Chemical Industry & Chemical Engineering Quarterly

Available on line at Association of the Chemical Engineers of Serbia AChE www.ache.org.rs/CICEQ

Chem. Ind. Chem. Eng. Q. 24 (2) 101−115 (2018) CI&CEQ

101

E. ABDUL JALEEL

K. APARNA

Dept. of Chemical Engineering, National Institute of Technology,

Calicut, Kerala, India

SCIENTIFIC PAPER

UDC 66.048.6:66

IDENTIFICATION OF A HEAT-INTEGRATED DISTILLATION COLUMN USING HYBRID SUPPORT VECTOR REGRESSION AND PARTICLE SWARM OPTIMIZATION

Article Highlights • Support vector regression (SVR) was used for identification of HIDC • Particle swarm optimization (PSO) performed the optimization of SVR • HYSYS software was used for generating data needed for identification • RMSE, R and MAE regression plots used for validating the performance • Hybrid PSO-SVR outperformed GA-SVR and BF-SVR Abstract

Distillation is the most commonly used method for separating fluid mixtures in oil and gas industries. It is a process that requires high energy usage. One of the efficient ways to save energy in a distillation column is by heat integration. One such type of distillation column is called a heat-integrated distillation column (HIDC). In HIDC, the prediction of mole fractions of the component in the product can be made using proper identification, or modeling, of the HIDC. However, nonlinear modeling of HIDC is a highly challenging task. Methods based on first principles are not sufficient for a highly nonlinear HIDC. Hence, a novel method for identification of HIDC using a non-parametric “support vector regression (SVR)” method for predicting benzene composition in benzene-toluene HIDC is proposed in this work. The data used for identification is generated using pro-cess simulation software HYSYS. 100 samples of data were used for training and 50 samples of data were employed for validating the model. Particle swarm optimization (PSO) was also incorporated with SVR for obtaining optimized parameters of SVR. The proposed model is compared with other SVR models optimized with optimization methods other than PSO. The proposed model showed better performance over others.

Keywords: SVR, HIDC, identification, PSO.

Distillation is a widely employed unit operation in petrochemical industries where 95% of liquid sep-aration is carried out using this process [1]. It causes significant energy consumption in refineries and pet-rochemical plants [2]. It accounts for 3% of world energy consumption [3]. Much research has been done to find methods for energy conservation since 1970 [4]. Process integration is an effective way to

Correspondence: E. Abdul Jaleel, Dept. of Chemical Engineering, National Institute of Technology, Calicut, Kerala 63601, India. E-mail: jaleeledathol@gmail.com Paper received: 18 November, 2016 Paper revised: 10 May, 2017 Paper accepted: 13 June, 2017

https://doi.org/10.2298/CICEQ161118023J

increase energy efficiency [5]. One of the methods developed is heat integration of two distillation columns [6]. Hence, the thermodynamic efficiency of the system can also be increased using this approach [7]. M. Nakaiwa et al. [4], Nakanishi et al. [8], Ponce [9] and Li et al. [10] proposed a heat integrated dis-tillation column (HIDC). Energy conservation was observed in all of the above proposed models when compared with conventional distillation columns.

The required purified products in HIDC, like in a conventional distillation column, can be obtained by controlling the product compositions [11,12]. The exact prediction of product compositions is necessary to preserve the products with essential purity and thereby, optimal control of compositions is also feasible [13]

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and realistic modeling of HIDC is needed. Two approaches are practiced for the modeling of these types of complexed systems. One method employed is to produce a model based on the first principle where model parameters are determined from the process descriptions [14]. An alternative approach to the first principle is developing the system model from the process input-output data. This type of modeling is called system identification or simply, identification.

Several researchers practiced using first prin-ciple-based modeling of the heat-integrated distil-lation column in research works [8,15-18]. Numerous assumptions are used for this kind of modeling, and it results in deviation from the actual process for the performance of the system [14]. Further, complex dif-ferential equations also appear when this kind of modeling is employed [14] and finding the solutions of these equations takes longer time [20]. These kinds of derived models based on the first principle are inadequate to represent nonlinear systems accurately [21,22]. The distillation column shows strong nonlin-ear dynamic characteristics during high purity oper-ations [21,23]. An HIDC process is a process with complex dynamics and is highly interactive [24]. Operating a HIDC is also more complicated than operating a conventional column [5]. Since it is a non-linear process, first principle-based modeling is inade-quate for system modeling.

