Nonlinear parameter estimation through particle swarm optimization

Parameter estimation procedures are very important in the chemical engineering field for development of mathematical models, since design, optimization and advanced control of chemical processes depend on model parameter values obtained from experimental data. Model nonlinearity makes the estimation of parameter and the statistical analysis of parameter estimates more difficult and more challenging. In this work, it is shown that many of these difficulties can be overcome with the use of heuristic optimization methods, such as the particle swarm optimization (PSO) method. Parameter estimation problems are solved here with PSO and it is shown that the PSO method is efficient for both minimization and construction of the confidence region of parameter estimates. Moreover, it is shown that the elliptical approximation of confidence regions of nonlinear model parameters can be very poor sometimes and that more accurate likelihood confidence regions can be constructed with PSO, allowing for more reliable statistical analysis of the significance of parameter estimates.     

Investigation of gas–solid bubbling fluidized beds using ECT with a modified Tikhonov regularization technique

Electrical capacitance tomography (ECT) provides a non‐intrusive means to visualize cross‐sectional material distribution of gas–solid bubbling fluidized beds. Successful application of ECT strongly depends on the image reconstruction algorithm used. For on‐line measurements of bubbling fluidized beds, employing an algorithm that can produce high‐quality images without extensive computation is necessary. Using the conventional Tikhonov regularization algorithm, image quality in the central area is basically satisfied but suffers from artifacts in the near‐wall region. To solve this problem, a similar division operation learned from linear back projection was introduced to modify the conventional Tikhonov algorithm. Both numerical simulations and experiments were performed to evaluate the modified technique. The results indicate that the artifacts can be effectively removed and the reconstructed image quality is similar to Landweber method with dozens of iterations. Furthermore, the modified Tikhonov technique shows high accuracy when obtaining important hydrodynamic parameters in gas–solid bubbling fluidized beds. © 2017 American Institute of Chemical Engineers AIChE J, 63: 29–41, 2018     

Robust PSD determination of micro and meso-pore adsorbents via novel modified U curve method

Proper estimation of pore size distribution (PSD) for various solid adsorbents is essential for efficient characterization of such materials. PSD determination is inherently an ill-posed problem which can be solved by resorting to Tikhonov regularization technique. Automatic selection of the optimal regularization level is crucial for efficient extraction of PSD from experimental adsorption or condensation isotherms. A detailed comparison of Leave One Out Cross Validation, L-curve, U-curve and modified L-curve criteria are presented using several real adsorption and condensation case studies. A modified U-curve criterion is also proposed and its superior performances are thoroughly validated. The new method employs a relatively simple minimization procedure, which is much easier than quite complex and tedious procedures used by conventional methods to compute their internal optimum parameters. Other practical issues, such as efficient computation of L-corner have been received proper attention in this article and a simple algorithm is presented for this purpose.     

粒子群优化算法在催化裂化模型参数估计中的应用

参数估计是化工模型工业应用中的重要课题,有相当的难度。针对催化裂化八集总模型的动力学参数估计问题,考察了不同类型优化算法的应用效果,结果表明,粒子群优化算法简单、容易实现,而且可以避免传统方法对初始值的依赖,并进一步提出用结合Levenberg-Marquardt算法的混合粒子群优化算法提高参数估计效果。工业实例表明,用混合粒子群优化算法得到的动力学参数可以保证模型的预测精度。     

复合粒子群优化算法在模型参数估计中的应用

化工非线性模型的参数估计是较为困难的寻优问题，经典方法常会陷入局部极值。粒子群算法操作简便、容易实现且全局搜索功能较强，适用于非线性参数估计。但其参数值的确定与问题相关，若设定不当，会严重影响全局搜索的性能。今提出引入遗传算法，在粒子群算法的搜索过程中，逐代优选参数，包括惯性权值，加速常数，以此构建为复合粒子群优化算法。分析与测试表明，其全局搜索性能有显著改善。进一步的工作又将两种粒子群算法成功地应用于重油热解模型的参数估计。采用复合粒子群优化算法估计参数构建的重油热解模型，其预报相对误差比常规粒子群优化算法降低了8.97％，比简单遗传算法降低了23.21％，效果明显。     

A developed smart technique to predict minimum miscible pressure—eor implications

