Iterative, Small-Signal L2 Stability Analysis of Nonlinear Constrained Systems

This paper provides a method to analyze the small-signal L2 gain of control-affine nonlinear systems on compact sets via iterative semi-definite programs (SDPs). First, a continuous piecewise affine (CPA) storage function and the corresponding upper bound on the L2 gain are found on a bounded, compact set's triangulation. Then, to ensure that the state does not escape this set, a (CPA) barrier function is found that is robust to small-signal inputs. Small-signal L2 stability then holds inside each sublevel set of the barrier function inside the set where the storage function was found. The bound on the inputs is also found while searching for a barrier function. The method's effectiveness is shown in a numerical example

Dynamic Energy Management

We present a unified method, based on convex optimization, for managing the power produced and consumed by a network of devices over time. We start with the simple setting of optimizing power flows in a static network, and then proceed to the case of optimizing dynamic power flows, i.e., power flows that change with time over a horizon. We leverage this to develop a real-time control strategy, model predictive control, which at each time step solves a dynamic power flow optimization problem, using forecasts of future quantities such as demands, capacities, or prices, to choose the current power flow values. Finally, we consider a useful extension of model predictive control that explicitly accounts for uncertainty in the forecasts. We mirror our framework with an object-oriented software implementation, an open-source Python library for planning and controlling power flows at any scale. We demonstrate our method with various examples. Appendices give more detail about the package, and describe some basic but very effective methods for constructing forecasts from historical data.Comment: 63 pages, 15 figures, accompanying open source librar

Perspective Cuts for the ACOPF with Generators

International audienceThe alternating current optimal power flow problem is a fundamental problem in the management of smart grids. In this paper we consider a variant which includes activation/deactivation of generators at some of the grid sites. We formulate the problem as a mathematical program, prove its NP-hardness w.r.t. ac-tivation/deactivation, and derive two perspective reformulations

Large-scale unit commitment under uncertainty: an updated literature survey

The Unit Commitment problem in energy management aims at finding the optimal production schedule of a set of generation units, while meeting various system-wide constraints. It has always been a large-scale, non-convex, difficult problem, especially in view of the fact that, due to operational requirements, it has to be solved in an unreasonably small time for its size. Recently, growing renewable energy shares have strongly increased the level of uncertainty in the system, making the (ideal) Unit Commitment model a large-scale, non-convex and uncertain (stochastic, robust, chance-constrained) program. We provide a survey of the literature on methods for the Uncertain Unit Commitment problem, in all its variants. We start with a review of the main contributions on solution methods for the deterministic versions of the problem, focussing on those based on mathematical programming techniques that are more relevant for the uncertain versions of the problem. We then present and categorize the approaches to the latter, while providing entry points to the relevant literature on optimization under uncertainty. This is an updated version of the paper "Large-scale Unit Commitment under uncertainty: a literature survey" that appeared in 4OR 13(2), 115--171 (2015); this version has over 170 more citations, most of which appeared in the last three years, proving how fast the literature on uncertain Unit Commitment evolves, and therefore the interest in this subject

Conic relaxations for power system state estimation with line measurements

This paper deals with the nonconvex power system state estimation (PSSE) problem, which plays a central role in the monitoring and operation of electric power networks. Given a set of noisy measurements, PSSE aims at estimating the vector of complex voltages at all buses of the network. This is a challenging task due to the inherent nonlinearity of power flows (PFs), for which the existing methods lack guaranteed convergence and theoretical analysis. Motivated by these limitations, we propose a novel convexification framework for the PSSE problem using semidefinite programming (SDP) and second-order cone programming (SOCP) relaxations. We first study a related PF problem as the noiseless counterpart, which is cast as a constrained minimization program by adding a suitably designed objective function. We study the performance of the proposed framework in the case where the set of measurements includes: 1) nodal voltage magnitudes and 2) branch active PFs over at least a spanning tree of the network. It is shown that the SDP and SOCP relaxations both recover the true PF solution as long as the voltage angle difference across each line of the network is not too large (e.g., less than 90^{\circ } for lossless networks). By capitalizing on this result, penalized SDP and SOCP problems are designed to solve the PSSE, where a penalty based on the weighted least absolute value is incorporated for fitting noisy measurements with possible bad data. Strong theoretical results are derived to quantify the optimal solution of the penalized SDP problem, which is shown to possess a dominant rank-one component formed by lifting the true voltage vector. An upper bound on the estimation error is also derived as a function of the noise power, which decreases exponentially fast as the number of measurements increases. Numerical results on benchmark systems, including a 9241-bus European system, are reported to corroborate the merits of the proposed convexification framework

Conic relaxations for power system state estimation with line measurements

This paper deals with the nonconvex power system state estimation (PSSE) problem, which plays a central role in the monitoring and operation of electric power networks. Given a set of noisy measurements, PSSE aims at estimating the vector of complex voltages at all buses of the network. This is a challenging task due to the inherent nonlinearity of power flows (PFs), for which the existing methods lack guaranteed convergence and theoretical analysis. Motivated by these limitations, we propose a novel convexification framework for the PSSE problem using semidefinite programming (SDP) and second-order cone programming (SOCP) relaxations. We first study a related PF problem as the noiseless counterpart, which is cast as a constrained minimization program by adding a suitably designed objective function. We study the performance of the proposed framework in the case where the set of measurements includes: 1) nodal voltage magnitudes and 2) branch active PFs over at least a spanning tree of the network. It is shown that the SDP and SOCP relaxations both recover the true PF solution as long as the voltage angle difference across each line of the network is not too large (e.g., less than 90^{\circ } for lossless networks). By capitalizing on this result, penalized SDP and SOCP problems are designed to solve the PSSE, where a penalty based on the weighted least absolute value is incorporated for fitting noisy measurements with possible bad data. Strong theoretical results are derived to quantify the optimal solution of the penalized SDP problem, which is shown to possess a dominant rank-one component formed by lifting the true voltage vector. An upper bound on the estimation error is also derived as a function of the noise power, which decreases exponentially fast as the number of measurements increases. Numerical results on benchmark systems, including a 9241-bus European system, are reported to corroborate the merits of the proposed convexification framework

Alternating Current Optimal Power Flow with Generator Selection

International audienceWe investigate a mixed-integer variant of the alternating current optimal flow problem. The binary variables activate and deactivate power generators installed at a subset of nodes of the electrical grid. We propose some formulations and a mixed-integer semidefinite programming relaxation, from which we derive two mixed-integer diagonally dominant programming approximation (inner and outer, the latter providing a relaxation). We discuss dimensionality reduction methods to extract solution vectors from solution matrices, and present some computational results showing how both our approximations provide tight bounds