Data-Driven-Based Intelligent Alarm Method of Ultra-Supercritical Thermal Power Units

Abstract

In order to ensure the safe operation of the ultra-supercritical thermal power units (USCTPUs), this paper proposes an intelligent alarm method to enhance the performance of the alarm system. Firstly, addressing the issues of slow response and high missed alarm rate (MAR) in traditional alarm systems, a threshold optimization method is proposed by integrating kernel density estimation (KDE) and convolution optimization algorithm (COA). Based on the traditional approach, the expected detection delay (EDD) indicator is introduced to better evaluate the response speed of the alarm system. By considering the false alarm rate (FAR), and EDD, a threshold optimization objective function is constructed, and the COA is employed to obtain the optimal alarm threshold. Secondly, to address the problem of excessive nuisance alarms, this paper reduces the number of nuisance alarms by introducing an adaptive delay factor into the existing system. Finally, simulation results demonstrate that the proposed method significantly reduces the MAR and EDD, improves the response speed and performance of the alarm system, and effectively reduces the number of nuisance alarms, thereby enhancing the quality of the alarms.

1. Introduction

Compared to traditional thermal power units, ultra-supercritical thermal power units (USCTPUs) exhibit higher thermal conversion efficiency and lower coal consumption, which can significantly reduce the pollution emissions of power plants, thereby enhancing the energy-saving and environmental protection effects of thermal power plants [1]. However, due to the complex system structure and increased coupling among equipment, USCTPUs are prone to experiencing abnormal operating conditions. The occurrence of malfunctions during operation not only reduces the production efficiency and economic benefits of the power plant, but can also lead to serious safety accidents [2]. In order to mitigate the adverse effects of malfunctions and prevent unit shutdown or equipment damage caused by malfunctions, it is necessary to utilize an alarm system to monitor critical equipment and parameters.

In the current industrial power generation field, alarm systems are widely used due to their critical role. However, the alarm system of USCTPUs faces issues such as slow system response, missed alarms, and excessive nuisance alarms. The main cause of slow system response and increased missed alarms is the improper setting of alarm thresholds. When the alarm threshold set by the system is large, it delays the generation of alarm signals, resulting in a slow system response and an increase in missed alarms, thus increasing the risk of equipment damage due to abnormal operating conditions. The excessive number of nuisance alarms is primarily due to the lack of a delay factor in traditional alarm systems. Due to the complexity of USCTPUs, there are strong correlations among alarm variables. The repeated occurrence of alarm signals for a specific monitoring parameter may be influenced by other parameters and does not necessarily indicate a fault. This leads to a significant number of nuisance alarms in the alarm information, which severely interferes with the judgment of operators [3,4]. Therefore, an efficient and intelligent alarm system is crucial for ensuring the secure and steady running of USCTPUs.

Currently, the commonly used method for optimizing alarm systems is to optimize and design alarm thresholds. The optimization and design methods for alarm thresholds can be categorized into model-based [5], experience-based [6], and data-driven approaches [7,8]. Reference [9] proposes a method for designing alarm thresholds applicable to condensate water pumps in thermal power plants and constructs a model estimator based on Bayesian filtering to maximize the absolute cumulative error of model parameters, thus obtaining the alarm thresholds, and the effectiveness of the proposed method is verified through experiments. In [10], expert knowledge is combined with data featuring reliability characteristics to build an optimized model for alarm threshold based on interval evidence inference, and the effectiveness of this method is demonstrated through experimentation. Unlike the limitations of the above modeling and knowledge-based approaches, data-driven methods do not require the establishment of complex mechanistic models or extensive empirical knowledge; they only necessitate the analysis of alarm data, making them simple and efficient, and widely applied in industrial fields. In [11], a multivariable alarm threshold optimization method was proposed to address the issue of high false alarm rate (FAR) resulting from unreasonable alarm threshold settings, and simulation results demonstrate that this method effectively reduces FAR. Reference [12] introduces a variable alarm threshold optimization method based on correlation and cluster analysis, establishing an objective function concerning FAR and missed alarm rate (MAR) with variable weight coefficients, and employs a numerical optimization algorithm (NOA) to solve the objective function. Reference [13] uses standardized Euclidean distance to cluster and group the variables, calculates the variable weights by entropy weighting method, establishes the threshold optimization objective function with variable weights, and finally uses ant colony optimization algorithm to solve the optimal alarm threshold, and the experimental results show that this method can effectively reduce the FAR. Reference [14] initially groups parameter variables, calculates original weights, establishes an optimization objective function concerning FAR, MAR, and original weights, and then utilizes particle swarm optimization algorithm (PSO) to solve the objective function; experimental results indicate that the proposed method reduces the number of alarms by 37.8% compared to the initial situation. Reference [15] proposes a dynamic variable alarm threshold optimization algorithm suitable for non-stationary processes under different operating conditions, significantly reducing FAR and ensuring the reliability of the alarm system. Reference [16] presents a process alarm threshold optimization method based on satisfaction optimization, constructing a weighted objective function for FAR and MAR, and solving the optimal threshold through NOA, further enhancing the alarm system performance. Reference [17] utilizes non-parametric statistical methods based on windowing and testing to adjust the Hotelling T² statistic and Q statistic, thereby generating second-level control limits for alarm thresholds, and the experimental results demonstrate that using these two methods can effectively reduce the FAR. Reference [18] addresses the lack of the ability in traditional alarm threshold optimization methods to adjust thresholds online, offering an adaptive optimization method for simplifying multi-layer Bayesian network alarm thresholds based on active transfer entropy, achieving a reduction in the sum of the average FAR from 28.5% to 19.7%. The aforementioned studies have further optimized alarm thresholds, to some extent reducing FAR; however, they overlook the issue that simply reducing FAR may lead to an increase in MAR, posing significant safety risks to production. Additionally, they neglect the problem of slow response in the alarm system. Furthermore, due to the absence of delay factors in traditional alarm systems, there is an issue of excessive nuisance alarms, preventing substantial improvement in the performance of alarm systems.

