Research article
Open Access
Particle Distribution Algorithm for Solving Optimal
Reactive Power Dispatch Problem
Lenin K1*, Ravindhranath Reddy B2, Surya Kalavathi M3
1Researcher, JNTU, Hyderabad, India.
2Deputy Executive Engineer, JNTU, Hyderabad, India.
3Professor, Department of Electrical and Electronics Engineering, JNTU, Hyderabad, India.
*Corresponding author: Lenin K, Researcher, JNTU, Hyderabad, 500 085, India, Tel: 91-9677350862; E-mail:
@
Received: January 06, 2017; Accepted: January 25, 2017; Published:January 31, 2017
Citation: Pandey A, Burse K (2017) Cascade Neuro-Fuzzy Architecture Based Mobile-Robot Navigation and Obstacle Avoidance in Static and Dynamic Environments. Int J Adv Robot Automn 2(1): 1-9. DOI:
10.15226/2473-3032/2/1/00118
Abstract
In this paper, a novel perception of the particle swarm
optimization (PSO) system called Particle Distribution Algorithm
(PDA) is proposed to solve Reactive power problem. In that swarm
individual particles are moved away, but discreetly characterize
each particle by a field or by distribution & its behaviour absolutely
dissimilar from the family of old PSO scheme. The projected PDA
has been tested in standard IEEE 30 bus test system and simulation
results show clearly the better performance of the proposed
algorithm in reducing the real power loss.
Keywords:
Particle Swarm Optimization; Particle Distribution; Optimal
Reactive Power; Transmission Loss;
Introduction
Various numerical methods like the gradient method [1, 2]
Newton method [3] and linear programming [4-7] have been
utilized to solve the optimal reactive power dispatch problem.
The problem of voltage stability and collapse play a key role
in power system planning and operation [8]. Enhancing the
voltage stability, voltage magnitudes within the limits alone will
not be a reliable indicator to indicate that, how far an operating
point is from the collapse point. The reactive power support and
voltage problems are internally related to each other.This paper
formulates by combining both the real power loss minimization
and maximization of Static Voltage Stability Margin (SVSM) as
the objectives. Numerous Evolutionary algorithms have been
already utilized to solve the reactive power flow problem [9-20].
This paper proposes Particle Distribution Algorithm (PDA) to
solve reactive power dispatch problem. Proposed algorithm has
better exploration and exploitation capabilities in searching the
global near optimal solution & each particle can be abstracted
by a field or distribution. The parameters of the distribution are
modernized by using a scheme alike to Thompson’s sampling,
leading to a completely new and unique perspective on particle
swarm systems [21-40]. Proposed PDA has been evaluated in
standard IEEE 30 bus test system. Simulation results show that
our proposed approach outperforms all the entitled reported
algorithms in minimization of real power loss and voltage profile
index is enhanced.
Voltage Stability Evaluation
Modal analysis for voltage stability evaluation
Modal analysis is one among best methods for voltage
stability enhancement in power systems. The steady state system
power flow equations are given by.
Where
ΔP = Incremental change in bus real power.
ΔQ = Incremental change in bus reactive Power injection
Δθ = incremental change in bus voltage angle.
ΔV = Incremental change in bus voltage Magnitude
Jpθ , JPV , JQθ , JQV jacobian matrix are the sub-matrixes of
the System voltage stability is affected by both P and Q.
To reduce (1), let ΔP = 0 , then.
Where
is called the reduced Jacobian matrix of the system.
Modes of Voltage instability
Voltage Stability characteristics of the system have been
identified by computing the Eigen values and Eigen vectors.
Let
Where,
ξ = right eigenvector matrix of JR
η = left eigenvector matrix of JR
^ = diagonal eigenvalue matrix of JR and
From (5) and (8), we have
or
Where ξi is the ith column right eigenvector and η the ith
row left eigenvector of JR.
λi is the ith Eigen value of JR.
The ith modal reactive power variation is,
where,
Where
ξji is the jth element of ξi
The corresponding ith modal voltage variation is
If | λi | =0 then the ith modal voltage will collapse
In (10), let ΔQ = ek where ek has all its elements zero except the kth one being 1. Then,
k th element of
V –Q sensitivity at bus k
Problem Formulation
The objectives of the reactive power dispatch problem is to
minimize the system real power loss and maximize the Static
Voltage Stability Margins (SVSM).
Minimization of Real Power Loss
Minimization of the real power loss (P
loss) in transmission
lines is mathematically stated as follows.
Where n is the number of transmission lines, gk is the
i and bus j, and θij is the voltage angle difference between bus i
and bus j.
Minimization of Voltage Deviation
Minimization of the voltage deviation magnitudes (VD) at
load buses is mathematically stated as follows.
Minimize VD =
Where nl is the number of load busses and Vk is the voltage
magnitude at bus k.
System Constraints
Objective functions are subjected to these constraints shown
below.
Load flow equality constraints:
where, nb is the number of buses, PG and QG are the real
and reactive power of the generator, PD and QD are the real and
reactive load of the generator, and Gij and Bij are the mutual
conductance and susceptance between bus i and bus j.
