Download VIII. Constraints For Interior Point Method

Performance Evaluation of Optimal Power Flow
by Using Interior Point Method
Wah Wah Lwin, Pyone Lai Swe, Zaw Htet Myint

Abstract— Optimal power flow, which is characterized as a
difficult optimization problem, involves the optimization of an
objective function that can take various forms, for example,
minimization of total production cost, and minimization of total
loss in transmission networks, subject to a set of physical and
operating constraints such as generation and load balance, bus
voltage limits, power flow equations, and active and reactive
power limits . The application of interior point method by using
matlab interior point solver (MIPS) in matpower to optimal
power flow can also reduce the power losses considerably and
can improve the voltage profile of the power systems. The IEEE
14 bus system is used to compare the performance of the
mentioned method and power system optimization techniques.
For the implementation of the system Matpower4.1 software
package and Matlab optimization toolbox packages are
employed.
Index Terms— Optimal Power Flow, Matlab Interior Point
Solver (MIPS), Karush-Kuhn-Tucker (KKT) conditions, IEEE
test system.
I. INTRODUCTION
The goal of OPF is to find the optimal settings of a given
power system network that optimize the system objective
functions such as total generation cost, system loss, bus
voltage deviation, emission of generating units, number of
control actions, and load shedding while satisfying its power
flow equations, system security, and equipment operating
limits. Different control variables, some of which are
generators’ real power outputs and voltages, transformer tap
changing settings, phase shifters, switched capacitors, and
reactors, are manipulated to achieve an optimal network
setting based on the problem formulation.
According to the selected objective functions, and
constraints, there are different mathematical formulations for
the OPF problem. They can be broadly classified as follows
(1) Linear problem in which objectives and constraints are
given in linear forms with continuous control variables.
(2) Nonlinear problem where either objectives or constraints
or both combined are nonlinear with continuous control
variables.
(3) Mixed - integer linear problems when control variables are
both discrete and continuous.
Wah Wah lwin, Department of Electrical Power Engineering,
Mandalay Technological University (e-mail: [email protected]).
Mandalay , Myanmar , Phone/ Mobile No:+9533224233
Pyone Lai Swe, Department of Electrical Power Engineering, Mandalay
Technological University, Mandalay , Myanmar, Phone/ Mobile
No.+95402590310
Zaw Htet Myint, , Department of Electrical Power Engineering,
Mandalay Technological University, Mandalay , Myanmar, Phone/ Mobile
No+95402679802.
Various techniques were developed to solve the OPF
problem. The algorithms may be classified into two groups:
(1) conventional optimization methods, (2) intelligence
search methods.
II. SOLUTION METHODS OF OPF
The problem of Optimal Power Flow is the allocation of
given load amongst the generating units in operation so that
the overall cost of generation is minimum. In OPF, the entire
set of equality and inequality constraints, all the necessary and
sufficient conditions of control parameters etc. must be
satisfied thoroughly. The OPF is important for operating and
planning engineers. The objective function can take various
forms such as fuel cost, transmission losses, and reactive
sources allocation. The objective function of interest in this
paper is designated as the minimization of the total production
cost of scheduled generating units. Such a minimization
problem is most used as it reflects current economic dispatch
practice and importantly, cost related aspect is always ranked
high among operational requirement in power systems.
Various techniques have been proposed to solve the OPF
problem for example, non-linear programming, quadratic
programming , linear programming, and interior point
methods Among these, the interior point method has been of
recent interest and will be employed as a major tool to solve
the OPF problem in this paper .[3]
III. INTERIOR POINT METHODS
The Interior Point Methods (IPM’s) have been used in the
past two decades to solve OPF problems and many
investigations and algorithms have been developed in this
sense. Initially, the IPM’s were used in the form of barrier
methods and later thanks to the multiple investigations
developed in this area such as the ellipsoid method and the
projective scaling method. These algorithms have grown in
popularity due to their performance in big dimensions
problems.
The interior point (IP) technique was purposed by
N.K.Karmarkar. It can solve a large-scale linear programming
problem by moving through the interior, rather than the
boundary as in the simplex method, of the feasible region to
find an optimal solution. The IP method was originally
proposed to solve linear programming problems; however,
later it was implemented to efficiently handle quadratic
programming problems.[4]
The interior point technique starts by determining an initial
solution using Mehrotra’s algorithm, which is used to locate a
feasible or near-feasible solution. There are then two
procedures to be performed in an iterative manner until the
1
optimal solution has been found. The former is the
determination of a search direction for each variable in the
search space by a Newton’s method. The latter is the
determination of a step length normally assigned a value as
close to unity as possible to accelerate solution convergence
while strictly maintaining primal and dual feasibility. A
calculated solution in each iteration will be checked for
optimality by the Karush-Kuhn-Tucker (KKT) conditions,
which consist of primal feasibility, dual feasibility and
complementary slackness.[5]
Currently the IPM’s are being applied extensively to Power
Systems Optimization due to their mathematical features over
other methods, the Primal Dual IPM is one of the IPM’s has
been applied to solve optimization problems in Power
Systems.[6]-[7]
IV. CONCEPT OF OPTIMAL POWER FLOW
The main objective of an OPF problem is to determine the
optimal setting of control variables in a power system network
to optimize an objective function while respecting a set of
physical and operating constraints such as generation and load
balance, bus voltage limits, power flow equations, and active
and reactive power limits. Generally, an OPF Problem can be
formulated as
NG
 (P
i 1
Gi
N
N
Pi (V ,  ) Vi ViYij cos( i   j   ij )
N
Qi (V ,  ) Vi ViYij sin(  i   j   ij )
(9)
j 1
Nonlinear equations (8) and (9) can be linearized by the
Taylor’s expansion using
P(V ,  )   J 11 J 12   
Q(V ,  )   J J  V 

