Anupm Kumri ,Mukun Mhw ,C.B.Mjumer ,Amit Aror ,Gurv Dixit

a Department of Chemical Engineering,Indian Institute of Technology,Roorkee,India

b DHC Trading India Private,Limited,Delhi,India

c Department of Chemical Engineering,Shaheed Bhagat Singh State University,Ferozepur,Punjab,India

d Gas Hydrate Research and Technology Centre,Panvel,Mumbai,India

ABSTRACT The gas hydrate formation in pipelines of industries and chemical plants can cause various operational damages and can increase economic risks.Hence,the knowledge of hydrate formation conditions has become a critical research study to overcome the problems arising from the formation of hydrates.In this study,we applied an algorithm to develop an LSSVM model to predict the formation temperature of natural gas hydrate for a comprehensive range of data points.Total 188 experimental data points were applied from the literature for the development of the LSSVM model.The input parameter was finalized based on the structure of hydrates by each gas species.The results obtained by the LSSVM model have good accuracy as compared with empirical correlations available in the literature.This model gave the squared correlation coefficient(R2),and root mean square error of 0.9901 and 0.59974,respectively.The composition of gases may affect the phase equilibrium condition of gas hydrates.The applied algorithm revealed that the developed LSSVM model could become a good alternative for calculating the formation temperature of hydrate for the range of all data sets.The results showed that the proposed LSSVM model could be applicable for the prediction of hydrate formation temperature for all data points.

Keywords:Gas hydrate LSSVM Correlation Hydrate formation temperature

1.Introduction

Gas hydrates are crystalline compounds generated when the molecules of gas entrapped inside the cages formed by the hydrogen bonded molecules of water[1].These gas hydrates are formed at low temperatures and high pressure considering the suitable size of the guest molecules,which can match the cavity's size[2].The formation of gas hydrates can lead to the blockage of equipment and gas pipelines.Several empirical correlations and thermodynamic models are widely applied for the prediction of formation condition of gas hydrates[3].The empirical correlations can be applied for the prediction of formation pressure or temperature only while the thermodynamic models can predict the formation pressure of hydrate with a give temperature and also vice versa[4].Gas hydrate can form three types of structure such as sI,sII and sH.Specific pressure-temperature conditions are required for the gas hydrate formation in pure water and sediments[5].The prognostication of formation temperature of natural gas hydrates of any composition through experimental methods is challenging and time taking.Many correlations and models are available in literature for the predictions of hydrate formation temperature,such as Hammerschmidt[6],Katz gravity[7]and K-Value methods[8],Makogon[9],Berge[10],Kobayashi[11],Motiee[12],Safamirzaei[13],Towler &Mokhtab[14],Ghayyem[15]and Salufu[16].One of the main disadvantages of the above correlations is that they have not considered the different structures of hydrate.The presence of a very negligible quantity of sII structure hydrate forming species gives the predomination to the sII structure in the gas hydrate phase.Hence structure differences have a very remarkable reaction on hydrate formation temperature.It could possibly be the main reason for the non-performance of the above correlations for the evaluation of formation temperature for the hydrates of the natural gas mixture[17,18].

The base of the thermodynamic model is Vander Waals and Platteeuw model[17,19].Parrish and Prausnitz[20]developed a thermodynamic model which fits with good error accuracy.Ng and Robinson[21],John and Holder[22],Chen and Guo[23]and John et al.[24]presented different corrections in Vander Waala and Platteew model for the evaluation of formation temperature of gas hydrates.These models assumed the activity coefficient of water equal to unity.The prediction of hydrate formation temperature has emerged as the best alternatives because of the increasing advancement in artificial intelligence[17].Artificial Neural Network(ANN)has been used nowadays for the prediction of hydrate formation temperature.But ANN is not a good alternative for the prediction due to random initialization and difference in the terminating norms throughout the optimization of the variables of the model[17].In this study,we applied an algorithm by using the Least Squares Support Vector Machine(LSSVM)model for the evaluation of formation temperature of the gas hydrate of natural gas mixtures.This model used the chemical properties of gases and types of structure of hydrates.

