Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

Solving Ordinary Differential Equations

Advanced Mathematical Methods for Scientists and Engineers

Given the square matrices $A, B, D, E$ and the matrix $C$ of conforming dimensions, we consider the linear matrix equation $A{\mathbf X} E+D{\mathbf X} B = C$ in the unknown matrix ${\mathbf X}$. Our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems, which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing.

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Computational Methods for Linear Matrix Equations

Short-term load forecasting is an essential instrument in power system planning, operation, and control. Many operating decisions are based on load forecasts, such as dispatch scheduling of generating capacity, reliability analysis, and maintenance planning for the generators. Overestimation of electricity demand will cause a conservative operation, which leads to the start-up of too many units or excessive energy purchase, thereby supplying an unnecessary level of reserve. On the other hand, underestimation may result in a risky operation, with insufficient preparation of spinning reserve, causing the system to operate in a vulnerable region to the disturbance. In this paper, semi-parametric additive models are proposed to estimate the relationships between demand and the driver variables. Specifically, the inputs for these models are calendar variables, lagged actual demand observations, and historical and forecast temperature traces for one or more sites in the target power system. In addition to point forecasts, prediction intervals are also estimated using a modified bootstrap method suitable for the complex seasonality seen in electricity demand data. The proposed methodology has been used to forecast the half-hourly electricity demand for up to seven days ahead for power systems in the Australian National Electricity Market. The performance of the methodology is validated via out-of-sample experiments with real data from the power system, as well as through on-site implementation by the system operator.

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Short-Term Load Forecasting Based on a Semi-Parametric Additive Model

Long-term electricity demand forecasting plays an important role in planning for future generation facilities and transmission augmentation. In a long-term context, planners must adopt a probabilistic view of potential peak demand levels. Therefore density forecasts (providing estimates of the full probability distributions of the possible future values of the demand) are more helpful than point forecasts, and are necessary for utilities to evaluate and hedge the financial risk accrued by demand variability and forecasting uncertainty. This paper proposes a new methodology to forecast the density of long-term peak electricity demand. Peak electricity demand in a given season is subject to a range of uncertainties, including underlying population growth, changing technology, economic conditions, prevailing weather conditions (and the timing of those conditions), as well as the general randomness inherent in individual usage. It is also subject to some known calendar effects due to the time of day, day of week, time of year, and public holidays. A comprehensive forecasting solution is described in this paper. First, semi-parametric additive models are used to estimate the relationships between demand and the driver variables, including temperatures, calendar effects and some demographic and economic variables. Then the demand distributions are forecasted by using a mixture of temperature simulation, assumed future economic scenarios, and residual bootstrapping. The temperature simulation is implemented through a new seasonal bootstrapping method with variable blocks. The proposed methodology has been used to forecast the probability distribution of annual and weekly peak electricity demand for South Australia since 2007. The performance of the methodology is evaluated by comparing the forecast results with the actual demand of the summer 2007-2008.

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Density Forecasting for Long-Term Peak Electricity Demand

24-h electrical load data—a sequential or partitioned time series?

Novel constructions of empirical controllability and observability
gramians for nonlinear systems are proposed for subsequent use in a balanced
truncation style of model reduction. The new gramians
are based on a generalisation of the fundamental solution for a
Linear Time-Varying system. Relationships between the given
gramians for nonlinear systems and the standard gramians for both
Linear Time-Invariant and Linear Time-Varying systems are
established as well as relationships to prior constructions proposed
for empirical gramians. Application of the new gramians is
illustrated through a sample test-system.

Empirical Balanced Truncation of Nonlinear Systems

Purpose – Nonlinear dynamical systems may, under certain conditions, be represented by a bilinear system. The paper is concerned with the construction of the controllability and observability gramians for the corresponding bilinear system. Such gramians form the core of model reduction schemes involving balancing.Design/methodology/approach – The paper examines certain properties of the bilinear system and identifies parameters that capture important information relating to the behaviour of the system.Findings – Novel approaches for the determination of approximate constant gramians for use in balancing‐type model reduction techniques are presented. Numerical examples are given which indicate the efficacy of the proposed formulations.Research limitations/implications – The systems under consideration are restricted to the so‐called weakly nonlinear systems, i.e. those without strong nonlinearities where the essential type of behaviour of the system is determined by its linear part.Practical implications –...

Nonlinear systems – algebraic gramians and model reduction

A novel forward backward iterative scheme for solving the three-dimensional (3-D) electric field integral equation is presented. This communication details how a naive extension of a 2-D forward backward algorithm to 3-D problems results in convergence difficulties due to spurious edge effects. The method proposed in this communication postulates the use of local "buffer regions" to suppress these unwanted effects and ensure stability. Results are presented illustrating the convergence of the algorithm when applied to scattering by a 15/spl lambda/ square metallic plate with an aperture and a metallic right-angled wedge.

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A novel iterative solution of the three dimensional electric field Integral equation

We present a method to compute efficiently solutions of systems of ordinary differential equations (ODEs) that possess highly oscillatory forcing terms. This approach is based on asymptotic expansions in inverse powers of the oscillatory parameter, and features two fundamental advantages with respect to standard numerical ODE solvers: first, the construction of the numerical solution is more efficient when the system is highly oscillatory, and, second, the cost of the computation is essentially independent of the oscillatory parameter. Numerical examples are provided, featuring the Van der Pol and Duffing oscillators and motivated by problems in electronic engineering.

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Marissa Condon

Papers

24-h electrical load data—a sequential or partitioned time series?

Empirical Balanced Truncation of Nonlinear Systems

Nonlinear systems – algebraic gramians and model reduction

A novel iterative solution of the three dimensional electric field Integral equation

On second-order differential equations with highly oscillatory forcing terms