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Transient stability assessment of systems comprising phase-shiftingFACTS devices by direct methods

Rafael Mihalic*, Uros Gabrijel

Faculty of Electrical Engineering, University of Ljubljana, Trzaska 25, Ljubljana 1000, Slovenia

Abstract

In this paper the concept of a control Lyapunov function is used to develop control strategies for thyristor-controlled phase-shifting

transformers (TCPSTs). Two typical representatives of TCPSTs were chosen for consideration: a Quadrature Boosting Transformer (QBT)

and a Phase Angle Regulator (PAR). For the case of the QBT we showed that the proposed control law can significantly improve the large

signal stability in comparison to the control-law-based on the traditionally used injection model proposed in various references. Even if the

system parameters are estimated very roughly, our proposed strategy gives better results than the traditional strategy. Also, some novel

energy functions, which consider the action of an optimally controlled TCPST in a single-machine infinite-bus (SMIB) system, are presented

and validated by a numerical example.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Thyristor-controlled phase-shifting transformer; Control strategy; Energy function; Direct method

1. Introduction

Limitations relating to the stability of an electric

power system, technical problems connected with long-

distance power transmission, as well as the limited

possibility for electric power flow redirection are reasons

why todays transmission facilities are, on average,

loaded less than they theoretically could be. The

expansion of electric power networks is becoming

increasingly problematic due to the publics refusal to

accept such solutions to the point where it is sometimes

practically impossible to obtain permission for new rightsof way. With existing knowledge the problem can, in

general, be solved using the so-called dynamic power

flow control method. Such a concept of electric power

system design and operation can be realised by applying

Flexible AC Transmission System (FACTS) devices. The

term FACTS is used to describe a broad spectrum of

devices. In this article the problem of large signal

(transient) stability enhancement, applying thyristor-con-

trolled phase-shifting transformers (TCPSTs), is

discussed. It is true that it is unusual to use TCPSTs

in such a way, but if TCPSTs are already present in the

system for other reasons (like the redirection of power

flows according to agreements in deregulated markets) it

possible to add a transient-stability enhancement

module to the TCPST controller and in this way increase

the transient-stability margin.

A power system undergoes large system parameter

changes during a disturbance. If this disturbance is not

cleared in time the system experiences an escape of one

or more generators. For TCPSTs there already exist

several control strategies [14] for power oscillation

damping and/or transient stability improvement. Based onfindings in [5] we can assume that damping will be more

effective if a regulator uses global information instead of

locally measured quantities. We obtain this global

information from the system that is reduced to a two-

machine equivalent and further into a single-machine

infinite-bus (SMIB) system. The latter is the system used

in this paper. One of the major tasks in transient stability

assessment is determining the critical clearing time

(CCT). In terms of improving the transient stability of a

power system we were particularly interested in the

maximum CCT that can be achieved through the correct

action of a TCPST with a given rating hence we used

a feedback design based on Lyapunovs method in thesame way as the authors in [5].

0142-0615/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijepes.2003.12.001

Electrical Power and Energy Systems 26 (2004) 445453www.elsevier.com/locate/ijepes

*Corresponding author. Tel.: 386-1-4768-438; fax: 386-1-4768-289.E-mail addresses: [email protected] (R. Mihalic); gaber@

leoants.fe.uni-lj.si (U. Gabrijel).
http://www.elsevier.com/locate/ijepeshttp://www.elsevier.com/locate/ijepes

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Lyapunov-like direct methods have been successfully

competing with conventional methods in the field of

transient stability assessment for quite some time now

[6,7]. Their advantages lie in the fast computation of the

CCT and the possibility of providing various types of

margins and real-time tools for sensitivity assessment and

preventive control [6]. Supposing that a TCPST already has

a transient stability enhancement module implemented in its

control unit, we are obviously presented with an obstacle

when trying to obtain the CCT of the given system by a

direct method. The same problem studied Hiskens, Hill [8],

and later Padiyar, Immanuel [9] who handled only Static

Var Compensator and presented structure-preserving energy

functions for this device. Recently Gabrijel and Mihalic

presented an energy function for an SSSC in a SMIB system

in [10]. We tried to expand this knowledge by carrying outnew energy functions for optimally controlled TCPSTs in a

SMIB system considering devices limitations.

