A Method for Completing Economic Load Dispatch Using the Technique of Narrowing Down Area

Economic load dispatch solutions based on published methods, both conventional and artificial, have been very well-formulated through point-to-point movement methodologies to reach a convergence point. Iteration always starts from the starting point to obtain the following solution point, leading to the convergence point. This paper presents a new method to solve economic load dispatch problems by narrowing the minimum and maximum power limits between generator units. This idea approximates the solution point with a tiny space formed by the very narrow power limits of each generator. The methodology used is the distance between the minimum and maximum power limits of each generator divided into several segments. Then, the best segment is determined by the minimum total cost calculated based on the center point of the segment. Continue to the following iteration process until the best segment is the smallest. This iteration process is another artificial method that works without calculus calculations, so it does not depend on the objective function. This method has been validated using two generator units with differentiable objective functions, with calculation accuracy less than 0.00001 MW of the power distance of the generator limit, and the iteration stops at the $23^{\mathrm {rd}}$ step. Furthermore, this method has been successfully applied to the nondifferentiable objective function, piecewise and valve point effects.


I. INTRODUCTION
Economic dispatch problem (EDP) practices are always an interesting study. The EDP can generally be differentiated into differentiable EDPs and nondifferentiable EDPs. The objective function in the form of differentiable EDP is continuous and differentiable, generally quadratic. In contrast, the objective function in nondifferentiable EDP can be either noncontinuous or non-differentiable. Examples of nondifferentiable EDP objective functions are non-smooth functions, having prohibited operation zones and steps. The development of this objective function has prompted experts to study it. Conceptually, EDP is a simple optimization problem in a power system. All committed generator units are assumed to collect in one bus with a single load. The consequence The associate editor coordinating the review of this manuscript and approving it for publication was Bilal Alatas . of this concept is that transmission loss has been neglected. Conventionally, the ELD (economic load dispatch) problem is a differentiable problem, and the methods for solving it have been widely published. The conventional ELD problem has been defined by [1], which involves equivalence and inequality constraints. However, the actual operation of the power plant can cause undifferentiated ELD problems, such as a non-smooth objective function and the presence of POZ in the fuel cost curve [2]- [4]. This makes the EDP problem even more complicated because it changes its objective function. This method consists of an artificial network and a Lagrange multiplier. The methodology uses the artificial network to control the Lagrange parameter. However, it can repair the computation time, but information regarding its accuracy has not been discussed [2]. A solution of the non-convex optimization for ELD problems using the Fuzzy Technique studied by [3]. This technique has been successfully applied to the ELD problems, but the formulations are enough complex, and applying to POZ needs to advance study. Economic Dispatch (ED) problem solving by considering valve point effect (VPE), transmission loss, and restricted operating zone (POZ) has been studied by [3] using a full mixed-integer linear programming (FMILP) formulation. In this study, a reformulation trick is used, by converting the objective function into a set of traceable quadratic constraints [4].
Experts have published a few methods for solving EDP based on calculus methods (CMs) and artificial intelligent methods (AIMs). CMs are specially used for differentiable EDPs, as studied by [5]- [9]. The methodology developed by [5]- [8] is based on the LaGrange technique with its completion through iterative steps as in the lambda iteration, gradient, and Newton methods. In addition to CMs, there is also a direct method studied by [9]. This methodology is based on deriving mathematics to obtain formulas with straightforward solutions. The advantages are that the method can be applied to large systems and has the shortest computation time. However, the method is limited to objective functions in quadratic form only. The accuracy of the CMs is so high (the certainty of the solution falls on the global minimum point) that they are often used to validate the other methods (or AIMs). The optimal point is achieved by deriving the LaGrange function so that the CMs cannot resolve the nondifferentiable EDP.
