Computational intelligent strategies to predict energy conservation

Applied Thermal Engineering 102 (2016) 432–446
Contents lists available at ScienceDirect
Applied Thermal Engineering
journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
Computational intelligent strategies to predict energy conservation
benefits in excess air controlled gas-fired systems
Alireza Bahadori a,⇑, Alireza Baghban b, Meysam Bahadori c, Moonyong Lee d, Zainal Ahmad e,
Mehrasa Zare f, Ehsan Abdollahi g
a
School of Environment, Science and Engineering, Southern Cross University, Lismore, NSW, Australia
Young Researcher and Elite Club, Fasa Branch, Islamic Azad University, Fasa, Iran
National Iranian Drilling Co, Kish, Iran
d
School of Chemical Engineering, Yeungnam University, Gyeungsan, Republic of Korea
e
School of Chemical Engineering, Universiti Sains Malaysia (USM), Engineering Campus, Seri Ampangan, 14300 Nibong Tebal, Penang, Malaysia
f
Salman Farsi University, Kazerun, Iran
g
Department of Gas Engineering, Ahwaz Faculty of Petroleum Engineering, Petroleum University of Technology (PUT), Ahwaz, Iran
b
c
h i g h l i g h t s
Intelligent models are applied to predict combustion efficiency of natural gas in gas-fired systems.
The developed models were examined using experimental data, and a reasonable match was attained.
The models have been tested and validated using around 100 data points.
Hybrid algorithm is used to determine parameters of membership function in ANFIS approach.
a r t i c l e
i n f o
Article history:
Received 11 September 2015
Revised 10 March 2016
Accepted 2 April 2016
Available online 2 April 2016
Keywords:
Conservation benefits
Combustion efficiency
Artificial neural network
Adaptive neuro fuzzy inference system
Support vector machine
a b s t r a c t
Gas-fired systems and boilers are the noteworthy energy consumers in industries for energy production.
The mostly applied factor in systems which contain boilers is the combustion efficiency. This parameter
has widely applied to determine important information about exhaust gas such as CO2 and O2. This
study plays importance on applying the predictive techniques based on the Adaptive Neuro-Fuzzy
Inference System (ANFIS), Multi-Layer Perceptron Artificial Neural Network (MLPANN), Support Vector
Machine (SVM), and simple correlation for predicting the combustion efficiency of natural gas at
extensive range of excess air fraction and stack temperature rise. The developed tools can be of great
assessment for engineers dealing with combustion to have a rapid check on combustion efficiency of
natural gas at broad range of applications without the requirement of any unit as pilot plant. The
Levenberg–Marquardt algorithm is employed to optimize the bias and weight values of the ANN model.
In addition, Hybrid algorithm (coupling of back propagation and least square methods) is used to
determine parameters of membership function in the ANFIS approach. Hyper variables of the SVM
technique (i.e., C, K and e) are determined using tried and error procedures. To that end, a database
including 72 data points has been collected from the energy management handbook. Approximations
are recognized to be in high approbation with reported data points. Furthermore, potential of aforementioned models has been evaluated through statistical analyses. The developed models were examined
using several data, and a reasonable match was attained showing a good potential for the proposed
predictive tools in estimation of the combustion efficiency of natural gas at extensive range of excess
air fraction and stack temperature rise.
Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction
⇑ Corresponding author.
E-mail address: Alireza.bahadori@scu.edu.au (A. Bahadori).
http://dx.doi.org/10.1016/j.applthermaleng.2016.04.005
1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.
Nowadays, the main part of global energy production is from
the combustion of fossil fuels [1–3]. The concentrations of carbon
dioxide in the atmosphere have been steadily increasing over the
A. Bahadori et al. / Applied Thermal Engineering 102 (2016) 432–446
433
Nomenclature
AAD
ANFIS
ANN
ARD
BP
C
Cal.
Exp.
FLS
GA
HGAPSO
ICA
K
Average Absolute Deviation
Adaptive Neuro Fuzzy Inference System
Artificial Neural Network
Average Relative Deviation
Back Propagation
penalty factor
calculated
experimental
Fuzzy Logic System
Genetic Algorithm
hybrid genetic algorithm and particle
optimization
imperialist competitive algorithm
kernel function parameter
swarm
LSSVM
MF
MLP
MRE
MSE
PSO
R2
RBF
RMSE
STD
SVM
TST
UPSO
e
Least Square Support Vector Machine
Membership Function
multi-layer perceptron
Mean Relative Error
Mean Square Error
particle swarm optimization
R squared
Radial Basis Function
Root Mean Square Error
Standard Deviation
Support Vector Machine
Twu–Sim–Tassone
unified particle swarm optimization
insensitive coefficient
Fig. 1. Construction of typical ANFIS.
years, mainly due to emission of CO2 from the burning of fossil
fuels. The ever increasing concentration of CO2 is starting to have
significant effects on the global climate and the effect will continue
to multiply if the problem is not addressed [4].
