Abstract
Optimal load dispatch (OLD) is an important engineering problem in power system optimization field due to its significance of reducing the amount of electric generation fuel and increasing benefit. In the paper, an improved cuckoo search algorithm (ICSA) is proposed for determining optimal generation of all available thermal generation units so that all constraints consisting of prohibited power zone (PPZ), real power balance (RPB), power generation limitations (PGL), ramp rate limits (RRL), and real power reserve (RPR) are completely satisfied. The proposed ICSA method performance is more robust than conventional Cuckoo search algorithm (CCSA) by applying new modifications. Compared to CCSA, the proposed ICSA approach can obtain high quality solutions and speed up the solution search ability. The ICSA robustness is verified on different systems with diversification of objective functions as well as the considered constraint set. The results from the proposed ICSA method are compared to other algorithms for comparison. The result comparison analysis indicates that the proposed ICSA approach is more robust than CCSA and other existing optimization approaches in finding solutions with significant quality and shortening simulation time. Consequently, it should lead to a conclusion that the proposed ICSA approach deserves to be applied for finding solutions of OLD problem in power system optimization field.
1. Introduction
Over the past decades, an enormous number of studies have concerned and solved different optimization operation problems in regard to electric grids by utilizing potential search ability of optimization approaches. Many concerned operation problems regarding distribution power network, transmission network, and different types of power plant as well as electric components in the network have been successfully solved. This study focuses on optimal load dispatch (OLD) problem with the task of allocating the generated power of all considered thermal generation units to reduce the cost of burnt fossil fuels. All physical and operational constraints are required to be exactly satisfied. If all generation units in each power plant and all power plants are working under the most appropriate schedule, total fuel cost of all units can be the smallest and consumers can get significant amount of revenue [1]. The achievement is thanks to the meaning of OLD problem.
A huge number of optimization approaches using mathematical programming have been widely applied so far for solving the considered OLD problem such as dynamic programming (DP) [2], lambda iteration method [3], Newton Raphson and Lagrangian multiplier (NRLM) method [4], and linear programming (LP) [5, 6]. Conventional methods focused on the systems with simple constraints and convex objective function where nonlinear constraints and the effects of valve loading process were not considered. The complex level of the constraints and objective has been mentioned in many articles. For example, the authors in [3] could only solve the OLD problem successfully by separating three-curve objective function into three different single curve objective functions. In [7], a more realistic representation of the electric generation fuel function corresponding to multi fossil fuel sources was introduced, in which the authors considered a discontinuous cost function and the effects of valve loading process (EoVLP) of thermal units. More complicated models were also introduced. For example, some thermal generation units were driven by burning multi fossil fuel sources (MFS) to generate electricity [8], or some operating conditions of generators including upper generation boundary, lower generation boundary, and prohibited power zones were added [9]. However, these methods have been only applied to the systems where the generation power-fuel cost characteristic of thermal generation units was mathematically modelled as the second order function and the effects of valve loading process were ignored [10].
Other artificial intelligence-based advanced methods have been recently applied for solving OLD problem. Evolutionary programming-based approaches (EP) [1, 11, 12] and genetic algorithm (GA) [13–15] were considered as fast algorithms because of their parallel search ability. In addition, GA and EP possessed good properties such as finding solutions nearby global optimum, capability of effectively handling nonlinear constraints, reliable search ability, and not many adjustment parameters [10]. Thus, it could become a suitable choice for successfully solving OLD problem. However, GA was prematurely convergent to local optimum solutions [16]. In this regard, simulated annealing approach (SA) [17] was a better probabilistic approach in finding solutions with appropriate fitness but there was a high possibility of easily converging to local optimal zones when coping with complicatedly constrained problem. It converges slower than GA and EP, though. Differential evolution algorithm (DE) [10, 18] also belongs to the same class as GA and EP. However, it is more popular thanks to simple structure with several adjustment parameters and high rate of success. DE has been more widely and successfully applied than SA and GA [19]. But its faster convergence manner led to the same drawback as GA like high rate of falling into local optimum and hardly ever toward promising zones quickly. In fact, these shortcomings could be tackled by setting population size to higher value. But high population size could suffer from long simulation time to calculate fitness function and evaluate quality of solutions [20]. Hopfield neural network (HNN) [21, 22] focused on optimizing energy function and was only successfully applied for optimization problems where objective functions were differentiable. HNN could be an appropriate method for large-scale systems with high number of generation units but it needed long simulation time and may also converge to local optimum solution zones [23]. Particle swam optimization (PSO) [16, 24] is a random search approach developed by behavior of a swarm or flock during food search process. In comparison with GA, PSO owned more advantages such as simpler implementation and few parameters with easy selection. However, the success rate of PSO was highly influenced by adjustment parameters and it copes with high rate to be trapped in many zones with local optimum solutions [25]. Harmony search (HS) [26] is a metaheuristic-based method inspired from music. Instead of using gradient search, HS has employed stochastic random search to exploit its potential ability. Thus, it tended to converge to local optimal zones rather than global optimal zones [27]. Biogeography-based optimization (BBO) [28] could compete with PSO and DE since its solutions were directly updated by migration from other existing solutions and its solutions directly shared their attributes with other solutions [29].
It is clear that each method has advantages as well as disadvantages for different applications for finding OLD problem solutions. Hence, another natural approach is to combine different methods to exploit the advantages of each method and enhance the overall searching capability. Several hybrid methods have been developed in such way including hybrid Genetic algorithm, Pattern Search and Sequential Quadratic Programming (GA–PS–SQP) method [30], hybrid Artificial Cooperative Search algorithm (HACSA) [31], hybrid PSO-SQP [32], and hybrid GA (HGA) [33]. Basically, these hybrid methods could deal with OLD problem more effectively than each member method. On the other hand, they could suffer from the difficulty of selecting many controllable parameters. In addition to such popular original algorithms and hybrid methods, there are many other original and improved methods that have been applied for solving the considered OLD problem. These methods are Symbiotic organisms search algorithm (SOS) [34] and its modified version (MSOS) [34], teaching activity and learning activity-based optimization (TLBO) [35], chemical reaction-based approach (CRBA) [36], enhanced particle swarm optimization (EPSO) [37], sequential quadratic technique-based cross entropy approach (CEA-SQT) [38], traverse search-based optimization approach (TSBO) [39], invasive weed approach (IWA) [40], Improved Differential evolution (IDE) [41], immune algorithm using power redistribution IAPR [42], Colonial competitive differential evolution (CCDE) [43], Chaotic Bat algorithm (CBA) [44], Exchange market algorithm (EMA) [45], adaptive search technique algorithm and differential evolution (GRASP-DE) [46], θ-Modified Bat Algorithm (θ-MBA) [47], Tournament-based harmony search algorithm (TBHSA) [48], New Modified β-Hill Climbing Local Search Algorithm (Mβ-HCLSA) [49], improved version of artificial bee colony algorithm (IABCA) [50], artificial cooperative search algorithm (ACSA) [51], and ameliorated grey wolf optimization algorithm (AGWOA) [52]. Among these methods, ACSA and AGWOA were the two latest methods, which were applied for OLD and published in early 2019. However, the demonstration of real performance of the two methods is still questionable. In fact, ACSA has been tested only on systems with small scale, single fuel and simple constraints such as generation limits and power balance. The largest scale system was considered in [51] to be 40-unit system. Unlike [51], different types of fuel cost function, complicated constraints, and large scale system with 140 units have been taken into account in [52]. Via comparisons with many existing methods, AGWOA has been stated to be the best one with many surprising results. Thus, the validation of reported solutions from the method must be verified and its strong search ability must be reevaluated. In the numerical results section, we will report the verification of the two questionable issues.
