This article presents new methods for robustly simulating process flowsheets containing nondifferentiable models, using recent advances in exact sensitivity analysis for nonsmooth functions. Among other benefits, this allows flowsheeting problems to be equipped with newly developed nonsmooth inside-out algorithms for nonideal vapor–liquid equilibrium calculations that converge reliability, even when the phase regime at the results of these calculations is unknown a priori. Furthermore, process models for inherently nonsmooth unit operations may be seamlessly integrated into process flowsheets, so long as computationally relevant generalized derivative information is computed correctly and communicated to the flowsheet convergence algorithm. These techniques may be used in either sequential-modular simulations or simulations in which the most challenging modules are solved using tailored external procedures, while the remaining flowsheet equations are solved simultaneously. This new nonsmooth flowsheeting strategy is capable of solving process simulation problems involving nonsmooth models more reliably and efficiently than the algorithms implemented in existing software, and, in some cases, allows for the solution of problems that are beyond the capabilities of classical approaches. As examples of the latter, it will be shown that the nonsmooth approach is particularly well-suited for highly accurate simulation of natural gas liquefaction processes, in which many nonsmooth modeling elements are present in combination with nonideal thermodynamic behavior and complex heat-transfer considerations.

•The moments of a particle size distribution (PSD) are only a summary description.•Still, they imply bounds on various industrially relevant descriptions of the PSD.•These bounds are calculated efficiently by solving convex optimization problems.•These bounds complement and validate distribution reconstruction methods.Many chemical engineering processes involve a population of particles with a distribution of sizes that changes over time. Because calculating the time evolution of the full particle size distribution (PSD) is computationally expensive, it is common to instead calculate the time evolution of only finitely many moments of the distribution. The problem with moments is that they provide only a summary description of the PSD. In particular, they do not contain enough information to answer industrially relevant questions such as: How many particles are there in the size range [a,b]? What is the shape of the distribution? What is its D10? While these questions cannot be answered exactly, in this paper, we demonstrate that one can efficiently calculate rigorous bounds on the answers by solving semidefinite programs. To the best of the authors’ knowledge this natural application of semidefinite programming to PSDs has, until now, gone unnoticed.Download high-res image (66KB)Download full-size image

•Phase changes in multistream heat exchangers can be detected with a nonsmooth model.•Flash calculations with single-phase outlets can be handled with a nonsmooth model.•Complex simulations of a natural gas liquefaction process can be performed.•Simulations do not require the solution of a difficult optimization problem.•Automatic differentiation outperforms finite differencing in the solution procedure.A new method for modeling phase changes in multistream heat exchangers (MHEXs) is presented. In many industrially relevant applications, streams in MHEXs will undergo phase changes between their inlet and outlet. In this model, nonsmooth equations are formulated which properly account for the existence or nonexistence of phases in heat integration, flash and physical property calculations in a MHEX. These new equations are used in conjunction with a recently developed nonsmooth model for MHEXs to create a compact equation system which can be used for the simulation and design of complex processes. Notably, this formulation does not involve the solution of a difficult optimization problem, since it avoids the use of either disjunctive or complementarity constraints. The robustness and functionality of the new formulation is illustrated through several simulations of the well-known Poly Refrigerant Integrated Cycle Operations (PRICO) process for liquefied natural gas production.

•Optimal shale oil and gas infrastructure investments in the U.S. are determined.•The investment horizon studied is 2015–2039.•Two scenario sets with different GDP and oil price assumptions are analyzed.•The importance of incorporating uncertainty into the model is discussed in detail.We present a comprehensive supply chain optimization model to determine optimal shale oil and gas infrastructure investments in the United States. The model encompasses multiple shale plays, commodities, plant locations, conversion technologies, transportation modes and both local and foreign markets. The dynamic evolution of supply, demand and price parameters and the uncertainty in parameter realizations are fully taken into account. Imposing two different scenario sets over a time horizon of twenty-five years, the model maximizes the expected net present value of the entire undertaking. We analyze the features of the optimal infrastructure investments and associated operating decisions, perform case studies which highlight the importance of incorporating uncertainty into the model and analyze the stability of the stochastic solutions as the degree of uncertainty changes. The overall opportunity set of investments is sparse, and there is a tendency for over-investment in new liquefied natural gas capacity when the uncertainties in future oil prices are not taken fully into account.

