power flow solution methods for ill- conditioned...

104

POWER FLOW SOLUTION METHODS FOR ILL-

CONDITIONED SYSTEMS

5.1 INTRODUCTION:

In the previous chapter power flow solution for well conditioned

power systems using Newton-Raphson method is presented. The

conclusions drawn from chapter 4 indicate that the standard Newton-

Raphson method fails to obtain the solution when the R/X ratio is

high. Researchers in the past indicated that the standard N-R method

failed to converge due to the following reasons.

1. Selection of reference slack bus

2. Existence of negative line reactance

3. High R/X ratio

4. Choosing initial values

In this chapter, load flow solution methods such as Iwamoto‘s

optimal multiplier method, Runge-Kutta method and Newton‘s

accelerated methods are implemented. The proposed methods are

tested with 11 and 13-bus Ill-Conditioned test systems which have the

characteristics as mentioned above. Solutions are obtained along with

the incorporation of the FACTS devices. Conclusions drawn from these

solutions are furnished.

5.2 RUNGE – KUTTA METHOD

5.2.1. MODELING

It is well known that the forward Euler method, even with the

variable time step, can be numerically unstable. Reference [52]

105

suggests that given analogy between the power flow equations (5.1)

and ODE (5.2), any well-assessed numerical method can be used to

integrate (5.2). It is thus intriguing to use some efficient integration

method for solving (5.1).It is observed that, the computation of f=-[g]-1g

implies the inversion of power flow Jacobain matrix, only explicit

integration methods are suitable and computationally efficient, since

one does not need to compute the Jacobian matrix of f.

F(x) =0 (5.1)

ii

x

i ffx1

(5.2)

)(.

xfx (5.3)

For the sake of example, we use classical fourth order Runge-

Kutta formula (RK4). A generic step of the RK4 is as follows:

K1=f(x(i)) (5.4)

K2=f(x(i)+0.5∆tk1)

K3=f(x(i)+0.5∆tk2)

K4=f(x(i)+∆tk3)

X(i+1)=x(i)+∆t(k1+ 2k2+ 2k3+ k4)/6

The time step ∆t can be adopted based on the estimated

truncation error of integration method [54]. For example, RK4 error

can be estimated based on half step method.

=max(abs(k2-x(i+1))) (5.5)

Then the time step ∆t can be computed based on the following

simple heuristic rules.

If >0.01 then ∆t=max(0.985∆t,0.75) (5.6)

106

If 0.01 then ∆t=min(1.015∆t,0.75)

Based on these rules, the time step is increased if the

truncation is greater than a given threshold and decreased if the

truncation error is lower than a given threshold. The minimum value

of time step is limited to 0.75. If the lower value of time step ∆t is not

limited, in the case of unsolvable power flow problems, the proposed

algorithm provides a solution close to the feasibility boundary of

power flow equations [53]. All thresholds and tuning parameters in

equation (5.6) have been determined based on heuristic criteria.

5.2.2. STEPS TO IMPLEMENT RK

1. Initialize x(0) and set iteration count i=1, Δt=1

2. Solve (5.4)

3. If ε is greater than max(abs(∆x(i))) , then stop the iterations.

Otherwise go to step 4

4. Update Δt using (5.5) and increment iteration count by 1.

5. If iteration count is more than maximum no.of iterations then

stop the iterations. Otherwise go to step (2)

107

5.2.3. FLOW CHART

Fig.5.1. Flowchart of the proposed RK4-based continuous Newton‘s

method for solving the power flow analysis.

5.2.4. NUMERICAL EXAMPLE

Let us consider the two variable problem given below. The problem

can be solved easily using the RK method.

24.2222 2

221

2

1 xxxx

(5.7)

64.022 2

221

2

1 xxxx

Where the initial estimate is

0.1

0.10

EX ,

And time interval is Δt=0.1

Using eq(5.4) , K1, K2, K3, K4 and L1, L 2, L 3, L 4 are obtained as below.

K1=0.24, K2=0.2877, K3=0.3442, K4=0.4758

No

No

Yes

Yes

Initialization

X(0)

,i=1,

∆t=1

Solve(5.4)

Є>max(abs(∆x(i)

))

Update ∆t using (5.6)

i=i+1

i> imax

Stop

108

L1=1.64, L 2=1.6161, L 3=1.5879, L 4=1.5221

The new estimate is therefore

0.8405)/6L 2L 2L t(Lxx

0.9670)/6k 2k 2k t(kxx

4321

(0)

2

(1)

2

4321

(0)

1

(1)

1

The same procedure is applied for the rest. The converged solution is

as follows: X1=0.8001 and X2=1.2000

5.3. DERIVATION OF THE PROPOSED OPTIMAL MULTIPLIER

Let us derive the optimal multiplier. Moving all the right-hand

side of Taylor series to the 1eft hand side, we have

Ys-y(xe)-J∆x-y(∆x)=0 (5.7)

In order to adjust the length of the correction vector ∆x, we multiply

the scalar quantity μ by ∆x. Then it follows that.

Ys-y(xe)-Jμ∆x-y(μ∆x)=0 (5.8)

In the above equation, μ in the third term can appear in front of J

being a scalar, and the forth term can become μ2y(∆x), that is

Ys-y(xe)-μj∆x-μ2y(∆x)=0 (5.9)

Here we define the vectors a,b,c for simplicity.

)(

.

.

.

,

.

.

.

),(

.

.

.

111

xy

c

c

cxJ

b

b

bxyy

a

a

a

nn

es

n

(5.10)

a+μb+μ2c=0 (5.11)

In order to determine the value of the μ in a least squared sense, the

following cost function is considered.

109

Minimize 2

1

2

2

1

n

i

iii cbaF (5.12)

The optimal solution μ* of the above equation can obtained by solving

the equation below.

0

F (5.13)

Namely,

03

3

2

210 gggg (5.14)

Where

n

i

iii

n

i

ii cabgbag1

2

1

1

0 2,

n

i

i

n

i

ii cgcbg1

2

3

1

2 2,3 (5.15)

It can be easily observed that the equation (5.14) is a scalar cubic

equation with respect to μ. This equation can be solved for optimal

value of μ.

5.3.3. Application of the Optimal Multiplier to the N-R Method

The most widely used AC load flow calculation method is the N-

R method, and our examination also revealed that the application of

the optimal multiplier to the N-R method was most effective. Thus, we

describe here how to apply the optimal multiplier to the N-R method.

If applied to the N-R method, the solution never diverges but

converges in such a manner that the value of the cost function always

decreases.

In the N-R method, the correction vector ∆x(r) is obtained

basically by triangulating the matrix J(r) in the following equation.

110

Ys-y(xe(r))=J(r)∆x(r) (5.16)

The quantities required for calculating the optimal multiplier μ(r) * are

given by (5.10) as below.

A(r)=ys-y(xe(r)) (5.17)

B(r)=-J(r)∆x(r)=-a(r) (5.18)

C(r)=-y(∆x(r)) (5.19)

Note the important fact that b(r) =-a(r) in (5.18). These calculations are

carried out automatically in the process of the N-R method, and thus

no additional calculations are required.

5.3.4. EXAMPLE

In order to illustrate the proposed method, first a simple example

using two variables is considered

Two Variables Example: Let us consider the problem mentioned

earlier in this chapter. Although the problem can be solved easily

using the RK method, it is used here just to demonstrate the

application procedure of the proposed optimal multiplier.

24.2222 2

221

2

1 xxxx

(5.20)

64.022 2

221

2

1 xxxx

Where the initial estimate is

0.1

0.10

EX

Using the NR method, Δx(0) is obtained as below.

28.0

16.0)0(X

A(0),b(0),c(0) are calculated from (5.21),(5.22),(5.23).

111

1504.0

2976.0,

64.1

24.0,

64.1

24.0)0()0()0( cba (5.21)

)0(

3

)0(

2

)0(

1

)0(

0 ,,, gggg are from (5.19),

2224.0,9542.0

1110.2,7472.2

)0(

3

)0(

2

)0(

1

)0(

0

gg

gg

(5.22)

The scalar cubic equation to be solved is (omitting (0) for simplicity).

07472.2111.29542.02224.0 23 (5.23)

Solving the equation, the optimal multiplier is obtained as follows.

8798.0*)0( (5.24)

The new estimate is therefore

2463.1

8592.0

28.0

16.0*8798.0

0.1

0.1

)0(*)0()0()1( xxx ee

(5.25)

Using )1(

ex , the next correction vector Δx(l) is obtained by the N-R

method, and μ(l)* is calculated by the proposed method, then

)1(*)1()1()2( xxx ee , and the same procedure is applied for the rest.

5.4. NR METHOD WITH ACCELERATED CONVERGENCE

5.4.1. TWO STEP ALGORITHM

This method is based on the numerical technique [63] and can be summarized mathematically as follows:

)()(1

nnnn xBxJxy

(5.26)

)()(1

nnn yByJ

(5.27)

112

nnnn Cyx )2(1

(5.28)

Where and C are positive constants, n is the norm of the vector

n and y is the intermediate solution vector.

5.4.2. THREE STEP ALGORITHM

This method is an extension of two step algorithm [63] and can be

summarized mathematically as follows:

)()(1

nnnn xBxJxy

(5.29)

)()(1

nnnn yByJyz

(5.30)

)()(1

nnn zBzJ

(5.31)

nnnn Czx )2(1

(5.32)

Where and C are positive constants, n is the norm of the vector

n and y and z are the intermediate solutions.

5.5. CASE STUDY WITH RK METHOD

In this section case study is presented for 13- and 11- Bus Ill-

Conditioned systems with and without devices. The Bus data and Line

data for the test systems are shown from Appendix-III to Appendix-IV.

The initial data of devices for 13- and 11- Bus Ill-Conditioned system

is as follows:

113

Initial Values for 13-bus Ill-Conditioned system with STATCOM

STATCOM is connected to bus No : 3

Converter‘s reactance (p.u.), Xvr = 10

Target nodal voltage magnitude (p.u.)=1

Initial source voltage magnitude (p.u.), Vvr = 1

Initial source voltage angle (deg)=0

Lower limit of source voltage magnitude (p.u.)=1.1

higher limit of source voltage magnitude (p.u.)=0.95

Initial Values for 13-bus Ill-Conditioned system with SVC

Susceptance Model

SVC is connected to bus No : 9

Initial SVC‘s susceptance value (p.u.)=0.02

Lower limit of variable susceptance (p.u.)=-0.25

Higher limit of variable susceptance (p.u)=0.25

Target nodal voltage magnitude to be controlled by SVC (p.u.)=1

Initial Values for 13-bus Ill-Conditioned system with SVC Firing

Angle Model


Capacitive reactance (p.u.)=1.07

Inductive reactance (p.u.)=0.288

Initial value of SVC‘s firing angle (Deg)=140

Lower limit of firing angle (Deg)=90

Higher limit of firing angle (Deg)=180


114

Initial Values for 13-bus Ill-Conditioned system with TCSC

Variable Impedance Power Flow Model

TCSC is connected between bus-10 and bus-11

Reactance of TCSC=-0.05

Lower reactance limit=-0.09

Higher reactance limit=0.09

Power flow direction is taken from sending end to receiving end

Active power flow to be controlled=0.1


Variable Firing Angle Power Flow Model


Capacitive reactance of TCSC (p.u.)=9.375

Inductive reactance of TCSC (p.u)=1.625e-1

Power flow direction is taken from receiving end to sending end

Target active power flow (p.u.)=0.13

Initial firing angle (deg)=140

Firing angle lower limit (deg)=60

Firing angle higher limit (deg)=180

Initial Values for 13-bus Ill-Conditioned system with UPFC

Shunt converter is connected to bus=7

Series converter is connected between bus-7 and bus-8

Inductive reactance of Shunt impedance (p.u.)=0.1

Inductive reactance of Series impedance (p.u.)=5



115

Target reactive power flow (p.u.)=0.02

Initial value of the series source voltage magnitude (p.u.)=0.04

Initial value of the series source voltage angle (rad.)=-pi/4

Lower limit of series source voltage magnitude (p.u.)=0.001

Higher limit of series source voltage magnitude (p.u.)=0.2

Initial value of the shunt source voltage magnitude (p.u.)=1

Initial value of the shunt source voltage angle (rad.)=0

Lower limit of shunt source voltage magnitude (p.u.)=0.95

Higher limit of shunt source voltage magnitude (p.u)=1.1

Target nodal voltage magnitude to be controlled by shunt

branch (p.u.)=1

Initial Values for 11-bus Ill-Conditioned system with STATCOM

STATCOM is connected to bus No : 4

Converter‘s reactance (p.u.), Xvr = 10

Target nodal voltage magnitude (p.u.)=1

Initial source voltage magnitude (p.u.), Vvr = 1

Initial source voltage angle (deg)=0

Lower limit of source voltage magnitude (p.u.)=1.1

higher limit of source voltage magnitude (p.u.)=0.95

Initial Values for 11-bus Ill-Conditioned system with SVC Susceptance Model SVC is connected to bus No : 4

