Strategy of changing cracking furnace feedstock based on improved group search optimization☆

Xiaoyu Nian,Zhenlei Wang*,Feng Qian

Key Laboratory of Advanced Control and Optimization for Chemical Processes(East China University of Science and Technology),Ministry of Education,Shanghai 200237,China

Keywords:Cracking furnace Scheduling of feedstock Group search optimizer Adaptive penalty function Double fitness values

ABSTRACT The scheduling process of cracking furnace feedstock is important in an ethylene plant.In this paper it is described as a constraint optimization problem.The constraints consist of the cycle of operation,maximum tube metal temperature,process time of each feedstock,and flow rate.A modified group search optimizer is proposed to deal with the optimization problem.Double fitness values are defined for every group.First,the factor of penalty function should be changed adaptively by the ratio of feasible and general solutions.Second,the“excellent”infeasible solution should be retained to guide the search.Some benchmark functions are used to evaluate the new algorithm.Finally,the proposed algorithm is used to optimize the scheduling process of cracking furnace feedstock.And the optimizing result is obtained.

1.Introduction

Ethylene(C2H4)is an important chemical raw material.Ethylene industry is the cornerstone of the petrochemical industry.Ethylene production indicates the level of petrochemical development[1].The ethylene cracking furnace is the key equipment in the ethylene industry and consumes as much as 50%to 60%energy of the plant.Therefore,optimizing and scheduling of cracking furnace are important because of high benefit[2,3].

Studies on cracking furnace scheduling mainly focus on the operation and optimization.Edwin and Balchen used dynamic optimization to determine the batch processing time and time-dependent operation trajectory of a cracking furnace[4].For multi-furnace scheduling,Jain and Grossmann developed a mixed-integer nonlinear programming(MINLP)model for cyclic scheduling that considers coil coking[5].Schulz et al.extended the MINLP model to ethane-fed ethylene plants with more focus on recycled ethane[6].A discrete-time MINLP model was recently developed to optimize cyclic furnace shutdowns and downstream separation units[7].Lim et al.integrated a neural network-based cracking simulation model into scheduling.The dynamic data of ethylene and propylene yields,tube metal temperature(TMT),and pressure drop could be obtained and used in the scheduling decisions[8].

However,feedstock scheduling in one operation cycle has not been considered yet.In many ethylene plants,the cracking furnace cracks a feedstock first,and then switches to others.The switch time and feed flow rate changing will decide the operating cycle and product yields.

To improve the benefit of cracking furnace,a model describing the scheduling process of feedstock is constructed in this study.The model considers the following factors:operating condition of each feedstock,operating cycle,maximum TMT,and process time of each feedstock.The goal is to maximize the benefit in an operating cycle.Software Coilsim1D is used to simulate the cracking furnace[9,10].The cracking reaction is simulated with series complex free radical reactions.

A modified group search optimizer(GSO)is proposed to solve the nonlinear constraint problem.This method searches for the smaller feasible region by adaptive penalty function and retains the “excellent”infeasible solution to improve the optimization accuracy.The algorithm is evaluated in some numeral simulations.Finally,the algorithm is used to the scheduling process of feedstock.

2.Description of the Ethylene Cracking Furnace

The structure of a typical cracking furnace is shown in Fig.1.The feeds(gas or liquid)and dilution stream are preheated in the convection section to form a mixture,gasified to the cracking temperature[11,12].The cracking reaction mainly occurs in the radiant section at a high temperature.Meanwhile,the coil of the radiant section and the transfer line exchange are coked[13].The mixture gas in the radiant section is cracked into C2H4,C3H6,C2H6and other by-products,such as cracked gasoline.

Fig.1.Structure of the cracking furnace.

During the cracking process,the growing coke on the inner surface of the coil and transfer line exchange reduces the efficiency of heat transfer,and consequently,increases TMT and export temperature of transfer line exchange.It also reduces the inner diameter and increases the pressure drop,decreasing the selectivity toward cracking.Therefore,the cracking furnace should be shut down from 50 days to 90 days to decoke.