The economic production of algal biofuels requires novel strategies, such as microbial consortia and synthetic ecologies, to boost the productivity of open pond systems. These strategies have not been fully explored partly due to the lack of reliable and predictive process models. This study uses genome-based metabolic networks to build a process model of a raceway pond. This process model is used as a discovery tool for novel process strategies. First, an algal monoculture with flue gas sparging is modeled. Then, an oleaginous yeast monoculture is modeled. The yeast monoculture is O2 limited and the presence of algae in the culture would result in better resource utilization. Next, an algal/fungal raceway pond with a feed of cellulosic glucose is explored. Finally, an oleaginous yeast that can consume a glucose/xylose mix, resulting from the hydrolysis of lignocellulosic waste, is modeled. This model predicts biomass and lipids productivities comparable to those reported in the literature. Assuming 50% yield loss due to contamination and invasion, a simple economic analysis shows that an algae/yeast coculture can produce biodiesel at competitive prices, $2.01 per liter for pure glucose and $1.44 per liter for the sugar mix, whereas the algae monoculture can do so only at very short distances from a flue gas source. This modeling framework will enable the use of optimization algorithms in the design of open pond systems in the near future and will allow the exploration of novel strategies in bioprocesses employing microbial communities.

In this work, equality-constrained bilevel optimization problems, arising from engineering design, economics, and operations research problems, are reformulated as an equivalent semi-infinite program (SIP) with implicit functions embedded, which are defined by the original equality constraints that model the system. Using recently developed theoretical tools for bounding implicit functions, a recently developed algorithm for global optimization of implicit functions, and a recently developed algorithm for solving standard SIPs with explicit functions to global optimality, a method for solving SIPs with implicit functions embedded is presented. The method is guaranteed to converge to ϵ-optimality in finitely many iterations given the existence of a Slater point arbitrarily close to a minimizer. Besides the Slater point assumption, it is assumed only that the functions are continuous and factorable and that the model equations are once continuously differentiable.

Campaign continuous manufacturing (CM), characterized by relatively short operational windows such as a few weeks, is being explored as an alternative to traditional batch-wise manufacturing in the pharmaceutical industry. However, optimal operation in campaign CM can be challenging because of the significance of startup and shutdown phases, which can negatively affect on-specification production and plant economy. In this paper, the effectiveness and computational tractability of several known optimization approaches when applied to campaign CM are investigated. Inspired by the turnpike property in optimal control, a new approach is then proposed, which aims to maximize on-specification production explicitly rather than minimizing the startup/shutdown times as commonly adopted in high-volume industries. A main contribution in the new approach is that the resulting optimization formulation is guaranteed to be differentiable, despite the underlying hybrid dynamic system. Thus, it can be solved reliably using gradient-based algorithms. Case studies are presented to demonstrate the effectiveness of the proposed approach.

Time-varying, or dynamic, experiments can produce richer data sets than sequences of steady-state experiments using less material and time. A case study demonstrating this concept for microreactor experiments is presented. Beginning with five kinetic model candidates for the reaction of phenylisocyanate with t-butanol, an initial dynamic experiment showed that two of the five models gave a similar quality of fit to the experimental data, whereas the remaining three gave significantly poorer fits. Next an optimal experiment was designed to discriminate between the remaining two models. This drove the two models to differ significantly in quality, leaving a single model and a set of kinetic parameter values that adequately described the data. This method can be applied to future kinetic studies to reduce material use and experimental time while validating a dynamic model of the physics and chemical kinetics.

This paper considers the global optimization of challenging stochastic or multiperiod mixed-integer nonconvex programs that arise from integrated process design and operation. The difficulties of the problems are large scale and the nonconvexity involved. Recently, a novel decomposition method, called nonconvex generalized Benders decomposition (NGBD), has been developed to solve this problem to global optimality finitely, and this method shows dramatic computational advantages over traditional branch-and-bound based global optimization methods because it can exploit well the decomposable structure of such problems. Since the convergence rate of NGBD is largely dependent on the tightness of the convex relaxations of the nonconvex functions, the efficiency of NGBD can be improved by generating tighter convex relaxations. Building on the success of piecewise linearization for bilinear programs in the process systems engineering literature, this paper develops a piecewise convex relaxation framework, which can yield tighter convex relaxations for factorable nonconvex programs, and integrates this framework into NGBD to expedite the solution. Case studies of a classical literature problem and an industry-level problem show that, while NGBD can solve problems that are intractable for a state-of-the-art global optimization solver, integrating the proposed piecewise convex relaxation into NGBD helps to reduce the solution time by up to an order of magnitude.