Moreover, air contamination, acid rain, and health risks can be
occurred due to Nitrogen oxides, Sulphur oxides, polycyclic aromatic hydrocarbons (PAHs), carbon monoxide emissions [5,6].
Hence, using new methods is crucial to attain high combustion
efficiency with the minimum quantity of contaminant emissions.
To that end, monitoring of combustion processes or adjusting of
the fuels can great help [4].
According to obtained results from researchers and engineers
who study on boiler in industry, excess air is an important parameter and it has great effect on thermal efficiency and operation of
boilers [4]. Due to the increase in the excess of air value in the furnace, the adiabatic flame temperature is diminished. Also, due to
the flue gas temperature reducing, the heat transfer coefficients
are increased quickly for all the convective equipment. Because
of rising of the flame temperature in the boiler, the more the concentration of oxygen in the main combustion zone, the more the
Fig. 2. Flowchart diagram of proposed models.
excess air ratio. It causes that the temperature in the boiler
radiation area is dropped [7,8].
The chemical reaction between oxygen and fuel is carried out
during combustion and the heat and light are released [9].
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Table 1
Tuned coefficients used in Eq. (16).
Coefficient
Optimal value
a1
a2
a3
a4
a5
a6
0.8969
3.743e05
0.0002801
5.208e08
3.275e06
2.684e08
hy o
þs
Ao ¼ 4:77 c þ
4 2
Ai Ao
X¼
Ao
Fig. 3. Flow diagram of SVM approach.
Table 2
Obtained values of MSE and R2 through raising the number of neurons.
Neuron
MSE
R2
1
2
3
4
5
0.000039
0.000002
0.000001
0.000001
0.000001
0.9853
0.9994
0.9995
0.9997
0.9998
Generally the fuel is a hydrocarbon and the required oxygen is
prepared from ambient air. To have complete combustion, the
sufficient oxygen is required for converting the fuel to carbon
dioxide and water [10]. If the fuel is partially combusted and other
products instead of CO2 and H2O such as carbon monoxide,
hydrogen, etc. are produced, incomplete combustion occurs.
Representation of complete combustion of typical carbon-based
fuel compound presents in Eq. (1) [10].
hy o
C c Hhy Oo Ss Nn þ c þ
þ s O2
4 2
n
hy
H2 O þ ðsÞSO2 þ
N2
! ðcÞCO2 þ
2
2
Air is formed approximately 20.9% oxygen on a dry base. Therefore, 4.77 mol of air is supplied from 1.0 mol of oxygen. Using this
strategy to methane, 9.54 mol of air is required for every mole of
methane. Eq. (2) denotes the calculation of theoretical required
air for carbon-based fuel, in moles of air per mole of fuel (Ao) and
this is derived following stoichiometry [9,10]:
ð1Þ
ð2Þ
ð3Þ
where in Eq. (3), Ao denotes the theoretical required air of fuel, in
moles of air per mole of fuel, Ai denotes the inlet air in moles of
air per mole of fuel and X represents the fraction of excess air.
The more combustion air must be used than theoretical
required air in a boiler for safeguarding that combustion is
completed and operation is safe. Due to the very low rate of air,
carbon monoxide will be released rapidly in the flue gas and, in
dangerous conditions, smoke will be appeared [11]. Simultaneously, the efficiency of boiler has very close relation with the
excess air rate. Hence, amount of excess air must be low in practice
for reducing the amount of extra air that is released heat at the
stack temperature [11,12].
The quantity of excess air in the flue gas and the stack
temperature rise above the inlet air temperature are important
to determine the efficiency of the combustion process. To control
the ratio of air-to-fuel, a widely used approach namely, excess
oxygen measured in the exhaust stack is applied. Moreover, to
guarantee the best performance, it should be considered the
importance of furnace operation. Also, achieving ideal control over
unnecessary losses can obtain through measuring of the excess air
continuously [13].
Taking into account the aforementioned subjects, evolving a
precise and simple-to-apply approach which is more simplify than
present methods for forecasting the efficiency of natural gas combustion as a function of excess air fraction and stack temperature
rise is essential. Bahadori and Vuthaluru developed a correlation
based on principle component analysis (PCA) and partial least
square (PLS) [14]. They estimated the efficiency of natural gas combustion as a function of excess air fraction and stack temperature
rise. While, this aforementioned approach is able to estimate efficiency of natural gas combustion accurately, using of obtainable
technique is usually limited to this case. It means that this model
requires some parameters which are tuned and they must be
determined according experimental data points. Actually, these
parameters are not generally acceptable in the absence of experimental data points. In such conditions, evolving and employing
wide-ranging models is desirable to estimate the efficiency of
natural gas combustion.
