A novel method for identification of critical points in flow sheet synthesis under uncertainty

Transcription

1 Ian David Lockhart Bogle and Michael Fairweather (Editors), Proceedings of the nd European Symposium on Computer Aided Process Engineering, 17-0 June 01, London. 01 Elsevier B.V. All rights reserved A novel method for identification of critical points in flow sheet synthesis under uncertainty Mihael asaš, Zdravko ravanja, Zorka ovak Pintarič University of Maribor, Faculty of Chemistry and Chemical Engineering, Smetanova 17, SI-000 Maribor, Slovenia Abstract A solving strategy for two-stage stochastic problems in chemical process synthesis under uncertainty is presented. A strategy is based on the formulation of significantly reduced deterministic equivalent which is solved at the union of the nominal point and the minimum set of critical points. The later define flexible values of the first-stage design variables in the flow sheet model, while the expected objective value of the second-stage (operating) variables is approximated at the nominal point. The main emphasize in this contribution is on a novel method for critical points identification. These are identified by solving max-min problem iteratively by connecting GAMS files to codes written in Fortran. Two examples show that minimum set of critical points can be found by using the proposed method. eywords: two-stage stochastic, deterministic equivalent, scenario reduction. 1. Introduction One of the approaches to solving a problem of chemical process synthesis under uncertainty is to transform the infinite two-stage stochastic problem into its deterministic equivalent. The main issue is the selection of discrete points or scenarios that yield the result close to the optimum of the original problem at reasonable computational effort. Applying rigorous methods, like Gaussian integration and Monte Carlo simulation (Acevedo and Pistikopoulos, 1998), to large-scale flow sheet models with many uncertain parameters usually leads to unmanageable set of scenarios. Several scenario reduction techniques have been developed in the past, and applied to engineering problems under uncertainty (e.g. Pineda and Conejo, 010; Dupačová et al., 003). aruppiah et al., 010 developed a special mixed-integer linear program (MILP) to determine a reduced set of scenarios so that the total probability of each uncertain parameter value over the reduced set of scenarios is equal as primarily defined. In our previous researches, we developed several heuristic approaches in which the expected objective value is approximated at few points or even at one single point (ovak Pintarič and ravanja, 004), while feasibility is achieved by using the critical points (ovak Pintarič and ravanja, 008). The conventional methods for identification of critical points often generate redundant points, therefore, a new enhanced method is presented in this contribution, which has a potential to be used for real industrial problems with huge number of uncertain parameters.. Formulation of significantly reduced deterministic equivalent.1. General formulation of the reduced deterministic equivalent Our approach is based on solving a significantly reduced multi-scenario deterministic equivalent problem (RDE) in which the expected objective value (EZ) is approximated

2 M. asaš et al. at the nominal point,, while feasibility is achieved by using a set of critical points, C. The latter are defined as those scenarios that determine optimal overdesign so that flexibility is achieved over the specified uncertainty region in a robust way. dyx EZ min f(,,, ) s s s.t. hdyx (,,, ) 0 s s gdyx (,,, ) 0 ss C s s gd ( y, x, ) s x,, y 0,1 (RDE) In (RDE), d and y are the vectors of the first-stage design and discrete topological variables, respectively. x represents a vector of the second-stage continuous variables, h, g, g d are vectors of equality, nonequality and design specification constraints. S is a set of scenarios consisting of critical points and nominal value. It is assumed that uncertain parameters,, are allowed to vary between predefined lower and upper bounds. Several methods have been proposed for identification of critical points but only a few of them, mostly the heuristic ones, are suitable for large-scale problems. Unfortunately, the heuristic methods often generate redundant critical points that unnecessarily increase the size of deterministic equivalent. A novel more precise method is presented in the continuation... A novel method for critical points identification The basic idea is that the addition of critical scenarios to the nominal point in the deterministic equivalent problem deteriorates the objective value. Starting with a singlescenario problem at the nominal point, a new critical point,, is sought at each major iteration so that the biggest rise of the first-stage design variables would be achieved. Among several points that meet this goal, we seek to find the one with the lowest value of the objective function, which leads to the max-min problem: Z max min f( d, x,, ) x, d s.t. hdx (,, ) 0 gdx (,, ) 0 k1,,..., g ( x, ) d k L U 1 x, d,,, The problem (P1) is solved through the two-level procedure. The second-level problem (P1 ) is solved directly by GAMS, while the first-level (P1 1 ) by GAMS s external equations written in Fortran. (P1) First level Z' max f ( ) impl L, x x U (P1 1 )