  21 22  

(10)
Transmission loss ( PL ) given in the Equation (7) can be
directly calculated from the power flow.
C. Inequality Constraints
The inequality restrictions are the power limits in the
branches of the system. These inequalities are represented as
follow:
(11)
h f (, Vm )  Ff (, Vm )  Fmax  0
(12)
h t (, Vm )  Ft (, Vm )  Fmax  0
D. Variables limits
The variables have upper and lower limits determined by
(13) which represent the operational limits of the system, in
terms of the buses voltage limits and the generators power
limits.
vim, min  v im  v im, max , i 1,...., n b
h(u , x)  0
(3)
pig, min  pig  pig, max , i 1,...., n g
(13)
q ig, min  q ig  q ig, max , i 1,...., n g
 ref   i   ref ,   ref
vector of control variables
vector of state variables
Objective function
set of equality constraints
set of inequality constraints
VI. INTERIOR POINT METHOD WITH MIPS SOLVER
Matpower includes a new primal-dual interior point solver
called MIPS, for Matlab Interior Point Solver.[1]
Some details on the primal-dual interior point algorithm
used by MIPS are as follow:
V. FORMULATION OF OPF
As has been discussed, the objective function considered in
this paper is to minimize the total production cost of
scheduled generating units. OPF formulation consists of three
main components: objective function, equality constraints,
and inequality constraints.
A. Objective Function
A. Problem Formulation and Lagrangian
Min f ( X )
(14)
G( X )  0
(15)
(16)
(17)
x
Subject to
H(X )  0
l  AX  u
Ng
i 1
(8)
j 1
(2)
MinFT   ( i P   i PGi   i )
(7)
i 1
Subject to g (u, x)  0
2
Gi
(6)
)   ( PDi )  PL  0
(1)
u
:
x
:
f (u,x) :
g (u,x) :
h (u,x) :
(5)
Q Gi  QDi  Q(V ,  , t )  0
MinimizeF (u, x)
Where;
(4)
Where
Ng
: number of generators in the system
α i , β i , γ i : fuel cost coefficient of the generator at bus i
PGi
PGi  PDi  P(V ,  , t )  0
: real power generation at bus i
B. Equality Constraints
The equality restrictions are the power balance equations in
every bus of the system. And the injections of active and
reactive power and are calculated through the power flow
equations.
X min  X  X max
The approach taken involves converting the ni inequality
constraints into equality constraints using a barrier function
and vector of positive slack variables Z.
ni