The most significant embranchment of computational intelligence paradigms is the Artificial Neural Network (ANN) that can
be a benefit application for phase equilibrium modeling. Artificial
neural networks are non-parametric and statistical modeling tools
that do not require any pre assumption between inputs and output
[15–17]. To approximate any non-linear systems, artificial neural
networks have these abilities with high interpolation capacity
[18,19]. In addition, the fuzzy logic system is one of the embranchment of computational intelligence paradigms that has been
created to category the concept of partial truth-truth values
between completely true and completely false [20,21]. Another
embranchment of computational intelligence paradigms was
created by combining of the learning capability of neural networks
[22–24] with the knowledge of Fuzzy Logic summarizes in the
ANFIS method which is abbreviate of adaptive neuro-fuzzy inference systems [25–27]. Recently, Support Vector Machines (SVMs)
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Fig. 4. Performance plot of ANN model.
Table 3
Weight and bias values of the ANN model.
Neuron
Input Layer
Output Layer
Weight
1
2
3
4
5
Bias
Weight
Bias
Stack temperature rise
Excess air fraction
b1
Combustion efficiency
b2
0.690115
8.9921
1.42943
47.792
1.52882
0.18594
3.38592
0.32808
33.80368
0.34798
1.6166
15.93858
0.210648
24.3715
0.160937
1.195993
0.708773
4.075821
0.0023
3.39472
0.52819
Table 4
Details of proposed intelligent models.
ANN
ANFIS
Type
Value/comment
Type
Value/comment
No. of training data
No. of testing data
Transfer function
Hidden layer neuron
Hidden layer
Optimization method
Epoch
18
54
Logsig
5
1
Levenberg–Marquardt
1000
No. of training data
No. of training data
Membership function
No. of MF parameters
No. of clusters
Optimization method
Initial step size
Step size increase
18
54
Gaussian
60
10
Hybrid
0.01
1.1
SVM
Type
Value/comment
No. of training data
No. of training data
Kernel function
C
18
54
RBF
1000
0.0005
0.0002
e
K
are a popular learning technique for classification, regression, and
other learning task [28]. Unlike traditional artificial neural network
technique, the quadratic programming (QP) with linear limitations
is formulated in SVM problems [29,30]. Moreover, simplifying the
optimization processes of the SVMs can be performed through a
modification version of the SVM namely Least Square Support Vector
Machine (LSSVM) [31–33]. So as to create optimum construction of
these intelligent schemes, their related parameters can be obtained
by using of different mathematical optimization techniques
including: Back Propagation (BP) [34], Genetic Algorithm [35],
Particle Swarm Optimization (PSO) [36,37], Hybrid Particle Swarm
Optimization and Genetic Algorithm (HPSOGA) [38], Unified Particle Swarm Optimization (UPSO) [22] and Imperialist Competitive
Algorithm (ICA) [39]. There are lots of studies regarding application
of these computational strategies in different fields. Esen et al.
applied two models based on the ANFIS and ANN for predicting
performance of a solar ground source heat pump system [40]. Their
results indicated better capability of the ANFIS against the ANN
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model. Moreover, Baghban et al. used the ANFIS, ANN, SVM, and
LSSVM techniques as modeling tools for different applications such
as predicting phase behavior of carbon dioxide in ionic liquids,
estimation of semi-clathrate hydrate conditions of seven gases in
TBAB, prediction thermal conductivity of supercritical CO2, and
approximation of air dew point temperature [41–44].
In this study, the Adaptive Neuro Fuzzy Inference System
(ANFIS), Multi-Layer Perceptron Artificial Neural Network
(MLPANN), Support Vector Machine (SVM), and simple-to-use
correlation have been developed to predict the efficiency of natural
gas combustion as a function of excess air fraction and stack
temperature rise. Obtained results indicate ability of the proposed
intelligent approaches for predicting of energy conservation
benefit with satisfactory accuracy in comparison with the experimental reported data in the database (see Appendix A) [45].
2. Theory
2.1. Adaptive Neuro Fuzzy Inference System (ANFIS)
In this part, the basic theory of the ANFIS is submitted. ANFIS
base on a five layers adaptive network-based fuzzy inference
system proposed by Jang and Sun [46]. A typical fuzzy inference
system based on subtractive clustering has been trained by both
back propagation and hybrid learning methods frequently [47].
In Artificial Neural Networks (ANN), the most commonly used
algorithms is Back Propagation (BP). However, other computational algorithm such as Genetic Algorithm, and Particle Swarm
Optimization can be applied for optimizing structure of the ANN.
The goal of these optimization algorithms is determine difference
between corresponding target value and output value in the training set of data which was introduced to the network is calculated
[44,48].
A combination of back propagation method and the least square
method is applied in the typical ANFIS in order to train the ANFIS to
follow a training set of data. The training process happens when
the difference between testing and training errors are in a definite
range. A typical ANFIS structure has shown in Fig. 1. The adapted
parameters by the ANFIS are located in premise and consequent
parameters while other parameters are constant in other layers.
