tcsc control design for transient stability improvement

Electric Power Systems Research 77 (2007) 470483

TCSC control design for transient stabgamTechnril 20006

Abstract

This pape SC),power system ansieby a variabl ownoptimal loca dentiof the use of 2006 Else

Keywords: T argin

1. Introdu

The demthe process of development of new infrastructure for power gen-eration and dispatch is restricted due to mainly economic andpartially environmental constraints. These result in the need forbetter utilization of the existing system. Flexible AC transmis-sion systemgoal. Thesand thus tocontrollersmore optio

In the scontrollingever, the otduring largto improveThe effectisient stabiltransient ststanding thpreserving

CorresponE-mail ad

rmintem srd to

of Lyapunov and depends on finding a suitable energy functionand identifying the unstable equilibrium point of the post-faultsystem. This is a rather difficult task especially in the case oflarge power systems. The trajectory sensitivity (TS) has been

0378-7796/$doi:10.1016/j(FACTS) controllers are potent tools to achieve thise devices help in pushing the system to their limitsattain higher operational efficiency. The use of theseincreases the flexibility of operation by providing

ns to the power system operators.teady state, FACTS controllers like TCSC help inand increasing the power flow through a line. How-her important aspect of these controllers is their usee disturbances like faults because of their capabilitythe transient stability condition of a power system.

veness of TCSC-controllers in enhancing the tran-ity limit has been studied in Ref. [1]. Evaluation ofability condition of the system is essential for under-e effects of application of FACTS devices. Structureenergy margin sensitivity has been used in Ref. [2]

ding author. Tel.: +91 512 2597179; fax: +91 512 2590063.dress: [email protected] (A. Ghosh).

proposed in Ref. [3] as an alternative to the TEF based methods.It helps to identify stressed systems for a set of contingenciesand is independent of unstable equilibrium point calculationsand model complexity. The use of TS in finding critical valuesof parameters has been discussed in Ref. [4]. The applicationof TS is extended to hybrid systems in Ref. [5]. The use of TSto investigate the Nordel Power Grid disturbance of 1 January1997 is discussed in Ref. [6].

Different control strategies for effective use of series con-nected FACTS devices are discussed in Refs. [1,7,8]. Abangbang control of TCSC is used in Ref. [1] to improve stabil-ity of power systems. In Ref. [7], a power flow control loop anda stability control loop have been used in unison to provide therequired control action for a TCSC. A control strategy based onenergy function method is derived in Ref. [8] which is shown tobe effective for both small signal and large signal disturbances.

In this paper, a dynamic simulation of the power system hasbeen carried out. Dynamics of the network have been consideredalong with the dynamics of the generators and exciters. Trajec-

see front matter 2006 Elsevier B.V. All rights reserved..epsr.2006.04.011multi-machine power system usinDheeman Chatterjee, Arind

Department of Electrical Engineering, Indian Institute ofReceived 14 May 2005; received in revised form 18 Ap

Available online 15 June 2

r discusses the effects of thyristor controlled series compensator (TC. Trajectory sensitivity analysis (TSA) has been used to measure the tr

e capacitor, the value of which changes with the firing angle. It is shtions of the TCSC-controller for different fault conditions can also be iTCSC with a suitable controller over fixed capacitor operation.

vier B.V. All rights reserved.

CSC; Trajectory sensitivity analysis; Dynamic simulation; Transient stability m

ction

and of electrical power is ever increasing. However,

to deteon sysstandaility improvement of atrajectory sensitivityGhosh ology, Kanpur 208 016, India06; accepted 24 April 2006

a series FACTS controller, on the transient stability of ant stability condition of the system. The TCSC is modeled

that TSA can be used in the design of the controller. Thefied with the help of TSA. The paper depicts the advantage

; Optimal location

e the effects of shunt and series FACTS controllerstability. Transient energy function (TEF) method is aol used for this purpose. It is based on direct method

D. Chatterjee, A. Ghosh / Electric Power Systems Research 77 (2007) 470483 471

Fig. 1. TCSC circuit and its equivalent.

tory sensitivity with respect to fault clearing time has been usedto assess transient stability margin of power systems. At first, acontroller has been designed for effective use of TCSC under var-ious fault conditions. TS has been used for finding suitable valuesof the controller parameters. The effects of placement of TCSC-controllers at various locations of a power system on the transientstability are studied. A comparison of the effects of TCSC-controller w

The TCSreactance mThe systembus WSCCimpedance

2. Modelin

2.1. Mode

The TCSof the TCS

XC = 1(Xwhere

1 = 2(

3 =

X

X

5 = 422 c

Let us denTCSC, wh

that in this work the TCSC is operated only in the capacitivemode. The capacitive reactance XFC of the TCSC is chosen ashalf of the reactance of the line in which the TCSC is placed andthe TCR reactance XP is chosen to be 1/3 of XFC.

2.2. Dynamic model of multi-machine power systems

Dynamics of the generators as well as transmission lines areconsidered while modeling the multi-machine power system.

2.2.1. Generator dynamicsThe generators are represented by the flux-decay model as

shown in Fig. 2(a). Also a simplified static exciter model withone gain and one time constant is considered (Fig. 2(b)). InFig. 2(a):

=[(

x) ]

genuatio

ri

ri

t=

Eqidt

=

. . . ,

fdi

t=

EqiV

. . . ,

is tchro

nt, Kith those of fixed capacitors is also carried out.C is represented by a fundamental frequency lumpedodel that varies with the change in the firing angle.under consideration is the three-machine, nine-system [9]. System loads are taken as constant

.

g of the TCSC and the power system

ling of TCSC

C model is given in Fig. 1. The overall reactance XCC is given in terms of the firing angle as [10]:

FC + 2) 45 XFC (1)

) + sin 2( )

, 2 = XFCXPXFC XP ,

FC

P, 4 = 3 tan [3( )] tan( ),

os2( )XP

ote the fundamental frequency capacitance of theich is equal to 1/(sXC), as Ctcsc. It is to be noted

