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Description
The Kuramoto model is an archetypal model for studying synchronization in groups
of nonidentical oscillators where oscillators are imbued with their own frequency and
coupled with other oscillators though a network of interactions. As the coupling
strength increases, there is a bifurcation to complete synchronization where all oscillators
move with the same frequency and show a collective rhythm. Kuramoto-like
dynamics are considered a relevant model for instabilities of the AC-power grid which
operates in synchrony under standard conditions but exhibits, in a state of failure,
segmentation of the grid into desynchronized clusters.
In this dissertation the minimum coupling strength required to ensure total frequency
synchronization in a Kuramoto system, called the critical coupling, is investigated.
For coupling strength below the critical coupling, clusters of oscillators form
where oscillators within a cluster are on average oscillating with the same long-term
frequency. A unified order parameter based approach is developed to create approximations
of the critical coupling. Some of the new approximations provide strict lower
bounds for the critical coupling. In addition, these approximations allow for predictions
of the partially synchronized clusters that emerge in the bifurcation from the
synchronized state.
Merging the order parameter approach with graph theoretical concepts leads to a
characterization of this bifurcation as a weighted graph partitioning problem on an
arbitrary networks which then leads to an optimization problem that can efficiently
estimate the partially synchronized clusters. Numerical experiments on random Kuramoto
systems show the high accuracy of these methods. An interpretation of the
methods in the context of power systems is provided.
of nonidentical oscillators where oscillators are imbued with their own frequency and
coupled with other oscillators though a network of interactions. As the coupling
strength increases, there is a bifurcation to complete synchronization where all oscillators
move with the same frequency and show a collective rhythm. Kuramoto-like
dynamics are considered a relevant model for instabilities of the AC-power grid which
operates in synchrony under standard conditions but exhibits, in a state of failure,
segmentation of the grid into desynchronized clusters.
In this dissertation the minimum coupling strength required to ensure total frequency
synchronization in a Kuramoto system, called the critical coupling, is investigated.
For coupling strength below the critical coupling, clusters of oscillators form
where oscillators within a cluster are on average oscillating with the same long-term
frequency. A unified order parameter based approach is developed to create approximations
of the critical coupling. Some of the new approximations provide strict lower
bounds for the critical coupling. In addition, these approximations allow for predictions
of the partially synchronized clusters that emerge in the bifurcation from the
synchronized state.
Merging the order parameter approach with graph theoretical concepts leads to a
characterization of this bifurcation as a weighted graph partitioning problem on an
arbitrary networks which then leads to an optimization problem that can efficiently
estimate the partially synchronized clusters. Numerical experiments on random Kuramoto
systems show the high accuracy of these methods. An interpretation of the
methods in the context of power systems is provided.
ContributorsGilg, Brady (Author) / Armbruster, Dieter (Thesis advisor) / Mittelmann, Hans (Committee member) / Scaglione, Anna (Committee member) / Strogatz, Steven (Committee member) / Welfert, Bruno (Committee member) / Arizona State University (Publisher)
Created2018
Description
For a (N+1)-bus power system, possibly 2N solutions exists. One of these solutions
is known as the high-voltage (HV) solution or operable solution. The rest of the solutions
are the low-voltage (LV), or large-angle, solutions.
In this report, a recently developed non-iterative algorithm for solving the power-
flow (PF) problem using the holomorphic embedding (HE) method is shown as
being capable of finding the HV solution, while avoiding converging to LV solutions
nearby which is a drawback to all other iterative solutions. The HE method provides a
novel non-iterative procedure to solve the PF problems by eliminating the
non-convergence and initial-estimate dependency issues appeared in the traditional
iterative methods. The detailed implementation of the HE method is discussed in the
report.
While published work focuses mainly on finding the HV PF solution, modified
holomorphically embedded formulations are proposed in this report to find the
LV/large-angle solutions of the PF problem. It is theoretically proven that the proposed
method is guaranteed to find a total number of 2N solutions to the PF problem
and if no solution exists, the algorithm is guaranteed to indicate such by the oscillations
in the maximal analytic continuation of the coefficients of the voltage power series
obtained.
After presenting the derivation of the LV/large-angle formulations for both PQ
and PV buses, numerical tests on the five-, seven- and 14-bus systems are conducted
to find all the solutions of the system of nonlinear PF equations for those systems using
the proposed HE method.
After completing the derivation to find all the PF solutions using the HE method, it
is shown that the proposed HE method can be used to find only the of interest PF solutions
(i.e. type-1 PF solutions with one positive real-part eigenvalue in the Jacobian
matrix), with a proper algorithm developed. The closet unstable equilibrium point
(UEP), one of the type-1 UEP’s, can be obtained by the proposed HE method with
limited dynamic models included.
The numerical performance as well as the robustness of the proposed HE method is
investigated and presented by implementing the algorithm on the problematic cases and
large-scale power system.
is known as the high-voltage (HV) solution or operable solution. The rest of the solutions
are the low-voltage (LV), or large-angle, solutions.
In this report, a recently developed non-iterative algorithm for solving the power-
flow (PF) problem using the holomorphic embedding (HE) method is shown as
being capable of finding the HV solution, while avoiding converging to LV solutions
nearby which is a drawback to all other iterative solutions. The HE method provides a
novel non-iterative procedure to solve the PF problems by eliminating the
non-convergence and initial-estimate dependency issues appeared in the traditional
iterative methods. The detailed implementation of the HE method is discussed in the
report.
While published work focuses mainly on finding the HV PF solution, modified
holomorphically embedded formulations are proposed in this report to find the
LV/large-angle solutions of the PF problem. It is theoretically proven that the proposed
method is guaranteed to find a total number of 2N solutions to the PF problem
and if no solution exists, the algorithm is guaranteed to indicate such by the oscillations
in the maximal analytic continuation of the coefficients of the voltage power series
obtained.
After presenting the derivation of the LV/large-angle formulations for both PQ
and PV buses, numerical tests on the five-, seven- and 14-bus systems are conducted
to find all the solutions of the system of nonlinear PF equations for those systems using
the proposed HE method.
After completing the derivation to find all the PF solutions using the HE method, it
is shown that the proposed HE method can be used to find only the of interest PF solutions
(i.e. type-1 PF solutions with one positive real-part eigenvalue in the Jacobian
matrix), with a proper algorithm developed. The closet unstable equilibrium point
(UEP), one of the type-1 UEP’s, can be obtained by the proposed HE method with
limited dynamic models included.
The numerical performance as well as the robustness of the proposed HE method is
investigated and presented by implementing the algorithm on the problematic cases and
large-scale power system.
ContributorsMine, Yō (Author) / Tylavsky, Daniel (Thesis advisor) / Armbruster, Dieter (Committee member) / Holbert, Keith E. (Committee member) / Sankar, Lalitha (Committee member) / Vittal, Vijay (Committee member) / Undrill, John (Committee member) / Arizona State University (Publisher)
Created2015
Description
A statistical method is proposed to learn what the diffusion coefficient is at any point in space of a cell membrane. The method used bayesian non-parametrics to learn this value. Learning the diffusion coefficient might be useful for understanding more about cellular dynamics.
ContributorsGallimore, Austin Lee (Author) / Presse, Steve (Thesis director) / Armbruster, Dieter (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2021-05