We present sufficient conditions under which thermal generators can be aggregated in mixed-integer linear programming (MILP) formulations of the unit commitment (UC) problem, while maintaining feasibility and optimality for the original disaggregated problem. Aggregating thermal generators with identical characteristics (e.g., minimum/maximum power output, minimum up/down time, and cost curves) into a single unit reduces redundancy in the search space induced by both exact symmetry (permutations of generator schedules) and certain classes of mutually nondominated solutions. We study the impact of aggregation on two large-scale UC instances: one from the academic literature and the other based on real-world operator data. Our computational tests demonstrate that, when present, identical generators can negatively affect the performance of modern MILP solvers on UC formulations. Furthermore, we show that our reformation of the UC MILP through aggregation is an effective method for mitigating this source of computational difficulty.
Electricity markets rely on the rapid solution of the optimal power flow (OPF) problem to determine generator power levels and set nodal prices. Traditionally, the OPF problem has been formulated using linearized, approximate models, ignoring nonlinear alternating current (AC) physics. These approaches do not guarantee global optimality or even feasibility in the real ACOPF problem. We introduce an outer-approximation approach to solve the ACOPF problem to global optimality based on alternating solution of upper- and lower-bounding problems. The lower-bounding problem is a piecewise relaxation based on strong second-order cone relaxations of the ACOPF, and these piecewise relaxations are selectively refined at each major iteration through increased variable domain partitioning. Our approach is able to efficiently solve all but one of the test cases considered to an optimality gap below 0.1%. Furthermore, this approach opens the door for global solution of MINLP problems with AC power flow equations.
Increasing penetration levels of renewables have transformed how power systems are operated. High levels of uncertainty in production make it increasingly difficulty to guarantee operational feasibility; instead, constraints may only be satisfied with high probability. We present a chance-constrained economic dispatch model that efficiently integrates energy storage and high renewable penetration to satisfy renewable portfolio requirements. Specifically, we require that wind energy contribute at least a prespecified proportion of the total demand and that the scheduled wind energy is deliverable with high probability. We develop an approximate partial sample average approximation (PSAA) framework to enable efficient solution of large-scale chance-constrained economic dispatch problems. Computational experiments on the IEEE-24 bus system show that the proposed PSAA approach is more accurate, closer to the prescribed satisfaction tolerance, and approximately 100 times faster than standard sample average approximation. Finally, the improved efficiency of our PSAA approach enables solution of a larger WECC-240 test system in minutes.
We describe pyomo.dae, an open source Python-based modeling framework that enables high-level abstract specification of optimization problems with differential and algebraic equations. The pyomo.dae framework is integrated with the Pyomo open source algebraic modeling language, and is available at http://www.pyomo.org. One key feature of pyomo.dae is that it does not restrict users to standard, predefined forms of differential equations, providing a high degree of modeling flexibility and the ability to express constraints that cannot be easily specified in other modeling frameworks. Other key features of pyomo.dae are the ability to specify optimization problems with high-order differential equations and partial differential equations, defined on restricted domain types, and the ability to automatically transform high-level abstract models into finite-dimensional algebraic problems that can be solved with off-the-shelf solvers. Moreover, pyomo.dae users can leverage existing capabilities of Pyomo to embed differential equation models within stochastic and integer programming models and mathematical programs with equilibrium constraint formulations. Collectively, these features enable the exploration of new modeling concepts, discretization schemes, and the benchmarking of state-of-the-art optimization solvers.
We describe experiments concerning enhancing a simple, yet effective method to compute high-accuracy prediction intervals (PIs) for day-ahead wide area wind power forecasts. The resulting PIs are useful for operators and traders, to improve reliability, anticipate threats, and increase situational awareness. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under Contract DE-AC04-94-AL85000. This work was funded by the Bonneville Power Administration (BPA).
