Optimization Models and Decomposition Approaches for the Power System Operation under Uncertainty

Abstract

Power systems consist of generation, transmission, and distribution systems indicating the electricity network from its generation to the loads. Among various optimization problems in the system, the unit commitment (UC) problem is a fundamental problem that aims to minimize operation costs while meeting electricity demand by coordinating generation resources. To ensure effective and reliable operation, various uncertain factors such as renewable generation, load demand, and contingencies should appropriately be considered. However, since the computational burden increases as uncertain factors are incorporated into the optimization models, efficient models and solution approaches are needed. Thus, this dissertation aims to propose optimization models and decomposition methods to solve optimization problems in power system operation under uncertainty.

First, we investigate a novel modeling approach in two-stage stochastic programming, which is a widely used framework in power system operations under uncertainty. We propose a partition-based risk-averse two-stage stochastic program that mitigates the drawbacks of the traditional two-stage stochastic programs with finite support. In the model, a set of scenarios is partitioned into several groups a second-stage cost is represented as the expectation of conditional value-at-risks of costs for each scenario group. We propose decomposition methods based on column-and-constraint generation to solve the model exactly for a given partition. In addition, a scenario partitioning method to enable the risk level of the model to be close to a given target is devised, and partitioning schemes for combining it with the proposed column-and-constraint generation algorithm are proposed. Numerical experiments were performed that demonstrated the effectiveness of the proposed partitioning schemes and the efficiency of the proposed solution approach.

Next, we address single-unit commitment (1UC) problems, which arise when an individual power producer bids a generator's schedule to the deregulated electricity market. Especially, we devise efficient dynamic programming algorithms to solve 1UC problems under stochastic electricity prices, by extending the results in the deterministic counterpart. Furthermore, leveraging the efficient algorithms on 1UC problems, we present two unit decomposition frameworks to solve the general UC problem under stochastic net load, which include a novel decomposition that has not been proposed. We propose a total of four solution approaches based on Lagrangian relaxation or column generation and analyze the dual bounds obtained from the methods. Through the numerical experiments, we demonstrate the efficiency of the proposed algorithms on 1UC problems. In addition, we analyze various unit decomposition methods for the stochastic UC problems and emphasize the scalability of the proposed novel column generation method with regard to the number of scenarios.

Finally, we study optimization models for microgrid operation under stochastic islanding and net load, where a microgrid is a localized power system with various distributed energy resources having the distinguishing feature that can be operated in an islanded or connected mode. Although multistage stochastic optimization models can address the dynamics and probabilistic nature of uncertainty, they suffer from the curse of dimensionality in that the number of scenarios grows exponentially with regard to the number of time periods. To overcome this drawback, we propose scalable optimization models to operate a microgrid under both uncertain factors. To deal with uncertain islanding events, a replanning procedure with models that consider at most one upcoming islanding event is proposed. To incorporate stochastic net load, the range of generation is determined in addition to commitment decisions to reduce the size of the model. Numerical experiments demonstrate that practical-sized instances can be solved using the proposed models, whereas they cannot be solved using the standard multistage model. The results also demonstrate the effectiveness of the solutions from the proposed models in an environment that both stochastic factors sequentially realize.