Calculation of frequency response of thyristor-controlled series capacitor

Calculation of Frequency Response of Thyristor-Controlled Series Capacitor

NAOTO KAKIMOTO and ANAN PHONGPHANPHANEEKyoto University, Japan

SUMMARY

In this paper, we propose a method of analyticallycalculating an impedance of a thyristor-controlled seriescapacitor (TCSC). The TCSC consists of a thyristor-con-trolled reactor (TCR) and a capacitor. If a small voltagecomponent of frequency f superimposes the TCSC voltageof frequency fo, then current components of frequency(nfo ± f) flow through the TCR, where n takes even numbers.We derive theoretical equations for them. In the derivation,we consider an influence of odd harmonics in the TCSCvoltage. We also consider deviations in its zero-crossingtimes due to the superimposed voltage. The current compo-nents flow through the capacitor and the transmission line,and produce new voltage components of frequency (nfo ±f). In order to count their influence, we introduce admit-tance matrices for the TCR, the capacitor, and the transmis-sion line. By solving a network equation consisting of thematrices, we can obtain the distribution of the voltage andcurrent components. We define the impedance of TCSC asa ratio of the voltage component of frequency f to thecurrent one of the same frequency. The impedance analyti-cally obtained corresponds well with that obtained by simu-lations. Its frequency responses vary with the firing methodof thyristors. © 2002 Wiley Periodicals, Inc. Electr Eng Jpn,139(3): 35�44, 2002; Published online in Wiley Inter-Science (www.interscience.wiley.com). DOI 10.1002/eej.1158

Key words: TCSC; frequency response; analyticalsolution.

1. Introduction

Thyristor-controlled series capacitors (TCSC) havebeen developed as a means of increasing the transmittingcapacity of power systems [1]. The TCSC is a series capaci-tor with a thyristor-controlled reactor (TCR) connected inparallel. With conventional series capacitors, there is a riskof torsional resonance (or subsynchronous resonance,

SSR), but the TCSC is designed to mitigate SSR. In thisconnection, analysis using EMTP and analog simulatorshas been carried out, as well as verification experiments onreal systems [2, 3]. In addition, analytical methods basedon Poincaré maps and dynamic phasors have been proposed[4�7].

However, it still remains unclear whether SSR relaxa-tion is related to the characteristics of the TCSC itself, orinstead to the control scheme. There are some reports in thisfield [8�10] with emphasis placed on the frequency re-sponse of TCSC impedance. In the previous studies, veryweak signals were injected into a system, and the timeresponse was calculated using the EMTP program or othertechniques; thus, the frequency response was found byfrequency scanning. However, the calculation takes muchtime, and variation of the firing angle and other parameterscannot be easily handled. In the present study, an analyticalmethod of determining the frequency response is proposed.However, the control system is disregarded in order toexamine the characteristics of the TCSC itself. It is knownthat the TCSC characteristics depend on the firing method[7]. In this study, an analytical solution is derived for threemethods based, respectively, on the current, voltage, andfundamental wave zero point.

An outline of the TCSC is given in Section 2. Theo-retical expressions for the current components of a thyris-tor-controlled reactor are obtained in Section 3. The TCSCadmittance matrix and its application to transmission sys-tem are explained in Section 4. Section 5 presents analyticalresults for frequency response, and a comparison withsimulation results.

2. Outline of TCSC

2.1 Operation of TCSC

A transmission system with a TCSC is shown in Fig.1. Here the transmission line resistance is ro = 0.02 pu, andthe inductance is Lo = 0.002387 pu. The per-unit system isbased on 500 kV, 1000 MVA. As regards the reactance of

© 2002 Wiley Periodicals, Inc.

Electrical Engineering in Japan, Vol. 139, No. 3, 2002Translated from Denki Gakkai Ronbunshi, Vol. 121-B, No. 3, March 2001, pp. 334�341

35

the transmission line, compensation by means of TCSC isassumed; in particular, the TCSC reactor is set to L =0.0001358 pu, and the series capacitance to C = 0.00884pu. The reactor has a resistance of 0.002 pu. The compen-sation rate is adjusted by the thyristor�s firing angle.

