The processing of a raw material is a phenomenon that varies its quantity and quality along a specific network and logics and logistics to transform it into final products. To capture the production framework in a mathematical programming model, a full space formulation integrating discrete design variables and quantity-quality relations gives rise to large scale non-convex mixed-integer nonlinear models, which are often difficult to solve. In order to overcome this problem, we propose a phenomenological decomposition heuristic to solve separately in a first stage the quantity and logic variables in a mixed-integer linear model, and in a second stage the quantity and quality variables in a nonlinear programming formulation. By considering different fuel demand scenarios, the problem becomes a two-stage stochastic programming model, where nonlinear models for each demand scenario are iteratively restricted by the process design results. Two examples demonstrate the tailor-made decomposition scheme to construct the complex oil-refinery process design in a quantitative manner.
Phenomenological Decomposition Heuristic for Process Design Synthesis of Oil-Refinery UnitsBrenno C. Menezes,*a Jeffrey D. Kelly,b Ignacio E. Grossmannc
aRefining Optimization, PETROBRAS Headquarters, Rio de Janeiro, 20231-030, BrazilbIndustrial Algorithms, 15 St. Andrews Road, Toronto, ON, M1P 4C3, CanadacDepartment of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA15213, United States.
AbstractWe propose a decomposition algorithm to solve the non-convex mixed-integer nonlinear process design synthesis of oil-refinery units in a two-stage stochastic programming model with a second stage complete recourse in crude-oil diet, processing, and fuel imports. The quantity-logic-quality phenomena or qualogistics problem is solved considering, in a first stage, quantity and logic variables in a mixed-integer linear model and, in a second stage, quantity and quality variables in a nonlinear programming formulation. Iteratively, nonlinear models of each product demand scenario are restricted by the process design results. An industrial-sized example demonstrates the tailor-made decomposition scheme to construct the complex oil-refinery process design framework in a quantitative manner.
1. IntroductionMulti-period and multi-scenario process design synthesis models integrating discrete, [0,1] or logic variables and quantity-quality relations gives rise to large scale non-convex mixed-integer nonlinear (MINLP) models often difficult to solve. In order to overcome this problem, a decomposition solution method denoted here as phenomenological decomposition heuristic (PDH) is proposed to avoid full space MINLP models, which are decomposed, partitioned, or separated into two simpler submodels namely logistics (quantity and logic) and quality (quantity and quality) optimization problems. The logistics model solves an mixed-integer linear (MILP) problem with quantity and logic variables subject to quantity and logic balances and constraints. Quality optimization solves an nonlinear (NLP) model for quantity and quality variables subject to quantity and quality balances and constraints after the logic variables have been fixed at the values obtained from the solution of the logistics optimization.
The proposed MILP-NLP decomposition resembles the well-studied approach suggested in Benders decomposition where “complicating” variables (in this case, binary) are fixed such that a simpler problem may be solved, which are later freed again for the new iteration (Geoffrion, 1972). A similar method has also been applied for a different purpose, namely that of integrating decentralized decision-making systems through a hierarchical decomposition heuristic (HDH) (Kelly and Zyngier, 2008). In the context of the integration between logistics and quality problems, the coordinator (logistics MILP subproblem) would send what we call logic pole-offers to the
2 B. Menezes et al.
cooperator (quality NLP subproblem), which in turn would send back logic pole-offsets to the coordinator. This procedure continues until convergence is achieved, hopefully providing at least a feasible MINLP solution.
A phenomenological decomposition example considering an MINLP crude-oil scheduling problem (Mouret et al., 2009) compares the full space solution and its decomposed MILP-NLP problem by neglecting the pooling or blending nonlinear constraints in the MILP model, and then composing the model in an NLP problem by relating quantity and quality variables for the binary results found in the MILP model. In their work, the full space solution becomes intractable for industrial-sized examples, but they are solved in MILP-NLP decomposition with objective function gap between both solutions lower than 4%. Only a small example considering low number of time slots is solved using the MINLP formulation, which yields the same result found in the decomposed solution, but with higher computational expense.
