Direct method for calculation of AC side harmonics
and interharmonics in an HVDC system
L.Hu and L.Ran
Abstract: AC side current harmonics in an HVDC system arc shown to be dependent on the
commutation overlap and thc DC side ripple. Interharmonics exist when the AC networks operate at
different frequencies. An analytical inethod is proposed to reprcscnt such effects in the AC sidc
harmonic calculation. The algorithm does not require iteration between the AC and D C sides. Thc
rcsult obtained using the analytical method is verified using time domain simulation.
1
Introduction
There is sustained interest in studying harmonic generation
in HVDC systems. On the AC side, harmonics in the
quasi-square current waveform arc well known [I]. It has
been shown that close coupling exists between the HVDC
system and the AC network, which should be includcd in ia
harmonic load flow study [2, 31. Other studies havc shown
the effects of AC supply unbalance, jittering of firing angle
and commutation overlap due to harmonics, and network
asymmetry [4-71.
The modulation theory has been used i n harmonic studies of HVDC systems [8, 91. Using switching functions
which describe the convcrter state, the harmonic current on
the AC side and the voltage on the D C side are calculated.
Thc advantages of a modulation-theory-based method
include an insight into thc mechanism of liarmonic generation and the casc of iniplenienlation. The accuracy of such
a method is usually low since the switching runctions are
often simplified a s being quasi-square with the commutation overlap ignored. The D C side ripple could be
accounted for by iteration, which however makes tlic
method complicated to apply, and the accuracy is still
limited,
A phenomenon associated with some HVDC systems is
the interharmonics when the AC networks operatc at dirferent frequencies. The modulation theory has also bccn
uscd to investigale the mechanism giving rise to interharmonics which feature non-integer harmonic ordcrs [IO, 1 I].
Existing methods need improvement in accuracy and are
not yet suitablc for direct calculation. If harmonics or intcrharmonics are to be calculated directly without iteration, it
is essential to properly represent all parts of the system.
This paper improves the modulation analysis of the AC
harmonics and interharmonics in an HVDC system. A
direct AC side method is proposed with the coinmulation
overlap included. It is assumed that the overlap is known
0IEE,2 W
IEE P ~ o c ~ ~ ~ ~oiiliiic
d i i & no.
s 2CHXKIh68
DOf 10.1049/ip-gld:2DOM)668
I'ap first irceived 301h Scplcinbcr 1999 kind iii rcviscd Ibmi 27th April 2000
I..Hu is with Ilic Business Innovation CciilIc, SlafFordshirc, S r12 OAR, IJK
1.Ran is wit11 1112 School of Engineering, Univcrsily of Nortlioinb~iatit Ncwcastle, NE1 BST, UK
when describing the method. Calculation or this operating
parameter is discussed regarding its correlation with the
harmonics. Harmonics and intcrharmonics associated with
the constant DC currcnt and its ripplc are analysed to illustratc that diffcrcnt AC and D C side iinpcdanccs should be
used in different cases. The calculation results arc compared with timc-domain simulation for the CICKE HVDC
benchmark system.
/
Fig. 1
t vac;:av v v v v v v v v v v v (t)
~
<im/;,riw~i~ioii
o/ iiii
~
~
~.
t\
~~ ~ ~ ~
j
Vac2_abc(t)
~~ ~~
f i V / K : iimi,oiiiarioii . S J N W ~ ~
2 Generation and different types of AC side
harmonics
Fig. I shows thc configuration or an I-IVDC systcni. First,
let LIS considcr that the AC networks at both cnds operate
at the same frcqucncy so that interharmonics will not be
gcnerated. Although thc final calculation is dircctly carried
out on the AC side, it is illustrative to considcr that the AC
side currcnt is the modulation product of the DC side current icci
(!) with the current switching functions sil,,(t).
Phase '(I' a t the sending end is uscd a s an example.
