Order‐reduction of parabolic PDEs with time‐varying domain using empirical eigenfunctions

A novel methodology for the order‐reduction of parabolic partial differential equation (PDE) systems with time‐varying domain is explored. In this method, a mapping functional is obtained, which relates the time‐evolution of the solution of a parabolic PDE with time‐varying domain to a fixed reference domain, while preserving space invariant properties of the initial solution ensemble. Subsequently, the Karhunen–Loève decomposition is applied to the solution ensemble on fixed spatial domain resulting in a set of optimal eigenfunctions. Further, the low dimensional set of empirical eigenfunctions is mapped on the original time‐varying domain by an appropriate mapping, resulting in the basis for the construction of the reduced‐order model of the parabolic PDE system with time‐varying domain. This methodology is used in three representative cases, one‐ and two‐dimensional (1‐D and 2‐D) models of nonlinear reaction‐diffusion systems with analytically defined domain evolutions, and the 2‐D model of the Czochralski crystal growth process with nontrivial geometry. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4142–4150, 2013     

Solving Multi‐objective MILP Problems in Process Synthesis using the Multi‐Criteria Branch and Bound Algorithm

The paper briefly describes the problem of process synthesis in the area of chemical engineering, and suggests its formulation as a Multi‐Objective Programming problem. Process synthesis optimization is usually modeled as Mixed Integer Linear Programming (MILP) or Mixed Integer Non‐Linear Programming (MINLP) with an economic objective function. We claim that incorporating more criteria (e.g., environmental criteria) in this kind of combinatorial optimization problem offers the decision makers the opportunity to refine their final decision by examining more than one solution (a set of efficient or Pareto optimal solutions instead of one optimal solution). For solving the multi‐objective process synthesis problem, an improved version of the Multi‐Criteria Branch and Bound (MCBB) algorithm, which has been developed by the same authors, is used. MCBB is a vector maximization algorithm capable of deriving all efficient points (supported and unsupported), for small and medium sized Multi‐Objective MILP problems. The application of MCBB in two examples from process synthesis is also presented.     

Feedback control of dissipative PDE systems using adaptive model reduction

The problem of feedback control of spatially distributed processes described by highly dissipative partial differential equations (PDEs) is considered. Typically, this problem is addressed through model reduction, where finite dimensional approximations to the original infinite dimensional PDE system are derived and used for controller design. The key step in this approach is the computation of basis functions that are subsequently utilized to obtain finite dimensional ordinary differential equation (ODE) models using the method of weighted residuals. A common approach to this task is the Karhunen‐Loève expansion combined with the method of snapshots. To circumvent the issue of a priori availability of a sufficiently large ensemble of PDE solution data, the focus is on the recursive computation of eigenfunctions as additional data from the process becomes available. Initially, an ensemble of eigenfunctions is constructed based on a relatively small number of snapshots, and the covariance matrix is computed. The dominant eigenspace of this matrix is then utilized to compute the empirical eigenfunctions required for model reduction. This dominant eigenspace is recomputed with the addition of each snapshot with possible increase or decrease in its dimensionality; due to its small dimensionality the computational burden is relatively small. The proposed approach is applied to representative examples of dissipative PDEs, with both linear and nonlinear spatial differential operators, to demonstrate its effectiveness of the proposed methodology. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

Integration of scheduling and control for batch processes using multi‐parametric model predictive control

Integration of scheduling and control results in Mixed Integer Nonlinear Programming (MINLP) which is computationally expensive. The online implementation of integrated scheduling and control requires repetitively solving the resulting MINLP at each time interval. (Zhuge and Ierapetritou, Ind Eng Chem Res. 2012;51:8550–8565) To address the online computation burden, we incorporare multi‐parametric Model Predictive Control (mp‐MPC) in the integration of scheduling and control. The proposed methodology involves the development of an integrated model using continuous‐time event‐point formulation for the scheduling level and the derived constraints from explicit MPC for the control level. Results of case studies of batch processes prove that the proposed approach guarantees efficient computation and thus facilitates the online implementation. © 2014 American Institute of Chemical Engineers AIChE J, 60: 3169–3183, 2014     

Explicit model predictive control of hybrid systems and multiparametric mixed integer polynomial programming

Hybrid systems are dynamical systems characterized by the simultaneous presence of discrete and continuous variables. Model‐based control of such systems is computationally demanding. To this effect, explicit controllers which provide control inputs as a set of functions of the state variables have been derived, using multiparametric programming mainly for the linear systems. Hybrid polynomial systems are considered resulting in a Mixed Integer Polynomial Programming problem. Treating the initial state of the system as a set of bounded parameters, the problem is reformulated as a multiparametric Mixed Integer Polynomial optimization (mp‐MIPOPT) problem. A novel algorithm for mp‐MIPOPT problems is proposed and the exact explicit control law for polynomial hybrid systems is computed. The key idea is the computation of the analytical solution of the optimality conditions while the binary variables are treated as relaxed parameters. Finally, using symbolic calculations exact nonconvex critical regions are computed. © 2016 The Authors AIChE Journal published by Wiley Periodicals, Inc. on behalf of American Institute of Chemical Engineers AIChE J, 62: 3441–3460, 2016     

