On the Tightening of the Standard SDP for Vertex Cover with ell1ell_1 Inequalities

We show that the integrality gap of the standard SDP for vc~on instances of $n$ vertices remains $2-o(1)$ even after the addition of emph{all} hypermetric inequalities. Our lower bound requires new insights into the structure of SDP solutions behaving like $ell_1$ metric spaces when one point is removed. We also show that the addition of all $ell_1$ inequalities eliminates any solutions that are not convex combination of integral solutions. Consequently, we provide the strongest possible separation between hypermetrics and $ell_1$ inequalities with respect to the tightening of the standard SDP for vc

Distributionally Robust and Structure Exploiting Algorithms for Power System Optimization Problems

The modern power systems are undergoing profound changes as the large-scale integration of renewable energy and increasingly close interconnection of regional power grids. The intermittent renewable sources are bringing significant uncertainties to system operation so that all the analysis and optimization tools for the power system steady-state operation must be able to consider and manage the uncertainties. The large-scale interconnection of power systems increases the difficulty in maintaining the synchronization of all generators and further raises the challenging problem of systematically design multiple local and wide-area controllers. In both steady-state and dynamical problems, the large-scale interconnection is increasing the problem scale and challenging the scalability of analysis, optimization and design algorithms. This thesis addresses the problems of power system operation optimization under uncertainties and control parameter optimization considering time delays. The contributions are as follows. This thesis proposes data-driven distributionally robust models and algorithms for unit commitment, energy-reserve-storage co-dispatch and optimal power flow problems based on novel ambiguity sets. The problem formulations minimize the expected operation costs corresponding to the worst-case distribution in the proposed ambiguity set while explicitly considers spinning reserve, wind curtailment, and load shedding. Distributionally robust chance constraints are employed to guarantee reserve adequacy and system steady-state security. The construction of ambiguity set is data-driven avoiding presumptions on the probability distributions of the uncertainties. The specific structures of the problem formulation are fully exploited to develop a scalable and efficient solution method. To improve the efficiency of the algorithms to solve the operation and control optimization problems, this thesis investigates computational techniques to exploit special problem structures, including sparsity, chordal sparsity, group symmetry and parallelizability. By doing so, this thesis proposes a sparsity-constrained OPF framework to solve the FACTS devices allocation problems, introduces a sparsity-exploiting moment-SOS approach to interval power flow (IPF) and multi-period optimal power flow (MOPF) problems, and develops a structure-exploiting delay-dependent stability analysis (DDSA) method for load frequency control (LFC). The power system stabilizers (PSS) and FACTS controllers can be employed improve system damping. However, when time delays are considered, it becomes more difficult to analyzing the stability and designing the controllers. This thesis further develops time-domain methods for analysis and synthesis of damping control systems involving time delays. We propose a model reduction procedure together with a condition to ensure the $\epsilon$-exponential stability of the full-order system only using the reduced close-loop system model, which provides a theoretical guarantee for using model reduction approaches. Then we formulate the damping control design as a nonlinear SDP minimizing a carefully defined $H_2$ performance metric. A path-following method is proposed to coordinately design multiple damping controllers