Research article
Open Access
Waning Of Real Power Loss By Modified Particle
Swarm Optimization
Lenin K^{1*}, Ravindhranath Reddy B^{2}, Surya Kalavathi M^{3}
^{1}Researcher, Jawaharlal Nehru Technological University, Kukatpally.
^{2}Deputy Executive Engineer, Jawaharlal Nehru Technological University, Kukatpally.
^{3}Professor of Electrical and Electronics Engineering, Jawaharlal Nehru Technological University, Kukatpally.
*Corresponding author: Lenin K, Researcher, Jawaharlal Nehru Technological University, Kukatpally, Hyderabad, 500 085, India, Tel: 91-9677350862; E-mail:
@
Received: December 03, 2016; Accepted: December 20, 2016; Published: January 11, 2017
Citation: Lenin K, Reddy BR, Kalavathi MS (2017) Waning Of Real Power Loss By Modified Particle Swarm Optimization. Int J Adv Robot Automn 2(1): 1-5.
Abstract
This paper presents Modified Particle Swarm Optimization
(MPSO) algorithm for solving reactive power problem. A nonlinear
decreasing weight factor used to change the fundamental ways
of Particle Swarm Optimization (PSO). To allow full play to the
approximation capability of the function of Back Propagation neural
network & overcome the main shortcomings of its liability to fall into
local extreme value this study proposed a concept of modified PSO
algorithm. Back Propagation network jointly to optimize the original
weight, threshold value of network and incorporating the PSO
algorithm into Back Propagation network to establish a Modified PSOBP
network system. Proposed method MPSO progresses convergence
speed and the capability to search optimal value. In order to evaluate
the proposed MPSO algorithm, it has been tested on IEEE 30 bus
system and compared to other reported standard algorithms and
simulation results show that (MPSO) is more efficient than other
algorithms in reducing the real power loss & voltage profiles are
within the limits.
Keywords: Modified Particle Swarm Optimization; Back
Propagation; optimal reactive power; Transmission loss;
Introduction
Reactive power problem is one of the difficult optimization
problems in power systems. Various mathematical techniques
have been adopted to solve this optimal reactive power dispatch
problem. These include the gradient method [1, 2], Newton
method [3] and linear programming [4-7].The gradient and
Newton methods suffer from the difficulty in handling inequality
constraints. To apply linear programming, the input- output
function is to be expressed as a set of linear functions which may
lead to loss of accuracy. Recently global optimization techniques
such as genetic algorithms have been proposed to solve the
reactive power flow problem [8, 9]. In recent years, the problem of
voltage stability and voltage collapse has become a major concern
in power system planning and operation. To enhance the voltage
stability, voltage magnitudes alone will not be a reliable indicator
of how far an operating point is from the collapse point [10]. The
reactive power support and voltage problems are intrinsically
related. Hence, this paper formulates the reactive power dispatch
as a multi-objective optimization problem with loss minimization
and maximization of Static Voltage Stability Margin (SVSM) as
the objectives. The Particle Swarm Optimization (PSO) [11-15]
was based on the optimal algorithm of swarm intelligence and it
guides optimal search through swarm intelligence producing by
the corporation and competition among particles. The artificial
Neural Network possesses the ability of processing database
in parallel, organizing and learning by itself. It has made great
achievements in many fields. Especially neural network technique
has been used in reliability and few researchers apply the
method [16-20] to predict failures. Theoretically, Network [21-
32] may approach any continual nonlinear function. However,
affected greatly by the sample, it may fall into local minimum
value, thus it can’t ensure that it converge the minimum value
in overall situations. In our paper PSO algorithm is introduced
into BP network optimization by using iterate algorithm of
particle swarm instead of gradient correction algorithm of
BP network. This method can shorten the time of training
network and improve the convergence speed of BP algorithm.
