Group Search Optimization Algorithm with Fuzzy Clustering for Generation Power Dispatch Problem
Amit kumar1, L.Lakshminarasimman2, and S.Velusami3
1PG Scholar, Department of Electrical Engineering, Annamalai University, Tamilnadu, India
2Associate Professor, Department of Electrical Engineering, Annamalai University, Tamilnadu, India
3Professor&Head, Department of Electrical Engineering, Annamalai University, Tamilnadu, India Email ID-1[email protected]
ABSTRACT: The economic emission dispatch (EED) assumes a lot of significance to meet the clean energy requirements of the society and simultaneously minimizes the cost of generation. In this paper focus on the work a fuzzy clustering-based group search optimization (FCGSO) algorithm has been proposed to solve the highly constrained EED problem involving conflicting objectives.
FCGSO uses an external repository to preserve nondominated particles found along the search process.
The proposed fuzzy clustering technique, manages the size of the repository within limits without destroying the characteristics of the Pareto front. In addition, the algorithm incorporates a fuzzy-based feedback mechanism and iteratively uses the information to determine the compromise solution. The algorithm’s performance has been examined over the standard IEEE 30 bus six – generator test system, whereby it generated a uniformly distributed Pareto front whose optimality has been authenticated by benchmarking against the ε – constraint method.
Keywords: Economic emission dispatch (EED), Group Search Optimization (GSO), Fuzzy Clustering.
1. INTRODUCTION
Electrical power dispatch is an essential function required in modern energy management systems to determine the optimal steady-state operation of electric power generation. The primary objective is to schedule the committed generating unit output, so as meet the load demand at minimum operating cost. [1] The economic emission dispatching option is an attractive short-term alternative in which the emission, in addition to the fuel cost objective, is to be minimized.[2] Thus, the economic emission dispatching problem can be handled as a GPD-objective optimization problem with non commensurable and contradictory objective. The generation of electricity from fossil fuels such as coal, oil and gas, releases several contaminants such as sulphur oxides (SOx), nitrogen oxides (NOx) and carbon dioxide into the atmosphere. The enactment of the ‘Clean Air Act Amendment of 1990’ and its acceptance by all nations forces the utilities to modify their operating strategies.
Environmental/economic dispatch (EED) is a multiobjective problem having conflicting objectives, as the minimization of
emission is contrary to the maintenance of cost economy [3]. There has been much research pertaining to the EED problem. A linear programming (LP)-based technique has been proposed in [4] which considers one objective at a time. However, the approach failed to give any information regarding the tradeoff front.
References [5]–[6] linearly combined different objectives through the weighted sum method to convert the multiobjective EED problem in single-objective optimization problem. These methods generate the nondominated solution by varying the weights, thus requiring multiple runs to generate the desired Pareto set of solutions.
Recently, a new optimization algorithm metaheuristic, Fuzzy clustering – based group search optimization (FCGSO), for constrained / unconstrained generation power dispatch (GPD) optimization. The frame of the proposed FCGSO metaheuristic encompasses various conventional multiobjective techniques, creatively modified, to make the overall approach competitive and compatible with the problem at hand. The metaheuristic has been applied to solve the EED problem, whereby it efficiently found the representative Pareto front and simultaneously determined the compromise solution by delving deeper into those regions having higher probability of its occurrence. The main purpose of considering the EED problem as the benchmark for testing the proposed FCGSO algorithm is related to its inherent requirement of real-time decision making which is mandatory for testing the effectiveness of the proposed approach. This paper has been prepared with the following objectives.
• To validate the efficacy and applicability of the proposed approach on GPD objective problems.
• To establish the strategic inclusion of decision making with the search process.
• To solve the highly constrained multi objective EED problem and obtain the Pareto front and the compromise solution.
The main purpose of considering the EED problem as the benchmark for testing the proposed FCGSO algorithm is related to its inherent requirement of real- time decision making which is mandatory for testing the effectiveness of the proposed approach. Exploratory potential required at various stages of the search is adjusted through the use of a self-adaptive mutation
Operator. Due to the imprecise nature of Decision Maker’s [7]-[8] judgment, the compromise solution is extracted using a fuzzy-based mechanism.