For modeling of these type of complex and non-linear systems, different types of identification non-parametric methods can be used, such as artificial neural network, fuzzy logic, etc. The neural network has the robust capability of approximating any func-tion, the ability for parallel processing, the ability to learn from data sets and the capacity to learn the system from input-output data. Due to the advantages mentioned above, it has been used for identification of different systems, including distillation. Although identification of many systems is carried out using the artificial neural network, it has some shortcomings like empirical and structural risks [25]. Moreover, the limited number of training samples causes over-fitting and leads to poor generalization in the case of the artificial neural network [26]. Due to all of these short-comings of the neural network, it is difficult to design a good model of the neural network and its variants, especially to those who have little experience and little prior knowledge. These limitations, when the traditional neural network is used, can be solved by support vector regression [27,28]. Support vector reg-ression is one of the applications of a support vector machine. A support vector machine is a statistical learning algorithm. It can be used for classification [29], fault diagnosis [25,30] and regression. A support

vector machine for data regression (SVR) is used as a powerful tool for learning in many applications [31- –35]. The greatest advantage of SVR is structural risk minimization, rather than empirical risk used in the neural network. Structural risk minimization minimizes the upper bound of generalization errors, rather than of empirical errors used in the neural network [32]. Due to structural risk minimization, the optimal struc-ture is achieved by SVR.

To obtain better accuracy for support vector reg-ression, the model parameters of SVR have to be chosen selectively. By trial and error, it takes a long time to achieve the best parameters. Hence, the para-meters of a SVR model can be optimized using an evolutionary algorithm like genetic algorithm approach [36], bacterial foraging [37], particle swarm optimization [38], etc. Yian et al. [25], Chen et al. [39] and Ustun et al. [40] used genetic algorithm (GA) for finding optimal parameters of SVR. Wu et al. emp-loyed bacterial foraging (BF) for optimization of a sup-port vector machine (SVM) [41]. Yang et al. practiced BF for choosing optimal parameters of SVR [42]. Yian et al. [25], Kong et al. [31] and Lou et al. [43] developed models using SVR optimized with PSO. These optimized SVR models are called PSO-SVR models. Among these algorithms, particle swarm opti-mization is more attractive because of its simple imp-lementation, real convergence to the right solution with less time meeting the requirement of the object-ive function [44]. It has good stochastic global opti-mization as well. PSO also has excellent computati-onal efficiency, requiring less memory space and less CPU speed and a lesser number of parameters to tune [45]. PSO can be programmed easily using fund-amental mathematical and logical operations. Many nonlinear and optimization problems can be efficiently solved using PSO [46]. PSO can achieve the best solution, even though it does not need any gradient information about the objective function. PSO also has good convergence property to satisfactory sol-utions [46]. Hence, PSO is used for optimizing para-meters of SVR in this work.

Any of the non-parametric methods above have not been applied in existing literature for identification of a heat-integrated distillation column. The objective of this work is to develop a support vector regression model for heat-integrated distillation.

MATERIALS AND METHODS

Heat-integrated distillation column

A distillation column consists of a feed section, reboiler and condenser. The separating agent is heat and the reboiler provides this heat. Supplied heat is

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lost at the condenser. Reboiler temperature is higher than the condenser temperature. Hence, heat added at a higher temperature is lost at the lower tempe-rature in the distillation column. So, thermal energy is lost in the condenser. Thermal energy is not reused in the conventional distillation column. In this case, energy is degraded from reboiler to condenser.

To overcome this energy degradation and to improve thermal efficiency, two methods are applied to the heat integrated distillation column: 1) Inter-coolers and inter-heat exchangers are used. 2) The distillation column is divided into two sections: a high pressure (HP) column and a low pressure (LP) column. To establish heat transfer between two parts,

operating pressure and temperature at the HP column should be higher than one in the LP column. There-fore, lower operating pressure and temperature is used in the LP section, whereas higher operating pressure and temperature is employed in the HP section. Reflux flow is carried out in the HP section, and vapor flow is held in the LP section. Therefore, the condenser in the HP section and the reboiler in the LP section can be avoided. The conceptual flow sheet of the heat-integrated distillation column used in this work is shown in Figure 1. The HYSYS flow sheet of the heat-integrated distillation column is illustrated in Figure 2.

Figure 1. Conceptual diagram of a heat integrated distillation column (HIDC).

Figure 2. HYSYS simulation diagram of a heat integrated distillation column (HIDC).

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In this work, a Peng-Robinson fluid package was used for simulation. The stream feed is fed to HP and LP columns. The streams D and B are the dis-tillate stream from the LP column and the bottom pro-duct from the HP column, respectively. Feed condi-tions and compositions are given in Figure 1. The feed consists of benzene (light component) and tolu-ene (heavy component). This HIDC was used for the separation of benzene and toluene. Feed stream pressures are at a higher pressure than therequired feed stage pressure. This pressure is reduced to the required levels using control valves V1 and V2.

Feed locations were chosen to satisfy minimum reboiler duty for the HP column and minimum duty for the auxiliary heater for the LP column. Heat duty versus feed location for the HP and the LP column is shown in Figure 3.

The degree of freedom of one was observed for both HP and LP column. Hence, a 0.001 mol fraction of benzene at the reboiler stage was specified for the HP column, and a 0.99 mol fraction of benzene was chosen at the condenser stage for the LP column.

