An adaptive discretization MINLP algorithm for optimal synthesis of decentralized energy supply systems

Decentralized energy supply systems (DESS) are highly integrated and complex systems designed to meet time-varying energy demands, e.g., heating, cooling, and electricity. The synthesis problem of DESS addresses combining various types of energy conversion units, choosing their sizing and operations to maximize an objective function, e.g., the net present value. In practice, investment costs and part-load performances are nonlinear. Thus, this optimization problem can be modeled as a nonconvex mixed-integer nonlinear programming (MINLP) problem. We present an adaptive discretization algorithm to solve such synthesis problems containing an iterative interaction between mixed-integer linear programs (MIPs) and nonlinear programs (NLPs). The proposed algorithm outperforms state-of-the-art MINLP solvers as well as linearization approaches with regard to solution quality and computation times on a test set obtained from real industrial data, which we made available online.     

Artificial lift infrastructure planning of shale gas horizontal wells considering endogenous and exogenous uncertainties

Artificial lift methods (ALMs) lift the accumulated fluids from horizontal shale-gas-producing wells and help sustain well performance. An artificial lift infrastructure plan includes the selection of ALMs and their operating schedule. This paper presents two discrete-time large-scale nonconvex mixed-integer nonlinear programming models to solve the artificial lift infrastructure planning problem. Two equivalent mixed-integer linear programming models are formulated using the special structure of the nonlinear terms. A set of valid inequalities is defined to tighten the models and shorten solution times to two orders of magnitude, considering well production limitations. We incorporate endogenous uncertainty in ALM-dependent production rates and exogenous uncertainty in shale gas prices into the models. For a hypothetical case study under only endogenous uncertainties, the value of the stochastic solution is 5%. For the same case study, the exogenous uncertainty in gas prices does not change the optimum solution.     

Optimal design of reverse-osmosis networks for wastewater treatment

An optimization study of reverse-osmosis networks (RON) for wastewater treatment has been carried out by describing the system as a nonconvex mixed-integer nonlinear problem (MINLP). A mixed-integer linear problem (MILP) is derived from the original nonlinear problem by the convex relaxation of the nonconvex terms in the MINLP to provide bounds for the global optimum. The MILP model is solved iteratively to supply different initial guesses for the nonconvex MINLP model. It is found that such a procedure is effective in finding local optimum solutions in reasonable time and overcoming possible convergence difficulties associated with MINLP local search methods. Examples of water desalination and wastewater treatment from the pulp and paper industry are considered as case studies to illustrate the proposed solution strategy.     

Using convex nonlinear relaxations in the global optimization of nonconvex generalized disjunctive programs

In this paper we present a framework to generate tight convex relaxations for nonconvex generalized disjunctive programs. The proposed methodology builds on our recent work on bilinear and concave generalized disjunctive programs for which tight linear relaxations can be generated, and extends its application to nonlinear relaxations. This is particularly important for those cases in which the convex envelopes of the nonconvex functions arising in the formulations are nonlinear (e.g. linear fractional terms). This extension is now possible by using the latest developments in disjunctive convex programming. We test the performance of the method in three typical process systems engineering problems, namely, the optimization of process networks, reactor networks and heat exchanger networks.     

Sustainable scheduling of batch processes under economic and environmental criteria with MINLP models and algorithms

We address the bi-criterion optimization of batch scheduling problems with economic and environmental concerns. The economic objective is expressed in terms of productivity, which is the profit rate with respect to the makespan. The environmental objective is evaluated by means of environmental impact per functional unit based on the life cycle assessment methodology. The bi-criterion optimization model is solved with the ε-constraint method. Each instance is formulated as a mixed-integer linear fractional program (MILFP), which is a special class of non-convex mixed-integer nonlinear programs. In order to globally optimize the resulting MILFPs effectively, we employ the tailored reformulation-linearization method and Dinkelbach's algorithm. The optimal solutions lead to a Pareto frontier that reveals the tradeoff between productivity and environmental impact per functional unit. To illustrate the application, we present two case studies on the short-term scheduling of multiproduct and multipurpose batch plants.     

