ADVANCES IN GLOBAL OPTIMIZATION FOR STANDARD, GENERALIZED, AND EXTENDED POOLING PROBLEMS WITH THE (EPA) COMPLEX EMISSIONS MODEL CONSTRAINTS

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1 ADVANCES IN GLOBAL OPTIMIZATION FOR STANDARD, GENERALIZED, AND EXTENDED POOLING PROBLEMS WITH THE (EPA) COMPLEX EMISSIONS MODEL CONSTRAINTS Ruth Misener, Chrysanthos E. Gounaris, and Christodoulos A. Floudas Department of Chemical Engineering, Princeton University Princeton, NJ Abstract We discuss recent advances in deterministic global optimization for (a) standard pooling problems, (b) generalized pooling problems, and (c) extended pooling problems. These three classes of pooling problems are ubiquitous in the chemical, petrochemical, manufacturing, supply chain, and wastewater treatment industries. The primary aim of pooling problems is to maximize the profit of a complex network consisting of refinery exit streams or wastewater streams, blending pools, and final products. The network is subject to constraints reflecting the material balances, and product quality restrictions. In (a), linear blending rules are employed, the nonconvexities are of the bilinear type, and the mathematical model is a nonconvex NLP. In (b), the existence of intermediate streams, the pools, and all connections are treated as discrete alternatives and the resulting model is a nonconvex MINLP. In (c), the EPA Complex Emissions Model (40CFR80.45, 2007), which legally certifies the emissions of reformulated and conventional gasoline, is introduced explicitly. Since several types of nonconvexities result from the mathematical formulation of the EPA model, we present novel theoretical and computational findings for piecewise-linear relaxations and the generation of convex envelopes using the edge-concave paradigm. We incorporate these relaxations into a global optimization algorithm and present computational results on the EPA toxics model. Keywords Pooling problems, Piecewise-linearization, EPA Complex Emissions Model Introduction In a petroleum refinery, final products are created by combining feed stocks emerging from distillation units, reformers, and catalytic crackers (DeWitt et al., 1989). Because of limited storage availability and transportation requirements, these streams are sent to common pools before being mixed into products (Visweswaran, 2009). A refinery uses the material from up to nine intermediate pools to create a plethora of final products such as three grades of gasoline, diesel fuel, aviation jet fuel and fuel oil (Rigby et al., 1995). Optimally combining intermediate stocks into final products is of a long-standing interest in the petroleum industry. As early as the 1950s, Exxon was using linear programming to improve blending schemes (Baker and Lasdon, 1985). The objective of these early models, like the more sophisticated models that followed, was to maximize profit while meeting product-specific constraints. Blending feed stocks became more challenging in the 1970s as recognition of environmental and health hazards limited the octane-enhancing additive tetra-ethyl lead (DeWitt et al., 1989; Meyer and Floudas, 2006). Other legislation, such as the Clean Air Act of 1970, restricted the sulfur content and volatility of gasoline. These environmental standards, coupled with limited availability of low-sulfur crude and new automobiles requiring high octane fuels (DeWitt et al., 1989), inspired extensive research interest into the pooling problem. The pooling problem involves a feed-forward network topology and a set of product quality restrictions. If intermediate storage pools were unnecessary, the problem could be expressed as a linear program, but monitoring pool composition requires nonconvex bilinear and, for large-scale problems, trilinear terms. In this paper, we review recent advances in deterministic global optimization for the standard pooling problem, which can be modeled as a nonconvex NLP and the generalized pooling problem, which can be modeled as a nonconvex MINLP. Finally, we introduce the extended pooling problem, Author to whom all correspondence should be addressed (floudas@titan.princeton.edu; Tel: (609) ; Fax: (609) ).

