Stochastic Optimization of a Natural Gas Liquefaction Process

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Article Cite This: Ind. Eng. Chem. Res. 2018, 57, 2200−2207

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Stochastic Optimization of a Natural Gas Liquefaction Process Considering Seawater Temperature Variation Based on Particle Swarm Optimization Hye Jin Yang,† Kyu Suk Hwang,† and Chang Jun Lee*,‡ †

Department of Polymer Science and Chemical Engineering, Pusan National University, 2, Busandaehak-ro 63beon-gil, Geumjeong-gu, Busan 46241, Republic of Korea ‡ Department of Safety Engineering, Pukyong National University, 45, Yongso-ro, Nam-gu, Busan 48513, Republic of Korea ABSTRACT: This paper presents a systematic stochastic optimization method for a dual mixed-refrigerant (DMR) process modeled with the simulator Aspen HYSYS. First, a base case design and an objective function are developed based on the simulator and an equation. Next, decision variables among many process variables are determined by a sensitivity analysis. Among the process variables, seawater temperature variation, which has a large impact on operation cost, is considered as a random variable. Since it is not possible to use a deterministic optimization solver for the simulator, a particle swarm optimization (PSO) technique, which employs a gradient-free optimization tool, is employed to solve a stochastic optimization problem. A case study shows the efficacy of the proposed algorithm. This method is general and can be applied to various processes modeled with commercial simulators. traditional cycles.6 It has been confirmed that a DMR process performs 15% better than C3 MR process and there are advantages of the flexibility of full gas turbine exploitation and the wider range of available temperatures.6,7 The DMR process consists of two mixed refrigerant cycles to improve the productivity of LNG. The first cycle is to precool the natural gas and condense the refrigerant of the second cycle, and its refrigerant is mainly mixed n-butane, ethane, and propane. The second cycle liquefies the natural gas to LNG, and its refrigerant consists of methane, ethane, propane, and nitrogen. The compositions of refrigerant and operational conditions for each cycle are different; thus they have to be optimized independently to achieve the most effective operational costs.1 LNG processes require very high energy and operational cost, since natural gas should be condensed into a liquid by reducing its temperature to around −160 °C. Therefore, the optimization of operational conditions for minimizing energy consumption is the most important point for LNG producers.1 However, the optimization problem of LNG processes is very challenging due to their high nonlinearities and various uncertain factors. There are many studies associated with the optimization of the liquefaction processes. In general, traditional approaches are to formulate the mathematical model of

1. INTRODUCTION Liquefied natural gas (LNG) has been widely used as a fuel for commercial, industrial, and residential purposes, since LNG has more economical advantages of storage and transportation than natural gas. Moreover, for offshore natural gas production, LNG floating production, storage, and off-loading units (FPSOs) have been actively developed and constructed for the past 2 decades. The advantages of LNG FPSOs are lower investment costs and a shorter construction period. Moreover, they can be redeployed to another gas field after the resources of the first gas field are exhausted.1−3 The liquefaction processes are the key section for determining construction and operational costs for LNG FPSOs. Therefore, they have to be properly constructed and operated for reducing risks and increasing project viability according to the special marine environments.4 There are three representative types of liquefaction processes for LNG FPSOs; the cascade, mixed refrigerant, and expander cycles.4 Most commercial liquefaction processes are based on these three types or the combination of these cycles. Representative processes are the pure-component cascade cycle, C3 MR (propane-precooled mixed-refrigerant) cycle, DMR cycle, single mixed-refrigerant cycle, mixed-fluid cascade process, compact LNG technology, integral incorporated cascade process, etc.5 Among various processes, C3 MR has been most widely used for a long time. However, a DMR process, which is one of the most advanced techniques, has been more widely used due to its better efficiency than other © 2018 American Chemical Society

Received: Revised: Accepted: Published: 2200

November 1, 2017 January 23, 2018 January 25, 2018 January 27, 2018 DOI: 10.1021/acs.iecr.7b04546 Ind. Eng. Chem. Res. 2018, 57, 2200−2207

