Calculation of Fundamental Impedance Characteristic of a TCSC using a Normalised Model

Transcription

1 INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 70, DECEMBER 7-9, Calculatio o Fudametal Impedace Characteristic o a TCSC usig a Normalised Model Amit K. Jidal, Aviash Joshi ad Aridam Ghosh Abstract-- The paper discusses a method o derivatio o udametal impedace characteristic o a Thyristor-Cotrolled Series Capacitor (TCSC) usig a ormalised model which is useul or desig purposes. This derivatio uses the state space model o a Fixed Capacitor-Thyristor Cotrolled Reactor (FC-TCR) circuit. It is assumed that FC-TCR circuit is supplied by a costat ac curret source. The steady-state waveorms ad the udametal TCSC characteristics are the derived by the step-by-step solutio o the state equatio usig MATLAB. I this derivatio we assume that TCR has iite Q actor. The model is the veriie d through simulatio usig simulik. Idex Terms-- Fudametal Characteristic, Normalisatio, State Trasitio Equatio, TCSC. T I. INTRODUCTION HE IMPEDANCE characteristic o a Thyristor- Cotrolled Series Capacitor (TCSC) is oliear i ature. Usually the TCSC cotrol systems employs a covetioal type cotroller to obtai the reactace order, which i tur, is coverted ito the correspodig irig agle o the TCR through a look-up table. The mai drawback o this approach is the absece o a closed-loop model through which the stability o the system ca be aalysed. The modelig o TCSC or a TCSC compesated trasmissio lie has bee a subject o much iterests i recet times [-]. The device is modeled usig the Poicare map i []. A discrete-time model based o the liearized behavior o the state trasitio equatios is proposed that ca also predict the shit i the zero-crossig o the lie curret (or capacitor voltage) very accurately [-]. However udametal characteristic o the TCSC has bee approximated as a capacitor that varies with the chage i its irig agle, by assumig the Q o the iductor to be iiite [4-5]. I [5] a modelig approach is preseted that icorporates a estimate o this characteristic based o Amit K. Jidal is with Electrical Egieerig Departmet at Idia Istitute o Techology, Kapur, 0806 INDIA (telephoe: , jidal@iitk.ac.i). Aviash Joshi is with Electrical Egieerig Departmet at Idia Istitute o Techology, Kapur, 0806 INDIA (telephoe: , ajoshi@iitk.ac.i). Aridam Ghosh is with Electrical Egieerig Departmet at Idia Istitute o Techology, Kapur, 0806 INDIA (telephoe: , e- mail: aghosh@iitk.ac.i). which a closed-loop cotrol system is desiged. The exact steady-state solutio o the state trasitio equatios o the TCSC circuit is give i [6]. I this paper, the steady-state equatios o TCSC have bee preseted i a ormalised orm, which is useul or desig purposes. A importat eature o this method is that the iite quality actor (Q) has bee icorporated i the equatios. The ormalised udametal requecy impedace characteristics have bee calculated by Fourier aalysis. We shall validate the ormulatio through simulatio results. II. STEADY-STATE OPERATION OF TCSC The udametal characteristic o a TCSC is derived whe the system is i the steady state. Let us assume that the TCSC is drive by a siusoidal curret source il as show i Fig., the resistace R P accouts or the iite Q- actor o the parallel iductor LP. The TCR curret ad capacitor voltage have bee deoted by i P ad v c respectively. Fig. Schematic diagram o a TCSC drive by a curret source. The steady-state TCSC voltage ad curret waveorms i the ormalised orm over a hal cycle are show i Fig., the other hal beig symmetrical. Note that i Fig. subscript deote the ormalised values o the correspodig variables. Let us deie the ollowig Istat θ 0: This is the positive goig zero-crossig istat o the capacitor voltage v c. The iductor curret at this istat is deied as i p (θ 0) I p0. The irig

