 Open Access
 Authors : Nabiya Ellahi
 Paper ID : IJERTV12IS060094
 Volume & Issue : Volume 12, Issue 06 (June 2023)
 Published (First Online): 26062023
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Comparative Analysis of Particle Swarm Optimization with Classical Methods for Load Flow Analysis
Nabiya Ellahi
Lab Engineer, Institute of Electrical, Electronics & Computer Engineering University of the Punjab
MS Electrical Engineering, University of Engineering and Technology Lahore, Pakistan
AbstractPower flow analysis are one of the main aspects for planning and operation of power system and its analysis. Main aim is to apply Power Flow Analysis provide information about system variables and these variables are complex voltage V, complex power P, and consequently currents, voltages in constant state. Load always remains stationary and it is the power that flows through transmission lines, due to which load flow analysis is preferred to be called as Power Flow Analysis. Through the load flow studies obtained parameters are the voltage magnitudes and angles at each bus in the stationary state. It is a necessity that the bus voltages should remain within a specified limit. Additionally, Particle Swarm Optimization Technique (PSO) is also utilized. Particle Swarm Optimization (PSO) is a computational method that optimizes a given problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. Classical iterative methods such as Newton Raphson method, and Gauss Siedel Method are also applied with artificial intelligence based algorithm Particle Swarm Optimization (PSO). Main aim of this research is to build algorithm to obtain optimized results. For reference comparison of PSO is made between Gauss Siedel and Newton Raphson and the results are also verified through Matlab Codes.
KeywordsParticle Swarm Optimization, Load Flow Analysis, Newton Raphson, Gauss Siedel

INTRODUCTION
Power flow analysis also known as load flow analysis. The study identifies the operational state of a system for given loading. Power flow study solves the system for a set nonlinear algebraic equations for the two unknown variables. To solve these parameters it is required to have fast, accurate and efficient numerical techniques[1 2]. The important information which we acquire from this analysis is

Magnitude of voltage

Phase angle of Voltage
The output of power flow analysis is the real and reactive power, slack bus power and line losses.
Moreover particle swarm optimization technique will be adopted to search for appropriate bus voltages and phase angles. PSO is an optimization technique in which particles change their position with time. In this system, particles fly around in multidimensional search space. Every particle in the swarm tries to look for best possible position which is linked as the best possible solution that has been so far attained by that particle. Particles have the capability to
change their position by forming communication with neighboring particles by utilizing the best position encountered by itself and its neighbors. There is another best value which is known as global best and is tracked by the PSO. This is the best possible value that has been obtained by any particle in the neighborhood. Implementation of PSO is quite convenient as only few parameters requires adjustment. PSO has been fortuitously applied to solve optimization problems in the area of electric power systems such as: economic dispatch, Reactive Power Control and Power Losses Reduction function, Optimal Power Flow (OPF), Power System Controller Design, artificial neural network training, generation expansion planning, load forecasting, feederswitch relocation problem and fuzzy system control. This technique was first introduced by Kennedy and Eberhart motivated by social behavior of swarms such as fish schools and bird flocking.


PROBLEM STATEMENT Determination of voltage magnitude and phase angle obtained at each bus Determine the active and reactive power flow in each power line. Each bus has four state variables

Voltage magnitude

Voltage phase angle

Real power injection

Reactive power injection


EXPLANATION
Each bus has either two of the four above variables described or given. It is a usual practice to consider first bus as slack bus. The angle of voltage of this bus basically serves as a reference for all other buses and their angular voltages. Angle designated to slack bus is generally 0 which is not considered important because the difference between voltage and angles determines the calculated values of Pi and Qi.[3] No defined mismatches are defined for the slack bus, voltage magnitude V is specified as the other known
quantity along with 1 = 0Â°. Therefore, there is no requirement to
include the slack bus in the powerflow problem[4]. Rest of the two
buses are described below

Regulated bus (generator bus, PV bus)

Generation model station buses.

Real power and magnitude of voltage are given.

Solution: Reactive power flow and angular voltage.


Load Bus (PQ bus)

Models load center buses

Active and reactive power injections are given (Negative values for loads).

