CHAPTER 4 GENETIC SEARCH FOR A MULTI-OBJECTIVE OPTIMAL POWER FLOW
Transcript of CHAPTER 4 GENETIC SEARCH FOR A MULTI-OBJECTIVE OPTIMAL POWER FLOW
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CHAPTER – 4
GENETIC SEARCH FOR A MULTI-OBJECTIVE OPTIMAL POWER
FLOW SOLUTION FROM A HIGH DENSITY CLUSTER
4.0 INTRODUCTION
A new algorithm is developed for the solution of multi-objective
optimal power flow problem. Two individual objective functions are
chosen: 1) minimization of Fuel Cost and 2) minimization of Power
Loss. The proposed algorithm is the application of a multi-objective
genetic algorithm (MOGA), using the combination of High Density
cluster and continuous genetic algorithm. The OPF is modeled as a
nonlinear, non-convex and large scale constrained problem with
continuous variables. The algorithm uses a local search method for
the search of Global optimum solution. Binary coded Genetic
algorithm is replaced with continuous genetic algorithm that uses
real values of generation instead of binary coded data. An attempt is
made to reduce the chromosome length.
4.1 OBJECTIVES OF PROPOSED METHODOLOGY
Chapter – 4 presents need for alternative methodologies that
can avoid all the difficulties in the various approaches and provide a
better OPF solution. In this work, a Multi Objective Genetic algorithm
combining with high density cluster algorithm is proposed. This
methodology has the following objectives:
Aims for incorporating a local search method within a genetic
algorithm that can overcome most of the obstacles that arise as a
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result of finite population size.
Aims for better improvements in speed and accuracy of solution.
Aims for consideration of large varieties of constraints and system
nonlinearities.
Aims for avoiding the blind search, encountering with infeasible
strings, and wastage of computational effort.
Aims for reduction in population size, number of populations in
order to make the computational effort simple and effective,
considering population is finite in contrast to assuming it to be as
infinite.
Aims for testing other types of Genetic Algorithm methods instead
of conventional GA that uses binary coded chromosomes.
Aims for a suitable local search method that can achieve a right
balance between global exploration and local exploitation
capabilities. These algorithms can produce solutions with high
accuracy.
Aims for identification and selection of proper control parameters
that influence exploitation of chromosomes and extraction of
global optimum solution.
Aims for improvements in coding and decoding of Chromosome
that minimizes the population size.
Aims for undertaking multi-objective OPF problem. By integrating
objective functions, other than cost objective function, it can be
said economical conditions can be studied together with system
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security constraints and other system requirements.
The elitism was implemented to increase performance of algorithm
and prevent loss of good quality solutions found during the search.
4.2 SALIENT FEATURES IN THE PROPOSED METHODOLOGY
This work aims for examining several issues that need to be
taken into consideration when designing, a genetic algorithm that
uses another search method as a local search tool. These issues
include the different approaches for employing local search
information that is useful for genetic algorithm searches for global
optimum solution. The salient features in this work are depicted in
Fig 4.1.
The algorithm proposed and developed elitist is based on a
population that contains all the primary non-dominated solutions
over the generations. This is due to fact that GA does not begin by a
random generation of population but with parent chromosome which
Fig: 4.1 Salient Features of the work
Uses an existing popular method as a local search method to obtain a
Global optimum solution
Salient features of Proposed OPF Algorithm
Under takes Multi-Objective OPF problem for minimization of fuel
cost and power loss
Uses continuous Genetic Algorithm to develop population around the
suboptimal solution typically a high density cluster
Develops MOGA to search for an optimal OPF solution for the said
Multi-objective OPF problem.
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is a sub optimal solution, obtained through popular existing OPF
methodology.
The size of chromosome is restricted to minimal in this method.
This is due to fact that it does not contain control variables like
voltage magnitudes, tap position etc. As the size of chromosome is
short, population size can be minimum.
4.3 STRUCTURE OF PROPOSED OPF SOLUTION METHODOLOGY
Inspired by the results of EGA method and to overcome the
general difficulties in GA or EGA approaches, a novel method is
proposed in this work. The method uses high density cluster DBSCAN
and Continuous GA algorithms. The new technique for the solution of
OPF based on Genetic search from a High Density Cluster named in
short form as “GSHDC” is proposed in this work. The objective of
GSHDC is to retain advantages of Mathematical Programming
techniques and to encounter the difficulties of evolutionary methods
like GA and PSO Methods.
Proposed GSHDC has three stages for Single Objective OPF
solution and has four stages for Multi-Objective OPF solution. Stages
1 and 2 are common for both.
Stages in GSHDC Algorithm for Single Objective OPF Solution:
In the first stage a suboptimal solution for OPF problem is
obtained by the conventional analytical method such as Interior
Point method OR by PSO Methods that considers equality
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constraints, transmission loss B-Coefficients and penalty factors.
This solution for OPF problem is treated as an approximate,
owing to the limitations in the methods. However this solution
shall give a better insight in to the exact solution as the OPF is
solved with the regular proven methods.
