# GROUP OF RECURRENT NEURAL NETWORKS

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55

### SYSTEM ENVIRONMENTS

BINDESHWAR SINGH

Assistant Professor, Kamla Nehru Institute of Technology (KNIT), Sultanpur-228118, U.P., India,

bindeshwar.singh2025@gmail.com

ABSTRACT

This paper presents the development of the differential Algebraic equation (DAE) model of various FACTS controllers such as TCSC and SVC for operation, control, planning &protection of power systems. Also this paper presents the current status on development of the DAE model of various FACTS controllers such as TCSC and SVC for operation, control, planning &protection of power systems. Authors strongly believe that this article will be very much useful to the researchers for finding out the relevant references in the field of the DAE model of FACTS controllers for operation, control, planning & protection of power systems.

Keywords:- Flexible AC Transmission Systems (FACTS), FACTS Controllers, Differential Algebraic

Equations (DAE) model, Thyristor Controlled Series Capacitor (TCSC), Static Var Compensator (SVC), Power Systems.

1. INTRODUCTION

Traditionally, the objective of the reactive power (VAR) planning problem is to provide a minimum number of new reactive power supplies to satisfy only the voltage feasibility constraints in normal and post-contingency states. Various researches have been carried out for this subject [1] and [2]. Recently, due to a necessity to include the voltage stability constraints, a few researches have been reported concerning new formulations considering the voltage stability problem [3] and [4], which provides more realistic solutions for the VAR planning problem. However, the obtained solutions are sometimes too expensive since they satisfy all of the specified feasibility and stability constraints. In [5], a new formulation and solution method are presented for the VAR planning problem including FACTS devices, taking into account the issues just mentioned. TCSC and SVC are used to keep bus voltages and to ensure the voltage stability margin. The TCSC model presented in [2]-[5] and SVC model presented in [6]-[21]. TCSC [2]-[5] and SVC [6]-[21] can be used for power flow control, loop flow control, load sharing among parallel

corridors, enhancement of transient stability, mitigation of system oscillations and voltage (reactive power) regulation. Performance analysis and control synthesis of TCSC and SVC requires its steady-state and dynamic models.

This paper is organized as follows: Section II discusses the DAE model of TCSC and SVC. Section III presents the conclusions of the paper. 2. DIFFERENTIAL ALGEBRAIC

EQUATION (DAE) MODEL OF TCSC and SVC

A. DAE model of TCSC

1. Fundamentals of TCSC:

(2)

56 operate stably at power levels well beyond those for which the system was originally intended without endangering system stability [3]. Apart from this, TCSC is also being used to mitigate SSR (Sub Synchronous Resonance).The TCSC module shown in Fig.5.

α

α

Fig. 5.TCSC module

The steady-state impedance of the TCSC is that of a parallel LC circuit, consisting of fixed capacitive impedance,XC, and a variable inductive impedance, XL(α), that is,

C L

L C TCSC

X X

X X X

− =

) (

) ( )

(

α α α

Where

∞ ≤ ≤

− −

= , ( )

2 sin 2 )

( α

α α

π π

α L L L

L X X X

X

L

XL =ω , and αis the delay angle measured from the crest of the capacitor voltage (or, equivalently, the zero crossing of the line current). The impedance of the TCSC by delay is shown in Fig. 6.

Fig.6.TCSC equivalent Reactance as a function of firing angle

2. TCSC Controller Model:

The structure of the TCSC is the same as that of a FC-TCR type SVC. The equivalent impedance of

the TCSC can be modeled using the following equations [4].

⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢

⎣ ⎡

− −

+ + − − =

) 2 tan 2 tan .( ) 1 (

) 2 / ( cos . . 4

sin . 1 1

2 2

2 2

2

σ σ π

σ π

σ σ

k k k

k k

k

X

XTCSC C

Where

=

α Firing angle delay (after forward vale voltage)

=

σ Conduction angle=2(π−α)and

=

k TCSC ratio = XC /XT

The TCSC can be continuously controlled in the capacitive or inductive zone by varying firing angle in a predetermined fashion thus avoiding steady state resonance region.

3. Incorporation of TCSC in Multi-machine Power Systems:

The block diagram representation of TCSC shown in Fig. 7.