HP and LP column parameter specifications

Cooling water at 25 °C is inexpensive and it can be used in the condenser for cooling purposes. A temperature difference of 20 °C is essential for heat transfer in the condenser [47]. Therefore, the reflux drum temperature in the LP column can be selected as 45 °C, for heat transfer to occur. To accomplish 45 °C at the reflux drum, a pressure of 34.47 kPa (0.34 atm) has to be chosen in the condenser of the LP column, as shown in Figure 1. To establish a pres-sure difference between top and bottom sections of column, 44.47 kPa (10 kPa greater than in the con-denser) is used in the bottom of the LP column. This pressure results in a temperature of 82.95 °C at the bottom of the LP column. The pressure difference of

10 kPa between the top and the bottom is equally distributed in each tray of the LP column. Since 20 trays were used in this work, a pressure drop of nearly 0.5 kPa occurs in each tray.

To achieve heat transfer between the bottom section of the LP column and the top section of the HP column, a sufficient differential temperature between these two sections is essential [47]. In this work, heat has to be transferred from the top section of the HP column to the bottom section of the LP column. Hence, the temperature in the top section of the HP column should be higher in value compared to the temperature in the bottom section of the LP col-umn. To attain higher temperature in the top of the HP column, a higher pressure value of 344.47 kPa or 3.4 atm (300 kPa greater than the bottom pressure of the LP column) is chosen at the top of the HP column. The temperature at the top of the HP column then reaches 128.5 °C. This differential temperature of 45.55 °C (128.5-82.9 °C) between the top of the HP column and the bottom of the LP column is enough for heat transfer. To accomplish a pressure difference between the top and bottom sections of the column, 3544.47 kPa (10 kPa greater than at the top of the column) is applied in the reboiler of the HP column. The liquid composition and temperature profile for LP and HP columns in a steady state are shown in Fig-ures 4 and 5.

Heat integration

Heat integration occurs between the top section of the HP column (344.47 kPa, 128.5 °C) and the bottom section of the LP column (44.47 kPa, 82.95 °C), as shown in Figure 1. Heat integration between the two columns is carried out through the heat exchanger. The top section of the HP column works as a condenser and the bottom section of the LP column works as a reboiler, as described in literature

Figure 3. Heat duty versus feed location: a) HP column; b) LP column.

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Figure 4. Composition and temperature profile of LP column: a) composition; b) temperature.

Figure 5. Composition and temperature profile of HP column: a) composition; b) temperature.

[16,18,19,47]. The operations in these two sections are as follows.

The process fluid at a temperature of 128.5 °C coming from the HP column enters the inlet part of the tube side of the heat exchanger. This process fluid leaves from the outlet part of the tube side of the

exchanger with a temperature of 124.6 °C. The tem-perature of this stream is again reduced using the auxiliary cooler. This cooler is used with lower duty compared to the condenser. The temperature of the process fluid is coming from the outlet of the cooler at 122.6 °C. This process fluid is refluxed to the top of

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the HP column through a drum. Hence, a decrease in temperature in the refluxed stream is obtained through heat integration, i.e., the condenser action is carried out through heat integration.

The process fluid at a temperature of 82.95 °C coming from the bottom of the LP column enters the inlet part of the shell side of the heat exchanger. This process fluid leaves from the outlet part of the shell side of the exchanger with a temperature of 128.3 °C. The temperature of this stream is again increased using the auxiliary heater. This heater is also used with lower duty compared to reboiler duty. The tempe-rature of process fluid coming from the outlet of this heater is 132.2 °C. This process fluid is fed to the bottom of the LP column to provide heat. As a separ-ate auxiliary cooler and heater are used, cooling required at the top of the HP column and heat needed at the bottom of the LP column can be individually controlled. Therefore, heat is supplied to the lower part of the LP column through heat integration, i.e. reboiler action is carried out at the bottom of the LP column through heat integration.

This type of heat integrated distillation column rivals the conventional column in terms of energy saving and total annual cost saving. Economic ana-lysis comparison of a conventional distillation column and a heat integrated distillation column is shown in Table 1. The total energy cost and TAC is lesser in HIDC compared to conventional distillation columns.

Support vector regression (SVR)

For a regression-based support vector, the goal is to find a function f that maps actual input to actual output in such a way that the predicted output has, at most, a deviation of ε from the actual output y with maximum flatness as possible. Errors which are less than ε are not taken into considerations, but at the same time, errors greater than є are not accepted. Let us suppose the observation samples consist of samples of ( 1_LPr , 1_D LPx ), ( 2 _LPr , 2 _D LPx ), ( 3 _LPr , 3 _D LPx ),…,( _n LPr , _Dn LPx ) and _i LPr ϵ Rn and

_Di LPx ϵ R. _i LPr represent the reflux rate samples used in the LP column and 1_D LPx represent the mole fraction of the benzene component in the dis-tillate in the LP column, corresponding to reflux rate samples _i LPr :

= +LPf kr d (1)