In the second stage, with the help of a newly proposed
continuous data chromosome a population is formed
surrounding the suboptimal solution that is obtained in the first
stage by GA and two individual high density clusters one for
minimum fuel cost and another for minimum power loss are then
created by DBSCAN algorithm. The high density cluster consists
of several suboptimal solutions, one of which can be the exact
one.
In the third stage, a genetic search is carried out for finding the
exact solution. The solution in the last stage is the exact one, as
is confirmed by the best Fitness Value. The proposed GSHDC
technique in contrast to GA method avoids the blind search,
encountering with infeasible strings, and wastage of
computational effort.
Stages in GSHDC Algorithm for Multi- Objective OPF Solution:
As mentioned earlier, the first two stages in single objective
GSHDC algorithm are common in case of multi-objective OPF
solution. In these stages, two individual high density clusters are
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formed. They consist number of high density core points which are
having best fitness function values.
In the third stage, each core point in each high density cluster is
assigned a membership function value. In minimum fuel cost
cluster, the OPF solution with minimum fuel cost amongst the
others is assigned membership function value of 1 and relatively
maximum fuel cost is assigned 0. The same procedure is followed
for minimum power loss cluster points.
In the final stage, a search is carried out for the exact multi
objective optimal solution using Multi Objective Genetic Algorithm
(MOGA) based on Pareto- Optimal.
Above stages are explained in detail as below:
Common Stages in GSHDC Algorithm for Single Objective &
Multi-Objective OPF Solution:
Stage-1: In the first stage a suboptimal solution for OPF problem is
obtained by any of the following local search methods 1) Modified
Penalty Factor Method 2) Primal-Dual Interior Point method and 3)
Particle Swarm Optimization Method that considers Lagrange
multipliers, equality constraints, transmission loss B-Coefficients and
penalty factors. Owing to the limitations of the methods as discussed
in Chapter-3, this solution is taken only as suboptimal or local
optimal. However this solution shall give a better insight in to the
exact solution as the OPF is solved with the regular mathematical
programming / intelligent approaches. Because of this reason, this
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OPF solution cannot be taken as a global one. However due to
consideration of constraints of control parameters this solution gives
a core point in the high density cluster. The OPF solution for
minimum fuel cost and minimum power loss is show in Fig. 4.2 and
4.3, which are initial core points for the two individual high density
clusters for minimum fuel cost and power losses.
`
Stage-2: Owing to the limitations of the local search methods, in the
second stage, two independent High Density Clusters, which consists
of other suboptimal data points in the vicinity of the first are formed
by using Continuous Genetic Algorithm and DBSCAN algorithm. This
is also illustrated in Figs 4.2 and 4.3. The active power generation of
individual generating units is taken as continuous control variables
and the suboptimal solution obtained in the first stage, is first
encoded into a chromosome. This chromosome is treated as one of
Fig: 4.2 High density Cluster for minimum fuel cost
Initial Core Point obtained
through suboptimal solution methods
Cluster for Minimum cost
Other high density cluster points for
minimum-cost
Fig: 4.3 High density cluster for minimum power loss
Other high density cluster points for
minimum-loss
Cluster for Minimum-
power loss
Initial Core Point obtained
through suboptimal solution
methods
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the core points and the Fitness Function (FF) value of this
chromosome is calculated and is termed as Eps. Then the selection,
crossover and mutation processes are carried out for generating new
population consisting of other core points (or say other suboptimal
solutions), subject to FF values of these chromosomes are within Eps.
This forms a high density cluster and thoroughly avoids noise points
and border points which are regarded as infeasible solutions. It can
be stated here that, the length of chromosome in the proposed
method is reduced due to non consideration of certain control
parameters. This reduces the size of population of the high density
cluster to a large extent. However, the constraints of the control
parameters are considered in the third stage before arriving to the
exact optimal solution.
Further Stages in GSHDC Algorithm for Single Objective OPF Solution:
Stage-3: In the third stage, a genetic search is carried out for
finding the exact solution. The solution in the last stage is the exact
one, as is confirmed by the best Fitness Value.
Further Stages in GSHDC Algorithm for Multi-Objective OPF Solution:
Stage-3: In the third stage, each core point in each high density
cluster is assigned a membership function value. The basis for
assigning membership function values are assigned is discussed
below.
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Membership function values
The fuzzy sets are defined by equations called membership
functions. These functions represent the degree of the membership in
some fuzzy sets using values from 0 to 1. The membership value 0
indicates, incompatibility with the sets, while value 1 means full
compatibility. By taking account of the minimum and maximum
values of each objective function together with the rate of increase of
membership satisfaction, the decision maker must detect
membership function ( )iF in a subjective manner. Here it is
assumed that ( )iF is a strictly monotonic decreasing and
continuous function defined as:
min
maxmin max
max min
max
1 ;
( ) ;
0 ;
i i
i ii i i i
i i
i i
F F
F FF F F F
F F
F F
(4.1)
The value of membership function suggests how far (the scale
from 0 to 1) a non–inferior (non-dominated) solution has satisfied the
Fi objective. The sum of membership function values ( )iF
(i = 1, 2,.…M) for all the objectives can be computed in order to
measure the accomplishment of each solution in satisfying the
objectives.
