Algebraic Variables Research Articles

Symbolic elimination in dynamic optimization based on block-triangular ordering

We consider dynamic optimization problems for systems described by differential-algebraic equations (DAEs). Such problems are usually solved by discretizing the full DAE. We propose techniques to symbolically eliminate many of the algebraic variables in a preprocessing step before discretization. These techniques are inspired by the causalization and tearing techniques often used when solving DAE initial value problems. Since sparsity is crucial for some dynamic optimization methods, we also propose a novel approach to preserving sparsity during this procedure. The proposed methods have been implemented in the open-source JModelica.org platform. We evaluate the performance of the methods on a suite of optimal control problems solved using direct collocation. We consider both computational time and probability of solving the problem in a timely manner. We demonstrate that the proposed methods often are an order of magnitude faster than the standard way of discretizing the full DAE, and also significantly increase probability of successful convergence.

A Multi-Dimensional Holomorphic Embedding Method to Solve AC Power Flows

It is well known that closed-form analytical solutions for AC power flow equations do not exist in general. This paper proposes a multi-dimensional holomorphic embedding method (MDHEM) to obtain an explicit approximate analytical AC power-flow solution by finding a physical germ solution and arbitrarily embedding each power, each load or groups of loads with respective scales. Based on the MDHEM, the complete approximate analytical solutions to the power flow equations in the high-dimensional space become achievable, since the voltage vector of each bus can be explicitly expressed by a convergent multivariate power series of all the loads. Unlike the traditional iterative methods for power flow calculation and inaccurate sensitivity analysis method for voltage control, the algebraic variables of a power system in all operating conditions can be prepared offline and evaluated online by only plugging in the values of any operating conditions into the scales of the non-linear multivariate power series. Case studies implemented on the 4-bus test system and the IEEE 14-bus standard system confirm the effectiveness of the proposed method.

Low-frequency oscillations of wind power systems caused by doubly-fed induction generators

The existing researches on the wind power systems do not specify whether of the low-frequency oscillations are caused by the synchronous generators or the wind turbine generators. In this paper, the low-frequency oscillation caused by the doubly-fed induction generators (DFIGs) is studied. The linearized state equations of the DFIGs with the stator voltage transient are newly proposed, where the stator voltages as algebraic variables are expressed by the state variables. The low-frequency oscillation modes, usually caused by the synchronous generators, are newly defined by the mechanical parameters of the DFIGs to derive the participation ratios and find the low-frequency oscillation modes caused by the DFIGs. The low-frequency oscillation modes caused by the DFIGs are validated by the dynamic response using the band-pass/low-pass filter. The multi-step derivatives are proposed to quantify the sensitivity of the oscillation modes caused by the DFIGs to the parameters of the DFIGs and the synchronous generators. The damping schemes are derived by comparing the sensitivity and the accuracy is validated. The numerical results on the IEEE RTS system and a regional system show the error due to neglect to the stator voltage transient. The low-frequency oscillations caused by the DFIG have longer period, and converge more slowly than those caused by the synchronous generators.

A quick method to solve the optimal coordinated voltage control problem based on reduction of system dimensions

This paper presents a new method to speed up the solving process of the optimal coordinated voltage control (OCVC) problem. To this end, after occurrence of a fault, the power system is initially partitioned into interior, boundary and exterior zones by spectral graph partitioning method. Partitioning is based on the fact that voltage instability just after the fault, is a local phenomenon, therefore the variation of boundary and exterior variables is smaller than that of interior ones. Then, the equations of boundary and exterior zones are linearized, the algebraic variables and equations of exterior zone are removed and a considerable decrease in the number of power system equations is obtained which is equal to a reduction in the system dimensions. As a result the determination process of the optimal control actions is fastened. Also, based on the monotonic behavior of long-term voltages, a new method of predicting voltage trajectory is proposed, in which two points are used: the first point referring to the system state just after applying the control actions, which is called jumping point and the final one referring to the steady state of the system. In a case study on the New England 39-bus power system, it was verified that using this method a remarkable increase in calculation speed can be obtained while retaining the system precision.

Intelligent integration strategies of wind farms in a super grid

In this article, we have an interested implementing new intelligent strategies based on fuzzy controllers, for a better integration of renewable energy in the powerful electrical networks, commonly called super grids. A preliminary investigation has allowed us to learn more about the impact of this integration on algebraic variables of an electrical network on one hand and on the modes of operation of power plants on the other. Indeed, if necessary to ensure the strategic balance production and consumption, especially during peak hours, power operators shall ensure that the integration of one or more wind farms does not affect any aspect of the quality Energy distributed to subscribers. Perfect stability of large power grids at times of integration of renewable energy sources is ensured through the use of new strategies constituting a potential support which acts to overcome the hazards may hinder the operation of the components basic of these grids.

