Golden jackal optimization for economic load dispatch problems with complex constraints

ABSTRACT


INTRODUCTION
The design and operation of power systems become more complex day by day.One of the primary key optimization challenges for the efficient and error-free functioning of power systems are economic load dispatch (ELD) problem.A power system's overall load demand is distributed across various generating units using the ELD problem to increase operational efficiency.ELD problem is a non-convex, non-linear complex optimization problem in power engineering.Traditionally, quadratic fuel functions are used in ELD formulation.While considering various real-time constraints such as valve point loading (VPL) effect and prohibited operating zone (POZ) the ELD problem complexity increases.Due to VPL effects and POZ, the search space for the answer will have discontinuities and the number of minimum points.Therefore, the ideal problem for the ELD issue is non-linear with discontinuities and calls for suitable solution methods.
Numerous optimization strategies, including mathematical programming techniques such as lamda iteration method, Lagrange multiplier method, newton method, gradient method, dynamic programming method, and heuristic algorithms such as simulated annealing, artificial bee colony algorithm, have been used to address the issue of economic load dispatch.Due to the problem's extremely non-linear characteristics, the

ELD PROBLEM FORMULATION
The main aim of the ELD problem is to optimize the output of the generators in the power system to meet the system load demand under various system constraints.This section explains the objectives and various system constraints in ELD problem.

Objective function
Traditionally, the fuel cost relation for a thermal generator is denoted by the quadratic and represented by (1): Where: a, b, c: the coefficient of fuel cost functions.Minimization of the fuel cost is the main objective of the ELD problem, and it is represented as (2): Where C is the scheduling cost of the system,   (  ) is the fuel cost function of  ℎ unit, N is the total number of generating units in the system, and   is the power output of the  ℎ unit.The consecutive valve opening in multivalve steam turbines ripples the generator's fuel cost curve.The fuel cost function should take this VPL effect to simulate a real and valuable ELD problem. Figure 1 shows the fuel cost function of a thermal generator for the two different cases.
here d and e are the coefficients related to the VPL effects of the thermal generators.In some cases, there are few generating units with different fuel sources.In (4) describes the fuel cost equation for such generating units.
The cost curve for a generator with k fuel alternatives is segregated into k discrete sections between upper and lower limits.In this,   ,   ,    are the cost coefficients of the  ℎ unit using fuel type k.Fuel cost function with many fuel alternatives and no VPL impacts and with VPL impacts depicted in Figure 2 and Figure 3

System constraints 2.2.1. Power balance restriction
The total summation of power output from the generators must be equal to the sum of the power needed and any transmission losses.This condition is given by ( 5): Here   is the system loss and   is the total load requirement.
The method-based B coefficient formula is adopted to calculate the system loss.  ,   ,   are the generator's loss coefficients.

Generator capacity constraints
Generators in the power system network can generate power between two extreme capacities.It is an inequality constraint.The limitation is depicted by (7): Where    and    are the upper and lower limits of the power generated by the i th generator.

Ramp rate limit constraints
Ramp rate constraints limit the operating range of generating units such that they can only operate continuously between two neighboring defined operating regions.The ramp rate limitation regulates all generating units' power output, which appears in (8): Where URi, and DRi, denotes upper and lower end of the generator limits.   is the initial power output of the i th generating unit.

Prohibited operating zone
The prohibited operating zones are caused by the working of the steam valves or vibrations in the shaft bearings.The practically feasible areas of unit i can be shown (9): Where  . and  . are the lower and upper limits of prohibited operating zones of i th generator, k is the number of prohibited zones.

Constraint handling mechanism
Constraint violations are handled using penalty-based approach.Thus, the overall fitness function combines both equality constraints and objective function and it can be defined as (10): When objective function includes VPL effect then the overall fitness function can be mentioned as (11): ) (11) Penalty factor in the above mentioned equations is a constant value and it is taken as 500.

GOLDEN JACKAL ALGORITHM
Chopra et al. [21], devised the swarm intelligence algorithm known as the golden jackal optimization algorithm; it imitates golden jackals' natural hunting techniques.Usually, male and female golden jackals hunt together.Three steps make up the golden jackal's hunting habit: i) searching and moving toward the prey; ii) getting close to the prey and agitating it until stops moving; and iii) bouncing towards the prey.A set of prey position matrices with random distributions are constructed during the initialization phase.

