Intelligent Diagnosis Method for Constrained Primary Frequency Regulation Capacity of Coal-Fired Units Based on ISO-MLRF

Abstract

To address the challenges of low diagnostic accuracy of constrained primary frequency regulating (PFR) capacity for coal-fired units due to complex and strongly coupled restricting factors, an intelligent diagnosis method based on an improved snake optimizer-based multi-label random forest classification algorithm is proposed. By analyzing the factors restricting PFR capability, a set of characterization parameters and constraint factors for unit regulating capacity is established. The snake optimizer is enhanced by introducing dynamic update mechanisms and novel search strategies to improve its convergence speed and accuracy. The improved algorithm is then applied to optimize the hyperparameters of the multi-label random forest algorithm, enabling online diagnosis of PFR capacity limitations. Simulation results demonstrate that the proposed algorithm exhibits superior convergence performance, with lower medians of false alarm rate and missing alarm rate across all labels, coupled with reduced result dispersion compared to alternative algorithms. Tests on real operational data show an average false alarm rate of 0.029% and an average missing alarm rate of 0.053 for all labels. The results indicate that the proposed method is feasible and effective, enabling accurate online diagnosis of constrained PFR capacity of coal-fired units.

1. Introduction

The increasing integration of high proportions of renewable energy and the widespread application of power electronic devices have led to reduced system inertia, weakened disturbance resistance, and accelerated frequency change rates, making the maintenance of system frequency stability an unprecedented challenge [1]. In grid frequency regulation tasks, PFR provides an automatic and rapid response to frequency fluctuations caused by load changes. It is the most fundamental and fastest automatic mechanism for maintaining real-time power balance and frequency stability in power systems. As the proportion of renewable energy sources further increases in the new electricity system, coal-fired power generation units are transitioning from the main power source to supporting and regulating sources, and undertaking grid frequency regulation tasks more frequently [2].

In actual operation, due to complex factors such as boiler inertia, regulation system performance, operating conditions, and equipment health status, many units cannot meet grid requirements for PFR in terms of response time, droop coefficient, actual action integral electricity, and other regulation capabilities, thereby affecting grid frequency stability [1,3]. Additionally, units with constrained regulation capabilities are subject to assessments under the power grid’s “Two Detailed Rules,” leading to economic losses for power plants.

Currently, there is constrained research on comprehensive diagnostic analysis of the constrained regulation capabilities of coal-fired units in PFR. Existing studies mainly focus on modeling the PFR process for coal-fired units and analyzing some constraining factors. In terms of mechanistic models, ref. [4] analyzes the impact of main steam pressure fluctuations on PFR, incorporating a boiler dynamic model that includes heat exchange processes into a generic unit PFR model, thereby improving model accuracy. Reference [5] establishes a mathematical model for extraction condensing steam turbines applicable to various PFR technologies and operating condition changes, analyzing the effects of different frequency regulation technologies and condition variations on unit PFR capabilities. Reference [6] examines the limiting mechanisms and influencing factors of unit PFR capabilities from three aspects: main steam valve opening, main steam pressure, and power limits, and builds an assessment model for PFR capabilities of deep peak-shaving units. Reference [7] establishes a unit PFR capability assessment model based on system identification technology, analyzing the decoupling of main steam pressure from unit power response. Reference [8] studies the influence of patterns of boiler heat storage coefficients and main steam pressure on unit PFR capabilities, establishing a PFR mechanistic model based on observer PID. In terms of data-driven models, ref. [9] develops a data-driven model for predicting PFR capabilities, with inputs including load commands, actual power, main steam pressure, valve opening, and rotational speed, aiding in the analysis of limiting factors for unit PFR capabilities. Reference [10] creates a fully data-driven model for online estimation of PFR capabilities in deep peak-shaving thermal power units using an LSTM neural network, enabling the assessment of unit PFR capabilities. While the aforementioned models for coal-fired unit PFR can identify deviations in unit PFR capabilities and analyze some limiting factors, they cannot achieve a comprehensive diagnosis of constrained PFR capabilities.

Thermal power units are highly complex nonlinear systems with strongly coupled dynamic variables, making it difficult to comprehensively analyze the reasons for constrained PFR capabilities through mechanistic modeling. Compared to mechanistic modeling approaches, data-driven methods can achieve comprehensive analysis of unit regulation capability constraints without prior knowledge [11]. Due to the strong correlations among unit equipment and the multiple, highly coupled causes of constrained PFR capabilities, each reason for constrained regulation can be treated as a label. Thus, the problem of diagnosing constrained regulation capabilities in coal-fired units can be transformed into a multi-label classification problem. Currently, methods for solving multi-label classification problems are mainly divided into two categories: problem transformation and algorithm adaptation. Mainstream algorithms in problem transformation include binary relevance, classifier chains [12], and label powerset [13]. Algorithm adaptation methods mainly include K-nearest neighbors (KNN) [14], random forests (RF) [15], gradient boosting, support vector machines (SVM), neural networks, etc. Among these, random forest is an ensemble classification model based on decision trees, with few hyperparameters and no need for extensive parameter tuning. Therefore, random forest is adopted as the diagnostic classification algorithm in this paper. Random forest is an ensemble of decision trees, and the number of decision trees, their depth, and other hyperparameters affect the final classification diagnostic results. Optimizing the hyperparameters of a random forest can improve the accuracy of classification diagnosis. Common algorithms for hyperparameter optimization include particle swarm optimization (PSO) [16], gray wolf optimizer (GWO) [17], genetic algorithm (GA) [18], snake optimizer (SO) [19], and Bayesian algorithm (BYS) [20].