In Part 1 of this paper, we presented a scenario-based multiperiod mixed-integer linear programming (MILP) formulation for a capacity planning problem in continuous pharmaceutical manufacturing under clinical trials uncertainty. The number of scenarios and, thus, the formulation size grows exponentially with the number of products. The model size easily becomes intractable for conventional algorithms for more than 8 products. However, industrial-scale problems often involve 10 or more products, and thus a scalable solution algorithm is essential to solve such large-scale problems in reasonable times. In this part of the paper, we develop a rigorous decomposition strategy that exploits the underlying problem structure. We demonstrate the effectiveness of the proposed algorithm using several examples containing up to 16 potential products and over 65 000 scenarios. With the proposed decomposition algorithm, the solution time scales linearly with the number of scenarios, whereby a 16-product example with over 65 million binary variables, nearly 240 million continuous variables, and over 250 million constraints was solved in less than 6 h of solver time.

Unlike traditional batch-based pharmaceutical manufacturing, where the active pharmaceutical ingredient (API) and the final drug product are often produced in different facilities at different locations, novel continuous pharmaceutical manufacturing strategies enable the production of both the API and the final drug product in the same integrated facility. The capacities of such integrated continuous facilities must be determined for potential products in the face of clinical trials uncertainty. Given a portfolio consisting of potential products in the development stage, the goal of capacity planning is to ensure the availability of enough production capacity to meet the projected demands of products, which vary from the launch to the peak-demand periods. To address this problem, we propose a multiscenario, multiperiod, mixed-integer linear programming (MILP) formulation that takes into account uncertainty in the outcome of clinical trials. We illustrate the proposed framework using several examples. The exponential increase in problem size with the number of products motivates us to develop an efficient solution method, which is discussed in Part 2 of this paper.

The pharmaceutical industry has historically benefited from high profit margins for their products, and over the years limited efforts have been made to change the main manufacturing concept from batch into continuous. However, over the past decade, as a result of an increased demand for more efficient and cost-effective processes, interest has grown in the application of continuous manufacturing to address economical and technical issues in the pharmaceutical field. This option is becoming more viable, particularly with the implementation of new process analytical technology (PAT). In this paper, we present a plant-wide mathematical model inspired by a recently developed continuous pharmaceutical pilot plant. This model is first used to simulate a base case that shows typical limitations in achieving simultaneously high productivity and quality. The main critical quality attribute considered is the purity of the final product. To alleviate the base case limitations and improve the pilot plant performance, the effects of several design parameters are investigated and the most critical are identified. In addition, alternative start-up scenarios are considered to improve the transient performance of the pilot plant, particularly time to steady state. The environmental footprint of the pilot plant is evaluated and shown to be low.

A techno-economic analysis of several process systems to convert coal and natural gas to electricity, methanol, diesel, and gasoline is presented. For these polygeneration systems, a wide range of product portfolios and market conditions are considered, including the implementation of a CO2 emissions tax policy and optional carbon capture and sequestration technology. A new strategy is proposed in which natural gas reforming is used to cool the gasifier, rather than steam generation. Simulations along with economic analyses show that this strategy provides increased energy efficiency and can be the optimal design choice in many market scenarios.

Several polygeneration process systems are presented which convert natural gas and coal to gasoline, diesel, methanol, and electricity. By using solid oxide fuel cells as the primary electricity generator, the presented systems improve upon a recently introduced concept by which natural gas is reformed inside the radiant cooler of a gasifier. Simulations and techno-economic analyses performed for a wide range of process configurations and market conditions show that this strategy results in significant efficiency and profitability improvements when CO2 capture and sequestration are employed. Market considerations for this analysis include variations in purchase prices of the coal and natural gas, sale prices of the products, and CO2 emission tax rates.Graphical abstractOptimal design choice as a function of CO2 tax and electricity sale price. Fuel and product prices are at base case values. Design choices include “Coal Only”, “External Reforming” and “Internal Reforming”; Gas Turbines or SOFCs; with or without Cabon Capture and Sequestration, and the amount of liquid fuels or electricity produced, expressed as a percentage of the output.Research highlights► Techno-economic analysis of 80 polygeneration system configurations. ► Natural gas is reformed in the radiant cooler of a coal gasifier. ► Solid oxide fuel cells (SOFCs) are used for power. ► Diesel, gasoline, and methanol are also produced. ► Market analysis considers price fluctuations and determination of optimal design.

