*WU-ABC Optimization*

In this study, we propose a solution through the WU-ABC algorithm, a hybrid of the weighting update method and the ABC algorithm, to solve the CEED optimization problem. The ABC algorithm is a swarm-based optimization method that simulates the foraging behavior of a bee swarm; it has recently been used in many studies to solve optimization problems [36,37]. However, it is difficult to solve the proposed CEED problem under emission constraints using the ABC algorithm alone. Therefore, the weighting update method is also employed to solve the emission problem by adjusting the penalty factor and EBDR capacity. The WU-ABC algorithm involves five main steps, as described below:

• *Initialization step*: In this step, the initial population of food source is placed randomly in a D-dimensional problem space.

$$\mathbf{x}\_{nm} = \mathbf{x}\_{nm}^{\min} + rand[0, 1] \cdot (\mathbf{x}\_{nm}^{\max} - \mathbf{x}\_{nm}^{\min}) \tag{35}$$

Equation (35) represents the *m*th random food source of dimension *n* in the CEED problem.

• *Employed bees step*: Each bee repeatedly explores a food source to find the optimal solution, and then it chooses a new optimal position (*vmn*) close to the reference position.

$$
\omega\_{mn} = \mathbf{x}\_{mn} + \phi\_{mn} \cdot (\mathbf{x}\_{mn} - \mathbf{x}\_{an}), \quad m \neq o \tag{36}
$$

When an employed bee finds a food source that is better than the reference one (*xmn*), the reference food source is superseded with the new candidate.

• *Onlooker bees step*: Onlooker bees look for new positions that are close to the old position through a greedy search method. The bees consider a fitness value, the amount of nectar, and probabilistically determine the space for their next exploration, as follows:

$$P\_m = \frac{fit\_m}{\sum\_{m=1}^{SN} fit\_m} \tag{37}$$

In Equation (37), *fitm* is a value that is proportional to the amount of nectar.


$$h(s+1) = h(s) \times \exp\left[\eta \times \left(\frac{em(s) - em^{\max}(s)}{em^{\max}(s)}\right)\right] \quad \text{if} \quad cm(s) > cm^{\max}(s) \tag{38}$$

After updating the penalty factor using Equation (38), optimization is performed again through the ABC algorithm.

Figure 3 summarizes the proposed CEED solution. The procedure is performed sequentially as follows:


**Figure 3.** Overall process of the combined economic emission dispatch (CEED) solution.