Fig.7.Block diagram representation of TCSC module

Let a TCSC be connected between bus k and bus m as shown in Fig. It has been assumed that the controller is lossless. The power-balance equation and

### B

TCSCare given as [4]

) sin( k m TCSC

m k k VV B

P = θ −θ (1)

) cos(

2

m k TCSC m k TCSC k

k V B VVB

Q = − θ −θ

(2)

) sin( m k

TCSC m k m VV B

P = θ −θ (3)

) cos(

2

k m TCSC m k TCSC m

m V B VV B

Q = − θ −θ

(3)

57

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎣ ⎡

− −

− −

− +

− −

− +

− − − −

− +

− − =

)) ( cos sin cos 4

) ( sin cos 4

) ( cos 2 sin

) ( cos 2 sin

) ( cos 2

) ( cos 2

) ( cos

) ( cos (

/ ) ( cos ) 1 2 (

2 2 3 2 4

2 4

4

2 4

α π α α

α π α

α π α

α π α

α π α

α π α

α π π

α π π

α π π

k k

k k

k k

k k

k k

k k

k k k X

k k

k B

C TCSC

(5) Equation (21) is obtained from (16).

There are number of control strategies for TCSC [4]

• Reactance control: BsetBTCSC=0

• Power control: PsetP=0

• Current control: IsetI =0

• Transmission angle control: δset −δ =0 Where the subscript “set” indicates set point. Any of the above mentioned control strategies can be used to achieve the objectives of TCSC. In this paper, the power control strategy has been used, the block diagram of which is shown in Fig.

The line power is monitored and compared to desired powerPset. The error is fed to proportional-integral (PI) controller. The output of PI controller is fed through a first order block to get the desired

α. The block diagram representation of TCSC with PI controller shown in Fig.8.

I

1

C

0

1TCSC

2TCSC

ref

### α

Fig.8. Block diagram representation of TCSC with PI controller

The controller equations are given as )

(

2 P P

s K

X set

I

TCSC = −

P K P K

X TCSC = I setI

2

(6)

1 1 1 1

2 1 1 1

c o c P c

set P c TCSC c

TCSC TCSC

T T

P K T

P K T X T X

X =− + + − +α

(7)

Linearization of (6) and (7) yields

1 2

1

1 1 1

2

TCSC TCSC P k TCSC

c c c

TCSC I k

X X K P

X

T T T

X K P

−∆ ∆ ∆

∆ = + −

∆ = − ∆

(8) & (9)

As the TCSC controller is assumed lossless, the real power at the two ends of the bus is same. The real power feedback is taken from bus k.

sin( ) sin( ) sin( ) cos( ) cos( )

k k mo TCSCo ko mo ko m TCSCo ko mo ko mo TCSC ko mo ko mo TCSCo ko mo k ko mo TCSCo ko mo m

P VV B V V B V V B

V V B V V B

θ θ θ θ θ θ

θ θ θ θ θ θ

∆ =∆ − + ∆ − + ∆ −

− ∆ − − ∆

Substituting value of

### P

k into (8) and (9)

1 2

1

1 1 1

2

sin( ) sin( ) sin( ) cos( ) cos( )

k mo TCSCo ko mo ko m TCSCo ko mo TCSC TCSC P

TCSC ko mo TCSC ko mo ko mo TCSCo ko mo k c c c

ko mo TCSCo ko mo m k mo TCSC

TCSC I

VV B V V B

X X K

X V V B V V B

T T T

V V B

VV B

X K

θ θ θ θ

θ θ θ θ θ

θ θ θ

∆ − + ∆ −

⎡ ⎤

−∆ ∆ ⎢ ⎥

∆ = + − + ∆ − + − ∆

⎢− − ∆ ⎥

⎣ ⎦

∆ =−

sin( ) sin( ) sin( ) cos( ) cos( )

o ko mo ko m TCSCo ko mo ko mo TCSC ko mo ko mo TCSCo ko mo k ko mo TCSCo ko mo m

V V B

V V B V V B

V V B

θ θ θ θ

θ θ θ θ θ

θ θ θ

− + ∆ −

⎡ ⎤

+ +

⎢ ⎥

⎢− − ∆ ⎥

⎣ ⎦

Writing above equations in matrix notation.

⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢

⎣ ⎡

∇ ∇ ∇ ∇

+ ∇ =

∇ •

m m k k

TCSC TCSC TCSC TCSC

V V B X A X

θ θ

Where

⎦ ⎤ ⎢

⎣ ⎡ ∇ ∇ = ∇

TCSC TCSC TCSC

X X X

2 1

(4)

58

[ ]

[ ]

1 1 1

1

1

1

1 ( )sin( ) 1

( )sin( ) 0

cos( )

sin( )

cos(

P ko mo ko mo

c c c

TCSC

I ko mo ko mo

TCSC

P

ko mo TCSCo ko mo c

P

mo TCSCo ko mo c

P

ko mo TCSCo ko mo c

K V V FFCONS

T T T

A

K V V FFCONS

A B C D

B

E F G H

K

A V V B

T K

B V B

T

K

C V V B

T θ θ θ θ θ θ θ θ θ θ − − ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ − − ⎥ ⎣ ⎦ ⎡ ⎤ = ⎢ − = − − = − − = [− − ] [ ] [ ] [ ] [ ] [ ] 1 ) sin( ) cos( ) sin( ) cos( ) sin( ) P

ko TCSCo ko mo c

I ko mo TCSCo ko mo I mo TCSCo ko mo I ko mo TCSCo ko mo

I ko TCSCo ko mo

K

D V B

T

E K V V B

F K V B

G K V V B

H K V B

θ θ θ θ θ θ θ θ θ θ − = − = − − = − − = − − − = − −

a. Details of Constant FFCONS:

Linearizing the equations (3.16) gives 1

( )

TCSC TCSC

B FFCONS X

∆ = ∆

Details of FFCONS are obtained as shown below. Linearization of (3.11) gives

2 2

1 1 1

( 1) ( 2) ( 1) sin( )

TCSC TCSC TCSCo TCSC

B FF X FF kπ k kπ kX X

∆ + ∆ = − − − ∆

or

2 2

1 1 1 2 ( 1) sin( )

1

TCSCo TCSC

TCSC TCSC

FF k k k kX X

B X FF π π ⎡− − − − ∆ ⎤ ∆ = ∆ ⎣ ⎦ or 1 ( ) TCSC TCSC

B FFCONS X

∆ = ∆

Where

2 2

1 1 2 ( 1) sin( )

1

TCSCo TCSC

FF k k k kX X

FFCONS FF π π ⎡− − − − ∆ ⎤ = ⎢ ⎥ ⎣ ⎦

FF1 and FF2 are computed 4 1 1 4 1 1 2 1 1 4

1 1 1

3 2

1 1

2 1

1 cos( )

cos( )

2 cos( )

2 )cos( )

2 sin( )cos( )cos( )

4 cos ( )sin( )

4 sin(

c TCSCo c TCSCo

TCSCo TCSCo c TCSCo TCSCo c

TCSCo TCSCo TCSCo c TCSCo TCSCo c

TCSCo

FF X k k kX

X k kX

k X k kX X

k X k kX X

k X X k kX X

k X k kX X

k X π π π π π π π π = − − − − − + − − − − −

− 1 1

2

1 1 1

)cos( )cos( )

2 sin( )cos( )cos( )

TCSCo TCSCo c TCSCo TCSCo TCSCo c

X k kX X

k X X k kX X

π π −

+ −

Linearizing the equations (3.12-3.15) gives

2

sin( ) sin( ) sin( )

cos( ) cos( )

2 cos( )

k k mo TCSCo ko mo ko m TCSCo ko mo ko mo TCSC ko mo ko mo TCSCo ko mo k ko mo TCSCo ko mo m

k ko TCSCo k ko TCSC k mo TCSCo ko mo ko m TCSCo

P V V B V V B V V B

V V B V V B

Q V B V V B VV B V V B

θ θ θ θ θ θ

θ θ θ θ θ θ

θ θ

∆ = ∆ − + ∆ − + ∆ −

+ − ∆ − − ∆

∆ = ∆ + ∆ −∆ − − ∆ cos( )

cos( ) sin( ) cos( )

sin( ) sin( ) sin( )

cos( )

ko mo ko mo TCSC ko mo ko mo TCSCo ko mo k ko mo TCSCo ko mo m

m k mo TCSCo mo ko ko m TCSCo mo ko ko mo TCSC mo ko ko mo TCSCo mo ko m

V V B V V B V V B

P V V B V V B V V B

V V B

θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ − − ∆ − + − ∆ − − ∆ ∆ = ∆ − + ∆ − + ∆ − + − ∆ − 2 cos( )

