1. Ingeniería

International Electrical Engineering Journal (IEEJ)
Vol. 5 (2014) No.12, pp. 1659-1664
ISSN 2078-2365
http://www.ieejournal.com/
Single objective optimization using PSO
with Interline Power Flow Controller
Praveen.J, B.Srinivasa Rao
[email protected], [email protected]
Abstract — Optimal Power Flow (OPF) problem was solving
from the decades with different objectives as the economy was
growing. In this paper cost of generation, transmission losses
and L-index are taken as three different objectives for optimal
operation of power system with and without FACTS device.
One of the VSC based multi type FACTS device namely an
Interline Power Flow Controller (IPFC) is placed into power
system network. The optimization problem has been solved by
using Particle Swarm Optimization (PSO) with power injection
model of the FACTS device. The proposed methodology is
tested on standard IEEE 30-bus test system and the results are
compared for single objective optimization with and without
FACTS device.
Index Terms— IPFC, PSO, OPF.
I. INTRODUCTION
The Optimal Power Flow (OPF) problem was first
formulated in 1960’s and Carpentier done major research in
1962 and then it found to be as not an easy problem since
optimal power flow itself defined as the combination of
economic dispatch i.e., minimization of cost function such as
operating cost by considering realistic or practical equality
and inequality constraints. In this paper single objective
optimization has been done for OPF i.e., cost, loss, and
L-index. When cost is taken as the objective cost should be
minimized as minimum as possible without violation of the
constraints and the same case with loss and L-index. The
control variables such as Pg’s (generated real powers), Vg’s
(voltages at generators), Tap’s (transformer tap settings) and
Qc’s (reactive power injections) are considered for optimal
operation of power system using Particle Swarm
Optimization (PSO).
PSO is superior in convergence characteristics and has
shorter execution time when compared to other Evolutionary
Programing (EP) and classical methods. Where as the
classical methods[1] like Gradient method, Quadratic
Programing (QP), Linear Programming(LP), Non-Linear
Programming (NLP), and Interior point method have the
drawback of dimensionality, slow convergence and thus
clsassical methods turned unfeasible to implement.
In [2] PSO was applied for optimal location of FACTS
devices cosnsidering cost of installation and system
loadability. Here FACTS devices were used to increase the
loadability of the system by controlling the electrical
parameters. Power electronic devices plays a major role in
FACTS devices here IPFC uses two Voltage Source
Converters (VSC) which will control the bi-directional power
flow between two lines.
Rest of the paper is organised into six sections. In
section-II, problem formulation for optimal power flow is
explained with objectives and constraints. The overview of
particle swarm optimization is given in section-III. The
section IV deals with the algorithm implimented for OPF
using PSO. A brief description about power injection model
of IPFC is given in section V. In section VI simulation results
are analysed and finally conclusions are drawn in section VII.
II. PROBLEM FORMULATION
The main goal of the OPF problem is to optimize a selected
objective function via optimal adjustment of the power
system control variables, subject to several equality and
inequality constraints [3]. The optimal power flow problem
can be mathematically formulated as follows:
Min F(x,u)
(1)
Subject to:
g(x,u) = 0
(2)
h min ≤ h(x,u) ≤ h max
(3)
Where:
F - objective function to be minimized
x - vector of dependent variables (state variables)
u - vector representing all control variables
g - equality constraints representing line flow equation
h - inequality constraints representing operating limits of
control variables.
A. Objective functions
1) Cost objective
Cost of generation is one of the major objective it is not
only economical but also increases the transmission
efficiency because losses will be decreased when cost of
generation is decreased which is generally observed.
Mathematical form for generation cost is stated as follows [4]
𝑁𝑔𝑒𝑛
2
𝐹1 (𝑋) = ∑𝑖=1 (𝑎𝑖 𝑃𝑔𝑖
+ 𝑏𝑖 𝑃𝑔𝑖 + 𝑐𝑖 )$/ℎ
(4)
Where
X=[Pg,Vg,Taps,Qc]1xn
(5)
Pg=[Pg1,Pg2,Pg3,…..Pg(Ngen-1)]1x(Ngen-1)
(6)
Vg=[Vg1,Vg2,Vg3,….VgNgen]1xNgen
(7)
Taps=[Tap1,Tap2,Tap3,….,TapNtran]1xNtran
(8)
Qc=[Qc1,Qc2,…Qc3,……QcNcap]1xNcap
(9)
N=(Ntran+Ncap+Ngen+(Ngen-1))
(10)
1659
Praveen. and Srinivasa
Single objective optimization using PSO with Interline Power Flow Controller
International Electrical Engineering Journal (IEEJ)
Vol. 5 (2014) No.12, pp. 1659-1664
ISSN 2078-2365
http://www.ieejournal.com/
Here F1(X) is the total fuel cost($/h), and ai,bi,ci are cost
coefficients of fuel of the ith unit. P gi, Vgi, are real power
generation and voltage magnitude of the ith generator
respectively, and Tapi is the tap of ith transformer, Qci is the
reactive power of the ith compensation capacitor, Ngen
indicates the number of generation units, Ntran indicates the
number of tap transformer and Ncap for number of
compensation capacitors.
