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Nonlinear Modal Interactions and Participation Factors Analysis via Normal Form

As noted before, when controlled generators are considered, the power systems are usually represented with rst order models. In designing and siting of these controls such as PSS, usually linear techniques such as state observability and controlloability are used. These techniques lead to the denition of some indices such as participation factors and residues, which enables one to design and optimally site the controls in the power system. As we saw in chapter 1, increasing stress in the system leads to nonlinear interactions of the fundamental modes. These interactions can aect these controls, hence the quest for nonlinear equivalence of those indices (e.g. participation factors). In this section, the extension of linear participation factors to the nonlinear one is presented. Also, some already-existing indices for estimating the eects of modal interactions are presented.

Selective Nonlinear Modal Interactions

In chapter 4, a method which facilitates the computation of the nonlinear coecients required for NF application has been developed. The method enables computing selectively, any desired term rapidly, by avoiding the usual Taylor expansion. However, much reduction is still needed as there are still too many terms being considered in the analysis. For specic NF studies, it is possible to use some selected terms instead of all the terms. The main goal of this section is to reduce the computational burden associated with NF application to power systems, especially when it is applied to understand the signicant modal interactions and the accompanying new frequencies. The section investigates further reduction of NF computation by considering fewer terms in the nonlinear approximation based on the information provided by the linear analysis. The analysis is then focused only on the considered terms. As stated before, analysis of nonlinear modal interaction is not new in power systems. What is new here is the selective application of NF to this analysis, which is made possible by the developed method.

Proposals for Selective NF Applications

Let us recall again from chapter 3 that the NF coecients (i.e., h2 and h3) needed in the NF solution (5.2) are given by where Cres is a residual term from second order transformation and is expressed as pqr is the original third order term. Our goal is to not compute all the coecients but only some and set the other h-coecients to zero. If modal interaction is the objective of study, whereby sources of unknown frequencies in time responses are explained; signicant reduction of NF computation can be achieved by careful selection of relevant hcoe cients.
Careful observation of (5.14) and (5.2) shows that the indices of the NF coecients are consistent with the indices of the mode combinations such that h2j kl = 0, h3j pqr = 0 implies that that mode combinations k +l, p +q +r have zero eects on the dynamics of state i. Therefore, to neglect the eect of a mode combination, the corresponding NF coecient can be set to zero. However, the challenge remains how to decide which coecients to set to zero. In this section, we propose two approaches that can be used to discriminate some h-coecients, thereby reducing further the NF computations:
• real modes/eigenvalues exemption and
• mode’s energy consideration.
They are discussed in the following subsections.

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Real Modes/Eigenvalues Exemption Proposal

Let us assume that we can compute all the coecients. Then, observation of (5.2) shows that there are interactions among the linear modes. Previous works on NF and spectral analysis prove that oscillatory modes can interact to produce new oscillations [31, 84]. However, there has not been any meaningful interpretation to interactions involving real modes or its physical phenomenon. The stability indices proposed in [55, 57] are based on the interactions associated to only oscillatory modes. With controls included in the models, there may be many of these real modes. Real modes are aperiodic and the actual interactions involving real modes may only aect the damping, but not alter the analysis of modal interaction. We propose to reduce NF computation by keeping all the linear modes in the linear part of the 3rd order approximate model, but considering only the interactions among oscillatory modes in the nonlinear part. The proposal is based on the interpretation of (5.2). Given that all modes are initially stable, (5.2) leads to the following deductions:
1. A combination of only real modes does not lead to a new frequency in the spectral.
2. A 2nd order combination of a real mode with an oscillatory mode does not lead to a new frequency in the spectral, rather a more damped version of the oscillatory mode which combined with the real mode.
3. A 3rd order combination of real and oscillatory modes may lead to a new frequency but this frequency must be the more damped version of a combination of two oscillatory modes already existing at 2nd order.

Table of contents :

Acknowledgement
Abstract
Resume
List of Tables
List of Figures
1 Introduction
1.1 General Context and Motivation
1.2 Tools for Power System Dynamic Performance Analysis
1.3 Modes of Oscillation
1.4 Modal Interactions and Nonlinear Modes
1.5 Normal Form Method
1.6 Objective and Scope of the Research
1.7 Contributions of this PhD Research Work
1.8 Thesis Outline
2 Literature Review 
2.1 Revisiting Linear Modal Analysis Tools
2.2 Basic Idea of Normal Form
2.3 Applications of Normal Form in Power Systems
2.4 Present Challenges with Normal Form Method
2.5 Power System Model Order Reductions
2.6 Summary
3 Normal Form for Power System Models 
3.1 Power System Models
3.2 General Normal Form Theory
3.3 Normal Form of Second Order System Models
3.4 Summary of Normal Form Steps in Power System
4 Developed Method for Facilitating Normal Form Applications 
4.1 Motivation for the Proposed Method
4.2 Computation of Nonlinear Coecients: 2nd Order Model
4.3 Computation of Nonlinear Coecients: 1st Order Model
4.4 Comments on the Computational Accuracy/Eciency
4.5 Summary
5 Applications of Normal Form in Power Systems 
5.1 Nonlinear Modal Interactions and Participation Factors Analysis via Normal Form
5.2 Selective Nonlinear Modal Interactions
5.3 Concept of Nonlinear Frequency
5.4 Detection of Frequency-Amplitude Shifts via Normal Form
5.5 Transient/Mode’s Stability Estimation using Normal Form
5.6 Summary
6 Conclusions and Future Works 
6.1 Conclusions
6.2 Future Works
Bibliography

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