To address the issues of slow response speed and high MAR in traditional alarm systems, an optimization method for alarm thresholds is proposed in this paper. Firstly, the data under both normal and abnormal operating conditions were fitted into probability density functions (PDF) using the kernel density estimation (KDE) approach. Then, based on the traditional threshold optimization methods, the detection delay expectation (EDD) to measure the agility of the response to the alarm system is considered, and the threshold optimization objective function is established with the FAR, MAR, and EDD as the optimization evaluation indexes. The convolution optimization algorithm (COA) is used to solve the problem, and the optimal alarm threshold is obtained. Furthermore, to address the issue of a high number of nuisance alarms, an adaptive alarm delay method is proposed. Based on the analysis of the alarm time interval data, an adaptive delay factor is introduced into the existing system to eliminate nuisance alarms. Finally, the effectiveness of the proposed methods is validated using historical operational data from 1000 MW and 660 MW USCTPU. The results demonstrate that the proposed method significantly reduces the MAR and EDD, improves the response speed and performance of the alarm system, effectively reduces the number of nuisance alarms, and enhances the quality of alarms.

The rest of the paper is organized as follows: Section 2 briefly introduces the relevant theoretical knowledge of KDE and COA. Section 3 presents the proposed alarm threshold optimization method and adaptive alarm delay method in detail. Section 4 applies the alarm threshold optimization method to 1000 MW and 660 MW USCTPU for case analysis and conducts an instance analysis of the adaptive alarm delay method to verify the performance of the proposed methods. Section 5 provides a summary of the entire paper.

2. Fundamental Theories

2.1. KDE

KDE is developed based on the statistical histogram, which differs from the traditional histogram estimation which yields a continuous PDF. In practical applications, KDE can estimate unknown variables by performing weighted averaging over fixed windows of data, thereby revealing the probability density distribution pattern. The estimated density function minimizes the integrated mean square error between the estimated and true density functions of the variables. Additionally, KDE allows for the study of distribution characteristics based on the data, making it suitable for data with arbitrary distribution properties. Therefore, selecting KDE to compute the PDF of the data in this paper facilitates the calculation of probability expressions for performance metrics [19]. It can be expressed as follows:

$$\widehat{H}\left(x\right)=\frac{1}{mu}{\displaystyle \sum _{p=1}^{m}K\left(\frac{x-x_p}{u}\right)}$$

(1)

where x is the estimation point, x_p is the monitoring data point, m is the data length, K( ) represents the kernel function, u is the width of the window, and $\widehat{H}\left(x\right)$ is the estimated value of the probability density function of the current monitoring data point. In KDE, the kernel function and the window width collectively decide on the precision of KDE. In this paper, we choose the Gaussian kernel function, and the window width u is obtained by minimizing the mean integrated square error (MISE) [20]. The definition of MISE is as follows:

$$\begin{array}{cc}\hfill MISE[\widehat{H}(x)]& =E\left({\displaystyle \int {{\displaystyle \left[\widehat{H}(x)-H(x)\right]}}^2}dx\right)\hfill \\ & ={{\displaystyle (mu)}}^{-1}\left[{\displaystyle \int {{\displaystyle K}}^2(s)}ds\right]+\frac{{{\displaystyle \left[{\displaystyle \int {{\displaystyle s}}^{2}K(s)}ds\right]}}^2}{4}·{\displaystyle \int {{\displaystyle \left[\frac{{{\displaystyle d}}^2}{dx}H(x)\right]}}^2}dx·{{\displaystyle u}}^4+o(\frac{1}{mu}+{{\displaystyle u}}^4)\hfill \end{array}$$

(2)

where H(x) is the real worth of the probability density function at the monitoring point, and o() denotes a higher-order infinitesimal term. After neglecting the infinitesimal term in (2), the asymptotic mean integrated square error (AMISE) is obtained [21], and it can be expressed as follows:

$$AMISE[\widehat{H}(x)]={{\displaystyle (mu)}}^{-1}\left[{\displaystyle \int {{\displaystyle K}}^2(s)}ds\right]+\frac{{{\displaystyle \left[{\displaystyle \int {{\displaystyle s}}^{2}K(s)}ds\right]}}^2}{4}·{\displaystyle \int {{\displaystyle \left[\frac{{{\displaystyle d}}^2}{dx}H(x)\right]}}^2}dx·{{\displaystyle u}}^4$$

(3)

When the derivative of AMISE is equal to 0, its value reaches the minimum. The optimal window width u_b is given by the following [22]:

$$u_b=\sqrt[5]{\frac{{\displaystyle \int {{\displaystyle K}}^2(s)ds}}{m{{\displaystyle \left[{\displaystyle \int {{\displaystyle s}}^{2}K(s)ds}\right]}}^2·{\displaystyle \int {{\displaystyle [\frac{{{\displaystyle d}}^2}{dx}H(x)]}}^{2}dx}}}$$

(4)

Finally, employing the Silverman’s rule of thumb [23] to simplify the calculation of (4), the u_b can be expressed as follows:

$$u_b=S\times {(\frac{4}{3}\times \frac{1}{m})}^{\frac{1}{5}}$$

(5)

where S is the standard deviation of the data.

2.2. COA

2.2.1. Convolutional Search Process

Step 1: Update the longitudinal convolution location. Firstly, define a longitudinal convolution as follows [24]:

$${{\displaystyle X}}_1^t={{\displaystyle X}}^t\ast [2\times rand(k,1)-{{\displaystyle I}}_a]$$

(6)

where rand(k,1) is a k × 1 matrix whose elements are random numbers between 0 and 1, k is the size of the convolution kernel, I_a is a k × 1 matrix whose elements are all one, t is the current iteration number, * denotes convolution operation, X^t is the location vector of the population in generation t, and ${{\displaystyle X}}_1^t$ is the location vector of the population in generation t after updating the longitudinal convolution location.

