Flexible Batch Process and Plant Design Using Mixed-Logic Dynamic

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Flexible Batch Process and Plant Design Using Mixed-Logic Dynamic Optimization: Single-Product Plants Marta Moreno-Benito, Antonio Espuña, and Luis Puigjaner* Department of Chemical Engineering, Universitat Politècnica de Catalunya, ETSEIB, Av. Diagonal 647, Barcelona, E-08028, Spain S Supporting Information *

ABSTRACT: Batch manufacturing plants require the frequent adaptation of production facilities to incorporate internal and external uncertainties. The introduction of flexibility in the design of general-purpose plants represents a cornerstone for the design and retrofit of batch processes along the plant’s lifecycle. Moreover, the integration of batch process development decisions during the plant design also contributes to handle uncertainty. This paper presents a novel optimization-based approach for the flexible design of batch plants by integrating process synthesis and plant allocation decisions, such as feedforward control trajectories of processing conditions or the selection of operating modes and equipment items. To do so, a modeling strategy based on Mixed-Logic Dynamic Optimization and two-stage stochastic formulation is used to maximize the expected profit taking into account several uncertain scenarios. Moreover, a heuristic algorithm is proposed to solve the resulting problem that is characterized by a great complexity, consisting of an iterative procedure with embedded deterministic solutions of the integrated problem. The proposed approach is illustrated through the design of a single-product batch plant to carry out a competitive reaction process. However, as Mauderli and Rippin6 enunciated, “the big advantage of a multipurpose batch plant, its flexibility, also poses the problem of making the best use of it”. In fact, several experts2,3 stress the importance of stimulating the design of versatile plants as well as exploiting their ability to agilely accommodate the periodical development of competitive and sustainable processes. Even though these two problems are characterized by a high complexity, they are also recurrent in most specialty chemicals companies and organizations considering to invest in a new batch production facility. In this context, this contribution has two principal targets. First, flexibility of the resulting batch plant design is pursued by incorporating uncertainty in product demand, which reflects changing market conditions and variations of the plausible customer orders. Second, the integration of batch process development decisions within the plant design problem is posed as a strategy to enhance plant flexibility through a better usage of the designed plant and to ensure the rapid and efficient process development in future scenarios. The resulting problem is defined as the optimal design of batch plants and the set of master recipes to be used in the uncertain probability space. By doing so, this problem aims to minimize the expectation of the decision criteria, to increase the accuracy of decisions, to define a system which guarantees a manageable response to changes in the business environment, and to improve the process performance in order to provide an optimal demand order satisfaction in the defined probability space.

1. INTRODUCTION Batch manufacturing is mostly devoted to the specialty chemicals industry and requires the frequent adaptation of production facilities to market fluctuations. The products to be manufactured in a batch plant near the end of its lifetime are most probably unknown at the time of its design. This is because many specialty products are subject to continuously changing market requirements and may be displaced by new product approvals or advancing technologies. In this context, the introduction of flexibility in the design of general-purpose plants represents a cornerstone for the design and retrofit of batch processes along the plant’s lifecycle. Particularly, plant flexibility pursues the ability of a specific design to deal with a set of uncertain parameters and conduct most of the processes in the future company’s portfolio with minimum plant modifications.1 The maximization of future flexibility at minimum investment during the solution of the plant design contributes to the rapid incorporation of forthcoming production changing requirements and being first in the market.2 Plant flexibility is also a crucial element to achieve long-term viability of new processes and to keep competitiveness in the market in front of shareholders, by facilitating the development of sustainable processes since the early moment of their conception.3 In most cases it is possible to retrofit process design to introduce new incentives if required, for example, waste reduction or energy saving;4 however, end-of-pipe process redesign may require bigger investments and operating costs than process design with such considerations at early stages. This way, significant economic leverage can be gained through the ability of flexible plants to incorporate both fast and high quality processes.5 Overall, batch industry encounters the need of methodologies, tools, and decision-support systems to design and exploit adaptable equipment. © XXXX American Chemical Society

Special Issue: Jaime Cerdá Festschrift Received: February 2, 2014 Revised: May 18, 2014 Accepted: June 4, 2014

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dx.doi.org/10.1021/ie500480b | Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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stochastic programming, fuzzy programming, and stochastic dynamic programming. Each of them has several advantages and drawbacks. The interested reader is addressed to the review by Sahinidis.1 Out of these approaches, the most extended one to address the design of chemical process systems under uncertainty is stochastic programming, which addresses optimization problems through the incorporation of uncertain parameters governed by known or estimated probability functions. Specifically, probability distribution functions that represent uncertain parameters can be defined by means of continuous functions or by a finite number of scenarios resulting from the discretization or sampling of the probability distribution. Accordingly, uncertain parameters are categorized as11 (i) stochastic uncertain parameters, described by a probabilistic distribution function, and (ii) deterministic uncertain parameters, modeled through a series of periods or scenarios with particular values and associated with a probability or weight. One of the most widespread stochastic methodologies to solve process engineering problems is the two-stage programming strategy12,13also referred to as programming with recoursesince it is considered to be very effective.14 This strategy differentiates between f irst-stage variables, which remain fixed once selected and commonly correspond to plant design decisions, and second-stage variables, which can be adapted as a function of uncertain parameter values and usually correspond to process operative decisions. The latter are originally interpreted as corrective actions or recourses against infeasibilities or inefficiencies that could arise due to a particular realization of the uncertainty. However, the role of second-stage variables is not only to make it possible to achieve feasibility but also to improve process performance. Additionally, depending on the targets pursued during the problem solution, other variations of stochastic programming have been proposed, like the robust stochastic programming and the probabilistic programming. Robust stochastic programming is basically an extension of the two-stage programming strategy which additionally captures risk assessment in the objective function. Specifically, risk is represented through a variability measure of the second-stage costs, for example, variance, and a risk tolerance. As a result, the search for robust design appears often as a multiple criteria decision problem, where trade-offs between expected performance and dispersion measures are found, rendering a compromise in the resulting robust optimal solutions. As for probabilistic programming, it prioritizes the system reliability. With this purpose, this strategy incorporates reliability constraints, which express the minimum requirement on the probability of satisfying the problem constraints. Several authors have applied two-stage programming strategies to solve the batch plant design problem under uncertainty, either using scenario-based deterministic uncertain values or probability distribution functions to represent the uncertain parameters (e.g., Ierapetritou and Pistikopoulos,11 Reinhart and Rippin,15 Shah and Pantelides,16 Cao and Yuan,17 Alonso-Ayuso et al.,18 Aguilar-Lasserre et al.,19 Pinto-Varela et al.,20 Wang et al.,21 and Moreno and Montagna,22 among many others). In most cases, uncertainty in product demand was considered, and some studies also include uncertainty in process parameters11 or economic data.18 Moreover, the problem is typically modeled as Mixed-Integer Linear Programming (MILP),11,20 where multiobjective decision criteria may be introduced, including several economic

With this purpose, a novel optimization-based approach is proposed, where the expected profit maximization is solved through a modeling strategy based on a Mixed-Logic Dynamic Optimization (MLDO) and two-stage stochastic formulation which takes into account several demand scenarios. The principal novelty of this methodology is the integration of all the alternatives for batch plant design and process development under uncertainty in a unique optimization model. Additionally, a solution strategy is proposed to handle the mathematical complexity of such a problem. In particular, the solution algorithm consists of a heuristic search procedure that computes the integrated MLDO problem for each uncertain scenario using a direct-simultaneous optimization method inside an iterative loop. In each iteration, the holistic formulation that takes into account simultaneously first-stage and second-stage decisions associated with equipment selection and sizing, batch process synthesis, and plant allocation subproblems is solved as a deterministic problem. Thus, the scope of this study assumes a given set of scenarios with an associated probability to represent the uncertainty. Summarizing, the objective of this contribution is to introduce the novel concepts and detailed solution approach of a rich integrated decision-making and to evaluate its potential to handle the size and the rate of nonlinear terms of the problem through the solution of the plant and process design of a single-product batch plant.

