Development of CO 2 emission model of an acid gas incinerator using Nelder–Mead least squares support vector regression

Abstract

This paper aims to develop a CO₂ emission model of an acid gas incinerator using Nelder–Mead least squares support vector regression (LS-SVR). The Malaysia Department of Environment is actively imposing the Clean Air Regulation to mandate heavy industries to comply with emission limits. One of the latest measures is to mandate the installation of analytical instrumentation known as a continuous emission monitoring system (CEMS) to report the emission level online to the Department of Environment office. As a hardware-based analyser, CEMS is expensive, maintenance intensive and often unreliable. Therefore, software predictive techniques are often preferred and considered a feasible alternative to replace the CEMS for regulatory compliance. The LS-SVR model is built based on the emissions from an acid gas incinerator that operates in a liquefied natural gas complex. Simulated annealing is first used to determine the initial hyper-parameters, which are further optimized based on the performance of the model using a Nelder–Mead simplex algorithm. The LS-SVR model is shown to outperform a benchmark model based on back-propagation neural networks in both training and testing data.

Keywords

Artificial neural networks industrial pollution predictive algorithms support vector machines

Introduction

Power plants, chemical plants, government utilities and petroleum refineries typically produce more than the plant’s capacity in order to meet demands. Hence these industries must continuously monitor their exhaust stacks for primary pollutants such as nitrogen oxides, carbon monoxide, sulphur dioxide and carbon dioxide under regulations promulgated under the New Source Performance Standards [US Environmental Protection Agency (EPA) PA 40 CFR Part 60) or the Clean Air Act Amendments Title IV (40 CFR Part 75), 2008] and the Malaysian Environmental Quality Act, 1978 (2000). A continuous emission monitoring system (CEMS) utilizes a sample probe, umbilical and sample conditioning system to extract a representative sample of the stack gas exhaust stream to provide a continuous flow for direct measurement of the pollutant concentration on individual analysers (Fan and Zheng, 2009). A data acquisition system (DAS) is typically used to collect, calculate emission rates, alarm and store historical data from these CEMS (Zheng and Tang, 2008). The promulgated regulations in US EPA 40 CFR covering the monitoring of combustion units for primary and other pollutants allow for the use of predictive emissions monitoring systems (PEMS) in lieu of CEMS. A PEMS is a sophisticated software-based sensor/prediction system that is directly interfaced with the process control system and inputs from the combustion or pollution control process. The US EPA (2011) have revealed the devastating effects of the continuous release of pollutants such as carbon dioxide, methane, nitrous dioxide and other greenhouse (heat-trapping) gases into the atmosphere. Climate change induced by these gases can cause damage to human health, agriculture, natural ecosystems, coastal areas and other climate sensitive systems. Currently, emission monitoring is done via analytical instruments, which are very expensive to install and maintain; the cost for an online NO_x analytical and monitoring system is around $100,000–200,000 and the cost for maintenance is approximately $15,000 per year (Dong and McAvoy, 1995). Several studies have been performed to develop predictive systems for industrial emissions. One of the earlier ideas was presented by Baines (1999), a consultant from Fisher-Rosemount Solutions. The author proposed the idea of predicting the stack contaminants in real time by correlating emissions with key unit parameters like fuel type, air and fuel flows, and combustion temperature. Chakravarthy et al. (2000) have developed a PEMS for industrial process heaters. The authors have used heuristic an optimizer genetic algorithm (GA) to tune the NO_x kinetic parameters. Deng and Stobart (2007) examined the idea of using a hybrid clustering technique that involves the modifications of the Fuzzy C-means method and using neural networks to achieve the best performance of the fuzzy system for a diesel engine emission model. Zheng et al. (2008) used a generalized regression neural network (GRNN) to establish a non-linear model between the parameters of the boiler (of 300 MW steam capacity) and the NO_x emissions. Later, the same authors have proposed the idea of replacing the existing GRNN PEMS with a least squares support vector regression (LS-SVR) model. The authors claim that this new algorithm yields better accuracy of a coal combustion unit (Zheng et al., 2010). The main objective of this paper is to develop a CO₂ emission model of an acid gas incinerator based on an LS-SVR algorithm with Nelder–Mead (NM) optimization of its hyper-parameters. This paper is organized as follows: Section 2 presents a brief literature on artificial neural networks (ANNs) and LS-SVR. Section 3 describes the modelling background in terms of the parameters used, a case study of an acid gas incinerator and the dataset used. Section 4 presents the results and analysis, and Section 5 gives a brief conclusion.