Initial SVC‘s susceptance value (p.u.)=0.02

Lower limit of variable susceptance (p.u.)=-0.25

Higher limit of variable susceptance (p.u)=0.25


116

Initial Values for 11-bus Ill-Conditioned system with SVC Firing

Angle Model


Capacitive reactance (p.u.)=1.07

Inductive reactance (p.u.)=0.288

Initial value of SVC‘s firing angle (Deg)=140

Lower limit of firing angle (Deg)=90

Higher limit of firing angle (Deg)=180



Variable Impedance Power Flow Model


Reactance of TCSC=-0.015

Lower reactance limit=-0.05

Higher reactance limit=0.05

Power flow direction is taken from sending end to receiving end

Active power flow to be controlled=0.21


Variable Firing Angle Power Flow Model


Capacitive reactance of TCSC (p.u.)=9.375e-3

Inductive reactance of TCSC (p.u)=1.625e-2


Target active power flow (p.u.)=-0.45

Initial firing angle (deg)=145

117

Firing angle lower limit (deg)=90

Firing angle higher limit (deg)=180

Initial Values for 11-bus Ill-Conditioned system with UPFC

Shunt converter is connected to bus=4

Series converter is connected between bus-4 and bus-5

Inductive reactance of Shunt impedance (p.u.)=0.1

Inductive reactance of Series impedance (p.u.)=5



Target reactive power flow (p.u.)=0.02

Initial value of the series source voltage magnitude (p.u.)=0.04

Initial value of the series source voltage angle (rad.)=-pi/4

Lower limit of series source voltage magnitude (p.u.)=0.001

Higher limit of series source voltage magnitude (p.u.)=0.2

Initial value of the shunt source voltage magnitude (p.u.)=1

Initial value of the shunt source voltage angle (rad.)=0

Lower limit of shunt source voltage magnitude (p.u.)=0.95

Higher limit of shunt source voltage magnitude (p.u)=1.1

Target nodal voltage magnitude to be controlled by shunt

branch (p.u.)=1

118

5.5.1 13-BUS ILL-CONDITIONED SYSTEM WITH OUT ANY DEVICE

Table 5.1: Voltage Magnitudes and Phase angles

Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.01139 1.54536

3 1.05712 2.43665

4 1.03562 2.49271

5 1 2.57593

6 1.037 9.77015

7 1.06322 8.98467

8 1.1 8.12085

9 0.943 14.3468

10 1.1 8.33322

11 1.01771 12.1034

12 1.06717 8.08853

13 1.04424 5.22269

Table 5.2: Complex Power Flows through Lines

From To

Sending end Power Receiving end Power

Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)

1 2 -326.269 -114.341 326.747 124.501

1 3 -498.743 -571.966 501.046 626.503

5 4 -6.2859e-5 -376.081 0.565812 389.475

4 3 -234.371 -155.671 -242.881 158.902

6 2 199.497 46.5908 -452.566 1.31683

6 7 109.629 -260.641 -126.59 268.868

8 3 73.9698 200.785 -899.607 -143.966

7 8 117.546 -259.824 -117.546 270.616

9 10 224.332 -730.658 -865.844 908.718

10 11 490.744 -619.639 -490.744 542.187

11 12 177.67 -309.111 -727.607 355.699

12 13 133.234 156.673 -631.442 -134.116

13 8 387.183 378.376 -387.183 -418.692

119

5.5.2 13-BUS ILL-CONDITIONED SYSTEM WITH STATCOM

The 13-bus Ill-Conditioned system is designed by Japanese

researchers which exhibits high degree of Ill-Conditionality, and it is

also reported in the literature that the conventional Newton-Raphson

method fails to converge for this system. The same system is

considered to demonstrate the effect of FACTS devices on the power

flow and convergence behavior.

The steady state mathematical model developed for STATCOM in

chapter-3, is incorporated into the RK method to test for the existence

of the solution for 13-bus Ill-Conditioned system. The STATCOM is

connected at bus no 3 to regulate the voltage to 1.0 per unit.. The

voltage profile before incorporation of the device is 1.05712 per unit.

The STATCOM reactance is taken as 10 per unit on 100 MVA base.

The initial source voltage angle is taken as 00. From the results

obtained, it is observed that the device is able to regulate the voltage

at bus no 3 to 1.0 per unit. The source voltage angle is 0.467 degrees.

The reactive power flow in the lines connected to the bus no 3 is

significantly affected. The real power flow in the lines with and without

the device remains the same. The real power mismatch variation is

large in the first three iterations. The problem is converged in 18

iterations without device and 21 iterations with device. The large

number of iterations for the solution may be due to system Ill-

Conditionality as well as non linearities introduced by the device.

120


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.01127 1.57937

3 1 2.67681

4 1.01258 2.63894

5 1 2.66899

6 1.037 9.98532

7 1.06327 9.24013

8 1.1 8.42893

9 0.943 14.6549

10 1.1 8.6413

11 1.01771 12.4115

12 1.06717 8.39662

13 1.04424 5.53078


From To



1 2 -333.203 -112.442 333.698 122.953

1 3 -491.795 32.2948 492.766 -9.29171

5 4 -0.0010128 -132.794 0.0715502 134.464

4 3 -223.589 89.0537 -217.872 -87.9476

6 2 206.665 47.1469 -459.487 2.83505

6 7 102.46 -260.592 -119.435 268.651

8 3 66.8144 470.035 -848.889 -379.817

7 8 110.391 -259.607 -110.391 270.166

9 10 224.332 -730.658 -865.843 908.718

10 11 490.743 -619.64 -490.743 542.187

11 12 177.669 -309.11 -727.606 355.698

12 13 133.234 156.673 -631.442 -134.116

13 8 387.183 378.377 -387.183 -418.693

121

5.5.3 13-BUS ILL-CONDITIONED SYSTEM WITH SVC-

SUCCEPTANCE MODEL

The steady state mathematical models developed for SVC in

susceptance and firing angle modes in chapter-3 are incorporated

into the Runge -Kutta algorithm to test for the existence of solution for

a 13-bus Ill-Conditioned system The SVC is connected to bus no 9.

The initial susceptance value is chosen as 0.02 per unit on 100 MVA

base. With a susceptance of 0.03 p.u in susceptance, it is revealed

from the study that reactive power flow variations are significant. The

solution is converged in 20 iterations whereas the problem is

converged in 18 iterations without device. This may be due to non-

linearities of the device model. The mismatch power variations are

large in first three iterations.

122


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.00684 1.5793

3 1.03845 2.53577

4 1.02166 2.58208

5 1 2.6334

6 1.037 9.89408

7 1.06297 9.12731

8 1.1 8.28632

9 1 13.9646

10 1.1 8.42415

11 1.04807 12.1153

12 1.08457 8.27655

13 1.08677 5.46832


From To



1 2 -329.307 -60.5005 329.755 70.0293

1 3 -500.958 -374.113 502.521 411.133

5 4 0.000170489 -228.74 0.209117 233.695

4 3 -113.983 -119.922 -117.232 121.893

6 2 268.783 56.9478 -392.1 -7.68607

6 7 110.779 -258.058 -118.932 266.076

8 3 290.85 288.966 -694.783 -223.539

7 8 114.413 -261.557 -114.413 272.376

9 10 345 -495.599 -682.346 595.714

10 11 494.797 -396.773 -494.797 346.907

11 12 305.294 -237.398 -592.194 275.93

12 13 261.064 -26.7997 -524.1 46.1061

13 8 391.818 86.1807 -391.818 -106.622

123

5.5.4 13-BUS ILL-CONDITIONED SYSTEM WITH SVC-FIRING

ANGLE MODEL

The SVC is connected to bus no 9. The initial values are chosen

as capacitive reactance is 1.07 p.u., inductive reactance is 0.288 p.u.

and firing angle is 140 degrees. From the results it is observed that

the reactive power flow variations are significant. The solution is

converged in 20 iterations whereas the problem is converged in 18

iterations without device. This may be due to non-linearities of the

device model. The mismatch power variations are large in first three

iterations.

124


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.00684 1.5793

3 1.03845 2.53577

4 1.02166 2.58208

5 1 2.63339

6 1.037 9.89407

7 1.06297 9.12729

8 1.1 8.2863

9 1 13.9646

10 1.1 8.42413

11 1.04807 12.1152

12 1.08457 8.27652

13 1.08677 5.4683


From To Sending end Power Receiving end Power


1 2 -329.307 -60.5005 329.755 70.0293

1 3 -500.957 -374.113 502.52 411.133

5 4 0.000503694 -228.74 0.208784 233.695

4 3 -113.982 -119.922 -117.232 121.893

6 2 268.782 56.9478 -392.1 -7.68612

6 7 110.781 -258.058 -118.934 266.076

8 3 290.849 288.966 -694.782 -223.539

7 8 114.414 -261.557 -114.414 272.376

9 10 344.999 -495.599 -682.345 595.714

10 11 494.796 -396.772 -494.796 346.906

11 12 305.292 -237.398 -592.193 275.929

12 13 261.062 -26.7988 -524.099 46.1051

13 8 391.817 86.1817 -391.817 -106.622

125

5.5.5 13-BUS ILL-CONDITIONED SYSTEM WITH TCSC VARIABLE

IMPEDANCE POWER FLOW MODEL

The steady state mathematical models developed for TCSC in

susceptance and firing angle modes in chapter-3 are incorporated into

the runge -kutta algorithm to test for the existence of solution for a

13-bus .The TCSC is connected between bus-10 and bus-11 with the

reactance of -0.05 p.u.and pre specified power flow in the line is set at

0.1 p.u. It is observed that with the incorporation of TCSC there is

much difference in real power flows through the lines where as

reactive power flows are not much affected. From the power mismatch

versus iterations it is observed that there is a large variation in

mismatch powers in the first two iterations and the variable

impedance power flow model exhibits oscillations between fourth and

fifth iterations.