Optimizing thermal generation commitments and dispatch in the presence of high penetrations of renewable resources such as solar energy requires a characterization of their stochastic properties. In this paper, we describe novel methods designed to create day-ahead, wide-area probabilistic solar power scenarios based only on historical forecasts and associated observations of solar power production. Each scenario represents a possible trajectory for solar power in next-day operations with an associated probability computed by algorithms that use historical forecast errors. Scenarios are created by segmentation of historic data, fitting non-parametric error distributions using epi-splines, and then computing specific quantiles from these distributions. Additionally, we address the challenge of establishing an upper bound on solar power output. Our specific application driver is for use in stochastic variants of core power systems operations optimization problems, e.g., unit commitment and economic dispatch. These problems require as input a range of possible future realizations of renewables production. However, the utility of such probabilistic scenarios extends to other contexts, e.g., operator and trader situational awareness. We compare the performance of our approach to a recently proposed method based on quantile regression, and demonstrate that our method performs comparably to this approach in terms of two widely used methods for assessing the quality of probabilistic scenarios: the Energy score and the Variogram score.
The solution of the Optimal Power Flow (OPF) and Unit Commitment (UC) problems (i.e., determining generator schedules and set points that satisfy demands) is critical for efficient and reliable operation of the electricity grid. For computational efficiency, the alternating current OPF (ACOPF) problem is usually formulated with a linearized transmission model, often referred to as the DCOPF problem. However, these linear approximations do not guarantee global optimality or even feasibility for the true nonlinear alternating current (AC) system. Nonlinear AC power flow models can and should be used to improve model fidelity, but successful global solution of problems with these models requires the availability of strong relaxations of the AC optimal power flow constraints. In this paper, we use McCormick envelopes to strengthen the well-known second-order cone (SOC) relaxation of the ACOPF problem. With this improved relaxation, we can further include tight bounds on the voltages at the reference bus, and this paper demonstrates the effectiveness of this for improved bounds tightening. We present results on the optimality gap of both the base SOC relaxation and our Strengthened SOC (SSOC) relaxation for the National Information and Communications Technology Australia (NICTA) Energy System Test Case Archive (NESTA). For the cases where the SOC relaxation yields an optimality gap more than 0.1 %, the SSOC relaxation with bounds tightening further reduces the optimality gap by an average of 67 % and ultimately reduces the optimality gap to less than 0.1 % for 58 % of all the NESTA cases considered. Stronger relaxations enable more efficient global solution of the ACOPF problem and can improve computational efficiency of MINLP problems with AC power flow constraints, e.g., unit commitment.
Forecasts of available wind power are critical in key electric power systems operations planning problems, including economic dispatch and unit commitment. Such forecasts are necessarily uncertain, limiting the reliability and cost-effectiveness of operations planning models based on a single deterministic or “point” forecast. A common approach to address this limitation involves the use of a number of probabilistic scenarios, each specifying a possible trajectory of wind power production, with associated probability. We present and analyze a novel method for generating probabilistic wind power scenarios, leveraging available historical information in the form of forecasted and corresponding observed wind power time series. We estimate nonparametric forecast error densities, specifically using epi-spline basis functions, allowing us to capture the skewed and nonparametric nature of error densities observed in real-world data. We then describe a method to generate probabilistic scenarios from these basis functions that allows users to control for the degree to which extreme errors are captured. We compare the performance of our approach to the current state-of-the-art considering publicly available data associated with the Bonneville Power Administration, analyzing aggregate production of a number of wind farms over a large geographic region. Finally, we discuss the advantages of our approach in the context of specific power systems operations planning problems: stochastic unit commitment and economic dispatch. Our methodology is embodied in the joint Sandia–University of California Davis Prescient software package for assessing and analyzing stochastic operations strategies.
We observe suitably located energy storage systems are able to collect significant revenue through spatiotemporal arbitrage in congested transmission networks. However, transmission capacity expansion can significantly reduce or eliminate this source of revenue. Investment decisions by merchant storage operators must, therefore, account for the consequences of potential investments in transmission capacity by central planners. This paper presents a tri-level model to co-optimize merchant electrochemical storage siting and sizing with centralized transmission expansion planning. The upper level takes the merchant storage owner's perspective and aims to maximize the lifetime profits of the storage, while ensuring a given rate of return on investments. The middle level optimizes centralized decisions about transmission expansion. The lower level simulates market clearing. The proposed model is recast as a bi-level equivalent, which is solved using the column-and-constraint generation technique. A case study based on a 240-bus, 448-line testbed of the Western Electricity Coordinating Council interconnection demonstrates the usefulness of the proposed tri-level model.