min  f ( X )    ln( Zm)
X
m 1


Subject to G ( X )  0
H(X )  Z  0
Z 0
(18)
(19)
(20)
2
As the parameter of perturbation approaches zero, the
solution to this problem approaches that of the original
problem.
For a given value of the Lagrangian for this equality
constrained problem is
ni
(21)
L ( X , Z ,  ,  )  f ( X )  T G ( X )   T ( H ( X )  Z )   ln( Zm)

m1
Taking the partial derivatives with respect to each of the
variables yields:
LXX X  GX   H X (  Z  (e   (H ( X )  Z  H X X )))  LX
T
T


T
LZ ( X , Z ,  ,  )    e [Z ]
L ( X , Z ,  ,  )  G T ( X )
T
L ( X , Z , ,  )  H ( X )  Z
T
T
H X [ Z ] 1 (e   H ( X ))  H X [ Z ] 1 [ Z ]    LX
T
(23)
X 
 Z 
F     F ( X , Z ,  ,  )
  
 
  
T
Where;
(26)
(28)
 Z  Z    Z  e
Z   Z   e   Z
(29)
     Z  (e   Z )
1
Solving the 4th row of (28) for ΔZ yields
H X X  Z  H ( X )  Z
(30)
Z  H ( X )  Z  H X X
Then, substituting (29) and (30) into the 1st row of (28)
results in
T
T
LXX X  G X   H X (  Z  (e   Z ))   LX
T
T
1
T
T
T
T
T
T
Combining (31) and the 3rd row of (28) results in a system of
equations of reduced size:
 M G X T  X    N 
(32)

G X
   

0      G ( X )
The Newton update can then be computed in the following
3 steps:
1. Compute ΔX and Δλ from (32).
2. Compute ΔZ from (30).
3. Compute Δμ from (29).
In order to maintain strict feasibility of the trial solution,
the algorithm truncates the Newton step by scaling the primal
and dual variables by αp and αd, respectively, where these
scale factors are computed as follows:
 Zm  
,1
 Zm  


 m  
,1
 d  min   min  
m  0
 m  

(33)
Zm  0
This set of equations can be simplified and reduced to a
smaller set of equations by solving explicitly for Δμ in terms
of ΔZ and for ΔZ in terms of ΔX. Taking the 2nd row of (28)
and solving for Δμ we get
T
M  f XX  G XX ( )  H XX (  )  H X [ Z ] 1 [  ]H X

(27)
T


 X 
LX


 

Z



Z


e

    
 G( X ) 
   
 T

 
T
   
 H ( X )  Z 
LXX X  G X   H X    LX
(31)
T
 P  min   min  
C. Newton Step
The first order optimality conditions are solved using
Newton's method. The Newton update step can be written as
follows:
HX
[Z ]
0
0
T
MX  G X    N
T
T

  f X T  GX T   H X T  
LX

 

[  ]Z  e
 Z  e   

F ( X , Z , ,  )  
 G( X )  

G( X )
 T
 

T
H (X )  Z

 H ( X )  Z  
T
T
N  f X  G X   H X   H X [ Z ] 1 (e   H ( X ))
where
GX
[ ]
0
0
0
I
0
T
T
(25)
0
T
N  LX  H X [Z ] 1 (e   H ( X ))
 0
 LXX

 0
G
 X
 H X
T
( LXX  H X [ Z ] 1 [  ]H X )X  G X  
T
Z 0
F
1
T
H X [ Z ] 1 [ Z ]  H X [ Z ] 1 [  ]H X X   LX
(22)
B. First Order Optimality Conditions
The first order optimality (Karush-Kuhn-Tucker) conditions
for this problem are satisfied when the partial derivatives of
the Lagrangian above are all set to zero:
(24)
F ( X , Z , ,  )  0
FZ
1
T
T
LXX ( X , Z ,  ,  )  f XX  G XX ( )  H XX (  )
X
T
M  LXX  H X [ Z ] 1 [  ]H X
1
And the Hessian of the Lagrangian with respect to X is
given by
F
T
L X  G X   H X   H X [ Z ] e  H X [ Z ]  H ( X ) 
T
LX ( X , Z ,  ,  )  f X  T G X   T H X

