Uncertainty quantification of power systems with uncertain power input


In the near future, a significant portion of electricity generated in the world may come from renewable sources. Contrary to production of conventional fossil fuels, production of renewable (i.e., wind and solar) energy typically exhibits spatio-temporal variations and is ubiquitously uncertain. Therefore, the increasing share of power generation from renewable sources poses a fundamental challenge to the existing power grid in its ability to cope with uncertainty in power generation. One way to address this challenge is to employ numerical models to quantify the uncertainty and its impact and to assist decision making under uncertainty.
The dynamics of a power grid is governed by a set of nonlinear differential algebraic equations (DAEs). The fluctuations, induced by renewable sources, are usually modeled as random input parameters with distribution and correlation function obtained from the observed or modeled data.
We presented a novel method to compute the probabilistic density function (PDF) from the SODEs describing the dynamics of a generator with uncertain power input. We used the PDF method in combination with the LED closure to obtain a deterministic equation for the joint PDF of the generator phase angle and angular speed of a generator. We solved the resulting equation numerically. The accuracy of the LED closure and numerical solutions was verified via comparison with MCS.
 Fig 1. Dynamics of stochastic system with nonequilibrium initial condition. (Top) joint PDF at times t = 0, 0.1, 0.3, 0.5, and 0.7, along with the trajectory of θ and ω (the equilibrium point (θeq,ωeq) is marked by the red square); (middle) snapshots of the marginal PDFs pθ and pω computed using MCS and the PETSc solver of the PDF equation; (bottom) temporal evolution of θ and ω.
Peng Wang, professor, school of mathematics and system science, Beihang University, E-mail:
P. Wang, D. A. Barajas-Solano, E. Constantinescu, S. Abhyankar, D. Ghosh, B. F. Smith, Z. Huang, and A. M. Tartakovsky, “Probabilistic density function method for stochastic ODEs of power systems with uncertain power input”, SIAM/ASA Journal of Uncertainty Quantification, 2015.