Eii

Theing eq

didt

=

2His

dd

T doid

i = 1,

TAidE

d

Pei =

i = 1,wherethe synconstaFig. 2. (a) Flux-decay model of generator and (b) st1 dixqi

Vi sin(i i) + jEqi ej(i/2) (2)

erator and exciter dynamics are described by follow-ns [11]:

s, i = 1, . . . , m (3)

Pmi Pei KDis

(ri s), i = 1, . . . , m (4)

xdixdi

Eqi +(

xdi

xdi 1)

Vi cos(i i) + Efdi ,

m (5)

Efdi + (Vrefi Vi)KAi , i = 1, . . . , m (6)

i

sin(i i)xdi

+ 0.5(

1xqi

1xdi

)V 2i sin 2(i i),

m (7)he angular position of the rotor, r the rotor speed, snous speed, m the number of machines, H the inertiaD the damping coefficient, Pm the mechanical power

atic exciter model.

472 D. Chatterjee, A. Ghosh / Electric Power Systems Research 77 (2007) 470483

input, Pe the electrical power output, xd and xq the d-axis andq-axis synchronous reactance, xd the d-axis transient reactance,T do the d-axis open circuit time constant, KA and TA the gainand time constant of the exciter, V the terminal voltage of themachine in per unit and is its angle.

Now if we replace rotor speed (ri ) with per unit speed devi-ation (denoted by ri ) in Eqs. (3) and (4) using the relations:

ri =ri s

sand

dridt

= 1s

dridt

Then, we get modified Eqs. (8) and (9) asdidt

= s

2Hidri

dt

2.2.2. NetwFirst let

shown in Fof the line,the TCSC (across thisof transmisceptance isrepresentstances corrgenerators;internal emseen that thDenoting thvc1 and vc2

e1 vc1 =

vc1 vs

vc2 e2 =

ddt

vc1 =1

C

ddt

vs = 1Ct

Fig. 3. SchemTCSC.

ddt

vc2 =1

Cch(i i2) (15)

For three-phase systems, using the notations:

eTabc = [ ea eb ec ], vTabc = [ va vb vc ], andiTabc = [ iaEqs. (10)(

e1abc vc1

vs

e2

bc =

c =

bc =

can

do-srma

[

]

R

L

]

c1d

c1q

]ri , i = 1, . . . , m (8)

= Pmi Pei KDiri , i = 1, . . . , m (9)

ork dynamicsus consider a single line containing the TCSC (asig. 3). R and L are the resistance and inductancerespectively. The variable capacitor Ctcsc representsas described in the previous subsection). The voltagecapacitor is denoted by vs. Nominal- representationsion lines is considered and hence line-charging sus-included. The capacitor Cch at each end of the line

half of the line-charging. L1 and L2 are the induc-esponding to the d-axis transient reactance of thee1 and e2 are the instantaneous values of generatorfs (Ei i in phasor form as per Eq. (2)). It can beere are three loop currents i1, i and i2 in this case.e voltages across the two line-charging capacitors by

, the following relations can be written from Fig. 3:

L1di1dt

(10)

vc2 = Ri + Ldidt

(11)

L2di2dt

(12)

ch(i1 i) (13)

csci (14)

atic diagram of a single line (nominal- representation) containing

vc1abc

vc2abc

ddt

vc1a

ddt

vsab

ddt

vc2a

Wea pseutransfo[e1d

e1q

]

[vc1d

vc1q

=[

[vc2d

vc2q

ddt

[v

vib ic ]

15) can be rewritten as

abc = L1 ddt i1abc (16)

abc vc2abc = Riabc + Lddt

iabc (17)

abc = L2 ddt i2abc (18)

1Cch

(i1abc iabc) (19)

1Ctcsc

iabc (20)

1Cch

(iabc i2abc) (21)

transform any three-phase variable in abc frame totationary dq0 frame [9,12]. Using this abc to dq0tion in Eqs. (16)(21), we get

vc1d

vc1q

]=[

0 L1L1 0

][i1d

i1q

]+ L1 ddt

[i1d

i1q

]

(22)

[vsd

vsq

][vc2d

vc2q

]

LR

][id

iq

]+ L d

dt

[id

iq

](23)

[e2d

e2q

]=[

0 L2L2 0

][i2d

i2q

]+ L2 ddt

[i2d

i2q

]

(24)

=

1Cch

0

01

Cch

[i1d idi1q iq

]

+[

0 0

][vc1d

vc1q

](25)


ddt

[vsd

vsq

]=

1Ctcsc

0

01

Ctcsc

[id

iq

]+[

0 0

][vsd

vsq

]

(26)

ddt

[vc2d

vc2q

]=

1Cch

0

01

Cch

[id i2diq i2q

]

+[

0 0

][vc2d

vc2q

](27)

In a mulwith variouloop currenAlso there. . ., Cr. Levc1 , vc2 , . .

ages (vc1 , vabove, we csystem nettype. The csenting thetions.

IntegratEfdi , i an(Eii) as pof this voltThen, theobtained froutputs of tare then usnext time stis given in

A symmThe locatiowhich is albe of self-cthe correspstudied und

Fig. 4.