The optimal design and operation of static polygeneration systems using coal and biomass to coproduce power, liquid fuels and chemicals is studied under different economic and policy scenarios. A mathematical model including material and energy balances, capital cost estimations, and economic analyses is proposed. Optimal product portfolios are obtained under different product price scenarios. The influence of different carbon tax policies on the optimal production strategy, such as the implementation of carbon capture and sequestration (CCS) or biomass usage, is also discussed.

The optimal design and operation of flexible polygeneration systems using coal and biomass to coproduce power, liquid fuels, and chemicals is studied. In flexible systems, the various product rates change throughout the lifetime of the plant in response to market conditions in different scenarios. A two-stage programming formulation is proposed to optimize simultaneously the design decision variables and the operational decision variables in all scenarios. The optimal product portfolios, equipment capacity usages, and CO2 emissions of flexible polygeneration systems under different market conditions are discussed. In general, flexible polygeneration systems achieve higher net present values than static ones for the same oil price and carbon tax. Higher net present values can be obtained with increasing operational flexibility.

A unique electricity generation process uses natural gas and solid oxide fuel cells at high electrical efficiency (74%HHV) and zero atmospheric emissions. The process contains a steam reformer heat-integrated with the fuel cells to provide the heat necessary for reforming. The fuel cells are powered with H2 and avoid carbon deposition issues. 100% CO2 capture is achieved downstream of the fuel cells with very little energy penalty using a multi-stage flash cascade process, where high-purity water is produced as a side product. Alternative reforming techniques such as CO2 reforming, autothermal reforming, and partial oxidation are considered. The capital and energy costs of the proposed process are considered to determine the levelized cost of electricity, which is low when compared to other similar carbon capture-enabled processes.

A dynamic, heterogeneous, two-dimensional model for packed-bed water gas shift reactors is presented. It can be applied to both high and low temperature shifts, and at scales ranging from industrial (for power plant applications) to small (such as automotive fuel cell applications). The model is suitable for any catalyst for which kinetic data are available, and shows excellent agreement with available experimental data for non-equilibrium conditions. The model is applied to an IGCC-TIGAS polygeneration plant to examine the dynamic behavior of the WGS units. The development of catalyst hot-spots is predicted during start-up or transition between steady states under certain conditions.

Recently, various alternatives to batteries, such as microfabricated fuel cell systems, have been proposed for portable power generation. In large-scale power production plants emphasis is placed on energy conversion efficiency. On the other hand, the intrinsic design objective for portable power generation devices is the energy density, i.e., the electrical energy generated from a given mass or volume of device and fuel cartridge. It is plausible to stipulate that an increase in the energy conversion efficiency of a system leads to an increase in energy density, but we demonstrate through theoretical analysis and case studies that the two metrics are not equivalent. In some cases, such as systems with a combination of fuels, maximizing efficiency leads to drastically different design, operation and performance than maximizing energy density. Another interesting observation is that, due to interaction between components, maximal component efficiency does not always imply maximal system efficiency.

•Scope: solutions of ordinary differential equations with absolute-value functions embedded.•Absolute-value arguments change sign only finitely often in any finite duration.•An absolute-value argument can only become identically zero when another argument crosses zero.•Numerically verifiable necessary conditions exist for absolute-value arguments becoming identically zero.We consider nonsmooth dynamic systems that are formulated as the unique solutions of ordinary differential equations (ODEs) with right-hand side functions that are finite compositions of analytic functions and absolute-value functions. Various non-Zenoness results are obtained for such solutions: in particular, any absolute-value function in the ODE right-hand side can only switch between its two linear pieces finitely many times on any finite duration, even when a discontinuous control input is included. These results are extended to obtain numerically verifiable necessary conditions for the emergence of “valley-tracing modes”, in which the argument of an absolute-value function is identically zero for a nonzero duration. Such valley-tracing modes can create theoretical and numerical complications during sensitivity analysis or optimization. We show that any valley-tracing mode must begin either at the initial time, or when another absolute-value function switches between its two linear pieces.

