Evaluation of two termination criteria in evolutionary algorithms for multi-objective optimization of complex chemical processes

chemical engineering research and design 1 2 4 ( 2 0 1 7 ) 58–65

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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd

Evaluation of two termination criteria in evolutionary algorithms for multi-objective optimization of complex chemical processes G.P. Rangaiah ∗ , Shivom Sharma, H.W. Lin Department of Chemical & Biomolecular Engineering, National University of Singapore, Engineering Drive 4, Singapore 117585, Singapore

a r t i c l e

i n f o

a b s t r a c t

Article history:

Multi-objective (or multi-criteria) optimization (MOO) is useful for gaining deeper insights

Received 23 February 2017

into trade-offs among objectives of interest and then selecting one of the many optimal

Accepted 30 May 2017

solutions found. It has attracted numerous applications in chemical engineering. Com-

Available online 8 June 2017

mon techniques for MOO are adaptations of stochastic global optimization methods, which

Keywords:

techniques have been used mostly with maximum number of generations (MNG) as the

Multi-objective optimization

termination criterion for stopping the iterative search. This criterion is arbitrary and compu-

include metaheuristics and evolutionary methods, for single-objective optimization. These

Chemical processes

tationally inefficient. Hence, this study investigates two termination criteria based on search

Evolutionary algorithms

progress (i.e., performance or improvement in solutions), for MOO of three complex chemi-

Genetic algorithms

cal processes modeled by process simulators, namely, Aspen Plus and Aspen HYSYS. They

Differential evolution

are Chi-Squared test based Termination Criterion (CSTC) and Steady-State Detection Termi-

Termination/stopping criterion

nation Criterion (SSDTC). Both these criteria are evaluated in two evolutionary algorithms for MOO. Results show that CSTC and SSDTC are successful in giving optimal solutions close to those after MNG but well before MNG. Of the two criteria, CSTC is more reliable and terminates the search earlier, thus reducing computational time substantially. © 2017 Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

1.

Introduction

Optimization of the design and operating conditions of industrial chemical plants is essential for improving their performance. In such cases, many optimization problems involve simultaneous minimization and/or maximization of more than one objective related

tive Optimization (SOO) are also summarized in Rangaiah et al. (2015). Majority of the optimization studies covered in these reviews have used an evolutionary algorithm such as the elitist Non-dominated Sorting Genetic Algorithm (NSGA-II), and they were programmed using programming languages such as FORTRAN and C++.

to economics, environment, energy and safety, and some of these objectives can be conflicting in nature. This type of optimization is

Although FORTRAN and C++ are used by researchers and engineers, they are more familiar with Microsoft Excel (MS Excel), which is widely used in both industry and academia. Hence, Sharma et al. (2012) imple-

referred to as multi-objective optimization (MOO), and its resulting optimal solutions are termed as Pareto-optimal front, which consists

mented binary-coded NSGA-II algorithm in MS Excel (Excel-based MOO or EMOO program), to provide a user-friendly and readily available plat-

of many non-dominated solutions or equally good solutions in terms of objectives used. More details on MOO and its various applications in chemical engineering can be found in Bhaskar et al. (2000), Zaman

form for performing MOO. An MOO problem can also be solved by deterministic optimization methods by first transforming it into a SOO

and Rangaiah (2009), Sharma and Rangaiah (2013a), and Rangaiah et al. (2015). Similarities and differences between MOO and Single Objec-

∗

problem, which is known as scalarization approach (Rangaiah et al., 2015). The resulting SOO problem has to be solved repeatedly for obtaining many non-dominated solutions. For example, ␧-constraint method

Corresponding author. E-mail addresses: [email protected], [email protected] (G.P. Rangaiah). http://dx.doi.org/10.1016/j.cherd.2017.05.030 0263-8762/© 2017 Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

