Investigation of Equilibrium Optimizer to Solve Economic Dispatch with Practical Constraints

— Presently power demand keep on increasing due to rapid changes occurs in power industry. For that purpose establishing new generating is costlier rather than effectively utilize the available generating stations. In order to properly planning of power system generation sharing Economic Dispatch (ED) plays vital role. In this paper, Equilibrium Optimizer (EO) is used to solve the ED problem with effect of valve-point, prohibited operating zones (POZs), ramp rate up/down limits and pollution like practical constraints. In order to analyze the capability of the EO algorithm, the algorithm is applied to four test systems with 6 unit systems and the results are compared with other optimization algorithms. The comparative results proven that EO is better optimization technique to solve ED problem with practical constraints.


I. INTRODUCTION
HE SYSTEM which can deals with generation, transmission nd distribution to supply the energy to the consumers on economic basis is known as power systems.Electric power demand is increasing in the current context due to developments in both the industrial and public sectors.The main source of this electricity is primarily thermal plants are likely to meet load demand.In general, the cost of generating for any thermal plant will be proportional to the cost of fuel.As a result, proper load sharing of generating units is essential to give lower generation costs.The Economic Dispatch (ED) problem is examined for this purpose in order to obtain optimal allocation of generation by all generating units while minimizing total fuel cost while meeting both the equality and inequality requirements.ED problems are typically complicated by practical limits of thermal units such as transmission network losses, valve-point loading, banned operating zones, and numerous fuels.The operational cost function is approximated by a single quadratic function in standard ED problems, and valve-point loading is neglected.
Typically, the Lambda Iteration approach is employed to solve the ED problem for proper thermal unit allocation at the lowest possible fuel cost.Therefore, proper distribution of producing units for large systems is problematic.To address this issue, researchers are looking for new methods that are similar to Particle Swarm Optimization (PSO) [1], Firefly Algorithm (FFA) [2], Quick Group Search Optimizer (QGSO) [3], Cuckoo Search Algorithm (CSA) [4] and Genetic Algorithm (GA).Because an ED problem in a practical power system is non-convex due to valvepoint loading, the applicability of traditional approaches is limited.Improved Differential Evolution (IDE) [5,6], Tournament-based Harmony Search (THS) [7], and Oppositional based Grey Wolf Optimization (OGWO) [8] methods are utilized to tackle the ED problem with valvepoint loading.
However, note the discontinuities in the turbine-generator set performance characteristics, which are caused by valvepoint (non-convex) loading in plants [9].For tackling the ED with valve point effect (EDVPE) problem, hybrid approaches such as modified Sub-Gradient (MSG) and Harmony Search Algorithms hybrid GA-NSO [10] and MSG-HS [11] and methods are utilized.
Furthermore, instability in generation at certain levels of unit loading may be induced by physical restrictions or flaws.This issue can be overcome by employing the prohibited operating zones (POZ) paradigm [9] and switching the unit's generating level between any two.Its ramp rate restrictions for concurrent periods must not be exceeded [9].Backtracking search algorithm (BSA) [12], PSO [13], Enhanced Random Drift PSO (RDPSO) [14], Exchange Market Algorithm (EMA) [15], Modified CSA [16] techniques are utilized to solve ELD with ramp rate limitations and POZs.
Despite the fact that ED reduces operational costs greatly, the environmental impact is still not addressed.By incorporating emission constraints into the ED problem, the problem is renamed the Economic Emission Dispatch (EED) Problem.For the EED problem with/without transmission losses, differential evolution (DE) [17], Glowworm Swarm Optimization (GSO) [18], MOEA [19], and Summation based Multi Objective DE (SMODE) [20] approaches are utilized.Approaches such as Multi objective BSA [21], multiobjective EA [22], new global PSO (NGPSO) [23] are used to combine fuel cost with emission as a specific objective problem.
Non-convex loading, emission, ramp rate restrictions, and POZs should all be considered while solving a practical ED problem, as it is extremely difficult to find an ideal solution.The Equilibrium Optimizer (EO) [24] approach was utilized in this article to tackle ED issues with numerous practical (5) Where Pimin and Pimax are the minimum and maximum real power generation limits of the ith generating unit.