2 64 NATIONAL POWER SYSTEMS CONFERENCE, NPSC 00 agle θ α /ω (i radias) is measured rom this istat. Istat θ : This is the istat at which the iductor curret goes to zero. Istat θ : This is the istat at which oe o the thyristor is ired. We the have θ θ 0 α /ω. Istat θ : This is the egative goig zero-crossig istat o the capacitor voltage. The iductor curret at this istat is deied as ip(θ ) Ip0. Further rom the steady-state waveorms (Fig. ), it ca be see that the ormalised capacitor voltage v c is a combiatio o the ollowig two sectios v c i L dθ whe oe o the thyristors is coductig, e.g., betwee θ ad θ. v c (i L i P )dθ whe oe o the thyristors is coductig, e.g., betwee θ 0 ad θ or θ ad θ. To obtai a expressio or the capacitor voltage, we must thereore combie the above two piecewise liear models together takig care o cotiuity o the state variables. I the derivatio preseted i this paper, we deie the istat θ 0 whe the capacitor voltage passes through zero as the origi (θ 0 0). Let us assume that the curret source is give by i L I si t m ( ω +φ) ω is the system requecy ad agle φ is the phase o the drivig curret. Fig. Typical TCSC voltage ad curret waveorms. From the steady-state characteristics o the capacitor voltage ad iductor curret (Fig. ), we ca surmise that the behavior o the capacitor voltage ca be completely determied oce the ollowig three quatities i ormalised orm are kow. The curret i the iductor I P0 at the start o the cycle i.e. positive zero crossig o v c. The istat θ at which the iductor curret I P goes to zero. The phase (φ) o the drivig curret. This is measured with respect to positive zero crossig o v c. I the ext sectios, aalytical expressios or v c ad i p durig the three itervals, i.e. 0 to θ, θ to θ ad θ to θ, are obtaied. From these expressios, three o-liear equatios or the three ukows give above are obtaied. These three oliear equatios are the solved usig solve routie o MATLAB to determie the ukows. Oce these three quatities are determied, the actual capacitor voltage waveorm (like the oe show i Fig. ) is computed. III. STATE SPACE EQUATIONS AND THEIR SOLUTION The waveorm give i Fig. depeds o the parameters o the circuit o Fig.. I order to id a geeral characteristic useul or desig o TCSC, the ollowig base quatities are chose. Base Curret, (Ib) Im A LP Base Impedace, Zb Ω C Base Frequecy, ω b ω 0 rad/sec L P C ad hece the ormalised requecy is ω ω /ω o. Usig above base quatities the state space equatios i ormalised orm are obtaied. From Fig. it ca be see that there are two piecewise liear modes o operatio o the TCSC whe oe thyristor is o ad whe both thyristors are o. We shall discuss these two modes separately. Mode (a): Whe oe Thyristor is ON To derive the system model or this mode i which oe o the thyristor is o, we deie a state vector as x T [v c i P ]. We ca write the state-space descriptio o the system as 0 x & x + il Ax + Bi () L Q 0 i L si(ω θ + φ). I above equatios the idepedet variable is θ ω 0 t ad suix deotes the ormalised values o the correspodig variables obtaied by dividig these by their base values. Q is the quality actor o the iductor L p at the base requecy. Give ay arbitrary iitial coditio x(θ i) at a time θ i, the solutio o the above equatio is o the orm ( ) Φ( θ θi ) x( θ i ) + Φ( θ τ ) B il ( τ ) θ x θ dτ () θ i the matrix Φ(θ) e Aθ is the state trasitio matrix (STM). Deiig θ θ - θ i, the elemets Φ i,j, i,, j, o the STM are λ θ λ θ Φ {( + ) e + ( ) e }