Various AIMs have been published as in [10]- [31]. Based on artificial intelligence, two concepts have been developed, namely, point-to-point and area reduction. A point-topoint AIM is a method in the optimization process through a point-to-point approach to reach a solution point. These methods include the neural network method (NNM), particle swarm optimization (PSO), and genetic algorithm (GA). For AIM-based area reduction, the solution is obtained by reducing the viable area to a very small value.
NNM is based on the process of the human brain of working in parallel and exchanging information via the synaptic connectors on neurons. These neurons encapsulate all the information that comes to them, and if the result is higher than the given potential called an action potential, the neurons send pulses through the axons to the next stage. An artificial neural network consists of simple computing units (artificial neurons), and each unit is connected to other units via heavy connectors. Then, these units calculate the total input weight and know the output using the squashing function or activation function. The applications of this neural network have been studied by [10], while applications to EDP have been presented by [11] and [12]. Although NNM can be applied to complete EDP, a high-speed processor is needed because artificial nerves work in parallel. In addition, a large memory is required for data storage. In addition, there are no specific rules for determining an artificial neural network structure; a suitable network structure is achieved through experience and trial and error. Therefore, the speed and memory of computers will be a problem if NNM is implemented in large systems.
PSO is also overgrowing, as studied by [13]- [17]. The methodology starts from a randomly selected initial spot (initial point); continues to scatter many candidates around it at a certain distance (step-length); then, the best candidate is selected as the next spot until a solution point is obtained. This method will not be accurate if the local minimum is close to the initial point chosen because the method can ascertain that the solution point falls on that local minimum. Accuracy is also dependent on the number of candidates; the optimal number of candidates is still uncertain. Parameter control is required in each iteration step, and the processed dataset is vast, which will influence the computation time when applied to large systems.
GA is a metaheuristic based on the process of natural selection. The applications for generating high-quality solutions in optimization and search problems rely on biological operators such as mutation, crossover, and selective operators. The method has been applied to solve EDP, as studied by [18]- [25]. Similar to PSO, GA is no guarantee of finding global maxima. A decent-sized population and many generations must be processed so that it is burdensome to work on computers to obtain solutions over a long time, especially for large systems. GA also needs fine-tuning all parameters in determining mutation, which is often just trial and error.
The other methods stress the complexity of the optimization problem presented by [27]- [31]. The model with the conditional value at risk recourse function [27] has solved the robust dynamic economic dispatch distribution problem. Inspiration by the moth flame moving toward the moon was developed by [28]. Optimizing an electric power system is very complex, so a heuristic method is needed to solve it. The basic heuristic optimization techniques for application to power systems have been presented by [29]. The complicated methods used to solve optimal power flow were studied by [30] and [31]. The Newton-Rapson method determines power flows based on PSO, which has been satisfactorily developed [26]. These methods are also artificial; they will face problems if applied to large systems. The other method is based on calculus, the interior point, presented by [31]. The interior algorithm has been developed through predictorcorrector and reduction techniques to work very fast and to be applicable for large systems. Nevertheless, the method cannot be applied to nondifferentiable optimizations. All methods mentioned above use a point-to-point optimization process to reach a convergence solution. The following are published based on reducing feasible areas developed by [32]- [35]. This method is commonly used to detect two-dimensional image objects in telecommunication, as studied by [32]- [34], with a coarse-to-fine technique. The feasibility area is divided into grids, and each grid is tracked to determine the best grid. Furthermore, this best grid reduces until a tiny grid is obtained, representing a convergent solution. The development of this multidimensional method was presented by [35]. The purpose of this development is to complete the EDP. The multidimensional feasibility area is formed by the minimum and maximum power limits of VOLUME 10, 2022 the generator units. The feasibility area is divided into several areas with the same dimensions. Each area represents a candidate with a coordinate point at the center. In addition to determining the best candidate, the candidate must also meet the power balance between total power generation and load. Fig. 1 shows a difference illustration between the point-topoint and reducing area methods. Based on the point-to-point method in Fig. 1a, the iteration step starts from the initial point (x 0 ), and through iteration steps, the solution points will lead to the convergence point (x op ). As the reducing area in Fig. 1b shows, the iteration step starts from the feasibility area and reduces to the best area-1 and best area-2. This method will successfully yield when the optimal points are in their respective best areas. This paper develops a new method, called the technique of narrowing down area (ToNDA), to work for all kinds of objective functions (differentiable or not). The methodology is to narrow the power distance of the generator limits (PDGL) of each generator, expressed in (1), until it approaches the solution point, P op , as illustrated in Fig. 2.