3. Application of trajectory sensitivity analysis intransient stability margin assessment

3.1. Trajec

Supposeset of diffe

x = f (t, x,where x isters. The separameters0. The equ[

f

x

]

x =

andeve

g nuramet to ie compu

x22

is smo thehe ce gecan b

on

ce. Tursior annsiti

i

sen

of tatives conjecto

aboariabstathe oationfor

n as mi=ti-machine power system there are several such liness interconnections giving rise to a number of suchts. Let us denote those currents by say i1, i2, . . ., ip.are line-charging capacitors denoted by say C1, C2,t the voltages across these capacitors be denoted by. , vcr . Taking these currents (i1, i2, . . ., ip) and volt-c2 , . . . , vcr ) as the state variables and proceeding asan get the equations for the complete multi-machine

work. Loads are considered as constant impedanceonstant values of resistances and inductances repre-loads are also included in the line dynamics equa-

ion of Eqs. (5), (6), (8) and (9) gives values of Eqi ,d ri . These are used to find generator internal emfer Eq. (2). In the next step, ei, the instantaneous value

age Eii, is used to solve network Eqs. (22)(27).generator terminal voltages and angle (Vii) areom these solved variables. Also the electrical powerhe generators, Pei are calculated using Eq. (7). Theseed in the solution of Eqs. (5), (6), (8) and (9) in theep. A block diagram representation of this procedureFig. 4.etrical fault is simulated in one of the lines at a time.n of the fault is chosen to be one end of the line,most equivalent to a fault at the bus. The fault maylearing type or it may have to be cleared by isolatingonding line from its two ends. System response iser both types of faults.

Block diagram representation of the power system model.

x =

wherejectory

Howby usinone parespec2). Ththen co

sens =

If close t

In te.g., th(r). Nowreferenthe excthe rotThe se

ij

=

Theinsteadthe relbility i

Tramationstate vticularknow tinformculatedis give

SN =tory sensitivity: computation and quantication

a multi-machine power system is represented by arential equations given as

), x(t0) = x0 (28)the state vector and is a vector of system parame-nsitivities of state trajectories with respect to systemcan be found by perturbing from its nominal valueations for trajectory sensitivity can be found as [3]:

x +[f

], x(t0) = 0 (29)

x/. Solution of (28) and (29) gives the state tra-trajectory sensitivity, respectively [3]:

r, the sensitivities can also be found in a simpler waymerical method. To describe this, let us choose onlyter, i.e., becomes a scalar and the sensitivities with

t are studied. Two values of are chosen (say 1 andrresponding state vectors x1 and x2, respectively areted. Now the sensitivity at 1 is defined as

x1 1 =

x

(30)

all, the numerical sensitivity is expected to be veryanalytically calculated trajectory sensitivity value.

ase of a power system, sensitivity of state variables,nerator rotor angle () and per unit speed deviatione computed as in (30) with respect to some parameter

e of the generators, say the jth one, is taken as thehen, the relative rotor angle of the ith machine (i.e.

on of the rotor angle of the ith machine with respect togle of the reference machine) is given by ij = i j.vity of ij with respect to is computed as

j

(31)

sitivity of relative rotor angle is considered herehe sensitivity of of an individual machine becauserotor angle is the relevant factor when angular sta-cerned.ry sensitivities (ij/ and ri /) give us infor-ut the effect of change of parameter on individualles and hence on the generators (to which the par-

e variable correspond) of the system. However, toverall system condition, we need to sum up all these. The norm of the sensitivities of ij and ri are cal-this. The sensitivity norm for an m machine system

1

[(ij

)2+(

ri

)2](32)


Fig

Next a nthe inversesystem momore resultvalues of Therefore,from instabtime (tcl) athe relative

3.2. Assesstrajectory s

The systbus systemone of thein three stesmall time.line. Simulcleared aftefor a longeThe fault mnot requirethe faultedfor fault in

Let us cthe base

Table 1AVariation of normalized with TCSC location: self-clearing fault

Fault near bus TCSC in line

45 57 46 69 78 89

5 0.6827 1.1081 1.0411 1.1363 0.9953 1.00656 1.0743 1.2423 0.6850 1.0727 0.9493 0.96038 1.0464 1.0346 1.0331 1.0795 1.0626 0.9462

Table 1BVariation of normalized with TCSC location: fault cleared by line isolation

Fault nearbus

Lineisolated

TCSC in line

45 57 46 69 78 89

.e. wrrespns. Tore,ion imor

ifferlts aclear

nea

lue oas 1nt stmea

r theare p) for which the value of are higher. It can be seen from1A and 1B that the preferable location of TCSC is linethe cases of fault near bus 5 or bus 8 whereas it is line. 5. Single line diagram of the WSCC nine-bus system.

ew term (ETA) is introduced [4]. is defined asof the maximum of SN, i.e., = 1/max(SN). As the

ves towards instability, the oscillation in TS will being in larger values of SN. This will result in smaller. Ideally should be zero at the point of instability.the value of gives us an indication of the distanceility. In this paper TS with respect to fault clearing

nd the corresponding has been used for assessingstability conditions of the system.

ment of transient stability margin usingensitivity

em studied in this paper is the WSCC 3 machine nine-shown in Fig. 5. A three-phase fault is simulated in

lines of the nine-bus system. The simulation is doneps. To start with, the pre-fault system is run for aThen, a symmetrical fault is applied at one end of a

ation of the faulted condition continues till the fault isr a time tcl. Then, the post-fault system is simulated

r time (say 5 s) to observe the nature of the transients.ay be of self-clearing type (i.e. isolation of line is

5

6

8

loop (ithe colocatioTherefconditof is for dthe faufaultslocatedThe vaTCSCtransie> 1.0that foTCSCof viewTables69 ind for fault clearance) or may be cleared by isolatingline. The trajectory sensitivity and are computeddifferent lines.onsider the value of for the case without TCSC asand denote it by 0. Next, a TCSC is placed in open

57 for fauThe res

fault of durof relativefor two cas

Fig. 6. Response of delta31 for fault in line 69 with TCSC in45 1.0754 1.0139 1.1567 1.0095 1.022057 0.7609 1.0974 1.2399 1.0487 1.0421

46 1.0955 1.2559 1.0203 0.9660 0.976469 1.0892 1.3018 0.7152 0.9777 0.9993

78 0.9708 0.9474 1.0825 1.1550 0.971689 0.9859 0.9668 1.0714 1.1534 0.9689

ithout any controller) in one of the lines at a time andonding values of are computed for different faulthese values are then normalized with respect to 0.