Then, compare the fitness values of X_t and ${{\displaystyle X}}_1^t$, and optimally replace the larger individual location in X_t as shown below:

$${{\displaystyle X}}_b^t=\left\{\begin{array}{l}{{\displaystyle X}}_{1b}^t, {{\displaystyle fitness}}_{{{\displaystyle X}}_{1b}^t}<{{\displaystyle fitness}}_{{{\displaystyle X}}_b^t}\\ {{\displaystyle X}}_b^t, {{\displaystyle fitness}}_{{{\displaystyle X}}_{1b}^t}\ge {{\displaystyle fitness}}_{{{\displaystyle X}}_b^t}\end{array}\right.$$

(7)

where $X_b^t$ is the location of the b-th individual, $X_{1b}^t$ is the location of the b-th individual after updating the longitudinal convolution location, and fitness is an adaptation function.

Step 2: Update the horizontal convolution location. Firstly, define a horizontal convolution as follows:

$${{\displaystyle X}}_2^t={{\displaystyle X}}^t\ast [2\times rand(1,k)-{{\displaystyle I}}_c]$$

(8)

where rand(1,k) is a 1 × k matrix whose elements are random numbers between 0 and 1, I_c is a 1 × k matrix whose elements are all one, and ${{\displaystyle X}}_2^t$ is the location vector of the population in generation t after updating the horizontal convolution location.

$${{\displaystyle X}}_b^t=\left\{\begin{array}{l}{{\displaystyle X}}_{2b}^t, {{\displaystyle fitness}}_{{{\displaystyle X}}_{2b}^t}<{{\displaystyle fitness}}_{{{\displaystyle X}}_b^t}\\ {{\displaystyle X}}_b^t, {{\displaystyle fitness}}_{{{\displaystyle X}}_{2b}^t}\ge {{\displaystyle fitness}}_{{{\displaystyle X}}_b^t}\end{array}\right.$$

(9)

where $X_{2b}^t$ is the b-th location of the individual after updating the horizontal convolution location.

Step 3: update the matrix convolution location. Firstly, define a matrix convolution as follows:

$${{\displaystyle X}}_3^t={{\displaystyle X}}^t\ast [2\times rand(k,k)-{{\displaystyle I}}_n]$$

(10)

where rand(k,k) is a k × k matrix whose elements are random numbers between 0 and 1, I_n is a k × k matrix whose elements are all one, and ${{\displaystyle X}}_3^t$ is the location vector for the population of the t-th generation after updating the matrix convolution location.

Then, compare the fitness values of X^t and ${{\displaystyle X}}_3^t$, and optimally replace the larger individual location in X^t as shown below:

$${{\displaystyle X}}_b^t=\left\{\begin{array}{l}{{\displaystyle X}}_{3b}^t, {{\displaystyle fitness}}_{{{\displaystyle X}}_{3b}^t}<{{\displaystyle fitness}}_{{{\displaystyle X}}_b^t}\\ {{\displaystyle X}}_b^t, {{\displaystyle fitness}}_{{{\displaystyle X}}_{3b}^t}\ge {{\displaystyle fitness}}_{{{\displaystyle X}}_b^t}\end{array}\right.$$

(11)

where $X_{3b}^t$ is the location of the b-th individual after updating the matrix convolution location.

Step 4: Update the integrated location. Add ${{\displaystyle X}}_1^t$, ${{\displaystyle X}}_2^t$, and ${{\displaystyle X}}_3^t$ together according to proportional weights, as shown below:

$$X_4^t=\frac{e_1\times X_1^t+e_2\times X_2^t+e_3\times X_3^t}{e_1+e_2+e_3}$$

(12)

where $X_4^t$ is the location vector for the population of the t-th generation after updating the integrated location, and e₁, e₂, and e₃ are random numbers between 0 and 1.

Then, compare the fitness values of X^t and $X_4^t$, and optimally replace the larger individual location in X^t as shown below.

$${{\displaystyle X}}_b^t=\left\{\begin{array}{l}{{\displaystyle X}}_{4b}^t, {{\displaystyle fitness}}_{{{\displaystyle X}}_{4b}^t}<{{\displaystyle fitness}}_{{{\displaystyle X}}_b^t}\\ {{\displaystyle X}}_b^t, {{\displaystyle fitness}}_{{{\displaystyle X}}_{4b}^t}\ge {{\displaystyle fitness}}_{{{\displaystyle X}}_b^t}\end{array}\right.$$

(13)

where $X_{4b}^t$ is the location of the b-th individual after updating the integrated location.

2.2.2. Result Enhancement

Select the best solution $X_{be}^t$ from X^t, and apply Gaussian mutation with non-inertial weight to $X_{be}^t$ in order to avoid falling into local optima during the iteration, as shown below:

$${{\displaystyle \widehat{X}}}_{be}^t=\left[1-{{\displaystyle \left(\frac{t}{{{\displaystyle t}}_{\max }}\right)}}^2\right]·{{\displaystyle X}}^t+randn·{{\displaystyle X}}_{be}^t$$

(14)

where ${\widehat{X}}_{be}^t$ is the best solution after enhanced perturbation, t_max is the maximum number of iterations, and randn is a random number. Finally, compare ${\widehat{X}}_{be}^t$ and $X_{be}^t$, and replace the individual location of $X_{be}^t$ on a merit basis:

$${{\displaystyle X}}_{be}^t=\left\{\begin{array}{l}{{\displaystyle \widehat{X}}}_{be}^t, {{\displaystyle fitness}}_{{{\displaystyle \widehat{X}}}_{be}^t}<{{\displaystyle fitness}}_{{{\displaystyle X}}_{be}^t}\\ {{\displaystyle X}}_{be}^t, {{\displaystyle fitness}}_{{{\displaystyle \widehat{X}}}_{be}^t}\ge {{\displaystyle fitness}}_{{{\displaystyle X}}_{be}^t}\end{array}\right.$$

(15)

