In general, the DED problem reduces the overall manufacturing cost for committed units over the course of an operational horizon, which is obtained using Eq. 1 as follows (Behnam et al., 2012): min   f = ∑ t = 1 T ∑ m = 1 N G a m + b m P G m , t + C m P G m , t 2 , (1) where f indicates the total cost (TC) of all thermal units, P G m , t represents the real power of the m th unit at the t th hour, and N G indicates the number of generators.

Modern large power-producing units that use multi-valve steam turbines exhibit non-convexity in the input–output characteristic functions, demonstrating a substantial variance in the characteristic curves. Valve-point effects and mixing of various fuels cause the former to be non-convex. When each steam valve starts to open, ripples appear at the valve point. When valve-point effects are included, the ED cost objective function is often characterized as a superposition of a sinusoidal function and a quadratic function as shown in Eq. 2. In the current research, a non-convex function is used to evaluate the influence of the valve points (Behnam et al., 2012). min   f = ∑ t = 1 T ∑ m = 1 N G a m + b m P G m , t + C m P G m , t 2 + d m × ⁡ sin e m P G m , t min − P G m , t , (2) where a m , b m , c m , d m & e m refer to the cost characteristics of the m th generator and P G m , t min indicates the minimum values of the real power of the m th unit at the t th hour.

The cost function of wind generation is subject to the investment cost of the equipment and the operation and maintenance (O&M) cost of the generated energy (Behnam et al., 2012) and is given using Eq. 3 as follows: F P w p = a I p P w p + G E P w p , (3) where P _wP denotes the wind power generation (kW), a indicates the annuitization coefficient, and I _p and G ^E represent the investment costs and O&M cost per unit generated ($/kW), respectively. However, the investment cost of land will be considered as constant. Only the O&M cost is considered in the current work (Behnam et al., 2012). It is assumed that the investment costs and O&M cost are considered as $1,400 and 1.6 cents per kW, respectively. In the current research work, the wind generation data on a location in the east coast of the United States of America are considered. It gives constant wind power at each and every hour (Augustine et al., 2020).

The comprehensive cost function is obtained using Eq. 4 as follows: TC = m i n f + F P w p , (4) where TC indicates the total cost.

The constraints considered in the current work are described in brief below (Faisal et al., 2020).

Individual krill constantly travel in an n-dimensional search area with a specific velocity to attain high krill density, and their orientation is controlled by the local impact supplied by neighboring krill and the goal effect. The i th krill velocity is given as follows (Gandomi and Alavi, 2012): N i k + 1 = N max ∑ j = 1 N N F ^ i j X ^ i j + 2 r a n d + I / I max X ^ i , b e s t F ^ i , b e s t + ω n N i k , (8) where N i k + 1 & N i k indicate the motion induced by other krill to the i th krill in the k + 1 th and k th generations, respectively, N max denotes the maximum induced speed, F ^ i j indicates the normalized fitness difference between the i th & j th krill, X ^ i j indicates the normalized position variation between the i th & j th krill; F w o r s t , F b e s t are the worst and best fitness values, respectively, and N N denotes the number of neighbors. Different methods may be used to choose a neighbor. It is easy to calculate the number of krill that are within a certain distance of each other using the neighborhood ratio. The sensing distance (d_s) surrounding an individual krill, as given in Figure 1, should be computed using the actual behavior of the individual krill, and the neighbors should be located. The sensing distance is calculated as follows: d s i = 1 5 N ∑ k = 1 N X i − X k , (9) where d s , i denotes the sensing distance of the i th krill. N indicates the number of krill. If the sensing distance between two krill is less than a certain value, then the two krill are considered neighbors.

It is divided into two sections, one for the present iteration and the other for prior versions. A good food location is a mix of food attraction and the local best food placement that is utilized to attract individual krill to the global ideal solution. The foraging action of the krill is given as follows (Gandomi and Alavi, 2012): F i k + 1 = V f 2 1 − I / I max F ^ i , b e s t X ^ i , b e s t + F ^ i , f o o d X ^ i , f o o d + ω f F i k , (10) where F i k + 1 & F i k denote the foraging motion of the i th krill, V f indicates the foraging speed, ω f indicates the inertia weight, F ^ i , b e s t & X ^ i , b e s t are the normalized fitness and position variation between the i th and best individual krill, respectively, and F ^ i , f o o d & X ^ i , f o o d are the normalized fitness and position variation between the i th krill and the center of food, respectively.