In this paper, we have proposed an improved cuckoo search algorithm (ICSA) for dealing with large scale OLD problem with the consideration of complicated constraints together with nondifferentiable fuel cost function. In the proposed ICSA approach, some new modifications have been performed on conventional cuckoo search algorithm (CCSA) to improve the quality of CCSA. The CCSA method was first developed in 2009 [53] for solving a set of popular benchmark functions and its highly superior performance over PSO and GA has attracted a huge number of researchers in learning and applying for different optimization problems in different fields. Furthermore, its improved variants are also an extremely vast number. In relation to OLD problem, CCSA has been applied and presented in [23, 54–57] meanwhile its improved methods consisting of modified cuckoo search algorithm (MCSA) and improved cuckoo search algorithm with one solution evaluation (OSE-CSA) have been, respectively, presented in [58, 59]. In [55], Basu has applied CCSA for solving OLD problem with 40-unit system with single fuel option and the effective of valve loading process, 20-unit system with single fuel and quadratic fuel cost function, and 10-unit system with multiple fuel sources and without the effects of valve loading process. The author has made a big effort in demonstrating the high potential search of CCSA by comparing with many popular metaheuristic algorithms but the shortcoming of the study was neglecting complicated constraints and large scale systems.
Studies in [56, 57] have dealt with OLD problem with two power systems considering simple constraints and small number of units. Only three simplest constraints, such as power balance, limitations of generation, and prohibited power zones, have been taken into account meanwhile the largest system was solved to be 6-unit system. Thus, there were few methods compared to CCSA and the real performance of CCSA was not shown persuasively in the studies. In [23], authors have applied CCSA for solving different systems with very complicated constraints, complicated characteristics of thermal generating units and high number of units. Among the mentioned studies regarding CCSA for OLD problem, authors in [23] could show the best view in evaluating the real performance of CCSA since there were six cases that were carried out and a huge number of methods were compared to CCSA. In spite of the real potential search ability, CCSA has been commented to be low convergence to global optimum and significantly improved better [58, 59]. MCSA in [58] has been proposed by using a new strategy for the second generation technique. The mutation operation in CCSA has been replaced with current-to-best/1 model of DE in [60]. MCSA has been applied for solving four systems with 3, 6, 15, and 40 units in which the most complicated constraint considered was prohibited power zone and only single fuel source was taken into account. MCSA method has been compared to other popular methods such as PSO, GA, and EP. But the comparisons with CCSA have not been carried out. Thus, the improvement of such proposed method in [58] was not proved persuasively. OSE-CSA in [59] has canceled one evaluation time in case that OSE-CSA has continued to improve solution quality. The improvement seemed to be appropriate for CCSA in dealing with OLD problem with complicated systems. CCSA in [23, 55] have been considered for comparison in [59] and they have been proved to be less effective than OSE-CSA. However, OSE-CSA has used one more control parameter, called one rank parameter, and it needed to be tuned thoroughly for obtaining high performance. In the proposed ICSA approach, we have focused on a new strategy of the second new solution generation in CCSA method. As shown in [58, 59], CCSA has become a strong search method thanks to the first new solution generation, which was performed by Levy flight technique while the second new solution generation could not take on local search function well. In the second update progress via mutation operator, two random old solutions are used to generate an increased step size. However, the manner can lead to new low quality solutions because the increased step will be very small when iterative algorithm is carrying out at the last iterations. In fact, current solutions at final several iterations tend to be close together and the different values between each two ones are very small, leading to a very small increased step. In order to tackle the disadvantage of the CCSA, we apply a new adaptive technique for improvement of solution quality. Firstly, we propose two ways for producing the increased step including two-solution-based increased step and four-solution-based increased step. The decision when which step size will be used is dependent on the result of comparison between fitness function ratio (FFR) and a predetermined parameter . FFR is defined as a ratio of deviation between fitness functions of the considered solution and the most promising solution to the fitness value of the best one meanwhile is a boundary to give the final decision for the selection of a used step size. At the beginning, is a fixed value for all solutions and then it will be adaptive based on the comparison between it and FFR. When FFR of a solution is less than , of the solution will be decreased equally to ninety percent of the previous value. Otherwise, the value of remains unchanged in case the FFR is equal to or higher than . The adaptive technique has a significantly important role in enhancing the potential search ability of the proposed method. This proposed method is investigated on six cases with different considered constraints, different types of fuel cost function, and large scale systems. The detail of the six cases is as follows:
Case 1. Four systems with single fuel source (SFS) and power loss (PL) constraint
Case 2. A 110-unit system with SFS
Case 3. Four systems with SFS and the effects of valve loading process (EoVLP)
Case 4. Two systems with SFS, and PPZ and RPR constraints
Case 5. A 15-unit system with SFS, and RRL, PPZ, and PL constraints
Case 6. Three systems with multiple fuel sources (MFS) and EoVLP
The achieved results in terms of minimum fuel cost, average fuel cost, maximum fuel cost, and standard deviation found by the proposed method compared to those obtained by others reveal that the method is very efficient for the OLD problem. In addition, the performance improvement of the proposed method over CCSA is also investigated via the comparison of the best solution and all trial runs. In summary, the main advantages of the proposed ICSA approach over CCSA as well as the main contribution of the study are as follows:(i)Based on fitness function of each considered solution, local search or global search is decided to be applied more effectively(ii)Find better solutions with smaller number of iterations and shorter execution time for each run(iii)Shorten simulation time for the whole search of each study case
However, the proposed method also copes with the same shortcomings as CSA. Although the shortcomings do not cause bad results for the proposed method, they make the proposed method be time consuming in tuning optimal parameter. The shortcomings are analyzed as follows:(i)Control parameter, probability of replacing control variables in each old solution, must be tuned in range between 0 and 1. There is no proper theory for determining the most effective values of the parameter. Thus, the performance of the proposed method must be tried by setting the parameter to values from 0.1 to 1.(ii)The method uses more computation steps for search process. Thus, the proposed method uses higher number of computation steps for each iteration. However, due to more effective search ability for each iteration, the proposed method can use smaller number of iterations but it finds more effective solutions.
