∑ i = 1 m     P i = P D + P L , i = 1 , 2 , ⋯ , m (4)

where, m is the sum of generating units in the system; P_D is the target value of the load in the power system; P_L is the loss value of the power grid. Under a certain distribution network range, when the load in the system is distributed centrally and the coverage is small, the network loss is negligible, so the formula (5) is adjusted to the following form:

∑ i = 1 m     P i = P D , i = 1 , ⋯ , m (5)

P i min ≤ P i ≤ P i max , i = 1 , 2 , ⋯ , m (6)

where P i max and P i min respectively represent the maximum and minimum operating power of the i-th unit, and P_i is the corresponding effective power of the i-th unit.

In summary, the mathematical model of the ELD problem considering the valve point effect, power balance constraint and operating power constraint is:

min F = ∑ i = 1 m F i ( P i ) = ∑ i = 1 m a i + b i P i + c i P i 2 + | d i sin ( e i ( P i min − P i ) ) | (7)

s .t .   { ∑ i = 1 m     P i = P D P i min ≤ P i ≤ P i max (8)

It can be seen from the above formula that the mathematical model of the ELD problem contains constraints of nonlinear equality and inequality constraints, the function variables are not differentiable and discontinuous, and the range of feasible solutions sought is non-convex. In order to solve the ELD problem more effectively, this paper proposes to improve the original two algorithms and combine them into a new optimization algorithm.

Considering the high complexity of the power balance constraint in the high-dimensional case, the paper proposes a strategy that takes both the operation constraint and the power balance constraint into account, as follows:

Step 1 Set the population serial number as i ( i = 1 , 2 , ⋯ , N ) , and N is the number of population.

Step 2 Calculate the sum of power P_sum of all units in the current population i and the total load value P_D of the system demand, and calculate the difference. If the difference is not 0, continue to the next step; if it is 0, skip to Step 5.

Step 3 Generate a random serial number r (r is any integer within [1, m]), and set the active power P_i of the unit r under this serial number to P i = P i − ( P s u m − P D ) .

Step 4 Check P_i's unit operation constraints, if it exceeds the upper and lower limits, set them as the upper and lower limits. Jump to Step 2. Re-verify whether the difference is 0.

Step 5 i = i + 1 Constraint and adjust the units of all populations in turn, until the difference between the P_sum of all populations and the total load value P_D of the system demand is 0.

W i , G + 1 = { X G − A ⋅ | X G − C ⋅ X * |                                                                 if         p < 0.5     and     | A | < 1 X r a n d − A ⋅ | X G − C ⋅ X r a n d |                               if         p < 0.5     and     | A | ≥ 1 D ⋅ e b l ⋅ cos ( 2 π l ) + X *                                                             if         p ≥ 0.5 (19)

where W_i,G+1 are a new generation of mutant populations. That is, W_i,G+1 is used instead of V_i,G+1 for the next step of crossover operation.

At the same time, improve the mutation operation of the whale optimization algorithm: the mutual coordination of the exploration and exploitation capabilities of WOA guides the whole algorithm. Among them, the exploration capability refers to the ability to search a wider range of regions and jump out of local optimum; the exploitation capability refers to the ability to perform local search on regions near known solutions, which is beneficial to accelerate the convergence speed.

From literature [8] and literature [12], it can be seen that the convergence factor a in the WOA algorithm has a crucial impact on the exploration and exploitation ability of the algorithm: the larger the convergence factor a, the stronger the global search performance of the algorithm; if the convergence factor a is smaller, the local search performance of the algorithm will be stronger and the convergence rate will be higher. In the standard WOA algorithm, as the iteration progresses, the linearly decreasing method of the convergence factor a allows the algorithm to have better global search capabilities in the early stage, but the convergence speed is too slow; and when the algorithm is about to reach the maximum number of iterations. The convergence factor a approaches the zero point, and its convergence speed has accelerated, but it has not been able to get rid of the dilemma of local optimality, and it has not achieved a good coordination of exploration and exploitation. Therefore, the convergence factor in the original WOA algorithm cannot effectively utilize the superiority of the algorithm.

