Prediction of temperature distribution in a furnace using the incremental deep extreme learning machine

Boiler description

In the present study, a 350 MW supercritical coal-fired boiler is selected as the research subject. The boiler has a π-shaped arrangement with a single furnace chamber and a double flue. The furnace is 58.3 m high and has a cross-sectional area of 14.627 m × 14.627 m. The horizontal flue is 5.32 m long and the depths of the front and rear tail flue are 6.05 m and 6.82 m, respectively. A new type of tangential combustion is adopted in the boiler. Six layers of pulverized coal air chambers (A∼F) are distributed in the main combustion area of the furnace, and each layer is arranged with four horizontal pulverized coal nozzles on the four walls of the water-cooled wall. Moreover, eight layers of the secondary air (AA, AB, BC, CC, DD, DE, EF, and FF) and four layers of separated over fire air (SOFA1∼SOFA4) enter the furnace through the nozzles in the four corners of the chamber. The boiler structure and the burner arrangement are schematically shown in Fig. 1.

The main operating parameters at the rated power of the boiler are shown in Table 1.

Framework of IDEM modeling

Modeling the temperature distribution in the furnace mainly consists of three steps, including CFD simulation, data sets classification and reconstruction, and IDELM modeling. The overall modeling process is presented in Fig. 2.

Step 1: Input parameters and boundary conditions are set according to the type of boiler and the unit load. Then CFD simulation is carried out to simulate the combustion s and flow process.

Step 2: Based on the secondary air distribution mode, the CFD datasets are divided into 6 subclasses. The K-Mean clustering method is used to find the benchmark working condition of each subclass. Representative samples of each working condition are selected and then the dataset is reconstructed using data obtained by solving the corresponding increments between other working conditions and the benchmark.

Step 3: Based on the reconstruction datasets of temperature increment of each working condition, prediction models of each subclass are established based on the DELM.

Classification of the working conditions

There are not only continuous variables among the inputs of the CFD model but also variables such as burner arrangement mode and distribution mode of the air dampers that have discrete nature. These variables affect the location of the center of the combustion flame, the temperature distribution, and the combustion products.

Although the volume of secondary air and SOFA are different under different loads, the flame center position and temperature distribution in the furnace are generally similar under the same air distribution patterns. Figure 3 indicates that there are two kinds of secondary air distribution: balanced mode and pagoda mode. Moreover, there are three SOFA distributions, including balanced mode, pagoda mode, and waist drum mode. In order to cover different air distribution modes, 120 working conditions are divided into six subclasses, in which each subclass has 20 datasets. The subclasses are listed in Table 4.

Selecting clustering centers

The training sets for each subclass is limited. In order to improve the prediction accuracy, a benchmark working condition was set for each subclass. Then the increment-based method was used to construct the temperature distribution model. Considering the high dimensionality, large number, and complex nature of the simulated experimental data, the K-means data clustering method was applied to select clustering centers. This method is an iterative clustering algorithm, which continues iterative calculations until the criterion function converges.

For each subclass, the temperature field data obtained from CFD simulation is extracted, and the distance function is used as the evaluation index for the similarity measure of the K-means clustering algorithm. Based on the minimum Euclidean distance between the benchmark working condition and other working conditions of this subclass, the benchmark working condition for each subclass is obtained. This can be mathematically expressed as follows: (1)${\displaystyle \mathrm{D}=\sum _{j=1}^k\sqrt{\sum _{i=1}^n{\left(x_i-y_i\right)}^2}}$

Where, k is the number of cluster centers, x_i is the temperature distribution value in the benchmark working condition, y_i is the temperature distribution value in other working conditions, and D is the sum of the Euclidean distance.

The final cluster center selection is shown in Table 5. The clustering results show that the benchmark working conditions of the six subclasses are 75% load, BCDEF mill operation, and SOFA is 0 tilt angle.

Data preprocessing

The studied boiler is equipped with six coal mills, and the pulverized coal enters the furnace through the burner nozzles on the four walls of the boiler. Three types of mill operation modes are designed for each load. Figure 4 shows the temperature distribution of YOZ planes in different coal mill operation modes at 100% load. Fig. 4A the case, in which ABCDE mills operate. In this case, the flame center is higher than others, while the temperature of the cold ash hopper is the highest. However, the lowest temperature occurs in the SOFA area. In Fig. 4B, the mill distribution range is large so the cold ash hopper area temperature and SOFA take the temperature between the middle. Fig. 4C shows that the flame center moves upward, the high-temperature area is large, and the highest temperature occurs in the SOFA area. Meanwhile, the lowest temperature occurs in the cold ash hopper area. It is inferred that the coal mill operation mode significantly affects the temperature field distribution.