1
T

XX
(34)
Resulting in the variable updates below.
X  X   P X
(35)
Z  Z   P Z
(36)
     d 
(37)
(38)
     d 
In MIPS, ξ is set to 0.99995.
MIPS uses the following rule to update γ at each iteration,
after updating Z and μ:
ZT
(39)
 
ni
In MIPS, σ is set to 0.1.
VII. APPILCATION OF INTERIOR POINT METHOD TO IEEE14
BUS SYSTEM
The one line diagram of IEEE 14 bus system is shown in
Figure 1. The system data is taken from MATPOWER.
MATPOWER is a package of Matlab m-files for solving
power flow and optimal power flow problems. It is intended
as a simulation tool for researchers and educators which will
be easy to use and modify. The line data, bus data and
generation data are given in matrix forms, respectively. The
data is on 100 MVA base.
A. Single Line Diagram
As shown in single line diagram, the generation power is
largest at Bus 1 and thus it is taken as slack bus. The
3
remaining generator buses are taken as voltage control buses.
There are four voltage control buses and ten load buses.
According to the data, four transformers and three reactive
power compensations are also placed in single line diagram.
The buses of IEEE14 bus system are interconnected to form a
network. There are 20 interconnected lines. The single line
diagram of IEEE14 bus system is described in Figure 1.
13
14
12
11
10
G1
G5
6
1
C
9
C
G4
8
C. Generator Data
The generator data for IEEE14 bus system is shown in
Table II. There are five generator buses in the system. Among
them Bus one is operates as slack bus and remaining buses are
voltage control buses. Bus 1 exhibits the largest generating
capacity and bus 2 exhibits the second largest generating
capacity. The powers are also shown with per unit values
based on 100 MVA. The maximum generating capacity of
remaining buses are set as 100 pu. The maximum and
minimum reactive powers are also shown in column 4 and 5.
They are the reactive power constrained of the system. The
status shown in column 8 are set as 1 for all generator. This
means that all generators are operating.
Table II. Generator Data for IEEE14 Bus System
7
5
Q
4
bus
2
G3
G2
Figure 1. Single Line Diagram for IEEE14 Bus System
B. Bus Data
The bus data for IEEE14 bus system is shown in Table I. In
bus type identification, bus type 3 stands for slack bus, 2 for
voltage control bus and 1 stands for load bus. P d and Qd are
the demand or load power at the corresponding buses. The
demand powers are given in per unit with 100 MVA base.
According to data, Bus 3 exhibits the largest load in terms of
both real and reactive power. Gs and Bs are the shunt
conductance and susceptance at the buses. These values are
set as zero except Bus 9 where the bus susceptance is assigned
as 19 p.u. The next column is area data and it is to be use for
interconnected system. In IEEE14 bus system all buses are
assigned as area 1 and thus the buses are located at the same
area. Voltage magnitude and angle at each bus are denoted by
Vm and Va in the table. Base kV is not defined in this system.
Zone is also assigned as 1. The last two columns describe
maximum and minimum voltages at the buses.
Table I. Bus Data for IEEE14 Bus System
Bu
s-i
t
y
p
e
s
a
r
e
a
Pd
Qd
s
Vm
1
3
0
0
0
0
1
2
2
22
13
0
0
3
2
94
19
0
4
1
48
-4
5
1
7.6
6
2
7
1
8
2
Va
ba
se
K
V
1.1
0
1
1
0
1
0
0
1.6
0
11
7.5
0
0
G B
0
0
zo
ne
V
V
max
min
0
1
1.1
0.9
-5
0
1
1.1
0.9
1
-13
0
1
1.1
0.9
1
1
-10
0
1
1.1
0.9
0
1
1
-9
0
1
1.1
0.9
0
0
1
1.1
-14
0
1
1.1
0.9
0
0
1
1.1
-13
0
1
1.1
0.9
0
9
1
30
17
0
0
1
9
1
1.1
-13
0
1
1.1
0.9
1
1.1
-15
0
1
1.1
0.9
10
1
9
5.8
0
0
1