Temperature distribution reconstruction in Czochralski crystal growth process

A mechanical geometric crystal growth model is developed to describe the crystal length and radius evolution. The crystal radius regulation is achieved by feedback linearization and accounts for parametric uncertainty in the crystal growth rate. The associated parabolic partial differential equation (PDE) model of heat conduction is considered over the time‐varying crystal domain and coupled with crystal growth dynamics. An appropriately defined infinite‐dimensional representation of the thermal evolution is derived considering slow time‐varying process effects. The computational framework of the Galerkin's method is used for parabolic PDE order reduction and observer synthesis for temperature distribution reconstruction over the entire crystal domain. It is shown that the proposed observer can be utilized to reconstruct temperature distribution from boundary temperature measurements. The developed observer is implemented on the finite‐element model of the process and demonstrates that despite parametric and geometric uncertainties present in the model, the temperature distribution is reconstructed with the high accuracy. © 2014 American Institute of Chemical Engineers AIChE J, 60: 2839–2852, 2014     

Symmetry breaking in MILP formulations for Unit Commitment problems

This paper addresses the study of symmetry in Unit Commitment (UC) problems solved by Mixed Integer Linear Programming (MILP) formulations, and using Linear Programming based Branch & Bound MILP solvers. We propose three sets of symmetry breaking constraints for UC MILP formulations exhibiting symmetry, and its impact on three UC MILP models are studied. The case studies involve the solution of 24 instances by three widely used models in the literature, with and without symmetry breaking constraints. The results show that problems that could not be solved to optimality within hours can be solved with a relatively small computational burden if the symmetry breaking constraints are assumed. The proposed symmetry breaking constraints are also compared with the symmetry breaking methods included in two MILP solvers, and the symmetry breaking constraints derived in this work have a distinct advantage over the methods in the MILP solvers.     

MILP scheduling model for multipurpose batch processes considering various intermediate storage policies

MILP (Mixed Integer Linear Programming) scheduling models for non-sequential multipurpose batch processes are presented. Operation sequences of products have to be made in each unit differently by considering production route of each product under a given intermediate storage policy to reduce idle time of units and to raise the efficiency of the process. We represent the starting and finishing time of a task in each unit with two coordinates for a given storage policy. One is based on products, and the other is based on operation sequences. Then, using binary variables and logical constraints, we match the variables used in the two coordinates into one. We suggest MILP models considering sequence dependent setup times to guarantee the optimality of the solutions. Two examples are presented to show the effectiveness of the suggested models.     

An MILP-based reordering algorithm for complex industrial scheduling and rescheduling

The challenging structure as well as the extensive economical potential of solving various scheduling problems has fascinated numerous researchers during the recent decades. Despite the significant progress made in the fields of operations research and process systems engineering (Puigjaner, Comp. Chem. Eng., 23 (1999): S929–S943; Shah, Proceedings of FOCAPO'98, Snowbird, Utah, USA, 1998) the complexity of many industrial-size scheduling problems means that a global optimal solution cannot be reached within a reasonable computational time. In these cases, the production schedule must be generated using e.g. some kind of sophisticated heuristics, which can often lead to suboptimal solutions. In this paper, we introduce a Mixed Integer Linear Programming (MILP) based algorithm, which can be efficiently used to improve an existing feasible, but non-optimal, production schedule or to reschedule jobs in the case of changed operational parameters. The algorithm has been successfully applied to certain scheduling problems in both the paper-converting and pharmaceutical industry.     

Exergy as an objective function to MINLP problems in process synthesis

The optimal layout of a chemical plant is determined by both economic and environmental aspects. The mathematical approach to this problem is given by representing the process through a superstructure, including any foreseeable process topology. It is described by integer variables, whereas real variables arise from the unit models and the thermodynamic calculations. By including process constraints and an objective function, a Mixed Integer Nonlinear Programming (MINLP) optimization problem is postulated. In this paper MINLP algorithms are combined with a technique for the exergy analysis of chemical processes for the preliminary screening of process alternatives. The exergy analysis as a screening technique is advantageous due to a significantly smaller amount of required data and computing time, compared with true optimization, including cost estimation at an earlier stage of process synthesis and analysis. Thus, a broader solution space can be examined. The subsequent cost analysis has then only to be focused on a reduced set of parameters.     

