An E ﬀ ective Method for Minimizing Electric Generation Costs of Thermal Systems with Complex Constraints and Large Scale

: In this paper, an improved antlion optimization algorithm (IALO) was proposed to search for promising solutions for optimal economic load dispatch (ELD) problems to minimize electrical generation fuel costs in power systems with thermal units and to ensure all constraints are within operating ranges. IALO can be more e ﬀ ective than the original method, called the antlion optimization algorithm (ALO), because of the high performance of the applied modiﬁcations on the new solutions searching process. In order to evaluate the abilities of the IALO method, we completed many tests on thermal generating systems including 10, 15, 20, 30, 60, 80, and 90 units with di ﬀ erent constraints and fuel-consuming characteristics. The results suggest that the o ﬀ ered method is superior to the ALO method with more stable search ability, faster convergence velocity, and shorter calculation times. Furthermore, the obtained results of the IALO method are much better than those of almost all the other methods used to solve problems for the same systems. As a result, IALO is suggested to be a highly e ﬀ ective method, and it can be applied to other problems in power systems instead of ALO, which has a lower performance.


Introduction
Economic load dispatch (ELD) is known as a means of lowering fuel costs for electricity generation in thermal power plants. In power system operation, the main purpose of the ELD problem is to allocate the power output of each power plant continuously, while observing the conditions such as fuel consumption characteristics, capacity of each generator, technical requirements in the system, and total demand of all loads. The objective function of this problem is to reduce the values of a quadratic function, a non-convex function, or a non-smooth function as much as possible. So, the complexity of the problem is partially dependent on the complexity level of the objective function. Basically, a solution can be regarded as an acceptable operation method if all constraints are exactly met. Thus, complex of constraints is also a challenge for the complexity level of the ELD problem. Due to these challenges, 1.
Present all the computation steps of the ALO in detail and analyze drawbacks of the method; 2.
Discuss improved versions of the ALOs that previous studies have proposed; 3.
Propose a highly effective modification to produce a promising solution search strategy; 4.
Combine our proposal with a modification that was proposed in a previous work. The combination of the highly effective proposed modification and the applied modification can support the proposed IALO method in finding optimal solutions effectively and quickly; 5.
Consider the different systems that contain complex objective functions and constraints; 6.
Compare our proposed IALO with classic ALO. Our proposed IALO can achieve much better results than those of the ALO with the same settings for control parameters. The proposed method is better or equal to other existing methods in comparing quality of solution, but it is more robust than these methods when comparing computational steps.
The remaining parts of the paper are organized as follows: Section 2 expresses the mathematical formulation of the ELD problem. Section 3 presents the ALO and IALO methods in detail. Section 4 shows the detailed implementation of IALO for the ELD problem. Section 5 presents simulation results and discussion. Section 6 summarize the achievement and conclusions of the study.

Objective Function of the ELD Problem with a Single Fuel Option
In the optimization operation of a system powered by thermal generation units (TGUs), a great concern in the optimal load dispatch problem is the technique to reduce fuel cost of the system. This problem can be described by using the following equation.
In accordance with this theory, the fuel cost function (FCF) of the ith generating unit is written as the single quadratic function form with a variable for real power output, which can be seen by the next equation: where α i , β i , and γ i are the known coefficients in function f i (p i ) of the ith TGU and p i is the power generated by the ith TGU. When TGUs consider VPEs, the FCF is more complicated, as given in the following model.
where δ i and ε i are the known coefficients in FCF of the ith TGU considering VPEs and p i,min is the minimum real power output of the ith TGU.

Objective Function of the ELD Problem with MFs
Because coefficients of each TGU are modified according to the different types of fuel being used, the fuel cost function form is the sum of different quadratic functions, as in (4). When VPEs are considered, the form is more complex with the presence of a sinusoidal function, and the objective function can be expressed as in (5): where α im , β im , and γ im are the known coefficients of FCF of the ith TGU corresponding to the mth fuel type for the case without VPEs; δ im and ε im are the known coefficients of FCF of the ith TGU corresponding to the mth fuel type for the case with VPEs; and p i,m,min and p i,m,max are the minimum and maximum power outputs of the ith TGU corresponding to the mth fuel type.