if the value of normalized in a particular operatings greater than 1.0, it indicates that the absolute valuee than 0. Table 1A shows the values of normalizedent combinations of TCSC and fault locations whenre of self-clearing type. Table 1B shows the same fored by line isolation. The faults are considered to ber one of the load buses (bus 5, 6 or 8) in all the cases.f tcl is taken as 0.15 s and the firing angle () of the

60. As described in the previous section, gives theability margin of the system. Therefore, a normalizedns that the transient stability margin is higher thansystem without TCSC. Further, those locations of

referable (from transient stability improvement pointlt near bus 6.ults are verified by PSCAD/EMTDC simulation. Aation 0.15 s is simulated in line 69. The responsesmachine angle delta31 (3 1) are shown in Fig. 6es: (a) the TCSC placed in line 46 and (b) the TCSC

: (a) line 64 and (b) line 57.


placed in line 57. The firing angle of TCSC is 160 in both thecases. It is clear that the maximum peak of the response is muchlower in case of TCSC in line 57, which indicates a highertransient stability margin. This is in accordance with the highervalue of normalized for the TCSC in line 57 than in line 46in the corresponding cases in Table 1B.

4. The control scheme and use of TSA in the choice ofcontrol parameters

4.1. The control scheme

In the next step, a controller is employed along with theTCSC. The block diagram of the control scheme used is shownin Fig. 7. The active power flow (P) through the line containingTCSC is taken as the control variable. It is compared with thereference value of active power flow (Pref) and the error is fedto a PI controller. The output of the PI is the firing angle of theTCSC, . Tthe capacit180). TheTCSC. Theas describethe line dyn

This schtype, becauhence the sand after tthe faulty lchange of tfore, a corcontrol actiThe currenframe, id anues in the pto calculate

i0 =

i2d0

Fig. 8

(a) Variation of normalized with controller constants; fault in bus 5,line 69; (b) variation of normalized with KI for different KP; fault

, TCSC in line 69; (c) variation of normalized with KI for differentt in bus 8, TCSC in line 57.his is passed through a limiter to keep it withinive operation zone of the TCSC (between 145 andoutput of the limiter is supplied to the firing circuit ofcapacitance value of the TCSC (Ctcsc) is computed

d in Section 2.1. This capacitance is then included inamics.eme is sufficient if the fault is only of self-clearingse there is no change in system configuration and

teady state power flow should remain the same beforehe fault. But when the fault is cleared by isolatingine, the system configuration changes resulting in ahe steady state power flow through the lines. There-responding change in Pref is needed. An additionalon is taken to achieve this, which is shown in Fig. 8.t flowing through the line (containing TCSC) in dqd iq, are used for this purpose. First, the current val-

re-fault steady state, denoted by id0 and iq0, are stored

+ i2q0 (33)

Fig. 7. Block diagram of control scheme.

. Block diagram of control scheme for power reference.

Fig. 9.TCSC inin bus 5KP; faul


Then, at every time step, the current values of id and iq aremeasured to obtain

i =

i2d + i2q (34)The change in i (denoted by i) is calculated as

i = i i0 (35)This i is fed through a proportional controller to produce thePref, the change in the power reference. Pref is then addedwith the pre-fault steady state value of power reference (Pref0)to get the new power reference (Pref).

4.2. Choice of controller constants using trajectorysensitivity analysis

The effect of the TCSC on the transient stability of the sys-tem depends largely on the proper functioning of the controller.Therefore, choice of suitable values of controller constants KPand KI is very important. TSA is used here for making thatchoice.

As shown in Tables 1A and 1B and discussed in Section3.2, TCSCthe most wforth we shin these twclearing faat a time, wputed for dare then no0. A threethe controland TCSCthree differin Fig. 9(b)values of binations (

Fig. 10. VariaTCSC in 69.

KI > 6.0). Fig. 9(c) is for a fault near bus 8 with TCSC in line 57.In this case, higher values can be obtained for combinations(KP = 0.5 athe TCSCment of thranges.

Next, wby line isolfound. Theof KP are shto clear thethe suitableKI > 3.0), (or KI > 6.0)and KI valwith TCSC

4.3. Effectcontroller c

Up to tcontroller c

ty cog timsituaontro(a)culaear b.8. T(b) id KP

nd 0.qui

maliighecons

ller clar trovin

omp

therof T

thoutor thfixeapacith s

ty co fot arean bthe-conratiocauses improvement of system stability conditionhen it is placed in line 69 or line 57. So, hence-all study the effect of TCSC (along with controller)o locations. First, let us consider the case of self-ult near one of the three load buses (bus 5, 6 or 8)ith tcl = 0.15 s. Trajectory sensitivity and are com-ifferent combinations of KP and KI. These valuesrmalized with respect to the corresponding value of-dimensional plot of variation of normalized withler constant is shown in Fig. 9(a). For the same faultlocations, the variation of normalized with KI forent values of KP (KP = 0.1, 0.5 and 0.8) are shown. It can be observed form the two figures that highercan be obtained with the controller constant com-

KP = 0.5 or 0.8 and KI < 1.0) or (KP = 0.1 or 0.5 and

tion of normalized with KI for different KP; fault in line 45 and

stabiliclearinferentwith cFig. 11a partifault nKP = 0Fig. 1169 an0.15 a

It isof nor(tcl). Htrollercontroparticuin imp

4.4. C

Anoeffectsi.e., wicapacias thefixed ction (wstabiliues ofof faul

It ctions,TCSCof opend 1.5 < KI < 2.0) or (KP = 0.8 and KI = 3.0). Hence,will be most effective in transient stability improve-e system when the controller constants are in these

e consider the cases in which the faults are clearedation. As before, values for different KP and KI arevariation of normalized with KI for different valuesown in Fig. 10 for a fault near bus 5 (line 45 isolatedfault) with the TCSC in line 69. It can be seen thatcombinations of constants in this case are (KP = 0.1,

