A Hybrid Sine Cosine Algorithm with SQP for Solving Convex and Nonconvex Economic Dispatch Problem

ED (Economic Dispatch) is one of the major problems of power system operation. The aim of ED problem is the efficient utilization of resources to provide the demanded power while generating cost turns out to be minimum and no constraint is violated either equality or inequality. The ED optimization problem, is necessary because of limited resources, high fuel cost and ever growing demand of power. This paper presents solution to convex and nonconvex ED problems using a novel HSCA (Hybrid Sine Cosine Algorithm). The proposed HSCA technique enhances the exploration capabilities of SCA (Sine Cosine Algorithm) by equipping it with mutation and crossover operators from DE(Differential Evolution) algorithm. DE algorithm introduces diversity in the operation of SCA enabling it to avoid local minima and premature convergence. To ensure precise and accurate optimum tracking results are finally refined by SQP (Sequential Quadratic Programming) algorithm. The high feasibility and applicability of proposed technique has been tested and validated on 13, 15 and 40 IEEE Standard test systems considering transmission losses and prohibited operating zones in “MATLAB 2014a”. Comparisons of results obtained from HSCA indicate significant improvement in convergence time and fuel cost as compared to the techniques reported in the literature.

. The valve-point loading effect is

A Hybrid Sine Cosine Algorithm with SQP for Solving Convex and Nonconvex Economic Dispatch Problem
due to multiple valve operations in thermal plants to change the fuel type according to the generation. And the valve operation makes the characteristic curve of the generating unit nonlinear and discontinuous. This effect is modeled [2] in the characteristic curve as cyclical sinusoidal function as shown in Fig. 1(a).
The discontinuity due to POZs in the fuel cost characteristics may be due to the resonant frequency created by the addition of all the frequencies of different parts of the machine, or due to the variation in shaft bearing affected by the valve operation or due to the faults in the auxiliary equipment of machine like boiler and feed pumps etc. Generally, the determination of prohibited zones is a difficult procedure accompanied by performance tests. In practical operation, the generation in these regions (POZs) is avoided as it would not be economical to use these zones of operation [3][4]. The objective function of cost which considers this constraint (POZs) is shown in Fig. 1

(b).
To tackle with the ED problem various approaches have been proposed in literature so far. These proposed methods cab be classified as classical and non-classical.
In classical methods, Base-Point and Participation Factors, λ-iteration method [5], linear programming, quadratic programming [6], branch and bound [7], gradient method [8] were included. These methods were able to compute the ED successfully but because of the discontinuities and nonlinearities introduced by valve-point loading and the POZs, the gradient-based methods were unsuccessful in achieving an optimum solution. The strength of classical methods was demonstrated for continuous and smooth objective functions but the difficulty faced in solution to discontinuous and non-convex problems established the need for further improvement. Many improvements in the classical methods were made in the last decade but these improvements required additional computations to address these complications.

PROBLEM FORMULATION
This section describes the formulation of ED problem. While, FC represent the total cost of fuel and the F i (P i ) represents the cost of i th the unit and "n" gives the total number of units. The expression for cost of fuel is represented as:

Description of ED Problem
Where P i gives the real power engendered from the i th unit and a i is the fuel-cost coefficient for that specified unit i in While "i" represents the committed generating unit and "d" and "e" represent the valve point loading coefficients.

Constraints
ED problem is about the minimization of fuel cost using fuel cost function (either convex or nonconvex) subjected to different constraints as follows: , and B 00 are the B-coefficients.

Capacity Limit Constraint
The real power generated from each generation unit must not violate its max   max i P and   min i P limit. Mathematically,

Prohibited Operating Zones
There are some regions in the range of operation of generating unit in which the system may lose stability if operated in these specified regions due to synchronization of all the components of the generating unit. These regions are named as prohibited operating zones. And are therefore avoided in practical generations.
Mathematically these are represented as follows: Where the  In more detail, the influence of CSA trigonometric functions in the specified range as in Fig. 2. i.e. [-2,2] is explained in Fig. 3. Fig. 3 clearly describes the effect of altering the range of cosine and sine that how this change necessitates an answer to update its next location inside or outside of the search space in between itself and next solution.
As we have discussed the performance measures of SCA, the technique has been hybridized with DE and SQP using the following steps to solve the concerned ED problem.
Step-1: Initialization: First, the population is initialized randomly in the lower and upper limit of generating units according to the following equation. P = P min + rand.(P max -P min ) (8) Step-2: Evaluation of Fitness Function: The population is updated according to the CSA equations as stated below:

FIG. 3. SINE, COSINE IN THE RANGE IN ["2,2] REPRESENTING HOW A SOLUTION GOES INSIDE OR OUTSIDE THE SPACE CONFINED
Step-3: Update the Population According to SCA: The population is updated according to the CSA Equations (10-11) as stated below: As Equations (10)(11) show, there are four random parameters r 1 , r 2 , r 3 , and r 4 .
The first random parameter r 1 prescribes the direction of movement which could be outside the destination and solution or inside it. This random parameter adaptively changes the range of cosine and sine functions to balance between exploration and exploitation using the equation as: Where "t" denotes the present iteration and "T" represents the max number of iteration and "a" is a constant. The second random parameter "r 2 " emphasizes that how long next movement would be towards or outwards the destination point. This movement inside or outside the search space is obtained by defining the random parameter "r 2 " in the range [0, 2]. The third random parameter "r 3 "is the random weight for the destination point distance stochastically. And the fourth random parameter "r 4 "is the switch between the sine and cosine function to be used for updating the population.
Step-4: Application of Mutation and Crossover: By applying the crossover and mutation, the updated population is modified further to introduce diversity and avoid local optima stagnation.