Power Balance Requirement
The active power generated by power sources has to be equal to the sum of the total load demand (p d ) and transmission loss (p l ). It can be seen by the function in the following formula.
where total real power loss is calculated by using the Kron's formula below: where B ij , B 0i , and B 00 are transmission loss coefficients. Power output limits requirement: Each TGU has serious limits within the upper level and lower level of generation. They must not exceed this segment as shown in the following rule.
where p i,max is the highest acceptable working power of the ith TGU.

POZ, SR, and RRL Constraints
Technically, TGUs are not allowed to work in restricted areas because negative impacts on these units will cause damage to steam turbines, even the bearings can be destroyed. Therefore, POZ is a constraint that must be strictly observed in the operation process of TGUs. The fuel-power curve of TGUs will be interrupted due to the presence of POZs, so the determination of the power position of the units in neighboring POZs is quite complicated. In addition, if a TGU in the system is stopped due to a breakdown, the power balance constraint must still be met. Therefore, each unit must reserve power to ensure stability for the system. The solution to this problem is called a spinning reserve. Another important constraint is the ramp rate limit (RRL), which does not allow the TGU to generate power with values beyond a predetermined space. This constraint considers increase and decrease limits for power output of the TGU. The details of the constraints are as follows: Prohibited operating zones: According to the POZ constraint, the TGU can operate at power ranges as shown in the following model.
where p l h and p u h are lower and upper power output of the ith at the hth zones; p u i,0 = p i,min considering h = 1 and p l i,nz+1 = p i,max ; nz is the number of POZs. Determining the power generated by the TGU beyond the POZs must be done first and then other constraints such as spinning reserve and online power balance can be considered.
Spinning reserve: To ensure energy security issues, the system always reserves a power under the following inequality: n i=1 pr i ≥ pr all (10) where pr i is the real power reserve of the ith TGU; and pr all is the required spinning reserve of the whole system. The spinning reserve is managed by the following equation: where Ω is the set of TGUs that can supply reserve power to loads. Ramp rate limit constraint: Increasing or lowering of the power generated by the TGU in the process of operation must be managed to keep the output within a permissible limit. If RUL is the ramp up limit and RDL is the ramp down limit of the ith TGU, the ramp rate constraint is defined as follows: where p i,0 is the initial power output from the previous operating hour of the ith TGU.

Population Initialization
Each antlion represents a solution and it must be born at the beginning by the following rule: where Bound lower and Bound upper , respectively, are the lowest values and highest values of control variables; N p is population; and ρ is a random number in the range of [0, 1]. Among the initialized solutions, the solution with the best fitness function is set to the best antlion, ALO best .

New Solution Update Process
In ALO, updating new solutions is the process of updating new positions of ants, which are the prey of antlions. The position of each ant is newly updated based on the following rule where XAB G s is created by using a random walk around the best antlion ALO best and XARW G s is determined by using a random walk around one antlion that is found by using a roulette wheel mechanism. XARW G s is determined by using the following steps: Step 1: Determine the minimum limit and maximum limit for the positions of each antlion s based on the following models: where ALO G RW is one antlion that is chosen by the roulette wheel mechanism; and w is obtained by one of the options below: where G max is the number of iterations Step 2: Determine step size for each antlion s by the function of the formula below: Step where G is the current iteration; Csum is the cumulative sum; t is a step size of random walk; and R(t) is determined by the following model.
where φ is a random positive number not higher than 1.
Step 3: Determine XARW G s by applying the formula below: For determining XAB G , the following steps are applied: Step 1: Determine the lower limit and upper limit for a random walk around the best antlion where ALO G−1 best is the best antlion in the previous iteration; and w is determined as mentioned above.
Step 2: Calculate step size for each antlion based on Equation (17

Selection Technique
All antlions are able to move to a new position of ants if the new positions are promising for finding prey. However, not every antlion moves, how many move depends on the potential of the new positions. In the case where the new position is more effective than the old one, the antlion will certainly move. This phenomenon is similar to the comparison between the quality of old solutions and new solutions for selection. The action is called selection technique, which can be accomplished by: The search procedure of ALO is presented in Figure 1.

