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CHAPTER 4

MODELLING OF UPFC AND STATCOM

FORMULTIMACHINE SYSTEM STABILITY

4.1 INTRODUCTION

Stability is a condition of equilibrium among opposing forces. The

method by which interconnected synchronous machines keep on synchronism

with another machine is through restoring forces, which acts when there are

forces tending to increase or decrease its speed on one or more machines with

respect to other machines. Instability in a power system is described by

depending upon the system configuration and operating mode. Generally, the

stability problem has been maintaining of synchronous operation. This aspect

of stability is influenced by the dynamics of machine rotor angles. But,

instability may also be encountered without loss of synchronism. For

example, a system can go unstable because of the collapse of load voltage.

Maintaining synchronism is not an issue in this instance; instead, the concern

is stability and control of voltages. In this work, the discussion is restricted to

voltage and rotor angle stability.

Rotor angle stability is the ability of interconnected synchronous

machines of a power system to remain in synchronism [2]. This stability

problem is concerned with the behavior of a synchronous machine after it has

been perturbed. Under steady state conditions, there is equilibrium between

the input mechanical torque and the developed electrical torque of each

machine. This equilibrium is upset during perturbation of the system. The

torque unbalance is caused by a change in load, generation or any other

network condition.

In any case, for the system to be stable all the machines must remain

operating in parallel and at the same speed. However, the statement declaring

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the power system to be stable is not meaningful unless the conditions under

which this stability has been examined are clearly stated. This includes the

operating conditions as well as the type of perturbation (which can be large or

small) given to the system.

Transient stability is the ability of the power system to maintain

synchronism when subjected to a large disturbance [2]. The resulting system

response involves large excursions of generator rotor angles and is influenced

by the nonlinear power-angle relationship. Stability depends on both the

initial operating state of the system and the severity of the disturbance. This

disturbance is usually so large that it alters the post disturbance equilibrium

conditions relative to those existing prior to the disturbance. The work

presented in this thesis is focused on the power system behavior when

subjected to large disturbances and the enhancement of this stability using

FACTS controller.

The most common form of instability between interconnected

generators is loss of synchronism, monotonically, in the first few seconds

following a fault due to lack of synchronizing torque and damping torque.

The first step in a stability study is to make a mathematical model of

the system. The elements included in the model are those affecting the

machine. The complexity of the model depends upon the type of stability

study. Generally, the components of the power system that influence the

electrical and mechanical torques of the machines are included in the model.

Such components are the loads and their characteristics, the network during

the disturbance and the parameters of synchronous machines (such as inertia

of the rotating mass). Thus, the basic requirements for these studies are initial

conditions of the power system prior to the start of the disturbance and the

mathematical description of the main components of the system that might

affect the behavior of synchronous machines.

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Generally,differential equations are used to describe the various

components. The system equations for small signal stability analysis are

usually nonlinear. The behavior of any dynamic system, such as a power

system, is described by a set of n first order non-linear differential equations

of the form given by (4.1).

t~ ,u~ ,x~f=•~x (4.1)

In Equation (4.1), f is a vector of nonlinear functions. The column

vector~x is referred to as a state vector and

~u is the vector of inputs to the

system. The study of dynamic behavior of the system is based on the nature of

these differential equations.

The first step is to do a load flow study as discussed [2], to obtain the

initial steady state conditions. After establishing initial conditions, a

mathematical model of the power system is formed as discussed in [4]. In this

model the effects of AVR and PSS are taken into account except the effects of

governor. The mathematical model obtained is set of non-linear differential

equations. Solving these equations by using Runge-Kutta method the state

variables are determined. After giving a large disturbance for a particular

period the system is again restored and the behavior of the system is studied at

the same time the effect of change in load and change in mechanical input

power of the system is also studied.

4.2 SYSTEM MODELLING

This section presents the mathematical models used for the power

system components such as generator, exciter and PSS. Mathematical

modeling of STATCOM and UPFC is also discussed in this section. Feed

Back Linearizing Controller (FBLC) modeling and implementation to

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STATCOM and UPFC is also described in this chapter. Further tuning of

FBLC is done using two intelligence techniques Bacterial Foraging Algorithm

and Differential Evolution is discussed.

4.2.1 Synchronous Machine Model

Synchronous machine is represented by means of the single-axis model

[4]. The state variables are rotor angle i , rotor angular velocity, and the

voltage proportional to main field flux linkage, 'qE . The sub transient

reactance, saturation and turbine governor dynamics are neglected.

In developing equations for a mathematical model of a multi-machine

power system, the following assumptions were made

1. Mechanical power input Pm is constant.

2. The mechanical rotor angle of a machine coincides with the angleofthe

voltage behind the transient reactance.