2 based on the EPA Complex Emissions Model (40CFR80.45, 2007), and discuss novel theoretical and computational findings for generating tight convex underestimators and incorporating the relaxations into a branch-and-bound scheme. Standard Pooling Problem In the standard pooling problem, the flow rates on a predetermined network structure of feed stocks, pooling tanks, and final products are optimized to maximize profit subject to quality constraints on the final product composition. Although the standard pooling problem considers only fuel qualities that blend linearly, bilinear terms arise from the quality balances about the pooling tanks (Floudas et al., 1989; Floudas and Aggarwal, 1990; Floudas, 2000; Tawarmalani and Sahinidis, 2002). Haverly (1978), who published the first algorithm to locally improve the classic pooling problem, observed that the solution to his algorithm depended on the starting point. But, in the high-throughput petroleum industry where a saving a fraction of a cent per gallon translates into large profits, even implementing local NLP solvers had a high impact. De- Witt et al. (1989) conservatively estimate that implementing the local optimizer OMEGA for gas blending improvements yielded 30 million dollars in annual revenue for Texaco. Interest in theoretically guaranteeing global optimality for the pooling problem led to a number of advances in the field of global optimization, especially with respect to biconvex and bilinear programming. Notable contributions have been made by Floudas et al. (1989) and Floudas and Aggarwal (1990), who designed a global search algorithm based on Generalized Benders Decomposition; Lodwick (1992), who determined implied variable bounds through preprocessing analysis; Floudas and Visweswaran (1990), Visweswaran and Floudas (1990), and Floudas and Visweswaran (1993), who developed the first rigorous deterministic global approach for biconvex and bilinear problems based on duality theory; Foulds et al. (1992), who implemented the bilinear envelopes of McCormick (1976) and Al-Khayyal and Falk (1983); Ben-Tal et al. (1994), who introduced the q-formulation; Adhya et al. (1999), who explored Lagrangian approaches; Quesada and Grossmann (1995), who used the reformulation-linearization technique of Sherali and Alameddine (1992) 1 ; Audet et al. (2000), who designed a branch and cut method for quadratic programs using four classes of linearizations; Tawarmalani and Sahinidis (2002), who proposed the pq-formulation; Varvarezos et al. (2008), who implemented additional refinery planning needs such as risk management into the optimization framework; Almutairi and Elhedhli (2009), who suggested a new Lagrangian relaxation for the pooling problem and demonstrated that their relaxation is often tighter than previously-developed Lagrangian relaxations; and Pham et al. (2009), who incorporated discretized pool qualities into an algorithm that quickly generates near-optimal results. To address the higher-order multilinear terms which arise in large-scale pooling problems, Meyer and Floudas (2003, 2004) derived explicit facets of the convex envelopes for trilinear monomials. Generalized Pooling Problem In recent years, researchers have studied a second class of pooling problems, known as generalized pooling problems. In the generalized pooling problem, inter-pool links are permitted and network components such as intermediate streams and pools are treated as discrete alternatives. The resulting nonconvex disjunctive program can be modeled as a MINLP. Figure 1 indicates the difficulty of the generalized pooling problem. Because each of the pipes depicted in Figure 1 may or may not be activated, the problem is combinatorially complex with respect to the binary decision variables and bilinear terms. The first researchers to consider the class of generalized pooling problems, Audet et al. (2004), considered a single, predetermined network topology of three feeds, two pools, and three products. Meyer and Floudas (2006) not only allowed flow between pools, but also substantially broadened the class of generalized pooling problems to include discrete decisions, such as whether to build a pool or pipeline. Meyer and Floudas (2006) considered an industrially-relevant topological superstructure, and optimized the network configuration using disjunctive programming. Although the specific test case in Meyer and Floudas (2006) applies to wastewater treatment plants, the bilinear terms of the wastewater treatment problem match those of the pooling problem. The industrial case study presented in Meyer and Floudas (2006) optimizes a network with seven sources, ten potential plants, one sink, and three relevant qualities. Figure 1: Representative superstructure for the generalized pooling problem Karuppiah and Grossmann (2006) studied a variant of the generalized pooling problem by optimizing the network topology of water systems. Using disjunctive programming, 1 Also see Sherali and Adams (1999) for a comprehensive study of the reformulation-linearization technique developed by Sherali and co-workers.

