Sustainable Operation Strategy for Wet Flue Gas Desulfurization at a Coal-Fired Power Plant via an Improved Many-Objective Optimization

Abstract

Coal-fired power plants account for a large share of the power generation market in China. The mainstream method of desulfurization employed in the coal-fired power generation sector now is wet flue gas desulfurization. This process is known to have a high cost and be energy-/materially intensive. Due to the complicated desulfurization mechanism, it is challenging to improve the overall sustainability profile involving energy-, cost-, and resource-relevant objectives via traditional mechanistic models. As such, the present study formulated a data-driven many-objective model for the sustainability of the desulfurization process. We preprocessed the actual operation data collected from the desulfurization tower in a domestic ultra-supercritical coal-fired power plant with a 600 MW unit. The extreme random forest algorithm was adopted to approximate the objective functions as prediction models for four objectives, namely, desulfurization efficiency, unit power consumption, limestone supply, and unit operation cost. Three metrics were utilized to evaluate the performance of prediction. Then, we incorporated differential evolution and non-dominated sorting genetic algorithm-III to optimize the multiple parameters and obtain the Pareto front. The results indicated that the correlation coefficient (R2) values of the prediction models were greater than 0.97. Compared with the original operation condition, the operation under optimized parameters could improve the desulfurization efficiency by 0.25% on average and reduce energy, cost, and slurry consumption significantly. This study would help develop operation strategies to improve the sustainability of coal-fired power plants.

1. Introduction

Growth in emerging and developing economies is paralleled by an increase in electricity demand. Simultaneously, advanced economies are promoting electrification to decarbonize their industrial, heating, and transportation sectors. This would result in a global power demand increment at a substantially greater rate of 3% annually over the 2023–2025 period and a global coal-fired electricity generation rise by 1.5% [1]. According to the energy market report by the International Energy Agency [1], China is expected to consume one-third of the world’s electricity by 2025. Hitherto, coal-fired electricity will remain dominant in the Chinese energy structure during the next decades [2]. As a typical nonrenewable resource, coal itself is a high-sulfur and high-pollution energy [3]. As shown in Figure 1, coal mining presents several emergency situations, such as methane emissions, aerological risks, sulfur pollution, groundwater contamination, and land degradation, each with significant impacts. Methane is a potent greenhouse gas that poses explosion risks and contributes to global warming, accounting for about 7% of total mining-related hazards. Aerological risks, including ventilation failures, affect air quality and safety, comprising around 20% of hazards. However, sulfur pollution is the most serious issue, responsible for 25–35% of environmental and health risks. It leads to the release of sulfur dioxide, causing acid rain and respiratory problems, and contributes to fine particulate matter formation. Addressing sulfur emissions is crucial for mitigating broader ecological damage and improving air quality. Therefore, the Chinese government promulgated a stringent SO₂ regulation in 2020 to restrict SO₂ emission standards from 100 mg/Nm³ to 35 mg/Nm³. Although clean coal technology and ultra-supercritical technology have witnessed significant progress, the desulfurization process nevertheless has substantial environmental burdens given the considerable base of pollutant emissions at coal-fired power plants. The significant removal of SO₂ is achieved at the expense of high energy consumption and a large quantity of limestone slurry. Therefore, further improving the desulfurization efficiency and lowering energy/resources would contribute to the sustainable transformation of coal-fired power plants significantly.

Wet flue gas desulfurization (WFGD) is a more mature, efficient, and widely applied desulfurization technique compared with the dry and semi-dry desulfurization technologies [4,5]. There has been a body of WFGD research to optimize the desulfurization efficiency and overall performance. One typical approach is enhancing desulfurizing performance from a physical mechanism perspective, for example, modifying the crucial components and developing new sorbent compositions. Previous studies have attempted to develop cost-effective sorbents to improve the performance of desulfurization. Reference [6] adopted fly ash and bottom ash as solid sorbents to reduce SO₂ load in untreated flue gas, and the life cycle assessment method demonstrated that this approach would lower the energy consumption and environmental impacts by 4.0–5.0% and 3.0–5.0%, respectively [6]. Ref. [7] studied the effect of ammonia and ammonium ions on the performance of the WFGD absorber. They found that the addition of ammonia has a persistent effect on the absorption slurry pH as well as fresh CaCO₃ dosing regulation system and further influences the SO₂ removal [7]. The modified structure of the absorber is also expected to promote the gas-liquid mass transfer and induce better efficiency. For example, ref. [8] added a flow pattern-controlling device with an aperture plate in the spraying tower to enhance the desulfurization efficiency [8]. Ref. [9] installed a sieve plate scrubber in the tower and increased the removal efficiency by approximately 6% [9]. However, a thorough understanding of the mechanisms of desulfurization requires sophisticated knowledge of the physicochemical absorption and device characteristics. In addition, these methods of promoting SO₂ removal efficiency need extra large amounts of economic cost.

