3602
Ind. Eng. Chem. Res. 2007, 46, 3602-3609
PROCESS DESIGN AND CONTROL Multi-objective Particle Swarm Optimization Hybrid Algorithm: An Application on Industrial Cracking Furnace Chengfei Li, Qunxiong Zhu,* and Zhiqiang Geng School of Information Science and Technology, Beijing UniVersity of Chemical Technology, Beijing 100029, People’s Republic of China
In this paper, a new multi-objective particle swarm optimization (MOPSO) procedure, based on the Pareto dominance hybrid algorithm, is proposed and applied in a naphtha industrial cracking furnace for the first time. Pareto dominance is incorporated into particle swarm optimization (PSO). Our algorithm takes the Pareto set as a repository of particles that is later used by other particles to guide their own flight. In addition, an MOPSO and artificial neural network (ANN) hybrid model is applied in the operation optimization of a naphtha industrial cracking furnace. Therein, sensitivity analysis is investigated and taken as the basis on which decision variables of multi-objective problem base. From both theoretical computation and practical application, the validity and reliability of proposed algorithm are verified by two test functions studied, and actual application example of the optimization of operation parameter of cracking process. Moreover, the yields of ethylene and propylene are improved. I. Introduction There are many approaches to solve multi-objective optimization problems. The traditional method takes the weighted average of all of the objectives and treats multi-objective optimization problems as a single objective problem. However, the solutions are dependent on the chosen weights, which are subject to individual perception and knowledge of the process. Therefore, this method is deficient and cannot solve real multiobjective optimization problems. Currently, the best way to solve multiple optimization is through the generation of Pareto-optimal sets, which provides a spectrum of tradeoffs of the competing objectives.1,2 All of the solutions in the Pareto-optimal set are equally good. Evolution algorithms, such as the genetic algorithms (GAs), have been determined to be efficient in generating a Paretooptimal set and are suitable for solving multi-objective optimization problems. Schaffer studied multi-objective optimization problems using GAs in the 1980s. GA operators, which are designed to be applied in binary numbers, require binary coding for the real values of decision variables. However, it has been noted that the representation of real numbers with binary coding leads to many difficulties, such as the fact that finite-length binary strings are unable to achieve high precision in the decision variables. Moreover, the transition to a neighboring point requires the alteration of many bits, which, in turn, hinders the gradual search in the continuous search space (knowns as the Hamming cliff problem).3 Although realcoded NSGA-II with a simulated binary crossover (SBX) operator currently is studied and is able to perform as good as, or even better than, the binary coded GAs,4 it is difficult for GA coding in the large-scale nonlinear constraint problem. Moreover, crossover operation search ability is relatively * To whom all correspondence should be addressed. E-mail address:
[email protected].
weak, which makes it difficult to reflect environmental individual density information by Pareto sorting. The search performance of GAs is restricted. The aforementioned disadvantage affects the NSGAII solution’s diversity and convergence. Particle swarm optimization (PSO) is an evolutionary computation technique that is based on socially organized colonies and were developed by Kenney and Eberhart in 1995.5,6 It is also simple and easy to be understood, and it has been applied in many optimization problems.7 Similar to the other evolutionary algorithm, PSO is similar to the GAs and can be used to solve the multi-objective optimization problems.8-11 In this paper, a modified multi-objective particle swarm optimization (MOPSO) is presented and applied to the multi-objective optimization study of a naphtha cracker unit. Synchronously, a cracking model is described, using an artificial neural network (ANN) model. The MOPSO hybrid algorithm is used as the basis for multi-objective optimization of an ethylene reactor. This paper is organized in the following four parts. Section I introduces the problem and the approach we are