To benchmark performance against state-of-the-art advancements, the snake optimizer (SO) proposed by Hussien [21] is incorporated. This selection is predicated on SO’s distinctive dynamic exploration-exploitation mechanism, which epitomizes the latest evolutionary trends in swarm intelligence. However, despite its efficacy, SO often faces challenges in convergence speed and stability when tackling high-dimensional feature selection for diagnostic tasks. Therefore, to address these limitations and further enhance model transparency, this study proposes a novel hybrid framework, ISOMLRF. By integrating the strengths of the improved snake optimizer with random forest, our method aims to achieve a superior balance between diagnostic accuracy and computational efficiency.

In summary, high renewable integration intensifies the demand for precise PFR; coal-fired units often fail to meet grid codes due to complex, coupled constraints. Existing diagnostic approaches face limitations: mechanism models struggle with system nonlinearity, while current data-driven methods lack comprehensive multi-factor analysis. To bridge this gap, this study reframes PFR diagnosis as a multi-label classification task. The main contributions of this paper are

⮚

A comprehensive set of parameters and constraint factors is systematically analyzed and established to address the strong coupling effects in coal-fired units.

⮚

An improved snake optimizer featuring dynamic update mechanisms and novel search strategies is proposed to optimize the hyperparameters of a multi-label random forest, significantly enhancing convergence speed and diagnostic accuracy.

⮚

The proposed method is validated using real-world operational data, demonstrating superior performance with low FAR and MAR, providing high-precision online diagnosis for constrained PFR capacity.

The remainder of this paper is organized as follows. Section 2 presents the mechanism analysis of constrained PFR capacity. Section 3 provides a detailed description of the proposed fault diagnosis method. Section 4 presents the simulation analysis of the proposed method. Section 5 concludes the paper and presents the limitations of the proposed study.

2. Analysis of Constrained PFR Capabilities

2.1. The Constrained Factors of PFR Capabilities

Based on refs. [22,23,24] and on-site expert experience, the main reasons for the constrained PFR capability of coal-fired units are illustrated in

Table 1. The main reasons for the constrained PFR capability of the unit.

Main Categories	Main Reasons
Operational condition	High load rate or low load rate
	Insufficient equipment reserve capacity
Boiler heat storage capacity	High thermal inertia of the boiler
	Steam-water system delay
	Improper coordinated control mode
Poor regulation system performance	Delayed governor response
	Low control valve accuracy
	Logic conflict between CCS and DEH
Improper control system parameter settings	Improper frequency regulation coefficient
	Stringent limiter settings
	Excessive deadband setting
Auxiliary equipment and systems	Slow response of the condensate pump
	Slow response of the feedwater or heating regulation
Equipment aging	Actuator aging
	Sensor accuracy degradation

.

The reasons can be summarized into the following categories: operational condition limitations, boiler heat storage capacity limitations, poor governor system performance, improper control system parameter settings, limitations of auxiliary equipment and supporting systems (for units adopting new PFR performance enhancement technologies), equipment aging issues, and others.

2.2. Characterization Parameters Selection of PFR Capability Diagnosis

The PFR capability of a coal-fired unit primarily encompasses three aspects: speed, capacity, and stability [25,26]. Speed refers to the unit’s responsiveness to grid frequency changes, characterized by parameters such as response delay time (T_d), time to reach 75% of the target load response (T₇₅), and time to reach 90% of the target load response (T₉₀). Capacity refers to the regulation capacity during the unit’s PFR process, characterized by parameters including: PFR load response (ΔP), speed deadband (i), and speed governing droop (K_p). Stability is primarily characterized by parameters such as settling time (T_s), contribution energy (W), and overshoot (σ). To meet the needs of diagnosing constrained frequency regulation capability, operational state characterization parameters during the PFR process are also introduced here, including valve actuation rate, main steam pressure change rate, and intermediate point temperature change rate. The main characterization parameters for a unit’s PFR capability are shown in

Table 2. Characteristic parameters of PFR capability [6,25,26].

No.	Characterization Parameters	Unit	Threshold
1	Response delay time	s	<3
2	Time to reach 90% of the target load response	s	<30
3	Deadband	r/min	±2
4	Speed governing droop	%	4~5
5	Load regulation quantity	%	6~10
6	Settling time	s	<60
7	Contribution energy	%	>90
8	Valve actuation rate	%/s	>5
9	Main steam pressure change rate	MPa/s	<0.8
10	Intermediate point temperature change rate	°C/s	<2
11	Power adjustment rate	%/s	>1
12	Initial load rate	%	-
13	Initial main steam pressure	MPa	-
14	Initial frequency deviation	Hz	-
15	Reserve capacity	%	>2
16	Frequency regulation power limit	%	>±5
17	Deviation between valve opening and command	%	<±2
18	Condensate flow regulation rate	s	<30

.

