The optimization is an important role in wide geographical distribution of electrical power market, finding the optimum solution for the operation and design of power systems has become a necessity with the increasing cost of raw materials, depleting energy resources and the ever growing demand for electrical energy. In this paper, the real coded biogeography based optimization is proposed to minimize the operating cost with optimal setting of equality and inequality constraints of thermal power system. The proposed technique aims to improve the real coded searing ability, unravel the prematurity of solution and enhance the population assortment of the biogeography based optimization algorithm by using adaptive Gaussian mutation. This algorithm is demonstrated on the standard IEEE30 bus system and the comparative results are made with existing population based methods.
1. Introduction
In competitive electrical power market, electrical energy must be offered at a least cost with high quality, which is very difficult task for market operator in deregulated power system. Optimal power flow (OPF) is the tool for solving these complicated problems. The main objective of optimal power flow is to obtain optimal operating schedule for each generator which minimizes the cost of production and satisfies the system equality and inequality constraints. The earlier researches are done in different methods of optimal power flow. The methods are Linear Programming in
[1]
, Nonlinear Programming in
[2]
, Quadratically convergent in
[3]
, Newton approach in
[4]
, Interior Point Method in
[5
,
6]
and PQ decomposition in
[7]
.
In deregulated power system, multiple transactions are done every hour and hence loads are varied. Optimal power flow is carried out dynamically based on load variation. Dynamic optimal power flow (DOPF) is discussed in
[8]
. Thermal power plants are major part of power generation in electric power sector, where power is generated by burning of fossil fuels. It releases polluted gases in the environment. In the concern of environmental awareness, pollution should be minimized which is achieved by combining cost and emission dispatch in a single objective function. Emission constrained economic dispatch is discussed in
[9]
.
Earlier conventional based optimal power flows have excellent convergence characteristics, but they could not perform well when deal with systems having nondifferentiable objective functions and practical constraints with some theoretical assumptions. So researchers concentrate towards evolutionary algorithms like as Genetic Algorithm
[10
,
11]
, Enhanced Genetic Algorithm
[12]
, Evolutionary Programming
[13]
, Tabu Search
[14]
, Simulated Annealing
[15]
, Particle Swarm Optimization
[16]
, Differential Evolution
[17]
, Modified Differential Evolution
[18]
, Modified Shuffle Frog Leaping Algorithm
[19]
, and Artificial Bee Colony Algorithm
[20]
. Non – Reliability is the disadvantage of these optimization techniques.
In
[21]
, Biogeographybased Optimization (BBO) algorithm was employed by Bhattacharya and Chattopadhyay for solving OPF problems. This approach is briefly discussed in next section. The probability based random mutation is applied in the BBO algorithm, so that the population are diverted at the end of the solution. This is the main drawbacks of the algorithm. It could be avoided by using Gaussian mutation in real coded biogeography based optimization (RCBBO). In this paper, RCBBO algorithm is discussed, which is applied to dynamic optimal power flow problem. The results are compared with existing methods.
2. Biogeography Based Optimization
The Biogeographybased Optimization (BBO) technique, which is proposed by Dan Simon
[22]
is a comprehensive algorithm for solving optimization problems and is based on the study of geographical distribution of species. The nature’s way of distributing species is known as Biogeography, and is analogous to general problem solutions. The BBO technique has two main operators, they are migration and mutation.
 2.1 Migration operator
Migration is the process that probabilistically modifies each individual in the habitat by sharing information with other individual solution. Geographical areas with high Habitat Suitability Index (HSI) are said to be well suitable for biological species. Suitability Index Variables (SIVs) are the variables that characterize the habitat of the species. Geographical areas with high HSI tends to have a large number of species, high emigration rate and low immigration rate. Therefore, habitats with high HSI tends to be more static in their species distribution compared to low HSI habitats. A habitat with high HSI is analogous to a good solution and a habitat with low HSI is analogous to a poor solution. The sharing of features of individuals in the habitat is done based on the migration rate. The immigration rate, λ