KP = 0.5, KI < 1.0 or KI > 3.0) and (KP = 0.8, KI < 1.0. Table 2 shows the most suitable combination of KPues found by this study for different fault locations

in line 57 or 69.

of fault clearing time on the relation of withonstants

his point, we have observed the effect of TCSC-onstants on the value of (and hence on the transientndition of the system) for a particular value of faulte (tcl). However, tcl may also be different in dif-

tions. Therefore, the same study, i.e., variation of ller constants, is carried out at different tcl values.

and (b) shows the plots of variation of with KI (atr KP) for different values of tcl. Fig. 11(a) is for aus 6 (line 46 isolated) with TCSC in line 57 and

he three plots correspond to tcl = 0.15, 0.20 and 0.25 s.s for a self-clearing fault near bus 8 with TCSC in line= 0.8. Here, the three plots correspond to tcl = 0.10,

20 s.te clear from the figures that the nature of variationzed remains the same for different fault durationsr values of are obtained for the same ranges of con-tants even when the fault duration varies. Therefore,onstants chosen on the basis of variation of at a

cl will be helpful in effective operation of the TCSCg stability margin even when the tcl is different.

arison with the effects of TCSC in open loop

issue of interest is the comparison between theCSC-controller and the TCSC used in open loop,

t a controller. In open loop, the TCSC acts as a fixedroughout the period of disturbance. Let us term this

d capacitor mode of operation. The effects of thisitor and the effects of a TCSC-controller combina-uitably chosen controller constants) on the transientndition are investigated and compared here. The val-r these two cases with different types and locationsgiven in Table 3.

e observed from the table that for all the fault loca-value of is higher in case of compensation bytroller combination than in the fixed capacitor moden, the difference being quite high in some of the cases.


Table 2Suitable controller constants and corresponding normalized for different fault and TCSC locations

TCSC in line Self-clearing fault Fault cleared by line isolation

Fault bus (KP, KI) Faulted line (KP, KI) 57 5 0.8, 9.00 1.6357 45 0.8, 9.00 1.4641

6 0.8, 0.12 1.7357 46 0.8, 9.00 1.827969 0.8, 9.00 1.9094

8 0.5, 1.80 1.0906 78 0.8, 1.80 0.975889 0.5, 1.20 1.0014

69 5 0.1, 6.00 1.2944 45 0.1, 6.00 1.297257 0.1, 6.00 1.4221

6 0.8, 9.00 1.3001 46 0.8, 9.00 1.2487

8 0.8, 1.80 1.1583 78 0.8, 1.80 1.256089 0.8, 0.60 1.2636

Fig. 11. Variafault near bus

Therefore,sient stabilTCSC-conFig. 9(b) an

Table 3Comparison o

Type of fault

Self-clearing

Faultwithlineiso-la-tiontion of normalized with controller constants at different fault clearing time: (a) fault n8, TCSC in line 69.

it can be concluded that the improvement in the tran-ity condition of the system is much more with thetroller combination. This can also be verified fromd (c) and Fig. 10, where for fixed capacitor (with-

out controlvalues withtions of KPin Fig. 9(b)

f for system compensated by fixed capacitor and TCSC-controller

Fault location TCSC in line

57

Fixed capacitor With contr

Bus 5 1.1081 1.6357Bus 6 1.2423 1.7357Bus 8 1.0346 1.0906

Line 45 1.0139 1.4641Line 57 Line 46 1.2559 1.8279Line 69 1.3018 1.9094Line 78 0.9474 0.9758Line 89 0.9668 1.0014ear bus 6, line 46 isolated, TCSC in line 57 and (b) self-clearing

ler) mode of operation is also plotted along with the controller. In Fig. 10, thevalues for all the combina-and KI are higher than the fixed capacitor , whereasand (c), the values with controller are higher than

69

oller Fixed capacitor With controller

1.1363 1.29441.0727 1.30011.0795 1.1583

1.1567 1.29721.2399 1.42211.0203 1.2487

1.1550 1.25601.1534 1.2636


fixed capacitor for most of the KP and KI combinations. Theseresults highlight the effectiveness of the application of a TCSCalong with a controller in improving the transient stability of apower system.

5. Optimization of controller constants and its effect ontransient stability

5.1. Optimization of controller constants using trajectorysensitivity analysis

A method to choose the TCSC-controller constants for tran-sient stability improvement has been described in the previoussection. The constants are chosen for either self-clearing faultor for fault cleared by line isolation. However, the type of thefault is generally not known beforehand. Therefore, the con-troller needs to be designed such that it can serve both the cases.This choice can be made by a compromise of the results for thetwo types of faults.

Fig. 12 shows a combined plot of the normalized for thetwo cases f

The solbus 5 andsubsequentcases is thethe commotwo types oor 0.5 or 0.

Howeveof KP andvery difficucombined pindex (OI)KP and KI,fault and fais defined a

OI = scf +2

Fig. 12. Variation of normalized with controller constants for different typesof faults.

is coma

nd Kf w

hat cOIlarKf nond l9 anon oSC

n clef fa

ied ozed ce coectipe o

variation of OI with controller constants.or different values of KP.id lines in the figure are for self-clearing fault nearthe dash-dotted lines are for a fault near bus 5 withremoval of line 45. The fault duration in both thesame, 0.15 s. It can be observed from the figure that

n zone of higher values of (between the plots for thef faults) are for controller constant values (KP = 0.18, KI > 6.0) and (KP = 0.5 or 0.8, KI < 1.0).r, with the increase in the number of combinationsKI and hence in the number of plots, it becomeslt to figure out the most suitable constants from thelots only. Therefore, a new term called optimizationis invoked. Suppose, for a particular combination ofscf and f are the values in case of self-clearingult cleared by line isolation, respectively. Then, OIs the arithmetic mean of scf and f:

f

OIused toof KP avalue ocase, twhichparticuplots obus 5 aline 6variatiand TCone ca

types ois carroptimiwith th

In Seach ty

Fig. 13. (a) Variation of normalized and OI with KI and (b)mputed for different combinations of KP and KI andke the choice of constants. Earlier, that combinationI was being considered as most suitable for which theas found to be maximum. Now, for this optimizationombination of KP and KI is considered suitable for

is maximum. Plot of variation of OI with KI for aP is shown in Fig. 13(a) along with the corresponding

rmalized for both types of faults. The fault is nearine 45 is isolated to clear the fault. The TCSC is ind KP is chosen as 0.1. Fig. 13(b) shows the plots off OI with KI for three different values of KP. Faultlocations are the same as before. From these figures,arly identify the optimized set of constants (for bothult) to be KP = 0.1 and KI = 6.00. The same methodut for other fault and TCSC locations also and theontroller constants found are shown in Table 4 alongrresponding normalized values.on 4.2, the best possible controller constants to servef fault were found separately (Table 2). Let us now