Other decomposition methods, applied in large scale MILP process industry problems (You et al., 2011; Corsano et al., 2014) and relying on relaxations and primal and dual bouding information, showed that the bi-level decomposition (Iyer and Grossmann, 1998) requires smaller computational times leading to solutions that are much closer to the global optimum when compared to the full space solution and to Lagrangean decomposition (Guignard and Kim, 1987). A cross-decomposition algorithm (Mitra et al., 2014), combining Benders and scenario based Lagrangean decomposition in two-stage stochastic MILP problems with complete recourse, demonstrates a reduction of iterations and stronger lower bounds compared to pure multi-cut Benders decomposition.
2. Problem statementThe full designed problem (FDP) addressed in this paper can be stated as follows. Given future product demand scenarios sc, the decomposed MINLP problem based on phenomenological or quantity-quality decomposition heuristic consists of determining the expansion of existing units and installation of new units in petroleum refineries, which are defined by binary variables yt related to the investment decision and continuous variables xt that are the size of the new capacities to be installed.
The objective function (1) maximizes net present value (NPV) and consists of cash
flows (CF) from operational gains ∑sc
π sc CFop(x tsc) and investment costs
CF¿ ( xt , y t ), where πsc is the probability for scenario sc with ∑sc
π sc=1.
(FDP)max NPV =∑t (∑sc
πsc CF op(x tsc)−CF ¿ ( x t , y t )) (1)
s . t . A t x t+Bt y t ≤ bt∨ t (2)
Dt y t ≤ d t∨t (3)
x tsc ≤ x t∨sc , t (4)
Phenomenological Decomposition Heuristic for Process Desing Synthesis of Oil-Refinery Units 3
htsc (x t
sc )=0∨ sc , t (5)
x t , xtsc∈R+¿ , y t∈ { 0,1}¿ (6)
The investment layer in constraints (2) and (3) controls the capacity expansion/installation that is linked to the operational layer by Eq (4), e.g. if capacity is expanded/installed, additional operational decisions over the scenarios are available in
x tsc. Eq (5) specifies linear and nonlinar operational constraints for each scenario sc.
The PDH decomposition of the FPD problem considering operational scenarios is a two-stage stochastic programming problem with the first stage problem (FS) comprising the investment and operational layers in an MILP model (logistics optimization), and the second stage problems (SSsc) in an NLP model (quality optimization), in which recourse in crude-oil diet, processing, and fuel imports permits the problem to match the different demand scenarios despite the investment decision in the first stage.
3. Phenomenological decomposition heuristic algorithmTo formulate the MILP and NLP objective functions, the problem’s time horizon comprises investment t and operational t0 time periods as seen in Figure 1, although the problem is only performed at each t. After project executions over each t with time duration of ∆Tt, a new production framework is considered within the following time
periods to determine new operational gains given by throughputs QFu , tsc of crude-oil,
imports, and sales (u∈UG) that is considered fixed within the investment time period t, otherwise there would be needed to run the problem every each t0, that results in a problem combinatorially explosive with small or no influence in the discrete decisions.
Prices pru ,t 0 are considered varying annually to improve the accurancy of the NPV
calculation.
The FS problem maximizes the NPV MILPfunction (7) considering gains from operational scenarios and costs from investments yu,t to be approved among the process units u (u∈UI). Constraints (8) to (12) are the capacity planning equations to add new
capacities QN u ,t until the next-to-last time period (t < tend) considering capital available
CAPt, lower and upper bounds of the additions (QN u ,tL
and QN u ,tU
), and varying and
fixed cost coefficients ∝u ,t and βu ,t , respectively. Eq (12) represents binary constraints to reduce the tree search related to investment in groups of units with the same functionality and addresses process unit sequence-dependency investment based on the possible connectivity in the oil-refinery design. Eq (13) is the linking constraint, where
unit throughputs QFu , tsc
despite sc is lower than its current capacity QCu , t. Eq (14)
represents linear operational constraints, by neglecting the nonlinear relations or linearizing them.