However, according to different system object, if basic particle
swarm algorithm is modified, the performance of BP network is
improved and its practicality is better. For balancing between the
global search and the local search and improving the precision
of the result, this study proposed the nonlinear strategies for
decreasing inertia weight based on the idea of the existing
linear decreasing inertia weight. The result of simulation shows
that the optimizing methods speed the rapidity of convergence
obviously and the precision of simulation increase greatly. The
performance of Modified Particle Swarm Optimization (MPSO)
has been evaluated in standard IEEE 30 bus test system and the
simulation results show the best performance of the proposed
algorithm.
Objective Function
Active power loss
The objective of the reactive power dispatch problem is to
minimize the active power loss and can be written in equations
as follows:
$$\text{F}={\text{P}}_{\text{L}}={\displaystyle \sum}_{\text{k}\in \text{Nbr}}{\text{g}}_{\text{k}}\left({\text{V}}_{\text{i}}^{2}+{\text{V}}_{\text{j}}^{2}-2{\text{V}}_{\text{i}}{\text{V}}_{\text{j}}{\text{cos\theta}}_{\text{ij}}\right)$$
Where F- objective function, P
_{L} – power loss, g
_{k} - conductance
of branch,V
_{i} and V
_{j} are voltages at buses i,j, Nbr- total number of
transmission lines in power systems.
Voltage profile improvement
To minimize the voltage deviation in PQ buses, the objective
function (F) can be written as:
$$F={P}_{L}+{\omega}_{v}\times VD$$
Where VD - voltage deviation, ${\omega}_{v}$
- is a weighting factor of
voltage deviation.
And the Voltage deviation given by:
$$VD={\displaystyle \sum}_{i=1}^{Npq}\left|{V}_{i}-1\right|$$
Where Npq- number of load buses
Equality Constraint
The equality constraint of the problem is indicated by the
power balance equation as follows:
$${P}_{G}={P}_{D}+{P}_{L}$$
Where P_{G}- total power generation, P_{D} - total power demand.
Inequality Constraints
The inequality constraint implies the limits on components in
the power system in addition to the limits created to make sure
system security. Upper and lower bounds on the active power of
slack bus (P_{g}), and reactive power of generators (Q_{g}) are written
as follows:
$${P}_{gslack}^{min}\le {P}_{gslack}\le {P}_{gslack}^{max}$$
$${Q}_{gi}^{min}\le {Q}_{gi}\le {Q}_{gi}^{max}\text{},\text{}i\in {N}_{g}$$
Upper and lower bounds on the bus voltage magnitudes (Vi)
is given by:
$${V}_{i}^{min}\le {V}_{i}\le {V}_{i}^{max}\text{},\text{}i\in N$$
Uper and lower bounds on the transformers tap ratios (Ti) is
given by:
$${T}_{i}^{min}\le {T}_{i}\le {T}_{i}^{max}\text{},i\in {N}_{T}$$
Upper and lower bounds on the compensators (Qc) is given
by:
$${Q}_{c}^{min}\le {Q}_{c}\le {Q}_{C}^{max}\text{},i\in {N}_{C}$$
Where N is the total number of buses, N
_{g} is the total number
of generators, N
_{T} is the total number of Transformers, N
_{c} is the
total number of shunt reactive compensators.
