Multiobjective optimization of an industrial grinding operation using elitist nondominated sorting genetic algorithm

The elitist version of nondominated sorting genetic algorithm (NSGA II) has been adapted to optimize the industrial grinding operation of a lead-zinc ore beneficiation plant. Two objective functions have been identified in this study: (i) throughput of the grinding operation is maximized to maximize productivity and (ii) percent passing of one of the most important size fractions is maximized to ensure smooth flotation operation following the grinding circuit. Simultaneously, it is also ensured that the grinding product meets all other quality requirements, to ensure least possible disturbance in the following flotation circuit, by keeping two other size classes and percent solid of the grinding product and recirculation load of the grinding circuit within the user specified bounds (constraints). Three decision variables used in this study are the solid ore flowrate and two water flowrates at two sumps, primary and secondary, each of them present in each of the two stage classification units. Nondominating (equally competitive) optimal solutions (Pareto sets) have been found out due to conflicting requirements between the two objectives without violating any of the constraints considered for this problem. Constraints are handled using a technique based on tournament selection operator of genetic algorithm which makes the process get rid of arbitrary tuning requirement of penalty parameters appearing in the popular penalty function based approaches for handling constraints. One of the Pareto points, along with some more higher level information, can be used as set points for the previously mentioned two objectives for optimal control of the grinding circuit. Implementation of the proposed technology shows huge industrial benefits.     

不确定条件下的湿法炼锌除铜过程机会约束优化控制

除铜过程是湿法炼锌净化工艺中的重要步骤,受生产环境多变、矿源多样、机理复杂等因素的影响,除铜过程存在不确定性,影响生产的稳定性和可靠性。针对除铜过程中入口溶液流量、底流返回量和入口铜离子浓度的不确定性,造成出口铜离子浓度不稳定的问题,研究不确定条件下的除铜过程机会约束优化控制方法。首先分析了除铜过程的不确定性,利用统计学方法分析不确定参数的分布特性,引入了机会约束的思想,将不确定条件下的除铜过程优化问题建模为机会约束优化问题。然后采用可行域映射方法,将机会约束优化问题转化为非线性规划问题。最后,使用序列二次规划求解该非线性规划问题。Monte Carlo仿真验证了该方法的有效性,可以提高系统的鲁棒性。     

A Comparative Study of Fuzzy Techniques to Handle Uncertainty: An Industrial Grinding Process

Fuzzy‐based approaches like fuzzy chance constrained programming (FCCP) and fuzzy expected value model (FEVM) have been applied to a multi‐objective optimization problem of the industrial grinding process to carry out the uncertainty analysis. Results are compared with respect to the power of risk averseness adopted in the approaches used. The extent of constraint satisfaction due to the presence of uncertain parameters can be accommodated assuming credibility of constraint satisfaction under the FCCP framework whereas the robust set of parameters in the FEVM approach is determined by considering the expectation terms for objectives and constraints. Nonlinear relation of uncertain parameters has been handled by adopting simulation‐based approaches while computing the credibility. These approaches are very generic and can be applied for the study of parametric sensitivity for any process model in a novel manner.     

Optimization of refinery hydrogen network based on chance constrained programming

Deterministic optimization approaches have been developed and used in the optimization of hydrogen network in refinery. However, uncertainties may have a large impact on the optimization of hydrogen network. Thus the consideration of uncertainties in optimization approaches is necessary for the optimization of hydrogen network. A novel chance constrained programming (CCP) approach for the optimization of hydrogen network in refinery under uncertainties is proposed. The stochastic properties of the uncertainties are explicitly considered in the problem formulation in which some input and state constraints are to be complied with predefined probability levels. The problem is then transformed to an equivalent deterministic mixed-integer nonlinear programming (MINLP) problem so that it can be solved by a MINLP solver. The solution of the optimization problem provides comprehensive information on the economic benefit under different confidence levels by satisfying process constraints. Based on this approach, an optimal and reliable decision can be made, and a suitable compensation between the profit and the probability of constraints violation can be achieved. The approach proposed in this paper makes better use of resources and can provide significant environmental and economic benefits. Finally, a case study from a refinery in China is presented to illustrate the applicability and efficiency of the developed approach.     