Table 4Optimized controller constants and corresponding for different fault and TCSC locations

TCSC in line

57

69

compare th values) gconstants f

(i) Fault iof faul

(ii) Fault iof faul

Naturallsame as thsituation isTable 4). A

(i) Fault isible cwhereaKP = 0KP = 0optimient fro= 1.7

(ii) Fault isible cwhereaKP = 0are KPresponconstawith li

It is cleamized consa comparatHowever, tplace of 1.the TCSC-faults whic

ffectnts o

us n

y chpowlizedindilicatver

e rot0.15

) sysnts ation. Simnear

wn fller i9. IwitSelf-clearing fault

Fault bus (KP, KI) 5 0.8, 9.00 1.6357

6 0.8, 9.00 1.7225

8 0.5, 1.80 1.0906

5 0.1, 6.00 1.2944

6 0.8, 9.00 1.3001

8 0.8, 1.80 1.1583

ose with the optimized constants (and correspondingiven in Table 4. In some cases, the best possible

or both type of fault are same. As for example:

n bus 5, line 45 isolated; TCSC in 57. For both typet, KP = 0.8, KI = 9.00.n bus 8, line 78 isolated; TCSC in 69. For both typet, KP = 0.8, KI = 1.80.

y the optimized constants in these cases are also thee individual best possible constants. However, thedifferent in some of the cases (shown in bold face infew of these cases are discussed below.

n bus 6, line 46 isolated; TCSC in 57. The best pos-onstants for self-clearing fault are KP = 0.8, KI = 0.12s for fault with line isolation the constants are

.8, KI = 9.00. The optimized constants in this case are

.8, KI = 9.00, i.e., in the case of self-clearing fault, thezed constants (corresponding = 1.7225) are differ-m the best possible constant values (corresponding

5.2. Econsta

Letsuitablof thenorma

whichon appcan berelativ5 (tcl =and (iiconstaoscillasystema faultare shocontroline 6system357).n bus 8, line 89 isolated; TCSC in 69. The best pos-onstants for self-clearing fault are KP = 0.8, KI = 1.80s for fault with line isolation the constants are

.8, KI = 0.60. The optimized constants in this case= 0.8, KI = 1.80. Here, the optimized constants (cor-ding = 1.2452) are different from the best possiblent values (corresponding = 1.2636) in case of faultne clearance.

r from the results that in some cases the use of opti-tants results in slightly lower value of and henceively less improvement of stability in one fault type.he difference (1.7225 in place of 1.7357 or 1.2452 in2636) is too small to be significant and in exchangecontroller is being designed to counter both types ofh is more suitable for practical application.

TCSC placof TCSC pwhich indibeneficial ttent with th(1.8279) thTable 4.

For furtto the criticFor a self-be 0.41 s. Smore thanlocation wthe systemclearing timtaken as 0.the two caFault cleared by line isolation

Faulted line (KP, KI) 45 0.8, 9.00 1.4641

46 0.8, 9.00 1.827969 0.8, 9.00 1.9094

78 0.5, 1.80 0.955989 0.5, 1.80 0.9908

45 0.1, 6.00 1.297257 0.1, 6.00 1.4221

46 0.8, 9.00 1.2487

78 0.8, 1.80 1.256089 0.8, 1.80 1.2452

of TCSC-controller with properly chosenn transient stability

ow investigate the effect of a TCSC-controller withosen constants on the transient stability conditioner system. It can be observed from Table 4 that the values are greater than 1.0 in most of the cases,

cates an improvement of transient stability marginion of a TCSC with properly chosen constants. Thisified from Fig. 14(a), which shows the variation ofor angle delta2-1 for a self-clearing fault near buss). Plots are shown for: (i) system without TCSC

tem with TCSC-controller in line 69. The controllerre the optimized values as given in Table 4. Reducedin case of system with TCSC means a more stableilarly, Fig. 14(b) shows the variation of delta2-1 for

bus 6 (tcl = 0.15 s) cleared by isolating line 46. Plotsor: (i) system without TCSC, (ii) system with TCSC-n line 57 and (iii) system with TCSC-controller int is clear from the figure that, in comparison to thehout TCSC, the oscillation is damped for both the

ements. However, the damping is much more in caselaced in line 57 than in case of TCSC in line 69cates that placement of TCSC in line 57 is morehan in line 69 for this fault location. This is consis-e higher value of normalized for TCSC in line 57an that for TCSC in line 69 (1.2487) as shown in

her investigation, the fault duration is increased upal clearing time (tcr) for the system without TCSC.

clearing fault in bus 5, this value of tcr is found too the system gets unstable for a fault clearing timethis tcr. However, if the TCSC is placed in suitableith properly chosen values of controller constants,continues to remain stable even for a higher faulte. This is shown in Fig. 15. The fault duration is

42 s (>tcr) and the relative rotor angle delta2-1 s forses, system without TCSC and system with TCSC-


Fig. 14. (a) Variation of relative rotor angle delta2-1 for a self-clearing fault near bus 5 and (b) variation of relative rotor angle delta2-1 for a fault in line 46.

controller at line 69 (with optimized constant values), are plot-ted. The relative rotor angle are found to diverge and becomeunbounded in case of system without TCSC whereas it remainsbounded aplots clearldesigned cosystem.