Discussions on the Improvement of IALO
Several improved versions of antlion optimization algorithms were introduced and presented in previous studies [46][47][48][49][50]. Each variant of the ALO was applied to minimize certain optimization problems. Hence, this section only considers the improved versions of the ALO without the relative problems. One is the enhanced antlion optimization algorithm (e-ALO) [46] in which the e-ALO replaced the uniform probability distribution function with a stochastic probability function. Consequently, the e-ALO could reach higher quality solutions than the Taguchi method (TM), Cat Swarm Optimization (CSO), artificial bee colony (ABC), genetic algorithm (GA), PSO, etc. In the next variation, the modified antlion optimization algorithm (MALO) [47] concentrated on the improvement of global search, and the balancing of local search and global search. Results from the optimal reactive power dispatch problem indicated that ALO should be replaced with MALO for reducing power losses. Another version of the ALO is named the opposition-based Lévy flight antlion optimizer (OB-LF-ALO) [48]. The authors of [48] showed that the ALO was prone to stagnate in local optimal zones. Thus, they replaced the uniform distribution-based random walk of the Lévy flight-based random walk for a better exploration ability. Therefore, the OB-LF-ALO could accelerate convergence by enhancing the original diversification and strongly exploiting it in the later iterations. According to the tournament selection-based antlion optimization algorithm (TALO) [49,50], the authors pointed out that the classical ALO has some major drawbacks, including many computation steps and dependence on randomization, which prevent it from becoming a powerful method for optimization problems. Thus, they abandoned the roulette wheel selection mechanism and employed a tournament selection mechanism. As a result, the method was demonstrated to be more effective than the ALO, GA, PSO, and firefly optimization algorithm (FOA). In summary, the ALO is a weak method for optimization problems and it should abandon its ineffective mechanisms and apply more robust ones.

Discussions on the improvement of IALO
Several improved versions of antlion optimization algorithms were introduced and presented in previous studies [46][47][48][49][50]. Each variant of the ALO was applied to minimize certain optimization problems. Hence, this section only considers the improved versions of the ALO without the relative problems. One is the enhanced antlion optimization algorithm (e-ALO) [46] in which the e-ALO replaced the uniform probability distribution function with a stochastic probability function. Consequently, the e-ALO could reach higher quality solutions than the Taguchi method (TM), Cat Swarm Optimization (CSO), artificial bee colony (ABC), genetic algorithm (GA), PSO, etc. In the next variation, the modified antlion optimization algorithm (MALO) [47] concentrated on the improvement of global search, and the balancing of local search and global search. Results from the optimal reactive power dispatch problem indicated that ALO should be replaced with MALO for reducing power losses. Another version of the ALO is named the opposition-based Lévy flight antlion optimizer (OB-LF-ALO) [48]. The authors of [48] showed that the ALO was prone to stagnate in local optimal zones. Thus, they replaced the uniform distribution-based random walk of the Lévy flight-based random walk for a better exploration ability. Therefore, the OB-LF-ALO could accelerate convergence by enhancing the original diversification and strongly exploiting it in the

The Proposed IALO Method
The proposed IALO method was developed based on the following points 1.
Divide the whole population into two different groups in which solutions with better fitness functions are grouped into a potential set and the remaining solutions are put into an ineffective set; 2.
The group classification criterion is based on the fitness function of each individual and average fitness function of all solutions; 3.
Solutions in the potential set are updated based on their own old positions and an updated step size; 4.
Solutions in the ineffective set are updated based on the best solution and an updated step size.
The details of the proposed modifications are as follows: Step 1: Sort solutions based on the fitness function so that the best solution with the smallest fitness function is put in the first position and the worst solution with the highest fitness function is put in the final position.
Step 2: Calculate the average fitness function of the whole population, Step 3: Select solutions for the potential set and the ineffective set based on Algorithm 1.