chemical engineering research and design 1 2 4 ( 2 0 1 7 ) 58–65

(belonging to the scalarization approach) was employed for deployment of a hydrogen supply chain (Almaraz et al., 2015), for optimization of forest-based biorefinery supply chains (Cambero et al., 2016), and for design and retrofitting of water networks (Sharma and Rangaiah, 2016). Evolutionary algorithms such as genetic algorithms and differential evolution, and other stochastic algorithms (e.g., teaching–learning based optimization) for MOO have been implemented and used with maximum number of generations (MNG) or maximum number of function evaluations (i.e., computational effort) as the termination criterion for stopping the search (e.g., Sharma et al., 2012; Patel and Savsani, 2014; de Faria et al., 2016; Patel and Padhiyar, 2017). In order to ensure that optimal solutions are found, a large MNG has to be chosen; moreover, suitable MNG will change from one run to another owing to the stochastic nature (arising from the use of random numbers) in evolutionary/stochastic algorithms. Hence, use of MNG is computationally inefficient for application problems, which require computational time of seconds to minutes for each evaluation of objective functions and constraints. There is a need for reliably terminating the search by the evolutionary algorithm right after convergence (i.e., not much improvement in the optimal solutions over a number of generations). For MOO, Trautmann et al. (2008, 2009) proposed a termination criterion using different performance metrics to monitor search progress; search is terminated if variations in the performance metrics are not significant over certain number of generations. Sharma and Rangaiah (2013b) developed an improvement-based search termination criterion, and used it in Multi-Objective Differential Evolution (MODE). After analyzing several performance metrics, they chose Generational Distance (GD) and Spread (SP) for implementing the Chi-Squared (␹2 ) test based Termination Criterion (CSTC) in Improved MODE (IMODE) program. They tested CSTC on constrained benchmark problems and then on alkylation, Williams-Otto and fermentation processes. Guerrero et al. (2010) proposed least square termination criterion, which utilizes mutual domination rate, hyper volume and epsilon indicator. Recently, Saxena et al. (2016) summarized the approaches for termination criterion used in around 10 papers, mostly from conferences, and identified their limitations. Then, they proposed an entropy-based termination criterion and evaluated its performance through extensive simulations. Abdul Kadhar and Baskar (2016) proposed a termination criterion based on variations of Tchebycheff objective, for use in decomposition-based evolutionary algorithms for MOO. After testing on benchmark problems, they assessed the stopping criterion on the optimal design of a PID controller for both stability and performance. Rhinehart (2014) proposed and tested a convergence criterion for SOO based on steady-state detection by comparing the objective function variance estimated using two different methods. For MOO, Wong et al. (2016) developed Steady-State Detection Termination Criterion (SSDTC) using GD, and implemented it in the EMOO program (having both real- and binary-coded NSGA-II). Then, they employed this program with both CSTC and SSDTC to solve mathematical test functions and shell-and-tube heat exchanger design problems. Results reported in Sharma and Rangaiah (2013b) and Wong et al. (2016) show the potential benefit of CSTC and SSDTC. However, these two promising termination criteria have not been applied and tested for MOO of complex chemical processes simulated with a process simulator such as Aspen Plus and Aspen HYSYS. Further, SSDTC has been used in the EMOO program only, and not in the IMODE program. Hence, the broad objectives of this study are to evaluate the effectiveness of CSTC and SSDTC for MOO of complex chemical processes simulated using a process simulator, and to assess the applicability of CSTC and SSDTC in two evolutionary algorithms. For this study, Dividing-Wall Column (DWC) design, cumene production process, and biodiesel production process are optimized using both EMOO and IMODE programs having CSTC and SSDTC besides MNG as termination

59

The next section briefly describes EMOO and IMODE programs as well as the two termination criteria (CSTC and SSDTC). Section 3 provides details on the assessment of the two termination criteria and the two MOO programs for DWC design, cumene production process and biodiesel production process, respectively. Conclusions for this study are provided in the final section.