3. Methodology

3.1. Snake Optimizer Algorithm

In 2022, Professor Abdelazim G. Hussien drew inspiration from the predatory, combat, and mating behaviors of snakes in nature to pioneer the snake optimizer (SO) algorithm. According to [21], SO equally divides the snake population into two groups: male and female, as shown in Equation (1). Females only fight or mate with males when the temperature (Temp) is low, and the food quantity (Q) is sufficient. Temp and Q are given by Equations (2) and (3), respectively.

$$N_{\mathrm{m}}\approx N/2,N_{\mathrm{f}}=N-N_{\mathrm{m}}$$

(1)

$$Temp=\exp \left({\displaystyle \frac{-t}{T}}\right)$$

(2)

$$Q=c_1\cdot \exp \left({\displaystyle \frac{t-T}{T}}\right)$$

(3)

Among them, N represents the total number of individuals, while N_m and N_f denote the number of male and female individuals, respectively. T and t denote the maximum number of iterations and the current iteration number, respectively, and c₁ takes the value of 0.5. Then, the positions of male snakes can be updated as follows [21].

$$X_{i,\mathrm{m}}\left(t+1\right)=X_{\mathrm{rand},\mathrm{m}}\left(t\right)\pm c_2\cdot \exp \left({\displaystyle \frac{-f_{\mathrm{rand},\mathrm{m}}}{f_{i,\mathrm{m}}}}\right)\cdot \left(\left(X_{\max }-X_{\min }\right)\cdot rand+X_{\min }\right)$$

(4)

Here, X_i,m and X_rand,m represent the positions of male snakes, with their fitness values denoted as f_i,m and f_rand,m. X_max and X_min represent the upper and lower bounds of the positions, rand is a random number between 0 and 1, and c₂ is set to 0.05. The position update method for female snakes is the same as that for male snakes.

If the food quantity is sufficient (Q > 0.25), the snake population enters the exploration stage. If Temp > 0.6, the snake population will move entirely toward the direction of the food according to the rule specified in Equation [21].

$$X_{i,j}\left(t+1\right)=X_{\mathrm{food}}\pm c_3\cdot Temp\cdot rand\cdot \left(X_{\mathrm{food}}-X_{i,j}\left(t\right)\right)$$

(5)

where X_i,j and X_food represent the position of the snake (male or female) and the position of the best individual, respectively, and c₃ is set to 2.

If the environmental temperature is low (Temp < 0.6), the snakes enter either combat or mating mode. The combat mode for males is described by Equation (6). Conversely, the mating mode is expressed by Equation (7). Finally, the worst-performing males and females are updated using Equations (8) and (9), and the algorithm continues to run until the termination conditions are met [21].

$$X_{i,\mathrm{m}}\left(t+1\right)=X_{i,\mathrm{m}}\left(t\right)+c_3\cdot \exp \left({\displaystyle \frac{-f_{\mathrm{best},\mathrm{f}}}{f_i}}\right)\cdot rand\cdot \left(Q\cdot X_{\mathrm{best},\mathrm{f}}-X_{i,\mathrm{m}}\left(t\right)\right)$$

(6)

$$X_{i,m}\left(t+1\right)=X_{i,\mathrm{m}}\left(t\right)+c_3\cdot \exp \left({\displaystyle \frac{-f_{i,\mathrm{f}}}{f_{i,\mathrm{m}}}}\right)\cdot rand\cdot \left(Q\cdot X_{i,\mathrm{f}}\left(t\right)-X_{i,\mathrm{m}}\left(t\right)\right)$$

(7)

$$X_{\mathrm{worst},\mathrm{m}}=X_{\min }+rand\cdot \left(X_{\max }-X_{\min }\right)$$

(8)

$$X_{\mathrm{worst},\mathrm{f}}=X_{\min }+rand\cdot \left(X_{\max }-X_{\min }\right)$$

(9)

where X_best,f represents the position of the best female snake. f_best,f and f_i represent the fitness values of the best female snake and the snake population, respectively. The update process for female snakes is the same as that for male snakes. X_worst,m and X_worst,f correspond to the positions of the worst-performing male and female snakes, respectively.

3.2. Improved Snake Optimizer Algorithm

Although the snake optimizer (SO) algorithm has made significant progress compared to previous algorithms, it still faces considerable challenges when dealing with complex and high-dimensional optimization problems. Due to insufficient population diversity in the SO algorithm, it encounters issues such as slow convergence speed and low convergence accuracy when solving complex optimization problems in engineering applications that involve different dimensions and multiple nonlinear constraints. To address these issues, this paper introduces dynamic update and search mechanisms to enhance the performance of the snake optimization algorithm.

(1)

Dynamic Update Mechanism

In the SO algorithm, the food quantity is crucial for determining whether the algorithm is in the exploration stage or the exploitation stage. Based on ref. [27], a disturbance factor is introduced into Equation (3) to achieve a new dynamic update for c₁. The new c₁ is calculated according to Equation (10).

$$c_1^{\mathrm{new}}=c_1+{\displaystyle \frac{1}{10}}\times \cos \left(r_1^4\times {\displaystyle \frac{\pi }{2}}\right)$$

(10)

Here, r₁ is a random number between 0 and 1.