The remaining parts of the paper are arranged as follows: Section 2 shows the objective and constraints of the considered OLD problem. CCSA and the proposed method are clearly explained in Section 3. Section 4 is in charge of presenting the implementation of ICSA method for the studied problem. The simulation results together with analysis and discussions are given in Section 5. Finally, conclusion is summarized in Section 6. In addition, appendix is also added for showing found solutions by the proposed ICSA approach for test cases.
2. Optimal Load Dispatch Problem Description
2.1. Fuel Cost Function Forms with Single Fuel Source
In the considered OLD problem, the optimal operation of a set of thermal generation units is concerned as the duty of reducing total cost of all the units, which can be seen by the following model:
In traditional OLD problem, fuel cost function of the generation unit is represented as the second order function with respect to real power output and coefficients as the model below [2]:
In addition, for the case considering the effects of valve loading process on thermal generation units, fuel cost becomes more complicated by adding sinusoidal term as below [12]:
Real Power Balance Constraint. Total real power demand of all loads in power system together with real power loss in all conductors must be equal to the generation from all available thermal generation units. The requirement is constrained by the following equality:where total real power loss, , is determined by Kron’s equation below:
Generation Boundary Constraint. For the purpose of economy and safe operation, each thermal generation unit is constrained by the lower generation bound and upper generation bound as the following model:
2.2. Fuel Cost Function Forms with Multi-Fuel Sources
In this section, fuel cost function of thermal generation units is mathematically modeled in terms of different forms from that in the section above due to the consideration of multi-fuel sources. Each type of fuel source is formed as each second order function and the fuel cost function form is the sum of different second order functions for the case of neglecting the effects of valve loading progress. But for the consideration case of the effects, the form is more complex with the presence of sinusoidal terms [15]. As a result, the forms of cost function can be expressed in Equation (7) [21] and Equation (8) [15]:
Cost function forms in Equations (7) and (8) are only included in objective function (1) meanwhile main constraints in formulas (4) and (6) must be always satisfied.
2.3. Prohibited Power Zone, Real Power Reserve, and Ramp Rate Limit Constraints
Prohibited power zones (PPZ) are different ranges of power in fuel cost function that thermal generation units are not allowed to work due to operation process of steam or gas valves in their shaft bearing. The power generation of units in the violated zones is harmful to gas or steam turbines even destroyed shaft bearing. Thus, the constraint is strictly observed. In the fuel-power characteristic curve of generation units, PPZ causes small violation zones and such curves become discontinuous. As considering PPZ constraint, the determination of power generation of units is more complex and equal to either lower bound or upper bound. Unlike PPZ constraint, RPR constraint is not related to fuel-power feature curve but it causes difficulty for optimization approaches in satisfying one more inequality constraint. Each generation unit among the set of available generation units must reserve real power so that the sum of real power from all generation units can be higher or equal to the requirement of power system for the purpose of stabilizing power system in case that there are some units stopping producing electricity. On the contrary to PPZ constraint, ramp rate limit (RRL) constraint does not allow power output of thermal generating units outside a predetermined range. The constraint considers maximum power change of each thermal generating unit as compared to the previous power value. Thus, optimal generation must satisfy the RRL constraint. The PPZ constraint, RPR constraint, and RRL constraint can be presented as follows:
Prohibited Power Zones. As considering PPZ constraint, valid working zones of each thermal generating unit are not continuous and its generation must be outside the violated zones as the following mathematical description:
As observing Equation (9), generation units cannot be operated within the violated zones except for starting point and end point. Consequently, the verification of PPZ constraint violation should be carried out first and then the correction should be done before dealing with other constraints such as real power reserve constraint and real power balance. Besides, if power output of all units can satisfy the PPZ constraint, generation limits in Equation (6) are also exactly met.
Real Power Reserve Constraint. Real power reserve in power system aims to enhance the ability of stability recovery of power system and avoid blackout. In order to get high enough power for requirement, all available units are constrained by the following inequality:where is the real power reserve contribution of the thermal generation unit and the determination of can be done by employing the two models below:
Equation (10) shows that the constraint of prohibited power zones is not included in the real power reserve constraints; however, prohibited power zones are always strictly considered and must be exactly satisfied.
Ramp Rate Limit (RRL) Constraint. In OLD problem, all considered thermal generating units are supposed to be under working status but previous active power of each thermal generating unit is not taken into account. Thus, increased or decreased power is not constrained. This assumption seems to be not practical until RRL constraint is considered. RRL constraint considers initial power output and the power change is supervised. Regulated power can be higher or lower than the initial value as long as it is within a predetermined range. Increased step size (ISS) and decreased step size (DSS) are given as input data and they are used to limit the change of power output of each thermal generating unit. The constraint can be mathematically expressed as the following formula [7]: where is the initial power output of the thermal generating unit before its power output is regulated; and are, respectively, maximum increased and decreased step sizes of the thermal generating unit.
3. The Proposed Cuckoo Search Algorithm
3.1. Classical Cuckoo Search Algorithm
In search technique of CCSA [53], a set of solutions is randomly generated within a predetermined range in the first step and then the quality of each one is ranked by computing value of fitness function. The most effective solution corresponding to the smallest value of fitness function is determined and then search procedure comes into a loop algorithm until the maximum iteration is reached. In the loop algorithm, two techniques updating new solutions two times (corresponding to two generations) are Lévy flights and mutation technique, which is called strange eggs identification technique. The two generations can produce promising quality solutions for CCSA. After each generation, CCSA will carry out comparing fitness of newly updated solutions and initial solutions for keeping better ones and abandoning worse ones. The most effective solution at last step of the loop search algorithm is determined and it is restored as one candidate solution for a study case. The detail of the two stages is as follows.
3.1.1. Lévy Flights Stage
This is the first calculation step in the loop algorithm and it also produces new solutions in the first generation for CCSA. New solution is created by the following model: where is the positive scaling factor and it is nearly set to different values for different problems in the studies [53, 62]. In the work, the most appropriate values for such factor can be chosen to be 0.25/0.5 for different systems.
