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Modal sensitivity and participation factors
Introduction
The modal analysis for linear models is an important tool to model characteristics as the stability of damping oscillations among others. These characteristic depend on the coupling between components whose parameters are supposed to be known without uncertainty, which is not always the case.
It is a common knowledge that there always exists a certain discrepancy between actual (real-operating) and nominal trajectories of any system. This discrepancy is partly due to various inherently approximation schemes in system identification, and partly due to possible further parameter variations stimulated by environmental changes as slow aging of components, variations in raw products or system components in process control applications, inaccuracy of measurement devices or methods, etc .(Eslami 1994).
The study of the changes in the eigenvalues due to parameter changes is important, not only for stability purposes but also for the control of the state variables. Fundaments on sensitivity analysis are hereafter presented.
Sensitivity matrices
Modal analysis is based on sensitivity analysis of the eigenvalues of the state matrix A to changes in the system parameters.
In the particular case of repeated eigenvalues, it is not possible to obtain the solution, because the right and left eigen-matrices are not orthogonal.
For the solution of the system, a so-called “confluent” Vandermonde matrix can be used but is not related with the sensitivity of the eigenvalues.
Participation factors
The PF are used in the modal control theory better known as sensitivity analysis. Participation factors focus their application in electrical power systems in which having a prior knowledge of the system is recommendable. As it is mentioned in (I.J. Perez-Arriaga 1981) “A crucial feature of the Dynamic stability problem (in the power systems context) and one that makes it particularly suitable for selective modal analysis (SMA), is that only a few, lightly damped, oscillatory modes are of interest, out of the hundreds of modes that the model may have”. It is also said that the SMA is not restricted to any particular kind of physical systems but it is focused on electrical power systems.
Considered a linear system with distinct eigenvalues, al. 1988) “The term (PF) reflects the activity of the k (eigenvalue) is excited”.
Per-unit representation
In power system analysis, it is usually convenient to use a per-unit model to normalize system units. Compared to the use of physical units (amperes, volts, ohms, webers, henrys,
Participation factors
etc.), the per-unit system offers computational simplicity by eliminating units and expressing system quantities as dimensionless ratios.
A well-chosen per-unit system can minimize computational effort, simplify evaluation and facilitate understanding of the characteristics. Some base quantities may be chosen independently and quite arbitrary, while others follow automatically depending on fundamental relationships between system variables. Normally, the base values are chosen so that the principal variables still be equal to one per-unit under the nominal condition.
The general representation can be, quantity in per unit actual quantity base value of quantity (2.65)
Usage and interpretation of sensitivity matrices and participation factors
The sensitivity matrices are a tool used for modal analysis. Their development responds to two objectives (Porter and Crossley 1972),
i. The selection of system parameters such as the eigenvalues and their associated eigenvectors be as insensitive as possible to changes in such parameters.
ii. The determination of good approximations of the eigenvalues and eigenvectors associated with various system components, without having to compute all the eigenvalues and eigenvectors.
Participation factor theory was developed to understand and reduce models with uncoupled dynamics; its use in the selective modal analysis is well justified and understood as presented in section 2.2.
The use of participation factors has changed since its first definition. For example in (Arabi et al. 1991), the PF matrix is used to show how important is a state variable for the eigenvalues or how important the eigenvalue is for the state variable. In (Hsu and Chen 1987) and (Ignacio J. Perez-Arriaga et al. 1990) they are used for the identification of the best location for stabilizers.
In both cases, for the PF theory to be applied some assumptions have to be verified. The following examples will present some cases where this approach can suffer of a lack of precision.
• Example 2.2: RLC circuit: time response and participation factor approximation Three cases are considered with different values for the element parameters. In the first case, the values are chosen in order to have real eigenvalues with a non-diagonal dominant matrix. In the second case, the eigenvalues are complex and the state matrix is non-diagonal dominant. In the third one the values are chosen to have complex eigenvalues and diagonal dominant matrix. As outputs, the selected states are the current in the inductance and the voltage in the capacitor.
Table of contents :
General introduction
Chapter 1 State of the art
1.1 Context and motivation
1.2 Small signal stability
1.2.1 Modal Analysis in power electrical systems
1.2.2 Modal Analysis of Multi-terminal DC grids
1.3 Conclusions
Chapter 2 Modal sensitivity and participation factors
2.1 Introduction
2.2 Sensitivity matrices
2.3 Participation factors
2.3.1 Change of basis
2.3.2 Per-unit representation
2.4 Usage and interpretation of sensitivity matrices and participation factors
2.5 Conclusions
Chapter 3 Sensitivity analysis
3.1 Introduction
3.2 New definition of parametric sensitivity matrix
3.2.1 Example 3.1: Simple Circuit sensitivity of the eigenvalues to the elements
3.2.2 Normalization
3.2.3 Example 3.2: Simple Circuit with real eigenvalues
3.2.4 Example 3.3: Simple Circuit zero in a diagonal element
3.2.5 Example 3.4: Simple Circuit with diagonal dominant elements
3.3 Remarks and uses.
3.4 Conclusions
Chapter 4 State-space modeling of HVDC systems
4.1 Introduction
4.2 Context of the HVDC development
4.3 Voltage source converter
4.3.1 General control scheme
4.3.2 Per-unit in HVDC grids
4.3.3 Reduced linear equivalent
4.3.4 Three terminal DC grid model
4.4 Modular Multi-level Converter reduced model
4.4.1 Simplification of the AAM
4.4.2 Control strategy
4.4.3 Linearization
4.4.4 One station MMC in a MTDC network
4.4.5 Two stations MMC in a HVDC
4.5 Conclusions
Chapter 5 Modal analysis on HVDC systems
5.1 Objectives of the analysis
5.2 Simplified equivalent DC link
5.3 Dynamic analysis of a three terminal MTDC
5.4 Dynamic analysis of HVDC system with MMC
5.4.1 One MMC connected to a fixed voltage source: energy controlled by DC power
5.4.2 One MMC connected to a fixed voltage source: DC power mode and energy controlled by AC power
5.4.3 One MMC connected to a variable DC bus – energy reference constant
5.4.4 Two terminal MMC DC link (MMC1 Power mode W controlled by dc Power, MMC2 voltage mode W controlled by Ac Power)
5.5 Conclusions
Chapter 6 General Conclusions and Perspectives
Scientific contribution
Bibliography
