Prediction of coal ash fusion temperatures using computational intelligence based models
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Abstract
In the coal-based combustion and gasification processes, the mineral matter contained in the coal (predominantly oxides), is left as an incombustible residue, termed ash. Commonly, ash deposits are formed on the heat absorbing surfaces of the exposed equipment of the combustion/gasification processes. These deposits lead to the occurrence of slagging or fouling and, consequently, reduced process efficiency. The ash fusion temperatures (AFTs) signify the temperature range over which the ash deposits are formed on the heat absorbing surfaces of the process equipment. Thus, for designing and operating the coal-based processes, it is important to have mathematical models predicting accurately the four types of AFTs namely initial deformation temperature, softening temperature, hemispherical temperature, and flow temperature. Several linear/nonlinear models with varying prediction accuracies and complexities are available for the AFT prediction. Their principal drawback is their applicability to the coals originating from a limited number of geographical regions. Accordingly, this study presents computational intelligence (CI) based nonlinear models to predict the four AFTs using the oxide composition of the coal ash as the model input. The CI methods used in the modeling are genetic programming (GP), artificial neural networks, and support vector regression. The notable features of this study are that the models with a better AFT prediction and generalization performance, a wider application potential, and reduced complexity, have been developed. Among the CI-based models, GP and MLP based models have yielded overall improved performance in predicting all four AFTs.
Keywords
Ash fusion temperature Artificial neural networks Support vector regression Genetic programming Data-driven modeling1 Introduction
Coal as a feedstock is used in processes such as combustion, gasification, and liquefaction. It is a complex substance mainly comprising carbon, hydrogen, nitrogen, sulfur, oxygen, and mineral matter that can be intrinsic and/or extraneous with differing form and composition (Ozbayoglu and Ozbayoglu 2006). Being a natural resource, coal exhibits a large variation in its composition.
- (1)
Initial deformation temperature (IDT): Temperature at which ash just begins to flow.
- (2)
Softening temperature (ST): Refers to the temperature at which the ash softens and becomes plastic.
- (3)
Hemispherical temperature (HT): Denotes the temperature yielding a hemispherically shaped droplet.
- (4)
Fluid temperature (FT): At this temperature, ash becomes a free-flowing fluid (Slegeir et al. 1988).
- (1)
They indicate the temperature range for a possible formation of the deposits on the heat adsorbing surfaces (Ozbayoglu and Ozbayoglu 2006).
- (2)
Provide important clues regarding the extent to which the ash agglomeration and clinkering are likely to occur within the combustor/gasifier (Alpern et al. 1984; Seggiani 1999; Van Dyk et al. 2001).
- (3)
They are of particular significance to the operation of all types of gasifiers (Bryers 1996; Wall et al. 1998). For instance, to allow continuous slug tapping, it is necessary that the operating temperature in the entrained flow gasifiers is above the flow temperature (Hurst et al. 1996). In the case of fluid-bed gasifiers, AFTs set the upper limit for the operating temperature at which the ash agglomeration is initiated (Song et al. 2010).
- (4)
The knowledge of AFTs is routinely utilized by the furnace and boiler operators and engineers in power generation stations for predicting the melting and sticking behavior of the coal ash (Seggiani and Pannocchia 2003).
Conventionally, characterization of the ash fusibility is conducted using ASTM D1857 procedure. It comprises monitoring cones or pyramids of ash—prepared in a muffle furnace at 815 °C—in an oven operated under a reducing atmosphere and whose temperature is continuously increased steadily past 1000 °C to as high as possible, preferably 1600 °C (2910 °F). It may be noted that for a given coal, the AFT analysis conducted by different laboratories may vary by ± 20–100 °C (Jak 2002; Winegartner and Rhodes 1975).
- (1)
Most of the existing models have been developed using data pertaining to coals from a single or a few geographical regions. Since coals from different regions/countries may exhibit significantly different chemical and physical characteristics, the AFT models of coals from a single or a few geographies possess limited applicability.
- (2)
Some of the existing AFT prediction models do consider coals from multiple geographical regions (see, for example, Seggiani 1999; Seggiani and Pannocchia 2003). These models also possess reasonably good prediction accuracies. However, they are based upon a large number of predictors (input variables) and as a result, suffer from the following undesirable characteristics: (a) the models are complex, which adversely affects their generalization ability, and (b) costly and tedious experimentation needed for compiling the predictor data.
