Upload
others
View
23
Download
0
Embed Size (px)
Citation preview
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
SVC is connected to bus No : 13
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
Target nodal voltage magnitude to be controlled by SVC (p.u.)=1
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
Initial Values for 13-bus Ill-Conditioned system with TCSC
Variable Firing Angle Power Flow Model
TCSC is connected between bus-7 and bus-8
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
Power flow direction is taken from receiving end to sending end
Target active power flow (p.u.)=0.4
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
Target nodal voltage magnitude to be controlled by SVC (p.u.)=1
116
Initial Values for 11-bus Ill-Conditioned system with SVC Firing
Angle Model
SVC is connected to bus No : 11
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
Target nodal voltage magnitude to be controlled by SVC (p.u.)=1
Initial Values for 11-bus Ill-Conditioned system with TCSC
Variable Impedance Power Flow Model
TCSC is connected between bus-4 and bus-6
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
Initial Values for 11-bus Ill-Conditioned system with TCSC
Variable Firing Angle Power Flow Model
TCSC is connected between bus-4 and bus-6
Capacitive reactance of TCSC (p.u.)=9.375e-3
Inductive reactance of TCSC (p.u)=1.625e-2
Power flow direction is taken from receiving end to sending end
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
Power flow direction is taken from receiving end to sending end
Target active power flow (p.u.)=0.4
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
Table 5.3: Voltage Magnitudes and Phase angles
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
Table 5.4: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
Table 5.5: Voltage Magnitudes and Phase angles
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
Table 5.6: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
Table 5.7: Voltage Magnitudes and Phase angles
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
Table 5.8: Complex Power Flows through Lines
From To Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
Table 5.9: Voltage Magnitudes and Phase angles
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
Table 5.10: Complex Power Flows through Lines
From To Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
Table 5.11: Voltage Magnitudes and Phase angles
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
Table 5.12: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
Table 5.13: Voltage Magnitudes and Phase angles
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
Table 5.14: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.5.9 11-BUS ILL-CONDITIONED SYSTEM WITH OUT ANY DEVICE
Table 5.15: Voltage Magnitudes and Phase angles
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
Table 5.16: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.5.10 11-BUS ILL-CONDITIONED SYSTEM WITH STATCOM
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
Table 5.17: Voltage Magnitudes and Phase angles
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
Table 5.18: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
Table 5.19: Voltage Magnitudes and Phase angles
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
Table 5.20: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
Table 5.21: Voltage Magnitudes and Phase angles
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
Table 5.22: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.5.13 11-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-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
Table 5.23: Voltage Magnitudes and Phase angles
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
Table 5.24: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.5.14 11-BUS ILL-CONDITIONED SYSTEM WITH TCSC FIRING
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
Table 5.25: Voltage Magnitudes and Phase angles
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
Table 5.26: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.5.15 11-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 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
Table 5.27: Voltage Magnitudes and Phase angles
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
Table 5.28: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.5.16 COMPARISON OF MAXIMUM POWER MISMATCH FOR 11-
BUS ILL-CONDITIONED SYSTEM
Fig 5.3 Comparison of maximum power mismatch using Runge-Kutta
method for 11-bus Ill-Conditioned System
146
5.6. CASE STUDY WITH OPTIMAL MULTIPLIER METHOD
In this section case study is presented for 13- and 11- Bus Ill-
Conditioned systems with and without devices. The initial data for the
devices is presented in section 5.3
5.6.1 13-BUS ILL-CONDITIONED SYSTEM WITH OUT ANY DEVICE
Table 5.29: Voltage Magnitudes and Phase angles
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
Table 5.30: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.6.2 13-BUS ILL-CONDITIONED SYSTEM WITH STATCOM
Table 5.31: Voltage Magnitudes and Phase angles
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
Table 5.32: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.6.3 13-BUS ILL-CONDITIONED SYSTEM WITH SVC-
SUCEPTANCE MODEL
Table 5.33: Voltage Magnitudes and Phase angles
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
Table 5.34: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.6.4 13-BUS ILL-CONDITIONED SYSTEM WITH SVC-FIRING
ANGLE MODEL
Table 5.35: Voltage Magnitudes and Phase angles
