Quasi-oppositional Grey Wolf Optimizer Algorithm for Economic Dispatch

Objectives: To minimize the fuel price of generator while satisfying different constraints. Valve point effects and multiple fuel option are also considered in some cases. Methods: Quasi-Oppositional Grey Wolf Optimizer algorithm is applied here for solving different economic dispatch problems. Grey Wolf Optimizer is a meta-heuristic method, motivated by social behaviour of grey wolves. Quasi-Oppositional learning is implemented in the present work within Grey Wolf Optimizer for improving the quality of solution in minimum time. Quasi-opposite numbers are used within the algorithm in place of normal random numbers for improving the convergence speed. Findings: The proposed technique is applied to six different systems to test the efficiency of the algorithm. Simulation results obtained by this method are compared with those obtained by some well-known optimization methods to show the robustness and superiority of this technique. Simulation results also show that the computational efficiency of Quasi-Oppositional Grey Wolf optimizer is better as compared to several previously developed optimization methods. Improvement: It is found that the convergence speed, success rate, efficiency and solution quality of the proposed algorithm is improved. *Author for correspondence


Introduction
Minimization of power generation cost of fossil fuel based plants is a big challenge for the power engineers. Therefore, in recent decades, researchers have given much attention for minimizing the cost of power production. The objective of Economic Load Dispatch (ELD) is to minimize the cost of power generation while satisfying various equality and inequality constraints. Many techniques have been developed to solve the economic dispatch problems. Erstwhile, many classical optimization methods like Linear programming 1 , Dynamic Programming 2 , Lagrangian method 3 etc. have been devel-OBL exploits the opposite numbers. By contrasting a number compared to the opposite number, a compact search space is required to obtain the correct solution. It has been demonstrated that a quasi-opposite number 38 is likely to be nearer to the solution as compared to an arbitrary number. It has additionally been demonstrated that a quasi-opposite number is typically nearer to the solution compared to an opposite number. As quasi-opposition based learning is proved to have improved computational efficiency, the present authors have adopted this methodology in GWO (QOGWO) for accelerating the speed of convergence of GWO to a greater extent. In this paper, the Quasi Oppositional Grey Wolf Optimizer (QOGWO) algorithm is used for solving various ELD problems and results obtained by QOGWO method are compared to other optimization techniques. The details of this proposed technique have been discussed in section 3.
Section 2 gives the problem formulation and brief description of various economic load dispatch problems. Section 3 describes a short description of GWO algorithm. Short description about QOGWO algorithm is explained in Section 4. Simulation results are discussed in Section 5. The conclusion is described in Section 6.

Quadratic Cost Function of ELD
The objective function 12 of economic dispatch problem for this case can be written as

Economic Dispatch with Valve Point Loading Effect
The overall objective function C T of ELD with valve point 12 can be expressed as follows where, E 1i and F 1i represents the coefficients of unit i reflecting the effect of valve point.

Fuel Price Function Considering the Effect of Valve -point and Multiple Fuel
In a network, if N represents number of generator and n F is the fuel option of individual unit, then the generator fuel price function considering the effect of valve point and multiple fuel can be represented by ( ) ( )

Real Power Constraint or Demand Constraint
The total generation must be equal to transmission loss and system demands. This can be represented as where, L W , D W represents the total transmission loss and total system demand respectively. The transmission loss W L can be calculated as

Generator Operating Limits Constraint
The power generated by individual generator must vary within it's maximum and minimum limit. Therefore, mathematically this may be written as where, min and max i W are the minimum and maximum real power output of the i th generator.

Ramp Rate Limit Constraints
In practical circumstances, the working range of each online unit may be confined by the ramp rate limit 12 . Depending on up ( i UR ) and down ( i DR ) ramp rate limits, the generation can be increased or decreased.

If generation increases
If generation decreases where, 0 i W represents power generation of i th unit at earlier hour.

Prohibited Zone Constraint
Each generator might have some zone of operation where operation is limited because of fault in the machines, steam valve operation, boilers, vibration in shafts etc 12

With Transmission Loss
Using equation (6) and (14) the modified equation may be written as W N is the same as mentioned in 31

Grey Wolf Optimizer
This section of the paper presents a newly developed optimization method called grey wolf optimizer, which has been proposed in 36 . Grey wolves usually live in a group of 5-12 members. Alpha is the group leader and their main their main tasks are decisions making, hunting etc. The second member in the hierarchy is Beta and it helps Alpha in order to make the decision. Delta and Omega are the lowest ranking grey wolves. Omega wolves are not an important individual, but entire group facing the fighting problem if they lose any omega. Deltas are responsible for watching the boundary of territory, carrying weak grey wolves, and help the Alpha at the time of hunting.