The ANFIS obligation is that it can only simulate single output
systems. Suppose a system with two input parameters and one
output. The Takagi–Sugeno simulation of such system is structured
by five layers. The detail of hybrid learning process presented here
Fig. 5. The performance of ANN and ANFIS predictive tools in comparison with experimental data for calculating combustion efficiency.
A. Bahadori et al. / Applied Thermal Engineering 102 (2016) 432–446
that all equation in this ANFIS modeling are proposed by Jang and
Sun [46].
The first layer equations for each node i:
o1i ¼ lAi ðxÞ; For i ¼ 1; 2 or o1i ¼ lBi2 ðyÞ; For i ¼ 3; 4
ð4Þ
In this equation x and y are input parameters for nodes and each
node can be parameterized with a function membership with
range of 0–1. For example the Gaussian membership function
[46,49]:
lA ðXÞ ¼
ðxc Þ2
i
e 2ri 2
ð5Þ
where c and r are the input membership function parameters.
In the second layer all of the nodes are fixed nodes labeled p
output shown the weight of a rule [50].
r2i ¼ xi ¼ lAi ðxÞlBi ðyÞ; For i ¼ 1; 2
ð6Þ
The third layer nodes labeled N are fixed and the equation of
this layer are:
r3i ¼ xi ¼
xi
; For i ¼ 1; 2
x1 þ x2
ð7Þ
437
The fourth layer membership function equation is:
r4i ¼ xi f i ¼ xi ðpi x þ qi y þ ri Þ; For i ¼ 1; 2
ð8Þ
where pi ; qi and r i are output membership function parameters. The
output f function is a constant or linear.
The single node in the final layer is a fixed node that computes
all nodes and summation of them.
r5i ¼
X
i
P
xf
xi f i ¼ Pi i i
i xi
ð9Þ
2.2. Artificial neural network
An interesting potential of the ANN is the Non-linear modeling
by back-propagation learning approach called multi-layer perceptron (MLP). Via the ANN, we are capable to make complex and
nonlinear relations between forecasters as input variables and
estimated variables which are the outputs of network. Also, we
can obtain precise and complicate information from typical databases. Detecting capability of the ANN leads to find all possible
interactions between input and output variables [51–55]. In the
Fig. 6. The performance of correlation and SVM predictive tools in comparison with experimental data for calculating combustion efficiency.
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ANN, input and output variables Communicate with each other
without needing any assumptions about mathematical demonstration of the system which should be simulated. Furthermore, capability of the ANN for generalizing leads to process incomplete and
noisy data that defines the problem to be modeled. There are three
layers including: input layer, hidden layer and output layer in the
MLP structure. Each layer consists of several nodes called neurons.
There are directed connections between each neuron in one layer
and neurons of the next layer. These neurons have unique weight
and bias and via the optimization algorithm, these values should
be optimized and updated. Also, through activation functions
which are used in each layer, the values of input weight and bias
are converted to output values [44,56].
There are several activation functions for converting values of
weigh and bias. Among them, hyperbolic tangent sigmoid function
(tansig), log-sigmoid transfer function (logsig) and linear transfer
function (purelin) are the mostly used transfer functions in many
practical applications.
In order to Back Propagation (BP) optimization algorithm which
commonly applied in the ANN, different mathematical optimization methods including: Genetic Algorithm [32], Particle Swarm
Optimization (PSO) [57], Hybrid Particle Swarm Optimization and
Genetic Algorithm (HPSOGA) [24], Unified Particle Swarm Optimization (UPSO) [22] and, Imperialist Competitive Algorithm
(ICA) [23,39] can be applied to create optimum configuration of
ANNs related parameters such as weights and biases.
2.3. Support Vector Machine (SVM)
Based on statistical learning theories, support vector machine
(SVM) is a global learning method and its construction firstly
was introduced by Vapnik [58]. Furthermore, it has been mostly
used to solve regression problems and the obtained result is often
more accurate and better than other machine learning methods
such as multi-layer perceptron artificial neural network and fuzzy
logic system [59–61]. Based on the principle of risk minimization
theories, SVM takes the fitting of training data and the complexity
of the training sample into consideration [62]. Moreover, it can
predict the regression problem by using a collection of kernel functions which are specified in a high dimensional feature space. Since
the optimality problem is convex, a unique solution is delivered.
This is an advantage of the SVM over Neural Networks, which have
multiple solutions associated with local minima [63]. To describe
regression with the SVMs, Support vector regression (SVR) is used
Fig. 7. Experimental versus predicted combustion efficiency by: (a) ANN and (b) ANFIS.
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in literature. Based on a given data bases (xi, yi)n, estimating with
SVR is applied to forecast a function, where xi and yi represents
the input and output vectors respectively and the total number
of data bases is represented by n, (in this work, the input vectors
(xi) refer to stack temperature rise and excess air fraction, whereas
the target value (yi) refers to combustion efficiency of natural gas).
