Algebraic surrogate-based process optimization using Bayesian symbolic learning

Here, we propose a strategy for the global optimization of process flowsheets, a fundamental problem in process systems engineering, based on algebraic surrogates that are built from rigorous simulations via Bayesian symbolic regression. The applied method provides a closed-form expression that can be optimized to global optimality using state-of-the-art solvers, where BARON or ANTIGONE were the solvers of choice. When predicting unseen test data, the algebraic models show a similar accuracy level compared to traditional surrogates based on Gaussian processes. However, they can be more easily optimized to global optimality due to their analytical closed-form structure, which allows the user to apply well-established global deterministic solvers. We show the capabilities of our approach in several case studies, ranging from process units to full flowsheets. The performance of our approach is assessed by comparing the CPU time for model building, the prediction accuracy of the identified model, and the CPU time for the subsequent optimization with a proven benchmark.     

AN APPROACH FOR ESTIMATION OF CRITICAL POINTS OF WATER SORPTION ISOTHERMS OF FOODS

Abstract

The problem of optimal storage conditions of dry foods with sigma-shaped Water Sorption Isotherm is investigated. It is assumed that the optimal value of water activity, A_o, corresponds to the critical points of the isotherm where its local slope (the first derivative) has a minimum value. Two approaches were tested. In the first one the experimental data were fitted by the modified GAB/BET model and the optimal point was calculated by differentiation of the resulting equation. In regression the second approach estimates of local slope of the isotherm between adjacent experimental points was calculated. The data were fitted by the proposed empirical model for the first derivative of the sigma-shaped isotherm. Then, the minimum value of the first derivative was computed. Comparative analysis of both methods based on experimental moisture sorption data for pistachio nuts at 10, 20 and 30°C revealed that the relative error in the optimal point by the second method is less than that by the first one.     

Water networks security: A two-stage mixed-integer stochastic program for sensor placement under uncertainty

This work describes a stochastic approach for the optimal placement of sensors in municipal water networks to detect maliciously injected contaminants. The model minimizes the expected fraction of the population at risk and the cost of the sensors. Our work explicitly includes uncertainties in the attack risk and population density, so that the resulting problem involves optimization under uncertainty. In our formulation, we include the location of a number of sensors as first stage decision variables of a two-stage mixed-integer stochastic linear problem; the second stage evaluates the population at risk for the scenario obtained in the first stage and that information is then used to modify the first stage decisions for the next iteration. Since the model is integer in the first stage, a generalized framework based on the stochastic decomposition algorithm allows us to solve the problem in a reasonable computational time. The paper describes the mixed-integer stochastic model and the algorithmic framework, and compares the deterministic and stochastic optimal solutions. The network used as our case study has been derived through the water network simulator EPANET 1.0; four acyclic water flow patterns are considered. Results show a significant effect of uncertainty in sensor placement and total cost.     

基于混合自动微分算法的缩聚反应过程优化

针对缩聚反应过程优化模型中的分枝结构模块给求取导函数带来的困难 ,在扩展自动微分算法基础上提出了一种混合自动微分算法 .该算法结合符号自动微分算法和数值自动微分算法求取函数的导函数 ,并利用导函数计算导数 .这种求导算法不仅具有很高的精度和效率 ,且在开放式方程模型体系下可被化工过程优化领域中的各类典型优化命题调用 .优化计算的结果显示了基于混合自动微分算法的缩聚反应优化在效率上的显著优势.     

Learning surrogate models for simulation‐based optimization

A central problem in modeling, namely that of learning an algebraic model from data obtained from simulations or experiments is addressed. A methodology that uses a small number of simulations or experiments to learn models that are as accurate and as simple as possible is proposed. The approach begins by building a low‐complexity surrogate model. The model is built using a best subset technique that leverages an integer programming formulation to allow for the efficient consideration of a large number of possible functional components in the model. The model is then improved systematically through the use of derivative‐free optimization solvers to adaptively sample new simulation or experimental points. Automated learning of algebraic models for optimization (ALAMO), the computational implementation of the proposed methodology, along with examples and extensive computational comparisons between ALAMO and a variety of machine learning techniques, including Latin hypercube sampling, simple least‐squares regression, and the lasso is described. © 2014 American Institute of Chemical Engineers AIChE J, 60: 2211–2227, 2014     