2. LITERATURE REVIEW 2.1. Flexibility in Batch Plant Design. Batch plant design consists of the development of general-purpose facilities where several products are produced by sharing the available resources.7 Essentially, this problem tackles the definition of the equipment, that is, number, type, capacity, and connectivity of equipment items, the assignment of process stages to appropriated equipment items, and the definition of operational decisions. This area of study has attracted much interest since the mid 1980s,2 as the large number of research contributions corroborate (see the reviews by Barbosa-Póvoa7 and Reklaitis8). To address the batch plant design at the early stages of the plant lifecycle, it is necessary to assume given product portfolios and processing schemes, even though some parameters may not be fully defined yet or may involve uncertainty9 associated with different causes: On the one hand, chemical plants are often characterized by internal plant parameters quite different from those considered at the design stage, such as kinetic constants, heat transfer coefficients, or machine efficiency and reliability. On the other, rapidly changing market environments will have an effect on the variability of external conditions like product demands, cancellations, and returns, raw material availability, prices of chemicals, or environmental parameters. Simplifying hypotheses are often used in the solution of the plant design problem in order to partly mitigate the effect of possible future variations in internal and external parameters. For instance, extreme or mean values or the application of safety factors, based on past experience and engineering judgment, are typically employed.10 The drawback of these simplifications is that the reliability of the designed system −understood as its capacity to feasibly operate in an uncertain environment− cannot be guaranteed within the actual uncertainty margin. As a result, numerous approaches have been proposed in the literature to consider different sources of uncertainty in batch plant design problems, with outstanding methodologies like the B



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in practice that the use of fixed or approximated recipes hinders the adaptation of recipe parameters according to the global targets, during the solution of batch plant design problem, which becomes an important limitation in the design of batch manufacturing facilities. Also for this reason, this problem has been generally addressed through divide and conquer strategies and hierarchical approaches, entailing the successive solution of independent subproblems with more manageable representations. However, a significant part of the interaction among the decisions made is lost using such approaches. Notable efforts have been devoted to solve the integrated batch process development and plant design problem in the literature. Particularly, several works have studied the incorporation of recipe modifications in allocation of batch plants through the definition of mathematical models, which represent the process performance in each task and should be related to the plant design problem either using simultaneous or iterative decision-making strategies. Different degrees of detail have been proposed to characterize the performance models, like time and size factor models,29 algebraic performance models,30 or dynamic performance models,24,31 with a predominance of the two first ones. In this regard, one of the most relevant contributions is the work by Iribarren et al.,32 who proposed a Mixed-Integer Nonlinear Programming (MINLP) formulation to address the combination of process synthesis decisions, for example, the selection of microorganisms responsible of the biological process or the selection of separation and purification alternatives; plant allocation decisions, for example, the definition of the operating mode and processing order for each product; and plant design decisions such as equipment sizing. The importance of this problem is the great variety of decisions, which were optimized simultaneously, even though the representation of the process performance in batch process stages was simplified through the use of time and size factors33 which depended on the synthesis decisions considered. Another crucial step ahead in the literature was given by Charalambides et al.34 and Sharif et al.,35 who incorporated process dynamics in the optimization of integrated problems. In fact, the relevance of such contributions is due to the representation of the chain of process stages by means of hybrid discrete/continuous models,36 which are dynamic models that represent batch process performance and batch events, as well as to the optimization of dynamic trajectories of the control profiles to adjust the processing conditions. This way, the integrated problem was formulated by means of a Dynamic Optimization (DO) formulation27,37 in the paper by Charalambides et al. 34 and a Mixed-Integer Dynamic Optimization (MIDO) one in the paper by Sharif et al.35 in which integer variables were used to represent discrete equipment sizing. These strategies were applied to the solution of single-product reaction-separation systems that included complex process features such as material recycles and shared intermediates. However, the task sequence was predefined as well as allocation decisions, assuming that one equipment item was dedicated to each task. Thus, the principal drawback is that alternative equipment and process configurations could not be selected to handle operational issues and to adapt the plant to changing frameworks. The simultaneous consideration of structural decisions together with dynamic performance models and discrete events was addressed by Oldenburg et al.38,39 To do so, these authors proposed the combination of DO with the so-called

contributions19 or their combination with environmental impact factors.21 However, most of these contributions assume the use of fixed recipes rather than defining optimal process parameters according to performance models, as seen in the following. 2.2. Integration of Batch Process Development. Plant design problem is usually preceded by the sequential settlement of product and process lifecycles in grassroots scenarios.23 However, the simultaneous solution of process synthesis, plant allocation, and plant design subproblems allows to account for the interactions between the degrees of freedom associated with each subproblem and represents a challenge to avoid suboptimal systems. The resulting problem is defined as the integrated solution of batch process development and plant design and is motivated by the need to incorporate scheduling and operational decisions in the solution of the optimal plant design such that the use of the installed resources to achieve the production targets can be determined. On the one hand, the major operational concerns during the solution of the plant design problem are related to the plant allocation problem with decisions such as the timing, the operating mode, the batch size, the storage policies, the material and/or energy amount transferred from stage to stage, and the duplication of units. On the other, all these degrees of freedom should ideally be specified according to the physicochemical phenomena and process performance which take place in each process stage2 in order to avoid the definition of unrealistic and suboptimal solutions of the plant utilization. Physicochemical phenomena and process performance basically depend on the process synthesis solution problem, which essentially defines several recipe parameters like the selection of chemicals involved in the process, the sequencing of batch operations and phases, and the optimization of set points and feed-forward trajectories of control variables, among others. To sum up, by integrating the solution of process synthesis and plant allocation into the plant design problem, it is possible to account for the trade-offs between structural and performance decisions, first studied by Barrera and Evans.24 In fact, as Shobrys and Shobrys25 enunciated, integrated approaches that combine both types of decisions may ensure fully functional and optimally operated process plants in both nominal regimes and changing framework. The solution of the integrated problem involves some modeling requirements. At first place, the batch nature of the process involves not only the need of matching equipment and task networks through a combinatorial assessment,26 but also the adjustment of the processing conditions along time in each process stage, requiring dynamic models to represent process performance and dynamic profiles of control variables.27 Process operations are also featured by discrete events that determine the batch operation and phase transitions.28 Moreover, qualitative information should be also covered in the optimization model, representing decisions like task selection, sequencing, and splitting, equipment assignments, or chemicals selection, among others. Finally, batch integrity should be guaranteed in all processing path alternatives through the synchronization of material transference between batch and semicontinuous plant elements. In particular, the plant elements to be synchronized and their sequence depend on the selected processing scheme and on the process efficiency achieved in each task. Given the complexity associated with the simultaneous consideration of all these modeling requirements, it is frequent C



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• Plant diagram: the plant diagram with potentially equipment units to be installed for each process stage, pipelines, and connection nodes like mixers and splitters; • Task network: potential and mandatory process stages, alternative chemicals involved in each process stage, for example, reactants, solvents, or catalysts, allowed technologies, and possible reuse of intermediates; • Batch operating procedures: allowed task-unit assignments, list of batch operations and phases within each unit procedure, phase to phase switching conditions, and set of limiting processing conditions for each unit; • Process dynamics: DAE systems to represent the process behavior in each unit procedure, initial conditions, and set of process and control variables; • Data related to performance evaluation: decision criteria and specific data to evaluate the objective function, for example, selling price of final product, direct cost of raw materials, investment, amortization, operating costs in processing units, environmental impact indicators. The goal is to determine the following: • Plant design decisions: selection and sizing of equipment units; • Synthesis of processing schemes decisions: selection of process stages and splitting into subtasks, technological specification, selection of chemicals involved−i.e. reactants, solvents, or catalysts−, reference trajectories of the feed-forward control variables, duration of batch phases composing each task, recirculation of intermediate flows, and material transfer synchronization between tasks−i.e. synchronization of flow rates, compositions, and starting and final times; • Allocation of manufacturing facilities decisions: taskunit assignment−i.e. unit procedure selection−, selection of processing and storage units, operating mode−i.e. single, series, or parallel operation−, and batch sizes; such that the adopted performance metrics are optimized. In broad terms, the goal is to provide (i) a flexible plant design which maximizes the objective function expectation and (ii) the set of optimal master recipes which define the optimal process synthesis and plant allocation decisions within the flexible plant in the forecasted demand space. The objective function is governed by the maximization of future performance at minimum investment cost, since it considers simultaneously the capital costs associated with plant investment and the operational costs associated with the master recipes in the predicted scenarios. Moreover, multiscenario probabilities are used to represent uncertainty in the product demand and those other parameters which are susceptible to vary. The solution of single-product plants is addressed in this contribution, although the problem can be extended to multiproduct batch systems. The proposed optimization-based approach to solve this problem is based on the principal steps42 summarized in Figure 1. In step (a), the required information is identified according to the above-mentioned problem statement. Steps (b) and (c) to define the SEN superstructure that contains all the processing alternatives and its formulation into a MLDO model under uncertainty are detailed in next section. The proposed methodology to solve the mathematical problem obtained corresponds to step (d) and is presented afterward in section 5.