2. Multi-objective optimization and termination criteria The two MOO algorithms with termination criteria were coded using Visual Basic for Applications (VBA) in MS Excel. Further, MS Excel worksheets were used as user interface for providing the input and getting the results. See Sharma et al. (2012) and Sharma and Rangaiah (2013b) for algorithm and details of EMOO and IMODE programs, respectively. The main differentiating factor between the two programs is the MOO algorithm; EMOO is based on NSGA-II (Deb et al., 2002) whereas IMODE is based on differential evolution with taboo list (Srinivas and Rangaiah, 2007). In IMODE, taboo list allows for reduction in computational time by skipping (i.e., without evaluating) generated individuals that are close to previously evaluated individuals. Values of algorithm parameters in EMOO and IMODE are as follows (Sharma and Rangaiah, 2013b; Wong et al., 2016). Population size of 100 and maximum number of generations of 150 are used in both the programs. Other choices in EMOO are as follows: real coding, simulated binary crossover (SBX) with probability of 0.5, non-uniform mutation with probability of 0.01 and tournament selection. In IMODE, initial value for both crossover probability and mutation factor is 0.5, and both these are self-adapted during the search. These parameter values/choices in EMOO and IMODE are kept the same for solving three chemical processes presented later, in order to test their robustness. There are several inequality constraints in the formulated MOO problems for all three case studies. These inequality constraints are handled using the feasibility approach (Deb et al., 2002), which has been successfully used in a wide range of problems by many researchers. Performance-based termination criteria require suitable metrics for monitoring the progress or improvement in the optimal solutions. In the case of SOO, the objective function value can be used for this. In the case of MOO, there are usually many non-dominated (optimal) solutions, and all of them need to be assessed using suitable performance metrics. After analyzing a number of performance metrics, Sharma and Rangaiah (2013b) chose GD and SP based on their effectiveness to monitor the convergence and distribution of non-dominated solutions, and for their computational simplicity. Usually, GD is used to check the convergence of nondominated solutions obtained to the true Pareto-optimal front (van Veldhuizen and Lamont, 1998). Since true Pareto-optimal front is yet to be found in applications, Sharma and Rangaiah (2013b) modified GD to avoid using the true Pareto-optimal front by calculating it based on non-dominated solutions obtained from two consecutive generations. Modified GD, GDm , is calculated as follows:

criteria. Effectiveness of these two termination criteria in both EMOO and IMODE programs is evaluated. Results of this study show the benefit of CSTC and SSDTC for MOO of chemical processes independent of process simulator and evolutionary algorithm employed. Significant contribution of this study is comprehensive evaluation of two alternate search termination criteria in two MOO algorithms/programs on three chemical processes simulated using commercial simulators.

di = minSi − Sj 2 with respect to j=1,2,. . ., Ncurr

GDm =

1 Nprev

Nprev  i=1

d2i

(1)

(2)

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chemical engineering research and design 1 2 4 ( 2 0 1 7 ) 58–65

Here, Nprev and Ncurr represent number of non-dominated solutions in the previous and current generations, respectively. Si represents ith solution from the previous generation, whereas Sj represents jth solution from the current generation. Thus, di is the Euclidean distance between ith solution from the previous generation to the nearest solution in the current generation. Initially, SP was proposed for bi-objective problems to measure the distribution of non-dominated solutions obtained (Deb et al., 2002). It was modified by Zhou et al. (2006) for use for more than two objectives. Similar to GD, Sharma and Rangaiah (2013b) modified SP to avoid using the true Pareto-optimal front. Modified SP, SPm , is calculated as follows:

works on the ratio of two variances, Ri , determined by different methods on the same set of data (Cao and Rhinehart, 1995).

k = min Sk − Sj  2 with respect to j = 1, 2, . . ., Ncurr (i = / j) (3)

generation, as follows. First, the filtered value of PM is updated using the current value of PM and a suitable filter coefficient by:

Ncurr SPm =

k=1

¯ |k − |

initial value for ␦2 f,prev . These initial values are used in the next

CSTC uses GDm and SPm as the performance metrics (PM) to monitor search progress in each generation and terminate the search if the variance of each PM is below a specified value, which indicates insignificant change in the optimal solutions (Sharma and Rangaiah, 2013b). ␹2 -test is performed on the latest ten generations (i.e., ␭CSTC = 10), as follows.