As the WFGD system is influenced by multiple operating parameters and inappropriate parameter settings may cause inefficient desulfurization and energy use [10], data-driven techniques have been promising options to enhance the sustainability of the WFGD system, irrespective of the underlying complicated absorption mechanism. Ref. [11] applied response surface methodology to optimize the inlet slurry concentration, absorption temperature, SO₂ concentration, and MnSO₄·H₂O amount for efficient SO₂ removal [11]. Ref. [12] employed the K-means and C-means algorithms as analytical methods and information entropy as an evaluation index to figure out the optimal conditions of operation of the WFGD system based on historical data [12]. Ref. [13] proposed a hybrid model that integrated an artificial neural network and particle swarm optimization to obtain optimized operating parameters [13]. This data-driven model decreased the SO₂ emissions by 30.79% while increasing total cost by merely less than 2%. Most recently, long short-term memory has allowed for the precise and timely prediction of inlet SO₂ concentration with minimum delay [14]. The fundamental concept behind the aforementioned data-driven models is to collect and reprocess previous WFGD system data to discover operation parameters that perform best in terms of sustainability. Data-driven methods, particularly machine learning, multi-variable statistics, and artificial intelligence, enable the optimization of desulfurization with less risk, less investment, and higher accuracy when compared with mechanism modeling [10].

Previous studies primarily developed bi-objective or one-single optimization models for the WFGD system involving multiple indicators, including economic cost and desulfurization efficiency. Optimization problems with less than two objectives are common in WFGD research [15,16,17]. For example, recently, the stability and economic performance were improved in the bi-objective optimization model using economic model predictive control [18]. However, the sustainability of desulfurization is related to many, often more than three, indicators such as energy consumption, limestone slurry usage, and SO₂ absorption efficiency. And very few studies have addressed the issue of many objectives in a WFGD system. To overcome this research gap, the present study proposed a many-objective model to enhance the sustainability of WFDG in terms of four objectives, i.e., desulfurization efficiency, unit power consumption, limestone supply, and unit operation cost. With the operation data collected from a coal-fired power plant at DTP Chaozhou Power Co., Ltd., we performed a data preprocessing and correlation analysis to identify relevant variables. The extreme random forest (ERF) was developed to approximate the objective functions as prediction models, revealing the relationship between system inputs and outputs. The performance of prediction was evaluated by three metrics. We integrated differential evolution (DE) into the non-dominated sorting genetic algorithm-III (NSGA-III) to enhance the search capability. The Pareto-optimal front was analyzed to estimate the sustainability improvement regarding the four objectives. To the best of our knowledge, this is the first attempt to optimize four objectives regarding the sustainability performance of the WFGD system.

The rest of this article is organized as follows: Section 2 describes the process of desulfurization and formulates the many-objective optimization model. Section 3 presents the prediction models based on ERF and many-objective optimization based on NSGA-III-DE. Section 4 illustrates the results of prediction and optimization for four sustainability objectives, followed by Section 5, concluding this study.

2. Process Description and Problem Formulation

2.1. Desulfurization Process Description

Figure 2 presents a schematic diagram of the WFGD process. The system comprises several critical components that work together to achieve effective desulfurization, including the oxidation fan, booster fan, circulation pump, slurry pump, and mixer. The all-blower supplies air to the absorber, creating the necessary flow for the flue gas. As the slurry droplets are sprayed from the atomizers, they mix with the raw flue gas, which flows upward from the bottom of the absorption tower. The SO₂ in the flue gas reacts with the CaCO₃ in the slurry droplets and the O₂ introduced by the blower. The absorber plays a central role in facilitating these reactions, ensuring efficient contact between the flue gas and the alkaline slurry. The gypsum bleed pump extracts the byproduct gypsum from the system, while the density meter monitors the slurry concentration, optimizing the absorption process. After the chemical reactions occur in the absorption tower, the remaining gas passes through mist eliminators to remove any liquid droplets before being released into the atmosphere via the chimney. The gas–gas heat exchanger (GGH) recovers heat from the flue gas, enhancing the system’s overall efficiency. The main chemical reactions [19] occur in the absorption tower, as shown below:

$$\mathrm{S}{\mathrm{O}}_2\left(\mathrm{g}\right)+{\mathrm{H}}_2\mathrm{O}\rightleftharpoons {\mathrm{H}}^{+}+\mathrm{HS}{\mathrm{O}}_3^{-}$$

(1)

$$\mathrm{CaC}{\mathrm{O}}_3+2{\mathrm{H}}^{+}\rightleftharpoons \mathrm{C}{\mathrm{a}}^{2+}+{\mathrm{H}}_2\mathrm{O}+\mathrm{C}{\mathrm{O}}_2\left(\mathrm{g}\right)$$

