Improved Artificial Cooperative Search Algorithm for Solving Non-convex Economic Dispatch Problems with Valve-point Effects

This paper presents Improved Artificial Cooperative Search (IACS) algorithm for solving economic dispatch problems considering the valve point effects, ramp rate limits, transmission losses and prohibited operation zones. In order to improve the solution quality and increase the search efficiency, a novel perturbation scheme called “Global best guided chaotic local search” is proposed and incorporated into ACS algorithm. The effectiveness of the proposed IACS algorithm has been benchmarked with twelve widely known optimization test problems. In order to assess the performance of the proposed algorithm on non-convex optimization problems, four case studies related to highly nonlinear economic dispatch problems have been solved . Results retrieved from IACS algorithm have been compared with literature approaches in terms of minimum, maximum and average generation cost values. Comparison results indicate that IACS produces more economical power load than those of other optimizers available in the literature.


Introduction
Economic load dispatch (ELD) problem plays an important role in power systems planning.ELD is a constrained optimization problem whose main objective is to minimize total fuel cost of generating units while satisfying an equality and a great deal of inequality constraints including discontinuous prohibited zones, generating unit constraints and ramp rate limits.The cost of power generation in fossil fuel plants is very high therefore optimum scheduling of generation units is needed to save possible amount of expenditure on power generation systems.Each power unit is represented by a quadratic cost function which becomes highly nonlinear, non-convex and discontinuous due to the effect of valve point loadings and prohibited operating zones.This functional behaviour generates multiple local optimum points in solution space and complicates the locating the global optimum of the ELD problem [1].Mathematical programming techniques [2][3][4][5][6][7] have been utilized to reach the optimum solution of the ELD problem however these kind of methods have not provided feasible solutions yet and they generally get trapped in local optimum points in the search space [8].Dynamic Programming method [9] succeeds to solve ELD problems and copes with the nonconvexities occurred by valve point effect, however this method incurs high computational burden and its performance deteriorates with increasing number of generation units.Besides, Newton based methods have had trouble in handling large number of inequality constraints objected to ELD problem [10].Due to their supreme capability on maintaining acceptable balance between exploration of the search domain and exploitation of the promising areas, metaheuristic methods such as Genetic algorithm (GA) [11][12][13], Gravitational Search Algorithm [14][15], Simulated Annealing (SA) [16][17][18], Particle Swarm Optimization (PSO) [19][20][21][22], Differential Evolution [23][24][25][26], Harmony Search (HS) [27][28], Artificial Bee Colony (ABC) [29][30], Firefly Algorithm (FA) [31][32][33][34][35], Teaching Learning based Optimization(TLBO) [36][37], Cuckoo Search (CS) [38][39] and Biogeography-Based Optimization (BBO) [8,[40][41][42][43]76] have been recruited for solving economic dispatch problems.From the literature survey, it is seen that metaheuristic algorithms applied on ELD problem have not guaranteed to find the global optimum of the problem, however they are capable of finding near-optimal solutions.Detailed explanation of some of the studies mentioned above is given in Table 1.
Table 1 Detailed explanation of some studies in the literature Ref.
Explanation [11] The Atavistic GA is applied to solve the ELD problem with valve point discontinuities.The algorithm is applied to a system with 13 generating units and found better solutions than traditional GA. [14] The GSA algorithm is utilized to solve the ELD problem.
The results showed that the algorithm is easy to implement, robust, gives more favourable solutions with less execution time. [16] The SA algorithm is applied to solve the ELD problem.The transmission losses are later added to the equation.The results are compared with those found by the dynamic programming of the ELD. [29] The ABC algorithm is utilized to solve the ELD problem with 10, 13, 15 and 40 generating units.The results are compared with that of the other techniques reported in the literature.The ABC algorithm found more favorable results.[31] The FA algorithm is utilized to solve the ELD problem.Many nonlinear characteristics of the generating units have been taken into account.The results showed that the FA algorithm finds better solutions than the others.[36] The TLBO algorithm is suggested to solve the ELD problem.The proposed methodology is similar to the other studies in the literature.The TLBO algorithm found more favorable results than the other algorithms.
In this article, Improved Artificial Cooperative Search (IACS) is presented for successful solution of non-convex economic dispatch problems.ACS is based on the interaction between prey and predator individuals of the population while they are migrating to find possible food resources.ACS has fewer control parameters and uses different mutation and crossover strategies than other optimization algorithms [44].In order to enhance the convergence capability of the ACS algorithm, a novel perturbation scheme called "Global best guided chaotic local search" is proposed in this study.The proposed scheme is based on the motivation of exploitation of the explored areas of the search domain and refines the so-far-obtained optimum solution by means of the global best solution vector, which guides the population individuals during iterations.By this scheme, mutated individuals move towards to

