Neighborhood evidence-driven artificial bee colony algorithm with application for optimal operation of wet flue gas desulfurization system

Abstract

Artificial bee colony (ABC) algorithm is a widely used swarm intelligence algorithm due to its simple structure, good robustness and strong exploration ability. However, the unbalanced exploration and exploitation capabilities restrict its performance. To tackle this problem, a Neighborhood Evidence-Driven Artificial Bee Colony (NEDABC) algorithm is proposed in the framework of theory of belief functions, integrating information fusion with ABC algorithm. In the onlooker bee phase, instead of being guided by the current optimal solution or specific elite solutions, onlooker bees select their search directions by fusing the comprehensive evidence provided by food sources within their neighborhood. Both short-range solutions and high-quality solutions can have a great impact on the selection of search direction. With this constructive search strategy, onlooker bees will enhance the exploration and exploitation of their nearby potential areas. Under the joint action of the population, the potential optimal region will soon be found and exploited. The proposed algorithm is tested on a set of benchmark functions and compared with several variants of ABC. Experimental results reveal its powerful abilities in achieving higher accuracy and faster convergency. In addition, the NEDABC algorithm is applied to operation optimization of a wet flue gas desulfurization system, demonstrating the practicability and efficiency of the proposed algorithm in engineering applications.

Keywords

artificial bee colony information fusion theory of belief functions optimization wet flue gas desulfurization

1. Introduction

In recent years, metaheuristic algorithms have gained significant attention in solving optimization problems, primarily due to their flexibility and robustness beyond traditional optimization methods, especially when addressing complex and large-scale challenges. Their property of being independent of the organizational structure of problem enables them to adapt to various domains, thus efficiently finding near-optimal solutions in a range of applications. In particular, metaheuristic optimization algorithms have been widely applied in critical fields such as energy systems,^1–3 power systems,^4,5 manufacturing,^6,7 and automatic control,^8,9 where optimization plays a crucial role in improving efficiency and reducing operational costs.

Inspired by the group behavior of nature, a variety of meta-heuristic swarm intelligence algorithms have been proposed, such as the particle swarm optimization (PSO),¹⁰ ant colony optimization (ACO),¹¹ whale optimization algorithm (WOA)¹² and artificial bee colony algorithm (ABC).¹³ Among them, the ABC algorithm has gained significant attention in the field of artificial intelligence due to its fewer control parameters and straightforward tuning process,^14–17 which contribute to its ease of implementation.

Numerous numerical experiments have demonstrated the powerful ability of ABC in optimizing continuous high-dimensional functions, showcasing its insensitivity to initial solutions, strong robustness, and less requirement for control parameters.^18,19 However, the original ABC was not perfect when it first came out. Exactly like other swarm intelligence algorithms, ABC possesses the ability to explore and exploit at the same time. Nevertheless, the initial completely random search approach leads to its slow convergence and poor exploitation capability. As a matter of fact, it is essential to enhance exploitation capability for any meta-heuristic swarm intelligent algorithm including ABC, as some variant ABC algorithms have done. In Zhu and Kwong,²⁰ inspired by PSO, the information of the current best solution is integrated into the search mechanism and results in a strong exploitation capability. In a similar way, Gao et al.²¹ proposed ABC/best by introducing differential evolutionary algorithm, where each bee searches only around the best solution of the previous iteration. Alrosan et al.²² proposed MeanABC, which compares each bee’s current food source with global best location and positive direction of its own best location to update the search equation. Compared with only using the current best solution to guide, these variants have better exploration ability. On this basis, in Zhou et al.²³ and Wang et al.²⁴ a modified ABC with neighborhood search is proposed by further combining the search mechanism, which considers the joint action of the current best solution, two random solutions and the solution to be evolved. Furthermore, Wang et al.²⁵ proposed multi-strategy ensemble artificial bee colony algorithm covering three ABC variants to achieve a tradeoff between exploration and exploitation according to their respective advantage. Yavuz and Aydın²⁶ proposed self-adaptive search equation-based artificial bee colony algorithm, and three different search strategies are used to find new food sources, thus improving the exploration and exploitation capabilities of the algorithm. With a distinct viewpoint from above, in Akay and Karaboga²⁷ and Zhou et al.²⁸, a number of dimensions are changed in a single search and a new stage is added to change all dimensions of solutions with a proper probability. Some other contributions can refer, for example, to Banharnsakun et al.,²⁹ Rahnema and Gharehchopogh³⁰ and Zhou et al.³¹.

All the aforementioned studies primarily focus on enhancing the exploitation ability without adequately addressing the balance with exploration. In order to strengthen the exploitation capability of ABC algorithm while maintaining its exploration capability, this study presents a neighborhood evidence-driven artificial bee colony (NEDABC) algorithm within the framework of theory of belief functions (Dempster-Shafer theory). In the NEDABC, each food source (solution) around the selected food source can provide a piece of evidence rather than just a distance, to guide the onlooker bee to choose further search directions. More comprehensive information about the selected food source can be obtained by fusing the pieces of evidence within the neighborhood. Based on the pooled evidence, a high-quality food source in the neighborhood will be selected to guide the evolution. With respect to exploration, this neighborhood evidence-driven search manner divides multiple neighborhoods in the solution space, ensuring comprehensive coverage of the solution space and allowing flexible use of all information from high-quality solutions. As for exploitation, since the high-quality solution guiding evolution comes from the neighborhood of current food source, which increases the probability of exploring the optimal solution in the neighborhood of current evolution neighborhood, thus speeding up the search and improving the accuracy of the optimal solution. As will be demonstrated, by comparing with other ABC variants, the NEDABC performs well on a set of benchmark functions, both in terms of convergence speed and accuracy.

Furthermore, as previously mentioned, optimization is crucial for energy systems. Due to the complexity of the systems and the randomness in population updates, the optimization process often requires a prolonged duration to converge and may become trapped in local optima. These limitations are not negligible for practical applications in the field, as they inevitably lead to extended waiting times for subsequent control processes and increased difficulty in target tracking. As a matter of fact, for the on-site application of engineering problems, faster convergence speed is a more worthy concern than better optimization accuracy. Therefore, this study also takes a wet flue gas desulfurization (WFGD) system in a coal-fired power plant as an example for operation optimization to illustrate the superiority and practicality of the NEDABC algorithm in real-world applications.

The rest of this paper is organized as follows. Section 2 briefly recalls the basic concepts of the ABC algorithm. The NEDABC are then proposed in Section 3. Next, some comparative experiments based on benchmark functions are conducted to verify the performance of NEDABC in Section 4. Section 5 carries out optimization based on real data of the WFGD system and compares the economics before and after optimization. Finally, the conclusions of this study are provided in Section 6.

2. Background of artificial bee colony algorithm

ABC algorithm proposed by Karaboga¹³ is a swarm-based intelligent algorithm that simulates the foraging process of honeybees. In ABC, the colony is divided into three differentially functional kinds of bees, namely employed bees, onlooker bees and scout bees. The schematic diagram of the original ABC is shown below.

As shown in Figure 1, the employed bees search for candidate food sources in the neighborhood of their corresponding food sources. Once a better food source is found, the employed bee will abandon the original food source and record the information about its corresponding food source for sharing in the hive. Onlooker bees select food sources based on information shared by employed bees and then search for candidate food sources within the neighborhood. Similar to the employed bees, greedy selection is performed again. When food sources have not been updated beyond search times, the corresponding employed bee will abandon the current food source and turn into a scout bee to randomly search for a new food source in the vicinity of the hive. After multiple rounds of the above process, the optimal food source, i.e., the optimal solution to the optimization problem, can finally be found.

Figure 1.

Schematic diagram of the original ABC.

The general structure of the artificial bee colony algorithm consists of four parts.³² They are initialization, employed bee, onlooker bee and scout bee phases.

$I n i t i a l i z a t i o n p h a s e .$ A population $S N$ of food sources $x_{i} = (x_{i, 1}, x_{i, 2}, \dots, x_{i, D}) \in R^{D}$ is initially generated in a completely random way according to

\begin{aligned} x_{i, j} = x_{j}^{m i n} + r a n d [0, 1] \cdot (x_{j}^{m a x} - x_{j}^{m i n}), \end{aligned}

(1)

where

i \in {1, 2, \dots, S N}, j \in {1, 2, \dots, D}

;

r a n d [0, 1]

is a uniformly distributed random number in the range

[0, 1]

;

x_{j}^{m a x}

and

x_{j}^{m i n}

represent the upper bound and the lower bound of the jth dimension respectively.

Generally, when considering the minimization problem, the fitness value corresponding to each food source can be calculated according to

\begin{aligned} f i t_{i} = {\begin{cases} \frac{1}{1 + f (x_{i})}, & if f (x_{i}) \geq 0 \\ 1 + | f (x_{i}) |, & otherwise \end{cases} \end{aligned}

(2)

where

f i t_{i}

represents the fitness value of the

i

th food source,

f (x_{i})

represents the objective function value of the

i

th food source.

$E m p l o y e d b e e p h a s e .$ Every employed bee corresponds to a food source, and searches around it for a new food source according to

\begin{aligned} v_{i, j} = x_{i, j} + Φ_{i, j} \cdot (x_{i, j} - x_{k, j}), \end{aligned}

(3)

where

x_{k}

(

k \neq i

) is a randomly selected food source to guide the search direction,

v_{i}

is the newly generated food source. And

Φ_{i, j}

is a random number between

[- 1, 1]

responsible for controlling the search step size. Unlike traditional neighborhood search methods constrained by fixed geometric radii, ABC’s search mechanism spans the entire population space. When

x_{k, j}

is distant from

x_{i, j}

and

Φ_{i, j}

approaches

\pm 1

, the search tends to be more exploratory on a global scale. Conversely, if

x_{k, j}

is closer to

x_{i, j}

Φ_{i, j}

is smaller, the search focuses more on local exploitation. Thus, adopting this search mechanism enables ABC algorithm to have the ability of global exploration and local exploitation.

If the fitness value of $v_{i}$ is greater than that of $x_{i}$ , $x_{i}$ will be substituted by $v_{i}$ . Otherwise $x_{i}$ will remain unchanged and its iteration counter $t r i a l$ will be increased by one. When $v_{i}$ becomes the new food source, the employed bee will share its information with onlooker bees of the current iteration.

$O n l o o k e r b e e p h a s e .$ Onlooker bees evaluate food sources based on information received from employed bees and select a food source according to the probability

\begin{aligned} p_{i} = \frac{f i t_{i}}{\sum_{j = 1}^{S N} f i t_{j}}, \end{aligned}

(4)

where

p_{i}

is the probability of the ith food source.

Similar to employed bee phase, the onlooker bees will search around these selected food sources according to Eq. (3) and determine whether to retain the newly created food sources.

$S c o u t b e e p h a s e .$ If the iteration counter of a food source exceeds the limit, the homologous employed bee will convert to scout bee and randomly generate a food source by Eq. (1).

3. The proposed algorithm: neighborhood evidence-driven artificial bee colony

3.1. Basic idea of NEDABC

Neighborhood Evidence-Driven Artificial Bee Colony (NEDABC) algorithm is an extension of the ABC algorithm within the framework of the theory of belief functions, which leverages the principles of evidential reasoning to perform credibility assessment of information and fusion of multi-source information in the neighborhood, thereby guiding the search evolution process of onlooker bees.