Based on its density,the feedstock can be divided into two types,light and heavy ones.The light feedstock used in ethylene plants includes liquefied petroleum gas(LPG),naphtha(NAP),and light naphtha,while the heavy cracking feedstock involves atmospheric gasoline oil and heavy vacuum gas oil(HVGO).Two kinds of feedstock are usually cracked in one operating cycle.The attribute of feedstock has a great effect on the yield of product and coking rate.For instance,the coking rate of light feedstock is slower than that of heavy feedstock.Normally,the yield of ethylene and propylene of heavy feedstock is higher than that of light feedstock.Compared with light feedstock,heavy feedstock is cheaper.

3.Statement f Problem

We assume that i types of feeds(i=1,2,…,NF)are present in a cycle in a cracking furnace.The cracking gas consists of j types of components(j=1,2,…,NC).The switching time from one feedstock to another is very short compared with the operation cycle,so it is ignored.The total cycle time is divided into k batch time(k=1,2,3,…,NB).The process time of the i th feedstock is ti,k.The flow rate of the i th feedstock is denoted by Fiand Fi∈[Floi,Fupi],the lower and upper limits.The problem is how to schedule the type of the feedstock to maximize the profit considering ti,kand total cycle time Tcycle.

Fig.2 shows the cyclic scheduling of one furnace,with batch processing time and Tcycledemonstrated.The four feeds designated as A,B,C and D are allocated into different batch slots for processing.

3.1.Cracking reaction model

Fig.2.Demonstration for cyclic scheduling of ethylene cracking furnaces.

Cracking reaction is very complex and the number of cracking products is more than 100.Many processing variables cannot be measured directly,such as the yield of cracking product,while some gas phase compositions,such as CH4,C2H4,C2H6,C3H8and C3H6,can be measured by a gas chromatograph.Cracking furnaces are operated in steady state.The operating variable cannot be changed in a wide scope,so it is impossible to obtain the optimization model directly from the operating data.In the schedule optimization of feedstock,an appropriate cracker model is very important.

Coilsim1D is a new proprietary model for predicting product yields[14],which consists of the extensive reaction network for steam cracking of hydrocarbons.This fundamental approach produces accurate simulation results for different types of feedstock.

In order to simulate cracking furnaces appropriately,several parameters of Coilsim1D model must be corrected based on the parameters and operating data of cracking furnace,such as the coil size and material,cracking severity[15].After correcting,the output of cracker model is similar with the furnace output.Then the operation variables can be changed in a wide scope and the model can simulate the thickness of coke in the inner coil,so the coking data can be recorded precisely during the feedstock switchover.

3.2.Objective function

The objective of scheduling optimization is to maximize profitability.The net profit in a cycle is calculated by

where Pnet?profitis the net profit,Pincomeis the total income from various products,Praw?materialis the cost of all raw materials,and Poperationis the operational cost.

Primary product yields,such as hydrogen,benzene,ethylene,propylene,and butadiene,are calculated by Coilsim1D.NAP and HVGO are used as the feedstock.The parameters of cracking NAP and HVGO are respectively listed in Tables 1 and 2.The yields of products with operating time are demonstrated in Figs.3-10.The profile of each component can be represented with an exponential model

where ai,j,bi,j,and ci,jare the parameters of the i th feedstock(i=1,2,…,NF)and the j th product(j=1,2,…,NC).

Table 1 Parameters of cracking NAP

Table 2 Parameters of cracking HVGO

The problem is transformed to optimize the feed assignment.With the integration of the yield with respect to batch time(ti,k),the batch production amount can be expressed as Eq.(3).Based on Eq.(2),the profit index can be written as Eq.(4).

where Criis the cost of raw material i,Cv is the operational cost,and yi,k=1 when the i th feedstock is processed in batch k(k=1,2,…,NB).Otherwise yi,k=0.The numerator of Eq.(4)represents the total net profit for the furnace in one cycle.

Fig.3.Profile of benzene yield of NAP.

Fig.4.Profile of ethylene yield of NAP.

Fig.5.Profile of propylene yield of NAP.

Fig.6.Profile of butadiene yield of NAP.

Fig.7.Profile of benzene yield of HVGO.

Fig.8.Profile of ethylene yield of HVGO.

Fig.9.Profile of propylene yield of HVGO.

Fig.10.Profile of butadiene yield of HVGO.