3 Karuppiah and Grossmann (2006) demonstrated substantial objective value improvement in optimizing integrated water systems rather than sequentially optimizing freshwater and wastewater systems. Although the combinatorial complexity of the generalized pooling problem leads to large models, both Meyer and Floudas (2006) and Karuppiah and Grossmann (2006) were able to solve industrially-relevant examples by incorporating piecewise-linear underestimators of bilinear terms into a global optimization algorithm. Based on these successes, Wicaksono and Karimi (2008) analyzed a variety of novel piecewise-linear underestimators of bilinear terms and showed that the relaxation schemes of Meyer and Floudas (2006) and Karuppiah and Grossmann (2006) can be improved using alternate mathematical representations. Extended Pooling Problem Environmental Protection Agency (EPA) Title 40 Code of Federal Regulations Part 80.45: Complex Emissions Model (40CFR80.45, 2007) codifies and legally certifies a mathematical model of reformulated gasoline (RFG) emissions based on the eleven fuel components recorded in Table 1. Final products exiting an oil refinery must comply with emissions standards, or upper bounds, on volatile organic (VOC MAX ), NO X (NOX MAX ) and toxics (TOX MAX ) emissions (40CFR80.41, 2008). Table 1: Fuel components in the EPA Complex Emissions Model bounded by the limits of RFG model accuracy. Var Fuel Quality Bounds Units 1 OXY oxygen wt% 2 SUL sulfur ppm 3 RVP Reid Vapor Press psi 4 E o F dist. frac vol% 5 E o F dist. frac vol% 6 ARO aromatics vol% 7 BEN benzene vol% 8 OLE olefins vol% 9 MTB MTBE wt% O 2 10 ETB ETBE wt% O 2 11 ETH ethanol wt% O 2 The extended pooling problem, which was introduced by Gounaris and Floudas (2007), incorporates Title 40 Code of Federal Regulations Part 80.45: Complex Emissions Model (40CFR80.45, 2007) and associated legislative bounds into the constraint set. The extended pooling problem restricts the volatile organic, NO X, and toxics emissions of RFG by appending three sets of emissions model equations and the following constraints to the standard pooling problem: VOC VOC MAX (1) NOX NOX MAX (2) TOX TOX MAX (3) where VOC MAX, NOX MAX, and TOX MAX are parameters satisfying applicable legislation. The equations that make up the EPA Complex Emissions Model are not only nonconvex, but also non-smooth. Figure 2 illustrates the non-smooth nature of exhaust benzene (BENZ), a component of toxics emissions, by plotting BENZ versus the fuel quality E300 with all other fuel qualities held constant. Additionally, the coefficients of the EPA model (40CFR80.45, 2007) change according to the time of year, region in the country, and the type of vehicle. In the following sections, both a MINLP formulation and a tight MILP relaxation of the extended pooling problem are presented. The MINLP representation of the toxics component of the EPA Complex Emissions Model is stated without explanation because an equivalent formulation of the emissions model was recently published (Furman and Androulakis, 2008). The MINLP representation and MILP relaxation of the extended pooling problem is then integrated into a global optimization algorithm and the extended pooling problem is solved using an example test case. Because of space constraints in this paper, only the toxics component of the emissions model is presented, and the complete the formulation, relaxation, and construction of a global optimization algorithm that includes the volatile organic and NO X emissions will be presented elsewhere. BENZ in mg/mile Nonlinear when E300 < E300 in volume % Constant when E300 > 95 Figure 2: BENZ vs. E300 with other fuel qualities constant Formulation of the Extended Pooling Problem Standard Portion of the Problem Table 2 defines the indices, variables, and parameters which formulate the extended pooling problem using a representation equivalent to the standard pooling problem p- formulation. The objective is to maximize profit or, equivalently, minimize negative profit: min i,l c i x i,l l,j d j y l,j i,j (d j c i ) z i,j. (4) Bounds limit the availability of each petroleum feed stock: A L i l x i,l + j Mass balances are defined around each pool: z i,j A U i i (5)