going to take. Section II introduces MOPSO briefly, and then it presents the description of the approach. Section III defines the modeling of the cracking reactor and the multi-optimization problems that we intend to solve. It then introduces MOPSO and the ANN hybrid algorithm. Finally, it presents the experimental results and discussions, some conclusions are established, and some possible paths for future research are provided. The conclusion is presented in Section IV. II. Algorithms 1. Introduction. PSO, similar to the other evolutionary algorithm, randomly generates an inertia swarm. It takes the random candidate solution as the inertia solution (called the “particle swarm”). Every particle moves iteratively at the random velocity in the problem search space. The particle always moves
10.1021/ie051084t CCC: $37.00 © 2007 American Chemical Society Published on Web 04/26/2007
Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007 3603
to the local best position and to the global best position. This process is described as the following formula:
Vk+1 ) ωVk + c1r1k (p1k - xk) + c2r2k (p2k - xk)
(1)
xk+1 ) xk + Vk+1
(2)
where ω is the inertia weight. The weight values experience changes between 0 and 1.2.11 The parameters c1 and c2 are called cognitive and social parameters, respectively; values of c1 ) c2 ) 2, c1 + c2 e 4 and c1 > c2 are proposed. The parameters r1 and r2 are random numbers that are uniformly distributed within [0,1].12 The parameters V and x are the particle current velocity and position, respectively, and k is the iteration time. The particle velocity is updated according to the current velocity and the parameters. The particle is allured to the best position p1 and the global position p2 by itself. The particle-updating position is dependent on the velocity Vk+1 and the current position xk. In the engineering, the object is to have the following model characteristics: complicated, highly dimensional, non-convex, nonlinear, nondifferentiable. There are numerous local minimums in the objective function. Moreover, there is more than one competing objective. Here, PSO is used in parallel with traditional industrial optimization methods, and PSO searched a much larger portion of the problem space than the traditional method in nature. PSO is shown to be robust, the algorithm is simple and is global optimization, so it is applied in the complicated optimization problems that require global optimization. It is good at identifying a promising region through global exploration search and it can locate a global optimum in a single run. In conclusion, in comparison with a conventional deterministic optimization approach, direct/stochastic methods such as PSO have the following advantages: (1) It does not require additional assumptions for optimization problems and calculating derivatives of the optimization function. Moreover, it particularly suited for problems with ill- or unknown structures.13 (2) The classical optimization algorithm converges quickly and has an exact solution; however, it is sensitive to the inertia solution, and generation of the local solution is easy. Some global optimization, such as GA and simulated annealing (which is limited, in regard to both mechanism and structure), encounters difficulty in optimizing a complicated object.14 Moreover, it managed to reach optimum fitness using fewer fitness evaluations and less computation time than the other algorithms. In comparsion with other genetic algorithms, the PSO algorithm is simple and easy to perform; in addition, it can search the solution in solution space, and the range of the searching the solution is wide. Trapping into the local minimum is especially difficult, and it is a parallel global stochastic algorithm.15-19 Non-Dominated Sorting in Genetic Algorithms ( NSGA)20 is a popular non-dominationally based genetic algorithm for multi-objective optimization. It is a very effective algorithm but has been generally criticized for its computational complexity, lack of elitism, and choice of optimal parameter value for sharing the parameter share. A modified version, NSGA-II,21 was developed, which has a better sorting algorithm, incorporates elitism, and does not require any sharing parameter to be chosen a priority. 2. Pareto Optimization.8 This section presents several definitions used in this work. Definition 1 involves the general multi-objective optimization problem. For this definition, one must find the vector b x* ) [x*1, x*1, ..., x*1]T which will satisfy the m inequality constraints gi(x b) g 0 (for i ) 1, 2, ..., m), the p
