The characteristic of coal consumption refers to the curve of coal consumption of a thermal power unit changing with the load. It is the critical basis for analyzing energy consumption and load optimization scheduling of a thermal power plant. When a load of a single unit decreases with the generation condition, its coal consumption rate will increase, and the formula is as follows: F P g x = a x + b x ∗ P g x + c x ∗ P g x 2 (1)

In Eq. 1, F P g x represents the coal consumption of the thermal power unit; a x , b x , c x represents the coal consumption characteristic parameters of the x unit; and P g x represents the x unit’s power.

The influence of the valve-point effect on UC should be considered in the unit operation process. The leakage of steam causes the valve-point effect at the opening moment of the regulating valve of the steam turbine, which is reflected as the pulsation influence at high load in the coal consumption characteristic curve of the unit. The formula is as follows: G P x = d x ∗ sin e x ∗ P min x − P x (2)

In Eq. 2, d x , and e x represent the valve-point coefficient P min x represents the lowest power value of the x unit.

The mathematical model of the objective function can be expressed as the compound superposition function in summary. In the model, the quadratic function and the sine function of the valve-point effect are set for solving the minimum coal consumption characteristic. It can be expressed as follows: min ⁡ f P x = ∑ x = 1 N F P x + G P x (3)

Capacity constraint function is the prerequisite for the standard and safe operation of the thermal power unit. Its formula is as follows: P min x ≤ P x ≤ P max x (4) Where P max x denotes the upper limit of capacity constraint and P min x the lower limit of capacity constraint. This paper ignores power flow loss and assumes that only thermal power units participate in power generation in the network. Load balance constraint means that the sum of the power of each unit needs to be consistent with the total load, and its formula is as follows: ∑ x = 1 N P x = L o a d (5)

In Eq. 5, L o a d represents the total load of the system.

The purpose of adding a penalty function is to consider some other constraints or ignored losses, and its formula is as follows: h P x = ε ∗ ∑ x = 1 N P x − L o a d (6)

In Eq. 6, ε represents the penalty coefficient which can be fixed or changed to an adaptive value according to the characteristics of the algorithm. Thus, the mathematical model of ELD can be expressed as follows: min ⁡ F = f P x + h P x (7)

Chemotaxis is to simulate the motion part of E. coli foraging behavior. The process includes forward and reverses in two parts. E. coli runs along the vector direction with a random vector until the fitness value cannot continue to be smaller. In (Long et al., 2020), the cost function of the A∗ algorithm is used to improve the chemotaxis process to solve the path planning problem under different working conditions effectively. In the process of chemotaxis, the step length C is an essential factor affecting the movement process of E. coli. The step length C of chemotaxis is a fixed value in the traditional BFO algorithm. It will bring some disadvantages. The minor C value can improve the search accuracy, but it will reduce the search efficiency of the algorithm and easy to fall into the local optimal. The larger value of C can improve the search speed of the algorithm, but it will reduce the accuracy of the search results and lead to search misjudgment. Therefore, the value of step size dramatically affects the excellence of the algorithm. Inspired by the fish swarm algorithm, (Yufang and Jianwen 2021) uses an exponential function to modify the step size. Adaptive modified step size is used to replace the traditional fixed value in this paper. C x = exp − N c ∗ N r e ∗ N e d − τ j + k − 1 ∗ N c l − 1 ∗ N r e ∗ N e d 1 α ∗ C (8)

In Eq. 8, N c , N r e , N e d respectively represent the number of chemotactic restrictions, replication restrictions, and dispersal restrictions in BFO algorithm. α denoted as the step coefficient. j, k and l represent the current times of chemotaxis, replication and dispersion respectively. τ is a dynamic change. The BFO algorithm can search for optimization with giant steps in the early stage of the search to accelerate the algorithm’s convergence through the adaptive step correction exponential formula. And search for optimization with a small step in the late stage of search to improve the accuracy of the algorithm.

