decomposition techniques for the real-time optimization of a propylene

Krist V. Gernaey, Jakob K. Huusom and Rafiqul Gani (Eds.), 12th International Symposium on Process Systems Engineering and 25th European Symposium on Computer Aided Process Engineering. 31 May – 4 June 2015, Copenhagen, Denmark © 2015 Elsevier B.V. All rights reserved.

DECOMPOSITION TECHNIQUES FOR THE REAL-TIME OPTIMIZATION OF A PROPYLENE PRODUCTION UNIT A.M. Acevedo P.a, J.E.A. Gracianoa, F.S. Liporace,b A.S. Vianna Jr.a, G.A.C. Le Rouxa a

Dept. of Chem. Eng., University of São Paulo, Av. Luciano Gualberto, Trav. 3, São Paulo, Brazil b CENPES/PETROBRAS S.A, Av. Horácio Macedo, 950, Rio de Janeiro, Brazil

Abstract In this work, An application of decomposition techniques on a propylene production unit is presented. It is shown that the traditional Lagrangian Relaxation technique and the alternative technique “Pricing Interprocess Streams Using Slack Auctions” (Rinaldo and Ungar, 2000) do not converge for the presented problem. The issues involved in the each decomposition approach are identified and discussed; then a modification of the Lagrangean Relaxation algorithm is proposed using a new updating constrain rule. This modified algorithm is able to overcome the issues and solve the decomposed problem properly. Keywords: real-time optimization, process optimization, decomposition techniques.

1. Introduction Chapter 2 Chemical process industry has a high demand for methods and tools to perform process optimization in order to enhance profitability. Due to the large products rate, high energy and raw materials consumption, making small improvements in process efficiency can be translated into significant gains in operating profit (Carvalho, 2007). An alternative to increase the profitability of continuous process is to implement the Real Time Optimization technique (RTO), which the main objective is to drive the plant at each instant of time, as close as possible to the optimal operating conditions (Sequeira et al., 2002). The classic RTO framework includes the following sequential steps: steady state detection, treatment of gross errors, data reconciliation, parameter estimation and economic optimization (White, 1997). The problem of economic optimization encompasses the whole refinery, however the modeling of a complete refinery is too complex to be made, and then it is more advantageous to use local optimizations (models for each division) than to optimize the entire refinery considering only one model (Friedman, 1995; Ungar and Rinaldo, 2000). For this purpose, one can use decomposition techniques. This work evaluates two classic decomposition techniques found in literature: Lagrangean Relaxation and Pricing Interprocess Stream Using Slack Auctions (Rinaldo and Ungar, 2000). The results show that these techniques present a number of difficulties to be implemented in a real industry problem, a propylene unit from REPLAN refinery by Petrobras S.A. These difficulties inspired a proposed modification in the Lagrangean Relaxation algorithm, which allows effectively decompose the process studied.

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1. Decomposition techniques for process optimization In order to implement the economical optimization of a refinery, Naysmith and Douglas (1995) presented two possibilities. The first one considers a centralized structure (a unique model for the whole plant) and the second one, a distributed structure. This last approach decomposes the centralized problem into some less complex smaller optimization problems, which requires a decomposition technique to specify, adequately, the prices for intermediate streams, thus each division of the unit can be independently optimized, looking for a feasible solution for the whole plant. For this purpose, a classic "Lagrangean Relaxation" technique, and an alternative approach proposed by Rinaldo and Ungar (2000), “Pricing Interprocess Streams Using Slack Auctions”, are applied and tested in our case study. 1.1. Lagrangean Relaxation decomposition (LR) LR decomposition is based on an iterative algorithm, which relaxes the complicating constraints (Conejo et al. 2006) of the original optimization, generating sub-problems that are coordinated by a master problem. Lagrange multipliers (λ) are used in the relaxation and represent the shadow price of intermediate products; as a consequence, each division of the plant could be individually optimized, aiming to obtain the best profit for the entire unit. Based in Conejo et al. (2006), the resolution algorithm for this technique is composed by the follows steps: at step 0, initial guesses for all the λ’s are specified, this information is used in step 1, where all the optimization sub-problems are solved independently (each sub-problem represents one division of the whole unit). In step 2, the λ’s are updated using the classic subgradient approach; finally in step 3 the convergence criterion is checked, then, if the difference between the λ’s of two consecutive iterations is less than an arbitrary tolerance (ε) the solution was found, otherwise return to step 1. More details and examples of this technique can be found in Conejo et al. (2006) and Guignard (2003). 1.2. Pricing Interprocess Stream Using Slack Auctions (PISUSA) In this technique, the complicating constraints are rewritten as resource constraints (inequality constraints) to be used as a criterion for the algorithm convergence. An new negative term is added to the economic objective function of each sub-problem, which is composed of a weight parameter (p) multiplied by the absolute value of the complicating constraints, enforcing their difference to decrease at each iteration. For economic purposes, p represents the price of the intermediate streams. The iterative algorithm starts at step 0, in which the initial values of p and complicating constrains variables are set. In step 1, the sub-problems are solved independently. In step 2, the algorithm convergence is checked, and the difference between complicating constraints variables are compared to the slack (S). If the convergence is not reached, in step 3 the reference values for the complicating constraints variables and the price value (p) are adjusted in accordance to the previous results, then go back to step 1. More details and examples can be found in Rinaldo and Ungar (2000).