126


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.01371 0.688673

3 1.04746 3.42129

4 1.00734 5.85342

5 0.99 5.89508

6 1.037 4.38148

7 1.10265 7.76577

8 1.1 7.76578

9 0.943 13.9917

10 1.1 7.97814

11 1.01771 11.7484

12 1.06717 7.73346

13 1.04424 4.86763

14 1.1036 7.76165


From To Sending end Power Receiving end Power


1 2 -150.559 -153.353 150.744 157.278

1 3 -679.204 -452.767 681.869 515.868

5 4 0.00100898 -181.263 0.133084 184.437

4 3 76.8295 -291.406 -535.958 315.891

6 2 19.5817 41.4593 -277.14 -30.8834

6 7 -9.74332 9.74075 -9.7261 -9.73234

8 3 -43.5795 242.142 -775.689 -201.985

7 8 115.64 261.6 -115.64 -272.46

9 10 224.331 -730.658 -865.842 908.717

14 11 490.742 88.91 -490.74 -70.42

11 12 177.668 -309.11 -727.606 355.698

12 13 133.233 156.674 -631.44 -134.117

13 8 387.182 378.378 -387.182 -418.693

127

5.5.6 13-BUS ILL-CONDITIONED SYSTEM WITH TCSC FIRING

ANGLE MODEL

The TCSC is connected between bus-7 and bus-8 with the

following initial conditions: inductive reactance is 0.1625 p.u.,

capacitive reactance is 9.375 p.u and pre specified power flow in the

line is set at 0.13 p.u. The firing angle is 139.99 degrees. It is

observed that with the incorporation of TCSC there is much difference

in real power flows through the lines where as reactive power flows are

not much affected. From the power mismatch versus iterations it is

observed that there is a large variation in mismatch powers in the first

two iterations.

128


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.00306 2.13459

3 1.03115 1.72718

4 1.03375 1.72803

5 1 1.80703

6 1.027 13.5622

7 1.10132 5.5941

8 1.1 5.5941

9 0.943 13.0062

10 1.07 6.90739

11 0.794414 11.8975

12 1.08588 6.19713

13 1.31386 3.54316

14 1.1018 5.59204


From To



1 2 -439.873 -7.07966 440.647 23.5304

1 3 -341.264 -309.589 342.113 329.694

5 4 -0.0005 -356.368 0.508494 368.395

4 3 -115.403 18.7212 -116.973 -18.6742

6 2 385.134 50.906 -502.523 38.3462

6 7 -4.85583 4.85558 -4.85147 -4.85348

8 3 129.446 318.211 -531.887 -275.816

14 8 -40.89 -13.03 40.89 21.01

9 10 362.165 -590.595 -670.387 726.617

10 11 492.922 -1987.33 -492.922 1438.05

11 12 362.774 -1387.9 -563.749 1947.86

12 13 231.937 -1698.05 -538.155 2072.76

13 8 344.817 -1879.42 -344.817 1562.16

129

5.5.7 13-BUS ILL-CONDITIONED SYSTEM WITH UPFC

The steady state mathematical models developed for UPFC in

chapter3 are incorporated into the runge -kutta algorithm to test for

existence of solution for a 13-bus Ill-Conditioned system .The UPFC is

incorporated between bus-7 and bus-8 with shunt converter

connected close to bus 7 and series converter connected between

buses 7 and 8 with inductive reactance of shunt converter taken as

0.1 p.u and inductive reactance of series converter taken as 5 p.u with

specified active and reactive power flows set at 0.4 and 0.02

respectively. The shunt converter is set to maintain a target voltage of

1p.u .From the results it is observed that UPFC holds its target values

with the given initial conditions. From the power flow results it is

observed that UPFC is the only FACTS device which affects both real

and reactive power flows through the lines .The maximum power

mismatch is less when compared to the other facts devices discussed.

130


Bus

No

Voltage

Magnitude

Voltage

Phase Angle

1 1 0

2 1.01351 2.08542

3 1.05793 1.83475

4 1.03594 1.89204

5 1 1.976

6 1.06309 12.8535

7 1 13.29

8 1.1 6.16569

9 0.943 12.3916

10 1.1 6.37806

11 1.01771 10.1483

12 1.06717 6.13338

13 1.04424 3.26753

14 1.1036 6.16156


From To



1 2 -440.031 -130.375 440.874 148.278

1 3 -382.584 -589.786 384.561 636.589

5 4 -9.6612e-5 -379.529 0.576266 393.17

4 3 -234.529 -159.219 -243.236 162.596

6 2 310.988 109 -567.22 -21.9326

6 7 -9.74338 9.74011 -9.72604 -9.7317

8 3 -43.5775 192.79 -783.749 -156.766

14 8 182.41 76.01 -176.56 -104.81

9 10 224.332 -730.658 -865.844 908.718

10 11 490.744 -619.64 -490.744 542.188

11 12 177.669 -309.111 -727.607 355.699

12 13 133.234 156.672 -631.442 -134.115

13 8 387.183 378.376 -387.183 -418.692

131

5.5.8 COMPARISON OF MAXIMUM POWER MISMATCH FOR 13-

BUS ILL-CONDITIONED SYSTEM

Fig 5.2 Comparison of maximum power mismatch using Runge-Kutta

method for 13-bus Ill-Conditioned System

132



Bus

No

Voltage

Magnitude

Voltage

Phase Angle

1 1.024 0

2 1 0.21514

3 1.00123 0.41897

4 1 0.35121

5 1.00282 0.51259

6 1.00012 0.36028

7 1 1.54531

8 1 3.53235

9 1.00083 3.63548

10 1 4.30636

11 1.00191 4.53677


From To



1 2 -54.463 348.204 54.4629 -339.84

2 3 -23.129 -7.9175 23.1286 8.00949

2 4 -31.331 28.6684 31.3992 -28.594

3 5 -10.332 -1.8113 10.3458 1.83107

4 5 -6.1496 -6.132 6.14955 6.1666

4 6 -8.9974 -6.7958 8.99744 6.79804

4 7 -16.252 12.5089 16.5089 -12.168

7 8 -16.511 50.0814 18.237 -49.479

8 9 -2.5987 -0.8943 2.59928 0.8997

8 10 -15.635 11.6015 15.7901 -11.389

10 11 -15.788 -5.6237 15.7956 5.69802

133


The steady state mathematical model developed for STATCOM

in chapter3 is incorporated into the runge -kutta algorithm to test for

the existence of solution for a 11-bus Ill-Conditioned system The

STATCOM is connected at bus no 4 to regulate the voltage to 1.0 per

unit.. The voltage profile before incorporation of the device is .9987

per unit. The STATCOM reactance is taken as 10 per unit on 100 MVA

base. The initial source voltage angle is taken as 00. From results

obtianed, it is observed that the device is able to regulate the voltage

at bus no 4 to 1.0 per unit. The source voltage angle is 10 degrees.

The reactive power flow in the lines connected to the bus no 4 is

significantly affected. The real power flow in the lines with and without

the device remains the same. The real power mismatch variation is

large in the first three iterations.

134


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21512

3 0.99999 0.40914

4 1 0.35612

5 1 0.54785

6 1.00012 0.36519

7 1 1.55029

8 1 3.53734

9 1.00083 3.64049

10 1 4.31146

11 1.00191 4.54192


From To



1 2 -54.457 348.204 54.4568 -339.84

2 3 -21.989 0.09746 21.9887 -0.023

2 4 -32.466 29.7089 32.5385 -29.629

3 5 -9.1889 6.22326 9.20403 -6.2011

4 5 -7.2859 0.01219 7.28594 0.01219

4 6 -8.9996 -6.7975 8.99959 6.79968

4 7 -16.253 12.5097 16.5099 -12.168

7 8 -16.511 50.0817 18.2371 -49.479

8 9 -2.5993 -0.8945 2.59988 0.89995

8 10 -15.637 11.6031 15.7923 -11.391

10 11 -15.792 -5.6253 15.7993 5.69968

135

5.5.11 11-BUS ILL-CONDITIONED SYSTEM WITH SVC

SUSCEPTANCE MODEL

The SVC is connected to bus no 4. The initial susceptance value

is chosen as 0.02 per unit on 100 MVA base. With a susceptance of

0.06 p.u in susceptance, it is revealed from the study that reactive

power flow variations are significant. The solution is converged in 16

iterations whereas the problem is converged in 14 iterations without

device. This may be due to non-linearities of the device model. The

mismatch power variations are large in first three iterations.

136


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21514

3 1.00123 0.41895

4 1 0.3512

5 1.00282 0.51255

6 1.00012 0.36027

7 1 1.54532

8 1 3.53243

9 1.00083 3.63555

10 1 4.30644

11 1 4.54823


From To



1 2 -54.461 348.204 54.4608 -339.84

2 3 -23.127 -7.9168 23.1268 8.00877

2 4 -31.33 28.6674 31.3981 -28.593

3 5 -10.332 -1.8111 10.345 1.83088

4 5 -6.1489 -6.1315 6.14887 6.16608

4 6 -8.9967 -6.7952 8.99667 6.79745

4 7 -16.252 12.5091 16.5092 -12.168

7 8 -16.511 50.0834 18.2378 -49.481

8 9 -2.5985 -0.8942 2.59905 0.89961

8 10 -15.635 11.6014 15.79 -11.389

10 11 -15.789 1.60893 15.7952 -1.5423

137

5.5.12 11-BUS ILL-CONDITIONED SYSTEM WITH SVC FIRING

ANGLE MODEL

The SVC is connected to bus no 11. The initial values are

chosen as capacitive reactance is 1.07 p.u., inductive reactance is

0.288 p.u. and firing angle is 140 degrees. From the results it is

observed that the reactive power flow variations are significant. The

solution is converged in 20 iterations whereas the problem is

converged in 18 iterations without device. This may be due to non-

linearities of the device model. The mismatch power variations are

large in first three iterations.

138


Bus

No

Voltage

Magnitude

Voltage

Phase Angle

1 1.024 0

2 1 0.21514

3 1.00123 0.41894

4 1 0.35121

5 1.00282 0.51254

6 1.00012 0.36028

7 1 1.54566

8 1 3.53272

9 1.00083 3.63583

10 1 4.30703

11 1 4.54897


From To



1 2 -54.461 348.204 54.4614 -339.84

2 3 -23.125 -7.916 23.1253 8.008

2 4 -31.333 28.6699 31.4008 -28.595

3 5 -10.331 -1.8109 10.3446 1.83065

4 5 -6.1479 -6.131 6.14786 6.16564

4 6 -8.9959 -6.7946 8.99591 6.79687

4 7 -16.257 12.5126 16.5138 -12.171

7 8 -16.511 50.082 18.2372 -49.479

8 9 -2.5983 -0.8941 2.59884 0.89951

8 10 -15.641 11.606 15.7963 -11.394

10 11 -15.797 1.60985 15.8041 -1.5431

139



The steady state mathematical models developed for TCSC in

susceptance and firing angle modes in chapter-3 are incorporated into

the runge -kutta algorithm to test for the existence of solution for a

13-bus .The TCSC is connected between bus-4 and bus-6 with the

reactance of -0.015 p.u. and pre specified power flow in the line is set

at 0.21 p.u. It is observed that with the incorporation of TCSC there is

much difference in real power flows through the lines where as

reactive power flows are not much affected. From the power mismatch

versus iterations it is observed that there is a large variation in

mismatch powers in the first two iterations and the variable

impedance power flow model exhibits oscillations between fourth and

fifth iterations.