3 INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 70, DECEMBER 7-9, λ θ λ θ { e e } Φ Φ λ θ λ θ { e e } Φ λ θ λ θ {( ) e + ( + ) e } Φ λ ( ); λ ( + ) Q 4 ; Q I the above expressio λ ad λ are the eigevalues o the matrix A. Mode (b): Whe both Thyristors are OFF I this mode the curret through the iductor is zero. We the have the ollowig irst order equatio or the capacitor voltage. c si ( ω θ + φ ) v & () Give ay arbitrary iitial coditio vc(θi) at a time θi, the solutio o this equatio is o the orm v c ( θ ) v ( θ ) + si ( ω θ + φ ) v θ c i c θi ( θ ) + { cos( ω θ + φ ) cos( ω θ + φ) } i ω i dθ IV. NONLINEAR EQUATIONS TO DETERMINE INITIAL CONDITIONS Based o equatios () ad (4) it is possible to obtai the equatios or the ukows φ (Fig. ), i p0 ad θ (Fig. ). For this purpose the positive goig zero crossig o v c is take as the time reerece. The state equatio () o T Mode (a) is solved with the iitial coditio, x 0 [0 -I p0 ]. It is assumed that thyristor T is coductig i the steady state. This mode eds whe the curret i parallel iductor L p goes to zero i.e. x(θ ) T [v c (θ ) 0]. The circuit operates i Mode (b) durig θ θ θ θ is the triggerig agle o thyristor T. I Mode (b), i p is zero ad the capacitor is charged by the iput curret source. The capacitor voltage chages rom iitial value o v c (θ ) to ial value o v c (θ ) as per equatio (4). At θ θ α /ω, the thyristor T is gated ad the circuit agai operates i Mode (a) with iitial coditio x T 0 [v c (θ ) 0]. Assumig symmetry, the ial coditio at θ θ π ω is x T 0 [0 I p0 ] i the steady state. The thyristor T is gated agai at agle α /ω rom the egative goig zero crossig o v c. Based o above sequece the ollowig three oliear equatios have bee obtaied or the ukows i p0, φ ad θ. λθ λ θ 0 I γ e + γ e K (5) { } p 0 + ; (4) γ θ 0 ; γ + K Φ + ( θ τ ) si ( ω τ φ ) ; dτ λ ( θ θ ) λ ( θ θ ) i p0 { e e } vc( θ ) + K4 θ K4 + θ λ { ( θ θ ) λ ( θ θ ) 0 γ e + γ e } Φ ( θ τ ) si ( ω τ φ) ω K K λθ λθ { e e } I dτ { cos( ω θ + φ ) cos( ω θ + φ )} θ θ θ 0 Φ Φ P0 + K + ( θ τ ) si ( ω τ + φ ) ( θ τ ) si ( ω τ + φ ) dτ; dτ + K V. CALCULATION OF INITIAL CONDITIONS USING NORMALISED VARIABLES The ukow iitial coditios I p0, θ ad φ i the ormalised equatios (5) to (7) are determied usig solve routie o MATLAB. Table I lists these iitial coditios or various values o α. The value o ω is chose as 0.5. The values or Q 0 are compared with correspodig values or Q 00. These ca be used to desig the parameters L p ad C. It ca be see rom Table I that or high Q, the value o φ is close to ± π /, while at low Q it varies with α. Also the magitude o I p0 is quite high ear resoace. No solutio or iitial coditios ca be obtaied i the rage <α < 7. The lower limit o α is reached whe the itervals θ ad θ (Fig. ) are equal. Due to resoace the iductor curret is the cotiuous. From table, the value o θ at this poit (α 90, Q 00) is.068 radia while θ α/ω.4 radia. However or the low values o Q, operatio below α 90 is also possible. Similar tables have also bee obtaied or dieret values o ω. Usig these iitial coditios, we ca completely determie the capacitor voltage waveorm by itegratig umerically () ad () or Mode (a) ad Mode (b) respectively or each value o α. α (deg) Table I Iitial coditios or dieret Q actors I p0 (pu) ω 0.5 Q 0 Q 00 θ φ I p0 θ (rad) (deg) (pu) (rad) φ (deg) (6) (7)

4 644 NATIONAL POWER SYSTEMS CONFERENCE, NPSC No Operatio Capac itor Regio Iduct or Regio The variatios o various values o Ip0 ad θ rom Table I with the irig agle have bee plotted i Fig. ad Fig. 4 respectively. From Fig. 4 it ca be see that the value o θ alls liearly with irig agle especially or high values o Q. Fig. 5 Capacitive operatio with Q 0 ad α 50. VI. STEADY-STATE WAVEFORMS USING NORMALISED MODEL Fig. 5 ad Fig. 6 show the ormalised capacitor voltage ad iductor curret waveorms uder steady-state over hal a cycle or α 50 ad or α 80 respectively, the other hal cycle beig symmetrical. The value o ω is chose as 0.5. It ca be see that or a low value o Q, the iductive operatio is possible eve or α 80. Fig. 7 shows the iductive operatio or Q 00. It ca be cocluded that or a high value o Q the iductor curret is almost cotiuous or a irig agle o 90. Fig. Variatios o Iductor curret (I p0 ) with α. Fig. 6 Iductive operatio with Q 0 ad α 80. Fig. 4 Variatios o θ with α.

5 INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 70, DECEMBER 7-9, Fig. 7 Iductive operatio with Q 00 ad α 90. VII. ESTIMATION OF NORMALISED IMPEDANCE CHARACTERISTIC The udametal compoet o the calculated capacitor voltage is obtaied umerically. The ormalised udametal capacitor voltage is the udametal requecy equivalet impedace o the TCSC or a particular irig agle. By this method the resistace ad reactace characteristics o the TCSC or two dieret values o Q are determied ad plotted i Fig. 8 ad Fig. 9 respectively. It ca be see that the magitude o udametal resistace ad reactace icreases rapidly as the irig agle approaches the value at which the parallel combiatio o L p ad C approaches resoace. This value is a uctio o ω. Fig. 8 Variatios i Fudametal Resistace with α. Fig. 9 Variatios i Fudametal Reactace with α.

