Convex Reformulations of Security and Uncertainty Constraints for Power System Optimization

A sustainable future with continuing technological and economic progress can only be achieved based on a renewable, reliable and affordable electricity supply. Renewable generation such as wind and solar power is inherently uncertain. Integrating it efficiently without compromising reliability is a key challenge to modern power system operations.

To address this issue, system operators and electricity markets increasingly rely on the optimal power flow (OPF), an optimization tool for identifying cost-efficient and secure dispatch decisions. To this end, various formulations of the OPF with different levels of modeling accuracy and complexity are used.

They differ in their representation of the nonlinear AC power flow equations and in the operational requirements they address. Most OPF algorithms include only parts of the numerous aspects of power system security and adequacy. As a result, there is still a lack of consensus on a fundamental definition of the target operating region (i.e., the feasible space), ensuring both power system security and adequacy.

There have been only few attempts to derive a comprehensive and tractable constraint set capturing the true target operating region for the various power flow modeling frameworks. This is primarily due to the immense complexity of the numerous requirements imposed on power system operations. Many power system security requirements such as dynamic security criteria cannot be directly expressed as linear or nonlinear constraints.

Static security constraints are in theory tractable. However, they often lead to a computationally prohibitive problem complexity. At the same time, higher shares of partially unpredictable renewable generation further call for modeling approaches that account for uncertainty. Additionally, the nonconvexity of the power flow equations still challenges current solution algorithms.

As a result, the electricity industry generally relies on linear approximations, which are computationally more robust. However, these approximations can be prone to severe errors. This in turn has prompted the development of more accurate relaxations and approximations. They preserve convexity and provide increased levels of computational robustness.

This thesis builds on these advancements and develops convex reformulations of security and uncertainty constraints with the aim of defining the target operating region for a wide range of OPF formulations. First, we address the challenge of translating static and dynamic security requirements to tractable constraints suitable for any OPF framework.

We derive decision trees to partition the nonconvex security region into subspaces and subsequently incorporate them in the optimization through integer programming and disjunctive constraints. This is the first work to combine decision tree based security assessments with integer programming, allowing us to bridge the gap between machine learning and traditional mathematical optimization.

We preserve the nonconvex reality of power systems in an efficient and robust way, even when using convex relaxations and approximations of the power flow equations. Larger regions of the secure operating region are accessible in the optimization, resulting in less conservative solutions than current approaches.

We show how the security constraints can be combined with linear and nonlinear formulations of the power flow equations, depending on their application in market or system operations. To address the uncertainty of renewable generation, we develop the first formulation of a chanceconstrained AC-OPF based on the convex second-order cone formulation of the power flow equations.

Second-order cone programming is one of the computationally most efficient methods for nonlinear systems. We leverage extended formulations of quadratic chance constraints to approximate the nonconvex confidence region more accurately – the portion of the operating region secure against the impact of forecast errors with a desired probability.

Coupled with anex-postAC feasibility recovery, our method identifies lower cost solutions in the confidence region which are not represented in current non convex formulations of the chance-constrained AC-OPF. To consider both security and uncertainty, we provide a unified definition of the target operating region of power systems.

For this purpose, we formulate the problem as a security- and chance-constrained OPF and propose two tractable and efficient approximations suitable for any OPF algorithm. Compared to solely considering preventive control actions, corrective control actions allow to enlarge the target operating region, thereby substantially reducing system costs.

They can be provided by controllable and fast-acting system components, such as high voltage direct current (HVDC) transmission lines. To this end, we develop optimization tools for incorporating and optimizing corrective control in response to forecast errors. We focus on HVDC lines as well as HVDC grids, which are very likely to emerge from already existing point-to-point links.

In both cases, optimized corrective control actions allow for a better coordination of generation resources and reduce the cost associated with an uncertain system operation.