Generator bus voltage (VGi) inequality constraint:
Load bus voltage (VLi) inequality constraint:
Switchable reactive power compensations (QCi) inequality
constraint:
Reactive power generation (QGi) inequality constraint:
Transformers tap setting (Ti) inequality constraint:
Transmission line flow (SLi) inequality constraint:
Where, nc, ng and nt are numbers of the switchable reactive power sources, generators and transformers.
Particle Distribution Algorithm (PDA)
In the basic particle swarm optimization (PSO) model, an
individual particle consists of: A current position, a current
velocity and a personal best found position. Each of these
components is required in order to decide the succeeding position
of the particle for the subsequent iteration. The particle’s position
in the succeeding iteration depends upon its present position and
the particle’s velocity. The modernized velocity, in turn, depends
on the particle’s current velocity, the particle’s personal best
found position and the population’s global best found position.
The representation of a swarm of particles “flying” through space
no longer aptly describes the high-level concept of the algorithm.
Rather, the algorithm now consists of a population of “particle
fields” which move throughout the space in a different way.
Because the “positions” of these “particle fields” are defined as
arbitrary distributions, “evaluating” a “particle field’s” existing
“position” is nondeterministic, and so these “particle fields”
do not certainly have to “move” to discover new points. These
“particle fields” remain “stationary” in the space until either the
individual’s personal best point changes, or the population’s
global best point changes. This population of “particle fields”
can, itself, be alleged as an arbitrary field of particles defined as
a mixture distribution made up of each individual distribution.
This population level distribution can be thought of as an
abstract depiction of a particle swarm, representing a probability
distribution of all possible particle locations for the consequent
iteration. With this new outlook, it is possible to discover new
directions in refining or altering the behaviour of the algorithm.
With these changes taken into account, we have now moved
away from the traditional PSO paradigm and arrived at a
new, distinct algorithm, which will be hereafter referred to as
Particle Distribution Algorithm (PDA). This algorithm consists
of a population of “particle field” individuals and a “point pool”
of candidate solution points. The population of particle field
individuals uses PSO principles to guide the exploration of the
solution space, which is carried out by creating and weighing,
the pool of candidate solution points. Similar to traditional PSO
algorithms, the PDA algorithm consists of an initialization phase
and a simulation phase which loops until some end criteria is
met, at which point the best solution found by the algorithm is
returned as output. A particle field individual is designated at
an arbitrary position from the population, according to some
weighting scheme. Then, the point is produced by sampling
the arbitrary distribution defined by the selected individual.
This arbitrary distribution is constructed using the individual’s
personal best, and the global best points.
Given a particle field with personal best point P
i
and for which the global best point is P
g the position of the candidate
solution point,
c is determined according to:
Once the candidate solution has been produced, the objective
function is weighed, using this produced point as its input, in
order to assign it a value. After each candidate solution in the
point pool has been produced, the second phase begins. In this
phase, the population of particle field individuals is modernized.
Each individual updates its own best found point using the set
of candidate solutions produced from its own distribution.
Each individual selects the best point from the set of associated
candidate solutions. If the best associated candidate solution
is better than the individual’s personal best found point, the
individual sets its personal best found point to be equal to that
candidate solution point. The pool of candidate solutions is
then “emptied”, and the simulation endures to the succeeding
iteration. Once the termination criteria have been met, the global
best found point is returned as output of the algorithm.
Input:
Function f ( ) to be optimized
Initialization range lbound, ubound
Particle field population size nPop
Candidate solution point pool size npool
Weighting function ù ( )
Output:
Point
representing best found solution
Method:
Generate particle field population P with size
Generate candidate solution point pool C with size
For each particle field
do
if
then
End if
End for
While termination criteria not met do
For each candidate solution point
do Select particle field
with probability
End for
For each particle field
do
Choose point
from
which minimizes
if
then
If
Then
End if
End if
End for
End
Simulation Results
The efficiency of the proposed Particle Distribution Algorithm
(PDA) is demonstrated by testing it on standard IEEE-30 bus
system. The IEEE-30 bus system has 6 generator buses, 24 load
buses and 41 transmission lines of which four branches are (6-9),
(6-10) , (4-12) and (28-27) - are with the tap setting transformers.
The lower voltage magnitude limits at all buses are 0.95 p.u. and
the upper limits are 1.1 for all the PV buses and 1.05 p.u. for all
the PQ buses and the reference bus. The simulation results have
been presented in Tables 1, 2, 3 &4. And in the Table 5 shows
the proposed algorithm powerfully reduces the real power losses
when compared to other given algorithms. The optimal values of
the control variables along with the minimum loss obtained are
given in Table 1. Corresponding to this control variable setting,
it was found that there are no limit violations in any of the state
variables.