For nonlinear support vector regression, the input data points are transformed to higher dimen-sional feature space, i.e., Φ: χ → ψ, so that ( 1_LPr , 2 _LPr ,…, _n LPr ) in the input space is trans-formed to high dimensional feature space as (ψ( 1_LPr ), ψ( 2 _LPr ),…,ψ( _n LPr )). The function which relates inputs (ψ( 1_LPr ), ψ( 2 _LPr ),…,ψ( _n LPr )) can be expressed by SVR, as given in Eq. (2):

( )ψ= +LPf k r d (2)

Table 1. Economic results of HIDC

Parameter Conventional distillation

column HP section of HIDC LP section of HIDC

HIDC (combined HP and LP)

No of trays 20 20 20 40 Feed Tray 11 11 10 - Feed rate, kmol/h 90.72 45.36 45.36 90.72 Temperature of condenser 45 - 45 45 Temperature of reboiler 82 165 - 165 Condenser duty, kW 1573.75 - 506.187 506.187 Reboiler duty, kW 1684.22 - 956.24 956.24 Area of condenser, m2 92.36 - 29.71 29.71 Area of reboiler, m2 68.96 19.81 - 19.81 Diameter of column, m 1.5 0.749 0.789 - Length of column, m 13.176 13.176 13.176 - Column cost, 104$ 21.4952 10.2479 10.8338 21.0817 Condenser cost, 104$ 13.8241 - 6.6135 6.6135 Reboiler Cost, 104$ 11.4330 5.0816 - 5.0816 Total capital cost, 104$ 46.7523 15.3295 17.4473 32.7768 Cooling water cost, 104$ 1.0528 - 0.3386 0.3386 LP stream vapor cost, 104$ 10.0985 - - - HP stream vapor cost, 104$ - 7.5441 - 7.5441 Total energy cost, 104$/y 11.1513 7.5441 0.3386 7.8827 Total annual cost (TAC), 104$/y 26.7354 12.6539 6.1544 18.8083

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For achieving maximum flatness, optimization can be expressed by Eq. (3):

minimize 212

k

subject to ( )

( )ξ

ξ

− Ψ − ≤

Ψ + − ≤

_ _

_ _

Di LP i LP

i LP Di LP

x k r d

k r d x, i = 1,2,…, n (3)

If some amount of training errors outside the ξ sensitive zone are allowed, slack variables δi and ∗δi are introduced, as described by eEq. (4):

minimize ( )∗

=+ δ + δ2

1

12

n

i ii

k C

subject to

( )( )

ξ

ξ−

∗−

∗

− Ψ − ≤ + δ Ψ + − ≤ + δ

δ ,δ ≥

_

_

0

Di LP i LP i

i LP Di LP i

i i

x k r d

k r d x (4)

i = 1,2,…,n

> 0C indicates the trade-off between the flat-ness of f and the amount up to which deviations larger than ξ can be tolerated. Optimization problem sol-ution by Lagrangian method is given by Eq. (5):

( )

( ) ( )

( )( )

( )( )

μ μ

μ μ

ξ ψ

ξ ψ

∗ ∗ ∗

∗ ∗ ∗

= =

=

∗ ∗

=

δ,δ ,β,β = +

+ β δ + δ − δ + δ −

− β + δ − + +

− β + δ − − −

2

1 1

_ _1

_ _1

1, , , ,

2n n

i i i i i i ii i

n

i i Di LP i LPin

i i Di LP i LPi

La k C k

C

x k r b

x k r d

(5)

La is Lagrangian and μ μ∗ ∗β ,β ,, ,i i i i are Lagrange multi-pliers and μ μ∗ ∗β ,β ,, ,i i i i > 0 .

By minimizing with respect to primal variables ( )∗δ , δ, , i ik d , Eqs. (6)–(9) are obtained:

( ) ( )

( ) ( )

ψ

ψ

∗

=

∗

=

∂ = − β − β = ∂

= β − β

_1

_1

0 0n

i i i LPi

n

i i i LPi

La k rk

k r

(6)

( )∗

=

∂ = β − β =∂

1

0 0n

i ii

Lad

(7)

∂ = − β − δ =∂δ

0 0i ii

La C (8)

∗ ∗∗

∂ = − β − δ =∂δ

0 0i ii

La C (9)

By substituting, optimization can be written as:

maximize

( )

( ) ( ) ( ) ( )( )

ξ

ψ ψ

∗

∗ ∗

= =

∗ ∗

=

β,β =

− (β + β + (β − β = − β − β β − β

_1 1

_ _, 1

) )

12

n n

i i Di LP i ii i

n

i i j j i LP j LPi j

K

x

r r

subject to ( )∗

=

β − β =

β ,β

1

0

Є 0,

n

i ii

i j C

, i = 1,2, …,n (10)

Regression function can be written as shown below:

( ) ( ) ( ) ( )( )ψ ψ∗

=

= + _1

β -β ,n

LP i i i LP LPi

f r r r d (11)

Predicted output (mole fraction of benzene com-position in the distillate in the LP column) concerning support vectors is given by Eq. (12):