Particle Swarm Optimization
Particle Swarm Optimization Algorithm (PSO) is a population
based optimization tool where the system is initialized with
a population of random particles and the algorithm searches
for optima by updating generations. Suppose in the D-dimensional
objects searching space, there is a community composed
of N particle. The “I” particle represent a D-dimensional vector,
${X}_{i}=\left({X}_{i1},{X}_{i2},\mathrm{..}{X}_{id}\right)$
It means that the “i” particle represents its
position in this space. Every position of particle “X” is a potential
solution. If we put “x” into objective function, we can know
the adaptive value. We can know whether the “x” is the optimal
answer based on the adaptive value. The speed of particle
is also a D-dimensional, it also recorded as
${v}_{i}=\left({v}_{i1},{v}_{i2},\mathrm{..}{v}_{id}\right)$
We record the particle I to the h times, the optimal position was
${P}_{i}=\left({P}_{i1},{P}_{i2},\mathrm{..}{P}_{id}\right)$
All the particles to the h times, the optimal
position was
${P}_{gd}=\left({P}_{i1},{P}_{i2},\mathrm{..}{P}_{id}\right)$
The basic formulas are as follows:
$${v}_{id}^{t+1}=w{v}_{id}^{t}+{c}_{1}{r}_{1}^{t}\left({P}_{id}^{t}-{x}_{id}^{t}\right)+{c}_{2}{r}_{2}^{t}\left({P}_{gd}^{t}-{x}_{id}^{t}\right)$$
$${x}_{id}^{t+1}={x}_{id}^{t}+{v}_{id}^{t+1}$$
Where,
${c}_{1}$
&$\text{}{c}_{2}$
: Speeding coefficient, adjusting the maxim
step length that flying the best particle in whole situation and the
individual best particle respectively. Appropriate c1 and c2 speed
up the convergence and avoid falling into partial optimality r1&r2:
Random number between 0 and 1, for controlling the weight of
speed W: Inertia factor. It was oriented toward overall searching
.We usually take the original value as 0.9 and make it to 0.1 with
the addition and reduction of the times of iteration. It mainly
used to total searching, making the searching space converge
to a certain space. Then we can get the solution in high degree
of accuracy by partial refined researching. With the increasing
number of dimension of problems, basic PSO algorithm is easily
falling into partial extreme value, thus influence the optimal
function of algorithm. Someone brought up with improved
algorithm. Many scholars’ research shows [11- 15] that “w” has a
great influence on the algorithm of particle swarm. When the “w”
is bigger, the algorithm has a strong ability in total searching and
when the “W” is smaller, it is good for partial searching. Therefore,
in recent years, some scholars brought up many schemes. LDW
(Linearly Decreasing Inertia Weight) is given by,
$$w={w}_{max}-\frac{t\times \left({w}_{max}-{w}_{min}\right)}{{t}_{max}}$$
Where
${w}_{max}$
&
${w}_{min}\text{}$
The maximum and minimum value of W, t : The step of iteration,
${t}_{max}$
The maximum iteration step.
However, there are still problems in equation (12), so in
the primary period of operation, if it detects the optimal point,
it wants to converge to the optimal point promptly. However,
the linear reduction slows down the speed of convergence of
algorithm. In the later period of function, with the reduction of
“w”, it may make the ability of total searching decline and the
variety awaken. Finally it may easily fall into partial optimum. In
this text, we use the PSO method of nonlinear variation weight
with momentum to improve this method is given by,
$$w={w}_{max}-\frac{t\times \left({w}_{max}-{w}_{min}\right)}{{2}^{\theta}{t}_{max}}$$
2
^{θ} is momentum, when in θ = t / t
_{max} ,t is smaller, 2
^{θ} is near
to 1 and w is near to
${w}_{max}$
., it ensure the ability of total searching.
With the increasing of t, w reduces in non linearity, ensuring the
searching ability in partial areas. In the later t = t
_{max} avoiding the
problems caused by the decrease of w. That is, the reduction
ability of total searching and the decline of variety.
Back Propagation (BP) Neural Network
The standard Back Propagation (BP) Neural Network consists
of input layer, one or several hidden layers and an input layer.
The node action function of BP neural network is generally
“S” function. Common activation function f (x) is derivable
sigmoid function:
$$f\left(x\right)=\frac{1}{1+{e}^{-x}}$$
Error function R is:
$$R=\frac{{{\displaystyle \sum}}^{\text{}}{\left({Y}_{mj}-{Y}_{j}\right)}^{2}}{2}\text{}\left(j=1,2,\mathrm{..},n\right)$$
In this formula,${Y}_{j}$
is expected out
${Y}_{mj}$
is actual output n issample length.
