An Efficient Margin and Sensitivity Analysis Based Method For Calculating Voltage Collapse

This paper concerns compute and exploiting the sensitivity of the loading margin to voltage collapse with respect to various parameters. The main idea of this paper is that after the loading margin has been computed for nominal parameters, the effect on the loading margin of altering the parameters can be predicted by Taylor series estimates. Loading margin is a fundamental measure of proximity to voltage collapse. Linear and quadratic estimates to the variation of the loading margin with respect to any system parameter or control are derived. The accuracy of the estimates over a useful range and the ease of obtaining the linear estimate suggest that this method will be of practical value in avoiding voltage collapse.

Extrapolation of the functional calculus of generalized Dirac operators and related embedding and Littlewood-Paley-type theorems. I

Several rather general sufficient conditions for the extrapolation of the calculus of generalized Dirac operators from L2 to Lp are established. As consequences, we obtain some embedding theorems, quadratic estimates and Littlewood–Paley theorems in terms of this calculus in Lebesgue spaces. Some further generalizations, utilised in Part II devoted to applications, which include the Kato square root model, are discussed. We use resolvent approach and show the irrelevance of the semigroup one. Auxiliary results include a high order counterpart of the Hilbert identity, the derivation of new forms of ‘off-diagonal’ estimates, and the study of the structure of the model in Lebesgue spaces and its interpolation properties. In particular, some coercivity conditions for forms in Banach spaces are used as a substitution of the ellipticity ones. Attention is devoted to the relations between the properties of perturbed and unperturbed generalized Dirac operators. We do not use any stability results.

Heat kernel estimates and functional calculi of $-b \Delta$

We show that the elliptic operator ${\mathcal L} = - b(x) \Delta$ has a bounded $H^\infty$ functional calculus in $L^p(\boldsymbol R^n), 1 < p < \infty$, where $b$ is a bounded measurable complex-valued function with positive real part. In the process, we prove quadratic estimates for ${\mathcal L}$, and obtain bounds with fast decay and Hölder continuity estimates for $k_t(x,y) b(y)$ and its gradient, where $k_t(x,y)$ is the heat kernel of $-b(x) \Delta$. This implies $L^p$ regularity of solutions to the parabolic equation $\partial_t u + {\mathcal L} u = 0$.