K. R. Padiyar Auth. Analysis Of Subsynchronous Resonance In Power Systems.pdf 6l4n6w

Note that L includes L', the transient inductance of the generator. If the resonance frequency fer defined by

fer

1

= 2Ti"fLC

is close to (fo - fm),the impedance is small.It is resistive when(fo - fm) = fer and capacitive when fer> (fo - fm),otherwise it is inductive (when fer < (fo - fm)). The d-q components of the currents are given by ( 4.49)

92

ANALYSIS OF SUBSYNCHRONOUS RESONANCE IN POWER SYSTEMS

From Eqs.(4.46)-(4.48), we have

i~ub(t) = -

2 A~ (w o - wm ) cos[(w o - wm)t wm sub

+ do -

¢lsub]

i~ub(t) =

2 A~ (w o - wm ) sin[(w o - wm)t + do - ¢lsub] wm sub Substituting Eqs. (4.50) and (4.51) in (4.49), we get

(4.50) (4.51)

(4.52) The component of the torque T:ub, due to subsynchronous frequency currents,is given by

Tesub

(Wo = E' tq'sub = - 2A(E')2 Z wm sub

) . (

Wm sm wmt

The supersynchronous frequency voltage components supersynchronous frequency currents given by

e~up

A..) + '!'sub

and

e~up

(4.53) results in

(4.54) .sup

tfj

=2

AE'

Z (wo+wm)sin[(wo+wm)t+do-¢lsup] Wm sup

(4.55)

where ZsupL¢lsup is the network impedance at supersynchronous frequency ,viewed from generator (internal) bus. The magnitude of Zsup tends to be large compared to Zsub and ¢lsup is usually positive(the impedance is inductive) . The quadrature axis components (i~UP)due to the supersynchronous currents can be obtained as (4.56) The component of the torque due to the supersynchronous currents,T:up is given by 'sup _ A(E')2 ( Tesup -_ E' Zq - 2 Z Wo Wm sup

. (wmt - '!'sup A..) + Wm ) sm

(4.57)

Note that both torque components ,T: ub and T:up have same frequency Wm ,the frequency of oscillation of the generator rotor. The damping torque coefficient TDe is given by

(E')2 [(wo - wm ) A.. (wo + wm ) A.. ] T De -- --2-Z COS'!'sub Z cos'!'sup wm sub sup

(4.58)

ANALYSIS OF SSR WITH FIXED SERIES COMPENSATION

93

The net damping tends to be negative due to the fact that Z.ub « Zsup and tP.ub ~ 0 while tPsup ~ 90 0 in series compensated system when a torsional oscillation frequency is approximately equal to the complement of the electrical resonance frequency ,that IS

10 - 1m ~

ler

(4.59)

FigA.5 shows the phasor diagram giving the position of the torque components in relation to the rotor velocity. What is interesting is that the supersynchronous frequency currents in the network give rise to positive damping torque(although of small amplitude).It is the subsynchronous frequency components of network currents that cause negative damping.The smaller the oscillation frequency ,higher is the negative damping. Thus the first torsional mode(with the smallest frequency) can cause the most severe problem if the network impedance is minimum at that mode.

tP.ub

Figure 4.5.

Phasor diagram

Remarks 1. In the literature,[Kilgore et al (1977)] the negative damping introduced by the torsional interaction is expressed as (4.60)

where G is the conductance of the electrical network viewed from the generator bus defined as G(wo - wm ) = Re[Z-l[j(wo - wm )]. H m is the modal inertia corresponding to frequency Wm .It is not difficult to see that the above expression is derived from Eq.4.58 by assuming E' ~ 1.0 and neglecting the positive damping introduced by the supersynchronous frequency currents.A better approximation to the damping (decrement) factor (J' e is (4.61)

94

ANALYSIS OF SUB SYNCHRONOUS RESONANCE IN POWER SYSTEMS

2. The negative damping, measured in of the decrement factor (u e ) due to the electrical system is also dependent on the modal inertia and the frequency in addition to the conductance of the network at the subsynchronous frequency. 3. If ive shunt compensation is used instead of the series compensation, it is possible to damp the torsional modes. This is due to the fact, the electrical resonant frequency shifts to supersynchronous range with fixed shunt capacitor.

4.3.3 An alternative derivation using immittance functions The electrical torque (~Te) as a function of the change in the per unit rotor speed (~Sm) can be derived from the knowledge of the impedance functions in the D-Q axes. For the simplified generator model considered in section 4.3.2, the change in the electrical torque can be expressed as (4.62) From the relation,

iQ cos 6 + i D sin 6

(4.63)

we can derive (4.64) where

There is no loss of generality in assuming the operating value of 6 = 60 as zero. Hence, Eq.(4.64) can be simplified as (4.65) At the generator internal bus, the following equation applies (assuming that the generator is connected to an infinite bus through a ive impedance) (4.66) where,

~eD

=E

I

sin60~Sm

Wo

+ -E WE

I

cos60~6

(4.67) (4.68)


95

AeD and AeQ are derived from the fact that the generator voltage (at the internal bus) has only a q-axis component (e q ) given by

and eq is related to eD and eQ by

eD = eq sin 15 eQ = eq cosJ Assuming 150 = 0 (as before), we get Wo

'

AeD = - E AJ

(4.69)

= E'ASm

(4.70)

WB

AeQ

YDD, YDQ, YQD and YQ Q are the ittance functions and can be expressed as (see chapter 3)

YDD(S) YDQ(S)

= YQQ(s) = ~[Y(s + jwo) + Y(s -

= -YQD(S) = ~[Y(S -

jwo)

jwo)]

(4.71)

+ Y(s + jwo)]

(4.72)

where

Y(S) = Z-l(s)

(4.73)

and Z(s) is the impedance function (per phase) viewed from the generator internal bus. Since, AJ

= WB ASm S

(4.74)

we can derive the expression for AiQ as Wo

AiQ(s) = [YQD(S)S

+ YQQ(s)]E ' ASm

(4.75)

Substituting Eqs.(4.71) and (4.72) in (4.75) we get E'

AiQ(s) = 2s [(s - jwo)Y(s - jwo)

+ (s + jwo)Y(s + jwo)]ASm

(4.76)

From Eqs.(4.62) and (4.65), we can derive

(E')2 - --[(wo-w )G(wo-w) -(wo+w)G(wo+w)] (4.77) 2w

96


where

G(wo - w) = Re[Y(j(wo - w))] G(WO + w) = Re[Y(j(wo + w))] The damping torque coefficient (TDe) evaluated for a particular torsional mode of frequency, Wm can be obtained by substituting w = Wm in Eq.(4.77). It can be observed that the expression for TDe given above is identical to that given in Eq.(4.58). Remarks 1. The derivation for TDe given here for the simplified generator model can be extended to the case of the detailed generator model. This can be done by noting that the total impedance viewed from the generator internal bus is the sum of generator and the network impedance. In d-q axes, the generator impedances are defined from Eq.(4.26). However, this approach is cumbersome and will not be pursued. Instead, it is convenient to derive the transfer function Ye(s) from the state equations of the electrical system. 2. In general, the electrical system equations can be expressed as

= :J:(t.l),

XE = [AE]XE + [BEM]YM YE = !:l.Te = [CMelxE

(4.78) (4.79)

where

x~ = [x~ x~ XN], yk = [!:l.d' From Eq.(4.74), we can write

YM = [

!:l.~m

!:l.Sm]

] = [

i ]

!:l.Sm

m(s)!:l.Sm

(4.80)

From Eqs. (4.78) to (4.80), we can derive (4.81 ) where

~1

m(s) = [ • ] The details of the various matrices are given in the next section. 4.4

4.4.1

STATE EQUATIONS AND EIGENVALUE ANALYSIS

Analysis of Induction Generator Effect

To isolate the Induction Generator Effect from Torsional Interactions, it is convenient to assume that the generator rotor speed is constant. Hence only the


97

electrical equations are to be considered for analysis.In the absence of magnetic saturation and neglecting excitation system(treating Eld as constant) the system equations are linear. The system equations are defined from putting together the equations for the synchronous machine (rotor electrical) and the external network. These equations are reproduced below from chapters 2 and 3. (4.82)

Ye = [CelXe XN = [AN1XN

(4.83)

+ [BNdEb + [BN2]UN2

YN = [CN1XN

(4.84) (4.85)

where

UN2 can be expressed as

(4.86) where [ p ] =

-sino] smo coso

[c~so

(4.87)

If the ac network is as shown in Fig.3.4 with a shunt capacitor connected at the generator terminals, it is possible to express

(4.88)

In this case, Eqs. (4.82) to (4.88) can be combined and expressed as XE = [AE] XE

+[

BOl ] Eld

+ [ B~l

] Eb

(4.89)

where

x1 =

[X!

XJvl

and (4.90) The stability of the system is determined by the location of the eigenvalues of matrix [AE]. The system is stable if the eigenvalues are stable. It is to be noted that when the generator rotor speed is constant, [Pl is a constant matrix. It can be shown that the eigenvalues of [AE] are independent of the value of o. Hence it is convenient to assume that 0 = 0 and in this case [Pl becomes a unit

98


matrix of order two. A Special Case When there is no shunt capacitor is connected at the generator terminals (as in Fig.4.1), Eq.(4.88) is not applicable. For the network of Fig.4.1, the following equations apply [

~c

D ] vCQ

[

=

[0 -wo] 0

(4.91)

Wo

~: ] =

WXL ] [ RL [

cosJ sin J

~d ] + XL lq

- sin J ] {[ cosJ

WB

Vc D ] vCQ

[

1,f ]+ Tt

+[

(4.92)

0 ] Eb} 1

where RL and XL are the series resistance and inductive reactance seen from the generator terminals.lt is to be noted that Eq.(4.92) is similar to the Eq.(3.54) in chapter 3. After substituting for the derivatives of the armature currents, it is possible to express Ue in the following form

Ue = [F][G]Xe

+ [F][P]{YN + [ ~b

]}

(4.93)

where

[F] = [/2 - XL [Ce][Bdtl

(4.94)

[G] = [ZL][Ce] + XL [Ce][Ae]

(4.95)

wB

WB

[ZL] = [ [YN] = [

l!:L -WXL

vCD ] VCQ

WXL] RL

(4.96)

= [CN]XN

(4.97)

In this case, [CN] is a unit matrix of order two. The network and the generator (electrical) equations can be combined to give (4.98) where

[AEl ] = [Ae] + [Bd[F][G] [AE2] = [Bd[F][P][CN] [A E3 ] = [BN][P]t[C e] [A E4 ] = [AN]


BN

= [ WBXC o

0

WBXC

Beb = [Bd[F][P] [

~

99

]

]

Example 1 Consider IEEE FBM, the data for which is given in Appendix A. The eigenvalues of 8X8 matrix [AE] are shown in Table 4.1. This also shows the eigenvalues for the case with the generator model (1.1).

Table 4.1 Eigenvalues of [Ae] Model (2.2)

Model (1.1)

Comments

-4.7112 ± j 616.5856 -2.7675 ± j 137.4162 -0.9665 -4.3197 -20.3930 -33.1394

-4.4189 ± j 612.3871 -3.2675 ± j 141.5400 -0.9589 -4.1916

Supersynch. mode Subsynch. mode

-

-

The result shows that there are two network modes (a) Subsynchronous and (b) Supersynchronous. The sum of the frequencies of the two modes add to 2fa as their frequencies are given by fa ± fer where , fer in the frequency of the series resonance in the network. The frequency (fer) can be approximately evaluated as c (4.99) fer = fa +

~ XL Xg

where Xg

I

I

= Xd~Xq

for machine model (1.1)

= x~ ~x~ for machine model (2.2) The difference in the real parts of the eigenvalues corresponding to supersynchronous and subsynchronous modes arises from the Induction Generator Effect. It is seen that model (1.1) gives optimistic results on the negative resistance introduced by IGE. The influence of the series compensation level on the IGE is shown in Table 4.2 which gives the network modes as a function of the series capacitor (xc).

100


I

Table 4.2 Network modes as a function of Xc

Xc

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

I Subsynchronous Mode I Supersynchronous Mode -4.0610 -3.8786 -3.6974 -3.5040 -3.2902 -3.0479 -2.7675 -2.4366 -2.0385

±j ±j ±j ±j ±j ±j ±j ±j ±j

286.5116 248.9655 220.1662 195.8918 174.5094 155.1826 137.4162 120.8897 105.3842

-4.5301 -4.5857 -4.6231 -4.6517 -4.6748 -4.6943 -4.7112 -4.7260 -4.7393

±j ±j ±j ±j ±j ±j ±j ±j ±j

467.4484 504.9972 533.8001 558.0792 579.4679 598.8043 616.5856 633.1358 648.6800

It is interesting to observe that the damping of the subsynchronous mode continues to decrease while that of the supersynchronous mode increases slowly as Xc is increased. In general, IGE is expected to cause self excitation only at high levels of series compensation. IGE is also a significant function of the quadrature axis (open circuit) time constant. The network modes as a function of Xc are shown in Table 4.3 when T~o = 0.085 s instead of 0.85 s. Table 4.3 Network modes as a function of Xc (T~o=0.085)

I

Xc

0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

I Subsychronous Mode I Supersysnchronous Mode I -2.7676 ± j -1.8881 ± j -1.0293 ± j -0.1341 ± j 0.8260 ± j 1.8699 ± j 3.0096 ± j 4.2469 ± j 5.5655 ± j

288.1034 251.3094 223.1370 199.4519 178.6663 159.9813 142.9413 127.2715 112.8049

-5.0799 -5.3567 -5.5425 -5.6841 -5.7989 -5.8955 -5.9789 -6.0522 -6.1176

± ± ± ± ± ± ± ± ±

j j j j j j j j j

465.7574 502.6424 530.9375 554.7892 575.8020 594.7985 612.2674 628.5270 643.7982

The results show self excitation occurring at Xc > 0.20. The negative damping of the subsynchronous mode goes on increasing with increase in Xc. 4.4.2

Analysis of torsional interaction (TI)

The analysis of Torsional Interaction requires the modelling of the mechanical system in addition to the electrical system. Also, since the overall model is nonlinear, the analysis requires the linearization of the models for the study of self excitation.


101

Mechanical System Equations The linearized mechanical system equations (using multimass or modal model) can be expressed as (see chapter 2)

XM = [AM]XM + [BM]~Te YM = [GM]XM

(4.100) (4.101)

where XM is the state vector of small perturbations in the mechanical system variables ( inclusive of the rotor system, turbine and governor). YM is defined as (4.102) The per unit electrical torque acting on the generator rotor is given by (4.103)

Te = t/idiq - t/iqid Linearizing Eq. (4.103), we get ~Te

=

iqo~t/id

= [iqo - idol [

+ t/ido~iq -

~~;

]

= [Get]xe

t/iqo~id

-

ido~t/iq

+ [-t/iqo t/ido] [

~~;

]

+ [-t/iqo t/ido] [Ge]Xe = [Gmelxe

where Xe =

~Xe

(4.104)

and (4.105)

The nonzero elements of [Get] are defined by Get (1) = iqo , Get (2) = -ido Electrical System Equations In the previous subsection, the excitation system was ignored. The linearized equations of the excitation system (including AVR, exciter and PSS) can be expressed as

Xr = [Ar ]Xr + [Br ]u e + [Brm]YM Yr = ~Efd = [Gr]Xr

(4.106) (4.107)

where U e = D..Ue. Combining Eq. (4.106) with linearized Eqs.(4.82) and (4.83), we get

[ !: ] = [~e

B~~r] [ ~:

]

+[

IJ;r2 ] U e +

[

B~m

] YM

(4.108)

This can be written as

Xer = [Aer]xer + [Bedu e + [Ber1]YM Yer = [D..id D..iq]t = [Ger]xer

(4.109) (4.110)

102


where

[C er ] = [Ce 0] If the generator is connected to an infinite bus through a series R, L, C network, Eq.(4.92) applies. It can be shown that the combined state equations (including the network model) can be obtained using Eqs. (4.109) and (4.110). In deriving the linearized state equations, both c5 and Sm are to be treated as variables. The final equations can be written as follows.

(4.111) where

xk

= [x!

x~ x~]

= [x~r

x~], XN

= ~XN

[AE] = [AEI AE3

AE2] AE4

[AEd = [Aer] + [Ber2 ] [F][G 1] AE2 = [Bed[F][P][CN] AE3 = [BN][p]t[Cer ] AE4 = [AN] The matrix [F] is defined by Eqn. (4.94). [GI] is defined by

[G I ] = [ZL][Cer ] + XL [Cer][Aer] wE

(4.112)

The matrix [BEM] can be expressed as (4.113) where (4.114)

BE2 has two columns. The first column is given by

Bb = [

Ber2F[DP]{CNXNo + [ BN[Dp]tCerXerO

The second column of [BE2] is given by

ib ]} 1

(4.115)


103

where

X ero and XNO are the operating values of vectors Xer and XN. It can be shown that the nonzero elements of [DAer] and [DZd are given by DAer(1,2) = -WB, DAer(2, 1) = WB DZL(l, 2) = XL, DZL(2, 1) = -XL Combined System Equations Equations (4.100) and (4.101) can be combined with Eq.(4.111) to give the total state equations. ATe can be expressed as (4.117) where

[CME] = [Cme 0 0] The final system equations can be expressed in the compact form (4.118) where x~ = (xk

x:W-]

The eigenvalues of the matrix [AT] indicate system stability at the operating point considered. Example 2 The IEEE FBM model is taken up for study. The data are given in Appendix A. The mechanical damping is assumed to be zero. The eigenvalues [AT] matrix are calculated for three cases, namely (i) No load (Pb = ·0, Qb = 0) (ii) Full load (Pb = 1, Qb = 0) (iii) Case (ii) with static exciter modelled with KA 200, TA 0.025 s The excitation system is modelled by

=

.

1

=

KAA\!t]

(4.119)

A\!t = Vdo AVd + Vqo AVq

(4.120)

AEJd = TA [-AE Jd vto

-

\!to

104


The system matrix [AT] is assembled as described above. The elements of the matrix are functions of the operating point (which needs to be computed from power flow analysis). The calculation of initial conditions of the system state variables is described in Appendix B. The eigenvalues of [AT] for the three cases considered are shown in Table 4.4. Table 4.4 Eigenvalues of [AT]

Full Load

(Pb=1.0)

-4.712 ± j 616.62 0.000 ± j 298.18 0.001 ± j 202.86 0.023 ± j 160.38 -2.899 ± j 136.93 0.046 ± j 127.18 0.020 ± j 99.71 -1.562 ± j 10.42 -33.141 -20.288 -1.216 -0.966

-

-4.712 ± j 616.64 0.000 ± j 298.18 0.000 ± 202.79 0.032 ± j 160.26 -2.981 ± j 136.69 0.080 ± j 127.27 0.057 ± j 99.99 -0.631 ± j 11.23 -32.943 -20.371 ·3.883 -0.268

-

I Full load

with AVR

-4.707 ± j 616.67 0.000 ± j 298.18 -0.001 ± j 202.80 0.027 ± j 160.26 -2.396 ± j 137.11 0.053 ± j 127.26 0.003 ± j 99.93 -0.120 ± j 11.56 -38.903 -20.289 -2.820 -19.920

± j 21.59

I

Comments Supersyn. Mode Torsional Mode 5 Torsional Mode 4 Torsional Mode 3 Subsyn. Mode Torsional Mode 2 Torsional Mode 1 (Swing) Mode 0

Exciter Mode

It is interesting to observe that while the network modes (particularly the subsynchronous) are damped, the modes 1 to 4 are undamped due to TI. The effect of full load is to marginally increase the negative damping of the critical torsional modes. However, the mechanical damping is least for the no load condition. Interestingly, the effect of the AVR is to marginally reduce the negative damping of the torsional modes while reducing significantly the damping of mode zero or the low frequency swing mode. This latter effect is well known and will be discussed in chapter 5. To summarize the results, it is adequate to study TI at no load conditions, neglecting AVR as this will not only simplify computations, but also represents the worst conditions as the mechanical damping is the least for the no load conditions.

Effect of Series Compensation The effect of series' compensation is considered by varying the compensation level from 10% to 90%. The eigenvalues of [AT] are computed for the no load conditions. The network and the torsional modes are shown in Table 4.5 and 4.6 for different values of xc. (from 0.05 to 0.45 pu) excluding for Xc = 0.35, which is already given in Table 4.4.

105


Table 4.5 Network and Torsional Modes xc=0.05

xc=O.lO

xc=0.15

xc=0.20

-4.532 ± j 467.47 -4.071 ± j 286.42 0.000 ± j 298.18 -0.011 ± j203.07 -0.013 ± j 160.62 -0.055 ± j 126.97 -0.031 ± j 99.15 -1.011 ± j 8.45

-4.587 ± j 505.02 -3.900 ± j248.77 0.000 ± j 298.18 -0.003 ± j 203.13 -0.012 ± j 160.63 -0.055± j 126.97 -0.031 ± j 99.19 -1.075 ± j 8.70

-4.624 ± j 533.83 -3.788 ± j219.70 0.000 ± j 298.18 0.059 ± j 203.32 -0.010 ± j 160.66 -0.054 ± j 126.97 -0.030 ± j 99.23 -1.146 ± j 8.98

-4.652 ± j 558.11 -3.880 ± j 196.24 0.000 ± j 298.18 0.331 ± j 202.39 -0.001 ± j 160.72 -0.054 ± j 126.98 -0.029 ± j 99.29 -1.228 ± j 9.28

Table 4.6 Network and Torsional Modes xc=0.25

xc=0.30

xc=0.40

xc=0.45

-4.675 ± j 579.50 -3.420 ± j 174.12 0.000 ± j 298.18 0.004 ± j 202.82 0.070 ± j 160.93 -0.053 ± j 126.99 -0.027 ± j 99.37 -1.323 ± j 9.62

-4.695 ± j 598.84 -3.553 ± j 155.54 0.000 ± j 298.18 -0.009 ± j 202.90 0.445 ± j 159.93 -0.049 ± j 127.Q2 -0.021 ± j 99.50 -1.433 ± j 10.00

-4.726 ± j 633.17 -2.682 ± j 120.11 0.000 ± j 298.18 -0.014 ± j 202.95 -0.008 ± j 160.47 0.038 ± j 126.72 0.075 ± j 100.28 -1.718 ± j 10.91

-4.740 ± j 648.71 -5.154 ± j 103.00 0.000 ± j 298.18 -0.015 ± j 202.96 -0.012 ± j 160.50 -0.047 ± j 126.88 3.030 ± j 101.59 -1.910 ± j 11.48

It is interesting to observe that whenever the subsynchronous network mode frequency is close to one of the torsional modes, the the corresponding torsional mode is destabilized. This is an illustration of the phenomenon of mode coupling discussed in chapter 1. It can also be observed that mode 1 is the most negatively damped torsional mode and this occurs when Xc = 0.45 (90 % compensation level). In general, it can be said that the lower torsional modes are the most affected at high compensation levels. Often, reducing the compensation level can solve the problem of TI, but this is not an economic solution. Another point worth noting from the eigenvalue results is that the mode five is not affected as its modal inertia is very high. Synchronizing and Damping Torque Analysis

The variation of the synchronizing torque coefficient Ts m , which is defined as (4.121) Tsm{w) = -~Im[Ym(jw)l WB

as a function of w is shown in Fig.4.6. Note that Ym{s) is defined from Eq.{4.28). In general, Zm{s) is obtained from (4.122) where

[CSM] = [0 I][CM]

106


400 200 E rn

I-

0

-200 -400 50

100

150

Figure 4.6.

200

Win Radls

250

300

350

400

Variation of TSm

In deriving Eq.(4.122) it is assumed that the prime mover model is neglected in deriving the Eqs.(4.100) and (4.101) and only state variables corresponding to the rotor system are considered. The zero crossing of TSm (w) occurs at the frequencies corresponding to the torsional modes. While this assumes that the interaction with the electrical system is ignored, it can be shown that the consideration of TSe does not significantly affect the frequencies. Incidentally, the modal inertia (Hi) for ith mode can also be determined from

Hi = _ WB dTs m (w = wd 4Wi

dw

(4.123)

where Wi is the modal frequency. It is observed from FigA.6 that the pole and zero near the frequency 298 rad/sec. are practically indistinguishable. One could say that due to the pole zero cancellation, the mode 5 is not observable in the electrical system. The computation of TSe and TDe are carried out from the knowledge of Ye(s) which can be based on either simplified machine model or detailed model. In simplified machine model, the flux decay and damper windings are neglected and the stator is represented by a speed dependent "

II

source behind a constant inductance (x~WB +Xq). The variations of TSe and TDe based on detailed machine model(2.2) are shown in Figures 4.7 and 4.8 respectively. It can be seen from FigA.7 that TSe has much smaller magnitude than TSm and hence the torsional mode frequencies are unaffected even in the presence of TI. The variations of the damping torque coefficient TDe computed for the simplified machine model is shown in FigA.9. It is observed that the simplified model gives a fairly accurate value of the negative damping introduced by the network corresponding to the frequency f = fo - fer, where fer is the series resonance frequency.


107

15 10 5 Q)

fJj

I-

0 -5 -10 -15 0

50

100

Figure 4.7.

150

200

Win radls

250

300

350

400

350

400

Variation of TSe [Machine model (2.2)]

150 100 Q)

0 I-

50

r

0 -50 0

V

50

100

Figure 4.8.

150

200

Win rad/s

250

300

Variation of TDe [Machine model (2.2)]

Example 3 The system considered here is taken from IEEE SBM the data for which is given in Appendix A. The rotor shaft has four masses. In this system. only one of the two parallel lines is series compensated. There are three torsional modes having frequencies - 154, 203 and 321 rad/sec. The eigenvalues corresponding to torsional and network modes for the series compensation level of 55 % of are shown in Table 4.7.

108


r

-10

I

-20 Q)

o

-30

1--40 -SO -60 -m~--~----~----~--~----~--~~---~I--~

o

SO

100

Figure 4.9.

1SO

200 Win radls

250

300

3SO

400

Variation of TDe [Simplified Machine model]

Table 4.7 Eigenvalues for IEEE SBM Eigenvalue

Comments

-0.0492 ± j 321.0514 -0.0429 ± j 203.3862 0.4492 ± j 154.9207 0.0899 ± j 8.4796 -15.5494 ± j 605.4505 -15.4192 ± j 148.7232

Torsional Mode 3 Mode 2 Mode 1 Mode 0 Supersyn. Mode Subsyn. Mode

It is observed that mode 1 is destabilized due to series compensation. Mode zero is destabilized due to the excitation system (IEEE Type I) considered with KA 400, TA 0.02 s, KE 1.0, TE 0.8, Kp 0.03, Tp 1.0

=

4.5

=

=

=

=

=

AN ALGORITHM FOR COMPUTING TORSIONAL MODES

It was mentioned previously that Eq .(4.31) can be used to compute system

eigenvalues. Often ,it is adequate to compute the eigenvalues corresponding to the torsional modes which can be destabilized by TI with series compensation, HVDC and FACTS controllers. The effect of the electrical system on the mechanical system is depicted in the block diagram shown in Fig 4.10. Here, the transfer function of the mechanical system is G(s) defined as,

G(s) = Zm(s) = -CSM [s1 - AMr l [BMJ

(4.124)

The transfer function H(s) of the electrical system is given by

H(s)

= Ye(s) = CME [s1 -

AEr l [BEM]m(s)

(4.125)

109


6.Tm

+

Mechanical System

E

6.Te

Electrical System

Figure 4.10. tems

Block diagram showing interaction between electrical and mechanical sys-

From Fig 4.10, the poles of the closed loop system satisfy the equation 1 + G(s)H(s) = 0

(4.126)

This is equivalent to the equation

F(s) = Ym(s)

+

Ye(s) = 0

(4.127)

This is a nonlinear equation which has to be solved by any convergent iterative method.Newton's method has quadratic convergence and is expected to provide solution with minimum number of iterations. According to this method, the algorithm for computing the estimate of the eigenvalue at the end of the kth iteration is given by (4.128)

where Sk-l is the estimate at the beginning of the iteration. F' is derivative of F with respect to s and has to be updated for each iteration. Since the eigenvalues corresponding to the torsional modes are close to the imaginary axis and also since the variations in the imaginary part of Ym (jWi) are quite large compared to the variations in Ye(s), it is possible to approximate the magnitude of F'(s) corresponding to any torsional mode i as (4.129)

110


The above equation follows from Eq. (4.123).The advantage of this equation is that the derivation of F(s) remains constant at all iterations and requires no additional computations. The efficacy of this algorithm is checked by considering the example of IEEE FBM, the results of which are given in Table 4.4. The initial values of the eigenvalues are taken to be the eigenvalues of the matrix [AM l, except for mode zero. For the latter,the initial value is assumed to be 6.28 rad/sec. The eigenvalues at the end of each iteration for the case (a) Pb = 0.0 are given in Table 4.8. Table 4.8 Torsional modes at each iteration (Pb=O.O) Initial Value

Iteration 1

Iteration 2

Iteration 3

0.0000 ± j 298.18 0.0000 ± j 202.85 0.0000 ± j 160.52 0.0000 ± j 126.99 0.0000 ± j 98.72 0.0000 ± j 6.28

0.0000 ± j 298.18 0.0009 ± j 202.86 0.0235 ± j 160.38 0.0386 ± j 127.17 0.0177 ± j 99.70 -3.1761 ± j 10.21

0.0000 ± j 298.18 0.0009 ± j 202.86 0.0237 ± j 160.38 0.0451 ± j127.18 0.0204 ± j 99.71 -1.5825 ± j 10.57

0.0000 ± j 298.18 0.0009 ± j 202.86 0.0237 ± j 160.38 0.0465 ± j 127.18 0.0205 ± j 99.71 -1.5423 ± j 10.43

The results for the case (b) Pb = 1.0 are given in Table 4.9. Table 4.9 Torsional modes at each iteration (Pb=1.0) Initial Value

Iteration 1

Iteration 2

Iteration 3

0.0000 ± j 298.18 0.0000 ± j 202.85 0.0000 ± j 160.52 0.0000 ± j 126.99 0.0000 ± j 98.72 0.0000 ± j 6.28

0.0000 ± j 298.18 0.0006 ± j 202.79 0.0314 ± j 160.26 0.0612 ± j 127.24 0.0510 ± j 99.96 -1.6845 ± j 12.85

0.0000 ± j 298.18 0.0006 ± j 202.79 0.0319 ± j 160.26 0.0751 ± j 127.26 0.05686 ± j 99.98 -0.6563 ± j 11.30

0.0000 ± j 298.18 0.0006 ± j 202.79 0.0319 ± j 160.26 0.0790 ± j 127.27 0.05719 ± j 99.99 -0.6277 ± j 11.23

For both cases, it is observed the eigenvalues converge to the exact values in less than 5 iterations. This is in spite of approximating the derivative of F(s) in of the modal inertia. Remarks 1. This is a powerful algorithm and requires the evaluation of Ym and Ye at

each iteration. Computation of Ym (s) is relatively simple. The evaluation of Ye(s) can be complex, but the overall computations are still much less than what is required for the evaluation of the eigenvalues of [AT 1matrix. 2. Simplification of Ye(s) may be feasible to reduce the complexity. For example, the consideration of a simplified model of the generator may be adequate.