Section 2 provides mathematical models for two typical

TCPST representatives referred to as a PAR (thyristor

controlled phase angular regulator) and a QBT (thyristor-

controlled quadrature boosting transformer). Inverse opti-

mal design is applied in Section 3 to obtain optimal control

strategies for both TCPST representatives. Section 4

presents new Lyapunov-like energy functions, which

incorporate the optimal action of a TCPST in a SMIB

system. The numerical calculations in Section 5 show some

examples of the use of the new Lyapunov functions for

determining the CCT, and point out the advantages of the

proposed control strategies.

2. Modeling a TCPST in a SMIB system

There are many possible realisations of phase-shifting

transformers. While in Europe (especially the UK) QBTs

are widely used, in the USA it is the PAR that is the most

common device. The PAR terminal voltage phasors are

separated without changing their magnitude (if losses are

neglected). The QBT also separates terminal voltagephasors but the injected voltage UT phase is fixed with

regard to the input voltage phasor UPST1(b ^908-Fig. 1).

In both cases the angle a of the terminal voltage phasors

separation may be assumed to be a controllable parameter.

Although at first sight they appear to be similar devices, a

PAR and a QBT differ in terms of their impact on power

flow. While the PAR is a symmetrical device (no impact of

orientationthe way its terminals are connected to the

systemon power flow), the QBT is not, and its orientation

has an impact on the power flow. In addition, the PARs

location in the corridor does not influence power flow, while

the QBTs location is important [3]. The model of a

transmission system with a TCPST, and phasor diagrams forboth a PAR and a QBT are presented in Fig. 1.

Referring to [3] we can reproduce the PAR equation for

the real power flow that serves us as a mathematical model.

PPAR P2 P1 U1U2

X1 X2sind a 1

Doing the same with the QBT in orientation 1 we obtain:

PQBT P1 P2 U1U2

X1

cos a X2 cos a

sind a 2

2.1. Optimal control design

The development of control laws for PARs and QBTs in

SMIB systems in the literature follows the well-adopted

procedure of inverse optimal control design [11,12]. The

design is based on a control Lyapunov function (CLF). By

definition, any Lyapunov function whose time derivative

can be rendered negative is a CLF [11].

The following considerations are based on Fig. 2, which

shows a SMIB system (with a TCPST) in a transient state

and with the generator represented by the classical model.

2.2. Optimal PAR control

In order to create a closed-loop system with desirable

stability properties we take the common Lyapunov

function (5) and find its time derivative (6) along the

trajectory of the swing Eq. (3). Where d0S represents

Fig. 1. (a) Network scheme, (b) Phasor diagram of a PAR, (c) Phasordiagram of a QBT.

Fig. 2. Model of TCPST in SMIB system.

R. Mihalic, U. Gabrijel / Electrical Power and Energy Systems 26 (2004) 445453446

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the stable equilibrium point.

MdDv

dt

Pm 2 Pe 2Ddd0

dt

;dd0

dt

Dv 3

Pe E0US

X0sind0 a 4

nd0;Dv 1

2MDv2 2 Pmd

02 d0s

2E0US

X0cos d0 2 cos d0s 5

_n 2 DvE0US

X0sind0 a2 sin d

2DDv2 6

Real power flow is defined by Eq. (4). The last termof Eq. (6)

provides damping to the system. Additional damping with a

PAR can be achieved if the term in the square brackets of Eq.

(6) retains a positive value for any value of input variable.

However, we can talk of an optimal control law if the

considered term retains an ideal maximum value for anyvalue of input variables. In this case the maximum

improvement in transient stability and damping is achieved.

Using Eq. (6), the following optimal strategy is obtained:

a signDvp

22 d0 7

where the controllable parameter a is supposedly unlimited.