Under the framework of Dempster-Shafer theory,^33–35 the most basic concept is the mass function. Given a frame of discernment $Ω = {ω_{1}, ω_{2}, \dots, ω_{c}}$ , which constitutes a finite and exhaustive set of all possible outcomes on a specific problem Q. And let $2^{Ω}$ be the set of all subsets of $Ω$ . Then mass function, also called basic belief assignment (BBA), can be defined as a mapping from $2^{Ω}$ to $[0, 1]$ such that

\begin{aligned} \sum_{A \subseteq Ω} m (A) = 1, \forall A \subseteq Ω \end{aligned}

(5)

Each

m (A)

describes the degree of belief in the variable

ω

obtaining the actual value

ω_{i} \in A

. For any subset

A \subset Ω

, if

m (A) > 0

, A is called a focal element of

Ω

. A mass function used to describe a given problem is called a piece of evidence.

Once two independent pieces of evidence $m_{1}$ and $m_{2}$ induced from different sources are available, the combination of mass functions can be utilized, which is formally known as evidence fusion in the Dempster-Shafer framework. The conjunctive combination of $m_{1}$ and $m_{2}$ , denoted by , yields the following unnormalized mass function, i.e., the mass assigned to the null set $m (\emptyset)$ is not necessarily zero.

$m (\emptyset)$ is regarded as the inconsistency in the fusion of different pieces of evidence, serving as a measure of the degree of conflict between the evidence. If necessary, the normality condition $m (\emptyset) = 0$ can be recovered by dividing each mass m₁₂(A) by 1-m₁₂ $(\emptyset)$ . This operation is called D-S fusion rule, noted by $\oplus$ :

Both operations are associative, commutative and admit the vacuous mass function as the only neutral element.

Motivated by the above concept about evidence, the NEDABC algorithm can be proposed. Each neighboring solution of the food source to be evolved is treated as an independent information source, providing a piece of directional information for the search of onlooker bees. By incorporating the processing approach of evidence in Dempster-Shafer theory, the algorithm can synthesize comprehensive information from neighborhood solutions, enabling adaptive decision-making for search direction selection. Specifically, in the onlooker bee phase, for a chosen dimension $j$ of each food source $x_{i}$ , the evidence $m_{i}$ it provides for the food source to be evolved can be constructed by

\begin{aligned} {\begin{cases} m_{i} (x_{i, j}) = p_{i} \\ m_{i} (Ω) = 1 - p_{i} \end{cases} \end{aligned}

(8)

where

m_{i} (x_{i, j})

denotes the degree of direct support for the search direction toward

x_{i}

. And

p_{i}

indicates the reliability of the

i

th food source, which can be obtained through statistical data analysis methods or directly set to 1, i.e., all food sources are considered equally reliable. As introduced previously

Ω

constitutes a finite and exhaustive set of all possible outcomes on a specific problem, and specifically for the optimization task of this study, it represents all possible values on one selected dimension. Thus

m_{i} (Ω)

represents the degree of retention of global uncertainty in current dimension, i.e.,

x_{i}

is not certain which way the search direction should be oriented. In this way, we can establish an evidence base:

\begin{aligned} B = {e_{i} | e_{i} = (x_{i}, m_{i}), i = 1, 2, \dots, S N} . \end{aligned}

(9)

For each food source undergoing evolution, the algorithm identifies its $K$ nearest neighbors within the solution space. Only those neighbors exhibiting superior fitness values relative to the current food source to be evolved are retained from the evidence base for subsequent evidence fusion. Crucially, the credibility of evidence provided by each neighbor should not be identical, as it depends on both the spatial proximity (distance) and optimization performance (fitness) of the neighboring solutions. To operationalize this, the evidence discounting mechanism in theory of belief functions is applied. Specifically speaking, for the value $x_{\cdot d}$ on the $d$ th dimension of a food source to be evolved, each neighbor in evidence base can provide the discounted evidence $m^{ϕ}$ of different credibility through the following discounting operation:

\begin{aligned} m^{ϕ} (x_{\cdot d} | e_{i}) = {\begin{cases} ϕ (r_{i}) m_{i} (x_{i, j}), & x_{\cdot d} = {x_{i, j}} \\ 1 - ϕ (r_{i}) (1 - m_{i} (Ω)), & x_{\cdot d} = Ω \end{cases} \end{aligned}

(10)

where

r_{i}

is a comprehensive ranking defined by the product of the fitness ranking of the

i

th food source and its distance ranking from the original one.

ϕ (\cdot)

is the decreasing discount function, that dynamically adjusts the relationship between the credibility of the evidence provided by the

i

th food source and its ranking

r_{i}

, which also reflects the degree of preservation of the initial evidence

m_{i}

. Thus,

ϕ (\cdot)

should satisfy the following conditions:

\begin{aligned} {\begin{cases} ϕ (r_{i}) \in [0, 1] \\ lim_{r_{i} \to \infty} ϕ (r_{i}) \to 0 . \end{cases} \end{aligned}

(11)

In theory of belief functions, there is a natural choice for $ϕ$ :

\begin{aligned} ϕ (d) = θ \exp (- γ d^{2}), \end{aligned}

(12)

where

θ

is a constant parameter within

[0, 1]

. Generally, it is not set too small because smaller values excessively weaken the impact of neighborhood evidence, leading to slower convergence, and is usually prefer to 0.95.^36,37 The significance of this choice is that even if the distance between the neighboring food source

x_{i}

and the evolving food source approaches 0, the information from

x_{i}

is not entirely transferred to the evolving food source, which is a consideration known as avoiding “over-reliance”.

d

is the Euclidean distance between the neighbor and the original one. And

γ > 0

is an important structure parameter that controls the attenuation gradient of the decreasing distance function. In NEDABC, evidently, the comprehensive ranking

r_{i}

serves as a critical determinant in the evidential credibility assessment process. For a high-quality, close proximity solution, which means it has a higher overall ranking (the higher the ranking, the smaller the

r_{i}

), it is reasonable to believe that it can provide more credible information to reflect the possible direction of evolution. On the contrary, a low comprehensive ranking implies a long distance or a small fitness. If the distance between the ith food source and the food source to be evolved is large and the fitness is small, the

ϕ (r_{i})

is close to zero, which means the direction of the ith food source is of less potential for evolution. In particular when

ϕ (r_{i}) \approx 0

m^{ϕ} (x_{\cdot d} | e_{i})

becomes the evidence of ignorance. By incorporating this mechanism, NEDABC algorithm can effectively balance exploitation of promising regions with exploration of uncertain domains, leveraging the theoretical foundations of Dempster-Shafer theory to enhance decision-making robustness. Based on this requirement,

ϕ (\cdot)

is redefined in NEDABC as

\begin{aligned} ϕ (r_{i}) = θ e^{- γ r_{i}}, \end{aligned}

(13)

where

θ

and

γ

serve the same roles as in Eq. (12) and will be determined in Section 4.

With D-S fusion rule (Eq. (7)), evidence in Eq. (10) can be fused as follows and the final fused evidence can tell onlooker bees exploring direction:

\begin{aligned} m (x_{\cdot d}) = \oplus_{i = 1}^{h} m^{ϕ} (x_{\cdot d} | e_{i}) . \end{aligned}

(14)

where

h

denotes the number of neighbors, among the

K

-nearest neighbors of the food source undergoing evolution, whose fitness values are higher than that of the current food source. Then the fused evidence provided by these h food sources is

m = {m (x_{1, j}), m (x_{2, j}), \dots, m (x_{h, j}), m (Ω)}

. Among the

h + 1

choices (including

Ω

), the probability of the sth food source being selected can be calculated by

\begin{aligned} p_{m_{s}} = \frac{m (x_{s, j})}{\sum_{i = 1}^{h} m (x_{i, j}) + m (Ω)} . \end{aligned}

(15)

Note that if $Ω$ is selected by an onlooker bee, the search direction is ignorant. In this case, the food sources within the neighborhood of the solution to be evolved may not provide better search directions than before. Consequently, the onlooker bee will shift its search towards the direction of the current global best solution. Once the search direction is determined, the onlooker bee will try to search for a better food source with a certain step in this direction. Inspired by Akay and Karaboga,²⁷ Zhou et al.²⁸ and Banharnsakun et al.,²⁹ the onlooker bees change more than one dimension in a single search in NEDABC. The search mechanism of the onlooker bee phase of NEDABC is defined according to

\begin{aligned} v_{i, d} = x_{i, j} + r a n d [0, 1] \cdot f i t_{s} \cdot (x_{i, j} - x_{s, j}), \end{aligned}

(16)

where

d = 1, 2, \dots, D

and

v_{i, d}

is the dth dimension of the newly generated food source

v_{i}

x_{i, j}

is the to-be-evolved food source position

i

in a selected dimension

j

f i t_{s}

is the fitness of the selected food source

x_{s}

. And

x_{s, j}

(

s \neq i

) is the jth dimension of the selected food source.

Different from Banharnsakun et al.,²⁹ Eq. (16) leverages the selected food source to determine both the search direction and step size, where the step size is modulated by ( $r a n d [0, 1] \cdot f i t_{s}$ ). $f i t_{s}$ can adaptively adjust the step size based on the quality of the selected food source. In the case of small $f i t_{s}$ , which usually appears in early iterations, onlooker bees will not rush to approach the selected food source, but search in that direction with a small step size. On the opposite, if $f i t_{s}$ is large, whether in early or late iterations, onlooker bees will take a large step size and try to search around it. In addition, onlooker bees will directly choose the current global best food source to guide the search if there is no better food source in their current neighborhood. This adaptive search strategy ensures a balanced trade-off between exploration and exploitation, leveraging the credibility of fused evidence to guide decision-making robustly.

3.2. Procedure and computational complexity analysis of NEDABC

Based on the idea and interpretations in Section 3.1, we can propose the NEDABC and summarize it in the Algorithm 1. In that, $F E s$ is the number of times the fitness evaluation function is used, and $m a x F E s$ represents the maximum FEs allowed in the optimization process, which is used as a stopping criterion. Additionally, the maximum number of iterations can also be used as a stopping criterion. However, since different variants of ABC algorithm perform different operations during a single iteration, some algorithms do not strictly utilize all three types of bees for optimization. To ensure fairness of subsequent comparison of algorithms, $m a x F E s$ is used as the stopping criterion in this study.

Using the notations of Algorithm 1, the computational complexity of NEDABC can be analyzed. The computational complexity of NEDABC is higher than that of the original ABC, which can be expressed by $O (m a x C y c l e \cdot S N \cdot D)$ .²⁹ The main difference between NEDABC and original ABC is that NEDABC performs evidence fusion and distance ranking in the onlooker bee phase. In the process of evidence fusion, the calculation is only carried out in $K$ nearest neighbors. Its computational complexity is less than that of distance ranking and sorting, which is $O (D \cdot S N + S N \cdot l o g (S N))$ . In conclusion, the computational complexity of NEDABC is $O (m a x C y c l e \cdot S N \cdot (D \cdot S N + S N \cdot l o g (S N)))$ . However, in actual operation, NEDABC can usually find the optimal solution before reaching the maximum number of iterations. Its fast convergence and high precision ensure that it has practical significance.

3.3. Some insights with discussions

In this section, we illustrate the search mechanism of NEDABC with an intuitive demonstration.