( ) ( ) ( )( )ψ ψ∗

=

= β − β +_ _ˆ ,D LP i i i LP LPi SV

x r r d (12)

In Eq. (12), ( ( ) ( )ψ ψ_ ,i LP LPr r ) is represented by the kernel function ( )_ ,i LP LPK r r . Predicted output using support vectors and the kernel function can be represented as per Eq. (13):

( ) ( )∗

=

= β − β +_ _ˆ ,D LP i i i LP LPi SV

x K r r d (13)

( ) ( )

( ) ( )

ξ

ξ

∗

<β <

∗

<β <

∗

= −

− β − β − +

+ −

− β − β −

_

_

_0

_ _r Є

_0

_ _r Є

1{ [

, ]

[

, ]}

i

j LP

j

j LP

Di LPC

j j j LP i LPSV

Di LPC

j j j LP i LPSV

d xs

K r r

x

K r r

(14)

where s is the number of support vectors in Eq. (14). The general structure of SVR for input samples of x(1), x(2),…,x(n) and support vectors x1,x2,…,xs is shown in Figure 6.

Denoting ∗β = β ,β[ ]T T Tn n , where βT

n = = β ,β , ...,β1 2[ ]n and ∗β T

n = ∗ ∗ ∗β ,β ,...,β1 2[ ]n and kernel matrix ×n nK where ( )=, _ _,i j i LP j LPK K r r , then the optimization problem (10) can be described by con-vex quadratic programming, as per Eq. (15):

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minimize ξξ

− − β β + β +−

_

_

12

Tn D LPT

n D LP

XK KXK K

(15)

Different type of kernel functions K such as poly-nomial, sigmoid kernel, and radial kernel can be used. In this work, the radial kernel is used, which is given by Eq. (16):

( ) ( )= − − σ2 2

_ _, exp /LP D LP LP D LPK R X R X (16)

Particle swarm optimization (PSO)

Particle swarm optimization is originally based on the social behavior of bird flocks and fish schools, to search for candidate positions. When the birds search for food, the current position of the bird which is nearest to the food is searched. Every individual bird or particle is associated with a particular position and velocity. In the searching space, information is shared between individuals and thereby it helps to get the optimal solution of the particle. Each particle updates their moving trajectory, i.e., current position and velocity based on its experience or the compa-nion’s experience. As this process repeats, ultimately the particles achieve the optimal positions.

The initial position and velocity of the particle are randomly initialized in particle swarm optimiz-ation. The performance of the particle can be eva-luated by finding the fitness value of the particle. It helps to get the information of the current position and velocity whether they are good or bad. The fitness value of each particle is calculated at every iteration. The best particle is the particle with minimum fitness value if the problem is a minimization problem,

whereas if the problem is a type of a maximization problem, the best particle is the particle with maxi-mum fitness value. In this work, the problem is of the minimization category. Hence, the particle with mini-mum fitness value is chosen as the best particle. There-fore, the optimal particle solution is the position of each particle with minimum fitness value from the first iteration to the current iteration, and the global optimal solution is the best global position of all the particles from the first iteration to the current iteration [44].

Suppose there are P particles in the swarm. Let =t

ppbest 1 2, ,..., ,...,t t t tp p pd pDpbest pbest pbest pbest be

the best previous position of a particle p at iteration t and =tgbest 1 2, ,..., ,...,t t t t

d Dgbest gbest gbest gbest be the best global position of all the particles at iteration t. Let ( )t

p pf X and ( )tgf G be the fitness function

value of the thp particle and global optimal fitness value, respectively at the tht iteration. For the thp particle, if the value of fitness function at the current iteration is smaller than the value of fitness function at previous −1t iteration, then the best position of the particle pbest will be substituted by particle location at the current iteration step. Otherwise, pbest value remains constant. The same procedure is used for calculating gbest global optimum position of the all the particles, as illustrated from the following Eqs. (17) and (18):

−

=

1

tp

tp

Xpbest

pbest ( ) ( )−< 1

otherwise

t tp p p pf X f pbest

(17)

−

=

1

t

t

Ggbest

gbest ( ) ( )−< 1

otherwise

t tp pf G f gbest

(18)

Figure 6. Output predicting using SVR.

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At iteration −1t , the position and velocity of the particle are updated as per Eqs. (19) and (20):

( ) ( )+ = + − + −11 1 2 2

k t t tp p p pV kV f r pbest X f r gbest X (19)

+ += +1 1t t tp p pX X V

Where tpV , +1t

pV , tpX , +1t

pX are velocity of the particle, updated velocity of the particle, position of the particle, updated position of the particle and iner-tia weight respectively. 1f and 2f are acceleration constants and 1f is called social constant and 2f is called cognitive constant. 1r and 2r are random values between 0 and 1. k is called inertia weight and its value decreases from a maximum value to a minimum value for improving the performance as the iteration is progressing. pX and pV are being kept within range [ minX maxX ] and [ minV maxV ], respect-ively, for each particle. The flow diagram of PSO is given in Figure 7.