The control law Eq. (7) makes _n negative definitive,

therefore, the total energy of the post-fault system will

decrease with time. If a maximum transfer is requested thena

should follow p=22 d0 and vice versa.The phasor diagram according to Fig. 1a implies that the

controllable parameter a is bounded within ^1808. Within

these extremes the injected voltage UT gains doublethe value

of the terminal voltage. Theoretically, this kind of system

could sustain a fault of infinitive time and continue to

preserve stability after the fault clearance. If a is limited

between 2 amin andamax (note:amin is a positive value), the

proposed control strategy can only be applied between limits.

In order to achieve the extreme power transfer across a broad

d0 interval, switching over between amax and amin is required

(e.g. A and B are the points for maximum and C is the

point for minimum power transfer-cf. Fig. 3). From anequal-area point of view the system will be stable if there is

always enough braking surface above the mechanical power

Pm opposing the accelerating surface below Pm-Fig. 3.

In Fig. 3 the optimal control law (7) is graphi-

cally represented. The operating area is determined by the

two bold curves. The upper curve represents the maximum

transfer characteristics of the PAR limited at ^p=3; while

the lower bold curve represents its minimum transfer

characteristics.

2.3. Optimal QBT control

The control law for a QBT is derived in more-or-less the

same way as with a PAR (Section A). The main difference is

in the nature of the QBT, which is a non-symmetrical

device. This creates some difficulties in the procedure of the

control strategy determination. Following the procedurefrom Section A we take the Lyapunov function Eq. (5) and

find its time derivative along the trajectory of the swing

Eq. (3) in which we use Eq. (8) as a term for the real power

flow Pe:

Pe E0US

X0

1

cos a X2cos a

sind0 a 8

The procedure yields the time derivative of the energy

function as follows:

_n 2DvG2DDv2 9

GE0US

X0

1

cosaX2 cosa

sind0 a2E0US

X0

1 X2sind0 10

Our objective is to make the term in the square brackets of

Eq. (9) as negative as possible. In this way the reduction of

the total system energy after the fault is cleared will be a

maximum. This extreme case can easily be obtained by

searching for the zeros in the derivative of Eq. (10) of the

argument a. This gives, after taking Dv into consideration,

the following control law:

a arctan 2tand0 signDvcos d0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 X

2

X0

1

cos2 d0s !

11

Omitting the influence ofDv; Eq. (11) can be visualised as

in Fig. 4 E0 US 1: These, and all subsequent figures,

are in the per-unit system.

If a QBT succeeds in attaining the optimal control

strategy, the real power transfer characteristics determined

by Eq. (8) appear as shown in Fig. 5 X10 0:3;X2 0:7:

In order to demonstrate the benefits of the proposed

control strategy a comparison has been made with the

strategy derived from the injection model and presented in

Ref. [1]. In our paper the strategy presented in Ref. [1]

is referred to as the classical control strategy. It assumest hat a QBT contr ol labl e parameter i s set t o i tsFig. 3. PAR transfer characteristics with limitations.

R. Mihalic, U. Gabrijel / Electrical Power and Energy Systems 26 (2004) 445453 447

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maximum and minimum limits, the switching point being

d0 p=2 a amax; d0, p=2 and a 2amin; d

0. p=2:

The maximum real power transfers in a SMIB system

with the QBT parameter limited to ^p=6 were compared

according to the proposed PNEW and the classical control

strategy POLD: A comparison of the extreme real power

transfer using both strategies is shown in Figs. 6 and 7. Fig. 7

displays the difference between PNEW and POLD for extreme

power transfers, depending on the location of the QBT

device. It is evident that the new control law can

significantly increase the transfer of the real power flow

through the given line in the vicinity ofd0 p=2; while at

lower angles the effect drops off quickly. When considering

the minimum power transfer for the QBT phase-shift-

limitation to ^p=6; in the d0 interval between 0 and p the a

is always set to its limit (cf. Fig. 4-for 0 , d0

, p mintransfer curve does not enter the area between limits). This

means that in this case there is no difference between the

classical and our proposed strategy for minimum powers at

0 , d0 , p: Therefore, we can conclude that the classical

control strategy is appropriate for oscillation damping (a

relatively small QBT rating required), while for transient

stability enhancement the proposed strategy can be

significantly more effective. Of course this also manifests

in a longer CCT, as will be shown later in Section 5.