GLCILO(h)
= %-n(t)iCkl(t)
(1)
With commutation overlap, the current switching ftl'utiction
for a 6-pulse converter is shown in Fig. 2. When the pliase
undcr consideration is not conducting, thc function is zero.
When the phase is conducting the whole DC current, the
function is + I or --I depending on the current direction.
During commutation while the current is transited, thc
switching function is determined by the varying current
conducted by this phase [I]. The switching function is
defined by two operating parameters of the HVDC system:
the liring-clclay angle rx and the overlap anglc LI. These can
be estimated 1Pom thc load condition [ I , 5 , 121. The Fourier
series of the switching fLu1ction is shown in the Appendix
(Scction 1 I).
The DC current is represented a s shown in Fig. 3. The
D C voltage is determined using the phase voltages and
329
~
~
~
~
~
~
~.
~~
vollagc switching runctions. At the sending end,
'U& I
(t) = S,, I ( t )U n c i ( t )+ S
+
.%I-C(~'),[)acl
il
J -0
( t) l J a d_b (t)
-<:(/,I
- '/'
- S,l-nhc(t)2),,l-n6c(t)
(2)
where , ~ , , ~ - ( , / ~ ~and
( f ) v,,.l~~,~jc(t)
are column vectors of thc
switching fhctions and phase voltages, respectively. The
voltage switching function is shown in Fig. 4 for phase 'N'.
During commutation, the DC bus potential equals the voltage average of the two phases involvcd; the switching function is 1/2 or -ID.
current switching function is an operation from the D C
side to the AC side, whereas the voltagc switching function
is from the AC side to the D C side. Both switching futictions can be denoted as the sum of the fundamcntal and
harmonic components. For the 12-pulse configuration,
some Fouricr temis will be cancelled and others doublcd.
The remaining terms are of the 12k f 1 hannonic order.
Therefore the switching functions can be expressed as
. s i I _ n ( t ) = 2.5il-a-I
( t )+ 2
1
,~zl_aLlzk*l
( t ) (3)
k T l , 2 , 3 , . ..
.StJI-a(t)
+
= 2,9?>lLd(t) 2
S~>lLaJ2k+l(t)
(4)
k=l , 2 , 3 , .. .
whcre s i l
and
l(t) are the fundamental components wh%Feas s i l 12kbl(~) and ,s,,~ 12/<+l(t)the harmonic
components for 6-pulsc converter.-The constant voltage
component in ~ ~ / ~ ,is( tmainly
)
the modulation result of
.svl~ohc~l(t)and vciClc,/,c(t).Thc D C voltages at both ends of
the line produce a constant current, which is modulated by
the currenl switching function to givc the fhdamental AC
currenl and the inherent harmonics. Harmonics are also
gencrated in the D C voltage by modulating s y l a/jc 12k*l(t)
with Y , , ~f,/lc(l).
~
The results are at the 12th, 24th, Ibth, 48th
ctc. harmoi?ic ordcrs, and D C current ripplc will then
result. Modulated back to thc AC side, such DC ripple will
add to thc harmonics at the Ilth, 13th, 23rd and 25th etc.
ordcrs.
Combining voltagc and current-switching functions,
Table I detincs the different types of components in the
AC side current. This paper next addresses such components and the methods to calculate than directly. For thc
conveniencc of calculation, the following analysis will be
inaiiily in the frequency domain.
