Power flow simulations form an essential tool for electricity network analysis but conventional models are designed to work on a separated transmission or distribution network only. The continuing growth of electricity consumption, demand side participation, and renewable resources makes the electricity networks co-dependent. Integrated models incorporate the coupling of the networks and interaction that they have on each other, representing the power flow within this changing environment accurately. Several numerical methods are available to solve the power flow problem on integrated networks. They can be categorized as a unified or as a splitting method and networks can be modeled as a homogeneous or hybrid network. In this paper, we review and assess these methods on the network models by running simulations on small test networks and comparing the outcome on their numerical performance, ie on convergence rate and CPU-time. The review shows that the convergence rate is comparable for most of the methods, but that hybrid networks have a slight advantage in computational time. Realistic network models, running on millions of buses and with large distribution networks, should give a better insight into the speed of the computations.
Thermal nitric-oxide (NOx) formation in industrial furnaces due to local overheating is a widely known problem. Various industries made significant investments to reduce thermal NOx by varying the operating conditions and designs of the furnace. It is difficult to find the optimal operating conditions that minimize NOx formation in the furnace by trial and error methods. The high temperature in the furnace complicates performing experiments in the furnace. Numerical modeling can provide significant information in such cases. Therefore, the objective of this paper is to obtain a numerical model of the furnace in such a way that the operating conditions can be varied and examined.In this paper, a three-dimensional steady-state finite element model for the anode baking industrial furnace is discussed. The COMSOL Multiphysics software is used for modeling the non-premixed turbulent combustion and the conjugate heat transfer to the insulation lining. The cfMesh software is used for obtaining the mesh. The results show that the simulated temperature agrees well with the measured data from our industrial partner in regions distant from the flames. The analysis shows that by decreasing the fuel mass flow rate and increasing the fuel pipe diameter by 45%, the peak in thermal NOx ppm generated in the furnace decreases by 42%. The model is limited by the use of a single-step chemistry mechanism with an eddy dissipation combustion model and a simplified approach for radiation, such as the P1 approximation model. The model can be further improved by considering a detailed chemistry mechanism model for combustion and a discrete ordinate model for radiation.
Optimization is an important tool for the operation of an energy system. Multi -carrier energy systems (MESs) have recently become more important. Load flow (LF) equations are used within optimization to determine if physical network limits are violated. Due to nonlinearities, the solvability of the OF problem and the convergence of the optimization algorithms are influenced by how the LF equations are included in the optimal flow (OF) problem. In addition, scaling greatly influences the practical solvability of OF problems. This paper considers two ways to include the LF equations within the OF problem for general MESs. In formulation I, optimization is over the combined control and state variables, with the LF equations included explicitly as equality constraints. In formulation II, optimization is over the control variables only. The state variables are solved from the LF equations in a separate subsystem, for given control variables. Hence, the LF equations are included only implicitly in formulation II. The two formulations are compared qualitatively, from a theoretical perspective and based on numerical experiments. Both formulation I and formulation II result in a solvable OF problem. Formulation I is easier to implement and more efficient in terms of CPU time. However, formulation II ensures feasibility and can be used for optimization in combination with dedicated load flow solvers. Both matrix scaling and per unit scaling can be used to solve the OF problem, but they are not equivalent.
The term smart grid is typically associated with power systems, but it can also be applied to gas and heat networks. There is no consensus on the definition of smart (gas) grids in the literature. We use the description provided in [27]: the smart gas grid concept is based on maximizing the efficiency of overall energy usage and taking full advantage of the exibility and all the opportunities that gas and the gas grid can offer. In this report we present the results of a six-months project on technology and policy aspects of smart gas grids, performed between October 2014 and April 2015. The project has led to a successful Energy System Integration: planning, operations, and societal embedding (ESI-pose) application in 2015, increasing the degree of collaboration between the faculty of Technology, Policy and Management (TPM) and Numerical Analysis section of Electrical Engineering, Mathematics and Computer Science (EEMSC) faculty.
Electrical power systems are complex systems and traditionally modeled in two separate systems. Power is generated at the transmission system and at several substations converted to the distribution systems. The increasing amount of generation produced at distribution level can eventually effect the transmission network. An integrated model of both systems can help studying these effects and prevent harmful events on the power system. Transmission and Distribution systems differ significantly from each other. Where transmission systems are assumed to be balanced and therefore modeled as a single-phase system, the distribution systems are in general unbalanced and should be modeled in three-phase. Furthermore, high R/X ratios of distribution lines, the lower voltage level, the radial structure and the presence of unbalanced loading lead to different solution techniques. Connecting these two systems pose complications for both the solution method and the connection method. In this report, we present several methods to solve the integrated Transmission-Distribution system. One of them is to omit the simplifications we can make in a transmission system and solve both systems as a three-phase system. Another method is to use a master-slave splitting approach and solve both systems iteratively, using a boundary state. A last method is building an interconnected network which solves the system at once, respecting both the transmission and distribution conditions. Some artificial currents and voltages have to be injected on the boundary. We compare the different methods on CPU-time, convergence, accuracy and complexity and present the preferable method for the specific network criteria.