5.3. Comp

A PI coin a multi-mmargin. TraPI controlltuning contsame systemare comparThe detaileare given in

Fig. 15. Rela0.42 s.

A comparison of the results for the two controllers is shown inFigs. 16 and 17. A self-clearing fault of duration 0.4 s is consid-ered in bus 6 with the TCSC being placed in line 57. Response

tive rotor angle delta2-1 is shown in Fig. 16(a) and (b)e twfrom

ontroriatio.

he nith t

ng linle. Hty forthe slledear flosend stable when a TCSC is placed in line 69. They show the effectiveness of the TCSC with suitablyntroller to improve transient stability condition of the

arison with a self-tuning controller

ntroller is applied here for effective use of a TCSCachine power system to improve transient stability

jectory sensitivity has been used for the tuning of theer. To assess the performance of this method, a self-roller is employed in place of the PI controller in the

under the same operating conditions and the resultsed. A pole-shift controller is used for this purpose.d equations of the pole-shift self-tuning controllerAppendix A.

of relawith thchosenshift cthe vaclosely

In tered wisolatiunstabstabili1 forcontroIt is clquite ctive rotor angle delta2-1 for a fault in bus 5, fault-clearing time Fig. 16. Relaline 57 with:o controllers. The parameters of the PI controller areTable 2. The choice of the parameters of the pole-

ller is discussed in Appendix A. It can be seen thatns of delta2-1 with the two controllers match quite

ext case, a fault of duration 0.35 s in bus-8 is consid-he TCSC placed in line 57. The fault is cleared bye 78. When there is no TCSC, the system becomesowever, if a TCSC is placed, the system maintainsboth the types of controllers. The response of delta2-ystem without TCSC and the system with TCSC

by the two controllers is shown in Fig. 17(a) and (b).rom the figure that the response of delta2-1 matchesly for the two controllers.tive rotor angle delta2-1 for a self-clearing fault in bus 6, TCSC in(a) pole-shift controller and (b) PI controller.


Fig. 17. Comcontrolled by:78, TCSC in

From ththat the pevery muchMoreover,ier than thea severe proin unstableoccur with

6. Discuss

This papon the varitem. The dnetwork ansensitivitythe systemthe sensitivand rotor sbility marg

The effethe choicehas been sfault in a pit is cleareters for thehas been dso that theverify theing methodsimulatedtrollers aremuch compties associaavoided.

If TCSC is used without any controller, the firing angleremains constant throughout and hence the capacitance value

bytor mwithf thethe

effeme

entd toA syn exe locain s

wled

autat Uctiv

dix

polix.n:

)y(k)y(k)l inpTheor no

) = 1) = bparison of improvements in delta2-1 by the application of TCSC(a) pole-shift controller and (b) PI controller; fault applied in lineline 57.

e results shown in Figs. 16 and 17 we can concluderformance of the PI controller tuned using TSA iscomparable with the pole-shift self-tuning controller.the design of the self-tuning controller is much trick-PI controller. Also the self-tuning controller can haveblem like estimation parameter divergence resultingclosed-loop system. Such problems however do nota simple controller like the PI.

ions and conclusions

er presents a study of the effects of a TCSC-controlleration of transient stability condition of a power sys-ynamics of the power system, including both thed the generators, have been simulated. Trajectory

offeredcapaciparedmost omakes

Theimprovplacemis founcases.

duratiosuitablto rem

Ackno

TheIllinoisconstru

Appen

Theappendequatio

A(z1

wherecontroinput.operat

A(z1

B(z1analysis is used to assess the stability condition of. The inverse of the maximum value of the norm ofities () of state variables like generator rotor anglepeed deviation is used as a measure of transient sta-in.ctiveness of the TCSC is found to be dependent onof the controller parameters. Trajectory sensitivity

hown to be a useful tool for making that choice. Aower system can be either of self-clearing type or

d by line isolation. The suitable controller parame-se two types of faults may be different. A techniqueiscussed to choose optimized controller parametersTCSC can be effective for both types of faults. Toperformance of this TSA based PI controller tun-, a self-tuning pole-shift controller has also been

and the results of control action of the two con-compared. It is found that the results are very

arable. However, with the use of PI, the complexi-ted with the pole-shift self-tuning controller can be

where ny(k) = a

+ Let us defin

= [ a1(k) = [ Eq. (38) thy(k) = TIt is to be nwhile the vthe regress

It is assuand can alseter vectorcalled systeit becomes fixed. The effect of TCSC in this fixedode on the system stability is also studied and com-the effects of TCSC-controllers. It is found that incases, a controller, with suitably chosen parameters,

TCSC more effective in improving system stability.ectiveness of the TCSC-controller (in terms ofnt of transient stability) is found to vary with itslocation in the system. The placement of the TCSCbe beneficial for system stability in majority of the

stem without TCSC becomes unstable when the faultceeds the critical clearing time. However, TCSC ination with chosen parameters may help the systemtable even under those conditions.

gement

hors are thankful to Prof. M.A. Pai of University ofrbana-Champaign for his valuable suggestions and

e criticism on the paper.