(i
((
3 AC current components corresponding to
constant DC current
The fundamental and inherent AC harmonics arc proportional to the constant DC current which is driven by the
voltage difference across the HVDC transmission line. The
constant component of the converter D C voltage is calculated using fundamental voltage-switching functions and
the supply phase voltages. At the sending cnd,
whcre
L, I is a rotating vector representing s y 1
I(t),
while Po,; represents vCaI (,(f). It can be arbihiirily
assumed that both vectors iotate in the anti-dockwisc
direction. The time-domain signal is the projection of the
rotating vector on lhe horizonlal axis. Other vectors are
similarly defined. Mathematical opcration on such rotating
(I
Fig.4
~~~/iri~~-.s~i~ii~/~i~~,~,/~iiici~~~i~
The switching functions also apply to a 12-pulse configuration which is assumed in the following description. The
Table 1: Components in the AC-side current
sd t)
Fundamental
Harmonic
Fundamental
S r a b c J ( t ) v a ~ c ( f ) :constant DC current ld,
AC side current: fundamental component
Sv-abc-12kl '(dVd,c(t): DC current ripple idc-q2k(t)
AC side current: major associated harmonics
SiLabc-l(t)/dc
SiLabc-I ( t)k/cdc12k(
t)
Harmonic
AC side current: inherent harmonics
AC side current: minor associated harmonics
S i a b c ~ 1 2 k(t)idc-12h(
~l
8
SI( t)
vab,(t)
SiLabc_12k~l(t)/dc
330
IEF P,or.-Gorer. Tramm />i.slri/>.. V d 147, No. 6 , N o v i n b o . 2000
vectors follows the complex number convention;
1 is
the conjugate of S,,,
Every term in eqn. 5 is cf6rkcd
from a modulation product in the time domain. For cxiimple, Svl (, I P,,1. I , is derived as follows regarciing a 12-pu1se
converter-configtiration.
(i
2S"l-@_l(t)7J<><,L@(t)
+
= 2C,l_l cos(w1t O7,I-LVlcos(w1t
+
$1)
= c , , ~ l v l [ c o s ( ~- I& - I )+c:os(2Ldlt+$Jl
+&L1)1
(6)
The first tenn corresponds to tlie A,,l (, I V,,(,-,, when considered in the frequency domain, whG& the second term
spins to zero a,mong tlie balanced tlircc phascs. 3,,1-~,-1
V,,,.,-,,,
V,,,,-/, and
I P(,cl-c
are equal because
the phase voltage and thc fun&iiiicntal switching function
are phase shifted by the same angle.
The voltage at the receiving end oT the DC line can be
expressed in the same way. The efrccl or commutation
reactance has already been incliidcd in the voltage-switching function [I]. In calculating tlie constant DC currcnt, the
DC-side network is a resistance Rch.. Considering that the
rectifier and inverter arc connected in opposite polarities
with respect to the DC line, the D C line current can be allculated as
si,,-,,-,,
Thus the fundamental AC current can bc dcrivcd in a
matrix format for both ends.
When large smoothing inductance is applied, the DC
current is close to its average value and the ripple can be
ignored. In such a case, the inherent harmonics shown in
eqn. 9 will give good accuracy. Tlie above analysis shows
that, given switching fhictions, tlie inherent harmonics can
be analysed on the AC side directly. Tlie switching fiinctions can be expressed in the systeiii-volkigc phase angle,
firing angle ancl coniniukition overlap a s shown in the
Appendix (Section I I), which ftirthcr specifies tlie cquations.
Associated harmonics due to DC current ripple
4
When the ripple in thc DC current is large, the additional
A C current harmonics could be fotind tising cqn. I, however the D C current ripple is unknown. The AC side harmd DC side ripple are physically the S a m currcnl
viewed from dilTcrent sidcs or thc convertcr. It is possible to
calculate the additional A C currcnt harmonics directly.
The harmonics oftlie voltagc and current switching ftinctions are grouped in pairs. Viewed from the DC side, the
D C current ripple is related to the harmonics in the DC
volvage. Thc I2ktli l~ariiionic:voltage is due to modulating
the supply voltagc with the (12k * I)th Iiariiionic pair in
the voltage switching function. The D C side 12kth harinonic current can he cxpressed a s follows for tlic sending
end:
IdcI-lSi,
= 31/1I _Iai, (SW
I -&-I
where Sil I represents the hndamcntal current switching
I is
function IbT phase '(I' at the sending end, whcrcas Si* <,~
for tlie receiving end.