The current il of the TCR varies in the following way:

Here v stands for the TCSC voltage. Figure 2 presentswaveforms of the voltage v and current il when a current iwith an amplitude of 1 pu and a frequency of 60 Hz flowsthrough the TCSC. The thyristor is fired according to thezero point of the TCSC voltage v. In the diagram, the firingangle φ is 160°.

The current il includes the fundamental frequency aswell as the third (90.7%), fifth (74.0%), seventh (53.0%),and other odd harmonics. The voltage v includes the funda-mental frequency as well as small harmonics, such as thethird (3.8%), fifth (1.9%), seventh (0.95%), and so on.

2.2 Impedance of TCSC

Consider the following potential difference betweenthe generator and infinite bus:

Suppose that a small frequency component f is superim-posed on the fundamental wave with frequency fo, ampli-tude ao, and phase θo. Let this small component haveamplitude ε and phase θ. Applying Fourier analysis to thevoltage v and current i of the steady-state TCSC, compo-nents vf and if at frequency f are extracted as follows:

where the subscripts s and c denote the sine and cosinecomponents, respectively. The impedance of the TCSC isdefined in terms of the variations ∆vf, ∆if from ε = 0:

where r(f) and x(f) are the resistance and reactance atfrequency f. The impedance varies with frequency, and thefrequency response of the impedance could be obtained byfinding the TCSC voltage/current waveforms at each fre-quency with respect to Eq. (2), and then employing the fastFourier transform. However, this would take an enormousamount of time. An analytical method of finding the imped-ance is presented below.

3. Derivation of Analytical Solution

First consider the voltage v. Since odd harmonics areincluded,

where ai is the amplitude, i is an odd number, and ε is theamplitude of the superimposed component. The variationof the TCR current due to the superimposed component isshown in Fig. 3. Here the fundamental wave amplitude a1

is assumed to be 1 pu, and harmonics to be zero. Now, theamplitude of the superimposed component ε is 0.01 pu, andthe frequency is 20 Hz. As is evident from the diagram, thecurrent components have frequencies of 20, 100, 140, 220,260 Hz, and so on, that is, (nfo ± f), with n being an evennumber, and fo = 60 Hz. This is equivalent to a superim-posed component of 100 or 140 Hz.

3.1 Calculation method

Consider the theoretical expression for the TCR cur-rent variation due to the superimposed voltage componentin Eq. (4). The current component at a frequency of (nfo ±f) Hz is

Fig. 1. Thyristor-controlled series capacitor (TCSC).

(1)

Fig. 2. Voltage and current of TCSC.

(2)

(3)

(4)

(5)

36

Here ωo = 2πfo and ω = 2πf, and T is a sufficiently longintegration period (see details in Appendix 1). The currentii is a pulsating current as shown in Fig. 2, and the integralsin Eqs. (5) and (6) for each pulse (k = 0, 1, 2, . . . ) give

Here t1 and t2 denote the times at which the current com-mences and terminates, respectively. Substituting Eq. (4)into Eq. (1), and integrating from t1 to t (t1≤ t ≤ t2) gives

3.2 Voltage zero point

Consider thyristor firing based on the zero point ofthe voltage v. Without any superimposed component, timepoints t0, t1, t2 when the voltage is zero are defined asfollows:

Here k = 0, 1, 2, . . . , and the superscript o pertains to thecase in which no voltage is superimposed. However, if thereis a superimposed voltage, the time points shift. Assumingt = t0

o + ∆t0 in Eq. (4), we obtain

Terms of second and higher order in ε and ∆t0 can beneglected. Assuming v = 0, the following expression can bederived:

Here h = ω /ωo. The time difference between point t1 (whencurrent begins to flow through the reactor) and t0 corre-sponds to the firing angle φ. Therefore,

The time point t2 also varies with t1, and this variation isdenoted by ∆t2.

3.3 In-phase component

First, we solve the integral in Eq. (7):

Here ω__

= nωo + ω. From the above,

Fig. 3. Variation of TCR current.