2 cos( ) cos( )

cos( ) sin( ) sin( )

ko mo TCSCo mo ko k

m mo TCSCo m mo TCSC k mo TCSCo mo ko ko m TCSCo mo ko ko mo TCSC mo ko ko mo TCSCo mo ko m ko mo TCSCo mo ko k

V V B

Q V B V V B V V B V V B

V V B V V B V V B

θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ − ∆ ∆ = ∆ + ∆ −∆ − − ∆ − − ∆ − + − ∆ − − ∆

The above equation can be written in matrix notation ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ∇ ∇ ∇ ∇ + ∇ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ∇ ∇ ∇ ∇ m m k k TCSC TCSC TCSC m m k k V V D X C Q P Q P θ θ 1

Incorporation of (25), (26), and (5) gives DAE model of multi-machine power system with TCSC incorporated in the system. After reordering, final form of DAE model with TCSC is given as

5 1 1 4 1 5 1 1 2 1 3 1 1 4 2 1

2 sin( )

sin( )

2 cos( )

2 sin( )

2 cos( )

2 sin( )

2 cos (

TCSCo c TCSCo TCSCo c TCSCo

TCSCo c TCSCo TCSCo TCSCo c TCSCo TCSCo c TCSCo

TCSCo TCSCo c TCSCo TCSCo

FF k B X k kX

kB X k kX

k B X k kX

k X B X k kX

k B X k kX

k X B X k kX

k X π π π π π π π π = − − − − − − − + − + − − 1 4 2 1 1 5

1 1 1

3

1 1 1

4 2

1

) cos( )

2 sin ( ) cos( )

2 sin( )cos( ) sin( )

8 sin( )cos( ) sin( )

4 cos ( ) cos(

TCSCo c TCSCo TCSCo TCSCo c TCSCo

TCSCo TCSCo TCSCo c TCSCo TCSCo TCSCo TCSCo c TCSCo

TCSCo TCSCo c

B X k kX

k X B X k kX

k X X B X k kX

k X X B X k kX

k X B X k

π π π π − + − − − + − + 1 4 2 1 1 2 2 1 1 3

1 1 1

2 2

1 1

2 2

1

)

4 sin ( ) cos( )

4 cos ( ) cos( )

4 sin( )cos( ) sin( )

2 cos ( ) cos( )

2 sin (

TCSCo TCSCo TCSCo c TCSCo TCSCo TCSCo c TCSCo TCSCo TCSCo TCSCo c TCSCo

TCSCo TCSCo c TCSCo TCS

kX

k X B X k kX

k X B X k kX

k X X B X k kX

k X B X k kX

k X π π π π π − + − − − − − + − − 1 3

1 1 1

) cos( )

2 sin( )cos( ) sin( )

Co TCSCo c TCSCo

TCSCo TCSCo TCSCo c TCSCo

B X k kX

k X X B X k kX

π

π −

(5)