where X i is the PDGL of Generator i.P i andP i are the maximum and minimum power limits of Generator-i. Meanwhile, feasible areas are formed by PDGL from all generators. In each iteration, the PDGL is divided into segments, and each segment boundary is represented as a solution point. Once the best candidate is obtained, the PDGL is narrowed to reduce that candidate. The iteration process continues to update the PDGL (after being narrowed down from the previous one) until the smallest segments are reached. The smallest space is formed in a feasible area from a very narrow PDGL. where the smallest space can be stated as a solution point (convergence point).
While the narrowing process can be explained in Fig. 3, if the PDGL of each generator is divided into k−1 segments, k potential solution points (P 1 , P 2 , . . . , P k−1 , P k ) of each generator are obtained, where the first point (P 1 ) is atP, and the endpoint (P k ) is atP. For n generators, the candidate is a vector of dimension n, where the elements are the potential solution points of each generator and are expressed in Equation (2).
where cd i is as a candidate and P i 1 is, The candidates formed must meet the balance of power, namely: where P D is total load. Then the best candidate (Cd b ) is selected when the total cost is least compared to the others. Area reduction is carried out by establishing a new generator power limit involving the left and right segments. However, the new minimum or maximum power limit should not violate the initial minimum or maximum power limit. After obtaining Cd b or point (P j+2 ) on the generator, as shown in Fig. 3, which is marked with x, the distance of the power limit is narrowed by involving the left and right segments. Narrowing the distance of the power limit aims to anticipate the possibility of the convergence point outside the new distance of generator limits, such as the convergent point (P op ) seen in Fig. 3. In this case, the convergence point is improbable to be outside the new distance of the generator limits. Next, the generator limits are updated through Equations (5) and (6).
whereP new andP new are the new generator limits,P old and P old are the old generator limits. Ra is a quantity to determine the size of the area reduction. The smaller the Ra value is, the larger the reduced area. For a very large Ra it will allow the solution point to be outside the minimized area, so it is not accurate. This inaccuracy is shown in Table 2 with a Vra value of more than 80%. It is certain that a small Ra value (≤50%) in the table can guarantee accuracy.
where TC is the total cost, P i is the active power generator, and F i is the function of fuel cost to generator output power and is expressed by (10).
Then, S 0 is defined as the initial space (initial feasible area) formed by generator limits from n generator units so that this space will have n dimensions, as written in (11).
The initial PDGL of the generator is defined in (1) for generator i expressed by (12).
If X 0 i is divided into k−1 segments, the number of k points will be obtained as the potential solutions that are spread evenly between the minimum and maximum limits. Wherê P 0 i andP 0 i are the maximum and minimum initial power limits. Based on the description, an initial segment length of generator i (Ls 0 i ) is written in (13) is obtained.
For the kth iteration step, equation (14) becomes, where P k,i 1 is the power generator 1, it is in segment i at the k iteration step which satisfies the power balance in equation (4) or (14).
This last equation ensures that the equality constraint in (9) is consistently met. In this case, the segments of an n th generator are not the same length because the value of P k,m n must meet (14). From the explanation above, some candidates will be obtained, namely, J c and expressed in (15).
Suppose Cd 0 1 is a potential candidate. If Generator 1 to Generator n−1 is set with their solution potential points at the minimum power limit, the solution potential point for Generator n is: So, Cd 0 1 is: where Cd 0 1 ∈ S 0 . Of all the candidates distributed to S 0 , then the best candidate is chosen with a minimum value of total cost (TC k b ) and it falls on Cb k b , expressed in (19). All the candidates are distributed to the appropriate areas.
while Cb k b can be determined by (19).