6 646 NATIONAL POWER SYSTEMS CONFERENCE, NPSC 00 Fig. 0 Simulik model o a TCSC. VIII. VERIFICATION OF NORMALISED MODEL USING SIMULINK I this sectio, the eicacy o the above method has bee validated through simulatio studies usig Power System Blockset. The simulik model o a TCSC or a irig agle o 50 is show i Fig. 0. A curret source o oe per uit is used to drive the TCSC. Two thyristors coected back to back i the series with a resistor R_P.0 Ω ad a iductor L_P.89 mh orm the TCR while capacitor C 79.5 µf is the ixed capacitor. For these parameters the base impedace (Z b ) is 0 Ω. Agai i this model ω is chose as 0.5 with system requecy ω 50 Hz. Various sesors ad scopes are coected to measure the voltage ad curret waveorms. Fig. shows the simulatio results or the capacitive operatio o TCSC over oe complete cycle uder steady state while Q is chose as 0. The voltage ad curret is ormalised usig their base quatities. The time axis ca also be ormalised by usig the relatio θ (ω/ω ) t. For time t 0.0 secod ad or ω 0.5, the value o equivalet θ will be.56 radias. Now comparig the waveorms i Fig. with Fig. 5 ( these are or hal a cycle) it ca be cocluded that the results with the simulik model are close to those obtaied usig the ormalised model. Other results have also bee veriied by takig dieret values o the parameters. Fig. Simulatio result with Q 0 ad α 50. IX. VERIFICATION OF NORMALISED IMPEDANCE CHARACTERISTICS The ormalised impedace characteristics calculated i sectio VII ca be compared with results give i [4]-[5], a ormula or overall TCSC reactace (Z) is give. The ormula i ormalised orm is as ollows.

7 INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 70, DECEMBER 7-9, ( + ) 4 5 Z + per uit (8) 4 5 ω ω ( π α) + si ( π α ) 4ω cos π ( ω ) ta π { ( π α) } ta( π α) ( π α) ( ω ) Here ω is the ormalised requecy ad α is the irig agle. The ormalised characteristics or the capacitive regio obtaied usig above ormula with ω 0.5 have bee show i Fig.. The dark lie shows the result obtaied usig (8). [] S. Jalali, I. Dobso, R. H. Lasseter ad G. Vekatarama, Switchig time biurcatios i a Thyristor Cotrolled Reactor, IEEE Tras. Circuits & Systems-: Fudametal Theory & Applicatios, Vol. 4, No., pp.09-8, March 996. [] A. Ghosh ad G. Ledwich, Modellig ad cotrol o thyristorcotrolled series compesators, Proc. IEE Geer. Trasm. Distrib.,Vol. 4, No., pp , May 995. [] A. Ghosh ad G. Ledwich, A discrete-time model o thyristorcotrolled series compesators, Electric Power Systems Research, Vol., pp. -8, 995. [4] N. Christl, R. Hedi, P. E. Krause ad S. M. McKea, Advaced Series Compesatio (ASC) with thyristor cotrolled impedace, CIGRE Regioal Meetig, Sessio 99, Paris. [5] P. C. Srivastava, A. Ghosh, S. V. Jayaram Kumar, Model-based cotrol desig o a TCSC-compesated power system, Electrical Power ad Eergy Systems, Vol., pp , 999. [6] A. Ghosh, A. Joshi ad M. K. Mishra, State space simulatio ad accurate determiatio o Fudametal Impedace Characteristics o a TCSC, IEEE Witer Power Meetig 00, Columbus, Ohio. Fig. Compariso o udametal reactace characteristics. The results obtaied i sectio VII have also bee show. It ca be see that or Q 00 ad above, the two match closely. However or Q 0, the ormula give above gives lower reactace. X. CONCLUSIONS A ormalised state trasitio model or the computatio o the TCSC characteristics has bee preseted. The model is lexible i which the eects o a iite quality actor (Q) ca be take ito accout. The udametal impedace characteristic o the TCSC is computed rom the waveorms obtaied by the ormalised model. The proposed model has bee veriied by compariso with results o system simulatio usig simulik. The udametal impedace characteristic has bee compared with a well kow aalytical ormula [5]. The proposed model gives results close to the aalytical oes, whe Q is high. It is show that the variatio o the equivalet impedace o the TCSC is strogly aected by Q. For a low value o Q, operatio below α 90 is also possible. XI. REFERENCES