In the case of a reverse orientation of a QBT the correct

transmission characteristics are the same as Eq. (8) except

that X1

0 should be replaced with X2

(and vice versa), dwith -

dand a with-a. It can be shown that in the case of reverse

orientation the correct control law follows from replacing

X1

0 with X2

in the term (11).

A special case occurs when the QBT is located at the grid

terminals, orientation 2 (this orientation is reasonable in this

case). In this situation it is possible to modify Eq. (8), which

can be rewritten as:

Pe E0US

X0tan a cos d0 sin d0 12

The control strategy can be obtained from Eq. (11) where

the reverse orientation and X2 0 is considered:

a amax; Dvcos d0$ 0

13a 2amin; Dvcos d

0, 0

Evidently this control law is the same as the classical

control strategy and hence it can be concluded that the

proposed and the classical control strategies do not differ at

the grid terminals (orientation 2parallel branch at grid).

The differences arise when the QBT is moved electrically

away from the grid (also shown in Fig. 7).

The rating of the device again defines the maximum and

minimum phase angle shift: amin and amax: The maximum

and minimum transfer characteristics for E0 1; US 1

and amin amax p=6 are shown in Fig. 8.

Presented control laws for TCPSTs are globally optimal,

however, for a practical application it is reasonable to use

locally measurable electric quantities. A good example ofFACTS control based on local measurements is presented in

[2] and references therein. In this case control laws are not

Fig. 4. Control law for a QBT.

Fig. 5. Extreme QBT transfer characteristics.

Fig. 6. Comparison of maximum QBT transfer characteristics.

Fig. 7. Difference in maximum transfer characteristics.


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globally optimal anymore. Furthermore, instead of

the generators internal electro-motive force and infinite

busbar voltage, voltage measurements from both ends of the

transmission path can be also used to obtain a local

optimum.

3. Developing energy function

According to the specific bangbang control strategy a

TCPST must jump between maximum and minimum

transfer characteristics. Thus, we can speculate that during

the first swing the TCPST control parameter is always set so

that the maximum transfer characteristic is attained. We

have used this fact to develop several energy functions that

can be used in any system that can be reduced to SMIBsystem without loosing the information on FACTS location

e.g. longitudinal or interconnected systems. Note that a

radial system topology can also be a consequence of a relay

protection response to a disturbance in meshed systems. The

proposed energy functions meet all the conditions for a

Lyapunov function.

3.1. PAR energy function

From the direct-method point of view it is sufficient to

know the behaviour of the TCPST for the period of the fault-

on. Hence for the purposes of CCT determination we can

omit the sign of Dv (it does not change during theacceleration), which affects the PAR control law. We only

have to examine the PAR maximum transfer characteristic

shown in Fig. 9. This characteristic is composed of the

intervals defined in Eq. (14).

Ped0 d0 interval

Pmaxe sind

0amax; 2R k2p,d

0,p

22amax k2p

Pmaxe ;

p

22amax k2p#d#

p

2amin k2p

P

max

e sind

02amin;

p

2 amin k2p,d

0#

2pk12

R14

where Pmaxe is defined as

E0US

X0 and

k

IntegerPartR d0

2p

; d0$R

IntegerPartR d0

2p

21; d0,R

15

Ifamin amax then R equals p=2; otherwise R is defined as

shown in Fig. 9 (the point where curves sind0 amax and

sind02amin meet).

Energy functions are always constructed for the post-

fault system. In accordance with the general agreement that

the first integral of the swing equation constitutes a proper

energy function we integrated the swing Eq. (3) over all theintervals of Eq. (14). The integration yields the following

energy function:

nPARd0;Dv nKEDv nPE PARd

0 16

nKEDv 1

2MDv2 17

The term (17) as a kinetic energy is also used later on in theQBT case.

nPE PARd0 nPE PARi21

nPE PARiAd02R k2p, d0 ,

p

22amax k2p

nPE PAR iBd0p

22 amax k2p# d

0#

p

2

amin k2p

nPE PAR iCd0p

2 amin

k2p, d0 # 2pk 12R

8>>>>>>>>>>>>>>>>>>>>>>>>>:

18

where:

nPE PAR iA 2

Pmd

02 ^

d

0S

2

P

max

e cosd

0

amax2 cosd0S amax 19

Fig. 8. Transmission characteristics-QBT at grid.