As formulated in Eq. (10), the following function is used as the
linear regression function:
where C is a positive parameter and represents a parameter of extra
capacity control that specifies the trade-off between the complication of model and the amount up to which an error larger than e is
tolerated. The term of Euclidean norm, ||x||2 represents the regularization parameter and the empirical risk is measured by Le which
denotes insensitive loss function. Accordingly, by minimizing a
function in Eq. (11) subject to Eq. (12), a non-linear regression
function is given at following:
f ðxÞ ¼ x /ðxÞ þ b
f ðx; a; a Þ ¼
ð10Þ
where /(x) is a nonlinear function similar to linear, polynomial,
radial basis and Sigmoid functions, the parameter of weight vector
and bias have been represented by b and x respectively that should
be predicted from the data set. As formulated in Eqs. (11) and (12),
through a non-linear mapping, linear regression is exhibited in a
high dimensional feature space and the bias and weight values
are predicted by minimizing the sum of the empirical risk and a
complexity term.
n
X
1
Le ðf ðxi Þ; yi Þ kxk2
2
i¼1
0
for jf ðxi Þ yi j < e
Le ðf ðxi Þ; yi Þ ¼
jf ðxi Þ yi j e otherwise
R¼C
ð11Þ
ð12Þ
X
ða a Þkðxi ; xÞ þ b
ð13Þ
with ai ai ¼ 0 and ai ai P 0 for i = 1, . . ., N and the kernel function k
(xi,x) defines the internal product in the D-dimensional feature
space
kðx; yÞ ¼
D
X
/j ðxÞ/i ðyÞ
ð14Þ
i¼1
Also, the coefficients ai ai are obtained by maximizing the
following form:
Rðaa Þ ¼
N
N
X
X
1 X
ðai þ ai Þ þ
yi ðai ai Þ ðai þ ai Þ
2
i¼1
i;j¼1
ðai þ ai Þkðxi ; xj Þ
Fig. 8. Experimental versus predicted combustion efficiency by: (a) correlation and (b) SVM.
!
ð15Þ
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A. Bahadori et al. / Applied Thermal Engineering 102 (2016) 432–446
On the other hand, one of the major disadvantages of the SVM is
the requirement to solve a large scale quadratic programming
problem [64]. To reduce the complexity of optimization process
and solving linear equations instead of quadratic programming
problems, a modified version of the traditional SVM called leastsquares SVM (LS-SVM), has been developed to overcome this
drawback [65,66].
2.4. Description of simple correlation
According to Eq. (16), easy-to-use correlation has been developed to calculate amount of combustion efficiency of natural gas
(g) as a function of stack temperature rise (DT) and excess air
fraction (E). The unit of stack temperature rise used in this
correlation is °C.
There are six tuned coefficients in this correlation. The notations of these coefficients are a1–a6 and they summarize in Table 1.
Through minimizing summation of squared errors (difference
between correlation output and experimental value), optimal
values of these coefficients are determined.
g ¼ a1 þ a2 E þ a3 DT þ a4 E2 þ a5 EDT þ a6 DT 2
ð16Þ
3. Development of models
The accurate experimental databases are needed to develop
computational models. In the current cooperation, the combustion
efficiency of natural gas, stack temperature rise and excess air
fraction which were reported in the previous published papers
were applied and presented in Table A1 (see Appendix A) [45].
The combustion efficiency ranges from the 0.655 to 0.88 and
the ranges of excess air fractions and stack temperature rises are
0–1 and 150–700 °F.
After identifying and assembling the data set, the following step
was the selection of input parameters, which are the model’s
independent variables. In this regard, the excess air fraction and
stack temperature are considered as input variables and combustion efficiency is introduced as target variable. The data set which
contains 72 data points, have been divided into training data set
(75% of total data points) and testing data set (25% of total data
points) randomly. It should be noted the test data set has been
used to investigate the efficiency of the proposed models.
In this research, three different computational paradigms
including the Adaptive Neuro Fuzzy Inference System (ANFIS),
Multilayer Perceptron Artificial Neural Network (MLP-ANN),
Support Vector Machine (SVM) and, a simple-to-apply correlation
Fig. 9. Regression plots prediction of combustion efficiency using: (a) ANN and (b) ANFIS models.
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A. Bahadori et al. / Applied Thermal Engineering 102 (2016) 432–446
have been developed to predict the combustion efficiency of
natural gas. There are three parameters that require to be set in
SVM model before training, namely, penalty factor (C), insensitive
coefficient (e) loss function and, corresponding parameter of the
kernel function (K). To select an appropriate type of kernel function
and its parameters, we should consider application-domain
knowledge which reflects distribution of input values of the
training data bases [35,36]. In addition, the ANFIS approach has
membership functions and its parameters should be determined
based of membership function type. The commonly and mostly
applicable membership functions are the Gaussian, Triangular mf,
Trapezoidal mf, Generalized bell mf, zmf, pimf and, smf. Moreover,
the multi-layer perceptron artificial neural network contains some
parameters, i.e. bias and weight parameters which should be
optimized according to applied optimization method and transfer
functions types. A schematic diagram of proposed models is shown
in Fig. 2.
