We use Melnikov's method to study the chaotic dynamics of an attitude transition maneuver of a to... more We use Melnikov's method to study the chaotic dynamics of an attitude transition maneuver of a torque-free rigid body in going from minor axis spin to major axis spin under the influence of small damping. The chaotic motion is caused by the breaking up of heteroclinic orbits in the unperturbed system due to oscillation of small perturbing sub-bodies inside the satellite. The system equations are derived and then transformed into a form suitable for the application of Melnikov's method. An analytical criterion for chaotic motion is derived in terms of the system parameters. This criterion is evaluated for its significance to the design of artificial satellites.
When power grids are heavily stressed with a bulk power transfer, it is useful to have a fast ind... more When power grids are heavily stressed with a bulk power transfer, it is useful to have a fast indication of the increased stress when multiple line outages occur. Reducing the bulk power transfer when the outages are severe could forestall further cascading of the outages. We show that synchrophasor measurements of voltage angles at all the area tie lines can be used to indicate the severity of multiple outages. These synchrophasor measurements are readily combined into an "area angle" that can quickly track the severity of multiple outages after they occur. We present a procedure to define thresholds for the area angle that relate to the maximum power that can be transferred through the area until a line limit is reached. Then in real time we would monitor the area angle and compare it to the thresholds when line outages occur to determine the urgency (or not) of actions to reduce the bulk transfer of power through the area. The procedure also identifies exceptional cases in which separate actions to resolve local power distribution problems are needed. We illustrate the thresholds and monitoring with the area angle across several states of Northwestern USA.
The loading margin to a saddle-node or fold bifurcation is a useful and basic index of proximity ... more The loading margin to a saddle-node or fold bifurcation is a useful and basic index of proximity to voltage collapse. Sensitivities of the loading margin can be used to select controls to avoid voltage collapse. This paper justifies the use of static power models to compute loading margins and their sensitivities. In particular, dynamic load models may be reduced to static equations and then simplified without affecting loading margins and their sensitivities. Alternatively, if dynamic models are not well known, the computations may be done with static models and conclusions drawn about a general and sensible class of underlying dynamic models. Explicit assumptions and mathematical underpinnings are presented.
Melnikov's method is used to analtically study chaotic dynamics in the attitude transition maneuv... more Melnikov's method is used to analtically study chaotic dynamics in the attitude transition maneuver of a torque-free rigid body in going from minor axis to major axis spin under the influence of viscous damping. The equations of motion are presented and then transformed into a form suitable for the application of Melnikov's method. The analytical criterion for chaos is in the form of an inequality that is a sufficient condition for chaotic dynamics in terms of the system parameters. The criterion is evaluated for its physical significance and for its application to the design of satellites.
Power system normal form analysis has developed coefficients and indices in modal coordinates to ... more Power system normal form analysis has developed coefficients and indices in modal coordinates to quantify nonlinear model interactions. The coefficients can become very large near a strong, nondiagonalizable resonance occurring in the power system linearization. Moreover, the changes in the coefficients when the power system equations are expressed in different coordinates or units show that the coefficients are not intrinsic. On a different topic, the paper suggests an explanation for the modal interaction that causes the subsynchronous resonance instability in power systems. The modal interaction is associated with a pair of strong resonances which arise as a perturbation of a weak resonance of complex eigenvalues. This idea is supported using results from the IEEE first benchmark subsynchronous resonance model and perturbation theory.
If an interarea oscillatory mode has insufficient damping, generator redispatch can be used to im... more If an interarea oscillatory mode has insufficient damping, generator redispatch can be used to improve its damping. We explain and apply a new analytic formula for the modal sensitivity to rank the best pairs of generators to redispatch. The formula requires some dynamic power system data and we show how to obtain that data from synchrophasor measurements. The application of the formula to damp interarea modes is explained and illustrated with interarea modes of the New England 10-machine power system.
Given a load power forecast, the sensitivity with respect to controls of a load power margin meas... more Given a load power forecast, the sensitivity with respect to controls of a load power margin measuring proximity to voltage collapse is computed. The computation is simple enough to contribute to the practical planning and control of a power system to avoid voltage collapse blackouts. The computation applies to avoiding saddle node bifurcation instability of a general dynamical system
Iee Proceedings Part a Physical Science Measurements and Instrumentation Management and Education Reviews, 1987
An electrical axes formulation of Kron's tensor analysis is presented and applied to the represen... more An electrical axes formulation of Kron's tensor analysis is presented and applied to the representation and digital simulation of AC/DC bridge convertor systems. The electrical, transformer, and switching connections of a general convertor system are represented as fixed and varying electrical axes so that the system equations may be automatically formed and solved. The resulting general switching circuit simulation is capable of analysing transients in 12-pulse convertor systems with nonideal loads. The circuit theory used in deriving the simulation is suggestive of useful methods of describing and visualising convertor system phenomena.
