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To keep the main system and the auxiliary equipment of production system of thermal power plants operating normally, it’s a necessary work to fully consider the periodical repeat features of thermal process when modelling the thermal process based on thermal state data of machine units. Since most equipment in power plant work in an electromagnetic coupling environment, a too high ambient temperature will affect the safe operation of units, so when studying the dynamic features of multimodal thermal process of power plants and their soft environment, the temperature distribution features of multimodal thermal process should be classified. To solve these questions, this paper established a model for the coupled temperature field during multimodal thermal process and studied its analytical solutions. At first, this paper proposed a multimodal modelling method for solving the distribution features of coupled temperature during multimodal thermal process, the method can quickly determine the optimal number of local models of multimodal thermal process of power plant units while reducing the computational load of the coupled temperature field model. Then, this paper introduced an improved niche method into the basic crossentropy optimization algorithm and used it to solve the layout optimization problem of coupled temperature field during multimodal thermal process. At last, the validity of the proposed method was verified by experimental results.
multimodal thermal process, coupled temperature field, analytical solution, thermal power plant, cross entropy
In a power system, the thermal power plants serving as power suppliers are one of the most important links [14]. Their function is to complete the energy form transformation of fuel from chemical energy, to the thermal potential energy of steam, then to mechanical energy, and finally to electric energy. Boiler, steam turbine, and generator are called the three main equipment in thermal power plants [511], other auxiliary equipment of production system includes the coal delivery system, the chemical treatment system of water, and the slurry discharge system, etc. These systems cooperate with the main system to complete the task of electricity production [1215]. To ensure these equipment to operate normally, it’s a necessary work to fully consider the periodical repeat features of thermal process of power plants [1619] when modelling the thermal process based on thermal state data of machine units, thereby promoting the units to develop toward the direction of multiparameter, multimodal, and highly automated.
Zhang et al. [20] introduced a novel monitoring method of multimodal dynamic processes based on sparse dynamic inner principal component analysis. In their work, by adopting the concept of intelligent synapses in continual learning, a loss of quadratic term was introduced to penalize the changes of moderelated parameters, and modified synaptic intelligence was proposed to estimate the importance of these parameters. Then, authors also discussed the features of the proposed method, including computational complexity, advantages and potential limitations. At last, compared with several advanced monitoring methods, the effectiveness and superiority of the proposed method were demonstrated by a continuous stirred tank heater case and a practical industrial system. Grigor’ev et al. [21] optimized the structure and parameters of an autonomous hybrid power plant operating from renewable energy sources (sun and wind) and methanol fuel cells using mathematical models of thermal processes, then they applied the developed models in practice and gave the results of simulation experiments of an indicated power plant. Weng et al. [22] pointed out that typical thermal processes are usually monotonically responsive and can be well described by firstorder plus dead time (FOPDT) model, thus the system identification of FOPDT is a major concern in the field of thermal process control, including coalfired power plants and gas turbines. However, step response based openloop experiment is sometimes not available due to the limitation of field operation, which necessitates the development of closedloop identification. Authors employed artificial intelligence to derive a new method for closedloop identification, and used Convolution Neural Network (CNN) to identify operation characteristics of devices in the closedloop condition. Their experimental results proved that the application of CNN had dramatically improved identification accuracy; when the size of training datasets reached 50,000, the identification accuracy was higher than 98%.
Most equipment in power plants work in an electromagnetic coupling environment, a too high ambient temperature will affect the safe operation of units, so when studying the dynamic features of multimodal thermal process of power plants and their soft environment, the temperature distribution features of multimodal thermal process should be classified. To solve these questions, this paper established a model for the coupled temperature field during multimodal thermal process and studied its analytical solutions.
There are a few shortcomings with existing modeling method of coupled temperature field for multimodal thermal process: 1) Termination conditions are unreasonably designed; 2) Feature information contained in sample set are not fully utilized, and it’s easy to pick noise data by mistake; 3) The validity evaluation index of multimodal data clustering is not effectively utilized, which may result in redundancy of the multimodal model.
To overcome these shortcomings, this paper proposed an optimized multimodel modeling method for analyzing distribution features of coupled temperature during multimode thermal process, the method can quickly determine the optimal number of local models of multimodal thermal process of power plant units while reducing the computational load of the coupled temperature field model.