There are several nonlinear update strategies in existing studies. In the literature [9], the original convergence factor is proposed to be replaced by a trigonometric function with the following formula:

a = 2 − 2 sin ( π 2 t T ) (20)

where, t and T represent the current number of iterations and the maximum number of iterations respectively, the same below. Literature [13] proposes the use of segmented convergence factor update strategy, the formula is as follows:

a = { 2 − ( t T ) 2                                                         t ≤ T / 2 1 − 2 ( t − t T ) T 2 + ( t − T 2 T 2 ) 2               t > T / 2   (21)

The literature [14] proposes a convergence factor update strategy in the form of a positive exponential with e as the base, the formula is as follows:

a = 2 − 2 ( e t T − 1 ) e − 1 (22)

This paper proposes an adaptive convergence factor update strategy in the form of a negative exponent with a constant as the base:

a = 2 ∗ μ − t T (23)

where, μ represents a constant, and its value is adjusted according to the actual situation of the calculation example. After experimental testing, μ = 1000 is the best.

The above four strategy functions are plotted in the same graph as the linear decreasing strategy, as shown in Figure 1 (take the number of iterations T = 2000).

It can be seen from Figure 1 that compared with the other strategy functions,

the negative exponential decreasing update strategy maintains a larger value of a in the early iterative stage of the algorithm, which gives the algorithm more chances to jump out of the local extremes; in the middle of the algorithm, the function monotonically decreases more than the other four strategies, and the convergence factor a can be plunged to a smaller value, thus effectively ensuring the high efficiency of convergence; while the search for the optimal solution in the later stage. The range of the optimal solution is basically determined, and the convergence factor a in this strategy is in a small value and decreases slowly, so the final convergence accuracy of the algorithm is guaranteed.

Improve the crossover operation of differential evolution: The size of the crossover factor CR largely determines the convergence performance of the differential evolution algorithm. It can be seen from formula (17) that the larger the CR value, the higher the contribution rate of V_i to U_i, which means that the population will contain more variant individuals after the crossover operation; the smaller the value of CR, the contribution rate of X_i to U_i, the lower the value, the poorer the algorithm’s ability to exploit other regions of the solution, and the poor convergence efficiency, but it is conducive to retain the original individual characteristics (maintaining the diversity of the species), which makes the algorithm have a local solution during higher success rate.

Therefore, if the algorithm want to maintain the population diversity more stably in the early stage, it should start from a small CR value to maintain the diversity and success rate in the early stage, and then gradually increase the CR to enhance the convergence rate in the later stage, so as to effectively reduce the number of falling into local optimum in the whole iteration, which is the more ideal CR change rule.

In addition, this paper proposes to screen mutated individuals in advance in the crossover operation, so as to provide more opportunities for superior individuals in the population; while inferior individuals are given a lower probability of selection and are gradually eliminated. Therefore, this paper proposes an elimination mechanism based on individual fitness (as an example of solving for the minimum):

C R = { C R min + ( C R max − C R min ) ⋅ f i − f a v f min − f a v           if   f i < f a v   C R min                                                                                                                                                                                                                                                                 otherwise (24)

CR_max and CR_min are the maximum and minimum values of CR, respectively. At the same time, each iteration calculates the average f_av of the individual fitness of all populations and finds the best fitness f_min. Taking finding the global minimum as an example, if the average f_av of the population fitness is lower than the individual’s current fitness f_i, this individual can be regarded as a dominant individual. The crossover rate CR of the individual will be adaptively adjusted according to the degree of close to optimal fitness. If the average fitness f_av is higher than the individual’s current fitness f_i, then this individual can be regarded as a disadvantaged individual. The individual’s CR will be reset to CR_min. Therefore, the superior individuals will continue to be retained to the next generation, and the inferior individuals will gradually decrease due to the low probability of being selected.

Under this method, individuals with better fitness will be retained, and CR will gradually approach the maximum parameter value CR_max, thus conforming to the best change law of CR.

The flow chart of the IDEWOA algorithm for solving the economic load distribution model of the power system is shown in Figure 2.