The six mills in the actual plant are arranged at different furnace heights, but such discrete variables cannot be used as input for data-driven modeling. In order to resolve this problem, the coal mill operation mode is converted into a flame center position correction factor. (2)${\displaystyle \mathrm{M}=A-B\left(x_r+\Delta x\right)}$ where M is the parameter that reflects the effect of the relative position of the highest temperature along the furnace height. A and B are empirical coefficients that depend on the fuel type and furnace structure, Δx denotes the relative position correction value of the highest point of the flame (Li, Yan & Liu, 2017). X_r is the relative height of the burner, which can be calculated in formula (3). (3)${\displaystyle x_r=\frac{\sum n_i{B}_j{H}_{ri}}{H_L\sum n_i{B}_i}}$ where H_L is the height of the furnace chamber, that is, the height from the bottom of the furnace or the middle plane of the cold ash hopper to the middle of the furnace exit smoke window middle height of the furnace chamber; H_r is the height of the burner arrangement, that is the height of the burner axis from the middle plane of the cold ash hopper; H_L is the amount of coal burned corresponding to the burner; H_ri is the height of the burner arrangement corresponding to the layer; n_i is the number of burners in the layer.

The variables in the 3D data set have different orders of magnitude. Table 6 lists the value range of each parameter.

The min-max normalization is carried out to preprocess the data. This can be mathematically expressed as follows: (4)${\displaystyle x_i^{\ast }=\frac{x_i-x_{\min }}{x_{\max }-x_{\min }}}$

where x_i is the original value, $x_i^{\ast }$ is the normalized value, x_max and x_min x_min is the maximum and minimum value.

Reconstruction of the dataset

The training data in machine learning should be selected in a way to balance the trade-off between computational complexity and accuracy. Based on the mesh independence test, a model with 2.8 million meshes is selected in the simulation, so each dataset has 2.8 million data. However, the huge dataset in data-driven modeling will increase the information redundancy and model complexity, thereby reducing the computational efficiency. Therefore, information extraction is the main challenge for a data-driven model.

The process of sample selection and dataset reconstruction is shown in Fig. 5. For 20 typical working conditions of each subclass, the increment between the data of 19 working conditions and the corresponding variables of the benchmark working condition is firstly calculated to obtain 19 new increment data sets. Similar to cross-validation, one dataset is selected as the test dataset, and the remaining 18 newly constructed datasets are randomly sampled. In the present study, 50,000 samples with the same spatial location are selected for each working condition. Finally, the selected samples are reconstructed into one dataset as the training and validation set.

IDELM modeling

Deep extreme learning machine (DELM), also known as multi-layers extreme learning machine (ML-ELM), is a deep neural network formed by stacking multiple extreme learning machine autoencoder (ELM-AE). Its structure is shown in Fig. 6. In this method, ELM-AE is initially used as the basic unit for unsupervised learning to train and learn the input data. Then input weights and bias (W, b) of ELM-AE are randomly generated and the implied layer matrix H is formulated in the form below: (5)${\displaystyle H=g\left(WX+b\right).}$

The loss function of the ELM network is defined as follows: (6)${\displaystyle min{L}_{ELM-AE}=\frac{1}{2}{∥\beta ∥}^2+\frac{C}{2}{∥X-H\beta ∥}^2.}$

DELM adds the restriction of the output weight regular term, which can prevent overfitting. The output weights β of ELM-AE can be calculated using the following expression: (7)${\displaystyle \beta =\left\{\begin{array}{c}\left(\frac{I}{C}+H^{T}H\right)H^{T}X,N\le n\hfill \\ H^T\left(\frac{I}{C}+H^{T}H\right)H^{T}X,N>n\hfill \end{array}\right.}$

Where C is the network regularization parameter, which is introduced to improve the generalization performance of the ELM-AE method; X is the input sample matrix; n is the number of neurons in the hidden layer; N is the number of input samples, and g() is the activation function. By training ELM-AE, unsupervised mapping of samples to depth features is realized.

The input weights of the nodes in each hidden layer are transpositions of the output weights between that layer and the previous layer forming the ELM-AE. Therefore, each layer can be implemented to extract the features of the previous layer. This can be expressed as follows: (8)${\displaystyle H^k=g{\left({\beta }^k\right)}^T{H}^{k-1},k>1.}$

Unlike other deep learning methods, DELM does not require fine-tuning. Both ELM-AE and the final DELM regression layers use the least-squares multiplication method and only one step of inverse calculation to obtain the updated weights. Consequently, DELM has fast training so it is an appropriate model for online modeling and real-time prediction of temperature fields.

In the present study, the DELM network is constructed using three hidden layers. It should be indicated that the higher the number of hidden layers, the more complex the network, and the higher the training time. To accurately predict temperature distribution, trial simulations were carried out and the number of nodes in the three hidden layers is set to 30, 50, and 50, respectively. Moreover, a tanh function was used as the network activation function.

The following 11 variables were selected as inputs: x-coordinate, y-coordinate, z-coordinate, load increment, total air volume increment, total coal volume increment, total primary air volume increment, total secondary air volume increment, secondary air temperature increment, coal mill increment (relative position of the flame center), and angle increment. In return, the difference in 3D temperature distribution between the predicted working conditions and the benchmark working condition was the output variable. After IDEM modeling, the incremental prediction of the temperature distribution for different working conditions was obtained separately. The actual temperature field can be obtained by superimposing the incremental output and the benchmark working condition.