On the stochastic simulation of particulate systems

The stochastic chemical kinetics approach provides one method of formulating the stochastic crystallization population balance equation (PBE). In this formulation, crystal nucleation and growth are modelled as sequential additions of solubilized ions or molecules (units) to either other units or an assembly of any number of units. Monte Carlo methods provide one means of solving this problem. In this paper, we assess the limitations of such methods by both (1) simulating models for isothermal and nonisothermal size-independent nucleation, growth and agglomeration; and (2) performing parameter estimation using these models. We also derive the macroscopic (deterministic) PBE from the stochastic formulation, and compare the numerical solutions of the stochastic and deterministic PBEs. The results demonstrate that even as we approach the thermodynamic limit, in which the deterministic model becomes valid, stochastic simulation provides a general, flexible solution technique for examining many possible mechanisms. Thus the stochastic simulation permits the user to focus more on modelling issues as opposed to solution techniques.     

化工过程系统优化中的一种混合求导算法

在符号微分法的基础上结合差商和自动微分技术提出一种实用高效的混合求导算法,提高了化工过程系统优化的效率.该算法将被求导对象的各个子模块按照是否适于用符号微分求导分为简单模块和复杂模块,对前者用符号微分求导,对后者用差商法求导.针对该算法的关键技术——子模块导数累积问题讨论了链式求导法和摄动传递法.对某厂精馏塔操作优化问题的分析和计算表明,引入混合求导算法提高了优化的效率.     

Three phases in the development of computer simulation of chemical engineering systems

Three phases in the development of computer simulation of chemical engineering systems are considered. In the first phase, the mass and heat balances are calculated using detailed mathematical models of apparatuses. In the second phase, the chemical engineering system is optimized in a (strictly) deterministic formulation. In the third phase, the optimal parameters of the chemical engineering system are chosen provided that the system is serviceable throughout the possible range of internal and external factors. The method used in the third phase is the flexibility analysis of chemical engineering systems.     

The computer-derivation of thermodynamic equationspart II. first and second derivatives for complex unrestricted systems

An algorithm, based on Jacobians and implemented on a microcomputer (IBM PC), is described for the derivation of expressions for first and second thermodynamic derivatives for complex systems, unrestricted, however, by (chemical) equilibrium constraints. The work has progressed through the design of two separate algorithms implemented by symbolic manipulation software. First, a thermodynamic derivation system, THERMODER1, was developed, aided by a symbolic mathematical system (muMATH-83^tm), together with its language of implementation (muSIMP), and certain user-defined functions. THERMODER1 is capable of deriving expressions for first and second thermodynamic derivatives for simple systems (2 degrees of freedom, unrestricted). A second algorithm, THERMODER2, also implemented by the symbolic mathematical system, has been developed for both simple and complex (unrestricted) systems. THERMODER1 has been described previously, and THERMODER2 is described here. These algorithms are intended to be extended ultimately to chemical systems subject to equilibrium constraints. Examples, using THERMODER2, are given.     

Deterministic optimization of the thermal Unit Commitment problem: A Branch and Cut search

This paper proposes a novel deterministic optimization approach for the Unit Commitment (UC) problem, involving thermal generating units. A mathematical programming model is first presented, which includes all the basic constraints and a set of binary variables for the on/off status of each generator at each time period, leading to a convex mixed-integer quadratic programming (MIQP) formulation. Then, an effective solution methodology based on valid integer cutting planes is proposed, and implemented through a Branch and Cut search for finding the global optimal solution. The application of the proposed approach is illustrated with several examples of different dimensions. Comparisons with other mathematical formulations are also presented.     

Iterative symbolic regression for learning transport equations

Computational fluid dynamics (CFD) analysis is widely used in chemical engineering. Although CFD calculations are accurate, the computational cost associated with complex systems makes it difficult to obtain empirical equations between system variables. Here, we combine active learning (AL) and symbolic regression (SR) to get a symbolic equation for system variables from CFD simulations. Gaussian process regression-based AL allows for automated selection of variables by selecting the most instructive points from the available range of possible parameters. The results from these experiments are then passed to SR to find empirical symbolic equations for CFD models. This approach is scalable and applicable for any desired number of CFD design parameters. To demonstrate the effectiveness, we use this method with two model systems. We recover an empirical equation for the pressure drop in a bent pipe and a new equation for predicting backflow in a heart valve under aortic insufficiency.     