Mathematical Modelling of Economic Dispatch Problems
Economic dispatch problem aims to find optimum combination of power generation units that minimizes total fuel cost while subjected to an equality and several inequality constraints.Economic dispatch, which is a sub division of Unit Commitment (UC) problems, is an example of nonlinear programming optimization due to nonlinear characteristic of power systems [31].Formulation of the ELD problem can be described as Minimize 2 1 1 ( ) where F is total generation cost to be minimized and Fi is the cost function of i th generator; power output of the i th generator is represented by Pi; ai, bi and ci are the coefficients pertaining to i th generator and N is the number of the on-line generators in the power generation system.Modelling valve point loadings is necessary to capture the losses incurred due to the throttling of partially open valves in electric power generators [45].Introducing valve point effects into economic dispatch problem makes the objective function non-convex owing to the contribution of ripplelike effect occurred in multi valve steam turbines.This feature enhances the non-linearity of the objective function and increases the chance to be getting stuck in the local optimum points in generation cost curve.Superposition of sinusoidal function and total cost function is formulized as where Pi,min corresponds to lower bound of the power generation for the i th generator; ei and fi are fuel cost coefficients of the i th generator that model the valve point effect in the generation cost curve.

Power balance constraints
where PD is total load demand and PL represents transmission losses of the power generation system.The Bcoefficient method [46], commonly used by the power industry to calculate transmission network losses, is formulated by the following expression

Operational limits
Power output of the each generator should be restricted between maximum and minimum limits.Following inequality constraint should be applied for each generation unit: P i,min  P i  P i,max (5) where Pi,min and Pi,max are minimum and maximum power output for i th generating unit, respectively.When it is to consider ramp rate limits of each generator, operation bounds are modified as follows: In Eq. ( 6), 0 i P is previous generator output power; URi and DRi are respectively up and down ramp limits of the i th generator in terms of MW/h.

