Measurements of Magnetic Field Pattern in a Short LHC Dipole Model

The magnetic field in superconducting accelerator magnets has a fine structure with longitudinal periodicity. This periodic pattern, with period identical to the cable twist pitch, is originated by uneven current distribution within the cable. Here we present results of measurements of the periodic pattern performed in an LHC dipole model. We report in particular the results obtained powering the magnet with simple current steps and typical operation cycles as will be used during accelerator operation. The main result of the analysis is the time variation of the amplitude of the periodic pattern, from which we infer the evolution of the current distribution in the cable. We discuss the dependence of the pattern amplitude on ramp and pre-cycle parameters

On global in time self-similar solutions of Smoluchowski equation with multiplicative kernel

We study the similarity solutions (SS) of Smoluchowski coagulation equation with multiplicative kernel $K(x,y)=(xy)^{s}$ for $s<\frac{1}{2}$. When $s<0$% , the SS consists of three regions with distinct asymptotic behaviours. The appropriate matching yields a global description of the solution consisting of a Gamma distribution tail, an intermediate region described by a lognormal distribution and a region of very fast decay of the solutions to zero near the origin. When $s\in \left( 0,\frac{1}{2}\right)$, the SS is unbounded at the origin. It also presents three regions: a Gamma distribution tail, an intermediate region of power-like (or Pareto distribution) decay and the region close to the origin where a singularity occurs. Finally, full numerical simulations of Smoluchowski equation serve to verify our theoretical results and show the convergence of solutions to the selfsimilar regime

Analytical Calculation of Current Distribution in Multistrand Superconducting Cables

In recent years the problem of current distribution in multistrand superconducting cables has received increasing attention for large scale superconductivity applications due to its effect on the stability of fusion magnets and the field quality of accelerator magnets. A modelling approach based on distributed parameters has revealed to be very effective in dealing with long cables made of some tens or hundreds of strands. In this paper we present a fully analytical solution equation for a distributed parameters model in cables made of an arbitrary number of strands, whose validity is subjected to symmetry conditions generally satisfied in practical cables. We give in particular analytical formulae of practical use for the estimation of the maximum strand currents, time constants and redistribution lengths as a function of the cable properties and the external voltage source

An Analytical Benchmark for the Calculation of Current Distribution in Superconducting Cables

The validation of numerical codes for the calculation of current distribution and AC loss in superconducting cables versus experimental results is essential, but could be affected by approximations in the electromagnetic model or incertitude in the evaluation of the model parameters. A preliminary validation of the codes by means of a comparison with analytical results can therefore be very useful, in order to distinguish among different error sources. We provide here a benchmark analytical solution for current distribution that applies to the case of a cable described using a distributed parameters electrical circuit model. The analytical solution of current distribution is valid for cables made of a generic number of strands, subjected to well defined symmetry and uniformity conditions in the electrical parameters. The closed form solution for the general case is rather complex to implement, and in this paper we give the analytical solutions for different simplified situations. In particular we examine the influence of different boundary conditions, the effect of a localised resistance in the middle of the cable such as in the case of quench and the effects of localized time dependent magnetic fluxes acting on the cable

Analysis of Electrical Coupling Parameters in Superconducting Cables

The analysis of current distribution and redistribution in superconducting cables requires the knowledge of the electric coupling among strands, and in particular the interstrand resistance and inductance values. In practice both parameters can have wide variations in cables commonly used such as Rutherford cables for accelerators or Cable-in-Conduits for fusion and SMES magnets. In this paper we describe a model of a multi-stage twisted cable with arbitrary geometry that can be used to study the range of interstrand resistances and inductances that is associated with variations of geometry. These variations can be due to cabling or compaction effects. To describe the variations from the nominal geometry we have adopted a cable model that resembles to the physical process of cabling and compaction. The inductance calculation part of the model is validated by comparison to semi-analytical results, showing excellent accuracy and execution speed

A General Model for Thermal, Hydraulic and Electric Analysis of Superconducting Cables

In this paper we describe a generic, multi-component and multi-channel model for the analysis of superconducting cables. The aim of the model is to treat in a general and consistent manner simultaneous thermal, electric and hydraulic transients in cables. The model is devised for most general situations, but reduces in limiting cases to most common approximations without loss of efficiency. We discuss here the governing equations, and we write them in a matrix form that is well adapted to numerical treatment. We finally demonstrate the model capability by comparison with published experimental data on current distribution in a two-strand cable

A continuum model for current distribution in Rutherford cables

An analysis of eddy currents induced in flat Rutherford-type cables by external time dependent magnetic fields has been performed. The induced currents generate in turn a secondary magnetic field which has a longitudinal periodicity (periodic pattern). The dependence of the amplitude of the pattern on the history of the cable excitation has been investigated. The study has been carried out with two different models for the simulation of current distribution in Rutherford cables, namely a network model, based on a lumped parameters circuit and a "continuum" model, based on a distributed parameters circuit. We show the results of simulations of the current distribution in the inner cable of a short LHC dipole model in different powering conditions and compare them to experimental data. (12 refs)

Handbook of linear data-driven predictive control:Theory, implementation and design

Data-driven predictive control (DPC) has gained an increased interest as an alternative to model predictive control in recent years, since it requires less system knowledge for implementation and reliable data is commonly available in smart engineering systems. Several data-driven predictive control algorithms have been developed recently, which largely follow similar approaches, but with specific formulations and tuning parameters. This review aims to provide a structured and accessible guide on linear data-driven predictive control methods and practices for people in both academia and the industry seeking to approach and explore this field. To do so, we first discuss standard methods, such as subspace predictive control (SPC), and data-enabled predictive control (DeePC), but we also include newer hybrid approaches to DPC, such as γ–data-driven predictive control and generalized data-driven predictive control. For all presented data-driven predictive controllers we provide a detailed analysis regarding the underlying theory, implementation details and design guidelines, including an overview of methods to guarantee closed-loop stability and promising extensions towards handling nonlinear systems. The performance of the reviewed DPC approaches is compared via simulations on two benchmark examples from the literature, allowing us to provide a comprehensive overview of the different techniques in the presence of noisy data.</p

Handbook of linear data-driven predictive control:Theory, implementation and design

Data-driven predictive control (DPC) has gained an increased interest as an alternative to model predictive control in recent years, since it requires less system knowledge for implementation and reliable data is commonly available in smart engineering systems. Several data-driven predictive control algorithms have been developed recently, which largely follow similar approaches, but with specific formulations and tuning parameters. This review aims to provide a structured and accessible guide on linear data-driven predictive control methods and practices for people in both academia and the industry seeking to approach and explore this field. To do so, we first discuss standard methods, such as subspace predictive control (SPC), and data-enabled predictive control (DeePC), but we also include newer hybrid approaches to DPC, such as γ–data-driven predictive control and generalized data-driven predictive control. For all presented data-driven predictive controllers we provide a detailed analysis regarding the underlying theory, implementation details and design guidelines, including an overview of methods to guarantee closed-loop stability and promising extensions towards handling nonlinear systems. The performance of the reviewed DPC approaches is compared via simulations on two benchmark examples from the literature, allowing us to provide a comprehensive overview of the different techniques in the presence of noisy data.</p