Part 2—The firings of many neurons and their density; the neural network its connections and field of firings

This paper is concerned with the firing of many neurons and the synthesis of these firings to develop functions and their transforms which relate chemical and electrical phenomena to the physical world. The density of such functions in the most general spaces that we encounter allows us to use linear combinations of them to approximate arbitrarily close to any phenomenon we encounter, imagine or think about. Absence of the technology needed to represent all the senses and the mathematical difficulty of making geometric representations of functions of a complex and of more general division algebra variables make it difficult to validate the mathematical outcome of this approach to neural firings. But we think that this problem will be solved in the not-too-distant future when at least the senses of smell, taste and touch would have been so mathematized that it is possible to instill these qualities in robots in some fashion.

On asymptotic expansion solvers for highly oscillatory semi-explicit DAEs

The paper is concerned with the discretization and solution of DAEs of index $1$ and subject to a highly oscillatory forcing term. Separate asymptotic expansions in inverse powers of the oscillatory parameter are constructed to approximate the differential and algebraic variables of the DAEs. The series are truncated to enable practical implementation. Numerical experiments are provided to illustrate the effectiveness of the method.

Understanding the Algebraic Variable: Comparative Study of Mexican and Spanish Students

Students' difficulty in learning and suitably understanding the concept of the algebraic variable has been studied with a number of tools and documented for several populations. Little research has been conducted, however, using the same tool to explore understanding of the notion among populations from different countries in an attempt to establish similarities and differences in levels of achievement and difficulties. This article discusses the findings of a survey applying a questionnaire designed for use with the 3UV model, a theoretical-methodological tool for analysing the understanding of algebraic variables and their various usages (as specific unknown, as general number, functional). Comparisons were drawn of the results of a survey of a population comprising 184 ninth-year students, 92 in Spain and 92 in Mexico, and 82 Mexican and 85 Spanish eleventh-year students, all from medium-lower income families and attending public schools. The results obtained revealed similarities and differences among the groups, providing evidence of the strengths and weaknesses of the education system of each country.

Differential elimination by differential specialization of Sylvester style matrices

Differential resultant formulas are defined for a system P of n ordinary Laurent differential polynomials in n−1 differential variables. These are determinants of coefficient matrices of an extended system of polynomials obtained from P through derivations and multiplications by Laurent monomials. To start, through derivations, a system ps(P) of L polynomials in L−1 algebraic variables is obtained, which is nonsparse in the order of derivation, as defined in this paper. This enables the use of existing formulas for the computation of algebraic resultants, of the multivariate sparse algebraic polynomials in ps(P), to obtain polynomials in the differential elimination ideal generated by P. If the system satisfies certain conditions, then the formulas obtained are multiples of sparse differential resultants, defined by Li, Yuan and Gao, and provide order and degree bounds in terms of mixed volumes in the generic case.

Extending explicit and linearly implicit ODE solvers for index-1 DAEs

Nonlinear differential-algebraic equations (DAE) are typically solved using implicit stiff solvers based on backward difference formula or RADAU formula, requiring a Newton–Raphson approach for the nonlinear equations or using Rosenbrock methods specifically designed for DAEs. Consistent initial conditions are essential for determining numeric solutions for systems of DAEs. Very few systems of DAEs can be solved using explicit ODE solvers. This paper applies a single-step approach to system initialization and simulation allowing for systems of DAEs to be solved using explicit (and linearly implicit) ODE solvers without a priori knowledge of the exact initial conditions for the algebraic variables. Along with using a combined process for initialization and simulation, many physical systems represented through large systems of DAEs can be solved in a more robust and efficient manner without the need for nonlinear solvers. The proposed approach extends the usability of explicit and linearly implicit ODE solvers and removes the requirement of Newton–Raphson type iteration.