Prohibited Zones
Due to the physical limitations of machine components or vibrations on the shaft, start and stop of coal mills that take place in their auxiliary parts, generator units have prohibited regions that make operating curves of the generator non-continuous [47].Power output of the generators must be avoided from these areas to satisfy operation constraints.Fig. 1 shows the characteristics of the cost curves with prohibited operating zones (POZ).Mathematical representation of the constraints can be given as ,min , Nature has always been an inspiration for many researchers.Nowadays, many optimization algorithms has been inspired from biological, physical or chemical systems [48].Most of the bioinspired algorithms have been derived from the swarm intelligence concept.Swarm Intelligence (SI) is a special kind of bio-inspired algorithm deals with the behaviour of collective, multiple agents interacting with each other by following some predetermined rules.
In SI algorithms, each agent may behave as an unintelligent entity, but whole system, consists of multiple agents, may show some kind of intelligent behaviour.Artificial Cooperative Search (ACS), developed by Civicioglu [44] to be used in solving real-valued numerical optimization problems, is a dual-population based swarm intelligence algorithm.In nature, there are such individuals those utilize mutualism based biological interaction locations in order to sustain for their lives.Organisms involved in a mutualism based interaction locations try to take benefit of these location points.In mutualism, two types of organism living in the same habitat aims to derive mutual benefits from each other.Besides, there is another term called "cooperation" which is interaction of homogenous living beings that adopt mutualism.ACS algorithm is conceptualized on aforementioned mutual and cooperation based biological interaction of two eusocial superorganisms living in the same habitat."Habitat" term mentioned above matches the "search domain" concept pertaining to the optimization problem.ACS algorithm is based on the interaction between two artificial superorganisms as they interact and migrate to variety of areas to find more fruitful habitat.In nature, amount of food that can be found in a habitat depends on yearly climate changes.For that reason, many superorganisms have developed seasonal migration behaviour to find better food sources.Many species are known to set up a group called "superorganism" prior to migration.After a superorganism is formed, individuals of the superorganism start to move to better food sources by means of forming groups.In addition, many superorganisms can divide into sub-groups (subsuperorganisms) prior to migration.Many swarms use explorers to discover a habitat.Explorers discover a possible migration area, then collect information about this new explored area and share the information with the superorganism they belong to.If the superorganism decides to migrate to a new explored area, it moves to discovered area, and then this exploration process starts again and proceeds until they find productive feeding areas.
In ACS algorithm, artificial superorganisms migrating to find possible fruitful areas refer to superorganisms with random solutions under given search space.ACS algorithm is composed of two superorganisms, namely α and β, those inherit subsuperorganisms equal to the population size (N).Subsuperorganisms consist of D individuals which correspond to the dimension of the optimization problem.Prey and Predator subsuperorganisms are determined by means of α and β superorganisms.In ACS algorithm, predator sub-superorganism individuals pursue prey sub-super organism individuals while they are migrating to find productive feeding areas (optimum point of the problem).The whole iteration process in ACS algorithm can be named as "coevolution" which is based on the two superorganisms looking for the optimum solution of the problem, maintaining cooperation based biological interaction between each other.Individuals of the i th sub-superorganisms of α and β are initialized with the equation below  i, j:g  rnd.(up j  low j )  low j  i, j:g  rnd.(up j  low j )  low j (11) where i  1,2, 3,..., N , j  1,2, 3,..., D and g  1,2, 3,...,maxcycle .
The rnd represents a random number selected from a uniform distribution in the range of [0,1].The g value counts the iteration number.Symbols up j and low j show the upper and lower bounds of the search space for j th dimension of the problem.Fitness values (productivity values) of the associated sub-superoganisms are calculated by using the following formula; Table 1 gives the pseudo-code of the Artificial Cooperative Search algorithm equipped with the evolutionary boundary constraint mechanism that will be explained in the upcoming sections.In Table1, there are some symbolizations and abbreviatons those ease the comprehension of the conceptual descriptions.For instance, rand(0,1) stands for the representation of a uniform random number defined in the range [0,1]; permute(.)function shuffles the row elements of the population individuals; X represents the biological interaction locations for Predator and Prey individuals; R is the scale factor that determines the biological interaction speed; rndint(1,Y) generates pseudo-random integers defined between 1 and Y; determination of the passive individuals is procured by M matrix which is comprised of integers 0 and 1.

Global best guided chaotic local search
In this section, a novel local search mechanism is proposed to refine the optimal solutions corresponding to the interaction locations between prey and predator individuals and avoid being trapped in local optimum points.Inspired by the search equations of Differential Evolution [49] and Artificial Bee Colony [50] algorithms, proposed perturbation scheme takes full advantage of global best (Gbest) vector of current population and probes around the Gbest solution to circumvent the local optimum solutions faced on the course of iterations.Proposed scheme can be described as where i = 1,2,3,..,N; j = 1,2,..,D and ch is chaotic variable generated by Logistic map [51].Chaos is a deterministic, randomlike mathematical phenomena occur in nonlinear systems and has a strong dependence on initial conditions [51,77].Effective and ergodic chaotic sequences can be generated by an ordinary chaotic map on the concept of the following equation Logistic map, which is one dimensional chaotic map and demonstrates that how complex behaviour arises from a simple deterministic system without need of any random sequence, is defined as where γ is a control parameter and ch is a chaotic variable as defined before.For initial conditions, ch0 should be in the range of (0,1) and . Chaotic behaviour of the generated sequence can be controlled by the control parameter γ, however Logistic map sequence is chaotic when γ=4.0.