Eqn. 8 implies a two-port model for the HVDC system.
Since the voltage and current arc hotli on the AC side, such
a model can be combincd with the AC systems to solve for
the ftindamenlal AC current. This is illustrated iii Fig. 5
with the admittance matrix dcfined by eqn. 8. Depending
on the DC-side resistance and switching functions, each
admittance clement is a constant complex number. Tlie
admiltancc matrix provides a relationship governing the
amplitudes and phase angles of the voltages and currents.
etjrl-T(
jacl_a_l
I $ system
sending
HVDC system
~~ Ai + IV<UI
-0
)
where YIl
imd Y12
are tlie admittance elements in
the DC network matrii at thc 12kth harmonic l'rcquency.
Modulation operation in thc hrackets is to derive the harmonic voltages at the sending and receiving ends of the DC
line. Contributions from thc three phascs are equal.
When the DC current ripple is modulated back to the
A C side using the rundaincntal current switching function,
harmonics of 12k I orders result. They are of the major
associated harmonics shown in Table I . For the (12k +
l)th A C harmonic,
L
(.
I -<,-I2 k + I
= s i 1 -a-1 L i r I - 1 Zk
Vac2-a
One of our purposes is to derive the inherent harmonics
in the AC sick current. Since tlic rclationship hetwecn thc
A C current and the switching function is a simple gain.
they are obtained by scaling thc current switching function
to give the same fundamental currcnt as calculated in the
above equivalent circuit.
t ' r , i ~ c - ( ; c ~ i i wTriiii.im
er^ 1-0 +- SZ,I -
I @system
receiving end
Fig. 5 ~iinlu~nurnoliiil,~~~c~ii~ii~~
rnolwtkl o / the Hvnr /inold<
IEE
I
iacz-a-1
admittance matrix
defined by eqn. 8
Vac1-a
ah
Di.stri/~.,Vol. /47, A'o 6. Noaciiiho. 21jljO
Viewed from the AC side, tlic current ripple is driven by
voltc~gc
sources which are the DC harmonic vollagcs modulated to the A C side using tlic ctirrent switching runclion.
On the AC side, the current ripple will cxpcricncc impcdiiiice or the network, and the DC and AC circuits should
be solved simultaneously. Corresponding to eqn. 1 1, Fig. 6
shows an equivalent circuit for the iixijor associated AC
harmonic. The AC networks at both ends are single-phase
models at the 12ktli harmonic frequency. Normally two
331
phases of a 6-pulse bridge arc conducting and each phasc
presents harmonic impedance to the D C currcnt ripple.
When the AC network is represented in a singlc phasc, it
thus sees half of the DC side impedance. The whole DC
network is represented as the 'intemal' impedance of the
harmonic voltage sourcc, with all the impcdancc scaled
down by a ratio of 2. The referrcd source voltages arc
+SuLa-i2k+l V a c L a )
Val:2-a_12k+l
=
(12)
3 '
-si1
-a-l (Su2.n-lPk--J
2
L
o
+So2-n-l2k+1
Vac2-o)
(13)
On condition that switching functions are based on real
delay angles and overlaps, the above calculation does not
need to be itcrativc. The only approximation is that, in
deriving the AC network model, two phases arc assumed
lo be conducting, whereas actually all three phases are conducting during commutation.
Vacl_a_12kil
Fig. 6
*-
vac2_a_12k+1
,!Qiiiwleii/ cixuhfiv //v iiiigor uvsociii/<d/ i w i i o i i i r
The major associated harmonic at the (12k ~. I)th ordcr
can be similarly derived using an equivalent circuit. However, as it is associated with the Samc D C current ripplc
i,,ci12k, it can be directly calculated once I<,(I ', 12k+l is
obtained. Comparison shows that the pair oT aGociated
harmonics is of tlie same amplitude but different phasc
angles.
