55

### UTILITIES OF DIFFERENTIAL ALGEBRAIC EQUATIONS

### (DAE) MODEL OF SVC AND TCSC FOR OPERATION,

### CONTROL, PLANNING & PROTECTION OF POWER

### 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:*

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*

*X _{L}* =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*

*are given as [4]*

_{TCSC})
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 _{ }

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: *B _{set}*−

*B*=0

_{TCSC}• Power control: *P _{set}* −

*P*=0

• Current control: *I _{set}* −

*I*=0

• Transmission angle control: δ*set* −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 power*Pset*. 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*

*K*

*s*

1

### 1

### 1

### +

*sT*

_{C}0

### α

### α∆

_{α}

1*TCSC*

*X*

2*TCSC*

*X*

*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* *set*− *I*

•

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*

*into (8) and (9)*

_{k}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

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* *k*π *k* *k*π *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 −

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* *TCSC*∇ *SYS* *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

### α

Fig. 2(a) SVC firing angle model

## α

60

min

*I*

_{I}

_{I}

_{set}max

*I*

max
*B*

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),*XL* =ω*L*
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}_{=}_{90}0_{, TCR conducts fully and the equivalent }

reactance

XTCR becomes XL, while at_{α}_{=}_{180}0_{, 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*
*X _{C}* =1

_{ω}

⎭ ⎬ ⎫ ⎩

⎨

⎧ − + −

=

*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 on*XSVC* .

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}_{=}_{115}0_{ , 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*

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*

*supplied by the SVC can be written as*

_{SVC}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

)
3*SVCo*
*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 *D*2* _{new}*.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

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*

*D _{SVC}* =

*+*

_{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* *svc*∇ *sys* *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.

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**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