Evolutionary boundary constraint handling scheme
Gandomi and Yang [52] developed an evolutionary scheme for boundary constraint handling.According to this proposed scheme, when population individuals goes beyond the prescribed boundaries, they are pushed into the related bounds of the optimization problem by means of a uniform random number and global best solution vector obtained so far.Proposed constraint handling scheme can be formulized as (1 ) where α and β are real valued number in the range of [0,1]; lowi and upi are the i th variable of the lower and upper bounds of the

Implementation of Improved Artificial Cooperative Search Algorithm for ELD Problem
The proposed IACS algorithm will be applied on economic dispatch problems considering valve point effects, ramp rate limits and prohibited operating zones those all make the objective function of the problem non-linear, non-convex, and noncontinuous.IACS is proposed for optimum scheduling of each power generation unit satisfying both equality and inequality constraints.Solution steps of the economic dispatch problem using IACS algorithm are given as follows; Step 1: Apply upper and lower bounds; define cost coefficients, transmission loss coefficients, prohibited operating zones, valve point coefficients and ramp limits for each generation unit; initialize the chaotic sequence by using Logistic map; determine population size and maximum number of generation Step 2: Initialize α and β superorganisms by random real valued numbers as described in Eq. (11).Remind prohibited zones by adjusting the numerical values of superorganism individuals (α and β) with using Eqs.( 9) and (10).Calculate the fitness values of the α and β superorganisms with considering the valve point effects, total energy demand constraints, and transmission losses given as the following equation; where ψ is a problem dependant penalty coefficient which penalizes infeasible solutions.Set iteration counter to 1 Step 3: Determine the predator individuals and their respective fitness values by following the procedure given in Table 1 in the lines between 12 and 16 Step 4: Determine prey individuals by implementing the procedure given in Table 1 within the lines between 17 and 18 Step 5: Calculate scale factor (R) by the rule given in Table 1 in the lines between 19 and 23.
Step 6: Apply binary valued integer map (M) to determine passive individuals with the decision rule stated in Table 1 within the lines between 24 and 45 Step 7: Calculate the biological interaction locations with the equation given in line 47 in Table 1 Step 8: Determine the best solution (Gbest) of the current population and fine-tune the biological interaction locations with Eq. ( 13).If mutated solution vectors are better than those of inferior solutions, update the perturbed solution vector.Increment the iteration counter.
Step 9: Update the biological interaction locations with the procedure given within the lines between 48 and 53.
Step 10: Apply evolutionary boundary constraint handling scheme defined in the lines between 55 and 63 in Table 1.Handle the prohibited zone constraints with ( 9) and (10), and process the selection update mechanism through the procedure given within the lines between 65 and 67 in Table 1 Step 11: Determine new superorganisms for next generations with the decision rule described in the lines between 68 and 72 in Table 1.
Step 12: Get the best fitness value of predator sub-superorganism.Retain the best fitness value and its corresponding design variables for next generations.
Step 13: Update the chaotic sequence generated by Logistic map as described in Eq. ( 15) and increment the iteration counter.

Simulation and Analysis
In this section, IACS algorithm has been assigned to 13-, 38-, 40-, and 140-unit generation systems to verify its applicability and feasibility on ELD problems.IACS is implemented using Java executing Pentium Core i5 CPU @ 2.5 GHz and 6.0 GB RAM on a personal computer.