Mesbah and Soroush[17]proposed a LSSVM model based on the hydrate structure and gave a vigorous algorithm with a squared correlation coefficient of 0.9918.Baghban et al.[18]applied the LSSVM model based on Katz Chart data points and predicted the hydrate formation temperature with a squared correlation coefficient of 0.9973 and minimum square error of 0.778634.Yarveicy and Ghiasi[25]proposed a model based on extremely randomized trees to predict the formation conditions hydrate of different gases formed in the pure water,salts,and alcohols.The suggested model gave the squared correlation coefficient of greater than 96 and percentage absolute relative deviation(%ARD)between 0.04 to 0.32.Mohammadi et al.[26]applied the LSSVM model for the prediction of hydrate phase equilibria for the CO2,CH4,or C2H6hydrate in the presence of porous media with a good value of squared correlation coefficient.This model has the ability to give a good prediction of phase equilibria condition of gas hydrates.In this study,the applied model is also compared with some existing correlations given in various works of literature.

2.LSSVM model

The Support Vector Machine(SVM)is called as a nonprobabilistic binary linear classifier and it is based on machine learning[27-29].This tool can be employed for the evaluation of variety of complex mathematical problems.The SVM algorithm requires convex optimization for evaluating the regression problems equipped with the inequality constraint to discover a support vector[17,30].SVM method needs a excessive computational load to resolve the inequality constraints and regression problems.Least Square Support Vector Machine(LSSVM)is a modified form of the support vector machine given by Suykens and Vandewalle[28].This model contains all the advantages of the SVM model,but in the LSSVM model,the inequality constraints is changes to equality constraints.This algorithm can operate huge sets of data with acceptable error accuracy[28,31,32].The variation between the experimental value and the predicted value is called as the regression error,and it is added as a supplementary constraint in the LSSVM model.the error is optimized during the calculation steps in the conventional SVM model,but the error is mathematically defined in the LSSVM model[17,28,32,33]A illustrative diagram of the applied LSSVM model is indicated in Fig.1.

2.1.Equations

Assume a data set(D),which can be represented by the following equation(1)with variables x and y

Equation(1)is approximated by the non-linear function of equation(2)

Where w∈Rnhis weight vector in the initial weight space,g(○):Rn→Rnhis the non-linear mapping function that forecast the inputs to a higher dimensional feature space in which linear regression is executed,b is the bias term and nhis the dimension of the space.The optimization problem in the algorithm of LSSVM model is evaluated by

The above cost function is subjected to the constraints given in equation(4)

In equation(4),t is the weight matrix transpose,ekis the error variables and μ≥0 is regularization constant.Put the(equation(4))in the(equation(3))and obtain the equation(5)

Here Lagrangian multipliers are βk∈R.The conditions needed for the optimization because of the lagrangian multipliers is given in equation(6)to equation(9).

Assume a linear regression relationship between the independent and dependent variables of the LSSVM,then equation(10)will form,which is used only for linear regression problems.

The Kernel functions should be included in equation(10)to increase the applicability of this function for non-linear regression problems.

The Kernel function ofK(xi,x)is given as the multiplication of vectors g(xi)and g(x)(equation(12))as the inner product

The RBF function(equation(13))and polynomial function(equation(14)are applied in the LSSVM model.

In above equations(13)and(14),σ2is the squared bandwidth which optimizes in between the optimization step,and d is the degree of the polynomial.

The following equations are used for the analysis of error

Where N is the number of data points and Eiand Piare the experimental and predicted values of the LSSVM model respectively[2,25].

2.2.Experimental data

The experimental data points for phase equilibrium of hydrate used for the LSSVM model were implemented from the Katz chart[34].The data points used in this study was in the range of 273.7 K to 303.1 K for temperatures and 0.6 MPa to 62.85 MPa for pressures,and gas gravities range between 0.57 to 0.86.The data points were divided into two subsets as testing and training datasets.

3.Results and discussions

The radial basis function(RBF)was utilized as the kernel function in the applied LSSVM model.The specification used in RBF were greatly disturbed by the quantity of support vectors.The training elapsed time and the quantity of support vectors are directly dependent on each other.The optimization algorithm was performed by genetic algorithm(GA)to evaluate the variables of the LSSVM model.The composition range of natural gas used in this study is shown in Table 1.Total number of data points used in the development of the LSSVM model was 298,from which 72 data points are used as the training data sets and remaining 226 data points are used as testing data sets.The error analysis of the applied model was investigated by MSE,RMSE and R2between the experimental and predicted values.The optimum value of number trees in the developed extra trees model was 4 which was found by applying trial and error method.The maximum depth of the trees was varied from 2 to 26 for the best results.The input parameters for the model were pressure and composition of gas mixture.By employing the optimization algorithm,the tuned values of γ and σ2was 1.12192 and 2318.67451 respectively.γ represents the relative weight of the summation of the regression errors and σ2is the squared bandwidth.The optimization of these parameters was performed during the learning process.The error analysis for the applied LSSVM model is given in Table 2.