Conventional methods usually pick two samples with the lowest similarity in the data set of multimodal thermal state as the initial cluster centers, different from these existing methods, the multimodal modelling method proposed in this paper based on fuzzy satisfaction clustering could fully consider the timevarying characteristics of multimodal thermal process. Starting from the internal feature information of the data of sample set, the proposed method determines the initial cluster center via Singular Value Decomposition (SVD) to attain ideal clustering results. That is, at first, this method calculates the unit eigenvector and standard deviation of the covariance matrix of multimodal thermal state data; then, based on the calculation results, it selects two sample points around the center point of sample set and takes them as the initial cluster centers. The proposed method can control the core features of the sample set of multimodal thermal state data in the data expansion direction, and quickly complete the classification of sample set by taking the iteration advantage of clustering analysis.
The original multimodal thermal state data matrix C composed of original unit thermal state data samples can be written as:
$C=\left[ \begin{matrix} {{\phi }_{1}}\left( 1 \right) & \cdots & {{\phi }_{K}}\left( 1 \right) & b\left( 1 \right) \\ {{\phi }_{1}}\left( 2 \right) & \cdots & {{\phi }_{K}}\left( 2 \right) & b\left( 2 \right) \\ \vdots & \ddots & \vdots & \vdots \\ {{\phi }_{1}}\left( M \right) & \cdots & {{\phi }_{K}}\left( M \right) & b\left( M \right) \\\end{matrix} \right]$ (1)
By performing centralization and scale transformation operations on C, we have:
${{C}^{*}}=\left[ \begin{matrix} {{{\tilde{\phi }}}_{1}}\left( 1 \right) & \cdots & {{{\tilde{\phi }}}_{K}}\left( 1 \right) & \tilde{b}\left( 1 \right) \\ {{{\tilde{\phi }}}_{1}}\left( 2 \right) & \cdots & {{{\tilde{\phi }}}_{K}}\left( 2 \right) & \tilde{b}\left( 2 \right) \\ \vdots & \ddots & \vdots & \vdots \\ {{{\tilde{\phi }}}_{1}}\left( M \right) & \cdots & {{{\tilde{\phi }}}_{K}}\left( M \right) & \tilde{b}\left( M \right) \\\end{matrix} \right]$ (2)
where,
$\begin{align} & {{{\tilde{\phi }}}_{j}}\left( l \right)=\frac{{{\phi }_{j}}{{n}_{{{\phi }_{j}}}}}{{{\phi }_{j,MAX}}},j=1,2,...,K;l=1,2,...,M; \\ & \tilde{b}\left( l \right)=\frac{b{{n}_{b}}}{{{b}_{MAX}}},l=1,2,...,M \\ & {{n}_{{{\phi }_{j}}}}=\frac{1}{M}\sum\limits_{i=1}^{M}{{{\phi }_{j}}\left( i \right)},j=1,2,...,K; \\ & {{n}_{b}}=\frac{1}{M}\sum\limits_{i=1}^{M}{b\left( i \right)} \\ & {{\phi }_{jMAX}}=\underset{l}{\mathop{\max }}\,\left( \left {{\phi }_{j}}\left( l \right){{n}_{{{\phi }_{j}}}} \right \right),j=1,2,...,K; \\ & {{b}_{MAX}}=\underset{l}{\mathop{\max }}\,\left( \left b\left( l \right){{n}_{b}} \right \right) \\\end{align}$
u_{0}=T_{1×(}_{K}_{+1) }is the center of the sample set of multimodal thermal state data C^{*}, u_{0}=[n_{ψ}_{1}, ..., n_{ψ}_{2}, n_{b}] is the center corresponding to the original multimodal thermal state data matrix C, then the covariance matrix G_{0 }corresponding to C^{* }can be calculated:
${{G}_{0}}=\frac{1}{M1}{{C}^{*}}^{T}{{C}^{*}}$ (3)
Assuming: E_{0}_{j }represents the unit eigenvector of G_{0}, μ_{0j}(j=1, 2, ..., K+1) represents the corresponding eigenvalues, through SVD, E_{0j} and μ_{0j}(j=1,2,..., K+1) can be calculated:
$\begin{align} & {{G}_{0}}= \\ & \left[ {{e}_{01}},...,{{e}_{0\left( K+1 \right)}} \right]diag\left( {{\mu }_{01}},...,{{\mu }_{0\left( K+1 \right)}} \right){{\left[ {{e}_{01}},...,{{e}_{0\left( K+1 \right)}} \right]}^{1}} \\\end{align}$ (4)
The eigenvalue μ_{0j} of G_{0} describes the variance of C* in direction E_{0j}. Assuming: μ_{0,max} represents the maximum eigenvalue of the sample set of multimodal thermal state data, E_{0,max }represents the principle eigenvector, the main direction of data expansion of sample set is the direction of E_{0,max}, in this direction, the variance ε^{2}_{0,max }around u_{0}^{*} is equal to μ_{0,max}.