Algorithm 1: Solution selection for the potential set and ineffective set
For ALOs is stored in the ineffective set end end Step 4: Count the number of close solution pairs (N csp ) based on the Algorithm 2 below: Algorithm 2: Calculation of the number of close solution pairs Step 5: Update new positions for solution in the potential group. Determine the ratio of the number of close solution pairs to the number of solution pairs (N sp ) as follows: Update new solutions by using the following equations: If R < 15%, the equation below is used to produce new solutions: In the case of R < 30%, the following model is used: For all other cases, each solution is newly changed by: Step max Step 6: Update new positions for solutions in the ineffective group by using the following equation: where ζ 4 is a random number within the range from 0 to 1; and χ is determined by [51]: where ϕ is selected to be 1.5 [51].

Selection of Decision Variables
According to the algorithm, initializing the first population and creating the new population are influenced by random elements. Therefore, if there is no means to collect all of the elements in the possible search space, that will easily lead to a bad result. To solve this problem, the variables are divided into two types: decision variable and dependent variable. This article proposes decision variables consisting of the power output from the second TGU to the last TGU (p 2 , p 3 , . . . , p n ), and only the power output of the first TGU (p 1 ) belongs to the dependent variable. Minimum limit of solution y min and maximum limit of solution y max are defined as follows: In the next step, each solution is initially created based on the minimum and maximum limits of solution by the following equation:

Handling Constraints
The problem takes five constraints into account, including power balance, power output limits, prohibited operating zone, spinning reserve, and ramp rate limit. The five constraints can be solved by the following methods.

Redefining the Maximum and Minimum Power Outputs Considering RRL and POZ Constraints
The ramp up limit and the ramp down limit are redefined considering the RRL constraint by the following formulas: The redefined boundaries continue to be redefined for the second time by using Equations (36) and (37) considering the POZ constraints.

Handling the POZ Constraint for Decision Variables
In order to enable the new solutions to avoid violating the POZ constraints, correction is required and implemented by using the following equation: In order to satisfy the power balance requirement shown in Formula (6), p 1 is regarded as a dependent variable. Thus, the violation of the power balance is converted into the violation of p 1 . Power outputs from the second to the last units are considered to be control variables and are provided by the duty of the proposed method, whereas p 1 can be obtained by using Formulas (6) and (7). The formula below can find p 1 exactly.
Calculating p 1 deals with the power balance constraint. However, sometimes it can violate other constraints such as power output limits and POZ. Power output of the first TGU is checked and violations of the upper and lower limits are penalized using the following equation: where ∆p 1,k is the penalty for violation of the upper bound and lower bound of p 1 . If ∆p 1,k is zero, it means there are no violations of the bounds; however, p 1 would continue to be checked for POZ constraints by the following model:

Handing the SR Constraint
After all the TGUs are checked and penalized, the sum of the reserve power from all TGUs is calculated and compared to the reserve power of power system. If the reserve power of power system is not provided exactly, the penalty is as follows:

Calculating the Fitness Function of the Solution
The sum of objective function and penalty factors are included in the fitness function for evaluating the quality of the solution. For the considered problem, the fitness function is as follows:

Handling Lower and Upper Limits
If new solutions are outside the permitted limits, they will be adjusted using the following equation:

Numerical Results
To demonstrate the effectiveness of the proposed IALO method, four main study cases are implemented as follows: (i) Consider MFs for each unit of a 10-unit system; Case 1: Consider four load cases of 2400, 2500, 2600, and 2700 MW without VPEs; Case 2: Consider one load case of 2700 MW with VPEs; Case 3: Consider one load case of 2700 MW with VPEs and constraints such as SR, POZ, and RRL.
(ii) Consider SF for each unit of a 20-unit system supplying a 2500 MW load considering power loss; (iii) Consider MFs and VPEs for the 80-unit system with a 21,600 MW load; (iv) Consider SF for each unit for the 15, 30, 60, and 90-unit systems with POZ constraints.
The proposed method together with the ALO were programmed in the Matlab platform and run on a PC (Processor with 2.7 GHz, RAM with 4.0 GB). In order to investigate the real performance of the IALO method, another comparison criterion was also considered to be the number of fitness evaluations, N fes , which is shown in the following equation: In the equation above, ω is the number of new solution generation times in each iteration. The IALO method has only one generation time for each iteration, so ω is 1 for IALO while ω is 2 for CSA [27] and RCSA [29]. For other methods such as FOA and IFOA in [37], N fes had another model as follows: Furthermore, we also computed the improvement of the proposed method over other compared methods by using the best cost of the compared methods and our proposed method. The improvement in % (ic) can be obtained by the following model: Best cost of another method − Best cost of IALO Best cost of another method × 100% In Equation (48), values of ic can be classified into three cases in which Case 1 corresponds to positive values, Case 2 corresponds to zero values, and Case 3 corresponds to negative values. If Case 1 occurs, the suggested method is more effective than the compared method. In the case of an ic with zero value, the suggested method is as good as other ones. In contrast, Case 3 indicates the suggested method is less effective than the compared methods if the compared methods have reported valid solutions and used smaller N fes values in Equation (47).

Comparison for Results from the 10-Unit System with Different Cost Function Characteristics
This system has ten thermal generating units with a non-smooth fuel cost function. There are three cases for this system consisting of MFs; MFs and VPEs; and MFs and VPEs under complex constraints. Data of the system are taken from [7] for Case 1 and [11] for Cases 2 and 3.