Comparative Study of the Leak Characteristics of Two Ceramic/Glass Composite Seals for Solid Oxide Fuel Cells

Solid oxide fuel cells (SOFCs) have the potential to play a significant role in a future clean energy economy. However, SOFCs still face major obstacles before they can be commercialized, with efficient sealing being among the most prominent. The present research focuses on the comparative study of microstructure, crystal phase evolution, and leak rates, for two ceramic/glass seals used in an SOFC. The leak test apparatus is a controlled facility designed to incorporate different mechanical loading, stack configurations, and thermal cycles. Simultaneous leak testing with an acoustic emission (AE) sensor was also used to identify any micro‐damage in seals. A two‐level factorial design was applied to the first sealing composition to identify the main and the interactive factors for leak rates. MINITAB^® was also used to determine a linear regression‐based leak rate model. The second seal formulation employed a more stable glass which led to reduced leak rates. Additional factors in a two‐level factorial design were investigated for the second seal formulation. Based on multiple experiments with different stack components, it was determined that the number of interfaces is most critical for leak rate, showing that even in the presence of thermal cycling, leakage is an interfacial dominated phenomenon.     

Optimizing inventory policies in process networks under uncertainty

We address the inventory planning problem in process networks under uncertainty through stochastic programming models. Inventory planning requires the formulation of multiperiod models to represent the time-varying conditions of industrial process, but multistage stochastic programming formulations are often too large to solve. We propose a policy-based approximation of the multistage stochastic model that avoids anticipativity by enforcing the same decision rule for all scenarios. The proposed formulation includes the logic that models inventory policies, and it is used to find the parameters that offer the best expected performance. We propose policies for inventory planning in process networks with arrangements of inventories in parallel and in series. We compare the inventory planning strategies obtained from the policy-based formulation and the analogous two-stage approximation of the multistage stochastic program. Sequential implementation of the planning strategies in receding horizon simulations shows the advantages of the policy-based model, despite the increase in computational complexity.     

Improved algorithm for calculation of binary liquid-liquid equilibrium compositions with NRTL model based on equal area rule

Typical methods for calculation of binary liquid-liquid equilibrium compositions such as surfactant systems need proper initial guesses and/or checking the sign of the second derivative of molar Gibbs energy change of mixing, ΔG. Eubank and Hall [1] have shown the equal area rule (EAR) applies to the composition derivative of the Gibbs energy of a binary system at fixed pressure and temperature. Methods based on EAR do not need to check the sign of the second derivative of ΔG because EAR is a necessary and sufficient condition for phase equilibrium. However, the algorithm proposed by Eubank and Hall needs a reasonable initial guess. Furthermore, it is not easy to apply the algorithm to activity coefficient models such as Non-Random Two Liquid (NRTL) because the first and second derivatives of ΔG as a function of composition have various shapes for some sets of NRTL parameters. In this work, we have developed an improved algorithm for calculation of binary liquid-liquid equilibrium compositions based on EAR considering the various shapes of NRTL model. This algorithm needs neither any initial guess nor checking the sign of the second derivative of ΔG.     

红外指纹区光谱结合多阶导数融合技术无损分类鞋底物证

为了提高检验效率,降低检验鉴定成本,实现对鞋底的快速无损分类。采用傅里叶变换红外指纹光谱及其多阶导数光谱对5类不同品牌共计50个样本的鞋底进行分析,并构建Bayes判别和支持向量机2种分类模型。结果表明,在鞋底鉴别过程中,基于原始数据、一阶导数数据和二阶导数数据建立的融合模型,初级融合模型的区分效果优于单一模型和中级融合模型,总体分类准确率能达到80 %以上。而基于初级模型进行的成分特征提取中,BDA结合原始数据结合一阶导数模型是最好的,总体分类准确率达到92 %。红外指纹光谱结合一阶求导、二阶求导构建不同的融合模型进行区分对比,选择最为有效的融合模型可实现对日常皮鞋、运动鞋鞋底快速的无损鉴别,对今后的治安工作作具有借鉴意义,不仅缩小排查范围,也为案件的快速侦破提供了一种新的方式。     

Interfacial Rheology Measured with a Spinning Drop Interfacial Rheometer: Particularities in More Realistic Surfactant–Oil–Water Systems Close to Optimum Formulation at HLDN = 0