Recent research efforts aimed at iteratively solving the Helmholtz equation have focused on incorporating deflation techniques for accelerating the convergence of Krylov subspace methods. In this work, we extend the two-level deflation method in [6] to a multilevel deflation method. By using higher-order deflation vectors, we show that up to the level where the coarse-grid linear systems remain indefinite, the near-zero eigenvalues of the these coarse-grid operators remain aligned with the fine-grid operator keeping the spectrum of the preconditioned system fixed away from the origin. Combining this with the well-known CSLP-preconditioner, we obtain a scalable solver with theoretical linear complexity for the highly indefinite Helmholtz equation. This can be attributed to a fixed number of iterations independent of the wave number and an optimal use of the CSLP-preconditioner. We approximate the CSLP-preconditioner, while allowing the complex shift to be small. The proposed configuration additionally shows very promising results for the more challenging Marmousi problem.
Distance fields finds a lot of applications recently in computational geometry. We propose an algorithm to extract the mean camber line of an airfoil using distance fields. This method does not require computing any geometric approximations in- side the airfoil geometry. It also does not need the airfoil coordinates to be evenly distributed on the top and bottom. All the computations are done on a simple Cartesian mesh and can be easily coupled with any existing computational fluid dynamics codes. We show the efficiency of the algorithm on a variety of airfoil geometries. We demonstrate the robustness and exibility of the algorithm for a wide range of airfoil geometries.
This report describes the research that started with the research of Floris Buwalda and continued with the work of Michael Mersie. We will mostly work with Numerical Mathematics to find a function that expresses the power that needs to be delivered in terms of the ship’s desired speed in relation to the water. In order to keep up with the digital age where contracts and transactions are calculated to fractions of cents, it is essential that this function is as accurate as possible. We would like to take this opportunity to thank Hans Huisman and MARIN for their time and for contributing necessary data to this research.
Energy systems are becoming more complex due to increased coupling between different networks, resulting in multi-carrier energy networks. Conventional models for the separate networks are not able to capture the full extent of the coupling. Recently, different models for multi-carrier networks have been proposed, either using the energy hub concept or using a case specific approach. Although the energy hub concept can be applied to a general integrated net- work, it is unclear how the energy hub should be represented in the graph of the multi-carrier network. This paper presents a graph-based framework for steady-state load ow models of multi-carrier energy systems. Furthermore, the effect of coupling on the integrated system of equations is investigated. The proposed framework is tested on two small multi-carrier net- works, for comparison with models in literature. Results show that our framework is applicable to a general system, and that it generalizes both the energy hub concept and the case specific approaches.
The success of the application of Microbial Enhanced Oil Recovery (MEOR) techniques depends on several factors such as individual reservoir characteristics and microbial activity. Since the quantification of the relations between the aforementioned parameters is difficult to obtain, the development of mathematical and numerical models predicting the bacterial population growth and in situ production of by-products is of vital importance to develop a proper field strategy [15]. In this work, we use a pore network model to study the hydrodynamic changes over time on a porous medium as a result of biofilm growth. We propose a new model in the microscopic scale for biofilm growth which allows the spreading of the biofilm over the network. This formalism for the biofilm growth leads to a new relation between the permeability and the amount of biomass in the network. These results could be up-scaled to the continuum-based oil reservoir scales.
We consider deflation-based pre-conditioning of the pressure equation for large-scale reservoir models with strong spatial variations in the permeabilities. The use of deflation techniques involves the search for good deflation vectors, which usually are problem-dependent. We propose the use of proper orthogonal decomposition (POD) to generate physics-based problem-specific deflation vectors. The use of POD to construct pre-conditioners has been attempted before but in those applications, a snap-shot-based reduced-order basis was used as pre-conditioner directly whereas we propose the use of basis vectors as deflation vectors. We investigate the effectiveness of the method with numerical experiments using the conjugate gradient iterative method in combination with Incomplete Cholesky preconditioning (ICCG) and POD-based deflation (DICCG). We consider incompressible and compressible single-phase flow in a layered model with large variations in the permeability coefficients, and the SPE10 benchmark model. We obtain an important reduction for the number of iterations with our proposed DICCG method in comparison with the ICCG method. In some test problems, we achieve convergence within one DICCG iteration. However, our method requires a number of preparatory reservoir simulations proportional to the number of wells and the solution of an eigenvalue problem to compute the deflation vectors. This overhead will be justified in case of a large number of subsequent simulations with different control settings as typically required in numerical optimization or sensitivity studies.