Let ξ_{1 }be equal to ε_{0,max}e_{0,max}, which can be used to describe the data expansion information of the sample set, then the two initial cluster centers u^{0}_{11} and u^{0}_{12} of C^{*} are:
$u_{11}^{0}={{\xi }_{1}},u_{12}^{0}={{\xi }_{1}}$ (5)
Based on C^{*}, u^{0}_{11}, and u^{0}_{12}, the GK algorithm can be used to cluster the unit state data. If the coupled temperature field model identified based on clustering results does not meet precision requirements, then the number of clustering categories d needs to be increased. Figure 1 shows the generation of new cluster center. Assuming: C_{q}* represents the worst multimodal thermal state data sample subset determined by current iteration; G_{q }represents its covariance matrix, u_{q }represents the cluster center, e_{i}_{1,max }represents the maximum unit eigenvector calculated by SVD, ε_{i}_{1,max }represents the corresponding standard deviation of the data, then delete u_{q}, and let ξ_{i} =ε_{i}_{1,max}e_{i}_{1,max}, and define a new cluster center as follows:
$u_{i1}^{0}={{u}_{q}}{{\xi }_{i}},u_{i2}^{0}={{u}_{q}}+{{\xi }_{i}}$ (6)
Figure 1. Generation of new cluster center
After clustering was finished, by transforming u_{q} into the coordinates of original unit thermal state data, the structure parameters (N, d_{i}, ε_{i}) of the coupled temperature field model could be determined further, wherein N is the number of local models of multimodal thermal state, namely d:
N=d (7)
d_{i} (i=1, 2, ..., N) is the center of the scheduling function, namely the fuzzy cluster center formed by transforming the first K items of u_{i} into the coordinates of original unit thermal state data, which is denoted as u_{iψ}.
$\begin{align} & {{u}_{i}}=\left[ {{u}_{i1}},...,{{u}_{iK}},{{u}_{ib}} \right]\Rightarrow {{d}_{i}}={{u}_{i\psi }} \\ & =\left[ {{\phi }_{1,MAX}}{{u}_{i1}}+{{n}_{\phi 1}},\phi {{K}_{L,MAX}}{{u}_{iK}}+{{n}_{\phi K}}, \right],i=1,2,...,d \\\end{align}$ (8)
Assuming: u_{kψ} (1=1, 2, ..., L) represents L centers closest to the ith center u_{iφ}, α_{d }is a constant used to adjust the width of the Gaussian function, then ε_{i} (i=1, 2, ..., N) is determined by the mean distance between u_{i} and the nearest L centers.
${{\varepsilon }_{i}}=\frac{1}{{{\alpha }_{d}}}\left( \frac{1}{L}\sum\limits_{I=1}^{K}{\left\ {{u}_{i\phi }}{{u}_{k\phi }} \right\} \right),i=1,2,...,d$ (9)
Define I_{p}=[1,1,...,1]^{T}, $I_p \in R^{\prime}$, the formula below gives the applicable domain of the ith local model:
${{\Lambda }_{i}}\in \left[ {{d}_{i}}{{\varepsilon }_{i}}{{I}_{p}},{{d}_{i}}+{{\varepsilon }_{i}}{{I}_{p}} \right],i=1,2,...,N$ (10)
Figure 2 shows the flow of the proposed modeling method.