Generalized Disjunctive Programming (GDP)40,41 to allow the direct incorporation of qualitative decisions and heuristic rules associated with such decisions into the mathematical model. Precisely, the importance of the work by Oldenburg et al.38,39 is related to the simultaneous representation of quantitative information about the process behavior by using hybrid discrete/continuous models as well as qualitative information regarding the structural alternatives in the equipment design by using logical equations. The resulting formulation became a Mixed-Logic Dynamic Optimization (MLDO) problem, where the process performance models depended on logical variables. However, the strategy was only applied to address the optimal configuration, sequencing, and operation of individual processing units, rather than the synthesis of chains of tasks with several processing route alternatives. A crucial issue to incorporate structural decisions into DO formulations in order to consider simultaneously process dynamics, discrete events, and alternative processing routes is the synchronization of the material transfer stages as a degree of freedom. In particular, it is necessary that material transference is synchronized between subsequent unit procedures in order to take into account batch integrity. For this reason, the chain of operations represented in the hybrid discrete/continuous models should be defined and ordered according to the selected process stages and task-unit assignments. However, to the authors’ knowledge, there are no available references yet in the context of batch process development or batch plant design that address this comprehensive problem as a whole. 2.3. Motivation of the Proposed Approach. Beyond the optimization of dynamic control profiles, transition times, and equipment sizing in a sequence of process stages, the emphasis in this paper is placed on the solution of the integrated problem considering structural decisions. This way, the processing route is posed as a degree of freedom in the studied system, by optimizing the number of subtasks for each unit operation, the selection among different units to carry on each task, and the definition of the operating mode as single unit, series, or parallel configuration. Moreover, consecutive tasks are synchronized by relating input and output transfer operations in equipment sequences which are not known beforehand. The objective is the consideration of the processing route selection as a second-stage decision or recourse in the plant in order to give response to uncertainty. In broad terms, this contribution proposes an optimization-based strategy to evaluate the potential of simultaneous batch plant design and process development to fully and optimally exploit the processing conditions and alternative processing routes in a plant to enhance its flexibility. The purpose is to evaluate the challenges of applying a fully integrated strategy, analyze the expected benefits of solving simultaneously all these degrees of freedom, justify the impact of their interaction on plant flexibility, and verify whether the formulation can be actually solved.

3. PROBLEM STATEMENT To address the simultaneous problem of plant design and process development under uncertainty, the following information is assumed to be known. Given the following: • Planning data: final product, intermediates, and raw materials, probability space of the demand of final product in the considered time horizon; D



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and process performance in each process stage, which may be composed of a series of batch operations or phases. Moreover, the model strategy contemplates that multistage models can be combined and organized according to each processing path alternative. This is because the coexisting use of multistage models allows a modular representation of batch plant elements and the adaptation of the order and timing of batch units with respect to the time horizon of the entire recipe. The resulting need to synchronize flow rates and compositions in material transfer operations can be also handled by means of the disjunctive equations of the MLDO model. In broad terms, the obtained MLDO model is characterized by the use of (i) Booleans to represent the qualitative information related to the process synthesis and plant allocation subproblems as well as logical propositions to define their combinatorial assessment based on the SEN representation, (ii) DAE to represent process dynamics associated with batch operations and, if required, transient regimes of semicontinuous systems, (iii) dynamic profiles of control variables to fully achieve optimality for the batch performance, (iv) discrete events representing phase and operation transitions in batch unit procedures, and (v) disjunctive equations to synchronize flow rates and compositions in material transfer operations. Regarding the degrees of freedom, these are classified in the

Figure 1. Main steps of the optimization-based approach to solve integrated batch plant design and process development under uncertainty.

next four categories according to their mathematical representation: • Dynamic control variables udyn k (t): flow rate profiles j Fm,k (t) in input and output pipelines m of unit j in material transfer operations k and internal control variables intjk(t) like the reaction temperature or reactant dosage in operation k of unit j; • Time-invariant or static decision variables ustat: duration tk of mathematical stage k, including all batch operations within the process; • Integer decision variables uint: number NBp of batches of product p, capacity Sizej of unit j; and • Boolean or logical decision variables uBool: process stage Boolean Zi to indicate whether task i is selected, equipment Boolean Yj to indicate whether equipment item j is selected, configuration Boolean Xiψ to indicate whether alternative ψ is selected in process stage i, taskunit assignment Boolean Wj,q to indicate whether unit procedure order q is assigned to unit j, technology Boolean Vjλ to indicate whether technological specification λ is selected in unit j, chemical compound Boolean Sjc to indicate whether reactant, solvent, or catalyst c is selected in unit j, and recirculation Boolean Rn to indicate whether intermediate flow in pipeline n is recirculated.

4. SUPERSTRUCTURE REPRESENTATION AND PROBLEM FORMULATION 4.1. Deterministic Problem. The formulation of the integrated problem that combines all the degrees of freedom of the plant design and process development problem is first considered without uncertainty. The mathematical modeling of this problem faces several requirements as discussed in section 2.2. At first place, the direct consideration of physical plant decisions in the problem is accomplished through the SEN representation43 of the superstructure of processing alternatives. This representation also provides great flexibility in the definition of material transfer states in connecting pipelines, which are not fixed to a predefined value but are subject to the selected processing route and the performance of the unit procedures in each solution. Moreover, the selection of a potential unit to be installed involves the definition of a set of physical restrictions, such as the volume or temperature upper bounds. To solve the integrated problem, it is necessary to consider the maximum number of potential units in parallel or series in the superstructure. The subsequent mathematical representation of the superstructure and all the associated decisions is based on a MLDO formulation with embedded multistage models. In particular, disjunctive equations are used to define structural alternatives and multistage models are used to represent batch procedures

In general terms, the deterministic MLDO model without uncertainty reads as E


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ukdyn(t ), u stat , u int , u Bool

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First-Stage and Second-Stage Decisions. The abovementioned decisions are divided into f irst-stage plant design decisions u1 ∈ U1, whose assigned value in the optimal solution is the same independently to the value of uncertain parameters, and second-stage process synthesis and plant allocation decisions u2 ∈ U2 or recourses, whose optimal value can be adapted to each realization of uncertain parameters ξδ. On the one hand, u1 involves the capacity Sizej of new equipment units. On the other, scenario-dependent decisions u2 comprise the remaining variables, namely: dynamic control profiles Fjm,k(t) and intjk(t), batch phase durations tl, number of batches NBp, and all logical decisions associated with the equipment selection Yj, process stage selection Zi, definition of processing mode Xiψ, task-unit assignment Wj,q, technology specification Vjλ, chemical compound selection Sjc, and recirculation option Rn. Second-stage decisions imply the embedded optimization of the master recipes for different values of uncertain parameters, adapting the future operation in the optimal plant. Objective Function. The objective function represents the expectation of the profit maximization and is also distributed into two terms Φ1 and Φ2 in the two-stage programming formulation. The first term Φ1 of the stochastic problem is composed of the capital costs for the installation of new units. This way, it only depends on first-stage decisions u1 and is not affected by uncertain parameters. Regarding the second term Φ2, it is computed from the product revenue, raw material expenses, and processing and occupation costs. In contrast to the first-stage case, second-stage economic terms additionally depend on process synthesis and allocation decisions u2, which are subject to the different values of uncertain parameters ξδ as mentioned above. Overall, the optimization of Φ2 in the second stage of the problem is constrained by the feasible operating region associated with first-stage decisions, involving a compromise between economic performance and plant flexibility. Shortfall Penalty. The problem constraints used to guarantee the accomplishment of the demand are relaxed by penalizing the product shortfall in the second-stage part Φ2 of the objective function. By doing so, it is possible to guarantee the best utilization of plant resources for the optimal demand satisfaction, which is understood as the demand that can be satisfied in each uncertain scenario pursuing the common good in the selection of equipment investments. The weight associated with the shortfall penalty in the objective function can modify the trade-off between the different costs. On the one hand, small penalty weights would lead to the prioritizing of the cost minimization by reducing the amount of processed material and aggravating the unfulfillment of the demand. On the other, large penalty weights would promote that the demand is fulfilled in the maximum number of uncertain scenarios. Provided that the fully demand satisfaction is feasible in all uncertain scenarios, large penalty weights lead to the same solutions than formulations with demand satisfaction constraining, and this is the reason why they are preferred in this study. Scenario-Based Uncertainty. The uncertain space of stochastic parameters ξδ is characterized by multiscenario deterministic values ξsδ|ξL ≤ ξsδ ≤ ξU, s ∈ {1,...,NS}, where each plausible scenario s ∈ {1,...,NS} is associated with a normalized probability ws. The scope of this study assumes that the scenario-based information is given in the problem statement but it could be also generated through sampling techniques from probability distribution functions. Sampling strategies are especially valuable to obtain a representative number of

Φ(zk(t ), yk (t ), ukdyn(t ), u stat , u int , u Bool , γ , ρ),

s.t. fk (zk̇ (t ), zk(t ), yk (t ), ukdyn(t ), u stat , u int , ρ) = 0,

t ∈ [0, 1],

∀ k ∈ K, l(z1̇ (0), z1(0)) = 0, gk (zk(t ), yk (t ), ukdyn(t ), u stat , u int , ρ) ≤ 0,

t ∈ [0, 1],

∀ k ∈ K, gke(zk(1), yk (1), ukdyn(1), u stat , u int , ρ) ≤ 0, zk + 1(0) − mk (zk(1)) = 0, γ = h(z|K |(1), y|K | (1),

