A Novel Hybrid Moth Flame Optimization with Sequential Quadratic Programming Algorithm for Solving Economic Load Dispatch Problem

The insufficiency of energy resources, increased cost of generation and rising load demand necessitate optimized economic dispatch. The real world ED (Economic Dispatch) is highly non-convex, nonlinear and discontinuous problem with different equality and inequality constraints. In this research paper, a novel hybrid MFO-SQP (Moth Flame Optimization with Sequential Quadratic Programming) is proposed to solve the ED problem. The MFO is stochastic searching algorithm minimizes by random search and SQP is definite in nature that refines the local search in vicinity of local minima. Proposed technique has been implemented on 6, 15 and 40 units test system with different constraints like valve point loading effect, transmission loss, prohibited zones, generator capacity limits and power balance. Results, obtained from proposed technique are compared with those of the techniques reported in the literature, are proven better in terms of fuel cost and convergence.


INTRODUCTION
In past many conventional optimization techniques are implemented on ED problem. These techniques are based on derivatives. For example, λ-iteration and gradient methods [1] use first order derivatives.
Classical techniques are not suitable for solving nonconvex ED problem due to local optimum entrapment and requirement of starting point for their working [2]. LP (Linear Programming) method [3] is found to be competent but it is limited to linear optimization problem only. DP (Dynamic Programming) has been used to solve the non-convex problem but it suffers from curse of technique has some advantages as well as some drawbacks. There is always the need of new techniques, which can solve problem better than the current optimization techniques. Intelligence) technique developed by Mirjalili [11]. This SI algorithm is made by motivation from navigation method of moths.

MFO algorithm is novel SI (Swarm
This technique has been implemented on ED problem.
As compared to other techniques, MFO reaches near global optimum and less global searching ability especially for large scale. To overcome the abovementioned drawbacks, SQP is proposed to hybridize with the MFO. In the start MFO will run and explore the search space to get optimal solution. This optimal solution of MFO will become the starting point of SQP.
SQP is used as local fine tune searcher for the search space explored by MFO. In this way, results obtained after SQP will be more refined. This hybrid algorithm is tested on 6, 15 and 40 units systems and results are compared with those obtained by using different algorithms reported in the literature.

PROBLEM FORMULATION
ED is generally a minimization problem with different constraints. Mathematically there is an objective function that is total cost of generation supplied to the load. It is given by [12].
F t represents the total fuel cost of generation and F i P i is the fuel cost of i th generator and P i is the active power generation of i th generator.
Objective function of ED problem has two types, convex and non-convex. Quadratic curve is used to represent the convex characteristics and modeled in [12].

Power Balance Constraint
Real power balance constraint is given by Equation

Generator Limits
Generation from generator must be in limits which is modeled as: P i,min is the minimum power generation capacity and P i,max is the maximum power generation capacity of generator.

Transmission Loss
Transmission loss in ED problem is calculated by using coefficient method which is described as follows: B  is a constant, B oi is a vector which have dimension equal to P i and B ij represents the loss coefficient matrix.

Prohibited Operated Zones
There are regions in the generator output which are prohibited. Vibration is produced if generator is operated in that region which ultimately results into the damage. These zones are modeled as follows: max , , In Equation (7), m is represents the number of prohibited zones, P i,j-iu is the lower limit of i th operating zone of i th generator and P i,mu is the upper limit of i th operating zone of i th generator. All above constraints are presented in [12].

MOTH FLAME OPTIMIZATION
MFO is a novel SI technique developed by Mirjalili [11]. to update line search which is described in [13].
In Equations (8-9) H K represents the Hessian matrix in the langrangian function, d K represents the direction search, P is the vector that represents the lagrangian multiplier and H K matrix is calculated by quasi-newton formulation In every iteration, sub-QP problem the direction d K is calculated and the solution found from the novel iterate as: For achieving a significant decrease in augmented lagrangian merit function the step length value  K is determined.

IMPLEMENTATION OF MFO-SQP ON ED PROBLEM
Step-1: An initial random population containing generation allocation is generated by selecting the number of moths Step-2: In second step fitness values of all moths are calculated using objective function ED problem.
Step-3: In this step iteration number is checked. If iteration number is one, then sort single moth population. Sorted moth population is assigned as flames. If iteration number is greater than one, then previous population will be taken in this iteration. In this way, double population will obtain in every iteration after the first iteration. To get double flames, double population of moths will be sorted and their fitness function is calculated.
Step-4: In this step updating procedure is applied to get new population of moths. First flame number is to be found by using following formula:

A Novel Hybrid Moth Flame Optimization with Sequential Quadratic Programming Algorithm for Solving Economic Load Dispatch Problem
N is maximum number of flames. Where In spiral function b is a constant which defines the shape of spiral. In ED problem value of b is taken 1 and t is a random number. Following formula is used to calculate t. t = (-1+a) random + 1 (22) In Equation (22) value of a is called convergence constant which is linearly decreases from -2 to -1.
Step-5: Inequality and equality constraints are checked in this step. if inequality constraint such as generator capacity limits, prohibited operated zones are violated then fix them. Equality constraint is handled by using penalty.
Step-6: After reaching maximum iterations best position obtained through MFO will be used as starting point of SQP. Now SQP optimize and produce best solution which will replace with the solution of MFO.

CASE STUDIES AND RESULTS
Proposed algorithm has been implemented on 6 Fig. 1 is showing the convergence characteristics of MFO-SQP at 300 iterations. Proposed algorithm converges to optimal point before reaching to stopping criteria.

Test System-2
In this system 15 units test system has taken with load demand of 2630 MW. For the sake of comparison convex