The replication process is a process that simulates biological evolution and survival of the fittest. The E. coli are arranged in ascending order according to the cumulative fitness value. The first half of the high-quality E. coli is copied instead of the second half of the poor E. coli. The total number is unchanged in this process. Although this method reduces the algorithm’s complexity, it also brings some disadvantages. The diversity of the population is greatly decreased to ensure the diversity of the population and ensure that high-quality bacteria individuals are not lost. (Jufeng et al., 2020) used single individual ranking and crossover operations to replace the cumulative health ranking method. In this paper, the Crisscross Algorithm (Xiongmin et al., 2022), (Shaowei et al., 2021) and (Anbo et al., 2022) is a novel population random search algorithm that is proposed to improve the replication process. The Crisscross Algorithm includes two parts, horizontal crisscross, and longitudinal crisscross. Compared with the previous generation, the method achieves the optimal effect through each iteration of the crisscross process. Horizontal crossover is like the crossover process in GA, but it also has a comparison process with the previous generation. M S h c x , d = r 1 ∗ X x , d + 1 − r 1 ∗ X y , d + c 1 ∗ X x , d − X y , d (9) M S h c y , d = r 1 ∗ X y , d + 1 − r 1 ∗ X x , d + c 1 ∗ X y , d − X x , d (10)

The parameters x and y in Eqs. 9, 10 represent individual bacteria, d represents dimension solved by the algorithm, and r 1 c 1 both represent random numbers. The former is between 0 and 1, and the latter is between −1 and 1. This formula represents the offspring of bacteria x and y after horizontal crossing in the d dimension. Longitudinal crossover is similar to the mutation process in a genetic algorithm, and longitudinal crossover is the crossover of different dimensions of the same bacterium. After each crossover, a progeny with different dimensions from the previous generation is produced. The progeny produced each time should be compared with the previous generation to retain the optimal value. M S v c x , d 1 = r ∗ X x , d 1 + 1 − r ∗ X x , d 2 (11)

According to Eq. 11, bacteria x can produce a progeny by crossing dimensions d 1 d 2 . The crisscross algorithm was used to cross-optimize the chemotactic population. Compared with the traditional sequencing and replication method, the optimization and replication process of the crisscross algorithm not only retained high-quality bacterial individuals but also ensured the diversity of the population.

The random dispersal optimization of E. coli individuals was carried out according to a fixed dispersal probability. In this process, certain high-quality individuals were also dispersed to random areas. Although the global search performance of the algorithm was ensured in principle, the fitness value of the algorithm would also deteriorate, which would decrease the efficiency of the algorithm. In this paper, we use the adaptive dispersal probability instead of the traditional fixed value to avoid falling into the local optimum and ensure the global search performance of the algorithm. P x = P e d ∗ J w o r s t − J x J w o r s t − J b e s t (12)

In Eq. 12, J w o r s t represents the worst fitness value, J b e s t represents the optimal fitness, and J x is the real-time fitness value of the xth bacterium. The dispersal probability of E. coli was modified adaptively by this fraction. The fitness value of bacteria individuals with good fitness was small, and the dispersal probability was reduced while the dispersal probability of individuals with poor fitness was increased. In this way, the loss of high-quality individuals is avoided and the efficiency and performance of the algorithm are guaranteed.

The parameters need to be initialized first in the algorithm. It includes the number of iterations maxgen, the dimension p of the search range, the number of bacteria s, the maximum number of chemotaxis N c , the maximum number of steps of one-way movement in chemotaxis operation N s , the maximum number of replication N r e , the maximum number of dispersal N e d , and the fixed probability of bacterial dispersal P e d , the number of attractive factors and the release speed d_attract, ommiga_attract, the number of repellant factors and the release speed h_repellant, ommiga_repellant. Then the population is initialized. The population of this algorithm is generated according to the lower limit of unit load plus the difference between the upper and lower limits of unit load multiplied by a random number. Initialize the population as a high-dimensional array. After the population initialization is completed, the cycle is carried out. The maximum number of dispersals, replication, and chemotaxis, are determined first, and the chemotaxis operation is carried out after the requirements are met. In the chemotaxis process in this paper, N c is 60; N s is 4. After determining the adaptive correction step and the repulsive attraction between the bacteria, the fitness value was calculated. The bacteria turn over and then proceed with dynamic steps in the direction of the randomly generated vector until the maximum swimming limit is reached or the fitness value is updated to the optimal value. During this period, the bacterial constraint should be considered, namely the upper and lower limits of the unit output constraint. After the chemotactic process, the fitness value is updated to enter the replication stage. During replication, bacteria are sorted according to their cumulative fitness values, and new populations are formed in ascending order. Replicate 2 times in total. The longitudinal and horizontal crossover operators were used to update the fitness value, retain the perfect result and eliminate the wrong result. Finally, the bacteria were dispersed according to the adaptive dispersal probability when it came to the dispersal stage. The dispersal probability of the perfect result was tiny, while the dispersal probability of the impaired result was extensive. After the dispersal, the bacteria died out, and the new bacteria re-determined the random position. The algorithm ends when the maximum number of dispels reaches 4, and the algorithm iterates 480 times in total. The flow chart of bacterial foraging algorithm is shown in Figure 1.