3. Loads are represented by passive impedance.

The resulting differential – algebraic equations for the ‘m’ machine,

‘n’ bus system with exciter model is given below as the state equations in p.u.

dididiqifdiqi ixxEE

dtdE

)'(''

T'doi (4.2)

midt

di

i ,.....,11 (4.3)

miiixxiETdt

djqididiqiqiqmi

i ,...,1/})''('{ (4.4)

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Where j - fHs

id - direct axis current

iq - quadrature axis current

xd - direct axis synchronous reactance

xd’ - direct axis transient reactance

xq - quadrature axis synchronous reactance

xq’ - quadrature axis transient reactance

Eq’ - Voltage proportional to main field flux linkage

T’do - direct axis open circuit time constant

Efd - Equivalent stator emf corresponding to field voltage

Hs - inertia constant of synchronous machine

Tm - Mechanical torqueof synchronous machine

Equation (4.4) has dimensions of torque in per-unit.When the stator

transients were neglected, the electrical torque became equal to the per-unit

power associated with the internal voltage source. The dynamic performances

of STATCOM and UPFC have been analyzed with different types of

disturbance thus damping has not been included in equation 4.4, because

STACOM and UPFC can improve the damping of the system. The system

data have been given in Appendix 3.

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4.2.2 Algebraic Equation The stator algebraic equation can be written in the form of

m1,...,i

ejEix-xE-

)eji)(ijx(ReV0

)2

-j'qiqi

'di

'qi

'di

)2

-j

qidi'disii

i

ii

(4.5)

Where 'diE = 0, since one axis model has been considered.

4.2.3 Excitation System and PSS Model

IEEE Type 1S [4] excitation system model is considered in the sample

system. The block diagram of the excitation system with PSS is shown in

Figure 4.1. The state equations are given below.

1V•

= (Vt-V1)/TR (4.6)

Efd = (KAVe- Efd)/TA (4.7)

3

•

V = {[KF(KAVe-Efd)/TA]-V3}/TF (4.8)

Where Ve - VREF+VF-V1-V3+ upss (4.9)

Vt - terminal voltage of synchronous machine

V1 - output signal of filter

V3 - output signal of stabilizing circuit

VR - regulator output signal

VF - supplementary stabilizing signal

VREF - regulator reference voltage

Ve - error voltage

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KA - gain of amplifier excitation system

KF - gain of stabilizing feedback

TF - time constant of stabilizing feedback

TA - time constant of amplifier of excitation system

TR - regulator time constant

upss - PSS output signal.

The PSS is represented by a washout filter and a cascade of lead lag

controllers of the following form

2

1

11

1 sTsT

sTwsTwKU stabpss (4.10)

Where, the rotor speed deviation is taken as the input to the PSS. Kstab, T1

and T2 are stabilizer gain and time constants respectively.

The parameters Kstab, T1 and T2 are to be determined to enhance the

system damping for the electromechanical mode.Practically, the washout

block has little phase compensation effect and its time constant Tw is fixed to

ten seconds in advance. The Exciter, PSS data for 3 machine, 9-bus system

are given in Table A 3.5 and Table A 3.6 of Appendix 3.

Figure 4.1Block diagram of IEEE Type-1s Excitation model with PSS

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VjQ-P

=y 2i

LiLiLi

4.2.4 Internal Node Model

This is a widely used reduced-order multi-machine model in first-

swing transient stability analysis. In this model, the loads are assumed to be

constant impedances and converted to admittances as [41]

n1,....,i (4.11)

Where Liy load admittance at ith bus, Vi – Voltage at ith bus

LiP - real power load at ith bus and LiQ - reactive power load at ith bus.

There is a negative sign for Liy , since loads are assumed as injected

quantities.

(4.12)

Where,

'

1

dijXDiagy i = 1,2,….., m (4.13)

m -no of machines

and 000

1yYY NN (4.14)

y y 0

y1NY

0

augY

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If transmission line resistances are neglected, then the network

admittance matrix is

ijN BjY (4.15)

Where Bij=suspectance betweenith and jth bus.

Adding Liy to the diagonal elements of the N1Y matrix and makes it )(12 LiNN yDiagYY .

The modified augmented Y matrix becomes

(4.16)

Figure 4.2 Augmented Y matrix with constant impedance

y y 0

y 2NY

0

augnewY

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The passive portion of the network is shown in Figure 4.2. The

network equations for the new augmented network can be written as

B

A

DC

BAA

VE

YYYY

0I (4.17)

where yYA ,

0YB y ,

0y-YC and 2DY NY

The n network buses can be eliminated, since there is no current

injection at these buses. Thus

ACD1

BAA EYYYYI (4.18)

AintA EI Y

where ,the elements of AI and AE are respectively,

QiDi2

j

qidii jIIejIIIi

and iii EE . (4.19)

Where i = 1,…,m.

intY = CD1

BA YYYY (4.20)

The elements of intY are ijijij jBGY . Since the network buses have

been eliminated, the internal nodes are such as 1,.., m, for ease of notation.

j

1i EI

m

jijY

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i= 1, ……, m (4.21)

Real electrical power out of the internal node ‘i’ from Figure 4.2 is given by

i*

iei IEReP

m

1j

*j

*ijiei EYeEReP i

m

1j

-jijijiei

ji eEBGeEReP j

m

1jjiijijei sincosEEBGReP jiji jj

(4.22)

4.3 INTERFACING OF STATCOM AND UPFC

For 3 machines, 9-bus system the STATCOM has been connected at

8th bus, which is a load bus and UPFC has been connected between line 7-8

which is shown in Figure 4.3 and 4.4

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Figure 4.3 Line diagrams of 3 machines, 9-bus system with STATCOM

Figure 4.4 Line diagrams of 3 machines, 9-bus system with UPFC

4.3.1 Modelling of UPFC for Multi Machine System

The mathematical model of the UPFC is derived here in the d-q

(synchronously rotating at the system angular frequency ) frame of

reference. This is followed by a detailed description of the conventional PI

control strategy employed for active and reactive power control using UPFC.