Miscible gas injection (MGI) processes such as miscible CO₂ flooding have been in use as attractive EOR options, especially in conventional oil reserves. Optimal design of MGI is strongly dependent on parameters such as gas–oil minimum miscibility pressure (MMP), which is normally determined through expensive and time‐consuming laboratory tests. Thus, developing a fast and reliable technique to predict gas–oil MMP is inevitable. To address this issue, a smart model is developed in this paper to forecast gas–oil MMP on the basis of a feed‐forward artificial neural network (FF‐ANN) combined with particle swarm optimisation (PSO). The MMP of a reservoir fluid was considered as a function of reservoir temperature and the compositions of oil and injected gas in the proposed model. Results of this study indicate that reservoir temperature among the input parameters selected for the PSO–ANN has the greatest impact on MMP value. The developed PSO–ANN model was examined using experimental data, and a reasonable match was attained showing a good potential for the proposed predictive tools in estimation of gas–oil MMP. Compared with other available methods, the proposed model is capable of forecasting oil–gas MMP more accurately in wide ranges of thermodynamic and process conditions. All predictive models used other than the PSO–ANN model failed in providing a good estimate of the oil–gas MMP of the hydrocarbon mixtures in Azadegan oilfield, Iran. © 2013 Canadian Society for Chemical Engineering     

Nonlinear dynamic data reconciliation and parameter estimation through particle swarm optimization: Application for an industrial polypropylene reactor

This work presents a procedure to solve nonlinear dynamic data reconciliation (NDDR) problems with simultaneous parameter estimation based on particle swarm optimization (PSO). The performance of the proposed procedure is compared to the performance of a standard Gauss-Newton (GN) scheme in a real industrial problem, as presented previously by Prata et al. [2006. Simultaneous data reconciliation and parameter estimation in bulk polypropylene polymerizations in real time. Macromolecular Symposia 243, 91-103; 2008. In-line monitoring of bulk polypropylene reactors based on data reconciliation procedures. Macromolecular Symposia 271, 26-37]. Both methods are used to solve the NDDR problem in an industrial bulk propylene polymerization process, using real data in real time for the simultaneous estimation of model parameters and process states. A phenomenological model of the real process, based on the detailed mass and energy balances and constituted by a set of algebraic-differential equations, was implemented and used for interpretation of the actual plant behavior in real time. The resulting nonlinear dynamic optimization problem was solved iteratively on a moving time window, in order to capture the current process behavior and allow for dynamic adaptation of model parameters. Obtained results indicate that the proposed PSO procedure can be implemented in real time, allowing for estimation of more reliable process states and model parameters and leading to much more robust and reproducible numerical performance.     

采用二阶导数阵作为正则化的电容成像图像重建算法

Tikhonov正则化方法是解决病态逆问题的常用方法,正则化项的引入能改善问题的病态性。利用Laplace算子对正则化项中所包含的图像信息进行锐化处理,可提高电容成像图像重建的质量。仿真结果表明,相对于标准形式的Tikhonov正则化方法,采用正则算子为二阶导数算子的正则化方法图像重建结果边缘及轮廓清晰,对于各种设定流型均具有良好的适应能力,且图像重建结果对初始设定解不敏感,图像重建结果质量更高。     

POD-based observer for estimation in Navier–Stokes flow

In this paper, we propose a POD-based technique that is suitable for the design of reliable observers for the estimation of velocity field and contaminant flow for Navier–Stokes flow. POD modes are constructed using the method snapshot. Karhunen–Loeve (Galerkin) projection to develop a reduced-order model obtained by projecting the velocity field onto the most important POD modes. The resulting finite-dimensional dynamical system is suitable for the design of nonlinear observers. The estimate of the velocity field is then used to estimate the concentration field of a contaminant from the 2D advection–diffusion equation. The prime application considered is the estimation of airflow and contaminant flow in building systems. A 2D simulation example is provided to demonstrate the applicability of the technique.     