3. Proposed Intelligent Alarm Method

3.1. Calculation of Performance Optimization Metric

For a univariate alarm system, one of the most critical performance metrics is accuracy, which can be evaluated using metrics such as FAR and MAR [25]. FAR refers to the probability of the system issuing an alarm signal when the process variable is operating normally. A high FAR can increase the number of alarms, significantly impacting the judgment and efficiency of the operators. MAR denotes the probability that the alarm system fails to respond with an alarm when the process variable experiences an anomaly. A high MAR can increase the potential risk of equipment failure, posing serious safety hazards to production. Therefore, the index value of FAR and MAR can reflect the performance of the designed alarm system. In alarm systems, FAR and MAR are inversely related, and this contradictory relationship is reflected in the receiver operating characteristic (ROC) curve, as shown in Figure 1. Assuming a process variable x(t) with known PDF for normal and abnormal operating conditions, PDF curve distributions for normal and fault data can be obtained, as shown in Figure 2. Here, the solid line represents the PDF curve distribution of normal operating condition data ${\widehat{H}}_1\left(x\right)$, while the dashed line represents the PDF curve distribution of abnormal operating condition data ${\widehat{H}}_2\left(x\right)$.

According to Figure 2, we can calculate p₁, p₂, v₁, and v₂, where p₁ + p₂ = v₁ + v₂ = 1. It can be expressed as follows:

$$\left\{\begin{array}{l}{{\displaystyle p}}_1={\displaystyle {\int }_{{{\displaystyle x}}_t}^{+\infty }{{\displaystyle \widehat{H}}}_1\left(x\right)dx}\\ {{\displaystyle p}}_2={\displaystyle {\int }_{-\infty }^{{{\displaystyle x}}_t}{{\displaystyle \widehat{H}}}_1\left(x\right)dx}\end{array}\right.$$

(16)

$$\left\{\begin{array}{l}{{\displaystyle v}}_1={\displaystyle {\int }_{-\infty }^{{{\displaystyle x}}_t}{{\displaystyle \widehat{H}}}_2\left(x\right)dx}\\ {{\displaystyle v}}_2={\displaystyle {\int }_{{{\displaystyle x}}_t}^{+\infty }{{\displaystyle \widehat{H}}}_2\left(x\right)dx}\end{array}\right.$$

(17)

Based on the definitions of FAR and MAR [25], the FAR and MAR can be expressed as follows:

$$FAR={{\displaystyle p}}_1={\displaystyle {\int }_{{{\displaystyle x}}_t}^{+\infty }{{\displaystyle \widehat{H}}}_1\left(x\right)dx}$$

(18)

$$MAR={{\displaystyle v}}_1={\displaystyle {\int }_{-\infty }^{{{\displaystyle x}}_t}{{\displaystyle \widehat{H}}}_2\left(x\right)dx}$$

(19)

Apart from accuracy, agility is another important performance metric for alarm systems. Agility is a measure of the response speed of the alarm system, typically characterized by EDD. Detection delay refers to the time difference between the occurrence of an anomaly and the triggering of the alarm state by the alarm system. A smaller value indicates a higher sensitivity and faster response speed of the alarm system. Therefore, minimizing the EDD as much as possible is essential to ensure the stable and reliable operation of USCTPUs. Assuming that the moment of failure occurrence is t_s during operation, and an alarm signal is sent out after h moments, with the signal being sent at time t_a, the detection delay at h moments can be expressed as follows:

$$\left\{\begin{array}{l}p(DD=0)=p\{{{\displaystyle t}}_{al}={{\displaystyle t}}_s\}={{\displaystyle v}}_2\\ p(DD=1)=p\{{{\displaystyle t}}_{unal}={{\displaystyle t}}_s\}·p\{{{\displaystyle t}}_{al}={{\displaystyle t}}_{s+1}\}={{\displaystyle v}}_1{{\displaystyle v}}_2\\ ⋮\\ p(DD=h)=p\{{{\displaystyle t}}_{unal}={{\displaystyle t}}_s\}\dots p\{{{\displaystyle t}}_{unal}={{\displaystyle t}}_{s+h-1}\}·p\{{{\displaystyle t}}_{al}={{\displaystyle t}}_{s+h}\}={{\displaystyle v}}_2{{\displaystyle v}}_1^h\end{array}\right.$$

(20)

where t_al is the time when an alarm signal is issued, and t_unal is the time when no alarm signal is issued.

Next, find the EDD [25], calculated as follows:

$$E(DD)={\displaystyle \sum _{h=0}^{\infty }h·p(DD=h)}=\frac{{{\displaystyle v}}_1}{{{\displaystyle v}}_2}=\frac{MAR}{1-MAR}$$

(21)

3.2. Design of Alarm Threshold Optimization Method

In order to find the optimal alarm threshold, the proposed threshold optimization method in this paper takes into account the FAR, MAR, and EDD. With the FAR, MAR, and EDD as optimization indicators, an alarm threshold optimization objective function is established, and the objective is to minimize the objective function using the COA for optimization calculation, thereby obtaining the optimal alarm threshold x_c. The flowchart of the threshold optimization algorithm is shown in Figure 3, the pseudocode is shown in Algorithm 1, and the objective function can be represented as follows:

$$F(x)={{\displaystyle o}}_1\frac{FAR}{FAR*}+{{\displaystyle o}}_2\frac{MAR}{MAR*}+{{\displaystyle o}}_3\frac{EDD}{EDD*}$$

(22)

where o₁, o₂, and o₃ are the design weighting coefficients, and FAR*, MAR*, and EDD* are the maximum acceptable FAR, MAR, and EDD.

Algorithm 1: The pseudocode of the alarm threshold optimization algorithm.