During the exploration stage, the position updates for male and female snakes as they search for food are calculated using Equation (4). In this paper, a disturbance factor is added to achieve a new dynamic update for c₂. The new c₂ is calculated according to Equation (11).

$$c_2^{\mathrm{new}}=c_2+{\displaystyle \frac{1}{1000}}\times \cos \left(r_2^4\times {\displaystyle \frac{\pi }{2}}\right)$$

(11)

Here, r₂ is a random number between 0 and 1.

During the exploitation stage, the positions of male and female snakes are calculated by Equations (5)–(7). A sine factor is introduced here to improve the algorithm’s convergence speed, and the new c₃ is calculated according to Equation (12).

$$c_3^{\mathrm{new}}=c_3-\sin \left({\left({\displaystyle \frac{t}{T}}\right)}^4\times {\displaystyle \frac{\pi }{2}}\right)$$

(12)

(2)

BPED Search Mechanism

To enhance population diversity in the SO algorithm and improve its optimization capability, an innovative strategy called Bidirectional Population Evolution Dynamics (BPED) is introduced, which replaces the random search during egg hatching in the original SO.

First, based on the widely observed Pareto principle in nature, the top 20% of individuals with the highest fitness in the population are retained and allowed to undergo natural variation. For this high-quality population composed of the top 20% of individuals, new mutated individuals, denoted as X′, are obtained according to Equations (13) and (14) based on [28].

$$X_{\mathrm{good},\mathrm{new}}\left(t\right)=X_q+w\cdot \left(X_{\mathrm{best}}-\varphi \cdot X_k\right)$$

(13)

$$X_{\mathrm{good},\mathrm{new}}\left(t+1\right)=\left\{\begin{array}{cc}{X}_{\mathrm{good},\mathrm{new}}& \mathrm{if} f_{\mathrm{good},\mathrm{new}}<f_i\\ X_i& \mathrm{else}\end{array}\right.$$

(14)

Here, X_q and X_k represent two high-quality individuals distinct from X_i, both belonging to the top 20% of the population; X_best denotes the fittest individual in the population, while w represents a mutation factor based on a sine function, with its exact expression given by Equation (15).

$$w={\displaystyle \frac{\sin \left(\frac{2\pi t+\pi Dim}{Dim}\right)\left(t+T\right)}{2T}}$$

(15)

Here, Dim represents the dimension of the data, T denotes the total number of iterations, and t indicates the current iteration count of the algorithm.

The second component of the BPED strategy involves mutation, elimination, and population migration, applying the PED strategy to the remaining 80% of individuals. This process generates new individuals, denoted as X′. The latter 80% of individuals are randomly divided into two groups. Individuals randomly assigned to the first group undergo variation around the best individual in the population, as shown in Equation (16).

$$X_{\mathrm{bad},\mathrm{new}}(t)=X_{\mathrm{best}}+\mathrm{sign}\left(rand-0.5\right)\cdot \left(l{b}_i^{ap}+rand\cdot \left(u{b}_i^{ap}-l{b}_i^{ap}\right)\right)$$

(16)

Among the remaining 80% of individuals, the other part will migrate according to Equation (17), relocating themselves near their initial positions to conduct further exploration.

$$X_{\mathrm{bad},\mathrm{new}}\left(t+1\right)=X_{\mathrm{bad},\mathrm{new}}\left(t\right)-2\cdot \mathrm{sign}\left(rand-0.5\right)\cdot \left(lb+rand\cdot \left(ub-lb\right)\right)$$

(17)

Through the two strategies proposed in this study, the original snake optimizer (SO) has been enhanced. These skillfully designed modifications have significantly improved the algorithm’s convergence speed, accuracy, and stability.

3.3. Algorithm Validation

In this study, four test functions (F1–F4) from the CEC2017 benchmark set are selected to validate the convergence performance of the ISO algorithm. To ensure a comprehensive evaluation, the performance of the proposed ISO is compared against four widely recognized benchmark optimizers: the standard snake optimizer (SO), whale optimization algorithm (WOA), gray wolf optimizer (GWO), and particle swarm optimization (PSO).

(1)

The WOA is a nature-inspired metaheuristic algorithm proposed by Mirjalili and Lewis [29]. It mimics the social behavior and unique hunting strategy of humpback whales, specifically the bubble-net feeding technique. The algorithm mathematically models three main phases: encircling prey, spiral bubble-net attacking, and searching for prey. The WOA updates the position of a search agent (whale) using the following key equations:

Whales circle the prey during the hunt. This behavior is formulated as

$$\overrightarrow{D}=\left|\overrightarrow{C}\cdot \overrightarrow{X^{*}}\left(t\right)-\overrightarrow{X}\left(t\right)\right|$$

(18)

$$\overrightarrow{X}\left(t+1\right)=\overrightarrow{X^{*}}\left(t\right)-\overrightarrow{A}\cdot \overrightarrow{D}$$

(19)

where $\overrightarrow{X^{*}}$ is the position of the best solution obtained so far (prey), and $\overrightarrow{X}$ is the position of the current whale.