Design,synthesis and scheduling of multipurpose batch plants via an effective continuous-time formulation

Design, synthesis and scheduling issues are considered simultaneously for multipurpose batch plants. An earlier proposed continuous-time formulation for scheduling is extended to incorporate design and synthesis. Processing recipes are represented by the State-Task Network (STN). The superstructure of all possible plant designs is constructed according to the potential availability of all processing/storage units. The proposed model takes into account the trade-offs between capital costs, revenues and operational flexibility. Computational studies are presented to illustrate the effectiveness of the proposed formulation. Both linear and nonlinear models are included, resulting in MILP and mixed-integer nonlinear programming (MINLP) problems, respectively. The MILP problems are solved using a branch and bound method. Globally optimal solutions are obtained for the nonconvex MINLP problems based on a key property that arises due to the special structure of the resulting models. Comparisons with earlier approaches are also presented.     

Simultaneous mixed-integer dynamic scheduling of processes and their energy systems

Increasingly volatile electricity prices make simultaneous scheduling optimization desirable for production processes and their energy systems. Simultaneous scheduling needs to account for both process dynamics and binary on/off-decisions in the energy system leading to challenging mixed-integer dynamic optimization problems. We propose an efficient scheduling formulation consisting of three parts: a linear scale-bridging model for the closed-loop process output dynamics, a data-driven model for the process energy demand, and a mixed-integer linear model for the energy system. Process dynamics is discretized by collocation yielding a mixed-integer linear programming (MILP) formulation. We apply the scheduling method to three case studies: a multiproduct reactor, a single-product reactor, and a single-product distillation column, demonstrating the applicability to multiple input multiple output processes. For the first two case studies, we can compare our approach to nonlinear optimization and capture 82% and 95% of the improvement. The MILP formulation achieves optimization runtimes sufficiently fast for real-time scheduling.     

An improved piecewise outer-approximation algorithm for the global optimization of MINLP models involving concave and bilinear terms

In this paper a new version of the Outer Approximation for Global Optimization Algorithm by Bergamini et al. [Bergamini, M.L., Aguirre, P., & Grossmann, I.E. (2005a). Logic based outer approximation for global optimization of synthesis of process networks. Computers and Chemical Engineering 29, 1914] is proposed, in order to speed up the convergence in nonconvex MINLP models that involve bilinear and concave terms. Bounding problems are constructed replacing these nonconvex terms by piecewise linear underestimators. These problems, which correspond to mixed-integer linear programs, are solved to generate approximate solutions with improved objective value. When no further feasible solution can be found, this guarantees that the upper bound cannot be improved in the nonconvex problem, thus providing a termination criterion. The new algorithm is applied to five different synthesis problems in the areas of water networks, heat exchanger networks and distillation sequences. The results show a significant reduction in the computational cost compared with the previous version of the algorithm.     

Global optimization for scheduling refinery crude oil operations

In this work we present an outer-approximation algorithm to obtain the global optimum of a nonconvex mixed-integer nonlinear programming (MINLP) model that is used to represent the scheduling of crude oil movement at the front-end of a petroleum refinery. The model relies on a continuous time representation making use of transfer events. The proposed algorithm focuses on effectively solving a mixed-integer linear programming (MILP) relaxation of the nonconvex MINLP to obtain a rigorous lower bound (LB) on the global optimum. Cutting planes derived by spatially decomposing the network are added to the MILP relaxation of the original nonconvex MINLP in order to reduce the solution time for the MILP relaxation. The solution of this relaxation is used as a heuristic to obtain a feasible solution to the MINLP which serves as an upper bound (UB). The lower and upper bounds are made to converge to within a specified tolerance in the proposed outer-approximation algorithm. On applying the proposed technique to test examples, significant savings are realized in the computational effort required to obtain provably global optimal solutions.     

A global optimization algorithm for nonconvex generalized disjunctive programming and applications to process systems

A global optimization algorithm for nonconvex Generalized Disjunctive Programming (GDP) problems is proposed in this paper. By making use of convex underestimating functions for bilinear, linear fractional and concave separable functions in the continuous variables, the convex hull of each nonlinear disjunction is constructed. The relaxed convex GDP problem is then solved in the first level of a two-level branch and bound algorithm, in which a discrete branch and bound search is performed on the disjunctions to predict lower bounds. In the second level, a spatial branch and bound method is used to solve nonconvex NLP problems for updating the upper bound. The proposed algorithm exploits the convex hull relaxation for the discrete search, and the fact that the spatial branch and bound is restricted to fixed discrete variables in order to predict tight lower bounds. Application of the proposed algorithm to several example problems is shown, as well as a comparison with other algorithms.     