Chart 1. Flowchart of MOPSO Hybrid Algorithm
equality constraints hi(x b) ) 0 (for i ) 1, 2, ..., p), and will optimize the vector function
B(x f b) ) [f1(x b), f2(x b), ..., fk(x b)]T where
b x ) [x1, x2, ..., xn]T is the vector of decision variables. Definition 2 involves Pareto optimality. A point b x* ∈ Ω is Pareto optimal if, for every b x ∈ Ω and I ) {1, 2, ..., k}, either ∀i∈I (fi(x b) ) fi(x b*)) or there is at least one i ∈ I such that fi(x b) > fi(x b*). Definition 3 involves Pareto dominance. A vector b u ) (u1, ..., uk) is said to dominate b V ) (V1, ..., Vk) (denoted by b ue b V if and only if u is partially less than V, i.e.,
∀ i ∈ {1, ..., k} (for ui e Vi) ∧ ∃ i ∈{1, ..., k} (for ui < Vi) Definition 4 involves the Pareto-optimal set. For a given multi-objective optimization problem (MOP) B(x), f the Paretooptimal set (Ρ*) is defined as
Ρ* :) {x∈ Ω| ∃ x′ ∈ Ω B(x′) f e B(x)} f
3604
Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007
Definition 5 involves the Pareto front. For a given MOP B(x) f and Pareto-optimal set Ρ*, the Pareto front (ΡF*) is defined as
ΡF* :) {u b ) Bf ) (f1(x), ..., fk(x))|x ∈ Ρ*} The real-life optimization problems faced in industry usually involve more than one competing objective. The solution of a MOP is obtained by generating a Pareto-optimal set, which provides a spectrum of tradeoffs of the competing objectives. A Pareto-optimal set provides a wide range of design and operational options to designers and practitioners, and, hence, enhances the possibility of finding more-efficient processes.21 3. Execution of the Algorithm. The flowchart of the MOPSO algorithm is presented in Chart 1. Execution of the algorithm requires eight steps: Step 1: Initialize the particle swarm. Designate the population volume (n), generate speed (Vi), and position (Xi) of each particle randomly. Step 2: Evaluate the fitness of each particle, which are recorded as the parameters Fitness1 and Fitness2. These values are determined using the two objective functions F1(x) and F2(x). Step 3: Calculate the best individual solutions pbest1 and pbest2, and then calculate the best global solutions gbest1 and gbest2. Step 4: Evaluate gbest with the average of gbest1 and gbest2. Evaluate pbest with the average of pbest1 and pbest2. Step 5: Put pbest1 and pbest2 into the Pareto set P and sort pbest1 and pbest2 in a Pareto sequence. Other bad Pareto data are discarded. Step 7: Update the particle position Xi and velocity Vi, according to eqs 1 and 2. Step 8: Compare the particle position Xi with the Pareto data. If the particle position belongs to the Pareto data, place it into the Pareto set P; otherwise, discard it. We update the Pareto until the iteration stops. If the termination conditions have been achieved (maximum number of cycles has been reached), then stop; otherwise, go to step 2. III. Examples 1. Comparison between New MOPSO and GA. We use the following test function to obtain results from new MOPSO and NSGAII algorithms. (a) Test Function 1.
Min f1 ) x2 Min f2 ) (x - 2)
Figure 1. Pareto fronts produced by new (+) MOPSO and (/) NSGAII algorithms: test function 1.
Figure 2. Pareto fronts produced by new (+) MOPSO and (/) NSGAII algorithms: test function 2.
(b) Test Function 2.
Min f1 ) 1 - e-4x1 sin6(6πx1)
[ ( )]
Min f2 ) g(x) 1 g(x) ) 1 + 9
f1(x)
2
g(x)
(∑ ) 6
xi
i)1
4
0.25
0 e xi e1 The Pareto curve is shown in Figure 2. In the simulation, 100 particles and 100 iterations are used. This figure shows that the number of MOPSO Pareto solutions is greater than the number of NSGAII Pareto solutions. The computation time in the new MOPSO algorithm is ∼16 min, and the computation time in NSGAII is ∼20 min in the same computer system scheme. The computation time in the new MOPSO algorithm is faster than that in the NSGAII algorithm. (c) Test Function 3.
Min f1(X) ) x1 2
x ∈ [-5,7] The Pareto curve is shown in Figure 1. In the simulation, 100 particles and 100 iterations are used. This figure shows that the diversity of the MOPSO’s Pareto solutions is similar to the diversity of the NSGAII’s Pareto solutions. The computation time in the new MOPSO algorithm is ∼3 min and the computation time in NSGAII is 7 min in the same computer system scheme. Both the solutions of MOPSO and the solutions of NSGAII are similar to the Pareto solutions.