Literature review

Artificial neural network (ANN)

An ANN is represented as non-linear interconnected layers of processing nodes, which are normally referred to as neurons, a term borrowed from neurobiology (Zhou et al., 2004). The most common class of ANN is the multilayered feedforward network, which primarily consists of an input layer, one or more hidden layers and an output layer. The input layer holds the data and distributes it to the network via interconnections (neurons) in the hidden layer(s), where they are processed by the activation function to produce an output signal. This type of network is also known as a multilayer perceptron (MLP) (Haykins, 1999); a typical MLP network is shown in Figure 1. Common transfer functions used are linear, log-sigmoid, and tan-sigmoid as shown in Equations (1), (2) and (3).

y = A

(1)

y = \frac{1}{1 + e^{- A}}

(2)

y = \frac{2}{1 + e^{- 2 A}} - 1

(3)

where A is the element activation function and is represented by Equation (4):

A = \sum_{i = 1}^{n} w_{i} x_{i}

(4)

The variable $w_{i}$ is the weight term and $x_{i}$ is the network input. The network uses these weights to identify the strength of the interconnections between neurons. These weights are adjusted throughout the learning process. For an MLP network, learning algorithms based on the gradient or Jacobian of the network error with respect to the weights are preferred because of their superior performance. A common Jacobian-based algorithm is the back-propagation algorithm. The back-propagation neural network (BPNN) has been widely used to develop soft sensors for prediction of NO_x (Tronci et al., 2002). Some of the weaknesses of BPNN are the need for numerous controlling parameters, difficulty in obtaining a stable solution and the danger of overfitting. The solution shown by Zheng et al. (2008) points to the fact that the BPNN is unreliable even if all of network objects are pre-determined.

Support vector regression (SVR)

The foundations of support vector machines (SVM) were laid by Vapnik (1995). Early SVMs were developed for classification problems; later they were extended to the domain of regression problems (Vapnik et al., 1997) and collectively known as support vector regression (SVR). The formulation for SVR begins with a set of data (Gunn, 1998):

D = {(x^{1}, y^{1}), \dots, (x^{l}, y^{l})}, x \in R^{n}, y \in R

(5)

which is to be approximated to a non-linear function:

f (x) = w^{T} ϕ (x) + b

(6)

Where $ϕ (\cdot) : R^{n} \to R^{n_{h}}$ is a mapping function to higher or infinite dimensional feature space. The mapping function is implicitly defined. The optimal regression function is given by the minimum of the functional:

Φ (w, ξ) = \frac{1}{2} | | w | |^{2} + C \sum_{i} (ξ_{i}^{-} + ξ_{i}^{+})

(7)

Where $C$ is a pre-specified value, and ${ξ_{i}}^{+}$ and ${ξ_{i}}^{-}$ are slack variables that represent the upper and lower constraints on the system outputs. Equation (7) is a quadratic programming (QP) problem that is solved first by introducing a Hessian matrix (H). If H is proven to be a positive semi-definite matrix, then $Φ (w, ξ)$ is a convex function and a conditional global minimizer exists. On the other hand, if H is found to be positive definite, then the solution to the convex minimization problem is global and unique (Vanderbei, 1998).