Accordingly, the main objective of this study is to develop AFT prediction models that are parsimonious (i.e., with lower complexity) and applicable to coals from a large number of geographical regions. Towards this objective, the present study reports the results of the development of computational intelligence (CI) based models for the prediction of IDT, ST, HT, and FT. The three CI paradigms used in this modeling are genetic programming (GP), multi-layer perceptron (MLP) neural network, and support vector regression (SVR). The results of the CI-based modeling of AFT prediction models indicate that the GP and MLP based models predicting IDT, ST, HT, and FT have outperformed the existing linear models with relatively wider applicability in terms of possessing better generalization capability. Also, the said GP and MLP based models require a lower number of predictors than the stated linear models thus reducing the effort and cost involved in compiling the predictor data.
The remainder of this paper is structured as follows. An overview of the existing models for AFT prediction is provided in Sect. 2. The necessity to develop data-driven nonlinear models is explained in Sect. 3. Next, a brief overview of the GP, MLP and SVR formalisms is provided in Sect. 4 titled “CI-based modelling.” The Sect. 5 titled “Results and discussion” presents the development of the CI-based models for the prediction of four AFTs. This section also provides a comparison of the prediction and generalization performance of the CI-based models. Finally, “Concluding remarks” summarize the principal findings of this study.
2 Models for predicting AFT
A representative compilation of AFT predicting correlations/models
No. | Authors | Type of model | Model inputs (composition of ash constituents and other parameters) | Predicted AFT | Coal region | Statistical analysis^{a} of model predictions |
---|---|---|---|---|---|---|
1 | Winegartner and Rhodes (1975) | Stepwise regression | Fe_{2}O_{3}%, FeO%, FeO × CaO%, bases/acids, silica value | ST | USA | CC > 0.7 |
2 | Gray (1987) | Multiple regression | Different combinations of metal oxides | Reducing IDT and HT | New Zealand | R^{2} > 53% |
3 | Rhinehart and Attar (1987) | Thermo-dynamic modelling | Metal and other (P_{2}O_{5}, TiO_{2}, SO_{3}) oxides | IDT, ST, FT | US coals | R^{2} > 0.67 |
4 | Kucukbayrak et al. (1993) | Least square regression analysis | Combinations of metal oxides | HT | Turkish lignite | CC > 0.26 |
5 | Yin et al. 1998 | Back-propagation neural network | SiO_{2}, Al_{2}O_{3}, Fe_{2}O_{3}, CaO, MgO, TiO_{2}, K_{2}O + Na_{2}O | ST | Chinese | Average fractional error: 0.049 |
6 | Kahraman et al. (1998) | Empirical models | Al_{2}O_{3}, Fe_{2}O_{3}, CaO | IDT, spherical, FT | Australian | R^{2} > 0.84 |
7 | Seggiani (1999) | Linear regression | 49 parameters comprising concentrations of nine metal oxides, their squares, and combinations of these values, base, acid, Silica value, and dolomite ratio. | Reducing IDT, ST, HT, and FT for coal and biomass ashes | American, Australian, African, German, Italian, Polish, Spanish, etc. | CC range: 0.84–0.92 |
8 | Lolja et al. (2002) | Linear regression | Metal oxides, acids, bases, crystal components and fluxing agents | IDT, ST, HT, FT | Albanian | 0.93 ≤ CC ≤ 0.95 |
9 | Jak (2002) | Thermo-dynamic modelling (FACT package) | SiO_{2}, Al_{2}O_{3}, Fe_{2}O_{3}, CaO | IDT, Spherical, HT, FT | Australian | Liquidus temperature and AFT strongly correlated |
10 | Seggiani and Pannocchia (2003) | Partial least squares regression | 11–13 parameters comprising concentrations of nine metal oxides in ash, and their various combinations | Reducing IDT, ST, HT, FT | American, Italian, Spanish, German, Australian, Polish, African, French, Albanian | CC (training set) range: 0.75–0.82; CC (validation set) range: 0.76–0.84 |
11 | Ozbayoglu and Ozbayoglu (2006) | Linear and nonlinear regression | Chemical composition of ash and coal parameters | ST, FT | Turkey | Regression coefficient > 0.93 |