3.1.2. Discovery of Alien Eggs Stage
The step plays a very important role for updating new solutions of the whole population. However, not every control variable in each old solution is newly updated and the decision of replacement is dependent on comparison criteria as the following equation:
3.2. Proposed Algorithm
In the part, a new variant of CCSA (ICSA) is constructed by applying three effective changes on the main functions of CCSA in order to shorten simulation time corresponding to reduction of iterations and find more promising solutions. The proposed amendments are explained in detail as follows:(i)Suggest one more equation producing updated step size in addition to existing one in CCSA(ii)Create a new selection standard by computing fitness function ratio and comparing with a predetermined parameter . Thus, thanks to the standard, the existing updated step size and additional update step size will be chosen more effectively(iii)Automatically change value of for the xth solution based on the result of comparing with the previous
Such three points are clarified by observing the following sections:
3.2.1. Strange Eggs Identification Technique (Mutation Technique)
The first proposed improvement in our proposed ICSA approach is to select a more suitable formula for producing new solutions with better fitness function value. In CCSA, Equation (16) below is used to produce a changing step nearby old solutions for all current solutions.
The use of Equation (16) aims to produce a random walk around old solutions in search zones with intent to find out promising solutions. In order to reduce the possibility of suffering the local trap and approach to other favorable zones for searching, we propose a new Equation (17). The formula is built by the idea of enlarging search zone with the use of two more available solutions. Obviously, the larger changing step can own higher performance in moving to other search spaces that the classical approach used in CCSA. The suggestion is mathematically expressed by the formula below:
The changing step obtained by using Eq. (17) is named four-point changing step. Now, two solutions, which are newly formed by using two different changing steps shown in formulas (16) and (17), are found by the two following methods:
It can be clearly observed that the distance between (old solution) and (new solution) is lower than that between and . This difference can contribute a highly efficient improvement to the proposed ICSA approach search ability.
For the CCSA case, if two solutions and are either slightly different or completely coincident, such newly updated solution does not have good chance to leave the current zone and approach to more promising zones. In another word, the new one is approximately coincident with the old one. As the search task is taking place at some last iterations, this phenomenon becomes much worse because all current solutions are lumped in a small zone and the capability of moving to other zones is impossible. As a result, the CCSA approach will work ineffectively and search strategy is time consuming until other runs are started.
Contrary to the two-point step size, the new proposed formula may produce a large enough length to escape the local optimum zone and reach new favorable zones. It explains why the four-point changing step has positive impact on the considered random walk rather than the two-point changing step.
3.2.2. New Standard for Choosing the Most Appropriate Changing Step
In this section, we extend our analysis to answer the question when to use the four-point step size. From Equations (18) and (19), two new solutions which are represented as the results of the two-point-based factor and the four-point step size can be illustrated by using Figure 1 corresponding to the search process at the first some iterations and Figure 2 corresponding to the last some iterations. For the sake of simplicity, we rewrite the two equations as follows:
Here we suppose that and are obtained by four exact solutions, , , and , and calculated as follows:
As mentioned above, the high changing step between new solution and old solution can help to explore new favorable zones. However, in optimization algorithms, searching steps cannot be arbitrarily large; otherwise the algorithm may diverge, in particular, for the cases that the considered solutions are not close together in solution search space. For example, at the beginning of loop algorithm with the first iterations in Figure 1, is a better choice than because it is kept in a sufficient limit and does not lead to a risk of divergence. In contrast, as many of current solutions are in different positions but their distance is not very short or approximately coincident such as at the last iterations in Figure 2, and have a very short distance but and have higher distance. According to the phenomenon in Figure 2, the proposed ICSA approach needs to produce a high changing step to move solutions to other search zones without local optimum. Hence, would be preferred to .
Based on the argument above, the determination of the condition for using either two-point changing step or four-point changing step is really crucial to the performance of the proposed ICSA approach in searching solutions of OLD problem. Here, the ratio of , which can be found by Equation (24), is suggested to be a suitable measurement for the selection of two options.
For a particular set of the current solutions, each individual depending on its will create a corresponding new solution by using either Equation (18) or (19). If the value of one current solution is smaller than the predetermined parameter , Equation (19) is applied for updating such considered solution . Otherwise, Equation (18) is a better option. The steps of the modified algorithm are similar to the conventional CSA, except that an additional step should be added at each iteration. In this step, the of all individual solutions should be calculated by utilizing Equation (24) and then the result of comparing the ratio with will be used to decide which updating formula should be selected. The whole description of the proposed standard and new mutation technique can be coded in Matlab program language by using Algorithm 1.
| If | |
| If∆FFRx< Tolx | |
| else | |
| end | |
| else | |
| End | |
| End |
3.2.3. Adjustment of Tolerance for Each Solution
As pointed out above, the proposed method needs assistances to determine the most appropriate step size for finding out favorable solution zones. The given aim can be reached if the selection of is reasonable; however, the range of this parameter is infinite and hard to select. Thus, the adaptation of tuning the parameter is really necessary. First of all, the comparison between and is carried out and then the adaptation will be determined based on the obtained result from the comparison. Results of comparison between the two parameters can be either is less than or is higher than . The case that two parameters are equal hardly ever occurs.
As the comer assumption happens (i.e., is less than ) at the considered time, the four-point step size will be employed for the solution. If remains unchanged at the previous value, the identification of improvement from such four-point step size or two-point step size is vague. Consequently, value of must be automatically reduced to a lower value in case that it has significant contribution to found promising solution of previous iteration. Clearly, the decrease of can enable the proposed method to jump out local optimal zone and approach more effective zones. By trial and error method, is selected to be a function of itself that is 0.9 of the previous value. Finally, the implementation of the proposed ICSA approach is presented in Algorithm 2.
| Produce initial population with solutions (, , …, ,, …, ) | |
| Calculate fitness function (, , …, ,, …, ) | |
| Go to the loop algorithm by setting | |
| While () | |
| %(i) The first newly produced solutions | |
| %(ii) Perform selection approach | |
| %(iii) The second newly produced solutions | |
| If | |
| If | |
| ; | |
| Else | |
| End | |
| Else | |
| End | |
| % (iv) Perform selection approach | |
| % (v) Determine the most effective solution and its fitness | |
| Determine with the smallest value and assign to | |
| If , perform step (i) and increase to . Otherwise, stop the loop algorithm and report both | |
| the smallest fitness together with . | |
| End while |
4. The Application of the Proposed ICSA for OLD Problem
The whole computation steps of the proposed ICSA approach for solving OLD problem are explained as follows.