Min f2(X) ) 1 - h(X) n
g(X) ) 1 + 10(n - 1)+
(xi2 - 10 cos(4πxi)), ∑ i)2
h(X) ) X ) (x1, x2, ..., xn)
x
(n ) 10)
f1(X) g(X) (for 0 e xi e 1)
The Pareto curve is shown in Figure 3. In the simulation, 100 particles and 100 iterations are used. This figure shows
Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007 3605
Figure 3. Pareto fronts produced by new (O) MOPSO and (/) NSGAII algorithms: test function 3.
Figure 5. Schematic diagram of the cracking furnace.
Figure 6. MOPSO scheme based on ANN.
Figure 4. Pareto fronts produced by new (O) MOPSO and (/) NSGAII algorithms: test function 4.
that the diversity of the MOPSO Pareto solutions is greater than the diversity of the NSGAII Pareto solutions. The computation time in the new MOPSO algorithm is ∼13 min, and the computation time in NSGAII is ∼17 min in the same computer system scheme. The computation time in the new MOPSO algorithm is faster than that in the NSGAII algorithm. (d) Test Function 4.
Min f1 ) x1 Min f2 ) 1 g)1+ xi ∈ [0,1]
9
Figure 7. Depth of cracking (m), relative to the yield of ethylene (C2H4) and the yield of propylene (C3H6).
() f1 g
2
n
∑xi
n - 1 i)2
(n ) 30)
The Pareto curve is shown in Figure 4. In the simulation, 100 particles and 100 iterations are used. This figure shows that the number of MOPSO Pareto solutions is more similar to the true Pareto solutions than the NSGAII Pareto solutions. The computation time in the new MOPSO algorithm is ∼10 min, and the computation time in the NSGAII algorithm is ∼15 min in the same computer system scheme. The computation time in the new MOPSO algorithm is faster than that in the NSGAII algorithm. Figures 1-4 show that the performance of the MOPSO algorithm is better than that of the NSGAII algorithm, with respect to Pareto solution diversity and computation time. 2. Algorithm Applied in the Industrial Cracking Furnace. Figure 5 has been taken from ref 22. The industrial ethylene reactor is a component in one of the key production processes
Figure 8. Ratio of gas to hydrocarbon (k), relative to the yield of ethylene (C2H4) and the yield of propylene (C3H6). Table 1. Depth of Cracking and the Yield of Ethylene and the Yield of Propylene depth of cracking, m
yield of ethylene (%)
yield of propylene (%)
0.5 0.52 0.54
31.6075 31.5849 31.5547
13.8102 13.8364 13.9507
in the petrochemical industry. The key component of an ethylene plant with huge economic impact is a cracking furnace. Any improvement in its production process can bring huge benefits to the industry.23 However, it is difficult to model the ethylene cracking process, which is highly dimensional, multimode, and nonlinear. Besides, it is a multi-objective optimization problem, in regard to seeking a range of optimal operation conditions. Figure 5 shows a schematic diagram of a cracking furnace. There are 20 factors that directly or indirectly affect the yields of
3606
Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007
Figure 9. Outlet temperature (T), relative to the yield of ethylene (C2H4) and the yield of propylene (C3H6).
Figure 12. Outlet pressure (P), relative to the yield of ethane (f-C2H4) and the yield of propylene (f-C3H6).
Figure 10. Outlet pressure (P), relative to the yield of ethylene (C2H4) and the yield of propylene (C3H6).
Figure 13. Coil outlet temperature (COT) of pyrolysis gas (T), relative to the yield of ethane (f-C2H4) and the yield of propylene (f-C3H6). Table 2. Ratio of Gas and Hydrocarbon to the Yield of Ethylene and the Yield of Propylene
Figure 11. Ratio of gas to hydrocarbon (k), relative to the yield of ethane (f-C2H4) and the yield of propylene (f-C3H6).