140


Bus

No

Voltage

Magnitude

Voltage

Phase Angle

1 1.024 0

2 1 0.2151

3 1.00123 0.41887

4 1 0.35114

5 1.00282 0.51246

6 0.99966 0.32517

7 1 1.54515

8 1 3.53218

9 1.00083 3.63528

10 1 4.30606

11 1.00191 4.5364

12 0.99966 0.32517


From To



1 2 -54.452 348.204 54.4522 -339.84

2 3 -23.122 -7.9152 23.1223 8.00718

2 4 -31.324 28.6619 31.392 -28.587

3 5 -10.33 -1.8107 10.343 1.83051

4 5 -6.1477 -6.1303 6.14767 6.16487

12 6 -9.0 6.81 9.0 -6.80

4 7 -16.251 12.508 16.5077 -12.167

7 8 -16.51 50.0811 18.2369 -49.479

8 9 -2.598 -0.894 2.59856 0.8994

8 10 -15.632 11.5995 15.7874 -11.387

10 11 -15.784 -5.6218 15.7913 5.69608

141


ANGLE MODEL

The TCSC is connected between bus-4 and bus-6 with the following

initial conditions: inductive reactance is 9.375e-3 p.u., capacitive

reactance is 1.625e-2 p.u and pre specified power flow in the line is

set at 0.45 p.u. The firing angle is 139.99 degrees. It is observed that

with the incorporation of TCSC there is much difference in real power

flows through the lines where as reactive power flows are not much

affected. From the power mismatch versus iterations it is observed

that there is a large variation in mismatch powers in the first two

iterations.

142


Bus

No

Voltage

Magnitude

Voltage

Phase Angle

1 1.024 0

2 1 0.21511

3 1.00123 0.41888

4 1 0.35114

5 1.00282 0.51247

6 0.99993 0.34604

7 1 1.54516

8 1 3.53218

9 1.00083 3.63528

10 1 4.30606

11 1.00191 4.53641

12 0.99993 0.34604


From To



1 2 -54.453 348.204 54.4529 -339.84

2 3 -23.122 -7.9152 23.1224 8.00717

2 4 -31.325 28.6626 31.3928 -28.588

3 5 -10.33 -1.8107 10.3431 1.8305

4 5 -6.1476 -6.1303 6.1476 6.16488

12 6 -9.0 6.8 9.0 -6.8

4 7 -16.251 12.508 16.5077 -12.167

7 8 -16.51 50.081 18.2369 -49.478

8 9 -2.598 -0.894 2.59856 0.8994

8 10 -15.632 11.5995 15.7874 -11.387

10 11 -15.784 -5.6218 15.7913 5.69608

143


The steady state mathematical models developed for UPFC in

chapter3 are incorporated into the runge -kutta algorithm to test for

existence of solution for a 11-bus Ill-Conditioned system .The UPFC is

incorporated between bus-4 and bus-5 with shunt converter

connected close to bus 4 and series converter connected between

buses 4 and 5 with inductive reactance of shunt converter taken as

0.1 p.u and inductive reactance of series converter taken as 5 p.u with

specified active and reactive power flows set at 0.4 and 0.02

respectively. The shunt converter is set to maintain a target voltage of

1 p.u. From the results it is observed that UPFC holds its target

values with the given initial conditions. From the power flow results it

is observed that UPFC is the only FACTS device which affects both

real and reactive power flows through the lines .The maximum power

mismatch is less when compared to the other facts devices discussed.