4.1. Handling Constraints and Randomly Producing Initial Population
As shown in Section 2, the considered OLD problem takes five following constraints into account:(i)Power balance constraint is shown in Equation (4)(ii)Power output limitation constraint is shown in Equation (6)(iii)Prohibited power zone constraint is shown in Equation (9)(iv)Real power reserve constraint is shown in Equation (10)(v)Ramp rate limit constraint is shown in Equation (13)
Among the five constraints, ramp rate limit, generation limit, and prohibited power zone seem to be more complicated than power balance and power reserve constraints. However, the three constraints can be solved more easily because each unit is constrained independently in the three constraints whereas power balance constraint and power reserve constraint consider all the thermal generating units simultaneously. Power reserve constraint can be handled by penalizing the total generation of all units while power balance constraint can be solved by penalizing one violated thermal generating unit. The whole computation procedure for solving all constraints and calculating fitness function of solutions is described in detail as follows:
Step 1. Redefine maximum and minimum power output of each thermal generating unit as considering PPZ and RRL constraints by using the following formulas: Among the four Equations, (25) and (26) are used first in order to redefine upper bound and lower bound for all thermal generating units as considering RRL constraint. The, the redefined bounds continue to be redefined for the second time by using (27) and (28) as considering PPZ constraints.
Step 2 (randomly produce initial population). For dealing with the power balance constraint, all available units are separated into two groups in which the first group with decision variables consists of the power output from the second unit to the last unit (P2, P3, , PN) meanwhile only the power output of the first unit () belongs to the second group with dependent variable. So, upper bound solution and lower bound solution must be defined as follows: Based on the upper bound solution and lower bound solution, each solution is initially produced by the following model:
Step 3. Handle prohibited power zone constraint for decision variables P2, P3, …, PN
After being randomly produced, there is a high possibility that decision variables fall into PPZ and they violate PPZ constraint. So, the verification of falling into PPZ and correction of the violation should be accomplished by using the following formula:
Step 4. Handle RPB constraint by calculating and penalizing if it violates constraints.
In this step, power balance constraint is exactly handled by calculating and penalizing dependent variable (). is obtained by using formulas (4) and (5) as follows:whereIn Equation (32), has been determined for the purpose of dealing with real power balance constraint. However, it is not sure that can satisfy upper bound and lower bound constraints and prohibited power zone constraints. So, must be checked and penalized.
Firstly, is checked and penalized for upper and lower bound constraints by the following model:In Equation (34), if the second case or the third case occurs, it means P1 has violated either lower bound or upper bound and it would be penalized by using either (∆P1,x= P1,min - P1,x) or (∆P1,x = P1,x - P1,max). Otherwise, if P1 has not violated the bound constraints (i.e., the first case in (34) happened), P1 would continue to be checked for PPZ constraint by the following model:
Step 5. Handle real power reserve constraint (10).
First of all, is determined by using (11) and (12) and then the solution will be checked and penalized if power output of all thermal generating units cannot satisfy RPR constraint. The penalty for violation of the constraint can be calculated by using equation (36).
As a result, real power reserve constraint can be solved by using the penalty method.
4.2. Calculate Fitness Function for Solutions
Fitness function of each solution is used to evaluate quality of solution. Normally, the function is the sum of objective function and penalty of violating constraints and is obtained by
4.3. The First Newly Updated Solutions by Lévy Flights Technique
In this section, the first newly updated solutions are performed by employing Lévy flights technique using Equation (14). However, each new solution can be out of their feasible operating zone such as PPZ and upper and lower limitations. When the power output violates its PPZ constraints, Equation (31) will be applied to tackle the constraint. Besides, the following equation will be employed when power output is higher or lower than their limitations.
After that, Equations (32)-(37) are performed for determining all variables and penalty terms. Finally, Equation (38) is employed to calculate fitness function.
4.4. The Second Newly Updated Solutions by Using Mutation Technique
The second newly updated solutions are accomplished as presented in Section 3 above. Similar to the task after doing the first update, each solution in the new population must satisfy PPZ constraint and upper and lower boundaries by considering Equations (31) and (38). Then, Equations (32)-(37) are performed for determining all variables and penalty terms. Finally, Equation (38) is employed to calculate fitness function and the solution with the best value is assigned to the best one, .
4.5. Criterion of Stopping the Loop Algorithm
In the loop algorithm of using the proposed ICSA approach, the solution search work is stopped in case that the predetermined maximum iterations is reached. For each search termination, the most effective solution is stored and another run continues to be accomplished until the predetermined number of runs is reached. After finishing the runs, the best one is found and reported. In addition, other values such as the fitness of the worst solution and average fitness of all solutions are also reported for comparing with other methods.
4.6. The Whole Iterative Process
The whole iterative algorithm for implementing the proposed ICSA approach for coping with OLD problem is described in detail in Figure 3.
5. Results and Discussions
The proposed ICSA approach performance has been investigated on six cases with different fuel options, different fuel characteristics, and complicated constraints. The details of the studied cases are presented as follows.
Case 1. Four systems with single fuel source (SFS) and power loss (PL) constraint Subcase 1.1: A 3-unit system [57] Subcase 1.2: A 6-unit system [57] Subcase 1.3: A 3-unit system [56] Subcase 1.4: A 6-unit system [56]
Case 2. A 110-unit system with SFS [57]
Case 3. Four systems with SFS and the effects of valve loading process (EoVLP) Subcase 3.1: A 3-unit system supplying to a load of 850 MW [58] Subcase 3.2.: A 13-unit system supplying to a load of 1,800 MW [1] Subcase 3.3: A 13-unit system supplying to a load of 2,520 MW [1] Subcase 3.4: A 40-unit system supplying to a load of 2,500 MW [1] Subcase 3.5: An 80-unit system supplying to a load of 4,100 MW [49]
Case 4. Two systems with SFS and PPZ and RPR constraints Subcase 4.1: A 60-unit system supplying to a 10,600 MW load [9] Subcase 4.2: A 90-unit system supplying to a 15,900 MW load [9]
Case 5. A 15-unit system with SFS and RRL, PPZ, and PL constraints [61]
Case 6. Three systems with multiple fuel sources (MFS) and EoVLP Subcase 6.1: An 80-unit system supplying to a 21,600 MW load [15] Subcase 6.2: A 160-unit system supplying to a 43,200 MW load [15] Subcase 6.3: A 320-unit system supplying to an 86,400 MW load [54]
For each considered case with each load case, the proposed ICSA approach is run 50 times on the program language of Matlab and a PC with 4 GB of RAM and 2.4 GHz processor. The selection of adjustment parameters including and is carefully considered to obtain the best optimal solutions meanwhile two others such as and are chosen corresponding to the scale of particular test system. 9 values with the change of 0.1 in the range are in turn selected for while is 0.01 at the beginning. The information including load demand, , , and the best is reported in Table 1.