3.3.Constraints

The thickness of the coke layer increases with time.When the TMT rises to the high limit,the furnace must be shut down to clean the coke.The parameters of TMT in NAP and HVGO calculated by Coilsim1D are listed in Table 3.The change in TMT is illustrated in Figs.11 and 12.Each curve can be fitted by an exponential form

Table 3 Parameters of TMT in NAP and HVGO

where ui,vi,and wiare the parameters corresponding to the i th feedstock.To ensure that the furnace runs safely,Eq.(5)should meet the following condition:

Before optimization,the number of batches needed for each furnace in one cycle is unknown,so it is just a heuristic integer.Some batch slots may not be utilized for cracking.Usage of pre-designated batch slots is completely determined by the optimization solutions.However,some constraints may help reduce the solution-searching space,

Eq.(7)indicates that the first batch slot should always be used for cracking a feed,otherwise the first batch processing time will be zero,which indicates that the second batch actually functions as the first one.Eq.(8)indicates that one batch slot could only be used for cracking one feed at most.

Fig.11.Profile of TMT of NAP with time.

Fig.12.Profile of TMT of HVGO with time.

For the production,the total cycle time(Tcycle)cannot be less than the pre-set cycle time(Ts),

Meanwhile,each feedstock processing time and flow rate should satisfy the constraints as follows.

4.Optimization

The constraint conditions ensure the feasibility of operation of changing feedstock and adaptability for the cracking furnace.However,the complex constraint problem is difficult to solve by typical optimization algorithm.Therefore,GSO,a kind of evolutionary algorithm based on the swarm intelligence,is proposed[16].

4.1.GSO

In an n-dimensional search space,the i th member at the k th searching iteration has a current position∈Rn,a head angle,and a head direction=,which can be calculated fromvia a polar to Cartesian coordinate transformation as follows.

Thus the scanning field of vision is amplified and the search region is generalized to an n-dimensional space characterized by the maximum pursuit angle θmax∈ Rn?1and maximum pursuit distance lmax=R,as illustrated in Fig.13.

In the GSO,a group consists of three types of members,namely,producers,scroungers,and rangers[17].At the k th iteration,producer Xpbehaves as follows.

(1)The producer will scan at 0°and then scan laterally by randomly sampling three points in the scanning field:One point at 0°

one point at the left hand side hypercube

and one point at the right side hypercube

where r1∈R is a normally distributed random number with a mean of0 and a standard deviation of 1,and r2∈Rn?1is a uniformly distributed random sequence in the range(0,1).

(2)The producer will find the best point with the best resource( fitness value).If the best point has a better resource than its current position,it will fly to this point or stay in its current position and turn its head to a new randomly generated angle

where αmax∈ R is the maximum turning angle.

(3)If the producer cannot find a better area after a iterations,it will turn its head back to 0°

where α∈R is a constant.

During each searching bout,a number of group members are selected as scroungers.The scroungers will keep searching for opportunities to join the resources found by the producer.At the k th iteration,the area copying the behavior of the i th scrounger can be modeled as a random walk toward the producer

where r3∈Rnis a uniform random sequence in the range(0,1).

Fig.13.Scanning field of 3D.

The rest of the group members will be dispersed from their current positions.At the k th iteration,a random head angle φiis generated using Eq.(18),and then a random distance is chosen

and moves to the new point

4.2.Constraints

To solve the nonlinear problem with constraints,the method of double fitness values that separates the index function from the constraint condition is proposed.The merit and defect of member i are decided by the objective function[ fitness(i)]and the constraint function[violation(i)].

To guarantee that the method can find the feasible region quickly,the adaptive penalty factor is used.The value of penalty factor is calculated from the following equation:

where p(i)is the penalty factor of member i,nFis the number of feasible members,nSis the number of the whole members,and C is constant.If the feasible region is small,nFis very small and may even be zero.The penalty should be momentarily increased,and the search should be close to the feasible region.However,if the feasible region is large,the ratio of feasible solutions will be great,the penalty will be decreased,and the search will be performed carefully in the feasible region.

The feasible member is unconditionally better than the infeasible one in the rule of double fitness values.The members with the minimum objective function in the infeasible region are usually ignored.However,these members are often the key factors to guide the optimization.In Fig.14,a and c are the feasible solutions,whereas b and d are the infeasible ones,supposing that a is the optimal solution.Although b is infeasible,it is closer to the optimal value than c and d.To solve this problem,the feature of rangers in GSO is used,and they search at the boundary of the infeasible region.Thus the members in the infeasible region can be retained to help determine the optimal solution.

