Abstract
This thesis is concerned with small-signal analysis in power systems. More precisely, this thesis focuses on state-space analysis of a system comprising multiple subsystems, which can be described using any choice of state variables and can even be represented as black boxes.
When a state-space model is represented in a black-box form, the internal structures and parameters are not revealed, and the state variables are not interpretable. Such a state-space model will not reveal more information than an encrypted black-box time domain EMT model or an impedance model. Thus, black-box state-space models are useful in stability assessment of multi-vendor systems, where sensitive information such as control structure and control parameters must remain hidden. Traditional modal analysis tools, such as participation factors and mode shapes, relate system modes with system state variables. These methods will not be useful when applied directly to black-box state-space models in which the state has no known physical meaning.
This thesis demonstrates the shortcomings of traditional modal analysis applied to state-space models with black-box subsystems. Furthermore, it demonstrates the shortcomings of traditional modal analysis in assessing the root cause of poorly damped system modes. To close these gaps, this thesis derives and defines a set of tools suited for black-box state-space models, which can assess the root causes of modes in state-space models regardless of whether the system representation is transparent or black-boxed. In state-space models with black-boxed subsystems, the modes should be assessed through the inputs and outputs of the subsystem. Consequently, this thesis defines the Mode-in-Output participation factors and the Input-in-Mode participation factors, inspired by modal observability and controllability, for these purposes. In addition, this thesis defines the State-Matrix-in-Mode Participation matrix (SMMP), which is the principal finding of this thesis.
The novel SMMP-matrix relates the contribution of each state matrix element to a specific mode. This is especially useful in modal analysis to determine the root cause of poorly damped system modes.The SMMP-matrix associated with complex modes consists of complex numbers, and the real parts give the contribution to the real part of the eigenvalue from each state matrix element, and the imaginary parts give the contribution to the imaginary part of the eigenvalue from each state matrix element.
Thus, the real part of the SMMP describes the contribution of each state matrix element to the stability of the system. Another important property of the SMMP-matrix is that the aggregation of all elements belonging to a subsystem is equivalent regardless of how the system is represented. This last property is central to the main topic of the thesis, which is the modal analysis of state-space models with black-box subsystems.
The choice of aggregation of the SMMP-matrix can be viewed as different resolutions of the assessment of the root cause of modes in large systems. The SMMP-matrix can be aggregated to the unit level (e.g., converters, generators, loads, etc.), or to subsystems containing multiple units, or to regions containing multiple subsystems. Consequently, the level of detail of the contribution to the modes depends on the choice of aggregation, which in turn relates to the level of transparency of the system model. For black-boxed subsystems, the best achievable resolution is at the subsystem level, but for transparent subsystems, the resolution can be chosen freely from the contribution from a single element all the way up to a region level.
Finally, the thesis proposes a new framework for categorising system modes in power systems dominated by power electronic converters. System modes should be categorised by the three properties: root cause, excitability/controllability, and observability. The three properties can be assessed by using the SMMP-matrix, Input-in-Mode participation factor, and Output-in-Mode participation factors defined in this thesis.
The tools and methods defined in this thesis have been developed to perform modal analysis in converter-dominated power system state-space models with black-boxed subsystems. However, the tools and methods are not limited to such models and will be applicable to traditional power system models with full transparency, as well as to any other system described by a linear state-space model. The potential applicability of the SMMP-matrix is even more general, as it can be applied to any eigenvalue problem of diagonalisable matrices where it is of interest to relate the matrix's eigenvalues and elements.