Adv Syst Sci Appl 2020; 04:60–69
Published online at https://ijassa.ipu.ru.

Multi-Criteria Estimation of Input Parameters in Natural Gas
Quality Analysis

Ivan Brokarev1⇤, Sergei Vaskovskii2

1
National University of Oil and Gas Gubkin University, Moscow, Russia. E-mail: brokarev.i@gubkin.ru

2
V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia. E-mail:

v63v@yandex.ru

Abstract: In this paper, we present a method for assessing input parameters in a statistical model
used for the quality analysis of the natural gas. The analysis is done by measuring physical
quantities of natural gas through the hierarchy analysis, compromise programming, correlation
analysis and assessing the practical possibility of measuring selected input physical quantities
by available means. The problems arising when selecting input parameters are also considered.
The results are compared with the previously obtained results of the model input parameters
correlation analysis in order to improve the developed system for natural gas quality analysis.
The proposed method is applied to assess input parameters for certain samples of natural gas
based mixtures. Based on the results of the multicriteria assessment, the speed of sound, thermal
conductivity and concentration of carbon dioxide were selected as input parameters for the
developed natural gas quality analysis system.

Keywords: multi-criteria estimation, analytic hierarchy process, correlation analysis, natural
gas quality analysis, neural network analysis

1. INTRODUCTION

To analyze the quality of natural gas is to determine the energy characteristics out of the
measured physical parameters of the gas. At present, this problem is usually solved in the
gas industry by traditional methods, but there are alternative approaches [1]. The essence
of alternative methods and systems based on them is to determine the required indicators
by a statistical model, where the measured physical parameters of natural gas are input
parameters. These models, mostly neural networks, seem to be the most effective means
to solve the problem in comparison with other predictive models [2]. However, if we use
artificial intelligence methods, we encounter a number of problems that need to be solved for
the developed method to succeed in analyzing the quality of natural gas. The main problems
of the approach are high computational costs, a large number of input physical parameters to
measure, no general algorithm to choose the architecture and parameters of the model, and
the model’s decrease in accuracy, when the parameters deviate from standard values. Some of
the listed problems are associated with suboptimal choice of input parameters for the models.
For instance, if the number of input variables is excessive, the time spent on the analysis
procedure increases, up to the point when it is impossible to analyze in real time. In addition,
some of the input parameters are uninformative, so the model becomes less accurate.

⇤Corresponding author: brokarev.i@gubkin.ru



MULTI-CRITERIA ESTIMATION OF INPUT PARAMETERS IN NATURAL GAS QUALITY ANALYSIS 61

In practice, we choose the input parameters for a statistical model by heuristic methods
and a complete enumeration of the available parameters. However, there are algorithms aimed
to optimize the set of input parameters and to choose the parameters more efficient.

This article discusses our methodology to select input parameters for a natural gas quality
analysis system; the methodology is based on a combination of correlation analysis and a
multi-criteria estimation method.

2. METHODOLOGY FOR MULTI-CRITERIA ESTIMATION OF INPUT PARAM-
ETERS

The authors of [2, 3] chose the measured physical parameters out of a relationship between
the parameter under consideration and the component composition of the gas. In addition,
the authors took into account if it is possible to measure this parameter with commercially
available and relatively inexpensive measuring instruments. To determine the relationship
between the parameters and the component composition of natural gas, the authors conducted
a correlation analysis between the physical parameters and simulated samples of natural gas
of various component composition. Based on the calculated Pearson correlation coefficients
for the simulated sample, the authors figured out if they chose the input parameters correctly.
If the parameter had a high correlation (the value of the correlation coefficient in the range
0.7 - 1 modulo) with certain gas components and at the same time low correlation (0 - 0.3
modulo) with other input parameters, then this parameter was added to a set of model input
parameters.

We propose to marry the described approach with multi-criteria estimation methods. The
first step of the proposed methodology is to select criteria to estimate the input parameters
for the model. The selected criteria are divided into three groups, the first of which is
the availability of technology to measure the parameter (0 if absent, 1 if available), high
correlation with the model output parameters (component composition), and low correlation
with other input physical parameters of the gas. The second group includes three criteria:
correlations with the main components of the equivalent pseudogas - methane (CH4),
propane (C3H8), and nitrogen (N2).

The above criteria are not equivalent. The most important criterion is the availability
of technology to measure the parameter. The second is the criterion for high correlation
with the model output parameters. Correlation criteria are expressed as Pearson correlation
coefficients r. The least important criterion is the one of low correlation between the input
parameters; this is necessary mainly to eliminate the possible multicollinearity of the input
parameters.