Fig.14.Profile of distribution of the solutions.

4.3.Algorithm steps

The flow diagram of the algorithm is shown in Fig.15.

Step1 Initialization.Population Xi(i=1,2,…,n)are produced randomly and the initial head angle is φi=[φi1,φi2,…,φin].The parameters such as a,αmax,and θmaxare preset.

Step2 The direction Di(φi)=[di1,di2,…,din]is calculated by Eqs.(10)to(12).

Step3 The fitness value f(Xi)ofeach memberis calculated and producer Xpis chosen.Then,the search for better producers is performed using Eqs.(13)to(15),and the location of producer is updated.

Step4 A total of 80%of the other members are set as scroungers.The scroungers search the optimal by Eq.(18).

Step5 The violate values v(Xi)of the population is calculated.The penalty factor is obtained by Eq.(23)to update v(Xi).

Step6 The better member by the rules of double fitness value is chosen.

Step7 A total of 20%of the other members are set as rangers,and they search at the boundary of infeasible region by Eqs.(19)and(20).

Step8 Whether the algorithm should end or not is determined.If it meets the end condition,the algorithm is stopped.Otherwise,the algorithm is repeated from Step 2.

Fig.15.Flow diagram of algorithm.

5.Numerical Simulation

To test the performance of the improved GSO, five benchmark functions are listed in Table 4.Each function is tested 30 times.Theoptimization result of the improved GSO is compared with that of the four typical algorithms,namely,adaptive DE(A-DDE)[18],differential evolution variants(DECVs)[19],self-adaptive penalty function(SAPF)[20],and adaptive trade off model(ARMES)[21].In this case,the preset parameters are as follows:initial population N=50,the max iteration is 1000,φ0= π/4,a=2,θmax= π/4,αmax= π/8,and the maximum pursuit distance lmaxis calculated from the following equation:

Table 4 Benchmark test functions

The results of best,worst,mean,and standard of the five benchmark functions through five algorithms are listed in Table 4.

Table 5 shows that the improved GSO proposed by this paper can find the optimal solution and has a better performance than the other four algorithms.The accuracy of SAPF is not good enough and it cannot find the optimal condition in the test of G2 and G4.The robustness of A-DDE and ARMES is not satisfactory.

6.Case Study

The case study is derived from the cracking furnace of an ethylene plant.Two kinds of feedstock are cracked in one cycle operation.Fourproducts are considered,namely,ethylene,propylene,butadiene,and benzene.Their prices are listed in Table 6.The price of feed stock and operational costs are listed in Table 7.The process of switching feedstock for a cracking furnace is studied to maximize profit.This model involves nonlinear object function and violation function,so it is complex and cannot be easily solved.Therefore,the improved GSO is used to solve the problem based on the previous model.Then,the best feedstock switchover can be achieved.The operational condition is presented in Table 8.

Table 5 Comparison of improved GSO with the other fore algorithms on benchmark functions

Table 6 Price of cracked products

Table 7 Price of feedstock and operational costs

Table 8 Parameters of operational condition

For feedstock NAP and HVGO,the feedstock is changed only once.The identified optimal solution is shown in Fig.16.NAP is cracked for 45 d and NVGO for28 d.The yields of ethylene,propylene,and butadiene are considerably increased.By contrast,the benzene yield is reduced.

The optimization results are shown in Table 9.In an ethylene plant,the cracking times of NAP and HVGO are 40 and 30 days,respectively.The corresponding flow rates are 23.1 and 28.5 t·h?1.The benefit is 47861 CNY·d?1.After optimization,the profit per day increases 3.67%.The whole cycle time is longer,so the operational efficiency is improved.

Table 9 Comparison of optimization result

7.Conclusions

When different types of feed stock are processed in the same furnace,the scheduling process is crucial.In this paper,the scheduling of cracking furnace feedstock is translated to a constrained optimization problem.The modified GSO is used to solve the problem.The method of double fitness values is employed to improve the ability dealing with constraints.The method is proven effective by numerical test.The case study demonstrates the performance of the algorithm.

Fig.16.Optimal scheduling solution for case study.