∀ k ∈ K,

∀ k ∈ {1, ..., |K | − 1},

u|dyn K | (1),

u stat , u int , ρ),

⎡ u Bool ⎢ ⎢ f d (z ̇ (t ), z (t ), y (t ), u dyn(t ), u stat , u int , ρ) = 0, k k k ⎢ k k ⎢ t ∈ [0, 1], ∀ k ∈ K , ⎢ l d(z1̇ (0), z1(0)) = 0, ⎢ ⎢ ⎢ g d (zk(t ), y (t ), ukdyn(t ), u stat , u int , ρ) ≤ 0, k k ⎢ t ∈ [0, 1], ∀ k ∈ K , ⎢ ⎢ d ,e dyn stat int ⎢ gk (zk(1), yk (1), uk (1), u , u , ρ) ≤ 0, ⎢ ∀ k ∈ K, ⎢ ⎢ zk + 1(0) − mkd (zk(1)) = 0, ⎢ ∀ k ∈ {1, ..., |K | − 1}, ⎢ ⎢ dyn d stat int ⎢⎣ γ = h (z|K |(1), y|K |(1), u|K | (1), u , u , ρ)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

⎡ ¬u Bool ⎢ ∨̲ ⎢ Bd (z ̇ (t ), z (t ), y (t ), u dyn(t ), u stat , u int , γ , ρ) k k k k ⎢ = 0, t ∈ [0, 1] ⎣ Ω(u Bool) = true

⎤ ⎥ ⎥, ⎥ ⎦ (1)

where zk(t), yk(t), and udyn k (t) are the differential, algebraic, and control variables along time for each stage k ∈ K, where K is the set of mathematical stages defined for each element in the model. ustat, uint, and uBool represent the time-invariant, integer, and Boolean decision variables, γ corresponds to time-invariant algebraic variables evaluated at the final time of last stage |K|, and ρ are the deterministic model parameters. Φ is the objective function, l defines the initial conditions, h is the set of equations to calculate time-invariant variables, and f k, gk, gek, and mk are the DAE system representing the process dynamics, the path constraints, the end-point constraints, and the stage-tostage continuity in stage k ∈ K, which are hold independently d to Boolean decisions uBool. Accordingly, ld, hd, f dk, gdk, gd,e k , and mk are the analogous functions that are active in the case that variable uBool is true and Bd defines the system in case that uBool is false. Finally, Ω is the set of logical propositions that infer qualitative knowledge by relating the logical variables uBool to each other. 4.2. Stochastic Problem. To effectively consider flexibility, a two-stage stochastic scheme1,12,13 that maximizes the future performance at minimum investment cost is proposed. For this purpose, the previous deterministic MLDO problem of eq 1 is extended taking into account the following considerations. F



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For this reason, the solution method proposed is based on the subsequent solution of the integrated problem for each uncertain scenario within an iterative procedure, in order to reduce the computational load. In particular, the heuristic proposed in this study allows to optimize simultaneously the first- and the second-stage decisions, that is, the optimal plant design and the master recipe, for each demand scenario individually and to guide the obtained plant solutions toward convergence through an iterative rule-based loop where each plant is re-evaluated for every uncertain scenario. Since each iteration is associated with a deterministic value of uncertain parameters, the problem can be solved in each step through the deterministic formulation of eq 1 using a direct-simultaneous approach. Both the general heuristic proposed as well as the guidelines for the integrated problem solution within each iteration are further detailed in this section. 5.1. Principal Steps. The proposed heuristic procedure to solve the two-stage MLDO stochastic programming problem of eq 2 involves four principal steps to address the last part (d) of the general scheme of Figure 1. The algorithm and its pseudocode are summarized in Figure 1 and basically consist of (1) Definition of uncertain scenarios. Each of the NS uncertain scenarios s ∈ {1,...,NS} is first identified and characterized by means of the forecasted value of stochastic parameters ξsδ and its probability ws. In this study, this information is assumed to be given in the problem statement, although the generation of uncertain cases could be also addressed through sampling techniques. In every case, the scenarios should be ordered from lowest to highest demand value ξsDem, as it is later justified. (2) Iterative optimization of the integrated problem for each uncertain scenario. The integrated optimization of firststage and second-stage is addressed in a iterative loop that considers each uncertain scenario subsequently. Essentially, the iterative loop uses the first-stage decisions of each optimal solution to define plants Pp, p ∈ {0,...,NP}, which are used as a base to be improved for each of the NS demand scenarios in next iterations. Overall, every iteration involves the creation and solution of a deterministic MLDO problem minimizing the function Φ = Φ1(u1,ρ) + Φ2(zk(t), yk(t), u1, u2, γ, ρ, ξsδ), which corresponds to the deterministic MLDO of eq 1. The set of plants Pp, p ∈ {0,...,Np} is upgraded if solutions with additional processing units j are found. It is emphasized that the optimization of process synthesis and allocation with plant design is integrated in this step, in order to take into account the best plant utilization, defined through the optimization of the master recipe, before posing a new plant solution and the computational load involved. A prespecified termination criteria should be defined, for example, no new solutions are found in the previous NT iterations. (3) Evaluation of the expected profit. The expected profit for each plant solution Pp, p ∈ {0,...,NP} generated in the previous step is computed by taking into account the uncertain values ξsδ and first-stage decisions up1 to calculate Q(up1,ξsδ) as well as the probabilities ws of all the scenarios s ∈ {1,...,NS}. Once more, the deterministic MLDO problem of eq 1 is solved in this step, now considering the equipment sizing as restrictions instead of degrees of freedom. This way, the second-stage process synthesis and plant allocation decisions are optimized for each demand scenario s ∈ {1,...,NS} with a fixed plant Pp, p ∈ {1,...,NP}. (4) Best plant selection. To determine the best plant out of the NP solutions obtained in step 2, these are ranked according

scenarios in problems with multivariate probability distributions. According to the deterministic MLDO problem of eq 1 and the previous considerations, the stochastic optimization problem under uncertainty reads as NS

minimize j u1∈ {Size }

Φ1(u1, ρ) +

∑ w sQ (u1, ξδs), s=1

Q (u1, ξδs) =

with

minimize

dyn

u 2 ∈ {uk (t ), u stat , u int , u Bool}\ u1

Φ2(z k(t ), yk (t ), u1, u 2 , γ , ρ , ξδs) s.t. fk (z k̇ (t ), z k(t ), yk (t ), u1, u 2 , ρ , ξδs) = 0, t ∈ [0, 1],

∀ k ∈ K,

l(z1̇ (0), z1(0)) = 0, gk (z k(t ), yk (t ), u1, u 2 , ρ , ξδs) ≤ 0,

t ∈ [0, 1],

∀ k ∈ K, gke(z k(1), yk (1), u1, u 2 , ρ , ξδs) ≤ 0, z k + 1(0) − mk (z k(1)) = 0,

∀ k ∈ K,

∀ k ∈ {1, ..., |K | − 1},

γ = h(z |K |(1), y|K |(1), u1, u 2 , ρ , ξδs), ⎡ ⎤ u Bool ⎢ ⎥ ⎢ ⎥ f kd (z k̇ (t ), z(t ), yk (t ), u1, u 2 , ρ , ξδs) = 0, ⎢ ⎥ ∈ ∀ ∈ [0, 1], , t k K ⎢ ⎥ ⎢ ⎥ d l (z1̇ (0), z1(0)) = 0, ⎢ ⎥ ⎢ ⎥ d s ⎢ gk (z k(t ), yk (t ), u1, u 2 , ρ , ξδ ) ≤ 0, t ∈ [0, 1], ⎥ ⎢ ⎥ ∀ k ∈ K, ⎢ ⎥ ⎢ d ,e ⎥ s ≤ ∀ ∈ ρ ξ ( (1), (1), , , , ) 0, , g z y u u k K 1 2 k δ ⎢ k ⎥ k ⎢ ⎥ d ⎢ z k + 1(0) − mk (z k(1)) = 0, ∀ k ∈ {1, ..., |K | − 1}, ⎥ ⎢ ⎥ ⎢⎣ ⎥⎦ γ = hd(z |K |(1), y|K |(1), u1, u 2 , ρ , ξδs) ⎡ ⎤ ¬u Bool ⎢ ⎥ d s ⎢ B (z ̇ (t ), z (t ), y (t ), u , u , γ , ρ , ξ ) = 0, ⎥ , 1 2 k k δ k ⎥ ∨̲ ⎢ ⎣ t ∈ [0, 1] ⎦ Ω(u Bool) = true ,

(2)

where Q(u1,ξsδ) denotes the optimal value of the second-stage problem for a given value of first-stage decisions u1 and uncertain parameters ξsδ.