Using an $ε$ -sensitive loss function, Figure 2,

L_{ε} (y) = {\begin{matrix} 0 & for | f (x) - y | < ε \\ | f (x) - y | - ε & otherwise \end{matrix}

(8)

Employing the Lagrangian duality theorem, the minimization problem becomes:

max W (α, α^{*}) = \max \sum_{i = 1}^{l} α_{i}^{*} (y_{i} - ε) - α (y_{i} + ε) - \frac{1}{2} \sum_{i = 1}^{l} . \sum_{j = 1}^{l} (α_{i}^{*} - α_{i}) (α_{j}^{*} - α_{j}) K (x_{i}, x_{j})

(9)

With constraints

\begin{matrix} 0 \leq α_{i}^{*}, α_{i} \leq C, i = 1, \dots, l \\ \sum_{i = 1}^{l} (α_{i}^{*} - α_{i}) = 0 \end{matrix}

(10)

Solving Equations (9) and (10) determines the Lagrange multipliers, ${α_{i}}^{*}, α_{i}$ , and the regression function is given by

f (x) = \sum_{SVs} ({\bar{α}}_{i}^{*} - {\bar{α}}_{i}) K (x_{i}, x_{j}) + \bar{b}

(11)

where

\begin{matrix} 〈 \bar{w}, x 〉 = \sum_{i = 1}^{l} ({\bar{α}}_{i}^{*} - {\bar{α}}_{i}) K (x_{i}, x_{j}) \\ \bar{b} = - \frac{1}{2} ({\bar{α}}_{i}^{*} - {\bar{α}}_{i}) (K (x_{i}, x_{r}) + K (x_{i}, x_{s})) \end{matrix}

(12)

Least squares support vector regression (LS-SVR)

The SVM formulation above leads to a QP problem with linear constraints. The size of the matrix involved in the QP problem is directly proportional to the size of the training data (Zheng et al., 2010). To reduce the complexity of the optimization problem, Suykens et al. (2002) introduced a modified version of the SVM called the least squares SVM (LS-SVM). This formulation results in a set of linear equations instead of a QP problem. The LS-SVM is used for both classification and regression problems. The formulation for LS-SVR is begun by taking a training set as in Equation (5) and estimate using a non-linear function as in Equation (6) where $ϕ (\cdot) : R^{n} \to R^{n_{h}}$ is a mapping function to a high dimensional and potentially infinite dimensional feature space; in this paper, there are 22 dimensions in the low dimensional space and 484 dimensions in high dimension space. Next, the optimization is formulated in the primal weight space:

min Φ (w, e) = \frac{1}{2} w^{T} w + γ \frac{1}{2} \sum_{i = 1}^{l} e_{i}^{2}

(13)

Subject to:

y_{i} = w^{T} ϕ (x_{i}) + b + e_{i} i = 1, \dots, l

(14)

The optimization formulation in Equation (13) is a ridge regression cost function formulated in feature space. Constructing the Lagrangian of the problem, the dual problem is derived:

L (w, b, e, α) = Φ (w, e) - \sum_{i = 1}^{l} α_{i} {w^{T} ϕ (x_{k}) + b + e_{i} - y_{i}}

(14)

The conditions for optimality are given by (Suykens et al., 2002):

\begin{matrix} \frac{\partial L}{\partial w} = 0 \to w = \sum_{i = 1}^{l} α_{i} ϕ (x_{i}) \\ \frac{\partial L}{\partial b} = 0 \to \sum_{i = 1}^{l} α_{i} = 0 \\ \frac{\partial L}{\partial e_{i}} = 0 \to α_{i} = γ e_{i} i = 1, . ., l \\ \frac{\partial L}{\partial α_{i}} = 0 \to w^{T} ϕ (x_{i}) + b + e_{i} - y_{i} i = 1, . ., l \end{matrix}

(15)

Upon elimination of the variables $w$ and $e$ and solving in $α$ and $b$ one gets the following solution in dual space:

[\begin{matrix} 0 & 1_{V}^{T} \\ 1_{v} & Ω + 1 / γ \end{matrix}] [\begin{matrix} b \\ α \end{matrix}] = [\begin{matrix} 0 \\ y \end{matrix}]

(16)

Where $y = [y_{1}; \dots; y_{l}]$ $I_{v} = [1;,,,; 1]$ , and $α = [α_{1}; \dots; α_{l}]$ . The ‘kernel trick’ (Cristianini and Shawe-Taylor, 2000) is applied here as shown:

\begin{matrix} Ω_{ij} = ϕ {(x_{i})}^{T} ϕ (x_{j}) \\ = K (x_{i}, x_{j}) i, j = 1, \dots l \end{matrix}

(17)

Hence the resulting LS-SVR model becomes:

f (x) = \sum_{i = 1}^{l} α_{i} K (x_{i}, x_{j}) + b

(18)

In this paper, a radial basis function (RBF) kernel function is used:

K (x_{i}, x_{j}) = exp (- \frac{| x - x_{i} |^{2}}{σ})

Where $α_{i}$ and $b$ are solutions to the linear system represented by Equation (16). The LS-SVR formulation can be used to handle large datasets with no dimensionality problem. In Equation (17), a kernel function is used to replace the high order mapping function. In this paper, the RBF kernel will be used exclusively for all computations involving kernel operations. In the case of the RBF kernel, only two hyper-parameters need to be tuned ( $γ$ , $σ$ ), which is fewer than the standard SVM.

Modelling background

Back-propagation neural network (BPNN) model

Two models will be developed in this paper using two distinctive algorithms. The first model is developed as a benchmark using a BPNN. The optimal network parameters are chosen by varying the number of layers and the number of hidden neurons per layer. The parameter that gives the best performance is chosen and shown below:

Number of layers=2;

Number of neurons (hidden layer)=20;

Transfer functions (input layer)=tan-sigmoid;

Transfer function (hidden layer)=linear;

Training algorithm=Levenberg–Marquardt.

Nelder–Mead (NM) least squares support vector regression (LS-SVR)

It is understood that the hyper-parameters involved for the standard SVM with RBF kernel are $σ$ , $C$ and $ε$ . On the other hand, the LS-SVR algorithm with RBF kernel involves only two hyper-parameters; $γ$ and $σ$ (Suykens et al., 2002). The common method for optimizing these hyper-parameters is to use the ‘grid-search’ method, which is a global exhaustive search technique. Based on trial-and-error formulation, the hyper-parameters are changed exponentially to find those that give the best performance for the model (Hsu et al., 2010). Grid-search is straightforward but naïve. There are other advanced methods, which give better performance results and save computational cost. One such method is the NM simplex algorithm, which will be used in this paper. This algorithm is nicknamed the ‘amoeba’ because of its biological-like search patterns. In two dimensions, it consists of a search triangle, ‘crawler’ or ‘simplex’, with three points that represent the highest (worst) point, next highest point and the lowest (best) point. The intuition is to move away from high point towards the low point. The simplex moves in several transformations that are known as ‘reflection’, ‘contraction’, ‘reflection and expansion’, and ‘multiple contractions’ to find the optimal value for the minimization problem (Press et al., 1998). The parameter settings for the simplex algorithm are shown below:

Expansion steps=2;

Size of initial simplex=1.2;

Contraction steps=0.5;

Reflection steps=1;

Shrinkage steps=0.5;

Number of optimization steps=15;

Number of function evaluations=50;

Stopping criterion based on value of function=1e-6;

Stopping criterion based on change in minimizer=1e-6.

The initial values of the hyper-parameters ( $γ$ and $σ$ ) are found using simulated annealing (SA). This technique is borrowed from metallurgy where it is a global optimizer for a large search space. SA is designed to find an optimized solution in a given time, rather than finding the best possible solution (Kirkpatrick, 1983); this saves computation time and is suitable for finding initial values for further optimizations. SA parameter settings are as follows:

Initial temperature=1;

Max number of function evaluations=40;

Number of steps at fixed temperature=20;

Energy tolerance=1e-45.