4.6

111

COUNTERMEASURES FOR SSR

The steady state problem should be avoided by proper planning. However it may not be possible to identify all critical operating conditions (which give rise to self excitation) while planning. The problem of transient torques is mainly solved by providing proper filtering (to limit the magnitudes of torques) and damping (with auxiliary devices).[IEEE (1980)] In addition to the devices required for damping SSR, it is also necessary to provide protective measures and relaying to monitor conditions that can lead to damage. 4.6.1

System planning considerations

There are several things that can be considered during system planning. These are listed below (i) Series versus Shunt Compensation The use of shunt compensators do not result in electrical resonant frequencies below synchronous frequency. Actually, it can be shown that resonance due to shunt capacitor is at supersynchronous frequencies. It can be shown, that supersynchronous frequency currents result in positive damping of torsional modes. As a matter of fact, the low (and resistive) network impedance at supersynchronous frequency, increases the damping. However, the shunt compensation cannot completely replace series compensation. The use of series compensation tends to be economical and more flexi ble. The location of series capacitors in a line is not critical (com pared to that of shunt compensation which should be located at the midpoint of the line for maximum effectiveness). It is advantageous to use both series and shunt compensation where possible. The controllable shunt compensation (using SVC) not only helps to regulate the line voltage but also reduce the level of series compensation required. (ii) Turbine-Generator Modifications There are limitations on what can be achieved here. For example, it is impractical (due to constraints on shaft and bearing size) to design machines for which the lowest torsional (mode) frequency is greater than synchronous frequency. The torsional mode frequencies of the turbine-generator can be varied only within small limits. However, this has not much impact on the SSR problem as changes in the network due to future growth/line outages affect the critical electrical resonance frequency. Another factor that has a critical bearing on SSR problem is the mechanical damping of torsional modes. It would be desirable to increase this damping but is considered as impractical. Another means of limiting shaft torques is through the use of slip couplings which has not been investigated. Pole-face Amortisseur Windings can be added to reduce the net negative resistance of the generator at subsynchronous frequencies (to control the induction generator effect). It is relatively inexpensive to install pole-

112


face amortisseur windings in new machines. However, retrofitting on old machines is not feasible. Also, the windings are not effective to control torsional interaction and transient torques. (iii) System Modifications The steady state SSR problem is mainly due to the coincidence of electrical resonant frequency (fer) with the complement of a torsional mode frequency (fo - 1m). it is thus possible to adjust the series compensation to avoid this coincidence. As the problem is more severe at lower frequency torsional modes, reducing the level of series compensation helps. This can also be done during system operation by bying some capacitor segments. It is also possible to have coordinated series capacitor use with loading. As the mechanical damping decreases with the load, (power output of the generator), it is practical to insert capacitors only when the loading exceeds the minimum loading. As series capacitors are mainly required at higher loadings, this procedure of coordinating the capacitor use with load is consistent with the dual objectives of improved stability limit and avoidance of steady state SSR problem. Edris (1990) has proposed a fully ive scheme consisting of inductive and capacitive elements, that can reduce the potential of SSR.The compensation is carried out phase-wise by three different combinations of inductive and capacitive elements. The three combinations have identical impedances at the operating frequency,but unequal impedances at other frequencies. This implies that synchronous currents have a negative sequence component and the net torsional interaction is reduced. (iv) Series Capacitor Protection and Reinsertion It is not economically feasible to use capacitors that can withstand fault currents. Thus, it is necessary to protect the capacitor banks against overcurrents (and voltages) by bying them. In earlier schemes of protection, a spark gap flashes over as the voltage crosses a predetermined level. However, from system stability considerations, it is also necessary to reinsert the capacitors within a few (3 to 5) cycles after fault clearing. This can be done by opening the by switch. This process of reinsertion of capacitors produces transient voltages across the capacitors. When these voltages exceed the spark gap setting, the gap may reflash resulting in failure to reinsert the capacitors. The setting of the spark gap determines the protective level of the capacitor. While higher gap setting is desirable for successful reinsertion, lower setting is necessary to reduce the transient torques during faulted period. A dual gap (or extended range dual gap) protection scheme helps, to achieve both objectives of capacitor by at lower fault levels (even for remote system fault) and successful reinsertion. In recent schemes, the capacitor overvoltages are limited by using a highly nonlinear metal oxide resistor in parallel with the capacitor. (see Fig. 4.11). The resistor is sized such that very small currents flow through it during normal capacitor voltages. During faults, the resistor provides parallel path for the


113

SERIES CAPACITOR

NONLINEAR RESISTOR

REACTOR

Figure 4.11.

Series capacitor protection

current and the voltage across the capacitor reaches a saturation level. Thus, the capacitor is automatically byed and reinserted without any hitch. A case study showed that a 15 MW-sec nonlinear resistor applied for 5 cycles reduces the capacitor voltage from 3.5 to 2.2 pu with a corresponding reduction in the peak shaft torque from 3.7 to 1. 7 pu.

4.6.2 Filtering schemes Static Blocking Filter (SBF}[Farmer et al (1977)] This is inserted in series with the generator step-up transformer winding on the neutral end of the transformer high voltage winding. (see Fig. 4.12) It can also be placed on the high voltage side of the transformer winding. The SBF is a three phase filter made up of separate filters connected in series. Each section of the filter is a high Q, parallel resonant circuit tuned to block electric currents at (complementary) frequencies corresponding to each of the torsional mode. Thus, this filter provides an excellent transient torque control if properly tuned. The static blocking filter was first installed at Navajo Generating station in Western U.S.A. in 1976. Static blocking filter can provide solution to both steady state and transient torque problems. Since each filter is tuned to protect an individual generating unit, the filter performance is not much affected by system changes. However, the tuning of the filter is affected by changes in the system frequency and temperature sensitivity of the filter capacitor. A disadvantage of the filters is that they take up space in the switch yard. Also there is the requirement of increased insulation level of the generator transformer. By Damping Filter The by damping filter is connected in parallel across the series capacitor in each phase (see Fig. 4.13). The filter is useful for countering induction generator effect as it can introduce significant positive resistance in the circuit for subsynchronous oscillation frequencies up to 90% of the system frequency. However, the effectiveness of the filter reduces at the higher frequencies. The damping resistor is connected in series with a parallel combination of a reactor and capacitor which is tuned at the system frequency. Thus the filter

114

ANALYSIS OF SUBSYNCHRONOUS RESONANCE IN POWER SYSTEMS TRANSFORMER NEUTRAL

CURRENT LIMmNG REACTOR

THYRlTE

BY SWITCH

Figure 4.12.

Static Blocking Filter

DAMPING RESISTOR

,-------'\

SERIES CAPACITOR

Figure 4.13.

By Damping Filter

has a very high impedance at the system frequency and the power losses in the resistor are limited (under normal conditions). The damping resistor becomes effective at subsynchronous frequencies. The damping filter can be expensive at high voltage levels but in distribution or subtransmission circuits where series capacitors are used (e.g. in resonant link fault current limiting circuits) the filter has been applied.

4.6.3 Damping schemes


v,

l-

Vrcf

+~ ~~----~

v,

115

AVR

+

Figure 4.14.

SEDC modal control structure

Supplementary Excitation Damping Control(SEDC)[Bowler and Baker(1981)] This is an extension of the PSS concept applied for damping torsional frequency oscillations. It is well known that PSS used for damping low frequency oscillations is deleterious for torsional (mode) oscillations. This arises from the fact that the net phase shift between the input speed signal and the electromagnetic torque is such that the damping torque is negative at torsional frequencies. For damping torsional modes, it is also necessary to utilize the concept of observability. The speed signal can be measured from both ends (a)from the rotating exciter and (b) the front (standard) end at H.P. turbine. Although all modes are theoretically observable at each rotor location, from a practical viewpoint, it is advisable to use both control signals (from the two measurements). Also it is impractical to use a single dynamic compensator for damping all the the modes. The control objectives are met by parallel processing each mode as shown in Fig. 4.14. Each mode is isolated from the relevant speed signal measurement, by band filtering at each torsional frequency. The necessary phase and gain compensation are provided individually for each component mode and the output of SEDC is obtained by summing the control signal for all the modes. The phase compensation designed at Navajo generating station in 1975, utilized two first order phase delay networks to allow compensation within ±240°. Such a large variation was found necessary to for the variability of phase produced by a band filter around its centre frequency and lack of knowledge of this frequency at the time of its manufacture. The band filter was designed with a sharp cutoff to eliminate interference between adjacent signal channels. A desirable feature in the design of the SEDC is its applicability

116

ANALYSIS OF SUB SYNCHRONOUS RESONANCE IN POWER SYSTEMS

~~r ~\~ o

0.2

0.4

0.15

;! ~>~ j

0.8

1

1.2

1."

1.6

1.8

2

(a}Fundamental frequency combined with DC

(b}Fundamental frequency combined with subsynchronous frequency Figure 4.15.

Effect of presence of DC and subsynchronous components

to turbo-generators of different design. The SEDC tuned for different centre frequencies (of the band filters) are required for different generators. The rating of SEDC required while acting alone, to overcome TI and avoid self excitation at Navajo was found to be equal to the rating of the unit. Hence, the SEDC was designed to supplement the Blocking Filters which were designed as the main countermeasure for SSR problem (both steady state and transient). The use of SEDC improves the margin of torsional stability at the extremes of operating condition. Tests carried out at Jim Bridger generating station showed that SEDC with optimized phase settings (for resonant conditions) provide greater damping as resonance is approached. NGH Damping Scheme This scheme was proposed by N.G. Hingorani to damp subsynchronous frequency currents in transmission lines and was first installed at Southern California Edison's Lugo substation in U.S.A. in 1984.[ Hingorani (1981), (1987}) The basic concept of NGH-SSR damping scheme can be explained with reference to Fig. 4.15. If a sinusoidal voltage of frequency fo is combined with a dc voltage, it is seen that, for the combined voltage, alternate half cycles are longer than the nominal half cycle period of ~. If a sinusoidal voltage of frequency fe < fo is considered instead of the dc voltage, again it is observed that the combined voltage has some half cycles which are longer than ~. Similarly any combination of a base signal of frequency fo with dc and subsynchronous frequencies would result in some half cycles longer than the nominal half cycle period. The voltage across the series capacitor is a combination of fundamental frequency, dc and subsynchronous frequency components. The basic principle of NGH damping scheme is to dissipate capacitor charges whenever the measured


117

SET TIME

RESISTOR

CAPACITOR VOLTAGE

SERIES CAPACITOR

Figure 4.16.

Basic NGH - SSR damping scheme

half cycle period exceeds the nominal value. This is done by inserting a resistor across the capacitor through thyristor switches (see Fig. 4.16). The thyristor stops conducting whenever the capacitor voltage (and the thyristor current) reaches zero. Thereafter, the measurement of half cycle period restarts from a new voltage zero. No thyristor fires for half cycles which are shorter than the set period. Two thyristors are needed for the two polarities. For high voltages, the thyristor shown in Fig. 4.16 actually represents a series string of thyristors. The resistor value is not critical although lower its ohmic value, the more effective it is (except when it is too low). It is recommended that the resistor's ohmic value within 5 to 10 percent of the capacitor ohmic value will give satisfactory results. The operation of the thyristor controller is independent for each phase. The controller is simple and does not require detection of specific subsynchronous signals. The control signal can be obtained at the platform level. Alternately, it can be generated as the difference of two voltage signals from the measuring devices (on each side of the capacitor) ofthe voltage with respect to ground. In this case, the control circuitry could be located at the ground level and firing pulses transmitted through optical links. The objectives of the NGH damping scheme are a) reduce transient torques b) suppress steady state (self excitation) SSR problem c) suppress offset of series capacitors d) protect the series capacitors The scheme may be designed for any or all of the above purposes. If transient torque control is of major concern, the set period can be larger than the nominal half cycle period so that in steady state (and with small perturbations) the thyristors will not fire. The thyristors will fire only during large disturbances such as faults followed by clearing. If steady state SSR problem is of major concern, the set period can be slightly less than the nominal half period; then the thyristor will conduct during steady state at the tail end of each half cycle

118


of the capacitor voltage; This will provide detuning effect against gradual build up of oscillations. There will be continuous power loss in this case, but is very small and of minor consequence. The thyristors can help to protect the capacitors (and themselves) by firing if the instantaneous forward voltage exceeds a set level. The resistor limits the discharge current from the capacitor. The capacitor and thyristor protection can also be implemented by employing nonlinear zinc oxide resistors across the thyristors. The protection level of the zinc oxide resistor can be selected such that it (in series with the linear resistor) provides adequate protection for the capacitor. The by switch helps to relieve the thyristor/resistor when their duty exceeds safe limits. The by switch is also required when capacitor is to be byed and reinserted under normal operating conditions. NGH scheme is a ive scheme-it does not require signals. The requirement of thyristors are well within the state of the art. For example, a 1000 A, 30 ohm capacitor may require a 80 kV thyristor string with a resistor duty of 20-50 MJ per phase. The studies carried out on NGH scheme indicate that it is an effective countermeasure for both steady state and transient SSR problems. However it was found that some undamping can result for torsional modes which are 'off tune' (not in resonance with the electrical system). It is suggested that the use of SEDC can help to overcome this problem. In summary, the studies show that NGH scheme with appropriate SEDC is beneficial in applying series compensation over a wider range (0-75%) whereas SSR problem limited the compensation level to 15%. Dynamic Stabilizer [Ramey et al (1981)] This is a device based on the use of a modulated shunt reactance connected to the isolated phase bus of a turbine-generator unit. The modulation of the reactance is done with the help of thyristor control which acts in response to a signal based on the generator rotor velocity deviation. See Fig. 4.17. The shunt susceptance of the TCR (Thyristor Controlled Reactor) is given by the relation

B (rr)

rr - sinrr = ---rrXL

(4.130)

where rr is the conduction angle, XL is the reactance (at fundamental frequency) of the shunt reactor. The conduction angle rr is related to the firing angle a by the relation (4.131) rr=2(rr-a) The firing angle a is modulated around a quiscent point in response to the oscillation of the generator rotor. In the absence of the rotor oscillations, the dynamic stabilizer appears as a continuous reactive load. The design of the dynamic stabilizer is such that it generates sufficient current of appropriate phase to compensate for the critical subsynchronous frequency currents in the generator armature due to the network resonance. The interesting thing to note about the operation of the dynamic stabilizer is that under the worst (resonant) conditions, even a small modulation in the


119

UNIT TRANSFORMER

G~~T_O_R____- r______________- .______--<~~

I

~

I

STABILIZER TRANSFORMER

FIRING ANGLE CONTROL

Figure 4.17.

Dynamic stabilizer

reactor current results in large oscillating current in the generator armature due to amplification occuring at resonance. The amplification would be maximum if the stabilizer is located at a point in the system when all inductive reactance is on one side of stabilizer while all the capacitive reactance is on other side. The advantages of dynamic stabilizer are 1. The connection as a shunt device eliminates the requirement to carry continuous generator load current (as compared to blocking filters) 2. The operation is not sensitive to variations in the system frequency or ambient temperature. 3. Maintenance requirements are expected to be minimal. The disadvantageous of dynamic stabilizer are 1. It does not provide protection against induction generator or transient torque

problems. The damping is provided only when rotor oscillations are present. 2. Stabilizer introduces harmonic currents which can be minimized by a) connecting the thyristor controlled reactors (TCR) in delta and b) providing two secondary windings, one connected in delta and other connected in star to eliminate 5th and 7th harmonics (This arrangement is termed as 12 pulse operation of TCR). The first dynamic stabilizer was installed at San Juan generating station. This is a 12 pulse TCR and has short time rating of about 20% of that of the generating unit.

120


The dynamic stabilizer is essentially similar to the Static Var Compensator (SVC). The only difference is that SVC also has fixed or thyristor switched capacitor and voltage control in addition to any auxiliary control. The damping of torsional oscillations can also be done by providing an auxiliary (supplementary) controller with appropriate design. As the SVC is normally installed at a location away from the generator, the required control signal can be synthesized from locally measured variables of line currents and voltages. The generator rotor velocity can also be sensed by computing the frequency of the (internal) generated voltage. [Padiyar and Varma (1990)] 4.7

TORSIONAL OSCillATIONS IN PARAllel CONNECTED TURBINE GENERATORS

In the analysis of SSR,only one (equivalent) turbine - generator (T-G) is considered.However when there are parallel connected T-G sets,it is not accurate to consider a single equivalent generator as the magnitudes of shaft torques are influenced not only by the common (or in-phase) modes of torsional oscillations,but also by the antiphase modes.lf there are N identical T-G sets opearating in parallel and each T-G set has M rotor masses,there is one set of M (common) modes in which the corresponding rotors in each unit oscillate in phase and have the same amplitudes. These modes are excited by disturbances external to the T-G sets. There will be (N-l) sets of identical M (antiphase) modes representing intermachine oscillations [Alden et al (1977)]. These are excited by disturbances within the N T-G sets such as unequal loading of the generators. If the T-G units are not identical,the antiphase modes will have a spread in frequency. In general,the common mode has a lower frequency compared to the antiphase mode. The difference in frequency is significant for mode '0' (or swing mode). The antiphase mode corresponding to mode '0' has been termed as intraplant mode in studies involving low frequency oscillations.Normally,intraplant modes do not pose stability problems but if speed input PSS is used,there have been instances of PSS destabilizing an intra plant mode.lt has been suggested by Schlief et al (1979) to utilize a control signal containing only common mode component of oscillation. The difference in the frequencies of the common and antiphase modes corresponding to torsional oscillations is negligible. However ,the antiphase modes have to be considered in the damping of torsional oscillations and design of control schemes for this purpose [Iravani (1989}].Taking speed signal from one T-G set as the control signal for an auxiliary SVC control may destabilize the antiphase mode. The design of notch filters can also be a problem with non-identical T-G sets connected in parallel,because of the spread in the frequencies.

5

5.1

INTERACTIONS WITH POWER SYSTEM STABILIZER

INTRODUCTION

Modern power systems are affected by the problem of spontaneous low frequency oscillations particularly when operating under stressed system conditions associated with increased loading on long transmission lines. These oscillations are due to swings of the generator rotors caused by disturbances. It is to be noted that random load fluctuations are always present and if the damping of the swings is inadequate (or even negative) the rotor oscillations are observed in the transmission lines as power oscillations which can limit the power transfer and threaten system security. The frequency of these oscillations usually lies in the range of 0.2 to 2.0 Hz. It can be shown that a N generator system has (N-l) modes of rotor swings. The lower the frequency of oscillations, more generators participate. At the lower end of frequency spectrum(0.2-0.8 Hz) the swing modes are termed as interarea modes. At the higher end (1.5-3.0 Hz) of the frequency spectrum, they are termed as intraplant modes and the oscillations are limited to the generators within a plant. In the intermediate range of frequencies (0.8-1.8 Hz) lie the local swing modes corresponding to oscillations among generators within an area. It is assumed that for the local mode oscillations, the generators within a plant are coherent and swing against generators in another plant in the same area. The damping of intraplant modes is not a problem as the amortisseur circuits K. R. Padiyar, Analysis of Subsynchronous Resonance in Power Systems © Kluwer Academic Publishers 1999

122


or eddy current effects in the solid rotor of a turbogenerator are effective. It is mainly the local or interarea modes which are not adequately damped. The problem is aggravated by the high gain AVR's and fast acting excitation systems using thyristor-controlled rectifiers. Such high performance excitation systems have become the norm in present day power systems as they are considered to be essential for maintaining steady state and transient stability of modern synchronous generators, apart from providing fast control of the terminal voltage. However, these excitation systems can result in negative damping of the swing modes. This problem has been extensively investigated. The major factors that contribute to the oscillatory instability are (a) loading of the generator or tie line (b) power transfer capability of transmission lines (c) power factor of the generator and (d) AVR gain. A cost effective and satisfactory solution to the problem of undamped low frequency oscillations is to provide Power System Stabilizers(PSS) which are supplementary controllers in excitation systems. The PSS has an input signal (u) derived from rotor speed, bus frequency, electrical power or a combination of these variables. The output of PSS (V.) is added to the summing junction at the input of AVR. The main objective of PSS is to provide additional damping torque without affecting the synchronizing torque at the critical oscillation frequencies. It was observed [Watson and Coultes (1973)] that torsional vibrations can occur on large thermal units using stabilizing signals based on shaft speed measurement. The problem was initially solved by locating the speed transducer at the location near the node of the first torsional mode (which was responsible for TI). However, this is not a general solution. Larsen and Swann (1981) have suggested the use of properly designed torsional filters for this purpose. The design of filters must ensure a sufficient margin of safety for torsional interaction, while at the same time have minimum phase shift at the low frequencies for good stabilizer performance. The design of the torsional filter is dependent on the nature of the PSS input signal. Speed signal results in maximum TI while power signal results in minimum TI. But power input stabilizers cause adverse effects on voltage during rapid load changes. The selection of the torsional filter parameters can be problematic as the filter introduces an additional mode (of oscillation) which can be destabilized. The present trend is to simplify the PSS design by choosing a composite input signal derived from the rotor speed and electric power. In this chapter, the TI with PSS is investigated in detail. The study is illustrated by an example adapted from the IEEE FBM. 5.2

BASIC CONCEPT IN THE APPLICATION OF PSS

The requirement for the application of PSS is best illustrated with reference to a single machine system shown in Fig.5.1. Considering model (1.0) for the synchronous machine, (neglecting damper windings) the linearized model ofthe system is shown in the block diagram given in Fig 5.2

123


Xe

\!tLB

C9

I

(\f\(\(\

Figure 5.1.

Figure 5.2.

Re MI\

~EbLO

A single machine system

Block diagram of the linearized system shown in Fig.5.l

In deriving the block diagram of Fig 5.2, the stator and network connected to the generator are assumed to be in steady state. The constants Kl to K6 are termed as Heffron Phillips constants and are dependent on the operating point. If the losses in the stator and network are neglected, (Re = 0) then the expressions for the Heffron Phillips constants are given below.

(5.1 )

124


(5.2) (5.3)

(5.4) [{5

X~ VqoEb sin 00 (xe + x~)vto

= _ Xq VdoEb cos 00 (Xe T.'

+ Xq)vto

1i6

Xe = (xe+x

Vqo ) d vto I

(5.5) (5.6)

where the subscript '0' indicates the operating value of the corresponding variable and Eq is defined from

(5.7) It can be shown that

(5.8) It is to be noted that ido is usually negative based on the convention used here (that'd' axis leads 'q' axis). For the typical range of the operating point (xe > 0,0 < 00 < 90 0 ), it can be shown that the Heffron-Phillips constants are positive except for [{5. [{5 is positive for low values of 00 and tends to be negative as 00 is increased. Increase in Xe has the impact of reducing [{5. If a single time constant excitation system is assumed, then

EXC(s) If TA

~

J{A

= 1 +sTA

(5.9)

0, then the perturbation in the electrical torque is given by,

(5.1O) In deriving this equation, the damping (D) is neglected. If it is assumed that

then Eq.(5.10) can be simplified to

(5.11)


125

The synchronizing and damping torque coefficients, Ts and TD are defined by ~Te(jw) _ T

~J (jw) -

jw T

+ wB

s

(5.12)

D

From Eqs.{5.11) and{5.12) we get (5.13)

(5.14) where

For the system to be stable, oscillation frequencies. For Ts

Ts

and

TD

should be positive for all

> 0, the following conditions apply

(5.15) (5.16) These conditions are generally satisfied. For the damping torque coefficient TD > 0, the following condition applies (5.17) As mentioned previously, J{5 can be negative under higher loading on the generator and increased external reactance Xe. Under these conditions, the inequality(5.17) can be violated and the system will experience oscillatory instability. For large values of K A , (5.17) can be simplified to K5

>

°

(5.18)

Hence for AVR with large gain, low frequency oscillations are negatively damped when the constraint (5.18) is violated. Actually there is an upper limit on J{A when J{5 < and is given by

°

(5.19) It is not practical to reduce the AVR gain to satisfy (5.19) as the gain is normally determined from the requirement of voltage regulation. Hence, PSS is applied to ensure that the net damping torque remains positive even when K5 < 0.

126 5.3


DESIGN OF PSS

The objective of PSS is to provide additional damping torque to overcome the negative damping introduced by the fast acting, high gain excitation system. This is to be achieved without reducing the synchronizing torque. The most appropriate control signal for this purpose is the speed signal (t::.Sm ). This can be obtained from the speed transducer mounted on the rotor shaft or alternatively as the frequency of the induced voltage in the generator stator. The voltage induced in the stator can be synthesized from the measurements of the terminal voltage and armature currents. The design of PSS depends on the transfer function GEP(s) defined as

GEP(s) = t::.Te(s) t::.Vs(s)

(5.20)

From Fig.5.2, GEP(s) can be derived as (5.21)

The design of PSS can be explained wth reference to Fig.5.3. Here t::.Teo is the component of the electrical torque in the absence of PSS and t::.Tep is the torque component introduced by PSS. KJe(s) is the transfer function ~(W when t::. Vs = O. If PSS is to introduce only a damping torque, then the required transfer function of PSS(s) is given by

Dp PSS(s) = GEP(s)

(5.22)

where Dp is the required damping torque (coefficient) from PSS. Since this is not realizable in practice, a practical PSS assumes a transfer function T(S) with the following structure.

PSS(s)

sTw

= 1 + sTw T(s)

(5.23)

where T(s) is the transfer function of a dynamic compensator. A typical T(s) is defined by T(s) = Ks(l + sTd(l + sT3) (5.24) (1 + sT2)(1 + sT4) where Ks is the PSS gain, Tw is the washout time constant, T J, T 2, T3 and T4 are selected to provide appropriate phase compensation in the range of frequencies ( typically from 0 to 3.5 Hz). Since GEP(s) introduces phase lag, the PSS must be designed to introduce phase lead. Hence TJ > T2 and T3 > T4 . A single leadlag stage of the dynamic compensator is usually adequate when static exciter is used. The block diagram of a practical PSS is shown in Fig. 5.4. Here u is the input (control) signal. The objective of the washout circuit is to act as a high filter, preventing dc and very low frequency component. The value

127


Figure 5.3.

Block diagram with PSS

of Tw is not very significant. It is chosen in the range of 10 to 20 seconds to ensure that PSS can act on the lowest frequency swing mode corresponding to an interarea mode. The output of PSS is limited to ensure that PSS does not counteract the voltage regulator action of AVR. The purpose of torsional filter is to overcome the TI introduced by PSS. Larsen and Swann( 1981) formulated rules for the selection of PSS parameters. The phase compensation for PSS is designed from the following criteria. (a) The compensated phase lag (phase of P(s)=GEP(s)PSS(s)) should through 90° at the frequency around 3.5 Hz (for frequency input signal, this can be reduced to 2.0 Hz) (b) The compensated phase lag at the local mode frequency should be below 45°, preferably near 20°.

U

·1

WASHOUT

DYNAMIC

~

T(s)

l+sT w

1

CIRCUIT

Figure 5.4.

·1 COMPENSATOR

TORSIONAL

u, ·1 ""T{.J

I

~

I

/ V,

FILTER

Block diagram of a practical PSS

(c) The gain of the compensator at high frequencies (this is proportional to ~;~!) should be minimized.