A

e-shift self-tuning controller [13] is discussed in thisSuppose the system is governed by the difference

= B(z1)u(k) + e(k) (36)is the sampled value of the system output, u(k) is theut, e(k) is a zero-mean, uncorrelated random noisepolynomials A and B are given in backward shifttation z1 as

+ a1z1 + + anzn,1z

1 + + bmzm (37)m. Combining (36) and (37) we get1y(k 1) any(k n) + b1u(k 1) + bmu(k m) + e(k) (38)e the following two vectors:

an b1 bm ]T,y(k 1) y(k n) u(k 1) u(k m) ]T

en can be written in a compact form as

(k) + e(k) = T(k) + e(k) (39)oted that the vector contains the system parameters,ector contains the past input and output and is calledion vector.med that the parameters of the system are unknowno vary with time. We can then estimate the param- on-line using the system input and output. This ism identification. In this we have used what is called


ed in l

the recursiters are comfollowing e

(k) = y(k(k) = (k

K(k) =

P(k) = [IIn the aboving old dat

Once thpole-shift c

u(k) = S(zR(z

where yrefare given b

S(z1) = sR(z1) = 1The controobtained by

A(z1)R(zwhere A anare obtaine

In (43),istic equati

e sta

) = A

0 < y onhe sis chond 0

ch isFig. 18. Identified system parameters when a fault is appli

ve least squares (RLS) method, where the parame-puted recursively at each sampling instant using the

quations [13]:

) T(k 1)(k), 1) + K(k)(k),P(k 1)(k)

+ T(k)P(k 1)(k) ,

K(k)T (k)]P(k 1) (40)

to mor

T (z1

wherepenalt

In torder ias 0.8 aof whie equation, is a scalar less than 1 used for discount-a.e parameters are estimated, they are then used in theontrol in which the control law is given by1)1) {yref(k) y(k)} (41)

is the reference value and the polynomials S and Ry

0 + s1z1 + + sn1z(n1),+ r1z1 + + rm1z(m1) (42)

ller parameters, i.e., polynomials S and R, are thenthe solution of the following equation:

1) + B(z1)S(z1) = T (z1) (43)d B are the estimates of the polynomials A and B andd from (40).the polynomial T is the closed-loop system character-on. We can define it by shifting the open-loop poles

0.8 and 0.9which is shfor this casand is remocan be obspre-fault, d

Reference

[1] K.R. Padment in pControl

[2] K.N. Shenergy mFACTS d

[3] M.J. Laument usi953958

[4] M.A. Paical systAntsaklications:

[5] I.A. HiskIEEE Tr204220ine 78 with TCSC in line 57.

ble locations as [13]: (z1) = 1 + a1z1 + + nanzn (44)

< 1 is called the pole-shift factor that determines thecontrol.mulation studies performed in Section 5.3, the systemsen withn = m = 3. The parameters and are chosen.985, respectively for the self-clearing fault, the resultshown in Fig. 16(a). These parameters are chosen as99 for fault and subsequent line removal, the result ofown in Fig. 17(a). Some of the identified parameterse are shown in Fig. 18. In this the fault is applied at 4 sved by line isolation at 4.35 s. Most of the parameterserved to vary in three distinct steps associated withuring-fault and post-fault conditions.

s

iyar, K. Uma Rao, Discrete control of TCSC for stability improve-ower systems, in: Proceedings of the Fourth IEEE Conference on

Applications, September 2829, 1995.ubhanga, A.M. Kulkarni, Application of structure preservingargin sensitivity to determine the effectiveness of shunt and seriesevices, IEEE Trans. Power Syst. 17 (3) (2002) 730738.fenberg, M.A. Pai, A new approach to dynamic security assess-ng trajectory sensitivities, IEEE Trans. Power Syst. 13 (3) (1998).i, T.B. Nguyen, Trajectory sensitivity theory in nonlinear dynam-ems: some power system application, in: N. Michel, D. Liu, P.J.s (Eds.), Stability and Control of Dynamical Systems with Appli-A Tribute to Anthony, Birkhauser, Boston, 2003.ens, M.A. Pai, Trajectory sensitivity analysis of hybrid systems,

ans. Circuits Syst. Part 1: Fundam. Theory Appl. 47 (2) (2000).


[6] I.A. Hiskens, M. Akke, Analysis of the Nordel Power Grid disturbance ofJanuary 1, 1997 using trajectory sensitivities, IEEE Trans. Power Syst. 14(3) (1999) 987994.

[7] A.D. Del Rosso, C.A. Canizares, V.M. Dona, A study of TCSC controllerdesign for power system stability improvement, IEEE Trans. Power Syst.18 (4) (2003).

[8] M. Noroozian, L. Angquist, M. Gandhari, G. Anderson, Improving powersystem dynamics by series-connected FACTS devices, IEEE Trans. PowerDeliv. 12 (4) (1997).

[9] P.W. Sauer, M.A. Pai, Power System Dynamics and Stability, Prentice Hall,Upper Saddle River, 1998.

[10] N. Christl, R. Hedin, P.E. Krause, S.M. McKenna, Advanced series com-pensation (ASC) with thyristor controlled impedance, in: Proceedings ofthe CIGRE General Session, Paris, 1992.

[11] P. Kundur, Power System Stability and Control, McGraw-Hill, New York,1994.

[12] K.R. Padiyar, Power System Dynamics, Stability and Control, BS Publi-cations, Hyderabad, 2002.

[13] A. Ghosh, G. Ledwich, O.P. Malik, G.S. Hope, Power system stabilizersbased on adaptive control techniques, IEEE Trans. Power Appl. Syst. PAS-103 (1984) 19831989.

TCSC control design for transient stability improvement of a multi-machine power system using trajectory sensitivityIntroductionModeling of the TCSC and the power systemModeling of TCSCDynamic model of multi-machine power systemsGenerator dynamicsNetwork dynamics

Application of trajectory sensitivity analysis in transient stability margin assessmentTrajectory sensitivity: computation and quantificationAssessment of transient stability margin using trajectory sensitivity

The control scheme and use of TSA in the choice of control parametersThe control schemeChoice of controller constants using trajectory sensitivity analysisEffect of fault clearing time on the relation of with controller constantsComparison with the effects of TCSC in open loop

Optimization of controller constants and its effect on transient stabilityOptimization of controller constants using trajectory sensitivity analysisEffect of TCSC-controller with properly chosen constants on transient stabilityComparison with a self-tuning controller

Discussions and conclusionsAcknowledgementAppendix AReferences

tcsc control design for transient stability improvement

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