Evaluation indicator

In this article, the IDELM algorithm was used to model six subclasses and obtain the temperature distribution library of this unit under different operating conditions. The IDELM-based prediction results are compared with the values obtained from CFD simulation and ELM, deep belief network (DBN), and deep neural network (DNN) algorithms. The generalization performance and prediction accuracy of algorithms are evaluated using the mean absolute error (MAE), symmetric mean absolute percentage error (SMAPE), and decision coefficient (R²). These indicators are defined as follows: (9)${\displaystyle MAE=\frac{1}{N}\sum _{i=1}^N\left|y_p\left(i\right)-y_c\left(i\right)\right|}$ (10)${\displaystyle SMAPE=\frac{1}{N}\sum _{i=1}^N\frac{\left|y_p\left(i\right)-y_c\left(i\right)\right|}{\left(y_p\left(i\right)+y_c\left(i\right)\right)/2}}$ (11)${\displaystyle R^2=1-\frac{\left[\sum _{i=1}^N{\left(y_p\left(i\right)-y_c\left(i\right)\right)}^2\right]/N}{\left[\sum _{i=1}^N{\left({\overline{y}}_p\left(i\right)-y_c\left(i\right)\right)}^2\right]/N}.}$ Where N is the number of samples in the test set; ${\overline{y}}_p$ represents average value of temperature 3D distribution, y_c is numerical simulation value of temperature 3D distribution; y_p indicates prediction of temperature 3D distribution value.

The MAE indicator reflects the average deviation degree between the CFD simulation data and the predicted data. Moreover, the SMAPE indicator is used to evaluate the goodness-of-fit of the model. The lower the values of MAE and SMAPE, the better the prediction performance. Finally, the R² represents the matching degree between the predicted and numerical simulation data. The closer its value to 1, the stronger the fitting ability of the model.

Analysis of different sample sizes

In this article, the random sampling method is used to select representative samples among 2.8 million data in each working condition. The distribution of samples in the three-dimensional furnace is shown in Fig. 7. It is observed that samples cover the whole spatial area of the furnace so the integrity of the information is guaranteed. In the main combustion zone and SOFA zone, the combustion reaction is complex and the temperature distribution varies greatly. Therefore, the selected samples in these zones are relatively dense. Three models with 30,000, 50,000, and 80,000 samples were analyzed respectively. 19 increment-based datasets were used as test sets in turn, the remaining 18 working conditions are randomly sampled, and the datasets were reconstructed. After IDELM modeling, the average values of the performance indexes were calculated. Table 7 shows that the performance indicators of the 50,000 samples are better than those with 30,000 and 80,000 samples. Accordingly, 50,000 samples were selected in all working conditions.

Comparative analysis of different algorithms

To verify the effectiveness of the temperature distribution model based on the proposed method, the DELM prediction results were compared with the results obtained from deep belief network (DBN), deep neural network (DNN), and ELM algorithms. The prediction results in subclass 2 are presented in Table 8. It is observed that the prediction results of the DELM-based model outperform the other models. The mean R² of the DELM-based prediction model is 0.86, indicating that the prediction model can accurately predict the temperature distribution in the furnace. On the other hand, the DELM-based prediction model has the smallest MAE value, and its SMAPE value is around 10%. Accordingly, it is concluded that the DELM-based temperature distribution prediction model has promising prediction accuracy and excellent generalization ability.

Figure 8 shows the error boxplot of the studied algorithms. It is observed that among the studied algorithms, the DELM algorithm has the lowest absolute error while having a tighter variation bandwidth. The variation of the predicted results using the DELM algorithm is consistent with that of the CFD simulation. It is concluded that the DELM-based model has a reasonable fitting effect and prediction ability.

The PB4 working condition is closest to the average performance of subclass 2. Figure 9 illustrates the PB4 prediction results of four algorithms. We can see that the prediction results of the four algorithm models are consistent with the trend of the CFD target value. The predicted value under the IDELM model is closer to CFD target value, and the results of the other three prediction models of temperature distribution are higher than the target data, and the DBN results have the most deviation.

Prediction analysis of different working conditions

To analyze the overall performance of the proposed model, the prediction results of the six subclasses are validated respectively. Table 9 shows that the mean R² of all subclasses is higher than 0.82, which has a good consistency with experimental data. Furthermore, it is found that Subclass 2 has the best model prediction with the highest average R² and the smallest MAE and SMAPE values. On the other hand, Subclass 3 has the smallest mean R² and Subclass 5 has the largest prediction error with a mean MAE of 109.76 and SMAPE of 11.02%. It is inferred that as the working condition is closer to the center of clustering, the prediction results improve and all poor predictions occur near 50% load. The furnace flame filling degree at a lower load is reduced and the temperature 3D distribution changes significantly relative to the benchmark operating conditions.