Stage-4: In the final stage, a search is carried out for the exact multi
objective optimal solution using a Multi Objective Genetic Algorithm
(MOGA)[115,116]. The accomplishment of each non-dominated
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solutions can be read with respect to all the K non-dominated
solutions by normalizing its accomplishment over the sum of the
accomplishments of K non-dominated solutions as follows:
1
1 1
( )
( )
Mk
i
ik
D K Mk
i
k i
F
F (4.2)
The function D in Eq.(4.2) can be treated as a membership function
for non-dominated solutions, in a fuzzy set and represented as fuzzy
cardinal priority ranking of the non-dominated solution. The solution
that attains the maximum membership k
D , in the fuzzy set so
obtained can be chosen as the ‘best’ solution or the one having the
highest cardinal priority ranking.
Max {k
D : k = 1, 2,…., K} (4.3)
Based on the above procedure the best solution is obtained from Non-
domination sort data which has a membership function value close
to D , which also satisfies the constraints and convergence of load
flow.
GSHDC Method is implemented for two single objectives and one
multi-objective covering 3-Test cases (OPF Methods) and 3-Case Studies
(IEEE Test Systems) as described below:
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Suboptimal solution is obtained for two individual
objectives and one Multi-objective:
Objective-1: Minimum Fuel Cost
Objective-2: Minimum Power Loss
Using the OPF solutions obtained through objective-1 & 2 as
parent chromosomes, population is generated for the multi-objective
OPF problem. This is referred as:
Objective-3: Multi-Objective which includes Objective-1 & Objective-2.
Sub optimal OPF solutions for Minimum Fuel Cost and
Minimum Power Loss are obtained through following methods.
Test-1: Suboptimal Solution for Minimum Fuel Cost and Minimum
Power Loss is obtained through IP method
Test-2: Suboptimal Solution for Minimum Fuel Cost and Minimum
Power Loss is obtained through PSO method.
Test-3 In addition to above two test cases, GSHDC is also
implemented with suboptimal solution obtained through
modified penalty factor method to test its effectiveness. This
case is referred as Test-3.
Two individual High Density Clusters for Minimum Fuel
Cost and Minimum Power Loss are formed using continuous
Genetic algorithm for the following three case studies:
Case-1: IEEE 14-Bus System, Case-2: IEEE 30-Bus System
Case-3: IEEE 57-Bus System
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Summary of results are furnished as per the format given below:
4.4 PROPOSED GSHDC ALGORITHM FOR SINGLE & MULTI-
OBJECTIVE OPTIMAL POWER FLOW SOLUTION
The GSHDC involves the following steps:
Stage-1: Steps involved in stage-1 (for obtaining Sub-Optimal
Solution):
Step-1: Read Data namely: Cost coefficients of all units, B-
coefficients, convergence tolerance, error, step size and
maximum allowed iterations, Population Size, Probability of
Cross-over, Probability of mutation, λmin and λmax. System bus
data, load data.
Sl.No Method Description
1 Test-1. objective-1, Case-1/ 2/ 3
GSHDC-IP method, Min. Fuel Cost,14/30/57-Bus System
2 Test-1. objective-2
Case-1/ 2/ 3
GSHDC-IP method, Min. Power Loss,
14/30/57-Bus System
3 Test-1. objective-3, Case-1/ 2/ 3
GSHDC-IP method, Both Min Fuel Cost& MinPower Loss, 14/30/57- Bus System
4 Test-2. objective-1,
Case-1/ 2/ 3
GSHDC-PSO method, Min. Fuel Cost,
14/30/57-Bus System
5 Test-2. objective-2 Case-1/ 2/ 3
GSHDC-PSO method, Min.Power Loss, 14/30/57-Bus System
6 Test-2. objective-3,
Case-1/ 2/ 3
GSHDC-PSO method, Both Min Fuel Cost &
MinPower Loss,14/30/57- Bus System
7 Test-3. objective-1,
Case- 2
GSHDC-Penalty Factor method, Min. Fuel Cost,
30-Bus System
8 Test-3. objective-2,
Case- 2
GSHDC-Penalty Factor method, Min. Fuel Cost,
30-Bus System
9 Test-3. objective-3,
Case-2
GSHDC- Penalty Factor method, Both Min.Fuel
Cost& MinPower Loss,30- Bus System
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Step-2: Obtain Sub Optimal Power Flow solution that is, generation
schedule for minimum fuel cost and minimum power loss
through any one of the suboptimal solution methods (Sections
4.6, 4.7 and 4.8) for a given System load. This OPF solution
is the initial core point in the respective high density cluster.
(End of the Process for obtaining sub optimal OPF solution)
Stage-2: Steps involved in stage-2 (for High Density Cluster
Formation):
Step-3: Develop method for coding and decoding of chromosome
(string) for initial core point of high density cluster. (Section
4.9.1).