12 | Liu et al. (2007) | Back propagation neural network-ant colony optimization | SiO_{2}, Al_{2}O_{3}, Fe_{2}O_{3}, CaO, MgO, TiO_{2}, K_{2}O + Na_{2}O | ST | Chinese | Average training (Test) error: 1.55 (1.85)% |
13 | Zhao et al. (2010) | Least-squares support vector regression | SiO_{2}, Al_{2}O_{3}, Fe_{2}O_{3}, CaO, MgO, TiO_{2}, K_{2}O, Na_{2}O, SO_{3} | ST | Chinese | CC = 0.927, MSE = 0.0128 |
14 | Gao et al. (2011) | Support vector regression by ACO algorithm | SiO_{2}, Al_{2}O_{3}, Fe_{2}O_{3}, CaO, MgO, TiO_{2}, K_{2}O + Na_{2}O | ST | China | MSE = 1.52, CC = 0.999 (training), and = 0.9716 (test) |
15 | Karimi et al. (2014) | Adaptive neuro fuzzy inference system | Different combinations of metal oxides | IDT, ST, FT | USA | CC = 0.97, 0.98 and 0.99, respectively |
16 | Miao et al. (2016) | Back-propagation neural net | SiO_{2}, Al_{2}O_{3}, Fe_{2}O_{3}, CaO, MgO | ST | Chinese | – |
3 Experimental data and need for nonlinear AFT models
- (1)
IDT Albania, Australia, China, Colombia, India, Indonesia, Russia, South Sumatra, South Africa, USA, Venezuela.
- (2)
ST Australia, China, Colombia, Indonesia, Russia, South Africa, South Sumatra, USA, Venezuela.
- (3)
HT Albania, Australia, Colombia, India, Indonesia, Russia, South Africa, South Sumatra, USA, Venezuela.
- (4)
FT Albania, Australia, China, Colombia, India, Indonesia, Russia, South Africa, South Sumatra, USA, Venezuela.
- (1)
The four panels (1a–c, g) of Fig. 1, show that there exists an approximately linear relation between the IDT and the weight percentages of SiO_{2}, Al_{2}O_{3}, Fe_{2}O_{3}, and K_{2}O + Na_{2}O; however, due to the presence of a significant scatter in the corresponding data, a similar conclusion cannot be drawn from the cross-plots (Fig. 1d–f) involving other three ash components (CaO, MgO, and TiO_{2}).
- (2)
Notwithstanding the high scatter seen in all panels of Fig. 2, there exists a high probability of linear dependencies between the softening temperature and Al_{2}O_{3} (Fig. 2b), Fe_{2}O_{3} (Fig. 2c), MgO (Fig. 2e), and TiO_{2} (Fig. 2f); whereas, possibly the individual relationships between ST and SiO_{2} (Fig. 2a), CaO (Fig. 2d) and K_{2}O +Na_{2}O (Fig. 2g) are nonlinear.
- (3)
Cross-plots in Fig. 3 suggest a high probability of nonlinear relationships between HT and weight percentages of three ash components, namely SiO_{2} (Fig. 3a), Al_{2}O_{3} (Fig. 3b) and CaO (Fig. 3d). Whereas, the individual relationships between HT and Fe_{2}O_{3} (Fig. 3c), MgO (Fig. 3e), TiO_{2} (Fig. 3f), and K_{2}O + Na_{2}O (Fig. 3g), could be linear.
- (4)
In Fig. 4, linear dependencies are indicated with a high probability between the flow temperature (FT) and SiO_{2} (Fig. 4a), Al_{2}O_{3} (Fig. 4b), Fe_{2}O_{3} (Fig. 4c), and K_{2}O + Na_{2}O (Fig. 4g). However, in the remaining three panels of Fig. 4, the relationships between FT and CaO (Fig. 4d), MgO (Fig. 4e) and TiO_{2} (Fig. 4f), appear to be nonlinear.
As can be seen from the above observations, there exist several probable cases of the nonlinear relationships between various AFTs and the weight percentages of the individual oxides present in the coal ashes. Thus, it is necessary to explore nonlinear models for the prediction of the four AFTs. Such models are expected to better capture the relationships between the AFTs and the seven oxides in the coal ash and, thereby, make more accurate predictions than the linear models. Towards this objective, in the present study, three computational intelligence (CI) based data-driven modeling formalisms (GP, ANN, and SVR) have been employed for the prediction of the four AFTs from the knowledge of the mineral composition (oxides) of the coal ashes. The objective of developing multiple models for each AFT is to afford a comparison of their prediction and generalization performances and thereby selecting the best prediction model.