2. Propylene production unit The flowsheet shown in Figure 1 illustrates the main features of the propylene production unit that is composed by three divisions: Depropanizer (D_1), De-ethanizer (D_2) e C3 splitter (D_3). This last one uses a vapor recompression assisted distillation (VRD). The unit is fed by a GLP stream supposed to contain 10 components, mainly propylene and propane (assumed to be 37% and 13% molar, respectively). The most relevant conditions of the columns are presented in Table 1.

Decomposition techniques for the real-time optimization of a propylene production unit 3

Column T_1 T_2 T_3

# of trays 51 61 197

Product Specification Bottom Top < 1% molar light comp. < 0.1% molar heavy comp. < 120 ppm of ethane < 35% molar of propylene < 5% molar of propylene > 95% molar of propylene

Column Capacity Max Min 23,804 17,312 10,865 4,939 10,061 4,573

Table 1. Operating conditions of the propylene operating unit The entire unit comprises: 6 valves (Vi=1,...,6), 4 condensers (HXi=1,...,4), 3 reboilers (RBi=1,...,3), 4 pumps (Bi=1,...,4), 8 splitters (SBi=1,...3; SP1; SRi=1,...,3;SC1), 2 mixers (Mi=1,2), 1 reflux tank (TA1) and 1 compressor (C1). The propylene production unit was modeled, simulated and optimized at steady-state condition using equation-oriented environment EMSO (Environment for Modeling, Simulation and Optimization, Soares and Secchi, 2003). The model was based on Mendoza et al. (2013), where the same assumptions were done. The optimization problem for this process has the following structure: Max P1*F9 + P2*F10 - P*F1 + P3*F14 + P4*F29 + P5*F26 - OCD_1 - OCD_2 - OCD_3 s.t. Model and operational constraints D_1 Model and operational constraints D_2 Model and operational constraints D_3 5D_1 = 5D_2 17D_2 = 17D_3

where P, P1, P2, P3, P4 e P5 are the product market prices of F1, F9, F10, F14, F29, e F26, respectively; OCD_1, OCD_2 and OCD_3 are operating costs of each division D_1, D_2 and D_3 and 5D_1 = 5D_2 and 17D_2 = 17D_3 are the complicating constraints.

Figure 1. Simplified diagram of the propylene production unit Solving the centralized structure, the gross operating profit (GOP) was increased by 4% from the operational condition, which is the decomposed optimization target.

3. Application to a propylene production unit In this section the selected approaches are presented and discussed, and then a novel algorithm is developed. 3.1. Application of Lagrangean Relaxation decomposition technique (LR). In Figure 2, the Lagrangean multipliers obtained at each iteration are presented. The results obtained in this case study show that there is no λ1 and λ2 (shadows prices of streams 5D_1; 5D_2 and 17D_2; 17D_3 respectively) able to converge the algorithm. This is due to two main causes identified by Rinaldo and Ungar (2000): Issue 1, Indifferent divisions: In this case, there is no linear dependence between the gradients of the complicating constraints and the gradient of objective function, unless

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the complicating constrains variables are equal to zero (e.g. 5D_1=0 and 5D_2=0) or λ’s are zero. However, these results are useless for the decomposition approach, because those assumptions do not have physical or economical mean.

Figure 2. Convergence problem of the LR techinique Issue 2, Divisions with linear objectives: In this case, the divisions D_1, D_2 and D_3 always propose the maximum or the minimum value for the intermediate streams, depending on the value of λ1 and λ2. In short, there is no λ1 and λ2 that can guarantee the constraints 5D_1 = 5D_2 and 17D_2 = 17D_3 satisfaction as shown in Figure 3, which presents the results for the intermediate streams flow rates using the centralized and distributed structure using the LR technique.