Systematic network synthesis and design: Problem formulation,superstructure generation,data management and solution

The developments obtained in recent years in the field of mathematical programming considerably reduced the computational time and resources needed to solve large and complex Mixed Integer Non Linear Programming (MINLP) problems. Nevertheless, the application of these methods in industrial practice is still limited by the complexity associated with the mathematical formulation of some problems. In particular, the tasks of design space definition and representation as superstructure, as well as the data collection, validation and handling may become too complex and cumbersome to execute, especially when large problems are considered. In an earlier work, we proposed a computer-aided framework for synthesis and design of process networks. In this contribution, we expand the framework by including methods and tools developed to structure, automate and simplify the mathematical formulation of the design problem. Furthermore, the models employed for the representation of the process alternatives included in the superstructure are refined, through the inclusion of the energy balance. Finally, the features of the framework are highlighted through the solution of two case studies focusing on food processing and biofuels.     

Spatially explicit multi-objective optimisation for design and planning of hybrid first and second generation biorefineries

Climate change mitigation has become a binding driver in biofuels production. First generation bioethanol, initially indicated as the most competitive option, is now incurring in ever increasing discredits forcing the transition towards more sustainable productions (i.e. second and third generation technologies). This paper addresses the strategic design and planning of corn grain- and stover-based bioethanol supply chains through first and second generation technologies. A Mixed Integer Linear Programming framework is proposed to optimise the environmental and financial performances simultaneously. Multi-period, multi-echelon and spatially explicit features are embodied within the formulation to steer decisions and investments through a global approach. A demonstrative case study is proposed involving the future Italian biomass-based ethanol production. Results show the effectiveness of the optimisation tool at providing decision makers with a quantitative analysis assessing the economic and environmental performance of different design configuration and their effect in terms of technologies, plant sizes and location, and raw materials.     

考虑不确定汽电需求的蒸汽动力系统优化设计

蒸汽动力系统优化设计不仅要保证系统全周期总费用最小,还要保证系统具有一定可操作性应对各种不确定变工况影响。本文提出了同时考虑汽电需求确定性多周期变化以及燃料、电力价格,汽电需求不确定性变化的蒸汽动力系统优化设计策略,专门提出不确定变化参数对优化目标及约束的影响以及数学表达,建立混合整数线性规划(MILP)优化模型进行求解,并应用于某石化企业蒸汽动力系统优化设计。实例设计结果表明,与按照传统策略设计结果相比,应用本文提出的考虑不确定参数的设计策略获得的优化方案,具体设计了针对不确定因素实现而对应的运行调度安排,降低年总费用,保证系统安全稳定运行,实现经济性和可操作性综合最优。     

A MILP model for N-dimensional allocation

This paper presents a Mixed Integer Linear Programming (MILP) model for the solution of N-dimensional allocation problems. The applicability of the model is presented and demonstrated through some illustrative examples with different numbers of dimensions. Several problems, previously presented in the literature, are solved using the proposed model, such as, one-dimensional scheduling problems, two-dimensional cutting problems, as well as plant layout problems and three-dimensional packing problems. Additionally, some problems in four dimensions are presented and solved using the considered model. The presented model is applicable to a wide variety of allocation problems as it offers a general framework for modelling allocation problems with any given number of continuous or discrete dimensions. The presented problems are formulated as MILP problems where the first four dimensions usually are continuous spatial and time dimensions. Additional dimensions are often of a discrete nature.     

Systematische Prozesssynthese mit Hilfe mathematischer Methoden

This paper presents a novel method for the process synthesis and optimisation, the so called Mixed Integer Non Linear Programming (MINLP). The speciality of this tool is that not only the operating parameters but also the structure of the process are simultaneously optimised. A necessary condition for this kind of optimisation is the definition of an appropriate superstructure, that contains all physically relevant processes including the optimal one. Therefore, it is advantageous to determine these superstructures based on the information relating to thermodynamic equilibrium. The methodology for the development of superstructures and the results of the optimisation are presented for several examples concerned with distillation, extractive distillation, azeotropic distillation, and reactive distillation.     

Optimal planning of artificial lift operations in a shale gas multiwell pad

Lifting operations play a key role to sustain shale gas wells productivity. Artificial Lift Systems (ALS) involve a wide range of techniques aimed at deliquifying wells. Although these operations are relatively mature for vertical wells in conventional reservoirs, enhancing productivity from shale gas wells has become a new frontier for unconventional production. The main contribution of this work is the development of the first Mixed Integer Linear Programming (MILP) formulation determining the integrated planning of several ALS in a multiwell pad. The model simultaneously manages ALS selection, investment and operational decisions, hedging against rapid decline curves in different wells. We introduce very detailed piecewise functions accounting for the ALS installation and removal times. The performance of the MILP is evaluated by solving five illustrative case studies.     