Case Study 1: 13-Unit Test System
This case deals with 13-generating units which takes into account valve effects and prohibited zones without considering transmission losses.As number of generation sites increases, complexity of the system is improved owing to the non-linear characteristic of valve point loading effects.This non-linear behavior increases the number of local optimum therefore finding global optimum of the problem significantly becomes a challenging process.In this case, total load demand of 1800.0MW and 2520.0MW test systems are studied.Due to the stochastic characteristic of metaheuristic algorithms, 50 trial runs have been performed along with 50,000 function evaluations for both IACS and ACS algorithms.Problem data for 1800.0MW test system can be found in Sinha et al. [59].Table 5 reports the statistical results obtained by HGA [60], FA [31], BF-NM [61], MDE [62], IPSO-TVAC [63], SDE [64], MsEBBO [8], MsEBBO/sin [8], MsEBBO/mig [8], and ACS-based algorithms for 1800.0MW test system.From Table 4, it is seen that minimum fuel cost value (17,954.091$/h) obtained by IACS is lower than those acquired by other methods available in the literature.Table 5 lists the best generation cost results of above-mentioned literature approaches and their corresponding power generation rates.Table 6 reports statistical analysis obtained by FAMPSO [65], HHS [66], ACO [1], ICA-PSO [67], IPSO-TVAC [63] and ACS-based algorithms for 2520.0MW test system.Table 6 clarifies that however little improvement have been made by IACS over ACS algorithm on minimum generation cost value, IACS not only attains the best result among the other algorithms but also it surpasses the remaining algorithms in terms of robustness.This behavior indicates the supremacy of the proposed algorithm.Table 7 lists the comparison of the optimal solution in the literature.Table 7 Best power outputs for the 13-unit system with total load of 2520.0MW

Case Study 2: 38-Unit Test System
A test system with 38-generation units is considered to test the actual performance of IACS algorithm on non-convex problems.Fuel costs are represented by quadratic cost functions, transmission losses are not taken into account and total power demand is set to 6000.0 MW for this case.System parameters are taken from Liang and Glover [9].Table 8 compares the optimum results extracted by DE/BBO [68], PSO-TVAC [69], NPSO [69] and ACS based algorithms.Table 10 gives the statistical analysis for aforementioned algorithms in terms of minimum, maximum and average cost values.As seen from Table 10, IACS algorithm not only finds lower fuel cost values than the other methods but also it is so robust and consistent such that the worst fuel cost value obtained by IACS is much better than the best fuel cost value acquired by ACS.Both algorithms shows similar convergence characteristics until 32,653 function evaluations then both of them remain in stagnation till the end of iterations.However, ACS algorithm is trapped in the local optimum with the corresponding fuel cost value of 9,417,205.08($/h).Fig. 3 illustrates the set of optimum solutions obtained after 50 algorithm run for this case.ICA-PSO [67] IPSO-TVAC [63] ACS

Case Study 3: 40-Unit Test System
In this section, the proposed algorithm has been applied on a power system consists of 40 generating units incorporating valve loading effects.Total power demand is set to 10,500 MW for this case.Input parameters of the cost functions for all generating units are referred to the case study given in Coelho and Mariani [5].50 trial runs along with 500,000 function evaluations have been performed due to the stochastic nature of the proposed metaheuristic algorithm.Optimal fuel cost values obtained by FA [31], PS [10], BBO [68], SOH-PSO [70] along with corresponding generator loads have been reported in Table 11.Table 12 gives the detailed comparison of the proposed algorithm and literature studies in terms of minimum, maximum and mean of the generation cost value.

Case Study 4: Large Scale Application on Korea Power System
To test the efficiency of the proposed algorithm on a large scale application, numerical experiments have been conducted on the Korean power system which consists of 140 thermal generating units with ramp rate limits, valve point effects and prohibited operating zones.Total power load for this system is set to 49,342 MW.Input data for this case is obtained from Park et al. [47].1,000,000 function evaluations have been made owing to the high dimensionality of the problem.Due to the stochastic discrepancy, both ACS and IACS have performed 50 consecutive algorithm runs.Park et al. [47] proposed four PSO based algorithms including CTPSO, CSPSO, COPSO, and CCPSO for successful solution of this case study.In addition, Dalvand et al. [71] propounded Group Search Optimizer (GSO) and Continuous Quick Group Search Optimizer(CQGSO) in order to solve large scale Korea power system problem.Coelho et al. [26] proposed Differential Evolution algorithm combined with truncated Levy flight random walks and population diversity measure (DEL) to solve this case study.