Step-4: Map cost of generation in to Fitness Function. (Section 4.9.2)
Step-5: Compute Fitness value of initial core point and assign it as
Eps (section 4.9.2)
(End of the Process for obtaining Initial Core Point in High
Density Cluster)
Steps involved in obtaining other Core Points in High Density
Cluster - generation of population
Step-6: Randomly generate population (other core points) in and
around initial core point of high density cluster (Section
4.9.3).
Step-7: Carryout Parent Selection Process from high density cluster
points (section 4.9.3)
Step-8: Carryout Cross over process. (Section 4.9.4)
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Step-9: Carryout Mutation Process. (Section 4.9.5)
(End of the Process for formation of High Density Cluster)
Step-10: For Single Objective OPF Solution GOTO Step-11
for Multi-Objective OPF solution GOTO Step-18
Single Objective OPF solution using individual high density clusters i.e.
minimum fuel cost and minimum power loss.
Stage-3: Steps involved in stage-3 (search for exact OPF solution)
are given below:
Step-10: Compute Fitness values of the newly generated population
(Section 4.9.2).
Step-11: Compare Fitness values of each chromosome with Eps.
Step-12: Store better fitness value chromosomes as high density
cluster points and place the rest in to noise cluster.
Steps 10 to 13 are repeated for all high density cluster points.
Exact OPF Solution
Step-13: Select the core point having highest fitness value, from a
high density cluster obtained from Step-13. (Section 4.9.5)
Step-14: For selected core point chromosome, run Load Flows to
check convergence and check equality and inequality
constraints satisfaction, limits for control parameters etc. as
per Eqns. (3.13) to (3.20). Also check for slack bus
generation limits.
Step-15: For the selected core point chromosome, if violation in limits
for constraints takes place, send this core point in to noise
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cluster and select another core point chromosome with next
highest fitness value. Go To Step-15. Else Go To next Step.
Step-16: The decoded core point chromosome is treated as OPF
solution.
Step-17: Print OPF results (Individual Generation Schedule for
Minimum Fuel Cost and Minimum Power Loss)
Multi – Objective OPF solution using individual high density
clusters i.e. minimum fuel cost and minimum power loss.
Stage-3: Steps involved in stage-3 are given below:
Step-18: Assign a membership function value to each core point in
each high density cluster (Section 4.3).
Stage-4: Steps involved in stage-4 are given below:
Step-19: Using MOGA Algorithm compute member ship function
values for k-non dominated solutions k
D as described in
Section 4.3.
Step-19: Obtain OPF solution for multi-objective problem, a
generation schedule having maximum membership function
value i.e. from Eq.4.3.
Max { k
D : k = 1, 2,…….., K.}
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4.5 MULTI-OBJECTIVE OPTIMAL POWER FLOW PROBLEM-
OBJECTIVE FUNCTIONS
The selected problem can be designated as a multi-criteria and
multi-objective optimization problem which requires simultaneous
optimization of two objectives with different individual optima.
Objective Function-1: Total Fuel Cost
Total Generation cost function is expressed as:
2
1
( )G
i i
N
G i i G i G
i
F P P p
(3.1)
The objective function is expressed as:
Min F (PG)= f1 (x,u)
Constraints are mentioned in the set of Eq. (3.7 -3.20)
Objective Function-2: Total Power Loss
The objective functions to be minimized are given by the sum of line
losses
1
lN
kL lk
P P (3.21)
Individual line losses 1kP can be expressed in terms of voltages and
phase angles as
2 2 2 cos( )k i j i j i jl
kP g V V VV (3.22)
The objective function can now be written as
Min 2 2
1
( 2 cos( )lN
L k i j i j i j
i
P g V V VV
(3.23)
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This is a quadratic form and is suitable for implementation
using the quadratic interior point method. The constraints are
equivalent to those specified in Section 3.1.1 for cost minimization,
with voltage and phase angle expressed in rectangular form.
4.6 SUB OPTIMAL SOLUTION FOR OPF PROBLEM USING
MODIFIED PENALTY FACTOR METHOD
The objective is to maximize profits and usage of the equipment in
service so as to achieve the greatest financial benefits.
Mathematically, the problem is defined as:
2
1
( )G
i i
N
T G G
ii i iC P P (3.1)
Subject to the energy balance equation
1 1( ) ( )G D
i i
N N
i G i D LP P P (3.12)
and the inequality constrains
min max , 1,...,i i iG G G GP P P i N (3.14)
Where i, i, i are the cost coefficients, PDi= load demand and PGi =
real power generation of ith Machine; NG = number of generation
buses and PL = transmission power loss. Most of the OPF problems
use a set of Lagrangian equations such as:
GG
G G
G G
NG
L L LG
G
dCdC dC1 1 1λ
P P PdP dP dP1 1 1
P P P
21
1 2
1 2
........
NG
NG
(4.4)
= incremental cost of received power units $/MWhr.
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The co-ordination equation of ith Machine is:
i
ii
G
dCL
dP
(i=1, 2, 3, 4. . . . . NG) (4.5)
Eq.(4.2) illustrates for economical operation of the system the
incremental cost equations for all the units in a plant should be equal
in value. The variable that describes this value is called Lambda (λ).