The equivalent circuit model of a power system equipped with a UPFC

is shown in figure A3.2 inAppendix 3. The series and shunt VSIs are

represented by controllable voltage sources Vc and Vp, respectively.Rp and Lp

represent the resistance and leakage reactance of the shunt transformer

respectively. Leakage reactance and resistance of series transformer have

been neglected.

The mathematical model of UPFC is derived by performing standard

d-q transformation [4] of the current through the shunt transformer and series

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transformer. They are as given below ( is the angular frequency of the

voltages and currents).

4.3.2 Modeling of Shunt Converter

The dynamic equations governing the instantaneous values of the

three-phase voltages across the two sides of STATCOM and the current

flowing into it are given by [40]

R + L i = V V (4.23)

= R + V V (4.24)

Where:i = i i i , V = [V V V ] , V = V V V

= 0 0

0 00 0

and = 0 0

0 00 0

Under the assumption that the system has no zero sequence

components, all currents and voltages can be uniquely represented by

equivalent space phasors and then transformed into the synchronous d-q-o

frame by applying the following transformation (q is the angle between the d-

axis and reference phase axis):

=

cos cos +

sin sin sin + (4.25)

Thus, the transformed dynamic equations are given by,

= R + V V + i (4.26a)

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= R + V V i (4.26b)

Where, is the angular frequency of the AC bus voltage.

4.3.3 Cascade Control Strategy for Shunt Converter

The conventional control strategy for this inverter concerns with the

control of ac-bus and dc-link voltage. The dual control objectives are met by

generating appropriate current reference (for d and q axis) and, then, by

regulating those currents. PI controllers are conventionally employed for both

the tasks while attempting to decouple the d and q axis current

regulators.In this study, the strategy adopted in [40] [41] for shunt current

control has been taken.The inverter current ( pi ) is split into real (in phase with

ac-bus voltage) and reactive components.The reference value for the real

current is decided so that the capacitor voltage is regulated by power

balance.The reference for reactive component is determined by ac-bus voltage

regulator.As per the strategy, the original currents in d-q frame )i ,i( pqpd are

now transformed into another frame, qd frame, where d axis coincides

with the ac-bus voltage (Vs), as shown in Figure 4.5.

Figure 4.5 Phasor diagram showing d-q and d’-q’ frame

Thus, in qd frame, the currents dpi and qpi represent the real and

reactive currents and they are given by:

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= cos + sin (4.27)

= cos sin (4.28)

Now, for current control, the same procedure as outlined in [57] has

been adopted by re-expressing the above differential equations as:

= R + V V + i (4.29)

= R + V i (4.30)

Where

= cos + sin (4.31)

= cos sin (4.32)

= +

The VSI controlled voltages are as follows:

V = L i + L u (4.33)

V = L i + V L u (4.34)

By putting the above expressions for dpV and qpV in equations (4.29)

and (4.30) the following set of decoupled equations are obtained.

= i + u (4.35)

= i + u (4.36)

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Conventionally, the control signals du and qu are determined by

linear PI controllers. The complete cascade control architecture is shown

below in Figure 4.6, where dpqiqpicpcitpt K,K,K,K,K,K,K and diK are the

respective gains of the PI controllers.

In this study, the above design has been used for demonstration of

STATCOM control.This approach leads to good control as illustrated by the

simulation results.

The final and important stage of the design of PI based STATCOM

involves tuning of parameters of STATCOM, which is posed as an

optimization problem. In this problem the optimal output gain K0 are

determined by maximizing the damping out of transient voltage oscillations of

the load bus voltage and dc capacitor voltage being controlled. This is in

effect carried out by minimizing Sum Squared Deviation (SSD) of the load

bus voltage and dc capacitor voltage being controlled from the desired value

through non-linear simulation of power system under typical operating

condition and disturbance. The non-linear simulation is carried out using a

Transient Stability Algorithm [4] employing a Runge-Kutta fourth order

method. To get the original currents it is again transformed to d -q

Figure 4.6 PI-Control Structure of STATCOM

4.3.4 Modeling of Series Converter

= + ( sin ) (4.37)

89

= + cos (4.39)

For fast voltage control, the net input power should instantaneously

meet the charging rate of the capacitor energy. Thus, by power balance,

= + + + ( + )

= Vdcidc= +

= + + + ( ) + (4.40)

An appropriate series voltage (both magnitude and phase) should be

injected for obtaining the commanded active and reactive power flow in the

transmission line, i.e., uu Q ,P in this control. The current references are

computed from the desired power references and are given by,

= (4.41)

= (4.42)

The power flow control is then realized by using appropriately

designed controllers to force the line currents to track their respective

reference values. Conventionally, two separate PI controllers are used for this

purpose. These controllers output give the amount of series injected voltages

)V ,V( cqcd . The corresponding control system diagram is shown in Figure 4.7.