5. PROPOSED HEURISTIC FOR SOLVING THE PROBLEM UNDER UNCERTAINTY The resulting MLDO model representing the problem of batch plant design and process development is nonlinear, nonconvex, and subject to a large combinatorial size due to the presence of integer and logical variables. The consideration of uncertainty further increases the size and complexity of the problem, due to the enclosed second-stage optimizations in the two-stage stochastic programming formulation of the problem according to eq 2. As a result, the number of equations, continuous and discrete variables, and nonlinear terms exploits, thus obstructing the optimization of the stochastic problem considering the full-space of uncertain scenarios at the same time. G



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Figure 2. Heuristic procedure for integrated flexible plant design and batch process development of eq 2. The set of decision variables for each plant solution Pp, p ∈ {1,...,NP} is represented by up1.

selected from the solutions for particular uncertain scenarios. However, the proposed heuristic still represents an important step beyond the commonly known as hierarchical decomposition approaches, where the different decisions are solved sequentially. In the proposed approach all the decisions are optimized simultaneously in each step of the iterative procedure, which enhances the performance of the solution approach and the quality of the decisions taken under a richer knowledge base. The optimization for each uncertain scenario is addressed through the direct-simultaneous method that is detailed in what follows. 5.2. Direct-Simultaneous Solution. The deterministic optimization problem (eq 1) is addressed recurrently in the heuristic approach, in order to solve the stochastic problem (eq 2) for particular values of uncertain parameters. Specifically, this happens in each iteration of the step 2 to generate the plant solutions and in each second-stage optimization in step 3 to calculate the expectation of the objective function. The optimization of the deterministic problem is addressed in every case through a classical solution approach to solve MLDO problems through the transformation of the mixedlogic formulation into a mixed-integer one.44 In particular, a

to the expected profit computed in step 3. So, the forefront solution is selected. Let us remark that some rules have been established to simplify the solution algorithm. In particular, the iterative loop in step 2 only considers the generation of new plants with larger capacity than the base solution in each iteration. Then, it can be assumed that plant Pp has the capacity to feasibly and optimally produce lower demands to the demand ξsDem that generated such solution. This demand scenario is represented by the parameter s0p for each plant Pp in the proposed algorithm (see Algorithm 7.1). Since the scenarios have been ordered by increasing demand value, only those cases s ∈ {s0p + 1,...,NS} which have a larger demand than the base scenario s0p are subject to require a plant extension and, thus, require being evaluated. Overall, the algorithm avoids screening all plant alternatives, and only finds the improvements over plant solutions calculated in previous steps. The combinatorial load is thus reduced and savings in the computational time associated with the remaining scenarios s ∈ {1,...,s0p} can be made. For the same reason, the optimization of the problem for each value of the uncertain parameters separately implies that global optimality cannot be guaranteed, as the final design is H



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direct-simultaneous method45,46 is used, consisting of the full discretization of the dynamic model to approximate both state and control variable profiles at the same time. The advantage of this method is that it does not require sensitivities, second order derivatives computation, and obligatory variable continuity to be solved. Overall, the developed strategy includes tools like the binary multiplication, Conjunctive Normal Form (CNF) reformulation, orthogonal collocation on finite elements, and the use of initial feasible solutions (IFSs) to initialize the search procedure. Reformulation of the MLDO into a MIDO. Three transformations are required to relax the original MLDO problem into a MIDO one: (1) First, Boolean variables uBool ∈{true, false} should be replaced by binaries ubin ∈ {0,1}. Thus, the vector of Boolean decision variables uBool = {Zi,Yj,Xiψ,Wj,q,Vjλ,Sjc,Rn} is transformed into a vector of binary variables ubin = {zi,yj,xiψ,wj,q,vjλ,sjc,rn}, whose 0 and 1 values correspond to prior false and true values, respectively. (2) Next, disjunctive equations should be transformed into mixed-integer ones. This transformation can be done through methodologies like the big-M,47 the Convex-Hull Relaxation (CHR),48 or the binary multiplication. The last one is based on the decomposition of differential and algebraic variables żk(t), zk(t), and yk(t), and time-invariant variables γ into contributions żk,i(t), zk,i(t), yk,i(t), and γi for each disjunctive term i ∈ ID of disjunctive equations. The decomposed variables are multiplied by the binary ubin i that corresponds to each disjunctive term. (3) Finally, logical propositions should be expressed as linear constraints. This transformation can be done systematically by formulating the CNF of the original logical equations to obtain an expression like C1 ∧ C2 ∧··· ∧ CN where Cn are the clauses that must be true in the problem, which are related by “and” operators (∧). This procedure involves the application of a series of logical operations.47 Once the CNF is obtained, each clause Cn is transformed into an algebraic equation by means of the existing analogy between logical and algebraic expressions, as it is reported by Raman and Grossmann.47 Such transformation should be applied to the set of logical propositions Ω of eq 1. MIDO Discretization. The resulting MIDO problem is fully discretized by approximating state and control variable profiles through a set of polynomials on finite times.45,46 In particular, this contribution uses the orthogonal collocation method introduced by Cuthrell and Biegler49 to solve optimal control problems which have discontinuous control profiles, as it is the case of the batch processes with phase transitions handled herein. The orthogonal collocation method consists of dividing the time axis into a number of intervals, termed finite elements e ∈ {1,...,Ne}, and specific time points within each interval, termed collocation points m ∈ {1,...,M}, and approximating the state and control variable profiles. The location of the collocation points m can be carried out by computing the roots of orthogonal polynomials, for example, roots of Hermite polynomials, Laguerre polynomials, or Legendre polynomials, among others. As for the approximation of the state and control variables, monomial basis representations can be used, which are defined through different forms, such as power series, Lagrange form representation, or Runge−Kutta equations. In this work, the collocation points are calculated using a shifted Legendre polynomial, whose roots provide the normalized locations of collocation points m, beyond the boundary points.

Moreover, Lagrange polynomials are implemented to approximate the differential zk(t), algebraic yk(t), and control udyn k (t) variables and the derivative żk(t) in finite elements e. Details of the stability, symmetry, and accuracy properties rendered by this strategy can be found in section 3.1 of the paper by Cuthrell and Biegler.49 MINLP Solution. The resulting MINLP problem obtained can be solved through a variety of algorithms available in literature and implemented into commercial software. For example, the modeling system GAMS50 provides an interface with several MINLP solvers. Particularly, the DICOPT solver is used in this study, which implements the Outer Approximation algorithm.51 Being a decomposition-based strategy, this algorithm guarantees only local optima due to the existence of nonconvex terms in the model, for example, bilinear functions associated with mixers. As a result, it is convenient to repeat the optimization procedure for several IFSs, in order to improve the possibilities to find a global optimum. Then, the problem is first solved with constant control profiles and fixed configurations chosen randomly to provide several IFSs to the MINLP solver. This small heuristic contributes to the identification of local optima and to the success of the integrated solution of process development subproblems.

6. NUMERICAL EXAMPLE This section presents an illustrative example, tackling the design of a flexible reactor network to handle the batch production of a specialty chemical product through a single-product campaign. The problem pursues the maximization of the expected profit under an uncertain demand and is addressed through the twosage stochastic minimization formulation of eq 2 and its solution through the heuristic procedure summarized in Figure 2. As a result, the flexible reactor system and the set of master recipes for the range of forecasted demands are obtained. To quantify the improvement of the quality of the solution, the problem is also solved through a decomposition method where the decision variables are optimized sequentially. The production of the desired product is carried out through a competitive reaction mechanism, the Denbigh reaction system,52 which is defined by

The kinetic and thermodynamic data including the activation energies Ea,r, standard kinetic constants k0,r, and the enthalpies of reaction Δhr, as well as other process parameters like the molar density ρ and the molecular weight MW, are available in Table S1 (see Supporting Information). These data have been adapted from Schweiger and Floudas53 taking into account a nominal temperature Tnom of 80 °C and using the reference heats of formation and combustion provided in Tables 2−220 and 2−221 from Perry and Gree.54 6.1. Problem Statement. The maximization of the expected profit for the production of specialty chemical S under uncertainty should be accomplished within a time horizon of 144 h. In particular, uncertainty is defined through five prespecified demand scenarios s ∈ {1,...,5}, which comprise an estimated demand of 21 tn of product S and variations of ±25% and ±50%, each of them with the same probability ws of I



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problem, it involves the total revenue of product S, the raw material expenses, total occupation and processing costs in batch units j ∈ J, and the penalty associated with product shortfall. Then, the formulation of the objective function in the problem reads as