Case 1: 10-Unit Systems with MFs
The suggested IALO method was tested on four sub-cases of load demands consisting of 2400, 2500, 2600, and 2700 MW. For each case, 100 trial runs were executed to obtain results and comparisons. In order to investigate the search potential, we compared the proposed method with other methods based on the obtained results (including the best cost, minimum cost, and average cost) and the number of fitness evaluations N fes (which was calculated by using Equation (46) with the presence of the population size and the number of iterations). The comparison of obtained results indicated the search performance of the methods, meanwhile the comparison of N fes indicated the search speed of methods. For better comparison of search performance, we converted better cost into improvement in %, which was calculated by using Equation (48). Basically, methods have a better chance to reach better results if they are run with larger population sizes and more iterations. We wanted to reflect the real performance of the proposed method and so we compared the obtained results; however, N fes was always compared to assure a fair comparison between the proposed method and other methods. In Table 1, a comparison is shown for the same system with four different load cases. We ignored the comparisons of the mean and maximum costs because there were four comparison cases. Furthermore, the proposed method was run by using smaller N fes than that of other methods. Therefore, if the proposed method can reach the same best cost as the other methods, the proposed method is more effective because it is faster. Observing the best cost indicates that the suggested method is better than the ALO method in terms of the best cost for all of load demand cases. In comparison with the ALO, the highest improvement level was 0.931%, corresponding to the case of a 2600 MW load demand, and the lowest level of improvement was 0.330% for a 2700 MW load demand. As compared to the other methods, the IALO method can reach the same result as other methods, excluding CSA [27], which suffers from a slightly higher cost than the proposed method. For example, in the case of a load demand of 2600 MW, the best fuel cost of the IALO was 574.381 $/h. It is smaller than the 574.410 ($/h) value for the CSA, being tantamount to 0.005%. Although, the proposed method cannot reach better cost values than the other remaining methods, the proposed method uses smaller N fes than all other methods excluding EALHN [9], which is not a population-based method. In fact, the proposed method produced about 750 fitness evaluations while the other methods produced from 2000 to 20,000 fitness evaluations. The search speed of the suggested method was 2.6-26 times faster than those of the other methods. The analysis of the best cost and search speed indicates that the application of the suggested method to solve a 10-unit system with multi-fuel options is absolutely effective. This system had ten generating units with the sum of several non-convex functions. The load demand of the test system was 2700 MW. The obtained results including the best cost, mean cost, worst cots, and standard deviation for 100 trial runs are shown in Table 2. In addition, N p , G max , N fes , and ic (%) are also reported for comparison in detail. The best cost value of the IALO method was the lowest among all compared methods, including PSO, FOA, IFOA, CSA, RCSA, and ALO. The highest improvement was 6.124% as compared to the FOA method and the lowest improvement was 0.004% as compared to the RCSA method. Clearly, FOA was not a powerful method for the system and it was modified and changed into IFOA with much better results with an ic of 0.007% even though the IFOA used only half of the number of fitness evaluations. CSA [27] used the highest number of fitness evaluations, at 10,000, among all compared methods, but it could not reach the best solution and, therefore, the proposed method reached improvement over that method with 0.005%. The proposed method reached the best solution, but it used the highest number of fitness evaluations, excluding the comparison with CSA. The results show that it is hard to improve the results for the system with a small number of fitness evaluations, similar to the other compared methods. However, the running time of the proposed method was small compared to those of the other methods. Consequently, the proposed method is a potential method for 10-unit system taking multi-fuel options and vale point loading effects into account. Next, we added several complex constraints such as spinning reserve, generating capacity, POZs, and ramp rate into the 10-unit system with MFs and VPEs. Table 3 lists the obtained results for a 2700 MW load demand and reports N fes and ic for better comparisons. The IALO method was better than RCGA, FOA, IFOA, and ALO methods because the best cost of the IALO method was the smallest. The highest and the lowest improvement of the suggested method over these methods was, respectively, 7.304% and 0.005%. The values of N fes from all methods indicates that the proposed method was one of the fastest methods with 3000 fitness evaluations, whereas the slowest method used 6000 fitness evaluations. However, the best cost of the IALO is slightly higher than that of the NRCGA method with −0.001%. This number shows that the best performance was not achieved by the proposed method, but our method is still a very efficient one. In fact, NRCGA reached the lowest cost, but the application of that method was only for the study cases with multiple fuels and VPEs, whereas the proposed method was implemented for different constraint types, different fuel types, and very large scale systems with 80 units and 90 units.

Implementation of the Suggested Method on a 20-Unit System Considering Transmission Loss
This section presents the comparison of the best cost, CPU, and N fes from the proposed IALO and other methods with a load power of 2500 MW. Data of this system was taken from [39]. The obtained results for this study are given in Table 4. When compared to IA [39], the suggested method was more powerful for all criteria, such as the best cost with an ic % of 0.016%, CPU, and N fes . The suggested method was much faster when comparing 4500 and 20,000 fitness evaluations. The outstanding performance cannot be seen when the proposed method was compared to the other remaining methods, but the proposed method was still evaluated to be more powerful. In fact, the best costs of RCSA [29], CBOA [41], and ED-SHO [45] were equal to that of the suggested method, but RCSA, CBOA, and ED-SHO methods used higher values for N fes equaling 10,000, 12,000, and 100,000, respectively. These values were much higher than the 4500 of the IALO. Other values such as the mean cost and the standard deviation reveal the stable search ability of the proposed method since the mean cost and the best cost of the proposed method are approximately the same as those of the other methods, and the standard deviation is nearly zero. The proposed method used the lowest number of fitness evaluations, but it reached the same best cost and the most stable ability. As a result, it can be concluded that the IALO is a potential method for the system with 20 units considering quadratic fuel cost function and power loss constraint.