Global optimization of multiphase flow networks using spline surrogate models

A general modelling framework for optimization of multiphase flow networks with discrete decision variables is presented. The framework is expressed with the graph and special attention is given to the convexity properties of the mathematical programming formulation that follows. Nonlinear pressure and temperature relations are modelled using multivariate splines, resulting in a mixed-integer nonlinear programming (MINLP) formulation with spline constraints. A global solution method is devised by combining the framework with a spline-compatible MINLP solver, recently presented in the literature. The solver is able to globally solve the nonconvex optimization problems. The new solution method is benchmarked with several local optimization methods on a set of three realistic subsea production optimization cases provided by the oil company BP.     

Multi‐period planning,design, and strategic models for long‐term,quality‐sensitive shale gas development

In this work we address the long‐term, quality‐sensitive shale gas development problem. This problem involves planning, design, and strategic decisions such as where, when, and how many shale gas wells to drill, where to lay out gathering pipelines, as well as which delivery agreements to arrange. Our objective is to use computational models to identify the most profitable shale gas development strategies. For this purpose we propose a large‐scale, nonconvex, mixed‐integer nonlinear programming model. We rely on generalized disjunctive programming to systematically derive the building blocks of this model. Based on a tailor‐designed solution strategy we identify near‐global solutions to the resulting large‐scale problems. Finally, we apply the proposed modeling framework to two case studies based on real data to quantify the value of optimization models for shale gas development. Our results suggest that the proposed models can increase upstream operators’ profitability by several million U.S. dollars. © 2016 American Institute of Chemical Engineers AIChE J, 62: 2296–2323, 2016     

Generalized capital investment planning of oil-refineries using MILP and sequence-dependent setups

Due to quantity times quality nonlinear terms inherent in the oil-refining industry, performing industrial-sized capital investment planning (CIP) in this field is traditionally done using linear (LP) or nonlinear (NLP) models whereby a gamut of scenarios are generated and manually searched to make expand and/or install decisions. Though mixed-integer nonlinear (MINLP) solvers have made significant advancements, they are often slow for large industrial applications in optimization; hence, we propose a more tractable approach to solve the CIP problem using a mixed-integer linear programming (MILP) model and input–output (Leontief) models, where the nonlinearities are approximated to linearized operations, activities, or modes in large-scaled flowsheet problems. To model the different types of CIP's known as revamping, retrofitting, and repairing, we unify the modeling by combining planning balances with scheduling concepts of sequence-dependent changeovers to represent the construction, commission, and correction stages explicitly in similar applications such as process design synthesis, asset allocation and utilization, and turnaround and inspection scheduling. Two motivating examples illustrate the modeling, and a retrofit example and an oil-refinery investment planning problem are also highlighted.     

Active constraint strategy for flexibility analysis in chemical processes

It is shown in this paper that by exploiting properties of limiting constraints for flexibility in a design, problems for flexibility analysis can be formulated as mixed-integer optimization problems. Formulations are derived when control variables are present or not, and when equalities are eliminated or handled explicitly. These formulations do not rely on the assumption that critical parameter values are vertices, nor do they require exhaustive vertex searches. The case of linear constraints reduces to standard MILP problems, while for the nonlinear case a novel active constraint strategy is proposed and its theoretical properties are analyzed. Examples are presented for both rigorous and screening calculations.     

Linear method for the design of shell and tube heat exchangers using the Bell–Delaware method

In this article, we present a rigorous reformulation of the Bell–Delaware model for the design optimization of shell and tube heat exchanger to obtain a linear model. We extend a previously presented methodology1,2 of rigorously reformulate the mixed-integer nonlinear programing Kern model and we add disjunctions to automatically choose the different correlations to calculate heat transfer coefficients and pressure drop under different flow regimes. The linear character of the formulation allows the identification of the global optimum, even using conventional optimization algorithms. The proposed mixed-integer linear programming formulation with the Bell–Delaware method is able to identify feasible solutions for the design of heat exchangers at a lower cost than those obtained through conventional design formulations in the literature. Comparisons with the Kern method also indicate an average 22% difference (usually lower) in area.     