4 CI-based models for AFT prediction
4.1 Genetic programming (GP)
Genetic Programming formalism belongs to a class known as “evolutionary algorithms” that follow the principal tenet—commonly paraphrased as “survival of the fittest”—of Darwin’s theory of evolution along with the genetic propagation of characteristics. Originally, GP was proposed (Koza 1992) to develop automatically the computer programs that would execute the pre-specified tasks. Genetic programming’s other application known as “symbolic regression (SR),” is of interest to this study. The novel feature of the GP-based SR is as follows: provided an example dataset containing the function inputs (predictors/independent variables) and the corresponding outputs (dependent variables), it has the ability of searching as also optimising the specific structure (form) of an appropriate linear/nonlinear data-fitting function, and all its associated parameters. And, unlike MLP neural networks and the SVR formalism, the GP-based SR performs the stated search and optimisation without resorting to any assumptions about the structure and/or parameters of the linear/nonlinear data-fitting function. A data-driven modeling problem to be solved by the GP-based SR is explained below.
- (1)
Generates an initial population of candidate expressions/models in a purely stochastic manner. That is, unlike similar techniques performing data-driven modelling, namely MLP neural network and SVR, the GPSR algorithm does not make any assumptions about the form and parameters of the data-fitting models/expressions.
- (2)
Invariably, GPSR-searched models are of lesser complexity (i.e. parsimonious) when compared with the corresponding MLP neural network and SVR models. Consequently, these models are easier to grasp and use in practice.
- (3)
The automatic search and optimization of the form and associated parameters of the linear/nonlinear data-fitting function, performed by the GPSR obviates the trial and error approach associated with the traditional linear/nonlinear regression analysis.
4.2 Multilayer perceptron (MLP) neural network
4.3 Support vector regression (SVR)
Support vector machine (SVM) is a statistical learning based formalism for conducting supervised nonlinear classification (Vapnik 1995). To perform the said classification, SVM first maps the coordinates of objects to be classified into a high-dimensional feature space by employing nonlinear functions called kernels or features. Next, two classes are separated in this high dimensional space using a linear classifier as done customarily. Support vector regression employs same principles, however, for performing a nonlinear regression, which is of interest to this study.
- (1)
It minimizes a quadratic function with a single minimum, which avoids the problems associated with finding a solution in the presence of multiple local minima.
- (2)
Guarantees (a) robustness of the solution, (b) good generalization ability, and sparseness of the regression function, and (c) an automatic control of the regression function’s complexity.
- (3)
An explicit knowledge of the support vectors, which play a major role in defining the regression function assists in the interpretation of the SVR-derived model in terms of the training data.
5 Results and discussion
5.1 Principal component analysis (PCA)
It is a requisite—while developing the data-driven models—to avoid correlated inputs (predictors) since these cause redundancy and an unnecessary increase in the computational load. Thus, the seven inputs \((x_{1} - x_{7} )\) of IDT, ST, HT, and FT prediction models were subjected to the principal component analysis (PCA) (Geladi and Kowalski 1986). This analysis performs a transformation to obtain linearly uncorrelated variables. Subsequently, only the first few principle components (PCs) that capture the maximum amount of variance in the data are chosen as model inputs (predictors), thus enabling a reduction in the dimensionality of the model’s input space. In the present study, seven PCs were extracted from the wt% values of the seven oxides in the coal ashes listed in Supplementary Material.
Variance captured by the individual principal components (PCs) in respect of IDT, ST, HT and FT data sets
Dataset | Percentage variance^{a} | ||||||
---|---|---|---|---|---|---|---|
PC _{1} | PC _{2} | PC _{3} | PC _{4} | PC _{5} | PC _{6} | PC _{7} | |
IDT | 55.2 | 14.3 | 10.8 | 9.4 | 6.2 | 4.1 | 0 |
ST | 34.1 | 25.0 | 17.3 | 12.5 | 5.7 | 4.6 | 0.8 |
HT | 38.1 | 20.9 | 16.8 | 13.2 | 6.5 | 4 | 0.6 |
FT | 43.1 | 19.2 | 13.4 | 11.4 | 7.8 | 5.2 | 0 |
The PCA-transformed variables were used as the inputs in developing the GP-, MLP-, and SVR-based IDT, ST, HT, and FT prediction models. For constructing and examining the generalization ability of these models, the experimental data set for each AFT was randomly partitioned in 70:20:10 ratio into training, test, and validation sets. While the first set was used in training the CI-based models, the test and the validation sets were respectively used in testing and validating the generalization capability of models.
