Two-stage quadratic games under uncertainty and their solution by progressive hedging algorithms

A model of a two-stage N-person noncooperative game under uncertainty is studied, in which at the first stage each player solves a quadratic program parameterized by other players’ decisions and then at the second stage the player solves a recourse quadratic program parameterized by the realization of a random vector, the second-stage decisions of other players, and the first-stage decisions of all players. The problem of finding a Nash equilibrium of this game is shown to be equivalent to a stochastic linear complementarity problem. A linearly convergent progressive hedging algorithm is proposed for finding a Nash equilibrium if the resulting complementarity problem is monotone. For the nonmonotone case, it is shown that, as long as the complementarity problem satisfies an additional elicitability condition, the progressive hedging algorithm can be modified to find a local Nash equilibrium at a linear rate. The elicitability condition is reminiscent of the sufficient second-order optimality condition in nonlinear programming. Various numerical experiments indicate that the progressive hedging algorithms are efficient for mid-sized problems. In particular, the numerical results include a comparison with the best response method that is commonly adopted in the literature

A prediction-correction ADMM for multistage stochastic variational inequalities

The multistage stochastic variational inequality is reformulated into a variational inequality with separable structure through introducing a new variable. The prediction-correction ADMM which was originally proposed in [B.-S. He, L.-Z. Liao and M.-J. Qian, J. Comput. Math., 24 (2006), 693--710] for solving deterministic variational inequalities in finite dimensional spaces is adapted to solve the multistage stochastic variational inequality. Weak convergence of the sequence generated by that algorithm is proved under the conditions of monotonicity and Lipschitz continuity. When the sample space is a finite set, the corresponding multistage stochastic variational inequality is defined on a finite dimensional Hilbert space and the strong convergence of the sequence naturally holds true. A numerical example in that case is given to show the efficiency of the algorithm

Study Of Stochastic Market Clearing Problems In Power Systems With High Renewable Integration

Integrating large-scale renewable energy resources into the power grid poses several operational and economic problems due to their inherently stochastic nature. The lack of predictability of renewable outputs deteriorates the power grid’s reliability. The power system operators have recognized this need to account for uncertainty in making operational decisions and forming electricity pricing. In this regard, this dissertation studies three aspects that aid large-scale renewable integration into power systems. 1. We develop a nonparametric change point-based statistical model to generate scenarios that accurately capture the renewable generation stochastic processes; 2. We design new pricing mechanisms derived from alternative stochastic programming formulations of the electricity market clearing problem under uncertainty; 3. We devise a novel approach to coordinate strategic operations of multiple noncooperative system operators. The current industry practices are based on deterministic models that do not account for the stochasticity of renewable energy. Therefore, the solutions obtained from these deterministic models will not provide accurate measurements. Stochastic programming (SP) can accommodate the stochasticity of renewable energy by considering a set of possible scenarios. However, the reliability of the SP model solution depends on the accuracy of the scenarios. We develop a nonparametric statistical simulation method to develop scenarios for wind generation using wind speed data. In this method, we address the nonstationarity issues that come with wind-speed time-series data using a nonparametric change point detection method. Using this approach, we retain the covariance structure of the original wind-speed time series in all the simulated series. With an accurate set of scenarios, we develop alternative two-stage SP models for the two-settlement electricity market clearing problem using different representations of the non-anticipativity constraints. Different forms of non-anticipativity constraints reveal different hidden dual information inside the canonical two-stage SP model, which we use to develop new pricing mechanisms. The new pricing mechanisms preserve properties of previously proposed pricing mechanisms, such as revenue adequacy in expectation and cost recovery in expectation. More importantly, our pricing mechanisms can guarantee cost recovery for every scenario. Furthermore, we develop bounds for the price distortion under every scenario instead of the expected distortion bounds. We demonstrate the differences in prices obtained from the alternative mechanisms through numerical experiments. Finally, we discuss the importance of distributed smart grid operations inside the power grid. We develop an information and electricity exchange system among multiple distribution systems. These distribution systems participate/compete in common markets cohere electricity is exchanged. We develop a standard Nash game treating each distribution system (DS) as an individual player who optimizes their strategies separately. We develop proximal best response (BR) schemes to solve this problem. We present results from numerical experiments conducted on three and six DS settings

The Mathematics and Statistics of Quantitative Risk Management

It was the aim of this workshop to gather a multidisciplinary and international group of scientists at the forefront of research in areas related to the mathematics and statistics of quantitative risk management. The main objectives of this workshop were to break down disciplinary barriers that often limit collaborative research in quantitative risk management, and to communicate the state of the art research from the different disciplines, and to point towards new directions of research

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

Transmission Expansion Planning for Large Power Systems

abstract: Transmission expansion planning (TEP) is a complex decision making process that requires comprehensive analysis to determine the time, location, and number of electric power transmission facilities that are needed in the future power grid. This dissertation investigates the topic of solving TEP problems for large power systems. The dissertation can be divided into two parts. The first part of this dissertation focuses on developing a more accurate network model for TEP study. First, a mixed-integer linear programming (MILP) based TEP model is proposed for solving multi-stage TEP problems. Compared with previous work, the proposed approach reduces the number of variables and constraints needed and improves the computational efficiency significantly. Second, the AC power flow model is applied to TEP models. Relaxations and reformulations are proposed to make the AC model based TEP problem solvable. Third, a convexified AC network model is proposed for TEP studies with reactive power and off-nominal bus voltage magnitudes included in the model. A MILP-based loss model and its relaxations are also investigated. The second part of this dissertation investigates the uncertainty modeling issues in the TEP problem. A two-stage stochastic TEP model is proposed and decomposition algorithms based on the L-shaped method and progressive hedging (PH) are developed to solve the stochastic model. Results indicate that the stochastic TEP model can give a more accurate estimation of the annual operating cost as compared to the deterministic TEP model which focuses only on the peak load.Dissertation/ThesisPh.D. Electrical Engineering 201