5.1. Obtained Results on Case 1 considering Four Systems with SFS and PL Constraint
In this section, we have implemented the proposed ICSA approach for solving four systems divided into four subcases. Tables 2 and 3 show the comparisons of obtained results from Subcases 1.1 and 1.2 and Subcases 1.3 and 1.4, respectively. As listed in Table 2, the proposed ICSA method and CCSA can find equal fuel cost for Subcases 1.1 whereas the reduction of fuel cost from the proposed ICSA method as compared to CCSA is clearer for Subcase 1.2. As shown in Table 3 for comparing the proposed ICSA and three methods consisting of CCSA, ABC, and FA, the minimum fuel cost of the proposed ICSA is approximately equal to that of these methods for Subcases 1.3 but much less than that of these methods for Subcase 1.4. Furthermore, the proposed ICSA has been run by setting and to 5 and 20 but these values were much higher for CCSA, ABC, and FA. They are 20 and 5000 for CCSA, 40 and 100 for ABC, and 20 and 5000 for FA. Consequently, the proposed method is very efficient for ase 1 with four subcases.
Optimal solutions obtained by ICSA for ase 1 are shown in Tables 16–18.
5.2. Obtained Results on Case 2 considering 110-Unit System with SFS
In this section, we have employed a very large scale system with 110 units but there were not challenges for objective function and complex constraints since EoVLP and constraints were not taken into account. Both CCSA and the proposed ICSA methods have been run for comparing with BBO, hybrid BBO and DE (DE/BBO), and Oppositional real coded chemical reaction optimization algorithm (ORCCROA) in [36], IWA in [40], and AGWO in [52]. As shown in Table 4, AGWO [52] has reached less fuel cost than ICSA; however, the exact fuel cost, which was recalculated by using reported solution, pointed out that the method has reached a very high fuel cost of $215740.4250. For comparison with other methods, ICSA has found less fuel cost than all these methods. Particularly, the reduction of generation fuel cost is significant as compared to BBO, DE/BBO, and CCSA. Execution time comparisons are also useful evidence for indicating the high performance of ICSA. Thus, it can conclude that ICSA is a strong method for ase 2.
Optimal solution obtained by ICSA for the case is shown in Table 19.
5.3. Obtained Results on Case 3 considering Four Systems with SFS and EoVLP
In this section, the real performance of the proposed ICSA approach has been investigated based on five subcases with the gradual increase of number of units. The smallest scale system considers 3 units but the largest scale system takes 80 units. In addition to the implementation of the proposed ICSA, we have also implemented CCSA for Subcase 3.1 and Subcase 3.5 for further comparison because CCSA has not been run for the two subcases so far.
Comparison of obtained results from Subcase 3.1 shown in Table 5 indicates that the proposed ICSA is superior to CCSA with lower fuel cost but it seems to be less effective than MCSA [58]. The minimum cost of MCSA reported in [58] is the smallest fuel cost but the recalculated cost is much higher than that of the proposed method. Furthermore, MCSA has been implemented by setting very high values to and .
Reports for Subcases 3.2 and 3.3 shown in Table 6 are the comparisons of the proposed ICSA approach and other methods such as conventional Evolution programming (CEP) [1], Fast EP (FEP) [1], improved FEP (IFEP) [1], DE [12], multiplier Lagrange-based genetic algorithm with (GA-MU) [15], QPSO [16], GA-PS-SQP [30], PSO-SQP [32], Mβ-HCLSA [49], IABCA [50], CCSA [59], OSE-CSA [59], SOS [34], MSOS [34], CEA-SQT [38], TSBO [39], IWA [40], and CBA [44]. As observed from the table, ICSA approach obtains better solutions than most methods excluding DE [10], CCSA [59], OSE-CSA [59], SOS [34], MSOS [34], CEA-SQT [38], TSBO [39], IWA [40], and CBA [44], especially Mβ-HCLSA [49] with lower cost, $17,960.97. However recalculated cost from reported solution of Mβ-HCLSA is $17,969.1. Besides, ICSA is very fast as compared to most methods where two other versions of Cuckoo search algorithm, CCSA and OSE-CSA, are also included except two methods in [34]. The processor of computer that all the methods run on is also reported in the final column. Clearly, ICSA approach is very efficient for the case with the 13-unit system where effects of valve loading process are considered.
In Subcase 3.4, the number of units is much larger than that of three subcases above, up to 40 units [1]. The obtained result comparisons with others are indicated in Table 7. Clearly, the minimum cost comparisons reveal that the proposed method is one of the leading methods due to the lowest cost except the comparison with CCSA [23], OSE-CSA [59], SOS [34], MSOS [34], EMA [45], θ-MBA [47], and AGWOA [52]. It is noted that AGWOA [52] has reported the best minimum cost with $121404.30 but recalculated minimum cost, which was obtained by substituting reported optimal generation of all thermal generating units, is $121,413.31. The average and the maximum costs from the proposed method do not belong to the leading method group; however, the execution time that ICSA approach takes is smaller than most ones except IAPR [42] and θ-MBA [47], which have been run on stronger computers. Note that MSOS [34] has been faster than the proposed method for Subcases 3.1 and 3.2 above but it is too slower than the proposed method for the case, namely, 18.13 seconds compared to 1.46 seconds. Compared to two other versions of Cuckoo search, CCSA [23] and OSE-CSA [59], the proposed method is also faster about three times although the processors are slightly different. For this case, θ-MBA [47] shows a very good performance; however, the method has not been tested on more complicated systems and larger scale and therefore more comparisons with the method must end. Clearly, the proposed method is still efficient for the case where large scale and effects of valve loading process are included.
Subcase 3.5 is the largest scale system with 80 units. In addition to the implementation of ICSA, CCSA has been also run for the Subcase 3.5 for further investigation of efficiency improvement of the proposed ICSA approach. The comparisons of minimum cost in Table 8 show that the proposed ICSA can find more optimal solution than Mβ-HCLSA [49], AGWOA [52], and CCSA. The proposed ICSA is also superior to CCSA in terms of more stable search ability and lower fluctuation since its average cost and maximum cost are less than those of CCSA. The outstanding figure cannot be reached as compared to AGWOA [52]; however, it is hard to conclude AGWOA [52] is superior to the proposed ICSA approach about more stable search ability and lower fluctuation. Actually, comparison of the values of population and iterations as well as execution time cannot be accomplished because the information was not reported in [52]. Thus, it can conclude that the proposed ICSA is effective for the subcase.
Optimal solutions obtained by ICSA for the case are shown in Tables 20–23.