(2)

$$\mathrm{HS}{\mathrm{O}}_3^{-}+1/2{\mathrm{O}}_2\rightleftharpoons {\mathrm{H}}^{+}+\mathrm{S}{\mathrm{O}}_4^{2-}$$

(3)

$$\mathrm{C}{\mathrm{a}}^{2+}+\mathrm{S}{\mathrm{O}}_4^{2+}+2{\mathrm{H}}_2\mathrm{O}\rightleftharpoons \mathrm{CaS}{\mathrm{O}}_4\cdot 2{\mathrm{H}}_2\mathrm{O}\left(\mathrm{s}\right)$$

(4)

$$\mathrm{S}{\mathrm{O}}_2\left(\mathrm{g}\right)+2{\mathrm{H}}_2\mathrm{O}+1/2{\mathrm{O}}_2+\mathrm{CaC}{\mathrm{O}}_3\rightleftharpoons \mathrm{CaS}{\mathrm{O}}_4\cdot 2{\mathrm{H}}_2\mathrm{O}\left(\mathrm{s}\right)+\mathrm{C}{\mathrm{O}}_2\left(\mathrm{g}\right)$$

(5)

From these chemical equations, it can be inferred that the desulfurization process could be affected by multiple factors such as SO₂ concentration, slurry concentration, air flow rate, slurry flow rate, gas temperature, and pH value of circulated slurry [20,21,22]. It should be noted that the WFGD system has uncontrollable parameters like inlet SO₂ concentration and unit load. The desulfurization process is usually regulated by controllable parameters, for example, the airflow rate and limestone supply rate in this study. Given the non-linearity and large lag of variables, the model predictive control [23,24] and improved PID control [25,26], as well as the derivative methods, have shown satisfactory control accuracy for reducing fluctuations of critical variables. However, it is challenging for operators to quickly and properly establish the operating parameter values due to the changeable operating conditions. The optimization of desulfurization is largely based on the empirical models, resulting in excessive released SO₂ and consumed energy. This would deteriorate expectations of the ultra-low emission and environmental burden of the desulfurization process. The sustainability of the desulfurization process in this study is mainly concerned with desulfurization efficiency, unit power consumption, limestone supply, and unit operating cost. The objective of improving sustainability can be realized by conducting data-driven optimization based on historical operating data.

2.2. Many-Objective Optimization Model Formulation

The objectives of optimization in this study include unit energy consumption, desulfurization efficiency, limestone slurry usage, and unit operating cost. We directly collected the original data of the former three objectives from the management information system of the 600 MW coal-fired power (Figure 3) unit in Chaozhou, China. The operating cost of the WFGD system is considered the energy- and resource-related fee, irrespective of sewage and labor costs. The resource cost mainly refers to the sorbent usage, more specifically, the limestone slurry fee. Electric energy costs stem from the operation of slurry and circulation pumps, booster and oxidation fans, and mixers. In most cases, the energy consumption of a WFGD system dominates the overall operating cost. These operating costs were closely related to the operating parameters [12]. To capture desulfurization performance and economic cost simultaneously, we normalized the operating cost to a unit mass of SO₂ removal, as expressed in Equation (6):

$$C={\displaystyle \frac{C_p+C_s}{m_{S{O}_2}}}$$

(6)

where C is the operating cost for a unit SO₂ removal under a specific period, with a smaller C implying more efficient and economic desulfurization. $m_{S{O}_2}$ is the mass of SO₂ absorbed under specific WFGD conditions, and C_p and C_s are the electric power cost and sorbent cost, respectively. Hereafter, the operating cost refers to the C in Equation (6) for simplicity.

Based on the optimization objectives, correlation analysis was conducted to identify the influential parameters, leading to the selection of the following input variables: inlet flue gas flow density, gypsum slurry pH, gypsum density, oxidation fan inlet flow, and inlet flue gas flow. The inlet flue gas flow density and inlet flue gas flow determine the SO₂ load that needs to be processed, which directly impacts desulfurization efficiency, limestone usage, and the sizing and configuration of equipment. Gypsum slurry pH is a critical factor influencing the efficiency of SO₂ absorption reactions. Gypsum density affects the fluidity of the slurry and the efficiency of gas–liquid contact. Additionally, the oxidation fan inlet flow regulates the amount of oxygen required during the oxidation process, ensuring that byproducts are fully converted to gypsum, which in turn influences system stability and operating costs.

According to the national sulfur dioxide emission legislation, the SO₂ density at the outlet of the WFGD system should be less than 35 mg/m³. This legislation was regarded as a constraint of the optimization model.

Table 1. Variables and objectives information of the optimization problem.