2. PROBLEM FORMULATION
The problem formulation of generation power dispatch (GPD) problem which is a mixed integer, non-linear optimization problem that includes both equality and inequality constraints. The objectives such as minimization of fuel cost, and emission dispatch have been considered while transmission loss has been minimized by including it in equality constraint. The mathematical formulations of the objective functions and system, and operational constraints are detailed in the following section.
In real world optimization problems need to deal with two or more conflicting objective. In mathematical terms optimization can be formulated as
Minimize:- Subjected to :-
(1) Where:- = objective fn.
X = decision vector that represent a sol.
= no. of object.
2.1Economic Dispatch:-To minimize the total generation cost. The generator cost curves are represented by quadratic function & total fuel cost F(PG) in ($/h) can be expressed as:-
(2) Where ai, bi, ci = fuel cost coefficient generator of ith
generator. NG = number of generators, PGi =Active Power generator .
2.2 Emission Dispatch: - To minimize the pollutant due to fossil fuel thermal unit, such as sulfur dioxides and nitrogen oxide etc. The total emission E (PG) in (ton/hr) can be expressed as:-
Where αi, i, i, δi and λi =emission coefficient of ith
2.3 Transmission loss: - The aim of energy-saving generation dispatch is also to minimize power transmission loss, and it can minimized by considering as third objective. The power loss is computed using the Newton Raphson Load Flow method.
2.3 GPD CONSTRAINTS:-
2.3.1Power Balance Constraints :- In the total power output of generator must equal to the sum of total power demand ( PD) plus power loss PLOSS.
= 0 (4) Where PD = Power Demand, PLOSS = Power Loss, PGi
= Generation power of ith generator. Transmission losses can be readily obtained by using B-coefficients.
2.3.2 Generation Constraints: - For stable operation the real power output of each generator is restricted by lower and upper limits as follows:-
≤ ≤ I = 1…..N (5) 3. GROUP SEARCH OPTIMIZATION
ALGORITHM
Group search optimization (GSO) is a novel stochastic optimization algorithm that was developed by (S. He et al. 2009). This algorithm is inspired by the foraging behaviour of animals. The entire population of the GSO algorithm is termed a group and each individual in the population is called a member. There are three types of members namely: producers, scroungers and rangers. In order to achieve this foraging task the producer- scrounger strategy is engaged. Producing signifies to the action of searching for food and scrounging means joining the group for foraging. Rangers perform random walks in the search space. According to the GSO algorithm only one member is chosen to be the producer and the remaining members are scroungers and rangers. The producer constantly looks and finds the resources and the scroungers just join the producer.
During iterations, the member that is found to have the best fitness value is chosen as the producer.
fig. (3.1) Basic flow chart of GSO
In the GSO algorithm, at the kth iteration the producer Xp behaves as follows. The producer will first scan at zero degree and then choose three random points.
i) A point at zero degree
ii) A point crosswise at the right hand side of the producer iii)A point crosswise at the left hand side of the producer.[10]
XZ = Xp
k+r1lmaxDp
k (Фk) (6)
Xr = Xpk
+ r1 lmax Dpk(Фk + r2Ɵmax/2) one point in the
right hand side (7)
Xl = Xpk
+ r1 lmax Dpk(Фk - r2Ɵmax/2) one point in the
left hand side (8) where r1 is where r1∈R1 is a normally distributed
random number with mean 0 and standard deviation 1 and r2∈Rn−1 is a uniformly distributed random sequence in the range (0,1) and Ɵmax is the maximum pursuit angle and Ɵmax ∈R1 and maximum pursuit distance lmax∈R1.
If the producer finds a better position than its current position then it will move to that point otherwise it will stay in its current position and turn its head angle using the formula,
Фk+1=Фk+r2αmax (9) Where αmax∈R1is the maximum turning angle.
In case the producer cannot find a better position after a iterations, then it will turn its head back to zero degree.
Фk+a=Фk (10) where a ∈ R1is a constant.