Sustainable ammonia production through process synthesis and global optimization

Current ammonia production technologies have a significant carbon footprint. In this study, we present a process synthesis and global optimization framework to discover the efficient utilization of renewable resources in ammonia production. Competing technologies are incorporated in a process superstructure where biomass, wind, and solar routes are compared with the natural gas-based reference case. A deterministic global optimization-based branch-and-bound algorithm is used to solve the resulting large-scale nonconvex mixed-integer nonlinear programming problem (MINLP). Case studies for Texas, California, and Iowa are conducted to examine the effects of different feedstock prices and availabilities. Results indicate that profitability of ammonia production is highly sensitive to feedstock and electricity prices, as well as greenhouse gas (GHG) restrictions. Under strict 75% GHG reductions, biomass to ammonia route is found to be competitive with natural gas route, whereas wind and solar to ammonia routes require further improvement to compete with those two routes. © 2018 American Institute of Chemical Engineers AIChE J, 65: e16498 2019     

Waste high-density polyethylene recycling process systems for mitigating plastic pollution through a sustainable design and synthesis paradigm

This article addresses the sustainable design and synthesis of open-loop recycling process of waste high-density polyethylene (HDPE) under both environmental and economic criteria. We develop by far the most comprehensive superstructure for producing monomers, aromatic mixtures, and fuels from waste HDPE. The superstructure optimization problem is then formulated as a multi-objective mixed-integer nonlinear fractional programming (MINFP) problem to simultaneously optimize the unit net present value (NPV) and unit life cycle environmental impacts. A tailored global optimization algorithm integrating the inexact parametric algorithm with the branch-and-refine algorithm is applied to efficiently solve the resulting nonconvex MINFP problem. Results show that the optimal unit NPV ranges from $107.2 to $151.3 per ton of HDPE treated. Moreover, the unit life cycle greenhouse gas emissions of the most environmentally friendly HDPE recycling process are 0.40 ton CO₂-eq per ton of HDPE treated, which is 63% of that of the most economically competitive process design.     

Operational strategy and planning for raw natural gas refining complexes: Process modeling and global optimization

Optimal operational strategy and planning of a raw natural gas refining complex (RNGRC) is very challenging since it involves highly nonlinear processes, complex thermodynamics, blending, and utility systems. In this article, we first propose a superstructure integrating a utility system for the RNGRC, involving multiple gas feedstocks, and different product specifications. Then, we develop a large‐scale nonconvex mixed‐integer nonlinear programming (MINLP) optimization model. The model incorporates rigorous process models for input and output relations based on fundamentals of thermodynamics and unit operations and accurate models for utility systems. To reduce the noncovex items in the proposed MINLP model, equivalent reformulation techniques are introduced. Finally, the reformulated nonconvex MINLP model is solved to global optimality using state of the art deterministic global optimization approaches. The computational results demonstrate that a significant profit increase is achieved using the proposed approach compared to that from the real operation. © 2016 American Institute of Chemical Engineers AIChE J, 63: 652–668, 2017     

Simultaneous mixed-integer dynamic optimization for integrated design and control

We consider strategies for integrated design and control through the robust and efficient solution of a mixed-integer dynamic optimization (MIDO) problem. The algorithm is based on the transformation of the MIDO problem into a mixed-integer nonlinear programming (MINLP) program. In this approach, both the manipulated and controlled variables are discretized using a simultaneous dynamic optimization approach. We also develop three MINLP formulations based on a nonconvex formulation, the conventional Big-M formulation and generalized disjunctive programming (GDP). In addition, we compare the outer approximation and NLP branch and bound algorithms on these formulations. This problem is applied to a system of two series connected continuous stirred tank reactors where a first-order reaction takes place. Our results demonstrate that the simultaneous MIDO approach is able to efficiently address the solution of the integrated design and control problem in a systematic way.     

Challenges in the application of mathematical programming in the enterprise-wide optimization of process industries

Enterprise-wide optimization (EWO) has become a major goal in the process industries due to the increasing pressures for remaining competitive in the global marketplace. EWO involves optimizing the supply, manufacturing and distribution activities of a company to reduce costs, inventories and environmental impact, and to maximize profits and responsiveness. Major operational items include planning, scheduling, real-time optimization and control. We provide an overview of EWO in terms of a mathematical programming framework. We first provide a brief overview of mathematical programming techniques (mixed-integer linear and nonlinear optimization methods), as well as decomposition methods, stochastic programming and modeling systems. We then address some of the major challenges involved in the modeling and solution of these problems. Finally, we describe several applications to show the potential of this area.