Article

Industrial & Engineering Chemistry Research

Figure 1. Diagram of DMR process including dual cycles taken from Cha et al.12

of our research, which is to investigate the optimal operational conditions for a DMR cycle based on a stochastic optimization problem including uncertain variables. In this work, a stochastic optimization research is proposed based on the combination of a commercial simulator and an external optimization solver. Aspen HYSYS is used as a simulator, and a particle swarm optimization (PSO) technique, which is a gradient-free optimization tool and available in MATLAB, is employed as an external solver. First, the set of decision variables are populated randomly and they pass to the simulator. A simulator tests them, and the results of the objective function according to the set of decision variables are evaluated. They pass to an external optimization solver, and a PSO updates decision variables. And they pass to the simulator again. These iterations are performed until the optimal solution is investigated. The key point of this work is the combination of Aspen HYSYS and MATLAB to exchange decision variables and build a stochastic optimization model considering random variables. This paper is organized as follows. Sections 2 and 3 explain the basic model and how to determine decision variables and formulate a stochastic problem. Section 4 explains the PSO briefly, and section 5 describes case studies. Section 6 discusses the results of our stochastic model compared with the results of deterministic problem. Finally, section 7 gives concluding remarks.

the objective function and constraints based on a prior process knowledge or design process models on conventional process simulators integrating with the optimization tools of them. Lee et al. and Jensen handled a single mixed refrigerant cycle to optimize the compressor power based on the mathematical model.8,9 Venkatarathnam and Timmerhaus studied the optimal operational conditions of a few cycles including a DMR cycle by a sequential quadratic programming (SQP) solver of Aspen PLUS, in order to maximize the exergy efficiency.10 Kim et al. formulated the objective function of C3 MR cycles based on the process knowledge, and Cha et al. and Chang et al. handled the reverse Brayton cycle by the formulation of mathematical model.11−13 These previous approaches are to provide the global deterministic optimal solution. However, there may be many uncertain parameters in a process, which may lead to the deterministically identified optimal solutions not including real situations. Also, it is hard for conventional process simulators to generate a stochastic optimization problem including uncertainties. For example, even if there are seawater temperature variations that may have a significant impact on operational costs, previous studies do not consider them. Lee et al. handled a stochastic optimization problem including seawater temperature variations, which are very sensitive to the compressor power consumption in case seawater is used as the coolant for the CO2 liquefaction process.14 The seawater temperature becomes a critical factor in decreasing the operational energy of the CO2 liquefaction process, and the seawater temperature varies seasonally and even daily and also differs according to the seashore.14 Thus, this point also should be considered as generating a stochastic optimization problem for a DMR cycle, in which seawater is used as the coolant. Stochastic optimization approaches optimize the expectation value of the objective function considering the uncertainties and have been applied to various problems in computer-aided molecular design and process design under uncertainties.16−19 Thus, they can provide a decision that will give the best performance on average and their solutions can provide more probable optimal solutons than those of deterministic optimization problems.15 This reason motivates the objective

2. STOCHASTIC OPTIMIZATION OF THE DMR PROCESS 2.1. Base Case Design of the DMR Process. In this work, the DMR process is tested to verify the efficacy of the proposed approach. Figure 1 shows a diagram of the DMR process, which consists of two mixed refrigerant cycles, five expansion valves, four heat exchangers, three compressors, and two phase separators.10 This process is modeled in Aspen HYSYS, and the Peng−Robinson equation is employed to calculate the thermodynamic properties. Figure 1 illustrates that the DMR includes two refrigerant cycles: The precooling refrigerant, which consists of ethane, nbutane, and propane, cools the natural gas, the main refrigerant, 2201

DOI: 10.1021/acs.iecr.7b04546 Ind. Eng. Chem. Res. 2018, 57, 2200−2207

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Industrial & Engineering Chemistry Research and itself by circulating in the precooler cold box.10 The main refrigerant, which consists of nitrogen, methane, ethane, and propane, is cooled by the precooling refrigerant. Subsequently, this liquefies and subcools the natural gas and cools itself.10 During refrigeration, both refrigerants absorb the heat of evaporation and reject the atmosphere through water intercoolers.6 The efficacy of the DMR is determined by how to determine the operational temperatures of refrigerants, since more works are needed to reject heat at higher temperatures.6 Thus, the operational conditions of two refrigerant cycles should be optimized independently. In this study, process conditions for developing a base case design of the DMR are taken from Venkatarathnam and Timmerhaus.10 The objective function is defined as the total energy cost to operate the compressors and pumps in the process. The objective function is as follows: ̇ ̇ ̇ ̇ ẆTotal = WCompressor1 + WCompressor2 + WCompressor3 + WPump1 + ẆPump2