4 x i,l y l,j = 0 l. (6) i j Supply limits define the capacity of each pool: x i,l S l l. (7) The quality balances about a pool are: C i,k x i,l j i i p l,k y l,j = 0 l, k. (8) Augmenting the p-formulation, Gounaris and Floudas (2007) define outflow rates (of j ) and qualities (u j,k ). The nonconvex EPA model is integrated into the extended pooling problem by calculating the relevant emissions using the fuel qualities (u j,k ) at each product outflow. The outflow rate is: of j = y l,j + z i,j j (9) l i and the quality balances at the final products outflow are: (u j,k ) (of j ) = l p l,k y l,j + i C i,k z i,j j, k. (10) Table 2: Pooling problem p-formulation Indices i feed stocks from refinery j final products k fuel qualities in EPA Model l pools Vars p l,k quality k of pool l x i,l flow from stock i to pool l y l,j flow from pool l to product j z i,j flow from feed i to product j of j outflow rate of product j u j,k quality k of product j y E300, j binary switch for product j y ARO, j binary switch for product j Params c i cost of feed stock i d j revenue from product j A L i - AU i availability bounds on feed i C i,k quality k of feed stock i Dj L - DU j demand bounds for product j Pj,k L - P j,k U bounds on quality k for product j volumetric capacity of pool l S l Finally, hard bounds to define tight variable limits: 0 (x i,l ) min{a U i, S l, j 0 (y l,j ) min{s l, D U j, i D U j } (11) A U i } (12) 0 (z i,j ) min{a U i, D U j } (13) min C i,k (p l,k ) maxc i,k (14) i i D L j (of j ) D U j (15) P L j,k (u j,k ) P U j,k (16) Equations (4) (16) define the standard component of the extended pooling problem. Nonconvexities in this component of the extended pooling problem arise from the bilinear terms in the quality balances around the pools (Eq. (8)) and the products (Eq. (10)). EPA Complex Emissions Portion of the Problem The following MINLP formulation of the toxics model is presented without explanation, but a detailed discussion of constructing an equivalent MINLP representation can be found in Furman and Androulakis (2008). MINLP formulations of the volatile organic and NO X models will be presented in a future publication. The toxics emissions model is a function of the eleven fuel components presented in Table 1. These components are, in turn, functions of the outflow fuel qualities u j, k. For the most part, the two are identical: OXY j = u j,1 j E200 j = u j,4 j OLE j = u j,8 j ETB j = u j,10 j SUL j = u j,2 j BEN j = u j,7 j MTB j = u j,9 j ETH j = u j,11 j, but the value of RVP depends on the time of year: RVP j = { u j, 3 Summer 8.7 Winter the value E300 j is set to 95 vol% when u j, 5 95: (17) j, (18) u j, 5 95 (u L j, 5 95) y E300, j (19) u j, 5 95 (u U j, 5 95) (1 y E300, j ) (20) E300 j 95 (u U j, 5 95) y E300, j (21) E300 j 95 (u L j, 5 95) y E300, j (22) E300 j u j, 5 (u U j, 5 u L j, 5) (1 y E300, j ) (23) E300 j u j, 5 (u L j, 5 u U j, 5) (1 y E300, j ), (24) and the value ARO j is set to 10 vol% when u j, 6 10: u j, 6 10 (u L j, 6 10) y ARO, j (25) u j, 6 10 (u U j, 6 10) (1 y ARO, j ) (26) ARO j 10 (u U j, 6 10) (1 y ARO, j ) (27) ARO j 10 (u L j, 6 10) (1 y ARO, j ) (28) ARO j u j,6 (u U j, 6 u L j, 6) y ARO, j (29) ARO j u j,6 (u L j, 6 uu j, 6 ) y ARO, j. (30)