Mutation:
In the mutation operator, in the current population, a mutated vector is generated for each targeted vector except for the running index.
Where r 1 and r 2 are the randomly initialized integers dissimilar to each other and from the running index "i" too. And SF is the scaling factor which is a randomly engendered number between (0,1) but may not be homogeneously disseminated in the range of (0,1).
Generally, the choice for the SF is between 0.4 and 1 but here in the proposed work, the dynamic behavior to the SF is applied by varying SF in a randomly generated number in the range of (0.5, 0) [34]. Here in this case due to the randomly scaled difference vector, there is a better chance of pointing the new mutated vector at the even better location until the true global optimum is achieved.
Crossover: After the application of mutation operator, the crossover operation is applied to upsurge the diversity of individuals. The previously generated vector is assorted with the vector produced after mutation to form a new trial vector w ji,g+1 .
Where i=1…N P and j=1 …Q; And "m" is a random number elected between 1 and N P which makes sure that the new trial vector takes the smallest parameter from the mutant vector. And C R is the crossover parameter defined by the user and reins the diversity of the individuals and helps the solution to avoid from local minimum [34].

Step-5: Verification of Equality and Inequality
Constraints:Check if the active power of any of the generating unit violates the limit or not. In case, power generation of any unit is less than the minimum limit then it is fixed at the minimum limit and if power generation of any unit exceeds the maximum limit then it is clamped at maximum limit. After the satisfaction of inequality constraint, the equality constraint is verified.
Step-6: Application of Sequential Quadratic Programming: After the completion of constraints handling, the SQP is applied to the best results obtained so far from the proposed algorithm. In the SQP the QP is solved in each step to improve line search which can be mathematically defined as:

Sequential Quadratic
i=m e +1… m Here, H k is Hessian Matrix and d k is direction search, P k is the real power vector andg i (P k ) represents the equality and inequality constraints in current iteration which is represented by "k". And m e is the equality constraint's quantity while "m" represents quantity of constraints. L(P,) = F(P) + g(P) T  Where λ represents the Lagrangian multiplier and H k is formulated with the help of Quasi-Newton. And, In each iteration, the direction of sub-problem is achieved and the new solution is found from new iteration as follows: For calculation of step length ( k ) the following formula is used as this step length is significant for the decrease in the Lagrangian merit function.
Step-7: Stopping Criteria: The same procedure from step-2 is repeated until the stopping criteria are achieved.
The SCA stops further computations if there comes no noticeable improvement or a maximum number of iteration is completed. In this work, the stopping criteria are maximum number of iterations.

RESULTS
The proposed HSCA is implemented to three ED problems to verify its feasibility. These three ED problems refer to

Test System-1
The load demand for 13-unit test system is 2520 MW.
The input data consisting of load demand, cost coefficients and generator limits are taken from [5].

Parameters
The Table 1 shows the parameters of mutation, crossover, and others used to optimize the results of 13-unit test systems.

Results
In Table 3 there is the statistical comparison of the proposed algorithm with other techniques and clearly shows that SCA minimizes the fuel cost as compared to other techniques mentioned in Table 3. Fig. 4 demonstrates the convergence curve of 13-unit test system solved using SCA and clearly describes that the system converges to optimum before stopping criteria.
In Fig. 5 the fuel cost of 13-unit test system is compared using a bar chart to simplify the analysis. Fig. 5 shows that lowest cost is obtained by using the proposed HSCA which in this case is 24164.06 US$//hr.

Test System-2
The load demand for 15-unit test system is 2630 MW.
The input data consisting of load demand, cost coefficients and generator limits are taken from [27]. The transmission losses are calculated using B-coefficients method. Table 4 shows the mutation, crossover and other parameters which are taken to optimize the results of 15-unit test system. In Table 6 there is the statistical comparison of the proposed algorithm with other techniques and clearly shows that SCA minimizes the fuel cost as compared to other techniques mentioned in Table 6. Fig. 6 shows the convergence curve of 15-unit test system solved using SCA and clearly describes that the system converges to optimum before stopping criteria.

Results
In Fig. 7 cost of fuel for 15-unit system is compared using a bar chart to simplify the analysis. Fig. 7 shows that the lowest cost is acquired by using the proposed "HSCA with SQP" which in this case is 32548.03 US$//hr.

Test System-3
The load demand for 40-unit test system is 10500 MW.
The input data consisting of load demand, cost coefficients and generator limits are taken from [43].

Parameters
The Table 7 shows the mutation, crossover and other parameters used to optimize the results of 40-unit test system. Table 8 shows the individual cost of each unit and total power corresponding to total fuel cost. Total power output comes to be 10500 MW and total fuel cost is 121983.5 $/ hr. Table 9 there is the statistical comparison of the proposed algorithm with other techniques and clearly shows that SCA minimizes the fuel cost as compared to other techniques mentioned in Table 9. Fig. 8 shows the convergence curve of 40-unit test system solved using SCA and clearly describes that the system converges to optimum before stopping criteria.

And, in
In Fig. 9 the fuel cost of 40-unit test system is compared using a bar chart to simplify the analysis. Fig. 9 shows that lowest cost is obtained by using the proposed HSCA which in this case is 121983.5 US$//hr.   Taxila, Pakistan, and its staff for all the resources provided.