0.2, as is summarized in Table S2 (see Supporting Information). Proposed Solution Method. The capacities of the processing units j ∈ J to be acquired constitute the key first-stage plant design decisions u1 ∈ U1 of the problem. Additionally, plant flexibility is faced through the second-stage decisions or recourses u2 ∈ U2, which involve the following: (1) Process synthesis decisions: splitting of reaction stage into subtasks, feed-forward reference trajectories of input and output flow rates and processing temperature at each operation in batch units, duration of batch operations, and material transfer synchronization between tasks−i.e. synchronization of flow rates, compositions, and starting and final times. (2) Allocation of manufacturing facilities decisions: selection of processing units and task-unit assignment, operating mode in a single unit, series, or parallel in-phase equipment configuration, and number of batches to be ordered in the considered time horizon. Decomposition Method. The sequential solution involves a first step to optimize the synthesis decisions, including the feedforward reference trajectories of input and output flow rates and temperature as well as the duration of batch operations. These decisions are computed for a set of given batch sizes Batch = {250 kg, 500 kg, 750 kg, 1000 kg, 1250 kg, 1500 kg, 1750 kg, 2000 kg, 2250 kg, 2500 kg} to be processed in a unique reaction stage. Then, the optimal solution for each batch size is set as a fixed recipe whose parameters cannot be modified in next steps. The first-stage equipment sizing under uncertainty is then solved considering the number of batches and the selection of the fixed recipe as second-stage recourses, which have an associated processing time, batch size, and processing, occupation, and raw material costs. 6.2. SEN Superstructure and Stochastic MLDO Formulation. The general superstructure with all the processing units and connecting pipelines which can be incorporated into the plant alternatives corresponds to the SEN network of Figure 3. Several operating modes can be

minimize j u1∈ {Size }

j + ∑ costamortization j

NS

s ), ∑ w sQ (u1, ξDem s=1

s withQ (u1 , ξDem )=

minimize

u 2 ∈ {ukdyn(t ), u stat , u int , u Bool}\ u1}

+

cost raw mat. +

j ∑ costoccupation j

j ∑ cost processing + Penalty − Revenue, j

(4)

where udyn k (t) are the dynamic control variables, comprising input and output flow rates Fj1,k(t) and Fj2,k(t) and reaction temperature θjk(t) in batch reactors j ∈ J. ustat are the timeinvariant decision variables, namely the duration of batch operations tl. uint represent other integer decisions apart from equipment capacities Sizej, particularly in this case the number of batches NBS. Finally, uBool are the Boolean variables, including the selection of the active batch units Yj out of the installed ones in each plant solution, the task-unit assignment Wj,q, and the equipment configuration Xψ1 . Additionally, differential and algebraic variables z(t) and y(t), deterministic problem parameters p, and uncertain demand ξsDem are part of the problem. The economic parameters required to evaluate each contribution can be found in Table S3 (see Supporting Information). In particular, it should be noted that the weight of the shortfall penalty is defined as twice the selling price of product S. This is a high value compared to the other economic parameters and thus ensures to prioritizing the demand fulfillment over cost reduction. 6.3. Problem Solution. During the solution of the twostage stochastic problem according to the proposed heuristic procedure of Figure 2, the deterministic problem posed in each iteration of step 2 is addressed using the direct-simultaneous approach detailed in section 5.2. Specifically, 32 finite elements and 3 collocation points are used in the problem discretization and the DICOPT solver is used in the MINLP solution, with CONOPT 3.15D and CPLEX 12.4 to handle the NLP and MIP subproblems in GAMS version 23.8.2. Additionally, several IFSs are computed by fixing some of the integer decisions and used as a feasible starting point in the search procedure, as described in section 5.2. The same strategy is applied in step 3 to evaluate the suitability of the plants Pp, p ∈ {1,...,N P } generated, now considering fixed equipment capacities. Specifically, the NS master recipes that maximize the profit in each demand scenario s ∈ {1,...,NS} are computed. Overall, the resulting MINLP problem is characterized by 102 637 equations, 98 333 variables, 345 650 nonzero elements, and 159 584 nonlinear terms. The tree of plant solutions obtained in the three first iterations of the step 2 of the heuristic Algorithm 7.1 is presented in Figure 4. For illustrative purposes, it only includes the first-stage decisions, even though second-stage decisions have been optimized simultaneously in each node, the last ones representing the optimal master recipe that generates each plant solution for the particular demand scenario s considered. The reader interested in further details of the complete solutions and Key Performance Indexes (KPIs) is addressed to the Supporting Information (Tables S4 and S5).

Figure 3. SEN superstructure of the example.

represented in such a scheme, including the operation in one single unit (configuration α), the series operation (configuration σ), and the parallel in-phase operation (configuration π) defined in the problem statement. Following the two-stage stochastic MLDO formulation of eq 2 to face the stochastic nature of the problem, the first term Φ1 of the optimization problem for the profit maximization includes total amortization costs, which only depend on firststage design decisions u1 ∈ U1. As for the second part Φ2 of the J



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Figure 4. Tree of first-stage plant solutions up to the three first iterations p = {0,1,2} of the numerical example and computational times required in each node for (i) the calculation of the IFS and (ii) the optimization of the complete MINLP problems using the IFS as initial feasible point.

P1 is here discussed, generated with a product demand ξ1Dem of 10.5 tn and being composed of one unit J1 with a capacity of 1 m3. The same plant P1 with no variations is maintained for ξ2Dem and ξ3Dem, adapting the master recipe to drive the optimal process in the aforesaid reaction unit J1. In contrast, extended systems P6 and P7 are obtained for the two higher demand values ξ4Dem and ξ5Dem, incorporating an additional unit J2 with a capacity of 1 m3 and 1.75 m3 respectively. As a result of enforcing the installation of J1 with a predefined capacity of 1 m3, the optimal profit is reduced in the other scenarios s ∈ {2,...,5} in this iteration p = 1 compared to p = 0. For example, the higher production demands ξ2Dem and ξ3Dem compared to ξ1Dem, for which this plant was designed, lead the process optimization to more extreme conditions because a larger number of batches (36 and 54 in scenarios s = 2 and 3 respectively, instead of 23 in s = 1) should be processed in the same time horizon within the same processing capacity. Therefore, the available batch processing time is reduced from 6.26 h (s = 1) to 4.00 h (s = 2) and 2.67 h (s = 3) per batch and thus reaction temperatures are forced to reach the upper bound of 110 °C to fulfill the demand more rapidly, which as higher temperature than the optimal one in iteration p = 0 (87.0 and 88.6 °C, respectively). The processing time reduction and temperature increase go in hand with a significant reduction in the selectivity of product S, becoming 3.7% lower in s = 2 and 14.2% lower in s = 3. This loss of efficiency in the process also causes the size of the batches processed in the same plant P1 to be reduced despite the total demand increases, obtaining batches of 457, 438, and 389 kg/ batch of product S in scenarios 1, 2, and 3, respectively. In contrast, the installation of a second unit J2 operating in parallel allows to keep a process efficiency similar to that obtained in the previous iteration in demand scenarios s = 4 and 5. For example, the selectivity is over the 0.590 in both cases, batch processing times are longer (4.97 h and 5.76 h in s = 4 and 5, respectively), and batch sizes are not reduced more than the 20% in both scenarios (905 and 1,260 kg/batch in s = 4 and 5, respectively). Nevertheless, the total profit

The computational times to calculate the IFSs and to solve the MINLP in each step are also shown in Figure 4. In general, computational times are much lower in the calculation of the IFSs due to the reduction of the number of integer decisions. It has been also proved that the computational times increase around 1 order of magnitude when the problem is solved without IFSs. First Iteration. All the first-stage plant solutions in the first iteration p = 0 correspond to the installation of a unique processing unit, with a suitable capacity to fulfill the production targets with zero product shortfall. While the demand value increases from ξ1Dem = 10.5tn to ξ5Dem = 31.5tn, so does the equipment capacity, with optimal values Sizej = {1 m3, 1.5 m3, 2 m3, 2.5 m3, 3 m3} in demand scenarios 1 to 5. Eventually, all scenarios require the same number of batches NBS = 23 and increasing batch sizes BatchS = {457, 685, 913, 1141, 1370} with units kg/batch in scenarios 1 to 5. In all cases, the full time horizon of 144 h is utilized, thus determining the processing time of each batch through its division by NBS. This is because larger utilization times allow (i) a reduction of the capacity requirements and, as a result, amortization costs, which are directly related to equipment capacity; (ii) to diminish heating requirements and, as a result, processing costs; and (iii) to increase selectivity and, as a result, to reduce raw material costs. In contrast, the optimal highest temperature, rises gradually following a monotonically increasing behavior as long as the batch size goes up. This increase has an effect on higher processing costs, but also on worse selectivity values. The lower the temperature profile, the better the selectivity of product S. This is because low temperatures favor reaction 1 of the Denbigh reaction system (see eq 3) compared to reaction 2 (since Ea,1 < Ea,2) and thus promote the production yield of intermediate R and product S in front of T. Next Iterations. Next, each of the obtained plant solutions Pp, p ∈ {1,...,NP}, NP = 5 enters the iterative loop in step 2 of the heuristic procedure, using each plant as a base case on which to evaluate the installation of additional equipment items. For illustrative purposes, iteration p = 1 with base plant K