Figure 2. Results of sensitivity analysis. X is the percentage change from a set point, and the Y axis indicates the value of an objective function. 0% on X-axis is a set point taken from Hwang et al.20

(1)

subject to ̇ WCompressor1 = f1 ·h(T1 ,P1) + f12 ·h(T12 ,P12)

temperature should be considered as an uncertain variable in this study. The daily temperature records, which are the most detailed information, for 1 year are implemented to construct a stochastic optimization problem. In the next section, a method for building a stochastic optimization problem will be introduced.

̇ WCompressor2 = f11 ·h(T11 ,P11) + f10 ·h(T10 ,P10) ̇ WCompressor3 = f13 ·h(T13 ,P13) + f26 ·h(T26 ,P26) ẆPump1 = fsw1 ·h(Tsw1,Psw1) + fsw2 ·h(Tsw2 ,Psw2) ẆPump2 = fsw4 ·h(Tsw4 ,Psw4) + fsw5 ·h(Tsw5 ,Psw5)

(2)

3. SAMPLE AVERAGE APPROXIMATION Stochastic optimization techniques have been applied to various problems.21−24 They investigate the optimal decision variables to minimize or maximize the expected value of the objective function under scenarios generated by random variables. One computational difficulty to solving stochastic optimization problems may be that many problems involve a huge amount of scenarios generated by the combination of uncertain variables. This makes it prohibitively difficult and expensive to calculate the expectation values of the objective function. To tackle this limitation, there have been two major approximation schemes. The first scheme is based on the decomposition of sample spaces, and the second one is based on sampling data points. 15 The first is to adopt a decomposition of feasible sample spaces.25 Subsequently, the sample spaces are converged into narrow spaces for ensuring optimal solutions. However, the availability of the convexity, monotonicity, and differentiability of an objective function and constraints should be guaranteed.25 In this study, a process modeling is performed based on Aspen HYSYS in which governing equations are not revealed; thus the convexity, monotonicity, and differentiability of constraints cannot be established. The second scheme is based on sample average approximation (SAA) in which scenarios are generated by the combinations of sampling points based on Monte Carlo sampling (MCs) techniques and the optimal solutions is to minimize or maximize the expected objective value. This can be formulated follows:

In eq 1, Ẇ Total is the total required power for LNG liquefaction. In eq 2, f, T, and P denote the flow rate, temperature, and pressure of the corresponding stream number, and the enthalpy h is a function of T and P. 2.2. Determining Decision Variables and Random Variable. To determine a set of decision variables, sensitivity analysis is subsequently performed to evaluate to what extent each decision variable has an effect on the objective function and identify which variables are significant or negligible. To consider the scales of all variables, the sample points of each manipulated variable along equally spaced intervals are selected and tested. The changes of the objective function value are investigated according to ±10% changes (−10%, −8%, −6%, −4%, −2%, 2%, 4%, 6%, 8%, and 10%) of a set point on a specified process variable while other process variables are fixed to an assumed set point. Figure 2 shows the results of sensitive analysis, and the impact of each manipulated variable is summarized in Table 1. It can be concluded that eight variables among all the manipulated variables have a significant effect on the objective function, while other variables have a negligible impact. Moreover, mole fractions of precooling refrigerant (ethane, n-butane, and propane) and main refrigerant (nitrogen, methane, ethane, and propane) have to be determined by an optimization. Two and three mole fractions of each refrigerant should be determined. Therefore, a total of 13 variables are selected as a set of decision variables to design a stochastic optimization problem. In the DMR process, the seawater temperature is the most significant factor for determining the operating cost. However, the seawater temperature varies seasonally and even daily. Thus, these temperature variations make the optimization of the DMR process difficult. To reflect these, the seawater

