end of the ith commutation period I,,M is obtained by considering the PSS and PT responses of the commutation circuit associated with the applied distortion:

Ia,M = A~z,pT+A~a,PSS(Wotz+po) - A i d c ( W o t z + P o )

(28 ) The resulting switching-instant variation is summarised in Table 2, where coot, has been replaced with q t . The switch- ing instants are modulated by applied distortion in a simi- lar manner to the commutation current partial transients.

A method of evaluating the second distortion transfer mechanism becomes apparent after considering the base DC voltage waveform. This waveform is the piecewise combination of the PSS waveforms associated with the base AC voltage vIv and is described by

VdC = Py-dCvlI, (29) $

where Yuy""-"" is a previously defined sampling function. In accordance with eqn. IO the change in the DC voltage is

dJ Fig. 8 shows the modification of the sampling function due to the variation of the end of commutation period switch-

ing instants. The dashed sinusoidal signal represents an end of commutation period switching instant modulation signal, while the solid sinusoidal signal is the same signal time delayed by dq. The time delay is required so that the value of the signal at (%ti + ,U,,) represents the modifi- cation of the switching instant.

. . . . . . . . . . . . . . U0

. . . . ~ ..... . . . . . . . . . . . . . . . . . . . .

. . ............. ~. .................... ..............

. . . . . . . . . . . .

. . . . . .

. . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 8 , Vod@ation of suntpling function Yc-dc N Sainpling [unction 11' h End of commutatio~switching instant modulation signal and sampling function AYIvYVk

In the general large signal case the modifications of the sampling function are fixed-amplitude pulses of height 0.5, and of width the variation of the switching instants, in other words a pulse- duration-modulation spectrum. How- ever, in the small signal sense, this spectrum reduces to that of an amplitude modulated impulse train. The height and area of the of the impulses shown in Fig. 8b are half the value of the sinusoidal signal. When the sinusoidal signal is positive the switching instant is delayed, whereas when it is negative the switching instant is advanced.

The sampling function AY7,y+'c is modelled as the sum of four amplitude modulated impulse trains which have the appropriate signs and time shifts. The transfer of distortion due to the variation of the firing instants is modelled in a similar way to the that described here.

5 Conclusions

A small-signal analytic model of an HVDC converter, which predicts the transfer of waveform distortion, has been developed. The analysis is based on a piecewise linear state-variable representation of the converter, and accounts for all aspects of converter operation which can be modelled in a linearised manner. The change in the converter waveforms, as a result of applied distortion, have been identified as being the result of two distinct waveform distortion transfer mechanisms. The first is associated with the change in the waveforms during the unmodified conduction periods, while the second is associated with the unchanged waveforms during the modification of the conduction periods. It is necessary to identify whether or not a change in a waveform voltage or current-time area is a linear function of the applied distortion, and, conse- quently, whether it is relevant to the development of a linear small-signal model.

Table 2: Variation of the commutation period beginning and ending switching instants (radians)

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