Variable	Notation	Unit	Value
Inlet flue gas flow density	x1	mg/m3	[611, 1332]
Gypsum slurry PH	x2	-	[4.5, 5.5]
Gypsum density	x3	kg/m3	[1212, 1259]
Oxidation fan inlet flow	x4	m3/h	[9736, 10,730]
Inlet flue gas flow	x5	104 m3/h	[172.2, 289.4]
Desulfurization efficiency	f1	-	[0.98, 1]
Unit power consumption	f2	kW	-
Limestone supply	f3	m3/min	-
Unit operating cost	f4	CNY/(h*t)	-

lists the relevant variables and indicators for optimization in this study. The many-objective optimization problem of improving sustainability can be formulated as below:

$$\left\{\begin{array}{l}\min \hspace{1em}\hspace{1em}f\left(\mathit{x}\right)=\left(-f_1\left(\mathit{x}\right),f_2\left(\mathit{x}\right),f_3\left(\mathit{x}\right),f_4\left(\mathit{x}\right)\right)\\ s.t.\hspace{1em}\hspace{1em}{f}_1\left(\mathit{x}\right)-98\%\ge 0\\ \hspace{1em}\hspace{1em}\hspace{1em}{f}_4\left(\mathit{x}\right)-35\le 0\\ \hspace{1em}\hspace{1em}\hspace{1em}\mathit{x}\in X\end{array}\right.$$

(7)

where X is the domain of x = [x₁, x₂, x₃, x₄]. The objective indicators were functions of operating variables. Considering the complexity relationship of the variables, the f₁(x), f₂(x), f₃(x), and f₄(x) are constructed via prediction models based on the ERF.

3. The Method for Optimization

3.1. Data Preprocessing

Given the equipment power fluctuation, sensor malfunction, and other unpredictable circumstances, the raw data on the WFGD system might suffer anomalies such as incomplete, duplicate, corrupted, or even incorrect data. Direct feeding of the raw data into predictive models or algorithms could induce unreliable and incorrect outcomes. The data cleaning techniques were applied to fix the anomalous data following the work by [27], including outlier detection, noise removal, and time alignment. Specifically, we deleted abnormal values, eliminated some incomplete records caused by network communication problems, and used empirical data or averages to fill in individually missing data. The optimization was performed on the MATLAB platform and ran on a personal computer with 16.0 GB RAM and an 8-core CPU under Windows 10.

Variables of the WFGD system differ in units and value ranges. For example, the pH values of the discharged gypsum slurry range from 4.5 to 5.5, while the values of inlet SO₂ density range from 800 to 1400 mg/m³. Normalizing and limiting the data to a specified range can eliminate the influence of dimensional differences and singular samples. It contributes to reducing training time, speeding up convergence, and improving model accuracy. We adopted the min-max normalization method in this study to normalize the raw data.

As Table 1 presented five variables of the system, we performed a data correlation analysis to identify the correlation among variables and objectives and reduce the dimension of state space. The Pearson correlation coefficient r, ranging from 0 to 1, was used to present the correlation.

3.2. Prediction Model Based on Extreme Random Forest

The four objectives to be optimized were predicted using ERF. The extreme random forest (ERF) algorithm is often chosen for prediction tasks because of its high efficiency in handling large datasets, its robustness to overfitting, and its ability to model complex, nonlinear relationships between variables. As an extension of random forest, ERF is essentially a bootstrapping aggregation (Bagging) algorithm and belongs to ensemble learning [28]. Compared to the algorithms used in other studies, such as the Ada-XGBoost-CatBoost combination model [29,30], ERF stands out because it maintains high performance even with a less carefully tuned hyperparameter setup. ERF’s simpler setup and faster training times make it a good choice for systems with varied input parameters and where expert tuning might not always be feasible. The Bagging sampling method randomly extracted n training samples from the original sample set, and k rounds of extraction were performed to obtain k training sets. Then, k models were trained using these training sets. Different from the random forest algorithm trained by bootstrap sampling dataset, the ERF uses all the original datasets to train each decision tree using the classification and regression tree (CART). Nodes of each CART tree in ERF are randomly split, and data features are also randomly selected. The basic idea of the ERF algorithm is presented in Figure 4.

Suppose M is the number of features and K is the number of decision trees. A decision tree is built using CART based on the original dataset. With the randomly selected m (m ≤ M) features, split values of each node among the m features are also randomly selected. For each of the m features and split values, the Gini index and information gain can be used to choose the best split. The splitting process discards the pruning operation. A repeat of the decision tree generation process for K times creates an ERF. For the regression or prediction problems, the mean of the prediction results of K decision trees was adopted as the final prediction result. As ERF fully employs all the data, including the out-of-bag samples, to train the base classifier, it could significantly reduce the bias of prediction. The random node split and feature selection in ERF enhance the generalization ability, computational efficiency, and robustness to noise and outliers.