5 Also, MTB j, ETB j, and ETH j represent three of the four oxygen components, so they are constrained by OXY j : OXY j MTB j + ETB j + ETH j j. (31) Note that y E300, j and y ARO, j are binary decision variables representing disjunctions in the EPA Complex Emissions Model (40CFR80.45, 2007) such as the one illustrated in Figure 2. These binary variables make the extended pooling problem an MINLP rather than a nonconvex NLP like the standard pooling problem. Toxics emissions (TOX j ), is expressed as the sum of six components: exhaust benzene (BENZ j ), formaldehyde (FORM j ), acetaldehyde (ACET j ), 1,3-butadiene (BUTA j ), nonexhaust benzene (NEBENZ j ), and polycyclic organic matter (POM j ). The POM model is a linear multiple of the volatile organic exhaust emissions model and is excluded from this paper because of space. The toxics emissions model without the POM contribution is: TOX j =BENZ j + FORM j + ACET j + BUTA j + 10 NEBENZ j j (32) The five components of the simplified toxics emission model are presented in Eq. (33) (41). The model coefficients, which vary according to the time of year, region of the country, and whether the vehicle is a high (e = 1) or low (e = 2) emitter, are presented in 40CFR80.45 (2007) and Furman and Androulakis (2008). BENZ j = 2 e=1 BENZ(b) w T e e be(b) exp{t BE, e, j }, (33) t BE, e, j = c BE e, 1OXY j + c BE e, 2SUL j + c BE e, 3E300 j + FORM j = c BE e, 4ARO j + c BE e, 5BEN j (34) 2 e=1 FORM(b) w T e e fe(b) exp{t F, e, j } (35) t F, e, j = c F e, 1 E300 j + c F e, 2 ARO j + c F e, 3 OLE j + ACET j = c F e, 4 MTB j (36) 2 e=1 ACET(b) w T e e ae(b) exp{t A, e, j } (37) t A, e, j = c A e, 1SUL j + c A e, 2RVP j + c A e, 3E300 j + BUTA j = c A e, 4ARO j + c A e, 5MTB j + c A e, 6ETB j + c A e, 7ETH j (38) 2 e=1 BUTA(b) w T e e de(b) exp{t BU, e, j } (39) t BU, e, j = c BU e, 1OXY j + c BU e, 2SUL j + c BU e, 3E200 j + c BU e, 4 E300 j + c BU e, 5 ARO j + c BU e, 6 OLE j (40) NEBENZ j =α 1 BEN j + α 2 RVP j BEN j + α 3 BEN j MTB j + α 4 RVP 2 j BEN j+ α 5 RVP j BEN j MTB j + α 6 RVP 3 j BEN j + α 7 RVP 2 j BEN j MTB j (41) Relaxation of the Extended Pooling Problem To construct a MILP relaxation of the MINLP representation of the extended pooling problem, Eq. (8), (10), (33), (35), (37), (39), and (41) must be underestimated. In underestimating Eq. (8), we follow Meyer and Floudas (2006) and Karuppiah and Grossmann (2006) in constructing piecewise-linear underestimators for the bilinear terms. We use the recent results of Gounaris et al. (2008) which extend the work of Wicaksono and Karimi (2008) to choose the best mathematical representation of the underestimator. Specifically, we use a representation denoted nf4r to relax Eq. (8) (Gounaris et al., 2008). The bilinear terms p l, k y l, j in Eq. (8) are replaced with a placeholder variable: p L l, k yl l, j wp, y l, j, k pu l, k yu l, j. (42) To construct piecewise-linear bilinear underestimators, each fuel quality p l, k is ab initio partitioned into N segments according to the variable choice of Gounaris et al. (2008): p l, k (n) = p L l, k + n N (pu l, k pl l, k ) l, k, n = 0,, N (43) and a binary variable λ l, k (n) is introduced to activate one and only one domain segment: λ l, k (n) = { 1 if p l, k (n 1) p l, k p l, k (n) 0 else l, k, n = 1,, N p l, k (n 1) λ l, k (n) p l, k l, k, n = 1,, N (44) p l, k (n) λ l, k (n) (45) λ l, k (n) = 1 l, k. (46) Continuous variable y l, j, k (n), n = 1, N is a place holder for the flow rate y l, j in the bilinear relaxations: N y l, j = yl, L j + y l, j, k (n) (47)