Solution: Magnitude of voltage and angle


TECHNIQUES ADOPTED FOR SOLVING POWER
FLOW PROBLEMS:
The functions Pi and Qi are nonlinear functions of the state
variable Vi and i. Therefore iterative techniques can be utilized to solve a power flow problem. The techniques which
are most commonly adopted for solving a powerflow problem are

NewtonRaphson Technique

GaussSiedel Power Technique


NEWTON RAPHSON TECHNIQUE
Newton Raphson is mathematically a better technique as compared to GaussSiedel Method due to its quadratic convergence. Its divergence is least with illconditioned problems. For prominent powersystems, the Newton Raphson method is found to be

More Efficient

More Practical
The number of iterations required to obtain a solution is independent of the system size, but more practical evaluations are required at each iteration. [5]
The power flow equation is given in polar form. Ii= Yij < + (1)
where Yij is the admittance of the matrix in between the
buses i and j and is the voltage at bus shows the angular voltage in between buses i and j and is the phase
angle at bus j. For the typical bus of the powersystem the
current entering at bus i is specified by

Find new estimates for the bus voltage magnitudes and angles

Repeat the process until the mismatch (residuals) are less than the specified accuracy.[67]
 Pi k) (5)
 Qi (k) (6)
this symbol stands for epsilon not which defines specified
accuracy of the system

Merits of Newton Raphson Technique:
This method is superior to Gauss Siedel and fast decoupled power flow solution due to following advantages and reasons


The solution is reached within minimum iterations which are usually not more than 3.
3 iterations are required to obtain the solution. In case of GaussSiedel Method minimum iterations should be 7 and with fast decoupled method minimum iterations should be 14.

The accuracy of this method is greater with a power mismatch of 2.5Ã—104.[8]

Demerits of Newton Raphson Method
The disadvantages of this method are summarized below.

/li>


It is a very lengthy method and requires large computer memory

Computer Programming is difficult
V. GAUSS SIEDEL TECHNIQUE:
In the power flow study it is important to solve the set of non linearized equations for two unspecified variables at each node. In the GaussSiedel method following equation is used to solve for Vi, and the iterative process becomes
Ii=
=1
Yij (2)
()
+yijV
(k) j
The above equation includes j at bus i. Expressing the equation in polar form.
Vi(k+1) =
ji (7)
Ii=  < + j (3)
Where yij shown in lowercase letters is the actual
i
admittance in per unit. P sch
i
and Q sch
are the net real and
The complex power will be expressed as
reactive power expressed in per unit.
=1
PijQi=Vi<
Yij  < + j (4)

Gauss Siedel Method:
In the above equation Pi is the active power of the system at bus i and Qi is the reactive power at bus i and j is the vector
operator which shifts the vector by an of 90 Â°.
In case of GaussSiedel method, an initial estimate of 1+j0
for unspecified voltage is satisfactory and the converged solution co relates with the actual operating states.
Real And Imaginary Components of Voltages;
(k+1) )2 + (f (k+1))2 = V 2 (8)

Newton Raphson Steps
(ei i i
(k+1) )2 = V 2 (f (k+1))2 (9)
The process for power flow solution by Newton Raphson method involves the following steps
(ei i i
(k+1) and (f (k+1)) are real and imaginary parts of the

Set Flat Start:
where ei
i
(k+1) of the iterative sequence.

Calculate power mismatch:

Form the Jacobian Matrix:

Find the Matrix Solution (choose a or b):
voltage Vi
Rate of Convergence:
The rate of convergence is increased by applying an acceleration factor to the approximate solution obtained from each iteration.
minimized later. Having that done, the iterations are initialized. The following process is accomplished to each particle of the swarm.
V
(k+1)
i
V
(k)
= i
+ (Vical
(k)
i

V (k)
) (10)
is the acceleration factor. Its value depend upon the
system. The range of 1.3 or 1.7 is satisfactory. It has the
advantage that it can improve the rate of convergence if
> 1


Application of Gauss Siedel Power Flow:
The consumer wants to know the voltage profile
The nodal voltages for a given load and generation schedule Types of Network Buses:

Load Bus:
Known real (P) and reactive (Q) power injections
Generator Bus:
Known real (P) power injection and the voltage magnitude (V)
Slack Bus:
Known voltage magnitude (V) and voltage angle () Must have one generator as the slack bus.
Takes up the power slack due to losses in the network. [9]


Solution by GaussSiedel
System Characteristics:

Since both components (V &) are specified for the slack bus,

There are 2(n1) variables which must be solved iteratively.

For load buses, the real and reactive powers are known: scheduled

The voltage magnitude and angle must be estimated

In per unit, the nominal voltage magnitude is 1pu.

The angles are generally contiguous, so an initial value of 0 degrees is appropriate.



