Flywheels are mechanical devices that store energy as the inertia of a rotating disk. Flywheel Energy Storage Systems (FESS) can combat the challenges of intermittency and unreliability that prevent effective integration of renewable energy sources into the electric grid. They have long lifespans, can undergo deep discharge without degradation, and are made of environmentally safe materials, however, their cost and storage capacity limit their large-scale deployment.

The use of optimization methods with mathematical models of the system can considerably shorten design time, and minimize costly `hardware-in-the-loop' design iterations. The energy capacity of FESS rotors can be improved by choosing the optimal rotor geometry, operation conditions, rotor materials and by tailoring the material properties. Depending on the complexity of the design goals, the model used to represent the system may range from a fairly simple analytical model to a complex 3D finite element model. In this thesis, an open-source optimization framework with shape and topology optimization capabilities was developed for the design of optimal FESS rotors. A suite of 1D, 2D axisymmetric and 3D linear elastic numerical rotor models were developed for use with the optimizer.

FESS are broadly categorized as low-speed metal rotor and high-speed composite rotor systems, and although both systems have been analyzed and optimized in literature, there is no consensus on which system is better suited to grid applications. The first contribution of this thesis was to perform a quantitative comparison of the two FESS technologies. Results showed that the total kinetic energy of both composite and metal constant thickness rotors was comparable. Multi-rim composite rotors with certain material sequences could outperform single-rim composite and metal flywheels in terms of total or specific energy, but offered no significant cost advantage over single rim metal rotors.

The second contribution of this thesis was to offer holistic metal rotor FESS design guidelines by establishing the correlation between rotor shape, speed and radius and their combined effect on FESS energy capacity. Choosing the best combination of rotor shape, speed and radius resulted in 21\% to 46\% improvements in the energy storage capacity of two different FESS designs, indicating a strong correlation between these parameters. A study on the self discharge of optimally shaped flywheels indicated that low-speed rotors with a large radius had a lower self discharge than high-speed rotors with a smaller radius and the same weight, which could be an important consideration during the FESS design process.

Stress-constrained topology optimization was used to further optimize energy storage characteristics of the FESS by using complex geometries that could not be analyzed with shape optimization. This thesis proposed a novel specific energy formulation with a global stress constraint, which allowed the optimizer to choose the topology volume fraction that led to the best specific energy improvement. The proposed formulation consistently achieved better design improvements than conventional kinetic energy formulations for various operating speeds and rotational symmetries. A post-optimality analysis on the effect of acceleration related stresses on the optimal topologies determined these to be significant only when considering extremely short duration charge-discharge cycles of less than 0.1 s.

Two approaches were used to improve the discreteness and convergence of the specific energy topology formulation. Local stress constraints with an Augmented Lagrangian formulation were shown to achieve designs with a more uniform stress distribution compared to P-norm aggregated global stress constraints, where undesirable local stress concentrations could be seen at narrow bottleneck regions. A modified robust approach improved design discreteness and allowed for 3D topology optimization with the specific energy formulation. Two distinct 3D rotor designs with similar energy capacities were seen to emerge when two different ranges of density filter radius were used for design, with one design being similar to shape optimized rotors, and the other design having spokes connecting the central shaft with an outer rim.

The developed optimization framework will serve as a comprehensive design tool for FESS rotors. The open-source nature of the tool will allow for further extensions to the library in terms of materials or analysis of non-linear or transient behaviour.