Fig. 9. PAR maximum transfer characteristics (a is bounded).


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nPE PARiB Pmaxe 2 Pm d

02

p

22 amax

nPE PAR iC 2Pm d02 p

2 amin

2 Pmaxe cosd

02amin

d0S is the PAR stable equilibrium point (SEP) at the

maximum transfer characteristic. It equals d0S2 amax:

d0S represents the SEP of a post-fault system without a

PAR. nPE i21 expresses the constant value of the

potential energy with which we enter the current

interval. The initial value of nPE i21 is zero. In Appendix

A it is shown that Eq. (16) is a Lyapunov function in

the usual sense.

3.2. QBT energy function

3.2.1. QBT energy function for the proposed new strategy

The considerations that are valid for the PAR

case (Section A) are also valid for a QBT. Starting with

an examination of Eq. (11) it was established that neglecting

Dv has no impact on (11) if we consider the fault-ontrajectory, hence we can rewrite the control law as:

a arctan 2tan d0 sec d0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 X2

X0

1

cos2d0

s !20

We again executed the first integral of motion and took into

account Eq. (20) for Pe: However, the analytical solution of

the referred integral could not be found. We thereforesuggest that the problem is handled numerically. Hence

energy function can be denoted as:

WQBTd0;Dv WKEDv WPE QBTd

0 21

WPE QBTd0 2Pmd

02 d0S

d0d0S

E0US

X0

1

cosaX2 cosa

sind0 add0 22

where a varies as described in Eq. (20). In the special case

when a QBT is located at the grid (orientation 2) it is

possible to construct a strict Lyapunov function. As inSection 3.1 we make a similar assumption that the device

follows the maximum transfer characteristic (Fig. 10). The

latter is based on the transfer characteristic Eq. (12) and the

proposed bang bang control law (13) with the influence of

Dvneglected.

After an examination ofFig. 10 we chose to carry out the

integration of the swing equation over the following two

intervals:

Ped0 d0 interval

E0US

X0sin d0 tan amaxcos d

0; d0 ,

p

2

E0US

X0sin d0 2 tan amincos d

0; d0 $

p

2

23

If we suppose that Pmaxe . Pm; the post-fault system yields

only one SEP. Pmaxe is defined as E0US=X

0: Otherwise we

could deal with two SEP cases, which is evident from Fig. 10

if mechanical power Pm is increased. After the integration

of Eq. (3), keeping Eq. (23) in mind, we get:

nQBTd0;Dv nKEDv nPE QBTd

0 24

nPE QBTd0 2Pmd

02 d0 S2Pmaxe cosd

02cos d0S

nPE QBT i21 Pmax

e amaxsind02

sin^d

0S

;

d0,p

2

nPE QBT i212Pmaxe tanamin sind

02 sin

p

2

d0$p

2

8>>>>>>>>>>>>>>>>>:

25

nPE QBT i21 represents the constant value of the potential

energy from the previous interval. Its initial value is zero.

3.2.2. QBT energy function for classical control strategy

The classical control strategy has the same effect on the

transmission characteristic as Eq. (13), therefore weconstruct the energy function in the same way as in Sections

3, 3.1, 3.2, 3.2.1, by integrating the swing Eq. (3) over two

intervals of Eq. (26):

PEd0 d0 intervals

E0US

X0

1

cos amax X2cos amax

sind0 amax; d0,

p

2

E0US

X0

1

cos amin X2cos amin

sind0 2 amin; d0$

p

2

26

Fig. 10. Maximum transfer characteristics-QBT at grid (a limitations

considered).