In order to examine the performance of abovementioned models, the Mean Squared Errors (MSEs), Mean Relative Errors (MREs),
Average Absolute Error (AAE), Standard Deviations (STD), Root
Mean Square Error (RMSE), and the correlation coefficient (R) have
been carried out between the experimental and predicted values.
The formulations of these statistical analyses have been shown in
Eqs. (17)–(22).
MSE ¼
N
1X
ðaexp acal Þ2
N i¼1
ð17Þ
MRE ¼
N
1X
jaexp acal j
N i¼1
aexp
ð18Þ
AAE ¼
N
1X
jaexp acal j
N i¼1
STDerror ¼
RMSE ¼
N
1 X
ðerror errorÞ2
N 1 i¼1
!0:5
N
1X
2
ðaexp acal Þ
N i¼1
ð19Þ
!0:5
Pn
i¼1 f½aexp aexp ½acal acal g
qP
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pn
n
i¼1 ½aexp aexp i¼1 ½acal acal ð20Þ
ð21Þ
ð22Þ
where aexp is the experimental data point and acal is the calculated
data point by models. Also, N denotes the number of data points.
Fig. 10. Regression plots prediction of combustion efficiency using: (a) correlation and (b) SVM models.
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A. Bahadori et al. / Applied Thermal Engineering 102 (2016) 432–446
Fig. 11. The percentage of relative deviation between model results and experimental values for: (a) ANN and (b) ANFIS models.
4. Results and discussions
In the current study, the Radial Basis Function (RBF) is applied
as the kernel function of SVM approach. Its parameters greatly
affect the number of support vectors and there is close relation
between its parameters and the performance of SVM and training
elapsed time. If the number of support vectors increase more than
enough, over fitting might be occur. However, three parameters of
the SVM algorithm (C, e and k) have been determined optimally by
evaluating them in a user-defined manner based on prior knowledge and experience. Flow diagram of the proposed SVM model
has been shown in Fig. 3. Firstly, we introduced these three parameters to algorithm and according to specified training data points,
the SVM is trained and extracts the outputs. Hence, the MSEs
between the outputs and corresponding target variable of training
data points (combustion efficiency) have been considered as stop
criterion of algorithm. Based on specific stop criterion that we
define it, the parameters of SVM are tuned.
Moreover, to design the ANN architecture, it is vital to note that
the optimal number of hidden layers and the number of neurons
throughout every layer should be used. Based on results obtained
from a guideline regarding overfitting problem, the number of data
points should be 10 times the number of connections in the ANN.
Hence, as presented in Table 2, through raising the number of hidden neurons, the values of R2 and MSE were determined. Results
showed that the optimal number of hidden neurons as gave the
minimum deviation should be considered 5 neurons in its hidden
layer. Owning to great capability of the Levenberg–Marquardt
algorithm, this technique was applied for training process of the
ANN model. Moreover, as indicated in Fig. 4, the performance of
training stage, which is the MSE, was decreased by raising epochs
and the best values obtained 0.000001. To determine optimal construction of the ANN, its parameters namely, bias and weight
should be optimized. Levenberg–Marquardt algorithm is used in
order to optimize these parameters. Table 3 lists the optimal values
of bias and weight for the ANN model.
In addition, another model based on the ANFIS was created in
order to combustion efficiency estimation. Owning to remarkable
features of the Gaussian function, it was employed as membership
function of the ANFIS. Based on 10 clusters which have been
considered for this system and the number of inputs and output
variables, the total number of membership function parameters
obtained 60. In more details, each membership functions have
two parameters and the number of input and output variables
were 2 (stack temperature rise and excess air fraction) and 1
(combustion efficiency) respectively. So, based on ten clusters
and three variables, the total number of membership function
parameters obtained from Eq. (23).
NT ¼ Nc NV Nmf
ð23Þ
where Nc , N V ; and N mf refer to the number of clusters, number of
variables and number of each membership function parameters.
A. Bahadori et al. / Applied Thermal Engineering 102 (2016) 432–446
443
Fig. 12. The percentage of relative deviation between model results and experimental values for: (a) correlation and (b) SVM models.
Table 5
Statistical analyses obtained from proposed models.