Chance constrained optimization of process systems under uncertainty: I. Strict monotonicity

An approach for chance constrained programming of large-scale nonlinear dynamic systems is presented. The stochastic property of the uncertainties is explicitly considered in the problem formulation in which some input and state constraints are to be complied with predefined probability levels. The method considers a nonlinear relation between the uncertain input and the constrained variables. It also involves efficient algorithms so as to compute the probabilities and, simultaneously, the gradients through integration by collocation in finite elements. The formulation of single or joint probability limits incorporates the issue of feasibility and the contemplation of trade-off between robustness and profitability regarding the objective function values. The approach is relevant to all cases when uncertainty can be described by any kind of joint correlated multivariate distribution function. Thus, chance constrained programming is a promising technique in solving optimization problems under uncertainty in system engineering. The potential and the efficiency of the presented systematic methodology, which assumes a strict monotonic relationship between the uncertain input and the uncertain constrained output, are illustrated with application to a reactive batch distillation processes under uncertainty.     

Nichtlineare stochastische Optimierung unter Unsicherheiten

Nonlinear Stochastic Optimization under Uncertainty Robust decision making under uncertainty is considered to be of fundamental importance in numerous disciplines and application areas. In dynamic chemical processes in particular there are parameters which are usually uncertain, but may have a large impact on equipment decisions, plant operability, and economic analysis. Thus the consideration of the stochastic property of the uncertainties in the optimization approach is necessary for robust process design and operation. As a part of it, efficient chance constrained programming has become an important field of research in process systems engineering. A new approach is presented and applied for stochastic optimization problems of batch distillation with a detailed dynamic process model.     

Multiobjective batch process design aiming at robust performances

The most common batch design approach in practice and literature is a deterministic one. However, given the uncertainties prevailing in early stages of process design, a deterministically calculated productivity is not sufficient to select one of the large number of optional designs. Therefore, we propose a Tabu Search multiobjective optimization framework, which allows to approximate the Pareto-optimal set of designs while considering uncertain variables in the initial recipe. As a novel technique, we include performance robustness as a separate objective function within the multiobjective optimization alongside with productivity of a design, thus obtaining not only designs with high productivity or solely robust designs, but both high productivity and robust designs in one set of solutions. We examined several robustness criteria as a possible quantification of performance deviations under uncertain recipe variables. The implementation of a Tabu Search framework in combination with Monte-Carlo simulation and Latin Hypercube sampling provides a huge flexibility in the problem specification, in particular in the definition of parameter uncertainties. As a result we successfully demonstrate that metaheuristic optimization techniques are capable to approximate the Pareto-optimal set under uncertainty and are able to capture potentially antagonistic solution qualities such as high productivity and robustness by multiobjective optimization. With the help of this approach, parameters can be identified that have to be put into the focus of process research and development efforts in order to obtain high performance batch process designs.     

Multiobjective optimization under uncertainty of the economic and life‐cycle environmental performance of industrial processes

The combined use of multiobjective optimization and life‐cycle assessment (LCA) has recently emerged as a useful tool for minimizing the environmental impact of industrial processes. The main limitation of this approach is that it requires large amounts of data that are typically affected by several uncertainty sources. We propose herein a systematic framework to handle these uncertainties that takes advantage of recent advances made in modeling of uncertain LCA data and in optimization under uncertainty. Our strategy is based on a stochastic, multiobjective, and multiscenario mixed‐integer nonlinear programming approach in which the uncertain parameters are described via scenarios. We investigate the use of two stochastic metrics: (1) the environmental impact in the worst case and (2) the environmental downside risk. We demonstrate the capabilities of our approach through its application to a generic complex industrial network in which we consider the uncertainty of some key life‐cycle inventory parameters. © 2014 American Institute of Chemical Engineers AIChE J, 60: 2098–2121, 2014     

Optimal design of chemical processes with chance constraints

Generally, chemical processes (CP) are designed with the use of inaccurate mathematical models. Therefore, it is important to create a chemical process that guarantees satisfaction of all design specifications either exactly or with some probability. The paper considers the issue of chemical process optimization when at the operation stage the design specification should be met with some probability and the control variables can be changed. We have developed a common approach for solving the broad class of optimization problems with normally distributed uncertain parameters. This class includes the one-stage and two-stage optimization problems with chance constraints. This approach is based on approximate transformation of chance constraints into deterministic ones.     