1 fN̂ (Q ) = N 2202

N

∑ f (Q ,εk) k=1

(3) DOI: 10.1021/acs.iecr.7b04546 Ind. Eng. Chem. Res. 2018, 57, 2200−2207

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Industrial & Engineering Chemistry Research Table 1. Results of Sensitivity Analysis

a

decision variables

set point ± 10%

absolute average slopea [W/%]

rank

flow rate of the precooling refrigerant ( f 9) flow rate of the main refrigerant ( f 20) pressure in the precooling refrigerant (P1) pressure in the precooling refrigerant (P6) pressure in the precooling refrigerant (P9) pressure in the main refrigerant (P10) pressure in the main refrigerant (P20) temperature in the precooling refrigerant (T9) temperature in the main refrigerant (T20) temperature of seawater (Tsw) temperature of NG after precooling (T23)

0.913 ± 0.0913 g mol/s 1.0 ± 0.1 g mol/s 19.2 ± 1.92 bar 2.8 ± 0.28 bar 7.6 ± 0.76 bar 48.6 ± 4.86 bar 3 ± 0.3 bar 308.8 ± 30.88 K 234.3 ± 23.43 K 240 ± 24.0 302.1 ± 30.21 K

72.80 71.41 74.51 5.35 0.0 48.32 0.0 313.816 124.03 342.32 K 0.0

5 6 4 8 7 2 3 1

This means the absolute average slope between total required power (Y) and percentage change from a set point (X) in Figure 2.

where f and εk are the objective function and each scenario, respectively. N and Q are the total number of scenarios, which is generated by the combination of uncertain variables, and a set of decision variables, respectively. This means that the stochastic optimal solution will show the best performance on average under all scenarios. In these techniques, the probability distributions governing uncertain variables are previously assumed and sampling points for them are selected to generate scenarios.15 However, this can suffer from the number of scenarios, since this can grow exponentially according to the number of uncertain variables. In addition, it is uncertain whether the solutions obtained from SAA are really global optimal ones. The stochastic optimal solution based on SAA provides more realistic solutions if sufficiently large scenarios can be used.26 Lee et al. employed SAA to generate a stochastic model for solving a catalyst optimization problem.15 Additionally, Chouinard et al. used SAA to solve the supply loops design problem under several scenarios for recovery, demand, and processing volumes.27 These studies show that SAA can be advantageously used for solving stochastic problems based on unknown or complex process models, requiring only calculations according to process input variables. In this study, SAA is used to design a stochastic optimization problem since mathematical equations governing a process in Aspen HYSYS are not revealed. This means the convexity, monotonicity, and differentiability of a process model are not available. SAA is only used for handling the expectation operator, and an additional approach has to be introduced to investigate a stochastic optimal solution. Since there is no way to generate a stochastic problem combining a process model on Aspen HYSYS with a gradient-based optimization model, PSO, which is one of heuristic optimization methods and has advantages in handling complex equations without evaluating derivative, is employed. PSO can be easily combined with a simulator and applied in handling complex equations or commercial simulators.

Furthermore, there are a relatively small number of parameters to be determined as being compared with other population-based sampling optimization techniques.30 Detailed steps of the PSO technique are described in the research of Schwaab et al.30 The PSO parameter values in this study are listed in Table 2. Table 2. PSO Parameter Values Used To Find a Stochastic Solutiona

a

parameter

value

number of iterations, i number of particles number of search dimensions weight parameters

498 200 12 2 × (0.95)i × (2r − 1)

r denotes a random variable between [0, 1].

5. OBJECTIVE FUNCTION In this study, the stochastic optimization problem is designed to find the optimal solutions minimizing the expected value of an objective function in all scenarios generated by a random variable. For example, if 200 sets of decision variables are generated randomly for PSO and daily average seawater temperatures are used, 73,000 simulations should be performed for one iteration and the expected value of all simulations can be evaluated by calculating the average value of 73,000 results. However, this number of simulations requires massive computational efforts and time. To reduce the number of scenarios, a logical method should be implemented. Also, in SAA, MCs technique, which chooses randomly sampling points of uncertain variables, is used. However, if a small sample size is selected, MCs may not have good uniformity.31 Hammersley points can be used to uniformly sample a high-dimensional hypercube, and it is reported that a Hammersley sequence sampling technique can reduce the number of required sample points for a given accuracy of estimates to about 3−100 times on average compared to MCs techniques.32 This is powerful for handling a large number of uncertain parameters without any correlations. However, since there is only one uncertain variable in this study, it is not necessary to employ Hammersley sequence sampling technique. This study selects the monthly average seawater temperatures, since these values are the best to reflect daily temperatures and a good uniformity can be achieved. This means that the representative values of daily seawater temperatures are introduced and the total number of

4. PSO PSO is a population-based sampling optimization technique developed by Eberhart and Kennedy.28 The PSO technique is initialized with a population of particles randomly positioned in n-dimensional search spaces, and each particle updates its position and velocity according to historical optimal positions.29 The most attractive feature of a PSO technique is that this does not need derivatives of mathematical equations; thus this can be easily applied to complex or unknown models. 2203

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Figure 3. Overall scheme of stochastic optimization procedure in this study.