Equality and inequality constraints are taken in to
consideration and the constraints are used in the optimization
for better opimization using penalty factor method. Here the
fitness of the objective will improved when ever there is less
penalty. Having no penalty shows that there is no violation of
constraints.
2) Transmission loss objective
When transmission loss is taken as objective, the aim of
the objective is to optimize loss or to minimize the loss
irrespective of the level of increase or decrease in the value of
other objectives without violating the system parameters.
The mathematical form for loss objective is given by
𝑁𝑙𝑖𝑛𝑒
𝑃𝑙𝑜𝑠𝑠 = ∑ (𝑉𝑖2 + 𝑉𝑗2 − 2𝑉𝑖 𝑉𝑗 cos Ѳ𝑖𝑗 )𝑀𝑊
(11)
𝑘=1
Here Ploss is the second objective function which is given
by F2(X) = Ploss
3) L-index objective
L-index indicates how nearer the system is to the voltage
collapse. [5] The range of L-index is between 0 and 1and it is
directly proportional to voltage collapse. Maximum value of
L-index is unity so lesser the value of L-index greater is the
system stability.
L-index for jth bus is given by (12)
𝑁𝑔𝑒𝑛
𝐿𝑗 = |1 − ∑𝑖=1 𝐹𝑗𝑖
𝑉𝑖
𝑉𝑗
| 𝑗 = 𝑁𝑔𝑒𝑛 + 1, … . 𝑛
(12)
B. Constraints
Both equality and inequality constraints are considered in
this paper. Penalty factor method is used to consider the
constraints.
Equality constraints
Total power transfer comes under equality constraints i.e.,
total power generation must be equal to the total demand
whether it is real or reactive power generation.
Inequality constraints
These are nothing but the operating constraints of various
elements in power system. Here the operating constraints are
voltage limit of all the buses, reactive power generation
limits, capacitor reactive power generation limits,
transformer tap settings and transmission line flow limits.
In some cases voltage stability is also taken as a constraint
but here it is taken as an objective. However voltage limits
are taken into consideration.
III. OVER VIEW OF PSO
In PSO the optimization is performed by initializing
random population with random solutions and searches for
optima by updating the current population. The current
population may be present nearer to local best or global best
but finally all the population should reach the global best. The
population will be generated until the global best solution is
reached.
In PSO there will be no cross over and mutation the
solution will be searched through hyper space.
PSO is based on two fundamental disciplines: social
science and computer science. The cornerstones of PSO can
be described as follows. [6, 7]
1) Social Concepts: It is known that “human intelligence
results from social interaction.” Assessment, comparison,
and imitation of others, as well as learning from experience
allow humans to adapt to the environment and determine
optimal patterns of behavior, attitudes, and suchlike. In
addition, a second fundamental social concept indicates that
“culture and cognition are inseparable consequences of
human sociality.” Culture is generated when individuals
become more similar due to mutual social learning. The
sweep of culture allows individuals to move towards more
adaptive patterns of behavior.
2) Swarm Intelligence Principles: Swarm Intelligence can be
described by considering five fundamental principles.
2.1) Proximity Principle: the population should be able to
carry out simple space and time computations.
2.2) Quality Principle: the population should be able to
respond to quality factors in the environment.
2.3) Diverse Response Principle: the population should not
commit its activity along excessively narrow channels.
2.4) Stability Principle: the population should not change its
mode of behavior every time the environment changes.
2.5) Adaptability Principle: the population should be able to
change its behavior mode when it is worth the
computational price.
Since the PSO is a population based EP, a minimal or
maxima search will be done using the randomly generated
population. For each randomly generated population a
different weight or inertia will be given and it is given by
w=(maxit-it)/maxit)
(13)
w is formulated depending on the user. When the user
gives fixed iterations, the above equation will be suitable and
if the iteration termination depends on value of the objective
specified, then max and min of the weight will be specified.
Velocity of the particles will be given by
Vit(t)=Vit(t-1)+c1rand1(Pit-Xit(t-1))+c2rand2(Pg-Xit(t-1)) (14)
Where‘t’ is the present iteration c1 and c2 are constants
range from 2 to 4 and rand1,rand2, are random values range
from 0 to 1.