ethylene and propylene: 4 naphtha feed flow groups, 4 highpressure dilute steam flow groups, 4 pipe inlet temperature groups in a radiant box, 4 coil outlet temperature (COT) groups, 1 fuel flow in the bottom of the cracking furnace, 1 fuel flow in the wall of the cracking furnace, 1 flue gas cross temperature, and 1 flue temperature in the hearth. The optimization problem in the complicated industry often faces more than one optimization object functions. Modeling the ethylene reactor is very complex with a free-radical mechanism model. Sundaram and Froment developed a freeradical scheme for a cracking model; however, there are 49 reactions.24 Moreover, the free-radical mechanism leads to some stiff differential equations, which are difficult to solve. As a substitute method, the artificial neural network (ANN) learning algorithm has been adopted to simulate the cracking reactor model. It is well-known that ANN is widely applied to simulate the nonlinear objectives, because it does not require much knowledge. This paper presents an ANN model that is used to simulate a yield predication model, and it presents a new MOPSO method to determine a better condition solution to improve the performance of the ethylene and propylene production units, based on a naphtha feed. The yields of ethylene and propylene are improved.
ratio of gas to hydrocarbon, k (w/w)
yield of ethylene (%)
yield of propylene (%)
0.48 0.50 0.52
30.5604 30.8926 31.0738
15.0100 14.6705 14.6650
Table 3. Outlet Temperature to the Yield of Ethylene and the Yield of Propylene outlet temperature, T (°C)
yield of ethylene (%)
yield of propylene (%)
820 825 830
31.5940 31.6957 31.7478
13.9090 13.657 13.8400
(a) Modeling the Yields of the Ethylene and Propylene Using Artificial Neural Network (ANN) Learning. The construct of an ANN is as follows. There are three neurons in the input layer (namely, one neuron is one pair of gas-tohydrocarbon ratio values, the second neuron is one pair of COT values of the pyrolysis gas, and the third neuron is one pair of outlet pressure values), and the two neurons in the output layer are the yield of the ethylene and the yield of the propylene. The training data contain 200 pairs of experimental data, and the test data contain 100 pairs of actual industrial data. We evaluate the function with ANN, then optimize the decision variables with MOPSO. The MOPSO hybrid model is as described in Figure 6. In this figure, the parameters k, T, and P represent the ratio of gas to hydrocarbon, the outlet temperature of the pyrolysis gas, and the outlet pressure, respectively. YC2H4 denotes the yield of ethylene, and YC3H6 denotes the yield of propylene.
Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007 3607 Table 4. Outlet Pressure and the Yield of Ethylene and the Yield of Propylene outlet pressure, P (kg/cm2)
yield of ethylene (%)
yield of propylene (%)
1.85 1.9 1.95 2
31.6295 31.6406 31.6369 31.6110
13.7928 13.7758 13.7694 13.7857
For the optimization study of the cracking reactor, the objectives are to (i) maximize the ethylene production and (ii) maximize the propylene production. Because propylene is taken as material in the next production process, the maximum propylene production is one of the optimization objectives. Moreover, the yield of ethylene increases as the yield of propylene decreases. The two optimization objectives are competitive. Therefore, the selectivity of ethylene and the yield rate of ethylene are chosen as the objectives, so that a nondominant set of solutions can be obtained from a multiobjective optimization. Accordingly, the multi-optimization problem was formulated to maximize the values:
F(x) ) Max (F1,F2) F1(x) ) Max YC2H4 F2(x) ) Max YC3H6 The variables that affect the cracking reactor performance and can be adjusted in an industrial reactor were chosen as the decision variables. In full load condition, the decision variables are the steam-to-naphtha ratio, the outlet temperature, and the outlet pressure. The ranges of decision variables used are as follows:
0.45 e k e 0.60 809 °C e T e 845 °C 1.78 kg/cm2 e P e 2.08 kg/cm2
(gauge pressure)
The choice of decision variables is depednent on sensitivity analysis, which is discussed in the next section. Bounds on the decision variables are selected according to industrial practice. A large steam-to-naphtha ratio increases the yield of ethylene, and also increases energy consumption and decreases productivity. On the other hand, a small steam-tonaphtha ratio increases the rate of coking and shortens the production run cycle. The outlet temperature in the reactor tubes directly affects the depth of cracking and coking, the selectivity of the cracking product, and the periods of production. A high outlet temperature has a tendency to cause rapid coking and increase the temperature of the ektexine tubes. Moreover, it shortens the production period. A low outlet temperature decreases the depth of cracking, which has a tendency to reduce the yield of ethylene. Decreasing the depth of cracking has a tendency to decrease the conversion of naphtha and the yield of ethylene and increase the production costs. However, the depth of cracking is a function of the outlet temperature, the depth of cracking is not defined as the decision variable. The outlet pressure constraint is guided by the suction pressure of the cracked-gas compressor. Because the cracked gas must pass through a transfer line exchanger for quenching after coming exiting the reactor, which involves an additional pressure drop, the outlet pressure of the reactor was limited to 2.2 kg/cm2 (gauge pressure).