Table 3. Results of Stochastic and Deterministic Optimization Conditions design variable

stochastic solution

deterministic solution

flow rate of precooling refrigerant ( f12) flow rate of main refrigerant ( f 26) pressure in precooling refrigerant (P1) pressure in precooling refrigerant (P9) pressure in main refrigerant (P13) temperature in precooling refrigerant (T12) temperature in main refrigerant (T26) mole fraction of ethane in precooling refrigerant mole fraction of propane in precooling refrigerant mole fraction of n-butane in precooling refrigerant mole fraction of nitrogen in main refrigerant mole fraction of methane in main refrigerant mole fraction of ethane in main refrigerant mole fraction of propane in main refrigerant

0.85 g mol/s 0.90 g mol/s 17.92 bar 2.99 bar 44.18 bar 297.25 K 230.34 K 0.248 0.073 0.678 0.325 0.397 0.076 0.201

0.80 g mol/s 0.90 g mol/s 17.0 bar 3.10 bar 42.14 bar 307.50 K 223.70 K 0.235 0.106 0.658 0.303 0.403 0.069 0.223

simulations can be cut off by 1/12. When 200 sets of decision variables are evaluated and updated, 2400 simulations should be performed for each iteration. The objective function of a stochastic optimization problem is as follows: min W (Q ) = [ẆTotal(Q ,ε)] =

hundred sets of operational conditions in the DMR process were randomly generated over ±10% changes of the nominal value on the set-point of decision variables and updated in each iteration step. Monthly average seawater temperatures were sampled; the total number of simulations was 2400 for one iteration. These simulations were performed on AMD Athlon 2.9 GHz. The CPU time was 2160 s for one iteration of a stochastic optimization problem. A drawback of this study is that a global stochastic optimal solution cannot be guaranteed. To overcome this limitation, many simulations have been tested, as many as possible. A deterministic solution based on a yearly average seawater temperature is also investigated by a SQP solver on Aspen HYSYS for the comparison. The stochastic and deterministic optimal solutions are listed in Table 3. Figure 4 shows the minimum operational conditions at each iteration. To verify the efficacy of stochastic optimization, the solutions of the stochastic and a deterministic optimization problems were tested again with daily average seawater temperatures. It is assumed that this process is installed around East Sea in Korea and actual seawater temperatures in 2016 are used.33 The stochastic and deterministic solutions are given below:

12 ⎤ 1 ⎡⎢ ∑ ẆTotal(Q ,εk)⎥ ⎥⎦ 12 ⎢⎣ k = 1

(4)

In eq 4, εk is the kth month average seawater temperature and W is the total operating cost under a set of decision variables, Q. [W (Q ,ε)] is the expected value of the objective function evaluated by monthly average seawater temperatures, which are representative values of random variables. In this study, 200 sets of decision variables are first randomly generated and their expected objective function values are evaluated. These are updated via PSO for each iteration, and the stochastic optimal solution is finally be investigated. In this case, 2400 simulations should be performed for each iteration. Figure 3 shows the overall scheme of a stochastic optimization procedure in this study.

6. COMPUTATIONAL RESULTS In this section, the results of stochastic optimization are verified. First, seawater temperatures were considered a random variable for building a stochastic optimization problem. Two2204

DOI: 10.1021/acs.iecr.7b04546 Ind. Eng. Chem. Res. 2018, 57, 2200−2207

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Figure 5. Comparison of total required power for DMR cycle with operational conditions obtained from deterministic solution and stochastic solution. The red and green lines indicate the values of an objective function for the stochastic and deterministic solutions. A blue line is the temperature of seawater.

Figure 4. Minimum value of an objective function for each iteration of a stochastic optimization.