1.1
-15
0
1
1.1
0.9
11
1
3.5
1.8
0
0
1
1.1
-15
0
1
1.1
0.9
12
1
6.1
1.6
0
0
1
1.1
-15
0
1
1.1
0.9
13
1
14
5.8
0
0
1
1.1
-15
0
1
1.1
0.9
14
1
15
5
0
0
1
1
-16
0
1
1.1
0.9
max
Qmin
-16.9
10
0
stat
us
1.06
100
1
332.4
0
Vg
P
1
2
40
42.4
50
-40
1.05
100
1
140
0
3
0
23.4
40
0
1.01
100
1
100
0
6
0
12.2
24
-6
1.07
100
1
100
0
8
0
17.4
24
-6
1.09
100
1
100
0
C
3
Qg
mbase
Pg
232
.4
Pmax
min
D. Branch Data
There are 20 interconnected lines in IEEE14 bus system
and the data for each line is described in Table III. The first
two column show the from bus and to bus as fbus and tbus
respectively. The next three columns describe the resistance,
reactance and susceptance of the respective line. The three
rates are also mentioned as A, B and C. The ratio in column 8
describe the voltage ratio between the two buses. According
to this ratio data, the transformers are placed between buses 4
and 7, 4 and 9 and 5 and 6 in single line diagram. The angle in
column 9 is set as 0 for all buses. The status in column 10
describe all buses are sound and in operation.
Table III. Branch Data for IEEE14 Bus System
C
ra
tio
an
g
st
at
us
0
0
0
1
0
0
0
0
1
0
0
0
0
1
9900
0
0
0
0
1
9900
0
0
0
0
1
0.0128
9900
0
0
0
0
1
0
9900
0
0
0
0
1
0.2091
0
9900
0
0
1
0
1
0
0.5562
0
9900
0
0
0
1
0
0.252
0
9900
0
0
1
0.
9
0
1
11
0.095
0.1989
0
9900
0
0
0
0
1
12
0.1229
0.2558
0
9900
0
0
0
0
1
6
13
0.0662
0.1303
0
9900
0
0
0
0
1
7
8
0
0.1762
0
9900
0
0
0
0
1
7
9
0
0.11
0
9900
0
0
0
0
1
9
10
0.0318
0.0845
0
9900
0
0
0
0
1
9
14
0.1271
0.2704
0
9900
0
0
0
0
1
10
11
0.0821
0.1921
0
9900
0
0
0
0
1
12
13
0.2209
0.1999
0
9900
0
0
0
0
1
13
14
0.1709
0.348
0
9900
0
0
0
0
1
F
bu
s
Tb
r
X
b
rate
A
1
2
0.0194
0.0592
0.0528
9900
0
1
5
0.054
0.223
0.0492
9900
2
3
0.047
0.198
0.0438
9900
2
4
0.0581
0.1763
0.034
2
5
0.057
0.1739
0.0346
3
4
0.067
0.171
4
5
0.0134
0.0421
4
7
0
4
9
5
6
6
6
us
B
4
E. Generator Cost Function
The generator cost function for IEEE14 bus system is shown
in Table IV.
Table IV. Generator Cost Function for IEEE14 Bus System
No
1
2
3
4
5
Bus
1
2
3
6
8
c2
0.04
0.25
0.01
0.01
0.01
c1
20
20
40
40
40
c0
0
0
0
0
0
VIII. CONSTRAINTS FOR INTERIOR POINT METHOD
In optimal power flow study with interior point method,
the constraints are important data. Three constraints can be
observed in IEEE14 bus system. Thus the inputs files are also
necessary for these constraints. The assigned constraints are
reactive power constraints and capacity constraints at
generator buses, and voltage constraints at all bus.
A. Reactive Power Constraints
The reactive power constraints are assigned at generator
buses and these data are arranged in ‘qconstr’ input file as
follow:
% Bus no Qmax Qmin
qconstr=[
1
10
2
3
6
8
50
40
24
24
0
-4
0
0
-6
-6
B. Capacity Constraints
The capacity constraints or real power constraints are
assigned at generator buses and these data are arranged in
‘pconstr’ input file as follow:
% Bus no Pmax
Pmin
pconstr=[
1
2
3
6
8
332.4
140
100
100
100
0
0
0
0
0
C. Voltage Constraints
The voltage constraints are assigned at all buses and these
data are arranged in ‘vconstr’. The ‘vconstr’ for IEEE14 bus
system are within +/- 6% at all buses.
for optimization of general case and not intended only for
power system optimization.
The research and modeling of the power flow and OPF
problems were done entirely using the MATPOWER toolbox
version 4.1. The toolbox is capable of modeling complex
networks, providing properties of buses, branches,
generators, generator costs and loads. These quantities are all
given in matrices that can be easily modified. The power flow