5.2 GP-based modelling
The four GP-based AFT models were developed using Eureqa Formulize software package (Schmidt and Lipson 2009). A notable feature of this package is that it is tailored to search and optimize models with a low complexity and, thereby, possessing the much-desired generalization ability. There are multiple procedural attributes that affect the final solution provided by the GP. These include sizes of the training, test and validation sets, choice of the operators, and input normalization schemes. To secure parsimonious models endowed with a good AFT prediction accuracy and generalization capability, several GP runs were conducted by imparting variations in each of the stated attributes. The best solution (possessing maximum fitness) secured in each run was recorded. From multiple such solutions, the ones fulfilling the following criteria were screened to choose an overall optimal model (Sharma and Tambe 2014): (a) high and comparable magnitudes of CCs, and small and comparable magnitudes of RMSE, and MAPE, pertaining to the model predictions in respect of the training, test, and validation set data, and (b) model should possess a low complexity (i.e., containing a small number of terms and parameters in its structure).
- (a)
Model for predicting Initial deformation temperature (IDT)
- (b)
Model for predicting softening temperature (ST)
- (c)
Model for predicting hemispherical temperature (HT)
- (d)
Model for predicting flow temperature (FT)
Statistical analysis of the prediction accuracy and generalization performance of the GP-, MLP-, and SVR based models predicting IDT, ST, HT, and FT
Type of AFT | Dataset^{a} | Computational intelligence based models | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
GP | MLP | SVR | ||||||||
CC | RMSE | MAPE (%) | CC | RMSE | MAPE (%) | CC | RMSE | MAPE (%) | ||
IDT | Training | 0.843 | 59.60 | 3.96 | 0.876 | 53.22 | 3.39 | 0.850 | 58.29 | 4.18 |
Test | 0.846 | 70.48 | 4.60 | 0.824 | 77.93 | 4.73 | 0.776 | 84.13 | 5.23 | |
Validation | 0.803 | 33.79 | 2.18 | 0.764 | 38.78 | 2.66 | 0.774 | 38.83 | 2.65 | |
ST | Training | 0.833 | 102.16 | 6.35 | 0.841 | 127.20 | 7.78 | 0.832 | 103.6 | 5.32 |
Test | 0.921 | 68.81 | 4.83 | 0.841 | 121.60 | 6.47 | 0.836 | 93.31 | 4.97 | |
Validation | 0.823 | 82.20 | 5.86 | 0.878 | 90.96 | 5.09 | 0.949 | 48.35 | 3.04 | |
HT | Training | 0.803 | 96.63 | 5.68 | 0.926 | 70.10 | 4.12 | 0.813 | 95.42 | 4.89 |
Test | 0.929 | 75.07 | 4.09 | 0.894 | 107.30 | 6.77 | 0.800 | 110.68 | 5.74 | |
Validation | 0.953 | 81.01 | 5.26 | 0.851 | 109.59 | 5.80 | 0.804 | 118.02 | 5.76 | |
FT | Training | 0.915 | 62.53 | 3.59 | 0.962 | 44.70 | 2.81 | 0.941 | 53.21 | 2.91 |
Test | 0.941 | 56.94 | 3.75 | 0.901 | 71.38 | 4.38 | 0.857 | 82.61 | 4.74 | |
Validation | 0.886 | 79.33 | 4.95 | 0.900 | 71.18 | 4.53 | 0.891 | 75.18 | 4.76 |
5.3 Multilayer perceptron (MLP) neural network based AFT models
Details of the MLP-based optimal models for the prediction of IDT, ST, HT, and FT
MLP model | Number of nodes in input layer (L) | Number of nodes in the hidden layer (M) | Error back propagation algorithm parameter | Transfer function for hidden nodes | Transfer function at output node | |
---|---|---|---|---|---|---|
Learning rate (η) | Momentum (μ) | |||||
IDT | 5 | 6 | 0.3 | 0.8 | Logistic sigmoid | Identity |
ST | 4 | 6 | 0.31 | 0.23 | Logistic sigmoid | Identity |
HT | 5 | 6 | 0.271 | 0.1809 | Logistic sigmoid | Identity |
FT | 5 | 6 | 0.27 | 0.16 | Logistic sigmoid | Identity |
5.4 SVR-based modelling
Details of the SVR-based models predicting IDT, ST, HT, and FT
SVR Model | Kernel gamma (γ) | Tube radius (ε) | Regularization parameter (C) | Kernel degree (δ) | Number of support vectors | Kernel function |
---|---|---|---|---|---|---|
IDT | 2 | 0.6 | 2 | 1 | 48 | ANOVA |
ST | 0.011 | 0.2 | 10 | – | 45 | RADIAL |