Three indicators, i.e., mean square error ($MSE$), mean absolute percentage error ($MAPE$), and correlation coefficient ($R^2$), were selected to evaluate the accuracy and performance of the model. $MSE$ is a measure of the degree of difference between the actual value and the predicted value, and the mean squared error can evaluate the degree of change in the data. $MAPE$ indicates the average deviation of the predicted results from the actual results. The smaller $MSE$ and $MAPE$ mean a higher accuracy of the model. $R^2$ reflects the degree of correlation between the predicted value and the real value, and closer to 1 suggests better performance of the algorithm. These indicators can be determined as below:

$$MSE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i = 1}^n}{\left(y_i-{\widehat{y}}_i\right)}^2$$

(8)

$$MAPE={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n|{\displaystyle \frac{y_i-{\widehat{y}}_i}{y_i}}|$$

(9)

$$R^2=1-{\displaystyle \frac{{\sum }_i^n{\left(y_i-{\widehat{y}}_i\right)}^2}{{\sum }_i^n{\left(y_i-{\overline{y}}_i\right)}^2}}$$

(10)

where y_i is the real value, ${\widehat{y}}_i$ is the predicted value of the model, $\stackrel{\_}{y}$ is the average of the real values, and n indicates the total amount of data.

3.3. Formatting of Mathematical Components

The increment of dimensions or objectives in optimization issues would raise the computational cost exponentially. NSGA-III is a common approach to address the many-objective optimization problem [31]. NSGA-III shares the same structure with NSGA-II but modifies the diversity operator using the reference-point strategy [32]. Different from the NSGA-II, depending on crowding distance, NSGA-III adaptively updates the reference points to maintain and improve the diversity among the population members. The procedure of this algorithm includes the genetic operator setting, determination of reference points, non-dominated sorting, normalization, and association operation.

The reference points in NSGA-III are distributed widely on the normalized hyper-plane to guarantee population diversity. According to the approach proposed by [33], the total number of reference points P is determined by Equation (11).

$$P=\left(\begin{array}{c}M+H-1\\ H\end{array}\right)$$

(11)

where M is the number of objectives, and H is the number of divisions for each objective. For example, in a tri-objective problem (M = 3) with three divisions (H = 3) of each objective, $P=\left(\begin{array}{c}3+3-1\\ 3\end{array}\right)=10$ reference points are created and distributed on the normalized hyperplane, as shown in Figure 5.

Based on the population P_t at the t-th generation, the crossover and mutation operation could generate a new population Q_t with the same amount of N individuals in the population P_t. There are different crossover and mutation operators in NSGA-III, for example, the simulated binary crossover technique [34] and polynomial mutation method [35]. Mixing the population Q_t and P_t forms the population R_t with 2N individuals, among which half of the individuals in R_t should be selected as offspring as the next generation population P_t+1. This selection process is based on the non-dominant sorting algorithm, and the corresponding pseudocode is presented below. The domination relationship between two individuals is determined by the prediction models for objectives. Pseudocode is summarized in Algorithm 1.

Algorithm 1: non-dominant sorting process to generate offspring
Input: population Rt.  Output: the set of fronts F = {F1, F2, …}.
1	For each individual i in Rt do
2	set k = 0 {initial front number}, n(i) = 0 {number of individuals dominating i}
3	set S(i) = empty {the set of individuals dominated by i}
4	For each individual j in Rt do
5	If i$<$j Then S(i) = S(i)$\cup${j}
6	Else if j$<$i Then n(i) = n(i) + 1
7	If n(i) = 0 Then F1 = F1$\cup${i}
8	i = 1
9	While Fi is not empty do
10	Q is empty set
11	For each individual i in Fi do
12	For each individual j in S(i) do
13	n(j) = n(j) − 1
14	If n(j) = 0 Then Q = Q$\cup${j}
15	i = i + 1, Fi = Q
16	End

The normalization of population members could ensure a fair comparison between individuals in a many-objective problem, as different objectives usually have different units and scales. Suppose the minimum values of M objective functions at the t-th generation, i.e., the ideal point is $\widehat{z}=\left(z_1^{mi{n}_2^{mi{n}_M^{min}}}\right)$. Then, the objective values are transformed by Equation (12):

$$f_i^{\prime }\left(x\right)=f_i\left(x\right)-z_i^{min}$$

(12)

Subsequently, the extreme point of each objective can be determined through the achievement scalarizing function as below:

$$z^{i,max\underset{s}{\mathrm{argmin}}{\left(s,w^i\right)}_t}$$

(13)

$$ASF\left(x,w\right)={{\mathit{max}}_{i=1}^{M}f}_i^{\prime }\left(x\right)/w_i$$