(6)

(7)

(8)

(9)

(10)

(11)

(12)

37

Substituting t1 = t1o + ∆t1 and t2 = t2

o + ∆t2, and consideringfirst-order terms with respect to ε, we obtain

Here φ__ = π − φ. For calculation details, see Appendix 2. The

first term ilsko on the right-hand side represents the case in

which no voltage is superimposed, the second term thevariation caused by ∆t1 (= ∆t0), and the third term thevariation caused by the superimposed voltage during [t1, t2].Thus, ∆t2 exerts no influence. The current ils in Eq. (5) canbe obtained by summing over the integration interval [0, T],and then multiplying by 2/T. The variation ∆ils caused bysuperimposed voltage is

Here

3.4 Out-of-phase component

Similarly, the integral in Eq. (8) can be solved as

follows:

Therefore,

Here, too, substituting t1 = t1o + ∆t1 and t2 = t2

o + ∆t2, andconsidering first-order terms with respect for ε, we obtain

The current ilc in Eq. (6) can be obtained by summing overthe integration interval [0, T], and then multiplying by 2/T.Its variation ∆ilc is

Here

However, at n = 0,

The variation of the current calculated by Eqs. (15) and (19)is shown in Fig. 3, exhibiting good agreement with thesimulation results.

Above, the voltage v was that given by Eq. (4). Let

the voltage be

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

38

Then the current variation is

Here

The derivation is similar to that of Eqs. (15) and (19), andhence is omitted here. It should be noted, however, that,comparing to Eq. (11),

Parameters g, b, s, u correspond to admittances that relatesuperimposed voltage to current variation.

Now, as follows from definitions (5) and (6), thecurrent component at a frequency of (nfo � f) Hz can becalculated by the following substitutions into Eqs. (15),(19), and (22):

3.5 Admittances for harmonics

Consider the case in which the frequency f of thesuperimposed component is a multiple of the fundamentalfrequency fo, that is, h = f/fo is an integer. The admittancescan be then derived as

The above expressions pertain to the current components atnfo, and h ± n is an even number. For n, h, h ± n = 0, see Eq.(20).

4. Circuit Equations

4.1 Admittance matrix

As shown in the previous section, the superimposedvoltage at frequency f gives rise to components in TCR

current at fi = nfo ± f. For example, when f = 20 Hz, thecurrent harmonics are fi = 20, 100, 140, 220, 260, . . . Hz.These components flow through the TCSC capacitor andtransmission lines, and produce corresponding voltagecomponents at the capacitor�s terminals, which should begiven consideration. Now let us denote the voltage andcurrent components in order of increasing frequency as v1,v2, v3, . . . and il1, il2, il3, . . .

Here the subscripts s and c denote, respectively, the sine andcosine components. The current components ili and voltagecomponents vj are interrelated as follows:

Here i = 1, 2, 3, . . . . Now,

where the coefficients g, b, s, u are as explained above.Equation (26) may be expressed in matrix terms as

On the other hand, for the capacitance,

With i = 1, 2, 3, . . . ,

In matrix representation, the above expression gives

Now consider the TCSC current components ii (i = 1, 2, 3,. . . ),

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

39

Therefore,

Here I = (i1, i2, i3, . . . )t. The above expression gives the

circuit equation for small voltage/current variations inTCSC.

4.2 Current zero point

Now consider the case of thyristor firing based on thecurrent zero point. Suppose that the current i is

Here ci (i = 1, 3, 5, . . .) is the amplitude of the harmonic.With t0 denoting the time point when the current is zero, thefollowing is true when no current is superimposed:

Supposing that t0 = t0o + ∆t0, the current zero point can

be expressed as

Here

ρ′1 = ρ1 cos ψ − ρ2 sin ψ, ρ′2 = ρ1 sin ψ + ρ2 cos ψ

The terms g1, b1, s1, u1 in Eqs. (15), (19), and (22) are relatedto ∆t1 (= ∆t0), so that sin hkπ → g1, b1 in Eq. (11), and coshkπ → s1, u1 in Eq. (23). Therefore, from Eq. (32),