Three different cases were selected to study the effect of physicochemical formulation on interfacial rheology properties of surfactant–oil–water (SOW) systems by increasing the complexity of the system from a basic case. This was performed by changing the normalized hydrophilic–lipophilic deviation (HLD_N) to attain the optimum formulation at HLD_N = 0. Two types of SOW systems were studied: the first one used an ionic surfactant with a salinity scan, and the second one a mixture of two nonionic surfactants in a formulation scan produced by changing their proportion. Both of them contained cyclohexane as a pure oil phase, without alcohol. Sec-butanol was then added as a co-surfactant with hardly any formulation influence on HLD_N. The complexity in interfacial rheology was then increased by changing the oil to a light crude with low asphaltene content. The interfacial rheology is also reported for a realistic system with a high asphaltene content comprised of crude oil diluted in cyclohexane with a conventional surfactant and a commercial demulsifier. The findings confirm that at optimum formulation and whatever the scanning variable (salinity, average ethylene oxide number in the nonionic surfactant mixture, or surfactant/demulsifier concentration), the interfacial tension, and interfacial elastic moduli E, E′, and E″ exhibit a deep minimum. These observations are related to the acceleration of the surfactant exchanges between the interface, oil, and water, near the optimum formulation. Several arguments are put forward to explain how these findings could contribute to the decrease in emulsion stability at HLD_N = 0.     

A reactive optimization strategy for the simultaneous planning,scheduling and control of short-period continuous reactors

The efficient and economic operation of processing systems ideally requires a simultaneous planning, scheduling and control framework. Even when the optimal simultaneous solution of this problem can result in large scale optimization problems, such a solution can represent economic advantages making feasible its computation using optimization decomposition and/or few operating scenarios. After reducing the complexity of the optimal simultaneous deterministic solution, it becomes feasible to take into account the effect of model and process uncertainties on the quality of the solution. In this work we consider those changes in product demands that take place once the process is already under continuous operation. Therefore, a reactive strategy is proposed to meet the new product demands. Based on an optimization formulation for handling the simultaneous planning, scheduling, and control problem of continuous reactors, we propose a heuristic strategy for dealing with unexpected events that may appear during operation of a plant. Such a strategy consists of the rescheduling of the products that remain to be manufactured after the given disturbance hits the process. Such reactive strategy for dealing with planning, scheduling and control problems under unforeseen events is tested using two continuous chemical reaction systems.     

Off-Line Detection of Multiple Change Points by the Filtered Derivative with p-Value Method

Abstract

This article deals with off-line detection of change points, for time series of independent observations, when the number of change points is unknown. We propose a sequential analysis method with linear time and memory complexity. Our method is based, on a filtered derivative method that detects the right change points as well as false ones. We improve the filtered derivative method by adding a second step in which we compute the p-values associated to every single potential change point. Then, we eliminate false alarms; that is, the change points that have p-values smaller than a given critical level. Next, we apply our method and penalized least squares criterion procedure to detect change points on simulated data sets and then we compare them. Eventually, we apply the filtered derivative with p-value method to the segmentation of heartbeat time series, and the detection of change points in the average daily volume of financial time series.     

Droplets diameter distribution using maximum entropy formulation combined with a new energy-based sub-model

The maximum entropy principle (MEP) is one of the first methods which have been used to predict droplet size and velocity distributions of liquid sprays. This method needs a mean droplets diameter as an input to predict the droplet size distribution. This paper presents a new sub-model based on the deterministic aspects of liquid atom-ization process independent of the experimental data to provide the mean droplets diameter for using in the maximum entropy formulation (MEF). For this purpose, a theoretical model based on the approach of energy conservation law entitled energy-based model (EBM) is presented. Based on this approach, atomization occurs due to the kinetic energy loss. Prediction of the combined model (MEF/EBM) is in good agreement with the avail-able experimental data. The energy-based model can be used as a fast and reliable enough model to obtain a good estimation of the mean droplets diameter of a spray and the combined model (MEF/EBM) can be used to wel predict the droplet size distribution at the primary breakup.     

Multivariate Time‐Series Analysis With Categorical and Continuous Variables in an Lstr Model

Abstract. We develop a methodology for multivariate time‐series analysis when our time‐series has components that are both continuous and categorical. Our specific contribution is a logistic smooth‐transition regression (LSTR) model, the transition variable of which is related to a categorical time‐series (LSTR‐C). This methodology is necessary for series that exhibit nonlinear behaviour dependent on a categorical time‐series. The estimation procedure is investigated both with simulation and an economic time‐series. We obtain superior or equivalent model fits as compared with another smooth‐transition regression model. Furthermore, even when the nonlinear behaviour of the time‐series is dependent on a continuous time‐series, we propose a simplification of the modelling process, which is the automatic formulation of the transition variable from the categorical time‐series. We are able to capture this nonlinear dependence on a continuous time‐series by using regression theory for categorical time‐series.