128


The first criteria is important to avoid destabilization of the intra-plant modes with higher frequencies. The highest frequency is assumed to be below 3.5 Hz and hence the speed or power input stabilizer is designed to ensure damping for the intraplant modes. The (bus) frequency input signal does not contain any significant component of the intraplant modes and hence in this, the phase compensation can be designed to damp only local modes ( in addition to interarea modes). It is also preferable to have compensated phase to be lagging at interarea modes so that PSS provides some synchronizing torque at these frequencies. The time constant of the washout circuit also can also affect the compensated phase lag. The third criteria is required to minimize the noise amplification through PSS. The plot of the phase angle of (1 + sTd/(1 + sT2 ) is shown in Fig.5.5 as a function of the frequency for different values of the centre frequency Ie which is defined as (5.25 ) The phase angle (¢) shown in Fig. 5.5 is calculated for the value of n=2 where 'n' is defined as (5.26)

1e=1

20

5 10

10

o~~~~~~~~~~~~

1~

1~

Figure 5.5.

1~

1~

1~

1~

Variation of phase angle of compensator

It can be shown that the maximum phase angle ¢max

= tan

-1

n- 1

(2,fii)

¢max

is given by (5.27)

and occurs at the frequency I = Ie The studies carried out by Larsen and Swann indicate that depending upon


129

the input signal used, PSS is to be tuned for a particular system condition which has the highest stabilizer loop gain and phase lag. Full load on the generator yeilds the highest loop gain. The operating condition also depends on the external reactance (xe). It is claimed that for speed and power input stabilizers, the strongest system (with lowest value of xe) results in highest loop gain and phase lag. For frequency input stabilizers, the highest loop gain occurs with weakest system (highest value of xe) To select the dc gain of PSS (1<.)' a root locus analysis is performed. A typical root loci for the two critical modes are shown in Fig.5.6. Here, there are two possibilities. In the first case, (shown in Fig.5.6(a)) the swing mode becomes unstable as the PSS gain is increased. In the second case, another mode, termed as the exciter mode becomes unstable as the gain is increased. The exciter mode originates from the dynamics of flux decay and the excitation system.

JW

JW

SWING MODE

EXC. MODE

SWING MODE

o (a) Swing Mode instability

Figure 5.6.

o

(J'

(b) Exciter mode instability

Root loci with variation in stabilizer gain

The optimal PSS gain is chosen as the gain that results in maximum damping of the least damped mode. (swing mode or exciter mode). For input signals other than the rotor speed, the block diagram shown in Fig.5.3 is not valid. In such cases, the stabilizer loop is to be replaced by what is shown in Fig.5.7.

130


Figure 5.7.

Stabilizer loop with arbitrary input u

Su (8) is defined as the input signal sensitivity factor and F Bu (8) is defined as the input signal factor. For power input stabilizer, (5.28)

F Bp (8)

R::

G EP(s)

(5.29)

For the general case, the plant transfer function in the stabilizer path is given by Pu(s) ~Tep(8) Su(8) PSSu (8)GEP(8) (5.30) ~Sm (8) 1 - F Bu (8) P S Su (8)

=

=

For speed input stabilizer, Su = 1.0, F Bu = O. The design criteria is unchanged for a PSS with arbitrary input except that P(s) defined in Eq.(5.30) is used for the design of phase compensation. It is to be noted that in general, the PSS gain also affects the phase compensation unless F Bu (8) is small at the frequencies considered. 5.4

TORSIONAL INTERACTION WITH PSS

The PSS provides phase lead at low frequencies, much below the first torsional mode frequency. At the first torsional mode and higher frequencies, the PSS provides a constant gain of (KT~I The plant transfer function GEP(s) has increased phase lag at the torsional mode frequencies. Hence there is a danger of instability at one or more of the torsional modes due to reduced gain margin because of PSS. One of the solution to this problem of TI is to provide a torsional filter which is primarily a low filter. A typical second order filter has the transfer function

¥.).

FILT(8) =

82

2 Wn

+ 2(W n 8 + W~

(5.31)

where ( and Wn are two tunable parameters that have to be selected. The choice of Wn requires the knowledge of the first torsional mode frequency. Although the torsional filter provides attenuation at the torsional frequencies, it also contributes additional phase lag. Hence the design of the filter becomes critical. It has also been reported that torsional filter contributes new exciter mode(s) which can be destabilized by increasing PSS gain, thereby limiting the


131

damping of the swing mode for which PSS is applied. Lawson, Swann and Wright (1978) have proposed a band reject filter instead of the low filter.The objective is to attenuate the signal through the PSS at torsional frequencies in order to minimize TI. Analytical investigations led to the development of a band reject filter capable of providing attenuations greater than 100: lover the torsional frequency range, causing less than 20 0 of phase lag at local mode frequencies and less than 5° at inter area mode frequencies .The band reject is made up of five stagger tuned stages in cascade.Each stage has the transfer function given by

F ILT{s)

=

s2 s2

+ 2(l W n S + w; + 2(2W n S + w~

The depth of the notch characteristic and the bandwidth of each stage is determined by (1 and (2 . An alternative to provide a complex torsional filter is to design a PSS which provides attenuation at the torsional mode frequencies. This is feasible if a control signal is used which does not require phase lead (may even require phase lag) at the frequencies for which PSS is expected to provide damping. If acceleration signal is used, it already has a 90 0 inherent phase lead and PSS may be designed with a phase lag. If the mechanical power is constant, the acceleration is proportional to the negative change in the electric power. Hence it is to be expected that the PSS with power input is also immune to TI. However, as mentioned previously, the power input stabilizer provides spurious signals during changes in the generator loading (and mechanical power) and these can have an adverse effect on the bus voltage. The accelerating power signal can be synthesized from the speed and the power signals as shown in Fig.5.8.

Figure 5.8.

Synthesis of accelerating power signal

There are variations in the synthesis of PSS input signal based on the speed and power signals, In one instance, equivalent speed signal is synthesized by combining speed signal with the integral of the power signal [Lee et al (1981)]. It is claimed that with such PSS input signal, the torsional filter design can be simplified and is independent of the generator torsional characteristics.

132


5.5

A CASE STUDY

5.5.1

Description

To illustrate the Torsional Interaction (TI) with Power System Stabilizer (PSS), a case study of a single machine connected to infinite bus is presented. The system studied is adapted from the IEEE First Benchmark Model (FBM) consisting of a turbine generator feeding power to an infinite bus via series capacitor compensated system. (see Fig.5.9) The machine data are given in the Appendix A.

Figure 5.9.

One line diagram of a study system

The network data are assumed as RE 0.02, XE 1.0, Xc 0.5 The machine is represented by model (1.1) and the subtransient quantities are neglected. The system is operated with VI: 1.0, Pg 1.0, Eb 1.0

=

=

=

=

=

=

5.5.2 Modeling of the excitation system including PSS A rotating AC exciter with thyristor controlled rectifier is considered. The excitation system is represented by a single time constant model described by dEJd

1

-;It = TA [-EJd + KA(v"eJ - \It + v.)]

(5.32)

The transfer functions of the PSS is assumed to be

PSS(s) =

sTw K. (1 1 + sTw (1

+ sTt}(1 + sT3 ) + sT2)(1 + ST4)

(s2

w~

+ 2(w n s + w~)

(5.33)

The block diagram of the PSS identifying the state variables is shown in Fig.5.10. The state equations for the PSS (ignoring the limiter) are given below dV.

dt =wnX l d~

dt

= Wn [( -2(Xl

n~

+ X2) + T2T4 (I<.Sm

(5.34) ~

- X 4) + T4 X3 - v.]

(5.35)

133


v.

Figure 5.10.

dX 2 1 - d = ~[-X2 t.£4

dX 3

dt =

Block diagram of PSS considered

+ (1-

1 T2 [-X3 dX 4

dt =

T3

-)[X3 T4

+ (1 1

Tl

+ -(K.Sm T2

Tl

X 4 )]]

(5.36)

,

T2 )(!i.Sm - X 4)]

Tw [-X4

.

+ !i.Sm ]

(5.37) (5.38)

The equations (5.32), (5.34) to (5.38) can be linearized and expressed as

Xr = [Ar]xr

+ [Br]u e + [Brm]YM

(5.39)

Yr = 6.Efd = [Cr]x r

(5.40)

where

x; = [6.Efd u!

6. V. 6.X 1 6.X 2 6.X3 6.X4 ] = [6. Vd 6. Vq ], yk = [6.15 6.Sm ]

and the nonzero elements of the matrix [Ad are given below. Ar(l, 1) = -fA' Ar(1,2) Ar(2, 3) = W n, Am(3,2) = -Wn, Ar(3,3) = -2(w n Ar(3,4) = W n, Ar(3,5) = ¥.-wn Ar(3,6) = -~~~:Wn Ar (4, 4) = Ar (4, 5) = (1 -

=!k

i.,

i.

¥.-)

134


Ar(4,6)=-i.(1-¥.)R, Ar(5,5)=-i 2 , Ar(5, 6) = -i (1Ar(6,6) = -fw The nonzero elements of [Brl and [Brml and [Crl are given by - -T KA~ - -T KA~ B r (1 , 1) vto' Br(1, 2) V to A A Brm (3,2) =wnR¥.K., Brm (4,2) = r:}J<.(IBrm (5,2) = ~(1Brm (6,2) =~, Cr (l) = 1.0

R)'

¥.),

R)'

5.5.3

Results

The excitation system parameters are KA = 200, TA = 0.025 s. The PSS parameters were selected as Tw = 10 s, Tl = T3 = 0.1 s, T2 = T4 = 0.05 s, K. = 5, (= 0.5, Wn = 20 The eigenvalues of the linearized system are shown in Table 5.1. Table 5.1 System Eigenvalues

I Without

PSS

-3.307±j620.611 -1.904 ± j133.225 -1.850 ± j298.169 -0.364 ± j202.805 -0.641 ± j160.373 -0.009 ± 127.254 -0.206 ± j99.505 0.429 ± j9.688 -23.313 ± j20.642 -3.135

PSS without torsional filter -3.303 ± j620.616 -2.155 ± j132.774 -1.850 ± j298.169 -0.322 ± j202.770 -0.619 ± j160.279 0.150 ± j127.449 0.218 ± j100.253 -1.376 ± j9.116 -15.731 ± j22.554 -3.111 -37.172 -15.200 -0.100

I PSS

with torsional filter

-3.307 ± j620.612 -1.900 ± j133.236 -1.851 ± j298.169 -0.364 ± j202.806 -0.641 ± j160.374 -0.012 ± j127.249 -0.217 ± j99.471 -2.593 ± jlO.062 -23.316 ± j24.953 -3.107 -31.666 -14.942 -0.100 -3.675±j15.269

I Comments I Supersyn. Subsyn. Mode 5 Mode 4 Mode 3 Mode 2 Mode 1 Mode 0 Exciter

It is seen that without PSS, the zeroth mode (swing mode) of frequency 9.688 rad/sec. is destabilized. The PSS without the torsional filter resulted in damping the swing mode but destabilizing the first two torsional mode. The torsional filter is able to damp the torsional modes although marginally. In any case, the presence of the torsional filter overcomes the destabilizing action of the PSS. The parameters of the filter, ( and Wn are to be carefully selected as they affect the stability of the swing mode and an exciter mode.( < 0.2 resulted in instability of an exciter mode. The increase in ( improved the damping of both exciter and swing modes. However for ( > 0.5, the damping of the swing mode starts to decrease. Keeping ( = 0.5, Wn was varied to study its impact on the exciter and swing modes. For Wn < 15, the exciter mode was found to be unstable. Application of a composite signal


135

Instead of using a torsional filter, the application of a composite signal consisting of speed and acceleration is tested. To simplify the analysis, in the diagram of PSS shown in Fig.5.4, the torsional filter is replaced by the block shown in Fig.5.1l. TF2 was selected as 0.001 s. TFI corresponds to the ratio of the acceleration to speed signal.

u-l--------~·IL-___l_lt_:_~_;_~__~~--------v.-$ Figure 5.11.

Transfer function of block replacing torsional filter

The increase in TFl resulted in damping of the torsional modes 1 and 2. However, there was a reduction in the damping of modes 0, 3 and 4. Increase in TFI beyond 0.028 s resulted in destabilization ofthe subsynchronous network mode. For TFl = 0.0175 s, the eigenvalues of the system are given in Table 5.2 where they are compared to the eigenvalues with the torsional filter(shown in Table 5.1). It is to be noted that the PSS parameters selected in this study are not the optimal values. The main objectives of this study is the evaluation of the effects of the torsional filter and the gain of the acceleration signal on the torsional and other modes. The following conclusions can be drawn from the study 1. The choice of the parameters of the torsional filter is critical as it can affect the stability of an exciter mode. 2. The torsional filter can not increase the damping of the torsional modes. It can at the best counter the destabilizing influence of the dynamic compensator of PSS. 3. The acceleration signal has a significant impact on the damping of the critical modes. The lower frequency torsional modes are damped. 4. If a series capacitor is used, the limit on the acceleration gain is determined from the stability of a network mode corresponding to subsynchronous frequency.

136


Table 5.2 System Eigenvalues with PSS Using speed and acceleration signal -3.322 ± j620.659 -1.008 ± j132.049 -1.850 ± j298.169 -0.184 ± j202.878 -0.369 ± j160.294 -0.413 ± 127.904 -0.979 ± jlO1.071 -1.217 ± j8.906 -18.372 ± j24.101 -3.113 -30.974 -15.572 -0.100 -1000.377

I PSS with

torsional filter

-3.307 ± j620.612 -1.900 ± j133.236 -1.850 ± j298.169 -0.364 ± j202.806 -0.641 ± j160.374 -0.012 ± j127.249 -0.217 ± j99.471 -2.593 ± jlO.062 -23.316 ± j24.953 -3.107 -31.666 -14.942 -0.100 -3.675±jI5.269

I Comments Supersyn. Mode Subsyn. Mode Mode 5 Mode 4 Mode 3 Mode 2 Mode 1 Mode 0 Exciter

6

6.1

INTERACTIONS WITH HVDC CONVERTER CONTROL

INTRODUCTION

HVDe transmission has been established as a viable alternative to ac transmission in long distance bulk power transmission. It is also uniquely suited for underwater cables and asynchronous interconnection, even when the nominal frequencies for the two areas being interconnected are identical. HVDe interties often dispense with the dc line and are termed as back to back links where the rectification (conversion from ac power to dc) and inversion (conversion from dc power to ac) are performed in the same converter station. With the advent of thyristor valves since early seventies, the application of HVDe transmission has increased since fast control of power in the HVDe link has become viable as thyristor valves don't have the problem of arc backs that plagued mercury arc valves. This implies that a HVDe link can be used to improve system stability and security by employing dc power modulation and rapid control of power during emergencies. Using a higher level, auxiliary controller with control signal derived from bus frequencies or power in a parallel ac line, it is possible to damp swing modes in the same manner as power system stabilizers (PSS) that are used in conjunction with excitation systems. While it was anticipated, as in the case of PSS, that high gain power modulation control can destabilize torsional modes, it was also discovered during field tests conducted at Square Butte HVDe terminal in October 1977, that adverse torsional interaction occurred with basic converter current controls [Bahrman K. R. Padiyar, Analysis of Subsynchronous Resonance in Power Systems © Kluwer Academic Publishers 1999

138


et al(1980)]. It was found, during the tests that a 11.5 Hz torsional mode became unstable upon switching out a parallel transmission line. The impact of this discovery is that the planning and design of HVDe converter controls must incorporate the possibility of adverse TI with HVDe systems. However, it is important to note that the problem of TI with HVDe systems is much less severe compared to the TI with fixed series capacitors. Also, the problem can be tackled easily by modification of the current controller or by adding an auxilliary controller to damp subsynchronous frequency oscillations. In this chapter, the description of HVDe system is given with special reference to converters and control. The modelling of HVDe system for SSR studies is presented along with an exposition of the basic characteristics of the HVDe system that results in TI. A case study is presented highlighting the factors that affect the torsional interactions. A simplified damping torque analysis is also presented to obtain insights into the mechanisms of TI with HVDe system. 6.2

6.2.1

HVDC CONVERTERS AND CONTROL

GeneraJ

A simple schematic diagram of a typical HVDe converter station is shown in Fig.6.1. The major components of a converter station are converters for the conversion of power from ac to dc (at a rectifier station) or dc to ac ( at an inverter station) and controls. A converter station can act either as a rectifier or inverter by suitable control. Modern converters are twelve pulse with two Graetz bridges connected in series with one bridge supplied by the secondary windings of the converter transformer connected in star while the second bridge is supplied by the set of secondary windings connected in delta. (see Fig.6.2). The objective of the twelve pulse operation of converters is to reduce the harmonics in ac current and dc voltage. Under ideal conditions, only characteristic harmonics are present, the order of which are given below hac

= 12n ± 1 hdc = 12n

where hac and hdc are the order of harmonics present in ac and dc side quantities, n is any positive integer. The direct component of the dc side voltage for a 12 pulse converter is twice the direct component of the dc output voltage of a Graetz bridge. Similarly, the fundamental component of the ac side current is twice the fundamental component of the ac current for each bridge. Hence, it is adequate to consider the analysis of a single bridge if harmonics are not of consequence.

6.2.2

AnaJysis of a Graetz Bridge


~ y

°D

139

CD

~

Q)

~

o?i

oD

CD

CD Pole 2

~Q)

I-

II -

Figure 6.1.

-

CD

CD

12 Pulse converter

Q)

Transformer

Q)

Smoothing reactors

@ Q) @

DC Filters Tuned AC Filters HP AC Filters

Schematic diagram of a typical HVDC converter station

Consider a Graetz bridge fed from three phase voltage sources connected in series with the leakage impedances of the converter transformer. (see Fig.6.3). It is assumed that the output terminals of the bridge are connected to a current source (not necessarily of constant magnitude) The following assumptions are made in the analysis 1. The valves can be modelled as ideal switches with zero impedance when on (conducting) and with infinite impedance when off(not conducting). 2. AC voltage at the converter bus is sinusoidal, balanced(with positive sequence only). 3. The leakage impedance of converter transformer in all the three phases are equal. 4. The valves are turned on and off instantaneously(negligible switching time). The switching in occurs at the instant the gate pulse is present if the thyristor (valve) is forward biased (anode is positive with respect to cathode)and the switching out occurs at the instant when the valve current goes zero. (It is

140


-~

~-'=-

D.C .. Terminals

-=...==-

ACBus

Figure 6.2.

A Twelve pulse converter unit

135 ~ ~~------~

~

r----~~------~~

~

~-------+---r--~

Figure 6.3.

Graetz bridge circuit

to be noted that the current flow in the valve is unidirectional- from anode to cathode). 5. The gate pulses in steady state are equidistant The numbering of the valves in Fig.6.3 is according to the sequence in which individual valves are fired. In steady state, the interval between consecutive firing pulses is 60° as there are six firing pulses in a cycle of the supply voltage. At any given time, there are at least two valves conducting, one each from the top and bottom valve groups in the bridge. Assume, for example, that valves 1 and 2 are conducting and valve 3 is turned


e ba

141

commutation voltage of valve 3

. . are valve currents 11 ' 13

:-r I

: il

<X

i3

I

I

I

:

I

I

=delay angle

13=n-<x= angle of advance u

=

r=/3- u=

j Figure 6.4.

overlap angle (commutation angle) extinction angle

Definitions of various angles

~------------------------------o

Figure 6.5.

Equivalent circuit of the Graetz bridge

on. It is to be noted that the commutation voltage across valve 3 ( voltage prior to its being turned on) is (eb - ea). The earliest instant of firing is theoretically the instant when voltage (eb - ea) crosses zero to become positive. The delay angle (a) is the angle by which the firing pulse is delayed with respect to the zero crossing of the commutation voltage (see Fig.6.4). For a rectifier, the nominal operating valve of a varies between 5° to 20°. For an inverter, the nominal valve of a varies from 140° to 160°. After the valve is turned on, the current in the valve cannot instantaneously transfer to valve 3 because of the leakage inductance Le. The duration for which 3 valves conduct (for example, 1,2 and 3) is termed as the overlap angle or commutation angle 'u'. It is assumed that u < 60°. In general, the bridge can be represented by its equivalent circuit as shown in Fig. 6.5. Here, the equivalent voltage source eeq and the inductances Leq depend upon the conduction pattern. When valves 1, 2 and 3 are conducting (for duration of u) then

(6.1)

142


L123

eq

= ~L 2 c

(6.2)

With valves 2 and 3 conducting (for a duration of 60-u), the equivalent parameters are given by 23 (6.3) eeq = eb - ec

= 2Lc

(6.4)

= J2vc sin wt

(6.5)

L;~ Defining

eba

where Vc is the rms line to line voltage at the converter bus, the average of eeq over the interval of 60 0 is given by (6.6) Substituting for (eb - ec) as

= J2Vc sin(wt + 60)

eb - ec

(6.7)

we get,

Vd = -3yr.;V 2 c cos a 7r

-

3 r.; [cosa-cos(a+u)] - y 2Vc -=------'--~

2

7r

(6.8)

During the period of commutation, neglecting variation in Id, the valve current i3 is determined from di3 (6.9) 2Lcdt = (eb - ea) Substituting from Eq.(6.5) in (6.9) and noting the boundary conditions (6.10)

(6.11) we derive

v

J2-c [cos a - cos (a Id = 2wL c

+ u) ]

(6.12)

Substituting Eq.(6.12) in (6.8), we have (6.13)

3V;V

where Xc = wL c, Vdo = c. Applying the concept of state space averaging, [Middlebrook (1988)] the equivalent circuit for the converter, ignoring harmonics, is derived as shown in Fig. 6.6. Here,

La _ 2Lc(60 - u) eq 60

+

3Lcu 120

(6.14)


L

a eq

143

R c

+

Equivalent circuit in average quantities

Figure 6.6.

Rc

3 = -Xc, 11"

Ed

= Vdo cos

(6.15)

Cl

Rc is termed as the commutation resistance and L~q is the avearge inductance. It is to be noted that Rc only indicates the voltage drop caused by commutation, but does not imply converter losses. (Actually we have ignored losses in our modelling of the converter and transformer). Remarks 1. The equivalent circuit shown in Fig. 6.6 is valid for the entire cycle of the supply voltage waveform due to symmetry. 2. The equivalent circuit shown in Fig. 6.5 can be applied for transient analysis. It is applicable with minor modifications even when the leakage impedences are not equal in the three phases [see Padiyar and Kothari (1987)] 3. The equivalent circuit of Fig 6.6 is applicable also for the analysis of TI. 4. While the dc current flow through the Graetz bridge is unidirectional, the average dc voltage can be reversed (in the case of inverter operation) by operating at delay angles in excess of 90 0 • For an inverter, it is convenient to express dc voltage as a voltage drop in the direction of the dc current (unlike in the case of the rectifier where dc voltage is expressed as a voltage rise in the direction of the dc current). Hence, Vdi (voltage across the inverter) can be expressed as (6.16) Since Id can be expressed as [see Eq.(6.12)]

Id Vdi

Vdoi [ = -cos Ii 2Rci

cos (Ji ]

(6.17)

can also be expressed as (6.18)

where Ii = 11" - Cli - Ui, is the extinction angle and ad vance at the inverter.

(Ji

=

11" -

Cli,

is the angle of

144


V cr Vr

~C¢ 1: n a r r

Vci

I I~I

I~I

I

7D-1 n. a. : 1 I

Figure 6.7.

v.I

I

Schematic of a DC link

Fui

Figure 6.B.

Steady state equivalent circuit of a two terminal HVDC link

6.2.3 Converter and system control Consider a two terminal HVDC link, a schematic of which is shown in Fig.6.7. Here, nr and nj are the nominal turns ratio of the converter transformers at the rectifier and inverter respectively. a r and aj are the off-nominal turns ratios. In per unit quantities, the dc voltage equations in steady state are given by (6.19) (6.20) (6.21) where

kr

=

3V2nr n br VBr , 7rVdB

nbr and nbj are the numbers of bridges connected in series at the rectifier and inverter respectively. VBr and VBi are the base AC voltages at the rectifier and inverter bus. Rd is the per unit line resistance of the DC line. ZdB is the base DC impedance given by ZdB = where PB is the base power. Eqs.(6.19) to (6.21) represent a steady state equivalent circuit shown in Fig. 6.8. Here, Edr and Edi are given by

Yj:

(6.22)


Edi

=

kiai Vi cOSTi

145

(6.23)

The expression for the dc current Id, is given by

(6.24) The control of power in a HVDC link can be achieved either through the control of current or voltage or both. For HVDC links other than back to back (BTB) links, the minimization of loss in the HVDC transmission line requires that the voltage be held constant at the maximum feasible limit. The voltage regulation function is usually assigned to the inverter while the current control function is assigned to the rectifier for the following reasons. 1. The increase of power in the link is achieved by reducing Cl'r which improves the power factor at the rectifier for increased loading thereby minimizing the reactive power consumption at the rectifier station. 2. The inverter can now be operated at minimum value of the extinction angle "Ii thereby minimizing the reactive power consumption at the inverter also. 3. The current during line faults are automatically limited with rectifier station in current control. 4. The operation at the minimum extinction angle at the inverter and current control at the rectifier results in better voltage regulation than the operation with minimum delay angle at the rectifier and current control at the inverter. The operation of the inverter at Constant Extinction Angle (CEA) and the rectifier at Constant Current (CC) under normal conditions results in reduced costs of the converter stations and losses. However, with weak AC system (indicated by low short circuit ratio (SCR) where SCR is defined as the ratio of short circuit level at the converter bus to the rated dc power), the operation of CEA control is problematic. Constant DC Voltage (CDCV) or Constant AC Voltage (CACV) control are the alternatives that could be used at the inverter. Under normal conditions, on-load tap changer at the inverter is used to maintain a constant dc voltage by controlling ai. The tap changer (control over a r )at the rectifier is used to maintain delay angle within certain limits (say, 10° to 20°) in order to maintain certain voltage margin for the purpose of the current control while ensuring that the power factor is reasonably high. However the control action of tap changer is relatively very slow and during a transient, a r and ai may be assumed to be constants. Under conditions of reduced ac voltage at the rectifier, it is necessary to shift the current control to the inverter to avoid run down of the HVDC link when the rectifier control (of Cl') hits the minimum limit. This implies that a current controller must also be provided at the inverter in addition to the CEA controller. A smooth transition from CEA to CC takes place whenever the link current starts falling. To avoid the inverter current controller clashing with current controller at the rectifier, the current reference at the inverter is kept below that at the rectifier by an amount called 'current margin'. This is typically about 10% of the rated current. The control characteristics for the rectifier and the inverter stations are shown in Fig. 6.9.

146


a b

c

A

f

e

- B

g Figure 6.9.

c

Converter control characteristics

Here the dc voltage is shown at the inverter station. The rectifier control characteristics is 'abc' while the inverter characteristics is 'efg'. The point of intersection of the two characteristics (point A) determines the mode of operation-the rectifier on CC control and the inverter on CEA control. Depending on the ceiling voltage of the rectifier, there are two other modes of the operation of the DC link. 1. With slight dip in the ac voltage, the point of interaction drifts to C which implies minimum 0' at the rectifier and minimum "( at the inverter. 2. With lower ac voltage at the rectifier, the mode of operation shifts to point 'B' which implies current control at the inverter and minimum 0' at the rectifier.

System Control Hierarchy The control functions required for a HYDC link are performed using a hierarchical control structure where (a) Yalve group control is at the lowest level where firing pulses required for a group of valves (six or twelve) are generated. (b) Pole control is at the intermediate level where current or extinction angle controller generates the required order of firing angle. (c)Master control is provided power order (Pre!) either manually or from load dispatch center and generates current order. The power modulation controller which responds to a control signal taken from frequency or ac line power also modifies the power order. It is necessary that the net power order is transmi tted over communication channels to the remote converter station. The pole control also incorporates pole and HYDC line protection whereas valve group control also oversees valve monitoring, firing logic through the optical interface. It also includes commutation failure protection, tap changer


Figure 6.10.

147

LV

firing

GATE

Pulse

Block diagram of the system control

control, converter start/stop sequences. A block diagram of the system control is shown in Fig.6.10. It is to be noted that the current reference is obtained by dividing the power order by the DC voltage. However the power controller is deliberately made slow by incorporating a time delay. The current reference is also modified by Voltage Dependent Current Order Limiter (VDCOL) which has the functions of protection during low system voltages and ensuring the recovery of the DC link following system faults. The LV gate is used to select the minimum of the firing angle orders generated by the current and extinction angle controllers. This enables a smooth transition between the two controllers as the ac voltage varies.

Firing angle control Modern HVDC converter firing angle control systems use phase locked loop techniques to synchronize firing angle pulses to the ac voltage. Early control schemes utilized individual valve firing based on direct comparison with its commutating ac voltage waveform. This scheme was termed Individual Phase Control (IPC) and had severe problems involving magnification of noncharacteristic harmonics (low order harmonic resonance) particularly when operating with low SCR. Subsequent developments led to the concept of Equidistant Pulse Control (EPC) where a local oscillator produces the firing pulses in a sequence such that in steady state, the phase shift between consecutive pulses is constant irrespective of the zero crossing of the commutation voltages. However, it is necessary to synchronize the local oscillator to the converter bus ac voltage by Phase-Locked Loop(PLL}.