P I cqi

Equations (4.39) & (4.40)

sK

K idpd

sK

K iqpq

P I

refP

refQ

refcqi

refcdi

cdi

+

_

_

+

cdV

cqV

90

Figure 4.7 PI Control of Series Converter

4.4 IMPLEMENTATION OF FBLC FOR UPFC IN SMIB

ANDMULTI MACHINE

4.4.1 Implementation of FBLC for UPFC in SMIB

In this section, the design steps for the feedback linearizing control of

UPFC have been presented followed by simulation results under various

transient disturbances. A brief review of nonlinear control using feedback

linearization is presented in the Appendix 2.

4.4.2 FBLC design

In UPFC control, there are four objectives. They are (i) active power

control, (ii) reactive power control, (iii) ac-bus voltage (Vs) control and (iv)

dc link voltage (Vdc) control. Tracking of active and reactive power are

indirectly translated to tracking of line currents to their respective reference

values computed from Pref and Qref.The differential equations of the line

currents (ibd, ibq) and Vdcare already derived. Thus, for control of Vs, its

differential equation need to be derived, i.e, Vsis taken as an additional state

in this control design.

Now, for the control design, the complete state space model is

expressed in the form of Equations. (A.1) and (A.2) in Appendix 1 as follows:

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X=

123456

= (4.41)

= =

=

=

=

=

= =

Thus,

= ( ) + (4.42)

= ( ) + (4.43)

= ( ) + (4.44)

= ( ) + (4.45)

= ( ) (4.46)

= ( ) + + + + ( ) (4.47)

Where:

1( ) +1

Lp(Vsd) (4.48)

92

2( ) +1

Lp(Vsq) (4.49)

=1

=

3( ) +1

Le(Vsd Vbd) (4.50)

4( ) +1

Le)(Vsq Vbq) (4.51)

=1

=

5( ) = 1 + 2 (4.52)

51( ) =1

C[ Vsd + Vsq + (Vsd Vud) + (Vsq Vuq) )]

1 =1

C[Vsd 1 + Vsq 2 + ( Vud + Vsd) 3f3 + ( Vuq + Vsq) 4)

+ ( dVsd + dVsq + (dVsd dVud)+ (dVsq dVuq)]

C2 = 51[(1/C( ))[( (Vsd + Vsq + (Vsd Vud)+ (Vsq Vuq) )]

= ( 1)

= ( 2)

=1

( + ) 3

=1

( + ) 4

93

6( ) =VsdVs

(dvtd Re f1 + xt1 f2 Re f3 + xt1 f4)VsqVs

(dVtq + Re f2 + xt1 f1 + xt1 f3 + Re f4)(4.53)

1( 1 + 1 1)

=1

( 2)

1( 3 + 3)

=1

( 4)

The outputs of the system are: , , ,and Vdc..That is

Proceeding with the exact steps as outlined in the Appendix, the following

can be derived: =

3( )4( )6( )5( )

+

0 0 00 0 0

(4.54)

= A(x) + E(x) u

94

= ( ) + ( ) (4.55)

Thus:

= ( ) ( ) + (4.56)

The non-singularity of E(x) can be observed by computing the

determinant of E(x).

E(x) is nonsingular in the operating ranges of and Vdc. For tracking

of and Vdc, the new control inputs v1, v2, v3 and v4 are selected as (by both

proportional and integral control):

=

+ +

+ +

+ +

+ + +

(4.57)

Where y1ref is the ,y2ref is the ,y3ref isthe , ac-bus reference

voltage (Vsref) and y4ref is the dc-bus reference voltage (Vdref) and e1 and e2

are error variables defined by

= (4.58)

= (4.59)

95

= (4.60)

= (4.61)

The gain parameters K11, K12, K21, K22,K23are determined by assigning

desired poles on the left half s-plane and, thus, asymptotic tracking control to

the reference can be achieved.

From u1 u2 u3 and u4, the control signals in d'-q' frames are determined by:

(4.62)

(4.63)

(4.64)

(4.65)

Again, from Vpd' and Vpq', the control signals in d-q frame isVpdand

Vpq. In the computer simulation studies presented in the followings, the

derivative dvtd’/dtappearing in the control design, is neglected in the control

computation. This leads to the assumption that the generator bus voltage Vtis

treated as a constant only for the control design.

In this section, the design steps for the feedback linearizing control of

UPFC have been presented followed by simulation results under various

transient disturbances. A brief review of nonlinear control using feedback

linearization is presented in the Appendix 2.

4.4.3 Implementation of FBLC for STATCOM in Multi Machine

As mentioned earlier, in the STATCOM control, there are two broad

objectives, i.e., ac-bus voltage )V( s and dc-bus voltage )V( dc control.In the

following control design,

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sV is taken as an additional state in addition to the other three states

,i,i( qpdp and )Vdc in the STATCOM modeling.