Figure 2 shows the distribution of some solutions (circles in the figure) in a contour plot of function values. In the global best guided algorithms, the current best solution will be a regular and decisive direction. The final search direction will be determined by its direction and that of a random solution to maintain the diversity. In general, this strategy will lead the current solution to the current best solution. Moreover, the random solution, such as the worse solutions in the figure, may lead to poor search direction and result in slow convergence. Differently, for algorithms with elite strategy, the elite solutions will guide the search. In the figure, the elite solution near the global optimal (solution 4) can provide valuable information. Nevertheless, because the population will approach to better solutions, the elite solutions usually converge gathered near the current best solution. Thus, solution 4 has a small probability to be chosen to guide the search. To ameliorate it, in the proposed NEDABC, the search direction will be determined by the comprehensive information of the $K$ nearest solutions (solutions in the sector surrounded by dotted lines in Figure 2). Under NEDABC’s strategy, poorer solutions will not be used to guide the search. In Figure 2, solution 1, solution 2 and solution 3 have the same comprehensive ranking. In other words, both the far elite solutions and the near better solutions can guide the search, which can not only keep the diversity of population, but also strengthen the exploitation capability. In this case, if solution 2 or solution 3 is selected, the current solution will approach to the global optimal. On the contrary, if solution 1 is selected, the current solution is likely to exploit the region of a local optimal. In addition, in the figure, all the 7 better solutions in the neighborhood have a corresponding probability of being selected.

Figure 2.

Determination of search direction.

To reveal the difference of search strategies in the evolution process, Figure 3 respectively shows the evolution process of a single solution and the current optimal solution of four representative variants under the two-dimensional Sphere function. Original ABC uses a completely random search strategy (ABC¹³). Individuals in the population show strong exploration ability without guidance, which is reflected in the large fluctuation of the curve in Figure 3(a). The slow convergence caused by this strategy is reflected in the large fluctuation of the curve of the current optimal solution in Figure 3(b). GABC applies the current best guidance strategy (GABC²⁰). It can be seen from Figure 3(a) that the solution converges quickly to the current optimal. However, when the current optimum is not in the region of global optimum, the algorithm will fall into local optimum. The whole population will not migrate until the employed bees find the global optimal area. MGABC adopts the strategy of elite solutions (MGABC²⁸). The existence of elite solutions balances the ability of exploration and exploitation to a certain extent, so that the algorithm can quickly and smoothly find the global optimal solution in Figure 3(b). For NEDABC, due to the full exploitation of a single solution in its neighborhood, that solution converges to the current optimal slowly during the evolution process (as Figure 3(a) shows). Therefore, NEDABC retains the advantage of the powerful exploration capability of the original ABC. In the meanwhile, under such a combination of individual behavior, the population exhibits good exploitation ability as well, which is reflected by the curve in Figure 3(b).

Figure 3.

Evolution process of 4 variants. (a) Evolution of a single solution, (b) Evolution of the current best solution.

4. Experimental validation based on benchmark functions

Validation of the proposed method is needed to be certified by benchmark functions. In this section, experiments are carried out to validate the efficiency of NEDABC. In Section 4.1, the effect of hyperparameters $γ$ and $K$ on the performance of NEDABC is revealed and tuned for both. And in Section 4.2, the performance of NEDABC and other ABC variants are compared under 16 well-known benchmark functions. All information about the benchmark functions is summarized in Table 1, including dimensions and variable ranges. Among them, $f_{1}$ , $f_{4}$ , $f_{9}$ , $f_{10}$ , $f_{11}$ , $f_{12}$ , $f_{13}$ , $f_{15}$ , $f_{16}$ are unimodal functions. $f_{2}$ , $f_{3}$ , $f_{5}$ , $f_{6}$ , $f_{7}$ , $f_{8}$ , $f_{14}$ are multimodal functions.

Table 1.
Benchmark functions.

Functions Name D Search range Global minimum

$f_{1}$ Sphere 30 and 50 [−100, 100] 0

$f_{2}$ Schaffer 2 [−100, 100] 0

$f_{3}$ Griewank 30 and 50 [−600, 600] 0

$f_{4}$ Rosenbrock 30 and 50 [−2, 2] 0

$f_{5}$ Rastrigin 30 and 50 [−5.12, 5.12] 0

$f_{6}$ Ackley 30 and 50 [−30, 30] 0

$f_{7}$ Levy 30 and 50 [−10, 10] 0

$f_{8}$ Dropwave 2 [−5.12, 5.12] −1

$f_{9}$ Schwefel 2.22 30 and 50 [−10, 10] 0

$f_{10}$ Schwefel 2.21 30 and 50 [−100, 100] 0

$f_{11}$ Schwefel 1.2 30 and 50 [−100, 100] 0

$f_{12}$ Quartic with noise 30 and 50 [−1.28, 1.28] 0

$f_{13}$ Step 30 and 50 [−100, 100] 0

$f_{14}$ Alpine 30 and 50 [−10, 10] 0

$f_{15}$ Sum squares 30 and 50 [−10, 10] 0

$f_{16}$ Sum power 30 and 50 [−1, 1] 0

Functions	Name	D	Search range	Global minimum
$f_{1}$	Sphere	30 and 50	[−100, 100]	0
$f_{2}$	Schaffer	2	[−100, 100]	0
$f_{3}$	Griewank	30 and 50	[−600, 600]	0
$f_{4}$	Rosenbrock	30 and 50	[−2, 2]	0
$f_{5}$	Rastrigin	30 and 50	[−5.12, 5.12]	0
$f_{6}$	Ackley	30 and 50	[−30, 30]	0
$f_{7}$	Levy	30 and 50	[−10, 10]	0
$f_{8}$	Dropwave	2	[−5.12, 5.12]	−1
$f_{9}$	Schwefel 2.22	30 and 50	[−10, 10]	0
$f_{10}$	Schwefel 2.21	30 and 50	[−100, 100]	0
$f_{11}$	Schwefel 1.2	30 and 50	[−100, 100]	0
$f_{12}$	Quartic with noise	30 and 50	[−1.28, 1.28]	0
$f_{13}$	Step	30 and 50	[−100, 100]	0
$f_{14}$	Alpine	30 and 50	[−10, 10]	0
$f_{15}$	Sum squares	30 and 50	[−10, 10]	0
$f_{16}$	Sum power	30 and 50	[−1, 1]	0

In the experiments, the maximum number of FEs, i.e., $m a x F E s$ , is set to 150000. The number of food sources $S N$ is set to 50. $l i m i t$ is set to $D * S N$ . $θ$ for NEDABC is set to 0.95.

4.1. Parameters tuning

The hyperparameters $γ$ and $K$ play crucial roles in the performance of NEDABC. In order to choose the optimal $γ$ and $K$ , in this section we will experimentally illustrate the impact of both on the performance of NEDABC, and then make a choice of the values based on experimental results.

4.1.1. Impact of $γ$ on performance of NEDABC

In Section 3.1, the effect of $γ$ has been briefly discussed. It controls the decay rate of the decreasing function. In the algorithm, $γ$ affects the discount rate of food sources based on their comprehensive ranking. In general, as shown in Figure 4, the larger $γ$ is, the steeper the curve of $ϕ (r_{i})$ is. In this situation, the mass of food source with a low comprehensive ranking is very small, which means it can only provide little useful information for the evolution of food sources. In NEDABC, to adaptively change the decreasing rate through the neighborhood information, the number of better food sources in the neighborhood, denoted as $h$ , will lead to a different $γ$ . Figure 5 demonstrates the relationship between $h$ and $ϕ (r_{i})$ while $g$ is set to $h / 2$ . According to this adaptive mechanism, the more better food sources in the neighborhood, the greater the probability difference between search directions. On the contrary, if there is only one better food source in the neighborhood, its direction is very likely to be selected by the onlooker bee.

Figure 4.

Decreasing rate of $ϕ (r_{i})$ under different $γ$ ( $γ = g^{2}$ ).

Figure 5.

Decreasing rate of $ϕ (r_{i})$ under different $h$ .

In order to find the appropriate value of $γ$ , experiments are carried out on the benchmark functions. In the experiments, the parameters are set as described before, and in addition $K$ is set to 10. Each function has been run independently for 30 times and the average results are shown in Table 2. In Table 2, the bold font represents the best result in this function. As can be seen in the table, a reasonably large $g$ can accelerate convergence of NEDABC, as it is a trade-off between weakening the impact of relatively poor food sources and maintaining a diversity of search directions. In order to intuitively compare these results, the last row of Table 2 shows the overall ranking of each group under all benchmark functions. For each benchmark function, the group with smallest result ranks 1 and the group with biggest result ranks 6. In this table, the group with $g = h / 2$ has the smallest sum of rankings, so it is considered as the best value and is chosen for the subsequent experimental setup.

Table 2.

Experimental results of NEDABC with different $γ$ .

Functions	$g = h / 1$	$g = h / 2$	$g = h / 3$	$g = h / 4$	$g = h / 5$	$g = h / 6$
$f_{1}$	2.23E−18	7.74E−18	9.70E−18	9.21E−18	9.22E−18	8.29E−18
$f_{2}$	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
$f_{3}$	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
$f_{4}$	6.70E−17	3.60E−17	1.82E−16	1.13E−16	8.09E−17	3.45E−17
$f_{5}$	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
$f_{6}$	2.07E−15	1.60E−15	2.43E−15	2.07E−15	1.84E−15	2.07E−15
$f_{7}$	1.12E−18	4.18E−18	6.35E−18	7.66E−18	6.54E−18	6.90E−18
$f_{8}$	−1.00E+00	−1.00E+00	−1.00E+00	−1.00E+00	−1.00E+00	−1.00E+00
$f_{9}$	1.00E−17	2.17E−17	2.31E−17	2.77E−17	2.58E−17	2.61E−17
$f_{10}$	3.20E−15	1.24E−16	1.82E−15	5.75E−14	2.62E−14	1.15E−14
$f_{11}$	1.71E−19	3.91E−19	4.29E−19	5.59E−19	5.06E−19	4.80E−19
$f_{12}$	7.89E−05	5.28E−05	6.20E−05	7.17E−05	1.10E−04	1.10E−04
$f_{13}$	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
$f_{14}$	8.40E−16	1.56E−17	2.34E−16	1.64E−17	1.80E−17	2.20E−17
$f_{15}$	1.60E−18	7.90E−18	7.85E−18	8.48E−18	8.20E−18	9.09E−18
$f_{16}$	3.46E−20	3.75E−20	3.31E−20	4.15E−20	4.01E−20	4.92E−20
Total ranking	33	25	44	56	51	51

Bold type indicate the best performance among all compared algorithms under the corresponding condition.

4.1.2. Impact of

K

on performance of NEDABC

The range of neighborhood also affects the performance of the algorithm. In order to figure out its effect and find the optimal $K$ , tests are performed on the benchmark functions with different $K$ . In the tests, parameter settings are the same as described before, and in addition $g$ is set to $h / 2$ . $K$ is set to be proportional to $S N$ with scale factors of 0.1, 0.2, 0.3, 0.5, 0.7 and 1, respectively. For each $K$ , each benchmark function runs 30 times independently. The results are presented in Table 3. As in Section 4.1.1, for each $K$ , its total ranking is displayed in Table 3.

Table 3.
Experimental results of NEDABC with different $K$ .