Proposed model

Data needed for identification of a heat-inte-grated distillation column is created from HYSYS soft-ware. This software is widely used in chemical and oil and gas refineries. Here, the output considered is the benzene composition. The manipulating variable is used to change the compositions. The reflux rate is used as the manipulating variable. Hence, this mani-pulating variable is used as the input variable for iden-tification. A selective excitation input signal has to be used for identification [48]. As the system is non-lin-ear, random excitation signals are adopted [49]. 150 samples of data are collected from the HYSYS. Out of 150 samples collected, 100 samples are employed for training, and 50 samples are selected for validation of the proposed model.

The model used for predicting output variable mole fractions of benzene compositions is the SVR model. SVR is trained using the training input data to develop an SVR model. This SVR model trained with

Figure 7. Flow diagram of PSO.

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data creates a function which relates the input vari-able reflux rate with the output variable mole fractions of the benzene composition. This developed model is also tested with validation input data to ensure the accuracy of the model. Usually, the accuracy of SVR mainly depends on the parameters of SVR, C, σ and ξ . Hence, these parameters are also optimized using a simple global optimization PSO algorithm in this work. Parameters of PSO utilized in this work are listed in Table 2.

Table 2. Parameters of PSO

Name Value

Number of particles 20

Maximum iteration 20

Maximum inertia weight 1.9

Minimum inertia weight 1.4

Social constant 2

Cognitive constant 2

After dividing the data, the training data is applied to the PSO algorithm to choose the best para-meters of SVR. The position variables of PSO are the solutions of parameters. During the training of PSO, each particle’s positions (solutions of the SVR) are applied to Eq. (15). Values of β and β ∗ are obtained by solving the Eq. (15). These values are used to cal-culate the predicted output, according to Eq. (12). Fit-ness values of the corresponding particle (solutions of parameters of SVR) are calculated based on the pre-dicted outputs values. Fitness value is calculated based on the root mean square error (RMSE).

From the calculations of fitness values of par-ticles, the pbest value of each particle and the gbest values of particles are calculated. Calculation of pbest and gbest are repeated until a maximum

number of iterations is achieved. Finally, the optimal position values of PSO (optimal solutions of paramet-ers of SVR) are obtained.

RESULTS AND DISCUSSION

The input-output data used for identification is illustrated in Figure 8. 150 samples of input (reflux rate) and 150 samples of output variables (mole fractions of benzene composition) are illustrated in Figure 8. Figure 8a represents the input data samples and Figure 8b illustrates the output data samples.

In this study, five solutions of PSO parameters were used for identification of a heat-integrated distillation column. These optimal values of parameters of SVR were obtained using PSO. These five solutions of parameters are given in Tables 3 and 4. Usually, minimum and maximum values of position variables (solutions of SVR parameters) are initialized during PSO training. In this case, zero was used as the minimum value of all variables. But different maxi-mum values of the parameters σ and C were used during training of SVR using PSO. In these cases, dif-ferent values of optimal parameters were obtained. The number of support vectors was also different for various optimal values of the parameters, as shown in Table 2. The minimum value of support vectors (5)

Figure 8. Input-output data used for identification; a) input data; b) output data.

Table 3. Number of support vectors for various optimized para-meters of SVR

Parameters of SVR No. of support vectors C σ ξn

48.1634 0.7672 0.0155 48

100 1.1602 0.0138 55

188.3618 0.8135 0.0222 6

320.1290 0.6008 0.0201 12

877.8386 1.5671 0.0226 5

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was observed in the parameter selection: 877.8386 (C), 1.5671 (σ ) and 0.0226 ( ξn ). The maximum number of support vectors (55) showed for the para-meter selection group: 100 (C), 1.1602( σ ) and 0.0138 ( ξn ).

Response of the proposed model

The response of the proposed model for training data and validation data is shown in Figures 9 and 10. In both cases, actual values and predicted output values are very close, as shown in Figures 9a and 10a. Errors between predicted output values and actual experimental values in validation data are illus-trated in Figure 10b. These errors were slight (between -0.0003 and 0.0003).

Figure 9 illustrates the predicting capability of SVR from the provided samples. Figure 10 shows the capacity of SVR to predict outputs close to the real value from unknown samples (samples which are not used for training).