HT | 0.098 | 0.24 | 0.5 | 2.7 | 41 | ANOVA |
FT | 1 | 0.09 | 1.1 | 1 | 60 | ANOVA |
5.5 Comparison of AFT prediction models
5.5.1 Initial deforming temperature prediction models
5.5.2 Softening temperature prediction models
- (1)
There exists a minor variation in the ST prediction accuracies and generalization capabilities of the three CI-based models.
- (2)Among the CI-based models, the overall prediction and generalization performance of the GP-based model is marginally better than the MLP- and SVR-based models. This observation is unambiguously supported by the lower scatter in the predictions of the GP-based model (see Fig. 9a) when compared with the predictions of the other two models.
5.5.3 Hemispherical temperature predicting models
- (1)The CC magnitude corresponding to the predictions of the training set outputs (termed “recall” ability) by the GP-model is lower (0.803) than that of the corresponding MLP (0.926) and SVR (0.813) based models. However, the CC magnitudes in respect of the GP-model predictions for the test and validation sets (0.929, 0.953) are higher than that of the MLP (0.894, 0.851) and SVR (0.800, 0.804) models. It may be noted that higher CC values pertaining to the test and validation set outputs are indicative of the better generalization ability possessed by the model, which is critically important in correctly predicting the HT values for an entirely new set of inputs. The above observations suggest that the GP model possesses better generalization ability than the MLP and SVR based models. This inference is also supported by the parity plots depicted in Fig. 10 where it is seen that although the predictions of the training set outputs by the GP model (shown by “diamond” symbol) exhibit a higher scatter relative to the HT predictions by the MLP and SVR models, the GP-model predictions pertaining to the test and validation set data exhibit lower scatter (better generalization) than the predictions by the other two models.
- (2)
The parity plots in respect of the hemispherical temperature predictions depict that the MLP-based model has yielded better prediction and generalization performance than the GP- and SVR-based models. The MLP model predictions of HT have also yielded higher (lower) magnitudes of the correlation coefficient (RMSE/MAPE) relative to the two other CI-based models.
5.5.4 Fluid Temperature predicting models
- (1)
[Experimental (A)—GP model predicted (B)] and [Experimental (A)—MLP model predicted (C)]
- (2)
[Experimental (A)—MLP model predicted (B)] and [Experimental (A)—SVR model predicted (C)]
- (3)
[Experimental (A)—SVR model predicted (B)] and [Experimental (A)—GP model predicted (C)]
Results of the Steiger’s Z-test (testing the null hypothesis H_{0}, CC_{AB} = CC_{AC})
Performance variable | Model pair (B–C) | CC _{ AB} | CC _{ AC} | CC _{ BC} | df ^{a} | z ^{a} | p value ^{a} | H _{0} |
---|---|---|---|---|---|---|---|---|
IDT | GP-MLP | 0.848 | 0.860 | 0.942 | 181 | − 0.993 | 0.321 | Accept |
MLP-SVR | 0.860 | 0.832 | 0.917 | 181 | 1.869 | 0.06 | Accept | |
SVR-GP | 0.832 | 0.848 | 0.925 | 181 | − 1.07 | 0.286 | Accept | |
ST | GP-MLP | 0.848 | 0.843 | 0.903 | 94 | 0.221 | 0.825 | Accept |
MLP-SVR | 0.843 | 0.841 | 0.964 | 94 | 0.148 | 0.883 | Accept | |
SVR-GP | 0.841 | 0.848 | 0.914 | 94 | − 0.332 | 0.74 | Accept | |
HT | GP-MLP | 0.844 | 0.909 | 0.862 | 79 | − 2.681 | 0.007 | Reject |
MLP-SVR | 0.909 | 0.802 | 0.820 | 79 | 3.723 | 0.0002 | Reject | |
SVR-GP | 0.802 | 0.844 | 0.892 | 79 | − 1.518 | 0.113 | Accept | |
FT | GP-MLP | 0.916 | 0.944 | 0.933 | 91 | − 2.354 | 0.019 | Reject |
MLP-SVR | 0.944 | 0.919 | 0.945 | 91 | 2.258 | 0.024 | Reject | |
SVR-GP | 0.916 | 0.919 | 0.945 | 91 | 0.265 | 0.791 | Accept |
- (1)
The performances of the GP, MLP, and SVR models in predicting the IDT and ST magnitudes are comparable.