6.3 6.3.1

MODELLING OF HVDC SYSTEM FOR STUDY OF TORSIONAL INTERACTIONS

Factors affecting TI with HVDe system

Based on the studies carried out, following the discovery of the TI phenomenon at the Square Butte HVDC link, following factors have a major impact on TI. [Piwko and Larsen (1982)]

148


AC line

GEN

~I---+--~ ~

INF. BUS

I Figure 6.11.

System configuration

1. The converter control has a major effect on the turbine-generators in the close vicinity of the HVDC converters. The current controller at the rectifier, with inverter on the voltage or excitation angle control has the maximum effect on the generator connected near the rectifier station. 2. The firing angle controller also has an impact with EPC control causing more negative damping of the torsional modes than IPC control scheme. The synchronizing circuit used with EPC does not have a significant effect although theoretically, EPC scheme with a synchronizing circuit of sufficiently large bandwidth is closer in its response to the firing pulse generation scheme with IPC. 3. The negative damping of the torsional modes is of concern mainly when their frequency lies in the range of 10 to 20 Hz. 4. The negative damping increases with the power level on the DC link. 5. For a given power level, the negative damping introduced by the dc system is sensitive to the level of the dc voltage. This effect can be traced to the changes in the operating values of the delay angle (Q) caused by the voltage variations. The negative damping is less for smaller values of the delay angle. 6. The negative damping of the torsional modes, introduced by the HVDC system is increased as the strength of the ac system adjacent to the converter reduces. The worst case is when the operation of the HVDC link is in a radial mode. Based on these considerations, the simplest system that exhibits all the characteristics of the torsional interactions is shown in Fig.6.11. Here, AC filters at the converter buses are represented as capacitors. The AC line is shown to be series compensated. This enables comparison of the TI to HVDC system with that due to the fixed series compensation. 6.3.2

Interface between AC and DC system

The generator and ac network modelling have been discussed in Chapter 2 and 3 respectively. The effect of the converter on the ac network is represented as current sources injected at the converter bus (see Fig.6.12).


149

leba

I I

I

AC

NETWORK

J

I I

I

Figure 6.12.

AC Network in

D

D

(a) Rectifier Figure 6.13.

Sequence

Q

Q

(b) Inverter

Q

Phasor diagram showing converter bus voltage and injected current

Here, iga is the current injected by the generator, while ira and iia are the currents injected at the rectifier and inverter buses respectively. The D-Q components of the current at the converter bus can be derived from the phasor diagram shown in Fig.6.13. Ir and Ii are currents injected into the AC network at the rectifier and the inverter bus respectively. - Ir is the load current at the rectifier bus and lags the voltage v,. by the (power factor) angle r. Ii leads the voltage V; by i as the inverter consumes reactive power while supplying active power to the AC network. The equations for the D-Q components of Ir and Ii are given by (6.25) (6.26)

liD = Ii sin(Ji

+ i)

(6.27)

IiQ = Ii cos(Ji

+ ;)

(6.28)

150


where

(6.29) (6.30) tan8 r

VrD = -v. rQ

(6.31)

ViD ViQ

(6.32)

tan8 i = -

(6.33) (6.34) Vdr

and

and

Idi

6.3.3

Vdi are the dc voltages at the rectifier and inverter respectively. are the dc currents at the rectifier and inverter.

Idr

Modelling of the DC network

The dc network is obtained by assembling the equivalent circuits of the converters (see Fig.6.6) and the dc transmission line which can be modelled as a 7r network. The dc network for a two terminal HYDe link is shown in Fig.6.14. Here Edr and Edi (in per unit) are defined as

Rer

L~qr

Ld

Vdr

Rei

~~

Id

Idr

V di

Cdi

Cdr

L~qi

+ Edi

Edr

Figure 6.14.

DC Network

(6.35) Edi

= kiai Vi cos f3i

(6.36)


151

is the delay angle at the rectifier and f3i is the angle of the advance at the rectifier. The state equations for the network shown in Fig.6.14 can be expressed as O'r

(6.37)

dId

dt =

1

L}-RdId

+ Vdr

1 -dVdr = -[Idr dt Cdr

dIdi

dt

(6.38)

- Vd;]

Id]

(6.39)

dVdi = _1 [I _ I .] dt C di d d.

(6.40)

1

= La . [-Rc;ldi eq.

+ Vdi

- Ed;]

(6.41)

Equations (6.37) to (6.41) after linearization can be expressed in the compact form as

dXd

where u~ = [~Edr 6.3.4

~Edi]'

dt = [Ad]Xd + [Bd]Ud

(6.42)

Yd = [Cd]Xd

(6.43)

Y~

= [~Idr

~ Vdr

~ldi

~ Vdi]

Modelling of converter control

It is assumed that the rectifier controls the current and the inverter regulates

the dc voltage or the extinction angle. The current (voltage) controller at the rectifier (inverter) is shown in Fig.6.15. Here the gain K is assumed to be positive. The negative sign associated with the gain K, implies that the delay angle ordered (0' R) at the rectifier or the angle of ad vance (f3R) ordered at the inverter is decreased whenever the error positive. It is to be noted that actual 0' at the rectifier or f3 at the inverter is also influenced by the change in the phase angle (~Jc) of the ac voltage at the converter bus - with EPC scheme of firing pulse generation. The switch 51 is assumed to be open when IPC firing scheme is assumed. The switch 52 is assumed to be open with pure EPC scheme whereas with 52 closed, the transfer function of the synchronizing circuit is also considered. Typically, SYNC(s) is given by 5Y NC(s)

e- STD

= 1 + s Ts

(6.44) (6.45 )

and B Ws is the band wid th of the synchronizing circuit. A typical value of B W. is about 4 Hz. At this frequency, the synchronizing circuit has little impact on

152


+ (~)

+

Figure 6.15.

Block diagram of the converter controller

TI. TD represents the average transport delay due to the sampling action of the frequency transducer used for synchronization. The inverter control can be used to regulate the extinction angle rather than dc voltage. The block diagram shown in Fig.6.15 also applies in this case except that the controller gain is positive(The angle of advance increases if "{rei is higher than measured value of the extinction angle). The linearized state equations for the converter controllers can be expressed as dXdc = dt

[Adc]Xdc + [ ] Udc Bdc

(6.46 )

Ydc = [Cdc]Xdc + [Ddc]Udc

(6.47)

where

6.3.5

Combining AC network and DC system equations

The linearized ac network equations, expressed in D-Q components can be written in the compact form as XN = [AN]XN

+ [BNd]UNd + [BNg]UNg

(6.48)

YNd = [CNd]XN

(6.49)

where U~d

= [~IrD

Y~d = [~v,.D

~IiD

~IiQ]

~ViD

~ViQ]

U~g = [~IgD

~IgQ]

~IrQ

~v,.Q

153


tl.IgD and tl.IgQ are the D and Q axis components of the generator (armature) current injected into the AC network at the generator terminals. The HVDC system is divided into two components - dc network and converter controller. Combining Eqns. (6.42), (6.43), (6.48) and (6.49), we have XND

= [AND]XND + [BND]UND + [BNg]UNg YND

= [CND]XND + [DND]UND

(6.50) (6.51)

where t xND

= [xNt

Y}vD = [Y}vD

U}vD

= [U}vd

[AND] = Block diag [AN

Ad Adc ]

[BND] = Block diag[BNd

Bd Bdc]

[CND] = Block diag[CNd Cd Cdc] [DND] = Block diag[[O] [BNg]t

[0] [0] Ddc]

= [BNg

[0] [0]]

Linearizing equations (6.25) to (6.34), it is possible to express UND as (6.52) Combining Eqns. (6.50) to (6.52), we get XND = [AND

+ BNDFND(I - DNDFND)-lCND]XND + [BNg]UNg

(6.53)

where I is a unit matrix of appropriate dimension. 6.4

ANALYSIS OF TORSIONAL INTERACTIONS - A SIMPLIFIED APPROACH

The phenomenon of TI with HVDC system can be explained with reference to a turbine generator connected radially to a HVDC converter (rectifier) station. The analysis is based on the concept of damping torque. The block diagram outlining the transfer functions from the generator rotor speed to the electrical torque is shown in Fig. 6 .16. The following transfer function can be defined from Fig. 6.16.

154


Figure 6.16.

Block diagram showing connection from rotor speed to electrical torque

IREG(s) is the transfer function of the current controller. The transfer function ST(s) is due to the ac transmission system and the HVDe system with no control. The overall transfer function from the rotor speed to the electrical torque is given by

~Te(s) ~w(s)

= ~Te(s) = ST(s) _ ~Sm(s)

SI(s)I REG(s).AT(s) 1 + IREG(s)AI(s)

= STT(s)

(6.54)

The real part of this transfer function evaluated at the torsional mode frequency (Wk) gives the damping torque coefficient (TDk) at that frequency. The decrement factor ((j ek) is calculated from

(6.55 ) where subscript k refers to the kth torsional mode. TDk is positive if the phase of STT(jwk) lies in the range of -90 0 to 90 0 . In deriving the transfer functions based on simplified models, the following assumptions are made 1. The turbine-generator is supplying only the HVDe system (as its only load) 2. The commutating voltage (at the rectifier bus) magnitude and phase angle closely follow the generator internal voltage and angle. 3. In the generator, (a) field flux decay is neglected


155

(b) damper windings are neglected. Based on the assumptions, we can express changes in the magnitude and phase of the voltage at the converter bus as (6.56) ~8r

= ~8 = WB -~Sm S

(6.57)

The change in Edr is obtained from linearizing Eq.(6.22) as (6.58) For convenience, the subscript 'r' has been dropped for the delay angle Q'. The change in the delay angle can be expressed as (6.59) where Q'R is the firing angle command of current regulator and kp(s) is given by kp (s) = 0 for IPe scheme of firing = 1 for EPe scheme of firing = 1 - SY NC(s) for EPe scheme with a synchronizing circuit whose transfer function is SY NC(s). Substituting Eqs.(6.56), (6.57) and (6.59) in (6.58), we get

Assuming that

~Edi

= 0, it is possible to write that (6.61 )

where Yd(s) is the ittance function of the dc network viewed from the rectifier internal bus. The dc power is given by (6.62) The generator electrical torque (neglecting losses) is given by (6.63) Based on the above equations, the transfer functions defined earlier are given as (6.64) AT(s) = -kra r Vro sin Q'o[Ido + Edro Yd(S)]

AI(s) = -kra r Vro sin Q'OYd(S)

(6.65)

SI(s) = krarVro[coSQ'o - kp(s}WB sinQ'o]Yd(s}

(6.66)

s

156


Effect of the Current Control on Damping Torque The influence of the current controller on the damping torque can be analyzed by considering only the second term on the R.H.S. of Eq.(6.54). This can be expressed as

SIT(s) = _ SI(s)AT(s) IREG(s)AI(s) = _ SI(s)AT(s) G ( ) AI(s) 1 + IREG(s)AI(s) AI(s) c s

(6.67)

where G c (s) is the transfer function of the closed loop current controller. For a suitably designed current controller, Gc(s) approaches unity with zero phase shift at low frequencies. Hence the damping contribution of SIT(s) is determined by the transfer function IT(s) defined by

IT( ) = _ SI(s)AT(s) s AI(s)

(6.68)

By substituting Eqs.(6.64) to (6.66) in (6.68), it can be seen that AT(s) and AI(s) nearly cancel to a constant, particularly at light loads (Ida::: 0) leaving the characteristics of SI(s) determine the damping action. For low frequencies below 20 Hz, the transfer function SI(s) is dominated by the integral characteristics (for kp = 1.0) resulting in a phase lag of 90 0 . It is also interesting to note that the integral term is proportional to sin 0'0 which implies that the magnitude of the negative damping increases with 0'0. As mentioned earlier for a well designed current regulator, Gc(s) has a phase shift between 0 and -90 0 . Thus a current regulator actually contributes negative damping particularly at higher frequencies. 6.5

6.5.1

A CASE STUDY

Description

The configuration of the system considered is shown in Fig.6.11. To simplify the analysis, the inverter dc voltage is assumed to be constant. The generator data is taken from IEEE FBM and is assumed to be rated at 600 MVA. The rating of the HVDe converter is assumed to be 500 MW. To study the torsional interactions in detail, the following cases are considered [Kothari (1985)] (A) Radial operation of the HVDe link (B) Non radial operation with the ac line parameters of RL = 0.08, XL = 0.54 and variation of percentage compensation. To examine the influence of various factors that affect TI, the following cases are also considered 1. Influence of the current controller (a) with EPe and (b) with IPe In both cases, the operating value of 0' is assumed to be 18 0 with the HVDe link operating at full load. 2. The effect of the discrete model of the converter versus the continuous model.


157

Since the controller action in the HVDC converter is of discrete nature and not continuous, a discrete model is described by Padiyar and Sachchidanand (1985). Here, the converter is described by the equation. (6.69)

where h = ;:'0 is the sampling interval. V is the rms ac line voltage. The sampling instant is assumed to coincide with the firing instant of a valve. The small perturbation model of the converter is given by

~Vd(k) =

-hI [v'2 Wo

cosao~V(k) - v'2 Vo sinao~a - 2Lc~Id(k + 1) + Lc~Id(k)] Wo

(6.70) where the subscript '0' denotes the value of the variable at the operating point. The discrete model of the converter is combined with models of other subsystems most of which are continuous. To ensure compatibility, it is necessary to transfer a continuous system model described by

x = [A]x + [B]u

(6.71)

to the following discrete model

x(k + 1)

= [A']x(k) + [B']u(k)

(6.72)

where

(6.73) The sampling interval for a twelve pulse converter is half the sampling interval for a six pulse converter, namely 30 0 (h = 0 ). To examine the factors affecting TI, the following parameters were considered (a) Current controller gain (b) HVDC link loading (c) firing angle (d) DC line length (e) Generator rating relative to HVDC link rating (f) AC line strength (g) Series compensation level of ac line. The base case controller parameters were assumed to be J{ = 85, Tl = 0.01 s, T2 = 0.005 s. The controller is assumed to be EPC unless stated to be otherwise. The mechanical damping in the generator rotor system is neglected.

6:

6.5.2

Results and discussion

The eigenvalues corresponding to the torsional modes for the radial operation of the HVDC link are shown in Table 6.1.

158


Table 6.1 Eigenvalues corresponding to torsional modes (Radial HVDC link) EPC Controller I Mode I IPC Controller I EPC Controller I with higher generator rating 1 2 3 4 5

0.128 ± j 98.5 -0.013± j 126 -0.11 ± j160.5 -0.077± j 202.8 0.0±j298

0.41 ±j98 0.074 ± j 126.9 -0.208 ± j 160 -0.139 ± j 202 0.0 ± j298

0.282 ± j 98 0.05 ± j 126 -0.126± j 160 -0.086± j202 0.0 ± j 298

The converter is represented by continuous model. The results given in Table 6.1 show that the EPC firing control scheme results in increased negative damping of the critical torsional modes compared to the IPC scheme. The effect of the higher generator rating (from 600 to 900 MVA) is to reduce the negative damping of the critical modes, although the damping of the higher torsional modes is reduced. The effect of the converter modelling is shown in Table 6.2. Table 6.2 Effect of converter model (Radial HVDC link) Continuous I Discrete I I Mode I converter model converter model 1 2 3 4 5

0.41 ± j 98.0 0.074± j 126.9 -0.208 ± j 160.0 -0.139± j 202.0 O.O±j 298

0.39 ± j98.6 0.087 ± j 126.9 -0.306± j 160.0 -0.162± j202 0.0 ± j 298

The com parisi on of the continuous and discrete time representation is done by converting discrete system eigenvalues to their equivalent continuous system eigenvalues. If Ad is an eigenvalue of the discrete system, then it is related to the equivalent (continuous) system eigenvalue Ac, by the following relation. (6.74) where h is the sampling period.


159

Effect of DC link loading Although not shown here, it was observed that the effect of the dc link loading is to unevenly influence the damping of the torsional modes. While the undamping of mode 1 continued to increase with loading the mode 2 damping changed from negative to positive. Mode 5 is unaffected due to its high inertia. This result is surprising as it is generally believed that effect of increase in loading is to decrease the damping (or increase negative damping). It was observed that the dc system has a dominant mode of oscillation whose frequency varies between 108 to 136 rad/sec. for the variation of dc power from 0.1 to 1.0 pu. The frequency of the dc system mode is around 126 rad/sec. when the damping of mode 2 (whose frequency is 126 rad/sec) undergoes a sudden change. It was also observed that the differences between the continuous and discrete time models are significant in predicting the stability of the torsional mode 2. In other cases, the differences are minor. In general, the continuous time model gives consistently pessimistic results. The above discussion shows that TI is also affected by mode coupling between a torsional mode and a critical dc system mode.

Effect of Firing Angle It was observed that from the results (not shown here) that the frequency of oscillation corresponding to the critical dc system mode increased from 72 to 168 rad/sec. as the firing angle is increased from 5° to 30°. It was also observed that whenever the frequency of oscillation of the dominant dc system mode is around a torsional mode frequency, there is a drastic change in the damping of that mode. The eigenvalues for 0: = r are given in Table 6.3. The mechanical damping is considered in this case.

Table 6.3 Eigenvalues for 0: = r, dc line length =2 and 3 pu (Radial HYDe link) Mode

EPe with 0: 7°

1 2 3 4 5 dc system mode

-0.516 ± j 98 -0.087±j 126 -0.59±j160 -0.25± j 202 -1.22±j 298 -20.48 ±87.8

=

I

EPe with dc line length 2 pu

EPe with dc line length = 3 pu

0.947 ± j98 -0.111 ± j 126 -0.612 ± j 160 -0.259 ± j 202 -1.22± j298 -5.67± j107

-0.979 ± j 97.6 -0.084 ± j 126 -0.579± j 160 -0.226± j202 -1.22± j298 -4.3 ± j92

=

Effect of Deline length It was observed that the damping of dominant mode of the dc system reduces as line length increases. The frequency of this mode also decreases with increase in the line length. Again due to mode coupling, there are drastic changes in the damping of torsional modes. Mode 1 experiences maximum destabilization for 2 pu line length (the line length is doubled compared to the base case). All the torsional modes are stable for dc line length of 3.0 pu.

160


The eigenvalues for dc line lengths of 2.0 and 3.0 pu are also shown in Table 6.3. The operating value of Q' for these cases is set at the base value of 18°. Effect of Controller Parameters It was observed that as the controller gain is increased, damping of the dominant dc system mode is reduced and the mode becomes unstable as the gain exceeds a certain limit (around 200). The frequency of this mode increases with the gain. As with other parameters, the damping of the torsional modes changes significantly when the frequency of the dominant dc system mode is around the torsional mode frequency. Keeping the gain and T2 constant and varying Tl also resulted in a similar behavior of the critical modes as with variation in the gain. Increase in Tl beyond 0.015 s resulted in instability of the dc system mode. Effect of AC Line Strength and Series Compensation It was observed that the negative damping of the critical modes (1 and 2) continued to increase as the reactance of the parallel ac line increased. On the other hand, the damping of modes 3 and 4 increased with increase in the line reactance. The variation in the ac line reactance does not have much effect on the dominant dc system mode. However, the reduction in ac line reactance (increase in the strength of the ac system) results in increase in the frequency of zeroth mode(swing mode). This is as expected. The presence of series compensation affects the torsional modes in the same manner as explained in chapter 4. The eigenvalues of the torsional modes (including mode zero) for the two cases (a) without series compensation (XL = 0.1) and (b) with series compensation(XL 0.54, Xc 0.412) are shown in Table 6.4.

=

=

Table 6.4 Torsional Modes for Non-Radial HVDC link) Mode

o 2

3 4 5

I Uncompensated Line XL = 0.1

Compensated 0.54, Xc 0.412 Line XL

-0.976 ± j 12.45 -0.15± j 99 -0.08 ± j126 -0.55± j 160 -0.20±j 202 -1.22 ±j298

-0.58± j 11.73 0.56 ± j 98 -0.001 ± j 127 -0.55±j 160 -0.26± j202 -1.22± j298

=

=

It is observed that mode 1 is unstable with series compensation and mode 2 is critically stable. Padiyar and Geetha(1995) report a case study of the analysis of torsional interactions in a multi terminal dc system; based on linearized system model. The major conclusions of this study are 1. A rectifier on current control contributes to maximum negative damping of torsional modes, when the generator is connected to the rectifier terminal.


161

The torsional mode with the lowest frequency (mode 1) is most affected. The voltage controller at the inverter contributes to the negative damping to some extent. 2. The torsional modes are better damped when the rectifier is chosen as Voltage Setting Terminal (VST) as opposed to the case when the inverter is chosen as VST. An Explanation of the Phenomenon of HVDe Torsional Interaction The torsional interactions with HVDC system have been previously explained in a simplistic manner, based on the fact that dc power is maintained constant and constant power type load introduces negative damping. However, this explanation does not explain why damping of higher frequency torsional modes increase with dc power. Also the effects of controller and dc line parameters are not explained satisfactorily. A better explanation is to consider the amplitude and phase modulation of the ac voltage at the rectifier bus, resulting from the torsional oscillations. This induces a voltage in the dc link, of the same frequency as the torsional mode, due to effective demodulation introduced by the converter. The effective impedance (also influenced by the converter controller) of the dc link determines the current oscillations in the dc link. These oscillations result in turn, the injection of sub and supersynchronous frequency currents in the ac system, that flow in the generator armature. In general, the subsynchronous frequency currents introduce negative damping torque while the supersynchronous frequency currents result in positive damping torque. Unlike in the case of series compensation, the magnitude of the two currents are equal. The net damping is determined from the phase of the two current components relative to the ac voltage components.

6.6

A SIMPLIFIED DAMPING TORQUE ANALYSIS

Section 6.4 presented an analysis of torsional interactions in HVDC systems by defining a set of transfer functions based on simplified generator and HVDC system models. The major assumptions are that (i) the turbine-generator is supplying only the HVDC ststem (radial operation) and (ii) the commutating voltage at the rectifier bus closely follows the generator internal voltage. In chapter 4, a simplified damping torque analysis is presented where the damping torque is made up of two components (i) due to the subsynchronous frequency armature current and (ii) due to the supersynchronous frequency armature current. A similar analysis is attempted here. Since HVDC is the only load on the generator, the armature currents are determined by (i) the dc link impedance and (ii) converter control. It will be assumed that the current controller is of pure integral type and the firing scheme is pure

EPC. The ittance function of the dc network (viewed from the rectifier internal bus) Yd(S) is assumed to be

(6.75)

162 m

6.S


_ _-+l.'-_A_Ss_1

B_:__+-+f'~ ~

1

R+sL

fu s

Figure 6.17.

Block diagram representing HVDC system

For pure EPe scheme, Eq.(6.60) can be expressed as (6.76) The transfer function of the current regulator is given by f{[

IREG(s) = - -

(6.77)

s

where f{[ is the integral gain. Eqs. (6.75) to (6.77) can be combined to represent the block diagram shown in Fig. 6.17. Here, A and B are defined as A

B

(6.78)

= wBkaVrosinao

The transfer function SAI(s) Fig.6.17 as

SAI(s)

= kaVro cos a o

=

between 6.Sm

(6.79) and

6.Id

is obtained from

As - B 6. I d(s) 6.Sm (s) - L(s2 + ¥s + w~)

(6.80)

where

Wn

=

1

/LC'

C = WB

f{[ B

=

1

f{[kaVro sin a o

(6.81)

The transfer function SA(s) relating 6.Sm and 6.aR is given by (6.82)

163


It is to be noted that the HVDe system has a predominant mode whose frequency is given by

(6.83)

Calculation of AC current If it is assumed that b.Sm (or b.w ) varies sinusoidally ( see Eq.(4.38) in chapter 4) as given by the expression

(6.84) the dc current in the link, b.1d can be expressed as (utilizing Eq.(6.79) ) (6.85) where (6.86) Neglecting overlap angle and harmonics, the magnitude of the ac current is directly proportional to the dc current. The phase of the ac current lags the phase of the converter bus voltage by 0: .Using 0: - f3 sequence networks, given the converter bus voltage as (6.87) the ac current ira is given by (6.88) the change in ira is given by the expression

Since for pure EPe scheme , (6.90) we can rewrite Eq.{6.89) as

b.ira(t) = kab.ldsin(wot

+6

0 -

0: 0

) -

ka1do cos{wot

+6

0 -

O:o)b.O:R

(6.91)

From Eq.{6.82) and (6.84), b.O: R can be expressed as A

uO:R

= -

[(JSAI{wm)A Wm

( cos wmt

A. ) + 'I'd

(6.92)

Utilizing Eq.{6.85) and (6.92),we can express b.i ra in of sub and supersynchronous frequency components as

164


-12S~--~1~O-----1~S~---2~O~---2~S~--~30------3~S-----4LO-----4~S~--~ so f (Hz)

Figure 6.18.

Variation of TD~b with

1m

Following the derivation given in section 4.3.2 , chapter 4,we can derive the expression for the damping torque coefficient as

TDe -- T,ub De

p + T'u De

(6.94)

where

,ub kaE' SAI(wm) [1 Ido/{/j A. T De = + COS,+"ub 2 Wm ,up _ kaE' SAI(wm) [1 Ido/{/l A. TDe - - - - cos,+"up 2 Wm

(6.95) (6.96) (6.97) (6.98)

TD~b and Tl>~P and TDe are computed using the above expressions for the HVDe system,having the data L = 2.7 H, R = 151.4 n, Vro = E' = 1.0 pu(500 kV), ka = 1.18 WB = 377,0'0 = 18 0 and Ido = 1.0 pu(1 kA) The damping torque coefficients are calculated for two values of the integral gain (/{/ ); (i) 85 rad/pu and (ii) 170 rad/pu. The variations of TD~b and Tl>~P and the total TDe as functions of 1m (the oscillation frequency in Hz) are plotted in Figures 6.18,6.19 and 6.20 respectively.

It is interesting to observe that while TD~b becomes positive as the frequency Tl>~P is negative for frequencies in the lower range. Tl>~P reaches a minimum value whenever

I is increased,

(6.99)


165

5

4 KI_ 85

3

.... KI _ 170

...•... -'.

a..

Ul

~

\-.. "

0

.

... ....

-1

-2

5

10

20

Figure 6.19.

25

f (Hz)

35

30

Variation of Tl>~P with

40

45

50

1m

e,-----.------.-----,------.-----.------.----~------._----,

4

2

o

~ -2

KI_85 .... KI _ 170

10

20

Figure 6.20.

25

f (Hz)

30

35

Variation of TDe with

40

45

50

1m

where Ide is defined by Eq.(6.83).The values of Ide are (i) 11.2 Hz and (ii) 16.4 Hz for values of Kr = 85 and 170 respectively. The crossover of TDe from negative to positive values also takes place around Ide. The analysis given in section 6.4 also gives the same results for TDe (the total damping torque ). In this case,TDe is divided into TDe

=

TDe!

+ TDe2

(6.100)

166


-----'--...........•...........•..........•.........•...•...•••.•.••.......•.•.....

2

KI_ 85 .... KI _ 170

10

15

20

Figure 6.21.

25

!(Hz)

30

40

35

Variation of TDel with

45

50

1m

where

ST(s) = (ka Vro)2[cOS 0'0

-

TDel = Re[SIT(jwm)] TDe2 = Re[ST(jw m )]

(6.101) (6.102)

sin 0'0 WB] cos O'oYd(S) s

(6.103)

Yd(S) = R+ sL

(6.104)

1

The variations of TDel and TDe2 as functions of 1m are plotted in Figures 6.21 and 6.22 respectively. It is interesting to observe that while TDe2 is positive throughout ,the variation of TDel is qualitatively similar to the variation of TD~b .It is to be noted that TDel gives the contribution from the current controller while TDe2 is the contribution of the HVDC link in the absence of the current controller. To summarize, the results of the simplified damping torque analysis indicate that (1) The negative damping introduced by the HVDC system is mainly due to the converter current controller. Even here, the damping is negative only for

1m < Ide'

(2) Ide is reduced by

(i) reducing the controller gain [{I (ii) increasing inductance L (increase in dc line length implies increase in L) (iii) reducing operating value of firing angle 0' (3) The peak value of negative damping is an order of magnitude less compared to the case with fixed series compensation. Remarks


167

12,-----.------.-----.------,-----.------.-----.------,-----,

N

.:3

I-

6

4

2

°5~----~10~--~1~5~--~20~--~2~5~----30~--~3~5~--~4~O-----4~5~--~50 f (Hz)

Figure 6.22.