785
here N indicates the prey population numbers, and n indicates dimension.The following mathematical ( 13) and ( 14) give a mathematical illustration of the hunting behavior of golden jackals.Exploration phase here t shows the current iteration,   () shows the location of the female,   () depicts the location of the male golden jackal.() shows the prey location vector, and  1 () and  2 () are the upgraded locations of both jackals.The prey escaping energy E is obtained as (15): where  1 expresses the diminishing energy of the prey.
where T represents the maximum iteration,  1 represents a constant of 1.5, and  0 represents the starting state of the energy.
where  represents a random value in [0,1]. expresses a random vector based on the levy distribution.
where ( + 1) is the revised location of the prey according to both golden jackals.The element rl used in the algorithm provides random movement and helps to avoid local optimal.

SIMULATION RESULTS AND DISCUSSIONS
The effectiveness of the proposed GJO algorithm is analyzed on six test systems with 6-unit, 13-unit, 10-unit, 40-unit, and 140-unit systems with different complex constraints of power systems such as transmission loss, POZ, VPL effects, ramp rate limit, and MFO.The program is developed on MATLAB 21a software and implemented on a personal computer with an Intel i7 processor and 4GB RAM.The efficacy of GJO on specified cases of ELD is compared with numerous algorithms in literatures.Constraints considered for the test system is shown in Table 1.For the simulation of GJO, the population size of 100 and maximum iterations of 500 are considered.

Case study 1
In this research case, a test system with six thermal generators with a load demand of 1263 MW is considered.Different power system constraints, such as transmission loss, generator capacity constraints, POZ, and ramp rate limits are considered.Various fuel cost coefficients and generator constraints are taken from [1].The developed GJO algorithm is applied for this 6-unit test case, and results are tabulated in Table 2. Results show that the proposed algorithm gives better optimal generation scheduling without violating the power system constraints considered.A comparison of obtained results with the other heuristic algorithms such as coulomb-franklin's algorithm (CFA), exchange market algorithm (EMA), krill herd algorithm (KHA), backtracking search algorithm (BSA), is shown in Figure 4. GJO algorithm gives the best optimal cost of 15441.9($/hr.), which is lower than the other algorithms.A convergence characteristic of the GJO algorithm for this test case is shown in Figure 5, and it can be seen that the GJO algorithm gives the best optimal solution in the early stage of iteration.

Case study 2
A case study was conducted on a 13-thermal unit system with valve-point loading effect.The system simulation data were taken from [7].The required power demand was 2520 MW [7].Table 3 compares the results obtained using the proposed GJO algorithm and other state-of-the-art algorithms namely hybrid stochastic search (HSS), tabu search algorithm (TSA), hybrid evolutionary programming-sequential programming (EP-SQP), and hybrid particle swarm optimization-sequential programming (PSO-SQP).From Figure 6, it can be observed that the minimum optimal cost attained by GJO method is lower than that of the other alternative algorithms.The GJO algorithm yielded a minimum fuel cost of 24164.02($/hr), next best fuel cost is obtained by the PSO-SQP algorithm which is 24261.05$/hr.Figure 7 shows the convergence curve for the GJO method in simulation of test case 2 and it shows the robustness of the GJO method.

Case study 3
In this test case, challenging test system having 10-units along with MFO is considered.Total power system load demand of 2700 MW is considered.Transmission loss, VPL effects and other constraints are not considered in this case.Various parameters and fuel types are considered from [22].ELD problem for this test case is simulated using GJO algorithm and results are tabulated in Table 4.  Obtained best fuel cost by GJO algorithm is contrasted with best results of other algorithms namely hopfield neural network (HNN), hybrid real coded genetic algorithm (HGA), multiplier updating method is combined with conventional genetic algorithm multiplier updating (CCGA-MU) in Figure 8.The fuel cost calculated using the GJO technique is 623.8086$/hr, with no limitation violations, indicating the suggested approach's excellent accuracy.