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The integration yields:

nQBTd0;Dv nKEDv nPE QBTd

0 27

nPE QBTd0 2Pmd

02 d0S nPE QBT i21

2

Pmaxe1 cosd

0amax2cosd

0Samax;

d0 ,p

2

Pmaxe2 cosd

02amin2cos

p

22amin

;

d0 $p

2

8>>>>>>>>>>>>>>>>>:

28

where:

Pmaxe1

E0US

X0

1

cos amaxX2cos amax

29

Pmaxe2

E0US

X0

1

cosaminX2cosamin

nPE QBT i21 is denominated in the same way as in Section

3.2.1.

4. Numerical examples

In order to demonstrate the effectiveness of the proposed

control laws and the usefulness of the developed energy

functions some numerical examples were made. The test

case is the one from [3], where it is thoroughly examined. Its

scheme is presented in Fig. 11 while some of the system data

is presented in Table 1. The QBT device in orientation 2 is

located at the Grid (BUS 3 node). The transmitted pre-fault

real power equals 0.9 pu of the generator rated power. For

the purposes of the digital simulation in the NETOMAC

(Siemens AG) environment both the generator and the

excitation control (Semipol) are modelled in detail. On the

other hand, the generator classical model served to the direct

method and the transmission lines were presented as a

reactance calculated from 100-km-long p-sections.

The disturbance in the system is represented by three-

phase faults in the outgoing line near the generator terminal

(BUS1) and as a result the faulty line is disconnected.

The results obtained by both the direct and conventional

(simulation) methods are presented in Table 2. We can see

two different CCTs obtained with the direct method, where

one fault-on simulation takes into account the effect of the

damper windings D 0: The CCTs calculated by direct

methods approach the simulation results (which are

considered a reference) in a few ms. These results are

very promising, especially if the simplifications made byapplying direct methods are kept in mind.

As shown, the estimates of the CCTs obtained using the

direct method are not conservative. The reason for this may

be the fact that in the case of the direct method application

the system resistances are not taken into consideration (they

cannot be unless an integration is carried out numerically

along the swing trajectory). In addition, the generator model

is far simpler than one used for simulation purposes.

Next,the effectiveness of theproposedQBTcontrol strategy

compared to the classical control strategy is demonstrated. If

the QBT is positioned at BUS3 (supposing orientation 2)

there is,of course, no difference between theapplied strategies.

If on the other hand the QBT is, with the specific limitsregarding controllable parameter a, placed at the BUS 2,

differences become evident. The results based on various QBT

ratings and orientations are presented in Table 3.

Deviations in percent were calculated as a CCT difference

between the proposed and the classical control strategies.

Discussing the results in Table 3 it is evident that a step

forward has been made in searching for an optimal control

strategy. The developed control law (12) not only improves

the transient stability but also provides us with a strategy to

maximise or minimise the transfer capability of the given

corridor with a QBT for any of the operating states.

Fig. 11. The test system.

Table 1

System data

Generator data Lines data Grid data

Pn 1500 MVA xd00 0:182 pu r 0:03 V/km Un 500 kV

xd0 0:270 pu x 0:33 V/km Pg 1350 MVA

Tm 6:6 s xd 1:47 pu c 12 nF/km

Td00 0:034 s xq

00 0:211 pu

Td0 2 s xq

0 0:636 pu

Tq00 0:041 s xq xd

Table 2Obtained CCT

Direct method Digital simulation

D 0:0167pu D 0 pu

CCT DCCT CCT DCCT CCT

QBTamax;min [deg] [ms] [%] [ms] [%] [ms]

19.2 107 7.0 102 2.0 100.0

40.4 167 11.3 157 4.7 150.0

51.5 209 4.5 195 2 2.5 200.0

PARamax;min [deg]

39 109 9.0 104 4.0 100.0

157 171 14.0 159 6.0 150.0322 235 17.5 214 7.0 200.0


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Calculations have shown that the proposed control

strategy is quite insensitive with regard to mistakes in

the evaluation of system reactances. In order to demonstrate

this, it was assumed that X2 was not known exactly. In the

calculations of the optimal a, 290% and 400% of X2deviations were considered. The results with the QBT in

orientation 2 are summarised in Table 4. As shown, the

CCTs are no longer optimal (as in Table 3), but they are still

better than when a classical control strategy is applied. This

has been anticipated because the classical control strategy

ignores X2: The differences Dwere calculated with regard to

the optimal CCT from Table 3.