Table 1: Results of PDA – ORPD optimal control variables
Control Variables |
Variable Setting |
V1
V2
V5
V8
V11
V13
T11
T12
T15
T36
Qc10
Qc12
Qc15
Qc17
Qc20
Qc23
Qc24
Qc29
Real power loss
SVSM |
1.047
1.046
1.041
1.030
1.004
1.031
1.00
1.00
1.01
1.01
3
3
2
0
2
3
3
2
4.2869
0.2479 |
Table 2: Results of PDA -Voltage Stability Control Reactive Power Dispatch
Optimal Control Variables
Control Variables |
Variable Setting |
V1
V2
V5
V8
V11
V13
T11
T12
T15
T36
Qc10
Qc12
Qc15
Qc17
Qc20
Qc23
Qc24
Qc29
Real power loss
SVSM |
1.049
1.047
1.044
1.032
1.005
1.034
0.090
0.090
0.090
0.090
3
3
2
3
0
2
2
3
4.9896
0.2484 |
Table 3: Voltage Stability under Contingency State
Sl.No |
Contingency |
ORPD Setting |
VSCRPD Setting |
1 |
28-27 |
0.1419 |
0.1434 |
2 |
4-12 |
0.1642 |
0.1650 |
3 |
1-3 |
0.1761 |
0.1772 |
4 |
2-4 |
0.2022 |
0.2043 |
Optimal Reactive Power Dispatch problem together with
voltage stability constraint problem was handled in this case
as a multi-objective optimization problem where both power
loss and maximum voltage stability margin of the system were
optimized simultaneously. Table 2 indicates the optimal values
of these control variables. Also it is found that there are no limit
violations of the state variables. It indicates the voltage stability
index has increased from 0.2479 to 0.2484, an advance in the
system voltage stability. To determine the voltage security of the
system, contingency analysis was conducted using the control
variable setting obtained in case 1 and case 2. The Eigen values
equivalents to the four critical contingencies are given in Table
3. From this result it is observed that the Eigen value has been
improved considerably for all contingencies in the second case.
Conclusion
In this paper, Particle Distribution Algorithm (PDA) has been
successfully applied to solve Optimal Reactive Power Dispatch
problem. The proposed algorithm has been tested on the standard
IEEE 30 bus system. Simulation results show the heftiness of
projected Particle Distribution Algorithm (PDA) method in
decreasing the real power loss & voltage profile index has been
enhanced. The control variables obtained after the optimization
by Particle Distribution Algorithm (PDA) is within the limits.
Table 4: Limit Violation Checking Of State Variables
State variables |
limits |
ORPD |
VSCRPD |
Lower |
upper |
Q1 |
-20 |
152 |
1.3422 |
-1.3269 |
Q2 |
-20 |
61 |
8.9900 |
9.8232 |
Q5 |
-15 |
49.92 |
25.920 |
26.001 |
Q8 |
-10 |
63.52 |
38.8200 |
40.802 |
Q11 |
-15 |
42 |
2.9300 |
5.002 |
Q13 |
-15 |
48 |
8.1025 |
6.033 |
V3 |
0.95 |
1.05 |
1.0372 |
1.0392 |
V4 |
0.95 |
1.05 |
1.0307 |
1.0328 |
V6 |
0.95 |
1.05 |
1.0282 |
1.0298 |
V7 |
0.95 |
1.05 |
1.0101 |
1.0152 |
V9 |
0.95 |
1.05 |
1.0462 |
1.0412 |
V10 |
0.95 |
1.05 |
1.0482 |
1.0498 |
V12 |
0.95 |
1.05 |
1.0400 |
1.0466 |
V14 |
0.95 |
1.05 |
1.0474 |
1.0443 |
V15 |
0.95 |
1.05 |
1.0457 |
1.0413 |
V16 |
0.95 |
1.05 |
1.0426 |
1.0405 |
V17 |
0.95 |
1.05 |
1.0382 |
1.0396 |
V18 |
0.95 |
1.05 |
1.0392 |
1.0400 |
V19 |
0.95 |
1.05 |
1.0381 |
1.0394 |
V20 |
0.95 |
1.05 |
1.0112 |
1.0194 |
V21 |
0.95 |
1.05 |
1.0435 |
1.0243 |
V22 |
0.95 |
1.05 |
1.0448 |
1.0396 |
V23 |
0.95 |
1.05 |
1.0472 |
1.0372 |
V24 |
0.95 |
1.05 |
1.0484 |
1.0372 |
V25 |
0.95 |
1.05 |
1.0142 |
1.0192 |
V26 |
0.95 |
1.05 |
1.0494 |
1.0422 |
V27 |
0.95 |
1.05 |
1.0472 |
1.0452 |
V28 |
0.95 |
1.05 |
1.0243 |
1.0283 |
V29 |
0.95 |
1.05 |
1.0439 |
1.0419 |
V30 |
0.95 |
1.05 |
1.0418 |
1.0397 |
Table 5: Comparison of Real Power Loss
Method |
Minimum loss |
Evolutionary programming [41] |
5.0159 |
Genetic algorithm [42] |
4.665 |
Real coded GA with Lindex as SVSM [43] |
4.568 |
Real coded genetic algorithm [44] |
4.5015 |
Proposed PDA method |
4.2869 |
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