Variation of TDe2 with

1m

1. It is interesting to observe from Figures 6.18 and 6.19 that (i) Tb~b is not always negative (ii)T};~P is not always positive These results are in contrast to the results with fixed series compensation (see chapter 4). 2. It can be inferred from the comparison of Figures 6.18 and 6.21, that the adverse effect of converter current controller on TI is primarily due to Tb~b. 6.7

CONTROL OF TORSIONAL INTERACTION

It has been brought out from the case study and the simplified analysis, that the adverse torsional interaction is much less compared to that caused by fixed series compensation. Even here, the potential of adverse interaction exists only if radial operation of HVDC link connected to a tubine - generator is envisaged. There are two ways by which the adverse TI can be overcome. 1. Modification of the converter control 2. Providing a Subsynchronous Damping Controller (SSDC). The first option is feasible only if the modification does not affect the usual functions of the controller during normal and abnormal (contingency) conditions.A major requirement of the controller is to ensure satisfactory transient response during (i) dc line faults and (ii) recovery from ac line faults. Whenever the first torsional mode of the turbine generator has frequency higher than 15 Hz,the TI problem is minimal and can be solved with minor modifications of the existing converter controls. The second approach of providing an SSDC to modulate the firing angle of the converter to damp subsynchronous oscillations is more flexible. The input signal can be taken from the rotor speed or bus frequency [Svensson and

168


Mortensen (1982) ].Alternatively, the signal representing the frequency of the generator internal voltage can be synthesized from voltage and current measurements [Piwko and Larsen (1982)]. The SSDC can be designed based on (a) narrow bandwidth approach or (b) wide bandwidth approach.ln the former,sharp band filters are used to isolate the torsional mode of concern and has the advantage of minimizing the interference with existing HVDC converter controls.However,narrow band filters require precise tuning in the field,which may pose a problem.lt can also introduce negative damping at frequencies on either side of the frequency for which the filter is tuned.The output of the SSDC goes directly to the firingangle controller to modulate the delay angle Q' .In deg a SSDC,care must be taken to ensure that it does not amplify characteristic or non-characteristic harmonics.

7

7.1

INTERACTIONS WITH SHUNT COMPENSATORS

INTRODUCTION

Shunt connected reactive power compensators are used for voltage control at load buses and transmission networks. Apart from maintaining satisfactory voltage profile under load variations, reactive power compensators are expected to control dynamic overvoltages, increase power transfer capacity and improve system stability depending on the speed of control. Mechanically Switched Capacitors (MSC) and reactors have been used primarily for slow control during load tracking. Apart from the slow speed, they also have the drawback of providing only discrete and stepped control. Synchronous condensers provide a smoother and continuous control, but suffer from the problems of maintainance, loss of synchronism under a large disturbance etc. The application of thyristor switches, originally used with HVDC converters, for control of current in a reactor and insertion or removal of a capacitor bank has resulted in the development of Static Var Compensator (SVC) since mid-seventies. SVC's are now viewed as first generation FACTS controllers which are expected to revolutionize power transmission in future. Static Var Compensators were initially used for power factor compensation of dynamic loads such as steel mills and arc furnaces. In recent years, SVC's have been used for the fast ractive power control at HVDC converter stations, where short circuit ratios (SCR) are low. They also have the capability of balancing the load currents by independent control over phase currents and hence are K. R. Padiyar, Analysis of Subsynchronous Resonance in Power Systems © Kluwer Academic Publishers 1999

170


also used with predominant single phase loads such as electric traction. In transmission networks, their use was initially for improving the power transfer capacity of long AC lines (e.g.735 kV AC lines in Hydro Quebec). By locating a SVC at the midpoint of a long line, the power capacity can be practically doubled. (subject to the availability of sufficient ratings of the equipment). This concept can be extended for the application of multiple SVCs either in a single line or the network. A major consideration is the choice of optimal locations as the effectiveness of a SVC is strongly dependent on its location. It would be desirable to have a SVC which can be relocated if the location originally selected does not remain optimal with growth in the system. The advantages of a SVC over Synchronous Condenser (SC) are (i) the speed of response and (ii)lack of moving parts. The response time can be as fast as 30 ms although it tends to increase as the strength of the ac system (measured by the short circuit level at the SVC bus) increases. Fortunately, this is not a problem as voltage regulation is improved with strong system conditions. The speed of response of a SVC enables it to be used for improvement of stability by incorporating a Supplementary Modulation Controller (SMC) which modulates reactive power output of the SVC in response to a control signal. This can be used to primarily damp interarea, low frequency power oscillations. By providing higher bandwidth controllers, it is possible to damp higher frequency subsynchronous oscillations.The concept of dynamic stabilizer described in chapter 4 and used for damping of SSR is an example of such a modulation controller. However, the dynamic stabilizer is a Thyristor Controlled Reactor (TCR) connected at the generating station and has no role for voltage control; whereas a SVC is primarily intended for voltage control at a location away from a generating station. Rostamkolai et al(1990) studied the influence of voltage control in a SVC on the stability of torsional modes of nearby turbine generators. They reported that under certain system conditions, a SVC can significantly reduce torsional damping of nearby T-G units. As a countermeasure to this problem, they suggested the use of a low filter in the SVC vol tage regulator. Recently, the availability of high power Gate-Turn-Off (GTO) thyristors has led to the development of an advanced type of SVC utilizing Voltage Source Converters (VSC) [Gyugyi (1979), Edwards et.al (1988), Schauder et al (1995)]. This device has been termed as Static Condenser (STATCON) or Static Compensator (STATCOM). It is a precursor to the application of Unified Power Flow Controller (UPFC) using VSCs for series and shunt compensation for power flow and voltage control. The advantage of a STATCON over SVC are (i) elimination of bulky ive elements such as reactors. (ii) compact and modular construction requiring less space (iii) better control characteristics and (iv) the possibility of controlling active power in addition to the reactive power by employing an energy source such as batteries or fuel cells. In this chapter, the analysis of torsional interactions with a shunt reactive


171

power compensator is presented. Both SVC and STATCON are considered. The analysis proceeds by modelling these devices for SSR studies. The controllers used in SVC and STATCON are described and modelled in detail. The study is based on small signal stability analysis and illustrated by examples.A simplified damping torque analysis is also presented to get an insight into the torsional interactions caused by the voltage control.

7.2

7.2.1

STATIC VAR COMPENSATOR

Description

There are two types of SVC: 1. Fixed Capacitor-Thyristor Controlled Reactor (FC- TCR) 2. Thyristor Switched Capacitor - Thyristor Controlled Reactor (TSC-TCR) The second type is more flexible than the first one and requires smaller rating of the reactor and consequently generates less harmonics. The schematic diagram of a TSC-TCR type SVC is shown in Fig.7.1.

SVC BUS

STEP DOWN TRANSFORMER

IN

I

PT

,--t---t--7-i CO NTRO L

HP TUNED TCR FILTER FILTER Figure 7.1.

TSC

A Typical SVC (TSC- TCR) configuration

This shows that the TCR and TSC are connected on the secondary side of a step-down transformer. Tuned and high filters are also connected in paral-

172


leI which provide capacitive reactive power at the fundamental frequency. The voltage signal is taken from the high voltage SVC bus using a potential transformer. The TSC is switched in using two thyristor switches(connected back to back) at the instant in a cycle when the voltage across the switch is minimum and positive. This results in minimum switching transients. In steady state, TSC does not generate any harmonics. To switch off a TSC, the gate pulses are blocked and the thyristors turns off when the currents through them fall below the holding currents. It is to be noted that the switches are made up of a series string of thyristor devices(termed as thyristor valve) as the voltage rating of a thyristor is not adequate for the voltage level required. However the voltage ratings of valves for a SVC are much less than the voltage ratings of a HVDC valve as a step down transformer is used in the case of SVC. To limit ~: in a TSC it is necessary to provide a small reactor in series with the capacitor. Thyristor Controlled Reactor [Miller (1982), Gyugyi(1988)) The current in a TCR can be continuously varied from zero( corresponding to zero conduction angle) to maximum(corresponding to conduction angle of 180 0 )by phase control in which the firing angle a( with respect to the zero crossing of the voltage) is varied from 180 0 to 90 0 • The instantaneous current iTCR over half a cycle is given by iTCR

=1;(cosa-coswt), a<wt
(7.1 )

where V is the rms voltage applied, XL is the fundamental frequency reactance ofthe reactor. The current waveform in a single phase TCR is shown in Fig.7.2.

v

Figure 7.2.

Voltage and Current waveforms in TCR

The current is non-sinusoidal and contains odd harmonics which are functions of the conduction angle (7.


173

The fundamental component of the TCR current h, is given by

(7.2) where B

=

TCR

sm (T rrX L

(T -

(7.3)

By connecting the TCR in delta, the triplen harmonics are eliminated on the line side. The harmonics present in the line current are of the order n

= 6k ± 1

(7.4)

where k is an integer. The phase and line current waveforms in a delta connected TCR are shown in Fig. 7.3 for 3 different values of a. It is assumed that the TCR currents are (i) alpha.100

(i) alpha.100

~

k~;J

k?:SJ v:sJ (ii) alpha.130

(iii)

~ o

200 phase. deg

(a) Phase Current

Figure 7.3.

J pvs;J (iii)

alph~1~

f\ 100

(ii) alpha.130

300

o

100

alpha.1~

200 phase. deg

300

(b) Line Current

Phase and line current waveforms in delta-connected TCR

identical in waveform and magnitude in the three phases and only phase shifted from each other by 120 0 • This is possible only if a is the same in all three phases. To limit the harmonics entering the system, some of the fixed capacitors are connected as series tuned filters. To reduce the harmonics further, it is possible to have twelve pulse configuration of TCR, in which there are two branches of TCR supplied by the two sets of secondaries of the step down transformer. One set of the secondary windings is connected in delta while the other set is connected in star. In this case, the line currents on the primary side will have harmonics of the order (7.5) n = 12k ± 1 where k is an integer. This happens because the harmonics of the order other than those defined in (7.5) , generated in the two TCR branches cancel each

174


other as they flow on the primary side. The advantages of TSC-TCR type SVC over FC-TCR type are (i) the reduction in the reactor size and consequently the harmonics generated (ii) greater flexibility in control and (iii) better performance under system fault conditions. The power losses in the quiescent operating condition (with SVC output current close to zero) also tend to be less with TSC-TCR type of SVC.

7.2.2

Control Characteristics and Power Angle Curve

Consider a long transmission line with SVC connected at the midpoint as shown in Fig.7.4. It is assumed that the voltage magnitudes at the sending and receiving ends are held constants at the nominal (rated) voltage (V). The line losses are neglected for convenience.

VLJ

V LO

p

P f---

Isvc SVC Figure 7.4.

A transmission line with SVC connected at midpoint

It can be shown that the voltage variation in the line( due to variation in J) is maximum at the midpoint. SVC helps to limit the variation by suitable control. The steady state control characteristics of SVC is shown in Fig.7.5 where AB is the control range. OA represents the characteristic where the SVC hits the capacitor limit. BC represents the SVC at its inductor limit. Note that SVC current is considered positive when SVC susceptance is inductive. Thus

Isvc

= -BsvcVsvc

(7.6)

A positive slope (in the range of 1-5 %) is given in the control range to (a) enable parallel operation of more than one SVC connected at the same or neighbouring buses and


(b) prevent

175

sve hitting the limits frequently. Vsvc

Ie Vrej

/B

A

/ / o Figure 7.5.

Isvc

Control characteristics of SVC

The steady state value of the sve bus voltage is determined from the intersection of the system characteristic and the control characteristic (see Fig. 7 .6).

Vsvc

Vsvco System characteristics

Isvc

Isvco

Figure 7.6.

Determination of operating point of SVC

The system characteristic is a straight line with negative slope and is defined by

Vsvc

= VTh -

XTdsvc

(7.7)

176


where VTh and XTh are the Thevenin voltage and reactance viewed from the SVC bus. For the system shown in Fig.7.4, we have

v - v. _ Vcos6/2 Th- mo- cos(j/2

XTh =

Zn T

tan (j/2,

(j=(31

(7.8) (7.9)

where Zn is the surge impedance defined by (7.10) Land C are positive sequence inductance and capacitance of the line per unit length, I is the line length. (3 is the phase constant defined as

(7.11 ) where f is the operating frequency. Expression for Voltage and Power (a) Control Range The SVC control range is described by Vsvc = Vrej

+ X.Isvc

(7.12)

where X. is the slope of the control characteristic. Combining Eqs.(7.7) and (7.12), we get (7.13) The expression for power flow in the line is given by

P = Vm Vsin6/2 Zn sin (j/2 With

Vrej

=

V,

it can be shown that

P where

P

is given by,

= kPo + (1 -

p, _ V 2 sin 6

o-Zn SIn '(j'

and

k

=

(7.14)

k)Pl

(7.15)

2 P1 -_ V sin6/2 Zn sin (j/2

(7.16)

X.

(7.17)

X. +XTh

Remarks 1. Po is the power flow in the line without SVC and P l is the power flow in the line where SVC maintains a constant voltage V at the midpoint (Xs=O)


2. k -+ 1 as Xs -+ 00 3. For small values of (), it can be assumed that sin () :::: (), cos!!.2 '" 1 v2 • ~ V2 • ~/ IntIs h · case, P,0 XL SInU, PI 2XL SInu 2

=

.

9

177

9

sm 2:::: 2'

=

where XL = (wL)1 is the total reactance of the line.

(b). At SVC limits When the SVC hits the limit it can be represented as a fixed susceptance (Bsve) where Bsve = Be at capacitive limit = -BL at inductive limit Substituting Isve from Eq.(7.6) in Eq.(7.7), we get

Vsve =

VTh (1 - XThBsve)

V cosJ/2 (1 - XThBsve) cos ()/2

(7.18)

The power flow in the line is given by

Pn sinJ (1 - XThBsve) sin () (7.19) where Pn = is termed as the Surge Impedence Load (SIL). Power Angle Curve for SVC The power angle curve for SVC is made up of 3 segments corresponding to (i) Bsve = -BL, (ii) control range (iii) Bsve = Be For typical value of parameters, the power (expressed in per unit of Pn ) as a function of J is shown in Fig.7.7. The power angle curve for the line without SVC is also shown iIi Fig.7.7 (curve b). P=

Po (1 - XThBsve)

V 2 sin J

i:

7.2.3 SVC Controller The block diagram of the basic SVC controller incorporating voltage regulator is shown in Fig.7.8. [Romegialli and Beeler (1981), IEEE (1994)] This shows that both voltage (Vsve) and current (Isve) signals are obtained from potential and current transformers and then rectified. The AC filter is basically a notch filter to eliminate the signal component of frequency corresponding to the parallel resonance in the system viewed from the SVC bus. The line capacitance in parallel with SVC capacitance can result in parallel resonance with the line inductance. The SVC voltage regulator has a tendency to destabilize this resonant mode of oscillation and the notch filter is aimed at overcoming this problem. As a matter of fact, any parallel resonance mode (of frequency below second harmonic) can have adverse interaction with SVC voltage regulator. If series capacitors are used along with SVC, then they can cause parallel resonance with a neighbouring shunt reactor. If the second (parallel resonance)

178


1.6,-----,------.-----,---,-------,------,------,--,-----,

/

1.4 1.2

80

Figure 7.7.

a

100

120

Plot of Power Vs J

mode has a lower frequency (say below 20 Hz), a high filter in addition to the notch filter has been suggested [ Larsen et al (1990)). The rectified signal is filtered. The dc side filters include both a low filter (to remove the ripple content) and notch filters tuned to the fundamental and second harmonic components. The notch filters are provided to avoid the adverse interactions of SVC caused by second harmonic positive sequence and fundamental frequency negative sequence voltages on the SVC bus. For example, second harmonic positive sequence voltages at the SVC bus cause a fundamental frequency component in the rectified signal that results in the modulation of SVC susceptance at the same frequency. This in turn (due to amplitude modulation) results in two components at side band frequencies (0,2f) in the SVC current. The dc component can result in unsymmetric saturation of the SVC transformer and consequent increase in the magnetization current containing even harmonics. It has been observed that this adverse harmonic interactions between the SVC and the network can result in large distortion of the SVC bus voltage and impaired operation of SVC (termed as second harmonic instability ). The auxiliary signals mentioned in Fig.7.8 are outputs from the Susceptance (or reactive power) Regulator (SR) and Supplementary Modulation Controller (SMC). The Susceptance Regulator is aimed at regulating the output of SVC in steady state such that the full dynamic range is available during transient disturbances. The output of Susceptance Regulator modifies the voltage reference VreJ in steady state. However its operation is deliberately made slow such that it does not affect the voltage regulator function during transients. In contrast to the Susceptance Regulator, the Supplementary Modu-


179

DC FILTER

V re

/+

------''-<.~[

E

-+--

+

V.(AUX. SIGNAL)

- - - GAIN REDUCTION

TSC BLOCKING

LINEARIZER

FIRING PULSES TO

TSC/

TCR AND TSC

TCR LOGIC

GPU

AUX.SIG.

Figure 7.S.

SVC Controller

lation Controller (SMC) is designed to improve the performance of SVC during transient conditions and not affect the steady state. SMC has control inputs from appropriate signals obtained from local measurements and provides a limited output at either the summing junction before the voltage regulator or after it. Thus it modulates directly either the SVC bus voltage or susceptance in order to damp oscillations and improve stability. The Gate Pulse Unit (GPU) for SVC produces firing pulses for TCR and TSC.

180


The logic determines whether a TSC is to be switched in or out. B;e!:R is calculated from B~e.jc and if B;e!:R :S 0, then a TSC is switched in. If B~e!:R ?: BL, then a TSC is switched out. In computing B;e!:R from B~e.jc the effect of leakage reactance of the step down transformer is to be considered as Bsvc is given by Bsvc = _ B,,(Bc - BTCR) = Bc - BTCR (7.20) Bc - B TCR - B" (1- Bc-BTCR) B"

where Bc is the capacitance of the fixed capacitor or TSC. B" leakage reactance of the step down transformer. If B" »Bc or BTCR, then Bsvc ~ (1

Bc

+ -B )Bc - [1 +

(2Bc - BL)

"

B"

= i", X"

JBTCR

is the

(7.21)

BL is the reciprocal of XL(reactance of TCR). Neglecting leakage reactance of the transformer, the variation of BTCR with variation of Bsvc for a case with two identical TSC's is shown in Fig.7.9. BTCR

BCl

Figure 7.9.

BC2 Bsvc

Variation of BTCR with Bsvc for TSC-TCR type SVC

This shows that BL (the rating of TCR) is slightly larger than the rating of either TSC and there is hysteresis in operation of the TSC. This is desirable as the switching of a TSC is not well defined if BCl (susceptance of a TSC) is exactly equal to B L. If BCl is greater than B L, then the operation of the SVC is degraded. For stability studies,it is not essential to model G PU. In this case, the modelling of the controller shown in Fig.7.8 can be simplified greatly by also assuming that SVC does not generate harmonics. The block diagram of the controller in this case is shown in Fig.7.IO. Here the voltage regulator is typically a PI controller as shown in Fig.7 .11. The proportional gain (l{p) may be set zero unless a faster response is required. The transfer function H m (s) represents a

181


Vsv...:c"------'+~ ~

Isvc

}+----f

l!,.e!

Bs

v.

Figure 7.10.

Block Diagram of SVC Voltage Control Bmax

Bmin

Kp 1 + sTp

Figure 7.11.

B re ! svc Bmin

Voltage Regulator for SVC

low filter as given below First order LP filter:

(7.22) Second order LP filter: 1 H (s) - ----.,-------,-". m - 1 + 2s(Tm + (sTm)2

(7.23)

182


Tm is typically 2 to 3 ms and ( = 0.7 The transport delay, Td is typically (7.24) where T is the period of the supply voltage. Td arises due to the discrete nature of the firing pulse. n represents the average delay in getting Bsvc from the instant of delivering the order and is given by

n=-T4

(7.25)

n

It is to be noted that represents maximum (average) delay when the B~eJR changes from maximum (Bd to zero. In Fig.7.8 or Fig.7.1O, the current signal Isvc is used to provide a positive slope for the control characteristic of SVC. However the current signal contains harmonics and there are measurement problems particularly when Isvc is close to zero (the normal operating point) . A solution to this problem is to take the signal from B~~c instead of Isvc . In this case, the block diagram ofFig.7.10 can be simplified as shown in Fig.7.12.

Bmin

Figure 7.12.

Simplified Block Diagram of SVC Voltage Control

Here, assuming Kp = 0, Fi.R=1

X3

(7.26)

and

TR= KR (7.27) KJ KR (typically) varies from 20 to 100, whereas TR varies between 20 to 150 ms. Control Instability and Gain Supervisor [Belanger et al (1984)] The transfer function between the change in the SVC susceptance 6.Bsvc


183

and the change in the sve voltage AVsvc is independent of frequency if only fundamental component of Vsvc is considered. Assuming the operating voltage at sve bus as unity, then AVsvC

~

XThABsvc

(7.28)

The loop gain of the control system shown in Fig.7.12 depends on XTh. The speed of the response depends on the loop gain and increases as XTh is increased. However for high values of XTh the system can be unstable. The voltage regulator of sve is designed to provide the fastest response corresponding to a contingency condition at which XTh is maximum (in the range of operating conditions considered) or the short circuit level at the sve bus is minimum. However, during abnormal conditions resulting in tripping of several transmission lines in the system, XTh will be higher than the design value and the sve controller can be unstable. In such cases, it is necessary to reduce the gain of sve (f{I) automatically by detecting instability. This is called as gain supervisor. The gain is restored to the normal value when the instability is not present. It is to be noted that although instability can be avoided by deg the voltage regulator gain corresponding to the lowest value of short circuit level (highest value of XTh) this is not an optimal choice as the response of sve will be slower under normal conditions. The transfer function ( ~BVSYC ) is in general, a function dependent on frequency svc if network (electromagnetic) transients are considered. The magnitude of the transfer function is maximum at frequency, Ir defined by ~

Ir = INp - 10

(7.29)

where INp is the frequency corresponding to the parallel resonance in the network and 10 is the operating frequency. The controller gain will be severely restricted even under normal conditions if Ir is within the controller bandwidth. Hence the practical solution to this problem is to provide a notch filter in the controller (on the input side) as mentioned earlier. Susceptance Regulator (SR) A typical SR is shown in Fig.7.13. Here the output of the voltage regulator, B~Vc and a set reference B'S'Vc are compared. If the error exceeds a threshold, it activates an integrator after an adjustable time delay (of the order of several seconds). The integrator output is hard limited by a non-wind up limiter and modifies the voltage reference. The SR has to be coordinated with other reactive power controllers in the vicinity such as HVDe power control, switched capacitor and reactor banks, tap changing transformers etc. Supplementary Modulation Controller(SMC}[Larsen et al(1993),(1996)] This controller modulates either Vsvc or Bsvc by processing a signal obtained from local measurements. The objective is to damp critical low frequency interarea or local modes that can affect stability during disturbances. The control

184


Vmax DEAD BAND BreI

svc

TIME DELAY

Vrel

J{SR

s

Vmin

Figure 7.13.

Susceptance Regulator

signals are selected on the basis of following criteria 1. The signal should predominantly contain the mode that needs to be damped. 2. The controller design based on a particular control signal should be simple and reject noise. 3. The controller should not destabilize modes that are otherwise stable 4. The controller should be effective and robust under widely varying operating conditions. The control signals that have been used or suggested include (1) line current (2) active and reactive power (3) bus frequency (4) Computed Rotor Frequency (CRF) and (5) Computed Thevenin voltage. The last two signals are synthesized from current and voltage measurements at the SVC location. A typical controller configuration of SMC is shown in Fig.7.14. FILTER

GAIN

WASHOUT

PHASE LEAD

V$

U ---+I

Figure 7.14.

Block Diagram of SMC

The washout circuit is designed to drive the SMC output to zero during steady state. The gain and phase compensation of SMC are chosen such that it improves the system response under a wide variety of operating conditions and different types of disturbances.

7.2.4

Modelling of SVC

In modelling SVC for the study of torsional interactions, the step down transformer can be considered as part of the AC network. The capacitor (fixed or thyristor switched) and the TCR are connected at the secondary bus whose


185

voltage is denoted as VTR (see Fig.7.15). The capacitor C, also includes the representation for harmonic filters. The current drawn by TCR is denoted as ITR and can be derived as output

AC NETWORK

Figure 7.15.

Representation of SVC in AC Network

from SVC. The D and Q-axis components of the current in the TCR are obtained from the following equations. (7.30)

(7.31) The variable BTCR is the output of the SVC controller. In modelling the controller, the susceptance regulator (SR) can be neglected as it is slow acting. The voltage control is modelled as shown in Fig.7.1 O. H m (s) is assumed to be the transfer function of a first order low filter, given in Eq.(7.22). Neglecting the proportional control in the voltage regulator shown in Fig.7.11, the following equations apply for the voltage controller, (7.32) . 1 X C2 = Tm [-XC2

.

1

+ Vsvc

XC3 = Tb [-XC3

+ XCI]

. 1 BTCR = -[-BTCR

Td

- Xslsvc]

+ XC3]

(7.33)

(7.34) (7.35)

In deriving the above equations, it is assumed that e- STd ~ (1+~Td). Also, the limiters are neglected.

186


Linearizing equations (7.30) to (7.35), the SVC mathematical model can be expressed in the compact form as

Xsv = [Asv]xsv

+ [BSVI]~ V. + [BSV2]USV

Ysv = [Csv]xsv

(7.36) (7.37)

where

xh = [~ITRD ~ITRQ ~XlC ~X2C ~X3C ~BTCR] Y~v = [~IrRD ~IrRQ], u~v = [~VTRD ~VTRQ] The nonzero elements of [Asv], [Bsv] and [Csv] are given below

Asv(l, 1) = -wBRTCRBTCR, Asv{l, 2) = -Wo Asv(l, 6) = WB(VTRD - RTCRIrRD) Asv(2, 1) = Wo, Asv(2,2) = -wBRTCRBTCR Asv (2,6) = WB(VTRQ - RTcRITRQ) Asv(3,4) = -J{[ Asv(4,3) =

-:~, 1

Asv(5,3)=n' 1

Asv(6,5) = Td

Asv(4,4) =

-;m 1

Asv(5,5)=-Tb 1

'

Asv(6,6) = -Td

'

Bsvd3) = J{[, Csv(l, 1) = Csv(2, 2) = 1 Bsv2(1, 1) = wBBTCR, BSV2(2, 2) = wBBTCR VTRD VTRQ BSV2 (4, 1) = To V ' BSV2(4,2) = To V m TR m TR In deriving the above equations following assumptions have been made. VTR, This is also equivalent to neglecting the leakage impedance of (i) Vsvc the step down transformer (ii) ~Isvc = ~B~~c = ~XCI

=

7.3

7.3.1

TORSIONAL INTERACTIONS WITH SVC

Factors affecting TI

Rostamkolai et al (1990), have carried out a detailed study of TI with SVC in connection with the planning for the installation of a SVC at Chester. The studies are based on damping torque analysis. They considered a simple two area system connected by a tie line. Each area has a generator operating in parallel with an external source (infinite bus). The SVC is located at the midpoint of the tie line. The major conclusions based on the study are gIven below. 1. The torsional interactions are primarily due to the voltage regulation function of SVC. The phase locking scheme for the generation of firing pulses had


187

no influence. 2. The adverse TI occurred with the generating unit in the sending area. 3. The level of adverse interaction increased with (a) increased generator output (b) increased tie line power flow (c) increased capacitive reactive power output from sve (d) decreased system strength in the sending area system Even when the sve output was zero, the presence of the voltage regulator affected the torsional damping. As with a PSS, the presence of SMe with the sve results in increased adverse influence on the torsional damping than the voltage regulation alone. This can be overcome by using a low filter in the measurement circuit. A bandwidth of 5 Hz for the filter is suggested. The inclusion of a similar low filter in the voltage regulator loop also reduced the TI caused by the voltage regulation function. A notch filter tuned to the frequency of the critical torsional mode is suggested as an alternative to the low filter. An important point to be noted is that the negative torsional damping introduced by the sve is less than the negative damping introduced by the HVDe converter (current) controller.

7.3.2 A Case Study To evaluate the torsional interactions with the voltage regulator of a sve, a single machine infinite bus (SMIB) system shown in Fig. 7.16 is studied [Geetha (1996)]. In this system, a sve is connected at the midpoint of a 400 kV, 600 km long line for voltage regulation and improvement of power transfer. To examine, the effectiveness of sve, the contingency condition of outage of a line is considered. The generator is supplying 800 MW power and sve output is 220 MVAR. The system frequency 50 Hz. Tl

Zs

Q+-~~ rel="nofollow">--I--I-+--~--t---iCJ~~ .

PoLO

I \j.JJJ

T2

svc Figure 1.16.