Figure 2. Flow of the proposed modeling method
This paper chose to use the crossentropy optimization algorithm to solve the layout optimization problem of coupled temperature field during multimodal thermal process. Taking the model constructed in the previous section as the subject, this paper introduced an improved niche method into the basic crossentropy optimization algorithm for the purpose of improving the validity of the analytical solutions of the coupled temperature filed model of multimodal thermal process. Figure 3 shows the flow of algorithm for solving the analytical solutions. Expressions of the species formation strategy and crowding strategy of the niche method are given below. Assuming: R_{i} represents the ith niche, E represents the rest population, a_{j }represents individuals in E, a_{best }represents the optimal individual in E, s represents niche radius, C represents the dimension of design space, DIS(x, y) represents the distance between individuals x and y, then there are:
${{R}_{i}}=\left. \left\{ DIS\left( {{a}_{best}},{{a}_{j}} \right)=\sqrt{\sum\limits_{c=1}^{C}{{{\left( a_{best}^{c}a_{j}^{c} \right)}^{2}}}} \right\}\le s \right\}$ (11)
$\underset{{{a}_{j}}\in {{D}_{i}}}{\mathop{\arg \min DIS}}\,\left( {{a}_{j}},{{a}_{best}} \right)=\sqrt{\sum\limits_{c=1}^{C}{{{\left( a_{best}^{c}a_{j}^{c} \right)}^{2}}}}$ (12)
Figure 3. Flow of algorithm for solving analytical solutions
Overall speaking, the attained analytical solutions of coupled temperature field model for multimodal thermal process face many challenges. The first is the determination of niche radius and the challenge of algorithm parameters, in the meantime, it also needs to bear a huge computational load and maintain population diversity. Therefore, at first, this paper performed adaptive clustering on niche radius, and adopted elite strategy and local search to complete the estimation of distribution parameters, at last, the crossover operator was adopted to generate new niche individuals. After optimization, the algorithm’s ability to solve analytical solutions of the temperature field model had been greatly enhanced.
The adaptive clustering process of niche radius can be divided into two steps: dynamic population division of niche radius, and population adjustment based on same scale. At first, the optimal individual was taken as the seed and other individuals were sorted according to their distance from the seed, then the adaptability values of neighbor individuals were compared. If a worse individual is encountered, then the adaptability value of the niche radius can be determined based on the distance between this individual and the seed. The following formula gives the judgment criterion:
$f\left( {{x}_{i}} \right)>f\left( {{x}_{i1}} \right)$ (13)
To balance the exploration and development of the population, the population divided into multiple clusters was adjusted based on the same scale, that is, if a cluster contains more individuals, then poor individuals contained in it are eliminated; if a cluster contains less individuals, then new niche individuals are produced.
The estimation of distribution parameters of the algorithm includes two parts: the estimation of position selection λ_{i} of the optimal individual, and the estimation of initial standard deviation ε^{0}_{i} and standard deviation ε^{0}_{i}(t>0). To improve the convergence speed of the algorithm, assuming: a_{ibest }represents the optimal individual in the ith niche, if λ_{i} is selected to be a_{ibest}, then there is:
${{\lambda }_{i}}={{a}_{i,best}}$ (14)
Assuming: β represents the variance coefficient associated with upper and lower limits, in order to attain effective solution of the algorithm in the first round of iteration, this paper introduced β to control the distribution of initial population, finally, effective sampling of the individuals had been completed.
$\varepsilon _{i,c}^{0}=\frac{1}{\beta }\times \left( vy\left( c \right)ky\left( c \right) \right)\left( c=1,2...C \right)$ (15)
To provide sufficient multimodal thermal state data sample information, the calculation of standard deviation ε^{0}_{i}(t>0) needs to be performed based on all individuals in the niche. Assuming: k_{i} represents all individuals in the niche, m_{i }represents the size of population in the ith niche, then there is:
$\varepsilon _{i}^{2}=\frac{\sum\nolimits_{m\in {{k}_{i}}}{{{\left( a{{\lambda }_{i}} \right)}^{2}}}}{{{m}_{i}}}$ (16)
The performance of multimodel modeling strategy has a great impact on the global peak number of analytical solutions of the temperature field model searched by the crossentropy optimization algorithm. At the same time, the diversity of population declined sharply during algorithm iteration. To avoid the loss of population diversity, this paper used the crossover operator between the optimal individuals in each niche to produce new niche individuals. Specifically, keeping the total population size ME unchanged was taken as the objective, by referring to the size of niche a_{best}, the rest MEm_{best} new individuals were produced. Assuming: a^{c}_{j} and a^{c}_{l} represent the optimal individual in the ith and jth niche, a^{c}_{i} represents new individuals generated by the crossover operator, C represents the dimension of design variable, then there is:
$a_{i}^{c}=a_{j}^{c}+rand\left( {} \right)\times \left( a_{l}^{c}a_{j}^{c} \right)\left( c=1,2,3...,C \right)$ (17)
Some data samples of multimodal thermal state were listed, it’s set that the maximum number of allowable local models of multimodal thermal state was 5. Figure 4 shows the initial cluster centers of multimodal thermal state data sample set determined by the proposed method and the cluster center attained based on GK clustering. The comparison given in this paper showed that the proposed method is effective and reasonable, it can control the main features of the sample dataset of multimodal thermal state in the data expansion direction, and ensure that the algorithm can find the optimal clustering center of the sample set quickly and accurately.