PARTICLE SWARM OPTIMIZATION (PSO):
In this technique the particles positions can assume continuous values within the limits specified in the input data. The rule function parameters will be minimized in the PSO algorithm which is defined as fitness. The fitness is defined as the sum of the buses apparent power. Each particle has a local fitness, value obtained by its local best. The global fitness is the fitness related to the best global of all the particles. The global fitness is fitness related to the best global of all particles. The current fitness is the fitness obtained by a particle at a given iteration. [10] The first step of algorithm is to generate the initial values to particle position. Velocities, local best parameter and global best parameter. The angle receives random initial value with in specified boundary. Before initialization of the module value of each particle, the bus type needs to be verified and related in the equation. In the case of a PQ bus, the voltage module receives a random value within the specified boundary, for a PV bus, the voltage module receives the related value specified in the input data. The initial velocities are null. The local best parameters receive the particles positions values and the global best parameters receive the first particle value, arbitrarily. The grades are initialized with high values in order to be

Description of PSO:
The particle swarm optimization algorithm is multiagent parallel search technique which maintains a swarm of particles and each particle represents a potential solution in the swarm. Each particle will keep a track of its coordinates in the given space which are linked with the best solution (fitness) that it has reached so far. This optimum value of particle is called pbest (local best).Another best value which is obtained by PSO will actually be the best position that is obtained by any particle in the whole swarm and that value is called gbest (global best).[11]
+1 = + 1 1( ) + 2
2( ) (11)
All the symbols are defined in detail in the next page
Fig1. Diagram of PSO:
Fig.1 Particle Swarm Optimization Flow Diagram

Parameters of PSO:
There are some parameters which are of paramount importance in determining efficiency of PSO while others have belittle effect. The important parameters of PSO are number of iterations, acceleration coefficients, swarm size, velocity components, acceleration coefficients and inertia weight.[12]

Swarm size
Swarm size is the number of particles n in swarm. A big swarm provides larger space to cover per iteration. A large number of particles will minimize the number of iteration needed to obtain good optimization result. In comparison, large amount of particles per iteration enhances the computation complexity per iteration. Therefore most of the PSO implementations use an interval of 20 to 60 for swarm size

Iteration number
The number of iterations is also necessary to obtain a good result in optimization. Too low number of iterations may stop the search process prematurely, while too large iterations have the consequences of adding unnecessary computational complexity and more time requirement
[1315] 
Velocity components
The velocity components are very important for updating particle velocity. There are three terms of particles velocity

The term is called inertia component that provides the
previous flight direction. This component represents as
moment which prevents to drastically change the direction of the particles and to direct towards the current direction.

The term 1 1 (ti k Xi k) is called social component
which measures the performance of particles i with respect to
their neighbors. The social components effect is that each particle flies toward the best position found by the particles in neighborhood.

The term 2 2 ( Xi k) is called cognitive
component which measures the performance of particles i
relative to past performance [1617]. iv). Acceleration Coefficient:
The accelerationcoefficient 1 and 2 with random values 1 and 2 maintain the influence of social and cognitive components of particles velocity. The 1 shows how much confidence a particle has in its neighbors, while 2 shows how
much confident a particle has in itself. There are some
properties of 1 and 2.

When 1 = 2=0 then all particles continue to fly at their
current speed until they hit the search spaces boundary and
velocity updated equation is calculated as +1 =

When 1 > 0and 2 = 0 then all particles will be attracted to
single point G best in the entire swarm and updated velocity
becomes +1 = + 1 1 ( Xi k )

When 2 > 0 and 1 = 0 then all particles are independent and updated velocity becomes +1 = + 2
2( )

When 1 = 2 all particles are attracted towards the average of and .[18]
When1 2then all particles are largely influenced by global
best position, which cause particles to run prematurely to
optima. In contrast, in scenario when 2 1, each particle is
strongly influenced by personal best position, which results
in
excessive wandering.
v) Inertia weight:
Inertia weight in PSO plays an important role because of its control on particle speed. Hence, a suitable selection of it is important. Its value is from 0.1 to 0.9
w = rand(1) (12) where w stands for inertia weight


Steps of the PSO Algorithm

Assign the PSO algorithms parameters.

Initialize the particles position as bus voltage.

Real part of bus voltage is between 0.9 to 1pu population size and bus voltage.

Imaginary part of the bus voltage are generated between 0.1 to 0 pu.


Set iterations.

Initialize local best particle as bus voltage.

Calculate objective function.

Calculate Fitness Function.


Check whether the particle is fit or not.

Go to global particles and check whether it is suitable as compared to the local particle.

Update velocities and particle positions.

Go to next particle.