Using SA alone is not very strong because of the stopping criterion (i.e. number of iterations). The combination of SA and NM is much stronger. SA is only used as a ‘cost effective’ method for locating the initial hyper-parameters. The performances of the models are gauged using standard performance functions in the form of correlation factor (R), root mean square error (RMSE) and accuracy.

RMSE = \sqrt{\frac{1}{n} \sum_{1 = 1}^{n} {({\hat{X}}_{i} - X_{i})}^{2}}

(19)

R = \frac{\sum_{i = 1}^{n} (X_{i} - \bar{X}) ({\hat{X}}_{i} - \bar{\hat{X}})}{\sqrt{\sum_{i = 1}^{n} {(X_{i} - \bar{X})}^{2} \sqrt{{({\hat{X}}_{i} - \bar{\hat{X}})}^{2}}}}

(20)

Accuracy = 100 - (\frac{1}{n} \sum_{i = 1}^{n} (\frac{{\hat{X}}_{i} - X_{i}}{({\hat{X}}_{i})}) \times 100)

(21)

Where $X_{i}$ is the predicted value, ${\hat{X}}_{i}$ is the true value and $n$ is the number of testing samples. An overall flow of the NM LS-SVR algorithm is presented in Figure 3.

Case study

In this paper, the CO₂ emission from a tangentially fired acid gas incinerator is modelled. This incinerator train is part of a liquefied natural gas (LNG) complex. The plant is divided into four sections consisting of upstream facilities, gas treating, liquefaction and storage/terminal. The first part of the gas treating section deals with the acid gas in the feed. CO₂ and H₂S are removed to meet product specification. An amine-based solvent is used in the acid gas removal process. Solvents with absorbed acid gases are regenerated in regenerator column at high temperature and low pressure. The acid gases are then incinerated before being discharged safely into the environment. In this LNG complex, acid gases are incinerated using two methods: one uses steam boilers and the other uses acid gas thermal incinerators (Jamaludin et al., 2005). The process flow scheme of the incinerator train is shown in Figure 4. The main constituents of the inputs to the incinerator are acid gas, flash gas, fuel gas and combustion air. The flue gases that are released into the stack consist of NO_x, SO₂, SO₃, CO, CO₂ and H₂S. In this paper, only CO₂ emission will be modelled based on the plant input and output process variable data. The data is obtained from the LNG complex incinerator train distributed control system historical dataset taken from April 2010 to November 2010, and contains the data from 29 input process variables and seven output process variables. The most relevant process variables are selected by reviewing the design and operational specifications in the incinerator material balance sheet. An initial correlation analysis was also performed to the dataset to determine the variables that have the strongest variance with each other.

As a result, 22 input variables and one output variable (i.e. flue gas CO₂ content) are chosen with 1044 samples, taken at 30 minutes apart for the whole month of September 2010. The number of data points for LS-SVR is limited by the computational ability of the machine that is used to run the model, mainly because the M-file implementation is memory intensive compared with C-Mex or C++ languages. Tables 1 and 2 outline the characteristics of the chosen input and output variables. The relationship between CO₂ and the input variables are not easily modelled mathematically (and it is beyond the scope of this paper) and any attempt on graphical plotting shows erratic distributions (which on some level demonstrates a highly non-linear relationship).