The uniform expression of weight modified formula of BP
algorithm is:
$${w}_{ij}^{(t+1)}={w}_{ij}\left(t\right)+\eta {\delta}_{pj}{o}_{pj}$$
The specific process of BP algorithm can be generalized as
follows:
a. Select n samples as a training set.
b. Initialize weight and biases value in neural network. The
initialized values are always random numbers between
(-1, 1). Every sample in the training set needs the
following processing:
c. According to the size of every connection weight, the data
of input layer are weighted and input into the activation
function of hidden layer and then new values are obtained.
According to the size of every connection weight, the new
values are weighted and input into the activation function
of output layer and the output results of output layer are
calculated.
d. If error exist between output result and desired result,
the calculation training is wrong.
e. Adjust weight and biases value.
f. According to new weight and biases values, the output
layer is calculated. The calculation doesn’t stop until the
training set meets the stopping condition.
Modified Particle Swarm Optimization (MPSO)
We apply Particle Swarm Optimization (PSO) to train BP
Network by optimizing the original weight and the threshold
value. When the algorithm ends, we can find the point near the
overall situation optimal point. In the particle swarm, every
particle’s position represents weights set among the BP network
during the resent iteration. The dimension of every particle is
decided by the number of the weight and the threshold value
serving as connecting bridge.
The concrete process of MPSO is narrated as follows:
a. Initialization:
${n}_{i}$
is the number of neurons in the hidden
layer no representing the number of neurons in input
layer. So, the dimension of particle swarm D is:
$$D={n}_{i}\times {n}_{h}+{n}_{h}\times {n}_{\xb0}+{n}_{h}++{n}_{\xb0}$$
b. Setting fitness function of particle swarm’ in this text,
we choose mean square error in BP Neural Network as
fitness function of particle swarm:
$$E=\frac{1}{M}{\displaystyle \sum}_{K}^{m}{\displaystyle \sum}_{j-1}^{{n}_{o}}\left({y}_{kj}-\stackrel{-}{{y}_{kj}}\right)$$
${y}_{kj}$
The output in theory based on sample K
$\stackrel{-}{{y}_{kj}}$
The virtual output based on sample K
M: The number of Neural Network
c. Using the improved particle swarm algorithm to optimize
the weight and the threshold value of BP network.
Coming to the optimal weight and the threshold value based
on equation given below,
$${g}_{best}=\left[\begin{array}{c}{h}_{1},{h}_{2},\mathrm{..},{h}_{{n}_{h}},{\text{o}}_{1},{\text{o}}_{2},\mathrm{..},{\text{o}}_{{\text{n}}_{\text{o}}},{\text{ih}}_{1},{\text{ih}}_{2},\mathrm{..},{\text{ih}}_{{\text{n}}_{\text{i}}\times {\text{n}}_{\text{h}}},\\ {\text{ho}}_{1},{\text{ho}}_{2},\mathrm{..},{\text{h}}_{{\text{o}}_{\text{h}}\times {\text{n}}_{\text{o}}}\end{array}\right]$$
${\text{h}}_{\text{i}}\left(=1,2,\mathrm{..},{\text{n}}_{\text{k}}\right)$The threshold value in the hidden layer
${\text{o}}_{\text{i}}\left(\text{i}=1,2,\mathrm{..},{\text{n}}_{\text{o}}\right)\text{}$
The threshold value in the output layer
${\text{ih}}_{\text{i}}\left(\text{i}1,2,\mathrm{..},{\text{n}}_{\text{i}}\times \right)$
The weight between the hidden layer and
the input layer
${\text{ho}}_{\text{i}}\left(\text{i}=1,2,\mathrm{..},{\text{n}}_{\text{h}}\times {\text{n}}_{\text{o}}\right)$
The weight between the
hidden layer and the output layer
d. Let the optimal weight and the threshold value be as the
original weight and then put them into Neural Network
for training. Adjusting the weight and the threshold value
based on BP algorithm until the function index of the
network’s Mean Square Error (MSE) < e. “e” is the preset expected index.