To simulate the spiral bubble-net attacking motion, a helix-shaped movement is formulated as

$$\overrightarrow{X}\left(t+1\right)=\overrightarrow{D^{\prime }}\cdot e^{bl}\cdot \cos \left(2\pi l\right)+\overrightarrow{X^{*}}\left(t\right)$$

(20)

where $\overrightarrow{D^{\prime }}=\left|\overrightarrow{X^{*}}\left(t\right)-\overrightarrow{X}\left(t\right)\right|$ represents the distance between the whale and the prey, b is a constant defining the shape of the logarithmic spiral, and l is a random number in [−1, 1].

(2)

The GWO algorithm is a population-based metaheuristic algorithm proposed by Mirjalili [30]. It mimics the social hierarchy and cooperative hunting mechanisms of gray wolves in nature. The algorithm mathematically models four types of agents: Alpha (α, the leader), Beta (β, the subordinates), Delta (δ, the followers), and Omega (ω, the lowest ranking). The optimization process relies on three main phases: encircling prey, hunting, and attacking. The GWO algorithm updates the position of a search agent (wolf) using the following key equations:

Wolves circle the prey during the hunt. This behavior is formulated as

$$\overrightarrow{D}=\left|\overrightarrow{C}\cdot {\overrightarrow{X}}_p\left(t\right)-\overrightarrow{X}\left(t\right)\right|$$

(21)

$$\overrightarrow{X}\left(t+1\right)={\overrightarrow{X}}_p\left(t\right)-\overrightarrow{A}\cdot \overrightarrow{D}$$

(22)

where ${\overrightarrow{X}}_p$ is the position of the prey (optimal solution), and $\overrightarrow{X}$ is the position of the current gray wolf.

Since the Alpha, Beta, and Delta wolves have better knowledge of the prey’s location, they guide the Omega wolves. The position update is calculated using these three leaders:

$$\overrightarrow{X}\left(t+1\right)={\displaystyle \frac{X_1+X_2+X_3}{3}}$$

(23)

where X₁, X₂, and X₃ are the updated positions influenced by Alpha, Beta, and Delta, respectively, calculated using the encircling equations.

(3)

The PSO algorithm was originally proposed by Kennedy and Eberhart [31]. The canonical PSO algorithm updates the velocity and position of each particle i in a D-dimensional search space using the following two equations.

The velocity of particle i at iteration t + 1 is calculated as

$$v_{i,d}^{t=1}=w\cdot v_{i,d}^t+c_1\cdot r_1\cdot \left(pbes{t}_{i,d}-x_{i,d}^t\right)+c_2\cdot r_2\left(gbes{t}_d-x_{i,d}^t\right)$$

(24)

The new position of particle i is determined by adding the updated velocity to its current position:

$$x_{i,d}^{t+1}=x_{i,d}^t+v_{i,d}^{t+1}$$

(25)

Here, $x_{i,d}^t$ and $v_{i,d}^t$ are the position and velocity of particle i in dimension d at iteration t; w is the inertial weight; c₁ and c₂ are acceleration coefficients; r₁ and r₂ are random numbers uniformly distributed in [0, 1]; $pbes{t}_{i,d}$ is the personal best position achieved by particle i so far; $gbes{t}_d$ is the global best position found by the entire swarm so far.

These three algorithms were selected because they represent the evolutionary trajectory of swarm intelligence algorithms and are commonly applied in hyperparameter tuning tasks for machine learning models, providing a robust baseline for comparison.

The convergence curves of each optimization algorithm are shown in Figure 1.

It can be seen that compared to the SO algorithm and other classical optimization algorithms, the ISO algorithm has achieved significant improvements in convergence speed, stability, and accuracy. Therefore, this paper applies the ISO algorithm to the hyperparameter optimization of the multi-label random forest.

3.4. Hyperparameter Optimization of Multi-Label Random Forest Based on ISO

While the standard snake optimizer (SO) demonstrates promising performance, it may encounter limitations such as premature convergence when handling complex feature spaces. To address this, we propose an improved snake optimizer (ISO). Subsequently, we integrate this enhanced optimizer with a multi-label random forest classifier, resulting in the ISOMLRF hybrid model, which serves as the primary experimental framework for this study.

The hyperparameter optimization process of the MLRF based on the improved snake optimizer (ISO) algorithm is illustrated in Figure 2. First, a feature parameter dataset and a multi-label dataset for model training are constructed. Next, the hyperparameters of the MLRF (including the number of trees, maximum depth, minimum leaf size, and feature selection ratio) are introduced from the iterative ISO algorithm, and multi-label decision trees are built accordingly. Finally, the MLRF classification model is trained using the dataset, and the training error (Hamming Loss) is computed as the fitness to iteratively optimize the hyperparameters.

3.5. Diagnostic Procedure for Constrained PFR Capability in Units

The diagnostic procedure for constrained PFR capability in coal-fired units, based on the ISOMLRF, is illustrated in Figure 3.

In practical applications, online diagnosis of constrained PFR capability can be achieved by identifying the constrained regulation process online, extracting the characterization parameters of the PFR capability, and utilizing the ISO-MLRF diagnostic model to obtain predicted classification labels.