Encircling
Hunting behaviour of the grey wolf is started by encircling the prey. For mathematically model of this behaviour following equations have been developed. can be formulated as follows Here the value of s  are decreased in the iterative process from 2 to 0.

Hunting
Alpha wolves are always guiding the other grey wolves at the time of hunting. But in an abstract place, it is very difficult to guess the prey's optimal location. According to the best search agent position, other search agents' positions are updated. This behaviour is mathematically modelled using the following equations

Exploitation
The hunting process of grey wolves has been stopped after attacking the prey. In order to mathematically model this,

Exploration
Grey Wolves wander from one place to another in order to search a prey and they unite again to assault the prey. To model this behaviour, P  is used having random values less than -1 or more than 1 or in order to separate the search agents from the prey.
• Update , s P   and S  • Calculate the fitness value of all modified sets.
• Update the values of G α • Repeat the steps (iv)-(vii) until the termination criteria are fulfilled otherwise stop the process.

Oppositional based Learning
Tizhoosh has proposed Oppositional based learning technique to enhance computational speed and quicken the rate of convergence of various optimization algorithms. For randomly generated population number, it is not possible to guess about the value optimal solution and that is why time required to reach optimal solution is large. In OBL, opposite numbers are introduced along with population numbers. It is found that OBL has the capability to reach optimal solution in minimum time due to introduction of these opposite numbers. Oppositional based learning deals with opposite numbers, quasi-opposite numbers and Quasi-reflected numbers.
If y is the real number between [ pq , pr ] then the opposite number of y may be defined as Here qr y is the quasi-reflected point.
The above-mentioned definitions can without much of a stretch may be reached out to larger dimensions.

Implementation of OBL in GWO Algorithm
Oppositional based learning technique is implemented in GWO algorithm in order to accelerating the convergence speed of GWO.QOGWO algorithm starts by choosing GWO variables such as search agent number, maximum iteration number. The population matrix U is generated according to the number of search agent and dimension of the problems. It is necessary to check the various constraints limits. The quasi-reflected sets are generated. After generating quasi-reflected set, the fitness function is calculated for initial population set and quasi-reflected set. On the basis of fitness, the best U sets are sorted out and an updated matrix is formed. After that the minimum value of fitness function is calculated and the correspond-  is calculated using equation (18) and (19). The position of each search agent is updated using equation (20)- (25). Again random number is generated. If the random number is less than jumping rate, quasi reflected sets are generated using the updated set. The constraints limits are checked again. The fitness value for each updated search agent is calculated if all constraints limits are satisfied otherwise quasi-reflected sets are again generated. On the basis of fitness, the best U sets are sorted out between updated set and quasi reflected set.
The process is terminated if maximum number of iteration is reached.

Algorithms for Quasi Reflected based Initialization
The steps are given below: Randomly generate initial population U in between maximum and minimum limits of decision variables and in between [0 1] generate a reflection weight μ.
Generate Quasi-Reflected Sets (QRS) for each initially generated population set U, using following procedure Sort out best A individuals on the basis of their fitness.
Store the best population sets.

Effect of Jumping Rate
Quasi reflected sets have been used in GWO algorithm for accelerating the convergence speed. But it has been found that if quasi reflected sets are generated in every steps, it may increase the simulation time. Therefore, in order to optimize the computational time, a control parameter called jumping rate 38 has been used. It is a control parameter whose value is set by the user in order to skip the creation of quasi reflected set at certain generation. The effect of this parameter in QOGWO algorithm has been explained in the flow chart of section 4.4.
The pseudo code is given below

Application of Quasi-oppositional Grey Wolf Optimizer Algorithm in ELD
The flow chart of QOGWO algorithm is described in Figure 1 which shows the application of QOGWO algorithm in ELD problems.

Simulation Results
The QOGWO algorithm has been applied to six different systems of ELD problem and the performance of this method is compared with different soft computing techniques like GA 4 , PSO 7 , RCCO 11 , ORCCRO 12 , TLBO 14 , OIWO 32 , DE/BBO 31 , EMA 15 , GWO 20 and so on. This algorithm has been coded in Matlab 9 and the program has been executed on a 2.40 GHz core i3 computer with 2 GB RAM.