IEEE Conference Record - Abstracts. PPPS-2001 Pulsed Power Plasma Science 2001. 28th IEEE International Conference on Plasma Science and 13th IEEE International Pulsed Power Conference (Cat. No.01CH37255), 2000
a See Friday morning mini-course poster session for more detail and discussion.
Over the last couple of years there has been significant activity in the development of Flexible ... more Over the last couple of years there has been significant activity in the development of Flexible AC Transmission Systems (FACTS). Much of this work has been directed towards advanced series compensation (ASC) systems based on a Thyristor Controlled Reactor (TCR) ...
The area angle is a scalar measure of power system area stress that responds to line outages with... more The area angle is a scalar measure of power system area stress that responds to line outages within the area and is a combination of synchrophasor measurements of voltage angles around the border of the area. Both idealized and practical examples are given to show that the variation of the area angle for single line outages can be approximately related to changes in the overall susceptance of the area and the line outage severity.
ICOPS 2000. IEEE Conference Record - Abstracts. 27th IEEE International Conference on Plasma Science (Cat. No.00CH37087), 2000
Summary form only given. Due to the increasing bandwidth of Traveling Wave Tube Amplifiers (TWTAs... more Summary form only given. Due to the increasing bandwidth of Traveling Wave Tube Amplifiers (TWTAs), broadband klystrons, and the possibility of electron-beam modulation using Field Emitter Arrays (FEAs), there has been a resurgence of interest in understanding the electron beam-wave interaction when the beam and wave initially have multifrequency content. In the case of the klystron, this corresponds to understanding how the electron beam propagates a multitonal velocity or density disturbance. In a distributed interaction device like the TWTA the beam dynamics, and hence radiation dynamics, are further complicated since a beam electron interacts with a forcing field on the circuit for its entire trip through the device. We choose the simpler device, an idealized broadband klystron model, to study harmonic and intermodulation distortions in the multifrequency beam-wave system. The beam dynamics are modeled by nonlinear fluid equations, and the question of how the beam propagates general (multitonal) input conditions is addressed. Although the question has been considered before in some generality, we simplify the beam model and consider specifically multitonal excitation as would be used in a communications application
We use Melnikov's method to study the chaotic dynamics of an attitude transition maneuver of a to... more We use Melnikov's method to study the chaotic dynamics of an attitude transition maneuver of a torque-free rigid body in going from minor axis spin to major axis spin under the influence of small damping. The chaotic motion is caused by the breaking up of heteroclinic orbits in the unperturbed system due to oscillation of small perturbing sub-bodies inside the satellite. The system equations are derived and then transformed into a form suitable for the application of Melnikov's method. An analytical criterion for chaotic motion is derived in terms of the system parameters. This criterion is evaluated for its significance to the design of artificial satellites.
When power grids are heavily stressed with a bulk power transfer, it is useful to have a fast ind... more When power grids are heavily stressed with a bulk power transfer, it is useful to have a fast indication of the increased stress when multiple line outages occur. Reducing the bulk power transfer when the outages are severe could forestall further cascading of the outages. We show that synchrophasor measurements of voltage angles at all the area tie lines can be used to indicate the severity of multiple outages. These synchrophasor measurements are readily combined into an "area angle" that can quickly track the severity of multiple outages after they occur. We present a procedure to define thresholds for the area angle that relate to the maximum power that can be transferred through the area until a line limit is reached. Then in real time we would monitor the area angle and compare it to the thresholds when line outages occur to determine the urgency (or not) of actions to reduce the bulk transfer of power through the area. The procedure also identifies exceptional cases in which separate actions to resolve local power distribution problems are needed. We illustrate the thresholds and monitoring with the area angle across several states of Northwestern USA.
The loading margin to a saddle-node or fold bifurcation is a useful and basic index of proximity ... more The loading margin to a saddle-node or fold bifurcation is a useful and basic index of proximity to voltage collapse. Sensitivities of the loading margin can be used to select controls to avoid voltage collapse. This paper justifies the use of static power models to compute loading margins and their sensitivities. In particular, dynamic load models may be reduced to static equations and then simplified without affecting loading margins and their sensitivities. Alternatively, if dynamic models are not well known, the computations may be done with static models and conclusions drawn about a general and sensible class of underlying dynamic models. Explicit assumptions and mathematical underpinnings are presented.