(14)

where z^i,max is the extreme objective vector, and w is the weighting vector. These $M$ extreme objective vectors can constitute a hyper-plane with ($M-1$) dimensions. Suppose a_i is the intercept of the i-th objective axis, and then the objective functions could be normalized as Equation (15).

$$f_i^n\left(x\right)={\displaystyle \frac{f_i^{\prime }\left(x\right)}{a_i-z_i^{min}\frac{f_i\left(x\right)-z_i^{min}}{a_i-z_i^{min}}}}$$

(15)

As solving a many-objective optimization problem usually involves high computational costs, the conventional NSGA-III algorithm has limited convergence capability [31,36]. Considering that the DE algorithm has good local search ability [37] while the NSGA algorithm has a strong global search capability [38], the integration of DE into NSGA-III is expected to improve searchability. DE improves NSGA-III by enhancing local search efficiency through its mutation and crossover mechanisms, which focus on refining candidate solutions in promising regions of the solution space. The vector-based mutation in DE, where solutions are adjusted based on the difference between population members, facilitates quicker convergence and more precise exploitation. Additionally, DE’s simpler operations reduce computational overhead, which is critical in many-objective problems with high evaluation costs. The combination of NSGA-III’s global search strength and DE’s local optimization capability ensures a balanced approach to exploration and exploitation, ultimately leading to improved convergence speed and reduced computational cost, making this hybrid approach more effective for complex optimization tasks. The procedure of the NSGA-III-DE algorithm is presented in Figure 6. Specifically, the mutation and crossover operations of DE were introduced into NSGA-III following the study by [39]. For an individual $X_i^t$ at the t-th generation, the mutation operator could generate a mutant individual. We adopted one commonly used mutation strategy, i.e., DE/rand/1, as follows.

$$V_i^t=X_{r1}^t+F\left(X_{r2}^t-X_{r3}^t\right)$$

(16)

where F denotes the scaling factor and its value range from 0 to 1, the subscript r₁, r₂, and r₃ are mutually different and uniformly selected from the set {1, 2, …, N}. Due to its higher operating efficiency, the binomial crossover operation is more frequently utilized than the exponential form [40]. Each dimension experiences the binomial crossover operation under the probability of C_r. For the j-th dimension, the new vector, also called the trial vector, is formulated as follows:

$$u_{ij}^t=\left\{\begin{array}{c}{v}_{ij}^t,ifrand\left(\mathrm{0,1}\right)\le C_r\\ x_{ij}^t,otherwise\end{array}\right.$$

(17)

where $v_{ij}^t$ is the i-th vector at the t-th generation, C_r ranges from 0 to 1, and rand (0, 1) is the random number generated from [0, 1]. Then, the fitness comparison between $X_i^t$ and $U_i^t$ determines whether to preserve the $X_i^t$ for the next generation.

3.4. The Evaluation of Pareto Front

The shape of the Pareto front offers critical information on the trade-off between different optimization objectives. The presumption that the Pareto front has been sufficiently populated is essential to this trade-off. The true Pareto front, however, is rarely known since the result of NSGA-III is merely a proximation of the true Pareto front [41]. The quality of such approximation depends on two aspects: (1) the closeness between the spots on the approximated front and the true Pareto front and (2) the approximated front’s point diversity, with greater diversity generally being preferable. One prevalent measurement for the quality of the Pareto front is the hypervolume of the front. The hypervolume metric estimates the volume of the objective space dominated by a set of Pareto-optimal solutions. It is defined as the volume of the region enclosed by the reference point and all the solutions on the non-dominated solution set. The hypervolume index is calculated as follows, and a greater value implies better approximation performance.

$$HV=\delta \left({\cup }_{i=1}^{\left|S\right|}{v}_i\right)$$

(18)

where δ denotes the Lebesgue measure for volume estimation, |S| represents the number of solutions in non-dominated sets, and v_i denotes the hypervolume formed by the reference point and the i-th solution of the approximation set. Another commonly used index is the spacing metric that measures the evenness of distribution of the Pareto-optimal solutions in the objective space [42]. This metric is defined as the mean distance between each of the solutions and its nearest neighbor in the set, as formulated in Equation (19).

$$spacing\left(S\right)=\sqrt{{\displaystyle \frac{1}{\left|S\right|-1}}{\sum }_{i=1}^{\left|S\right|}{\left(\stackrel{\_}{d}-d_i\right)}^2}$$

(19)

where S is the solution set, $d_i={{\mathit{min}}_{\left(s_i,s_j\right)\in S,s_i\ne s_j}‖F\left(s_i\right)-F\left(s_j\right)‖}_1$ is the l₁ distance between a point $s_i\in S$ and the closest point s_j at the approximated Pareto front, and $\stackrel{\_}{d}$ is the average of d_i.