- (2)
In the case of HT and FT predictions, the performance of the MLP based models is better than that of the the GP and SVR models.
- (1)
All the four GP-based models (Eqs. 14, 20, 25, and 31) are nonlinear. As stated earlier, depending upon the relationship that exists between the inputs and the output, the GP method can search and optimize an appropriate linear or a nonlinear model from the example input–output data. The nonlinear forms fitted by the GP for predicting all four AFTs are indicative that the relationships between the AFTs and concentrations of seven oxides in coal ashes are nonlinear.
- (2)
A comparison of the performance of the CI-based models with the existing high performing ones with relatively wider applicability (Seggiani and Pannocchia 2003) indicate that (a) the GP and MLP based models predicting all four AFTs have outperformed the existing ones in terms of possessing better generalization capability, and (b) the GP and MLP based models require lower number of predictors (= 7) than those needed by the models proposed in Seggiani and Pannocchia (2003) that consider 13, 11, 11 and 12 predictors, respectively for IDT, ST, HT, and FT prediction models. The lower number of predictors used by the GP-, MLP-, and SVR-based models have reduced the effort and cost involved in compiling the predictor data.
- (3)
The RMSE which has the same units as the model predicted output is a measure of how close the model predicted values are to the corresponding experimentally measured ones. It is an absolute measure of the data fit and can be interpreted as the standard deviation of the unexplained variance. It has been observed that the reproducibility of the AFT magnitudes measured by the same analyst and using the same instrument varies between 30 and 50 °C. The corresponding variation between the measurements done at different laboratories is ~ 50–80 °C (Seggiani and Pannocchia 2003). In Table 3, it is seen that the RMSE magnitudes in respect of the test and validation set predictions by the GP-based models predicting IDT, ST, HT, and FT vary between 33.79 and 82.20. These magnitudes are a measure of the generalization ability of the models. Considering the extent of the inherent variability in the experimental measurements of the AFT values, the stated RMSE magnitudes can be considered as reasonable and, therefore, indicative of good prediction and generalization performance of the GP-based models.
- (4)
Among the three types of CI-based models, the GP-based ones due to their compact size, and ease of evaluation are more convenient to use and deploy in the practical applications. However, in situations when the highest AFT prediction accuracy is required then the MLP based models should be used preferentially for the prediction of HT and FT.
6 Conclusions
- (1)
They are predominantly linear models although a detailed scrutiny of the data indicates that the relationships between the AFTs and the weight percentages of some of the mineral matter constituents could be nonlinear.
- (2)
The models are developed using data of coals belonging to a limited number of geographical regions and, therefore, do not have wider applicability since coal properties differ widely depending upon coal’s geographical origin,
Notes
Acknowledgements
This study was partly supported by the Council of Scientific and Industrial Research (CSIR), Government of India, New Delhi, under Network Project (TAPCOAL). We are thankful to Mr. Akshay Tharval, Dwarkadas J. Sanghvi College of Engineering, Mumbai, India, and Mr. Niket Jakhotiya, VNIT, Nagpur, India for the assistance in compiling the AFT data during their internship at CSIR-NCL, Pune, India.
Compliance with ethical standards
Conflict of interest
Authors declare that they have no conflict of interest.
Supplementary material
References
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