5.4. Obtained Results on Case 4 with Two Systems considering SFS, and PPZ, and RPR Constraints
In this section, two test systems with SFS and PPZ and RPR constraints are considered. The test system size is up to 60 and 90 units for Subcases 4.1 and 4.2, respectively. Comparison for the cases is only performed with two Genetic algorithms consisting of GA and IGA-MU in [9] and two other versions of Cuckoo search algorithms including CCSA [23] and OSE-CSA [59] and presented in Table 9. Other studies have tended to ignore such complicated constraints of PPZ and RPR. The costs reported in Table 9 indicate that ICSA approach can obtain more effective solution than CCSA and OSE-CSA because it has reached lower minimum cost than the two ones. Furthermore, the proposed method also takes shorter computation time for the two cases from about two times to about three times although the processor of the proposed method is slightly stronger. The mean costs of ICSA approach are much less than those from IGA-MU and GA and slightly higher than those from OSE-CSA but there is a trade-off between the proposed method and CCSA for the two subcases. In fact, the proposed method obtains higher mean cost for Subcase 4.1 but lower cost for Subcase 4.2. Although GA methods have been run on a weak computer with 0.7 GHz of the processor compared to that with 2.4 GHz in the study, their execution times are significantly higher, namely, 563.81 seconds (GA) and 162.58 seconds (IGA-MU) compared to 0.9153 seconds of ICSA approach for Subcase 4.1 and 940.93 seconds (GA) and 255.45 seconds (IGA-MU) compared to 1.5892 seconds (the proposed method). The analysis can point out that ICSA approach is more efficient than these compared methods in terms of optimal solutions and execution time.
Optimal solutions obtained by ICSA for Subcase 4.2 are shown in Table 24.
5.5. Obtained Results on Case 5 with a 15-Unit System considering SFS and RRL, PPZ, and PL Constraints
In this section, a 15-unit system considering RRL, PPZ, and PL constraints is considered to be solved for finding optimal solution. For efficiency investigation of the proposed ICSA, we have also implemented CCSA for comparison. As listed in Table 10, the proposed ICSA is the most effective method with the smallest fuel cost. The comparisons of control parameters as well as CPU time are also good evidence to confirm the strong search of the proposed ICSA approach since it has been run by smaller values of control parameter and faster execution time as compared to all methods excluding CCSA.
Optimal solution obtained by ICSA for the case is shown in Table 25.
5.6. Obtained Results on Case 6 with Three Systems considering MFS and EoVLP
In this section, three test systems with the challenge on objective function including multi-fossil fuel sources and effects of valve loading process are considered. The scale is up to 80 units, 160 units, and 320 units for Subcases 6.1, 6.2, and 6.3, respectively.
Comparison for Subcase 6.1 reported in Table 11 reveals that the proposed method is the best method in terms of the lowest best cost, the lowest mean cost, and the lowest standard deviation and the fastest execution time. The processor from this proposed method is about four times stronger than CGA-MU and IGA-MU but the speed is from ten times to 35 times faster than these methods. Compared to CCSA and OSE-CSA, the proposed method is about two times faster but the processor is slightly stronger.
Comparison for Subcase 6.2 is reported in Table 12. Clearly, the proposed ICSA approach obtains better values of the best, mean, and worst costs than most methods except MSOS [34] where the best cost difference is about $ 0.24. However, the proposed method is the second fastest one with 11.19 seconds where the first fastest one, CBA [44], has spent 5.71 seconds. Clearly, MSOS is better than the proposed ICSA approach with respect to slightly less best cost but is worse than the proposed method in terms of execution time, namely, 25.72 seconds compared to 11.19 seconds while CBA [44] is faster than the proposed method but obtains significantly worse costs. The analysis can conclude that the proposed ICSA approach is very powerful for the subcase with 160 units.
Table 13 presents the comparison of three other methods including CCSA [54], SOS [34], and MSOS [34] accompanied with the proposed method for Subcase 6.3. The obtained result comparisons imply that ICSA approach can obtain better values of the best and standard deviation costs than CCSA and SOS but obtains slightly higher cost than MSOS by approximately $ 0.22. Besides, the execution time from ICSA approach is much shorter than others; especially, it is higher than five times faster than MSOS. The four methods have been run on approximately strong computers. Briefly, the proposed ICSA approach can find and converge to more favorable solution than other methods with shorter CPU time except the comparison with MSOS, which had better solution but spent higher than five times execution times. Consequently, the proposed method is a very promising optimization algorithm for Subcase 6.3, a system up to 320 units and with multi-fossil fuel sources and effects of valve loading process.
Optimal solution obtained by ICSA for Subcase 6.3 is shown in Table 26.
5.7. The Improvement of ICSA Approach Performance
5.7.1. The Outstanding Improvement over CCSA
In this section, the performance improvement of ICSA over CCSA has been investigated by analyzing obtained results and set control parameters. Table 14 has been formed by adding reduction cost, improvement level of the best cost, execution time, and control parameters consisting of and . Among the compared factors, reduction cost is the deviation of the cost of CCSA and that of ICSA whereas the improvement level is the ratio of the reduction cost to the cost of CCSA. The reduction costs indicate the proposed method could find either equal quality of solutions or higher quality of solutions than CCSA for all study cases. The reduction cost is from $0 to $42.839 corresponding to the improvement level that is from 0 to 0.8968%. The saving cost is not too much for one hour but the operation in one day, one month, or one year is very high. However, it should be noted that CCSA has been run by setting much higher population size and iterations for many cases excluding study cases implemented in the study such as ase 2, Subcase 3.1, Subcase 3.5, and ase 5. For instance, ICSA has used = 10 and = 15 for Subcase 1.3, and = 10 and = 25 for Subcase 1.4 whereas CCSA has been run by setting = 20 and = 5000 for the two subcases. Similarly, CCSA has been run for Subcases 3.2, 3.3, and 3.4 with much higher number of iterations. For the last subcase, CCSA has been run by setting = 320 and = 1,200 but those of ICSA have been 10 and 9,000. Due to the higher value of control parameters, CCSA has tended to spend more time in finding such high quality solutions for almost all study cases. Execution time of ICSA is less than 18 seconds while that of CCSA is up to higher 75 seconds. It is clear that the proposed ICSA could find better optimal solutions than CCSA for such considered systems. For further investigation of performance comparison, the best runs over 50 runs and fuel cost values of 50 runs obtained by CCSA and the proposed ICSA for ase 2, Subcase 3.1, Subcase 3.5, and ase 5 have been plotted in from Figures 4–11. The best run curves show the faster search of the proposed ICSA method whereas 50 values of fuel cost indicate that the proposed ICSA can find many solutions with better quality. Clearly, the proposed ICSA is outstanding in terms of stabilization of solution search and faster convergence. As a result, it can conclude that the proposed ICSA approach is more effective than CCSA in solving OLD problem with considered systems.