Next, S 0 can be reduced to S 1 (a new space), where S 1 ∈ S 0 . S 0 is obtained by narrowing the limits of generators and is formulated in (21) and (22).
whereP k i andP k i are the minimum and maximum power limits of the iteration k for Generator i. The greater the value of µ is, the slower the narrowing of the generator power limit, where: From (12), (22), and (23), the value of X 0 can be determined, namely: So, the area reduction speed (Vra) is: The process of narrowing the generator limit continues until it becomes tiny, namely, X small . For example, in the k th step, the space of the feasible area, S k , can already be considered to represent the solution point, where the value of X small is calculated by (26).
where ε is a very small number as the convergence. Algorithm steps can explain detailed procedures of ToNDA through a simple example of EDP consisting of two generator units. VOLUME 10, 2022 Total cost in (27), power of the generator limits in (28) and (29), and equality constraint in (30).
10 ≤ P 2 ≤ 50 (29) The first step is to determine the desired number of segments (for example, four segments or k = 5 and µ = 1), obtaining 16 rectangles and 25 points as much as possible, as shown in Fig. 4. Two horizontal axes present the generator power of P 1 (first generator) and P 2 (second generator). In contrast, the vertical axis presents the total cost. The ABCD rectangle (area 2500 MW 2 ) is S 0 from the initial minimum and maximum power limits of the generators. The PB-line is to be formed by equality constraint stating as power balance, and all candidates must lie on this line. From Fig. 4, there are five potential candidates of 25 points in space S 0 (PB line), i.e., Cd 1 , Cd 2 , Cd 3 , Cd 4 , and Cd 5 . From the potential candidates are chosen the best candidate that has a minimum total cost, where the minimum cost falls on the candidate Cd 4 (said to be the first best candidate, BC 1 ). Then, the PDGLs are narrowed based on (21) and (22) to be A 1 B 1 C 1 D 1 , as shown in Fig. 4 and Fig. 5. With the same procedures, five new candidates are obtained, and the best candidate is BC 2 and the new area (A 2 B 2 C 2 D 2 ), where the new area is less than the old area (A 1 B 1 C 1 D 1 ). The results of all steps are loaded in Table 1. The total cost going down along with narrowing feasible area from The iteration will stop after reaching the convergence point with the largest PDGL value of less than 0.00001 MW, namely, 50 MW from Generator-1. In this simulation, convergence is reached in the 23 rd step with an accuracy of 5 digits of decimal number accuracy and a total cost of $5445. 83333. Accurate solution points fall in P 1 = 7166762 ± 0.0000060 MW and P 2 = 38.33238 ± 0.0000048 MW   When comparing the ToNDA method and the calculus method for 5 digits of decimal number accuracy, the same results were obtained, namely, P b 1 = 71.66667 MW, P b 2 = 38.333343 MW, and TC = $5445.83333.
Vra affects the number of iterations, as shown in Table 2, where the larger Vra is, the fewer iteration steps. However, an area reduction of more than 80% causes the calculation to be inaccurate. From Table 2, the smaller area reduction will guarantee the accuracy, but the iteration step required is getting larger. For example, a Vra value of 5% requires 301 iteration steps. This paper suggests the Vra value of 50% as a reconciliation so that the accuracy of the calculation results is guaranteed.