The Study System

188


Damping Torque Analysis Combining the equations for the electrical system including the generator, excitation system (including PSS), the AC network and the SVC, the overall state equations can be expressed as

XF;

= [AElxE + [BEMl

[

:L ]

(7.38)

= [CMElxE

(7.39)

= [AElxE + [B~Mltl6 + [BEMltlSm

(7.40)

tlTe Expressing (7.37) as

XF;

where [B~Ml and [BEMl are the first and second columns of [BEMl respectively, and noting that WB (7.41) tl6(s) = -tlSm s

we can derive the transfer function between tlTe and tlSm as (7.42) where

mt(s) =

[WB

s

Il

The damping torque coefficient TDe is obtained as a function of frequency as

TDe

tlTeUW)

= Re[ tlSmUw) 1

(7.43)

The net damping torque coefficient (TD) is defined as (see chapter 4) (7.44)

TD = TDm +TDe

where TDm = Re[YmUw)l The net damping torque coefficient evaluated at the six torsional modes including mode zero, are shown in Table 7.1. Table 7.1 Damping torque coefficient ( TD ) for torsional modes with SVC

I Without voltage I With voltage I I Mode I inFrequency rad/s controller controller 0 1 2 3 4 5

5 99 127 160 203 298

59.53 -9.04 0.70 12.25 2.78 125.77

83.89 -9.51 1.04 13.03 3.97 150.30


189

This shows that even without the voltage regulator, mode 1 is unstable. This is due to the interaction of PSS while has no torsional filter. The effect of the SVC voltage controller is to increase damping of all the torsional modes (particularly mode zero) except mode 1. The mode 1 damping becomes more negative due to SVC voltage regulator action. An eigenvalue analysis is also performed by constructing the system matrix [AT] defined by (7.45) where [AM], [EM] and [eM] matrices are defined in chapter 2. The eigenvalues corresponding to the torsional modes evaluated at the operating point are shown in Table 7.2. Table 7.2 Eigenvalues corresponding to the torsional modes with SVC Mode

Without PSS Without voltage controller

With PSS Without voltage controller

With PSS and voltage controller

0 1 2 3 4 5

0.585 ± j4. 711 -0.021 ± j98.732 -0.007 ± j126.99 -0.058 ± j160.51 -0.032 ± j202.83 -0.167 ± j298.18

-0.475 ± j4.886 0.076 ± j98.817 -0.0006 ± j127.00 -0.043 ± j160.53 -0.016 ± j202.85 -0.167 ± j298.18

-0.698 ± j5.411 0.080 ± j98.798 -0.0008 ± j127.00 -0.045 ± j160.52 -0.023 ± j202.84 -0.167 ± j298.18

These results are in agreement with those shown in Table 7.1. It is observed that PSS while improving the damping of mode zero, destablizes other torsional modes (particularly modes 1 and 2). The mode five is unaffected due to its high inertia. 7.4

7.4.1

STATIC CONDENSER(STATCON)

Description

The static condenser is an advanced form of SVC using Voltage Source Converters (VSC). It is similar to the Synchronous Condenser (SC) in the sense that the control of the reactive current drawn by the device is achieved by controlling the magnitude of the voltage behind a series react ace (primarily the leakage reactance of the step down transformer). However, it is much faster in operation than SC and has no moving parts. The control characteristic of a STATCON is shown in Fig.7.17. It is to be noted that the reactive current output is not dependent on the bus voltage even at the limits. Also, a transient overload capacity in the capacitive range can be built in by proper choice of device ratings. Thus, in comparison with the control characteristics of a SVC (see Fig.7.5), the control characteristics of a STATCON are definitely superior. This implies that for identical

190


VSTAT

Transient Overload Capability

Cap. Figure 7.17.

Ind.

~TAT

Control Characteristics of STATCON

ratings, a STATCON provides better dynamic voltage than a SVC. In addition, a STATCON has the following advantages over a SVC. 1. A STATCON has smaller size due to reduced number and size of ive components. 2. It can also supply active power during contigencies if an energy source such as battery or fuel cell is connected on the dc side of the VSC. The drawbacks of STATCON are 1. Generation of voltage harmonics. A six pulse VSC generates a fifth harmonic of 20 % of the fundamental. The harmonics can be reduced by deploying several converters in multipulse operation. 48 pulse STATCON have been built and operated in Japan [Mori et al (1993)] and U.S.A.[Schauder et al (1995)] 2. The operation of multipulse converters require complex magnetics. If the converters are connected in series on the AC side (as it is usually done), the addition of voltage outputs of the various converters requires a summing transformer which needs to be carefully designed. The Pulse Width Modulation (PWM) of converters is still not feasible for high power converters that employ GTO thyristors as switches. Also, requirements of voltage ratings may necessitate series connection of GTO's which is quite complex. The development of efficient higher power, high frequency power semiconductor switches (such as IGBT, MCT) are still under way. A compromise appears to be the operation of multilevel converters that use more than two levels for voltage control, harmonic reduction. Three level converters have been used in a recent UPFC project which is made up of shunt and series converters. 3. Voltage source converters require power semiconductor switches that have inherent turn-off capability such as GTO's. Conventional thyristors used in SVC's tend to be economical and available in higher power ratings than GTO's.


191

However, with the developments in power semiconductor technology, the costs are expected to come down even at higher power levels.

7.4.2 Analysis of STATCON A six pulse STATCON is shown in Fig.7.18.

R

ib

Xs fVVY\

ic

Xs

n vb "v -

VC"v~

Figure 7.18.

Six pulse STATCON

Since STATCON is basically a reactive power controller, the direct voltage on the de side of the converter can be sustained by a capacitor. In the analysis of the circuit shown in Fig.7.18, the following assumptions are made. 1. The system voltages are balanced and sinusoidal. 2. The switches are assumed to be ideal. 3. The 'bottom' switch of the converter leg turns on immediately after the 'top'switch is turned off and vice-versa. (In practice a small delay is provided to prevent both switches of a leg being on at the same time) 4. The transformer impedances are equal in all phases. The converter output (phase) voltages can be expressed as

(7.46)

where Sa, Sb and Sc are termed as switching functions and shown in Fig.7.19. In steady state, they are symmetrical, and satisfy the following relation (7.4 7)

192


11"

3"

4,5,3

Figure 7.19.

1,5,3

,5,6

1,2,6

11"

11"

3"

3"

4,2,6

Switching Functions

4,2,3

4,5,3

1,5,3

(Sa, Sb ,Sc ) for

1,5,6

1,2,6

ON SWITCHES

6-pulse STATCON

Since the switches 1 to 6 are assumed to be ideal, they are lossless and from conservation of energy principle, we have, (7.48) Substituting Eq.{7.46) in (7.48), we get (7.49) For analysis involving subsynchronous frequencies, it is in order to ignore harmonics generated by the converter. The fundamental components of the switch-

193


ing functions can be written as Sal Sbl ScI

= ksin(wot + a + 0)

= ksin(wot + a + 0 _ 271") = ksin(wot + a + 0 +

}

~)

(7.50)

The system voltages are assumed to be (from assumption 1)

Ii

Vb

= V sin(wot + 0) } = /IVsin(wot + 0 - 2;))

Vc

= ~V sin(wot + 0 + 2;))

Va

(7.51)

It is to be noted that a is the angle by which the fundamental component of the

converter output voltages lead the STATCON bus voltages. Since harmonics are ignored, Eq.(7.50) applies even when multiphase converters are considered. In general, the constant 'k' is given by

k=v'6e 'Tr

(7.52)

6

where p is the pulse number. It is assumed that the basic converters (bridges) are connected in parallel on the dc side and in series on the ac side (which is the normal case). The equivalent circuit representation (on the ac side) of the STATCON is shown in Fig.7.20. Rs

'--~~~----~I V~ • iQ + j in

Figure 7.20.

Equivalent circuit of STATCON

Equations in D-Q Reference Frame Applying Kron's transformation (defined in chapter 3) with respect to a synchronously rotating reference frame, the following equations can be written down for iD, iQ and Vd X, din dt

-- = -

WB

R .

ltD -

Wo X .

WB

stQ

+ Vn - en

(7.53)

194


X$ diQ . - - d = -R$zQ t

WB

be dVde

WB

dt

Wo

.

+ -X$ZD + vQ WB

eQ

.

= -gpVde - Zde

(7.54) (7.55)

In of D-Q components, Eq.(7.48) can be expressed as (7.56) Substituting the fundamental components of the switching functions (in place of Sa, Sb and Se) in Eqns. (7.46) and (7.49), we get eD = ksin(a + 8)Vde

(7.57)

+ 8)Vde

(7.58)

+ kcos(a + 8)iQ]

(7.59)

eQ = kcos(a ide = -[ksin(a + 8)iD It is to be noted that

tan8= VD, VQ

V=./Vb+V~

V

(7.60)

From control viewpoint, it is convenient to define active(ip) and rective (iR) currents drawn by the STATCON as ip =

vDiD

+ vQiQ

V

.. . = ZD sm 8 + zQ cos 8

iR = vDiQ V - VQiD = -ZD . cos 8 + i Q sin8

(7.61 ) (7.62)

The reactive current iR is positive when STATCON is operating in the inductive region and negative when STATCON is operating in the capacitive region. 7.4.3 STATCON Control The primary control loop for a STATCON is the reactive current loop. Closed loop control is necessary as the reactive current depends not only on the control parameters (k and a ) but also on the system voltage magnitude and angle (V and 8). Schauder and Mehta (1993) define two types of control structures for STATCON. These are-: 1. Type I: Here both k and a are used as control variables to control ip and iR. The reactive current reference is usually set by a voltage regulator which maintains the bus voltage within limits. This type of controller is required,when active power exchange is desired. The control of k is usually achieved using PWM techniques. 2. Type II: When PWM techniques are to be avoided in the interests of reducing switching losses (thereby improving efficiency), the reactive current control can be achieved by varying a alone. The block diagram of this controller is shown


195

- - IsrATCON

Regulator Reactive Current Controller

«order

(a) Overall controller sTw r - - - -I ~-wr;-

o 90 ~-I--

i

I10rder

RlCL_'-o{

_90 0

(b )Reactive current controller Figure 7.21.

Type II controller for STATCON

The steady state values of the variables, iR ,ip and Vdc are shown in Fig. 7.22 for typical values of the STATCON parameters. It can be shown that these variables are independent of the operating values of (). It is assumed that variation of O! ( even in a narrow range) results in complete control of iR. It is interesting to observe that Vdc has minor variation over the entire range of operation. Vdc is higher in the capacitive region (for negative O! and iR) compared to the inductive region (for positive O! and iR). Schauder and Mehta (1993) have demonstrated that a pure P-I controller for reactive current regulation is destabilizing when operating in the inductive region. This result is also confirmed by Padiyar and Kulkarni (1997) who have suggested fuzzy control of reactive current which is more robust than the nonlinear controller suggested by Schauder and Mehta (1993).

196


0.8

1 8 I'--

~

04

0.2

~

~

0~--------------~~--------------------4

.~

"~-0.2 -0.4

-0.6

~

_0.8"-----'----'-----'------'-----'----~---'--------'-----'-----:c'

-0.15

-a.4

-0.3

-0.2

-0.1

Figure 7.22.

0

a

0.1

0.2

0.3

0.4

0.15

(degrees)

Steady state characteristics

However, in what follows, the reactive current control is assumed to be a P-I control for simplicity. It has been observed that with an outer loop of voltage regulation, the reactive current control structure is not crucial. 7.5

TORSIONAL INTERACTIONS WITH STATCON

The analysis of TI with STATCO N is presented here with small signal model of the system including STATCON.Both damping torque and eigenvalue analysis can be carried out using the system model,in a similar manner to the analysis with SVC.The analysis is illustrated with an example of a SMIB system shown in Fig 7.16. 7.5.1

Modelling for SSR analysis

Substituting Eqs.(7.57) to (7.59) in Eqs.(7.53) to (7.55) ,we can express the STATCON equations in the state space form. (7.63) where

VdS

XSE = [iD iQ USE = [VSD vSQ]t The subscript's' has been added to indicate the voltage at the STATCON bus. The matrices [ASE] and [ESE] are defined below.

[ASE] =

[

-~

x,

Wo ~k sin(a + fJ)

-Wo ~

- X, ~kcos(a

+ fJ)

-x;-ksin(a+fJ) -x;-kcos(a + fJ) _WB9p

be

1


[!

[BSE]

197

fl

Linearizing Eq.(7.63), we get,

+

XSE = [A sE ] XSE

[BSE]~a

+

where XSE = ~XSE and USE = ~USE [B SE ] and [B SE ] are defined below -

[BSe] =

~[iD cos(a

~[1

[ x,

[BSe] =

_

X,

+ B)

v,' _!!!...a.eDv'51 x, v,'

Cl

= -iDcos(a+B)

(7.65)

- iQ sin(a + B)]

kVdc cos(a+9)v'51]

kC,WBV.Q b c V,2

where

1

WBkVdC;~s(a+9) WBkvdc sin(a+9)

[

(7.64)

[BSe]USE

!!!...a. [1

x,

!!!...a. e 51 v ,D V/ _ kVdCSIn(a+9)v'D]

x,

v,'

kC1WRV,Q

bcV,'

1

(7.66)

+ iQsin(a+B)

In deriving the above expressions, the following equations are utilized (7.67) (7.68) Modelling of STATCON Control The linearized equations of STATCON controllers including reactive and voltage control loops can be expressed as Xse = [Ase] Xse

+ [B~cl [ ~~~

]

~a

+ =

[BscluSE

(7.69)

[Cse]xse

(7.70)

Equations (7.64) and (7.69) can be combined by eliminating and noting that

~a

from Eq.(7.70)

(7.71) where the nonzero elements of [CSE ] are CSE(l,l) CSE(2,2) 1

=

=

198


The combined equations ( including the controller) for STATCON can be expressed as

[Aslxs + [Bslus Ys = [Cslxs

Xs

(7.72) (7.73)

where

BSls~se]

,[Bsl = [

= [!~~]

