We give a short and elementary introduction to Lie group methods. A selection of applications of Lie group integrators are discussed. Finally, a family of symplectic integrators on cotangent bundles of Lie groups is presented and the notion of discrete gradient methods is generalised to Lie groups.
In this article, a unified approach to obtain symplectic integrators on T * G from Lie group integrators on a Lie group G is presented. The approach is worked out in detail for symplectic integrators based on Runge-Kutta-Munthe-Kaas methods and Crouch-Grossman methods. These methods can be interpreted as symplectic partitioned Runge-Kutta methods extended to the Lie group setting in two different ways. In both cases, we show that it is possible to obtain symplectic integrators of arbitrarily high order by this approach.
Symplectic integration of autonomous Hamiltonian systems is a wellknown field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of dimensions. We show that one possible extension of symplectic methods in the autonomous setting to the non-autonomous setting is obtained by using canonical transformations. Many existing methods fit into this framework. We also perform experiments which indicate that for exponential integrators, the canonical and symmetric properties are important for good long time behaviour. In particular, the theoretical and numerical results support the well documented fact from the literature that exponential integrators for non-autonomous linear problems have superior accuracy compared to general ODE schemes.
This study shows modelling developed during the first year of the SmartNet project. In particular, it presents a mathematical model for aggregation of curtailable generation and sheddable loads. The model determines the quantity and the cost of the flexibility provided by the flexible resources based on their physical and dynamic behaviours. The model also proposes a bidding strategy in order to translate the aggregated behaviour into market bids. 2 Flexibility intervals 2.1 Flexibility of a single device 2.1.1 Curtailable generation: A collection of wind generators, numbered from 1 to n G would be a good example of curtailable 24th International Conference & Exhibition on Electricity Distribution (CIRED)
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