Functions $K = 5$ $K = 10$ $K = 15$ $K = 25$ $K = 35$ $K = 50$

$f_{1}$ 7.67E−18 7.27E−18 7.35E−18 7.67E−18 7.59E−18 8.25E−18

$f_{2}$ 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00

$f_{3}$ 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00

$f_{4}$ 1.14E−17 1.85E−17 1.89E−17 1.57E−17 6.02E−17 6.79E−17

$f_{5}$ 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00

$f_{6}$ 1.84E−15 1.84E−15 1.60E−15 2.19E−15 2.19E−15 2.19E−15

$f_{7}$ 5.68E−18 4.94E−18 5.31E−18 5.85E−18 5.04E−18 4.86E−18

$f_{8}$ −1.00E+00 −1.00E+00 −1.00E+00 −1.00E+00 −1.00E+00 −1.00E+00

$f_{9}$ 2.11E−17 2.09E−17 1.93E−17 2.23E−17 2.01E−17 1.90E−17

$f_{10}$ 3.75E−17 2.03E−15 1.19E−13 8.37E−13 2.34E−15 4.01E−14

$f_{11}$ 5.75E−19 3.10E−19 4.54E−19 4.47E−19 3.94E−19 4.04E−19

$f_{12}$ 6.34E−05 5.12E−05 4.93E−05 4.49E−05 4.50E−05 5.87E−05

$f_{13}$ 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00

$f_{14}$ 4.53E−16 2.42E−16 2.36E−16 1.19E−17 2.39E−16 2.38E−16

$f_{15}$ 8.91E−18 7.35E−18 8.08E−18 7.48E−18 7.89E−18 6.95E−18

$f_{16}$ 2.57E−20 2.46E−20 5.33E−20 4.05E−20 3.34E−20 2.87E−20

Total ranking 50 33 44 48 42 43

Functions	$K = 5$	$K = 10$	$K = 15$	$K = 25$	$K = 35$	$K = 50$
$f_{1}$	7.67E−18	7.27E−18	7.35E−18	7.67E−18	7.59E−18	8.25E−18
$f_{2}$	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
$f_{3}$	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
$f_{4}$	1.14E−17	1.85E−17	1.89E−17	1.57E−17	6.02E−17	6.79E−17
$f_{5}$	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
$f_{6}$	1.84E−15	1.84E−15	1.60E−15	2.19E−15	2.19E−15	2.19E−15
$f_{7}$	5.68E−18	4.94E−18	5.31E−18	5.85E−18	5.04E−18	4.86E−18
$f_{8}$	−1.00E+00	−1.00E+00	−1.00E+00	−1.00E+00	−1.00E+00	−1.00E+00
$f_{9}$	2.11E−17	2.09E−17	1.93E−17	2.23E−17	2.01E−17	1.90E−17
$f_{10}$	3.75E−17	2.03E−15	1.19E−13	8.37E−13	2.34E−15	4.01E−14
$f_{11}$	5.75E−19	3.10E−19	4.54E−19	4.47E−19	3.94E−19	4.04E−19
$f_{12}$	6.34E−05	5.12E−05	4.93E−05	4.49E−05	4.50E−05	5.87E−05
$f_{13}$	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
$f_{14}$	4.53E−16	2.42E−16	2.36E−16	1.19E−17	2.39E−16	2.38E−16
$f_{15}$	8.91E−18	7.35E−18	8.08E−18	7.48E−18	7.89E−18	6.95E−18
$f_{16}$	2.57E−20	2.46E−20	5.33E−20	4.05E−20	3.34E−20	2.87E−20
Total ranking	50	33	44	48	42	43

Bold type indicate the best performance among all compared algorithms under the corresponding condition.

As shown in Table 3, NEDABC has the best performance when $K$ is set to $0.2 * S N$ , which is 10. In terms of the search mechanism of NEDABC, a larger $K$ increases the probability that onlooker bees will be attracted to food sources with larger fitness values, which can compensate for their distance from original food source and improve their comprehensive ranking. However, too large a $K$ exposes the algorithm to the risk of being too greedy, as in the case of those variants with elite populations. On the contrary, a smaller $K$ can increase the exploitation ability of the population when the function can quickly find the global optimal neighborhood, but leads to insufficient neighborhood information and slow convergence. Based on the above experimental results, we chose $K = 10$ for the subsequent experimental setup.

4.2. Comparison of other ABC variants

In this subsection, NEDABC is compared with other ABC variants based on the performance on the benchmark functions. The involved algorithms are listed as follows:

–
ABC/best²¹: A global best ABC
–
GABC²⁰: Gbest-guided ABC
–
MABC-NS²³: Modified ABC with neighborhood search
–
MEABC²⁵: Multi-strategy ensemble ABC
–
MGABC²⁸: ABC with multi-elite guidance
–
ABC¹³: The original ABC

The values of hyperparameters of other ABC variants are tuned on the 16 benchmark functions used in this study, following the tuning procedures described in their respective original papers. The parameter settings that achieved the best overall performance across the test functions are selected for subsequent comparative experiments. Specifically, for GABC and MEABC, $C$ is set to 1.5. For MABC-NS, $P$ is set to 0.1. For MGABC, $q$ is set to 0.1, $M R$ is set to 0.5, and $p$ is set to =0.1. For NEDABC, $K$ is set to 10, $g$ is set to $h / 2$ ( $γ = g^{2}$ ). Apart from the hyperparameters that are unique to each algorithm, other common parameters, such as the maximum number of function evaluations ( $m a x F E s$ ), the number of food sources ( $S N$ ), and the abandonment limit ( $l i m i t$ ) are set identically across all algorithms, specifically to 150,000, 50, and $D \times S N$ , respectively. Except for $D = 2$ in $f_{2}$ and $f_{8}$ , all the algorithms are tested in the case of $D = 30$ and $D = 50$ respectively. In addition, the convergence speed and standard deviation of the algorithms are compared in this paper. For all the algorithms, each benchmark function is run for 30 times independently.

Table 4 lists the results of the algorithms. It can be seen from Table 4 that NEDABC performs best on 13 out of the 16 benchmark functions and can find the global optimum consistently among 5 benchmark functions. And NEDABC achieves the lowest error values on Schaffer function ( $f_{2}$ ), Griewank function ( $f_{3}$ ), and Rastrigin function ( $f_{4}$ ), which are highly multimodal with numerous local optima. This suggests that NEDABC algorithm enhances exploration capability through neighborhood evidence fusion, which avoids falling into local optima and premature convergence by adaptively balancing global and local search directions. It is worth mentioning that in Rosenbrock function ( $f_{4}$ ), NEDABC achieves much better results than others. The global optimum of this function lies in a narrow banana-shaped valley, and algorithms with more exploration capabilities usually give better results on this problem.²³ The superior performance of NEDABC on this function indicates that its search strategy effectively adapts to different function characteristics, dynamically balancing exploration and exploitation capabilities. Furthermore, in the experimental results, EDABC consistently found solutions near the global optimum for all functions, without cases where a single poor result significantly affected the average performance. This demonstrates the robustness and stability of the algorithm.

Table 4.
Comparative results: $D = 2$ for $f_{2}$ , $f_{8}$ and $D = 30$ for other functions.

Functions ABC/best GABC MABC-NS MEABC MGABC ABC NEDABC

$f_{1}$ 5.89E−16 5.27E−16 1.91E−17 5.40E−16 9.98E−18 3.29E−12 6.79E−18

$f_{2}$ 0.00E+00 0.00E+00 0.00E+00 1.69E−06 0.00E+00 3.46E−10 0.00E+00

$f_{3}$ 2.47E−04 7.08E−12 0.00E+00 5.63E−16 0.00E+00 2.45E−07 0.00E+00

$f_{4}$ 2.31E+01 1.84E+01 2.13E+01 1.73E+01 2.77E+01 1.39E+01 1.22E−17

$f_{5}$ 5.12E−14 2.79E−13 4.97E−01 1.89E−14 0.00E+00 3.32E−02 0.00E+00

$f_{6}$ 4.10E−14 4.71E−10 2.66E−15 4.30E−13 8.88E−16 5.02E−06 2.55E−15

$f_{7}$ 5.89E−16 5.28E−16 1.04E−01 5.47E−16 5.96E−16 7.73E−14 4.93E−18

$f_{8}$ −1.00E+00 −1.00E+00 −1.00E+00 −1.00E+00 −1.00E+00 −1.00E+00 −1.00E+00

$f_{9}$ 1.70E−15 8.63E−11 4.19E−17 2.95E−14 3.47E−17 4.68E−07 2.16E−17

$f_{10}$ 4.37E+00 8.00E+00 4.27E−17 8.49E+00 9.53E−09 1.92E+01 1.24E−16

$f_{11}$ 1.13E+04 1.18E+04 1.30E−17 1.04E+04 8.64E−18 1.01E+04 4.12E−19

$f_{12}$ 2.25E−02 2.79E−02 1.50E−04 2.93E−02 7.92E−05 6.51E−02 3.55E−05

$f_{13}$ 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00

$f_{14}$ 2.65E−15 2.48E−05 4.55E−17 6.20E−14 3.68E−17 2.33E−04 1.27E−17

$f_{15}$ 6.40E−16 5.32E−16 2.07E−17 5.72E−16 9.76E−18 4.19E−13 6.43E−18

$f_{16}$ 2.37E−17 2.09E−17 1.89E−18 2.73E−17 1.75E−19 2.76E−17 2.56E−20

Total ranking 67 66 48 70 35 88 18

Bold type indicate the best performance among all compared algorithms under the corresponding condition.

In order to make the results more intuitive, for the 6 benchmark functions where the global optimal can be found, Table 5 shows the number of times that the global optimal is found in the 30 times and the average number of iterations cost for each algorithm. As shown in Table 5, NEDABC has the ability to find global optimal in a few iterations in the first five functions. For Alpine function ( $f_{14}$ ), only NEDABC finds its global optimum in 4 out of 30 runs, despite all algorithms facing difficulties. This highlights its enhanced exploration ability in rugged, multi-local-optima landscapes through diversified neighborhood sampling to some extent. However, it is undeniable that its success rate is only 13.3%, which exposes that there is still room for improvement of NEDABC algorithm in non-smooth oscillating function optimization problems.

Table 5.
Success rounds and average iterations of compared algorithms.

ABC/best GABC MABC-NS MEABC MGABC ABC NEDABC

Accept SR iters SR iters SR iters SR iters SR iters SR iters SR iters

$f_{2}$ 0 30 213.9 30 283.1 30 126.4 14 219.6 30 114.0 23 574.7 30 99.5

$f_{3}$ 0 6 859.7 0 30 82.1 9 843.8 30 51.7 0 30 36.5

$f_{5}$ 0 5 653.4 0 28 325.8 20 779.5 30 50.4 0 30 32.2

$f_{8}$ −1 30 118.1 30 152.5 30 39.8 28 118.5 30 46.2 30 282.1 30 50.8

$f_{13}$ 0 30 132.9 30 167.8 30 19.2 30 148.4 30 11.0 30 202.0 30 7.2

$f_{14}$ 0 0 0 0 0 0 0 4 66.75

Although the value of $m a x F E s$ is set the same in the experiments, the speed of convergence to the final result is different for each algorithm. Figure 6 presents the convergence curves of the compared algorithms on 6 benchmark functions. It can be seen from Figure 6 that NEDABC, MGABC and MABC-NS converge faster than the other four algorithms, because their food sources change multiple dimensions in a single search. In any case, compared with MGABC and MABC-NS, NEDABC still has an advantage in convergence speed. In addition, in terms of convergence accuracy, NEDABC shows superior or near-optimal precision on all six benchmark functions. Notably, on the Sum Power function ( $f_{16}$ ), NEDABC demonstrates significantly higher convergence accuracy compared to the next-best MGABC and MABC-NS, with MGABC even experiencing prolonged stagnation in local regions. This indicates that when dealing with high-order power functions with flat region traps, the NEDABC algorithm still maintains a good balance of exploitation and exploration capabilities, enabling efficient and rapid convergence.