Table 1

Selected input process variables

Process variables	Engineering units	Operation units
A-91921 sour feed gas flow	T/D	Incinerator unit 2
Fuel gas to A-91921	%	Incinerator unit 2
Fuel gas to incinerator A-91921	Density	Incinerator unit 2
K-91921 suction air flow	T/D	Incinerator unit 2
A-91921 stack exh. temp.	°C	Incinerator unit 2
A-91921 stack exh. Temp.	°C	Incinerator unit 2
H₂O analyser A	ppm	N/A
H₂O analyser B	ppm	N/A
CO₂ analyser	%	N/A
H₂S analyser	ppm	N/A
C-91101 feed gas flow	T/D	Flash gas unit
CO₂ in wet feed gas fr. C-91101	ppm	Flash gas unit
H2S in wet feed gas fr. C-91101	ppm	Flash gas unit
C-91101 treated NG temp.	°C	Flash gas unit
C-91101 treated NG temp.	T/D	Flash gas unit
Lean slvt to C-91101 flow	T/D	Flash gas unit
C-91103 solvent reflux flow	T/D	Sour water unit
HP F/G to KG-91320 flow	T/D	Gas turbine unit
MR IGV position feedback	N/A	Gas turbine unit
MR turbine speed	N/A	Gas turbine unit
MR comb. reference temp.	°C	Gas turbine unit
MR MW actual power	MW	Gas turbine unit

Table 2

Output process variables (only CO₂ is selected for this paper)

Process variables	Engineering units	Operation units
A-91921 flue gas O₂ content	mol%	Incinerator common stacks
A-91921 flue gas NO_x content	mg/nm³	Incinerator common stacks
A-91921 flue gas SO₂ content	ppmv	Incinerator common stacks
A-91921 CO content	mg/m³	Incinerator common stacks
A-91921 CO₂ content	%	Incinerator common stacks
A-91921 flue gas H₂S content	ppm	Incinerator common stacks
SO₃ content	ppm	Incinerator common stacks

However, the relationship can be proven by correlation analysis and by studying the process flow and material balance of the feed and product gases. Before modelling, data is pre-processed to remove bad inputs and non-values. For ANN modelling, 70% of samples are used for training, 15% for validation and 15% for testing; the SVR model uses 80% for training and cross-validation, and 20% for testing.

Results and analysis

The individual model performance of BPNN and LS-SVR will be presented in this section. Tables 3 and 4 presents a comparison of the performance of the models in terms of correlation, root means squared and accuracy of the predicted values. The performance of both models in terms of computational time is also shown. Figures 5, 6, 7 and 8 show distribution of real and predicted results. This will be followed by a brief analysis of the results obtained. The performance of the LS-SVR model depends greatly upon the values of the hyper-parameters selected. The initial values for the hyper-parameters are: $γ$ =34.84 and $σ$ =11.89, and was computed using SA. These values were then used for further optimization using the NM simplex algorithm. The objective function is to minimize model error (MSE) based on several constraints such as the maximum number of optimization steps, the maximum number of function evaluations, the stopping criterion based on the relative change in value of the function in each step and the stopping criterion based on the change in the minimizer in each step. Table 5 shows the simplex computation for this model. The optimized values for the hyper-parameters are:

γ = 32.96

σ = 11.43

Table 3

Performance of the back-propagation neural network model

	RMSE	R	Accuracy	Time
Training data	0.4216	0.8549	92.71%	21.0 s
Testing data	0.3787	0.8919	93.20%

RMSE, root mean square error.

Table 4

Performance of least squares support vector regression model

	RMSE	R	Accuracy	Time
Training data	0.0806	0.9955	98.76%	65.1 s
Testing data	0.3090	0.9330	94.64%

RMSE, root mean square error.

Table 5

Hyper-parameters optimization steps

Iteration	Function evaluation	Min f(x)	Log(γ)	Log(σ)	Procedure
1	3	1.413835e-001	3.5508	2.4753	Initial
2	5	1.413835e-001	3.5508	2.4753	Contract inside
3	7	1.413835e-001	3.5508	2.4753	Contract outside
4	9	1.413835e-001	3.5508	2.4753	Contract inside
5	10	1.413835e-001	3.5508	2.4753	Reflect
6	12	1.413835e-001	3.5508	2.4753	Contract inside
7	14	1.413835e-001	3.5508	2.4753	Contract outside
8	16	1.413835e-001	3.5508	2.4753	Contract inside
9	17	1.413835e-001	3.5508	2.4753	Reflect
10	18	1.413835e-001	3.5508	2.4753	Reflect
11	20	1.413835e-001	3.5508	2.4753	Contract inside
12	22	1.413835e-001	3.5508	2.4753	Contract inside
13	23	1.413835e-001	3.5508	2.4753	Reflect
14	25	1.413743e-001	3.4954	2.4364	Reflect
15	27	1.413743e-001	3.4954	2.4364	Contract inside