Simultaneous robust data reconciliation and gross error detection through particle swarm optimization for an industrial polypropylene reactor

In a previous study, a nonlinear dynamic data reconciliation procedure (NDDR) based on the particle swarm optimization (PSO) method was developed and validated in line and in real time with actual industrial data obtained for an industrial polypropylene reactor (Prata et al., 2009, Prata et al., 2008b). The procedure is modified to allow for robust implementation of the NDDR problem with simultaneous detection of gross errors and estimation of model parameters. The negative effects of the less frequent gross errors are eliminated with the implementation of the Welsch robust estimator, avoiding the computation of biased estimates and implementation of iterative procedures for detection and removal of gross errors. The performance of the proposed procedure was tested in line and in real time in an industrial bulk propylene polymerization process. A phenomenological model of the real process, based on the detailed mass and energy balances and constituted by a set of algebraic-differential equations, was implemented and used for interpretation of the actual plant behavior. The resulting nonlinear dynamic optimization problem was solved iteratively on a moving time window, in order to capture the current process behavior and allow for dynamic adaptation of model parameters. Results indicate that the proposed procedure, based on the combination of the PSO method and the robust Welsch estimator, can be implemented in real time in real industrial environments, allowing for the simultaneous detection of gross errors and estimation of process states and model parameters, leading to more robust and reproducible numerical performance.     

On the feasibility and reliability of nonlinear kinetic parameter estimation for a multi-component photocatalytic process

Nonlinear kinetic parameter estimation plays an essential role in kinetic study in reaction engineering. In the present study, the feasibility and reliability of the simultaneous parameter estimation problem is investigated for a multi-component photocatalytic process. The kinetic model is given by the L-H equation, and the estimation problem is solved by a hybrid genetic-simplex optimization method. Here, the genetic algorithm is applied to find out, roughly, the location of the global optimal point, and the simplex algorithm is subsequently adopted for accurate convergence. In applying this technique to a real system and analyzing its reliability, it is shown that this approach results in a reliable estimation for a rather wide range of parameter value, and that all parameters can be estimated simultaneously. Using this approach, one can estimate kinetic parameters for all components from data measured in only one time experiment.     

Estimation of Coefficients of Fluidized Bed Drying through the PSO and GA Metaheuristic Approaches

This article shows the results of heat and mass transfer coefficients estimation in a fluidized bed drying obtained through two independent metaheuristics, particle swarm optimization (PSO) and genetic algorithms (GAs). Upon estimating parameters, we aimed to minimize errors between the experimental data provided by Vitor (Modeling of Biomass Drying in Fluidized Bed, D.Sc. Thesis, 2003) and those calculated through a three-phase drying differential-algebraic model. The computational results showed that the two metaheuristics chosen were suitable to estimate the drying parameters proposed here. When the two metaheuristics are compared, the PSO shows slightly better results in much shorter computational times. The coefficient of heat transfer estimated here is compared to results obtained from other experiments and proves to be quite adequate.     

PARAMETER ESTIMATION OF MONOD KINETICS WITHOUT BIOMASS DATA AND INITIAL SUBSTRATE CONCENTRATION

The uniqueness of the parameters in a Monod kinetics model estimated from substrate depletion and product growth data were analyzed using nonlinear regression. The initial substrate and biomass concentrations were considered as unknown (unmeasured) parameters. Simulated data containing known measurement errors were generated first using specified parameters and then these data were used for the analysis. Sensitivity coefficients were determined by differentiating the original differential equations of the Monod kinetics. A procedure for determining the scale factors for the parameters (used in the nonlinear regression method) has also been developed. The number of parameters that can be determined uniquely depends on the region of substrate depletion (first order, mixed order or zero order). The availability of product data leads to additional estimation of parameters in the mixed order region. The number of parameters that could be estimated are identified qualitatively by the sensitivity coefficient analysis and quantitatively by analyzing the orthonormal eigenvectors of the Gauss-Newton matrix in the nonlinear regression algorithm. Experimental data on aerobic and anareobic substrate depletion were used to estimate the parameters and validate the analysis.     

Multi-stage economic model predictive control for a gold cyanidation leaching process under uncertainty

In order to dynamically operate the gold cyanidation leaching process (GCLP) under uncertainty, a multi-stage economic model predictive control (EMPC) is proposed for GCLP for the transient and steady-state economic optimization. The proposed multi-stage EMPC is composed of two steps. In the first step, the unmeasurable uncertain parameters are estimated by using Tikhonov regularization based method, so as to avoid amplification and propagation of the noise measurements into the estimation. Based on the estimated results, the scenario tree for multi-stage EMPC is generated from the historical data using a data-driven approach, and the control inputs are obtained from solving the resulting large nonlinear programming problem (NLP) at each sampling point. The resulting uncertainty model and the probability of each scenario are more consistent with the actual industrial GCLP, and the solutions are less conservative. The efficiency of the proposed multi-stage EMPC is verified through a simulated industrial GCLP. Compared with other EMPC methods, including classic EMPC and multi-stage EMPC with box uncertainty region, the proposed method can reduce the economic cost while accounting for the constraints at the same time.     