cost problem is solved using the Newton-Raphson and
Lambda iteration method, while the optimization problem has
several different solvers. The majority of the produced code
focused on building the corresponding matrices in the
problems for MATPOWER to solve.
In the simulation of interior point method for optimal
power flow for IEEE14 bus system, the optimal point is
reached within 0.3seconds. At the end of simulation, the total
generation cost is obtained as 8081.53 $/hr and the total loss
is 9.287 MW. Total transmission efficiency is 96.54 %. The
resulting system summary is shown in Table V.
Table V. OPF result with Interior Point Method for IEEE 14
Bus System
How many?
Buses
14
Generators
Committed
Gens
Loads
Fixed
5
System Summary
How much?
P (MW)
Total Gen
Capacity
772.4
On-line Capacity
Generation
5
(actual)
11
Load
11
Fixed
Q (MVAr)
-52 to 148
772.4
-52 to 148
268.3
259
259
67.6
73.5
73.5
Dispatchable
0
Dispatchable
-0.0 of -0.0
-0.0
Shunts
1
Shunt (inj)
-0.0
20.7
Branches
20
9.29
39.16
Transformers
3
-
24.3
Inter-ties
0
Losses (I^2 * Z)
Branch Charging
(inj)
Total Inter-tie
Flow
0.0
0.0
Areas
1
According to the assigned parameters, the parameters
operated at maximum and minimum values are also generated
by the program. Table 6 shows these minimum and maximum
operating points. The magnitudes as well as the location are
also described in this table.
Table VI. Result Data For Maximum and Minimun Values
IX. PERFORMANCE EVALUATION OF IEEE 14 BUS SYSTEM
The application of interior point method to IEEE14 bus
system is carried out by matlab interior point solver (MIPS) in
Matpower4.1 optimization toolbox. This software package in
intended for optimization of power system thus it is suitable
for this paper. In Matpower, ipm_opf_solver command is
used for interior point method of optimal power flow. With
Matlab software, interior point method can be executed using
ipm command. But Matlab optimization toolbox is intended
Voltage
Magnitude
Voltage
Angle
P Losses
(I^2*R)
Minimum
Maximum
0.94p.u.
@ bus 69
1.060 p.u.
at bus 40
-28.26 deg
@ bus 63
8.47 deg
at bus 34
-
6.05 MW
at line 43-47
5
Q Losses
(I^2*X)
Lambda P
-0.0 $/MWh
at bus 1
-0.0 $/MWh
at bus 1
38.31MVAr
at line 43-47
157.55$/MWh
at bus 69
499.9 $/MWh
at bus 69
Lambda Q
At the bus data, the voltage magnitudes and angles, real and
reactive power generation and demand and real and reactive
power cost at each bus are described. The slack bus generates
maximum real power with 194.33 MW and there is no
reactive power generation at slack bus. At the generator of bus
6, there is only reactive power generation with 11.55 MVAR.
The real and reactive power generation of each generators are
within the assigned limits. The cost of real power is maximum
at Bus 14 and it is minimum at Bus 1.
With the interior point method, the bus voltage profile is
very good at all bus. The minimum bus voltage is 1.014 pu at
bus 4. The bus voltage magnitude is at upper limit for Bus 1, 6
and 8. All bus voltages are also within the assigned constraint
values. The maximum bus voltage phase angle can be
observed at bus 14 where the angle is – 14.274. The
maximum voltage phase angle is also acceptable for system
stability.
There are 20 branches in IEEE14 bus system. In line data,
real and reactive power flow and losses on the lines are
described. Optimal line flow is intended for the minimum
power loss on the line.
According to line flow results, the maximum power flow
can be observed at line 1-2 and minimum line flow can be