System 1
A 3 generator 20 system having load demand of 850 MW and 1050 MW is considered in this case. The transmission loss has not been considered here. The best results obtained by QGWO, GWO 19 , GA 20 , PSO 20 , ABC 20 and ALO 20 are presented in Table1 and Table 2. The cost convergence characteristics for 3 generators system are    shown in Figure 2 and Figure 3. Best, worst and average fuel price obtained by QOGWO, Lambda iteration 20 , GA 20 , PSO 20 , ABC 20 , GWO 19 and ALO 20 are shown in Table 3 and Table 4.

System II
In this system, a 5 generator system is considered. The Fuel price characteristic is quadratic in nature. The sys-tem demand is 730MW. The data required for this case is taken from a paper written by Kamboj et al. 20 20 and ALO 20 are displayed in Table 6. Figure 4 shows the fuel price convergence curve.  Table 4. Performance analysis of different methods for system I taken after 50trails (Load=1050MW) Figure 4. Convergence characteristics of system II.   Table 6. Performance analysis of different methods for system II taken after 50trails (Load demand=730 MW)

System III
A 6 generators system having the system demand of 1263 MW is considered in this case. The prohibited zones, ramp rate limit and transmission loss are also considered here. Necessary data for this system is taken from a paper written by Roy et al. 33 The results obtained by QOGWO, GWO, GA 33

System IV
A 38 generators unit, 13 with quadratic fuel cost function is considered in this case. The system demand is 6000MW. The transmission loss has not been considered for this case. The best result obtained by QOGWO method is shown in Table 9. The average, worst and best fuel price obtained by QOGWO, GWO, BBO 13 , DE/BBO 26 and RCCRO 11 is shown in Table 10. Figure 6 presents the convergence characteristic for 38 generators system.

System VI
In this system, a 160 generators having multiple fuel options is considered. The load demand for this case is 43200 MW. The transmission loss is not considered here. The input data for this system is taken from a paper written by A.K. Barisal and R.C. Prusty 32 . The results of QOGWO and GWO techniques is shown in Table 13. Table 14 describes the minimum, mean and worst cost of  fuel obtained by various methods like ORCCRO 12 , BBO 13 ,  DE/BBO 31 , ED-DE 32 , IGA-MU 32 , CGA-MU 32 OIWO 32 , GWO and QOGWO method. The fuel price convergence curve of this system is shown in Figure 8.

Robustness and Solution Quality
The best results obtained by QOGWO algorithm are presented in Tables 1, 3, 5,7,9,11 and 13. It is found that QOGWO method gives better results as compared to other well-known optimization techniques. The best, average and worst values for different optimization techniques are presented in Tables 2, 4 The performance of QOGWO algorithm is judged after running the program for 50 numbers of trials. Out of 50 trails, QOGWO hits the minimum solution 48 times for system I, 47 times for system II, 50 times for system III, 50 times for system IV, 50 times for system V and 49 times for system VI. Therefore, it is found that the success rate of QOGWO is 92%, 90 %, 96 %, 100%, 100%, 100% and 98% respectively. Therefore, from these simulation results, it is seen that QOGWO algorithm shows better performance in terms of robustness and as well as solution quality when compared with GWO and other previously developed optimization techniques.

Computational Efficiency
The QOGWO algorithm takes less time to reach minimum solution as compared to other meta-heuristic optimization techniques. From tables 2, 4, 6, 8, 10, 12 and 14 it is found that, the computational efficiency of QOGWO is better as compared to recently developed optimization techniques. It is also seen that when oppositional based learning is applied in GWO algorithm then the convergence rate becomes faster as compared to other techniques such as GWO, BBO, DE/BBO, RCCRO, EMA and so on.

Statistical Analysis
In recent years, various statistical methods 39,40 has been used for finding out robustness of different algorithms. In this paper, Friedman test and Quade test are chosen to assess the solution quality of QOGWO algorithm as compared to GWO and other recently developed optimization techniques. Table 16 describes the statistical analysis of QOGWO, GWO, DE/BBO, BBO algorithms. Table 16 shows that F-statistic (Chi-Square) value is 9 and Q-statistic value is 15. It is found that F-statistic value is greater than its corresponding critical chi-square value (7.82) and Q-statistic value is also greater than its critical value (4.76). It is also found that p-values obtained by Friedman test and Quade test are less as compared to p-value at 5% significance level. Therefore, there is major dissimilarity between the algorithms. Depending on the average errors evaluated for different cases, the algorithms are ranked and results are shown in Table 16. Therefore, it is clear that rank achieved by QOGWO algorithm is minimum. The average errors of different algorithms are shown in Table 17. Therefore, it may be concluded that in terms of quality solution the QOGWO algorithm gives  Table 17. Average errors obtained in system II, system III and system IV