Var PM1 , PM2 , . . ., PM␭CSTC (␭CSTC−1 ) ␦2PM

P (PM) = ␹2 [Chi (PM) , ␭CSTC − 1)]

PMf,curr = ␭1 PMcurr + (1 − ␭1 ) PMf,prev

(5)

(6)

Here ␦PM refers to the user-defined tolerance value for the standard deviation of PM, and P(PM) is the probability that ␹2 test is supporting the hypothesis that the variance of PM is less than ␦2PM . Sharma and Rangaiah (2013b) recommended the user-defined tolerance values to be: ␦GDm = 0.0003 and ␦SPm = 0.1. The evolutionary algorithm will terminate if ␹2 -test indicates that the probability of both P(GDm ) and P(SPm ) is greater than 0.99 in the same generation. SSDTC was formulated by Rhinehart (2014) to determine the convergence of stochastic SOO algorithms. In the testing in the leapfrogging algorithm, it works by detecting steady state in the worst player’s objective function value, and the termination is prompted when steady state is detected. Rhinehart (2014) concluded that SSDTC performed with better precision, and it is computationally efficient. Steady-state detection

(7)

The above value will be used in the next generation. For the current generation, variance of filtered values of the metric, PM is found by:



␯2f,curr = ␭2 PMcurr − PMf,prev

2

+ (1 − ␭2 ) ␯2f,prev

(8)

Then, variance of original (i.e., unfiltered) values of PM is estimated by: 2

¯ curr − PM ¯ prev ) + (1 − ␭3 )␦2 ␦2f,curr = ␭3 (PM f,prev

(9)

Finally, Ri is calculated using: Ri =

Termination criteria

Chi(PM) =

value of both PMcurr and PMf,prev . The variance is used as the initial value for ␯2f,prev , and twice the variance is used as the

(4)

¯ Ncurr 

Here k is the Euclidean distance of solution Sk in the current generation to the nearest solution in the set S of Ncurr non¯ is dominated solutions obtained in the current generation.  the mean of k for set S. Thus, SPm is based on non-dominated solutions in the current generation only. In the engineering optimization problems, different objective functions have different magnitudes, and so it is very important to normalize them suitably before calculating values of multi-objective performance metrics such as GDm and SPm . In this study, for calculating performance metrics, objective functions are normalized using the extreme values of each objective function from the non-dominated solutions obtained in the previous and current generations of the stochastic MOO search, and so each objective will be between 0 and 1.

2.1.

In the following equations for calculating Ri , PM refers to the mean value of calculated PM such as GDm and SPm in the latest 10 generations. Subscript f refers to the filtered value, and the subscripts curr and prev refer to the current and previous generations, respectively. ␭1 , ␭2 and ␭3 are user-defined filter coefficients. Initially, mean and variance of calculated PM for the latest 10 generations are determined. The mean is used as the initial

2 (2 − ␭1 )f,curr

␦2f,curr

In the above equations, the user-defined parameters are specified as ␭1 = ␭2 = ␭3 = 0.1 and Rcrit = 0.9, as suggested by Rhinehart (2014) and successfully used by Wong et al. (2016). Wong et al. (2016) tested both GDm and SPm for detecting steady state in SSDTC. They found that SPm was very fluctuating even when it has reached convergence, which made steady-state detection very difficult. Hence, they used only GDm as PM in SSDTC for termination after 45 generations due to potential premature termination. Results of Wong et al. show that SSDTC terminates reliably for four constrained benchmark functions. In the present study, effectiveness of both SSDTC and CSTC is investigated in both EMOO and IMODE for optimizing complex chemical processes simulated using Aspen Plus and Aspen HYSYS.

3.

Applications, results and discussion

3.1.

Application to dividing-wall column design

Distillation processes are one of the most energy-intensive separation processes, whose energy requirement increase with decreasing relative volatility while decreasing energy efficiency (Olujic et al., 2009). Huge amount of energy consumption in these processes has become a major concern for sustainable development of chemical processes. Also, distillation columns are large equipment with high capital investment (Dejanovic et al., 2010). Hence, there has been