Network
ANN
Analysis
Training
Testing
ANFIS
Training
Testing
MSE
R2
MRE%
RMSE
STD
0.000001
0.9998
0.071294
0.000760
0.000502
0.000002
0.9996
0.142110
0.001428
0.000920
0.000000
1.000
0.026332
0.000283
0.000194
0.000004
0.9985
0.168241
0.002071
0.001595
Network
SVM
Analysis
Training
Testing
Correlation
Total Data
MSE
R2
MRE%
RMSE
STD
0.000000
1.0000
0.062801
0.000498
0.000017
0.000000
1.0000
0.062140
0.000494
0.000029
0.000002
0.9992
0.121108
0.001437
0.001108
Optimizing these parameters was carried out by hybrid algorithm (combination of least square and back propagation) in the
ANFIS. The stop criterion of this optimization technique was the
RMSEs between the actual and predicted combustion efficiency.
Details of aforementioned models have been presented in Table 4.
As a visual demonstration, the experimental against predicted
combustion efficiencies by aforementioned models have been
shown as function of the stack temperature rise and excess air
fraction in Figs. 5 and 6. The more excess air fractions and stack
temperature rises the less the combustion efficiencies. A comparison was shown in Figs. 7 and 8 between the experimental and
predicted combustion efficiently using the ANN, SVM, and ANFIS
models for both training and testing stages and a model based on
simple correlation. The strength of the developed models was
remarkably confirmed by the close match between the predicted
and experimental data points in assessing the predictive tools for
the training and testing steps.
A popular technique, which is commonly used in order to
modeling, is the regression analysis. Hence, Figs. 9 and 10 have
been prepared to show the regression analysis for the present
models. The R2 value is an indication of the relationship between
the model outputs and actual values. If R2 = 1, this indicates that
there is an exact linear relationship between the model outputs
and actual values. If R2 is close to zero, then there is no linear
relationship between the model outputs and actual values. A
close-fitting of data points about the 45° line for the models
indicate their accuracy. With the aim of above point, there are
great linear relationships between the output of models and actual
values. According to statistical viewpoints, Eqs. (24)–(27) are
generated as the linear regressions for test data set of the ANFIS,
ANN and SVM models and total data set of simple correlation
model respectively.
444
A. Bahadori et al. / Applied Thermal Engineering 102 (2016) 432–446
Table 6
Comparison of the proposed models with Bahadori and Vathaluru correlation [14].
Excess air,
fraction
Stack temperature rise,
°C
Reported combustion efficiency,
fraction
Bahadori and Vuthaluru
model
ANN
model
ANFIS
model
SVM
model
Correlation
0
0
0
0.2
0.2
0.2
0.2
0.4
0.4
0.4
0.4
0.6
0.6
0.6
0.6
0.8
0.8
0.8
0.8
0.8
1
1
1
1
1
1
1
65.5
204.4
371.1
93.3
232.2
315.5
371.1
65.5
176.6
232.2
343.3
93.3
204.4
315.5
371.1
65.5
121.1
260
315.5
371.1
65.5
121.1
148.8
204.4
232.2
315.5
371.1
0.88
0.84
0.789
0.864
0.814
0.784
0.764
0.867
0.82
0.7995
0.751
0.8495
0.796
0.742
0.717
0.8595
0.828
0.75
0.72
0.69
0.8525
0.819
0.8005
0.769
0.751
0.7
0.655
0.8792
0.8384
0.7898
0.8636
0.8144
0.7841
0.76338
0.8678
0.8216
0.798
0.7514
0.8497
0.7953
0.7417
0.7167
0.8582
0.8272
0.7509
0.72049
0.6898
0.8541
0.8173
0.8006
0.7684
0.7522
0.6989
0.6572
0.8815
0.8386
0.7895
0.8640
0.8127
0.7836
0.7644
0.8675
0.8214
0.7986
0.7509
0.8497
0.7959
0.7425
0.7149
0.8585
0.8273
0.7503
0.7200
0.6900
0.8529
0.8188
0.8019
0.7684
0.7517
0.6992
0.6550
0.8780
0.8396
0.7894
0.8641
0.8128
0.7838
0.7652
0.8684
0.8200
0.7995
0.7509
0.8497
0.7959
0.7418
0.7170
0.8530
0.8280
0.7501
0.7177
0.6900
0.8526
0.8187
0.8009
0.7686
0.7517
0.7000
0.6550
0.8784
0.8385
0.7893
0.8636
0.8144
0.7844
0.7642
0.8683
0.8219
0.7984
0.7510
0.8498
0.7959
0.7414
0.7139
0.8579
0.8275
0.7508
0.7198
0.6887
0.8527
0.8187
0.8016
0.7673
0.7501
0.6982
0.6635
0.8795
0.8395
0.7895
0.8635
0.8135
0.7845
0.7645
0.8665
0.8195
0.799
0.7515
0.849
0.7964
0.7425
0.7175
0.859
0.8275
0.7505
0.7205
0.6905
0.852
0.8185
0.8
0.7695
0.7515
0.7005
0.6555
Table A1
Experimental data used in this study [45].