5.7.2. The Improvement of Results over Other Methods
In this article, we have tested ICSA approach on 6 cases with 16 systems with different fuel cost forms, different constraints, and different scale systems from 3 units to 320 units. We have compared the yielded results from ICSA approach and other existing ones for evaluating the efficiency of ICSA approach. In subsections above, we have shown yielded results from ICSA approach and compared these results to those of other ones. However, the demonstration had not been very good for observing and comparing to lead to a conclusion. Thus, in the subsection we have summarized the result comparisons of the proposed and other ones. Table 15 has reported the reduction cost (in $) of ICSA approach compared to other ones. In addition, we have converted the reduction cost into improvement level (in %) for better comparison. The improvement has been shown from the lowest level to the highest level in terms of reduction cost and improvement percentage. In addition, we have also given the slowest and the fastest execution time of other compared methods together with that of the proposed method. The table implies that ICSA approach can find better optimal solutions with less fuel cost up to $0.52 for Subcase 1.3, $75.229 for Subcase 1.4, $17751.49 for ase 2, $118.1256 for Subcase 3.1, $84.38 for Subcase 3.2, $91.13 for Subcase 3.3, $1211.81 for Subcase 3.4, $34.12 for Subcase 3.5, $0.4329 for Subcase 4.1, $3.0227 for Subcase 4.2, $407.002 for ase 5, $1.8183 for Subcase 6.1, $139.65 for Subcase 6.2, and $118.17 for Subcase 6.3. These reduction costs are equivalent to improvement level (IL) of 0.0329%, 0.8968%, 8.2282%, 1.4143%, 0.47%, 0.38%, 0.99%, 0.01405%, 0.0003%, 0.002%, 1.2291%, 0.04%, 1.38%, and 0.59%. These quantitative comparisons reveal that larger scale systems can lead to better reduction cost but the improvement level is not high because total cost of compared methods tends to be large for large scale systems. Furthermore, very large scale systems with nondifferentiable objective have been normally solved by strong methods. In fact, systems in ase 3 have the same characteristic with single fuel and effects of valve loading process but Subcase 3.4 is a larger scale system with 40 units while Subcases 3.1, 3.2, and 3.3 are constructed by 3 units, 13 units, and 13 units. So, the improvement percentage of Subcase 3.4 can be up to 0.99% whilst that of Subcases 3.2 and 3.3 is 0.47% and 0.38%, respectively. Subcase 3.5 is with the largest system, 80 units, but the reduction cost is not much, only $34.12 because compared methods with ICSA are either state-of-the-art ones or improved ones. Also, Subcase 4.1 and Subcase 4.2 have considered single fuel and PPZ and spinning reserve constraints but Subcase 4.2 is larger scale system with 90 units and Subcase 4.1 is only with 60 units. Thus, the improvement of Subcase 4.1 is lower with 0.0003% but that of Subcase 4.2 is 0.002%. Similarly, systems in ases 6 have the same feature with multi-fossil fuel sources and effects of valve loading process but they are, respectively, constructed by 80, 160, and 320 units. As a result, the improvements of Subcase 6.2, 1.38%, and Subcase 6.3, 0.59%, are much higher than Subcase 6.1, 0.04%. However, Subcase 6.2 with smaller number of units but getting higher improvement is easily understood because there were nine compared methods but only three compared methods are considered for Subcase 6.3. In general, the improvement is not high; it is about under one dollar, several dollars, tens of dollars, and over one thousand dollars per hour; however, the saving cost will be significant if the operation is considered to be one month with 720 hours or one year with 8760 hours.
Execution time comparison can be evaluated by observing the fastest and the slowest compared methods in Table 15. These execution times of the proposed method are approximately equal to that of the fastest methods for Subcases 3.2, 3.3, and 3.4, and much shorter than other fastest compared methods for other cases, especially for Subcases 6.1 and 6.3. The fastest method for Subcase 6.2 is CBA [44] showing 5.7 seconds while that of the proposed method is 11.19 seconds. However, it cannot conclude that CBA is more effective than the proposed method because the proposed method could find better optimal solution with less fuel cost by $21.30. As considering execution time of the slowest compared methods, it can point out that ICSA is a very fast optimization tool since the execution time of these methods is 294.96 and 34.25 seconds for Subcases 3.2 and 3.3, 1167.35 seconds for Subcase 3.4, 563.81 seconds for Subcase 4.1, 940.93 seconds for Subcase 4.2, 1279.7 seconds for ase 5, 309.41 seconds for Subcase 6.1, 75.42 seconds for Subcase 6.2, and 96.41 seconds for Subcase 6.3 while the execution time of the proposed method for these cases is, respectively, 0.95, 1.46, 0.91, 1.5892, 4.6, 8.4828, 11.19, and 17.1384 seconds. It is clearly shown that ICSA is very fast as compared to these methods.
In summary, the proposed method has found approximately high quality solutions with several standard state-of-the-art meta-heuristic algorithms and improved versions of them together with other old methods. In addition, the proposed method could improve result better than approximately all methods with faster execution time. Compared to other methods with the fastest convergence speed and high quality solutions, the proposed method has been as fast as for some cases and much faster for other cases. The comparison with the slowest methods could show that the proposed method was extremely powerful since it was up to nearly one thousand times faster. Consequently, the proposed ICSA approach can be one of the strongest optimization tools for OLD problem.
6. Conclusions
This paper has proposed a good ICSA method for solving OLD problem in which many test systems with different objective functions and complicated constraints from simple to complex have been used as studied cases. The proposed ICSA method has been developed by performing several modifications on the second solution update of CCSA, which contained several drawbacks to global convergence and fast manner. The OLD problem has covered from single fuel to multi-fossil fuels, from quadratic objective function to nonconvex objective function in addition to PPZ, RPR, and RRL constraints. Many existing optimization algorithms have been concerned in aim to compare the performance and give the final conclusion on the proposed method. There have been six main cases with sixteen subcases. The evaluations have been made at the end of each study case. Clearly, the proposed ICSA approach has yielded more effective optimal solutions with faster execution time than almost all methods. Consequently, it can be concluded that the proposed method is much more superior to CCSA and is a very promising method for solving OLD problem.
Appendix
Nomenclature
| , , , , : | Fuel cost function coefficients of the thermal generation unit |
| , , : | Power loss matrix coefficients |
| , : | The values of fitness of solution and the so-far most effective solution among the current set of solutions |
| : | Number of fuels burnt in the thermal generation unit |
| : | Number of all available thermal generation units |
| : | Number of prohibited power zones of the thermal generation unit |
| , : | Upper and lower limits of the thermal generation unit corresponding to the kth PPZ |
| : | Probability of replacing control variables in each old solution |
| : | Real power demand of all loads in system |
| , : | The highest and lowest real power outputs of the ith thermal generation unit |
| , : | The highest and lowest real power outputs of the thermal generation unit corresponding to the fuel type |
| : | Maximum real power reserve contribution of the thermal generation unit |
| , , , : | Randomly mixed solutions from the set of current solutions |
| , : | The old solution x and the most effective solution |
| : | Real power reserve requirement of system |
| , , , , : | Fuel cost function coefficients of the thermal generation unit corresponding to the fuel type |
| , , , , , : | Random numbers between 0 and 1 |
| : | Number of generation units considering PPZ constraint. |
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflicts of interest.