The present particle updation will be done by using the
velocity
1660
Praveen. and Srinivasa
Single objective optimization using PSO with Interline Power Flow Controller
International Electrical Engineering Journal (IEEJ)
Vol. 5 (2014) No.12, pp. 1659-1664
ISSN 2078-2365
http://www.ieejournal.com/
The position of each particle is determined by the vector
Xi ∈ Rn and its movement by the velocity of the particle
Vit∈Rn.
Xit(t)=Xit(t)+Vit(t)
(15)
̅
̅ )∗
𝑆𝑖𝑠𝑒
= −2𝑉̅𝑖 (𝐼𝑠𝑒
∗
̅
̅
̅
𝑆𝑗𝑠𝑒 = 𝑉𝑗 (𝐼𝑠𝑒 )
̅ = 𝑉̅𝑘 (𝐼𝑠𝑒
̅ )∗
𝑆𝑘𝑠𝑒
(17)
IV. ALGORITHM OF OPF USING PSO
The basic PSO will be applicable to the power system
problem depending on the control variables and constraints.
The PSO was modified according the present power system
problem [8]
Procedure for the proposed PSO algorithm for Optimal
Power Flow is described as follows:
1. Initiate the parameters for the power system and
PSO.
2. Generate the random solution using the nth particle
and Xn position and Vn velocity.
3. Load flows will be performed using Newton
Raphson method for computing Optimal Power
Flow and to compute the control variables of IPFC.
4. Computing the objective function (cost,
loss,L-index), F for all the particles.
5. Compute the local best for nth particle, such that
Fa1<Fai, n>1, then set Lbest=Xa1 and search the global
best value (Gbest), with its location.
6. Compute the global best position until local best
becomes global best.
7. Bring up to date the inertia by using the equation
number (13)
8. Calculate the new particle velocity and location by
using equations (14, 15).
9. If the condition to stop the iteration is satisfied go to
next step else go to step2 (here stopping criteria is
taken by number of iterations)
10. Produce the optimal parameters of power system,
IPFC.
11. Stop
Fig. 1 Equivalent circuit for two converters IPFC
Therefore, the above ‘r’ and ‘ 𝛾 ’ values will help in
calculating the individual voltage and reactive power control.
If the voltage is only controlled then no need to inject the
angle and if the angle is only controlled no need to inject the
voltage. Depending on the system demand both the variable
has to be controlled in any manner. In this case the
controlling parameters are adjusted using PSO and the best
value is chosen. [9-13]
𝑃𝑖𝑠𝑒 = −2𝑟𝑉𝑖2 𝐵𝑠𝑒 sin(𝛾)
𝑄𝑖𝑠𝑒 = −2𝑟𝑉𝑖2 𝐵𝑠𝑒 𝑐𝑜𝑠(𝛾)
𝑃𝑗𝑠𝑒 = 𝑟𝑉𝑖 𝑉𝑗 𝐵𝑠𝑒 sin(𝛿𝑖𝑗 + 𝛾)
𝑄𝑗𝑠𝑒 = 𝑟𝑉𝑖 𝑉𝑗 𝐵𝑠𝑒 cos(𝛿𝑖𝑗 + 𝛾)
𝑃𝑘𝑠𝑒 = 𝑟𝑉𝑖 𝑉𝑘 𝐵𝑠𝑒 sin(𝛿𝑖𝑘 + 𝛾)
𝑃𝑘𝑠𝑒 = 𝑟𝑉𝑖 𝑉𝑘 𝐵𝑠𝑒 sin(𝛿𝑖𝑘 + 𝛾)
(18)
V. POWER INJECTION MODEL OF IPFC
The power injection modeling of FACTS devices is a
common technique which was generally followed for
incorporating FACTS devices in power system planning. It is
simple and easier to consider the effect of a FACTS device
once the modeling is completed, and it can be easily
implement in programming.
The jacobian elements are used to consider the affect
where the IPFC’s jacobian elements are added to the present
jacobian elements and the power injection will be calculated
at the three different buses where the IPFC was placed.
Figure (1) shows the series injection of voltages representing
the IPFC in the lines ‘ij’ and ‘ik’
𝑉̅𝑠𝑒 = 𝑟𝑉𝑖 𝑒 𝑗𝛾
(16)
From equation (16) the value of ‘r’ indicates the factor of
voltage or in other word at what percentage we can inject the
voltage should be given with this value and ‘𝛾’ indicates the
angle of the voltage injected.