59 U E

v z X

X

D D

C G

C K

P K

B B

A P

A A

P A X

X

TCSC

t new t

new TCSC

new new

t

new t new t TCSC t

new new

t

TCSC

⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢

⎣ ⎡ + ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢

⎣ ⎡

∇ ∇ ∇

⎥ ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢ ⎢

⎣ ⎡

=

⎥ ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢ ⎢

⎣ ⎡ ∇

• •

0 0

0 0

csc _ 2 csc _ 1 1

4 1

csc 4 2

csc 1

csc csc

2

2 csc 1 mod 1

Equation (27) can be written as

U E X

A

XSYS TCSC= SYS TCSCSYS TCSC+ TCSC

∇ • _ _ _

The System matrix with TCSC given as ) * )) ( ( *

( 2 4 3

1

_TCSC TC TC TC TC

SYS A A inv A A

A = −

Where

⎥ ⎦ ⎤ ⎢

⎣ ⎡ =

TCSC t

t TC

A P

P A A

csc 2

csc 1 mod 1 1

⎥ ⎦ ⎤ ⎢

⎣ ⎡ =

new t new t

new new TC

B B

A A

A

csc 1 csc

2 2

⎥ ⎦ ⎤ ⎢

⎣ ⎡ =

TCSC t TC

C G

P K A

1 csc 4 2 3

⎥ ⎦ ⎤ ⎢

⎣ ⎡ =

csc _ 2 csc _ 1

4 1

4

t new t

new

new new

TC D D

C K

A

B. DAE model of SVC

1. Fundamentals of SVC :

Static VAR Compensator (SVC) is a shunt connected FACTS controller whose main functionality is to regulate the voltage at a given bus by controlling its equivalent reactance. Basically it consists of a fixed capacitor (FC) and a thyristor controlled reactor (TCR). Generally they are two configurations of the SVC.

a) SVC total susceptance model. A changing susceptance Bsvc represents the fundamental frequency equivalent susceptance of all shunt modules making up the SVC as shown in Fig. 2(a).

b) SVC firing angle model. The equivalent reactance XSVC, which is function of a changing firing angle α, is made up of the parallel combination of a thyristor controlled reactor (TCR) equivalent admittance and a fixed capacitive reactance as shown in Fig. 2 (b). This model provides information on the SVC firing angle required to achieve a given level of compensation.

### α

Fig. 2(a) SVC firing angle model

## α

(6)

60

min

set

max

max

min

### B

Fig.3. steady-state and dynamic voltage/current Characteristics of the SVC

SVC firing angle model is implemented in this paper. Thus, the model can be developed with respect to a sinusoidal voltage, differential and algebraic equations can be written as

The fundamental frequency TCR equivalent reactance XTCR

σ σ

π

sin

− = L TCR

X X

Where σ =2(π−α),XLL And in terms of firing angle

α α

π π

2 sin ) (

2 − +

= L

TCR

X X

σ and αare conduction and firing angles respectively.

At α=900, TCR conducts fully and the equivalent

reactance

XTCR becomes XL, while atα=1800, TCR is

blocked and its

equivalent reactance becomes infinite.

The SVC effective reactance XSVC is determined by the parallel combination of XC and XTCR

L C

L C SVC

X X

X X X

π α α

π π α

− +

− =

] 2 sin ) ( 2 [ ) (

Where

C XC =1ω

⎭ ⎬ ⎫ ⎩

⎧ − + −

=

L C C k k

X X X V Q

π

α α

π ) sin2

( 2 [

2

The SVC equivalent reactance is given above equation. It is shown in Fig. that the SVC equivalent susceptance(BSVC =−1/XSVC)profile, as function of firing angle, does not present discontinuities, i.e., BSVC varies in a continuous, smooth fashion in both operative regions. Hence,

linearization of the SVC power flow equations, based on BSVC with respect to firing angle, will exhibit a better numerical behavior than the linearized model based onXSVC .

Fig.4. SVC equivalent susceptance profile The initialization of the SVC variables based on the initial values of ac variables and the characteristic of the equivalent susceptance (Fig.), thus the impedance is initialized at the resonance point XTCR =XC, i.e. QSVC=0, corresponding to firing angle α=1150 , for chosen parameters of L

and C i.e. XL =0.1134Ωand XC =0.2267Ω. 2. Proposed SVC power flow model:

The proposed model takes firing angle as the state variable in power flow formulation. From above equation the SVC linearized power flow equation can be written as

) ( ) ( 2

) (

] 1 2 [cos 2 0

0

0 i

k i

L k i

k k

X V Q

P

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∇ ∇ ⎥ ⎥ ⎦ ⎤ ⎢

⎢ ⎣ ⎡

− =

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ∇

α θ α

π

At the end of iteration i, the variable firing angle α

is updated according to

) ( ) 1 ( )

(i α i α i

α = − +∇

SVC SVC k

(7)