Robust Fail-Safe Iteration Free Solvers for Battery Models

Mathematical models of batteries1, 3, 4 are dynamic systems. which are typically solved by using a finite difference or other discretization technique in the spatial direction and solving the resulting system of nonlinear differential algebraic equations (DAEs) in time2. The resulting DAEs are stiff, often times ill-conditioned and many standard solvers fail to solve them. The complications in solving the system of DAEs also include the inability of the solvers to find consistent initial conditions for the algebraic variables5,6. Most stiff solvers include a Newton-Raphson type iterative methods to solve these stiff equations. (DASKR, RADAU, IDA7). In this work, we propose a single-step iteration free approach to system initialization and simulation allowing for systems of DAEs to be solved using explicit ODE solvers without exact knowledge of the initial conditions for the algebraic variables. We will present this new solver which is guaranteed to not fail for any meaningful physical situations. Acknowledgements The work presented herein was funded in part by the Advanced Research Projects Agency – Energy (ARPA-E), U.S. Department of Energy, under Award Number DE-AR0000275. References Methekar, R. N., Ramadesigan, V., Pirkle, J. C. and Subramanian, V. R. A perturbation approach for consistent initialization of index-1 explicit differential-algebraic equations arising from battery model simulations. Computers & Chemical Engineering 2011; 35(11): 2227-2234.Michelsen, M. L. Application of Semi-Implicit Runge-Kutta Methods for Integration of Ordinary and Partial-Differential Equations. Chem Eng J Bioch Eng 1977; 14(2): 107-112.M. Doyle, T. F. Fuller and J. Newman, J. Electrochem. Soc., 140, 1526 (1993).T. F. Fuller, M. Doyle and J. Newman, J. Electrochem. Soc., 141, 1 (1994).Cash, J. R. Efficient numerical methods for the solution of stiff initial-value problems and differential algebraic equations. P Roy Soc a-Math Phy 2003; 459(2032): 797-815.E. Hairer, G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential–algebraic Problems, second ed., Springer-Verlag, 1996.IDA,[http://computation.llnl.gov/casc/sundials/main.html]

Decentralised control of parallel inverters connected to microgrid using the derivative‐free non‐linear Kalman filter

Decentralised control for parallel inverters connected to the power grid is developed using differential flatness theory and the derivative-free nonlinear Kalman filter. It is proven that the model of the inverters, is a differentially flat one. This means that all its state variables and the control inputs can be written as differential functions of a single algebraic variable which is the flat output. By exploiting differential flatness properties it is shown that the multiple inverters model can be transformed into a set of local inverter models which are decoupled and linearized. For each local inverter the design of a state feedback controller becomes possible. Such a controller processes measurements not only coming from the individual inverter but also coming from other inverters which are connected to the grid. Moreover, to estimate the non-measurable state variables of each local inverter, the derivative-free nonlinear Kalman filter is used. Furthermore, by redesigning the aforementioned filter as a disturbance observer it becomes also possible to estimate and compensate for disturbance terms that affect each local inverter.

Differential elimination by differential specialization of Sylvester style matrices

Differential resultant formulas are defined, for a system P of n ordinary Laurent differential polynomials in n−1 differential variables. These are determinants of coefficient matrices of an extended system of polynomials obtained from P through derivations and multiplications by Laurent monomials. To start, through derivations, a system ps(P) of L polynomials in L− 1 algebraic variables is obtained, which is non sparse in the order of derivation. This enables the use of existing formulas for the computation of algebraic resultants, of the multivariate sparse algebraic polynomials in ps(P), to obtain polynomials in the differential elimination ideal generated by P. The formulas obtained are multiples of the sparse differential resultant defined by Li, Yuan and Gao, and provide order and degree bounds in terms of mixed volumes in the generic case.

Developing students’ functional thinking in algebra through different visualisations of a growing pattern’s structure

Spatial visualisation of geometric patterns and their generalisation have become a recognised pathway to developing students’ functional thinking and understanding of variables in algebra. This design-based research project investigated upper primary students’ development of explicit generalisation of functional relationships and their representation descriptively, graphically and symbolically. Ten teachers and their classes were involved in a sequence of tasks involving growing patterns and geometric structures over 1 year. This article focuses on two aspects of the study: visualising the structure of a geometric pattern in different ways and using this to generalise the functional relationship between two quantifiable aspects (variables). It was found that in an initial assessment task (n = 222), students’ initial visualisations could be categorised according to different types and some of these were more likely to lead either to recursive or explicit generalisation. In a later task, a small number of students demonstrated the ability to find more than one way to visualise the same geometric structure and thus represent their explicit generalisations as different but equivalent symbolic equations (using pronumerals). Implications for the teaching of functional thinking in middle-school algebra are discussed.

Generalized friezes and a modified Caldero–Chapoton map depending on a rigid object

AbstractThe (usual) Caldero–Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it mapsreachableindecomposable objects to the corresponding cluster variables in a cluster algebra. This formalizes the idea that the cluster category is acategorificationof the cluster algebra. The definition of the Caldero–Chapoton map requires the category to be 2-Calabi-Yau, and the map depends on a cluster-tilting object in the category. We study a modified version of the Caldero–Chapoton map which requires only that the category have a Serre functor and depends only on a rigid object in the category. It is well known that the usual Caldero–Chapoton map gives rise to so-calledfriezes, for instance, Conway–Coxeter friezes. We show that the modified Caldero–Chapoton map gives rise to what we callgeneralized friezesand that, for cluster categories of Dynkin typeA, it recovers the generalized friezes introduced by combinatorial means in recent work by the authors and Bessenrodt.