Figure 6.
The convergence curves of 6 benchmark functions at $D = 30$ . (a) Sphere Function, (b) Ackley Funcion, (c) Schwefel 2.22 Funcion, (d) Alpine Funcion, (e) Sum squares Funcion, (f) Sum power Funcion.

To verify the stability of the NEDABC, Table 6 records the standard deviation of each algorithm on all benchmark functions after 30 runs. It can be found that the standard deviation of NEDABC is basically in the same order of magnitude as that in Table 4, which indicates its strong stability. Besides, NEDABC exhibits the lowest standard deviation values on 11/16 functions and has the best overall ranking, demonstrating minimal outcome variance across 30 independent runs. It is notable that in the benchmark functions NEDABC achieves 0 standard deviation in the benchmark functions Schaffer ( $f_{2}$ ), Griewank( $f_{3}$ ), Rastrigin ( $f_{5}$ ), Dropwave ( $f_{8}$ ), and Step ( $f_{13}$ ), that NEDABC successfully reaches the global optimum (as reported in Table 4), NEDABC achieves 0 standard deviation. This means that across 30 independent runs with randomly initialized populations, NEDABC consistently obtain stable and accurate optimization results, demonstrating insensitivity to initial population conditions and strong robustness. Furthermore, while the performance of NEDABC algorithm on Quartic with noise function ( $f_{12}$ ) appears suboptimal compared to its results on the other 15 benchmark functions, it still demonstrated the highest convergence accuracy and smallest standard deviation compared to the other 6 comparison algorithms across 30 independent trials. This observation indicates that the neighborhood search and evidence discounting mechanism in NEDABC can filter the noise neighbors to a certain extent, thereby endowing NEDABC with superior noise robustness relative to other ABC variants. However, the residual performance gap suggests there is still room for further enhancement.

Table 6.
Comparative results of standard deviation.

Functions ABC/best GABC MABC-NS MEABC MGABC ABC NEDABC

$f_{1}$ 1.03E−16 6.99E−17 5.35E−18 7.46E−17 3.98E−18 2.85E−12 3.31E−18

$f_{2}$ 0.00E+00 0.00E+00 0.00E+00 5.50E−06 0.00E+00 1.34E−09 0.00E+00

$f_{3}$ 1.33E−03 2.82E−08 0.00E+00 1.96E−15 0.00E+00 1.13E−06 0.00E+00

$f_{4}$ 1.07E+01 5.03E+00 3.27E−01 4.87E+00 2.34E−01 5.06E+00 1.31E−17

$f_{5}$ 2.69E−14 5.12E−13 2.34E+00 2.68E−14 0.00E+00 1.79E−01 0.00E+00

$f_{6}$ 3.32E−15 1.69E−10 1.78E−15 9.82E−14 0.00E+00 1.90E−06 1.77E−15

$f_{7}$ 9.91E−17 8.90E−17 6.57E−02 8.74E−17 9.95E−17 7.03E−14 2.76E−18

$f_{8}$ 0.00E+00 0.00E+00 0.00E+00 2.56E−11 0.00E+00 0.00E+00 0.00E+00

$f_{9}$ 1.62E−16 2.65E−11 8.26E−18 1.20E−14 6.45E−18 1.35E−07 8.80E−18

$f_{10}$ 7.14E−01 1.42E+00 7.10E−18 1.16E+00 1.40E−08 3.19E+00 4.32E−16

$f_{11}$ 3.15E+03 2.40E+03 4.84E−18 2.15E+03 2.90E−18 1.55E+03 3.36E−19

$f_{12}$ 7.00E−03 6.20E−02 8.65E−05 5.40E−03 7.26E−05 1.42E−02 2.35E−05

$f_{13}$ 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00

$f_{14}$ 1.09E−15 2.70E−05 7.81E−18 4.17E−14 6.21E−18 2.71E−04 1.01E−17

$f_{15}$ 9.45E−17 9.32E−17 6.13E−18 1.05E−16 3.14E−18 2.74E−13 4.08E−18

$f_{16}$ 9.11E−18 8.30E−18 1.11E−18 1.12E−17 1.96E−19 1.05E−17 3.13E−20

Total ranking 69 70 44 75 27 90 23

Bold type indicate the best performance among all compared algorithms under the corresponding condition.

In order to further investigate the effects of different degrees of data fluctuations on algorithm performance, additional perturbation experiments were conducted on three representative benchmark functions: the unimodal function Rosenbrock $f_{4}$ , the multimodal functions Griewank $f_{3}$ and Rastrigin $f_{5}$ . Three types of noise were introduced: Gaussian noise with a mean of 0 and standard deviations of 0.01 and 0.1, as well as uniformly distributed noise within [0, 1]. Each experiment was independently repeated 30 times to ensure statistical reliability. The results under different noise conditions for each algorithm are summarized in Table 7. It can be observed that under Gaussian noise conditions, the NEDABC algorithm demonstrates consistent convergence accuracy across both unimodal and multimodal functions. In the presence of mild noise (Gaussian noise with a standard deviation of 0.01), the convergence accuracy ranges between $4.0 \times 10^{- 2}$ and $4.5 \times 10^{- 2}$ . When the standard deviation increases to 0.1, the accuracy drops to a range between $4.0 \times 10^{- 1}$ and $4.5 \times 10^{- 1}$ . Despite the differing characteristics of the original functions, the NEDABC algorithm achieves stable and reliable convergence. Notably, for the Rosenbrock function $f_{4}$ , while most comparison algorithms yield results exceeding 10, which is much larger than the actual global optimum, NEDABC still delivers outputs with accuracy comparable to those achieved on other functions. Additionally, some comparison algorithms, such as ABC/best and MEABC, achieve better performance than NEDABC on certain functions under Gaussian noise. However, due to their inconsistent performance across different functions, they do not outperform NEDABC in overall ranking. Interestingly, the original ABC algorithm ranks higher than most ABC variants on the Rastrigin function $f_{5}$ and Rosenbrock function $f_{4}$ , suggesting that it possesses relatively good robustness to Gaussian noise. Under uniformly distributed noise, the NEDABC algorithm ranks first across all three benchmark functions, indicating strong stability in the presence of uniform disturbances. Overall, NEDABC algorithm outperforms the six other comparison algorithms, demonstrating superior robustness and stable performance across varying noise conditions.

Table 7.
Comparative results under different noise perturbations.

Functions ABC/best GABC MABC-NS MEABC MGABC ABC NEDABC

$f_{3} - N (0, {0.01}^{2})$ 2.43E−02 5.07E−02 4.18E−02 2.47E−02 4.37E−02 7.44E−01 4.35E−02

$f_{3} - N (0, {0.1}^{2})$ 8.28E−01 8.69E−01 4.36E−01 8.62E−01 4.40E−01 9.84E−01 4.19E−01

$f_{3} - U (0, 1)$ 1.07E+00 1.09E+00 1.10E−03 1.09E+00 3.81E−04 1.19E+00 9.94E−05

$f_{4} - N (0, {0.01}^{2})$ 1.99E+02 1.56E+02 2.74E+02 1.56E+02 2.78E+02 1.21E+02 4.15E−02

$f_{4} - N (0, {0.1}^{2})$ 2.19E+02 1.83E+02 2.82E+02 1.65E+02 2.81E+02 1.13E+02 4.17E−01

$f_{4} - U (0, 1)$ 2.25E+02 1.71E+02 2.83E+02 1.85E+02 2.85E+02 1.25E+02 2.08E−04

$f_{5} - N (0, {0.01}^{2})$ 1.62E−02 1.89E−02 4.15E−02 1.34E−02 4.24E−02 8.60E−03 4.30E−02

$f_{5} - N (0, {0.1}^{2})$ 1.84E−01 1.48E−01 4.21E−01 1.47E−01 4.20E−01 9.17E−02 4.34E−01

$f_{5} - U (0, 1)$ 7.11E−02 1.19E−01 1.10E−03 1.04E−01 3.55E−04 3.09E−01 8.83E−05

Total ranking 35 42 41 31 43 37 24

Bold type indicate the best performance among all compared algorithms under the corresponding condition.

To evaluate the scalability and stability of the NEDABC algorithm, the same set of experiments was conducted with $D = 50$ . Table 8 presents the average results over 30 independent runs for each algorithm. The results of NEDABC show slight fluctuations as the dimensionality increases. However, NEDABC still maintains a precision below $10^{- 17}$ on 11/14 functions (excluding $f_{2}$ and $f_{8}$ , which have fixed dimensions), which is consistent with its performance at $D = 30$ . The only major difference is in the result of Schwefel 2.21 function ( $f_{10}$ ), which may be caused by a single unsuccessful run. Nevertheless, the overall impact of increased dimensionality on NEDABC remains minimal. Notably, NEDABC’s overall ranking even improves by one position compared to that in Table 4, which shows that the NEDABC algorithm has strong dimensionality scalability.

Table 8.
Comparative results: $D = 50$ for 14 functions.

Functions ABC/best GABC MABC-NS MEABC MGABC ABC NEDABC

$f_{1}$ 2.92E−15 2.40E−10 2.21E−17 4.95E−14 1.40E−17 8.18E−07 8.59E−18

$f_{3}$ 4.29E−09 3.73E−04 0.00E+00 3.61E−09 0.00E+00 1.77E−04 0.00E+00

$f_{4}$ 5.92E+01 5.11E+01 4.16E+01 4.99E+01 4.82E+01 4.57E+01 4.14E−17

$f_{5}$ 4.68E−13 4.61E−01 0.00E+00 1.00E−09 0.00E+00 4.42E+00 0.00E+00

$f_{6}$ 3.71E−08 3.64E−05 2.43E−15 7.47E−07 8.88E−16 8.09E−03 2.55E−15

$f_{7}$ 1.38E−15 2.32E−11 3.16E−01 5.09E−15 3.20E+00 1.06E−07 6.82E−18

$f_{9}$ 1.56E−08 1.10E−05 4.31E−17 1.44E−07 4.14E−17 1.04E−03 2.43E−17

$f_{10}$ 3.04E+01 3.87E+01 4.44E−17 3.76E+01 4.01E−17 4.97E+01 2.37E−13

$f_{11}$ 4.28E+04 4.30E+04 1.48E−17 3.86E+04 1.26E−17 4.12E+04 3.77E−19

$f_{12}$ 7.91E−02 1.03E−01 1.86E−04 1.08E−01 1.08E−04 2.10E−01 3.93E−05

$f_{13}$ 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00 0.00E+00

$f_{14}$ 8.83E−08 2.12E−03 4.34E−17 6.28E−07 4.03E−17 3.85E−02 1.48E−17

$f_{15}$ 1.93E−15 8.01E−11 2.08E−17 1.80E−14 1.35E−17 6.18E−07 7.19E−18

$f_{16}$ 2.92E−17 3.03E−17 4.42E−18 3.63E−17 7.62E−19 1.07E−15 1.19E−19

Total ranking 57 77 36 64 30 83 18

Bold type indicate the best performance among all compared algorithms under the corresponding condition.