144


Bus

No

Voltage

Magnitude

Voltage

Phase Angle

1 1.024 0

2 1 0.21512

3 1.00216 0.47259

4 1 0.3245

5 1.00561 0.58581

6 1.0008 0.298

7 1 1.511

8 1 3.498

9 1.00083 3.6011

10 1 4.27188

11 1.00191 4.50223

12 1.00561 0.58581


From To



1 2 -54.456 348.204 54.4561 -339.84

2 3 -29.243 -13.97 29.2427 14.1319

2 4 -25.207 6.58678 25.233 -6.5588

3 5 -16.45 -7.9355 16.4907 7.99536

12 5 239.72 337.2 -241.47 -337.99

4 6 -8.9949 -6.7939 8.99492 6.79612

4 7 -16.238 13.3487 16.5078 -12.991

7 8 -16.51 50.0806 18.2367 -49.478

8 9 -2.598 -0.894 2.59856 0.8994

8 10 -15.632 11.5995 15.7874 -11.387

10 11 -15.784 -5.6218 15.7913 5.69608

145

2 4 6 8 10 12 14 16

0

2

4

6

8

10

12

14

x 10-3

----------> No.of Iterations

----

----

-->

Ma

xim

um

Po

we

r M

ism

atc

h

Comparison of Maximum Power Mismatch

Without Device

With STATCOM

With SVC-B

With SVC-FA

With TCSC-POWER

With TCSC-FA

With UPFC



Fig 5.3 Comparison of maximum power mismatch using Runge-Kutta


146

5.6. CASE STUDY WITH OPTIMAL MULTIPLIER METHOD


Conditioned systems with and without devices. The initial data for the

devices is presented in section 5.3



Bus

No

Voltage

Magnitude

Voltage

Phase Angle

1 1 0

2 1.00686 1.57394

3 1.03847 2.52084

4 1.02167 2.56719

5 1 2.61853

6 1.037 9.86015

7 1.06296 9.08705

8 1.1 8.23779

9 0.943 14.383

10 1.1 8.36942

11 1.04853 12.0287

12 1.08454 8.22685

13 1.08624 5.44713


From To



1 2 -328.219 -60.7999 328.665 70.271

1 3 -498.135 -374.601 499.689 411.389

5 4 0.000161 -228.838 0.209306 233.797

4 3 -113.985 -120.022 -117.237 121.995

6 2 267.658 56.8606 -391.011 -7.92423

6 7 111.906 -258.066 -120.057 266.11

8 3 287.987 288.719 -691.96 -223.876

7 8 115.537 -261.591 -115.537 272.447

9 10 362.17 -730.657 -678.298 908.718

10 11 490.747 -393.099 -490.747 344.087

11 12 301.121 -234.461 -588.177 272.205

12 13 257.061 -23.0888 -519.982 41.9829

13 8 387.831 90.1689 -387.831 -110.324

147



Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.00684 1.57931

3 1.03845 2.53579

4 1.02166 2.5821

5 1 2.63341

6 1.037 9.89413

7 1.06297 9.12736

8 1.1 8.28637

9 1 13.9647

10 1.1 8.4242

11 1.04807 12.1153

12 1.08457 8.27659

13 1.08678 5.46835


From To



1 2 -329.309 -60.5011 329.757 70.03

1 3 -500.961 -374.117 502.524 411.137

5 4 0.000247 -228.742 0.209045 233.697

4 3 -113.983 -119.923 -117.232 121.894

6 2 268.784 56.9477 -392.102 -7.6855

6 7 110.78 -258.058 -118.933 266.076

8 3 290.853 288.964 -694.787 -223.537

7 8 114.413 -261.557 -114.413 272.376

9 10 345.004 -495.599 -682.35 595.715

10 11 494.799 -396.767 -494.799 346.901

11 12 305.296 -237.398 -592.198 275.929

12 13 261.065 -26.7971 -524.102 46.1036

13 8 391.821 86.1774 -391.821 -106.618

148


SUCEPTANCE MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.00684 1.5793

3 1.03845 2.53576

4 1.02166 2.58207

5 1 2.63338

6 1.037 9.89407

7 1.06297 9.1273

8 1.1 8.2863

9 1 13.9646

10 1.1 8.42412

11 1.04807 12.1152

12 1.08457 8.27652

13 1.08677 5.46832


From To



1 2 -329.307 -60.5016 329.755 70.0304

1 3 -500.957 -374.118 502.52 411.138

5 4 -0.000166 -228.742 0.2095 233.697

4 3 -113.983 -119.924 -117.232 121.894

6 2 268.782 56.9476 -392.1 -7.68588

6 7 110.78 -258.058 -118.933 266.076

8 3 290.849 288.964 -694.783 -223.537

7 8 114.414 -261.557 -114.414 272.376

9 10 344.998 -495.599 -682.344 595.714

10 11 494.794 -396.761 -494.794 346.897

11 12 305.29 -237.394 -592.192 275.924

12 13 261.06 -26.7918 -524.096 46.0978

13 8 391.815 86.183 -391.815 -106.624

149


ANGLE MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.00684 1.5793

3 1.03845 2.53577

4 1.02166 2.58208

5 1 2.63339

6 1.037 9.89409

7 1.06297 9.12732

8 1.1 8.28633

9 1 13.9646

10 1.1 8.42415

11 1.04807 12.1153

12 1.08457 8.27654

13 1.08677 5.46833


From To



1 2 -329.307 -60.5014 329.756 70.0303

1 3 -500.958 -374.117 502.521 411.138

5 4 -0.00027 -228.742 0.209564 233.697

4 3 -113.983 -119.924 -117.232 121.894

6 2 268.783 56.9476 -392.1 -7.68578

6 7 110.78 -258.058 -118.933 266.076

8 3 290.851 288.964 -694.784 -223.537

7 8 114.413 -261.557 -114.413 272.376

9 10 345 -495.599 -682.346 595.714

10 11 494.796 -396.763 -494.796 346.898

11 12 305.292 -237.395 -592.194 275.926

12 13 261.062 -26.7936 -524.098 46.0998

13 8 391.817 86.1811 -391.817 -106.622

150




Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.00713 1.49462

3 1.03838 2.58238

4 1.02163 2.62847

5 1 2.67965

6 1.037 9.35874

7 1.11657 8.30504

8 1.1 8.43803

9 0.943 14.582

10 1.1 8.57132

11 1.04841 12.2391

12 1.08454 8.4274

13 1.08639 5.64001

14 1.1 8.43803


From To



1 2 -312.113 -65.1355 312.519 73.7763

1 3 -509.771 -372.592 511.366 410.348

5 4 -0.0135 -228.439 0.222223 233.38

4 3 -113.989 -119.613 -117.203 121.574

6 2 251.022 55.6527 -374.899 -11.3965

6 7 128.416 -773.283 -132.552 835.106

8 3 299.803 289.744 -703.614 -222.47

7 8 115.64 262.60 -115.64 -272.46

9 10 361.908 -730.671 -678.039 908.677

10 11 490.747 88.91 -490.51 -70.42

11 12 302.243 -235.251 -589.258 273.206

12 13 258.138 -24.0856 -521.09 43.0899

13 8 388.903 89.0966 -388.903 -109.328

151


ANGLE MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.0047 2.12702

3 1.03928 1.9212

4 1.02199 1.96907

5 1 2.02131

6 1.037 13.3742

7 1.03745 13.3721

8 1.1 6.28942

9 0.943 12.4418

10 1.1 6.42173

11 1.04848 10.0845

12 1.08454 6.27861

13 1.0863 3.49577

14 1.1 6.28942


From To



1 2 -439.95 26.48 440.72 - 43.00

1 3 -384.49 392.36 385.70 - 420.93

5 4 0.03 232.25 000.19 - 237.36

4 3 -114.03 123.51 -117.44 - 125.60

6 2 294.39 -59.2 -417.81 - 19.08

6 7 84.16 103.4 -92.69 -105.36

8 3 172.91 -280.76 -578.25 236.55

14 8 88.35 100.79 -88.35 -109.64

9 10 362.74 730.63 -678.86 - 908.81

10 11 491.19 393.50 -491.19 - 344.40

11 12 301.58 234.78 -588.62 - 272.61

12 13 257.50 23.49 -520.43 - 42.43

13 8 388.27 -89.73 -388.27 109.92

152



Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.0047 2.12704

3 1.04531 1.88657

4 1.01362 3.54304

5 1 3.57547

6 1.037 13.3743

7 1 13.5126

8 1.1 6.23556

9 0.943 12.3731

10 1.1 6.36645

11 1.04859 10.0219

12 1.08453 6.22448

13 1.08618 3.44815

14 1.1018 6.2835


From To



1 2 -439.95 -26.4832 440.727 42.995

1 3 -382.658 -456.305 384.077 489.889

5 4 -0.0169 -143.862 0.0997 145.822

4 3 90.1896 -231.961 -320.599 245.139

6 2 383.865 69.4044 -502.807 19.0848

6 7 -4.85583 4.85583 -4.85146 -4.85374

8 3 171.945 252.292 -579.684 -211.147

14 8 1770.9 1116.1 -1726.4 -1462.5

9 10 361.547 -730.69 -677.682 908.621

10 11 490.265 -392.663 -490.265 343.752

11 12 300.624 -234.111 -587.699 271.762

12 13 256.585 -22.6476 -519.492 41.4932

13 8 387.356 90.6431 -387.356 -110.765

153

5.6.8 Comparison of Maximum Power Mismatch for 13-Bus Ill-Conditioned System

Fig 5.4 Comparison of maximum power mismatch using

Iwamoto method for 13-bus Ill-Conditioned System

154

5.6.9 11-BUS ILL-CONDITIONED SYSTEM WITHOUT ANY DEVICE


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21518

3 1.00123 0.41907

4 1 0.35128

5 1.00282 0.5127

6 1.00012 0.36035

7 1 1.54555

8 1 3.53274

9 1.00083 3.6359

10 1 4.3069

11 1.00191 4.53737


From To



1 2 -54.473 348.204 54.4728 -339.84

2 3 -23.135 -7.9198 23.135 8.01185

2 4 -31.338 28.6743 31.4056 -28.6

3 5 -10.335 -1.8119 10.3485 1.83165

4 5 -6.1515 -6.1337 6.15152 6.16835

4 6 -9 -6.7978 9 6.8

4 7 -16.254 12.5107 16.5113 -12.169

7 8 -16.512 50.0853 18.2385 -49.483

8 9 -2.5995 -0.8946 2.6 0.9

8 10 -15.638 11.6037 15.7932 -11.391

10 11 -15.793 -5.6256 15.8 5.7

155



Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21482

3 0.99998 0.40878

4 1 0.35563

5 1 0.54772

6 1.00012 0.3647

7 1 1.54575

8 1 3.52349

9 1.00083 3.62665

10 1 4.2963

11 1.00191 4.52677


From To

Sendng end Power Recevng end Power


1 2 -54.381 348.204 54.381 -339.84

2 3 -21.982 0.13919 21.9819 -0.0648

2 4 -32.422 29.6688 32.4946 -29.589

3 5 -9.2199 6.20878 9.23511 -6.1865

4 5 -7.2993 0.01224 7.29929 0.01224

4 6 -9 -6.7978 9.00002 6.80001

4 7 -16.198 12.4667 16.4535 -12.128

7 8 -16.437 49.8458 18.1477 -49.249

8 9 -2.5995 -0.8946 2.6 0.89999

8 10 -15.611 11.5833 15.7655 -11.372

10 11 -15.793 -5.6254 15.7999 5.6998

156


SUSCEPTANCE MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.2098

3 0.99999 0.39789

4 1 0.34726

5 1 0.5326

6 1.00012 0.35604

7 1 1.51933

8 1 3.47719

9 1.0008 3.57702

10 1 4.2315

11 1.00185 4.45455


From To



1 2 -53.111 348.199 53.1106 -339.85

2 3 -21.316 0.10599 21.3158 -0.036

2 4 -31.651 28.9615 31.7201 -28.886

3 5 -8.929 6.03612 8.94327 -6.0152

4 5 -7.0429 0.01139 7.04291 0.01139

4 6 -8.7094 -6.5784 8.70944 6.58047

4 7 -15.954 12.2751 16.2021 -11.946

7 8 -16.281 49.3418 17.9568 -48.757

8 9 -2.5156 -0.8659 2.51607 0.87096

8 10 -15.239 11.3036 15.3864 -11.102

10 11 -15.283 -5.4466 15.2901 5.51621

157


ANGLE MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21517

3 1.00123 0.41906

4 1 0.35126

5 1.00282 0.51269

6 1.00012 0.36033

7 1 1.54539

8 1 3.53223

9 1.00083 3.63538

10 1 4.30636

11 1 4.54822


From To



1 2 -54.47 348.204 54.4704 -339.84

2 3 -23.135 -7.9198 23.1348 8.01188

2 4 -31.336 28.6727 31.4038 -28.598

3 5 -10.335 -1.8119 10.3483 1.83168

4 5 -6.1517 -6.1337 6.15169 6.16832

4 6 -9 -6.7978 9 6.8

4 7 -16.252 12.5093 16.5094 -12.168

7 8 -16.509 50.0763 18.2351 -49.474

8 9 -2.5995 -0.8946 2.6 0.9

8 10 -15.637 11.6033 15.7926 -11.391

10 11 -15.793 1.60939 15.7997 -1.5427

158




Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21518

3 1.00123 0.41906

4 1 0.35127

5 1.00282 0.5127

6 0.99966 0.32545

7 1 1.54546

8 1 3.53247

9 1.00083 3.63563

10 1 4.3066

11 1.00191 4.53707

12 0.99966 0.32545


From To



1 2 -54.472 348.204 54.4715 -339.84

2 3 -23.135 -7.9198 23.1349 8.01187

2 4 -31.337 28.6734 31.4046 -28.599

3 5 -10.335 -1.8119 10.3484 1.83167

4 5 -6.1516 -6.1337 6.15162 6.16833

12 6 -9.01 6.81 9.01 -6.81

4 7 -16.253 12.5099 16.5102 -12.169

7 8 -16.51 50.0807 18.2367 -49.478

8 9 -2.5995 -0.8946 2.6 0.9

8 10 -15.637 11.6033 15.7926 -11.391

10 11 -15.793 -5.6256 15.8 5.7

159


ANGLE MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21534

3 1.00123 0.41925

4 1 0.35157

5 1.00282 0.51292

6 0.99993 0.34646

7 1 1.54832

8 1 3.54102

9 1.00083 3.64418

10 1 4.31603

11 1.00191 4.54651

12 0.99993 0.34646


From To



1 2 -54.513 348.204 54.5126 -339.84

2 3 -23.138 -7.9193 23.1379 8.0114

2 4 -31.369 28.7029 31.4369 -28.628

3 5 -10.338 -1.8114 10.3514 1.83121

4 5 -6.1486 -6.1341 6.14861 6.16879

12 6 -9.0 6.8 9.0 -6.8

4 7 -16.288 12.5371 16.5459 -12.194

7 8 -16.555 50.2249 18.2914 -49.619

8 9 -2.5995 -0.8946 2.6 0.9

8 10 -15.655 11.6166 15.8107 -11.404

10 11 -15.793 -5.6256 15.8 5.7

160

5.6.15 11-BUS ILL-CONDITIONED SYSTEM WITH UPFC MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21533

3 1.00216 0.47294

4 1 0.32493

5 1.00561 0.58622

6 1.00012 0.33401

7 1 1.52166

8 1 3.51431

9 1.00083 3.61747

10 1 4.28931

11 1.00191 4.51979

12 1.00561 0.58622


From To



1 2 -54.509 348.204 54.509 -339.84

2 3 -29.259 -13.978 29.2592 14.14

2 4 -25.244 23.0876 25.2878 -23.039

3 5 -16.459 -7.9401 16.5 8

12 5 239.72 337.23 -241.47 -338.02

4 6 -9 -6.7978 9 6.8

4 7 -16.287 12.5368 16.5455 -12.194

7 8 -16.555 50.2236 18.2909 -49.618

8 9 -2.5995 -0.8946 2.6 0.9

8 10 -15.655 11.6165 15.8105 -11.404

10 11 -15.793 -5.6256 15.8 5.7

161

1 2 3 4 5 6 7 8

0

0.02

0.04

0.06

0.08

0.1

----------> No.of Iterations

----

----

-->

Ma

xim

um

Po

we

r M

ism

atc

h


Without Device

With STATCOM

With SVC-B

With SVC-FA

With TCSC-POWER

With TCSC-FA

With UPFC



Fig 5.5 Comparison of maximum power mismatch using Iwamoto method for 11-bus Ill-Conditioned System

162

The 13 and 11-bus Ill-Conditioned systems discussed above are

tested with the same initial conditions using iwamoto optimal

multiplier method with the incorporation of FACTS devices .The

placement of the devices remains same as in the case with Runge-

kutta method. All the facts devices (STATCOM,SVC,TCSC and UPFC)

hold their functional capabilities with the incorporation of optimal

multiplier method for the steady state mathematical models presented

in chapter 3. The results for voltage magnitudes and phase angles

along with the power flows are tabulated above .The value of optimal

multiplier μ stays nearer to one with the incorporation of devices.

From the results obtained, it is observed that the number of iterations

taken to converge is less when compared with runge- kutta method.

The converging behavior of TCSC variable impedance power model is

oscillatory.