The Implementation of the Suggested Method on an 80-Unit System
In order to demonstrate the efficiency of the suggested method, a larger system with 80 thermal generating units with multi-fuel options and vale point loading effects was considered [11]. The load demand of the system was 21,600 MW. The study case was an expanded case of the 10-unit system with MFs and VPEs in Section 5.1.2 above. Compared methods had the same challenge of a discontinuous objective function and a high number of control variables. The comparisons with CSA, RCSA, and ALO are shown in Table 5. Because it is a complicated study case, the improvement of the proposed method over its conventional method can be seen clearly. The ic over ALO was 0.194%, but this value is much smaller compared to those of CSA and RCSA, where the values of ic were only 0.01% and 0.005, respectively. The insignificant improvement can be understood because the two methods were run by using 132,000 fitness evaluations while the proposed method was run by using only 80,000 evaluations. For this case, the proposed method was the fastest method and reached the best solution. So, it is a very efficient method for a large-scale, 80-unit system considering MFs and VPEs.  [6]. SR of the system was 200 MW for the case with 15 units and its increase was directly proportional to the number of units for the other remaining systems. Control parameters selection for ALO, the proposed method, and other compared methods are given in Table 6. Obtained results are reported in Table 7. Table 7 indicates that the IALO was highly superior to ALO method in terms of the best cost with very high improvement levels, such as 0.43% for a 15-unit system, 0.19% for a 30-unit system, 0.11% for a 60-unit system, and 0.15% for a 90-unit system. The saving cost percentage was significant for the systems because the two methods had the same control parameters selection, as shown in Table 6. The comparisons with CSA and RCSA do not show a benefit of the proposed method, since IALO was only slightly better than these methods for the case of a 90-unit system, whereas the other cases were similar, excluding the comparison with RCSA for a 15-unit system. The RCSA showed effective cost but did not show optimal generation for verification. The N p , G max , and N fes in Table 6 indicate that the proposed method is much faster than CSA and RCSA for the 15-and 30-unit systems. The proposed method used 2000 and 20,000 fitness evaluations for the 15-and 30-unit systems, respectively, but CSA and RCSA used 12,000 and 48,000 fitness evaluations, respectively. However, for the two large-scale systems with 60 units and 90 units, the proposed method was slower because it used 80,000 and 160,000 fitness evaluations, respectively, whereas CSA and RCSA used 60,000 and 72,000 fitness evaluations, respectively. The N fes and the best cost reveal that the proposed method was very effective for small-scale systems with 15 TGUs and 30 TGUs, but was less effective for large-scale systems with 60 and 90 TGUs.