61 3. SVC CONTROLLER MODEL:

Fig.5. Block diagram of SVC

The state equations of the SVC can be written from above figure

[ ]

1 3 1

2 , 1

3 2 , 1 3

1

(1 )

( )

1

( )

SVC SVC SVC SVC m

SVC I ref SVC SVC

SVC SVC P ref SVC SVC SVC c

X V KX X

T

X K V X

X X K V X X

T

= + −

= −

⎡ ⎤

= + − −

The reactive power

### Q

SVCsupplied by the SVC can be written as

2 3 SVC SVC SVC

Q =V X

Linearization of above equations ( )-( ) yields

[ ]

1 3 3 1

2 , 1

3 2 , 1 3

2

3 3

1 (1 ) )

( )

1

( )

2

SVC SVC SVCo SVCo SVC SVC m

SVC I ref SVC SVC

SVC SVC P ref SVC SVC SVC c

SVC SVCo SVC SVCo SVCo SVC

X V KX V K X X

T

X K V X

X X K V X X

T

Q V V X V X

∆ = ∆ + + ∆ − ∆

∆ = ∆ − ∆

⎡ ⎤

∆ = ∆ + ∆ − ∆ − ∆

∆ = ∆ + ∆

Where “∆” denotes perturbed value and subscript “o” denotes the nominal value. The above equations are linearized, reordered and then expressed as

SVC

### ]

SVCo m

SVC SVC SVC

c c c

P I

m SVCo m

SVC SVC SVC

V KX

T

X X X

T T T

K K

T KV T

X X X

⎥ ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢ ⎢

⎣ ⎡ +

+ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢

⎣ ⎡

∇ ∇ ∇

⎥ ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢ ⎢

⎣ ⎡

− −

− −

= ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢

⎣ ⎡

∇ ∇ ∇

• • •

0 0 1 ( 1

1 1

0 0 0 1

) 3

3 2 1

3 2 1

Above equation can be written as SVC SVC SVC AVC

SVC A X B V

X = ∇ + ∇

∇ •

Where

⎥ ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢ ⎢

⎣ ⎡

− −

− −

=

c c c

P I

m SVCo m

SVC

T T T

K K

T KV T

A

1 1

0 0 0 1

And

⎥ ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢ ⎢

+

=

0 0 1 ( 1

) 3SVCo m

SVC

KX T

B

4. INCORPORATION OF SVC IN MULTI-MACHINE POWER SYSTEMS:

In its simplest form SVC is composed of FC-TCR configuration as shown in Fig.2. The SVC is connected to a coupling transformer that is connected directly to the ac bus whose voltage is to be regulated. The effective reactance of the FC-TCR is varied by firing angle control of the thyristors. The firing angle can be controlled through a PI controller in such a way that the voltage of the bus where the SVC is connected is maintained at the desired reference value.

The SVC can be connected at either the existing load bus or at a new bus that is created between two buses. As DAE model is based on power-balance, rewriting of the power-balance equations at the buses with SVC connected in the system requires modification of D2new.When SVC is

connected at specified load buses, and gets modified as given below

n m

i

Y V V V Q Q

n m

i

Y V V V P P

ik k i ik k n

k i i Li SVCi

ik k i ik k n

k i i Li SVCi

... ... ,... 1

0 ) sin(

) (

..., ,... 1

0 ) cos(

) (

1 1

+ =

= − − −

+ + =

= − − −

+

### ∑

= =

α θ θ

α θ θ

(8)

62 0

0

2 1

2 1

= ∇ + ∇ + ∇

= ∇ + ∇ + ∇ + ∇

l SVC g SVC

SVC

l g

SVC SVC l SVC

V D V D X D or

V D V D X D V C

Where

2 2 C D

DSVC = SVC+

The incorporation of the SVC into DAE model of multi-machine power system is done on the same lines as explained in [2] given as follows:

U E

v z X

X

D D D G

C K P K

B P A P

A A

P A

X X

SVC

svc new svc new SVC

new new svc

svcnew svc

SVC SVC

new new

SVC

SVC

⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢

⎣ ⎡

+ ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢

⎣ ⎡

∇ ∇ ∇

⎥ ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢ ⎢

⎣ ⎡

=

⎥ ⎥ ⎥ ⎥ ⎥

⎦ ⎤

⎢ ⎢ ⎢ ⎢ ⎢

⎣ ⎡ ∇

• •

0 0 0

0 0

_ 2 _ 1 1

4 1 4 2

3 2

3 2

1 mod 1

The state equation for the system with SVC is then given as follows:

U E X

A

Xsys svc = sys svcsys svc+ SVC

∇ • _ _ _

The System matrix with SVC given as ) * ) ( ( *

( 2 4 3

1

_SVC SV SV SV SV

SYS A A inv A A

A = −

Where 1mod 1 1

2

2 3

2 3

2 4

3 1

1 4

4

1 _ 2 _

svc SV

svc SVC new new SV

svc svcnew svc SV

SVC new new SV

new svc new svc

A P

A

P A

A A

A

P B

K P

A

G D

K C

A

D D

⎡ ⎤

= ⎢ ⎥

⎣ ⎦

⎡ ⎤

= ⎢ ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦

⎡ ⎤

= ⎢ ⎥

⎣ ⎦

5. CONCLUSIONS

This paper presents the development of the differential Algebraic equation (DAE) model of various FACTS controllers such as TCSC and SVC for operation, control, planning & protection of power systems. Also this paper presents the current status on development of the DAE model of various FACTS controllers such as TCSC and SVC for operation, control, planning &protection of power systems. The Proposed model of TCSC and SVC also can be used for the steady–state analysis (i.e. low frequency analysis) such as placement and coordination of FACTS controllers in power systems from different angle such as power system oscillations enhancement, voltage stability enhancement, increase the available transfer capacity, increase the loadability of power systems, and decrease the active and reactive power losses, etc.

ACKNOWLEDGMENT

The authors would like to thanks Dr. S. C. Srivastava, and Dr. S. N. Singh, Indian Institute of Technology, Kanpur, U.P., India, and Dr. K.S. Verma, and Dr. Deependra Singh, Kamla Nehru Institute of Technology, Sultanpur, U.P., India, for their valuables suggestions regarding placement and coordination techniques for FACTS controllers form voltage stability, and voltage security point of view in multi-machine power systems environments.

REFERENCES

[1] Kundur P., “Inter-area Oscillations in Power System,” IEEE Power Engineering Society, pp. 13-16, October 1994.

[2] A. Kazemi, and B. Badrzadeh, “Modeling and Simulation of SVC and TCSC to Study Their Limits on Maximum Loadalibility Point,” Electrical Power & Energy Systems, Vol. 26, pp. 619-626, 2004.

[3] Claudio A. Canizares, and Zeno T. Faur, “ Analysis of SVC and TCSC Controllers in Voltage Collapse,” IEEE Trans on Power Systems, Vol. 14, No. 1, February 1999. [4] A. M. Simoes, D. C. Savelli,P. C. Pellanda,

N. Martins, and P. Apkarian ,“Robust Design of a TCSC Oscillation Damping Controller in a Weak 500-kV Interconnection Considering Multiple Power Flow Scenarios and External Disturbances,” IEEE Trans on Power Systems, Vol. 24, No.1, February 2009.

[5] B. Chaudhuri, and B. C. Pal, “Robust Damping of Multiple Swing Modes Employing Global Stabilizing Signals with a TCSC,” IEEE Trans on Power Systems, Vol. 19, No.1, February 1999.

[6] Y. Mansour, W. Xu. F. Alvarado, and C. Rinzin, “SVC Placement Using Critical Modes of Voltage Stability,” IEEE Trans. on Power Systems, Vol. 9, pp. 757-762, May 1994.

[7] Cigre Working Group, “Modeling of Static VAR Systems for Systems Analysis,” Electra, Vol. 51, pp.45-74, 1977.

[8] Malihe M. Farsangi, Hossein Mezamabadi-pour, “Placement of SVCs and Selection of Stabilizing Signals in Power Systems, “IEEE Trans. on Power Systems, Vol. 22, No. 3, August 2007.