In the process of solving the ELD problem by ICSBFO, the control variable is the initial population of E. coli, and the dependent variable is the coal consumption of the unit. Use the algorithm to solve Eq. 7, first, it is necessary to tune the relevant parameters of the ICSBFO algorithm in solving the ELD problem. The initial population of E. coli is generated with the upper and lower limits of the power load, and the number of units is set as the solution dimension. Then we need to determine the load and solve the ELD model. The fitness value in the solving process is the objective function proposed above after considering the penalty coefficient. If there is no new load command or percentage load requirement, each unit’s optimal load distribution and optimal coal consumption can be output.

In Case 1, taking 10 units in a power plant (Basu 2016) as an example, the coal consumption characteristic parameters of the unit and the upper and lower limits of the unit load are shown in the following Table 1:

In the case of a 10-unit medium-sized power plant, this paper uses the improved crisscross hybrid bacterial foraging algorithm (ICSBFO), the crisscross algorithm (CSO) (Meng et al., 2015), the bacterial foraging algorithm (BFO), and the particle swarm algorithm (PSO) (Hatata and Hafez 2019) to solve the 10-unit load optimization distribution model of a medium-sized power plant. Since different unit parameters and valve point effect parameters greatly influence the results of unit load economic dispatch, this paper uses the methods proposed by them in their respective articles to simulate the unit parameters of the same case. In the process of PSO testing, a mutation strategy is introduced to improve the problem that PSO is easy to fall into the local optimum. Since the test algorithms all contain a random search mechanism, each group is tested 30 times in this experiment, and the experimental results are taken as the average value of the tests. During the test, the population size was 50, the dimension was 10, the load was 2,700 MW, and the total number of iterations of various algorithms is 480.

In the process of dealing with ELD problems, PSO is widely used because of its fast convergence speed, but its disadvantage is poor robustness. CSO shows good convergence efficiency and accuracy in the ELD process due to its independent update iterations in the horizontal and vertical directions, and a simplified algorithm process characterizes it. It can be seen from Figure 2 that the improved bacterial foraging algorithm has faster convergence speed and convergence results than the previous two, which is due to the improvement of the convergence speed by the adaptive correction step size and the mixing in the replication process. Compared with traditional BFO, improvement optimization is more significant. In taking the average value of multiple experiments, we also found that ICSBFO has better robustness and the smallest variance of its optimal coal consumption. Research on the high-quality robustness of ICSBFO is expected to be carried out in the future.

It can be seen from the above Table 2 that the optimal coal consumption of ICSBFO is the smallest. Compared with the CSO, the coal consumption reduced by the ICSBFO algorithm is 4.6317 g · k W − 1 · h − 1 . The percentage reduction is 0.707%. Compared with the PSO, the coal consumption optimization results are more prominent, decreasing 24.0534 g · k W − 1 · h − 1 . The percentage reduction is 3.671%. Compared with the unimproved traditional BFO algorithm, the effect is more prominent, ICSBFO reduces coal consumption by 47.5621 g · k W − 1 · h − 1 . The percentage reduction is 7.26%. The resulting significant reduction in coal consumption can improve the efficiency of the power plant, reduce the economic cost of the power plant and reduce the pollution to the environment. (From the load distribution of each unit solved by the ICSBFO algorithm, since the upper and lower limits of a load of units 9 and 10 are both high, the outputs of units 9 and 10 are the most, which are 413 and 479 MW respectively.)