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Fig. 1 – HYSYS block flow diagram for simulating DWC as two inter-connected columns. growing research for either improving or redesigning the system, to achieve sustainability while maintaining economic feasibility. This includes techniques such as heat pumps, heat integration and DWC. Of these, DWC is the focus of this study as it is gaining popularity in the recent years. The reason for this trend is that DWC is superior for separation of mixtures of three or more components, as it can minimize entropy of mixing that can occur in the conventional column configurations (Dejanovic et al., 2010). According to Kolbe and Wenzel (2003), DWC configuration can reduce 30% equipment cost and 40% land area in addition to 36% energy savings. It is also useful for retrofitting existing columns (Premkumar and Rangaiah, 2009; Lee et al., 2016). The present study considers MOO of DWC for separating a ternary alkane mixture of n-pentane, n-hexane and n-heptane, based on process data in Chew et al. (2014), who investigated the potential of heat pumps for DWCs but they did not conduct MOO. For the present study, DWC without any heat pump was simulated using Aspen HYSYS version 8.6, and then optimized using EMOO and IMODE programs. DWC has a vertical wall within a main column; stages located on one-side of the wall act as the pre-fractionator whereas a side product (besides distillate and bottoms products) is taken out on the other side of the wall. It is simulated as two inter-connected distillation columns as shown in Fig. 1 along with data on feed and three product streams; in this simulation, C1 is the pre-fractionator and C2 is the main column. The objective functions used in this design optimization are simultaneous minimization of both Total Capital Cost (TCC) and Utility Cost (UC), which are expected to be conflicting. Both these objectives are important, and the trade-off among the many optimal solutions found by MOO will be useful for choosing the most suitable optimal solution. Decision variables (DVs) and constraints are summarized in Table 1. Feed stage is specified as a fraction of pre-fractionator stages, counted from the top stage of the pre-fractionator; similarly, side product stage is given as a fraction of number of stages in the column middle (i.e., on the other side of the wall, where side product is withdrawn). Constraint on column height is due to structural integrity and challenges in erection, and con-

Table 1 – DVs and constraints for DWC design. DVs and their ranges

Constraints

10 < pre-fractionator stages < 90 10 < number of stages above the dividing wall < 30 −4 < difference between number of stages in the column middle and pre-fractionator stages < 10 10 < number of stages below the dividing column < 30 0.1 < feed stage < 0.9 0.1 < side product stage < 0.9 0.5 < reflux ratio < 1.5 250 < draw rate < 350 kgmol/h

Column height < 170 ft LMTDCondenser > 10 ◦ C LMTDReboiler > 10 ◦ C

straints on log mean temperature difference (LMTD) for the condenser and the reboiler are to ensure sufficient driving force for heat transfer (Table 1). For optimizing the DWC design, 5 runs of each program (EMOO and IMODE) were performed to ensure reliability of the results obtained. Pareto-optimal fronts, obtained in one selected run of each program, are plotted in Fig. 2. It can be seen that the non-dominated solutions found after satisfying SSDTC, CSTC and MNG are comparable and acceptable for industrial use. Although conflicting trend of TCC and UC is expected, the quantitative trade-off and many optimal solutions in Fig. 2 are useful to the engineers. DWC design corresponding to one of the optimal solutions with TCC in the range 4.8–4.9 million US$ is a good choice for implementation. Although there have been many studies on DWC in the literature, Fig. 2 shows MOO results for DWC design for the first time. Performance of CSTC and SSDTC in both EMOO and IMODE programs is discussed for the three applications together, later in Section 3.4.

3.2.

Application to cumene production process

Cumene (i.e., isopropyl benzene) is an important intermediate for the production of phenol and acetone, which are used to manufacture crucial polymers such as

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Table 2 – DVs and constraints for cumene production process (Flegiel et al., 2014). DVs and their ranges

Constraints

325 < benzene flow rate < 335 kmol/h 290 < reactor inlet temperature < 305 ◦ C 365 < reactor outlet temperature < 375 ◦ C 1160 < expander outlet pressure < 1260 kPa 250 < valve outlet pressure < 300 kPa 20 < cooler outlet temperature < 45 ◦ C 8000 < number of reactor tubes < 12,000 0.38 < column C1 reflux ratio < 0.48 0.0004 < column C1 bottom benzene mole fraction < 0.0006 165 < column C2 reflux rate < 180 kmol/h

LMTDFEHE > 10 ◦ C LMTDcooler > 10 ◦ C LMTDC1,reboiler > 10 ◦ C LMTDC2,reboiler > 10 ◦ C Tcooler > 2 ◦ C TC1,condenser outlet > 7 ◦ C F1 stream propene content < 0.1 Product purity > 99.85 mol% Benzene recycle flow to feed ratio < 4 CPR > 37,500 kg/h