Based on the results of the previous studies and analysis [4-8], we choose the following
physical parameters as the input parameters: speed of sound (c), thermal conductivity (�),
concentration of carbon dioxide (CO2), dynamic viscosity (⌘), and dielectric permittivity (").

To form an optimal set of input physical parameters of natural gas, we propose to use
the analytic hierarchy process (AHP) [9]. This method allows you to find such an alternative,
that is best consistent with how you understand the essence of the problem and the problem
requirements. The main advantage of the analytic hierarchy process is high versatility - the
method can be used to solve a wide variety of problems, regardless of the field of application.
The disadvantage of the analytic hierarchy process is a large amount of information needed
to be obtained from experts. However, we compensate for this disadvantage if we calculate
the correlation coefficients and ask an expert to assess only an availability of technology to
measure the investigated parameter.

It should be noted that we could represent our system hierarchically to describe
how changing priorities at higher levels influences the priorities of lower level elements.
Hierarchies provide more detailed information about the structure and function of the system
at the lower levels. To satisfy the constraints on level elements, it is best to reproduce the
constrains at the next higher level. Hierarchically structured natural systems are much more

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62 I. BROKAREV, S. VASKOVSKII

efficient. Hierarchies are stable in the sense that small changes have little effect, and flexible
in the sense that if we add elements to a well-structured hierarchy, we do not destroy the
hierarchy’s characteristics.

The first stage of the AHP is to build a hierarchical structure that includes the goal, criteria,
alternatives, and other relevant factors. The structure to choose the optimal set is shown in
Fig. 1.

Fig. 2.1. The structure of how the optimal set of the model input parameters is formed as we analyze the
hierarchies.

The next step is to build a matrix of pairwise criteria comparisons. We compare each
criterion relative to all others. To conduct paired comparisons, T. Saaty [9] developed a scale
of relative importance, presented in table 1.

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MULTI-CRITERIA ESTIMATION OF INPUT PARAMETERS IN NATURAL GAS QUALITY ANALYSIS 63

Table 2.1. Scale of relative importance

Level of importance index Level of importance Essence of the level of
importance

0 Incomparable Difficult to compare
criteria

1 Equal importance Two criteria contribute
equally

3 Moderate superiority One criteria has moder-
ate advantage over the other

5 Strong advantage One criteria has strong
advantage over the other

7 Substantial advantage One criteria is so much
superior over the other, that
the advantage is substantial

9 Very strong advantage One criteria is obvi-
ously superior over the
other

2,4,6,8 Intermediate solutions
between two importance
criteria

Applied in edge cases
where it is difficult to deter-
mine what criteria wins

According to the list of important criteria given in the introduction, we pairwise-
compared the criteria on a qualitative scale and subsequently converted the criteria to points.
The matrix of pairwise comparisons is compiled as we calculate the coefficients aij - the ratio
of the weight of criterion i to criterion j. The matrix of pairwise comparisons of criteria is
shown in Table 2.

Table 2.2. Matrix of pair-wise criteria comparisons

aij Tekh. r(CH4) r(C3H8) r(N2) r(c) r(�) r(CO2) r(⌘) r(")
Tekh. 1 2 2 2 3 3 3 3 3
r(CH4) 1/2 1 1 1 2 2 2 2 2
r(C3H8) 1/2 1 1 1 2 2 2 2 2
r(N2) 1/2 1 1 1 2 2 2 2 2
r(c) 1/3 1/2 1/2 1/2 1 1 1 1 1
r(�) 1/3 1/2 1/2 1/2 1 1 1 1 1
r(CO2) 1/3 1/2 1/2 1/2 1 1 1 1 1
r(⌘) 1/3 1/2 1/2 1/2 1 1 1 1 1
r(") 1/3 1/2 1/2 1/2 1 1 1 1 1

Then the matrix of pairwise criteria comparisons is normalized. For this, the new element
of the matrix Aij is calculated as the ratio of the original element aij to the sum of the elements
of the corresponding column. The average values for each row (Avgi) are found as well; the
value is called the criterion weighted column for the goal. The normalized matrix of pairwise
comparisons of criteria is shown in Table 3.