Algorithm 1: Threshold optimization algorithm

Input: Process variable x(t)

Parameter: xt, dim, ub, lb, tmax, a

Output: FARbe, MARbe, EDDbe, xc

1: normal data = find x(t) ≤ xt, abnormal data = find x(t) > xt

2: normal PDF = PDF(normal data), abnormal PDF = PDF(abnormal data),

3: Function F(x) = fun(x)

4:  define o1, o2, o3, FAR*, MAR*, and EDD*

5:  compute FAR, MAR, EDD

6:  construct F(x)

7: end

8: Initial populations

9: repeat

10: for t = 1:tmax

11:  for i = 1:a

12:    vertical convolution position update

13:    horizontal convolution position update

14:    square convolution position update

15:    comprehensive location update

16:  end for

17:  calculate the best position

18:  for j = 1:dim

19:    solution quality enhancement

20:  end for

21:  update the global best location

22: end for

23: until Convergence

24: return xc, FARbe, MARbe, EDDbe

3.3. Design of Alarm Adaptive Delay Algorithm

The principle of the alarm delay method is that the system starts alarming when a variable exceeds d consecutive alarm signals, and the system stops alarming when there are d consecutive non-alarm signals. The alarm adaptive delay method designed in this paper can dynamically adjust the size of the delay factor d based on the actual alarm data and the mean and variance of the alarm duration. The steps of the alarm adaptive delay method are as follows:

Step 1: Based on the original alarm threshold, the alarm interval data E processed by the delay factor d can be obtained, where E = [E₁,E₂,E₃,…,E_r], and E_i is the time interval of the ith alarm (i = 1,2,…,r).

Step 2: Based on the alarm interval data E, calculate the total mean E_avg and the standard deviation S_E of E.

$$E_{avg}=\frac{{\displaystyle \sum _{i=1}^r{E}_i}}{r}$$

(23)

$$S_E=\sqrt{\frac{{\displaystyle \sum _{i=1}^r{(E_i-E_{avg})}^2}}{r-1}}$$

(24)

Step 3: Based on the E_avg and S_E, calculate the coefficient of variation CV_d for the alarm interval data processed by the delay factor d [26].

$$C{V}_d=\frac{S_E}{E_{avg}}$$

(25)

Step 4: Set the standard coefficient of variation CV_st to one, and compare the calculated CV_d with the CV_st. If CV_d > CV_st, update the delay factor d = d + CV_d, and return to step 1; if CV_d < CV_st, end the algorithm process, and obtain the final adaptive delay factor.

4. Experimental Analysis

4.1. Performance Validation of COA

In order to fully evaluate the performance of the COA, this paper compared it with seven intelligent optimization algorithms, including the subtraction-based averaging optimization (SABO) [27], optical microscope algorithm (OMA) [28], rimmed ice optimization (RIME) [29], as well as the classical genetic algorithm (GA), particle swarm optimization (PSO), whale optimization algorithm (WOA), and sparrow search optimization algorithm (SSA). The experimental dataset consisted of 12 test functions from the CEC 2017 benchmark. To make sure that the experiments are fair, the population size for all the algorithms was set to 20, the maximum number of iterations was set to 500, and the dimensionality was set to 100. Moreover, each algorithm was run five times separately, and the average value was taken as the final result to reduce random errors.

To visually observe the performance of the COA and the aforementioned seven optimization algorithms during the iteration process, an analysis of the iterative convergence curves of the eight optimization algorithms on test functions was conducted, as shown in Figure 4. Specifically, results from six challenging functions were selected for demonstration.

Figure 4 clearly demonstrates the variations in the fitness values during the optimization iteration process of the COA, SABO, OMA, RIME, GA, PSO, WOA, and SSA algorithms. From the convergence curves in Figure 4, it is evident that the COA consistently reaches the optimal value at the fastest rate during the iteration process of each test function, maintaining a leading convergence speed compared to the other seven algorithms. Furthermore, when reaching the maximum number of iterations, the fitness value of the COA is significantly lower than that of the other seven algorithms, indicating that the convergence accuracy of the COA exceeds that of the other algorithms. It can be observed that the COA possesses superior optimization capabilities, as well as stronger abilities in both local and global search.

Furthermore, to further validate the performance of the COA, the Friedman test was employed to compare the statistical differences between each group of results. The rank values were compared to evaluate the performance of the different algorithms, where smaller rank values indicate higher performance [30]. The Friedman rank and ranking results for each algorithm on the test functions are presented in

Table 1. Friedman rank sum ranking of 12 algorithms on different test functions.

Function	COA	PSO	SSA	WOA	GA	OMA	RIME	SABO	COA Ranking
F1: Shifted and Rotated Bent Cigar Function	1	2	7	4.8	8	4.2	3	6	1
F2: Shifted and Rotated Sums of Different Power Functions	1.2	1.8	7.2	5.2	7.8	4	3	5.8	1
F3: Shifted and Rotated Zakharov Function	1	4.6	4.2	6.6	7.2	3.2	2	7.2	1
F5: Shifted and Rotated Rastrigin’s Function	1	2.2	6.4	5.6	8	4.4	2.8	5.6	1
F8: Shifted and Rotated Lunacek BiRastrigin Function	1.2	3.2	7	5.6	7.8	3.8	1.8	5.6	1
F9: Shifted and Rotated Levy Function	1	5	2.6	5.8	7.8	5.4	2.6	5.8	1
F12: Hybrid Function 2 (N = 3)	1	2	7	4.8	8	4.2	3	6	1
F16: Hybrid Function 6 (N = 4)	1.4	1.6	8	6.2	6.8	4	3	5	1
F20: Hybrid Function 6 (N = 6)	1.4	1.6	5.2	6	6.8	6.2	3	5.8	1
F21: Composition Function 1 (N = 3)	1	4	7.4	5.8	6.6	2.8	2.2	6.2	1
F24: Composition Function 4 (N = 4)	1	3.6	7.4	5.2	7.6	3.8	2	5.4	1
F30: Composition Function 10 (N = 3)	1	2	7.2	4.4	7.8	4.8	3	5.8	1
Average	1.1	3	7	5.5	7.5	4.2	2.6	5.9	1

.

From Table 1, it can be observed that in the Friedman test conducted on each test function, the COA consistently achieved the smallest rank. The average rankings of the 8 algorithms were COA, RIME, PSO, OMA, WOA, SABO, SSA, and GA. The COA ranked first, indicating that the COA holds a statistical and performance advantage over the other algorithms.