Therefore the harmonic current at each characteristic frcquency may be contributed from diffcrent modulation
mechanisms. The calculation is performed in the frequency
domain with correct phase angles, so that the total harmonics in the AC sidc current can be obtained using superposition. The phase difference betwecn two AC systems
will affect the D C currcnt ripple and thus the AC-side harmomcs.
An HVDC system is usually regardcd as a currcnt harmonic sourcc on the AC side [ I , 131. However, the above
analysis shows that the harmonics associated with the D C
current ripple should not bc simply considered as such,
because the ripplc depcnds on the AC-side impedance. It is
appropriate to represent thc HVDC system as a voltage
harmonic source with internal impedance of the DC network. This is advantagcous when the DC-side harmonic
impedance is low, thus the DC current ripple is largc, e.g.
in some back to back schemes and systems with sub-sca
cables [13].
5
Calculation of the interharmonics
The above analysis can be extended to the case when thc
two AC nctworks operatc at different frequencies. The
switching functions at the two ends will have different fundamental and harmonic frequencics. The calculation of the
inhcrcnt AC side harmonics corresponding to the constant
DC current is the same as in Section 3, but the DC ripple is
differcnt and contains harmonic components or both systen1 frequencies. Intcrharmonics at one cnd are always
related lo the ripple associated with tlie D C voltage harmonics at the other end [IO, It]. Becausc the harmonic frequencies are different at the two terminals of thc DC
network, they have to he calculated separately.
...........................
HVDC system
'acl_a-l2k+l
DCneiwork
"acl~qiZk+l
Fig.7 /?quiiri/etit ciwuii ~
J
;
,
=!=
-
/he
F hconioiiir cowevpn,nr/infito
dip
DC
ciiweiit
rQip/e,/roiii /lie smw e i d
The same method can be used to calculate the major associated harmonics at the rcceiving end directly from its current switching function and the sending cnd results
obtained from Fig. 6. For each ripplc frequency on the DC
side, only one equivalent circuit nccds to be solved.
In addition to thc major associated harmonics, the DC
current ripple can also modulate with the harmonics of the
currcnt switching runction to cause minor associatcd
harmonics on the AC side. The D C current ripplc is
unchanged, so the AC networks should still be represented
at the same frequency a s for the DC side network.
Considcr the components of the current switching function
at orders 12n + I, the minor associated AC harmonics arc
or the orders 12k + 12n t 1 and 12k 12n + I,and can be
calculated using an cquivalcnt circuit similar to Fig. 6.
Altcrnativcly, they can bc directly calculated from
-
~ O C I ~ < ! ~ l 2 Ik. +
I n c l ~ n _ 1 2 k i - ( . L 2 7 L ~=
l)
silLa_1271il '
'
StI-n-1
L,l-a-l2k+l
(15)
Consider the sending end: Fig. 7 shows the equivalent
circuit used to calculatc the AC sidc hai-tnonics associatcd
with the D C current ripple caused by the vohage harinonics o l the same end. The harmonics are of characteristic frequencies of tlie sending system and are not interharmonics.
Corresponding to a DC-side harmonic number of 12k, the
frequency which is uscd to dctermine tlie sending end AC
network (single phase), D C network and recciving end AC
network (siaglc phase) is
where j'i is the fundamental frequcncy at the sending end.
The harmonic voltage source in the equivabnt circuit for
the AC-side harmonic ordcr 12k + 1 is the same as in
eqn. 12. This corresponds to the major associated harmonics, which can then be used to calculate ininor associated
harmonics, as shown in cqns. 15 and 16.
Intcrharinonics at the AC side of the sending cnd arc
caused by the ripple associated with thc DC voltagc harmonics of the rcceiving end. Fig. 8 shows the equivalent
circuit for calculation. Thc AC and D C nctworks are now
dcterinined by the harmonic frequency of the receiving system, which is shown below. The voltage source is referred
I I X P~.o.-CCW
Tror,s,ii.