Table 2.3. Normalized matrix of pair-wise criteria comparisons

Aij Tekh. r(CH4) r(C3H8) r(N2) r(c) r(�) r(CO2) r(⌘) r(") Avgi
Tekh. 0.240 0.267 0.267 0.267 0.214 0.214 0.214 0.214 0.214 0.235
r(CH4) 0.120 0.133 0.133 0.133 0.143 0.143 0.143 0.143 0.143 0.136
r(C3H8) 0.120 0.133 0.133 0.133 0.143 0.143 0.143 0.143 0.143 0.136
r(N2) 0.120 0.133 0.133 0.133 0.143 0.143 0.143 0.143 0.143 0.136
r(c) 0.080 0.067 0.067 0.067 0.071 0.071 0.071 0.071 0.071 0.070
r(�) 0.080 0.067 0.067 0.067 0.071 0.071 0.071 0.071 0.071 0.070
r(CO2) 0.080 0.067 0.067 0.067 0.071 0.071 0.071 0.071 0.071 0.070
r(⌘) 0.080 0.067 0.067 0.067 0.071 0.071 0.071 0.071 0.071 0.070
r(") 0.080 0.067 0.067 0.067 0.071 0.071 0.071 0.071 0.071 0.070

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64 I. BROKAREV, S. VASKOVSKII

Ones we determined the weights of the criteria, we can proceed to the next step – to
determine the weights of the alternatives. For this, it is proposed to use the method of
compromise programming [10]. This is a multi-criteria optimization method, where a solution
is determined to minimize the distance from the target point to a set of effective solutions. The
first step in this method is to calculate the value of the criteria for each alternative. The value
of the criterion for the presence of the technology to measure a parameter is equal to zero
if the technology is absent and to unity if the technology is present. The rest of the criteria
values were calculated as Pearson’s correlation coefficients modulo for a sample of natural
gas with the following concentration ranges for each component: 85 - 100% for methane, 0
- 5% for nitrogen, carbon dioxide and propane. Table 4 shows the values of the criteria for
each alternative.

Table 2.4. Values of criteria for each alternative

Oki c � CO2 ⌘ "
Tekh. 1 1 1 0 0
r(CH4) 0.899 0.848 0.628 0.785 0.117
r(C3H8) 0.477 0.557 0.053 0.339 0.906
r(N2) 0.787 0.764 0.083 0.750 0.129
r(c) 1 0.992 0.233 0.629 0.451
r(�) 0.992 1 0.141 0.535 0.554
r(CO2) 0.233 0.141 1 0.541 0.445
r(⌘) 0.629 0.535 0.541 1 0.403
r(") 0.451 0.554 0.445 0.403 1

At the next step, the proximity degree values of the criteria are calculated. For this, the
criteria are divided into two categories for minimum and maximum. In this problem, it is
necessary both to maximize the availability of measurement technology and correlation with
the output parameters, and to minimize the correlation between the input parameters. For the
minimum criteria, the proximity degree of the each alternative according to the each criterion
oki is calculated by the formula:

oki =
max(Oki)�Oki

max(Oki)�min(Oki)
, i = 1...n (2.1)

where Oki is an estimated value of alternative i over criteria k, min and max are min and
max operations, n is the number of alternatives.

For the max criteria, we compute the proximity degree of alternatives as:

oki =
Oki �min(Oki)

max(Oki)�min(Oki)
, i = 1...n (2.2)

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MULTI-CRITERIA ESTIMATION OF INPUT PARAMETERS IN NATURAL GAS QUALITY ANALYSIS 65

The computation results for the proximity degree of alternatives over the criteria are
shown in Table 5.

Table 2.5. Proximity degree of alternatives over the criteria

oki c � CO2 ⌘ "
Tekh. 1 1 1 0 0
r(CH4) 1 0.935 0.653 0.854 0
r(C3H8) 0.497 0.591 0 0.335 1
r(N2) 1 0.967 0 0.947 0.065
r(c) 0 0.010 1 0.484 0.716
r(�) 0.009 0 1 0.541 0.519
r(CO2) 0.893 1 0 0.534 0.646
r(⌘) 0.621 0.779 0.769 0 1
r(") 0.920 0.747 0.930 1 0

After we determined the proximity degree of alternatives, it is necessary to determine how
to assess each alternative according to the criteria. We need to take into account the criteria’
significance calculated above. For this, the set of criteria G for each pair of alternatives is
divided into two subsets: G1 is a subset of criteria, where the considered i-th alternative is
superior to the j-th; G2 is a subset of criteria, where the considered i-th alternative is inferior
to the j-th one. For example, the first alternative (the speed of sound c) exceeds the fifth
(dielectric permittivity ") according to the first, second, fourth, fifth, sixth, and eighth criteria,
but is inferior in all others. The general formula to calculate �ij estimates is as follows:

�ij =

P
k2G1

Avgk ⇤ (oki � okj)

P
k2G2

Avgk ⇤ (okj � oki)
(2.3)

Here Avgk are the values of criteria weights from Table 2; oki and okj -are the degrees of
proximity of i and j alternatives over k-th criteria from Table 4.