The penalty factor of ith Machine represented as Li is defined as :
1
1
i
iL
G
LP
P
(i=1, 2, 3, 4. . . . .NG) (4.6)
Where
i
L
G
P
P is the incremental transmission loss of ith Machine. The
total plant losses in terms of loss coefficients B as:
1 1
G G
i j
N N
L G ij G
i j
P P B P
(4.7)
The penalty factor represents the power losses incurred in
transmitting the power to the load demand buses. The operating cost
of some generating units, which are distantly located from a load, may
have a higher operating cost than some other more costly units
located nearby, due to transmission losses.
For NG generator bus system loss coefficient Bij – matrix is n x n
symmetric matrix given as:
11 12 1
21 22 2
1 2
n
n
n n nn
i j
B B B
B B BB
B B B
(4.8)
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The loss coefficients are used to calculate the penalty factors. Loss
coefficients describe the system losses for a given load.
The iterative method solves for each unit generation using the
incremental cost functions, multiplied by penalty factors. For a n-
generator bus system these set of n-equations are given by:
ii G iP (for i=1, 2,…….. NG) (4.9)
Set of linear coordinate equations can be written as:
( )ii Gi iL P (for i=1, 2,…….. NG) (4.10)
Eq.( 4.10) must satisfy for minimum cost of generation. The set of n-
coordinate equations (4.9) can be grouped in to n-1 equations by
eliminating λ from (4.9) and setting ith and i+1th equations equal to
each other as:
11 11 1( ) ( ) ( ) ( )
i ii G i G i ii i i iL P L P L L (4.11)
For NG – unit system the additional NG th equation is:
Gi
N
L Di 1
P P
G
P (4.12)
The complete set of linear equations in matrix form as:
1
2
2 2 1 1
1 1 2 2
3 3 2 22 2 3 3
1 11 1
0 0 0
0 0
0 0
1 1 1 1
G G G G
NG
G
G
NG NG NG NGN N N N
G
D
L LPL L
L LPL L
L LL L
PP
(4.13)
in condensed form Eq.(4.13) is:
[A][P] = [B] (4.14)
The new generation schedule can be obtained by using following
equation.
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[P] = [A]-1[B] (4.15)
The new generation schedule is used to calculate new penalty factors
using Eq. (4.6). Eqs. (4.6) and (4.15) are repeatedly solved until a
convergence criterion is satisfied. The algorithm for modified penalty
factors is presented below.
4.6.1. Algorithm Modified Penalty Factor Method
Step-1: Read System Line data comprising resistance, reactance and
charging admittance of various transmission lines.
Step-2: By representing loads as lumped admittances, formulate Bus
Admittance Matrix (YBus).
Step-3: Using partitioned YBus matrices, compute B-coefficients.
Step-4: Read NG -Generator cost coefficients βi and γi, Generator
Power limits, System load at various buses and total plant
load. Define convergence error € based on accuracy required.
Step-5: For a given system load, perform Load-Flow study by Newton-
Raphson method.
Step-6: Using the power generation schedule compute penalty factor
Li values.
Step-7: Using Li values and cost coefficients βi and γi for i=1, 2,…, NG,
compute A and B matrices.
Step-8: Compute new generation schedule matrix [P] that
containing,Gi new
P for i=1, 2,…, NG., using Eq.(4.15).
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Step-9: Check , ,max,minG G Gi new ii
P P P for i=1, 2,…, NG. If ,Gi new
P with in limit
then Go To Step-12. Else Go To Step-10.
Step-10: Set the limit violated unit generation to either,Gi new
P =,minGi
P or
,Gi newP =
,maxGiP as the case may be.
Step-11: Repeat from Step-6.
Step-12: Check for,Gi new
P -,Gi old
P ≤ €. If converged with in tolerance Go To
Step-14.
Step-13: Repeat from Step-6.
Step-14: Print Generation Schedule.
The mathematical programming method described above is
simple and capable of handling nonlinear incremental cost functions.
The number of iterations required in the solution ranges three to five.
Number of iterations are power system size independent but
dependent on tolerance value and non linearity of cost functions. The
developed method is fast and can be used with less CPU time and
memory. Complexities in representation of cost functions reduce
effectiveness of this method.
4.7 SUB OPTIMAL SOLUTION FOR OPF PROBLEM USING
PRIMAL DUAL INTERIOR POINT METHOD
OPF solution using Primal Dual Interior Point Method (PDIPM)
is presented in Section 3.3.5 in Chapter 3. For the sake of continuity,
the algorithm is reproduced below. The solution is obtained using
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PDIPM is only taken as sub-optimal, owing to the disadvantages
presented. However, in this work PDIPM is taken as local search
method as it is giving best insight for initial chromosome or core point
for a high density cluster.
4.7.1 Algorithm of PDIPM For The Solution Of OPF Problem
The PDIPM algorithm (Section 3.3.5.2) applied to the OPF
problem is summarized step-by-step as follows.
Step 1 Read relevant input data.
Step 2 Perform a base case power flow by a power flow
subroutine.
Step 3 Establish an OPF model.
Step 4 Compute Eqs. (3.105) – (3.107).