The physical spread of individual krill is considered a random phenomenon. This movement can be characterized by utilizing a random directional vector and a maximum diffusion speed. It is possible to put it this way using Eq. 11 (Gandomi and Alavi, 2012) as follows: D i = D max δ . (11)

D i indicates the physical diffusion of the i th krill in the k th generation, D max denotes the maximum diffusion speed, and δ indicates a random directional vector.

When a krill approaches closer to the global optimum solution, less random orientation is required. To account for this, a new term is introduced to the physical diffusion formula. Foraging motion and motion caused by other individual krill have a diminishing influence with time (iterations). A random vector of physical diffusion is shown in Eq. 8 and does not steadily decrease with the increase in the number of iterations. This results in the addition of an additional term Eq. 9 to Eq. 8. Random speed is reduced linearly with time and is given below: D i = 1 − I / I max D max δ . (12)

Each krill’s position is revised as given below: X i t + Δ t = X i t + C t ∑ k = 1 C V U L k − L L k N i + F i + D i . (13)

Step 1:

KH variables and max generations (G _max ) are initialized.

Step 2:

The unknown variable of this ED problem is generator-active power output. In the KH, all these variables constitute the individual position of several chromosomes that indicate a complete solution set. The position of any chromosome, X_k , is represented using Eq. 14 as follows: X k = P g 1,1 , k m , P g 1,2 , k m , … … , P g 1 , t , k g , . (14)

The complete search space for population NP is expressed using Eq. 15 as follows: X k m = P g 1,1 , k m P g 1,2 , k m ⋯ P g 1 , t , k g P g 2,1 , k m P g 2,2 , k m ⋯ P g 2 , t , k m ⋮ ⋮ ⋱ ⋮ P g N g , 1 , k m P g N g , 2 , k m ⋯ P g N g , t , k m   k = 1,2 … N P g = 1,2 … G max . (15)

Step 3:

The fitness of each krill can be evaluated by applying Eq. 12. F = f + w P P G 1 − P G 1 lim 2 . (16)

Step 4:

The three movements mentioned in Eqs 8, 10, 12 are applied, and the newly created individual krill are updated using Eq. 13.

Step 5:

If the current limitations of a variable are exceeded, it will be set to an explicit low or high value.

Step 6:

If utmost generations have been reached, the process is stopped, and the optimum result from the previous generation is used as the best solution. Otherwise, we proceed to step 3.

The flowchart of the proposed algorithm is given in Figure 2.

The quality of the answer and the pace at which the algorithm converges are both heavily influenced by the choice of appropriate algorithmic parameters. In order to achieve the optimal population size for various tests, the minimal cost for various populations was computed. The best value for the population size achieved for all the test cases is 30. In addition, multiple trials have been conducted to identify an appropriate number of maximum iterations. The associated test cases indicate that the best value for the maximum number of iterations is 500. A total of 20 independent trials were conducted for each system to verify the reliability and efficacy of the proposed KH method. The KH method input variables used in this simulation research are considered from the study by Gandomi and Alavi (2012), which are follows: maximum induced speed N ^max = 0.01; foraging speed (V _f ) = 0.05; and maximum diffusion speed D ^max = 0.01. It is important to note that the inertia weights (w _n , w _f ) are initially set at 0.9 to highlight the capacity of the search process to explore the search space, but these values are gradually decreased to 0.1 at the conclusion of the search process to maximize the amount of space that can be explored.

It contains five generators, and the full data on the system are obtained from the study by Faisal et al. (2020), which consist of the cost coefficients of generators and their limits, PoZ limits, and load in every period. For a five-unit system, two test scenarios were considered. In scenario 1, only transmission loss (TL) was considered. In scenario 2, TL with PoZ was considered. In the above two scenarios, the best combination of real powers obtained with the proposed KH method is given in Table 1. Table 2 represents statistical values obtained for the two test scenarios with the KH method compared with the artificial immune system (AIS) (Hemamalini and Simon, 2011), bare-bones PSO (Zhang et al., 2014), imperialist competitive algorithm (ICA) (Mohammadi-Ivatloo et al., 2012a), enhanced adaptive PSO (Niknam and Golestaneh, 2012), time-varying acceleration-improved PSO TVA-IPSO (Mohammadi-Ivatloo et al., 2012b), immunity GA (IGA) (Mohammadi-Ivatloo et al., 2013), HIGA (Mohammadi-Ivatloo et al., 2013), memory-based global DE (MBGDE) (Zou et al., 2018), IWO (Zhi-xin et al., 2019), CMIWO (Zhi-xin et al., 2019), MILP-IPM (Granelli et al., 1989), and BBOSB (Pandi and Panigrahi, 2011). These results proved that the KH method achieved the best results in the two scenarios compared to other methods. The convergence curves obtained with the proposed KH algorithm are given in Figure 3 and Figure 4, respectively, and it is shown that the KH technique converged quickly to the optimal solution.