Fig. 2 – Non-dominated solutions found after satisfying CSTC, SSDTC and MNG, for TCC versus UC of DWC design using EMOO (top plot) and IMODE (bottom plot) programs. polyamides (nylons), polycarbonates and phenolic resins (www.ineos.com/businesses/ineos-phenol/products). Flegiel et al. (2014) studied MOO of typical cumene production process with modifications for energy efficiency; however, they employed only EMOO program with MNG and not a performance-based termination criterion. In this process, feed is preheated by the reactor outlet stream, and then it enters the cumene reactor. The reactor outlet stream passes through an expander, heat exchanger and expansion valve before flowing into a flash tank for removing inerts. The liquid stream from the flash tank goes to the benzene column to recover and recycle unreacted benzene. The bottom stream from the benzene column is sent to the cumene column to produce cumene of specified purity. See Flegiel et al. (2014) for a detailed flow diagram of cumene production process with all important stream data. In the present study, cumene production process is simulated using Aspen HYSYS version 8.6, and then optimized using both EMOO and IMODE programs with CSTC and SSDTC besides MNG. For this optimization, the objective functions are simultaneous minimization of TCC and Material Loss (ML), with both normalised by cumene production rate (CPR). Here, TCC is an economic criterion and ML is an environmental criterion; both these together indicate the sustainability of the process. Table 2 summarizes DVs and constraints for the formulated MOO problem. The constraints are same as those in Flegiel et al. (2014), except for one additional constraint on product purity (>99.85 mol%) to allow optimizer to search for more solutions. Cumene production process was also optimized 5 times using EMOO and IMODE programs. Non-dominated solutions for TCC versus ML found after satisfying CSTC, SSDTC and MNG, in one run of these two programs (Fig. 3) shows that the solutions obtained after satisfying CSTC are marginally inferior compared to the other solutions. An optimal solution

Fig. 3 – Non-dominated solutions found after satisfying CSTC, SSDTC and MNG, for TCC/CPR versus ML/CPR of cumene process design using EMOO (top plot) and IMODE (bottom plot) programs. near to the corner, where TCC/CPR is about 2700 US$/(kg/h), is suitable for cumene process design.

3.3.

Application to biodiesel production process

Biodiesel is a renewable fuel that has gained significant interest recently. It is cleaner than petroleum diesel as it emits lower amount of carbon monoxide, particulate matter and unburned hydrocarbons (Morais et al., 2010). Biodiesel can be produced from Waste Cooking Oil (WCO), which is much cheaper than vegetable oil and is an environmental-friendly

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Table 3 – DVs and constraints for biodiesel production process (Patle et al., 2014). DVs and their ranges

Constraints

96 < WCO < 120 kt/annum 55 < TRFFA < 65 ◦ C 45 < TRTRANS1 < 60 ◦ C 45 < TRTRANS2 < 60 ◦ C 45 < TRTRANS3 < 60 ◦ C 1.5 < (residence time)RFFA < 2.5 h 1.5 < (residence time)RTRANS1 < 2.5 h

(Mass purity)Biodiesel > 0.99 (Mass purity)Glycerol > 0.95 TFRAC–1 < 150 ◦ C TFRAC–2 < 200 ◦ C TFRAC–3 < 250 ◦ C TFRAC–4 < 150 ◦ C (Methanol recovery)FRAC–1 > 0.98 (Methanol recovery)FRAC–2 > 0.98 (Methanol recovery)FRAC–3 > 0.98 (Methanol recovery)FRAC–4 > 0.98

1.5 < (residence time)RTRANS2 < 2.5 h 1.5 < (residence time)RTRANS3 < 2.5 h 2 < (feed stage)FRAC–1 < 7 2 < (feed stage)FRAC–2 < 9 2 < (feed stage)FRAC–3 < 10 2 < (feed stage)FRAC–4 < 9