_{k}
and the emigration rate, μ
_{k}
are functions of the number of species in the habitat. When there are no species in a habitat, the immigration rate of the habitat is maximal. The immigration rate, λ
_{k}
can be formulated as:
Where I is maximum possible immigration rate, k is number of species of k
^{th}
individual and n is maximum number of species. The emigration rate, μ
_{k}
can be formulated as:
Where E is maximum possible emigration rate.
 2.2 Mutation operator
The process of mutation tends to increase diversity among the individuals in the habitat to get better solution. Due to natural events, HSI of habitat is changed drastically. It causes a species count differ from its equilibrium value. Each species count is associated with probability (P
_{i}
). Individual’s solution is mutated with other solution if the probability is very low. So mutation rate of individual solution is calculated by using species count probability.
Where M
_{i}
is the mutation rate, M
_{max}
is the maximum mutation rate which is user defined parameter, and P
_{max}
is the maximum probability of species count.
In BBO, mutation characteristic function is given by:
where
X_{i}
is the decision variable;
and
are the lower and upper limits of the decision variable, respectively.
The advantages of BBO are that using of probabilistic migration can create the better solutions from the poor ones by sharing more information. For the meantime, it would not loss good solutions at the progress. The main drawback of BBO technique is that the migration operator fails to improve the exploration ability and the diversity of the population.
3. Real Coded BiogeographyBased Optimization
Real Coded Biogeographybased Optimization (RCBBO) is an extension of BBO where individuals are directly encrypted by a floating point for the continuous optimization problems.
In BBO, individuals are represented by a D–dimensional integer vector, whereas in RCBBO individuals are represented by a D–dimensional real parameter vector. In Real Coded Biogeography Based Optimization technique, the assortment of the population is improved and its searching ability is enhanced by integrating the mutation operator with BBO technique. Mutation operator is intended to expose liabilities belonging to the matching fault class. Real coded biogeography based optimization is discussed in
[24]
, where Gaussian mutation is used probabilistically based to modify the original BBO technique.
In this paper, Gaussian mutation operator is applied to improve the worst half of the individuals in the population. Adaptive mutation probability is used to prevent premature convergence and produce a smooth convergence. This method of mutation can be easily used for realcoded variables which have been widely used in Evolutionary Programming (EP) and it is able to carry out local search as well as global search.
The Gaussian mutation characteristic function is given by:
where
represents the Gaussian random variable with mean μ and variance σ
^{2}
. The values of mean and variance are considerd 0 and 1, respectively
[24]
.
Generally, a probabilitybased mutation operation is known to improve the convergence characteristics. Therefore, adaptive Gaussian mutation is applied in the present work to improve the solution of worst half set of habitats in the population.
In Eq. (5), μ = 0, and
σ_{i}
is found using the following Eq. [27]:
where
β
is the scaling factor or mutation probability, F
_{i}
is the fittness value of ith individual, and f
_{min}
is the minimum fitness value of the habitat set in the population.
Adaptive mutation probability is given by
where
β
_{max}
= 1 ,
β
_{min}
= 0.005 , T
_{max}
is the maximum iteration, and T is the current iteration. The main difference between Evolutionary Programming (EP) and Real Coded Biogeographybased Optimization (RCBBO) is that it makes use of migration operator, which utilizes the information of population effectively and the adaptive mutation balances the exploitation and exploration ability of the RCBBO technique.
4. Problem Formulation
Generally, an OPF problem is a largescale, highly constrained nonlinear optimization problem. It may be defined as
where f is the objective function to be minimized, x and u are the vectors of dependent and independent control variables, respectively, g is the equality constraint, and h is the operating constraint.
The vector of dependent variables can be represented as:
where
P
_{G1}
denotes the slack bus power,