Simulation Results
Validity of proposed Modified particle swarm optimization
(MPSO) algorithm has been verified by testing in IEEE 30-bus, 41
branch system and it has 6 generator-bus voltage magnitudes, 4
transformer-tap settings, and 2 bus shunt reactive compensators.
Bus 1 is taken as slack bus and 2, 5, 8, 11 and 13 are considered
as PV generator buses and others are PQ load buses. Control
variables limits are given in Table 1. In Table 2 the power limits
of generators buses are listed.
Table 3 shows the proposed MPSO approach successfully
kept the control variables within limits.Table 4 narrates about
the performance of the proposed MPSO algorithm. Figure 1 show
about the voltage deviations during the iterations and Table 5
Table 1: Primary Variable Limits (Pu)
Variables |
Min. |
Max. |
Category |
Generator Bus |
0.90 |
1.11 |
Continuous |
Load Bus |
0.91 |
1.01 |
Continuous |
Transformer-Tap |
0.92 |
1.01 |
Discrete |
Shunt Reactive Compensator |
-0.10 |
0.30 |
Discrete |
Table 2: Generators Power Limits
Bus |
Pg |
Pgmin |
Pgmax |
Qgmin |
1 |
96.00 |
49 |
200 |
-19 |
2 |
79.00 |
18 |
79 |
-19 |
5 |
49.00 |
14 |
49 |
-11 |
8 |
21.00 |
11 |
31 |
-14 |
11 |
21.00 |
11 |
28 |
-12 |
13 |
21.00 |
11 |
39 |
-14 |
Table 3: After optimization values of control variables
Control Variables |
MPSO |
V1 |
1.0541 |
V2 |
1.0452 |
V5 |
1.0263 |
V8 |
1.0374 |
V11 |
1.0745 |
V13 |
1.0575 |
T4,12 |
0.00 |
T6,9 |
0.01 |
T6,10 |
0.90 |
T28,27 |
0.91 |
Q10 |
0.10 |
Q24 |
0.10 |
Real power loss |
4.2809 |
Voltage deviation |
0.9085 |
Table 4: Performance of MPSO algorithm
Iterations |
32 |
Time taken (secs) |
8.91 |
Real power loss |
4.2809 |
Table 5: Comparison of results
Techniques |
Real power loss (MW) |
SGA(Wu et al., 1998) [33] |
4.98 |
PSO(Zhao et al., 2005) [34] |
4.9262 |
LP(Mahadevan et al., 2010) [35] |
5.988 |
EP(Mahadevan et al., 2010) [35] |
4.963 |
CGA(Mahadevan et al., 2010) [35] |
4.980 |
AGA(Mahadevan et al., 2010) [35] |
4.926 |
CLPSO(Mahadevan et al., 2010) [35] |
4.7208 |
HSA (Khazali et al., 2011) [36] |
4.7624 |
BB-BC (Sakthivel et al., 2013) [37] |
4.690 |
MCS(Tejaswini sharma et al.,2016) [38] |
4.87231 |
Proposed MPSO |
4.2809 |
Figure 1:Voltage Deviation (VD) Characteristics
list out the overall comparison of the results of optimal solution
obtained by various methods.
Conclusion
In this paper an inventive approach Modified Particle Swarm
Optimization (MPSO) algorithm is used to solve reactive power
problem, by considering various constraints. The efficiency of the
projected Modified particle swarm optimization (MPSO) method
is demonstrated by testing it on standard IEEE 30-bus system
and the real power loss has been considerably reduced when
compared to other reported standard algorithms & voltage
profile are well within the specified limits .
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