option since WCO disposal can be detrimental to the environment (Sharma and Rangaiah, 2013c; Patle et al., 2014). In this study, biodiesel production from WCO by process 1 in Patle et al. (2014) is optimized for two objectives, by both EMOO and IMODE having CSTC and SSDTC besides MNG. See Patle et al. for a detailed flow diagram of this process with important stream data. Firstly, WCO is processed in an esterification reactor where free fatty acids react with methanol in the presence of acid catalyst. Then, one column is used to remove the catalyst, and another one is used to recover and recycle the unreacted methanol. After that, the treated oil is reacted with methanol in three trans-esterification reactors. In the downstream separation, one distillation column is used to separate glycerol and methanol, and another column to separate biodiesel and methanol. The unreacted methanol is recycled back to the trans-esterification reactors, whereas glycerol is taken out as a side product. Finally, neutralization and water wash units are used to produce fuel-grade biodiesel. Patle et al. (2014) have optimized the biodiesel process outlined above, using EMOO with only MNG as the stopping criterion. In the present study, the process is simulated in Aspen Plus version 8.6, and then optimized for simultaneously maximizing profit and minimizing total heat duty required in the entire process. Total heat duty is directly related to carbon

Fig. 4 – Non-dominated solutions found after satisfying CSTC, SSDTC and MNG, for profit versus total heat duty of biodiesel process design using EMOO (top plot) and IMODE (bottom plot) programs.

dioxide emissions; its trade-off with profit is relevant for developing sustainable processes. DVs and constraints for biodiesel production process are summarised in Table 3. Similar to the previous applications, biodiesel production process was optimized 5 times using EMOO or IMODE (i.e., ten runs in total). Non-dominated solutions obtained after satisfying CSTC, SSDTC and MNG in one run of each program are plotted in Fig. 4. Non-dominated solutions found after satisfying CSTC are slightly worse than those after satisfying SSDTC and MNG, irrespective of whether EMOO or IMODE is employed for optimization. Non-dominated solutions between profit and total heat duty are practically linear. In the absence of carbon dioxide tax on emissions, an optimal

Table 4 – Effect of termination criterion on TGN for the three applications by MOO using EMOO and IMODE programs; see Table 6 for the corresponding normalized GD values. Application

DWC

EMOO IMODE

Cumene process

EMOO

Biodiesel process

EMOO

IMODE

IMODE

Run 1

Run 2

Run 3

Run 4

Run 5

CSTC SSDTC CSTC SSDTC

87 126 88 141

94 NS 95 121

77 140 133 111

44 NS 135 105

75 NS 111 NS

CSTC SSDTC CSTC SSDTC

51 116 114 120

77 103 76 121

82 NS 74 93

61 129 122 NS

91 NS 104 115

CSTC SSDTC CSTC SSDTC

36 135 42 95

54 134 76 99

58 143 67 131

46 NS 57 125

38 NS 44 110

Note: NS in this table indicates that the termination criterion was not satisfied within MNG = 150.

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Table 5 – Values Ri in runs for cumene process where SSDTC was not satisfied. Program and run

Ri

Generation number

EMOO run 3 EMOO run 5 IMODE run 4

0.90079 0.92047 0.94663

130 120 141

solution towards high profit is likely to be chosen for implementation.

3.4.

Comparison between termination criteria

Termination Generation Number (TGN), where either CSTC or SSDTC is satisfied, is shown in Table 4, which covers the optimization of the three applications by both EMOO and IMODE. Values of TGN in this table indicate that CSTC terminates the search earlier than SSDTC, in almost all cases studied with EMOO and IMODE for the three applications; the two exceptions are Runs 3 and 4 of IMODE for DWC design. Further, CSTC is consistent in terminating the search without any failure compared to SSDTC, which was not satisfied in 2–4 out of 10 runs for each application (indicated as NS in Table 4). This non-satisfaction may be due to the use of low Rcrit of 0.9, which decreases the possibility of termination. This can be observed from Ri values near to Rcrit , shown in Table 5 for the three runs that failed to terminate by SSDTC for cumene process optimization. It seems SSDTC is very sensitive to small fluctuations in objective function values. Normalized GD value of non-dominated solutions found, after satisfying the particular termination criterion in a run, is the Euclidean distance (in the normalized objective function space) between these solutions and the solutions in the true Pareto-optimal front. For finding the true Pareto-optimal front, non-dominated solutions obtained in all ten runs (five each with EMOO and IMODE) at MNG are combined and sorted according to the non-dominance; this is carried out using an Excel-based program described in Sharma et al. (2017). This same procedure is followed for finding the true Pareto-optimal front of each application in this study. There are 88, 115 and 207 non-dominated solutions in the true Pareto-optimal front for DWC design, cumene and biodiesel processes, respectively. Normalized GD values at TGN in Table 4 (i.e., after satisfying CSTC and SSDTC) are shown in Table 6. These are all very small