III. ALGORITHM AND FLOWCHART OF ToNDA
The ToNDA described above can be applied to the various objective functions, differentiable and nondifferentiable. The procedure steps of solving EDP can be explained in detail by the flowchart in Fig. 6. The calculation process starts with determining the segment distance of the limits of each power generator. The generator range (between the max limit and the min limit) is divided into several segments/areas. Each segment is represented by a point located in the middle of the area to determine the best area indicated with the lowest cost. The, a reduced area is formed involving the best area and the neighboring area.Then, the process is continued by choosing potential candidates who had to meet equality and inequality constraints and then determining the best candidate with the minimum total cost. Finally, the power limits of generators are narrowed based on the point of the best candidate, where the position of the point is the center. Convergence will be reached if the maximum distance of the power limits of the generator is smallest and satisfies (24). In this paper, the best very small area that is considered to represent the point of convergence is the area with a size of 0.012accuracy of 0.001.
The algorithm of the ToNDA method is written step by step as: 1. Start 2. Input data (number of generators, cost characteristics, generator power limit) 3. Set the maximum and minimum power limits of each generator. 4. Calculate the PDGL for each generator using (12). 5. Set the value of k 6. The PDGL is divided into k−1 to obtain segment; for example, k = 5 to obtain 4 segment 7. Calculate length segment through (13) 8. Determine k potential solution points that fall on the segment boundaries in the PDGL for each generator. 9. Determine candidates that meet balance of power in equation (4) or (14). Candidates for each generator are determined from the smallest k sequence starting from the first generator to the (n−1) generator. Candidates for the nth generator are determined later through equation (15). 10. Determine the best candidate with the lowest total cost of generation from all candidates. 11. Set the Vra (recommended 50%). 12. Create a new PDGL with the best candidate as the center involving neighboring segments. 13. If the minimum limit obtained is less than the initial minimum limit, then the minimum limit is set equal to the initial minimum. 14. If the maximum limit obtained is greater than the initial maximum limit, then the maximum limit is set equal to the initial maximum limit. 15. If the largest PDGL has been reduced to be the smallest value (for example, 0.00001 MW), then the convergence point is reached, and go to Step 16. Otherwise, go back to Step 6. 16. Finish

IV. SIMULATION AND ANALYSIS
The ToNDA work performance and this method were tested with the EDP with a fuel cost model in the form of a piecewise quadratic cost function and valve point effect. Simulation for VOLUME 10, 2022 EDP with a piecewise quadratic cost function uses a 10-unit system. In comparison, simulation for VPE uses two size systems, a 13-unit system, and a 40-unit system.

A. EDP WITH PIECEWISE QUADRATIC FUEL COST FUNCTION
The mathematical model of the fuel cost function is in piecewise quadratic forms depicted in Fig. 7 and formulated in (31). The case study uses a 10-unit system with three fuel types obtained from [36]. The ToNDA successfully simulated the case for three load demands of 2500 MW, 2600 MW, and 2700 MW, and the detailed results are presented in Table 3.
The simulation uses the Laptop Asus core i3 using the Fortran Programming Language. For an error rate of 0.0001 MW, the iteration stops at the 19 th step with a CPU time of 3.74 seconds. Calculation results of the EDP for the other methods are obtained from [36], namely, Sequential Approach with Matrix Framework (SAMF), Hierarchical Method (HM),   Hopfield Neural Network (HNN), Adaptive Hopfield Neural Network (AHNN), and Hybrid Genetic Algorithm (HGA). A detailed comparison of the results is presented in Table 4. Table 4 shows that the reference methods give almost no different results for load demands of 2500 MW, 2600 MW, or 2700 MW. The ToNDA method provides better results than the results of the reference methods. When referring to the SAMF results, the ToNDA method provides a lower total cost with an average fuel savings of 9.06 $/h or 1.55% (detailed in Table 5).
Fuel savings will increase when the load demand is greater; the percentage of fuel savings increases from 1.12% (load demand of 2500 MW) to 1.80% (load demand of 2700 MW). This fuel savings shows that the ToNDA method is always accurate even though the load demand is increasing.
