Unified formulation of a family of iterative solvers for power systems analysis

This paper illustrates the construction of a new class of iterative solvers for power flow calculations based on the method of Alternating Search Directions. This method is fit to the particular algebraic structure of the power flow problem resulting from the combination of a globally linear set of equations and nonlinear local relations imposed by power conversion devices, such as loads and generators. The choice of the search directions is shown to be crucial for improving the overall robustness of the solver. A noteworthy advantage is that constant search directions yield stationary methods that, in contrast with Newton or Quasi-Newton methods, do not require the evaluation of the Jacobian matrix. Such directions can be elected to enforce the convergence to the high voltage operative solution. The method is explained through an intuitive example illustrating how the proposed generalized formulation is able to include other nonlinear solvers that are classically used for power flow analysis, thus offering a unified view on the topic. Numerical experiments are performed on publicly available benchmarks for large distribution and transmission systems.Peer ReviewedPostprint (author's final draft

Algebraic and parametric solvers for the power flow problem: towards real-time and accuracy-guaranteed simulation of electric systems

The final publication is available at Springer via http://dx.doi.org/10.1007/s11831-017-9223-6The power flow model performs the analysis of electric distribution and transmission systems. With this statement at hand, in this work we present a summary of those solvers for the power flow equations, in both algebraic and parametric version. The application of the Alternating Search Direction method to the power flow problem is also detailed. This results in a family of iterative solvers that combined with Proper Generalized Decomposition technique allows to solve the parametric version of the equations. Once the solution is computed using this strategy, analyzing the network state or solving optimization problems, with inclusion of generation in real-time, becomes a straightforward procedure since the parametric solution is available. Complementing this approach, an error strategy is implemented at each step of the iterative solver. Thus, error indicators are used as an stopping criteria controlling the accuracy of the approximation during the construction process. The application of these methods to the model IEEE 57-bus network is taken as a numerical illustration.Peer ReviewedPostprint (author's final draft

An iterative multi-fidelity approach for model order reduction of multi-dimensional input parametric PDE systems

We propose a parametric sampling strategy for the reduction of large-scale PDE systems with multidimensional input parametric spaces by leveraging models of different fidelity. The design of this methodology allows a user to adaptively sample points ad hoc from a discrete training set with no prior requirement of error estimators. It is achieved by exploiting low-fidelity models throughout the parametric space to sample points using an efficient sampling strategy, and at the sampled parametric points, high-fidelity models are evaluated to recover the reduced basis functions. The low-fidelity models are then adapted with the reduced order models ( ROMs) built by projection onto the subspace spanned by the recovered basis functions. The process continues until the low-fidelity model can represent the high-fidelity model adequately for all the parameters in the parametric space. Since the proposed methodology leverages the use of low-fidelity models to assimilate the solution database, it significantly reduces the computational cost in the offline stage. The highlight of this article is to present the construction of the initial low-fidelity model, and a sampling strategy based on the discrete empirical interpolation method (DEIM). We test this approach on a 2D steady-state heat conduction problem for two different input parameters and make a qualitative comparison with the classical greedy reduced basis method (RBM), and further test on a 9-dimensional parametric non-coercive elliptic problem and analyze the computational performance based on different tuning of greedy selection of points

Simulation numérique 3D de la coextrusion des fluides polymériques et de l'effet d'enrobage

L'ensemble des travaux présentés dans cette thèse porte sur la simulation numérique des procédés de coextrusion par un modèle d'écoulement stratifié basé sur la méthode du champ de phase. L'avantage technologique offert par la coextrusion réside dans la possibilité de combiner des matériaux ayant des propriétés physiques très spécifiques dans un produit unique. Toutefois, les différences rhéologiques entre les divers matériaux sont elles-mêmes responsables d'un phénomène de distorsion de l'interface séparant deux couches adjacents. Les données expérimentales en coextrusion bicouches montrent que, en raison des différences de viscosité et d'élasticité entre le deux composants, le fluide le moins visqueux encapsule le fluide plus visqueux et le passage d'une configuration stratifiée à une encapsulée comporte une perte de qualité du produit final. Ce phénomène, dit d'enrobage représente donc un sujet de très grande actualité dans la recherche industrielle et la compréhension des mécanismes le générant sera utile pour l'amélioration des procédés de mise en forme des polymères. La nature intrinsèquement tridimensionnelle de l'enrobage a requis le développement d'un code pour la simulation tridimensionnelle basée sur la méthode des volumes finis pour la discrétisation des équations de Navier-Stokes pour les écoulement incompressibles et isothermes couplées avec une loi constitutive différentielle non linéaire (modèles de Giesekus ou PTT). La présence de deux fluides est prise en compte par une équation scalaire supplémentaire décrivant l'évolution de l'interface sur un maillage fixe. Cette équation offre une interprétation physique précise car elle est dérivée de la thermodynamique de séparation de phase d'un fluide binaire. Le modèle proposé est validé par confrontation avec les résultats expérimentaux et numériques disponibles dans la littérature. Une étude numérique de la coextrusion en filière rectangulaire est effectuée afin de mettre en évidence les facteurs influençant l'enrobage et la nature de son origineThe objective of the present work is the analysis of coextrusion processes by numerical simulation based on phase-field modeling of stratified confined flows. The study of such flows is motivated by the presence of complex phenomena appearing in a vast range of industrial operational coextrusion conditions due to the differences in the components properties and their viscoelastic behavior. The basic idea in coextrusion is to combine several layers of different polymers in a common die, to form a unique product with enhanced properties. However, the existence of fluid stratification in the die is responsible of a severe distortion of the interface between the fluid components, causing a loss of efficiency for the whole process. Experimental data show that, even if a stratified initial configuration is imposed at the die entry, one fluid eventually encapsulates the other in most of the flow condition analyzed. The intrinsically three-dimensional nature of this phenomenon has required the development of a three-dimensional flow solver based on the finite volume discretization of the Navier-Stokes equations for incompressible and isothermal flow, together with differential nonlinear constitutive equations (Giesekus, PTT models). The presence of two fluid phases is taken into account by a phase field model that implies the solution of an additional scalar equation to describe the evolution of the interface on a fixed Eulerian grid. This model, unlike others of the same family, has a thermodynamic derivation and can be physically interpreted. The proposed method is tested against experimental data and solutions already available in literature and a study of coextrusion in rectangular dies is performed to identify the dependence of encapsulation on the flow parametersST ETIENNE-Bib. électronique (422189901) / SudocSudocFranceF