8VVs (4.66)

Now, for the control design, the complete state space model is

expressed in the form of equations (A.1) and (A.2) as follows.

qp

dp

dc

qp

dp

VV

uu

VVii

xxxx

2

1

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3

2

1

,

ux (4.67)

= ( ) + (4.68)

= ( ) + (4.69)

(x)fx 33 (4.70)

= ( ) + + (4.71)

where,

( )= + + (4.72)

= = (4.73)

( ) (4.74)

( ) = (4.75)

8(x)f4 V (4.76)

97

, (4.77)

The outputs of the system are sV and Vdc

Thus, y1and y2 =Vdc

Proceeding with the exact steps as outlined in the Appendix 2, the

following can be derived.

= ( )

( ) ( ) + ( ) + ( )+

+ +

=A(x) +E(x) (4.78)

=1

2

where,

= (4.79)

(4.80)

= ( + ) (4.81)

= (4.82)

Thus, = ( ) ( ) + (4.83)

The no singularity of )x(E can be observed by computing the

determinant of )x(E , which is:

98

| | (4.84)

It is known that the magnitude of current 'pdi is very small such that

2 'pdpiR2 is almost negligible compared to Vs.Now it is readily seen that

E(x)is nonsingular in the operating ranges of sV and dcV .

For tracking of and , the new control inputs and are selected

as (by both proportional and integral control):

=+ +

+ + + (4.85)

Where y1refis the Ac bus reference voltage ( ) and y2ref is the dc bus

reference voltage ( )

and e1 and e2are error variables defined by :

= and

= (4.86)

From equation (4.86), the error dynamics are given by :

+ + = 0 (4.87)

+ + + = 0 (4.88)

The gain parameters ,K,K,K,K 22211211 and 23K are determined by

assigning desired poles on the left-half s-plane and, thus, asymptotic tracking

control to the reference can be achieved.From 1u and 2u , the control signals in

qd frame are determined by,

99

` and

` (4.89)

Again, from dpV and qpV , the control signals in qd frame i.e., pdV and

pqV , are obtained by making use of equations (4.31) and (4.32).

4.5 SAMPLE SYSTEM AND RESULTS FOR SMIB USING UPFC

The performance of the UPFC with PI controller and FBLC for

synchronous generator stabilization is evaluated by MATLAB simulation

studies. In this simulation UPFC has been connected to load bus of SMIB.

The system data are provided in Appendix 3. The following case studies are

undertaken for evaluating the performance of the proposed controller.

Case 1

The synchronous generator is assumed to operate at P = 1.2 p.u. and Q

= 0.85 p.u.A 3-phase fault occurs near the infinite bus for duration of 100 ms.

Figure 4.8 shows the transient response of the system with

conventional PI based UPFC and FBLC based UPFC. The transient

performances of the rotor angle, rotor speed deviation are shown for three

phase fault when the generator is operating at P = 1.2 p.u. and Q = 0.85 p.u. A

comparison of the system responses for a 3-phase fault at infinite bus which is

cleared after 100 ms is also shown. This study clearly indicates better

stabilizing properties of UPFC, particularly the restoration of bus voltages to

the pre-disturbance value and the performance of dc capacitor voltage Vdc for

the above case. The transient oscillations in rotor angle and speed

deviationexhibit good damping behaviour for FBLC compared to cascade PI

controllers which has been presented. This is possible because of non-linear

100

control of bus voltage, resulting in better power modulation, by FBLC

controller for stabilizing the synchronous generator.

Fig 4.8 Comparison of transient performances for Case 1 with PI based

UPFC ( ) and FBLC based UPFC ( )

Case 2

101

For this case The synchronous generator is assumed to operate at a

load of P=1.0 p.u. and Q = 0.5 p.u.The load values have been reduced by 50%

for duration of 100 ms.

The transient oscillations in rotor angle and speed exhibit good

damping behaviour for FBLC compared to cascade PI controllers which has

been presented in Figure 4.9. This is possible because of non-linear control of

bus voltage, by FBLC controller for stabilizing the synchronous generator.

The single line diagram of this system and the data of this system is given in

Appendix 3.

Figure 4.9 Comparison of transient performances for Case 2 with PI based UPFC () and FBLC based UPFC ( )

102

4.6 SAMPLE SYSTEM AND RESULTS FOR MULTI

MACHINEUSING STATCOM AND UPFC

A multi-machine system namely 3 machines, 9-bus system [4] is

considered for analysis. The single line diagram of this system is shown in

Figure 4.3 and 4.4 and the data of this system is given in

Appendix 2. In this system number of machines m = 3 and number of busn =

9. The system is subjected to various disturbances in order to identify the

critical cases. From the investigations the following cases are considered to be

critical cases. Initially base case load flow was run for the sample 9-bus

system, which results are given in Table 2.8 of Appendix 3. After getting the

initial values from load flow, the non-linear simulation is carried using

MATLAB.

Case 1 A three phase fault of 0.25 second duration is created at the middle of

the transmission line connecting bus 5 and bus 7. The performance of the

conventional PI controller with STATCOM and without STATCOM in the

network isanalysed.

The Figure 4.10 represents voltage at 8th bus with PI based STATCOM

and FBLC based STATCOM for the variation of dc capacitor voltage of

STATCOM. The dc capacitor voltage is a very important factor for successful

operation of the shunt converter.