During iterations a number of group members are selected as scroungers. The scroungers will keep searching for opportunities to join the resources found by the producer. At the kth iteration, the area copying behaviour of the ith scrounger can be modelled as a random walk toward the producer
Xik+1=xik+r3(xpk–xi
k) (11) Where ∈Rn is a uniform random sequence in the range (0,1). In case the scroungers or the rangers finds a better position than the producer then in the next iteration the one that has better position switches its role and becomes the producer. This mechanism helps the entire group to escape from being struck at local minima.
Random walks, which are thought to be the most efficient searching method for randomly distributed resources are employed by the rangers. At the kth iteration, it generates a random head angle ϕi using (5);
and then it chooses a random distance.
lik+1=α.r1lmax (12) and move to the new point
xik+1
= xik
+ 1iDik(Фk+1) (13) Different strategies were adopted by the animals to restrict their searches. GSO algorithm uses a strategy called as bounded search space. According to this strategy if any member is outside the search space it will turn back into the search space by setting the variables that violated the boundary criteria.
3.2 FUZZY CLUSTERING
Fuzzy clustering is a process of assigning these membership levels, and then using them to assign data elements to one or more clusters. Fuzzy clustering has been used widely in pattern recognition, image processing, and data analysis.
An improved fuzzy clustering algorithm was developed based on the conventional fuzzy c-means (FCM) to obtain better quality clustering results. The update equations for the membership and the cluster center are derived from the alternating optimization algorithm.
The goal of the clustering algorithm is to identify the cluster centers and the membership values by solving an optimization problem. Alternating optimization is a popular mathematical tool for the regular objective function-based fuzzy clustering algorithms.
3.1 FUZZY DECISION MAKING
Fig. 3.2
A fuzzy membership function has been used to simulate the DM’s preference and identify the best compromise solution which is later presented to the decision maker. Decision making is performed as soon as the repository gets filled. Membership value of each individual lying in the repository is computed using the membership function defined in the following way:
1, if Fi ≤ Fi min Fimax – Fi if Fi max
< Fi < Fimax
ki = Fimax– Fimin
, 0 if Fi≥ Fi max
(14) Where Ki stands for the membership value of the ith
function (Fi) . In the proposed approach, the values of Fmax and Fmin are calculated from the results obtained by optimizing the objectivesone at a time. However, Fmax
and Fmin could also be selected among the particles present in the repository by adaptively updating their values to the value of the boundary particles present there at any instant.
For each member k present in the repository, the normalized membership value (δ[k]) is calculated using:
(δ [k]) = (15)
Where m stands for the number of nondominated
solutions and n for the number of objectives. The membership function, characterizes the degree of realization of an objective function between 0 and 1, where δi =1 signifies completely satisfactory and δi = 0 as unsatisfactory. The membership function here represents a form of decision making criteria that is adaptive and changes with the available decision alternatives. The notionmimics the process followed in real-life decision making. When provided with a list of decision alternatives, the decision maker chooses the best solution from among them, and in the process ranks the alternatives on the basis of their relative advantages and disadvantages compared with other alternatives available to him/her. This process of decision making is incorporated in the proposed approach via the fuzzy membership function.
The FCGSO uses these membership values, via its memory component, in order to iteratively search in
preferable regions to mine out better compromise solution.
4. FLOW CHART PROPOSED FCGSO ALGORITHM
Start
Initialize GSO with random population position
For each group of i
Initialize each group of memory
Evaluate objectives eq.(i) Next group
Is group non- dominated
Store into repository
If i= popsize
For each group i
Update population position Xik+1
= Xik
+ rand (XPK – Xik
)
Next group
Update position Maintain group in search
space
Fuzzy membership value
allocation Update group’s memory Mutation operator
Is group non- dominated
Is nrep>
repsize If i=
popsize
Is termination
criteria Store into repository
End Pruning by fuzzy clustering procedure YES
NO
I = i+1 NO
YES
NO
YES NO
NO
YES YES
i = 1
Fig 4.1 Flow Char of Proposed FCGSO
4.1 FCPSO IMPLEMENTATION
The approach presented in this study is simulated on the standard IEEE 30-bus six -generator test system [13]
(Fig. 5). The power system is connected through 41 transmission lines and the total system demand amounts to 2.834 p.u. Fuel cost and NOX emission coefficients are provided in Tables I.