5. Practical case: optimal operation of WFGD system

Functions	ABC/best	GABC	MABC-NS	MEABC	MGABC	ABC	NEDABC
$f_{1}$	5.89E−16	5.27E−16	1.91E−17	5.40E−16	9.98E−18	3.29E−12	6.79E−18
$f_{2}$	0.00E+00	0.00E+00	0.00E+00	1.69E−06	0.00E+00	3.46E−10	0.00E+00
$f_{3}$	2.47E−04	7.08E−12	0.00E+00	5.63E−16	0.00E+00	2.45E−07	0.00E+00
$f_{4}$	2.31E+01	1.84E+01	2.13E+01	1.73E+01	2.77E+01	1.39E+01	1.22E−17
$f_{5}$	5.12E−14	2.79E−13	4.97E−01	1.89E−14	0.00E+00	3.32E−02	0.00E+00
$f_{6}$	4.10E−14	4.71E−10	2.66E−15	4.30E−13	8.88E−16	5.02E−06	2.55E−15
$f_{7}$	5.89E−16	5.28E−16	1.04E−01	5.47E−16	5.96E−16	7.73E−14	4.93E−18
$f_{8}$	−1.00E+00	−1.00E+00	−1.00E+00	−1.00E+00	−1.00E+00	−1.00E+00	−1.00E+00
$f_{9}$	1.70E−15	8.63E−11	4.19E−17	2.95E−14	3.47E−17	4.68E−07	2.16E−17
$f_{10}$	4.37E+00	8.00E+00	4.27E−17	8.49E+00	9.53E−09	1.92E+01	1.24E−16
$f_{11}$	1.13E+04	1.18E+04	1.30E−17	1.04E+04	8.64E−18	1.01E+04	4.12E−19
$f_{12}$	2.25E−02	2.79E−02	1.50E−04	2.93E−02	7.92E−05	6.51E−02	3.55E−05
$f_{13}$	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
$f_{14}$	2.65E−15	2.48E−05	4.55E−17	6.20E−14	3.68E−17	2.33E−04	1.27E−17
$f_{15}$	6.40E−16	5.32E−16	2.07E−17	5.72E−16	9.76E−18	4.19E−13	6.43E−18
$f_{16}$	2.37E−17	2.09E−17	1.89E−18	2.73E−17	1.75E−19	2.76E−17	2.56E−20
Total ranking	67	66	48	70	35	88	18

		ABC/best	GABC	MABC-NS	MEABC	MGABC	ABC	NEDABC
$f_{2}$	0	30	213.9	30	283.1	30	126.4	14	219.6	30	114.0	23	574.7	30	99.5
$f_{3}$	0	6	859.7	0		30	82.1	9	843.8	30	51.7	0		30	36.5
$f_{5}$	0	5	653.4	0		28	325.8	20	779.5	30	50.4	0		30	32.2
$f_{8}$	−1	30	118.1	30	152.5	30	39.8	28	118.5	30	46.2	30	282.1	30	50.8
$f_{13}$	0	30	132.9	30	167.8	30	19.2	30	148.4	30	11.0	30	202.0	30	7.2
$f_{14}$	0	0		0		0		0		0		0		4	66.75

Functions	ABC/best	GABC	MABC-NS	MEABC	MGABC	ABC	NEDABC
$f_{1}$	1.03E−16	6.99E−17	5.35E−18	7.46E−17	3.98E−18	2.85E−12	3.31E−18
$f_{2}$	0.00E+00	0.00E+00	0.00E+00	5.50E−06	0.00E+00	1.34E−09	0.00E+00
$f_{3}$	1.33E−03	2.82E−08	0.00E+00	1.96E−15	0.00E+00	1.13E−06	0.00E+00
$f_{4}$	1.07E+01	5.03E+00	3.27E−01	4.87E+00	2.34E−01	5.06E+00	1.31E−17
$f_{5}$	2.69E−14	5.12E−13	2.34E+00	2.68E−14	0.00E+00	1.79E−01	0.00E+00
$f_{6}$	3.32E−15	1.69E−10	1.78E−15	9.82E−14	0.00E+00	1.90E−06	1.77E−15
$f_{7}$	9.91E−17	8.90E−17	6.57E−02	8.74E−17	9.95E−17	7.03E−14	2.76E−18
$f_{8}$	0.00E+00	0.00E+00	0.00E+00	2.56E−11	0.00E+00	0.00E+00	0.00E+00
$f_{9}$	1.62E−16	2.65E−11	8.26E−18	1.20E−14	6.45E−18	1.35E−07	8.80E−18
$f_{10}$	7.14E−01	1.42E+00	7.10E−18	1.16E+00	1.40E−08	3.19E+00	4.32E−16
$f_{11}$	3.15E+03	2.40E+03	4.84E−18	2.15E+03	2.90E−18	1.55E+03	3.36E−19
$f_{12}$	7.00E−03	6.20E−02	8.65E−05	5.40E−03	7.26E−05	1.42E−02	2.35E−05
$f_{13}$	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
$f_{14}$	1.09E−15	2.70E−05	7.81E−18	4.17E−14	6.21E−18	2.71E−04	1.01E−17
$f_{15}$	9.45E−17	9.32E−17	6.13E−18	1.05E−16	3.14E−18	2.74E−13	4.08E−18
$f_{16}$	9.11E−18	8.30E−18	1.11E−18	1.12E−17	1.96E−19	1.05E−17	3.13E−20
Total ranking	69	70	44	75	27	90	23

Functions	ABC/best	GABC	MABC-NS	MEABC	MGABC	ABC	NEDABC
$f_{3} - N (0, {0.01}^{2})$	2.43E−02	5.07E−02	4.18E−02	2.47E−02	4.37E−02	7.44E−01	4.35E−02
$f_{3} - N (0, {0.1}^{2})$	8.28E−01	8.69E−01	4.36E−01	8.62E−01	4.40E−01	9.84E−01	4.19E−01
$f_{3} - U (0, 1)$	1.07E+00	1.09E+00	1.10E−03	1.09E+00	3.81E−04	1.19E+00	9.94E−05
$f_{4} - N (0, {0.01}^{2})$	1.99E+02	1.56E+02	2.74E+02	1.56E+02	2.78E+02	1.21E+02	4.15E−02
$f_{4} - N (0, {0.1}^{2})$	2.19E+02	1.83E+02	2.82E+02	1.65E+02	2.81E+02	1.13E+02	4.17E−01
$f_{4} - U (0, 1)$	2.25E+02	1.71E+02	2.83E+02	1.85E+02	2.85E+02	1.25E+02	2.08E−04
$f_{5} - N (0, {0.01}^{2})$	1.62E−02	1.89E−02	4.15E−02	1.34E−02	4.24E−02	8.60E−03	4.30E−02
$f_{5} - N (0, {0.1}^{2})$	1.84E−01	1.48E−01	4.21E−01	1.47E−01	4.20E−01	9.17E−02	4.34E−01
$f_{5} - U (0, 1)$	7.11E−02	1.19E−01	1.10E−03	1.04E−01	3.55E−04	3.09E−01	8.83E−05
Total ranking	35	42	41	31	43	37	24

Functions	ABC/best	GABC	MABC-NS	MEABC	MGABC	ABC	NEDABC
$f_{1}$	2.92E−15	2.40E−10	2.21E−17	4.95E−14	1.40E−17	8.18E−07	8.59E−18
$f_{3}$	4.29E−09	3.73E−04	0.00E+00	3.61E−09	0.00E+00	1.77E−04	0.00E+00
$f_{4}$	5.92E+01	5.11E+01	4.16E+01	4.99E+01	4.82E+01	4.57E+01	4.14E−17
$f_{5}$	4.68E−13	4.61E−01	0.00E+00	1.00E−09	0.00E+00	4.42E+00	0.00E+00
$f_{6}$	3.71E−08	3.64E−05	2.43E−15	7.47E−07	8.88E−16	8.09E−03	2.55E−15
$f_{7}$	1.38E−15	2.32E−11	3.16E−01	5.09E−15	3.20E+00	1.06E−07	6.82E−18
$f_{9}$	1.56E−08	1.10E−05	4.31E−17	1.44E−07	4.14E−17	1.04E−03	2.43E−17
$f_{10}$	3.04E+01	3.87E+01	4.44E−17	3.76E+01	4.01E−17	4.97E+01	2.37E−13
$f_{11}$	4.28E+04	4.30E+04	1.48E−17	3.86E+04	1.26E−17	4.12E+04	3.77E−19
$f_{12}$	7.91E−02	1.03E−01	1.86E−04	1.08E−01	1.08E−04	2.10E−01	3.93E−05
$f_{13}$	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00	0.00E+00
$f_{14}$	8.83E−08	2.12E−03	4.34E−17	6.28E−07	4.03E−17	3.85E−02	1.48E−17
$f_{15}$	1.93E−15	8.01E−11	2.08E−17	1.80E−14	1.35E−17	6.18E−07	7.19E−18
$f_{16}$	2.92E−17	3.03E−17	4.42E−18	3.63E−17	7.62E−19	1.07E−15	1.19E−19
Total ranking	57	77	36	64	30	83	18

The wet flue gas desulfurization (WFGD) system is the most widely used desulfurization technology in coal-fired power plants.³⁸ However, its significant material and energy consumption inevitably leads to high operational costs. Optimizing its operating parameters can reduce the waste of resources in this system under the premise of meeting environmental requirements. Due to the complexity of the desulfurization mechanism, frequent unit load fluctuations, and severe perturbations, it is challenging to determine the optimal operating parameters based on traditional mechanism models. Therefore, in this section we propose an operation optimization method for WFGD system based on data-driven model and NEDABC algorithm, which is used to illustrate the practicality of NEDABC algorithm.

5.1. System description

A typical WFGD system is presented in Figure 7. The primary desulfurization reactions take place in the absorption tower, utilizing limestone slurry as the desulfurization absorber. The raw flue gas enters the absorption tower from the bottom after being pressurized by the booster fan, and flows through the absorption tower from the bottom up. The on-site configured limestone slurry is drawn from the slurry tank to the slurry pool at the bottom of the absorption tower, and then sent to the spray layer by the slurry recirculating pump to be sprayed from top to bottom. Limestone slurry and rising flue gas flow in the reverse direction simultaneously with the heat exchange and chemical reaction, so as to remove SO $_{2}$ in the flue gas. The purified flue gas is discharged through the stack after liquid droplets are removed by the mist eliminator. The reactant CaSO $_{3}$ back in the slurry tank is forcibly oxidized by air blown in from the oxidation fan to generate CaSO $_{4}$ , and ultimately, gypsum is produced and discharged as a by-product. The overall chemical reaction of the above desulphurization process is summarized as follows:

\begin{aligned} 2 {CaCO}_{3} (\underline{s}) + 2 {SO}_{2} + O_{2} + 4 H_{2} O \to 2 {CaSO}_{4} \cdot 2 H_{2} O (\underline{s}) + 2 {CO}_{2} \end{aligned}

(17)

in which

(\underline{s})

denotes the solid state.

Figure 7.

Schematic diagram of WFGD system.

The desulfurization process is accompanied by a significant amount of energy and material consumption, with the primary energy-consuming equipment such as slurry circulation pumps and oxidation fans, etc., and the material consumption mainly from limestone raw material consumption and water consumption. Studies have indicated that various factors can affect the efficiency of desulfurization reaction, such as flue gas flow rate, SO $_{2}$ concentration of the original flue gas, limestone slurry circulation volume and concentration, and slurry pH value.^39,40 In recent years, with a large number of new energy generation such as wind and solar power connected to the grid, coal-fired units need to operate flexibly under a wide load range to meet peak-shaving demands. This variability of conditions makes it challenging for operators to accurately determine operating parameter settings promptly, which can lead to high desulfurization costs or non-compliant SO $_{2}$ emissions.