Melnikov's method is used to analtically study chaotic dynamics in the attitude transition maneuv... more Melnikov's method is used to analtically study chaotic dynamics in the attitude transition maneuver of a torque-free rigid body in going from minor axis to major axis spin under the influence of viscous damping. The equations of motion are presented and then transformed into a form suitable for the application of Melnikov's method. The analytical criterion for chaos is in the form of an inequality that is a sufficient condition for chaotic dynamics in terms of the system parameters. The criterion is evaluated for its physical significance and for its application to the design of satellites.
Power system normal form analysis has developed coefficients and indices in modal coordinates to ... more Power system normal form analysis has developed coefficients and indices in modal coordinates to quantify nonlinear model interactions. The coefficients can become very large near a strong, nondiagonalizable resonance occurring in the power system linearization. Moreover, the changes in the coefficients when the power system equations are expressed in different coordinates or units show that the coefficients are not intrinsic. On a different topic, the paper suggests an explanation for the modal interaction that causes the subsynchronous resonance instability in power systems. The modal interaction is associated with a pair of strong resonances which arise as a perturbation of a weak resonance of complex eigenvalues. This idea is supported using results from the IEEE first benchmark subsynchronous resonance model and perturbation theory.
If an interarea oscillatory mode has insufficient damping, generator redispatch can be used to im... more If an interarea oscillatory mode has insufficient damping, generator redispatch can be used to improve its damping. We explain and apply a new analytic formula for the modal sensitivity to rank the best pairs of generators to redispatch. The formula requires some dynamic power system data and we show how to obtain that data from synchrophasor measurements. The application of the formula to damp interarea modes is explained and illustrated with interarea modes of the New England 10-machine power system.
Given a load power forecast, the sensitivity with respect to controls of a load power margin meas... more Given a load power forecast, the sensitivity with respect to controls of a load power margin measuring proximity to voltage collapse is computed. The computation is simple enough to contribute to the practical planning and control of a power system to avoid voltage collapse blackouts. The computation applies to avoiding saddle node bifurcation instability of a general dynamical system
Iee Proceedings Part a Physical Science Measurements and Instrumentation Management and Education Reviews, 1987
An electrical axes formulation of Kron's tensor analysis is presented and applied to the represen... more An electrical axes formulation of Kron's tensor analysis is presented and applied to the representation and digital simulation of AC/DC bridge convertor systems. The electrical, transformer, and switching connections of a general convertor system are represented as fixed and varying electrical axes so that the system equations may be automatically formed and solved. The resulting general switching circuit simulation is capable of analysing transients in 12-pulse convertor systems with nonideal loads. The circuit theory used in deriving the simulation is suggestive of useful methods of describing and visualising convertor system phenomena.
IEEE Conference Record - Abstracts. PPPS-2001 Pulsed Power Plasma Science 2001. 28th IEEE International Conference on Plasma Science and 13th IEEE International Pulsed Power Conference (Cat. No.01CH37255), 2000
a See Friday morning mini-course poster session for more detail and discussion.
Over the last couple of years there has been significant activity in the development of Flexible ... more Over the last couple of years there has been significant activity in the development of Flexible AC Transmission Systems (FACTS). Much of this work has been directed towards advanced series compensation (ASC) systems based on a Thyristor Controlled Reactor (TCR) ...
The area angle is a scalar measure of power system area stress that responds to line outages with... more The area angle is a scalar measure of power system area stress that responds to line outages within the area and is a combination of synchrophasor measurements of voltage angles around the border of the area. Both idealized and practical examples are given to show that the variation of the area angle for single line outages can be approximately related to changes in the overall susceptance of the area and the line outage severity.
ICOPS 2000. IEEE Conference Record - Abstracts. 27th IEEE International Conference on Plasma Science (Cat. No.00CH37087), 2000
Summary form only given. Due to the increasing bandwidth of Traveling Wave Tube Amplifiers (TWTAs... more Summary form only given. Due to the increasing bandwidth of Traveling Wave Tube Amplifiers (TWTAs), broadband klystrons, and the possibility of electron-beam modulation using Field Emitter Arrays (FEAs), there has been a resurgence of interest in understanding the electron beam-wave interaction when the beam and wave initially have multifrequency content. In the case of the klystron, this corresponds to understanding how the electron beam propagates a multitonal velocity or density disturbance. In a distributed interaction device like the TWTA the beam dynamics, and hence radiation dynamics, are further complicated since a beam electron interacts with a forcing field on the circuit for its entire trip through the device. We choose the simpler device, an idealized broadband klystron model, to study harmonic and intermodulation distortions in the multifrequency beam-wave system. The beam dynamics are modeled by nonlinear fluid equations, and the question of how the beam propagates general (multitonal) input conditions is addressed. Although the question has been considered before in some generality, we simplify the beam model and consider specifically multitonal excitation as would be used in a communications application
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