(b) Sensitivity Analysis of the Steam Cracker. Sensitivity analysis of the steam cracker is taken as the basis on which we select the decision variables in the subsequent optimization study. It is observed from Table 1 and Figure 7 that, when the steamto-naphtha ratio, outlet temperature, and outlet pressure are kept constant and the depth of cracking is permitted to vary from 0.45 to 0.6, the yield of ethylene increases while the yield of propylene decreases. Furthermore, the increase in ethylene yield is greater than the decrease in propylene yield. Therefore, the combined yield of ethylene and propylene decreases. Table 2 and Figure 8 show that, when the depth of cracking, outlet temperature, and outlet pressure are kept constant (m ) 0.6000, T ) 840.1733 °C, P ) 1.78 kg/cm2, respectively) and the steam-to-naphtha ratio (k) is allowed to vary from 0.45 to 0.6, the yield of ethylene increases while the yield of propylene decreases. The sum of the yields of ethylene and propylene increase as k varies from 0.45 to 0.5, and the sum of the yields has a maximum at 0.5. The sum of the yields then decreases when k varies from 0.5 to 0.6. Table 3 and Figure 9 show that, when the depth of cracking, outlet pressure, and steam-to-naphtha ratio are kept constant and the outlet temperature is permitted to vary from 809 °C to 845 °C (m ) 0.6000, P ) 1.78 kg/cm2, k ) 0.5), the yield of ethylene increases while the yield of propylene decreases. The sum of the yields of ethylene and propylene increases and receives a maximum value at 840 °C, then the sum of the yields decreases. Table 4 and Figure 10 show a clear increase in the yield of ethylene with increasing outlet pressure from 1.78 to 1.93, while the yield of propylene decreases. The yield of ethylene decreases as the outlet pressure varies from 1.93 kg/cm2 to 2.08 kg/cm2, while the yield of propylene increases. At the same time, the depth of cracking, the outlet temperature, and the stream-tonaphtha ratio remain constant (m ) 0.6000, T ) 840.1733 °C, k ) 0.5) and the yield of ethylene increases while the yield of propylene decreases. The sum of the yields of ethylene and propylene increases and reaches a maximum value at 1.78 kg/ cm2, then the sum of the yields decreases. In the aforementioned sensitivity analysis, note that the ratio of gas to hydrocarbon, the COT value of the pyrolysis gas, the cracking depth, and the outlet pressure restrict each other, and, as a result, the yields of ethylene and propylene change in opposite directions. However, the depth of cracking is a function of the outlet temperature; thus, the depth of cracking is not defined as a decision variable. Therefore, the ratio of gas to hydrocarbon, the COT of the pyrolysis gas, and the outlet pressure are considered to be the decision variables of this optimization problem. (See Table 5.) (c) Optimization of the Naphtha Cracking Unit. In the preceding section, the objectives, decision variables, and constraints have been defined. The data in Table 6 are the Pareto-optimal results generated from the proposed algorithm, and they are compared with the industrial actual data. Figures 11-13 show the optimization operation conditions, relative to the yield of ethene and propylene. Compared with actual industrial data (see Table 6, which lists the optimization results), the Pareto-optimal set is consistent with the industrial requirements. The optimum operation conditions should ensure the production of the highest yield rate of the ethylene and the propylene. For example, the ratio of gas to hydrocarbon, the COT of the pyrolysis gas, and the outlet pressure from optimization are better than the actual industrial data; when the yield rate of ethylene and propylene are at 31.06 and 14.98,
3608
Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007
Table 5. Design and Operating Conditions for the Industrial Reactor constraint variable variable outlet pressure, P outlet temperature, T ratio of gas to hydrocarbon, k run result yield rate of ethene, Y1 yield rate of propylene, Y2
1.78 kg/cm2 e P e 2.08 kg/cm2 809 °C e T e 845 °C 0.45 e k e 0.6
1.78 kg/cm2 843.62°C 0.52
Y1 > 27.55 wt % Y2 > 13.63 wt %
30.54 wt % 14.54 wt %
Table 6. Result of the Optimization