Since �ii = 1, the assessment matrix for the alternative is the one in Table 6.

Table 2.6. The assessment matrix for the alternatives

�ij c � CO2 ⌘ "
c 1 1.277 6.391 5.207 4.046
� 0.783 1 5.667 4.230 4.391
CO2 0.156 0.176 1 0.850 1.452
⌘ 0.192 0.236 1.177 1 2.036
" 0.247 0.228 0.688 0.491 1

The final step of methodology proposed is to calculate the eigenvector of the assessment
matrix for the alternatives and subsequently normalize the matrix to determine the required
vector of weights for the alternatives. The resulting vector looks like this: [0.412; 0.351;
0.074; 0.095; 0.068]. Based on the obtained vector, it can be concluded that the first (speed
of sound) and second (thermal conductivity) alternatives are superior.

Next, it is necessary to evaluate the data obtained for consistency of the alternatives
assessment matrix and to reassess the alternatives. For this, the following parameters were
calculated: the maximum (main) eigenvalue �max, the concordance index (CI) and the
concordance ratio (CR).

We calculate the maximum eigenvalue. Then we use the matrix for evaluating alternatives
as follows: the sum of each column of the matrix is multiplied by the corresponding element
of the weight vector of the alternatives, and then the resulting numbers are summed up.
The closer the maximum eigenvalue is to the number of alternatives (the dimension of the
alternative assessment matrix) n, the more consistent the result is. For the obtained assessment

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66 I. BROKAREV, S. VASKOVSKII

matrix for the alternatives and the vector of alternatives’ weights �max is 5.106 for five
alternatives.

The results are then compared with the randomly scored data for a scoring matrix of the
same dimension. For this, the consistency index is calculated using the formula:

CI =
�max � n

n� 1
(2.4)

The consistency index for the received data is 0.026.
The consistency index of a randomly generated inversely symmetric matrix with the

corresponding reciprocal values of the elements is called the random index (RI), and the value
of the random index increases as the order of the matrix increases. Table 7 shows the order of
the matrix and the average values of the random index, determined for random samples [9].

Table 2.7. Dependence of the random index on the order of the matrix.

Matrix
order

1 2 3 4 5 6 7 8 9 10

Average
RI

0.00 0.00 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49

For the analyzed matrix dimension, the random index is 1.12 according to Table 6.
The ratio of the consistency index to the mean random index for a matrix of the same

order is called the consistency ratio.

CR =
CI

RI
(2.5)

A value of this parameter that is less than or equal to 0.1 is considered acceptable. If the
consistency ratio is 0, it means that the data is completely consistent. If the concordance ratio
exceeds 0.1, then we need to revise the estimates and recalculate the concordance parameters.
In the case study, this is not required, since the agreement ratio is 0.024. Based on the
calculated characteristics, it follows that the selected estimates of alternatives are consistent.

3. CONFIRMATION OF THE MULTI-CRITERIA ESTIMATION RESULTS BY
NEURAL NETWORK ANALYSIS

To confirm the results of the multi-criteria estimation of the input parameters for the neural
network model, we performed a neural network analysis. For our experiment, we simulated
a neural network model on the calculated data. The developed recurrent neural network [11]
was trained, tested, and simulated with different sets of input parameters to determine the
required component composition of gas mixtures. For five sets of input parameters, ten sets
of parameters are possible, three in each. All possible sets of input parameters for this case
are shown in Table 8.

Table 3.8. Possible sets of input parameters for the considered case

Set
number

1 2 3 4 5 6 7 8 9 10

Input
param-
eters

c, �,
CO2

c, �, ⌘ c, �, " c,
CO2,
⌘

c,
CO2,
"

c, ⌘, " �,
CO2,
⌘

�,
CO2,
"

�, ⌘,
"

CO2,
⌘, "

For each option from the set, the accuracy characteristics to determine the required
parameters were calculated: maximum absolute deviation (MaxAE), mean absolute deviation
(MAE), maximum relative deviation (MaxAPE), mean relative deviation (MAPE) according
to the formulas presented below.