I ~ ~ . D b f r M . . Vol. 147. No. 6. Nosonlm 2000
to the sending end whcrc thc intcrharmonics are calculated.
The current switching function of the sending end is used.
where J; is the fundamental frequency at the rccciving cnd.
Depending on the switching lunctions of both ends, the
voltage source will be at interharmonic frequencies. A siniilar proccdurc can be applied to derive intcrharmonics at
the receiving end. The AC and DC nctworks should be
appropriately represented, a s shown in this paper.
laclLa~i2k+l
\
\
\\
'
,
' /
DCnetwork
I
111)system
receiving end
-
"ac2~a~12k+i
Fig.8 Equiinlwit circuitJijr [lie iiwrliiivtiiuiiii
Interaction between harmonics and overlap
The mcthod has so far bccn described based on the
assumption that the commutation overlap is known. If the
DC current is small, the overlap can be easily estiinatcd
using the average DC current [I]. Alternatively, a measured
overlap angle can bc used in online calculation. In either
case, the method proposed is applied without iteration.
Commutation generally takes placc whcn the D C current
is at a minimum. When the DC ripple is large, the harmonics to be calculated will allect the overlap. In order to
achieve high accuracy, an iterative proccdtirc is necessary to
calculate this operating parameter. The method proposcd
in this paper can be implemcnteed in each step of the itcration, in which the A C and DC sides are always proccsscd
simultaneously. Given the overlap, the proposed method
will readily produce the AC harmonics and liencc ii DC
ripple, and the estimated overlap can then be updated.
Other methods ciin also bc developed to calculatc the ovcrlap [5, 141.
7
I
HVDC system
sending end
6
400011
. . . . . . . ~ - . ~ ~ ~ ~ ~ ~ . ~ ~ ~ ~ ~
:---
'
current control is applied to the rectilier to givc an average
D C currcnt of 1800A. A fiindamental load flow study gives
the operating parameters, as shown in Table 2.
Fig. 9 shows the simulated A C current; commutation
overlap is evident. Table 3 compares the calculated harmonics at the sending end with simulation, and close agreement is observed. Table 3 also shows the erfect ofchaiigiag
the sending-end commutation overlap by 10%, assuming
that the constant DC current is unchanged. The mcthod
can be applied to account for system uncertainties.
Verification of the proposed method
Two studies are performed to verify the proposed method:
the CIGRE Bcnchmark HVDC system, and a 49Hd51 Hz
asynchronous interconnection. In each case, the steadystate waveforms of the AC side current are obtained using
the SABER simulator and an FFT is applied to obtain the
harmonic content. The results are compared with those calculatcd using the analytical method, which is iniplenientccd
in MATLAB.
Details of the 12-pulsc, monopolar ClGRE Bcnchmark
HVDC system can bc found in [12]. At the rccciving end,
the extinction angle of the inverter is set to 18". Constant
Table 2: Operating parameters of the HVDC system (case 1)
Parameter
Sending end
Receiving end
AC side phase voltage,
187.7kV LO",50Hz 140.8kV L40". 50Hz
RMS
Firing-delay angle
10.28"
137.2'
Commutation overlar,
23.46"
14.8"
-400011
0.760 0.765
Fig.9
I
I
0.770 0.775 0.780 0.785 0.790 0.795 0.800
time, s
. S i i i i i i b t ~ ~ ( l / l ~ . \ciirvent
i~le (mw I )
Table 3: Harmonic components of the sending-end AC-side
current (case I )