Ys

,Us

[Csl

=

~~~

]

= USE

[CSE [0]]

For the STATCON controller shown in Fig.7.23,

Figure 7.23.

Block diagram of STATCON controller considered

we have

xse =

[~Xl ~X2 ~X3 ~x4lt

0 0 0

1 -Tm -

J{2

-J{2

-Tm

[Asel

-J{/ I

[

-J{~J{2 -

~1

0 [~ ~l V.Tm

[B~e]

0 0

vsQ

-V,T m

0

-

V6 Tm

.J!..LJ:L

V,Tm

0


(...!!..UL V,T m

[

[Bscl

~) Tm

-

0

199

(v.r.:-~) V, m Tm

0

.£:L

~

0

0

Tm

Tm

1

where

---vr= - V; - ---vr-

C2

C3

=

i.Q

iRV.D

i.D

iRV.Q

V; -

It is to be noted that in Fig.7 .23, the slope of the voltage controller characteristic is denoted by K. (instead of X.).

7.5.2 A Case Study The system considered is identical to that in Fig.7.16, except that STATCON replaces the SVC.The operating conditions are assumed to be same as those given in section 7.3.2, with STATCON supplying 220 MVAR. Two cases are considered in the study (i) with reactive current control ( without voltage control) and (ii) with reactive current and voltage control. The reactive current controller gains are chosen as Kl = 0.05 and K2 = 2.0 . The voltage controller gains are assumed to be the same as in the case of SVC voltage regulator (Kp = 0.0 KJ = -300 ) Table 7.3 gives the values of the net damping torque coefficient (TD) for the two cases considered. Table 7.3 Values of damping torque coefficient ( TD ) with STATCON

I current With reactive I With voltage I I Mode I inFrequency rad/s control only control 0 1 2 3 4 5

5 99 127 160 203 298

64.676 -0.354 12.003 17.658 1.464 115.71

86.19 -2.424 15.177 19.917 1.100 115.72

It is observed that with STATCON also, the torsional mode 1 is destabilized ; (particularly with voltage control), the comparison with SVC (see Table 7.1 ) shows that STATCON controller results in better damping of modes 1, 2 and 3. The eigenvalues for the torsional modes, with STATCON are shown in Table 7.4. Table 7.4 Eigenvalues corresponding to the torsional modes with STATCON

200


With reactive current I With voltage control I I Mode I control only 0 1 2 3 4 5

-0.522 ± j4.946 0.0038 ± j98.830 -0.0097 ± j127.008 -0.0606 ± j160.57 -0.0084 ± j202.91 -0.167 ± j298.18

-0.716 ± j5.426 0.0206 ± j98.789 -0.0124 ± j127.00 -0.0680 ± j160.59 -0.0062 ± j202.92 -0.167 ± j298.18

These results are in agreement with the results shown in Table 7.3, which indicates that the damping torque analysis is generally accurate in predicting the stability of the torsional modes. To summarize the results , it can be said that the voltage regulator used with SVC and STATCON marginally increases the negative damping of mode 1. 7.6

A SIMPLIFIED ANALYSIS OF TORSIONAL INTERACTION WITH VOLTAGE CONTROLLER

It is observed from the studies presented in sections 7.3 and 7.5,that both SVC and STATCON voltage regulators result in Tl.It has been mentioned previously that fixed shunt capacitors do not contribute to TI,unlike series capacitors.Hence,it is worth investigating what really causes TI with shunt FACTS controllers employing voltage regulators. An attempt is made here to provide an insight into the problem by a simplifed analysis of a SMIB system.lt was already mentioned in chapters 4 and 6 that such a simplified analysis is successful in highlighting the factors that are responsible for TI. Consider a simplifed SMIB system shown in Fig.7.24. The generator is repre-

R

CD

L

v., LB

R

L

C

Figure 7.24.

A simplified system

sented by a voltage source of constant magnitude Eg behind a constant inductance.Bus 1 in Fig.7.24 is the internal bus of the generator. The inductance L includes the generator and transformer inductances in addition to the line


201

inductance.A SVC or STATCOM is assumed to be connected at the electrical centre. The capacitance C includes the compensator capacitance in addition to the line and any fixed capacitor. The following assumptions are made in the analysis. (i) The output of the SVC or STATCOM is assumed to be a controllable reactive current. (ii) The time delays associated with the measurement of input variables for the voltage controller,and gate pulse generator are ignored for simplicity. (iii) The initial value of the reactive current IRo drawn by the compensator in Fig.7.24 is assumed to be zero. It is to be noted that even if IRo i- 0 ,it can be modelled as a part of C. It can be shown that the ittance function of the network viewed from the generator internal bus,can be expressed as

y s _ (s2 LC + sRC + 1) () - (R+sL)(s2LC +sRC +2)

(7.74)

The reactive current IR is assumed to be the output of the integral type voltage regulator as shown in Fig.7.25.

Vrej

Figure 7.25.

+

--""-+I

A simplified representation of the compensator controller

To simplify the analysis,the slope of the control characteristic is neglected. It has been observed from results of a case study that the influence of the positive slope of the control characteristic is to marginally reduce TI. From Fig.7.25,we have, (7.75) where ~V. is the variation in the compensator bus voltage (see Fig.7.24). Since IR is assumed to be in quadrature with the bus voltage Vs, and lagging the voltage by 90 0 , we can derive

= IRQ =

I RD

Linearizing Eqs.(7.76) and (7.77) and assuming ~IRD

=

~IRQ

(7.76) (7.77)

cos () IR sin ()

IR

IRo

= O,we get

-cos()o~IR

(7.78)

= sin ()o~IR

(7.79)

202


Substituting for L:!.IR from Eq.(7.75) in the above equations and noting that (7.80) we finally obtain,

] = [ L:!.IRD L:!.IRQ

1([ [

s

-cose o ] sin eo

[ . e sm 0

(7.81 )

There is no loss of generality in assuming eo = 0 .With this assumption,Eq.(7.81) reduces to (7.82) The equivalent impedance of the compensator per phase can be derived by applying the expression (3.85) given in chapter 3 . This is given by 1([ . ) Z eq (JW = ----

(7.83)

W -Wo

The above expression shows that the compensator offers positive resistance to subsynchronous frequency currents while offering negative resistance to supersynchronous frequency currents.The magnitude of the resistance is proportional to the integral gain of the regulator. The magnitude of the resistance is higher for lower torsional modes. Eg = Eb = For values of XL = 0.5 ,R = .05, Wn = l.5wB, (w n = l.0 and IB = 50H z ,the damping torque TDe is calculated from the following expressIOn

,k)

TDe

=

1

2w [(w - wo)G(w - wo)

+

(w

+ wo)G(w + wo)]

(7.84)

where G is the conductance of the network viewed from the generator internal bus,eveluated at the appropriate frequencies. The variation of TDe with the frequency (f ) is shown in Fig.7.26 for two different values of 1([ ( 100 and 300 ). It is observed that the magnitude of TDe decreases sharply (for 1([ = 100) as the frequency ( of the torsional mode)is increased.For 1([ = 300,the decrease in the negative damping of TDe with the frequency is less abrupt. The variation in TDe for the case when Wn = 2WB (lower value of C)is shown in Fig.7.27. This figure clearly shows the higher negative damping introduced by the voltage regulator at higher values of the gain for frequencies above 10 Hz. However,the negative damping increases slightly as C is increased (compare with Fig.7.26). Remarks l. Although the analysis is extremely simplified, the negative damping caused


203

0,-----,-----,------.-----,-----.------,-----, -0.5 -1

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

-1.5

~

-2

..

......

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

KI.100 . .. KI. 300

'

-2.5 -3

-3.5 ~5~----1~0----~15~--~2~0----~2~5----~~~--~375-----740· f (Hz)

Variation of TDe with frequency (w n

Figure 7.26.

= 1.5w B)

0.-----,-----,------.-----.-----.------,-----, -0.5

../ ......., ..........-"............, ......, .........................................

-1

-1.5 G)

~

KI.100

••••••

....

-2 •••• •·

KI.~O

-3.5 ~5L-----1~0----~15~----roL-----2~5----~30~--~3L5----~40

f (Hz)

Figure 7.27.

Variation of TDe with frequency (w n = 2WB)

by the voltage regulator of the compensator is clearly brought out.The effects of the operating value of the shunt capacitive compensation and the torsional mode frequency on the negative damping are illustrated. 2.It is interesting to note that,unlike with fixed series compensation, the shunt compensator results in negative damping caused by supersynchronous frequency currents ( due to the effective negative resistance at supersynchronous frequencies) .

204


3. The system impedance seen by the compensator is given by

R+sL Zth(S) = s2 LC + sRC + 2

(7.85)

For small values of R, the impedance is maximum for w ~ ,,;2w n . It is shown by Padiyar and Kulkarni (1997) that the transfer function relating AVs and AIR is given by

AVs(s) AIR(S)

j[Zth(S+jwo) - Zth(S-jw o)] 2

(7.86)

It can also be shown that for R ~ 0 ,the above transfer function in nonminimum phase type ( with a real zero in the RHP).It is not dificult to see that a P-I type voltage regulator can destabilize the electromagnetic (network) mode at high values of the controller gain. For example, with Wn = 1.5wB and J{/ = 300 , a network mode having a frequency about 2/0 is unstable.However,this instability is not evident in the computation of TDe. For avoiding interactions of the voltage controller with electromagnetic (network) modes,it is essential to provide a notch filter at the input to the controller.

8

8.1

INTERACTIONS WITH SERIES COMPENSATORS

INTRODUCTION

In chapter 4, we discussed the SSR phenomenon and torsional interaction with fixed series compensation. The problem of torsional interactions is most severe with fixed series compensation. Although countermeasures have been proposed and applied, the SSR problem has deterred extensive application of series compensation in transmission networks to increase power transfer capacity. The application of thyristor controllers in Static Var Compensators for voltage control was followed by controlled series compensation for power flow control. There are already three prototype projects in U.S.A. which have demonstrated successful implementation of Thyristor Controlled Series Compensator (TCSC) concept. TCSC is an important member of the family of FACTS controllers. It is important to study the SSR characteristics of TCSC, a variable impedance controller. The analysis based on damping torque concept [Hedin et al (1995)) and the experimental data based on SSR performance tests [Piwko at al (1996)) have indicated that TCSC does not normally contribute to SSR. It is even claimed by Piwko et ai, that TCSC is SSR neutral and can reduce SSR effects due to nearby fixed series capacitors. Just as voltage source converters can be used to provide advanced voltage control ( in STATCON) they can also be used to provide advanced power flow control by injecting reactive voltage of variable magnitude in series with transmission lines. Such a device is termed as Static Synchronous Series CompenK. R. Padiyar, Analysis of Subsynchronous Resonance in Power Systems © Kluwer Academic Publishers 1999

206


sator (SSSC). This FACTS device can be considered as a special case of Unified Power Flow Controller (UPFC) with only series element which injects a voltage in quadrature with the current. The voltage can be either lagging the current (inductive) or leading the current (capacitive). It is to be noted that the injected voltage is assumed to be a rise in the direction of the line current. The injection of reactive voltage ensures that no active power exchange takes place with SSSC except for the power drawn from the system to compensate for the losses. SSSC has some advantages over TCSC, one of them being control over power flow even when the phase angle across the transmission line is zero. It implies that power reversal is also possible by varying the voltage. On the other hand, SSSC requires voltage modulation to provide synchronizing and damping torque. In this chapter SSR characteristics of both TCSC and SSSC are examined from small signal analysis. The SSR performance of both FACTS controllers is also studied from digital simulation. 8.2 B.2.1

THYRISTOR CONTROllED SERIES COMPENSATOR General

Series Capacitors have been used in long distance EHV transmission lines for increasing power transfer. The use of series capacitors is generally the most economic solution for enhancing power flow. However, the problem of SSR has deterred system planners from going in a big way for series compensation. While the use of shunt capacitors don't have the problem of SSR, they have drawbacks oftheir effectiveness being dependent largely on their location. Even when a shunt capacitor is located at the midpoint of a long line, it requires much larger rating to achieve the same level of increase in power transfer as a series capacitor. It can be shown that the ratio of the two ratings is given by (8.1 ) where Q.e and Q.h are the ratings of the series and the shunt capacitor respectively, 6max is the maximum angular difference between the two ends of the line. For 6max in the range of 30° - 40°, Q.e varies from 7 % to 13 % of Q.h. Although the series capacitors tend to be twice as costly as shunt capacitors (per unit var), they are still cheaper to use. In addition, the location of a series capacitor is not critical. The use of thyristor control to provide variable series compensation makes it attractive to employ series capacitors in long lines. A major advantage is that the SSR problem (Torsional Interaction) is significantly reduced. The feasibility of fast control of thyristor valves enables the improvement of stability and damping of oscillations using appropriate control strategies. The first demonstration project ofTCSC was commissioned in 1991 at a 345 kV


207

BREAKER

Figure 8.1.

Single line diagram of a TCSC

Kanawha River Substation in West Virginia, U.S.A. under American Electric Power Company. This was a test installation of thyristor switches in one phase for rapid switching of series capacitor segments and was supplied by ABB, Sweden. In October 1992, the first three phase TCSC was installed at 230 kV Kayenta Substation in Arizona under Western Area Power istration (WAPA). Here a 15 n capacitor bank is connected in parallel with a TCR and permits a smooth and rapid control of (capacitive) reactance between 15 and 60 n through phase control of TCR. ( a varying between 145 0 to 180 0 ) • A large prototype three phase TCSC was installed in 1993 at 500 kV Slatt Substation in Oregon under Bonneville Power istration (BPA). The project was sponsored by Electric Power Research Institute (EPRI) and the equipment was developed by General Electric in U.S.A. Here, six modules of TCSC are connected in series and controlled to provide a variation in impedance from +1.4 n to -16 n.

8.2.2 Operation of TCSC A single line diagram of a TCSC is shown in Fig.8.1 which shows two modules connected in series. There can be one or more modules depending on the requirement. To reduce the costs, TCSC may be used in conjunction with fixed series capacitors. Each module has three operating modes (see Fig.8.2)

208


(a) Byed: Here the thyristor valves are gated for 180 0 conduction (in each direction) and the current flow in the reactor is continuous and sinusoidal. The net reactance of the module is slightly inductive as the susceptance of the reactor is larger than that of the capacitor. During this mode, most of the line current is flowing through the reactor and thyristor valves with some current flowing through the capacitor. This mode is used mainly for protecting the capacitor against overvoltages (during transient overcurrents in the line). This mode is also termed as TSR (Thyristor Switched Reactor) mode. (b) Inserted with Thyristor Valve Blocked In this operating mode no current flows through the valves with the blocking of gate pulses. Here, the TCSC reactance is same as that of the fixed capacitor and there is no difference in the performance of TCSC in this mode with that of a fixed capacitor. Hence this operating mode is generally avoided. This mode is also termed as waiting mode. (c) Inserted with Vernier Control In this operating mode, the thyristor valves are gated in the region of (Q'min < Q' < 180 0 ) such that they conduct for the part of a cycle. The effective value of TCSC reactance ( in the capacitive region) increases as Q' is reduced below 180 0 and maximum for Q' = Q'min (which may be 3 times Xc ). Q'min is above the value of Q' corresponding to the parallel resonance ofTCR and the capacitor ( at fundamental frequency ). Generally, vernier control is used only in the capacitive region and not in the inductive region.

8.2.3 Analysis of TCSC To understand the vernier control operation of TCSC, it is necessary to analyze the TCSC circuit. (see Fig.8.3). For simplicity, it is assumed that the line current is specified and can be viewed as a current source. The equations are

dvc . C dt = l$. () t - lTU

=

(8.2)

=

U 1 when the switch S is closed and U 0 when it is open. The current in the thyristor switch and the inductor, iT, is zero at the instant when the switch is opened. Hence, iT is described by

(8.3)

=

=

Note that when U 0, and the initial current iT 0, it remains at the zero value until S is turned on and U = 1. If the line current i$ is defined by,

i$(t)

= 1m coswt

(8.4)


-

(a) Byed

(b) Thyristor blocked

(c}Vernier operation Figure 8.2.

Operating modes in a TCSC

209

210


--+

S

c

Vc

'-----------t- 1----------'

Figure 8.3.

The TCSC Circuit

the equations (8.2) and (8.3) can be solved if the switching instants are known. The switch, 'S' is turned on twice in a cycle ( of the line current) at the instants (assuming equidistant gating pulses)

= =!-

tl t 3 -!!..::..I!.. w

where 0

< /3 < /3max

}

(8.5)

and /3 is the angle of advance defined by

/3=1f'-a

(8.6)

The thyristor switch turns off at the instants t2 and t4 given by

(8.7) where 0'1 and 0'2 are the conduction angles in the two halves of the cycle. In steady state, 0'1 = 0'2 = 0' with half wave symmetry and 0'

= 2/3

(8.8)


211

The typical waveforms of iT and Vc in steady state are shown in Fig.8.4. In steady state, iT is given by .

K2

cos (3

K - 1

cos K

ZT{t) = -2--1m[coswt - --(3 coswrt], wr

where K = ~ =

Vrx; X";'

Wr

-(3 ::; wt ::; (3

(8.9)

VLC

=

In steady state, the capacitor voltage at the instant wt = -(3 is given by Im X c [Sill . (3 - Kcos (3 tan K(3] VCl = -2-K

1

-

(8.10)

It can be shown that when wt = (3, iT = 0 and

(8.11)

VC{wt = (3) = VC2 = -VCl The expression for vc{t) is given by

:s :s

VC{t) = I,,'~f [- sinwt + KC~~S,.(3{3 sinwrt], -(3 wt (3 } vc{t) = VC2 + ImXc[sinwt - sin (3], (3 < wt < 7r - (3

(8.12)

The voltage across the capacitor is not sinusoidal in steady-state. The fundamental component of Vc is given by

VCF = -41~ vc{t) sinwtd{wt) 7r

(8.13)

0

The above equation follows from the fact that Vc has odd symmetry about the axis wt = O. The ratio of VCF to 1m is the equivalent reactance of TCSC and is given by

x

- VCF _ X _

TCSC -

1m -

+

C

4xl;

xl;

(Xc - XL)

(2(3 + sin 2(3) 7r

cos 2 (3 [K tan K(3 - tan (3]

-:-::-::-~--:--:-::----:- .:....-----~

(Xc - XL) (K2 - 1)

7r

(8.14)

The capacitor voltage contains odd harmonics of the order

n=2k-1,

k = 1,2 .......

(8.15)

The magnitude of the harmonic voltage, VCn is given by

100 VCn

VCF

= 0.77 [sin (3(1 n

(l-n)

_ 0.77 cos(3 [sin(3(K n COSK(3 (K - n)

n)

+ sin(3(l + n)] (l+n)

n) + sin(3(K + n)] (K

+ n)

(8.16)

212


0.8

0.6 0.4

0.2

-

!!

...

.

-0.2

-0.4 -0.6

-0.8 -50

350

(a) iT vs wt(degrees)

u

>

.,1

(b) Vc vs wt( degrees) Figure 8.4.

The variation of

The Waveforms of iT(t) and vc(t)

Xr;;c as

a function of f3 is shown in Fig.8.5. Eq.(8.14)

shows that there is parallel resonance of the capacitor and TCR at fundamental frequency, corresponding to the values of f3 defined by 7rW

f3re. = (2m - 1)-2-' Wr

m = 1,2 .....

(8.17)


By appropriately choosing the value of K. =

Wr W

213

it is possible to ensure that there

is only one resonance point for the range 0 < (i < 90 0 • Near the resonance, TCSC has very high impedance and this will result in a very large voltage drop. Hence, it is necessary to operate the TCSC such that XTCSC is not more than Xc a limit (between 2 and 3). The typical value of (ire. = 35 0 •

Figure B.S.

The variation of XTcsc as a function of (i

8.2.4 Control of TCSC The control of TCSC also includes protective functions (protective by). The control functions are partitioned into two levels - common ( to all modules) and the module(Ievel). Commands for the control flow from the common level to the module levels while the status information is sent back from each module level. Module Control Functions There are three basic functions at each module level. These are (a) reactance control (b) SSR damping control (involving modulation of the reactance) and (c) By (for protection) The controller also ensures that the transients associated with mode transitions are minimized. The module controller executes the ordered change to reactance within one half cycle. This includes bying, reinsertion and setting the vernier without overshoot. The protective by (TSR mode) is initiated in response to (i) line overcurrent

214


(ii) arrester overcurrent (iii) arrester energy exceeding a limit. The line overcurrent protection detects fault currents in the line and rapidly implements thyristor by to reduce duty on metal oxide varistor (MOV) and capacitor. The by is performed on all the three phases simultaneously. When the line current returns and stays within limits for a preset time, the by is removed. The arrester overcurrent protection detects overcurrents in the arrester and implements thyristor by in the affected phase(s) to reduce varistor duty. The by is removed when the arrester currents reduce and stay below limits for a preset time. The arrester energy protection initiates a by when f i 2 dt in the arrester exceeds its rating. In this case, the by is not removed automatically because of the long thermal time constants associated with excessive arrester energy. Common Control Functions The common level receives signals of line current and TCSC voltage to generate signals for closed-loop control functions. It also receives commands from energy management center for setting power order. The major control functions in a TCSC are described below. Power Scheduling Control The simplest type of power scheduling control adjusts the reactance order (or setpoint) slowly to meet the required steady-state power flow requirements of the transmission network. The adjustment may be done manually or by a slow acting loop. An alternative approach is to use a closed-loop current control in which the measured line current is compared to a reference current (which may be derived from the knowledge of the required power level). An interesting approach to power scheduling is one where during a transient, the line in which TCSC is situated carries the required power so that the power flow in parallel paths is kept consant. This is equivalent to maintaining the angular difference across the line a constant and has been termed as Constant Angle (CA) control [Padiyar and Uma Rao (1995)]. Assuming the voltage magnitudes at the two ends of the line are regulated, maintaining constant angle is equivalent to maintaining constant voltage difference between two ends of the line. Both CC (Constant Current) and CA controllers can be of PI type with dynamic compensation for improving the response. The steady state control characteristics of both CC and CA control are shown in Fig.8.6 (a) and (b) respectively. Assuming VTCSC to be positive in the capacitive region, the characteristics have three segments OA, AB and BC. The control range is AB. OA

215


h C

C B

Iref

""'/'---~A

/

/

/

/

/

/

/ VTCSC

(a) CC CONTROL Figure 8.6.

/

/

/

o

VTCSC

(b) CA CONTROL

Control Characteristics

and BC correspond to the limits on XTcsc. In Fig.5.23 (b), the control range AB is described by the equation (8.18)

where h is the magnitude of the line current, XLR is the net line reactance ( taking into the fixed series compensation if any), VLO is the constant (regulated) voltage drop across the line (including TCSC). Thus, the slope of the line AB is XLR. Line OA in Fig.5.23 (b), corresponds to the lower limit on TCSC reactance while BC corresponds to the higher limit on TCSC reactance. Power Swing Damping Control (PSDC) This is designed to modulate the TCSC reactance in response to an appropriately chosen control signal derived from local measurements. The objective is to damp low frequency swing modes (corresponding to oscillation of generator rotors) of frequencies in the range of 0.2 to 2.0 Hz. One of the signal that is easily accessible is the line current magnitude. Alternatively, the signal corresponding to the frequency of Thevenin (equivalent) voltage of the system across the TCSC can be used. This signal can be synthesized from the knowledge of voltage and current measurements. Transient Stability Control (TSC) This is generally a discrete control in response to the detection of a major system disturbance. The discrete or bang-bang control of TCSC in response to signals from locally measured variables is described by Padiyar and Uma Rao (1997). The controller is activated immediately after a major disturbance such as clearing of a fault and is deactivated when the magnitude of frequency deviation is below a threshold. This type of control is beneficial not only in reducing the first swing but also for damping subsequent swings. Subsynchronous Damping Control (SSDC) The use of vernier control mode at the module level by setting the reactance

216


setpoint at the requisite (minimum) level is often adequate to damp subsynchronous oscillations caused by series resonance in the line and sustained due to torsional interaction. However in some cases, the constant reactance control may not be adequate. In such cases, a damping control is added. The control signal is based on the synthesis of speed of remote turbo-generators. The control signal can be derived from the locally measured current and voltage signals. The coordination of control actions of all modules in a TCSC is carried out by devising a suitable logic. For example, at Slatt substation, the highest priority is given to the need to tackle SSR, which determines the minimum number of modules to be inserted and their minimum reactance. The power scheduling control has the next priority. Even here, there are two options - one based on minimizing the losses and other based on maximizing smooth operation (avoiding stepped variation in reactance order). It is to be noted that vernier operation results in increased losses in a module. Power swing damping control has the next priority in modulating the set point for reactance. This can be replaced by transient stability control wherever required. Under normal operational conditions, all the modules may not be required. To ensure long term duty for each module in a TCSC, the control logic also incorporates a rotation feature,according to which, the module in series are rotated if no insert/by operations occur for some preset time (sayan hour). The rotation is performed without changing the net reactance.

8.3

MODELLING OF TCSC FOR SSR STUDIES

TCSC is modelled asa variable inductive reactance (of TCR) in parallel with a fixed capacitor. The TCR susceptance (corresponding to fundamental frequency of the line current) is a function of the firing angle Q', or angle of advance, j3 and can be computed from Eq.(8.14) as

1

BTCR(Q')

= -X - X C

1 TCSC

()

(8.19)

Q'

It is to be noted that the above expression is different than the one derived

in chapter 7 for TCR connected across a sinusoidal voltage source. The difference arises from the fact that the TCR in a TCSC is connected across a non-sinusoidal voltage (across the capacitor Xc). For SSR studies it is not necessary to model the gate pulse unit and the generation of gate pulses. It is adequate to assume that the desired value of TCSC reactance is implemented within a well defined time frame. This delay can be modelled by first order lag as shown in Fig.8.7. The value of TTCSC varies from 15 to 20 ms. Xrej is determined by the power scheduling controller or in its absence, by manual control based on order from load dispatch centre. The block diagram of constant current (CC) or constant angle (CA) controller is shown in Fig. 8.8. Tm is the time constant of first order low filter as-


217

aux Xmax

XreJ

;-----+--~ l+.1'TCSC

+

XTCSC

Xmin

Xmod

MOD.

CaNT

u

Figure 8.7.

Block diagram of TCSC

X

re ! ;---+-

m

Figure 8.8.

Block diagram of CC or CA controller

sociated with the measurement of line current h and the TCSC voltage. S=O for CC control and S = -Xl for CA control. X LR is the net reactance of line LR given by (8.20) where XLine is the line reactance and XFC is the reactance of the fixed series capacitor if any. Generally, TCSC will be used in conjunction with fixed series capacitor to minimize the overall cost of compensation while providing effective control for stability improvement. The regulator block diagram is shown in Fig.8.9. This consists of mainly PI controller and phase lead circuit if required. f{p, the proportional gain can be set to zero if only integral control is used. The gain f{ [ is positive in the case of current control and negative in case of CA control. In the latter case Ire! is actually the voltage reference divided by X. Hence positive error signal implies the net voltage drop in the line is less than the reference and XTCSC (assumed

218


Ie

r

1 + sTl 1 + sT2

f{/ -

s

j

+

E

+

f{p '--

Figure 8.9.

1 + sTc

Block diagram of the Regulator

to be positive in capacitive region) is to be reduced. On the other hand for current control, if the error is positive, the controller has to increase XTC sc to raise the line current to reduce the error. The modulation controller is designed to damp power swings in the line by using a control signal derived from local measurement. A convenient signal to use is the magnitude of the line current. The control configuration is similar to the SMC used in SVC. Xaux could represent the increase in the reactance order required for mitigating SSR or improving transient stability. The limits on the total reactance order Xdes are based on the TCSC capability. For a single module,the operational region of TCSC in the VTCSC - h plane is shown in Fig.8.10(a). The operation region in (x).Fc~C) - h plane shown is in Fig.8.IO(b). The voltage and reactances are considered to be positive when TCSC is in the capacitive region. The line OA in Fig.8.10( a) corresponds to the constant reactance of TCSC at its limit X maxO . This is selected based on the TCSC design and should be such that the TCSC does not operate close to the resonance point which would be inherently unstable. A typical value of XmaxO (in pu of Xc) is 3.0. Line AB corresponds to maximum voltage rating of TCSC. Line OB corresponds to the operation of TCSC with thyristors blocked. Here X TCSC = 1 pu. The line OC has negative slope corresponding to Xby (which is negative) in the by mode of TCSC. Line OD corresponds to the upper limit on the inductive reactance (X minO ). Line DE corresponds to the voltage limit of TCSC in the inductive region. Line EC corresponds to the limit imposed on the thyristor current in the inductive vernier mode. Under normal conditions, the TCSC operates only in the first quadrant of both VTcsc - h plane and XTCSC - h plane. The operation at constant maximum voltage (across TCSC) implies that X TCSC is reduced in inverse ratio of the line current. The equations for TCSC can be written down from the block diagrams shown in Fig.8.7 to Fig.8.9. For transient simulation,it is necessary to model the operation of firing pulse generator also.ln this case, the block diagram gi ven in Fig. 8.7 is modified as shown in Fig.8.Il. Here Xdes is the input signal to a linearization block which computes O:order based on Eq.(8.14).The operating mode selector block is used


B

A

VTCSC

o

c D

E

(a) V-I plane

Xmaxo XTCSC

Xc 1.0

X mino

(b) X-I plane Figure 8.10.

TCSC capability curves for a module

219

220


Linearization

Blocking By Mode Mode

Synchronization Signal

block

aorder

Operating

I----+---{ ode Selector

Figure B.11.

Firing Pulse Generator

Firing Pulses for TCR

Extended TCSC control structure model for transient simulation

for the protection of the capacitor or metal oxide varistor (MOV) during short circuit conditions.By mode is activated whenever the line current or MOV current exceeds limits specified.The by mode fires all valves at a = 90° resulting in full conduction of TCR. The blocking mode operates when the capacitor is to be inserted and avoid vernier mode operation. The firing pulse generator has to be synchronized with either the capacitor voltage or the line current.It is observed from transient simulation [Pilot to et al (1996) , Angquist et al (1996)] that synchronization with the line current is definitely superior as the current has negligible harmonics.Angquist et al (1996) suggest using directly the capacitor voltage boost control (during vernier operation) instead of impedance control based on firing angle, as the steady state characteristic of TCSC (impedance) is highly nonlinear, particularly when the natural frequency (w r ) is high ( say, around 5 pu) .It is to be noted that if only one resonance value of f3 is to be permitted in the range o < f3 < 90°,then Wr < 3 (see Eq.(8.17)).Typical values of Wr (used in the Slatt project) is around 2.75 pu. There is danger of high capacitor voltage and harmonic instability [Jalali and Lasseter (1992)] if the operating value of f3 strays near f3res. This underlines the need for direct voltage control of the capacitor. Additional advantage of the voltage control is claimed to be linearity. Angquist,Ingestrom and Othman (1994) describe a control method for TCSC ,termed as Synchronous Voltage Reversal (SVR) scheme. This is explained below. Synchronous Voltage Reversal The control of TCSC can be viewed as the injection of controlled voltage source in series with the transmission line in which TCSC is connected. The constant impedance control can be viewed as injecting a voltage source whose magnitude is proportional to the line current magnitude. When the thyristor valves do not conduct, the voltage across the capacitor is sinusoidal and lags the line current ( assumed to be sinusoidal) by 90°. The triggering of a thyristor valve results in the flow of current in the inductance. The thyristor valve turns off when the current goes to zero. For small values of f3,the current pulse is narrow. At the end of the current pulse, the capacitor

221


voltage is reversed. Neglecting losses and assuming the inductance to be small (or the initial capacitor voltage is high), the capacitor voltage at the end of conduction of the thyristor is exactly equal to the negative of the inital capacitor voltage (vGo) and a total charge of Qr = 2CvGo es through the inductance.This is an approximation whose validity assumes that Qr dominates over the charge provided by the line current. With this assumption, the conduction angle of the thyristor is assumed to be (J'

=

rrwB

(8.21)

The influence of the line current is to reduce the conduction angle. However, the transient in the TCSC circuit caused by the thyristor turn-on can still be characterized as capacitor voltage reversal. Fig.8.12 shows the current pulse in the thyristor and the line current ( which is assumed to be constant during the conduction of the thyristor). :

_IL

----.;-l7l~'--' :

i

-----;---

: :

:

IT

__

.~-

(a) line and thyristor curren'ts

--, (b) ac1:Uai voltage reversal

boost:

(c) equivalent instantaneous voltage reversal

Figure 8.12.

Voltage reversal across the capacitor

It is to be noted that the line current is at its peak when the capacitor voltage es through zero. Fig .8.12 (b) and (c) shows the capacitor voltage reversal. While the actual voltage reversal takes finite time, it can be approximated as instantaneous as shown in Fig.8.12 (c) for simplifying the analysis. Fig. 8.13 shows the approximate steady state voltage waveform which is sinusoidal neglecting the voltage boost that occurs periodically at the interval of where T is the period of the fundamental component of the line current. The presence of subsynchronous frequency component in the line current would result in extra offsets in the capacitor voltage.If only conventional capacitor is used, the period (or interval) between voltage crossings is not equal (see chapter 4 where NGH damping is described). However in a TCSC, to using the concept of Synchronous Voltage Reversal ( SVR) implies that zero

t

t

222


0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -2

-1

0

2

3

4

5

(a) iL vs wt (radians) 1.5,---,------,-----.----.-------.-------,-----,

0.5

0

-0.5

-1

-1.5 -2

-1

0

2

3

4

5

(b) Vc vs wt (radians) Figure 8.13.

Idealized (approximate) waveforms

crossings of the capacitor voltage are equidistant ( and synchronized to the line current). In addition, the voltage boost can also be controlled either in relation to the line current magnitude (constant reactance control) or as part of higher level control scheme to provide damping of interarea low frequency oscillations and improve transient stability.


223

Remarks 1. The SVR scheme is similar to the vernier control scheme described earlier. It must be noted that SVR scheme is primarily designed for TCSC schemes where the vernier control is not provided. The thyristor control is intended to either switch in (insert) or out (by) the series capacitore.In such schemes,the in thyristors. On inductance value is kept low and just enough to limit the the other hand with schemes where vernier operation is provided, the inductance value is higher to prevent the multiple resonance points in the entire range of f3 varying from 0 to 90 0 • 2. It will be shown ( in the next section) how the concept of SVR can result in preventing or mitigating torsional interactions in a range of subsynchronous frequencies.

*

8.4

MITIGATION OF SSR WITH TCSC

The operation of TCSC using constant reactance control (XTcsc > Xc) using vernier mode or SVR is sufficient in most of the cases to avoid torsional interaction which is a serious problem with fixed series compensation. This feature of TCSC is due to the following characteristics (of TCSC ). 1. For the same compensation level that results in fer = fo - fm (where fer and fm are the electrical resonance, and the critical torsional mode frequency ) with a fixed series compensation, a TCSC avoids the resonance condition by detuning ( which results from different frequency response characteristics than a fixed series capacitor) .In other words, although the reactance of a TCSC and fixed capacitor may be identical at the fundamental frequency, their impedances are vastly different at subsynchronous frequencies). 2. For sufficiently large values of XTSCSC (~ 2Xc),which also imply conduction angle exceeding a minimum limit, TCSC appears inductive at subsynchronous frequencies of interest. This also means that the resonance frequency in the elctrical network is reduced for the same level of compensation ( compared to fixed series compensation) .Hedin et al(1995) mention that in a particular case,fer reduced from 40 Hz to 23 Hz. 3. Although the resistance of the TCR is small, the effective resistance that comes in series with the transmission line is high at subsynchronous frequencies. Also, the resistance increases with the frequency which implies that high compensation levels with TCSC are permissible without worrying about torsional interaction. Although the objective of NGH damping is also to increase the effective series resistance of the transmission line, it does not have the advantage of inductive characteristics at the subsynchronous frequency. While increased resistance is beneficial under worst case conditions (fer ~ fo - fm)it can increase negative damping in the off tune conditions. The frequency characteristics of a TCSC will be derived, for small perturbations.For simplicity ,it is assumed that the voltage reversals are instantaneous. Consider that time is discretized. Consider an interval of (tk ,tk+l ) where the sampling instants tk and tk+l are selected as midpoints between voltage

224


reversals. For, tk 2: t

< tk+t, we have

For tk+t < t < we can express

dvc . C(8.22) =ZL dt tk+l also, the above equation applies. Linearizing, Eq.(8.22),

~vc (k

~vc(k

1

+ -) = 2

~vc(k)

1 1t(k+~)

+-

C

1

(8.23)

~idt)dt

1

tk

1

+ 1) = ~v~(k + 2) + C

t (k+1)

(8.24)

~idt)dt

t(k+~)

where ~vc (k+~) and ~v~ (k+~) are the perturbations in the capacitor voltage just prior to and immediately after the (instantaneous) voltage reversal. Assuming losses in the circuit, we can write (8.25) where D is the loss factor (D < 1) Substituting Eq. (8.25) in (8.24) and using Eq.(8.23}, we get ~vc(k+1)

1

=

1

-D~vc(k)-D[~q(k+2)-~q(k)1+~q(k+1)-~q(k+2) (8.26)

where ~q is the change due to the line current Taking z-transform of Eq.(8.26) we can derive

~iL.

(zt - D)(zt - 1)

~Vc(z)

(z

~Q(z)

(8.27)

+ D)

In continuous time, the transfer function relating ~ Vc and ~Q using Laplace variable is obtained from Eq.(8.27) by substituting z esT, where T Wo is the operating radian frequency. Thus

= :0'

=

~Vc(s)

(e~ - D)(e~ - 1)

~Q(s)

(e::+D)

The frequency response is obtained by substituting s

(8.28)

= jw in the above expres-

SlOn.

Since,

~Q(s)

= ~h(s)

(8.29)

sC We can express the apparent impedance of TCSC as .

Zapp(Jw)

~Vc(jw)

.

i.!:!..!!.

Wo (e 2 ",0

= AI (. ) = -JXc LJ.

L JW

W

~

-

D)(e 2 ",0 -1)

.i!::!.2!.

(e"'o

+ D)

(8.30)


225

o.45,---.,---,---,-----r---r----,-----,--,----.,-------, 0.4 D.095

0.35

.... D.0.98

0.3

10 .25 Z

a: 0.2 0.15 0.1 0.05

.' . ........................... . .... .

~~-~5-~10~~,5~~~L--2~5--~~-~35--4~0--4L5-~~

, (Hz)

(a) 3.5 r - - , - - , - - - - . - - - , - - - - , - - - - - , - - , - - - - , - - - . , - - - ,

3

D .095

.... D. 0.98 2.5

0.5

~~~~-~10~~,~5-~2=0-~25~~~7-~3=5--4~0~~4~5-~~ '(Hz)

(b) Figure 8.14.

Real and imaginary parts of apparent impedance of TCSC

The real and imaginary parts of the apparent impedance are shown in Fig. 8.14 for Wo = 377 and two values of D=0.95 and 0.98. It is seen that the apparent resistance increases with frequency and higher for lower value of D (increased losses). The apparent reactance is inductive and is unaffected by the difference in the losses (for realistic values). Example 1 The system considered in this example is adapted from IEEE FBM. The gen-

226


XTC

BTCR Figure 8.15.

Series compensation with fixed and thyristor controlled series capacitor

erator data is same as given in Appendix A. The generator is connected to an infinite bus through a series compensated line (see Fig.5.9). The resistance (RE) and inductive reactance (X E ) external to the generator are assumed to be RE 0.02, XE 1 Three cases are considered namely, (a) The series compensation consists of a fixed capacitor (Xpc) of value equal to 0.63 . (b) The series compensation is made up of a fixed capacitor (Xpc = 0.3) and a TCSC connected in series. The TCSC has a capacitor (XTC = 0.15) in parallel with a TCR (see Fig.8.15).The TCSC operates with XTCSC = 2.2XTC (c) The series compensation is made up of a fixed capacitor (Xpc = 0.48) and TCSC in series. The parameters of TCSC are given in (b). The operating values of XTCSC = 1.1XTc. It is to be noted that in both cases (a) and (b), the level of series compensation is same, namely 63% of the total external inductive reactance. For case (c) , the level of series compensation when the TCSC is blocked is also 63%. With constant reactance control of TCSC,it is reasonably accurate to ignore the model of the controller and represent TCSC as a fixed series capacitor (XTC) in parallel with a fixed reactor whose value is determined from the operating value of XTCSC ( see Eq.(8.19)). The linearized system model is obtained at the operating point : Eb 1.0 ,Pg 1.0 ,Vg 1.0 .The synchronous machine is represented by model(1.1). The mechanical damping is considered. The static exciter is considered with J{A 200 ,TA 0.025 s .PSS is also modelled as described in chapter 5 with the parameters. J{s = 4.5, Tl = T3 = 0.10, T2 = T4 = 0.045, (= 0.6, Wn = 22, Tw = 10

=

=

=

=

=

=

The eigenvalues correponding to the torsional modes are given in table 8.1 for the three cases considered. It is observed that with fixed compensation, mode 1 is destabilized as fer ~ fo - f m .However ,for cases (b) and (c), all the torsional modes (including mode 1) are well damped .. What is interesting is


Line current magnitude

1.1

227

Torque (GEN-LPB shaft)

1.4 j 1.2 Ii 1

::J

Ii

0.9

0

2 3 Time in sec. Figure B.16.

0.8 0.6

4

0

2 3 Time in sec.

4

Simulation results with TCSC and series capacitor

that in case (c), even with a large share of fixed compensation,the presence of TCSC operating in the vernier mode is adequate to mitigate SSR. The transient simulation of the system is also carried out using MATLAB/ SIMULINK package.The disturbance considered is a 0.02 pu step reduction in the infinite bus voltage, initiated at t = 0.3 s .The results for the case (c) are shown in Fig. 8.16. Here, the TCSC is initially blocked.ln this condition, there is no difference between fixed and thyristor controlled series capacitor. As the first torsional mode is unstable (see Table 8.1 ,column 1),both line current magnitude and the shaft ( GEN-LPB) torque grow until at t=l.Os, the TCSC is switched in to vernier mode with XTCSC = 1.lpu(of XTC ).Since the system is stable for this condition,the oscillations decay.

Table B.l.

Mode

5 4 3 2 1 0

Eigenvalues correponding to torsional modes ( Example 1)

(a) Fixed Series Capacitor (XFC= 0.63) -1.8504 ± j298.1694 -0.3659 ± j202.8451 -0.6476 ± j160.4749 -0.0689 ± j126.9443 2.9112 ± jlO1.3693 -4.1142 ±j9.1245

(b)

FSC

+ TCSC

(c)

FSC

+ TCSC

(XFC= 0.30)

(XFC= 0.48)

-1.8504 ± j298.1695 -0.3829 ± j202.4152 -0.6461 ± j160.5797 -0.0678 ± j126.9239 -0.2469 ± j98.7561 -3.7992 ± j9.4279

-1.8504 ± j298.1694 -0.3663 ± j202.9363 -0.6466 ± j160.4856 -0.1238 ± j127.4551 -0.3285 ± j99.3661 -4.2688 ± j9.2858

Example 2 [Geetha (1996)] Here, IEEE SBM system is studied. Three control strategies are considered

228


(a) constant impedance ( reactance) control (b) constant current control and (c) constant angle control. For cases (b) and (c),the controller is modelled as shown in Fig. 8.8. The net series compensation level of 50% is assumed. This is made of (A) Fixed series capacitor only (XFC = 0.5XLl) (B) TCSC and fixed series capacitor (XFC = 0.25XLl,XTCSC = 1.667XTc,XTC = 0.I5XLr) Th exciter data is same as in Example 1. The PSS considered is assumed to have the transfer function PSS(S) =

1

sTw

+ sTw

/{s 1 + sTl 1 + sT2

PSS Data: Tw = 10 s, Tl = 0.048 s, T2 = 0.032 s, /{& = 12 The generator is represented by model (1.I).The regulator in Fig. 8.8 is assumed to be a pure integral type.TTCSc is assumed to be 0.005 sand Tm is neglected. For the operating condition,Pg = 1.0 ,Vg = 1.0 and a power factor =0.9 lagging at the generator terminals,the system is linearized and the eigenvalues corresponding to the torsional modes are given in Tables 8.2 and 8.3.Table 8.2 shows the results for the case of (A) fixed series compensation alone and (B) fixed series capacitor with TCSC opearating at constant reactance control.The effect of PSS (without torsional filter) on the damping of the first torsional mode is also demonstrated.

Table 8.2.

Eigenvalues correponding to torsional modes ( Example 2)with fixed capacitor and constant reactance control of TCSC

Capacitor I I Mode I Fixed(XFCSeries = 0.5XLl) 3 2 1 0

-0.0479 ± j321.051 0.0068 ± j203.365 0.3467 ± j155.412 -2.8531 ±jlO.129

FSC + TCSC Without PSS -0.0491 ± j321.051 -0.0477 ± j203.406 -0.0486 ± j155.064 0.413 ± j9.447

FSC + TCSC With PSS -0.0479 ± j321.051 -0.0217 ± j203.413 0.1998 ±jI55.126 -2.9777 ± jl0.323

Table 8.3 shows the results for the cases with (b) constant current and (c) constant angle control.It is observed that while the constant current controller increases the damping of the swing mode,it increases the negative damping of mode 1. The constant angle control improves the damping of both modes'I' and '0' (swing mode).The PSS signal is taken from generator rotor speed which results in torsional interaction unless countered by providing a torsional filter ( not considered here).