Investigation of the Real Performance of IALO and ALO
The results from the suggested method and ALO method are summarized in Table 8. In all the cases of Table 8, the best cost of the IALO was always smaller than that of the ALO. Moreover, the time to run the program of the IALO method was always lower than that of the ALO method for all cases. The ic of 0.43% was the highest improvement of IALO over ALO, corresponding to the case of 15-units with POZ for a 2650 MW load demand. In contrast, 0.0001% was the smallest improvement of the suggested IALO over the ALO, corresponding to the case of a 20-unit system considering transmission loses for a 2500 MW load demand. However, in this case, ALO used N fes = 9000 while IALO used only half of that quantity, i.e., 4500. To demonstrate the superiority of the IALO more persuasively, the convergence characteristics of the IALO and ALO are plotted in Figures 2-6 for different cases from the sections above. Through the figures, it can be seen that the IALO converged faster than the ALO after half of the number of iterations, or even fewer than half. For instance, Figure 2 shows the IALO reached a solution close to the best solution at the fifteenth iteration, whereas the solution of the ALO at the fiftieth iteration was far from the obtained solution of the IALO. In Figure 3, the solution of the IALO at the sixtieth iteration was much more effective than the solution of the ALO at the 200th iteration. Furthermore, Figures 4-6 indicate that the IALO was at least three times faster than the ALO, and the best solution of the ALO was much worse that the solution of the IALO at one third the number of iterations.     In order to investigate the efficiency of finding optimal solutions, the fitness of the best solution of successful runs was also plotted in Figures 7-11 for comparison. Fitness values of the IALO are black points while those of the ALO are red points. All the figures have the same characteristic that red points have very large fluctuations, but black points have very tiny deviations, even though they are on a line with the same fitness value. This manner implies that approximately all runs of the IALO are close to the best solution, but the ALO fails to reach the same achievement. Consequently, the performance improvement of the suggested IALO over the ALO is significant when dealing with these considered study cases.
Optimal generations of the studied cases found by the IALO are reported in Appendix A.      In order to investigate the efficiency of finding optimal solutions, the fitness of the best solution of successful runs was also plotted in Figures 7-11 for comparison. Fitness values of the IALO are black points while those of the ALO are red points. All the figures have the same characteristic that red points have very large fluctuations, but black points have very tiny deviations, even though they are on a line with the same fitness value. This manner implies that approximately all runs of the IALO are close to the best solution, but the ALO fails to reach the same achievement. Consequently, the performance improvement of the suggested IALO over the ALO is significant when dealing with these considered study cases.
Optimal generations of the studied cases found by the IALO are reported in Appendix A.         Appl. Sci. 2020, 10, x FOR PEER REVIEW 23 of 29 Figure 11. Fitness of 100 runs from the ALO and IALO for the system with 80 units, multi fuels, vale effects, and a 21,600 MW load.

Conclusions
In this paper, the proposed IALO was applied to find the best generation of available units with an intent to reduce the total generation cost of the whole system. Different studied cases with different types of systems and constraints were implemented for investigating the outstanding efficiency of the IALO over its conventional form. Systems with 10,15,20,30,60,80, and 90 units were considered. The used comparison criteria comprised the best solution, the search speed, the improvement percentage, and the number of fitness evaluations. After comparing the proposed method with its conventional method and other methods, the advantages and disadvantages of the proposed method can be summarized as follows: Figure 11. Fitness of 100 runs from the ALO and IALO for the system with 80 units, multi fuels, vale effects, and a 21,600 MW load.

Conclusions
In this paper, the proposed IALO was applied to find the best generation of available units with an intent to reduce the total generation cost of the whole system. Different studied cases with different types of systems and constraints were implemented for investigating the outstanding efficiency of the IALO over its conventional form. Systems with 10, 15, 20, 30, 60, 80, and 90 units were considered. The used comparison criteria comprised the best solution, the search speed, the improvement percentage, and the number of fitness evaluations. After comparing the proposed method with its conventional method and other methods, the advantages and disadvantages of the proposed method can be summarized as follows: (1) The proposed method can reach better results with faster speed than all compared methods for small-scale and large-scale systems with convex fuel cost functions and not very complicated constraints. In fact, for the system with 10 units and multi fuels, the proposed method could reach the best cost very quickly. Similarly, for 15, 30, 60, and 90-unit systems with POZ constraints and a convex fuel cost function, the proposed method is also superior to other compared methods. However, the outstanding performance cannot be reached for the same 10-unit system but with many complicated constraints and a nonconvex fuel cost function. (2) The proposed method can reach highly stable search ability since most of runs have the same or approximately the same solutions with the best run. This enables the proposed method to reach a very small fluctuation and find a high-quality solution easily. (3) The selection of population size and the number of iterations is simple but the conditions to apply Equations (27)- (29) are not easily determined. We had to try different systems to determine the most appropriate conditions for using these equations. For other problems, the conditions may be different from the 15% and 30% values found in this problem. (4) The proposed method can reach much better results than the ALO even when other conditions, different from 15% and 30%, are executed. So, ALO should be replaced with the proposed method for the ELD problem.
IALO is suggested as a highly efficient method in power systems, and it can be used to address other problems in wind power plants, solar power plants, and conventional power plants.