B. EDVPE
The ED problem cost objective function, considering the valve-point effects (EDVPE).Figure 2 shows the valve-point effect is incorporated in classic ED problem by superimposing the sine component model on the quadratic cost curve.NO ,CO andSO emissions which are usually represented by separate quadratic functions.Nevertheless, by combining all the pollutants as single emission introducing exponential function to the quadratic emission function as given in Equation ( 11) for overall emission level of the pollutants.
Weighted sum method: The fuel cost and emission objective problem is converted into single objective CEED problem given by Equation ( 12) by assuming weighting factor proportion to the importance of the objective.
III. EQUILIBRIUM OPTIMIZER A. Faramarzi proposes an Equilibrium Optimizer (EO).In EO, each solution with its position acts as a search agents.

A. Initialization and evaluation of the functions
The initial positions were randomly determined according to the number of particles in the search space

B. Equilibrium pool and candidates (Ceq)
The state of equilibrium is known as the final state of EO convergence.In EO these candidates are selected four best particles based on their fitness value during the entire optimization process with other particles whose positions are in the mean of the four best particles described above.Such five particles were called candidates for equilibrium and used to construct an equilibrium pool.

C. Exponential term (F)
The key updating rule for concentration is controlled by the exponential term F To achieve convergence by increasing the quest pace and improving discovery and exploitation capabilities, t 0 is modelled as (17) Now equation ( 15) can be rewritten as

D. Generation rate (G)
Generation rate is the most significant parameter used in equilibrium algorithm to increase the process of exploitation.
-k(t -t 0 0 Where Finally, EO's updating law shall be as follows: IV. RESULTS

A. Classical ED
In order to evaluate solution of classical ED problem generator practical constraints are not considered like valvepoint effect, ramp limits and prohibited operating zones.But the transmission losses, equality and inequality constraints are considered for 6-unit test system [2] with power demand of 800 MW.The fuel cost function is convex function follows as Equation 1.
The simulation and comparison results for the given test system are presented in Table 1.Among all the methods, the proposed EO method is better interms of optimal cost and convergence characteristics shown in figure 5 for 25 trails.6.The sinusoidal term added to convex cost function and resulting becomes non-convex cost function.The test system comprising six generating units [5] meets a power demand of 283.4 MW and includes valve-point effect and transmission loss.
The best and comparison results for test system are presented in table 2. Among all the methods the proposed EO method is better in terms of optimal cost and convergence characteristics shown in figure 6 for 25 trails.[11] a load demand of 1263 MW and includes loss, POZs and ramp up/down limits.The optimal and comparison results for EDRPOZ problem are presented in Table 3.The total operating cost during practical constraints is 15442.6753($/hr) and it is found to be lesser than the other methods reported in the literature.The convergence characteristics of 6 unit system with proposed method for EDRPOZ are shown in Figure 7 for 25 trails.The IEEE 30-bus 6-Unit system is considered as test system.Data is taken [16] with power demand of 283.4 MW.Table 4 represents the optimal results for test system for minimizing the cost, emission and combined economic emission with the help of EO.A new optimization algorithm called Equilibrium Optimizer (EO) is used in this paper to solve the economic dispatch problem with realistic restrictions such as valve point effect, ramp rate up/down limits, prohibited operating zones and pollution.The EO has been used to evaluate four separate 6unit test systems.The findings were consistent with other methods listed in the literature and showed that EO had quick convergence speed, better fuel cost outcomes, prevailing computational performance and more cognizant achievement.The suggested algorithm would be a viable solution to solve ED problem practical constraints.The proposed methodology is a potential approach in large-scale framework to solve complex non-smooth optimization problems.
Fig.1 Convex Fuel cost function The most simplified cost function of each generating unit i, can be represented as a quadratic function as: ( Minimize n T i i i1

Fig. 9
Fig. 9 Convergence characteristics of EED to emission minimization

Table 2 :
Comparison results for 6-unit to EDVPE

Table 5 :
Comparison results for 6-unit system to EED COST MINIMIZATIONFig.8Convergencecharacteristics of EED to cost minimization