in the range of 10−6 –10−4 . These values and Figs. 2–4 indicate that the non-dominated solutions found after satisfying CSTC and SSDTC are close to the true Pareto-optimal front. As expected, normalized GD value decreases with increasing TGN irrespective of the termination criterion, program and application. One exception to this is Run 1 of EMOO for biodiesel process (with GD of 1.61 × 10−4 and 1.92 × 10−4 for CSTC and SSDTC, respectively); this may be due to the change in the number of solutions and their distribution along the Paretooptimal front. Results in Figs. 2–4, Tables 4 and 6 indicate that CSTC is reliable in terminating the search and providing optimal solutions comparable to those after MNG but in fewer generations compared to SSDTC and MNG. Similar results were obtained for mathematical problems and heat exchanger design by Wong et al. (2016), who used only EMOO program with CSTC and SSDTC. Results in Table 4 indicate that CSTC can terminate the search by EMOO and IMODE much earlier than the given MNG of 150. Computational time taken for optimizing DWC, cumene process and biodiesel process by either EMOO or IMODE until MNG is 1, 2 and 3 days, respectively. Hence, CSTC is useful in significantly reducing the computational effort required for MOO of complex chemical processes modeled using process simulators, by stochastic global optimizers. Further work is required to enhance reliability of SSDTC.

4.

Conclusions

This study investigates the effectiveness of two termination criteria in evolutionary algorithms for MOO of three complex chemical processes simulated using Aspen HYSYS and Aspen Plus. The MOO was performed using the elitist nondominated sorting genetic algorithm (EMOO) and improved multi-objective differential evolution (IMODE). Both these programs include two search termination criteria, CSTC and SSDTC, based on the improvement in non-dominated solutions found with the progression of search. Results show that CSTC and SSDTC can timely terminate the search by both EMOO and IMODE programs well before the MNG, for optimizing the three complex chemical processes. Hence, CSTC and SSDTC can avoid unnecessary computations, where improvement in the non-dominated solutions is not significant. CSTC is found to be more reliable compared to SSDTC as it terminated both the programs in all runs for the three appli-

Table 6 – Effect of termination criterion on normalized GD values at TGN (in Table 4), for the three applications by MOO using EMOO and IMODE programs. Application

DWC

EMOO IMODE

Cumene process

EMOO IMODE

Biodiesel process

EMOO IMODE

Run 1

Run 2

Run 3

Run 4

Run 5

CSTC SSDTC CSTC SSDTC

1.07E − 4 7.87E − 5 8.09E − 5 6.62E − 5

7.03E − 5 NS 5.63E − 5 4.64E − 5

5.76E − 5 7.69E − 6 8.61E − 5 2.00E − 3

2.93E − 4 NS 6.60E − 5 1.15E − 4

1.15E − 4 NS 1.58E − 5 NS

CSTC SSDTC CSTC SSDTC

4.94E − 4 2.18E − 4 3.44E − 4 1.63E − 4

3.29E − 4 1.42E − 4 3.04E − 4 4.85E − 5

2.22E − 4 NS 2.77E − 4 2.81E − 5

2.84E − 4 3.61E − 5 1.83E − 4 NS

3.32E − 4 NS 1.91E − 4 4.03E − 5

CSTC SSDTC CSTC SSDTC

1.61E − 4 1.92E − 4 3.27E − 4 3.04E − 4

2.60E − 4 1.32E − 4 1.05E − 4 5.77E − 5

1.73E − 4 1.60E − 4 2.25E − 4 6.88E − 5

1.92E − 4 NS 1.82E − 4 6.43E − 5

3.12E − 4 NS 1.49E − 4 4.81E − 5

Note: NS in this table indicates that the termination criterion was not satisfied within MNG = 150.

chemical engineering research and design 1 2 4 ( 2 0 1 7 ) 58–65

cations. Finally, the non-dominated solutions obtained, after the termination of both programs using CSTC and SSDTC, are visually and quantitatively comparable to the Pareto-optimal front obtained at MNG. These confirm the effectiveness and usefulness of the two termination criteria in evolutionary algorithms for MOO of complex chemical processes.

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