5. RESULT AND DISCUSSION 5.1 GENERAL
In order to demonstrate the capability of the FCGSO based GPD proposed in this work, a modified IEEE 30- bus six-generator test system from the literature is used for the simulation study. The case study adjusts the minimization of fuel cost, emission dispatch and power loss output and improve the GPD problem. The FCGSO algorithm problems have been implemented using the Matlab1 programming package.
Numerous solutions have been performed with constraint strategy and the best obtained values were utilized to ensure the benchmarking of the proposed FCGSO approach against the very optimal solutions.
Best fuel cost and best emission dispatch of IEEE 30- bus system:-
Fig.5.1 Simulated plot of the standard IEEE 30-bus six -generator test system
5.2 COMPARISON OF RESULTS
RESULT COMPARISON TABLE (1)
Genera tor no.
NSGA
[19]
NPGA
[21]
SPEA
[22]
GSO
[1]
FCGSO
PROPO SED
PG 1 0.1168 0.1245 0.1086 0.1144 0.1147
PG 2 0.3165 0.2792 0.3056 0.3109 0.3033
PG 3 0.5441 0.6284 0.5818 0.5966 0.5973
PG 4 0.9447 1.0264 0.9846 0.9772 0.9819
PG 5 0.5498 0.4693 0.5288 0.5141 0.5128
PG 6 0.3964 0.3993 0.3584 0.3518 0.3551
Cost 608.24 5
608.14 7
607.80 7
607.39 0
607.381
Emissio n
0.2166 4
0.2236 4
0.2201 5
0.2196 0.21996
CONCLUSIONS
In this work, a group search optimization algorithm has been applied to solve generation power dispatch problem. The GPD problem has been formulated as a constrained optimization problem where several objective functions have been considered to minimize fuel cost, emission dispatch and power loss. The proposed approach has been tested and examined on the standard IEEE 30-bus test system. The simulation results demonstrate the effectiveness and robustness of the proposed algorithm to solve GPD problem.
Moreover, the results of the proposed FCGSO algorithm have been compared to those reported in the literature. The comparison confirms the effectiveness and the superiority of the proposed FCGSO approach over the classical and heuristic techniques in terms of solution quality. In comparison the other methods, the FCGSO method can avoid the shortcoming such as the premature convergence of PSO method and can discover higher quality solution for the problems
studied in this work. The FCGSO has been successfully implemented to solve GPD problem by considering continuous, discrete and binary variables. The minimum loss obtained by FCGSO is lesser than the minimum loss obtained by other algorithms. The minimum loss value obtained by the proposed FCGSO is always closer to the average loss value. The robustness of FCGSO is displayed by its convergence characteristics. It is observed from the repeated trail runs that FCGSO converges to near optimal solution with high success rate. The computational results show that the FCGSO can be used for solving the GPD problems successfully.
NOMENCLATURE
ai ,b i,ci,di ,ei - Fuel cost coefficients for the ith generator with valve –point effect.
αi , i , i, δi, λi - Emission coefficient for the ith generator.
Amax - Maximum number of iteration of scanning.
Crp, Crsl - Constants for controlling ramping rate of the hyperbolic tangent function.
Cp, max, CP, min - Upper and lower bounds of CPK
. Meq - Number of equality constrints.
Mineq - Number of inequality constraints.
MKing - Number of infeasible members in the gth group at kthiteration.
Mobj - Number of GPD objective functions.
Mp - Population size of each searching group.
NG - Number of generating units in the system.
NL - Number of branches in the system.
PD - Total active power demand.
PGi - Active power generation of generation i.
PGi, max - Maximum active power output of generation i
PGi , min - Minimum active power output of generation i
PLoss - Total power loss of transmission network.
UB - Upper bound for variable vector XKgj . Ug - Upper expectation limit for the gth objective player.
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