Fig. 2 Equivalent circuit for Power injection model of IPFC
VI. SIMULATION RESULTS
A. Test system data:
MATLAB simulation has been performed on IEEE 30 bus
test system to analyze the quality of PSO for OPF with and
without IPFC. The test system consists of 6 generators,30
1661
Praveen. and Srinivasa
Single objective optimization using PSO with Interline Power Flow Controller
International Electrical Engineering Journal (IEEJ)
Vol. 5 (2014) No.12, pp. 1659-1664
ISSN 2078-2365
http://www.ieejournal.com/
buses,41 transmission lines and 2 shunt reactors, 4 tap
changing transformer.
Base case: In the base case for the given values of IEEE 30
bus system data Newton Raphson load flow is performed
cost, loss and L-index are calculated and the values are
tabulated in Table-1.
Table I Objectives variation before and after optimization
Base case
Cost objective
Loss objective
L-index objective
Cost($/hr)
Loss(MW)
900.6540
800.8353
814.8308
820.8913
5.35
6.79
3.75
5.92
L-index
0.0819
0.0608
0.0457
0.0264
Case 1: single objective optimization without FACTS: In this
case three different objectives such as cost, loss and L-index
are considered for optimization with all constraints. The
results for the Case1 are tabulated in Table-1 it is very clear
when the cost is compared with the base case value it is
decreased by 11.08% that is from 900.6540$ to 800.8353$,
but the loss is increased by 21.21%.When Loss is taken as
objective it is decreased by 29.91% that is from 5.35MW to
3.75MW but the cost is increased by 1.72% when compared
to cost objective. When L-index is taken as objective it is
decreased by 67.76% that is from 0.0819 to 0.0264 but the
cost and loss are increased when compared to base case.
Case2: single objective optimization with FACTS: In this
case PSO is performed with IPFC and the results are
tabulated in Table-2. When IPFC is placed the cost is
decreased by 11.64% that is from 900.6540$
795.7405$.When Loss is considered it is decreased by
51.77% that is from5.35 to 2.58MW and when L-index is
considered it is decreased by 72% that is from 0.0819 to
0.0222 but the loss is increased by 28% w.r.t base case.
Fig. 3 Comparison of cost with and without FACTS
Fig. 4 Comparison of L-index with and without FACTS
Table II Objectives variation before and after placing IPFC
Cost($/hr) Loss(MW) L-index
Cost objective
795.7405
5.64
0.0516
Loss objective
818.7866
2.58
0.0393
L-index objective
825.0000
7.47
0.0222
Figure(3) to Figure (5) shows the convergence
characteristics of three objectives before and after placing
FACTS device where it clearly shows that after placing the
FACTS device the objectives cost, loss and L-index are
having considerable fall. Figure (6) shows voltage profile for
different objectives at all the load buses and in Figure (7)
percentage of line loading has been considered.
As real power is directly proportional to the load angle,
if the losses are decreased the load angle will also decreased
which is clearly shown in Figure (8) where there is a
considerable development for the load angles when loss is
taken as an objective.
Fig. 5 Comparison of loss with and without FACTS
1662
Praveen. and Srinivasa
Single objective optimization using PSO with Interline Power Flow Controller
International Electrical Engineering Journal (IEEJ)
Vol. 5 (2014) No.12, pp. 1659-1664
ISSN 2078-2365
http://www.ieejournal.com/
COST AS OBJECTIVE
LOSS AS OBJECTIVE
L-INDEX AS OBJECTIVE
magnitude of voltage in (p.u)
1.06
1.05
1.04
1.03
1.02
1.01
1
0.99
0.98
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
load buses
Fig. 6 Voltage Profile for Different Objectives
90
COST AS OBJECTIVE
80
LOSS AS OBJECTIVE
70
L-INDEX AS OBJECTIVE
% line loading
60
50
40
30
20
10
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
line number
Fig. 7 Percentage of line loading for different objectives
Bus number
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
0
load angle in p.u.
-0.05
-0.1
-0.15
-0.2
COST AS OBJECTIVE
LOSS AS OBJECTIVE
-0.25
L-INDEX AS OBJECTIVE
Fig. 8 Load Angles for Different Objectives
1663
Praveen. and Srinivasa
Single objective optimization using PSO with Interline Power Flow Controller
International Electrical Engineering Journal (IEEJ)
Vol. XX (2014) No.X, pp. XX-XX
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VII. CONCLUSION
In this paper particle swarm optimization method has been
applied successfully for single objective optimization of cost,
loss, and L-index and the values are highly satisfied when
compared to base case and previous methods. The detailed
procedure for OPF using PSO with facts device is described.
The results are obtained for the three objectives in IEEE 30
bus test system and they are compared with and without
FACTS devices. From the two cases it was found that all the
three objectives have been well optimized when IPFC is
placed in the test system.
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Praveen.J,B.Srinivasa Rao
Single objective optimization using PSO with Interline Power Flow Controller