Kinetic parameter estimation from TGA: Optimal design of TGA experiments

This work presents a general methodology to determine kinetic models of solid thermal decomposition with thermogravimetric analysis (TGA) instruments. The goal is to determine a simple and robust kinetic model for a given solid with the minimum of TGA experiments. From this last point of view, this work can be seen as an attempt to find the optimal design of TGA experiments for kinetic modelling. Two computation tools were developed. The first is a nonlinear parameter estimation procedure for identifying parameters in nonlinear dynamical models. The second tool computes the thermogravimetric experiment (here, the programmed temperature profile applied to the thermobalance) required in order to identify the best kinetic parameters, i.e. parameters with a higher statistical reliability. The combination of the two tools can be integrated in an iterative approach generally called sequential strategy. The application concerns the thermal degradation of cardboard in a Setaram TGA instrument and the results that are presented demonstrate the improvements in the kinetic parameter estimation process.     

基于杂交粒子群算法的多元线性回归参数估计及预测区间研究

为了提高多元线性回归分析模型预测混凝土28 d抗压强度的准确性和可靠性,采用杂交粒子群优化算法估算模型系数,依据正态分布和t分布求出预测点置信度为95%的预测区间。实验结果表明:此模型的预测精度优于传统基于最小二乘估算的回归分析模型,且预测结果可以是相关量的取值范围,扩大了相关量的适用范围,提高了预测的可靠性。     

电容层析成像技术在线测量气固流化床空隙率的研究

基于电容层析成像技术，提出了一种在线测量气固流化床空隙率的新方法。建立了相应的12电极电容层析成像气固流化床空隙率测量系统，可同时实现气固 流化床空隙率分布的在线显示和整体空隙率测量。选择加权反投影算法进行图像重建以保证空隙率分布显示的实时性和有效性。采用Tikhonov正则化原理和ART算法相结合的组合型新图像重建算法来实现整体空隙率的测量。Tikhonov正则化原理用于克服图像重建过程中的不适定问题，ART算法用于提高最终重建图像的质量。研究表明以上提出的空隙率测量新方法是有效的。空隙率分布在线测量的速度可达25幅／秒以上，整体空隙率测量的最大误差可小于5％。     

Real-time adaptive input design for the determination of competitive adsorption isotherms in liquid chromatography

The adaptive input design (also called online redesign of experiments) for parameter estimation is very effective for the compensation of uncertainties in nonlinear processes. Moreover, it enables substantial savings in experimental effort and greater reliability in modeling.We present theoretical details and experimental results from the real-time adaptive optimal input design for parameter estimation. The case study considers separation of three benzoate by reverse phase liquid chromatography. Following a receding horizon scheme, adaptive D-optimal input designs are generated for a precise determination of competitive adsorption isotherm parameters. Moreover, numerical techniques for the regularization of arising ill-posed problems, e.g. due to scarce measurements, lack of prior information about parameters, low sensitivities and parameter correlations are discussed. The estimated parameter values are successfully validated by Frontal Analysis and the benefits of optimal input designs are highlighted when compared to various standard/heuristic input designs in terms of parameter accuracy and precision.     

Gas Analysis by In Situ Combustion in Heavy‐Oil Recovery Process: Experimental and Modeling Studies

Enormous efforts have been made to facilitate produced‐gas analyses by in situ combustion implication in heavy‐oil recovery processes. Robust intelligence‐based approaches such as artificial neural network (ANN) and hybrid methods were accomplished to monitor CO₂/O₂/CO. Implemented optimization approaches like particle swarm optimization (PSO) and hybrid approach focused on pinpointing accurate interconnection weights through the proposed ANN model. Solutions acquired from the developed approaches were compared with the pertinent experimental in situ combustion data samples. Implication of hybrid genetic algorithm and PSO in gas analysis estimation can lead to more reliable in situ combustion quality predictions, simulation design, and further plans of heavy‐oil recovery methods.     

A method for computing the partial derivatives of experimental data

This article describes a procedure for obtaining the partial derivatives of experimental data that depend on two independent variables. The starting equation is an ill‐posed integral equation of the first kind. Tikhonov regularization is used to keep noise amplification under control. Implementation of the computation steps is described and the performance of the procedure is demonstrated by four practical examples. © 2010 American Institute of Chemical Engineers AIChE J, 2010