4. Result and Discussion

4.1. Data Description and Prediction Results

The operation data of the WFGD system are comprised of a time series of variables and objectives (except for the operating cost) obtained from the management information system of the coal-fired power plant. A sample in the dataset can be expressed as [X_i, F_i] = [x_i1, x_i2, x_i3, x_i4, x_i5, f_i1, f_i2, f_i3, f_i4], and the definitions of variables are presented in Table 1. The average unit price of electric power and sorbent were collected from the Guangdong Power Grid Co., Ltd. and dealers. The operation cost of each sample was calculated based on Equation (6). The training set and test set for the ERF algorithm contain 75% and 25% of the total 1441 samples, respectively. To obtain a preliminary understanding of the relationship between system inputs and outputs, we performed a correlation analysis with the preprocessed dataset to examine if the input variables are relevant to the system outputs. Pearson correlation coefficients between variables are presented in Figure 7. The variables considered in the WFGD system are relevant to the four objectives to be optimized.

We trained the ERF-based prediction model for desulfurization efficiency, unit power consumption, and limestone supply, irrespective of the unit operation cost, since it can be deduced by the former three objectives. Figure 8 compared the prediction values with the true values of 144 test samples. The majority of the predicted points were quite consistent with the original counterparts in terms of three optimization objectives. Three indicators, namely $MSE$, $MAPE$, and $R^2$, were used to evaluate the prediction performance using all the testing data. As can be seen from

Table 2. Performance evaluation of the prediction models.

Objectives	MSE	MAPE	${\mathit{R}}^2$
Desulfurization efficiency (f1)	0.004	0.038	0.978
Unit power consumption (f2)	1.881	0.496	0.999
Limestone supply (f3)	0.002	0.019	0.980

, the ERF-based prediction model enabled an accurate prediction with small errors. The values of $R^2$ are over 0.97, implying a high degree of consistency between the predicted value and the real value.

4.2. Results of Many-Objective Optimization Using NSGA-III-DE

Based on the prediction models above, we developed the NSGA-III-DE algorithm to optimize the desulfurization efficiency, power consumption, limestone usage, and operation cost of the WFGD system. The size of the initial population for the NSGA-III section was set to 50, and the maximum generation was 2000, based on balancing computational efficiency with solution diversity. A population size of 50 ensures sufficient diversity in the solution space without imposing excessive computational overhead, a common approach in similar optimization problems. The maximum of 2000 generations allows adequate iterations for the algorithm to converge to an optimal Pareto front, a parameter often recommended for complex many-objective tasks where more iterations are required to ensure convergence. In the DE section, crossover and mutation rates were both set to 0.2 to achieve moderate exploration and prevent excessive disruption of promising solutions, following standard practices in evolutionary algorithms. The scaling factor in the crossover operator was set to 0.5, which provides sufficient diversity in mutation without being too aggressive, a value typically chosen to balance exploration and exploitation in differential evolution algorithms. Considering the difficulty of displaying a four-dimensional Pareto-optimal solution in three-dimensional space, we used the color to represent the fourth dimension. Figure 9 illustrates the Pareto-optimal front of this many-objective optimization. As evident in this figure, each point implied a non-dominated optimal solution, and the points were distributed in the three-dimensional space rather than a curved surface in three-objective problems. The points or solutions with higher desulfurization efficiency, lower unit power consumption, and lower limestone consumption tend to have higher unit operation costs.

Table 3. Objective values of the pre- and post-optimization.

Objectives	Pre-Optimization	Post-Optimization	Improvement	Unit
Desulfurization efficiency (f1)	97.89%	98.14%	0.25%	-
Unit power consumption (f2)	125.51	91.49	34.02	kW
Limestone supply (f3)	0.471	0.229	0.242	m3/min
Unit operation cost (f4)	3700.00	2906.81	793.19	CNY/(h*t)

and Figure 10 illustrate the pre- and post-optimization results of the objectives. The objective values of pre-optimization referred to the average operating performance obtained from the initial data, while that of the post-optimization were the mean values of 35 solutions in the Pareto front. With the optimized operating condition, the desulfurization efficiency would be improved by 0.25%. Unit power consumption and limestone can be reduced by 34.02 kW and 0.242 m³/min, respectively. During one single hour, the cost of absorbing one ton of SO2 would save 793.19 CNY on average. It should be noted that the improvements in Table 3 were the average of optimal solutions rather than one specific operating condition. The actual improvement under one operating state would be less than some parts of the post-improvement values in Table 3. Every single point in Figure 9 suggests one operating condition; the operating improvement strategy in practice depends on the preference of the objectives. Practically, the desulfurization efficiency of over 97% is sufficient to satisfy the emission standard required by the Environmental Protection Bureau, namely, less than 35 mg/m³ of SO₂ emission at the outlet. Thus, engineers at coal-fired power plants can offer more preferences on other objectives like energy consumption and operating costs. The average values of Spacing and HV are 0.013 and 1.746, respectively. The Spacing value was relatively low, indicating a concentrated set of solutions with small gaps between neighboring solutions, while the hypervolume value typically indicates a specific coverage of the solutions on the Pareto front. It suggests that the set of solutions captures a portion of the optimal trade-off region in the objective space.