Harmonic
Simulation
Calculation
Overlap,
+IO%
Overlap,
-10%
Ist, 50Hz
1723.3A
1719.31A
1721.30A
1719.11A
11th. 550Hz
59.84A
62.72A
50.97A
75.69A
13th, 650Hz
37.81A
38.51A
29.67A
49.32A
23th. 1150Hz
15.96A
14.62A
12.97A
14.73A
25th. 1250Hz
13.5A
14.41A
10.78A
15.74A
3.95A
35th. 1750Hz
5.30A
5.60A
6.08A
37th. 1850Hz
5.20A
5.85A
5.39A
4.12A
47th. 2350Hz
1.53A
1.89A
3.07A
3.60A
49th, 2450Hz
1.56A
1.52A
3.11A
2.17A
In the second study, the ClGRE system is modified to
show interharmonics. It is assumed that the sending system
operates at 49 N x whereas the receiving system opemtes at
5 I Hx. The I .OH DC induct" is rcduccd to a quarter.
This results in a higher DC current ripple. Fig. 10 shows
the simulated DC current at the sending sidc. The peak lo
peak ripple is ovcr 55A; about 3'%, and a complex modulation effect is seen.
1840 11
I
Fig. I 1 shows the AC-side current waveroms at the
sending and receiving ends. T a b b 4 shows the calcukated
harmonics at the sending end as coinpared with the simulation. In order to achieve the required frequency resolution,
the simulation waveform was sampled over 1s. Harmonics
and interharmonics at the receiving end can bc similarly
calcukaled. Table 4 compares amplitude only. The phase
angle can be verified by reconstructing the current waveform from its harmonic content. Fig. 12 shows the reconstructed waveforms, which should be compared with
Fig. 11.
tcms because the filters become lcss effective. The distribution of harmonics and interharmonics in the AC network
can be subsequently calculated.
Conclusions
8
This paper presents an improved method of calculating the
AC side hannonics and intcrhainionics in an HVDC system. Equivalent circuits are obtained from the modulation
theory. The switching functions employed are derived from
the actual commutation process so that the effects of the
commutation overlap are included. Using the proposed
method, calculation is carried out on the AC side directly,
without iteration with the D C side. The algorithm is implemented in MATLAB. The accuracy of the method has
been verified using time-domain simulation.
The proposed method provides further insight into the
mechanism of harmonic generation in an HVDC system. It
allows powcr system engineers to predict the effects on harmonics and intcrharmonics when the system conditions are
changed.
9
0.770 0.775 0.780 0.785 0.7’90 0.795 0.800
time. s
S i a l r t ~AC.si(C
i
w r m /r?ac 2)
0.760 0.765
Fig. 11
Table 4: Harmonics/interharmonics of the sending-end ACside currents (case 2)
Frequency
Modulation terms
Simulation
Calculation
12f1-11f1,etc.
1720.89A
1721.30A
539Hz
Ilfl,
12f1-f1,etc.
60.24A
61.99A
563Hz
12f2- fIr etc.
637Hz
13f1, 12f1+ fi, etc.
49Hz
f1,
12f2+ fl, etc.
661 Hz
1127Hz
23f1, 24f1 - fl! etc.
1175Hz
244 - fl, etc.
1225Hz
254, 24f1 + fl, etc.
1273Hz
1715Hz
2.78A
2.92A
38.95A
40.72A
2.51A
2.88A
16.13A
15.44A
3.36A
3.85A
13.61A
13.53A
24f2 + f,, etc.
3.26A
3.82A
35f1. 36f1 - f1, etc.
5.24A
6.04A
1813Hz
37f1, 36f1 +
4 , etc.
5.04A
5.46A
2303Hz
47f1. 48f1 - 4, etc.
1.56A
2.29A
2401Hr
49f1. 486
+ f,, etc.
1.59A
2.01A
-4000 I
0
Fig.12
I
0.01
Xaori,s~v,,.um/ AC
0.02
time, s
0.03
The authors would like to thank Prof. R.E. Morrison for
his advice.