5. Conclusions

In this paper new control laws based on a Control

Lyapunov Function for a PAR and a QBT have been

presented. They prove that we can squeeze the

maximum out of a device in any given situation and

can therefore be called optimal. For a QBT we showed

that the proposed strategy is more effective in the area of

large transmission angles than the classical control

strategy, while for small angles the differences are of

minor importance. Hence it can be concluded that for

oscillation damping purposes there is practically no

difference between the classical and proposed strategies.

On the other hand, for transient stability purposes the new

strategy has advantages. Furthermore, it has also been

shown that the proposed QBT control strategy is robustwith regard to the selection of inappropriate parameters.

For SMIB systems with TCPSTs new energy functions

have been constructed that take the effect of FACTS

devices and their control (optimal and classical) into

consideration. Any multimachine system with a TCPST that

can be reduced to a SMIB without losing the information

about the TCPST location can be treated with developed

energy functions (e.g. remote generation or interconnec-

tion). In this case a system is first reduced into a twomachine equivalent and then further reduced into a SMIB

system [6].

The results of test calculations are promising, especially

if the simplifications made by applying direct methods are

kept in mind.

Future research work in this field should also endeavour

to construct structure-preserving energy functions that are

more versatile.

Acknowledgements

The authors gratefully acknowledge the JuniorResearcher Scholarship awarded to U. Gabrijel by Ministry

of Education, Science and Sport of the Republic of Slovenia.

Appendix A

Proving a Lyapunov function for a PAR mainly follows a

procedure from the literature [2]. A function must satisfy

three conditions to be recognised as a Lyapunov function:

(a1) it must have the same stationary points as the maximum

transfer characteristic (14); (a2) it is positive definite in the

vicinity of one of the equilibrium points; (a3) its derivative

is not positive.(a1) Satisfying the first condition can be checked by the

determination of the nd0;Dv gradient. Assuming that

Pm , Pmaxe we only check the functions on lateral intervals

(note:nPE PAR iA;iC are constant):

gradnPAR

n

Dv

n

d0

2664

3775

nKE

Dv

nPEd0

2664

3775 A1

(a1-1) First interval 2R , d0 ,p

22 amax verification:

gradnPAR int: 1 MDv

2Pm 2 Pmaxe sind

0 amax

" #A2

Table 3

CCT obtained applying different control strategies

amax;min [deg] Orientation 1 Orientation 2

Classical Proposed Classical Proposed

CCT [ms] D[%] CCT [ms] D[%]

19.2 88 93 25.4 96 98 22.0

28.8 79 96 217.7 102 107 24.7

40.4 27 97 272.2 103 117 212.8

51.5 97 81 123 234.1

Table 4

CCT obtained by applying false X2 in optimal a calculation

amax;min [deg] 10% X2 500% X2

CCT [ms] D[%] CCT [ms] D[%]

19.2 96 22.0 97 21.0

28.8 103 23.7 105 21.9

40.4 107 28.5 112 24.251.5 107 213.0 109 211.4


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The gradient is equal to zero at the stationary point whereDv 0 and the electrical power is equal to the mechanical

power given by d0

1

d0S: This point is the equilibrium point

of the maximum transfer characteristic.

(a1-2) Third intervalp

2 amin , d

0# 2p2R verifica-

tion:

gradnPAR int:3 MDv

2Pm 2 Pmaxe sind

02 amin

" #A3

The gradient Eq. (A3) is equal to zero at the stationary point

where Dv 0 and the electrical power is equal to the

mechanical power given by d02 d0U: As can be visualised

from Fig. 9 this is also an equilibrium point of the maximum

transfer characteristic.