Therefore, the optimization objective of the wet flue gas desulfurization (WFGD) system is to reduce the desulfurization costs as much as possible under the premise of complying with the SO $_{2}$ emission standard. This can be achieved by establishing data-driven models for the WFGD system, including a model of SO $_{2}$ concentration in the outlet flue gas and a model of the desulphurization cost. On this basis, NEDABC algorithm can be employed to solve the model, ensuring optimal performance and cost-effectiveness.

The data for this study were sourced from the distributed control system (DCS) of a 1000MW power plant’s WFGD system, which includes five slurry circulation pumps and five oxidation fans. The dataset spans six months from January 1, 2022, to June 30, 2022, with samples taken every minute, covering all operating conditions and comprising a total of 259,200 data points. Table 9 lists all variables used in this study.

Table 9.

Summary of desulfurization system variables.

ID	Variable	Abbreviation	Maximum value	Minimum value	Unit
$1$	unit load	P	1056.72	362.42	MW
$2$	inlet flue gas flow rate	Q $_{in}$	4942158	2604752	m $^{3}$ /h
$3$	inlet flue gas SO $_{2}$ concentration	C $_{in}$	3038.92	1086.33	mg/m $^{3}$
$4$	pH value	pH	8.24	3.69	/
$5$	slurry level height	H	9.13	7.42	m
$6$	flow rate of limestone slurry supply	q	50.00	3.00	m $^{3}$ /s
$7$	liquid-to-gas ratio	L/G	13.23	4.26	L/m $^{3}$
$8$	unit cost of desulphurization	I $_{S O_{2}}$	0.836	0.199	yuan/kg
$9$	outlet flue gas SO $_{2}$ concentration	C $_{out}$	19.58	0.08	mg/m $^{3}$

5.2. Modeling and optimization

The optimization of WFGD system involves two key factors: whether the SO $_{2}$ concentration of outlet flue gas meets emission standards, and whether the system’s economic performance is optimal, that is, the unit desulfurization cost is high or low. The objective of this study is to find the most cost-effective operational strategy under various operating conditions, while ensuring that the outlet flue gas SO $_{2}$ concentration meets the required constraints.

The operating cost of the WFGD system primarily consists of the electricity consumption cost of each sub-system equipment, the absorbent (i.e., limestone) consumption cost, the environmental tax, and the water consumption cost. Among them, the first two items account for about 80% of the total cost, indicating significant potential for energy savings. Water consumption constitutes only 6% of the total cost, with over 90% of it being water vapor carried away by desulfurized flue gas,⁴¹ leaving little room for optimization; thus, it is not considered in this study. The unit desulfurization cost is the expense consumed to remove 1 kg of SO $_{2}$ under the ultra-low emission standard, and the lower the unit desulfurization cost is, the better the economy of WFGD system is. The formula for calculating unit desulfurization cost is as follows:

\begin{aligned} I_{S O_{2}} = \frac{C_{e} + C_{a} + C_{t}}{m_{S O_{2}}}, \end{aligned}

(18)

where

C_{e}

C_{a}

and

C_{t}

represent the electricity consumption cost, the absorbent consumption cost and the environmental tax, respectively, and they are calculated as shown in Eqs. (19)–(21).

m_{S O_{2}}

represents the amount of SO

_{2}

removed by the WFGD system per hour.

\begin{aligned} C_{e} = \sum \sqrt{3} U I \cos ϕ \cdot c_{e}, \end{aligned}

(19)

\begin{aligned} C_{a} = m_{a} \cdot c_{a}, \end{aligned}

(20)

\begin{aligned} C_{t} = λ \cdot c_{t} \cdot \frac{C_{out} Q}{10^{6} \cdot 0.95}, \\ {\begin{cases} λ = 1, & 0.5 \cdot 35 < C_{out} \leq 35 \\ λ = 0.5, & 0.3 \cdot 35 < C_{out} \leq 0.5 \cdot 35 \\ λ = 0.25, & C_{out} \leq 0.3 \cdot 35 \end{cases} \end{aligned}

(21)

where,

U, I, \cos ϕ

and

c_{e}

represent voltage, current, power factor, and unit price of electricity, respectively;

m_{a}, c_{a}

denote limestone consumption and limestone unit price;

λ, c_{t}, C_{out}

and

Q

are the environmental tax reduction coefficient, environmental tax unit price, outlet flue gas SO

_{2}

concentration and flue gas flow rate, respectively.

Based on the above calculation of unit desulfurization cost, the objective function can be established as follows:

\begin{aligned} min I_{S O_{2}} = \frac{C_{e} + C_{a} + C_{t}}{m_{S O_{2}}} . \end{aligned}

(22)

Considering the ultra-low emission requirements and the operating range of the operating variables, the constraints are determined

{\begin{cases} C_{out} = f (P, Q_{in}, C_{in}, p H, H, q, L / G), \\ C_{out} \leq 35, \\ η \geq 90 %, \\ 4.5 \leq p H \leq 6.0, \\ 7.5 \leq H \leq 9.0, \\ 0 \leq q \leq 50.0, \\ 1 \cdot q_{re} / Q_{i n} \leq L / G \leq 5 \cdot q_{re} / Q_{i n}, \end{cases}

where

η

represents desulphurization efficiency,

q_{re}

represents the flow rate of a single slurry recirculation pump.

To derive the objective function and the outlet flue gas SO $_{2}$ concentration in the constraints for a certain operating condition, it is imperative to establish predictive models for these quantities. In this study, the evidential regression (EVREG) model we previously proposed, which has been validated in building energy consumption prediction,⁴² is adopted to model the WFGD system. EVREG model is a data-driven model based on theory of belief functions.⁴³ The evidence base is constructed based on historical operational data and is constructed using the same methodology as used in the NEDABC algorithm proposed in this study. Each nearest neighbor of the sample to be predicted in the evidence base can provide a piece of evidence for its output, with reliability of the evidence varying with distance between the nearest neighbor and the sample to be predicted. The reliability of evidence from different sources is discounted by using Eq. (10), and the evidence from $K$ nearest neighbors is fused according to Eq. (7). After the final mass function is calculated, EVREG can provide the expected output of the quantity to be predicted:

\begin{aligned} \hat{y} = \sum_{i = 1}^{K} m (y = y_{i}) \cdot y_{i} + m (y = Ω) \cdot \frac{y_{min} + y_{max}}{2}, \end{aligned}

(23)

For the EVREG model, there are two important parameters to be learned: neighborhood size $K$ and structural parameter $γ$ which controls the discounted decay rate. Specifically, empirical studies suggest that $K$ typically lies within the range of 3–40. Thus, $[\underline{K}, \bar{K}]$ can be set for $K$ , iterating over all possible values within this range and learning the corresponding $γ$ for each $K$ . The optimal combination is then selected based on the best predictive performance. For the choice of $γ$ , as suggested in Petit-Renaud and Denoeux,⁴³ leave-one-out (LOO) method is applied to optimize $γ$ by minimizing the following criterion.

\begin{aligned} C V (γ) = \frac{1}{N} \sum_{i = 1}^{N} (y_{i} - \hat{y_{i}} [x_{i}, T^{- i}, γ])^{2}, \end{aligned}

(24)

where

x_{i}

is the input parameters of the training sample and

y_{i}

is the output corresponding to

x_{i}

, which is the variable to be predicted.

\hat{y_{i}} [x_{i}, T^{- i}, γ]

is the prediction about

y_{i}

based on the training set without example

(x_{i}, y_{i})

. The estimator

\hat{γ}

of parameter

γ

is then obtained by minimizing this criterion.

\begin{aligned} \hat{γ} = \underset{γ}{\arg min} C V (γ) . \end{aligned}

(25)

This model proved to be particularly effective in dealing with imprecision and uncertainty of the data, improving the accuracy of prediction results. In order to assess the accuracy of the model, the mean absolute error (MAE), mean absolute percentage error (MAPE) and root mean square error (RMSE) were used as evaluation metrics, mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean square error (RMSE) are used as evaluation metrics, and calculated as shown in Eqs. (26)–(28).

\begin{aligned} MAE = \frac{1}{n} \sum_{i = 1}^{n} | \hat{y_{i}} - y_{i} |, \end{aligned}

(26)

\begin{aligned} MAPE = \frac{1}{n} \sum_{i = 1}^{n} \frac{| \hat{y_{i}} - y_{i} |}{y_{i}}, \end{aligned}

(27)

\begin{aligned} RMSE = \sqrt{\frac{\sum_{i = 1}^{n} (y_{i} - \hat{y_{i}})^{2}}{n}}, \end{aligned}

(28)

where

n

is the number of samples,

y_{i}

and

\hat{y_{i}}

are the real value and predicted value of building energy consumption respectively. All metrics measure the difference between the predicted value and the true value. The smaller the metrics, the better the accuracy of the model.

On this basis, the overall flow of WFGD system operation optimization can be obtained, as shown in Figure 8. It consists of the following steps: (1) Collect historical operational data of WFGD system, perform data pre-processing, and obtain steady-state data samples for modeling; (2) Based on evidential regression model, establish prediction models for the unit desulfurization cost and outlet flue gas SO $_{2}$ concentration of WFGD system; (3) Determine the boundary conditions for the optimization condition, that is, unit load and inlet flue gas parameters. Initialize the population ${x_{i} = ({pH}_{i}, H_{i}, q_{i}, L / G_{i}) | i = 1, 2, \dots, S N}$ , and calculate the fitness using Eqs. (22) and (2) based on the models established in step (2); (4) Based on the evolutionary mechanism proposed in Algorithm 1, update the population and calculate fitness until the iteration termination conditions are satisfied. In this study, the iteration ends when either the variation in the best fitness of NEDABC is less than 10⁻⁶, or the maximum number of function evaluations $m a x F E s$ is reached, thus the optimal operating parameters under current condition can be obtained.

Figure 8.

Workflow of WFGD system operation optimization.

5.3. Results and discussion

Due to environmental or human interference and equipment failures, anomalous or missing data are inevitably present in the DCS database and they were removed through statistical analysis. The quartiles range rule was adopted to determine whether the data are outliers. In addition, due to significant fluctuations during load adjustment processes, operational parameters at these times cannot accurately reflect the true state of the desulfurization system. Therefore, the steady state detection algorithm based on the sliding window standard deviation is employed to extract steady state data. Given the excessive number of samples, density biased sampling (DBS)⁴⁴ is used to obtain a moderate-sized dataset that covers the full range of operating conditions. We ultimately obtained 22,889 steady state data sets with steady periods lasting over 20 minute, and extracted 8,000 sample sets for modeling.

The evidential regression model was adopted to predict unit desulfurization cost and outlet SO $_{2}$ concentration. For the obtained steady-state samples, the training and test sets were divided in a ratio of 0.8:0.2. It should be noted that in this study, the electricity price was set at 0.5 yuan per kWh, and the limestone price was set at 80 yuan per ton.