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
Table 5.36: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.6.5 13-BUS ILL-CONDITIONED SYSTEM WITH TCSC VARIABLE
IMPEDANCE POWER FLOW MODEL
Table 5.37: Voltage Magnitudes and Phase angles
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
Table 5.38: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.6.6 13-BUS ILL-CONDITIONED SYSTEM WITH TCSC FIRING
ANGLE MODEL
Table 5.39: Voltage Magnitudes and Phase angles
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
Table 5.40: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.6.7 13-BUS ILL-CONDITIONED SYSTEM WITH UPFC
Table 5.41: Voltage Magnitudes and Phase angles
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
Table 5.42: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
Table 5.43: Voltage Magnitudes and Phase angles
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
Table 5.44: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.6.10 11-BUS ILL-CONDITIONED SYSTEM WITH STATCOM
Table 5.45: Voltage Magnitudes and Phase angles
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
Table 5.46: Complex Power Flows through Lines
From To
Sendng end Power Recevng end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.6.11 11-BUS ILL-CONDITIONED SYSTEM WITH SVC
SUSCEPTANCE MODEL
Table 5.47: Voltage Magnitudes and Phase angles
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
Table 5.48: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.6.12 11-BUS ILL-CONDITIONED SYSTEM WITH SVC FIRING
ANGLE MODEL
Table 5.49: Voltage Magnitudes and Phase angles
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
Table 5.50: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.6.13 11-BUS ILL-CONDITIONED SYSTEM WITH TCSC VARIABLE
IMPEDANCE POWER FLOW MODEL
Table 5.51: Voltage Magnitudes and Phase angles
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
Table 5.52: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.6.14 11-BUS ILL-CONDITIONED SYSTEM WITH TCSC FIRING
ANGLE MODEL
Table 5.53: Voltage Magnitudes and Phase angles
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
Table 5.54: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
Table 5.55: Voltage Magnitudes and Phase angles
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
Table 5.56: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
Comparison of Maximum Power Mismatch
Without Device
With STATCOM
With SVC-B
With SVC-FA
With TCSC-POWER
With TCSC-FA
With UPFC
5.6.16 COMPARISON OF MAXIMUM POWER MISMATCH FOR 11-
BUS ILL-CONDITIONED SYSTEM
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
In this section case study is presented for 13- and 11- Bus Ill-
Conditioned systems with and without devices. The initial data for the
devices is presented in section 5.3
5.7.1 13-BUS ILL-CONDITIONED SYSTEM WITH OUT ANY DEVICE
Table 5.57: 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 9.77016
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
Table 5.58: 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 -114.34 326.747 124.501
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
5.7.2 13-BUS ILL-CONDITIONED SYSTEM WITH STATCOM
Table 5.59: Voltage Magnitudes and Phase angles
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
Table 5.60: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
Table 5.61: Voltage Magnitudes and Phase angles
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
Table 5.62: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.7.4 13-BUS ILL-CONDITIONED SYSTEM WITH SVC-FIRING
ANGLE MODEL
Table 5.63: Voltage Magnitudes and Phase angles
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
Table 5.64: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.7.5 13-BUS ILL-CONDITIONED SYSTEM WITH TCSC VARIABLE
IMPEDANCE POWER FLOW MODEL
Table 5.65: Voltage Magnitudes and Phase angles
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
Table 5.66: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.7.6 13-BUS ILL-CONDITIONED SYSTEM WITH TCSC FIRING
ANGLE MODEL
Table 5.67: Voltage Magnitudes and Phase angles
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
Table 5.68: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.7.7 13-BUS ILL-CONDITIONED SYSTEM WITH UPFC
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
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
Comparison of Maximum Power Mismatch
SIMPLE MOD2
MOD2 WITH SSC
MOD2 WITH SVC-B
MOD2 WITH SVC-F
MOD2 WITH TCSC-P
MOD2 WITH TCSC-F
MOD2 WITH UPFC
5.7.8 COMPARISON OF MAXIMUM POWER MISMATCH FOR 13-
bus ILL-CONDITIONED SYSTEM
Fig 5.6 Comparison of maximum power mismatch using 2 Step
method for 13-bus Ill-Conditioned System
171
5.7.9 11-BUS ILL-CONDITIONED SYSTEM WITHOUT ANY DEVICE
Table 5.71: Voltage Magnitudes and Phase angles
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
Table 5.72: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.7.10 11-BUS ILL-CONDITIONED SYSTEM WITH STATCOM
Table 5.73: Voltage Magnitudes and Phase angles
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
Table 5.74: Complex Power Flows through Lines
From To
Sendng end Power Recevng end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.7.11 11-BUS ILL-CONDITIONED SYSTEM WITH SVC
SUSCEPTANCE MODEL
Table 5.75: Voltage Magnitudes and Phase angles
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
Table 5.76: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.7.12 11-BUS ILL-CONDITIONED SYSTEM WITH SVC FIRING
ANGLE MODEL
Table 5.77: Voltage Magnitudes and Phase angles
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
