Hence, we consider a family nonlinear dynamic systems, parameterized by the slow variation in demand, driven by a white noise input. Therefore, we can no longer think of the system state as sitting precisely on a (deterministic) stable equilibrium. Rather, one has a diffusion process, in which the state wanders in the basin of attraction of the deterministically stable point, with the possibility of stochastic exit from this basin. Results of Wentzell-Freidlin show that in the limit of small noise, the expected exit time and points of exit are closely related to a quasi-potential function. Roughly speaking, this quasi-potential may be thought of as a "cost of control" in an optimal control problem, in which the stochastic input tries to drive the state out of the basin with least energy.
In the power system application, we will show that this quasi-potential can be identified in closed form, and that it has close association to the interconnection structure of the network. Moreover, we observe that as the slowly moving average demand in a network is increased parametrically, the closest saddle point that allows exit from the stable basin tends to drop to lower and lower potential. The nature of the states at this saddle point suggest a stochastic loss of stability mechanism in which electric voltage magnitudes in the network rapidly decline. This correlates well to a physically observed loss of stability phenomena known as "voltage collapse." The close association of the quasi-potential to the network interconnection structure gives insights into effective load reduction and grid re-enforcement strategies for avoiding this type of catastrophic failure.
Time and Place: Wed., Sept. 27 3:30-4:30 pm in 4610 Engr. Hall.
SYSTEMS SEMINAR WEB PAGE: http://www.cae.wisc.edu/~gubner/seminar/