Figure 4.10 Comparison of transient response of voltage at8th bus for case 1, With PI STATCOM ( ), with FBLC STATCOM ( )

103

Case 2

The load values have been reduced by 50% for duration of 100

ms.. The performance of the conventional PI controller with UPFC and

without UPFC in the network is shown in Figure 4.11,Figure 4.12 and Figure

4.13 respectively and represents the relative rotor angle oscillation,the speed

deviation and changes in voltage at 8th bus, the variation of dc capacitor

voltage of UPFC. The dc capacitor voltage is a very important factor for

successful operation of the shunt converter.

Figure 4.11 Comparison of transient response of relative rotor angle for case 2, without UPFC ( ), with PI based UPFC ( )

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Figure 4.12 Comparison of transient response of voltage at 8th bus for case 2, without UPFC ( ), with PI based UPFC ( )

Figure 4.13 Comparison of transient response of speed deviation for case 3, without UPFC ( ), with PI controller based UPFC ( )

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Case 3:The mechanical torque input of generator 1 is suddenly

increased by 50% for 100 ms duration. The oscillations are presented in the

figure below respectively for without with STATCOM PI control and FBLC

control schemes. The superiority of FBLC over PI is well marked in damping

oscillations in Figures 4.14 and 4.15

Figure 4.14 Comparison of transient response of relative rotor angle for case 3, FBLC based STATCOM ( ), With PI based STATCOM ( )

Figure 4.15 Comparison of transient response of speed deviation For case 3, FBLC based STATCOM ( ), With PI based STATCOM ( )

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4.7 BACTERIAL FORAGING OPTIMIZATION ALGORITHM

In this sectionBacteria Foraging Optimization Algorithm, is used for

tuning the parameters of FBLC controller. The simulation is done for FBLC

controller with STATCOM and the results are discussed. BFOA [68]

proposed by Passino, is a new comer to the family of nature-inspired

optimization algorithms. For over the last five decades, optimization

algorithms like Genetic Algorithms, Evolutionary Programming and

Evolutionary Strategies which draw their inspiration from evolution and

natural genetics, have been dominating the realm of optimization algorithms.

Recently natural swarm inspired algorithms like Particle Swarm

Optimization, Ant Colony Optimization have found their way into this

domain and proved their effectiveness. Following the same trend of swarm-

based algorithms, Passino[71] [72] proposed the BFOA.

Bacterialforaging optimization algorithm has been widely accepted as

a global optimization algorithm of current interest for distributed optimization

and control. BFOA is inspired by the social foraging behavior of Escherichia

coli. BFOA has already drawn the attention of researchers because of its

efficiency in solving real-world optimization problems arising in several

application domains. The biology behind the foraging strategy of ‘E, coli’is,

emulated in an extraordinary manner and used as a simple optimization

algorithm.

Application of group foraging strategy of a swarm of ‘E, coli’bacteria

in multi-optimal function optimization is the key idea of the new algorithm.

Bacteria search for nutrients in a manner to maximize energy obtained per

unit time. Individual bacterium also communicates with others by sending

signals. A bacterium takes foraging decisions after considering two previous

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factors. The process, in which a bacterium moves by taking small steps while

searching for nutrients, is called chemotaxis and key idea of BFOA is

mimicking chemotactic movement of virtual bacteria in the problem search

space.

It possess four process

Figure 4.16Four Process In BFOA

4.7.1 Chemotaxis

The process, in which a bacterium moves by taking small steps while

searching for nutrients, is called chemotaxis. This process simulates the

movement of an ‘E, coli’cell through swimming and tumbling via flagella.

Biologically an ‘E, coli’bacterium can move in two different ways. It can

swim for a period of time in the same direction or it may tumble, and alternate

between these two modes of operation for the entire lifetime.

CHEMOTAXIS

SWARMING

REPRODUCTION

ELIMINATION & DISPERSAL

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Figure 4.17 SWIM & TUMBLE OF A BACTERIUM

During foraging of the real bacteria, locomotion is achieved by a set of

tensile flagella. Flagella help an ‘E, coli’bacterium to tumble or swim, which

are two basic operations performed by a bacterium at the time of foraging.

When they rotate the flagella in the clockwise direction, each flagellum pulls

on the cell. That results in the moving of flagella independently and finally

the bacterium tumbles with lesser number of tumbling whereas in a harmful

place it tumbles frequently to find a nutrient gradient. Moving the flagella in

the counterclockwise direction helps the bacterium to swim at a very fast rate.

In the above-mentioned algorithm the bacteria undergoes chemotaxis, where

they like to move towards a nutrient gradient and avoid noxious environment.

Figure 4.17 depicts how clockwise and counter clockwise movement of a

bacterium take place in a nutrient solution.

4.7.2 Swarming

An interesting group behavior has been observed for several motile

species of bacteria including ‘E, coli’andS. typhimurium, where intricate and

stable spatio-temporal patterns (swarms) are formed in semisolid nutrient

medium. A group of ‘E, coli’ cells arrange themselves in a traveling ring by

SWIM TUMBLE

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moving up the nutrient gradient when placed amidst a semisolid matrix with a

single nutrient chemo-effecter. The cells when stimulated by a high level of

succinate, release an attractant aspertate, which helps them to aggregate into

groups and thus move as concentric patterns of swarms with high bacterial

density.