4.2. Performance Analysis of Threshold Optimization

4.2.1. Data Validation of 1000 MW USCTPU

To verify the intelligent alarm capability of the proposed alarm threshold optimization method during the normal operation of the unit, experimental data from a 1000 MW-level secondary reheating ultra-supercritical DC boiler system in a certain thermal power plant were collected. The data are the variations in a high-pressure feed pump BV winding temperature over time, with an initial alarm threshold set at 99.78. The sample data curve is shown in Figure 5.

Firstly, the process values of the BV winding temperature data of the high-pressure feedwater pump were separately collected under normal and abnormal working conditions based on the alarm threshold, resulting in two groups of data. Then, according to Equation (5), the optimal window widths for the two groups of data were calculated as u_b1 = 3.4 and u_b2 = 1.4, respectively. Subsequently, Equation (1) was used to fit the PDF curves of the normal and abnormal states, as shown in Figure 6.

According to Figure 6, the PDFs of the normal and abnormal state data can be obtained, denoted as L(x) and R(x), respectively. It can be expressed as follows:

$$L(x)=\frac{1}{32753\times 3.4}{\displaystyle \sum _{q=1}^{32753}\frac{1}{\sqrt{2\pi }}}{{{\displaystyle e}}^{-\left(\frac{x-x_q}{\sqrt{2}\times 3.4}\right)}}^2$$

(26)

$$R(x)=\frac{1}{1918\times 1.4}{\displaystyle \sum _{d=1}^{1918}\frac{1}{\sqrt{2\pi }}}{{{\displaystyle e}}^{-\left(\frac{x-x_d}{\sqrt{2}\times 1.4}\right)}}^2$$

(27)

Based on Equations (26) and (27), we can further derive the expressions for the FAR, MAR, and EDD, as expressed below

$$FAR={\displaystyle {\int }_{{{\displaystyle x}}_b}^{+\infty }\frac{1}{32753\times 3.4}{\displaystyle \sum _{q=1}^{32753}\frac{1}{\sqrt{2\pi }}}{{{\displaystyle e}}^{-\left(\frac{x-x_q}{\sqrt{2}\times 3.4}\right)}}^{2}dx}$$

(28)

$$MAR={\displaystyle {\int }_{-\infty }^{{{\displaystyle x}}_b}\frac{1}{1918\times 1.4}{\displaystyle \sum _{d=1}^{1918}\frac{1}{\sqrt{2\pi }}}{{{\displaystyle e}}^{-\left(\frac{x-x_d}{\sqrt{2}\times 1.4}\right)}}^{2}dx}$$

(29)

$$EDD=\frac{{\displaystyle {\int }_{-\infty }^{{{\displaystyle x}}_c}\frac{1}{1918\times 1.4}{\displaystyle \sum _{d=1}^{1918}\frac{1}{\sqrt{2\pi }}}{{{\displaystyle e}}^{-\left(\frac{x-x_d}{\sqrt{2}\times 1.4}\right)}}^{2}dx}}{1-{\displaystyle {\int }_{-\infty }^{{{\displaystyle x}}_c}\frac{1}{1918\times 1.4}{\displaystyle \sum _{d=1}^{1918}\frac{1}{\sqrt{2\pi }}}{{{\displaystyle e}}^{-\left(\frac{x-x_d}{\sqrt{2}\times 1.4}\right)}}^{2}dx}}$$

(30)

Next, by substituting the expressions for the FAR, MAR, and EDD into Equation (22), we obtain the objective function F(x) for optimizing the alarm threshold, where the FAR* = 0.02, MAR* = 0.02, and EDD* = 0.04. The optimization is performed using the COA; at this point, the variable search range is set to [95, 105], and the fitness value curve during the optimization process with increasing iteration numbers as well as the curve showing the variation in the objective function values under different alarm thresholds, are illustrated in Figure 7.

According to Figure 7a, it can be observed that when using the COA to optimize the objective function, the minimum value of the function is reached after 6 iterations, with a value of 0.371, which remains unchanged in the subsequent iterations. Additionally, as shown in Figure 7b, the optimal alarm threshold corresponding to the minimum function value is x_c = 98.333. Furthermore, based on Equations (28)–(30), the optimal values for the FAR, MAR, and EDD are calculated to be FAR = 0.0079, MAR = 0.0068, and EDD = 0.0069, respectively.

Finally, in order to better validate the effectiveness of the proposed threshold optimization method in this paper, the threshold optimization method proposed in this paper is compared with the 3σ method commonly used in the industry, NOA threshold optimization method, PSO threshold optimization method, and the optimization index results of the original system alarm threshold. The results are shown in

Table 2. Comparison of indicator results.

Index	Method	FAR	MAR	EDD	Alarm Threshold
1	Original system	0.0046	0.0423	0.044	99.78
2	3σ	0.0029	0.0538	0.0568	106.11
3	KDE + NOA	0.0073	0.01	0.0101	98.5888
4	KDE + PSO	0.0072	0.0103	0.0104	98.6119
5	KDE + COA	0.0079	0.0098	0.0069	98.3333

.

From the results for the first and fifth methods in Table 2, it is evident that compared to the original system before optimization, although the proposed threshold optimization method in this paper has resulted in a slight increase in the FAR by only 0.0033, the obtained FAR remains within an acceptable range. Furthermore, the original values of the MAR and EDD exceeded the maximum acceptable range, posing safety hazards to production and reducing the response speed of the alarm system. However, after optimization using the proposed method, both the MAR and EDD are reduced to approximately one-sixth of their original values, and the optimized results fall within the maximum acceptable range. This achievement demonstrates that the proposed threshold optimization method significantly reduces the MAR and EDD without significantly increasing the FAR, thereby effectively mitigating the safety hazards associated with excessive false negatives and improving the response speed and performance of the alarm system. Although using the 3σ method can greatly reduce the FAR, the MAR and EDD are much higher than the threshold optimization method in this paper. Additionally, from the results for the third, fourth, and fifth methods in Table 2, it can be inferred that the utilization of the COA method in this paper further reduces the MAR and EDD, indicating that the COA method employed in the optimization process exhibits superior performance.