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deterioration with regard to the previous iteration is still significant, with a percentage of the 7.7% and 6.5% in s = 4 and 5, respectively. This is because the economic savings obtained thanks to the better processing conditions are countervailed by the higher amortization costs. Evaluation of Plant Alternatives. To determine the best flexible plant alternative out of the NP plants generated, their total profit expectancy is computed for each plant solution Pp, p ∈ {1,...,NP}, taking into account the optimal master recipes for each demand scenario s ∈ {1,...,5}. The function reads as Evp =

NS

s ) ∑ w sv(Pp , ξDem s=1

(5)

where Epv represents the expected value of v, referred to the total profit or other KPIs, for example, product revenue, raw material expenses, total processing, occupation, and amortization costs, and shortfall penalty. The standard deviation σpv of the profit is also computed as an indicative measure of the risk associated with each plant solution Pp, according to the function σvp =

1 NS

NS

s ) − Evp) ∑ (v(Pp , ξDem s=1

(6)

6.4. Results and Discussion. General Overview of Plant Solutions. The expectancies of the profit and other KPIs for each plant solution are summarized in Figure 5, where the solution of the sequential methodology is also presented. Two differentiated plant behaviors to deal with uncertainty are identified in Figure 5a. On the one hand, the expected profit rises from plants P1 to P5, composed of a single unit J1, as long as the reactor size increases from 1 m3 to 3 m3. Such gradual improvement is due to a reduction in the total costs. This way, raw material, processing, and occupation costs decrease, as is illustrated in Figure 5b, and consequently the total cost is reduced globally, despite the increase in amortization costs. On the other hand, plants P6 to P9, which comprise two processing units, provide similar results among them, since the cost contributions find a compromise through the adaptation of allocation and processing decisions, leading to similar global outputs in the different solutions. Finally, plant P1 is distinguished from the other solutions due to the large shortfall penalty of 2248€ associated with the largest demand scenario ξ5Dem, reflected in a expectancy of 450€. The economic impact of this contribution to the objective function in problems with high penalty weight p̂penalty (in this example, twice the selling price p̂S of product S) guarantees the fulfillment of the complete demand in all other solutions. Plant Selection. According to the expected profit maximization objective, the best plant solution is plant P3, with a value of 5036€ within a range of [2140€, 7804€] and σ3Profit of 2005€. At first place, this flexible plant solution represents an improvement of the 35.8% compared to the solution obtained with the sequential approach. The last one has a value of 3709€ within a range of [1607€, 6039€] (see Figure 5a). Such improvement is principally related to the solution of partial subproblems successively. For instance, during the optimizations of the production of single batches with predefined batch sizes to define the fixed recipes, any upper bound for the time horizon cannot be considered because it is unknown, even though it is a determinant factor to fulfill the total demand in the next step.

Figure 5. Expectation Epv for each plant solution Pp in the numerical example, as well as for the sequential solution, referred to (a) total profit, (b) raw material expenses, total processing, occupation, and amortization costs in processing units, and shortfall penalty.

As it can be observed in Figure 5a, the value of the expected profit in plant solution P3 is very close to the value of other solutions like P4 and P5. This fact indicates that a similar average performance can be achieved by different plants for a given structure, provided that the processing capacity in the solution space fulfills the production levels in all plausible scenarios. Clearly, a factor of paramount importance to achieve such flexibility in several plant solutions is the support of plant design by the dynamic optimization of master recipes, involving process synthesis and plant allocation decisions. Let us bear in mind that a huge uncertainty margin with equivalent probability weights ws in all scenarios is considered in this case, ranging from −50% to +50% of the mean product demand. Therefore, the proposed methodology is able to provide solutions that fulfill the product demand in most of the screened situations. To fully reflect the details regarding the adaptability patterns obtained within the best plant solution compared to some other solutions, an isoline graphic is provided in Figure 6, where the behavior of selected plants along the different demand scenarios is illustrated through the representation of the profit L



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enhancement of plant flexibility, the adaptation of the tworeactor network P7 is presented in Figure 8. Specifically, the profiles and control variables of scenarios 1 and 5 are shown, being characterized by a different operating mode, namely single-unit operation α in s = 1 and parallel in-phase operation π in s = 5. Additionally, the capacity of processing units in this plant P7 is asymmetrical with 1 m3 and 1.75 m3 in reactors J1 and J2. As a result, occupation costs can be reduced through the selection of the unit with the most suitable capacity according to the production needs of each demand scenario, such that occupation costs can be reduced. This way, the production level in the two lowest demands ξ1Dem and ξ2Dem can be fulfilled within a single reactor operating at low temperatures (50 and 69.8 °C respectively) and high batch processing times (11.04 h and 7.20 h). It is possible that solutions with more than one processing units would be the optimal ones if the amortization load were reduced by considering longer amortization periods, since the availability of several processing units with different sizes involves savings in occupation costs through the utilization of a lower number of units with the most suitable size for each demand scenario. The details of plant solution P7 are shown in Table S7 (see Supporting Information). The stochastic solution of this particular example corresponds to the optimal solution with the estimated mean value of the uncertain parameter, namely a product demand ξ3Dem of 21 tn. One of the principal reasons is the equivalent probability weight of 0.2 that defines each uncertain scenario. In contrast, if the nonequivalent and asymmetrical probability framework defined by ws = {0.050, 0.100, 0.200, 0.450, 0.200} for ordered scenarios s = 1 to 5 is considered, the stochastic design corresponds to plant solution P4 which is computed with a demand value different to the nominal one, namely ξ4Dem with a value of 26.25 tn. The resulting expected profit of this solution achieves a value of 5990€, which slightly improves the expected profit of plant solutions P3 and P4 with 5973€ and 5951€, respectively. Moreover, the solution strategy can be also exploited to evaluate other optimization criteria. For instance, if risk is evaluated through the standard deviation of the total expenses with the original uncertainty prediction, that is, equivalent probability weights of 0.2, the best less risky solution of those evaluated would be P5, with a deviation σ5costtotal of 1074€ and expected profit of 4967€. The solution P5 is obtained with the largest demand ξ5Dem with a value of 31.5 tn. Final Remarks. Beyond the numerical results obtained, which are subject to the specific economic data as well as to the kinetic and thermodynamic information, the adaptation ability shown by the proposed optimization-based strategy is highlighted in this example. Even though the apparent similarity in the performance of most of the plant solutions which should be inferred from the results of Figure 5, each plant is characterized by a different behavior as it is shown in the diverse trajectory profiles of the feed-forward control variables and operating modes presented in Figures 7 and 8 and in the spectrum of cost values in Tables S6 and S7 (see Supporting Information). The improvement of the plant performance obtained with the proposed methodology is also shown in Figure 6 compared to the sequential solution solved using a series of optimal but fixed recipes whose synthesis and allocation parameters cannot be modified. Additionally, meeting the demand is accomplished in most of the cases, except for P1 in demand scenario 5. Such satisfactory ratio of demand fulfillment solutions is due to the adaptation of process synthesis and allocation decisions, which

Figure 6. Evolution of the profit per unit of final product along increasing demand scenarios for selected plant solutions P1, P2, P3, P5, and P7, as well as for the sequential solution. In bold, selected plant P3.

per unit of final product. Therein, the compromise of solution P3 (bold line) in the whole range of the considered demands can be observed. That is to say, plant P3 provides one of the highest profits per unit of final product for high demands like ξ5Dem, which is not excessively deteriorated for low demands like ξ1Dem. For example, on the one hand, solutions P1 and P2, with the smallest processing capacities, perform poorly for high demands, but their behavior is neither extraordinary in low ones, for which they are the optimal solution. On the other hand, plants with higher capacity like P5 show a small improvement in high demands, although this solution also provides an inferior profit in low ones due to the high amortization expenses that oversize small production requirements. Figure 6 also presents the plant behavior defined through the sequential decomposition approach, illustrating the significant improvement of the solution obtained through the proposed methodology integrating process development decisions in the plant design. The profiles of process and control variables associated with the optimal master recipes of scenarios 1, 3, and 5 in the best flexible plant P3 are also presented in Figure 7. It can be observed that batch cycle times and processing conditions are adapted in order to provide a compromise between high product selectivity and low occupation costs. For example, reaction temperature profiles goes up gradually along the different demand scenarios (from a constant profile set in 50 °C in s = 1, as is shown in Figure 7a1, to a dynamic profile with a highest temperature of 110 °C in s = 5, as is presented in Figure 7c1). At the same time, batch processing time is reduced (from 11.12 h/batch in s = 1 to 3.89 h/batch in s = 5) in order to produce larger number of batches (from 12 for s = 1 to 37 in s = 5) with a similar batch size (between 851 and 926 kg/ batch). However, the selectivity is reduced, which introduces proportionally larger raw material costs. The characterization of the five master recipes for all uncertain scenarios is also summarized in Table S6 (see Supporting Information). Plant solutions with two processing units J1 and J2 do not improve the global solution, as is illustrated in Figure 5a. This is principally due to their higher investment costs. For instance, plant solution P6 with two units has a total capacity (∑jSizej = 2 m3) equivalent to plant solution P3 with one unti (∑jSizej = 2 m3). However, the sum of amortization costs increases from 859€ in plant P3 to 1215€ in plant 6, as is shown in Figure 5b. To illustrate the effect of second stage decisions in the M