Step 5 Calculate search directions with Eqs. (3.112) – (3.114).
Step 6 Compute primal, dual and actual step-lengths with Eqs.
(3.115) – (3.118).
Step 7 Update the solution vectors with Eqs. (3.119) – (3.121).
Step 8 Check if the optimality conditions are satisfied by Eqs.
(3.98) – (3.110) and if μ ≤ ε (ε = 0.001 is chosen).
If yes, go to the next step. Otherwise go to step 4.
Step 9 Perform the power flow subroutine.
Step 10 Check if there are any violations in Eqs. (3.13) – (3.20). If
no, go to the next step; otherwise, go to step 4.
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Step 11 Check if a change in the objective function is less than or
equal to the prespecified tolerance. If yes, go to the next
step; otherwise, go to step 4.
Step 12 Print and display an optimal power flow solution.
Disadvantages of PDIPM
Limitation due to starting and terminating conditions
Infeasible solution if step size is chosen improperly
4.8 SUB OPTIMAL SOLUTION FOR OPF PROBLEM USING
PARTICLE SWARM OPTIMIZATION METHOD
OPF solution using Particle Swarm Optimization (PSO) method
is presented in Section 3.4.2 in Chapter 3. For the sake of continuity,
the algorithm is reproduced below. The solution is obtained using PSO
is only taken as sub-optimal, owing to the disadvantages presented.
However, in this work PDIPM is taken as local search method as it is
giving best insight for initial chromosome or core point for a high
density cluster.
4.8.1 PSO Algorithm
Description of basic elements required for the development of Solution
Algorithm is presented in section 3.4.2.2.
In order to make uniform search in the initial stages and very
local search in later stages, an annealing procedure is followed. A
decrement function for decreasing the inertia weight given as w(t)=
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w(t-1), is a decrement constant smaller than but close to 1 , is
considered here.
Feasibility checks, for imposition of procedure of the particle
positions, after the position updating to prevent the particles from
flying outside the feasible search space.
The particle velocity in the kth dimension is limited by some
maximum value, vk max. With this limit, enhancement of local
exploration space is achieved and it realistically simulates the
incremental changes of human learning. In order to ensure
uniform velocity through all dimensions, the maximum velocity in
the kth dimension is given as :
max max min( ) /k k kv x x N (3.112)
In PSO algorithm, the population has n particles and each particle is
an m – dimensional vector, where m is the number of optimized
parameters. Incorporating the above modifications, the computational
flow of PSO technique can be described in the following steps.
Step 1 (Initialization)
Set the time counter t = 0 and generate randomly n particles,
[ (0), 1,... ]jX j n , where , 1 ,(0) [ (0),..., (0)]j j j mX x x .
, (0)j kx is generated by randomly selecting a value with uniform
probability over the kth optimized parameter search space
min max[ , ]k kx x .
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Similarly, generate randomly initial velocities of all
particles,[ (0), 1,... ]jV j n , where , 1 ,(0) [ (0),..., (0)]j j j mV v v .
, (0)j kv is generated by randomly selecting a value with uniform
probability over the kth dimension max max[ , ]k kv v .
Each particle in the initial population is evaluated using the
objective function J.
For each particle, set *(0) (0)j jX X and * , 1,...,j jJ j nJ . Search
for the best value of the objective function bestJ .
Set the particle associated with bestJ as the global best, **(0)X , with
an objective function of **J .
Set the initial value of the inertia weight (0)w .
Step 2 (Time updating)
Update the time counter t = t + 1.
Step 3 (Weight updating)
Update the inertia weight ( ) ( 1)w t w t .
Step 4 (Velocity updating)
Using the global best and individual best of each particle, the
jth particle velocity in the kth dimension is updated according to the
following equation:
*
, , 1 1 , ,( ) ( ) ( 1) ( ( 1) ( 1))j k j k j k j kv t w t v t c r x t x t
**
2 2 , ,( ( 1) ( 1))j k j kc r x t x t (3.113)
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Where 1c and 2c are positive constants and 1r and
2r are uniformly
distributed random numbers in [0, 1]. It is worth mentioning that the
second term represents the cognitive part of PSO where the particle
changes its velocity based on its own thinking and memory. The third
term represents the social part of PSO where the particle changes its
velocity based on the social-psychological adaptation of knowledge. If
a particle violates the velocity limits, set its velocity equal to the limit.
Step 5 (Position updating)
Based on the updated velocities, each particle changes its
position according to the following equation:
, , ,( ) ( ) ( 1)j k j k j kx t v t x t (3.114)
If a particle violates its position limits in any dimension, set its
position at proper limit.
Step 6 (Individual best updating)
Each particle is evaluated according to its updated position. If
* , 1,...,j jJ J j n , then update individual best as *( ) ( )j jX t X t and
*
j jJ J and go to step 7; else go to step 7.
Step 7 (Global best updating)
Search for the minimum value minJ among *jJ , where min is the
index of the particle with minimum objective function, i.e.
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min { ; 1,..., }j j n . If **
minJ J , then update global best as
**
min( ) ( )X t X t and **
minJ J and go to step 8 ; else go to step 8.