163

5.7. Case Study with 2-Step Method






Bus

No

Voltage

Magnitude

Voltage

Phase Angle

1 1 0

2 1.01139 1.54536

3 1.05712 2.43665

4 1.03562 2.49271

5 1 2.57593

6 1.037 9.77016

7 1.06322 8.98467

8 1.1 8.12085

9 0.943 14.3468

10 1.1 8.33321

11 1.01771 12.1034

12 1.06717 8.08853

13 1.04424 5.22269


From To



1 2 -326.269 -114.34 326.747 124.501

1 3 -498.743 -571.97 501.046 626.505

5 4 3.45E-09 -376.08 0.56575 389.477

4 3 -234.371 -155.67 -242.88 158.903

6 2 199.497 46.5907 -452.57 1.31694

6 7 109.629 -260.64 -126.59 268.868

8 3 73.9698 200.784 -899.61 -143.97

7 8 117.547 -259.82 -117.55 270.616

9 10 224.333 -730.66 -865.84 908.718

10 11 490.744 -619.64 -490.74 542.185

11 12 177.67 -309.11 -727.61 355.699

12 13 133.234 156.675 -631.44 -134.12

13 8 387.183 378.376 -387.18 -418.69

164



Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.01139 1.54508

3 1.05712 2.43587

4 1.03562 2.49193

5 1 2.57516

6 1.037 9.76839

7 1.06322 8.98258

8 1.1 8.11832

9 0.943 14.1523

10 1.1 8.13873

11 1.05557 11.7735

12 1.06123 7.99802

13 1 5.09347


From To



1 2 -326.212 -114.36 326.69 124.514

1 3 -498.593 -571.99 500.896 626.518

5 4 1.43E-10 -376.09 0.56577 389.482

4 3 -234.371 -155.68 -242.88 158.908

6 2 199.439 46.5863 -452.51 1.30457

6 7 109.688 -260.64 -126.65 268.87

8 3 73.8181 200.772 -899.46 -143.98

7 8 117.605 -259.83 -117.61 270.619

9 10 224.333 -730.66 -865.84 908.718

10 11 490.744 -341.38 -490.74 297.075

11 12 157.729 -44.06 -725.29 73.3777

12 13 136.957 432.954 -610.97 -388.86

13 8 386.973 656.449 -386.97 -743.55

165

5.7.3. 13-BUS ILL-CONDITIONED SYSTEM WITH SVC

SUSCEPTANCE MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.01139 1.54508

3 1.05712 2.43587

4 1.03562 2.49193

5 1 2.57516

6 1.037 9.76839

7 1.06322 8.98258

8 1.1 8.11832

9 0.943 14.1523

10 1.1 8.13873

11 1.05557 11.7735

12 1.06123 7.99802

13 1 5.09347


From To



1 2 -326.212 -114.36 326.69 124.514

1 3 -498.593 -571.99 500.896 626.518

5 4 1.39e-14 -376.09 0.56577 389.482

4 3 -234.371 -155.68 -242.88 158.908

6 2 199.439 46.5863 -452.51 1.30457

6 7 109.688 -260.64 -126.65 268.87

8 3 73.8181 200.772 -899.46 -143.98

7 8 117.605 -259.83 -117.61 270.619

9 10 224.333 -730.66 -865.84 908.718

10 11 490.744 -341.38 -490.74 297.075

11 12 157.729 -44.06 -725.29 73.3777

12 13 136.957 432.954 -610.97 -388.86

13 8 386.973 656.449 -386.97 -743.55

166


ANGLE MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.01139 1.54508

3 1.05712 2.43587

4 1.03562 2.49193

5 1 2.57516

6 1.037 9.76839

7 1.06322 8.98258

8 1.1 8.11832

9 0.943 14.1523

10 1.1 8.13873

11 1.05557 11.7735

12 1.06123 7.99802

13 1 5.09347


From To



1 2 -326.212 -114.36 326.69 124.514

1 3 -498.593 -571.99 500.896 626.518

5 4 -1.5e-12 -376.09 0.56577 389.482

4 3 -234.371 -155.68 -242.88 158.908

6 2 199.439 46.5863 -452.51 1.30457

6 7 109.688 -260.64 -126.65 268.87

8 3 73.8181 200.772 -899.46 -143.98

7 8 117.605 -259.83 -117.61 270.619

9 10 224.333 -730.66 -865.84 908.718

10 11 490.744 -341.38 -490.74 297.075

11 12 157.729 -44.06 -725.29 73.3777

12 13 136.957 432.954 -610.97 -388.86

13 8 386.973 656.449 -386.97 -743.55

167




Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.00686 1.57402

3 1.03847 2.52106

4 1.02167 2.56741

5 1 2.61874

6 1.037 9.86065

7 1.06296 9.08764

8 1.1 8.23851

9 0.943 15.5561

10 1.1 9.54247

11 1.09342 11.6469

12 1.08426 8.11928

13 1.04599 5.33991

14 1.09342 11.6469


From To



1 2 -328.235 -60.796 328.68 70.2675

1 3 -498.176 -374.59 499.73 411.385

5 4 8.15e-05 -228.84 0.20938 233.796

4 3 -113.985 -120.02 -117.24 121.994

6 2 267.674 56.8619 -391.03 -7.9208

6 7 111.89 -258.07 -120.04 266.11

8 3 288.03 288.723 -692 -223.87

7 8 115.519 -261.59 -115.52 272.446

9 10 362.169 -730.66 -678.3 908.718

14 11 590.75 249.81 -590.75 -219.21

11 12 288.911 50.8531 -588.63 -23.44

12 13 257.662 272.412 -510.43 -244.28

13 8 387.89 366.814 -387.89 -405.89

168


ANGLE MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.0047 2.12702

3 1.03928 1.91869

4 1.022 1.96633

5 1 2.01842

6 1.037 13.3742

7 1.03745 13.3721

8 1.1 6.28162

9 0.943 12.4269

10 1.1 6.41325

11 1.04853 10.0725

12 1.08454 6.27068

13 1.08624 3.49096

14 1.1 6.28162


From To



1 2 -439.946 -26.484 440.723 42.996

1 3 -384.013 -392.42 385.219 420.971

5 4 0.0002238 -232.26 0.21556 237.371

4 3 -114.063 -123.52 -117.41 125.612

6 2 383.861 69.4039 -502.8 19.0839

6 7 -4.29787 -4.3047 -4.3087 4.30657

8 3 172.454 280.733 -577.8 -236.59

14 8 96.02 40.10 -96.1 -54.0

9 10 362.171 -730.66 -678.3 908.718

10 11 490.748 -393.1 -490.75 344.087

11 12 301.121 -234.46 -588.18 272.205

12 13 257.062 -23.089 -519.98 41.9832

13 8 387.831 90.1685 -387.83 -110.32

169


The Problem is converged in 17 iterations Table 5.69: Voltage Magnitudes and Phase angles

Bus

No

Voltage

Magnitude

Voltage

Phase Angle

1 1 0

2 1.0047 2.12704

3 1.0453 1.89138

4 1.022 1.9765

5 1 2. 023

6 1.037 13.3743

7 1 13.25081

8 1.1 6.25081

9 0.943 12.402

10 1.1 6.38301

11 1.04849 10.0452

12 0.9532 6.23998

13 1.17129 3.45763

14 1.1018 6.24875

Table 5.70: Complex Power flow through Lines

From To



1 2 -439.95 -26.483 440.727 42.995

1 3 -383.579 -456.19 385 489.826

5 4 0.0132289 -143.8 0.06948 145.756

4 3 90.1616 -231.97 -320.57 245.146

6 2 383.865 69.4044 -502.81 19.0848

6 7 -4.85583 4.85583 -4.8515 -4.8537

8 3 172.842 252.336 -580.57 -211.05

14 8 1444 1133.4 -1400.7 -1393.8

9 10 362.652 -730.63 -678.78 908.794

10 11 491.119 -393.44 -491.12 344.346

11 12 301.504 -234.73 -588.55 272.547

12 13 257.429 -23.429 -520.36 42.361

13 8 388.197 89.8026 -388.2 -109.98

170

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

-------->No.of iterations

----

----

>M

ax

.Po

we

r M

ism

atc

h


SIMPLE MOD2

MOD2 WITH SSC

MOD2 WITH SVC-B

MOD2 WITH SVC-F

MOD2 WITH TCSC-P

MOD2 WITH TCSC-F

MOD2 WITH UPFC


bus ILL-CONDITIONED SYSTEM

Fig 5.6 Comparison of maximum power mismatch using 2 Step


171



Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21518

3 1.00123 0.41906

4 1 0.35127

5 1.00282 0.5127

6 1.00012 0.36034

7 1 1.54546

8 1 3.53246

9 1.00083 3.63562

10 1 4.30659

11 1.00191 4.53707


From To



1 2 -54.472 348.204 54.4715 -339.84

2 3 -23.135 -7.9198 23.1349 8.01187

2 4 -31.337 28.6734 31.4046 -28.599

3 5 -10.335 -1.8119 10.3484 1.83167

4 5 -6.1516 -6.1337 6.15162 6.16833

4 6 -9 -6.7978 9 6.8

4 7 -16.253 12.5099 16.5102 -12.168

7 8 -16.51 50.0806 18.2367 -49.478

8 9 -2.5995 -0.8946 2.6 0.9

8 10 -15.637 11.6033 15.7926 -11.391

10 11 -15.793 -5.6256 15.8 5.7

172



Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21518

3 1.00123 0.41906

4 1 0.35127

5 1.00282 0.5127

6 1.00012 0.36034

7 1 1.54546

8 1 3.53246

9 1.00083 3.63562

10 1 4.30659

11 1.00191 4.53707


From To



1 2 -54.472 348.204 54.4715 -339.84

2 3 -23.135 -7.9198 23.1349 8.01187

2 4 -31.337 28.6734 31.4046 -28.599

3 5 -10.335 -1.8119 10.3484 1.83167

4 5 -6.1516 -6.1337 6.15162 6.16833

4 6 -9 -6.7978 9 6.8

4 7 -16.253 12.5099 16.5102 -12.169

7 8 -16.51 50.0806 18.2367 -49.478

8 9 -2.5995 -0.8946 2.6 0.9

8 10 -15.637 11.6033 15.7926 -11.391

10 11 -15.793 -5.6256 15.8 5.7

173


SUSCEPTANCE MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21518

3 1 0.42157

4 1 0.35014

5 1.00197 0.50492

6 1.00012 0.35922

7 1 1.54434

8 1 3.53134

9 1.00083 3.6345

10 1 4.30547

11 1.00191 4.53594


From To



1 2 -54.471 348.204 54.4707 -339.84

2 3 -23.392 0.04213 23.3916 0.04213

2 4 -31.079 28.4372 31.146 -28.364

3 5 -10.592 -3.6816 10.607 3.70429

4 5 -5.893 -4.2714 5.89301 4.29571

4 6 -9 -6.7978 9 6.8

4 7 -16.253 12.5099 16.5102 -12.168

7 8 -16.51 50.0806 18.2367 -49.478

8 9 -2.5995 -0.8946 2.6 0.9

8 10 -15.637 11.6033 15.7926 -11.391

10 11 -15.793 -5.6256 15.8 5.7

174


ANGLE MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21518

3 1.00123 0.41906

4 1 0.35127

5 1.00282 0.5127

6 1.00012 0.36035

7 1 1.54551

8 1 3.53259

9 1.00083 3.63575

10 1 4.30676

11 1 4.54863


From To



1 2 -54.472 348.204 54.4721 -339.84

2 3 -23.135 -7.9198 23.1349 8.01186

2 4 -31.337 28.6739 31.4052 -28.599

3 5 -10.335 -1.8119 10.3484 1.83166

4 5 -6.1516 -6.1337 6.15157 6.16834

4 6 -9 -6.7978 9 6.8

4 7 -16.254 12.5104 16.5108 -12.169

7 8 -16.511 50.0825 18.2374 -49.48

8 9 -2.5995 -0.8946 2.6 0.9

8 10 -15.638 11.6039 15.7933 -11.392

10 11 -15.793 1.60943 15.8 -1.5427

175




Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21518

3 1.00123 0.41906

4 1 0.35127

5 1.00282 0.5127

6 0.99966 0.32547

7 1 1.54546

8 1 3.53247

9 1.00083 3.63563

10 1 4.3066

11 1.00191 4.53708

12 0.99966 0.32547


From To



1 2 -54.472 348.204 54.4715 -339.84

2 3 -23.135 -7.9198 23.1349 8.01187

2 4 -31.337 28.6734 31.4047 -28.599

3 5 -10.335 -1.8119 10.3484 1.83167

4 5 -6.1516 -6.1337 6.15162 6.16833

12 6 -9 6.81 9 -6.8

4 7 -16.253 12.5099 16.5102 -12.169

7 8 -16.51 50.0807 18.2368 -49.478

8 9 -2.5995 -0.8946 2.6 0.9

8 10 -15.637 11.6033 15.7926 -11.391

10 11 -15.793 -5.6256 15.8 5.7

176


ANGLE MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21518

3 1.00123 0.41907

4 1 0.35128

5 1.00282 0.5127

6 0.99993 0.34617

7 1 1.54555

8 1 3.53275

9 1.00083 3.6359

10 1 4.30691

11 1.00191 4.53738

12 0.99993 0.34617


From To



1 2 -54.473 348.204 54.4728 -339.84

2 3 -23.135 -7.9198 23.135 8.01185

2 4 -31.338 28.6743 31.4056 -28.6

3 5 -10.335 -1.8119 10.3485 1.83165

4 5 -6.1515 -6.1337 6.15152 6.16835

12 6 -9 6.8 9 -6.8

4 7 -16.254 12.5108 16.5113 -12.169

7 8 -16.512 50.0855 18.2386 -49.483

8 9 -2.5995 -0.8946 2.6 0.9

8 10 -15.638 11.6038 15.7932 -11.391

10 11 -15.793 -5.6256 15.8 5.7

177



Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21535

3 1.00216 0.47297

4 1 0.32494

5 1.00561 0.58625

6 1.0008 0.2984

7 1 1.51423

8 1 3.5071

9 1.00083 3.61026

10 1 4.28214

11 1.00191 4.51262

12 1.00561 0.58625


From To



1 2 -54.516 348.204 54.5162 -339.84

2 3 -29.259 -13.978 29.2592 14.14

2 4 -25.25 6.61413 25.276 -6.586

3 5 -16.459 -7.9401 16.5 8

12 5 239.72 337.23 -241.47 -338.02

4 6 -9 -6.7978 9 6.8

4 7 -16.276 13.3792 16.547 -13.019

7 8 -16.556 50.2294 18.2931 -49.623

8 9 -2.5995 -0.8946 2.6 0.9

8 10 -15.656 11.617 15.8112 -11.404

10 11 -15.793 -5.6256 15.8 5.7

178

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03


----

----

>M

ax

.Po

we

r M

ism

atc

h


SIMPLE MOD2

MOD2 WITH SSC

MOD2 WITH SVC-B

MOD2 WITH SVC-F

MOD2 WITH TCSC-P

MOD2 WITH TCSC-F

MOD2 WITH UPFC



Fig 5.7 Comparison of maximum power mismatch using 2 Step method for 11-bus Ill-Conditioned System