Figure 5 shows the variation of validity index under the condition of different category numbers. When the initial threshold of the performance index was given a small value, although the validity index of clustering reached local minimum value at a category number of 3, the modelling accuracy at this time was not satisfactory enough. When the initial threshold of the performance index was adjusted to a larger value, the modelling accuracy met requirements at a category number of 5, the validity index of clustering was ideal, and the modelling process terminated. To ensure successful modeling, the threshold of performance index should be adjusted smaller referring to the validity index of clustering, otherwise ideal model accuracy won’t be realized even if the number of local model reaches the preset maximum value. Table 1 gives the specific system model parameters.
Figure 4. Clustering of nonlinear data samples of multimodal thermal state
Figure 5. Validity index under different category numbers
Table 1. System model parameters
Model number 
Parameters of scheduling function 
Parameters of local models 

d_{i}_{1} 
d_{i}_{2} 
ε_{i} 
χ_{i}_{0} 
χ_{i}_{1} 
χ_{i}_{2} 

1 
3.9521 
4.1753 
1.3453 
5.0854 
0.4736 
0.3319 
2 
1.5987 
3.6758 
1.2736 
8.8123 
1.9357 
0.7618 
3 
3.5329 
1.7284 
1.2459 
7.2617 
0.5723 
1.4317 
4 
1.6874 
1.5923 
1.1736 
10.4358 
2.1736 
1.9524 
Table 2. Comparison of system modeling accuracy
Model 
Number of local models 
Model error 
Piecewise affine model 
7 
0.081 
TS fuzzy model 
11 
0.0145 
Local model network 
5/6 
0.113/0.103 
The proposed method 
5/6 
0.062/0.035 
Table 2 compares the accuracy of the proposed model and other models in terms of the target multimodal thermal process system, as can be known from the table, the proposed model exhibited obvious advantages in terms of accuracy and local model number.
For the purpose of fair comparison, a same initial population size and a same calculation times of maximum objective function were set for all algorithms used in this study to solve the analytical solutions of the layout optimization problem of coupled temperature field. The proportion of elite samples in the multimodal thermal state sample dataset was set to 0.1, and the settings of other parameters are given in Table 3.
To evaluate the advantage of the proposed method in solving the analytical solutions of the layout optimization problem of coupled temperature field, this paper compared it with other algorithms. The selected reference algorithms include several classic optimization algorithms, such as ant colony optimization (ACO) algorithm, particle swarm optimization (PSO) algorithm, bacteria foraging optimization (BFO) algorithm, firefly algorithm (FA), artificial fish swarm algorithm (AFSA), and artificial bee colony (ABC) algorithm. Moreover, this paper also compared the proposed algorithm 1 (before adaptive clustering of niche radius) with the proposed algorithm 2 (before introducing elite strategy and local search to perform distribution parameter estimation) to verify the improvement effect of the conventional niche algorithm.
Table 3. Model parameter settings
Sample No. 