(a2) Satisfying the second condition for a Lyapunov

function can be checked by determining the Hessian matrixfor the first SEP:

HPAR

2nPAR

Dv2

2nPAR

d0 Dv

2nPAR

d0 Dv

2nPARd02

26664

37775

M 0

0 Pmaxe cosd0

amax

" #A4

Silvesters theorem says that this matrix is positive definite

when the diagonal elements are greater than zero: hence

M. 0 (which is always true) and in the long term (A4)must be greater than zero, which holds true for the first

stationary point d0S:

(a3) The third condition can be checked by determining

_n dn=dtalong the trajectory of the swing Eq. (3). First we

express the derivative as:

_nPAR dn

dt

dnKEdt

dnPE PAR

dtA5

and calculate each component respectively:

dnKEdt

nKEDv

dDv

dtMDv

dDv

dt M

dDv

dt

Dv A6

The factor in the square brackets corresponds to left-handside of Eq. (3). Replacing it by the right-hand side and

considering Eq. (14) as Pe yieldsdnKE

dt:

Similarly, the derivative of the potential energy can be

calculated (note: nPE PARi A;B;C are constant):

dnPE PARdt

nPE PAR

d0

dd0

dtA7

Summing the results of Eqs. (A6) and (A7) gives:

_nPAR dnPAR

dt 2D Dv2 A8

whichrepresentsthetotalsystemenergydecay.Itcorresponds

to the natural damping and its positive damping coefficient.

Therefore the _nPAR derivative is negative and hence nPARmeets all three conditions for a Lyapunov function.

QBT Lyapunov functions Eqs. (24) and (27) can be

verified in a similar way.

References

[1] Noroozian M, Angquist L, Ghandari M, Andersson G. Improving

Power System Dynamics by Series-Connected FACTS Devices. IEEE

Trans Power Delivery 1997;12(4):163541.

[2] Machowski J, Bialek JW, Bumby JR. Power system dynamics and

stability. Wiley; 1997.

[3] Yong Hua Song Allan T, editor. Flexible ac transmission systems

(FACTS), vol. 30. London: IEE, cop; 1999. p. 443505. (IEE power

and energy series).

[4] Mihalic R, Zunko P. Phase-shifting transformer with fixed phase

between terminal voltage and voltage boost-tool for transient stabilitymargin enhancement. IEE Proc., Gener. Transm. Distrib. 1995;

142(3):25762.

[5] Ghandhari M, Andersson G, Pavella M, Ernst D. A control strategy for

controllable series capacitor in electric power systems. Automatica

2001;37:157583.

[6] Pavella M, Murthy PG. Transient stability of power systems. Wiley;

1994.

[7] Peter Sauer W, Pai MA. Pai, power system dynamics and stability.

Prentice Hall; 1998.

[8] Hiskens IA, Hill DJ. Incorporation of SVCs into energy function

methods. Trans Power Syst 1992;7(1).

[9] Padiyar KR, Immanuel V. Modelling of SVC for stability evaluation

using structure preserving energy function. Electric Power Energy

Syst 1994;16(5).

[10] Gabrijel U, Mihalic R. Direct methods for transient stability

assessment in power systems comprising controllable series devices.

IEEE Trans Power Syst 2002;September:111622.

[11] Sepulchre R, Jankovic M, Kokotovic PV. Constructive nonlinear

control. Springer; 1997.

[12] Freeman RA, Kokotovic PV. Robust nonlinear control design.

Boston: Birkhauser; 1996.

Uros Gabrijel (1972) received his Dr Sc Degree in electrical

engineering from the University of Ljubljana, Slovenia, in 2003.

After a year with a local electric energy distribution company he joined

the LPEE Laboratory at the Faculty of Electrical Engineering in

Ljubljana where he is currently working as a researcher. His areas of

interest include system analysis and FACTS devices.

Rafael Mihalic (1961) received his diploma Engineer, MSc and Dr Sc

degrees from The University of Ljubljana, Slovenia, in 1986, 1989, and

1993, respectively. After finishing his basic technical education in 1986

he became a teaching assistant at the Department of Power Systems and

Devices of the Faculty for Electrical and Computer Engineering in

Ljubljana. Between 1988 and 1991 he was a member of the Siemens

Power Transmission and Distribution Group, Erlangen, Germany.

Since 2000 he has been an associate professor at the University of

Ljubljana. His areas of interest include system analysis and FACTS

devices. Prof. Mihalic is an IEEE, NY and Cigre, Paris member.


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