10
I
2
3
4
5
6
7
References
KIMBARK, E.W.: ‘Direct currcnt transmission - vol. 1’ wilcy Interscicncc, New York, 1971) pp, 295-327, pp. 71-123
XIA, D., and HEYDT, G.T.: ‘Harmonic power flow studies, part I
Formulation and solution’,EEIEEEEEEEEEEEEEEEEEEETr1ifl.Y. P u n w A p p v Syst., 1982, 102,
(6), pp, 1257-1265
XIA, D., and HEYDT, G.T.: 'Harmonic power flow studies, parl I1
implcnienlation and praclical application’, 1
S y ~ t . ,1982, 102, (6), pp. 12661270
WOOD, A.R., and ARRILLAGA, J.: ‘The frequency dependent
impedance of an HVDC convcrlcr’, IEEE Ti”. Poii:er Deliv., 1995,
140, (6), pp. 1635-1641
WOOD. A.R., and ARRILLAGA, J.: ‘HVDC converter waveform
dislorlion: a fscqucncy domain analysis’, IEE Pruc. Gmer. Trmrm.
Di.st~.i/i.,1995, 142, (I), pp. 88-96
XIA, D., SHEN, Z . , and LlAO, Q.: ‘Solution of noncharacteristic
liarmonics caused by multiple factors in HVDC transmission systcms’.
P~-occcdiogsof the IEEE international conferencc on Hurmonics in
p u i w .~y.s/eriis,
Nashvillc, USA, 1988, pp, 222-228
HLJ, L., and MORRISON, R.E.: ‘The use of modulation theory to
calculale the harmonic dislortion in HVDC systems opcrating on km
1lIlba~allCedSUppl)r’, IE,?E T?’<lIlS.POoil’e~SJWt.,1997, 12, (2), pp. 973oxn
8 r S H l D , M.H., and MASWOOD, R.: ‘Ane!ysis of three phase ACI
IIC cunvcrlcrs under unbalanced supply conditions’, IEEE T,WISlid
App/., 1988, 24, (3), pp. 449455
9 HU, L., and YACAMINI, R.: ‘Harmonic lransfer through converters
and HVDC links’, /EEE Trum Power Eklec/i.az., 1992, 7, (3), pp, 5 1 6
525
10 DELANEY, E.J., and MORRISON, R E . : ‘The calculation of harmonic atid interharmonic distortion in current source converter systems’. Proceedings of the IEEE intcrnational conference on ifurniunks
in po~vi’r.sy~/ein.s,Atlanta, USA, 1992, pp. 251-255
I I HU, L., and YACAMINI, K.: ‘Calculation of harmonics and interharmonics in HVDC sclicmes with low DC side impedance’, IEE
Pror.. C: G e i w Trmsiii. Uis/~.iIi.,1993, 140, (6), pp, 469476
I ? ARRILLAGA, J., and SMITH, B.C.: ‘AC-DC powcr syslcm analysis’ (IEE Power & Energy Series, London, 1998)
13 ARRILLAOA, J.: ‘High voltage direct current transmission’ (Peler
Pcrcgrinus, IEE Powcr Engineering Scries 8, London, 1983) pp, 35-50,
pp. 6 6 7 5
14 XU, W., DRAKOS, J.E.. MANSOUR, Y., and CHANG, A.: ‘A
hasc converter niodcl for harmonic analysis of HVDC systems’,
77mis. POIIPI.
Dclii’., 1994, 9, (3), pp. 17261731
-1
0.04
11 Appendix: Fourier expansion of switching
functions
,si<li,WI.LW~
As the fundamental fkequencics deviate from 50Hz, more
harmonics and interharmonics penetrate into the AC sys334
Acknowledgment
For a 6-ptilsc converter, the current and voltage switching
functions, as shown in Figs. 2 and 4 are expressed as follows. The Fourier series expressions are valid for both the
16l<P,nc.-Gc,,cr.. ’ h m m Di.wil>.,V d . 147. N o
6. Nov‘mher 2OlNJ
and for the voltage switching function,
f I)($
1 /,
-)
2
is the phasc of the AC side voltagc, a is thc firing delay
angle aiid LI is thc coninititation ovcrlap.
Oi,_ljk+l
= (Gk:
- <V
-
335