It contains six generators, and the whole data for this system are obtained from the study by Faisal et al. (2020), which include the cost characteristics of generators, generator limitations, and load demand in each period. Table 3 shows the optimum set of real powers found for this system using the KH algorithm, and a graphical illustration is given in Figure 5. Table 4 compares these findings with those of BA (Faisal et al., 2020) and DBA (Faisal et al., 2020), and these results show that the proposed technique is a better procedure to identify the best solutions to such complicated DED challenges, with the lowest, average, and highest costs, as shown in the table. Using the formulation of KH, a minimum value of 262,186.5 ($/day) was attained, demonstrating the extraordinary nature of the KH technique. In addition, the KH cost characteristics are given in Figure 6. This graph shows that the proposed KH converged very quickly, which indicates that it achieved optimum results in less time.

It this test case, the valve-point loading effect and generation limits were considered. The complete test data on this system are obtained from the study by Harish et al. (2016), which include the cost coefficients of thermal limits and their limits and the load in every period. Table 5 presents the optimum combination of real powers found for this system using the KH method. The statistical values achieved with the present method are compared with those obtained by the AIS (Zhang et al., 2014), BBPSO-DCS (Zhang et al., 2014), ICA (Mohammadi-Ivatloo et al., 2012a), TVAC-IPSO (Mohammadi-Ivatloo et al., 2012b), chaotic differential bee colony optimization (CDBCO) (Lu et al., 2014), crisscross optimization algorithm (CSO) (Meng et al., 2015), EAPSO (Niknam and Golestaneh, 2012), IGA (Mohammadi-Ivatloo et al., 2013), MBGDE (Zou et al., 2018), IWO (Zhi-xin et al., 2019), and CMIWO (Zhi-xin et al., 2019) and are given in Table 6. It is observed that the proposed method provided a superior strategy for identifying solutions. In addition, the KH cost characteristics are given in Figure 7, and it is shown that the proposed KH converged very quickly, which indicates that it achieved optimum results in less execution time (ET).

The efficiency of the KH algorithm is determined by examining 15-unit systems while addressing the DED issue with the inclusion of wind power. This system obtained complete data from the study by Hu et al. (2016). Table 7 shows the optimum set of real powers derived using the KH algorithm for this system in the first scenario (without wind power). In the second scenario involving renewables, the KH algorithm was employed in the same unit system incorporating one wind farm for a 24-h period. The best combination of real power values obtained in this case is given in Table 8. Table 9 presents the statistical results obtained for this system with and without the inclusion of wind power, compared with other methods published in the literature. This includes the lowest, average, and highest cost, along with the corresponding ET, and it is proven that the suggested technique produced better optimal results than the BA (Faisal et al., 2020) and DBA methods (Faisal et al., 2020). The cost characteristics of the KH technique are given in Figure 8. Overall, these results show that the KH algorithm is effective in the presence of renewable energy sources as well.

This research simulated the scheduling of a 30-unit system without taking into account TL and PoZs to demonstrate the dispatch performance of the KH technique on large-scale systems. The data for the 30-unit system are obtained by multiplying the information for a 10-unit system by 3, and the data of a 10-unit system are obtained from the study by Zhi-xin et al. (2019). Table 10 displays the most cost-effective dispatch solution produced by the KH algorithm and also displays the ideal solution of the KH algorithm, which confirms that the self-adaptive repair approach is effective since it fulfills the power demand balance, generating capacity restrictions. Table 11 compares the optimum dispatch results obtained by the KH algorithm with those obtained by other techniques from the literature. The suggested KH technique achieves the lowest generation costs compared to state-of-the-art approaches. The KH cost characteristics are given in Figure 9. The graph shows that the proposed KH converged very quickly, which shows that the KH technique achieved optimum results in less time. As a result, the suggested KH algorithm provides superior performance in terms of lowering generating costs during the dispatch of large-scale systems.

A well-known one-sample t-test (Mowafaq, 2022) is performed on all the test systems to evaluate the stability of the suggested KH technique for solving the DED issue. The significance level used in this test is 0.05. Statistical analysis provides us the data to keep or discard any hypothesis. The comparison of statistical data using significance analysis may be used to reject or retain any hypothesis. If the statistical value is larger than the threshold of significance, the hypothesis is accepted; otherwise, it is rejected. The results of the all-sample t-test are shown in Table 12 to demonstrate the validity of the null hypothesis.