Optimization problem with normally distributed uncertain parameters

The issue of chemical process optimization when at the operation stage the design specification should be met with some probability and the control variables can be changed has been considered. A common approach for solving the broad class of optimization problems with normally distributed uncertain parameters were developed. This class includes the one‐stage and two‐stage optimization problems with chance constraints. This approach is based on approximate transformation of chance constraints into deterministic ones. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2471–2484, 2013     

Distributionally robust chance constrained programming with generative adversarial networks (GANs)

This paper presents a novel deep learning based data-driven optimization method. A novel generative adversarial network (GAN) based data-driven distributionally robust chance constrained programming framework is proposed. GAN is applied to fully extract distributional information from historical data in a nonparametric and unsupervised way without a priori approximation or assumption. Since GAN utilizes deep neural networks, complicated data distributions and modes can be learned, and it can model uncertainty efficiently and accurately. Distributionally robust chance constrained programming takes into consideration ambiguous probability distributions of uncertain parameters. To tackle the computational challenges, sample average approximation method is adopted, and the required data samples are generated by GAN in an end-to-end way through the differentiable networks. The proposed framework is then applied to supply chain optimization under demand uncertainty. The applicability of the proposed approach is illustrated through a county-level case study of a spatially explicit biofuel supply chain in Illinois.     

Multiobjective optimization of an industrial wiped film poly(ethylene terephthalate) reactor: some further insights

Multiobjective optimization of an industrial third-stage, wiped-film poly(ethylene terephthalate) reactor is carried out, using a pre-validated model. The two objective functions minimized are the acid and vinyl end group concentrations in the product. These are two of the undesirable side products produced in the reactor. The optimization problem incorporates an end-point constraint to produce polymer having a desired value of the degree of polymerization (DP). In addition, the concentration of the di-ethylene glycol end group in the product is constrained to lie within a certain range of values. The possible decision variables for the problem are the reactor pressure, temperature, catalyst concentration, residence time of the reaction mass in the reactor and the speed of rotation of the agitator. The nondominated sorting genetic algorithm (NSGA) is used to solve this multiobjective optimization problem. It is found that this algorithm is unable to converge to the correct solution(s) when two or more decision variables are used, and we need to run the code several times over (with different values of the computational variable, S_r, the seed for generating the random numbers) to obtain the solutions. In fact, this is an excellent test problem for future multiobjective optimization algorithms. It is found that when temperature is kept constant, Pareto optimal solutions are obtained, while, when the temperature is included as a decision variable, a global unique optimal point is obtained.     

Optimal operation of distillation processes under uncertain inflows accumulated in a feed tank

A new method of optimizing the operation of distillation processes with uncertain inflow streams from upstream plants is discussed. The case considered is the streams that will be first accumulated in a tank before feeding to the column. To minimize the total amount of operating energy while keeping a stable column operation under these inflows, a novel decomposed optimization strategy consisting of two steps was used. For an optimal planning of the dynamic operation, a smooth feed flow policy to the column is developed in the first step by stochastic optimization under chance constraints by ensuring a predefined probability of holding the tank level inside the desired range. An easy-to-use method developed computes the maximum reachable probability of holding the constraints so that a feasible solution of the chance constrained problem can be guaranteed. Since the uncertainty in the inflow stream variability is absorbed in the stochastic optimization over the tank, the operation of the dowstream distillation column is deterministic. Therefore, in the second step, the reflux and reboiler duty policies of the column are developed by deterministic dynamic optimization. The optimal overall strategy is obtained by the maximized smoothness of the feed flow to the distillation column. The approach is applied to a pilot column, and the developed operating policies are implemented on the real plant by experiment.     

几种典型水泥粉磨系统的比较

水泥粉磨系统已由早先的球磨机系统逐步发展到现在的球磨机+辊压机系统和立磨系统。通过分析流程和主要配置及运行参数,归类出比较典型的七种水泥粉磨系统的技术性能持点。其中,立磨终粉磨系统流程最简单,能耗最低,是水泥粉磨方案的首选;辊压机+球磨(带涡流选粉机)组成的联合粉磨系统及立磨和球磨组成的联合粉磨系统流程相近,能耗相近,可作为水泥粉磨系统的优选方案;辊压机和球磨机组成的开流及半开流系统缺点最多,应尽量避免选用。     