Case study 4
This system considers the test case 3 along with VPL effects.Coefficients of the VPL effects and other data are referred from [22].The power requirement of 2700 MW is considered.The suggested GJO algorithm's simulated best results are tabulated in Table 5. Results given by the other methods namely improved genetic algorithm with multiplier updating method (IGA-MU), CCGA-MU method, BSA, and coulomb-franklin's algorithm (CFA) are compared with the outcome of GJO method as shown in Figure 9.The GJO algorithm yielded a minimum fuel cost of 623.849 ($/hr), and it is lower than the results of other compared algorithms.In this analysis, a 40-unit system with valve-point loading effect and transmission loss is considered.The total load demand for this case study is 10,500 MW.The system's fuel cost, VPL, and loss coefficients are taken from [32].Table 6 shows the optimal power generation schedule obtained from different algorithms, including the proposed GJO algorithm.The simulation results indicate that the GJO algorithm produces the best feasible solution for this 40-unit test system.To validate the superiority and robustness of the GJO algorithm, its numerical results are compared with those of other algorithms, including GA-API [33], quasi-oppositional teaching learning based optimization (QTLBO) [34], oppositional real coded chemical reaction optimization (ORCCRO) [35], invasive weed optimization (OIWO) [32], teaching learning based optimization (TLBO) [34], shuffled differential evolution (SDE) [36], KHA [37].The comparison is shown graphically in Figure 10.The results show that the GJO algorithm produces better results than the others.Figure 11 shows the convergence characteristics of the fuel cost graph for the GJO algorithm.The graph shows that the GJO algorithm converges quickly and achieves the best optimal generator scheduling in the early stages of iteration.

Case study 6
To assess the efficacy of the suggested GJO method in solving large-scale power systems, this case study considers a system with 140 power-generating stations.The fuel cost characteristic coefficients and other data for thermal, gas, nuclear, and oil power plants are cited from [38].In this test case, VPL effects are taken into consideration for the thermal generating units.The total load demand required to be met for the test system is 49342 MW.The best power generation scheduling for the case study with the GJO techbique is  7 (in Appendix).It can be noticed that the best total fuel cost obtained using GJO for the large-scale power system is 1559703.40($/hr), which is the lowest among the other heuristic approaches compared such as shuffled differential evolution (SDE) [36], improved particle swarm optimization (IPSO) [38], grey wolf optimization (GWO) [39], artificial algae algorithm (AAA) [40], opposition-based krill herd algorithm (OKHA) [41] and KHA [41].It is graphically compared in the Figure 12.The convergence behaviour for the 140 unit power system is exhibited in Figure 13, and it depicts the robustness of the proposed GJO method.

CONCLUSION
In this research work, a comparatively new and an efficient algorithm named GJO which is based on the cooperative hunting nature of golden jackals is developed to solve ELD problems.The developed algorithm is applied to six different ELD problems in the power system with various real time complex constraints, such as VPL, POZs, MFO, and transmission loss.The outcomes show that the suggested GJO can ensure better quality solutions and has good robustness in optimizing generation scheduling of ELD problems while meeting the different constraints rather than the other compared algorithms.The overall research shows that the GJO algorithm is a competing algorithm for finding the best optimal generation scheduling for ELD problems.Further, performance of proposed GJO algorithm for DEED and economic emission dispatch problem, hybridization of GJO along with other algorithms can be explored to further improve the search ability in future works.

Figure 2 .
Figure 2. Fuel cost function with multiple fuels Figure 3. Fuel cost function with multiple fuels and valve point effect

Figure 4 . 1 Figure 5 .
Figure 4. Performance of different heuristic approaches for case study 1

Figure 6 . 2 Figure 7 .
Figure 6.Performance of different heuristic approaches for case study 2

Table 1 .
Test cases and considered constraints

Table 2 .
Best generation schedule of different methods for case study 1

Table 4 .
Best generation schedule of different algorithms for case study 3

Table 5 .
Best generation schedule of different algorithms for case study 4

Table 6 .
Best generation schedule of different algorithms for case study 5

Table 7 .
Best generation schedule of GJO algorithms for case study 6 Golden jackal optimization for economic load dispatch problems with … (Ramamoorthi Ragunathan) 791 APPENDIX