6 0 y l, j, k (n) (yl, U j yl, L j) λ l, k (n). (48) The final relaxation of the Eq. (8) bilinear terms l, j, k is: 2 NEBENZ j MTB 2 j 2 NEBENZ j BEN 2 j = 0 0 j (58) = 0 0 j (59) w p, y l, j, k y L l, j p l, k + y l, j, k (n) w p, y l, j, k y U l, j p l, k + p l, k (n 1) (49) {p l, k (n) (50) ( y l, j, k (n) (yl, U j yl l, j ) λ l, k(n))} w p, y l, j, k yl, L j p l, k + {p l, k (n 1) (51) ( y l, j, k (n) (yl, U j yl, L j) λ l, k (n))} w p, y l, j, k yl, U j p l, k + p l, k (n) y l, j, k (n). (52) Because the bilinear terms in Eq. (10) are closely related to the ones in Eq. (8), we chose not to construct a piecewiselinear relaxation of the (u j,k ) (of j ) terms. Instead, we replaced each of the bilinear terms in Eq. (10) with the continuous variable wj, k and underestimated the terms using the envelopes of McCormick (1976): wj, k u L j,k of j + u j,k ofj L ul j,k ofl j (53) wj, k u U j,k of j + u j,k ofj U u U j,k ofj U (54) wj, k u L j,k of j + u j,k ofj U u L j,k ofj U (55) wj, k u U j,k of j + u j,k ofj L u U j,k ofj L (56) j, k We underestimated the convex Eqs. (33), (35), (37), and (39) using outer approximation. We partitioned the continuous variables t BE, e, j, t A, e, j, t F, e, j, and t BU, e, j and constructed supporting hyperplanes at each partition point. Equation (41), representing nonexhaust benzene emissions, is the final nonconvex equation in the extended pooling problem. Although Eq. (41) is not edge-concave, we use the paradigm of edge-concavity to efficiently generate a tight lower bound on NEBENZ. Tardella (1988/89, 2003, 2008) introduced edge-concave functions, a class of functions that admit a vertex polyhedral convex envelope. Using the theoretical results of Tardella (2003), Meyer and Floudas (2005) developed an algorithm to generate the convex envelope of any three-dimensional edge-concave function. According to Tardella (2003), a function on a box is edgeconcave if and only if it is componentwise concave, that is: 2 NEBENZ j RVP 2 j = 2α 4 BEN j + 6α 6 RVP j BEN j + 2α 7 MTB j BEN j 0 j (57) Since Eq. (58) and (59) are always true, the remaining task is to see when Eq. (57) is negative. Because 2 NEBENZ j RVP 2 j is edge-concave when: 0, Eq. (41) is not edge-concave. However: NEBENZ j α 4 RVP 2 j BEN j (60) α 4 = α α 6 RVP L j + α 7 MTB L j, (61) so we underestimate Eq. (60) using the algorithm of Meyer and Floudas (2006), which results in the facets of the convex envelope, and the remainder (α 4 RVP 2 j BEN j ) using the recursive arithmetic techniques of Maranas and Floudas (1995) and Ryoo and Sahinidis (2001). We will not explicitly state the linear equations describing the convex envelope of Eq. (60) because they change as bounds are tightened within a global optimization algorithm, but the method to develop the convex envelope can be found in Meyer and Floudas (2006). NEBENZ has a global minimum of 0, but the underestimate attained using only recursive arithmetic techniques is in Region 1 and in Region 2 (Maranas and Floudas, 1995; Ryoo and Sahinidis, 2001). Combining recursive techniques with the edge-concave algorithm of Meyer and Floudas (2005) improves the lower bound by more than 20%, to in Region 1 and in Region 2. The relaxations in this section are substituted for the nonlinear components presented in the Eq. (4) (41). The next section presents an extended pooling problem test case that integrates the MINLP formulation and MILP relaxation into a global optimization algorithm. Example of the Extended Pooling Problem Table 3: Cost (c i ) and Availability (A L i & AU i ) of Feed i i c i A L i A U i We use the topology illustrated in Figure 3 and defined by the parameters listed in Tables 3 6 as an extended pooling problem test case. The test case has four grades of petroleum feed stocks, three fuel additives, one pool, and two final products. The four feed stocks represent the characteristics of four intermediate stocks leaving the distillation, reforming, or catalytic cracking units of a refinery. Each feed stock is estimated to have a market value based on its composition. The pool has capacity S 1 = 300. In reality, transportation considerations may require additives to be mixed into the gasoline at a distribution station (e.g., ethanol is rarely transported by pipeline with other

