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1. 1. Fuzzy adaptive gravitational search algorithm PREPARED BY K.GAUTHAM REDDY 2011A8PS364G
2. 2. Bidding  Agents submit bids (Quantity and cost) to either buy or sell energy.  Independent System Operator (ISO)  Market clearing price  Uniform pricing or pay as bid
3. 3. Strategic bidding  Knowing their own costs, technical constraints and     their expectation of rival and market behavior, suppliers face the problem of constructing the best optimal bid. Three basic approaches: i) Based on the estimation of market clearing price ii) Estimation of rival’s bidding behavior and iii)On game theory
4. 4. Fuzzy adaptive GSA Mathematical formulation  Consider total of ‘m’ suppliers  Uniform pricing method is followed  The jth supplier bid with linear supply curve denoted by Gj (Pj ) = aj + bj Pj Pj is the active power output, ajand bj are nonnegative bidding coefﬁcients of the jth supplier.
5. 5.
6. 6.  When we solve the above equation we get the solutions as  Cost function :Cj (Pj ) = ej Pj + fj Pj2 , where ej and fj are the cost coefﬁcients of the jth supplier.
7. 7. Profit maximization  Hence our main objective is to maximize profits which is the difference between the selling price and the production price which is as follows  The objective is to determine bidding coefficients aj and bj so as to maximize F(aj,bj) subject to equations 5 and 6.  Keep one constant and vary other as they are interdependent
8. 8. Gravitational search algorithm:  Follows two basic laws i) Law of gravity ii) Law of motion.  Agents are considered as objects and their performance is measured by their masses.  Lighter masses gravitate towards the heavier masses (which signify good solutions)  The position of the masses correlates to the solution space in the search domain while the masses characterize the fitness space.
9. 9.  As the iterations increase, and gravitational interactions occur, it is expected that the masses would conglomerate at its fittest position and provide an optimal solution to the problem.  Now, consider a system with N agents (masses), the position of the ith agent is deﬁned by: Xi = (x1 , . . . , xd , . . . , xn ) for i = 1, 2, . . . , N where xd presents the position with N agents (masses), the position of the ith agent in the dth dimension and n is the space dimension
10. 10.  At a specific time ‘t’ we define the force acting on mass ‘i’ from mass ‘j’ as following: where Maj is the active gravitational mass related to agent j, Mpi is the passive gravitational mass related to agent i, G(t) is gravitational constant at time t, ε is a small constant and Rij(t) is the Euclidian distance between two agents i and j.
11. 11.  The total force acting on each mass i is given in a stochastic form as the following  where rand(wj) ∈ [0, 1] is a randomly assigned weight. Consequently, the acceleration of each of the masses, is then as follows. where Mii is the inertial mass of ith agent.
12. 12.  The next velocity of an agent is considered as a fraction of its current velocity added to its acceleration. Therefore, its position and its velocity could be calculated as follows: vi (t + 1) = randi × vi (t) + ai (t) xd (t + 1) = xd(t) + vd(t + 1) where randi is a uniform random variable in the interval [0,1].
13. 13.  The gravitational constant, G, is initialized at the beginning and will be reduced with time to control the search accuracy. Hence, G is a function of the initial value (G0) and time (t): Here G0 is set to 100.
14. 14.  Fuzzification:  Inputs : (i) normalized fitness value (NFV) (ii) current gravitational constant (G) Outputs: The correction of the gravitational constant (dG).
15. 15.  Input variables represented by three linguistic values, S (small), M (medium) and L (large) where as output variable (G) is presented in three fuzzy sets of linguistic values; NE (negative), ZE (zero) and PE (positive) with associated triangular membership functions.
16. 16.  The value of the parameter ‘G’ is large at the beginning of the search process and gradually it becomes small as the iterations are increasing. The change in gravitational constant (dG) is small and requires both positive and negative corrections.  After we get a new value of G, GSA until iteration reaches their maximum limit. Return the best fitness (optimal bid value bj) computed at final iteration as a global fitness. Using bj values, calculate MCP