This thesis contributes to the field of modelling and simulation of hybrid dynamical systems by developing,
• a language for the description of hybrid dynamical models, SCHEMA, that capturesassumptions made during model development,
• a new numerical method for the solution of hybrid dynamical models.
Hybrid dynamical behaviour has only recently been recognised as a mathematical framework that needs to be incorporated into process modelling software. The hybrid dynamical framework unites descriptions of continuous dynamic behaviour and discrete dynamic behaviour. However, existing languages focus on the equation set of the model and interaction between the continuous and discrete description.
Recent research has proposed modelling tools that assist the modeller in a phenomenological description of processes. A conceptual description is developed from which the equations are generated as an artifact. Previous modelling tools have not explicitly recognised modelling assumptions and their role in transforming the model during development. SCHEMA, a System for Chemical Engineering Model Adaptation, allows a more natural description of hybrid dynamical behaviour from an assumption-based framework. Recognising that assumptions have limited ranges of validity, discontinuities are introduced to move between alternative model descriptions, extending the overall validity of the hybrid model.
The equations generated from SCHEMA must, in most applications, be solved using an appropriate numerical method. Many solvers for hybrid dynamical systems are now available but all rely on the backwards-differentiation formula, or BDF method. BDF methods restart with low order and so it is proposed that a method that starts at higher order may offer advantages for highly discontinuous systems. The second objective of this thesis was to develop a Runge-Kutta method capable of solving hybrid dynamical problems and to compare its performance to that of traditional BDF approaches.
SCHEMA is developed and implemented such that models developed with the language increase in information richness as model equations are simplified. The approach to hybrid model description advocated in this thesis is demonstrated with several examples. In particular, a model of a batch prefermenter, studied experimentally during the thesis, is developed using the language to demonstrate the language’s use on industrially relevant problems.
SCHEMA makes unique contributions in its approach to modelling instantaneous changes of variables within hybrid models. In contrast to previous languages, instantaneous changes are modelled in a distinct mode within the model. These modes are developed in a manner analogous to continuous modes, considering conservation balances and any additional constitutive equations. The language leads to new approaches to traditional problems. An alternative description of initial values of a model is proposed by their inclusion as a mode within a hybrid model. The initialisation problem is explicitly recognised as part of model development allowing and prompts the modeller to consider which constitutive equations should hold during initialisation.
A significant contribution of this thesis is the new Runge-Kutta method it develops. An Explicit Stage, Diagonally Implicit Runge-Kutta (ESDIRK) method, it achieves higher stage order than normally possible for diagonally implicit methods allowing solution of index 2 differential algebraic equation sets. Implementation as a hybrid solver is also discussed.
The new method proves to be a viable alternative to existing numerical methods and may be considered when BDF methods prove inadequate. The ability to solve discontinuous index 2 problems without symbolic differentiation exists when future research yields an appropriate consistent initialisation method.
Implementation of the numerical solver within Daesim required extension of the simulator’s existing language to describe discontinuities. The proposed language allows hybrid behaviour to be embedded within hybrid behaviour. Using these extensions, SCHEMA files can be compiled to generate Daesim input. This was demonstrated using the batch prefer- menter model, the Daesim extensions allowing description of the discontinuous control of theprocess.
In summary, SCHEMA offers a new approach to the description of process models and further research is recommended into the use of the language to document model families and their development. The numerical method developed in the thesis provides an alternative to the traditional BDF method which was previously the only viable alternative in solving hybrid dynamical process engineering problems.