Performance analysis

Performance analysis of the system is carried out through four degrees of measure: 1) regression plot; 2) root mean square error (RMSE); 3) correlation coefficient (R); 4) mean average error (MAE). The

equations for RMSE, MAE and R are given by Eqs. (21)-(23). In Eqs. (21)-(23), ( )y k , ( )y k , ( )y k and

( )y k are actual or experimental output value, pre-dicted output value, average value of actual or expe-rimental output values, and average value of pre-dicted output values, respectively:

( ) ( )( )=

= −2

1

1 ˆN

iRMSE y k y k

N (21)

( ) ( )=

= −1

1 ˆN

iMAE y k y k

N (22)

( ) ( )( ) ( ) ( )( )( ) ( )( ) ( ) ( )( )

=

= =

− −=

− −

1

22

1 1

ˆ ˆ

ˆ ˆ

N

iN N

i i

y k y k y k y kR

y k y k y k y k (23)

Regression plot shows a linear relation between the predicted output value and the actual value. The predicted output values are also displayed in the regression plot. Actual values are positioned along the x-axis and predicted output values along the y-axis. The equation relating predicted output value and the real value is given by Eq. (24):

Predicted output value = Target value + Constant (24)

Table 4. Table of comparison of statistical criteria for different values of optimized parameters

Parameters of SVR Training data Validation data

C σ ξn RMSE MAE R RMSE MAE R

48.1637 0.7672 0.0155 0.0013 0.0011 0.9998 0.0010 0.00082 0.9999

100 1.1602 0.0138 0.0013 0.0010 0.9999 0.00085 0.00065 0.9995

188.3618 0.8135 0.0222 0.0018 0.0015 0.9998 0.0016 0.0013 0.9998

320.1290 0.6008 0.0201 0.0017 0.0014 0.9981 0.0016 0.0013 0.9998

877.8386 1.5671 0.0226 0.0018 0.0015 0.9997 0.0016 0.0013 0.9998

Figure 9. Training data response: (a) Actual versus predicted; (b) Regression plot.

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If the predicted output values are closer to the regression line, the model will be more accurate. Pre-dicted data points of PSO-SVR in Figure 9b and also in Figure 10c are lying very close to the regression line. It also shows the accuracy of the proposed SVR model.

RMSE is also used for the performance analysis of the model. RMSE value is between zero and one. RMSE value of zero indicates perfect matching of the predicted model and the actual model of the system. RMSE value of one represents greater mismatching between the predicted model and the real one.

RMSE for both training and validation data are small in all SVR models with the mentioned SVR parameter group in Table 4. The lowest value of RMSE (0.0013 for training and 0.00085 for validation) for both training (0.0013) and validation (0.00085) observed in the parameters group 100 (C), 1.1602 (σ ) and 0.0138( ξn ). MAE can also be used for per-formance analysis. MAE (0.0010 for training and 0.0065 for validation) has a lower value in the SVR model with the parameter group 100 (C), 1.1602 (σ ) and 0.0138 ( ξn ).

Another performance parameter is R-value. The R-value is between 0 and 1. R-value of one repre-sents the perfect matching of the predicted model and the actual model. R-value of zero represents no rel-ation between the predicted model and the actual

model of the system. Hence, as R-value gets closer to one, the predicted model will be closer to the real model of the system. R values are close to one (0.9998, 0.9999, 0.9998, 0.9981 and 0.9997 for train-ing data, 0.9999, 0.9995, 0.9998 and 0.9998 for valid-ation data) in all the SVR models with optimized para-meters, shown in Table 3.

Comparison of PSO-SVR with artificial neural network

To compare the performance of PSO-SVR with the artificial neural network (ANN), model responses, regression plots of both PSO-SVR and the neural net-work for training data are plotted in Figure 9. The Levenberg-Marquardt algorithm is used for the train-ing neural network. The model response of PSO-SVR model is very close to the real sample values com-pared with the neural network in Figure 9a. The data points of the SVR model are also nearer to the reg-ression line compared with ANN output values in Figure 9b. Validation data comparison of both ANN and PSO-SVR is shown in Figure 10. The model res-ponse and regression response of validation data of PSO-SVR is better compared with ANN, as shown in Figure 10b and c. Figure 10b represents the error between real values and model output values which is less in PSO-SVR compared to ANN.

RMSE and R values, for both training and valid-ation data in the case of PSO-SVR and ANN, are

Figure 10. Validation data response: a) actual versus predicted; b) error in the predicted outputs of validation data; c) regression plot.

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shown in Table 5. RMSE values of the proposed model (0.0013 for training data and 0.0016) are com-paratively lesser than ANN (0.0157 for training data and 0.0137 for validation data). High value of R (0.99979 for training data and 0.99981 for validation data) is observed for PSO-SVR compared with ANN (0.98375 for training data and 0.98574 for validation data). The discussions above reveal that the perform-ance of PSO-SVR is better compared to ANN.

Table 5. Table of comparison of PSO-SVR and ANN

Algorithm Training data Validation data

RMSE R RMSE R

PSO-SVR 0.0018 0.99979 0.0016 0.99981

ANN 0.0157 0.98375 0.0137 0.98574

The ability to predict any model from a small number of samples with desired accuracy is an imp-ortant characteristic of SVR. From Tables 3 and 4, It is understood that SVR can predict the whole system using five support vectors or five numbers of samples (last row in Tables 3 and 4). But prediction of the entire system using such a small number of samples (5) is not possible with the neural network. To illus-trate the generalization property of SVR, RMSE of validation data for models trained with 50 and 100 numbers of sample data are shown in Table 6. RMSE in both cases (50 numbers of samples for training and 100 numbers of samples for training) are similar. But RMSE values differ more in ANN models.