After all particles go to next iteration. End finish this process till last iteration
Fig. 2 Flow Chart of Particle Swarm Optimization Technique: Fig 3. Single Line Diagram For Load Flow Analysis:
Initialize Current Position as Global Position
Iteration=Iteration+1
Particle=Particle+1
Initialize Local Best Position
Calculating Objective Function
Calculating Fitness Function
if P P(Fitness k+1) < P(Fitness k) Yes
no
G Best = P Best
no
Calculate Fitness Function
Calculate Objective Function
Check Global Best
if G Best Fitness < P Best fitness Yes
Update Velocities and Particle Position
Next iteration till end of Iteration
Fig.2 Flow Chart of Particle Swarm Optimization Technique
Fig.3 Single Line Diagram for Load Flow Analysis
A. Single Line Diagram and Computation of Load Flow Analysis
It is 26 bus system Single Line diagram. Voltages and phase angles on each bus are calculated using two techniques of power flow which are Gauss Siedel and Newton Raphson. SLD was simulated on Software which is Power World Simulator (PWS) and also the bus voltages were verified and phase angles by using Matlab Code. 45 iterations were done for Gauss Siedel and 5 iterations for NewtonRaphson to achieve accuracy[19]. The results of simulations and Matlab code matched and accuracy level was also attained.
Fig 4. Simulation of Single Line Diagram In PWS (Power World Simulator):
GaussSiedel and Newton Raphson technique are shown below:
A. Particle Swarm Optimization Results
Table below depicts results obtained from Particle Swarm Optimization technique
Tab.1 PSO Code Results
720MW
1.00 pu145Mvar
4.0Mvar
15Mvar
108Mvar
Bus No Bus Voltages
1) 1<0
2) 0.9999<0.7735
3) 1<5.8137
4) 0.9999<9.2767
5) 1<4.318
6) 0.96362<4.613
7) 0.97707<4.69
8) 0.98366<1.057
9) 0.936<9.8668
10) 0.9613<10.882
11) 0.9035<2.416
12) 0.95086<6.582
13) 0.98448<7.57
14) 0.97204<5.0343
15) 0.9830<16.183
16 0.96611<14.102
17 0.9695<10.729
18 0.984<1.8577
19 0.9921<7.6163
20 0.9896<9.15110
21 0.9444<4.1227
22 0.94445<8.813
23 0.92476<3.335
24 0.9502<9.675
25 0.9898<5.35
26 0.9999<6.1712
20MW
A
153MW
0.00 Deg
51MW
41Mvar
A
slack
1
22MW
A
79MW
A
173Mvar
3
1.00 pu
Amps
67Mvar
Amps
300MW
71Mvar
A
Amps
1.00Apmpsu
1.01 Deg 2
A
A
Amps
4.64 Deg
A
A
31MW
0.98 pu
1.86 Deg
1.00 pu
1.66 Deg5.0 var
40MW
20Mvar
Amps
A
Amps
Amps
Amps
8 0.98 pu 64MW
Amps
15Mvar
18 5
26 0.98pu
60MW 1.00 pu 3.46 Deg
7 100MW
3.60
Deg
50Mvar
13 0.99 pu
A A 47Mvar
1.97 Deg
188Mvar
25MW 2.0Mvar 24MW
4.84 Deg
0.97 pu
Amps Amps 30Mvar
50MW
A
Amps
A
A Amps 10Mvar
12Mvar
2.58 Deg
6
1.9 Mvar
A
15Mvar 0.96 pu
75MW 5.85 Deg
9
Amps
1.00 pu
3.98 Deg
A 4
Amps A
0.98 pu
A
Amps
A
Amps

pu
A
A 89MW
Amps
5.46 Deg
6.32 Deg
29Mvar Amps
50Mvar
48MvaAmrps 12 14 16

pu1.4Mvar Amps 76MW A
4.5 Mvar
89MW A 0.98pu
11 Amps
Amps
A
1.9 Mvar
5.09 Deg A
3.35 Deg 0.93 p2u8MW
13Mvar
A
Amps 19
A

pAumps
Amps
Amps
A
A
A
Amps
7.14
Deg 25
A 6.6
8 Deg

pu 10
Amps
55MW
12Mvar Amps 54MW
Amps 27Mvar
25MW A
15Mvar Amps
25MW
A
A Amps
27Mvar
45MW
5.95 Deg
A
78MW

pu
A
0.93 p2u3
Amps
24
A
0.93 p2u2Mvar
7.74 Deg
A
Amps
Amps
48MW
38Mvar
70MW
0.5 Mvar 5.98 Deg
15
Amps
7.49 Deg
46MW 21
23Mvar
Amps
A
Amps 0.95 pu
A
Amps
27Mvar
20
A
31Mvar
0.95 pu
6.05 Deg
A
22 6.85 Deg
0.95 pu
6.43 Deg
17 0.96 pu
Amps
A
A
Amps
Amps
A
Amps
5.15 Deg
Amps
Fig.4 Simulation of Single Line Diagram in Power World Simulator


