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Keywords = backward difference formula (BDF)

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11 pages, 1085 KiB  
Article
Methods for Parameter Estimation in Wine Fermentation Models
by Robert Coleman, James Nelson and Roger Boulton
Fermentation 2024, 10(8), 386; https://doi.org/10.3390/fermentation10080386 - 27 Jul 2024
Cited by 1 | Viewed by 1619
Abstract
The estimation of parameters in a wine fermentation model provides the opportunity to predict the rate and concentration outcomes, to strategically intervene to change the conditions, and to forecast the rates of heat and carbon dioxide release. The chosen parameters of the fermentation [...] Read more.
The estimation of parameters in a wine fermentation model provides the opportunity to predict the rate and concentration outcomes, to strategically intervene to change the conditions, and to forecast the rates of heat and carbon dioxide release. The chosen parameters of the fermentation model are the initial assimilable nitrogen concentration and yeast properties (lag time, viability constant, and specific maintenance rate). This work evaluates six methods for parameter estimation: Bard, Bayesian Optimization, Particle Swarm Optimization, Differential Evolution, Genetic Evolution, and a modified Direct Grid Search technique. The benefits and drawbacks of the parameter computational methods are discussed, as well as a comparison of numerical integration methods (Euler, Runge–Kutta, backward differential formula (BDF), and Adams/BDF). A test set of density-time data for five white and five red commercial wine fermentations across vintage, grape cultivar, fermentation temperature, inoculated yeast strain, and fermentor size was used to evaluate the parameter estimation methods. A Canonical Variate Analysis shows that the estimation methods are not significantly different from each other while, in the parameter space, each of the fermentations were significantly different from each other. Full article
(This article belongs to the Special Issue Modeling, Control and Optimization of Wine Fermentation)
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18 pages, 18820 KiB  
Article
A Frequency/Phase/Amplitude Estimator for Three-Phase Applications Operating at a Low Sampling Rate
by Abdullah M. Abusorrah and Hamed Sepahvand
Mathematics 2024, 12(3), 363; https://doi.org/10.3390/math12030363 - 23 Jan 2024
Cited by 1 | Viewed by 1163
Abstract
A frequency/phase/amplitude estimator is arguably one of the most crucial components in the control and synchronization of grid-connected equipment. Such an estimator may also be useful for monitoring and protection purposes in power systems. In this paper, an open-loop (and therefore unconditionally stable) [...] Read more.
A frequency/phase/amplitude estimator is arguably one of the most crucial components in the control and synchronization of grid-connected equipment. Such an estimator may also be useful for monitoring and protection purposes in power systems. In this paper, an open-loop (and therefore unconditionally stable) estimator for the accurate and rapid extraction of the grid voltage phase, frequency, and amplitude is presented. In designing the proposed technique, special focus is given to applications operating at a low sampling rate. Examples of such applications include high-power converters, where both switching and sampling frequencies are very low. The study concludes with a comprehensive evaluation of the proposed estimator, demonstrating its effectiveness in accurately and swiftly estimating the fundamental parameters of grid voltage under low sampling rates. It highlights the estimator’s enhanced performance in scenarios of distorted grid conditions and its superiority in filtering capabilities compared to traditional methods. These findings underline the estimator’s potential for broad applicability in power system monitoring, protection, and control. Full article
(This article belongs to the Special Issue Mathematical Applications in Electrical Engineering)
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30 pages, 863 KiB  
Article
Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems
by Jens Lang and Bernhard A. Schmitt
Algorithms 2022, 15(9), 310; https://doi.org/10.3390/a15090310 - 29 Aug 2022
Cited by 3 | Viewed by 2183
Abstract
This paper is concerned with the construction and convergence analysis of novel implicit Peer triplets of two-step nature with four stages for nonlinear ODE constrained optimal control problems. We combine the property of superconvergence of some standard Peer method for inner grid points [...] Read more.
This paper is concerned with the construction and convergence analysis of novel implicit Peer triplets of two-step nature with four stages for nonlinear ODE constrained optimal control problems. We combine the property of superconvergence of some standard Peer method for inner grid points with carefully designed starting and end methods to achieve order four for the state variables and order three for the adjoint variables in a first-discretize-then-optimize approach together with A-stability. The notion triplets emphasize that these three different Peer methods have to satisfy additional matching conditions. Four such Peer triplets of practical interest are constructed. In addition, as a benchmark method, the well-known backward differentiation formula BDF4, which is only A(73.35∘)-stable, is extended to a special Peer triplet to supply an adjoint consistent method of higher order and BDF type with equidistant nodes. Within the class of Peer triplets, we found a diagonally implicit A(84∘)-stable method with nodes symmetric in [0, 1] to a common center that performs equally well. Numerical tests with four well established optimal control problems confirm the theoretical findings also concerning A-stability. Full article
(This article belongs to the Section Analysis of Algorithms and Complexity Theory)
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18 pages, 1793 KiB  
Article
Second-Order Time Stepping Scheme Combined with a Mixed Element Method for a 2D Nonlinear Fourth-Order Fractional Integro-Differential Equations
by Deng Wang, Yang Liu, Hong Li and Zhichao Fang
Fractal Fract. 2022, 6(4), 201; https://doi.org/10.3390/fractalfract6040201 - 2 Apr 2022
Cited by 4 | Viewed by 2841
Abstract
In this article, we study a class of two-dimensional nonlinear fourth-order partial differential equation models with the Riemann–Liouville fractional integral term by using a mixed element method in space and the second-order backward difference formula (BDF2) with the weighted and shifted Grünwald integral [...] Read more.
In this article, we study a class of two-dimensional nonlinear fourth-order partial differential equation models with the Riemann–Liouville fractional integral term by using a mixed element method in space and the second-order backward difference formula (BDF2) with the weighted and shifted Grünwald integral (WSGI) formula in time. We introduce an auxiliary variable to transform the nonlinear fourth-order model into a low-order coupled system including two second-order equations and then discretize the resulting equations by the combined method between the BDF2 with the WSGI formula and the mixed finite element method. Further, we derive stability and error results for the fully discrete scheme. Finally, we develop two numerical examples to verify the theoretical results. Full article
(This article belongs to the Special Issue Novel Numerical Solutions of Fractional PDEs)
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