Conclusion
This paper introduces Improved Artificial Cooperative Search (IACS) algorithm for successful solution of economic dispatch problems regarding valve point effects, ramp rate limits, transmission losses and prohibited operation zones those all make the operation cost curve non-continuous, non-convex, and highly non-linear.ACS is dual-population based metaheuristic algorithm, which is based on the interaction between two superorganisms as they are migrating and searching more fruitful areas for subsistence.ACS has a fewer control parameters and uses advanced perturbation strategies that ease its implementation on any optimization problem.In order to improve the solution quality and increase convergence rate, a novel local search strategy is incorporated into ACS algorithm.In addition, an evolutionary boundary constraint-handling scheme is utilized to restrict population individuals into the prescribed limits.In order to verify the applicability of the proposed algorithm on multi-dimensional optimization test problems, IACS is benchmarked with twelve widely known benchmark functions.The results obtained from IACS, ACS and some other metaheuristics in the literature showed that IACS performs better than the others.Also, IACS converges faster than the other algorithms.Then, IACS has been applied to 13

Figure 1 .
Figure 1.Fuel cost curve considering prohibited operating zones

Figure 3 .
Figure 3. Set of optimum results obtained by IACS for 38-unit test system

Figure 4 .
Figure 4. Sequence of optimum results obtained by IACS for 40-unit test system

3. Artificial Cooperative Search Algorithm 3.1. Fundamentals of the Artificial Cooperative Search
..., xi is the i th mutable decision variable of the related optimization problem.In the view of ACS algorithm, xi is the biological interaction location where prey individuals are pursued by predator individuals aiming for finding more suitable areas for subsistence.

4. Experimental Studies on Improved Artificial Cooperative Search Algorithm
In this section, twelve widely known 50 dimensional optimization test functions have been carried out in order to assess the performance of the IACS algorithm over new emerged International Journal of Intelligent Systems and Applications in Engineering IJISAE, 2018, 6(3), 228-241 | 232 Table 3 gives the statistical results for all mentioned optimizers in terms of mean and standard deviation values.IACS algorithm finds global optimum of Sphere, Rastrigin, Griewank and Step functions in each algorithm run and outperforms other algorithms with regards to statistical results it attains over 11 out of 12 test functions.Concerning the best results obtained after consecutive algorithms runs, convergence performance of all above-mentioned algorithms have been compared with IACS algorithm in Fig. 2. It is clear that IACS is more quicker than other algorithms since in each test function, except for Pathologic function in Fig. 2(k), IACS is getting closer to the optimum point while others remain stagnant and are far away from the optimum.

Table 2 .
Mathematical representations of the benchmark functions

Table 4
Statistical analysis for the 13-unit test system with total load demand of 1800 MW N/A means "not available"

Table 5
Best power outputs for the 13-unit test system with total load demand of 1800 MW

Table 8
Comparison of the best results for 38-unit system

Table 10
Statistical results and comparison for the 38 unit system.
N/A means "not available"

Table 11
Table 12 clarifies that proposed IACS algorithm surpasses other remaining methods in terms of minimum fuel cost values.Table 13 reports the convergence frequency of the best results Optimal dispatch results for 40 unit systems for total power demand of 10,500 MW

Table 12
Statistical results for the 40 unit test system Table 14 compares the optimum solutions acquired by literature studies discussed above with the best results of ACS and IACS algorithms.As reported in Table 14, IACS algorithm surpasses other algorithms concerning the minimum fuel cost values.Table 15 gives the best results obtained by both ACS and IACS algorithms.According to the Table 15, IACS algorithm succeeds in finding much better generation cost values than ACS algorithm with the minimum fuel cost of 1,657,956.80$/h.Although both algorithms follow similar trends until the end of iterations, ACS is first to saturate and get trapped in a local optimum solution.After all, it can be concluded that IACS proves its efficiency in solving high dimensional optimization problems.

Table 14
Statistical results for the 140 unit Korean power generation system International Journal of Intelligent Systems and Applications in Engineering IJISAE, 2018, 6(3), 228-241 | 239

Table 15
Best results of the ACS and IACS algorithms for 140-unit Korean power generation system -, 38-, 40-and 140-unit systems and obtained results over 50 algorithm runs have been compared with recently published ED optimizers.Outcomes of the comparisons indicate that IACS algorithm is very effective and efficient in finding the optimum solution of the economic dispatch problem and can be nominated as an alternative method for solving non-linear ELD problems as well as multi-dimensional real world optimization problems.For a