Boundary control synthesis for a lithium‐ion battery thermal regulation problem

The thermal regulation problem for a lithium ion (Li‐ion) battery with boundary control actuation is considered. The model of the transient temperature dynamics of the battery is given by a nonhomogeneous parabolic partial differential equation (PDE) on a two‐dimensional spatial domain which accounts for the time‐varying heat generation during the battery discharge cycle. The spatial domain is given as a disk with radial and angular coordinates which captures the nonradially symmetric heat‐transfer phenomena due to the application of the control input along a portion of the spatial domain boundary. The Li‐ion battery model is formulated within an appropriately defined infinite‐dimensional function space setting which is suitable for spectral controller synthesis. The key challenges in the output feedback model‐based controller design addressed in this work are: the dependence of the state on time‐varying system parameters, the restriction of the input along a portion of the battery domain boundary, the observer‐based optimal boundary control design where the separation principle is utilized to demonstrate the stability of the closed loop system, and the realization of the outback feedback control problem based on state measurement and interpolation of the temperature field. Numerical results for simulation case studies are presented. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3782–3796, 2013     

Heat exchanger network design of large-scale industrial site with layout inspired constraints

This paper presents a systematic approach for the synthesis of the heat recovery network in total site using a Mixed Integer Linear Programming model. This model returns a near-to-optimal network configuration with minimum utility cost while allows to select geographically closest matches. The Heat Load Distribution is the subproblem of the network design and has been reported to be quite expensive to solve for large-scale problems. The computational complexity of HLD resides in the number of streams and the feasible networks. An additional challenge, raising particularly in industrial problems, has been the intermediate heat transfer network which aggravates the combinatorial complexity. The presented methodology deals with those difficulties by priority consideration based on the location of process units. It helps significantly reducing the computational time and also comes with a realistic network sketch with respect to the plant layout. Several examples are discussed along with a real industrial case study.     

The effect of uncertainty on the optimal closed-loop supply chain planning under different partnerships structure

In a global economy, the key to success is providing products around the world at the right time in the right quantity and quality, at a low cost. Efficient supply chains have an important role in guaranteeing this success. Optimized planning of such structures is required and uncertainties regarding product demands and prices, amongst other supply chain conditions, should also be considered. In this paper, we look into supply chain planning decisions that account for uncertainty on product portfolios demand and prices. A multi-period planning model is developed where the supply chain operational decisions on supply, production, transportation, and distribution at the actual period consider the uncertainty on products’ demand and prices. Different decision scenarios, involving the evaluation of the supply chain economical performance, are analyzed (e.g. global operating costs/profit realized) for different criteria on the importance of the partners within the global chain (i.e. partners’ structure). A Mixed Integer Linear Programming (MILP) formulation is formulated for each planning scenario and the optimal solution is reached using a standard Branch and Bound (B&B) procedure. The final results provide details on the supply chain partners production, transportation and inventory, at each planning period, while accounting for the importance of each partner in the global chain as well as demand/price uncertainties. The applicability of the developed formulation is illustrated through the solution of a real case-study involving an industrial chain in the pharmaceutical sector.     

A COMBINED MILP AND LOGIC-BASED APPROACH TO THE SYNTHESIS OF OPERATING PROCEDURES FOR BATCH PLANTS

Synthesizing a set of operating procedures for the safe and efficient transient operation of chemical plants is a difficult problem owing to the enormous number of possible combinations of actions in a typical plant. In most current industrial plant design practice, there are no formal methods for systematically transforming process specifications into operating procedures for the plant operators and into sequence control instructions for the control computers. There is much scope for a formalized computer-based procedure synthesis methodology to assist the design engineer/plant operator with both the formulation and assessment of procedures off-line and, eventually, with the on-line problem of procedure synthesis in response to unexpected situations

A recently developed approach for operating procedure synthesis for multipurpose batch plants is considered. The modelling formalism used includes the separate definition of process operations, as State Task Networks, and of physical plant, at the level of detail of a piping and instrumentation diagram. In this paper, a subgoaling procedure is developed using the State Task Network representation which decomposes the procedure synthesis goals into simpler subgoals by means of an efficient Mixed Integer Linear Programming (MILP) technique. Detailed control sequences are then generated for each subgoal using a set of rules and algorithms specific for each type of subgoal. The procedure sequences thus generated are validated by simulation on a plant model with checking of physical and operational constraints at each new plant state. We have found that this hierarchical approach to the procedure synthesis problem greatly reduces the problem complexity

The usefulness of the general approach and of the subgoaling procedure in particular are demonstrated through a multiproduct batch plant example.