NSGAII yield rate ethylene propylene decision variable ratio of gas to hydrocarbon, k COT of pyrolysis gas outlet pressure, P
Max
Min
30.613 16.766
28.134 14.126
0.5934 844.801 2.065
0.4578 811.201 1.689
New Multiobjective PSO yield rate ethylene 31.503 propylene 17.165 decision variable ratio of gas to hydrocarbon, k 0.5994 COT of pyrolysis gas 844.486 outlet pressure, P 2.037
present running setpoint
27.764 14.635 0.4617 810.768 1.633
respectively, the ratio of gas to hydrocarbon is ∼0.51, the COT of the pyrolysis gas is ∼842.46 °C, and the outlet pressure is 1.75. In addition, it provides optional support for the decisionmaker. IV. Conclusion A hybrid multi-objective particel swarm optimization (MOPSO) procedure, based on a Pareto set, is presented to handle multi-objective problems in industrial processes, such as optimization of the ethylene and propylene yield rates. The ratio of gas to hydrocarbon, the coil outlet temperature (COT) of the pyrolysis gas, and the outlet pressure are considered to be the decision variables of multi-objective problems, using sensitivity analysis. The proposed algorithm has better convergence and diversity Pareto solutions than the NSGAII algorithm. The naphtha industrial cracking process is simulated using an artificial neural network (ANN) model, instead of a free-radical mechanism model. Comparison with the industrial data shows that the algorithm is feasible. This investigation has provided a wide option of optimal operational conditions for the ethylene reactor. Nomenclature ω ) inertia weight c1 ) cognitive parameters c2 ) social parameters f ) function g, h ) constraints conditions m ) depth of cracking k ) ratio of gas and hydrocarbon P ) outlet pressure P ) Pareto optimal set ΡF ) Pareto front r1, r2 ) random numbers uniformly distributed within [0,1] T ) coil outlet temperature (COT) of the pyrolysis gas p1 ) best particle position p2 ) global particle position
Vk ) particle current velocity Vk+1 ) next iteration velocity u, V ) Pareto sets Vlim ) individual xk ) current position of particle xk+1 ) next position of particle x ) population set b x* ) Pareto optimal point b x ) vector of decision variables YC3H6, J2 ) yield of propylene YC2H4, J1 ) yield of ethylene Subscript i ) index Accents f ) vector signal * ) Pareto optimal point Acknowledgment Partially supported by Sinopec Science & Technology Development (Project No. E03007). Literature Cited (1) Deb, K. Multi-objectiVe Optimization Using EVolutionary Algorithms; Wiley: New York, 2001. (2) Chankong, V.; Haimes, Y. Y. MultiobjectiVe Decision Making: Theory and Methodology; North Holland Series in System Science and Engineering, 8; North-Holland: Amsterdam, 1983. (3) Deb, K.; Pratap, A.; Agarwal, S.; Meyarivan, T. A Fast and Elitist Multi-Objective Genetic Algorithm: NSGA-II. IEEE Trans. EVol. Comput. 2002, 6 (2), 182. (4) Yee, A. K. Y.; Ray, A. K.; Rangaiah, G. P. Multi-objective optimization of an industrial styrene reactor. Comput. Chem. Eng. 2003, 27 (1), 111. (5) Kennedy, J.; Eberhart, R. C. Particle Swarm Optimization. Proc. IEEE Int. Conf. Neural Networks 1995, 1942. (6) Kennedy, J.; Eberhart, R. Particle swarm optimization: developments, applications and resources. Proc. Congr. EVol. Comput. 2001, 81. (7) Shi, Y.; Eberhart, R. A Modified Particle Swarm Optimizer. Proc. Congr. EVol. Comput. 1998, 3, 4. (8) Coello Coello, C. A.; Lechuga, M. S. MOPSO: A proposal for multiple objective particle swarm optimization. Proc. Congr. EVol. Comput. 2002, 5 (1), 1051. (9) Hu, X.; Eberhart, R. multiobjective optimization using dynamic neighbourhood particle swarm optimization. Proc. Congr. EVol. Comput. 2002, 5 (2), 1677. (10) Hui, X.; Eberhart, R. C.; Shi, Y. Particle swarm with extended memory for multi-objective optimization. Proc. 2003 IEEE Swarm Intell. Symp. 2003, 4, 193. (11) Shi, Y.; Eberhart, R. Parameter Selection in Particle Swarm Optimization. In Proceedings of the 7th Annual Conference on EVolutionary Programming, 1998; p 591. (12) Clerc, M.; Kennedy, J. The particle swarmsExplosion, stability, and convergence in a multidimentional complex space. IEEE Trans. EVol. Comput. 2002, 6 (2), 58. (13) Bhutani, N.; Ray, A. K.; Rangaiah, G. P. Modeling, Simulation, and Multi-objective Optimization of an Industrial Hydrocracking Unit. Ind. Eng. Chem. Res. 2006, 45, 1354.