Table 1 Composition range of natural gas.

For evaluating the performance of applied LSSVM model,the same data points were also used to see the accuracy performance of the most popular available correlations.These available correlations such as Motiee[12],Safamirzaei[13],Towler&Mokhtab[14],Ghayyem[15]and Salufu[16]have been applied for the prediction of the hydrate formation temperature.The error analysis for theperformance of available correlations and the LSSVM model was checked by root mean square error(RMSE)(Table 3).

Table 2 The error analysis for the applied LSSVM model.

Table 3 Error analysis for LSSVM and available correlations.

The comparison between the experimental and estimated values by the LSSVM model and available correlations is indicated in Fig.2.As shown in Fig.2,the deviations between the experimental hydrate phase equilibrium data and predicted data are negligible for the LSSVM model as compared to the other correlations.The best fitted correlation for the natural gas hydrate mixture was Motiee correlations.The order of the error accuracy among the LSSVM model and correlations is LSSVM Model ＜Motiee ＜Safamirzaei ＜Ghayyem ＜Salufu ＜Towler&Mokhtab.

Fig.3 demonstrates the comparison of the experimental temperature vs predicted temperature by the correlations and the applied LSSVM model by correlation coefficient(R2).As shown in Fig.3,the obtained correlation coefficient for the LSSVM model was 0.9901 which is less than the correlation coefficient obtained by the correlations.The value of correlations coefficient in increasing order is LSSVM Model ＞Motiee ＞Ghayyem ＞Safamirzaei ＞Towler&Mokhtab ＞Salufu.There is a change for the order as compared to error accuracy.Correlation coefficient measures the strength of the connection between two variables.

The deviation in relative error between experimental data and predicted data is given in Fig.4.This analysis shows the relative deviation between the experimental temperature and predicted temperature.The relative deviation of predicted temperature from the experimental temperature is lower for data obtained by the LSSVM model as compared to the predicted temperature by correlations.For the LSSVM model,some predicted data points best fitted with the experimental data but some predicted data points have not given good accuracy with the experimental data.It shows that the LSSVM model best fitted with the experimental data as compared to the correlations.

The applied LSSVM model gives predictions with good accuracy as compared to the experimental data.Also,the root mean square error(RMSE)and correlation coefficient(R2)were found to be 0.59974 and 0.9901 respectively.The different compositions of gases may affect the stability conditions of gas hydrates.In presence of H2S,the gas hydrates can form at high temperatures and low pressures.Hence,there is need to develop LSSVM model by considering the composition of gases.The correlations applied in this study(Motiee,Safamirzaei,Towler and Mokhtab,Ghayyem and Salufu)gave serious deviations from experimental data because these applied correlations are depended on the gravity of gases not on the composition of gases.

4.Conclusions

In the present study,a mathematical tool was applied for the prediction of formation temperature of natural gas hydrates with the help of the least square support vector machine(LSSVM).The input parameter was selected by considering the type of the structure of hydrates.The statistical parameter showed that the LSSVM algorithm can give better predictions for natural gas hydrates.The applied LSSVM model gave lower MSE(0.43958)as compared with the results of previous studies.The lower MSE indicates the better accuracy of the applied LSSVM model.Additionally,the applied model was also compared with the prediction results of empirical correlations available in the literatures.This LSSVM model gave good predictions of hydrate formation temperatures of hydrates of gas mixtures.The genetic algorithm was also applied to optimize the parameters(σ2and γ)of LSSVM model.This model gives low mean square error(MSE)as compared to the MSE obtained by the correlations.This study can help in the estimation of hydrate formation temperature for the engineers with low dependent parameters which will be a helpful tool for oil and gas industry as well as chemical plants wherever there is possibility of hydrate formation in pipelines.

Acknowledgements

The authors would like to acknowledge research grants from Gas Hydrate Research and Technology Centre(GHRTC),Panvel,Mumbai,India and Ministry of Human Resource Development(MHRD),India.

Abbreviations