The prediction results of the model on the test set are shown in Figure 9. Figure 9(a) shows the comparison between the predicted and actual values of the unit desulfurization cost, as well as the corresponding errors. It can be seen that the two align very well, with prediction errors ranging between [ $- 0.06$ , 0.06]. The model’s MAE is 0.0054, MAPE is 1.47%, and RMSE is 0.0076. Figure 9(b) illustrates the comparison between the predicted and actual values of outlet flue gas SO $_{2}$ concentration, along with the corresponding errors. It can be observed that the overall error of the model is small, and the fluctuations in the predicted values are similar to those in the measured values, with most relative prediction errors falling between [ $- 0.1$ , 0.1]. However, it can be observed that some predicted outlet concentrations exhibit relatively large errors. This is primarily because these concentration values deviate from the distribution range of most training samples. In other words, the output values of their nearest neighbors in the training set differ significantly from the true values of the test samples, leading to larger prediction errors. Considering the characteristics of the WFGD system, it may be attributed to sudden changes in system or equipment operating conditions, resulting in reduced desulfurization efficiency and a temporary spike in outlet flue gas SO $_{2}$ concentration. While the sample distribution of unit desulfurization cost is more uniform, the prediction accuracy of the model is much higher. Nevertheless, overall, the model still demonstrates strong performance in predicting outlet concentrations, with an MAE of 0.3954, a MAPE of 8.45%, and an RMSE of 0.8495. These results indicate that both models achieve satisfactory prediction accuracy, making them sufficiently reliable to support subsequent optimization tasks.

Figure 9.

Comparison of prediction results and actual value. (a) Prediction of unit desulfurization cost, (b) Prediction of outlet flue gas SO $_{2}$ concentration.

In the actual operation of coal-fired power plant units, the unit load and coal quality conditions are constantly changing, which leads to continuous variations in the flow rate and SO $_{2}$ concentration of the flue gas entering WFGD system. Therefore, in this section, we divide the operating conditions based on the unit load, inlet flue gas flow and SO $_{2}$ concentration, and use the NEDABC algorithm proposed in Section 3 to solve the problem. It should be noted that in this case, the convergence condition was set to a maximum $F E S$ of 5000. And in practical applications, convergence can be determined based on whether the change in the fitness value during iterations is sufficiently small.

When the unit operates at 55%, 75%, and 95% load levels, two sets of typical operating conditions were selected for optimization under each load. The proposed NEDABC algorithm is compared with the above six comparison algorithms under these six typical operating conditions. Each algorithm was independently run 30 times with the same hyperparameter settings as described in Section 4. Table 10 presents the average optimization results over 30 independent runs for six typical operating conditions, Table 11 lists the corresponding standard deviations, and Figure 10 gives the convergence curves for each algorithm, where the working conditions a-e in the tables correspond to those in Figure 10. It can be observed that the proposed NEDABC algorithm consistently achieves the lowest minimum value among all algorithms across six operating conditions, and its overall ranking is higher than that of the other algorithms. Combined with Figure 10, it can be found that, the variation in the best fitness of NEDABC is less than 10⁻⁶ when the $F E s$ is less than 1500, indicating that the convergence criterion for optimization has been satisfied. This implies that the NEDABC optimization algorithm is capable of achieving effective convergence after no more than fifteen iterations. It is also worth noting that, unlike the tests on benchmark functions, the convergence speed of all algorithms improves significantly in the WFGD system optimization problem. While NEDABC still converges slightly faster than the other algorithms, the performance gap is obviously not as large as that observed on benchmark functions. This phenomenon is considered to have two main reasons: 1) The WFGD optimization problem considers only four optimization parameters, selected based on actual system characteristics, which is a significantly lower dimensionality than the 30 or 50 dimensions used in benchmark function tests. 2) In order to ensure reliable prediction performance of the EVREG model for desulfurization cost and outlet SO $_{2}$ concentration, all input variables were normalized to the interval of [0,1], thereby constraining the search space and accelerating convergence. Nevertheless, as shown in Table 11, NEDABC achieves the lowest standard deviations in most cases and the highest overall ranking, demonstrating superior stability and consistency compared to other algorithms. This robustness is particularly critical in real-world engineering applications. Therefore, the proposed NEDABC algorithm has better utility in the field and is able to determine the optimal operating parameters promptly and reliably.

Figure 10.

The convergence curves of the best fitness. (a-b) Convergence curves at 55% load, (c-d) Convergence curves at 75% load, (e-f) Convergence curves at 95% load.

Table 10.

Comparison results under 6 working conditions.

Working conditions	ABC/best	GABC	MABC-NS	MEABC	MGABC	ABC	NEDABC
$a$	0.3042	0.3043	0.3042	0.3043	0.3042	0.3042	0.3042
$b$	0.2939	0.2939	0.2939	0.2939	0.2939	0.2939	0.2939
$c$	0.2854	0.2851	0.2860	0.2849	0.2855	0.2854	0.2849
$d$	0.2280	0.2280	0.2280	0.2280	0.2280	0.2280	0.2280
$e$	0.2725	0.2725	0.2725	0.2725	0.2725	0.2725	0.2725
$f$	0.2652	0.2652	0.2652	0.2652	0.2652	0.2652	0.2652
Total ranking	8	8	10	7	9	8	6

Bold type indicate the best performance among all compared algorithms under the corresponding condition.

Table 11.

Comparative results of standard deviation.

Working conditions	ABC/best	GABC	MABC-NS	MEABC	MGABC	ABC	NEDABC
$a$	5.57E−04	4.79E−04	3.52E−04	3.93E−04	4.53E−04	8.58E−04	1.58E−04
$b$	1.78E−06	3.57E−06	1.70E−06	1.72E−06	3.22E−06	3.05E−06	1.62E−06
$c$	0.0024	0.0019	0.0030	0.0024	0.0022	0.0024	0.0017
$d$	2.93E−07	4.24E−08	0.00E+00	2.93E−17	3.41E−11	1.26E−06	7.12E−11
$e$	4.76E−07	2.48E−07	4.07E−13	4.76E−06	1.86E−09	1.30E−07	1.19E−08
$f$	1.90E−09	7.69E−07	6.86E−12	4.90E−08	3.65E−12	1.23E−06	4.55E−12
Total ranking	30	30	14	24	19	34	12

Bold type indicate the best performance among all compared algorithms under the corresponding condition.

The changes in operating variables before and after optimization under these six operating conditions are displayed in Table 12. Taking the optimization results under 75% load as an example, it can be observed that under both sets of inlet flue gas parameters, the changes in pH and liquid level height before and after optimization are not significant, but the slurry supply flow rate is noticeably reduced, and the liquid-gas ratio is also slightly decreased, with the number of slurry circulation pumps on reduced to 2. After optimization, the outlet flue gas SO $_{2}$ concentration slightly increases but still meets the ultra-low emission requirements. The unit desulfurization cost decreases by 0.1034 and 0.0215 yuan/kg under the two conditions, representing reductions of 33.19 % and 7.04%, respectively, significantly improving the system’s economic efficiency.

Table 12.

Comparison of variables before and after optimization under different working conditions.

	Flue gas parameters			Operating variable
Unit load	Q $_{in}$	C $_{in}$	Before/After	pH	H	q	L/G	Unit cost	C $_{out}$
$55 %$	3313560	1515	Before	5.65	8.14	18.34	6.16	0.390	4.75
			After	5.63	8.09	4.67	5.77	0.304	5.79
	3521120	1860	Before	5.67	8.09	18.25	8.69	0.405	6.34
			After	5.62	8.02	5.61	6.05	0.294	6.77
$75 %$	3875500	2500	Before	5.19	7.87	12.17	10.53	0.305	4.78
			After	5.53	8.09	6.40	5.27	0.285	9.87
	4100438	1960	Before	5.55	8.08	20.60	7.46	0.331	5.82
			After	5.53	8.09	6.05	5.15	0.228	7.80
$95 %$	4435904	2244	Before	5.67	8.02	29.16	6.90	0.294	6.27
			After	5.61	7.93	12.42	6.95	0.273	9.73
	4547445	2560	Before	5.56	8.04	40.61	6.73	0.290	7.43
			After	5.60	8.04	29.19	6.89	0.265	7.94

From the optimization results under the three loads, the pH and liquid level height are generally maintained between 5–6 and 7.5–8.5 m, respectively. There is not much room for optimization, and the values show no significant changes before and after optimization. The main optimized variables are the slurry flow rate and liquid-gas ratio, especially the slurry flow rate shows particularly large optimization potential. This indicates that during actual operation of WFGD system, the consumption of limestone raw material is much higher than the amount required for desulfurization, which may be related to the production of gypsum as a byproduct. After optimization, the unit desulfurization cost are significantly reduced, allowing the system to operate more economically while still meeting environmental requirements.

6. Conclusions

This paper proposes a neighborhood evidence-driven artificial bee colony (NEDABC) algorithm. The aim of this new algorithm is to accelerate convergence and enhance the exploitation capability while maintaining the strong exploration capability. In the proposed NEDABC, onlooker bees select their search directions by considering the comprehensive information about food sources in their neighborhood, which is obtained by evidence fusion. The information about a food source includes its fitness value and its distance from other food sources in the neighborhood. For a single onlooker bee, all the better food sources in its neighborhood may become its search direction, and it is most likely to select the nearest food source with the highest fitness value. Under this mechanism, the neighborhood of each onlooker bee will be fully exploited. In the meantime, as there is no better food source in the neighborhood, the onlooker bee will approach the current optimal food source, thus accelerating the evolution of the global optimal food source. Results of numerical experiments on a set of benchmark functions demonstrate that the modification of NEDABC brings higher accuracy and faster convergence compared to other ABC variants. Meanwhile, the WFGD system of a 1000 MW power plant is used as a case study for optimization, and the comparison with ABC variant algorithms highlights the practicability and reliability of the proposed method for real-world application. Optimal values for key operating parameters, such as limestone slurry flow rate and liquid-gas ratio are obtained under different typical operating conditions. Furthermore, the comparison with field operating data confirmed the economic benefits of the optimization scheme, which provides a foundation for formulating efficient operational strategies for WFGD system.

Footnotes

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 52076037) and the Scientific and Technological Innovation Project of Carbon Emission Peak and Carbon Neutrality of Jiangsu Province (Grant No. BE2023090).

Credit author statement

Chao Liu: Methodology, Formal analysis, Data curation, Writing-Original draft. Yao Fu: Conceptualization, Writing-Original draft, Data curation. Zhi-gang Su: Resources, Project administration, Supervision, Funding acquisition.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China (Grant No. 52076037) and the Scientific and Technological Innovation Project of Carbon Emission Peak and Carbon Neutrality of Jiangsu Province (Grant No. BE2023090).

Conflict of interest

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Availability of data and material

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

ORCID iD

Chao Liu

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		ABC/best		GABC		MABC-NS		MEABC		MGABC		ABC		NEDABC
	Accept	SR	iters	SR	iters	SR	iters	SR	iters	SR	iters	SR	iters	SR	iters
$f_{2}$	0	30	213.9	30	283.1	30	126.4	14	219.6	30	114.0	23	574.7	30	99.5
$f_{3}$	0	6	859.7	0		30	82.1	9	843.8	30	51.7	0		30	36.5
$f_{5}$	0	5	653.4	0		28	325.8	20	779.5	30	50.4	0		30	32.2
$f_{8}$	−1	30	118.1	30	152.5	30	39.8	28	118.5	30	46.2	30	282.1	30	50.8
$f_{13}$	0	30	132.9	30	167.8	30	19.2	30	148.4	30	11.0	30	202.0	30	7.2
$f_{14}$	0	0		0		0		0		0		0		4	66.75

Neighborhood evidence-driven artificial bee colony algorithm with application for optimal operation of wet flue gas desulfurization system

Abstract

Keywords

1. Introduction

2. Background of artificial bee colony algorithm

3.1. Basic idea of NEDABC

3.3. Some insights with discussions

4.1.1. Impact of γ on performance of NEDABC

5.1. System description

Footnotes

Acknowledgements

Credit author statement

Funding

Conflict of interest

Availability of data and material

ORCID iD

References

4.1.1. Impact of $γ$ on performance of NEDABC