Step 8 (Stopping criteria)
If one of the stopping criteria is satisfied then stop; else go to
step 2.
4.8.2 Disadvantages of PSO Method
The candidate solutions in PSO are coded as a set of real
numbers. But, most of the control variables such as transformer
taps settings and switchable shunt capacitors change in discrete
manner. Real coding of these variables represents a limitation of
PSO methods as simple round-off calculations may lead to
significant errors.
Slow convergence in refined search stage (weak local search
ability).
4.9 DESCRIPTION OF GSHDC ALGORITHM
The following sub sections provide the description of steps in
continuous GAs that are implemented in GSHDC algorithm.
4.9.1 Formation of Chromosome
Chromosomes in Binary Genetic Algorithms (BGAs) were
generated using binary representation of variables describing possible
solutions. Many studies recommend the use of real numbers, instead
of binary coded values to represent the possible solution
chromosomes for optimizing functions with inherently continuous
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domains such as OPF problem. In Continuous Genetic Algorithms
(CGAs) a real valued chromosome represents the parameters defining
solution in actual form.
Ex: If the chromosome has ‘n’ variables (an n-dimensional
optimization problem) given by1 2, ,....,
NGG G GP P P that is active power
generations of NG generators in power system, then the chromosome
is written as an array with (1×n) elements so that
Chromosome = [1 2, ,....,
NGG G GP P P ]
In this case, the variable values are represented as floating – point
numbers. Each chromosome has a cost found by evaluating the cost
function f as the variables1 2, ,....,
NGG G GP P P .
Cost = f (chromosome) = f (1 2, ,....,
NGG G GP P P ) =
Subject to constraints,min ,maxG G Gi i i
P P P . Since f is a function
of1 2, ,....,
NGG G GP P P , the clear choice for the variables is:
Chromosome = [1 2, ,....,
NGG G GP P P ]
For the present problem of OPF, The structure of the chromosomes
used in conventional BGAs such as Extended Genetic Algorithm
(EGA) method in [103] and in this proposed GSHDC method are
shown in the Figs.4.4 and 4.5 respectively. The interested reader can
refer [103] for complete information for string formation. The
difference in string lengths of chromosomes can be noted in both the
methods.
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4.9.2 Fitness and Cost Functions
The Cost Function is defined as: 2
1
( )G
i i
N
T G G
ii i iC P P
. Our
objective is to search GiP for i=1, 2,…,NG, generations in their
admissible limits so as to min( CT ) is obtained. The cost function of
initial core point in High Density Cluster is obtained through one of
the suboptimal solution methods. The value of the cost is then
mapped into a fitness value so as to fit in the genetic algorithm. The
fitness value of core point is taken as Eps. To minimize the cost is
equivalent to getting a maximum fitness value in the searching
process. A core point chromosome that has lower cost function
should be assigned a larger fitness value (f). The objective of the OPF
has to be changed to the maximization of fitness to be used in the
roulette wheel as follows:
1GP : GNGP
1GU : GNG
U t1 : t NG bs
h1 : bsh NG
Unit active Power Outputs
Generator bus voltage magnitudes
Transformer tap settings
Bus shunt admittances
Fig: 4.4 String Structure in Binary Genetic Algorithms [103]
1GP
2GP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
GNGP
Unit active Power Outputs
Fig. 4.5 String Structure in GSHDC Method
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Fitness Value of ith chromosome fitness (i)= Eps - fi : if fi ≤; Eps ; for
i=1,2,…NG. ; = Zero other wise. Thus the GA tries to generate better
offspring to improve the fitness.
Because only active power generations are used in the fitness, the
reactive power levels and voltage constraints are scheduled in the
Power Flow Study. It can be understood that active power limits are
checked using GA procedure and the other constraints are checked
using an efficient power flow study.
4.9.3 Parent Selection
The sub optimal solution obtained through local search method
is the core point in the high density cluster. This OPF solution is a
chromosome and its structure is formed as discussed earlier. By
increasing or decreasing power generation for a constant load
demand, a population consists of certain number of chromosomes
having better FF values is then generated randomly. The
chromosomes in this population having better FF values are then
chosen as Parent Chromosomes for generating next off springs with
the help of Genetic operators. A Blending Method is used for crossover
operation and polynomial mutation is used to produce off springs.
4.9.4 Crossover
Crossover operator is believed to be the main operator that
creates the search of power of GAs in optimization problems [8].
Crossover operators have two important functions:
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First function is it searches the initial population that is initial
random strings containing problem variables to come up with a
good solution and
Second function is to combine good portions of these strings to
form even better solutions.
Since the use of real value representation is proposed in this work,
the search for a better crossover operator is on. Blending Method in
[113] used in this work. To generate a new offspring variable value,
PGinew from a combination of corresponding variable values in parent
chromosomes.
PGinew = PGi
male + (1- ) PGifemale for i=1,2,……n (4.16)
is the random variable chosen on the interval [0, 1] PGimale and
PGifemale are the incremented and decremented value respectively
around the suboptimal solution obtained through local search
method.