Unified formulation of a family of iterative solvers for power system analysis

This paper illustrates the construction of a new class of iterative solvers for power flow calculations based on the method of Alternating Search Directions. This method is fit to the particular algebraic structure of the power flow problem resulting from the combination of a globally linear set of equations and nonlinear local relations imposed by power conversion devices, such as loads and generators. The choice of the search directions is shown to be crucial for improving the overall robustness of the solver. A noteworthy advantage is that constant search directions yield stationary methods that, in contrast with Newton or Quasi-Newton methods, do not require the evaluation of the Jacobian matrix. Such directions can be elected to enforce the convergence to the high voltage operative solution. The method is explained through an intuitive example illustrating how the proposed generalized formulation is able to include other nonlinear solvers that are classically used for power flow analysis, thus offering a unified view on the topic. Numerical experiments are performed on publicly available benchmarks for large distribution and transmission systems

Non-intrusive Sparse Subspace Learning for Parametrized Problems

We discuss the use of hierarchical collocation to approximate the numerical solution of parametric models. With respect to traditional projection-based reduced order modeling, the use of a collocation enables non-intrusive approach based on sparse adaptive sampling of the parametric space. This allows to recover the low-dimensional structure of the parametric solution subspace while also learning the functional dependency from the parameters in explicit form. A sparse low-rank approximate tensor representation of the parametric solution can be built through an incremental strategy that only needs to have access to the output of a deterministic solver. Non-intrusiveness makes this approach straightforwardly applicable to challenging problems characterized by nonlinearity or non affine weak forms. As we show in the various examples presented in the paper, the method can be interfaced with no particular effort to existing third party simulation software making the proposed approach particularly appealing and adapted to practical engineering problems of industrial interest

Residual-based adaptivity for two-phase flow simulation in porous media using Physics-informed Neural Networks

This paper aims to provide a machine learning framework to simulate two-phase flow in porous media. The proposed algorithm is based on Physics-informed neural networks (PINN). A novel residual-based adaptive PINN is developed and compared with the residual-based adaptive refinement (RAR) method and with PINN with fixed collocation points. The proposed algorithm is expected to have great potential to be applied to different fields where adaptivity is needed. In this paper, we focus on the two-phase flow in porous media problem. We provide two numerical examples to show the effectiveness of the new algorithm. It is found that adaptivity is essential to capture moving flow fronts. We show how the results obtained through this approach are more accurate than using RAR method or PINN with fixed collocation points, while having a comparable computational cost

Sensitivity analysis using Physics-informed neural networks

The paper's goal is to provide a simple unified approach to perform sensitivity analysis using Physics-informed neural networks (PINN). The main idea lies in adding a new term in the loss function that regularizes the solution in a small neighborhood near the nominal value of the parameter of interest. The added term represents the derivative of the loss function with respect to the parameter of interest. The result of this modification is a solution to the problem along with the derivative of the solution with respect to the parameter of interest (the sensitivity). We call the new technique to perform sensitivity analysis within this context SA-PINN. We show the effectiveness of the technique using 3 examples: the first one is a simple 1D advection-diffusion problem to show the methodology, the second is a 2D Poisson's problem with 9 parameters of interest and the last one is a transient two-phase flow in porous media problem.Comment: 22 pages, 11 figure

Tensor representation of non-linear models using cross approximations

This is a post-peer-review, pre-copyedit version of an article published in Journal of scientific computing. The final authenticated version is available online at: http://dx.doi.org/10.1007/s10915-019-00917-2Tensor representations allow compact storage and efficient manipulation of multi-dimensional data. Based on these, tensor methods build low-rank subspaces for the solution of multi-dimensional and multi-parametric models. However, tensor methods cannot always be implemented efficiently, specially when dealing with non-linear models. In this paper, we discuss the importance of achieving a tensor representation of the model itself for the efficiency of tensor-based algorithms. We investigate the adequacy of interpolation rather than projection-based approaches as a means to enforce such tensor representation, and propose the use of cross approximations for models in moderate dimension. Finally, linearization of tensor problems is analyzed and several strategies for the tensor subspace construction are proposed.Peer ReviewedPostprint (author's final draft

Advanced parametric space-frequency separated representations in structural dynamics: A harmonic–modal hybrid approach

This paper is concerned with the solution to structural dynamics equations. The technique here presented is closely related to Harmonic Analysis, and therefore it is only concerned with the long-term forced response. Proper Generalized Decomposition (PGD) is used to compute space-frequency separated representations by considering the frequency as an extra coordinate. This formulation constitutes an alternative to classical methods such as Modal Analysis and it is especially advantageous when parametrized structural dynamics equations are of interest. In such case, there is no need to solve the parametrized eigenvalue problem and the space-time solution can be recovered with a Fourier inverse transform. The PGD solution is valid for any forcing term that can be written as a combination of the considered frequencies. Finally, the solution is available for any value of the parameter. When the problem involves frequency-dependent parameters the proposed technique provides a specially suitable method that becomes computationally more efficient when it is combined with a modal representation