4.3. Discussion of the Results

The proposed extreme random forest (ERF) and NSGA-III-DE algorithms have successfully completed the many-objective optimization of the coal mine desulfurization process, addressing key objectives such as desulfurization efficiency, unit power consumption, limestone supply, and operation cost. The results clearly demonstrate that ERF provided highly accurate predictions of the critical performance metrics, with high R² values (over 0.97) indicating strong consistency between predicted and real values across the test samples. This reliable prediction framework facilitated the effective implementation of the NSGA-III-DE algorithm, which identified a set of Pareto-optimal solutions that trade-off between the multiple conflicting objectives.

In practical applications, the integration of ERF and NSGA-III-DE holds significant potential for coal-fired power plants. The optimization results suggest that the desulfurization efficiency can be increased by 0.25%, while power consumption and limestone supply can be substantially reduced, leading to an average cost saving of 793.19 CNY per hour for SO₂ absorption. These improvements could enhance the economic feasibility of the wet flue gas desulfurization (WFGD) system, particularly in regions where operating costs are a major concern. Furthermore, the ability to reduce energy consumption aligns well with the global push for more energy-efficient and sustainable industrial processes, offering potential environmental benefits.

However, there are several practical considerations and challenges that should be noted. First, while the economic benefits are evident, the feasibility of implementing the proposed optimization methodology on a large scale depends on factors such as the initial investment in computational infrastructure, the adaptability of the current systems, and the potential variability in input data. For instance, the system’s performance may vary depending on fluctuations in the price of electricity or limestone, which could affect the cost-saving estimates. Additionally, operational constraints in real-world coal-fired plants, such as maintenance schedules or unplanned outages, could limit the continuous application of the optimized solutions.

Moreover, while the optimization has led to clear improvements, it is crucial to balance desulfurization efficiency with regulatory compliance. As indicated in the results, the 97% desulfurization efficiency exceeds the threshold required by environmental regulations, meaning that further improvements in this area might not be as critical. Therefore, plant operators could prioritize optimizing other objectives like power consumption and operational costs. Lastly, it is important to recognize that the proposed NSGA-III-DE approach, while computationally efficient, may still require significant computational resources, particularly when scaling to larger datasets or real-time applications.

In conclusion, the combination of ERF and NSGA-III-DE provides a powerful tool for optimizing the performance and cost-efficiency of coal mine desulfurization processes. The practical application of this methodology in coal-fired power plants shows promising potential for cost savings and energy efficiency, but its success will depend on careful implementation and consideration of the operational and economic factors in a real-world setting.

5. Conclusions

The present study demonstrates significant scientific novelty by developing a many-objective optimization model specifically designed to enhance the operational sustainability of the WFGD system at a 600 MW coal-fired power plant in Chaozhou, China. A major contribution of this work lies in the integration of the extreme random forest (ERF) prediction model with the NSGA-III-DE algorithm, which provides a novel approach for tackling the complex, nonlinear relationships among key variables and objectives such as desulfurization efficiency, energy consumption, limestone usage, and operational cost. The adoption of the ERF model, with its superior prediction accuracy (as shown by metrics like MSE, MAPE, and R²), offers a robust foundation for optimization, highlighting the strength of this data-driven approach in handling high-dimensional and intricate datasets.

From a practical standpoint, the proposed optimization methodology has considerable potential for improving the real-world sustainability of coal-fired power plants. This study demonstrates that the optimized solutions on the Pareto front can lead to substantial reductions in energy consumption and operational costs, thus contributing to the economic and environmental performance of the WFGD system. Furthermore, the method’s adaptability in response to changes such as equipment upgrades or variations in coal quality underscores its practical significance for dynamic operational environments.

Despite the success of this offline and static optimization based on historical data, one limitation of this study is the absence of real-time, online optimization capabilities. Addressing this limitation in future research could further enhance the system’s sustainability performance by allowing continuous adaptation to operational fluctuations. Additionally, the computational efficiency of the many-objective optimization process remains an area for improvement, particularly for real-time applications. The challenge of selecting the most appropriate solution from the diverse set of optimal solutions on the Pareto front also suggests a need for developing more refined multi-criteria decision-making rules in future studies.

In summary, this study provides a novel and practical framework for improving the sustainability of desulfurization processes in coal-fired power plants, with potential applications extending to other stages of flue gas treatment, such as dust elimination and denitrification. The proposed optimization approach not only advances the field scientifically but also offers actionable strategies for enhancing the energy and resource efficiency of WFGD systems in practice.