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MULTI-CRITERIA ESTIMATION OF INPUT PARAMETERS IN NATURAL GAS QUALITY ANALYSIS 67

Table 3.9. Simulation results of the developed neural network model for various sets of input parameters

Component Characteristic Set number
1 2 3 4 5 6 7 8 9 10

MaxAE, % 0.425 0.696 0.751 0.516 0.567 0.832 0.524 0.535 0.556 0.987
Methane MAE, % 0.008 0.011 0.012 0.009 0.010 0.015 0.009 0.009 0.010 0.018

MaxAPE, % 0.451 0.714 0.789 0.562 0.579 0.875 0.542 0.551 0.561 0.991
MAPE, % 0.009 0.011 0.012 0.010 0.011 0.016 0.010 0.011 0.010 0.018
MaxAE, % 0.285 0.482 0.621 0.378 0.415 0.781 0.394 0.419 0.402 0.871

Nitrogen MAE, % 0.011 0.013 0.016 0.012 0.013 0.018 0.013 0.014 0.013 0.018
MaxAPE, % 0.308 0.482 0.568 0.376 0.421 0.790 0.382 0.407 0.399 0.862
MAPE, % 0.012 0.017 0.018 0.012 0.013 0.017 0.012 0.012 0.012 0.018
MaxAE, % 0.256 0.521 0.535 0.279 0.293 0.541 0.311 0.321 0.361 0.671

Propane MAE, % 0.007 0.012 0.012 0.007 0.007 0.012 0.008 0.008 0.009 0.014
MaxAPE, % 0.235 0.539 0.546 0.291 0.318 0.567 0.309 0.311 0.354 0.649
MAPE, % 0.006 0.011 0.011 0.007 0.007 0.012 0.007 0.007 0.008 0.013

MAE =
1

n

nX

i=1

|Youtput � Ytarget| (3.6)

MAPE =
100%

n

nX

i=1

|Youtput � Ytarget

Ytarget
| (3.7)

MaxAE = max |Youtput � Ytarget| (3.8)

MaxAPE = 100%max |Youtput � Ytarget

Ytarget
| (3.9)

Here Youtput – the value obtained from the statistical model, Ytarget – the initial values of
the target parameters, n – sample size, max – operator to compute maximum value.

The proposed calculation was performed in Matlab 2019b [12] with a plug-in for NIST
REFPROP [13]. The data included 150000 gas mixtures with the ranges described above and
the calculated physical input parameters under study. The simulation results of the developed
neural network model are shown in Table 9. The accuracy of carbon dioxide characteristics
are not shown, since the concentration of this component is an input parameter and is
considered known.

From the results of the neural network analysis, it can be concluded that the best accuracy
is in the first set; the set contains the alternatives prevailing from the multi-criteria estimation.
At the same time, the worst accuracy is in the tenth set with the least weighty alternatives.
The complete ranking of all sets of input parameters is shown in Table 10. This procedure was
carried out using expert estimations to clarify the calculated accuracy characteristics. Based
on the results obtained, it follows that the results of the multi-criteria estimation and neural
network analysis coincide.

Table 3.10. Ranged sets of input parameters

Set
number

1 2 3 4 5 6 7 8 9 10

Set
rank

I VII VIII II V IX III IV VI X

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68 I. BROKAREV, S. VASKOVSKII

4. CONCLUSION

This paper investigates how to choose input parameters for the statistical model to analyze
the quality of natural gas. It is shown that the proposed technique based on multi-criteria
estimation methods allows us to choose the quantitative and qualitative values of the criteria
for each input physical parameter. Also, the proposed methodology makes it possible to
choose the parameters before training the statistical model. This significantly reduces the
computational and time costs; this is extremely useful when we analyze hierarchies in real
time.

Note, that our technique showed the speed of sound and thermal conductivity to dominate
over the rest of the input parameters. These results partially coincide with previous studies,
where only the correlation analysis of input parameters was used [2, 3].

It was also found that the concentration of carbon dioxide, inferior to most of the
considered alternatives, might not be used in further studies as an input parameter of the
statistical model, although it is needed to reduce the amount of unknown concentrations of
the components of the analyzed gas.

The results of our multi-criteria estimation of the input parameters are confirmed by the
results of our neural network analysis carried out on various sets of input parameters.

To investigate this area further, we need to apply our system to analyze the quality of
natural gas [14] in laboratory and industrial conditions. We plan to use a recurrent neural
network model as the main model to determine the required quality characteristics of natural
gas by analogy with the neural network model of a multilayer perceptron, which was used
earlier [15].

ACKNOWLEDGEMENTS

The authors would like to thank the international cooperation projects between BRISK
II TA and Erasmus + 2017-1-SE01-KA107-034292 Staff Mobility for the opportunity to
conduct this study.

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MULTI-CRITERIA ESTIMATION OF INPUT PARAMETERS IN NATURAL GAS QUALITY ANALYSIS 69

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Copyright c� 2020 ASSA. Adv Syst Sci Appl (2020)


	Introduction
	Methodology for multi-criteria estimation of input parameters
	Confirmation of the multi-criteria estimation results by neural network analysis
	Conclusion