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Figure 7. Control and process variable profiles in master recipes to be implemented in plant P3, which is composed of one reactor J1 of 2 m3, for the demand scenarios 1, 3, and 5 in the example.

synthesis and plant allocation decisions, such as reference trajectories of the feed-forward control or the selection of the operating mode.

permit operation in optimal conditions according to the tradeoff between economic performance and plant flexibility considered in the objective function. Even in the case of plant solution P1, which is the one with smaller capacity, that is, one reactor J1 with a capacity of 1 m3, the complete demand is fulfilled for all the demand values except for 31.5 tn (ξ5Dem), with a production level over 98% with respect to the design demand of 10.5 tn (ξ1Dem). This way, the integrated flexible plant design and process development proposed in this paper is proven to have a crucial role in the assessment of potential plant flexibility through the simultaneous solution of process

7. CONCLUSIONS The fast development of sustainable processes and their agile introduction into production systems are crucial elements for competitiveness in specialty chemical industry. To give a response to these challenges, this paper has proposed the application of a novel optimization-based approach to tackle the problem of flexible batch plant design and process N



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Figure 8. Control and process variable profiles in master recipes to be implemented in plant P7, which is composed of two reactors J1 and J2 of 1 m3 and 1.75 m3, respectively, for the demand scenarios 1 and 5 in the example.

development, seeking a feasible and efficient operation in changing frameworks. The proposed optimization-based approach relies on a twostage stochastic MLDO formulation and a heuristic search procedure which successfully evaluates the optimization problem for particular values of uncertain parameters iteratively. In particular, the deterministic optimization problem evaluated at each node of the search procedure optimizes simultaneously the first-stage plant design and the second-stage

performance and structural process development decisions, which contrasts to the division of the decision-making procedure commonly applied in decomposition and hierarchical approaches. Thus, the heuristic procedure relies on the representation of uncertain parameters through a series of plausible scenarios, which can also include uncertain values of internal model parameters in future studies in order to reflect process variability and model inaccuracy. O



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guarantee a representative number of scenarios, to generate a wider map of solutions, and to further move toward the global solution. Finally, the study of the effect of the penalty weight within the objective function is posed as an interesting problem to analyze the changes in the compromise between the diverse cost contributions of the objective function.

The strategy is illustrated through the design of a singleproduct batch plant under demand uncertainty for implementing a competitive reactions system in a grassroots scenario. This example demonstrates that an important role is played by process development decisions in the flexible plant design. Both performance and structural degrees of freedom, like the reference trajectories for the feed-forward control variables or the selection of the operating mode, permit the adaptation of master recipes, in order that the entire range of uncertain demands can be fulfilled in every plant solution. Overall, the holistic evaluation of plant design integrating process synthesis and plant allocation decisions allows full exploitation of the degrees of freedom which otherwise would be fixed in predefined recipes. Additionally, the results indicate that physical plant restrictions barely represent a determinant factor in the plant performance, provided that reasonable demand levels are defined. The analysis of the numerical results obtained in the example evidence that the apparent similarity in the behavior of the different plant solutions is precisely achieved thanks to the ability of the proposed approach to adapt the process synthesis and plant allocation degrees of freedom, even considering a wide demand uncertainty space, between the −50% and the +50% of an estimated value. The exception are those plants of smaller processing capacities which are unable to fulfill larger demands and have a huge load in shortfall penalties. Some of the crucial degrees of freedom are for example the optimal trajectories of the feed-forward control variables, the operating modes, and the equipment unit selection for each realization of the demand. Thus, these results show the importance of including structural decisions to select the processing route within the final plant design. This statement is reinforced by the improvement over the 35% with regard to the solution obtained using a sequential decomposition strategy that does not consider the variation of these variables in the plant design. The relevant results obtained motivate the continuation of the research in several directions. First, the novel approach presented here should be refined to better treat computational aspects, with the purpose of handling comfortably a larger number of equations and variables. The authors recommend focusing on techniques to improve the efficiency of the deterministic problem optimization addressed in each iteration. In particular, the use of stochastic and hybrid optimization methods should be considered. Additionally, computational times could be reduced through the inheritance of solutions obtained in previous iterations to be used as initial feasible solutions in the next nodes of the search algorithm. Moreover, the integrated MLDO problem formulation addressed in each iteration could be simplified: the results obtained indicate that the dynamic control profiles in material transfer operation could be approximated to constant functions without damaging the quality of the solution. This way, a significant number of equations associated with the synchronization of unit procedures would be eliminated from the formulation, reducing part of the problem complexity. By doing so, the novel concepts and rich integrated decisionmaking approach here proposed can be extended to more complex batch systems, like multiproduct batch plants, as well as real industrial case studies, which will be the subject of the next papers dealing with the improvement of the computational efficiency. Additionally, the consideration of a larger number of scenarios and the introduction of sampling techniques in the first step of the heuristic procedure would contribute to

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ASSOCIATED CONTENT

* Supporting Information S

[Process parameters (Table S1), ordered demand scenarios (Table S2), and economic parameters (Table S3) of the numerical example, solutions generated in selected iterations of the proposed heuristic in the numerical example (Tables S4 and S5), and KPIs of the optimal master recipes in selected plant solutions of the numerical example (Tables S6 and S7).] This material is available free of charge via the Internet at http:// pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund for supporting the present research by project SIGERA (DPI2012-37154-C02-01).

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NOTATION

Acronyms

CHR = convex-hull relaxation CNF = conjunctive normal form DAE = differential algebraic equations DO = dynamic optimization GDP = generalized disjunctive programming IFS = initial feasible solution KPI = key performance index MIDO = mixed-integer dynamic optimization MILP = mixed-integer linear programming MINLP = mixed-integer nonlinear programming MLDO = mixed-logic dynamic optimization mSTN = maximal state-task network NLP = nonlinear programming PSE = process systems engineering RTN = resource-task network SEN = state-equipment network STN = state-task network Indices

s = demand scenario p = plant solution k = mathematical stage, corresponding to batch phases and operations m = pipeline n = recirculation pipeline j = equipment item i = process stage or task ψ = operating mode q = processing order λ = technological specification c = chemical compound P



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General Parameters

NS = number of demand scenarios NP = number of plant solutions ρ = deterministic parameters ξδ = stochastic parameters ξsδ = deterministic realization of the parameter δ in the uncertain scenario s ξsDem = deterministic realization of the demand Dem in the uncertain scenario s ws = probability associated with uncertain scenario s s0p = uncertain scenario which provided the particular plant solution p

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Xiψ = Boolean for operating mode ψ selection in process stage i Wj,q = Boolean for order q assignment to unit j Vjλ = Boolean for technology λ selection in unit j Sjc = Boolean for chemical compound c selection in unit j Rn = Boolean for recirculation of intermediate flow in pipeline n selection

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General Variables

zk(t) = differential process variables in mathematical stage k yk(t) = algebraic process variables in mathematical stage k udyn k (t) = dynamic control variables in mathematical stage k ustat = time-invariant or static continuous control variables uint = integer control variables uBool = logical or Boolean decisions ubin = binary decisions u1 = first-stage decision variables u2 = second-stage decision variables γ = algebraic time-invariant variables Epv = expectation of variable v in the plant solution p σpv = standard deviation of variable v in the plant solution p General Functions

f k = DAE system in mathematical stage k gk = path constraints in mathematical stage k gek = end-point constraints in mathematical stage k m k = stage-to-stage continuity between consecutive mathematical stages k and k+1 l = set of relations that define initial conditions h = algebraic equations evaluated at the final time f dk = DAE system in mathematical stage k in disjunctive equations gdk = path constraints in mathematical stage k in disjunctive equations gd,e k = end-point constraints in mathematical stage k in disjunctive equations m kd = stage-to-stage continuity between consecutive mathematical stages k and k+1 in disjunctive equations ld = set of relations that define initial conditions in disjunctive equations hd = algebraic equations evaluated at the final time in disjunctive equations Bd = equations system in false term of disjunctive equations Ω = logical propositions Φ = objective function Φ1 = first-stage objective function Φ2 = second-stage objective function Decision Variables of the Problem

Fjm,k(t) = flow rate profile in pipeline m of unit j in material transfer operation k intjk(t) = Internal control variables in operation k of unit j tk = duration of mathematical stage k NBp = number of batches of product p Batch = production size associated with each batch of product Sizej = capacity of unit j Zi = Boolean for process stage i selection Yj = Boolean for equipment item j selection Q



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Flexible Batch Process and Plant Design Using Mixed-Logic Dynamic

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