Q d ∗ = argQ max [ẆTotal(Q ,(ε))] deterministic solution Q s ∗ = argQ max [ẆTotal(Q ,ε)]

hypothesis value, the confidence level was fixed to 99%, i.e., the T-score, t0.01/2 , was 2.58 at a 99% confidence level. The value of H0 was −57.082 ± 6.888 and the minimum difference was − 50.194, which means that a stochastic optimal solution can save at least 4.5% of a total operating cost as compared with a deterministic one. Therefore, the stochastic solution achieved a significant improvement. It is worth nothing that this difference shows the efficacy of the stochastic solution. Figure 6 shows the difference between the operating costs of stochastic and deterministic cases in all scenarios.

stochastic solution (5)

As presented in Table 4, the total average operating cost of the stochastic optimal solution is lower than that of the Table 4. Comparison of Stochastic and Deterministic Optimization Solutions result average value of objective function maximum value of objective function minimum value of objective function standard deviation

stochastic solution

deterministic solution

10101.058 [W] 10887.670 [W]

10158.140 [W] 11135.632 [W]

9732.049 [W]

9771.243 [W]

318.661

369.669

deterministic optimal solution. Figure 5 shows the results of stochastic and deterministic optimal solutions under daily average seawater temperatures. Since a deterministic problem ignores a random variable, the operating cost is biased upward when considering random variables. To verify the benefit of a stochastic solution, the value of the stochastic solution (VSS) is introduced as follows: VSS = ε[ẆTotal(Q s *,ε)] − ε[ẆTotal(Q d *,ε)] = 10101.058 − 10158.140 = −57.082

(6) Figure 6. Differences of total power requirement between stochastic and deterministic solutions. The negative value means that a stochastic solution provides a better result than a deterministic one.

In this study, VSS is negative, which indicates that the average performance of a stochastic solution is better than that of a deterministic solution. To calculate the exact difference between the two solutions, a t-test, which is used to evaluate the statistical difference in the case of paired samples, was employed. −t0.01/2 ≤ t =

VSS − H0 sd N

7. CONCLUSION Previous research on DMR processes has overlooked seawater temperature variations, which are the most significant factor among process variables. In this study, the stochastic optimization problem including seawater temperature variations was considered and solved. First, the DMR process was modeled on Aspen HYSYS and the objective function

≤ t0.01/2 (7)

H0 and sd are the hypothesis value and the standard deviation of the difference between the two solutions, respectively. Additionally, N is the number of scenarios. To obtain an acceptable 2205