KBM磨内改造技术及其应用

KBM磨内改造技术将康毕登磨与高产磨、高细磨、筛分磨技术优化组合在一起，以专有技术装置代替原有的普通隔仓板及细磨仓少部分衬板，并辅以合理的球段方案，优化磨机仓内配置、对开流磨进行增产、节能、提质、增效改造，可使其产量增加30％以上，电耗下降25％以上，水泥强度提高约5MPa，产品比表面积增加50-100m^2／kg，且颗粒分布均匀。另外，改造投资低，改造工期短。     

Polynomial chaos‐based robust design of systems with probabilistic uncertainties

A new algorithm is proposed for the design of nonlinear dynamical systems with probabilistic uncertainties. The dependence of the design objective and constraints on uncertainties is quantified by the polynomial chaos expansions (PCEs), while the relationships between the design parameters and the design objective/constraints are parameterized by Legendre polynomials. In two case studies, the polynomial chaos‐based algorithm reduces the number of system evaluations required by optimization by an order of magnitude. Quantifying the dependence on uncertain parameters via the PCEs and including the quantification in design optimization simultaneously improved the distribution of the performance index and the probability of constraint fulfillment. © 2016 American Institute of Chemical Engineers AIChE J, 62: 3310–3318, 2016     

Optimal robust optimization approximation for chance constrained optimization problem

Chance constraints are useful for modeling solution reliability in optimization under uncertainty. In general, solving chance constrained optimization problems is challenging and the existing methods for solving a chance constrained optimization problem largely rely on solving an approximation problem. Among the various approximation methods, robust optimization can provide safe and tractable analytical approximation. In this paper, we address the question of what is the optimal (least conservative) robust optimization approximation for the chance constrained optimization problems. A novel algorithm is proposed to find the smallest possible uncertainty set size that leads to the optimal robust optimization approximation. The proposed method first identifies the maximum set size that leads to feasible robust optimization problems and then identifies the best set size that leads to the desired probability of constraint satisfaction. Effectiveness of the proposed algorithm is demonstrated through a portfolio optimization problem, a production planning and a process scheduling problem.     

Nonlinear model of pneumatic conveying dryer for economic process control

Abstract

This paper presents a nonlinear dynamic model, suitable for economic process control of pneumatic conveying dryer for drying of food grains. The dynamic model is developed by reshaping the process equations derived for the batch drying, dilute phase, and a negative-pressure conveying system. The dynamic model parameters are identified by numerically solving a nonlinear least squares optimization problem, subject to a set of differential and algebraic equality constraints that describe the system dynamics and bounds in the parameters. A detailed parametric uncertainty and sensitivity analysis are performed providing valuable insight into the influence of critical model parameters on observables, the interplay among various parameter-state-measured disturbances, and quantifying uncertainties in the model. Further, different process economic performance and product quality indicator of uncertain dryer model are studied. The model validation study as performed with the underlying process shows a very good agreement in understanding necessary dynamic characteristics and interplay between the various parameter of interest.     

Probability bounds analysis for nonlinear dynamic process models

Dynamic process models frequently involve uncertain parameters and inputs. Propagating these uncertainties rigorously through a mathematical model to determine their effect on system states and outputs is a challenging problem. In this work, we describe a new approach, based on the use of Taylor model methods, for the rigorous propagation of uncertainties through nonlinear systems of ordinary differential equations (ODEs). We concentrate on uncertainties whose distribution is not known precisely, but can be bounded by a probability box (p‐box), and show how to use p‐boxes in the context of Taylor models. This allows us to obtain p‐box representations of the uncertainties in the state variable outputs of a nonlinear ODE model. Examples having two to three uncertain parameters or initial states and focused on reaction process dynamics are used to demonstrate the potential of this approach. Using this method, rigorous probability bounds can be determined at a computational cost that is significantly less than that required by Monte Carlo analysis. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

A global optimization algorithm (GOP) for certain classes of nonconvex NLPs—II. Application of theory and test problems

In Part I (Floudas and Visweswaran, Computers chem. Engng 14, 1397, 1990), a deterministic global optimization approach was proposed for solving certain classes of nonconvex optimization problems. An algorithm, GOP, was presented for the rigorous solution of the problem through a series of primal and relaxed dual problems until the upper and lower bounds from these problems converged to an -global optimum. In this paper, theoretical results are presented for several classes of mathematical programming problems that include: (i) the general quadratic programming problem; (ii) quadratic programming problems with quadratic constraints; (iii) pooling and blending problems; and (iv) unconstrained and constrained optimization problems with polynomial terms in the objective function and/or constraints. For each class, a few examples are presented illustrating the approach.