4.2.2. Data Validation of 660 MW USCTPU

To validate the applicability and scalability of the threshold method on the different types of USCTPUs, experimental data from the 660 MW ultra-supercritical variable-pressure direct current boiler system of a thermal power plant were collected. The data depict the variation in temperature over time in the non-drive section of the high-pressure feedwater pump A. The initial alarm threshold is set at 62, and the sample data curve is shown in Figure 8.

To validate the effectiveness of the threshold optimization method proposed in this paper on the 660 MW USCTPU, the optimization indicators resulting from the threshold optimization method proposed in this paper are compared with those under the original system alarm thresholds, where the FAR* = 0.02, MAR* = 0.02, and EDD* = 0.04. The final results are shown in

Table 3. The performance indexes of the threshold optimization method are compared with the original system and the traditional threshold optimization method.

Index	Method	FAR	MAR	EDD	Alarm Threshold
1	Original system	0.12189	0.074827	0.080879	62
2	KDE + COA	0.15697	0.015809	0.016063	61.2357
3	3σ	0.038	0.1538	0.1818	65.46
4	KDE + NOA	0.155	0.055	0.058	61.465
5	KDE + PSO	0.1546	0.067	0.072	61.33

.

From Table 3, it can be observed that compared to the results of the original system before optimization, the threshold optimization method proposed in this paper increases the FAR by 0.035, but it remains within the maximum acceptable range without affecting the safe operation. However, both the MAR and EDD exceeded the maximum acceptable range before optimization, posing safety hazards to production. After optimization using the method proposed in this paper, the MAR and EDD are significantly reduced and are within the maximum acceptable range. Compared with the 3σ method, the MAR and EDD optimized by the threshold method proposed in this paper are 1/10 of the 3σ method, indicating that using the threshold method can improve the response speed of the alarm system. Additionally, from the results for the second, fourth, and fifth methods in Table 3, it can be inferred that the utilization of the COA method in this paper further reduces the MAR and EDD, indicating that the COA method employed in the optimization process exhibits superior performance. This demonstrates that the proposed threshold optimization method also exhibits a superior performance on the 660 MW USCTPU.

4.2.3. Performance Verification of the Proposed Threshold Optimization Method under System Fault Conditions

To verify the robustness of the threshold method proposed in this paper, it is applied when the system experiences faults. Fault data from a 1000 MW-level secondary reheating ultra-supercritical direct current boiler system were collected. The fault observed was an elevated temperature in the winding 1 of the high-pressure feedwater pump AV. The sample data operational curve is shown in Figure 9.

From Figure 9, it is evident that the system experienced a fault at sample point 13,967, after which the temperature remained above the alarm threshold set in the original system. Subsequently, the threshold optimization method proposed in this paper was applied to optimize the alarm thresholds of the original system. The optimized results were then compared with the optimization indicator results under the original system’s alarm thresholds, where the FAR* = 0.02, MAR* = 0.002, and EDD* = 0.002. The comparative results are presented in

Table 4. Comparison of performance indicators under fault conditions.

Index	Method	FAR	MAR	EDD	Alarm Threshold
1	Original system	0.0094	0.004	0.004	71
2	KDE + COA	0.0129	0.00146	0.00146	70.4312
3	3σ	0.005	0.035	0.036	75.472
4	KDE + NOA	0.011	0.018	0.018	70.465
5	KDE + PSO	0.012	0.0174	0.018	71.3324

.

From Table 4, it can be observed that after optimizing the alarm thresholds using the method proposed in this paper under fault conditions, the MAR and EDD are significantly reduced and controlled within the safety range. Although there is an increase in the FAR, it is also within the safety range. Overall, the threshold optimization method proposed in this paper demonstrates good performance even under fault conditions. As shown by the second, third, fourth, and fifth methods in Table 4, the results obtained in this section are similar to those in Section 4.2.1 and Section 4.2.3. This indicates that when the system fails, compared with traditional threshold design methods, the proposed threshold method is the optimal solution.

4.2.4. Real-Time Analysis of the Proposed Threshold Optimization Method

To validate the effectiveness of the threshold optimization method on real-time operational data, the algorithm was deployed onto a user interface within an online platform. The threshold optimization algorithm was integrated as a functional module on the user interface, enabling all functionalities of the optimization algorithm. Real-time operational data stored in the Distributed Control System (DCS) were inputted into the threshold optimization module through the database on the online platform to obtain the optimized performance metric results. This integration ensured seamless connectivity with existing alarm systems. The results before and after the optimization were visualized on the platform’s graphical interface.

The real-time operational data of the high-pressure main steam temperature from a 1000 MW-level secondary reheating ultra-supercritical direct current boiler system were taken as the research subject. The sampling frequency was set at 1 s, and 9.5 h of real-time operational data were collected. Among them, the alarm threshold is 556 °C. The real-time operational data curve is depicted in Figure 10.

The performance comparison results of the alarm threshold before and after optimization using the threshold optimization algorithm module proposed in this article are displayed in the graphical configuration of the online platform, as shown in Figure 11. From Figure 11, it can be observed that during real-time operation, the threshold optimization algorithm proposed in this paper still achieves a further reduction in the MAR and EDD without significantly increasing the FAR. This demonstrates the effectiveness of the threshold optimization algorithm proposed in this paper during the real-time operation of the unit.

4.3. Performance Analysis of Alarm Adaptive Delay Algorithm

4.3.1. Data Verification of 1000 MW USCTPU

To validate the suppression capability of nuisance alarms during the normal operation of a 1000 WM USCTPU using the proposed adaptive alarm delay method, the reheat desuperheating water flow data in a 1000 MW-level secondary reheat ultra-supercritical DC boiler system were selected as the research subject, with its operational curve depicted in Figure 12. It can be observed from Figure 12 that the flow data contain numerous sample values fluctuating around the alarm threshold. Whenever the sample values of the flow data fluctuate below the alarm threshold, an alarm is triggered, leading to repetitive alarm signals, known as nuisance alarms. These alarms are often not caused by variable anomalies but rather by strong correlations among alarm variables, with the repeated occurrence of alarm signals potentially influenced by other parameters rather than indicating a malfunction. Upon statistical analysis, the original number of alarms reached 1647.