As a master student in AppliedMathematics at the Delft University of Technology I am highly educated in Numerical Analysis. My interest in this field even mademe choose elective courses such as Advanced Numerical Methods, Applied Finite Elements and Computational Fluid Dynamics. In my search for a challenging graduationproject I chose a research proposal on the material point method, an extension of the finite element method that is well-suited for problems involving large deformations. The proposal met al my personal requirements, since it contained a mathematical basis I am already familiar with and the opportunity to gain knowledge about a relatively new numerical method. Furthermore, it included an internship at Deltares
Multilevel sequentially semiseparable (MSSS) matrices form a class of structured matrices that have low-rank off-diagonal structure, which allows the matrix-matrix operations to be performed in linear computational complexity. MSSS preconditioners are computed by replacing the Schur complements in the block LU factorization of the global linear system by MSSS matrix approximations with low off-diagonal rank. In this manuscript, we analyze the convergence properties of such preconditioners. We show that the spectrum of the preconditioned system is contained in a circle centered at (1, 0) and give an analytic bound of the radius of this circle. This radius can be made arbitrarily small by properly setting a parameter in the MSSS preconditioner. Our results apply to a wide class of linear systems. The system matrix can be either symmetric or unsymmetric, definite or indefinite. We demonstrate our analysis by numerical experiments.
In this manuscript, we study preconditioning techniques for optimal in-domain control of the Navier-Stokes equation, where the control only acts on a few parts of the domain. Optimization and linearization of the optimal in-domain control problem results in a generalized linear saddle-point system. The Schur complement for the generalized saddle-point system is very difficult or even impossible to approximate. This prohibits satisfactory performance of the standard block preconditioners. We apply the multilevel sequentially semiseparable (MSSS) preconditioner to the underlying system. Compared with standard block preconditioning techniques, the MSSS preconditioner computes an approximate factorization of the global generalized saddle-point matrix up to a prescribed accuracy in linear computational complexity. This in turn gives parameter independent convergence for MSSS preconditioned Krylov solvers. We use a simple wind farm control example to illustrate the performance of the MSSS preconditioner. We also compare with the performance of the state-of-the-art preconditioning techniques. Our results show the superiority of the MSSS preconditioning techniques to standard block preconditioning techniques for optimal in-domain control of the Navier-Stokes equation.
One-dimensional models for multiphase flow in pipelines are commonly discretised using first-order Finite Volume (FV) schemes, often combined with implicit time-integration methods. While robust, these methods introduce much numerical diffusion depending on the number of grid points. In this paper we propose a high-order, space-time Discontinuous Galerkin (DG) Finite Element method with h-adaptivity to improve the efficiency of one-dimensional multiphase flow simulations. For smooth initial boundary value problems we show that the DG method converges with the theoretical rate and that the growth rate and phase shift of small, harmonic perturbations exhibit superconvergence. We employ two techniques to accurately and efficiently represent discontinuities. Firstly artificial diffusion in the neighbourhood of a discontinuity suppresses spurious oscillations. Secondly local mesh refinement allows for a sharper representation of the discontinuity while keeping the amount of work required to obtain a solution relatively low. The proposed DG method is shown to be superior to FV.
The boundary element method (BEM) is widely used in fast numerical solvers for concentrated elastic contact problems arising from the wheel-rail contact in the railway industry. In this paper we extend the range of applicability of BEM by computing the influence coefficients (ICs) numerically. These ICs represent the Green’s function of the problem, i.e. the surface deformation due to unit loads. They are not analytically available when the half-space is invalid, for instance in conformal contact. An elastic model is proposed to compute these ICs numerically, by the finite element method (FEM). We present a detailed investigation to find proper strategies of FEM meshing and element types, considering accuracy and computational cost. Moreover, the effects of computed ICs to contact solutions are examined for a Cattaneo shift contact problem. The work in this paper provides a guidance to study fast solvers for the conformal contact.
This paper discusses the solution of large-scale linear matrix equations using the Induced Dimension reduction method (IDR(s)). IDR(s) was originally presented to solve system of linear equations, and is based on the IDR(s) theorem. We generalize the IDR(s) theorem to solve linear problems in any finite-dimensional space. This generalization allows us to develop IDR(s) algorithms to approximate the solution of linear matrix equations. The IDR(s) method presented here has two main advantages; firstly, it does not require the computation of inverses of any matrix, and secondly, it allows incorporation of preconditioners. Additionally, we present a simple preconditioner to solve the Sylvester equation based on a fixed point iteration. Several numerical examples illustrate the performance of IDR(s) for solving linear matrix equations. We also present the software implementation.
Classical point-mechanics is derived from three principles —called axioms— that are based on observations of simple kinematical phenomena. Predefined concepts of ‘force’ and ‘mass’ are not required. The concept ’mass’ and corresponding concepts of momentum and energy follow from the first and second axiom. Together with the third axiom, a basic way for constructing equations of motion is derived, more or less equivalent to Gauss’ principle of least constraint.