Table 6. RMSE of validation data for model trained with differ-ent number of training data samples

Algorithm No. of the samples of training data

50 100

SVR 0.0023 0.0016

ANN 0.1111 0.0137

Comparison of PSO-SVR model with GA-SVR, BF SVR

To compare the accuracy of optimized para-meters of SVR, PSO-SVR is compared with a genetic algorithm-based SVR (GA-SVR) and a bacterial for-aging (BF)-based SVR (BF-SVR). In GA-SVR, the optimization of SVR parameters is carried out by the

genetic algorithm. Optimization of SVR parameters is employed using a bacterial foraging global optimiz-ation algorithm in BF-SVR. Statistical criteria RMSE and R are used for comparison. In this case, the mini-mum and maximum values of the variable used and the maximum number of iterations (20) are the same for the three optimization techniques. Optimized para-meter values and RMSE and R-values for both training and validation are shown in Table 7. RMSE values show little values for PSO-SVR for both train-ing (0.0018) and validation (0.0016) data compared with GA-PSO and BF-SVR. RMSE for GA is 0.0042 for training and 0.0043 for validation data. Similarly, RMSE values of BF-SVR are higher for training (0.0040) and validation (0.0040) compared with PSO- -SVR. The R-value has the higher value for both train-ing (0.99979) and validation (0.99981) data for PSO- -SVR compared to other models. From the statistical criteria above of SVR models, the better capability of the PSO-SVR model to predict the outputs from the input data is understood.

CONCLUSIONS

In this study, identification of a heat-integrated distillation column was carried out using optimized sup-port vector regression. Parameters of SVR were opti-mized using particle swarm optimization. The results showed high accuracy of the PSO-SVR model com-pared with GA-SVR and BF-SVR models. The pro-posed model was also compared with a neural net-work model and showed better performance over the artificial neural network (ANN). Five groups of opti-mized parameters were obtained, keeping the differ-ent maximum value of optimized parameters during the optimization stage. For various parameter groups, different number of support vectors was observed. The model accuracy was found to be nearly the same for these different numbers of support vectors. Statistical criteria was used for both training and validation data. The result showed the SVR capability to predict mole fractions of benzene compositions (output) from the manipulated variable reflux rate (input) with high accur-acy. This work can be extended to provide more accur-acy when using modified or improved PSO for opti-mization.

Table 7. Table of comparison of different algorithms for performance analysis

Algorithm Parameters of SVR Training data Validation data

C σ ξn RMSE R RMSE R

PSO-SVR 877.8386 1.5671 0.0226 0.0018 0.99979 0.0016 0.99981

BF-SVR 825.5206 1.1045 0.0452 0.0040 0.99935 0.0040 0.99925

GA-SVR 857.3357 1.1089 0.0456 0.0042 0.99919 0.0043 0.99894

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E. ABDUL JALEEL K. APARNA

Dept. of Chemical Engineering, National Institute of Technology,

Calicut, Kerala, India

NAUČNI RAD

IDENTIFIKACIJA TOPLOTNO INTEGRISANE DESTILACIONE KOLONE KORIŠĆENJEM HIBRIDNE VEKTORSKI PODRŽANE REGRESIJE I OPTIMIZACIJE ROJA ČESTICA

Destilacija je najčešće korišćena metoda za razdvajanje tečnosti u industriji nafte i gasa. To je proces koji koristi mnogo energije. Jedan od efikasnih načina za uštedu energije u destilacionoj koloni je toplotna integracija. Ovakva vrsta destilacione kolone se naziva toplotno integrisana destilaciona kolona (TIDK). U slušaju TIDK, predviđanje sastava proizvoda može se postići korišćenjem odgovarajuće identifikacije ili modela TIDK. Međutim, nelinearno modelovanje TIDK-a je izuzetno težak zadatak. Metode zasnovane na prvim principima nisu dovoljne za jako nelinearnu TIDK. Stoga se u ovom radu pred-laže novi metod za identifikaciju TIDK korišćenjem neparametričke metode “vektorski podržane regresije (VPR)” za predviđanje koncentracije benzena u koloni za razdva-janje benzena od toluena. Podaci koji su korišćeni za identifikaciju generisani su pomoću softvera za simulaciju procesa HYSYS. Za obuku je korišćen je set od 100 podataka, dok je 50 podataka korišćeno za validaciju modela. Optimizacija roja čestica (ORČ) je, takođe, kombinovana sa VPR radi dobijanja optimizovanih parametara VPR. Predloženi model su upoređeni sa VPR modelima optimizovanim drugim metodama optimizacije (različte od ORČ). Predloženi model pokazao je bolje performanse u odnosu na druge.

Ključne reči: vektorski podržana regresija, toplotno integrisana destilaciona kolona, Identifikacija, optimizacija roja čestica.