179

5.8. CASE STUDY WITH 3-STEP METHOD






Bus

No

Voltage

Magnitude

Voltage

Phase Angle

1 1 0

2 1.01139 1.54536

3 1.05712 2.43665

4 1.03562 2.49271

5 1 2.57593

6 1.037 9.77016

7 1.06322 8.98467

8 1.1 8.12085

9 0.943 14.3468

10 1.1 8.33321

11 1.01771 12.1034

12 1.06717 8.08853

13 1.04424 5.22269


From To



1 2 -326.269 -114.34 326.747 124.501

1 3 -498.743 -571.97 501.046 626.505

5 4 3.45E-09 -376.08 0.56575 389.477

4 3 -234.371 -155.67 -242.88 158.903

6 2 199.497 46.5907 -452.57 1.31694

6 7 109.629 -260.64 -126.59 268.868

8 3 73.9698 200.784 -899.61 -143.97

7 8 117.547 -259.82 -117.55 270.616

9 10 224.333 -730.66 -865.84 908.718

10 11 490.744 -619.64 -490.74 542.185

11 12 177.67 -309.11 -727.61 355.699

12 13 133.234 156.675 -631.44 -134.12

13 8 387.183 378.376 -387.18 -418.69

180



Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.00687 1.57128

3 1.03848 2.51344

4 1.02168 2.5598

5 1 2.61115

6 1.037 9.84332

7 1.06296 9.06708

8 1.1 8.21372

9 0.943 13.9724

10 1.1 7.95877

11 1.12459 11.3702

12 1.07426 8.0224

13 1 5.19763


From To



1 2 -327.679 -60.948 328.123 70.3903

1 3 -496.734 -374.84 498.283 411.513

5 4 2.46E-12 -228.89 0.20955 233.846

4 3 -113.986 -120.07 -117.24 122.045

6 2 267.099 56.8177 -390.47 -8.0424

6 7 112.464 -258.07 -120.61 266.127

8 3 286.567 288.598 -690.56 -224.04

7 8 116.094 -261.61 -116.09 272.483

9 10 362.166 -730.66 -678.29 908.718

10 11 490.744 165.703 -490.74 -198.96

11 12 280.128 329.579 -585.25 -289.7

12 13 259.434 533.516 -497.85 -478.2

13 8 385.853 656.509 -385.85 -743.49

181


SUCEPTANCE MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.00551 1.53373

3 1.05706 2.47072

4 1.0356 2.52669

5 1 2.60987

6 1 10.1056

7 1.04162 9.16321

8 1.1 8.23158

9 0.943 14.4575

10 1.1 8.44395

11 1.01771 12.2142

12 1.06717 8.19926

13 1.04424 5.33342


From To



1 2 -318.766 -45.561 319.18 54.3747

1 3 -505.303 -570.86 507.627 625.899

5 4 7.63E-14 -375.86 0.56509 389.24

4 3 -234.361 -155.44 -242.86 158.666

6 2 201.26 -21.51 -443.54 69.984

6 7 117.74 -394.55 -132.88 413.067

8 3 80.6198 201.349 -906.15 -143.19

7 8 124.197 -404.39 -124.2 429.128

9 10 224.333 -730.66 -865.84 908.718

10 11 490.744 -619.64 -490.74 542.185

11 12 177.67 -309.11 -727.61 355.699

12 13 133.234 156.675 -631.44 -134.12

13 8 387.183 378.376 -387.18 -418.69

182


ANGLE MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.01139 1.54539

3 1.05712 2.43675

4 1.03562 2.4928

5 1 2.57603

6 1.037 9.77037

7 1.06322 8.98492

8 1.1 8.12115

9 0.943 14.3574

10 1.1 8.34379

11 1 12.1212

12 1.06716 8.09352

13 1.04593 5.22803


From To



1 2 -326.276 -114.34 326.754 124.5

1 3 -498.761 -571.97 501.064 626.503

5 4 5.43E-10 -376.08 0.56575 389.476

4 3 -234.371 -155.67 -242.88 158.902

6 2 199.505 46.5913 -452.57 1.31845

6 7 109.622 -260.64 -126.58 268.868

8 3 73.9882 200.786 -899.63 -143.96

7 8 117.539 -259.82 -117.54 270.615

9 10 224.333 -730.66 -865.84 908.718

10 11 483.128 -749.27 -483.13 650.735

11 12 175.026 -409.67 -715.26 468.262

12 13 133.17 144.27 -632.19 -122.2

13 8 387.136 367.247 -387.14 -406.29

183




Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.00916 0.73359

3 1.05825 3.31837

4 1.08315 5.31751

5 1 5.50328

6 1.037 4.57289

7 1.10132 7.65436

8 1.1 7.65436

9 0.943 20.4973

10 1.1 14.4838

11 1.10101 10.9992

12 1.08421 7.5168

13 1.03917 4.73847

14 1.1018 7.6523


From To



1 2 -156.68 -99.373 156.818 102.299

1 3 -670.833 -568.01 673.924 641.183

5 4 -0.0011 -877.96 3.08437 950.958

4 3 160.982 178.536 -410.2 -164.48

6 2 91.9188 51.4603 -219.45 -39.667

6 7 -4.85583 4.85583 -4.8515 -4.8537

8 3 172.278 191.278 -585.13 -155.3

7 8 115.52 261.59 -115.52 -272.45

9 10 362.159 -730.66 -678.29 908.717

14 11 490.741 89.49218 -490.74 -70.993

11 12 286.788 101.584 -588.58 -73.486

12 13 257.632 322.434 -508.61 -290.59

13 8 387.66 411.537 -387.66 -455.94

184


ANGLE MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.00894 0.8359

3 1.04412 3.30122

4 1.0396 5.35546

5 1 5.44764

6 1.037 5.21403

7 1.10132 7.65759

8 1.1 7.6576

9 0.943 13.8028

10 1.1 7.78921

11 1.04853 11.4484

12 1.08454 7.64665

13 1.08624 4.86697

14 1.1018 7.65553


From To



1 2 -177.661 -95.541 177.824 98.9997

1 3 -652.655 -420.08 655.065 477.127

5 4 -0.0002 -418.11 0.69948 434.669

4 3 152.117 -42.007 -388.24 51.8803

6 2 113.246 51.2508 -240.43 -36.395

6 7 350.46 -93.2 -357.84 79.70

8 3 172.445 257.891 -579.71 -216.12

14 8 353.63 -83.90 -353.63 164.69

9 10 362.164 -730.66 -678.29 908.717

10 11 490.742 -393.09 -490.74 344.084

11 12 301.116 -234.46 -588.17 272.2

12 13 257.056 -23.084 -519.98 41.9777

13 8 387.826 90.1738 -387.83 -110.33

185



Bus No

Voltage Magnitude

Voltage Phase Angle

1 1 0

2 1.00472 2.12202

3 1.04087 1.91731

4 1.022 1.96305

5 1 2.01593

6 1.037 13.7492

7 1 13.5126

8 1.1 6.2974

9 0.943 12.4378

10 1.1 6.4086

11 1.04857 10.0854

12 1.08453 6.264

13 1.0862 3.4868

14 1.1 6.27503

Table 5.98: Complex Power flow through Lines

From To



1 2 -438.9 26.8 439.7 -43.3

1 3 -383.6 392.5 384.8 -421.0

5 4 0 232.3 0.2 -237.4

4 3 -121.682 19.32 -113.89 -19.3722

6 2 382.8 -66.5279 -501.8 -18.8

6 7 -3.2 -355.3 -4 342.6

8 3 172.1 -280.7 -577.4 236.6

14 8 1773.7 1115.7 -1729.1 -1462.9

9 10 361.781 -730.68 -677.91 908.657

10 11 490.446 -392.83 -490.45 343.878

11 12 300.81 -234.24 -587.88 271.928

12 13 256.764 -22.813 -519.68 41.6765

13 8 387.534 90.4656 -387.53 -110.6

186

1 2 3 4 5 6 7 8 9 10

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08


----

----

>M

ax

.Po

we

r M

ism

atc

h


SIMPLE MOD3

MOD3 WITH SSC

MOD3 WITH SVC-B

MOD3 WITH SVC-F

MOD3 WITH TCSC-P

MOD3 WITH TCSC-F

MOD3 WITH UPFC



Fig 5.8 Comparison of maximum power mismatch using 3 Step method for 13-bus Ill-Conditioned System