4.7.3 Reproduction

When they get food in sufficient, they are increased in length and in

presence of suitable temperature they break in the middle to from an exact

replica of itself. This phenomenon inspired Passino to introduce an event of

reproduction in BFOA. The least healthy bacteria eventually die while each of

the healthier bacteria (those yielding lower value of the objective function)

asexually split into two bacteria, which are then placed in the same location.

This keeps the swarm size constant.

4.7.4 Elimination and Dispersal

Due to the occurrence of sudden environmental changes or attack, the

chemotactic progress may be destroyed and a group of bacteria may move to

some other places or some other may be introduced in the swarm of concern.

This constitutes the event of elimination-dispersal in the real bacterial

population, where all the bacteria in a region are killed or a group is dispersed

into a new part of the environment.

Gradual or sudden changes in the local environment where a bacterium

population lives may occur due to various reasons e.g. a significant local rise

of temperature may kill a group of bacteria that are currently in a region with

a high concentration of nutrient gradients. Events can take place in such a

fashion that all the bacteria in a region are killed or a group is dispersed into a

new location. To simulate this phenomenon in BFOA some bacteria are

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liquidated at random with a very small probability while the new

replacements are randomly initialized over the search space.

In a three machine nine bus system, the 3 phase symmetrical fault

occurs in the line connecting the buses 5 and 7, for the duration of 0.25s. The

fault is created at 1s and removed at 1.25s. STATCOM is connected at the 8th

bus. The controllers used are PI controller and FBLC. First PI controller is

used and then FBLC is used to improve the output. The following combined

output waveforms show the comparison of transient responses of STATCOM

in presence of two different controllers.

In order to get better output, the control parameters of FBLC are tuned

by using Bacterial Foraging Optimization Algorithm (BFOA). There are five

control parameters namely K11, K12, K21, K22 & K23.

These control parameters are determined by using two poles namely

Pole1 and Pole2. The formula for calculating the control parameters are given

below:

K11 = 2 x Pole1

K12 = Pole1 x Pole1

K21 = 3 x Pole2

K22 = 3 x Pole2 x Pole2

K23 = Pole2 x Pole2 x Pole2

These two poles are determined by using BFOA. The value for Pole1

is 0.03 and the value for Pole2 is -28.

For the same duration of 0.25s three phase symmetrical fault is created

at multi-machine system as that of previous case and STATCOM is connected

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at 8th bus. Here FBLC controller is only used and is tuned by using BFOA to

get better output. The simulations are carried out using MATLAB software.

The following combined output waveforms show the comparison of

transient responses of STATCOM in presence of generalized FBLC and

FBLC tuned by BFOA.

The figure 4.18 shows the comparison of rotor angle curves of

generator 1. This output waveforms clearly show that the critical clearing time

(CCT) is 4.5s in presence of STATCOM with FBLC and CCT is 4.3s in

presence of STATCOM with FBLC when tuned by BFOA. Thus the CCT got

reduced by using BFOA and hence the transient stability of the multi-machine

power system got enhanced.

Similarly the comparison of rotor angle curves of generator 2 and

generator 3 are shown in the figures 4.19 and 4.20 respectively

Then the voltage curves at the 8th bus are compared and are shown in

the Figure 4.21 Finally the DC voltage curves of STATCOM with FBLC and

then tuned by BFOA are compared.

From all the above waveforms it is clearly understood that by using

STATCOM three phase symmetrical fault gets cleared at faster rate and also

CCT in presence of FBLC tuned by BFOA is less than that of generalized

FBLC.

Hence STATCOM with FBLC tuned by BFOA damp out the rotor

oscillations quickly and enhance the transient stability of the multi-machine

power system at a faster rate.

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Figure 4.18 Comparisons of Rotor Angle Curves of Generator 1


----- : STATCOM with FBLC

___ : STATCOM with FBLC tuned by BFOA



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Figure 4.21 Comparisons of Voltage Curves at 8th Bus





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4.8 DIFFERENTIAL EVOLUTION

The constants of the controller for STATCOM can be tuned by

different evolutionary algorithms for the purpose of improved performance.

Differential evolution (DE) algorithm is arguable one of the most powerful

stochastic real parameter optimization algorithm in current use. DE operates

through similar computational steps as employed by a standard evolutionary

algorithm (EA). How- ever, unlike traditional EAs, the DE-variants perturb

the current- generation population members with the scaled differences of

randomly selected and distinct population members.

Therefore, no separate probability distribution has to be used for

generating the offspring. Since its inception in 1995, DE has drawn the

attention of many researchers all over the world resulting in a lot of variants

of the basic algorithm with improved performance. This section presents a

detailed review of the basic concepts of DE and a survey of its major variants,

its application tomulti objective, constrained, large scale, and

uncertainoptimization problems,

andthetheoreticalstudiesconductedonDEsofar.Also, it provides an overview of

the significant engineering applications that have benefited from the powerful

nature of DE.

Differential evolution is an attractive optimization tool because of the

following reasons

Compared to most other EAs, DE is much simpler to implement end

easy to code in any programming language.

DEexhibitsmuch better

The number of control parameters in DE is very few (Cr, F, and NP in

classical DE).