observed at line 12-13. The maximum real power losses
occurred at line 1-2 where the losses are 2.902 MW. At line
4-7, 4-9, 5-6, 7-8 and 7-9 the real power loss is zero. The
reactive power losses are rather large compared to real power
losses. The maximum reactive power loss occurred at line 1-2
and minimum reactive power losses can be observed at line
12-13. According to the result, the real power losses on the
line is 9.287 MW and reactive power losses is 39.16 MVAR.
These losses are smaller compared to other methods.
According to simulation results of bus data, all bus voltages
are within regulation limits. Thus interior point method can
solve optimal problem as well as voltage regulation problem.
In power generation, generator at bus 4 is operating at upper
bound of real power and generator at bus 1 is operating at
upper bound of reactive power.
X. COMPARISON WITH OTHER POWER SYETEM
OPTIMIZATION METHODS
To examine the performance of the interior point method in
OPF problem, the same case (IEEE14 bus system) is also
solved by Newton-Raphson and Lambda Iteration methods
which are used for power system optimization . The results for
important data by each method are shown in table.
Table VII. Comparison with Other Power System
Optimization Methods
SN
1
2
Parameter
Generation Cost
($/h)
Power Loss
Linear
Programming
Lambda
Newton-
Iteration
Raphson
8081.53
8498.72
9459.50
9.287
11.14
15.992
(MW)
3
4
5
6
7
Transmission
Efficiency (%)
Voltage
Regulation (%)
96.54 %
95.88 %.
94.185
%.
6%
12.1 %
14 %
Max Voltage 1.06 at buses
(pu)
Min. Voltage
(pu)
Max Voltage
angle (deg)
1,6,8
1.014 at bus 4
- 14.274.
1.0 at bus1
1.0 at bus
1
0.879 at bus 0.860 at bus
14
-16.819
14
- 19.657
According to the results, solving interior point method with
MIPS solver can provide the best performance in
optimization as well as voltage profile. It can also handle all
assigned constraints. In Newton-Raphson and Lambda
Iteration methods, the constraints are not satisfied.
XI. CONCLUSION
This paper presented interior point method with matlab
interior point solver to optimal power flow of IEEE 14 bus
system. These interior point method captures both the voltage
phase angles and magnitudes, which are coupled through
equations for active and reactive power. Experimental
comparisons with AC solutions on a variety of standard IEEE
and MatPower benchmarks shows that the interior point
method are highly accurate for active and reactive power,
phase angles and voltage magnitudes. The interior point
method can be efficiently used as a building block for
optimization problems involving constraints on real and
reactive power flow and voltage magnitudes.
ACKNOWLEDGEMENTS
The author wishes to express her deep gratitude to his
Excellency, Minister Dr. Ko Ko Oo, Ministry of Science and
Technology, for opening the Master of Engineering course at
Mandalay Technological University. The author is very
thankful to Dr. Myint Thein, Pro-Rector of Mandalay
Technological University, for his in valuable permission and
kind support in carrying out this research work. The author
wishes to record her thanks to Dr. Khin Thuzar Soe, Associate
Professor and Head, Department of Electrical Power
Engineering, Mandalay Technological University, for his
guidance, suggestions and necessary advice. The author is
deeply indebted to her supervisor, Dr. Pyone Lai Swe,
Lecture, Department of Electrical Power Engineering,
Mandalay Technological University, for his careful guidance,
necessary advice and encouragement.
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