Using this blending method the offspring variables inherit that
property from their parent’s variables, and their value always fall
between the values in parent’s variables.
Ex: Let the suboptimal solution through a local search method for
two generations of 2-Generator problem gave PG1 = 100 MW and PG2 =
200 MW say (Initial Chromosome). The boundary for incremented
values PGimale (Male) are taken as 102 and 202 MWs and the boundary
for decremented values PGifemale (Female) are taken as 98 and 198
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MWs. Now the population can be generated using different values of
and using (4.1) as:
With = 0, Parent -1= (PG1, PG2) = (98 MW, 198 MW)
With = 0.1, Parent -2= (PG1, PG2) = (98.4 MW, 198.4 MW)
With = 0.25, Parent -3= (PG1, PG2) = (99 MW, 199 MW)
With = 0.5, Parent -4= (PG1, PG2) = (100 MW, 200 MW)
With = 0.75, Parent -5= (PG1, PG2) = (101 MW, 201 MW)
With =1 Parent -6= (PG1, PG2) = (102 MW, 202 MW)
Chromosome-1= [98 MW, {any PG2 value obtained from different values
of }]
Chromosome--2= [98.4 MW, {any PG2 value obtained from different
values of }]
…… … …. … … … … … … … … … …
….. …. … ….. …. …. ….. ….. ….. ….. ….. ….
Chromosome--n= [102 MW, {any PG2 value obtained from different
values of }]
Since, applying these blending methods does not result in introducing
values beyond extreme values of that variable in the initial
population, some extrapolating blending have been suggested in
the literature. However, these methods could generate value could
generate value outside the acceptable range for a parameter. Then the
offspring must be discarded and another selected.
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4.9.5 Mutation
Mutation Operator has two important roles:
The first role is to introduce unexplored genetic chromosomes to
the population.
Second role is to maintain the diversity of the chromosomes in a
population over the generations, preventing premature
convergence of the GA to suboptimal local optima.
Mutation in BGAs is rare compared to crossover, resulting in
lower probability of mutation. As the mutation probability for all the
bits in a binary string is the same, it will lead to higher probability of
small changes and lower probability of large changes.
One way to implement mutation in CGAs is make it to work
similar to binary BGAs. In this way if a variable PGi in the
chromosome has been chosen for the mutation, a random number F
is selected between [ 0,1] then the new variable PGi mutation would be:
PGi mutation = PGi crossover + F (PGi male - PGifemale ) for i =1, 2,…n (4.17)
Where PGi crossover is ith variable of the parent chromosome
selected for mutation, PGi male is the upper limit and PGifemale is the
lower limit for the PGi crossover.
Ex: Let F = 0.2; PGi crossover = Parent -3= (PG1, PG2) = (99 MW, 199 MW);
PGimale = [102,202] and PGi
female = [98,198].
PGi mutation = [{99, 199 MW}] + 0.2[{102-98}, {202-198}] = [99,199] +
[{0.8}, {0.8}] = [99.8, 199.8]
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for different values of F different chromosomes for mutation can be
obtained.
4.9.6 GA Search for the exact OPF solution-Convergence Criterion
After crossover and mutation, Load flow study is run. If load
flow converges and slack bus generation is checked for limits for any
generation, the minimum generation cost amongst all (3.20). If any
violation of constraints, is observed the next core chromosome in the
list is selected. The Process is repeated until the desired chromosome
which satisfies all constraints is selected.
4.10 ADVANTAGES OF GSHDC OVER THE OTHERS
The advantages of GSHDC method over the other methodologies
are given below:
Length of Chromosome is reduced and hence the size of
population is reduced.
Number of generations is reduced. This makes the computational
effort simple and effective.
The problem of use of specific mutation or crossover operators is
avoided. This makes the OPF as another simple GA search
problem.
Blind search is avoided.
The process begins with no insignificant chromosomes.
System nonlinearities are somewhat considered as the initial
chromosome is obtained from the mathematical programming of
nonlinear equations.
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4.11 CONCLUSIONS
In this Chapter, a new algorithm for the solution of optimal
power flow problem was presented. The algorithm was unfolded into
three stages. In the first, a suboptimal solution was obtained by the
following local search methods: 1) Modified Penalty Factor method 2)
Primal Dual Interior Point 3) Particle Swarm Optimization method.
Owing to limitations in these methods, in the second, a High Density
Cluster, which consists of other suboptimal data points in the vicinity
of the first were formed by using Continuous Genetic Algorithm. In the
final stage, a search was carried out for the exact optimal solution
from a high density small size population High Density Cluster. The
final optimal solution thoroughly satisfies the well defined fitness
function. This work mainly undertakes the primary goal of OPF that
is to minimize the cost of generation for meeting a load demand while
maintaining the security of the system. System security can be
thoroughly maintained when each device of a power system, work in
desired operation range under steady state conditions.
This work aims for examining several issues that need to be
taken into consideration when designing genetic algorithm that uses
another search method as a local search tool. These issues include
the different approaches for employing local search information that
is useful for a genetic algorithm searches for global optimum solution.