To test the influence of different loads on the performance and solution quality of the algorithm, this paper uses 90%, 80%, and 70% of the rated load to solve the load optimization distribution model of the 10-unit case of the medium-sized power plant. This experiment compares the ICSBFO and unimproved traditional BFO.

Affected by the valve-point effect, traditional methods often perform poorly in low-load optimization of processing units. For example, in the traditional BFO algorithm, it can be seen from the above Figure 3 that the optimization of coal consumption by BFO in reducing the load is not very obvious. In this process, the coal consumption at 70% load is higher than 80% load, which shows that BFO can no longer effectively optimize the ELD problem at low load. However, ICSBFO can solve the problem of adapting to low-load optimization. As seen from the above Figure 3, as the load rate decreases, the convergence speed of ICSBFO gradually slows down, but the optimization result of ICSBFO is still significantly better than that of BFO. It can be seen from Table 3 that under the condition of 90% load rate, under the load of 2430 MW, the optimized reduction is 66.2689 g · k W − 1 · h − 1 . The percentage reduction is 12.624%. Under the load rate of 80%, the load of 2160MW, the coal consumption is saved by 90.4748 g · k W − 1 · h − 1 .The percentage reduction is 21.183%. Under the load rate of 70% and the load capacity of 1890MW, the optimized amount is 220.1997 g · k W − 1 · h − 1 . The percentage reduction is 65.357%. From the analysis of the load optimization distribution of different units, the ICSBFO algorithm can give full play to the output advantages of different units. In this case, the two units G9 and G10, with higher upper and lower load limits, have the most output under different load ratios. From this, it can also be concluded that ICSBFO has an excellent performance in optimizing the load of 10 units. By comparing different load rates, it can be found that under high load rates, the optimized amount of ICSBFO is small, and as the load rate decreases, the optimized coal consumption gradually increases. The traditional BFO algorithm has a poor unit optimization effect in the case of low load, but ICSBFO still has an excellent optimization effect in the case of low load rate. Therefore, the algorithm is expected to have good application scenarios during the low-peak electricity consumption period in spring and autumn.

In case 2, taking 3 units of a miniature thermal power plant as an example, the coal consumption characteristic parameters of the units and the upper and lower limits of unit load are shown in the following Table 4:

In the case of a small power plant with 3 units, this paper uses ICSBFO, BFO and PSO to solve and compare the load distribution optimization model of the same unit data. In this experiment, each group was tested 30 times, and the average was taken. The population size was 50, the dimension was 3, the load was 900 MW, and the total number of iterations of various algorithms is 480.

This experiment sets the number of units as the solution dimension. The crisscross algorithm used in this paper in the replication process, its horizontal crossover operator is the crossover of dimensions, while the dimension of this experimental case is 3, and the crossover dimension is too small, resulting in The offspring after crossover are highly similar to the previous generation, so the horizontal crossover operator is omitted in this experiment, and only the vertical crossover operator is used to improve the replication process, so as to ensure the diversity of the population. Although the energy consumption coefficients of the three units in this experiment are different, the upper and lower limits of the unit load are the same. It can be seen from Figure 4 that the result of ICSBFO is significantly better than that of PSO and BFO, but the convergence rate will be slightly lower. Under the condition of 900 MW load, the optimal coal consumption of ICSBFO is 975.23 g · k W − 1 · h − 1 , which is 1.37% lower than that of BFO algorithm and 1.06% lower than that of PSO. Combining the low load rate in the 10-unit case and the 3-unit case, it can be concluded that the traditional bacterial foraging algorithm generally performs in processing the optimization data of small-scale units, and the solution time is slightly longer. However, the improved bacterial foraging algorithm can overcome such problems. Given this characteristic, follow-up research will be carried out on the case of large-scale power plant units. In addition, in the future work, we will also study the dynamic ELD problem with multiple constraints, not limited to the valve-point effect constraint.