Calculation times of maximum objective function 
Population size 
Number of correct predictions 
Coefficient of variance 
1 
5.1E+4 
90 
30 
1/35 
2 
2.1E+4 
110 
30 
1/35 
3 
2.1E+4 
310 
30 
1/35 
4 
4.1E+4 
310 
30 
1/35 
5 
2.1E+4 
10 
30 
1/35 
6 
2.1E+4 
210 
30 
1/35 
7 
4.1E+4 
210 
60 
1/35 
8 
4.1E+4 
210 
110 
1/35 
9 
4.1E+4 
210 
110 
1/35 
Table 4. Optimal analytical solutions of different algorithms
Algorithm 
Unit 1 
Unit 2 

Measuring point 1 
Measuring point 2 
Measuring point 1 
Measuring point 2 
Measuring point 3 

ACO 
2.0915 
2.7358 
0.0209 
0.0112 
0.257 
PSO 
0.0035 
0.5137 
0.0000 
0.1436 
0.0241 
BFO 
0.1723 
1.4465 
0.0223 
0.0145 
0.0135 
FA 
1.5734 
1.4578 
0.0157 
0.0085 
0.0023 
AFSA 
0.4712 
0.0239 
0.2675 
0.2736 
0.1935 
ABC 
0.2357 
0.5671 
0.0453 
0.0106 
3.50E06 
The proposed algorithm 1 
1.2735 
2.4941 
0.0213 
0.0229 
0.0127 
2.3675 
1.1445 
0.0025 
0.0045 
4.472E02 

0.0134 
0.1024 
2.53E05 
3.89E06 
0.1153 

The proposed algorithm 2 
1.2257 
0.8938 
0.0046 
0.0125 
0.00589 
0.2512 
0.3857 
0.0083 
0.0113 
0.0176 

0.3359 
2.0592 
0.0142 
4.937E05 
0.0145 

The proposed algorithm 
2.5359 
1.3605 
0.0165 
0.0053 
0.0123 
2.7512 
0.1574 
0.0089 
0.0114 
0.0037 

0.3157 
1.1731 
0.0251 
0.0067 
0.0019 
Table 4 gives the statistical results and the optimal results of the proposed algorithm and reference algorithms in solving the analytical solutions of the layout optimization problem of coupled temperature field, as can be seen from the data in the table, the proposed algorithm performed better than other reference algorithms.
The population size of the improved niche algorithm used in the paper was respectively adjusted to 100, 200, and 400 to solve the temperature field layout optimization problem of all equipment in the target unit, and four optimal threshold values of 2.6, 2.7, 2.8, and 2.9 were set. Figure 6 shows the number of optimal analytical solutions attained by the proposed algorithm. As can be seen from the figure, when the population size of the niche algorithm was set as 200 or 400, the proposed algorithm gave more competitive advantages of multimodal optimization performance of unit thermal process state in terms of solving analytical solutions of the layout optimization problem of couple temperature field. When the optimal threshold was 2.7, the proposed algorithm could get more than 130 different optimal analytical solutions in a single iteration, which had verified that the proposed algorithm attained more excellent multimodal optimization ability in case of larger population size.
Figure 6. Number of optimal analytical solutions attained by the proposed algorithm
This paper modeled the coupled temperature field in multimodal thermal process and studied its analytical solutions. At first, an improved modelling method was proposed for analyzing the distribution features of coupled temperature in multimodal thermal process, the method can quickly determine the optimal number of local models of multimodal thermal process of power plant units while reducing the computational load of the coupled temperature field model. Then, an improved niche method was introduced into the basic crossentropy optimization algorithm for the purpose of solving the layout optimization problem of coupled temperature field in multimodal thermal process.
In the experiment, the clustering of nonlinear data samples of multimodal thermal process state was given, and the results proved the proposed modeling method is reasonable and effective. Then, the variation of validity index under the condition of different category numbers was given; for the target multimodal thermal process system, the modeling accuracy of the proposed model and other models was compared, and the results verified the obvious superiority of the proposed model in terms of accuracy and number of local models. Moreover, the optimal results of the proposed algorithm and a few reference algorithms in solving the layout optimization problem of coupled temperature field were shown, and the results demonstrated that the proposed algorithm had outperformed other reference algorithms. At last, this paper analyzed the number of optimal analytical solutions attained by the proposed algorithm and the conclusion indicated that the proposed algorithm attained more excellent multimodal optimization ability in case of larger population size.
The construction and research of “New Medicine” curriculum system under the background of artificial intelligence (Project Numbe 202101040001).
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