(FS)max NPV MILP=∑t (∑t0=t i
t 0=tf
∑sc
∑u∈U G
πsc pru , t0QFu ,t
sc
(1+irt )t0
− ∑u∈U I
∝u ,t QN u ,t+ βu , t yu ,t
(1+ir t )∑
t
∆t−∆t1 )(7)
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s . t . ∑u∈U I
∝u ,t QN u , t+βu ,t yu , t ≤CAPt∨t <t end (8)
QCu , t+1=QCu ,t+QNu , t∨u∈U I , t<t end (9)
QN u ,tL yu , t ≤QN u ,t ≤ QN u ,t
U yu , t∨u∈U I , t<t end (10)
∑t<tend
yu , t ≤1∨u∈U I (11)
Du , t yu ,t ≤ du ,t∨u∈U I , t< tend(12)
QFu , tsc ≤QCu ,t∨u∈U E , sc , t (13)
~hu ,t
sc (xu , tsc )=0∨u∈U E , sc , t (14)
QCu , t ,QN u , t ,QF u ,tsc , xu ,t
sc ∈R+¿ , yu, t∈ { 0,1}¿ (15)
Figure 1. PDH algorithm flowchart.
The second stage SSsc problem of each scenario maximizes the NPV NLPsc function (16)
for fixed investment decisions QN u ,tand yu , t and unit throughputs upper bounds lower
than the capacities from the FS problem, so it means QFu , tsc ≤Q Cu ,t for existing units
(u∈UE) (both previously existing and those installed). Eq (17) represents nonlinear equations for crude-oil dieting, processing, and blending that can be found in Menezes et al. (2013) and Menezes et al. (2014).
( SSsc ) max NPV NLPsc =∑
t (∑t 0=ti
t0=t f
∑u∈U G
pru ,t0QFu ,t
sc
(1+ir t)t0
−∑u∈U I
∝u ,t QN u ,t +βu ,t yu ,t
(1+ir t)∑
t
∆t−∆t1 )∨ sc(16)
s . t . hu ,tsc ( xu ,t
sc )=0∨u∈U E , sc , t (17)
QFu , tsc , xu , t
sc ∈R+¿ ¿ (18)
Figure 2 shows the PDH algorithm flow. Single-period NLP operational planning problem for the existing units generates optimized initial points considering profit as goal in a warm-start phase. The details of the algorithm can be found in Menezes (2014).
Phenomenological Decomposition Heuristic for Process Desing Synthesis of Oil-Refinery Units 5
Figure 2. PDH algorithm flowchart.
4. Illustrative exampleWe apply the proposed decomposition to capacity planning of multi-site refineries in São Paulo State supply chain in Brazil, where the investments in new capacities for both expansion and installation of units are defined in the first stage and the full operation including the nonlinearities are defined in the second stage with complete recourse that brings new yields, rates, etc. to the first stage.
As shown in Table 1, the decomposition solution converged after 4 iterations, when the investment results between consecutive iterations are coincident (4th iteration in Table 2)
within a 5% gap between the NPV MILP (FS) and the average-NPV NLPsc (SSsc) solutions.
The indices r, u, and n in Table 2 mean refinery, unit, and number of the unit, respectively. The full space MINLP problem was tested in several solvers, but it get infeasible in the root relaxation. Table 3 shows the model statistics of the problems. Note the number of equations and variables in the NLP problems change over the time because of different investment decisions. The NLP solver used is CONOPT and the MILP solver is CPLEX.
Table 1. São Paulo State refineries example: MILP and NLP solutions (in billions of U.S. dollars).
iteration →
Optimality gap (%)
sc=1
sc=2
sc=3
gap (%)
1st
2nd
3rd
4th
8,063
0.9573
6,722
6,567
6,099
7,812
0.9937
8,429
8,287
7,777
7,893
0.9982
7,856
7,719
7,099
7,934
0.9914
7,866
7,730
7,139
4.519.9 4.5 4.2
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Table 2. Capacity expansions (exp) and installations (ins) (in 1,000 m3/d).
5. ConclusionWe have described how to iteratively solve a qualogistics or MINLP capital investment problem using MILP and NLP sub-solvers configured in a coordinated manner that resulted in a MILP-NLP solutions gap within 5%. This same technique can be applied to any advanced planning and scheduling MINLP problem in the process industries given our assertion that these problems can be phenomenologically modeled using the QLQP (quantity-logic-quality phenomena) attributes to concatenate the sub-problems. The major advantage of the PDH approach is that each sub-problem can be isolated and thoroughly investigated to debug inconsistencies and unexpected solutions when they exist. Existing MINLP and global optimizers are treated as black-boxes and if reliable and relevant solutions are not obtained which is usually the case in practice, then little insight and analysis is afforded back to the development and/or deployment user.
Phenomenological Decomposition Heuristic for Process Desing Synthesis of Oil-Refinery Units 7
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