Here

On the other hand, the terms g2, b2, s2, u2 are related to thesuperimposed voltage during [t1, t2], and hence sin 2πft

→ g2, b2 in Eq. (4), and cos 2πft → s2, u2 in Eq. (21).Supposing that the voltage is

the current variation caused by the superimposed voltage is

From Eqs. (33) and (35), the following corresponds to Eq.(26):

Here

Equation (36) can be represented by the matrix relation

Combining this with Eq. (29) gives

4.3 Application to power system

If there is a component superimposed on the voltagedifference vo between the generator terminals and the infi-nite bus as in Eq. (2), weak components i1, i2, . . . and v1,v2, . . . appear in the TCSC. Representing the superimposedvoltage in Eq. (2) by the vector ∆Vo (similarly to ∆I, ∆V),we have

Here

and [x] stands for the largest integer smaller than a realnumber x. The small variation ∆I of the transmission linecurrent is determined as

Here

The parameters ro and Lo are, respectively, the resistanceand inductance of the transmission line. Variations ∆I and∆V can be found by combining Eq. (41) with Eq. (30) or(38).

5. Results Obtained

The characteristics of the TCSC frequency responseare presented in Fig. 4; the simulation results are shown by

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

40

solid lines. The frequency variation is assumed to be within

1 Hz. The constants of the transmission line and TCSC are

as in Section 2.1.Thyristor firing is based on the TCSC voltage v,

current i, and fundamental wave, with the firing angle being

160°. However, in current-based control, the thyristor isfired at 70° from the current zero point because v and i differin phase by 90° as shown in Fig. 2.

The frequency response depends heavily on the firingcontrol. The analytical solution obtained by Eqs. (30) and(41) is shown in panel (a); the value of β in Eq. (16) was0.89. The analytical solution obtained by Eqs. (38) and (41)is shown in panel (b); the value of β in Eq. (34) was �20.37.Panel (c) pertains to firing control based on the fundamentalwave of the TCSC voltage (or current). When finding ananalytical solution, β was set to 0 in Eqs. (16) and (34); bothexpressions gave the same result. As is evident from Fig. 4,the analytical solutions for all control methods agree wellwith the simulations. When varying phase θ in Eq. (40), theimpedance changes at 0, 60, and 120 Hz, but there is noinfluence at other frequencies. This is explained by Eq.(22).

6. Conclusions

In this study we have proposed an analytical methodfor determining the frequency response of a thyristor-con-trolled series capacitor. The results of the study can besummarized as follows.

(1) If a small voltage at frequency f is superimposedon TCSC voltage v, current components at (nfo ± f) flow inthe thyristor-controlled reactor (TCR); theoretical expres-sions for these components were derived.

(2) Not only the fundamental wave of the voltage vbut also its odd harmonics were taken into account.

(3) Theoretical expressions were obtained not onlyfor voltage-based firing control but also for current-basedand fundamental wave-based control.

(4) Circuit equations were obtained with regard forthe influence of current components at (nfo ± f) flowingthrough capacitor and transmission line. Solving theseequations provides the distributions of the TCSC voltageand current.

(5) Analytical results for the TCSC frequency re-sponse obtained by the proposed method exhibit goodagreement with the numerical simulation results for allkinds of thyristor firing control.

The proposed method may be helpful in the analysisof power systems containing TCSC.

Acknowledgment

The authors thank Professor T. Takuma for valuableadvice.

Fig. 4. Frequency response of TCSC.

41

REFERENCES

1. Urbanek J, Piwko RJ, Larsen EV et al. Thyristorcontrolled series compensation prototype installationat the Slatt 500kV substation. IEEE Trans PowerDelivery 1993;8:1460�1469.

2. Nyati S, Wegner CA, Delmerico RW, Piwko RJ,Baker DH, Edris A. Effectiveness of thyristor con-trolled series capacitor in enhancing power systemdynamics: An analog simulation study. IEEE TransPower Delivery 1994;9:1018�1027.