Matlab Results:
Matlab results obtained after implementation of Particle Swarm optimization and comparative analysis between

Comparative Analysis of Results Between Gauss Siedel and Newton Raphson Power Fow Techniques

GaussSiedel 45 Iteration Result:
Table below shows the results obtained from Gauss Siedel Analysis
Tab 2 Gauss Siedel Iteration Results

Newton Raphson 5th Iteration Result:

Table below shows the results obtained from Newton Raphson Analysis
Tab. 3 Newton Raphson Iteration Results
Bus No 
Voltages 
1 
1<0 
2 
1<0.55575 
3 
0.9999<1.90266 
4 
1<2.5848 
5 
0.9999<1.0428 
6 
0.9769<2.317 
7 
0.986<1.16226 
8 
0.9886<1.6693 
9 
0.9712<3.47 
10 
0.96966<3.9146 
11 
0.96527<4.8612 
12 
0.98463<2.8699 
13 
0.99448<2.27 
14 
0.98963<2.623 
15 
0.98454<2.9926 
16 
0.97743<3.5134 
17 
0.958826<3.905 
18 
0.98887<1.0777 
19 
0.9276<6.804 
20 
0.9620<4.3818 
21 
0.9395<5.8829 
22 
0.94876<5.243 
23 
0.88612<9.7588 
24 
0.91277<7.5797 
25 
0.79314<13.384 
26 
0.9999<6.206 
Bus No 
Voltages 
1 
1<0 
2 
0.99998<1.35 
3 
0.9999<3.853 
4 
1<5.6353 
5 
1<3.537 
6 
0.9791<4.6397 
7 
0.98004<3.609 
8 
0.9858<4.7419 
9 
0.96855<6.7713 
10 
0.96986<6.8216 
11 
0.97763<7.4405 
12 
0.9835<5.78305 
13 
0.99434<4.262 
14 
0.98928<4.8766 
15 
0.98283<5.3184 
16 
0.97831<5.8020 
17 
0.96424<5.248 
18 
0.99056<1.938 
19 
0.93267<9.3421 
20 
0.963<7.158 
21 
0.9428<8.318 
22 
0.9509<8.107 
23 
0.89129<12.4005 
24 
0.91677<10.228 
25 
0.8024<15.918 
26 
0.9999<8.493 
X. Conclusion
Matlab code did 45 iterations for Gauss Siedel to obtain the results. Most desired optimum results should be close to 1. In Gauss Siedel Results magnitudes of voltages were 0.8 on some buses. While in Newton Raphson code did 5 iterations and these results were more precise than Gauss Siedel. In PSO number of particles are hundred and we obtained the desired results in just 20 iterations and the results were approximately near to one and also comparable with Newton Raphson. PSO is preferable as compared to NewtonRaphson because in very less time it give results and it does not require large memory as for Newton Raphson. It takes less runtime [20]. It just involves 2 loops. One for iterations and one for particle while for Newton Raphson each iteration has 4 loops and each loop runs for 25 times to make jacobian matrix. And one extra loop for mismatch calculation. One extra loop is used to calculate fitness function. And we can get suitable answers by running the program twice or thrice so that a best particle would be picked up but in PSO results are somewhat different every time because of the random samples but every time the values are acceptable because they are close to expected results obtained from NewtonRaphson. As our objective was to get results in less time. So by setting 20 iterations in PSO case optimum results were obtained. This scheme has been duly and successfully implemented. So it has been shown that mathematical conversion of heuristic technique can be easily translated to the Load Flow Problem. When this technique was compared to Newton Raphson the results were although similar to each other and the results obtained by this technique were more appropriate.
ACKNOWLEDGMENT
I am thankful to all Professors and other faculty staff of Institute of Electrical, Electronics & Computer Engineering, University of the Punjab, Lahore for their constant support and encouragement. I also want to thank my parents and family members without their constant support and guidance this could not have been possible, for they taught me the worth of hard work. They gave me the encouragement and motivation to complete this task with utmost dedication. I would also like to thank to my siblings and wellwishers.
Last but not the least, I would like to render my heartiest thanks to my friend whos ever helping nature and suggestions had helped me to complete this present work.
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