Firstly, based on the pre-set alarm threshold and the original alarm data, the time interval data for each alarm are obtained. Then, the alarm system is optimized using an alarm adaptive delay method according to the alarm time interval data. The changes in the coefficient of variation CV and the delay factor d are shown in Figure 13.

From Figure 13a, it can be seen that the value of the CV is 0.994 after the 14th update, which is less than the CV_st. This indicates that the setting of the delay factor is reasonable at this point, and the algorithm meets the stopping update condition. Figure 13b shows that the optimal delay factor after the optimization using the alarm adaptive delay method is d = 22.

In order to visually demonstrate the effect of setting the delay factor on suppressing nuisance alarms, a comparison is made between the alarm signal data before and after setting the delay factor. The alarm signal values are recorded as 1 when an alarm occurs, and 0 otherwise. The results are shown in Figure 14.

As shown in Figure 14a, without setting an adaptive delay factor in the traditional alarm system, the number of alarms is 1647, severely affecting the operation of the staff. However, in Figure 14b, due to the reasonable setting of the adaptive delay factor, the number of alarms is reduced to 27, which is 1/61 of the original number of alarms. Therefore, adding an appropriate delay factor in the alarm system can significantly reduce the number of nuisance alarms and improve the quality of the alarm signals.

4.3.2. Data Verification of 660 MW USCTPU

The expansion data of the gas exhaust expansion section in the 660 MW ultra-supercritical direct current boiler system are used as the research object to verify the effectiveness and universality of the proposed adaptive delayed alarm method on the different types of USCTPU. The operational curves are depicted in Figure 15.

The alarm signal data before and after setting the delay factor are compared, and the results are shown in Figure 16. As shown in Figure 16a, the number of alarms without setting the delay factor is 1121, while after setting the adaptive delay factor, the number of alarms decreases to 41. It can be seen that the adaptive delay alarm method proposed in this paper can greatly reduce the number of disturbance alarms. This also verifies the effectiveness and universality of the proposed delay alarm method on the different types of ultra-supercritical thermal power units.

4.3.3. Performance Verification under System Fault

To further investigate the effectiveness and reliability of the proposed adaptive delay alarm method when the system experiences faults, data from a 1000 MW USCTPU experiencing a fault are used as the research subject. The fault scenario involves the RSV2 opening too slowly, and the operational curve during the fault occurrence is depicted in Figure 17.

From Figure 17, it can be observed that during the fault occurrence, the opening speed of RSV2 is too slow, below 0.1, resulting in 836 alarms. Subsequently, when applying the adaptive delay alarm method during the fault occurrence, the generated alarm signals are shown in Figure 18. It can be seen from Figure 18 that the number of alarms after setting the adaptive delay factor is 118. Compared to the original system, the number of alarms has been significantly reduced. Therefore, it can be concluded that under fault conditions, the adaptive delay alarm method proposed in this paper still performs well, further validating the effectiveness and reliability of the proposed adaptive delay alarm method.

4.3.4. Real-Time Analysis of the Proposed Adaptive Delay Alarm Method

To verify the effectiveness of the proposed adaptive delay alarm method validated on real-time operational data, the adaptive delay alarm method proposed in this paper is deployed onto an online platform. The adaptive delay alarm algorithm and the alarm signal generation algorithm’s code are integrated into different modules, replacing the entire functionality of the algorithms. Real-time operational data of a 1000 MW USCTPU DCS, stored in the online platform’s database, are utilized, with the measurement point being the high-pressure main steam flow rate. The original alarm threshold is set to 250. The sampling period is 1 s, and 4 h of real-time operational data are captured as the test data. The real-time operational curve is depicted in Figure 19.

The real-time operational data are inputted into the alarm signal generation module, and based on the alarm threshold, the number of alarms is determined. After statistical analysis, it is found that a total of 184 alarms occurred within the 4 h period. Subsequently, the real-time operational data are processed using the adaptive delay alarm module. The processed alarm signals are depicted in Figure 20. From Figure 20, it can be observed that after applying the delay factor, the number of alarms reduces to 27. This indicates that the adaptive delay alarm method proposed in this paper exhibits a superior performance even during real-time system operation, effectively reducing the number of alarms.

5. Conclusions

This paper has proposed an intelligent alarm method, which includes the optimization method of alarm threshold based on KDE and COA. Unlike traditional methods, this method has taken the MAR as the optimization objective, aiming to significantly reduce the MAR while controlling the FAR within a reasonable range. It also considers the EDD of the alarm system and combines the FAR, MAR, and EDD to construct the objective function for threshold optimization. Finally, the COA has been used to optimize and solve the objective function, obtaining the optimized alarm threshold. The results have shown that the proposed threshold optimization method can significantly reduce the MAR and EDD, almost without increasing the FAR, thus eliminating the safety hazards that a high MAR may bring and improving the safety of USCTPUs.

Furthermore, to address the issue of a high number of nuisance alarms, this paper has introduced an alarm adaptive delay method. Initially, based on the pre-set alarm threshold and original alarm data, the time intervals between each alarm are obtained. Subsequently, the delay factor is adaptively updated based on the mean and variance of these alarm time intervals. The results have indicated that the inclusion of the delay factor can significantly reduce the number of nuisance alarms and enhance the quality of alarms.

The intelligent alarm method proposed in this paper is designed for single-variable alarm systems, but in practical production operations, multi-variable alarm situations are more common. Once an abnormality occurs in one variable, it may cause simultaneous abnormalities in other related variables, leading to alarm surges, which are highly detrimental to safe production operations. Therefore, future work will focus on studying the correlation of variables in multi-variable alarm systems. We will determine the topology of variables by analyzing causal relationships among them, thereby more accurately identifying the source of alarms.