3 A novel method for identification of critical points 3 Second level Z min f( dx,, ) x, d s.t. hdx (,, ) 0 gdx (,, ) 0 k1,,..., (P1 ) gd ( x, ) k 1 x,, In the first iteration (=1), a single-scenario problem (P1 ) is solved at the nominal point, 1 =, minimizing the original objective variable Z. The solution obtained is usually inflexible, and the objective value represents a lower bound for minimization problem. The marginal values of are transferred through the external GAMS functions to the first level (P1 1 ), where an implicit function of the objective value Z vs is derived. This function is maximized, thus generating a new point +1 that maximizes design variables. The identified point is transferred to the second-level model, and added to the nominal point and those critical points generated in previous iterations. The external routine written in Fortran iterates between both levels until the objective values equalize (Z = Z ). The number of scenarios in the problem (P1 ) grows by one at each major iteration, while the objective value Z rises as well. If the converged objective value in current iteration is larger than the one obtained in the previous iteration, the procedure repeats in order to generate a new critical point which is then added to the problem (P1 ). The complete minimum set of critical points is found when the objective value Z is not growing anymore. A pleasant fact is that the final solution of iterative procedure corresponds to the solution of the problem (RDE). 3. Examples 3.1. Small theoretical example The deterministic model of the first example (E1) comprises two state variables (x 1, x ), two design variables (d 1, d ), and two uncertain parameters ( 1, ) with the nominal values ( 1, ) = (3,3), and lower and upper bounds 1 and 5. o probability distribution is provided. The design obtained is inflexible for specified deviations of uncertain parameters. The solution of one-scenario nominal problem is given in the first column of Table 1. Z min 3x x d d s.t. d x x d x x x x1, x R, d1, d 0 1, / x 30 1 In the next step, a multi-scenario deterministic equivalent (E1 DE ) is formulated. This problem comprises four vertices, so the problem (E1 DE ) is firstly solved over the union of the nominal point and all vertices, ss =(3,3), (1,1), (1,5), (5,1), (5,5), yielding a feasible design for specified deviations (second column of Table 1). (E1)

4 4 M. asaš et al. Z min 3x x d d s s s d1 x1 x 1 s s s s d 1/ x1 ( x) 0.51 s s ( x1 ) x 30 s s 1,, 1, 0, s.t. x x R d d s S (E1 DE ) When applying the novel two-level method, one critical point is identified in the second main iteration, and deterministic equivalent is solved over two scenarios, ss=(3,3), (5,5). The results obtained are equal to those obtained by using all vertices, however, the size of the problem is halved. It should be noted, that conventional methods, e.g. the sequential enumeration of vertices, identify two critical points (1,5) and (5,5), from which the first one is redundant (compare to example P7 in ovak Pintarič and ravanja, 008). Table 1: Results of the first example ominal point ominal point + 4 vertices ominal point + 1 critical vertex Z 494,6 047,8 047,8 d 1 3,484 4,896 4,896 d 7,806 1,701 1,701 o. of constraints o. of variables HE design example A heat exchanger network (HE) under study is shown in Fig. 1. The problem comprises 4 uncertain parameters, T 3, T 5, CF(C1) and CF(C), which are defined by Beta distribution function, and form 16 vertices. The goal is to design flexible HE for prespecified deviations of uncertain parameters. Figure 1. HE design The problem is firstly solved at the nominal point yielding inflexible design (first column of Table ). ext, a rigorous Gaussian quadrature is used to solving a multiscenario deterministic equivalent at 5 4 = 65 quadrature points and 16 vertices. This method produces flexible design (second column of Table ) in which the total HE area (A) is larger than in inflexible design. Finally, the proposed two-level approach is used which identifies two critical points, and deterministic equivalent is solved over the set of three scenarios: nominal point and two critical points. HE design obtained in this way (column 3) is flexible, the objective value is close to the Gaussian one, and the

5 A novel method for identification of critical points 5 computational effort is orders of magnitude lower. ote, that three critical points are identified by using sequential enumeration of vertices or other heuristic methods. Table : Results of HE example ominal point Gaussian Quadrature Reduced determ. equivalent Cost (EUR/year) 46,145 48,471 48,48 A 1 (m ) A (m ) A 3 (m ) A 4 (m ) A 5 (m ) A (m ) o. of scenarios o. of variables o. of constraints CPU (s) Conclusion A formulation of significantly reduced deterministic equivalent of a two-stage stochastic problem is presented. The set of scenarios in the reduced equivalent is defined as the union of the nominal point and critical points. The expected objective value is approximated at the nominal point, while critical points assure feasibility of the firststage design variables over the prespecified deviations of uncertain parameters. A novel method for efficient identification of critical points is presented, which starts with solving the one-scenario problem, and then adds critical points to the gradually growing multi-scenario problem. Additional critical point is identified at each major iteration by solving max-min problem using external functions in Gams. Two examples illustrate that the proposed method generates smaller set of critical points than other methods. References J. Acevedo, E.. Pistikopoulos, 1998, Stochastic optimization based algorithms for process synthesis under uncertainty, Computers & Chemical Engineering,, 4/5, J. Dupačová,. Gröwe-uska, W. Römisch, 003. Scenario reduction in stochastic programming: An approach using probability metrics. Mathematical Programming, A 95, R. aruppiah, M. Martín, I. E. Grossmann, 010, A simple heuristic for reducing the number of scenarios in two-stage stochastic programming, Computers and Chemical Engineering, 34, 8, Z. ovak Pintarič, Z. ravanja, 004, A strategy for MILP synthesis of flexible and operable processes, Computers and Chemical Engineering, 8, 6/7, Z. ovak Pintarič, Z. ravanja, 008, Identification of critical points for the design and synthesis of flexible processes, Computers and Chemical Engineering, 3, 7, S. Pineda, A. J. Conejo, 010, Scenario reduction for risk averse electricity trading, IET Generation, Transmission & Distribution, 4, 6,