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The space complexity of DE is low and this feature helps in extending

DE for handling large scale and expensive optimization problems.

Differential evolution uses three main control parameters to optimize

the objective function used. They are Cr which is the crossover ratio, F which

is the mutation constant and NP which denotes the number of population

members.

4.8.1 Stages of Differential Evolution

The main stages of DE are as follows and shown in Figure 4.23

Initialization of vectors

Difference vector based mutation

Crossover/Recombination

Selection

Figure 4.22 Stages of Differential Evolution Algorithm

4.8.2 Initialization of the Parameter Vectors

DE searches for a global optimum point in a D-dimensional population

of NP D dimensional real-valued parameter vectors. Each vector, also known

as genome/chromosome, forms a candidate solution to the multidimensional

optimization problem.

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4.8.3 Mutation with Difference Vectors

Biologically, “mutation” means a sudden change in the gene

characteristics of a chromosome. In the context of the evolutionary computing

paradigm, however, mutation is also seen as a change or perturbation with a

random element. In DE-literature, a parent vector from the current generation

is calledtargetvector,amutantvectorobtained through the differential mutation

operation is known as donor vector and fin0 ally an offspring formed by

recombining the donor with the target vector is called trial vector.

4.8.4 Crossover

To enhance the potential diversity of the population, a crossover

operation comes into play after generating the donor vector through mutation.

There are two kinds of crossover strategies usually used

Exponential Crossover

Binary Crossover

4.8.5 Selection

To keep the population size constant over subsequent generations, the

next step of the algorithm calls for selection to determine whether the target or

the trial vector survives to the next generation

4.9 IMPLEMENTATION OF DIFFERENTIAL EVOLUTION TO

DETERMINE FBLC CONSTANTS

The FBLC controller uses pole tracking algorithm to determine the

error constants. The two poles used here are poledc and polev. These are

determined using the differential evolution algorithm. From the poles

determined the error constants for the FBLC constants are calculated. The

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ranges for both the pole values are specified in the DE algorithm. An initial

value may be assigned. The range for each pole is determined using trial and

error method. Each pole value is determined using differential evolution

following the above algorithm. The poledc value is for the DC part of the

STATCOM while the polev value is for the AC part of the STATCOM. The

tuned FBLC values result in better settling time than the normal pole values.

In a three machine nine bus system, the 3 phase symmetrical fault

occurs in the line connecting the buses 5 and 7, for the duration of 0.25s. The

fault is created at 1s and removed at 1.25s. FACTS device is connected at the

8th bus. The controller used is FBLC. The following combined output

waveforms show the comparison of transient responses with and without

FACTS devices. The simulations are carried out using MATLAB software.

The figure 4.23 shows the comparison of rotor angle curves of

generators 1 and 2. This output waveforms clearly shown that the critical

clearing time (CCT) is 6.0s in the absence of any FACTS devices and CCT is

5.5s in presence of STATCOM. Thus the CCT get reduced by using FACTS

devices and hencethe transient stability of the multi-machine power system

got enhanced.

Then the comparison of speed curves of the generators 1&2, 2&3 and

3&1 are shown in the figures 4.23, 4.24 and 4.25 respectively. The voltage

curves at the 8th bus are compared and are shown in the figure 4.26.From all

the above curves it was clearly understood that by using FACTS devices three

phase symmetrical fault get cleared at faster rate. Hence STATCOM with PI

controller damp out the rotor oscillations quickly and enhance the transient

stability.

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Figure 4.23 Comparisons of Rotor Angle Curves of Generators 1 & 2


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Figure 4.26 Comparisons of Dc Voltage Curves

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4.10 SUMMARY

Modeling of STATCOM and UPFC is successfully done in this

chapter. A simple mathematical model of a STATCOM and UPFC has been

systematically derived. To study in detail the effectiveness of STATCOM and

UPFC, a Single Machine Infinite Bus system and standard 3 machine 9 bus

system is considered. As illustrated by simulation studies, the damping of the

electromechanical oscillations of the synchronous generator in a simple power

system is better with FBLC based STATCOM and UPFC over conventional

PI controller.hm.

The proposed FBLC for UPFC is proved to be very effective and

robust in damping power system oscillations and thereby enhancing system

transient stability. As illustrated by computer simulation studies, the superior

damping of the electromechanical oscillations of the synchronous generator

provided by this proposed control strategy over the conventional cascade

control approach has been established for a variety of severe transient

disturbances.

Both the STATCOM and the UPFC using the proposed control scheme

provides significant improvement in damping out the electromechanical

oscillations of the generators. The oscillations show significant reductions in

their first and subsequent swings for this controller, in comparison to

conventional PI controllers. The FBLC tuned by BFOA and DE gives better

performance over the other controllers. The electromechanical oscillations

have damped quickly. The settling time for the generators 1, 2 and 3 has

reduced from 5.5, 5 and 5 seconds to 4.5, 5.5 and 4.5 seconds respectively for

FBLC controller and 4.3, 5.2 and 4.2seconds for FBLC tuned by BFOA.

Finally several faults and load disturbance results have been presented to

highlight the effectiveness of proposed FACTS controllers.