3. Piwko RJ, Wegner CA, Kinney SJ, Eden JD. Subsyn-chronous resonance performance tests of the Slattthyristor-controlled series capacitor. IEEE TransPower Delivery 1996;11:1112�1119.

4. Othman HA, Ängquist L. Analytical modelling ofthyristor-controlled series capacitors for SSR studies.IEEE Trans Power Syst 1996;11:119�127.

5. Rjaraman R, Dobson I, Lasseter RH, Shern Y. Com-puting the damping of subsynchronous oscillationsdue to a thyristor controlled series capacitor. IEEETrans Power Delivery 1996;11:1120�1127.

6. Perkins BK, Iravani MR. Dynamic modeling of aTCSC with application to SSR analysis. IEEE TransPower Syst 1997;12:1619�1625.

7. Mattavelli P, Stankovi{ AM, Verghese GC. SSRanalysis with dynamic phasor model of thyristor-con-trolled series capacitor. IEEE Trans Power Syst1999;14:200�208.

8. Hedin RA, Weiss S, Torgerson D, Eilts LE. SSRcharacteristics of alternative types of series compen-sation schemes. IEEE Trans Power Syst1995;10:845�850.

9. Zhu W, Spee R, Mohler RR, Alexander GC. AnEMTP study of SSR mitigation using the thyristorcontrolled series capacitor. IEEE Trans Power Deliv-ery 1995;10:1479�1485.

10. Kakimoto et al. Explanation on mechanism of SSRprevention using TCSC. Trans IEE Japan 1997;117-B:168�175.

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APPENDIX

1. Derivation of Eq. (14)

In Eq. (13), the first term in braces is

The second term in braces is

The third term in braces is

In Eqs. (42) to (44), the first term is a stationary termcorresponding to the first term in Eq. (14). The third termin Eq. (42) cancels the second term in Eqs. (43) and (44),thus becoming zero. The second term in Eq. (42) can berewritten as

The term including k is periodic, and summation gives

Here m = [2foT]. This value may be neglected if T in Eq. (5)is large enough. Thus, the second term in Eq. (14) can beobtained. If f is an integer, then T = 1, and the aboveexpression becomes zero.

Now, the fourth term in Eq. (13) is

(42)

(43)

(44)

(45)

(46)

42

The fifth term in Eq. (15) is

The sixth term in Eq. (13) is

The sum of terms including k in Eqs. (47) and (49) can beneglected as in Eq. (46). Thus, the third term of Eq. (14)can be obtained from Eq. (47).

2. Calculation of Harmonics

The amplitudes ai of odd voltage harmonics are re-quired in order to calculate β in Eq. (16). Supposing thevoltage to be

the reactor current is

Therefore,

and

From the above, the n-th harmonic of reactor current can befound:

Here Il = (il1, il3, il5, . . . )t, V = (a1, a3, a5, . . . )

t, and

Since ic = Cdv/dt, the capacitor current is

Here Ic = (ic1, ic3, ic5, . . . )t, Yc = diag(nωoC).

(47)

(48)

(49)

43

The TCSC current i is the sum of il and ic, and hence

Here I = (i1, i3, i5, . . . )t, and V = (a1, a3, a5, . . .)

t. Forexample, if a TCSC is driven by a fundamental-wave cur-

rent source, the amplitude ai for all harmonics can be foundby solving Eq. (50) with i3, i5, . . . = 0. The current harmonicsare required in order to calculate β in Eq. (34); for thispurpose, Eq. (50) may be solved in combination with thecircuit equations of the transmission system.

AUTHORS (from left to right)

Naoto Kakimoto (member) completed his doctorate in electrical engineering at Kyoto University in 1980 and joined thefaculty of Hiroshima University as a research associate. He moved to Kyoto University as a lecturer in 1986, and has been anassociate professor since 1999. His research interest is power system stability. He holds a D.Eng. degree, and is a member ofIEEE.

Anan Phongphanphanee (member) graduated from Chulalongkorn University, Thailand, in 1996, and enrolled in the M.E.program at Kyoto University in 1999. His research interest is TCSC.

(50)

44