Figure 3. Flow rates of the intermediate streams obtained by LR approach Considering that the feed flow rate (D_1) is fixed at 21,456 kg/h, it can be observed that the intermediate product flow rate (5D_1 - ▴mark) does not swicht between the upper and lower bounds, due to the purity specification of the top product of T_1. 3.2. Pricing interprocess stream using slack auctions The values obtained for the intermediate products (5D_2 and 17D_3) are p1 = 7 Cr$/kg and p2 = 9.7 Cr$/kg, respectively (an arbitrary scale is assumed for confidentiality reasons). Also in this case, the algorithm is unable to converge. Those prices for the intermediate streams are larger than the market prices of the final products in each division, which are 1.6; 2.4; 2.7; 6.27 and 1.6 Cr$/kg for the streams F9, F10, F14, F26 and F27, respectively. In this method, the slack term included in the objective function of each sub-problem always tries to maximize the product that has a price defined by the market, since the intermediate stream has a negative weight (price multiplied by a negative signal) in the objective function. The only way this approach can converge is by setting high prices values (p). However, when this setting is implemented the price (p) loses all its economical meaning, seeing that the intermediate product price is higher than the market price of the final products. Finally, this method is not effective for the studied problem (joint production).

Decomposition techniques for the real-time optimization of a propylene production unit 5 3.3. Modification of the Lagrangean Relaxation algorithm (proposed approach) In this section the classic LR algorithm is modified to overcome the convergence problem observed in the previous cases. For this reason, a new variable called availability of intermediate product (AIP) is added to the LR algorithm. The modifications of the classic LR algorithm (see Section 2.1) can be defined as follows: at step 0, the AIP variables are initialized as the maximum feed flow rate that can be processed in respectively division; at step 1, a resource constraint is added to the feed stream of each division, which for our case study is given by 5D_2(v) ≤ AIP1(v) for division D_2 and 17D_3(v) ≤ AIP2(v) for division D_3, where v is the iteration number. The new resource constraint is not considered in D_1, because the feed stream F1 is fixed; at step 2, in addition to updating the multipliers (λ’s), the AIP’s are updated, as follows: AIP1(v+1) ← 5D_1(v) and AIP2(v+1) ← 17D_2(v). This ensures that in the next iteration the feed flow rate of each division is physically available; in addition, the step 3 maintains the same formulation than the classic LR algorithm. Figure 4 depicts the results obtained by the new approach. Note that after 9 iterations the algorithm converges, and the shadow prices for the intermediate streams are properly determined (1.85 Cr$/kg for 5D_1 and 5D_2 streams; 2.8 Cr$ for 17D_2 and 17D_3 streams).

Figure 4. Convergence of the Lagrange multipliers for the proposed algorithm In comparison with the results obtained in the optimization of the centralized structure, it can be observed in Figure 5 that the flow rates of the intermediate streams are similar, which means that the proposed algorithm optimizes the whole plant using a decomposition technique.

Figure 5. Flow rate values obtained by the proposed algorithm and the centralized approach Table 2 presents some relevant information about the optimization problems of each approach disscused in the present paper. It can be observed that the distributed approaches reduces the problem complexity. Furthermore, each sub-problem generated by the decomposition thecniques can be optimzated in parallel, using different

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processors and reducing the total amount of time expended by the whole process optimization (Darby and White, 1988). Optimization Structure # Variables # Equations # Inequalities GOP (Cr$/h)

LR

Centralized 13,640 13,634 22 71,622.92

PISUSA

Proposed approach

D_1 2,393

D_2 2,746

D_3 8,510

D_1 D_2 D_3 D_1 2,393 2,746 8,510 2,393

D_2 2,746

D_3 8,510

2,391 6

2,743 10 ---

8,507 10

2,391 2,743 8,507 2,391 2,743 8,507 6 10 10 6 11 11 --71,754.71

Table 2. Relevant information used in each approach The diference obtained between the GOP of the centralized and distributed structure is of 0.184%. This difference is due to non-liniearity of the optimization problem studied, which results in different local optima solutions.

4. Conclusions Three decomposition techniques were applied to the propylene unit at REPLAN refinery (Petrobras), which is a large scale optimization problem. In the first approach the whole unit was optimized with a centralized structure. Then, two selected decomposition methods were applied, which presented different issues, and they were not able to converge to feasible solution. Those problems inspired the proposition of a modified LR algorithm, adding a new variable called AIP, the novel method guarantees a feasible solution for the distributed structure, which is slight better than the local optimum obtained by the centralized problem. This proposed modification is promising to be studied and tested in other processes with similar characteristics.

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