3.6. Evaluation of Model Effectiveness

Multi-label classification algorithms are typically evaluated using the following metrics [12]:

(1)

Hamming Loss

$$F={\displaystyle \frac{1}{nL}}{\displaystyle {\sum }_{i=1}^n{\displaystyle {\sum }_{j=1}^L〚{\mathrm{y}}_{predict ij}\ne y_{test ij}〛}}$$

(26)

where n is the number of samples in the test set; L is the number of labels; y_predictij is the predicted value; y_testij is the test/true value.

(2)

Macro-averaging F1 Score

$$F={\displaystyle \frac{1}{L}}{\displaystyle {\sum }_{j=1}^L\left(2\times \frac{T{P}_j}{T{P}_j+F{P}_j+F{N}_j}\times \frac{T{P}_j}{T{P}_j+F{N}_j}\right)}$$

(27)

(3)

Alarm Rates

False Alarm Rate (FAR):

$$FAR={\displaystyle \frac{FP}{TN+FP}}$$

(28)

Missed Alarm Rate (MAR):

$$MAR={\displaystyle \frac{FN}{TP+FN}}$$

(29)

where TN is the number of correctly predicted normal samples; TP refers to the number of correctly predicted abnormal samples (True Positives); FN refers to the number of incorrectly predicted normal samples (False Negatives); FP refers to the number of incorrectly predicted abnormal samples (False Positives).

3.7. Experimental Platform and Implementation Details

To ensure computational efficiency, all experiments were performed on a workstation configured with an Intel Core i7-14700KF processor (up to 5.6 GHz), an NVIDIA RTX 4090 graphics card (24 GB GDDR6X), and 32 GB of DDR5 RAM, operating on Windows 11. The proposed methodology was developed within the MATLAB R2024a environment. Specifically, we leveraged the Parallel Computing Toolbox to accelerate computations and the Statistics and Machine Learning Toolbox for data analysis.

4. Experimental Result and Analysis

4.1. Simulation Data

4.1.1. Data Generation

To rigorously evaluate the performance of the proposed ISO-MLRF algorithm, two sets of multi-label synthetic datasets were generated following the standard procedures established in [32]. The generation process was implemented using MATLAB R2024a and consisted of the following steps:

Initially, the algorithm defines a D-dimensional space and randomly generates K hypercubes, where K is the number of labels. Each hypercube is defined by its center coordinates and a side length, determining its boundaries. To create a synthetic data point, the algorithm randomly selects a subset of these hypercubes. It then samples a point within the geometric intersection or proximity of the selected hypercubes.

The core mechanism relies on geometric overlap: if a point falls inside a specific hypercube’s boundaries, it is assigned the corresponding label (value 1); otherwise, it is 0. By controlling the centers and sizes of the hypercubes, the method regulates label co-occurrence and correlation. For instance, overlapping hypercubes generate positive examples for multiple labels simultaneously, effectively simulating real-world multi-label scenarios where features trigger multiple outcomes.

Finally, Gaussian noise is added to the points to increase data realism and complexity. The output is a dataset of feature vectors paired with a binary label vector indicating the presence or absence of each label. This method ensures a balanced and controllable generation of complex label dependencies.

To validate the classification performance of the ISO-MLRF algorithm, this paper generates two sets of simulation datasets. Each dataset contains 5000 samples: dataset 1 includes 20 feature parameters and three labels, and dataset 2 includes 30 feature parameters and 10 labels.

4.1.2. Simulation Analysis

To evaluate the effectiveness of the proposed ISOMLRF algorithm, we conducted a comparative study against one baseline and three benchmark algorithms. Specifically, the comparison includes the standard snake optimizer algorithm (SOMLRF) as the baseline, and the whale optimization algorithm (WOAMLRF), the gray wolf optimizer algorithm (GWOMLRF), and the particle swarm optimization algorithm (PSOMLRF) as benchmarks. For dataset 1 and dataset 2, the iterative convergence curves of the five hyperparameter optimization algorithms on the training set are shown in Figure 4.

It can be seen that for dataset 1, the iteration count and Hamming Loss at convergence for the ISOMLRF algorithm proposed in this paper are both lower than those of the other algorithms, indicating that this algorithm offers superior performance in optimizing random forest hyperparameters. The convergence patterns of the various algorithms on the training set of dataset 2 are similar to those of dataset 1.

The five trained optimized classification diagnostic models and the unoptimized MLRF model were simultaneously applied to the test datasets. The Hamming Loss of the six multi-label classification diagnostic algorithms on the test set is shown in

Table 3. Hamming loss.

Test Data	ISO	SO	WOA	GWO	PSO	MLRF
data 1	0.141	0.153	0.157	0.159	0.155	0.234
data 2	0.160	0.177	0.171	0.170	0.170	0.254

, while the macro-averaging F1 Score and the average model diagnosis time are presented in

Table 4. Macro-average F1 score.

Test Data	ISO	SO	WOA	GWO	PSO	MLRF
data 1	0.861	0.846	0.841	0.843	0.845	0.767
data 2	0.841	0.825	0.826	0.832	0.831	0.746

and

Table 5. Average diagnosis time of the model.