229


Table B.3.

Eigenvalues correponding to torsional modes,TCSC with CC and CA controller ( Example 2)

I Mode I 3 2 1 0

Figure B.17.

With CA (I(I

-0.0478 -0.0462 -0.0048 -3.076

With CC

= -52) ± j321.051 ± j203.429 ± j155.163 ± j12.643

(/(1=5.0) -0.0478 ± j321.051 -0.0209 ± j203.412 0.2068 ± j155.125 -4.199 ± j9.526

Reactive voltage injection in the middle of a transmission line

An alternative to torsional filter is to use a composite control signal ( derived from rotor speed and electrical power) as discussed in chapter 5. 8.5

B.5.1

STATIC SYNCHRONOUS SERIES COMPENSATOR (SSSC)

Description

The SSSC is similar to STATCON (or STATCOM) in the sense that both devices use voltage source converter (VSC).While a STATCON is shunt connected device where the voltage output of VSC is controlled to regulate the reactive current drawn (or injected) by the STATCON, SSSC is used for the injection of a controllable reactive voltage source in series with the transmission line in which it is connected. The major objective of SSSC is to control or regulate the power flow in the line in which it is connected while a STATCON is used to regulate the voltage at the bus where it is connected. The influence of injecting a reactive voltage on the power flow can be analyzed by considering a symmetric loss less line shown in Fig.8.17. Here,the voltage at the two ends of the line are assumed to have the same magnitude V for simplicity and the reactive voltage Vq is injected at a location midway between the two ends of the line. It can be shown that the line current magnitude is given by

I _ 2V sino/2 L -

XL

+

~ XL

(8.31 )

230


I\!f\/\

Transmission Line

VSC

Figure 8.18.

SSSC

The power flow in the line P is given by

P

=Vh

cos (J/2)

= V XLsinJ + -VV q cos(J/2) XL 2

(8.32)

The above expression can be compared with that in the case of series compensation , given by (8.33) where k. e is the degree of series compensation. It is observed that for small values of J,the increase in the power flow introduced by SSSC is constant and nearly independent of J.On the other hand,the power flow increased by series compensation is a percentage of the power flow in the uncompensated line.Thus,at J = 0, the series compensation has no effect on the power flow. The schematic diagram of a SSSC is shown in Fig.8.18. The dc voltage in the VSC can be maintained by a capacitor just as in the case of a STATCON.However,the control of reactive volage output of SSSC is complicated unlike in the case of a STATCON.This is due to the fact that the magnitude of the output voltage is directly proportional to the capacitor voltage ( with Type II controller) which has to vary over a wide range in the case of a SSSC. In a STATCON,the capacitor voltage variation is marginal over the entire range of reactive current output. Type I control, with control over both magnitude and phase of the output voltage has greater control flexibility but results in increased losses,harmonics and complex circuitry over Type II control. Just as in the case of a STATCON,it is possible to exchange active power by connecting the SSSC to an energy source or a sink.Thus , it is possible to compensate for the line resistance by injecting a voltage source in the line in phase with the line current. This enables increase in the ratio of the transmission line which tends to decrease as series compensation is increased. Increasing ratio has the advantage of (a) increasing power transfer capability and (b) reducing the reactive power requirements at the receiving end of the line [Gyugyi

i

i

231


Transmission line

I

!VIV\

VSC 2

t

series Transformer

shunt Transformer

~

VSC 1

T

t

control Figure 8.19.

UPFC

et al(1997)]. It is possible to combine a STATCON and SSSC connected at the same location to form Unified Power Flow Controller (UPFC). See Fig.8.19. This arrangement eliminates the need for a separate energy source as VSC1 and VSC2 can exchange active power through the dc link.Thus ,a UPFC can be used to regulate three quantities in the transmission line - active power and reactive power (or voltages) at the two ends of a line. To summarize, the SSSC has the following unique features which result in certain advantages over TCSC. 1. SSSC can generate a controllable reactive voltage over an identical capacitive and inductive range independent of the magnitude of the line current. 2. With an external dc power supply,SSSC can supply power to compensate the resistance in a transmission line. The objective is to keep effective ratio high,independent of the degree of compensation. 3. It has better control over the line flow compared to TCSC.The power flow can even be reversed if required. 4. Unlike series compensation, an ideal synchronous (fundamental frequency) voltage source does not cause electrical resonance in the line and hence the SSSC is expected to be SSR neutral.

i

8.5.2 Control of SSSC The voltage injected by SSSC is predominantly reactive (in quadrature with the line current).As the losses in SSSC have to be supplied by the system, there is also a small (negative) component of the voltage source in phase with the line current (v,,). The reactive voltage order is set by the power controller as shown in Fig.8.20. The regulator shown in Fig.8.20 can be a P-I controller.

232


+

E

REG

k~ 1+STw

Figure 8.20.

Power controller

An auxiliary signal derived from the phase of the line current (¢) is used to stabilize the control loop.lt is to be noted that a power controller cannot be used when there is no parallel path.ln this case, Vqo may be set at a value that varies slowly depending on the loading conditions.To improve system stability,a damping controller is also provided. There are two controller structures,as with a STATCON,that can be used for SSSC. Type I controller Here,both magnitude and phase of the converter output voltage are controlled. The capacitor is maintained at a constant voltage by controlling the voltage component in phase with the line current. The reactive voltage reference is obtained by summing Vqo with ~Vq , (output from the damping controller). (see Fig.8.21). The capacitor voltage reference can be varied (depending on reactive voltage reference) to give reduced harmonics. [Hatziadoniu and Funk (1996)]. The outputs of the controller are real and reactive voltage orders. The magnitude and phase angle orders are obtained as follows (8.34)

,=

Vqref t an -1 v"ref

(8.35)


Modulation

+

Damping

Signal

233

+ Vqref

Controller

Vdcref

V pref

+

REG

Figure 8.21.

Type I controller

where"Y is the angle of the reference voltage vector in relation to the line current vector. The absolute phase of the reference voltage vector (with reference to a synchronous reference frame) is given by (8.36) where ¢J = tan- 1 ~ is the phase of the line current vector. The magnitude control is normally achieved by Pulse Width Modulation (PWM) using three level converters. However, it is also possible to vary the magnitude by employing two square wave converters with a phase displacement as shown in Fig. 8.22. Assuming each converter is 12 pulse, the magnitude (V) of the resultant voltage injected is given by V =

4J6

--Vdc 7r

cos B

(8.37)

This magnitude can be varied by changing B. Type II Controller If no energy source is connected to the dc side of the converter, then only the reactive voltage control is required. This can be done by controlling the phase angle of the injected voltage whose magnitude is dependent on the capacitor voltage. This is termed as type II controller and is shown in Fig. 8.23. The operation of Type II controller, with no external dc power supply, is dependent

234


D

-

-

-

- - - -, V ,

,,

, ,,

I

Q Figure 8.22. VqreJ

Magnitude control

+ REG

Figure 8.23.

Type II controller

on the charging of the capacitor by the line current. Hence, this controller may not be satisfactory at low power level. 8.6

TORSIONAL INTERACTIONS WITH SSSC

If it is assumed that a SSSC injects an ideal synchronous voltage source (at fundamental frequency) whose magnitude and phase are independent of the line current, then, it does not contribute to series resonance in the ac line. Hence, it can be said that such an ideal SSSC is SSR neutral. However an ideal SSSC is not realistic, as flow of subsynchronous frequency currents in the line can affect the capacitor voltage which in turn can modify the injected voltage. Instead of injecting a constant synchronous voltage, the voltage vector can be kept continuously and instantaneously in quadrature with the line current vector by suitable control (discussed in the previous section). This ensures that the instantaneous total power at the terminals of the SSSC will be maintained at zero and would imply constant voltage across the dc capacitor. Such a SSSC is termed as an " energy neutral" device by Gyugyi et al (1997).


235

They also postulate that an energy neutral device cannot contribute to SSR which essentially involves oscillation of energy (and power) between a generator and energy storage components in the network connected to the generator. Example 3 [Kulkarni (1997)] The torsional interactions with a SSSC are illustrated by taking up a modified IEEE FBM system considered in Example 1. Here, the compensation level is assumed to be 50 % in all the cases studied. Three cases are considered. (i) The series compensator is a fixed capacitor (ii) The compensator is made up of a fixed capacitor (35 %) and a SSSC (15 %). (iii) The compensator is only a SSSC (50 %) The controller was assumed to be type I with Vqref = constant = Vqo. For small signal analysis, the dynamics of the controller and VSC are neglected (as it would be very fast) and the injected voltage is assumed to be of constant magnitude. However the phase of the voltage is variable as it depends on the phase of the line current. Neglecting losses in the SSSC, the D and Q axis components of the injected voltage are given by VqD VqQ

=

Vq

=-

cos¢

(8.38)

sin ¢

(8.39)

Vq

where tan ¢ = ~. The generator is represented by model (1.1). The excitation system data are same as in Example 2. PSS is considered without a torsional filter. The self (mechanical) damping ofrotors is considered, but mutual (shaft) damping are neglected. The operating data is same as in Example 1. The eigenvalues corresponding to the torsional modes and network modes are shown in Table 8.4 for all the three cases considered.

Table 8.4.

Network and torsional modes

Case (i)

Case (ii)

Case (iii)

-3.3038 ± j620.62 -2.1695 ± j132.85 -0.3635 ± j298.18 -0.0332 ± j202.78 -0.2894 ± j160.29 0.1990 ± j127.41 0.3162 ± j100.14 -3.2981 ± jl0.620

-2.8071 ± j569.20 -3.9558 ± j160.83 -0.3635 ± j298.18 -0.0401 ± j202.72 1.4692 ± j160.41 0.0053 ± j127.08 0.1881 ± j99.540 -2.6031 ± j9.3567

-4.4018 ± j289.17 -0.3635 ± j298.18 0.0083 ± j202.96 -0.2576 ± j160.59 -0.0062 ± j127.03 0.1114 ±j99.115 -1.7396 ± j7.5314

-

I Comments Supersyn. Mode Subsyn.Mode Mode 5 Mode 4 Mode 3 Mode 2 Mode 1 Mode 0

It is observed that the first two torsional modes are destablized in case (i) with fixed series capacitor. Actually, while mode 2 is destabilized on of

236


I

·I,-------,w.¥.1 '1,-------, +~ ~ D

Figure 8.24.

Table 8.5.

Damping controller

Network and torsional modes (case (ii) with damping controller)

Without AVR

With AVR

With PSS

I Comments

-2.0607 ± j573.18 -1.5177 ± j168.32 -0.3637 ± j298.17 -0.5517 ± j204.88 -1.2192 ± j155.24 -0.0815 ± j126. 73 -0.7096 ± j97.492 -0.8603 ± j7.8698

-2.0548 ± j573.20 -1.3784 ± j168.51 -0.3637 ± j298.17 -0.5551 ± j204.90 -1.0572 ±jI55.32 -0.0766 ± j126. 73 -0.6917 ± j97.474 -0.1081 ± j8.5879

-2.0513 ± j573.21 -1.4031 ± j168.34 -0.3637 ± j298.17 -0.5132 ± j204.85 -1.0241 ± j155.43 -0.0588 ± j126. 75 -0.3943 ± j97.821 -3.1524 ± j9.0127

Supersyn Subsyn Mode 5 Mode 4 Mode 3 Mode 2 Mode 1 Mode 0

TI with series capacitor, mode 1 is destabilized due to PSS (without torsional filter). It is observed that with the introduction of SSSC (cases (ii) and (iii)) the negative damping of both modes 1 and 2 are reduced compared to case (i). It is interesting to observe that the subsynchronous network mode frequency (fo - fer) is increased with increase in the compensation level of SSSC. The resonance frequency (fer) reduces as the fixed series compensation is reduced. But it is interesting to note that fer > 0 even for case (iii) without fixed capacitor. This is because, a SSSC appears as variable capacitor with reduced impedance at subsynchronous frequencies. Mode 4 is destabilized in case (iii) due to the action of PSS. Because of resonance, mode 3 is also destabilized in case (ii). This shows that proper care has to be taken in the choice of compensation level. However, if a wide band damping controller is utilized (shown in Fig. 8.24) all the modes can be damped. The data on the controller parameters are l.0 D 5.0,]{ 5.0, Ta 0.02, n 0.001, Tw The eigenvalues corresponding to the torsional and network modes, with damping controller included, are given in Table 8.5. This table also shows the eigenvalues for the condition (i) neglecting AVR and (ii) neglecting PSS, but including AVR. The response of the system with a combination of fixed capacitor and SSSC (case (ii)) is obtained by digital simulation to the disturbance of a step increase of 0.05 pu in the (mechanical) torque produced by the HP turbine. The varia-

=

=

=

=

=


237

85r-----~----~----~----~------~----~----,

84.5 84

~

83.5

~

83

a:

I

~82.5

82

'" 81.6 81 80.5

800~----~----~2~----3~----~4----~5~----~6----~7 time (seconds)

m

'" .-'

O.90'------~------,2"------~3----~~----~----~8~----.J7

time (seconds)

0.9

w

~08

~ a: f2 0.7

§

U O.8

<>

0.5

0.4 '--~:----------=2--------:3':-------:4-------5;:---------:6':----' time (seconds)

Figure 8.25.

Response for step change in mechanical input of HP turbine

tions in the generator rotor angle and the shaft torque (GEN-LPB) are shown in Fig.8.25. The damping controller is considered. There are no oscillations in the rotor angle (6) and the steady state is reached quickly. The shaft torque takes some time to reach steady state, but the torsional oscillations are eefectively damped. It is also observed that the SSSC

238


controller regulates the dc capacitor voltage at 0.75 pu indicating that the SSSC is operating as an energy neutral device.

Appendix A Data on IEEE Benchmark Models

A.1

IEEE FIRST BENCHMARK MODEL ( FBM )

IEEE FBM was created by the IEEE Working Group on Subsynchronous Resonance in 1977 for the purpose of establishing a benchmark model which can be used as a test bench for the comparison of different methods of computer based analysis and simulation. The system consists of a single generator connected to an infinite bus through a single series compensated line as shown in Fig A .1.

XT

R

X A C S Y S INF

BUS

Figure A.I.

IEEE FBM system diagram

Table A.l gives the network impedances in per unit on the generator MVA base of 892.4 MVA. Table A.l Network Impedances Parameter

Positive Sequence

Zero Sequence

R

0.02 0.14 0.50 0.06 0.35

0.50 0.14 1.56 0.06 0.35

XT XL X SYS Xc

Table A.2 gives the synchronous machine parameters.

240


Table A.2 Synchronous machine parameters for IEEE FBM

I Reactance I Value I Time Constant I Value I Xaa Xd

x'd x"d Xq

x'q x"q

0.130 1.790 0.169 0.135 1.710 0.228 0.200

T~o

Til

dO

T~o Til qO

4.300 0.032 0.850 0.050

The reactances are in per unit on the generator base and the time constants are in seconds. From the specified open circuit time constants, the short circuit time constants can be derived from Eqs.(2.74) to (2.77) in chapter 2. These values are given below. T~ = 0.4000, T~' = 0.0259 T~ = 0.1073, T~' = 0.0463 The rotor model of the FBM is shown in Fig . A.2. This is typical of large

668BBB Figure A.2.

Rotor model for FBM

turbine-generator which has several turbine sections modelled separately. The data are given in Table A.3. Table A.3 Shaft inertias and spring constants for the FBM in per unit on the machine base Inertia HP turbine IP turbine LPA turbine LPB turbine Generator Exciter

Inertia I Shaft Section I Spring Constant I Constant (H) K in p.u T /rad 0.092897 0.155589 0.858670 0.884215 0.868495 0.0342165

HP - IP IP - LPA LPA - LPB LPB - GEN GEN - EXE

19.303 34.929 52.038 70.858 2.82

APPENDIX A: DATA ON IEEE BENCHMARK MODELS

241

The damping data is not provided as part of FBM data.The damping is neglected in the case studies given in chapter 4. Whenever the damping is considered, the following data are assumed. Self damping: DHP DIP DLA DLB 0.2 Mutual damping: DHI DIA DAB DBG 0.3, DGE 0.005

=

A.2

=

=

=

=

=

=

=

IEEE SECOND BENCHMARK MODEL ( SBM )

The system diagram for SBM is shown in Fig. A.3. RI

XLI

Xc

INF BUS

Figure A.3.

Second Benchmark Model system

The value of the capacitive reactance, Xc is not specified explicitly. It is specified as a variable taking on values from 10% to 90% of the series inductive reactance of the same line.The network impedance data are given in Table A.4. Table A.4 Network Impedances in per unit based on 100 MVA for SBM Parameter RT XT

RI XLI R2 XL2 Rsys XSYS

I Positive Sequence I Zero Sequence 0.0002 0.0200 0.0074 0.0800 0.0067 0.0739 0.0014 0.0300

0.0002 0.0200 0.0220 0.2400 0.0186 0.2100 0.0014 0.0300

All data are given on a 100 MVA base and the line impedances are on a 500 kV base. The generator is rated at 600 MVA.The reactances and time constants are given in Table A.5.

242


Table A.5 Synchronous machine parameters for IEEE SBM

I Reactance I Value I Time Constant I Value I 0.0045 0.140 1.650 0.250 0.200 1.590 0.460 0.200

Ra Xaa Xd

x'd x"d Xq

x'q x"q

4.500 0.040 0.550 0.090

T~o Til dO T~o Til qO

The rotor model is shown in Fig. A.4.

Figure A.4.

Rotor model for SSM

It has four masses including the exciter. The data are given in Table A.6.

Table A.6 SBM rotor model data Inertia I Damping I I Mass I Ibm - ft Ibf-ft-sec/rad 2

EXC GEN LP HP

1383 176204 310729 49912

4.3 547.9 966.2 155.2

Shaft Section

Spring Constant Ibf-ft/ rad

EXC-GEN LP-GEN HP-LP

4.39 x 10 6 97.97x 10 6 50.12x 10 6

The rotor mode shapes are given in Table A.7 and the computed modal quantities are given in Table A.8. Table A.7 Rotor mode shapes for SBM

I Rotor I Mode 1 I Mode 2 I Mode 3 I EXC GEN LP HP

1.307 1.000 -0.354 -1.365

1.683 1.000 -1.345 4.813

-102.600 1.000 -0.1180 0.0544

APPENDIX A: DATA ON IEEE BENCHMARK MODELS

Table A,8 Computed modal quantities for SBM

I Mode 1 2 3

fk

Uk

Hk

Hz

rad/s

seconds

24,65 32,39 51.10

0,05 0,05 0,05

1.55 9,39 74,80

243

Appendix B Calculation of Initial Conditions

In general,the inital conditions (equilibrium values) ofthe system state variables are obtained by solving the algebraic equations

(B.I) where Ue is the input vector when the system is in equilibrium.lt includes parameters such as input mechanical torque (Tm ) ,infinite bus voltage (Eb), voltage reference to the AVR ( Vre ! ). In power sysem studies,the operating point is established by conducting a power flow analysis;the output from which gives the active power (P), reactive power (Q),voltage magnitude (V) and angle (0) at each bus including the generator bus. This is the starting point for the calculation of the initial conditions of the state variables. Synchronous Generator 1. The armature current (fa ) is calculated from

ra = f a LA. = PVg -L-O jQg 'f'

9

(B.2)

9

(the subscript 'g' refers to the generator terminal bus) 2. Compute Eq and J from

(B.3) where Eq is the voltage behind Xq 3. Compute id ,iq , Vd , and Vq from

Vd

=

sin(J - c/J) iq = fa cos(J - c/J) = -Vg sin(J - Og)

id

Vq

-fa

= Vgcos(J -

Og)

(BA) (B.5) (B.6) (B.7)

246


4. Compute Efd ,E~ ,E~ from EJd = Eq - (Xd - Xq)id E~

= E Jd + (Xd - X~)id E~

= -(xq -

x~)iq

(B.8) (B.9) (B.lO)

5. Compute t/Jd and t/Jq from t/Jd = Xdid

+ E Jd

t/Jq = xqiq

(B.ll) (B.12)

6. Compute t/JJ ,t/Jh ,t/Jg and t/Jk from t/Jf = t/Jd

X'

+ (Xd - d xdI)EJd t/Jh = t/Jd t/Jg = t/Jq t/Jk = t/Jq

(B.13)

(B.14) (B.15) (B.16)

7. Calculate VreJ and Tm from (B.17)

(B.18) For the rotor system,the slips of the various masses (Sm, SHP, SIP etc) cannot be calculated from the equations describing them.They are assumed to be zero if the rotor speed (w o ) is same as the base speed (w B)' The equilibrium values of the shaft torques can be calculated from the state equations for the rotor system.For example,for the four mass system shown in Fig.2.l6 (chapter 2),the initial condtions of the shaft torques are given by

TLG THI

= Te, TIL = TLG - FLpTm = TIL - F[pTm = FHpTm

(B.19) (B.20)

If, instead of shaft torques,the modal angles are used as state variables, their initial conditions can be calculated from the state equations. For example,for the system shown in Fig.2.l6, we have, <11 = (q~ FTm - Te)/ J{1

(B.2l)

<12 = (q~FTm - Te)/ J{2

(B.22)

<13 = (q~FTm - Te)/ J{3

(B.23) (B.24)

<10 =
where ql, q2 and q3 are the columns of the [QJ matrix corresponding to torsional modes 1,2 and 3. [FJ is a row vector defined by

F= [FHP ,FIP ,FLP ,OJ

(B.25)

APPENDIX 8: CALCULATION OF INITIAL CONDITIONS

247

f{l ,f{2 and f{3 are the modal spring constants. Electrical Network The network equations are given by

(B.26) If the matrix [AN] is nonsingular,the inital conditions of the network state variables are given by (B.27)

The input vector UN includes variables such as D-Q components of the generator (armature) currents and infinite bus voltages. For a particular case of a simple network made up of a series R- L-C network connected between a genearator and an infinite bus (with no shunt branch),there are only two state variables VCD and vCQ .The initial values of those are computed from the complex equation (VCQ

where Xc = Wo --L-c

+ JVCD) =

-jXc(iQ

+ jiD)

(B.28)

and (B.29)

Appendix C Abbreviations

ac : alternating current AGC : automatic generation control AVR : automatic voltage regulator CA : constant angle CACV : constant ac voltage (control) CC : constant current (control) CDCV : constant dc voltage (control) CEA : constant extinction angle (control) dc : direct current EPC : equidistant pulse control ESS : excitation system stabilizer FACTS: flexible ac transmission system FCL : fault current limiter GPU : gate pulse unit GTO : gate turn-off thyristor H P : high (filter) HP : high pressure (turbine) HVDC : high voltage direct current Hz : hertz IGBT : insulated gate bipolar transistor IGE : induction generator effect IP : intermediate pressure (turbine) IPC : individual phase control KVL : Kirchoff's voltage law LP : low (filter) LP : low pressure (turbine) MCT : metal oxide semiconductor controlled thyristor MOV : metal - oxide varistor PLL : phase-locked loop PSDC : power swing damping controller

250


PSS : power system stabilizer PWM : pulse width modulation rad : radian rms : root mean square s: second SC : synchronous condenser SCR : short circuit ratio SEDC : supplementary excitation damping controller SMC : supplementary modulation controller SR : susceptance regulator SSO : subsynchronous oscillation SSR : subsynchronous resonance SSSC : static synchronous series compensator STATCOM : static synchronous compensator STATCON : static condenser SVC : static var compensator SVR ; synchronous voltage reversal TCBR : thyristor controlled braking resistor TAR : thyristor controlled phase angle regulator TCR : thyristor controlled reactor TCSC : thyristor controlled series compensator T-G : turbine generator TI : torsional interaction TSC : thyristor switched capacitor UPFC : unified power flow controller VDCOL : voltage dependent current order limiter VSC : voltage source converter VST : voltage setting terminal

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Index

Acceleration signal, 131 Analysis of Graetz Bridge, 138 Analysis of Induction Generator Effect(IGE),83 Analysis of TCSC, 208 Analysis of Torsional Interaction(TI), 87 Antiphase mode, 120 Apparent impedance of TCSC, 225 Automatic Generation Control (AGC), 7 Automatic Voltage Regulators (AVR), 7 Back To Back (BTB) links, 145 Band reject filter, 131 By Damping Filter, 113 By, 213 Calculation of initial conditions, 245 Clarke's transformation, 68 Combined system equations, 103 Common mode, 120 Commutation voltage, 141 Composite signal, 134 Constant AC Voltage (CACV), 145 Constant Angle (CA) control, 214 Constant Current (CC), 145 Constant DC Voltage (CDCV), 145 Constant Extinction Angle (CEA), 145 Control of SSSC, 231 Converter control, 144 Countermeasures for SSR, 111 D-axis equivalent circuit, 37 D-Q Axes impedances, 77 D-Q components, 65 D-Q reference frame, 193 Damping torque analysis, 14 Delay angle, 141 Design of PSS, 126 Dynamic stabilizer, 118 Eigenvalue analysis, 96 Electrical analogue for the torsional system, 46 Electrical system equations, 101

Electro-hydraulic governor, 56 Electromagnetic Transients Program, 59 Electromagnetic transients, 78 Energy neutral, 234 Equivalent circuits, 35 Excitation control system, 41 Exciter mode, 129 Extinction angle, 143 FACTS (Flexible AC Transmission System), 8

Fault Current Limiter, 11 Firing angle control, 147 Gain supervisor, 182 Gate Pulse Unit (GPU), 179 Graetz bridge, 138 GTO thyristors, 190 HVDC converter station, 138 HVDC transmission, 137 IEEE First Benchmark Model (FBM), 239 IEEE Second Benchmark Model (SBM), 241 IGBT,190 Immittance functions, 94 Impedance functions, 75 Implicit algorithm, 79 Induction Generator Effect(IGE), 5 Interarea mode, 121 Interface between AC and DC system, 148 Intraplant mode, 121 Kron's transformation, 65 Local swing mode, 121 Lumped multi mass model, 44 Master control, 146 MCT,190 Mechanical system equations, 101 Mechanically switched capacitors (MSC), 169 Metal Oxide Varistor (MOV), 214 Mitigation of SSR, 223 Modal damping, 49 Modal inertia, 48

262


Modal spring constant, 48 Mode coupling, 13 Mode shapes, 51 Modelling of TCSC, 216 Modelling of transmission network, 64 Module control functions, 213 NGH Damping Scheme, 116 NGH damping, 8 Numerical integration, 78 Operational impedances, 35 Overlap angle, 141 Park's transformation, 22 Per unit quantities, 30 Phase-Locked Loop(PLL)., 147 Pole control, 146 Power angle curve, 174 Power invariant transformation, 28 Power scheduling control, 214 Power Swing Damping Control (PSDC), 215 Power System Stabilizers(PSS), 4, 122 Protective by, 213 Pulse Width Modulation (PWM), 190 Q-axis equivalent circuit, 37 Self excitation, 4 Sequence networks or and (3,69 Series capacitor protection, 112 Short Circuit Ratio (SCR), 145 Speed governor, 55 SSR neutral, 234 Static Blocking Filter (SBF), 113 Static Compensator (STATCOM), 170 Static Condenser (STATCON), 170, 10 Static Synchronous Compensator (STATCOM), 10 Static Synchronous Series Compensator (SSSC), 10, 229 Static Var Compensator (SVC), 10, 169,4 Steam turbine, 55 Subsynchronous Damping Control (SSDC), 215 Subsynchronous Damping Controller (SSDC),167

Subsynchronous oscillations, 3 SubSynchronous Resonance(SSR), 2 Supplementary Excitation Damping Control(SEDC), 114 Supplementary Modulation Controller (SMC),170 Susceptance Regulator (SR), 183 SVC Controller, 177 Synchronizing and damping torque, 88 Synchronizing circuit, 151 Synchronous Condenser (SC), 189 Synchronous machine model, 18 Synchronous Voltage Reversal (SVR), 220 System control hierarchy, 146 Thyristor Controlled Phase Angle Regulator (TAR),10 Thyristor Controlled Reactor (TCR), 8, 170 Thyristor controlled series capacitor, 206 Thyristor Controlled Series Compensator (TCSC), I, 205 Thyristor Switched Capacitor (TSC), 171 Torque angle loop, 15 Torsional filter, 130 Torsional interaction with PSS, 130 Torsional interaction with voltage controller, 200 Torsional interaction, 5 Torsional interactions with SSSC, 234 Transient Stability Control (TSC), 215 Transient torques, 5 Transmission lines, 64 Trapezoidal rule, 79 TSR mode, 213 Turbine generator mechanical system, 43 Unified Power Flow Controller (UPFC), 10, 231 Valve group control, 146 Vernier control, 208 Voltage Dependent Current Order Limiter (VDCOL), 147 Voltage Source Converters (VSC), 189 Waiting mode, 208

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