Highlights•Scenario-based stochastic formulation for design and operation under uncertainty.•Global optimization of large-scale nonconvex MINLPs via NGBD.•NGBD for non-separable functions and continuous recourse variables.•Piecewise convex relaxation to yield tighter bounds and expedite convergence.•Two industrial problems showing advantages of the stochastic formulation and NGBD.This paper is concerned with integrated design and operation of energy systems that are subject to significant uncertainties. The problem is cast as a two-stage stochastic programming problem, which can be transformed into a large-scale nonconvex mixed-integer nonlinear programming problem (MINLP). The MINLP exhibits a decomposable structure that can be exploited by nonconvex generalized Benders decomposition (NGBD) for efficient global optimization. This paper extends the NGBD method developed by the authors recently, such that the method can handle non-separable functions and integer operational decisions. Both the standard NGBD algorithm and an enhanced one with piecewise convex relaxations are discussed. The advantages of the proposed formulation and solution method are demonstrated through case studies of two industrial energy systems, a natural gas production network and a polygeneration plant. The first example shows that the two-stage stochastic programming formulation can result in better expected economic performance than the deterministic formulation, and that NGBD is more efficient than a state-of-the-art global optimization solver. The second example shows that the integration of piecewise convex relaxations can improve the efficiency of NGBD by at least an order of magnitude.

This paper is concerned with integrated design and operation of energy systems that are subject to significant uncertainties. The problem is cast as a two-stage stochastic nonconvex mixed-integer nonlinear program, in which the first and second stages include design decisions and operational decisions, respectively. By exploiting the separable and decomposable structure of the problem, an efficient global optimization method, called nonconvex generalized Benders decomposition (NGBD), is developed based on convex relaxation and generalized Benders decomposition. The efficiency of NGBD can be further improved via the notion of piecewise convex relaxations. The advantages of the proposed formulation and solution method are demonstrated through case studies of two industrial energy systems, a natural gas production network and a polygeneration plant. The first example shows that the stochastic programming formulation can result in better expected economic performance than the deterministic formulation, and that the NGBD solution method is dramatically more efficient than a state-of-the-art global optimization solver, especially for large numbers of scenarios. The second example further shows that the integration of piecewise convex relaxations can improve the efficiency of NGBD by at least an order of magnitude.

We consider the problem of computing reachable sets for continuous-time nonlinear continuous-stirred tank reactors (CSTRs) using differential inequalities method. Existing comparison-based methods yield a conservative enclosure for the reachable set due to the non-quasi-monotonic property for CSTR systems. The method proposed in Scott and Barton [2013] is effective; however, it requires a nontrivial a priori enclosure for the reachable set, which is difficult to construct for CSTR reaction systems. We propose a linear transformation to map the dynamics of the CSTR system onto a sparse state-space. Exploiting the sparsity, we construct a priori enclosure for CSTR systems. The application of the proposed method in designing a fault detection procedure is discussed, and its efficacy illustrated on an example problem.

In this paper, the problem of optimal crude oil procurement combined with refinery operations is addressed to obtain an ɛ–global optimal solution. Rather than the traditional planning and scheduling methodologies relying on linear programming (LP), a nonlinear model using Geddes fractional index (FI) is employed to describe the behavior of the crude distillation unit (CDU) and integrated with the entire plant-wide model. Although this representation provides more accurate prediction of the real production of the refinery than the conventional fixed yield approach, its global optimization becomes more difficult owing to the existence of many nonlinear, non-convex terms. To overcome this challenge, advanced interval reduction techniques are developed and combined with state-of-the-art global optimization software to obtain an e–global optimal solution more efficiently. The optimization and comparison are conducted to show the effectiveness of the proposed approach.

This work considers the computation of time-varying enclosures of the reachable sets of nonlinear control systems via the solution of an initial value problem in ordinary differential equations (ODEs) with linear programs (LPs) embedded. To ensure the numerical tractability of such a formulation, the properties of the ODEs with LPs embedded are discussed including existence and uniqueness of the solutions of the initial value problem in ODEs with LPs embedded. This formulation is then applied to the computation of rigorous componentwise time-varying bounds on the states of a nonlinear control system. The bounding theory used in this work exploits physical information to yield tight bounds on the states; this work develops a new implementation of this theory. Finally, the tightness of the bounds are demonstrated for a model of a reacting chemical system with uncertain rate parameters.

AbstractA novel method based on the generalized gradient and nonsmooth optimization techniques called bundle methods is introduced to optimize the performance of a class of dynamic systems whose governing equations change depending on the values of the parameters, controls and the current state of the system.