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(7) Husnil, Y. A.; Lee, M. Synthesis of an optimizing control structure for dual mixed refrigerant process. J. Chem. Eng. Jpn. 2014, 47, 678−686. (8) Lee, G. C.; Smith, R.; Zhu, X. X. Optimal synthesis of mixed refrigerant systems for low temperature processes. Ind. Eng. Chem. Res. 2002, 41, 5016−5028. (9) Jensen, J. B. Optimal operation of refrigeration cycles. Ph.D. Thesis, Faculty of Natural Sciences and Technology, Department of Chemical Engineering, NTNU, Tronheim, Norway, 2008. (10) Venkatarathnam, G.; Timmerhaus, K. D. Cryogenic Mixed Refrigerant Processes; Springer: New York, 2008. (11) Kim, H. J.; Park, C. K.; Lee, J. Y.; Kim, W. B. Optimal degree of superheating of the liquefaction process for application of LNG FPSO. Proceedings of Annual Spring Meeting, The Society of Naval Architecture of Korea; 2010; pp 961−965. (12) Cha, J. H.; Lee, J. C.; Roh, M. I.; Lee, K. Y. Determination of the optimal operational condition of the Hamworthy mark I cycle for LNG-FPSO. J. Soc. Nav. Archit. Kor. 2010, 47, 733−742. (13) Chang, H. M.; Chung, M. J.; Kim, M. J.; Park, S. B. Thermodynamic design of methane liquefaction system based on reversed-Brayton cycle. Cryogenics 2009, 49, 226−234. (14) Lee, S. G.; Choi, G. B.; Lee, J. M. Optimal design and operational conditions of the CO2 liquefaction process, considering variations in cooling water temperature. Cryogenics 2009, 49, 226−234. (15) Lee, S. G.; Choi, G.; Lee, J. M. Stochastic nonlinear optimization for robust design of catalysts. Ind. Eng. Chem. Res. 2015, 54 (51), 12855−12866. (16) Halemane, K. P.; Grossmann, I. E. Optimal process design under uncertainty. AIChE J. 1983, 29, 425−433. (17) Maranas, C. D. Optimal molecular design under property prediction uncertainty. AIChE J. 1997, 43, 1250−1264. (18) Tayal, M.; Diwekar, U. Novel sampling approach to optimal molecular design under uncertainty. AIChE J. 2001, 47, 609−628. (19) Shastri, Y.; Diwekar, U. An efficient algorithm for large scale stochastic nonlinear programming problems. Comput. Chem. Eng. 2006, 30, 864−877. (20) Hwang, J. H.; Roh, M. I.; Lee, K. Y. Determination of the optimal operational conditions of the dual mixed refrigerant cycle for the LNG FPSO topside liquefaction process. Comput. Chem. Eng. 2013, 49, 25−36. (21) L’Ecuyer, P.; Haurie, A. Preventive replacement for multicomponent systems: An opportunistic discrete-time dynamic programming model. IEEE Trans. Reliab. 1983, R-32, 117−118. (22) Maranas, C. D. Optimal molecular design under property prediction uncertainty. AIChE J. 1997, 43, 1250−1264. (23) Batcho, P. F.; Case, D. A.; Schlick, T. Optimized particle-mesh Ewald/multiple-time step integration for molecular dynamics simulations. J. Chem. Phys. 2001, 115, 4003−4018. (24) Shaitan, K. V.; Tourleigh, Y. V.; Golik, D. N.; Kirpichnikov, M. P. Computer-aided molecular design of nanocontainers for inclusion and targeted delivery of bioactive compounds. J. Drug Delivery Sci. Technol. 2006, 16, 253−258. (25) Chiang, M. Balancing transport and physical layers in wireless multihop networks: Jointly optimal congestion control and power control. IEEE J. Sel. Areas Commun. 2005, 23, 104−116. (26) Jirutitijaroen, P.; Singh, C. Reliability constrained multi-area adequacy planning using stochastic programming with sample-average approximations. IEEE Trans. Power Syst. 2008, 23, 504−513. (27) Chouinard, M.; D’Amours, S.; Aït-Kadi, D. A stochastic programming approach for designing supply loops. Int. J. Prod. Econ. 2008, 113, 657−677. (28) Eberhart, R.; Kennedy, J. A new optimizer using particle swarm theory. MHS ’95, Proceedings of the Sixth International Symposium on Micro Machine and Human Science; IEEE, 1995; pp 39−43, DOI: 10.1109/MHS.1995.494215. (29) Chen, W. N.; Zhang, J.; Lin, Y.; Chen, N.; Zhan, Z. H.; Chung, H. S.; Li, Y.; Shi, Y. H. Particle swarm optimization with an aging leader and challengers. IEEE Trans. Evol. Comput. 2013, 17, 241−258.

evaluating the total energy, including the compressors and pumps in the process, was defined. Next, sensitivity analysis was performed to determine a set of decision variables that have a large impact on an objective function. The main goal of this study is to find the optimal set of decision variables that lead to minimization of the objective function. To generate the stochastic optimization problem, the SAA technique in which the expected objective value in all possible scenarios was employed, since the process model on Aspen HYSYS is not revealed. In this study, monthly average seawater temperatures were sampled to reduce the number of scenarios. To solve this stochastic problem, the PSO technique, which can easily be applied without derivatives of mathematical equations, was employed. To verify the efficacy of the proposed method, VSS was evaluated under daily average seawater temperatures. The result shows that the average performance of a stochastic solution was improved. Therefore, this study can contribute to optimizing processes that include random variables as well as achieve significant improvement. In addition, this study shows how to generate the stochastic optimization problem and combine the simulation results based on commercial simulators with the PSO technique.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +82-51-629-6465. Fax: +82-51-629-6463. ORCID

Chang Jun Lee: 0000-0003-2275-165X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by the Young Researcher Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (Grant NRF2015R1C1A1A02036567).

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DOI: 10.1021/acs.iecr.7b04546 Ind. Eng. Chem. Res. 2018, 57, 2200−2207

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DOI: 10.1021/acs.iecr.7b04546 Ind. Eng. Chem. Res. 2018, 57, 2200−2207