Table 5.78: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.7.13 11-BUS ILL-CONDITIONED SYSTEM WITH TCSC VARIABLE
IMPEDANCE POWER FLOW MODEL
Table 5.79: Voltage Magnitudes and Phase angles
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
Table 5.80: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.7.14 11-BUS ILL-CONDITIONED SYSTEM WITH TCSC FIRING
ANGLE MODEL
Table 5.81: Voltage Magnitudes and Phase angles
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
Table 5.82: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.7.15 11-BUS ILL-CONDITIONED SYSTEM WITH UPFC MODEL
Table 5.83: Voltage Magnitudes and Phase angles
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
Table 5.84: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
-------->No.of iterations
----
----
>M
ax
.Po
we
r M
ism
atc
h
Comparison of Maximum Power Mismatch
SIMPLE MOD2
MOD2 WITH SSC
MOD2 WITH SVC-B
MOD2 WITH SVC-F
MOD2 WITH TCSC-P
MOD2 WITH TCSC-F
MOD2 WITH UPFC
5.7.16 COMPARISON OF MAXIMUM POWER MISMATCH FOR 11-
BUS ILL-CONDITIONED SYSTEM
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
In this section case study is presented for 13- and 11- Bus Ill-
Conditioned systems with and without devices. The initial data for the
devices is presented in section 5.3
5.8.1 13-BUS ILL-CONDITIONED SYSTEM WITH OUT ANY DEVICE
Table 5.85: 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 9.77016
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
Table 5.86: 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 -114.34 326.747 124.501
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
5.8.2 13-BUS ILL-CONDITIONED SYSTEM WITH STATCOM
Table 5.87: Voltage Magnitudes and Phase angles
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
Table 5.88: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.8.3 13-BUS ILL-CONDITIONED SYSTEM WITH SVC-
SUCEPTANCE MODEL
Table 5.89: Voltage Magnitudes and Phase angles
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
Table 5.90: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.8.4 13-BUS ILL-CONDITIONED SYSTEM WITH SVC-FIRING
ANGLE MODEL
Table 5.91: Voltage Magnitudes and Phase angles
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
Table 5.92: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.8.5 13-BUS ILL-CONDITIONED SYSTEM WITH TCSC VARIABLE
IMPEDANCE POWER FLOW MODEL
Table 5.93: Voltage Magnitudes and Phase angles
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
Table 5.94: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.8.6 13-BUS ILL-CONDITIONED SYSTEM WITH TCSC FIRING
ANGLE MODEL
Table 5.95: Voltage Magnitudes and Phase angles
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
Table 5.96: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.8.7 13-BUS ILL-CONDITIONED SYSTEM WITH UPFC
Table 5.97: Voltage Magnitudes and Phase angles
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
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
-------->No.of iterations
----
----
>M
ax
.Po
we
r M
ism
atc
h
Comparison of Maximum Power Mismatch
SIMPLE MOD3
MOD3 WITH SSC
MOD3 WITH SVC-B
MOD3 WITH SVC-F
MOD3 WITH TCSC-P
MOD3 WITH TCSC-F
MOD3 WITH UPFC
5.8.8 COMPARISON OF MAXIMUM POWER MISMATCH FOR 13-
BUS ILL-CONDITIONED SYSTEM
Fig 5.8 Comparison of maximum power mismatch using 3 Step method for 13-bus Ill-Conditioned System
187
5.8.9 11-BUS ILL-CONDITIONED SYSTEM WITHOUT ANY DEVICE
Table 5.99: Voltage Magnitudes and Phase angles
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
Table 5.100: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.8.10 11-BUS ILL-CONDITIONED SYSTEM WITH STATCOM
Table 5.101: Voltage Magnitudes and Phase angles
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
Table 5.102: Complex Power Flows through Lines
From To
Sendng end Power Recevng end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.8.11 11-BUS ILL-CONDITIONED SYSTEM WITH SVC
SUSCEPTANCE MODEL
Table 5.103: Voltage Magnitudes and Phase angles
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
Table 5.104: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.8.12 11-BUS ILL-CONDITIONED SYSTEM WITH SVC FIRING
ANGLE MODEL
Table 5.105: Voltage Magnitudes and Phase angles
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
Table 5.106: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.8.13 11-BUS ILL-CONDITIONED SYSTEM WITH TCSC VARIABLE
IMPEDANCE POWER FLOW MODEL
Table 5.107: Voltage Magnitudes and Phase angles
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
Table 5.108: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.8.14 11-BUS ILL-CONDITIONED SYSTEM WITH TCSC FIRING
ANGLE MODEL
Table 5.109: Voltage Magnitudes and Phase angles
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
Table 5.110: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
5.8.15 11-BUS ILL-CONDITIONED SYSTEM WITH UPFC MODEL
Table 5.111: Voltage Magnitudes and Phase angles
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
Table 5.112: Complex Power Flows through Lines
From To
Sending end Power Receiving end Power
Real(MW) Reactive(MVAR) Real(MW) Reactive(MVAR)
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
-------->No.of iterations
----
----
>M
ax
.Po
we
r M
ism
atc
h
Comparison of Maximum Power Mismatch
SIMPLE MOD3
MOD3 WITH SSC
MOD3 WITH SVC-B
MOD3 WITH SVC-F
MOD3 WITH TCSC-P
MOD3 WITH TCSC-F
MOD3 WITH UPFC
5.8.16 COMPARISON OF MAXIMUM POWER MISMATCH FOR 11-
BUS ILL-CONDITIONED SYSTEM
Fig 5.9 Comparison of maximum power mismatch using 3 Step
method for 11-bus Ill-Conditioned System
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.