V_{L}
denotes the load bus voltage,
Q_{G}
denotes the reactive power output of the generator,
S_{L}
denotes the transmission line flow,
N_{g}
is the number of voltagecontrolled buses, N
_{pq}
the number of load buses, and N
_{l}
is the number of transmission lines.
The vector of independent control variables can be represented as:
where N
_{t}
and N
_{C}
are the number of tapchanging transformers and shunt VAR compensators, respectively;
P_{G}
is the active power output of generators;
V_{G}
is the voltage at the voltagecontrolled bus; T is the tap setting of the tapchanging transformer; and
Q_{C}
is the output of shunt compensating devices.
 4.1 Objective function
This paper discusses about two different objective functions and combined both functions into single objective function to prove the effectiveness of the proposed technique based on RCBBO. The objective functions are discussed below:
 4.1.1 Minimization of fuel cost
This objective function aims to minimize the total fuel cost for the operation and planning of power systems under varying loads. The objective function is formulated as:
Where FC is the total fuel cost, N
_{g}
is the number of generators. The fuel cost function for the operation of Power Systems can be expressed as:
Where P
_{Gi}
is the real power output of an i
^{th}
generator and a
_{i}
, b
_{i}
and c
_{i}
are the fuel cost coefficients.
 4.1.2 Minimization of environmental pollution
The main goal of this objective function is to minimize the environmental pollution caused by the operation of thermal power systems. The objective function is formulated as:
Where
Em
is the total emission generation. The emission function can be expressed as:
Where α
_{i}
, β
_{i}
and γ
_{I}
are the emission coefficients of the i
^{th}
unit.
 4.1.3 Minimization of total cost
The objective functions are combined and formulated into a single optimization problem by introducing the Price Penalty Factor ‘h’ as follows:
The procedure of price penalty factor calculation is discussed in
[25]
.
 4.2. Constraints
 4.2.1 Equality constraints
The equality constraints are the power flow equations given by:
Where P
_{Gi}
& Q
_{Gi}
are the injected active and reactive power at i
^{th}
bus, P
_{Di}
& Q
_{Di}
is the demanded active and reactive power at i
^{th}
bus, Y
_{ij}
is the admittance between bus i and j, θ
_{ij}
is the load angle between bus i and j, δ
_{i}
is the phase angle of voltage at i
^{th}
bus and NB is the total number of buses.
 4.2.2 Inequality constraints
These constraints are the set of continuous and discrete constraints that represent the system operational and security limits as follows:
(a)
Generator constraints:
the generator active and reactive power outputs are restricted by their upper and lower limits.
Where P
_{Gi,min}
& P
_{Gi,max}
are the minimum and maximum value of real power generation at i
^{th}
generator bus, Q
_{Gi,min}
& Q
_{Gi,max}
are the minimum and maximum value of reactive power generation at i
^{th}
generator bus.
(b)
Security constraints:
these include the limits on the load bus voltage and transmission line flow limits:
Where
V
_{i,min}
&
V
_{i,max}
are the minimum and maximum value of magnitude of voltage at i
^{th}
load bus and
N _{pq}
is the number of load bus.
The power flow limit on transmission line is restricted by
Where
is the maximum rating of k
^{th}
transmission line.
The most common method for handling the inequality constraints is to make use of a penalty function. The original constrained optimization problem is transformed to an unconstrained one by penalizing the inequality constraints.
Finally, the dynamic optimal power flow objective function is combined with constraints as
Where
λ_{Pg}
,
λ_{Qg}
,
λ_{V}
&
λ_{Pf}
are the penalty factors.
 4.3 Algorithm
The steps for solving the OPF problem using RCBBO is as follows:
Step 1
: Initialization
Habitat modification probability (P
_{mod}
), minimum and maximum values of adaptive mutation probability (β
_{min}
and β
_{max}
), maximum immigration and emigration rates for each island, maximum species count (P), and maximum iterations are initialized.
Step 2
: Generate SIVs for the habitat randomly within the feasible region.
Individuals (control variables) in the habitats are initialized as:
where i = 1, 2… P, and j = 1, 2… N
_{var}
; N
_{var}
is the number of control variables;
and
are the lower and upper limits of jth control variable.
Step 3
: Perform load flow analysis using NewtonRaphson method and determine the dependent variables. Compute the fitness value (HSI) for each habitat set.
Step 4
: Based on the HSI value, elite habitats are identified.
Step 5
: Iterative algorithm for optimization:

(i) Perform migration operation on SIVs of each nonelite habitat selected for migration.

(ii) Calculate immigration and emigration rates for each habitat set, using Eqs. (1) and (2).

(iii) Update the habitat set after migration operation.

(iv) Recalculate the HSI value of modified habitat set; feasibility of the solution is verified and habitat set sorted based on new HSI value.

(iv) Perform mutation operation on the worst half set of population by Gaussian adaptive mutation using Eqs. (57)

(v) Compute the fitness value (HSI) for each habitat set after mutation operation and verify the feasibility of the solution.

(vi) Sort the habitat set based on new HSI value.

(vii) Stop the iteration counter if the maximum number of iterations is reached.
Step 6
: Finally SIVs should satisfy the objective function as well as constraints of the problem.
5. Simulation Results
The proposed Real Coded Biogeographybased algorithm for solving dynamic OPF problem has been applied to the IEEE 30bus test system. The numerical results are presented in this section. The results obtained by the proposed approach are compared with the results found by alternative populationbased algorithms reported in the literature recently. Power flow calculations by NewtonRaphson method were performed using the software package MATPOWER 4.1
[26]
.
The IEEE30 bus system has six generators at buses 1, 2, 5, 8, 11 and 13, and four tap changing transformers. The total system demand is 283.4MW for the active power, and 126.2 MVAR for the reactive power at 100 MVA base. Bus 1 is taken as the slack bus. The fuel cost and emission coefficients for IEEE30 bus is given in appendix.
The optimal control parameters for the algorithm are chosen from number of simulation results. They are: habitat size=50, habitat modification probability = 1, immigration probability = 1, step size for numerical integration = 1, maximum immigration and emigration rate = 1, mutation probability = 0.005 and maximum number of iterations = 200. The results show the corresponding objective functions for 50 independent trails.
In the subsequent paragraphs, we discuss the results obtained by the proposed RCBBO algorithm and existing BBO algorithm
[22]
with regard to each objective function of the OPF problem for standard system demand. The optimal settings of control parameters are given in
Table 1
. The bolded values represent the optimal value of respective objective functions.
Simulation results for minimization of fuel cost and emission
Simulation results for minimization of fuel cost and emission
The robustness of the proposed RCBBO algorithm is compared with different optimization techniques, for the objective function of minimization of fuel cost is presented in
Table 2
. The first two rows mentioned in the table are obtained by our own implementation of algorithms. Best fuel cost obtained by the proposed RCBBO was 799.0908$/h, which is lesser than minimum fuel cost obtained using BBO algorithm and solution reported in
[12
,
16

21]
. Convergence characteristics of optimization methods, considered in this work are depicted in
Fig. 1
, which indicates premature convergence in BBO and smooth convergence in RCBBO.
Comparison of results for minimization of fuel cost
Comparison of results for minimization of fuel cost
Convergence characteristics for objective function  minimization of fuel cost
The robustness of the RCBBO algorithm is compared with BBO algorithm for the objective function of minimization of emission, in
Table 3
. Convergence characteristics of proposed RCBBO algorithm and BBO algorithm for this objective function are depicted in
Fig. 2
.
Comparison of results for minimization of emission
Comparison of results for minimization of emission
Convergence characteristics for objective function  minimization of emission
Simulation results obtained by proposed RCBBO and BBO algorithm for minimization of total cost are presented in
Table 4
. Best total cost obtained by the proposed RCBBO was 1519.556$/h, which is lesser than minimum total cost obtained using BBO algorithm and solution reported in
[27]
.
Comparison of results for minimization of total cost
Comparison of results for minimization of total cost
For 24 hours load pattern, solution for dynamic optimal power flow is obtained by proposed RCBBO and BBO, are presented in
Tables 5
and
Table 6
respectively. The price penalty factor for the system demand of 283.4MW is 2.0534 and 1.7916, for all other demands. Total cost obtained for 24 hours by the proposed RCBBO is 23168.753$, which is 20$ lesser than total cost obtained using BBO algorithm. From the results, RCBBO based DOPF is perceived which provides higher lead to terms of accuracy and reliability.
Result obtained for DOPF using RCBBO method
Result obtained for DOPF using RCBBO method
Result obtained for DOPF using BBO method
Result obtained for DOPF using BBO method
6. Conclusion
In this paper, real coded biogeography based optimization algorithm is developed and successfully applied to solve the environmental constrained dynamic optimal power flow problems. This approach is tested and examined with combined multi objective functions including the generator constraints and security constraints to show its effectiveness using the IEEE 30bus system. The results obtained from the RCBBO approach are compared with those reported in the recent literature. The superiority and solution quality of the proposed method are found better than other techniques. According to the results obtained, the RCBBO algorithm has a simple framework and quick convergence characteristic and, therefore, can be used to solve the OPF problem in largescale power systems with several thousands of buses utilizing the strength of parallel computing.
BIO
A. Ramesh Kumar received his B.E degree in electrical and electronics engineering from Dr. Sivanthi Aditanar College of Engineering and M.E degree in power system engineering from Annamalai University. His research interests are power system optimization, deregulated power system and flexible AC transmission systems.
L. Premalatha received her Ph.D. in Electrical Engineering from Anna University, Chennai, India. She is currently working as Professor in Anand Institute of Higher Technology, Chennai, India. Her current research interests include power quality, nonlinear dynamic systems and control, electromagnetic compatibility and renewable energy systems.
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