(9)

63 Controllers”, IEEE Trans. on Power Delivery, Vol. 18, No.3, July 2003.

[10] N. Martins and L. T. G. Lima, “Determination of Suitable Locations for Power System Stabilizers and Static Var Compensators for Damping Electro-mechanical Oscillation in large Scale Power Systems,” IEEE Trans. on Power Systems, Vol. 5, No.4, pp. 1455-1469, November 1990.

[11] M. K. Verma, and S. C. Srivastava, “Optimal Placement of SVC for Static and Dynamic Voltage Security Enhancement,” International Journal of Emerging Electric Power Systems, Vol.2, issue-2, 2005.

[12] J.G. Singh, S. N. Singh, and S. C. Srivastava, “Placement of FACTS Controllers for Enhancing Power System Loadability”, IEEE Trans. on Power Delivery, Vol. 12, No.3, July 2006.

[13] R. K. Verma, “Control of Static VAR systems for Improvement of Dynamic Stability and Damping of Torsional Oscillations,” Ph. D. Thesis, IIT Kanpur, April 1998

[14] Roberto Mingues, Federico Milano, Rafael Zarate-Minano, and Antonio J. Conejo, “Optimal Network Placement of SVC Devices,” IEEE Trans. on Power Systems, Vol.22, No.4, November 2007.

[15] C. S. Chang, and J. S. Huang, “Optimal SVC Placement for Voltage Stability Reinforcements,” Electric Power System Research, Vol.42, pp.165-172, 1997.

[16] R. M. Hamouda, M. R. Iravani, and R. Hacham, “Coordinated Static VAR Compensators for Damping Power System Oscillations,” IEEE Trans. on Power Systems, Vol. PWRS-2, No.4, November 1987.

[17] Yuan-Lin Chen, “Weak Bus-Oriented Optimal Multi-Objective VAR Planning,” IEEE Trans on Power Systems, Vol. 11, No.4, November 1996.

[18] Ying-Yi Hong, and Chen-Ching Liu, “A heuristic and Algorithmic Approach to VAR Planning,” IEEE Trans on Power Systems, Vol. 7, No.2, May 1992.

[19] Y. Chang, “Design of HVDC and SVC Coordinate Damping Controller Based on Wide Area Signal,” International Journal of

Emerging Electric Power Systems, Vol. 7, Issue 4, 2006.

[20] C. P. Gupta, “Voltage Stability Margin Enhancement using FACTS controllers,” Ph. D. Thesis, IIT Kanpur, October, 2000. [21] Karim Sebaa, and Mohamed Boudour,

“Power System Dynamic Stability Enhancement via Coordinated Design of PSSs and SVC –based Controllers Using Hierarchical Real Coded NSGA-II,” Power &Energy Society General Meeting – Conversion & Delivery of Electrical Energy in the 21st Century 2008, IEEE, 20-24 July, 2008, pp1-8.

AUTHOR PROFILE:

Bindeshwar Singh was born in Deoria, U.P., India, in 1975. He received the B.E. degree in electrical engineering from the Deen Dayal Uphadhayay University of Gorakhpur, Gorakhpur, U.P., India, in 1999, and M. Tech. in electrical engineering (Power Systems) from the Indian Institute of Technology (IITR), Roorkee, Uttaranchal, India, in 2001. He is now a Ph. D. student at Gautam Buddha Technical University, Lucknow, U.P., India. In 2001, he joined the Department of Electrical Engineering, Madan Mohan Malviya Engineering College, Gorakhpur, as an Adoc. Lecturer. In 2002, he joined the Department of Electrical Engineering, Dr. Kedar Nath Modi Institute of Engineering & Technology, Modinagar, Ghaziabad, U.P., India, as a Sr. Lecturer and `subsequently became an Asst. Prof. & Head in 2003. In 2007, he joined the Department of Electrical & Electronics Engineering, Krishna Engineering College, Ghaziabad, U.P., India, as an Asst. Prof. and subsequently became an Associate Professor in 2008. Presently, he is an Assistant Professor with Department of Electrical Engineering, Kamla Nehru Institute of Technology, Sultanpur, U.P., India, where he has been since August’2009. His research interests are in Placement and Coordination of FACTS controllers in multi-machine power systems and Power system Engg.

Mobile: 09473795769

Figure

References

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