187



Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21518

3 1.00123 0.41906

4 1 0.35127

5 1.00282 0.5127

6 1.00012 0.36034

7 1 1.54546

8 1 3.53246

9 1.00083 3.63562

10 1 4.30659

11 1.00191 4.53707


From To



1 2 -54.472 348.204 54.4715 -339.84

2 3 -23.135 -7.9198 23.1349 8.01187

2 4 -31.337 28.6734 31.4046 -28.599

3 5 -10.335 -1.8119 10.3484 1.83167

4 5 -6.1516 -6.1337 6.15162 6.16833

4 6 -9 -6.7978 9 6.8

4 7 -16.253 12.5099 16.5102 -12.169

7 8 -16.51 50.0806 18.2367 -49.478

8 9 -2.5995 -0.8946 2.6 0.9

8 10 -15.637 11.6033 15.7926 -11.391

10 11 -15.793 -5.6256 15.8 5.7

188



Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21518

3 1.00123 0.41906

4 1 0.35127

5 1.00282 0.5127

6 1.00012 0.36034

7 1 1.54546

8 1 3.53246

9 1.00083 3.63562

10 1 4.30659

11 1.00191 4.53707


From To



1 2 -54.472 348.204 54.4716 -339.84

2 3 -23.135 -7.9198 23.1349 8.01187

2 4 -31.337 28.6734 31.4047 -28.599

3 5 -10.335 -1.8119 10.3484 1.83167

4 5 -6.1516 -6.1337 6.15161 6.16833

4 6 -9 -6.7978 9 6.8

4 7 -16.253 12.5099 16.5102 -12.168

7 8 -16.51 50.0806 18.2367 -49.478

8 9 -2.5995 -0.8946 2.6 0.9

8 10 -15.637 11.6033 15.7926 -11.391

10 11 -15.793 -5.6256 15.8 5.7

189


SUSCEPTANCE MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21518

3 1.00123 0.41906

4 1 0.35127

5 1.00282 0.5127

6 1 0.63097

7 1 1.54546

8 1 3.53246

9 1.00083 3.63562

10 1 4.30659

11 1.00191 4.53707


From To



1 2 -54.472 348.204 54.4715 -339.84

2 3 -23.135 -7.9198 23.1349 8.01187

2 4 -31.337 28.6734 31.4046 -28.599

3 5 -10.335 -1.8119 10.3484 1.83167

4 5 -6.1516 -6.1337 6.15162 6.16833

4 6 -277.37 0.67703 277.371 0.67703

4 7 -16.253 12.5099 16.5102 -12.168

7 8 -16.51 50.0806 18.2367 -49.478

8 9 -2.5995 -0.8946 2.6 0.9

8 10 -15.637 11.6033 15.7926 -11.391

10 11 -15.793 -5.6256 15.8 5.7

190


ANGLE MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21517

3 1.00123 0.41905

4 1 0.35126

5 1.00282 0.51269

6 1.00012 0.36033

7 1 1.54535

8 1 3.53217

9 1.00083 3.63533

10 1 4.30622

11 1 4.56122


From To



1 2 -54.47 348.204 54.4701 -339.84

2 3 -23.135 -7.9198 23.1348 8.01189

2 4 -31.335 28.6722 31.4034 -28.598

3 5 -10.335 -1.8119 10.3483 1.83169

4 5 -6.1517 -6.1337 6.15174 6.16831

4 6 -9 -6.7978 9 6.8

4 7 -16.252 12.5088 16.5088 -12.167

7 8 -16.509 50.076 18.235 -49.474

8 9 -2.5995 -0.8946 2.6 0.9

8 10 -15.636 11.602 15.7908 -11.39

10 11 -16.651 1.69871 16.6579 -1.6246

191




Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21518

3 1.00123 0.41906

4 1 0.35127

5 1.00282 0.5127

6 0.99805 0.20358

7 1 1.54545

8 1 3.53242

9 1.00083 3.63557

10 1 4.30654

11 1.00191 4.53702

12 0.99805 0.20358


From To



1 2 -54.4717 348.204 54.4717 -339.841

2 3 -23.1349 -7.91978 23.1349 8.01186

2 4 -31.3368 28.6736 31.4048 -28.5991

3 5 -10.3349 -1.81186 10.3484 1.83167

4 5 -6.1516 -6.13367 6.1516 6.16833

12 6 -71.68 54.44 71.68 -54.15

4 7 -16.2528 12.5097 16.51 -12.1683

7 8 -16.5099 50.0797 18.2364 -49.4772

8 9 -2.59946 -0.894575 2.6 0.9

8 10 -15.6371 11.6032 15.7925 -11.3909

10 11 -15.7926 -5.62563 15.8 5.7

192


ANGLE MODEL


Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21518

3 1.00123 0.41906

4 1 0.35127

5 1.00282 0.5127

6 0.99993 0.34616

7 1 1.54544

8 1 3.53241

9 1.00083 3.63557

10 1 4.30654

11 1.00191 4.53701

12 0.99993 0.34616


From To



1 2 -54.4713 348.204 54.4713 -339.841

2 3 -23.1349 -7.91979 23.1349 8.01187

2 4 -31.3364 28.6732 31.4044 -28.5987

3 5 -10.3349 -1.81187 10.3484 1.83167

4 5 -6.15164 -6.13367 6.15164 6.16833

12 6 -9 6.8 9 -6.8

4 7 -16.2528 12.5097 16.51 -12.1683

7 8 -16.5099 50.0797 18.2364 -49.4771

8 9 -2.59946 -0.89457 2.6 0.9

8 10 -15.6371 11.6032 15.7925 -11.3909

10 11 -15.7926 -5.62563 15.8 5.7

193



Bus No

Voltage Magnitude

Voltage Phase Angle

1 1.024 0

2 1 0.21516

3 1.00216 0.47277

4 1 0.32456

5 1.00561 0.58606

6 1.00012 0.29803

7 1 1.51076

8 1 3.49667

9 1.00083 3.59983

10 1 4.27064

11 1.00191 4.50111

12 1.00561 0.58606


From To



1 2 -54.4666 348.204 54.4666 -339.841

2 3 -29.2592 -13.9781 29.2592 14.14

2 4 -25.2086 6.57562 25.2342 -6.54758

3 5 -16.4592 -7.94005 16.5 8

12 5 239.68 337.23 -242.43 -338.03

4 6 -9 -6.79776 9 6.8

4 7 -16.2342 13.3463 16.5039 -12.9883

7 8 -16.5016 50.0529 18.2262 -49.451

8 9 -2.59946 -0.894575 2.6 0.9

8 10 -15.6339 11.6008 15.7891 -11.3885

10 11 -15.7926 -5.62563 15.8 5.7

194

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

0

2

4

6

8

10

12

14x 10

-4


----

----

>M

ax

.Po

we

r M

ism

atc

h


SIMPLE MOD3

MOD3 WITH SSC

MOD3 WITH SVC-B

MOD3 WITH SVC-F

MOD3 WITH TCSC-P

MOD3 WITH TCSC-F

MOD3 WITH UPFC



Fig 5.9 Comparison of maximum power mismatch using 3 Step


From the Newton‘s accelerated convergence method solution

with the incorporation of FACTS devices for 13 and 11-bus Ill-

Conditioned systems, it is observed from the simulation studies that

the converging behavior is oscillatory in both cases for two step and

three step algorithms, and the number of iterations is less compared

to runge-kutta and iwamoto optimal multiplier methods.

195

5.9 DISCUSSION OF RESULTS

The 11 and 13 bus ill-conditioned systems developed by

Japanese researchers which shows high degree of ill conditionality,

fails to converge with conventional Newton-Raphson method. The load

flow solution is obtained with RK method, optimal multiplier method

and Newton‘s modified methods such as two step and three step

algorithms. The above systems are also tested with the incorporation

of FACTS devices on load flow solution and convergence behavior.

RK method takes more number of iterations for convergence

compared to optimal multiplier method and modified Newton‘s

methods. In case of RK method, the number of iterations for

convergence remains almost same with shunt devices, but reactive

power flows through the lines is affected significantly. It is also

observed that SVC models improve maximum system voltage and

reduce voltage phase angles at all the buses as compared to

STATCOM.

TCSC power flow model takes more number of iterations

compared to any other FACTS devices. It is also observed that real

power flows through the lines are significantly affected. TCSC

increases the bus voltages between which it is connected and also

reduces corresponding voltage phase angles. In case of firing angle

model, the voltage phase angles at all the buses are reduced.

UPFC model takes less number of iterations compared to other

FACTS devices for convergence. It improves maximum system voltage

and reduces voltage phase angles at all buses. Similar observations

are noticed in case of 11 bus ill-conditioned systems for all devices

with less number of iterations as compared to 13 bus ill-conditioned

system.

Iwamoto method takes less number of iterations compared to RK

method for all devices. The shunt devices increase voltage phase

angles almost at all buses. Voltage magnitudes remains same for the

196

shunt devices. TCSC power flow model increases maximum system

voltage compared to all other FACTS devices. TCSC power flow model

shows considerable changes in real power flows as compared to shunt

devices. TCSC firing angle model reduces voltage phase angles at all

buses. It provides significant changes in real power flows as compared

to power flow model. UPFC increases maximum system voltage as

compared to the shunt devices and significant changes are observed

in case of real and reactive powers. Similar observations are noticed

with 11 bus system but number of iterations is reduced as compared

to 13 bus ill-conditioned system.

The Newton‘s modified methods take less number of

iterations compared to above two methods. The reactive power flow

changes through the lines are significant in case of the shunt devices

as compared to without device. Voltage phase angles as well as the

magnitudes are same for shunt connected devices. TCSC power flow

model shows significant changes in real power flows, voltage

magnitudes and slight increase in voltage phase angles at almost all

buses. Incase of TCSC firing angle model, the phase angles are

decreased at all buses. TCSC firing angle method takes more number

of iterations compared to other FACTS devices. In case of UPFC, the

maximum system voltage increases and shows significant changes in

real and reactive power flows.

5.10 CONCLUSIONS

Load flow solution is obtained for 11 and 13 bus ill-conditioned

systems with RK method, optimal multiplier method and modified

Newton‘s method even with the incorporation of FACTS devices. From

the results it can be concluded that RK method takes more number of

iterations to converge as compared to other methods. Newton‘s

modified method takes less number of iterations. In all the above

discussed methods, shunt devices show significant changes in

reactive power flows through the lines and almost same bus voltages.

197

The Series device TCSC shows significant changes in real power flows

with considerable changes in voltage phase angles.

In all the above discussed methods, TCSC models take

more no. of iterations for convergence compared to other FACTS

devices and UPFC improves maximum system voltage with significant

changes in real and reactive power flows.

198

CONCLUSIONS

Electrical power system deals with transmission and

distribution of electrical energy, associated with the unique feature of

control of the flow or demand of energy at desired nodes through out

the power network. Power flow techniques provide the basic

calculation procedure in order to determine the characteristics of

power system under steady state operating mode. Power flow studies

are usually conducted for planning purpose, or to obtain system

behavior in order to predict the loading of lines and equipment.

Increased use of transmission facilities due to higher industrial

output, power industry has powered the momentum for exploring the

new ways of maximizing power transfers in existing transmission

facilities and FACTS concepts have provided the answer for this

option.

The power flow analysis is one of the most important problems in

power system studies. Power flow studies are required for the

estimation of steady state conditions of the system. The power flow

solutions can be carried out either through Gauss Siedel method or

Newton-Raphson method. The Guass-Seidal method is quite easy to

implement but programming aspects of Newton-Raphson method are

a little bit complex. The convergence of Newton-Raphson method is

quadratic and the number of iterations is almost independent of

system size. The conventional Newton-Raphson method fails to

converge in the case of Ill-Conditioned systems. Non linear

199

programming techniques are needed for the solution of Ill-Conditioned

systems.

The work reported in this thesis is aimed at obtaining the power

flow solutions for Ill-Conditioned systems embedded with FACTS

devices such as STATCOM, SVC, TCSC, and UPFC. The steady state

mathematical models of the STATCOM, SVC, TCSC, and UPFC are

developed in Chapter 3. The same models are utilized for case studies

in Chapter 4. The IEEE 14 bus test system is considered for study

and the Ill-Condition is created by increasing the normal line

resistance to 4 times from the base case. The conventional Newton-

Raphson method fails to converge when the line resistance is

increased beyond 3.5 times with out any FACTS devices.

The same study is carried out with the above mentioned FACTS

devices. The critical analysis of the results has revealed that the

number of iterations for convergence is increased with the

incorporation of FACTS devices due to the additional nonlinearities

due to presence of FACTS devices.

In Chapter 5 the studies are carried out on 11-bus and 13-bus Ill-

Conditioned systems which exhibit high degree of Ill-Conditionality. It

has not been reported in the literature about the effect of FACTS

devices on the said Ill-Conditioned systems.

Generally the Runge-Kutta method is used to obtain the

numerical solution of non linear differential equations. The power flow

Jacobian also comprises of partial derivatives, so it is appropriate to

use Runge-kutta method for obtaining the solution for power flow

200

problem. Along with this method, Iwamoto‘s optimal multiplier method

and Newton‘s accelerated convergence methods are utilized to obtain

the power flow solution.

The solution results indicate that the convergence behavior of RK

fourth order method and Iwamoto‘s optimal multiplier method is the

same for both the systems. The number of iterations for convergence

is around 16 with out devices. The convergence behavior of RK fourth

order method with the incorporation of FACTS is becoming slow for

both the 11 and 13-bus systems, where as the Iwamoto‘s Optimal

Multiplier method provides quicker convergence. Usually FACTS

device models increase the nonlinearities of the system and this may

be the reason for slower convergence of RK fourth order method. The

Iwamoto‘s Optimal Multiplier method provides quicker solution under

high nonlinear conditions.

General observations noticed in the simulation studies are

a) the degree of non linearity of the system increases with the

incorporation of FACTS device and is reflected in the speed of

convergence, whereas the convergence is quicker without FACTS

devices. b) Choosing of initial values of FACTS devices has significant

effect on the accuracy as well convergence. c) Different initial values

may lead to multiple load flow solutions and some of them may lead to

divergence. All the solutions may not be practically operational.

201

SCOPE OF FUTURE WORK: The details of mathematical modeling of

FACTS devices can be improved by considering their actual operating

behavior and these models may be integrated into the power flow

methods. The different control strategies of FACTS devices can be

incorporated for examining their impact on the converging behavior

and accuracy.

power flow solution methods for ill- conditioned...

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