Generalized friezes and a modified Caldero–Chapoton map depending on a rigid object

AbstractThe (usual) Caldero–Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps reachable indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalizes the idea that the cluster category is a categorification of the cluster algebra. The definition of the Caldero–Chapoton map requires the category to be 2-Calabi-Yau, and the map depends on a cluster-tilting object in the category. We study a modified version of the Caldero–Chapoton map which requires only that the category have a Serre functor and depends only on a rigid object in the category. It is well known that the usual Caldero–Chapoton map gives rise to so-called friezes, for instance, Conway–Coxeter friezes. We show that the modified Caldero–Chapoton map gives rise to what we call generalized friezes and that, for cluster categories of Dynkin type A, it recovers the generalized friezes introduced by combinatorial means in recent work by the authors and Bessenrodt.

FRIEZES OF TYPE D

In this article, we establish a relation between the values of a frieze of type D˜n and some values of an SL2-tiling t associated with a particular quiver of type A˜2n−1. This relation allows us to compute, independently of each other, all the cluster variables in the cluster algebra associated with a quiver Q of type D˜n.

Nonlinear Feedback Control of Three-Phase Inverter Systems Using the Derivative-Free Nonlinear Kalman Filter

The use of three-phase voltage inverters (DC to AC converters) is frequently met in the electric power system, such as in the connection of photovoltaics with the rest of the grid. The paper proposes a nonlinear feedback control method for three-phase inverters, which is based on differential flatness theory and a new nonlinear filtering method under the name Derivative-free nonlinear Kalman Filter. First, it is shown that the inverter’s dynamic model is a differentially flat one. This means that all its state variables and the control inputs can be written as functions of a single algebraic variable which is the flat output. By exploiting differential flatness properties it is shown that the inverter’s model can be transformed to the linear canonical (Brunovsky’s) form. For the latter description the design of a state feedback controller becomes possible, e.g. using pole placement methods. Moreover, to estimate the non-measurable state variables of the linearized equivalent of the inverter, the Derivative-free nonlinear Kalman Filter is used. This consists of the Kalman Filter recursion applied on the linearized inverter’s model and of an inverse transformation that is based on differential flatness theory, which enables to compute estimates of the state variables of the initial nonlinear system. Furthermore, by redesigning the aforementioned filter as a disturbance observer it becomes also possible to estimate disturbance terms that affect the inverter and subsequently to compensate for them. The performance and disturbance rejection capability of the proposed nonlinear feedback control scheme is evaluated through simulation experiments.

Adjoint Sensitivity Analysis for Nonsmooth Differential-Algebraic Equation Systems

An adjoint sensitivity method is presented for nonsmooth semiexplicit differential-algebraic equation systems (NDAEs) of index 1. The adjoint sensitivity system is derived for a generalization of Mayer-type functionals subject to NDAE constraints. Opposed to classical Mayer-type functionals that depend on the differential and algebraic variables evaluated only at the initial and final times, the generalization depends additionally on the evaluation at multiple interior time points. The derived adjoint sensitivity method extends and builds on the method for smooth DAEs by Cao et al. [SIAM J. Sci. Comput., 24 (2003), pp. 1076--1089] and on the method for nonsmooth ODEs by Ruban [J. Comput. Systems Sci. Internat., 36 (1997), pp. 536--542].

Operational cost optimization of a full-scale SWRO system under multi-parameter variable conditions

In this work, an operational optimization designed to reduce the operational cost of a full-scale seawater reverse osmosis (SWRO) system was studied under variable operating conditions. To increase operational flexibility and cost savings potential, the storage tank was used as the buffer between freshwater production and supply. With a well-developed mathematical model, which included reverse osmosis (RO) process equations, the storage tank equations and operational cost equations, the optimal problem was formulated in the form of nonlinear DAOPs (differential algebraic optimization problems). To solve the problem effectively, a simultaneous approach was introduced in which the differential and algebraic variables were fully discretized. Then nonlinear solver of IPOPT was used to solve the discretized large-scale nonlinear programming (NLP) problem. Computational results show that the operational optimization has significant potential of more than 26% cost saving over conventional method. Based on the successful solution, the impacts of variable parameters, such as feed temperature, seawater salinity, electricity price and freshwater demand were analyzed in detail. The effect of these parameters on operational cost savings and corresponding key performance indicators was determined, which enhances the understanding of the SWRO process and its optimal control.