7 gasoline components), but the test case described in this study simplifies the problem by assuming that additives are blended into the final products at the refinery. Table 4: Quality Bounds k on Product j (P L j, k & P U j, k ) k P1, L k P2, L k P1, U k P2, U k the continuous variables t BE, e, j, t A, e, j, t F, e, j, and t BU, e, j are partitioned into 32 segments to create an outer approximation for Eq. (33), (35), (37), and (41). Table 6: Quality k of Raw Material i (C j, k ) k C 1, k C 2, k C 3, k C 4, k C 5, k C 6, k C 7, k After solving both the full MILP relaxation to develop a lower bound and locally solving the original MINLP representation to obtain an upper bound, we perform strong branching on the product flow rates (of j ). After each strong branching step, the intermediate flow rates y l, j are optimally tightened by replacing the objective function in Eq. (4) with y l, j l, j and maximizing and minimizing the temporary objective function. We also optimally tighten the quality variables (p l, k & u j, k ) satisfying: P i P C i, k x i, l j y p l, j l, k (62) p l, k P i {zi, j C i, k}+ P l {p l, k y l, j } of j u j, k 0.005, (63) u j, k Figure 3: Topology for the extended pooling problem We developed a C++ program that interfaces with the linear solver CPLEX (ILOG, 2005) to minimize the MILP relaxations of the extended pooling problem. The upper bounds are obtained by making system calls to the local nonlinear solver MINOS (Murtagh et al., 2004) through the modeling language GAMS (Brooke et al., 2005). Note that system calls to GAMS slow the program substantially and an interface with an open-source nonlinear solver to generate upper bounds can lead to significant improvements. Table 5: Price (d j ) and Demand (Dj L & DU j ) of Product j j d j Dj L Dj U Using the upper and lower bounding strategies, we designed a global optimization algorithm to reach ǫ- convergence. Each of the p l, k y l, j terms in Eq. 8, are underestimated using Eq. (42) (52) with N = 16 segments and i.e., the variables deviating more than 0.5% from their value in relation to the other variables. For a toxics standard TOX MAX = 61, the optimality gap reduces to 0.49% (LB = ) after seconds on a Pentium 4 running Linux. Table 7 displays the variable values at this solution. Conclusion This paper has reviewed the recent advances in deterministic global optimization for (a) standard pooling problems, (b) generalized pooling problems, and (c) extended pooling problems. Although small to medium-sized standard pooling problems have been successfully addressed by a number of researchers, industrially-sized pooling problems, the combinatorially-complex generalized pooling problem, and the newly-introduced extended pooling problem offer interesting areas of research. Solving these pooling problems will help improve today s energy systems in a costeffective and environmentally conscious manner. Although these problems have proved challenging, new global optimization methods, such as the relaxation techniques presented in this paper, may lead to major improvements.

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