Excess air 0%
Excess air 20%
Excess air 40%
Excess air 60%
Excess air 80%
Excess air 100%
DT (°F)
g
DT (°F)
g
DT (°F)
g
DT (°F)
g
DT (°F)
g
DT (°F)
g
150
200
250
300
350
400
450
500
550
600
650
700
0.88
0.87
0.8625
0.855
0.848
0.84
0.83
0.821
0.812
0.806
0.799
0.789
150
200
250
300
350
400
450
500
550
600
650
700
0.874
0.864
0.854
0.844
0.834
0.824
0.814
0.804
0.794
0.784
0.774
0.764
150
200
250
300
350
400
450
500
550
600
650
700
0.867
0.857
0.845
0.833
0.82
0.81
0.7995
0.788
0.774
0.761
0.751
0.74
150
200
250
300
350
400
450
500
550
600
650
700
0.8625
0.8495
0.837
0.822
0.809
0.796
0.783
0.77
0.754
0.742
0.73
0.717
150
200
250
300
350
400
450
500
550
600
650
700
0.8595
0.8415
0.828
0.812
0.796
0.7805
0.766
0.75
0.735
0.72
0.705
0.69
150
200
250
300
350
400
450
500
550
600
650
700
0.8525
0.836
0.819
0.8005
0.784
0.769
0.751
0.735
0.717
0.7
0.682
0.655
DT: stack temperature rise, g: combustion efficiency.
y ¼ 1:0035 x 0:002; R2 ¼ 0:9985
ð24Þ
2
ð25Þ
2
y ¼ 0:9924 x þ 0:006; R ¼ 1:0000
ð26Þ
y ¼ 0:999 x þ 0:00007; R2 ¼ 0:9992
ð27Þ
y ¼ 0:9861 x þ 0:011; R ¼ 0:9996
Some popular statistical techniques such as the Mean Squared
Errors (MSEs), Mean Absolute Errors (MAEs), Standard Deviations
(STD), Root Mean Square Error (RMSE), and R-Squared (R2) have
been used to examine the models. The percentage of relative
deviations between the experimental and predicted combustion
efficiency by the models have been shown in Figs. 11 and 12. These
figures indicated behavior of models in terms of accuracy. Deviation ranges of SVM approach have been shorter than other predictive models. The percentage of Average Relative Deviations (ARDs)
of testing data set for the ANN and ANFIS models are 0.142110 and
0.168241 respectively. In addition, these values obtained 0.062140
and 0.121108 for the SVM model and simple correlation respectively. The values of Mean Square Errors (MSEs), Average Relative
Deviations (ARDs), Standard Deviations (STD), Root Mean Square
Errors (RMSEs) and, R-squared (R2) have been summarized in
Table 5 for both testing and training data set of aforementioned
models. Moreover, as presented in Table 6, we compared the combustion efficiency values at different excess air fractions and stack
temperature rises that are determined from the models used in
this study with the Bahadori and Vuthaluru correlation that was
only previous work regarding estimation of natural gas combustion efficiency. Accordingly, all of models can predict combustion
efficiency of natural gas with high satisfactory precision.
5. Conclusion
Predicting energy conservation benefits in excess air controlled
gas-fired systems is as an interesting subject that was well organized in this contribution. The present study plays importance on
applying the computational models such as the Adaptive Neuro
Fuzzy Inference System (ANFIS), Multi-Layer Perceptron Artificial
A. Bahadori et al. / Applied Thermal Engineering 102 (2016) 432–446
Neural Network (MLPANN), Support Vector Machine (SVM) and,
simple correlation based of data gathered from the energy
management handbook. These aforementioned strategies can be
of great assessment in practice for chemical engineers to have a
rapid prediction of natural gas combustion efficiency at wide
ranges of applications. The present tools can predict natural gas
combustion efficiency as function of stack temperature rise and
excess air fraction. The Levenberg–Marquardt algorithm indicated
satisfactory performance to optimize the bias and weight values of
the ANN model. Hybrid algorithm which is the combination of back
propagation and least square was used as great optimization tool
to determine membership function parameters of the ANFIS
approach. In addition, based on experience knowledge, the optimal
values of C, e and, K were determined in the SVM approach.
Approximations are recognized to be in high approbation with
experimental data points. Based on figures accompanied with the
statistical analyses, although the SVM approach achieves much
better in accuracy than the other proposed models in order to
combustion efficiency estimation, other approaches also had great
accuracy. In addition, results obtained from the present models
and previous work carried out by Baharori and Vuthaluru confirmed their acceptable precisions. These artificial intelligence
methods promise noteworthy advantages in order to combustion
efficiency of natural gas modeling. Moreover, unlike complex
mathematical techniques for predicting combustion efficiency,
the proposed models are simple-to-apply and would be of great
help for petroleum engineers especially those dealing with the
gas-fired systems.
Appendix A
Experimental data bases that used in this study are presented in
this section. The stack temperature rise (DT), excess air fraction (E)
and combustion efficiently of natural gas (g) are presented in
Table A1.
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