Ind. Eng. Chem. Res., Vol. 46, No. 11, 2007 3609 (14) Mostaghim, S.; Teich, U. Strategies for Finding Good Local Guides in Multi-objective Particle Swarm Optimization (MOPSO). Proc. 2003 IEEE Swarm Intell. Symp. 2003, 26. (15) Noel, M. M.; Jannett, T. C. Simulation of a New Hybrid Particle Swarm Optimization Algorithm. Proceedings of the Thirty-Sixth Southeastern Symposium on System Theory; IEEE: Piscataway, NJ, 2004; p 150. (16) Schaffer, J. D. Multiple objective optimization with vector evaluated genetic algorithms. Genetic algorithms and their application. In Proceedings of the First International Conference on Genetic Algorithms; Morgan Kaufmann Publishers: San Mateo, CA, 1985; p 93. (17) Chen, Q.; Zhang, Y. X.; Ren, B. Study on Optimizing Parameters of the Nonlinear Model of Subsidence Based on Modified Floating-Coding GA. J. Highway Transport. Res. DeV. 2005, 22 (2), 75. (18) Juang, C.-F. A hybrid of genetic algorithm and particle swarm optimization for recurrent network design. IEEE Trans. Syst., Man, Cybernet.sPart: Cybernet. 2004, 34, 2. (19) Laskari, E. C.; Parsopoulos, K. E.; Vrahais, M. N. Particle Swarm Optimization for minimax problems. In Proceedings of the IEEE 2002 Congress on EVolutionary Computation; 2002; p 1582. (20) Srinivas, N.; Deb, K. Multi-objective Function Optimization Using Non-Dominated Sorting Genetic Algorithms. EVol. Comput. 1994, 2, 221.
(21) Yen, G. G.; Lu, H. Dynamic multiobjective evolutionary algorithm: adaptive cellbased rank and density estimation. IEEE Trans. EVol. Comput. 2003, 7, 253. (22) Geng, Z. Q.; Zhu, Q. X. Multiscale Nonlinear Principal Component Analysis (NLPCA) and Its Application for Chemical Process Monitoring. Ind. Eng. Chem. Res. 2005, 44 (10), 3585. (23) Tarafder, A.; Lee, B. C. S.; Ray, A. K.; Rangaiah, G. P. Multiobjective Optimization of an Industrial Ethylene Reactor Using a Nondominated Sortiing Genetic Algorithm. Ind. Eng. Chem. Res. 2005, 44, 124. (24) Sundaram, K. M.; Froment, G. F. Modeling of Thermal Cracking Kinetics. 3. Radical Mechanisms for the Pyrolysis of Simple Paraffins, Olefins, and Their Mixtures. Ind. Eng. Chem. Fundam. 1978, 17, 174.
ReceiVed for reView September 27, 2005 ReVised manuscript receiVed February 28, 2007 Accepted March 23, 2007 IE051084T