Test Data	ISO	SO	WOA	GWO	PSO	MLRF
data 1	0.746	0.761	0.783	0.743	0.739	0.767
data 2	0.760	0.771	0.792	0.832	0.775	0.787

, respectively.

It can be observed that for dataset 1 and dataset 2, which have different numbers of features and labels, the ISO-MLRF model achieves the smallest Hamming Loss on the test sets, with values of 0.141 and 0.160, respectively. It also yields the highest macro-averaging F1 Scores, at 0.861 and 0.841, respectively. The average diagnosis times are 0.746 ms and 0.760 ms, respectively. This demonstrates that the ISO-MLRF model has superior classification and diagnostic performance.

To better compare the classification effectiveness of the different diagnostic algorithms, Figure 5a,c present the FAR and MAR for the classification diagnosis of dataset 1. Similarly, Figure 5b,d show the FAR and MAR for the classification diagnosis of dataset 2.

It can be seen that, for both dataset 1 and dataset 2, the ISOMLRF algorithm exhibits lower and more stable false alarm rates and missed alarm rates compared to other algorithms. Consequently, it can more effectively achieve the classification and diagnosis of the simulated datasets.

4.2. Plant Data

4.2.1. Data Acquisition

This study utilizes historical DCS monitoring data from a 660 MW ultra-supercritical coal-fired unit, with a 1 s sampling interval. Directly sampled features include parameters such as power, grid frequency, rotational speed, main steam pressure, main steam temperature, intermediate point pressure, intermediate point temperature, valve opening, and AGC command.

The sliding window method is used to extract data segments (60 s) corresponding to PFR events. For each PFR segment, process features are calculated, including response delay time, load response time, pressure change rate, intermediate point temperature change rate, power change rate, and valve actuation rate. Figure 6 shows the curves for ramp-up and ramp-down segments of extracted PFR process data.

Using PFR performance indicators (response delay time, load response time, regulation accuracy, and contribution rate), PFR data segments with constrained capability are selected. Corresponding datasets for characterizing the unit’s PFR capability and a multi-label dataset for constrained regulation capability are constructed separately. The label data represent the reasons for constrained PFR capability of the unit, with each label value being “0” or “1”. A value of “0” indicates that the specific reason for constrained capability corresponding to that label did not occur, while “1” indicates that the label is a cause of the unit’s constrained PFR capability.

When collected data significantly deviates from the normal range or contains gaps, data cleaning and gap filling are required. Considering the impact of thermal inertia on the boiler, this study directly fills missing data gaps using forward fill to ensure data integrity. Methods for detecting outliers in the data include the median absolute deviation (MAD) method, the standard deviation method (3σ), and the percentile method. Given sufficient data volume and an adequate number of feature types, this study employs the standard deviation method to detect anomalous data.

4.2.2. Multi-Label Classification Dataset

Based on the analysis in Section 2.1 and Section 2.2, a dataset for constrained PFR capability in coal-fired units is constructed. The dataset consists of 1000 samples, containing 18 feature parameters and 15 labels for constrained PFR capability. The generated characterization dataset for the unit’s PFR capability is shown in

Table 6. Characterization parameters dataset of PFR capability for coal-fired unit.

No.	Response Delay Time	Settling Time	90% Target Load Response Time	…	Valve Opening Deviation	Delay Time
1	3.35	47.46	34.41	…	0.039	12.62
2	2.22	46.88	39.83	…	0.040	19.18
3	3.94	66.19	33.82	…	0.004	42.99
4	2.13	52.52	41.43	…	0.041	45.42
…	…	…	…	…	…	…
1000	3.94	30.57	24.87		0.010	43.69

, and the multi-label dataset for constrained PFR capability is shown in

Table 7. Multiple-label dataset of constrained PFR capability.

No.	Governor Response Delay		Poor Valve Control Accuracy		Excessive Dead Zone		…		Actuator Aging
	PV	AV	PV	AV	PV	AV	PV	AV	PV	AV
1	0	0	0	0	0	0	…	…	1	0
2	0	0	0	0	1	1	…	…	0	0
3	1	1	0	0	0	0	…	…	0	0
4	0	0	0	0	1	1	…	…	0	0
…	…	…	…	…	…	…	…	…	…	…
1000	0	0	0	0	0	0	…	…	0	0

Note: PV—predicted value, AV—actual value.

.

4.2.3. Results Analysis

To ensure reproducibility and provide a clear overview of the optimization process,

Table 8. The specific hyperparameters.

Hyperparameter	Description	Search Range	Optimal Value
numTrees	Number of trees	[10, 200]	182
maxDepth	Maximum depth	[10, 30]	20
minLeafSize	Minimum leaf size	[1, 30]	15
numFeatures	Feature selection ratio	[0.1, 0.8]	0.583

summarizes the specific hyperparameters considered, their respective search ranges, and the final optimal values identified by the optimization process. These tuned parameters were subsequently used for all experiments reported in this study.

To illustrate the predictive accuracy of the classification for each label in the test set, the confusion matrix is shown in Figure 7. It can be observed that the number of samples where the “Negative class” is predicted as the “Positive class” (False Positives) is one, and the total number of samples where the “Positive class” is predicted as the “Negative class” (False Negatives) is 28. This indicates that the classification algorithm has high accuracy. The low number of false positives is due to the relatively small number of “Positive class” samples for each label.