Based on Table 4, the proposed method has a performance improvement of 10.97 $/h or 1.7586% against the best comparison method (PSO or HGA), which costs 623.81. EDP simulation with VPE uses the IEEE testing system. The simulations carried out are for two systems with different sizes, namely, a system of 13 units and 40 units [37] with a demand load of 1800 MW and 10500 MW, respectively. Data of the 13-unit system and ToNDA results are presented in Table 6. The optimal power of generators (P opt ) is not out of power limits (P andP). Under this optimal condition, the VPE is 13.9829 $/h caused by the second unit of 12.3762 $/h. The optimal condition, fuel, and VPE costs must be minimized. The coefficient e is so large that the cost of VPE is minimized as low as possible. Therefore, the high fuel units will be optimal at the minimum power limit, the VPE cost is zero (5, 6, and 7 units) or above the minimum power limit, and the VPE cost is close to zero (4 units). The lowest fuel units will be optimal around the maximum limit, and the VPE cost is close to zero (unit 1).
Balance power between generating and load demand is shown by the IPSO-TVAC, HCRO-DE, DSD, and ToNDA methods. There are four different solution points shown in Table 7, namely, the results of QPSO (SSA, IPSO-TVAC, HCRO-DE, and DSD) and (HGA and FAPSO-VDE) ToNDA. The result of the SSA method is smallest compared to the IPSO-TVAC, HCRO-DE, and DSD methods, but the power is not balanced. The DSD method shows the best results (total cost of $17963. 29     is almost equal to the others, except for the QPSO method (18398.848 $/h). For quadratic and the results of various fuel types in Table 4, the proposed method is competitive, which gives 1.7586% better results than the best comparison method (PSO or HGA), which offers a cost of 623.81 $/h). However, for the case of VPE (in Table 7), the proposed method differs only in the results of 0.05072% worse than the best method (IPSO-TVAC)

2) THE LARGE UNIT SYSTEM
The test results for a 40-unit system with VPE are presented in Table 8. The lowest fuel units fall in the minimum power limit (units 10, 11, 12, 27, 28, and 29). The optimal power (P opt ) does not violate the power limits. The total cost of the ToNDA method (121412.87 $/h) is slightly larger than the others, as shown in Table 9.
In Tables 8 and Table 9, the proposed method is superior to QPSO and HGA, while when compared to DSD, there is a difference of 0.00028%. When the proposed method is compared with the method from the latest reference [4] which uses full mixed-integer linear programming (FMILP) formula, the results are also not significantly different. Figure 8 shows the execution times for various numbers of generators. From the time, it looks quite feasible to apply to the generator schedule for the next 1 hour, where generally the generator schedule is 1 hour ahead.

V. CONCLUSION
A new paradigm based on narrowing the area in solving any EDP problem has been described in this paper through the ToNDA method. The methodology developed narrows the feasible area at each iteration step to obtain a tiny feasible area (as the solution point). The feasible area is evenly traced to ensure very high accuracy. The advantage is that it does not depend on the forms of the objective function. All economic load dispatch problems can be solved satisfactorily, regardless of whether the economic load dispatch problems are differentiable.
The simulation of two generators (differentiable EDP) shows that the results are close to the results of the calculus method in step 14, with a limit range of 0.09766 MW or a feasible area size of 0.0000596 MW 2 (it is a tiny area that can be considered a convergence point).
The ToNDA method successfully applies to the nondifferentiable EDP, piecewise fuel cost, and fuel cost with the valve point effect. From the simulation results of EDP with a piecewise quadratic fuel cost function for ten units, the ToNDA can save 9.06 $/h of fuel. or 1.55% compared to the SAMF, HM, HNN, AHNN, and HGA) methods.
While the EDP simulation results with VPE are 13 units, the proposed method shows almost the same results as other methods, although the solution points are different. This method always balances the power generated and the load demand. This power balance must be maintained because a slightly unbalanced power can reduce fuel costs due to VPE.
The ToNDA method has also been successfully applied to large systems (40 generator units) in the case of the valve point effect. Moreover, this method still consistently shows optimal results compared to the other eight methods.