The CO₂ emission as shown in the above graphs presents a highly non-linear distribution; hence it takes a model with the capabilities to map these non-linearities with precision and accuracy. As shown in Tables 3 and 4, the performance of the LS-SVR model is clearly superior in both training and testing data compared with the BPNN model. The graphs also show that the LS-SVR model is able to track and predict the CO₂ emission with much better accuracy than BPNN model. The accuracy achieved by both models is above 90%, which satisfies the conditions under the US EPA Performance Specification 16 for Predictive Emission Monitoring Systems (2008). The models developed can be used in real world applications. LS-SVR is a much more ‘transparent’ algorithm, where the resultant model has weights and biases that were optimized to suit the particular dataset. Another plausible explanation is that, with BPNN, the initial weights are chosen randomly and hence the performance varies significantly with the number of simulations as shown by Zheng et al. (2008). In contrast, LS-SVR initial parameters are optimized via SA. The solutions provided by SVMs are unique and global in a dual space problem; this is achieved by solving the Karush–Kuhn Tucker (KKT) system at its core. The generalization ability of LS-SVM could be further improved by increasing the number of samples used for validation and reducing the stopping criteria when training using cross-validation. SVM algorithms are a kernel-based method; hence the complexity is in the order of $O (n^{3})$ , which means that the computational power and required running time is higher than the neural network-based method.

The LS-SVR solution comes from solving the linear system in Equation (16) by matrix inversion; this is a fairly tedious process to be solved numerically and results in a longer overall computation time, as shown in Table 4. Hence, the usability of LS-SVR depends heavily upon the number of samples used and the processing power of the machine that it currently runs on. To overcome this, several fixes have been proposed such as using quadratic Renyi entropy to select the most significant support vectors (Jiang et al., 2008), sequential learning LS-SVM (Jung and Polani, 2008), and using analytic QP and sparseness (Platt, 1999). One of the weaknesses of the NM optimizer as shown in Table 5 is that it only searches one better point (i.e. min f(x)=1.413743 e-001) in 15 iterations. This could be improved by increasing the expansion sizes that includes the contraction, reflection and shrinkage. In the future, a more complex optimizer will be considered to perform the hyper-parameters optimization. In Table 3, it can be seen that the testing performance of BPNN is better than training, which shows a good generalization ability, but to reduce overgeneralization, the number of training samples could be improved, while reducing validation and testing samples.

Conclusion and recommendations

In a nutshell, this paper has brought into focus the ability to model the CO₂ emission from an acid gas incinerator using an NM LS-SVR algorithm. The modelling was done using a series of steps that includes optimizing initial hyper-parameters and further optimization using a NM simplex algorithm. As a result, the LS-SVR model performs significantly better for both training and testing data compared with the benchmark BPNN model.

Additionally, the ability to track and predict future values of CO₂ emission with remarkable accuracy has also been demonstrated by the LS-SVR model. The superiority of the LS-SVR model can be accounted for by the ability to reach a global and unique solution for the minimization of a convex function. The proposed algorithm is robust in terms of the class of the problem that it can handle, be it classification or complex regression with multiple dimensions. However, one of the drawbacks of LS-SVR includes heavy computational dependencies when it comes to large datasets. Nevertheless, this could be improved using newer techniques. Overall, the primary objective of the paper has been achieved, and in the future, a hybrid model that involves SVMs with complex optimization will be developed in order to improve further the predictive accuracy and computational efficiencies.

Footnotes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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