To compare the multi-label classification prediction performance of various algorithms,

Table 9. The performance of the classification diagnosis algorithm for the coal-fired unit’s dataset.

Test Data	ISO	SO	WOA	GWO	POS	MLRF
FAR	0.029%	0.029%	0.055%	0.055%	0.029%	0.071%
MAR	0.053	0.073	0.053	0.062	0.061	0.083
Testing Time(s)	0.0094	0.0129	0.0119	0.0115	0.0112	0.0058
Training Time(s)	12.5	48.3	45.2	42.0	41.5	5.5

presents the false alarm rates, missed alarm rates, training time, and testing time of the six algorithms for the classification diagnosis of the unit’s field dataset.

As illustrated in Table 9, the primary advantage of ISOMLRF lies in its ability to balance precision and robustness. Compared to SOMLRF, our method reduces the average missing alarm rate by 27% (from 0.073 to 0.053). This improvement is attributed to the dynamic update mechanisms in ISO, which effectively prevent the model from falling into local optima—a common issue in traditional swarm intelligence algorithms.

Although the training time of ISOMLRF is slightly higher than that of the static MLRF due to hyperparameter optimization, it remains substantially lower than that of SOMLRF and PSOMLRF (12.5 s vs. 48.3 s and 41.5 s). This indicates that the proposed algorithm offers a more efficient convergence rate, making it more suitable for practical engineering applications where both diagnostic accuracy and computational cost are critical.

More importantly, in the context of practical application, the average FAR of 0.029% achieved by ISOMLRF is substantially lower than the 1–5% range typically reported in the recent literature for similar coal-fired unit fault diagnosis tasks [33]. This confirms that our method not only optimizes hyperparameters effectively but also enhances the model’s robustness against noisy operational data.

Furthermore, previous studies [6,8] primarily relied on physical modeling, which often fails to capture the complex interdependencies among multiple constraints. In contrast, our ISOMLRF algorithm, by utilizing a multi-label random forest optimized by an improved snake optimizer, provides a more holistic diagnosis.

Despite the superior diagnostic accuracy demonstrated by the proposed ISOMLRF algorithm, several limitations must be acknowledged. Firstly, its computational burden during the training phase is higher than that of static machine learning models (MLRF, SVM) due to the iterative optimization of hyperparameters. Furthermore, as an ensemble learning model, the inherent ‘black-box’ nature of the random forest limits the interpretability of the diagnosis results for field engineers. Future work will focus on optimizing the algorithm’s computational efficiency and exploring hybrid approaches to enhance model transparency.

To provide deeper insights into the model’s decision-making logic, Figure 8 illustrates the calculated feature importance scores for each specific fault label. The bar chart clearly highlights the primary driving parameters behind every diagnostic category, making the complex internal correlations within the ISO-MLRF model transparent. This granular visualization not only validates the physical consistency of the algorithm but also empowers plant operators by identifying the exact key indicators to monitor for specific PFR constraints, thereby significantly enhancing the interpretability and operational transparency of the diagnostic system.

5. Conclusions

(1)

The underlying constraints limiting the PFR capability of coal-fired units were analyzed. By integrating current standards with DCS measurement points, a set of quantitative characterization parameters was established, including response time, settling time, and 90% target time. Furthermore, a comprehensive set of limiting factors affecting PFR capability was identified.

(2)

The snake optimizer algorithm was enhanced by introducing a dynamic update mechanism and a novel search strategy, effectively improving its convergence speed and accuracy. Subsequently, a multi-label random forest classification algorithm based on the improved snake optimizer (ISO-MLRF) was proposed. This algorithm optimizes the hyperparameters of the random forest, thereby significantly improving classification accuracy.

(3)

The model was validated using both synthetic multi-label simulation datasets and actual operational data from a 660 MW ultra-supercritical unit. Results on the simulation datasets demonstrate that, compared to other benchmark algorithms, the ISO-MLRF algorithm achieves the highest convergence accuracy with superior convergence speed. Results on the actual unit dataset show that ISO-MLRF attains higher multi-label classification accuracy, with an average false alarm rate of 0.029% and an average missed alarm rate of 5.3% per label, both lower than or equal to those of comparative algorithms. Additionally, the average diagnosis time for a single sample is 9.4 ms, satisfying the requirements for online diagnosis.

(4)

Deploying the proposed multi-label random forest classification algorithm on the PFR capability monitoring and diagnosis platform enables real-time online diagnosis of constrained PFR capability. This provides crucial technical support for enhancing the regulation capability of source-side units and optimizing grid-side dispatch.

(5)

The future scope of this research is outlined. First, although ISO-MLRF performs well, its computational burden during offline training is relatively high; future work will focus on developing lightweight versions for edge computing devices. Second, the current model relies heavily on historical data quality; integrating physics-informed constraints into the data-driven framework could enhance its generalization under extreme or unseen conditions. Third, the application of this diagnostic framework will be extended to other types of power generation units, such as gas turbines and energy storage systems, to verify its universality.