Sample-Based Power Flow Approximations: Computational Methods, Analysis, And Applications
Author(s)
Buason, Paprapee
Advisor(s)
Molzahn, Daniel K.
Editor(s)
Collections
Supplementary to:
Permanent Link
Abstract
The non-convex nature of the power flow equations poses a challenge for solving various power system optimization and control problems. To address these challenges, linear approximations are often employed. However, the accuracy of these linearizations can vary depending on the specific characteristics of the power system and the operational range in which they are applied. Existing linearizations typically rely on general assumptions that apply to broad classes of systems, which can limit their accuracy and result in constraint violations when applied to specific systems.
In contrast to these existing approaches, we introduce "conservative linear approximations" of the power flow equations. These conservative linearizations intentionally overestimate or underestimate quantities of interest, aiming to make algorithms more tractable while avoiding constraint violations. We compute these conservative linear approximations through a sample-based method, involving the solution of a constrained linear regression problem.
Additionally, we introduce a class of approximations based on rational functions with linear numerators and denominators. This choice is motivated by the resulting linear inequality constraints, making these approximations well-suited for optimization formulations, while still providing enhanced accuracy compared to linear functions.
We enhance the conservativeness and accuracy of our approximations through an iterative sampling method, optimizing these functions with respect to the relevant quantities. We also conduct a sample-complexity analysis. To further develop our approach, we establish an "importance sampling" method for constructing linear and conservative linear approximations. This method's objective is to efficiently improve approximation quality by selecting the most informative samples. It does so by drawing samples from a relatively low-dimensional subspace exhibiting high curvature. This approach allows us to obtain highly accurate linear approximations with significantly fewer samples than random selection. By examining the relationships between the voltage magnitudes and the active and reactive power injections, we characterize the performance of our proposed power flow approximations for a range of test cases.
Furthermore, we examine applications of conservative linear approximations to prove their effectiveness in an optimal sensor placement problem that we formulate as a bilevel program. In the optimal sensor placement problem, our goal is to place a minimal number of sensors and avoid false sensor alarms in the upper level while the lower level ensures that these sensors will detect any voltage violations. We replace the nonlinear power flow equations with conservative linear approximations to make the bilevel problem tractable. With conservative linear approximations, we can ensure that the resulting sensor locations and thresholds are sufficient to identify any constraint violations. Additionally, we apply various problem reformulations to significantly improve computational tractability while simultaneously ensuring an appropriate placement of sensors. Lastly, we improve the quality of the results via an approximate gradient descent method that adjusts the sensor thresholds. We demonstrate the effectiveness of our proposed method for several test cases, including a system with multiple switching configurations.
Numerical tests demonstrate that our power flow approximations enhance accuracy compared to other linear approximations and prove effective in optimization problems. In future research, we plan to leverage machine learning techniques for our power flow approximations, extend these methods to other parameters (e.g., current flows), and apply them to additional applications like capacity expansion planning problems.
Sponsor
Date
2023-12-07
Extent
Resource Type
Text
Resource Subtype
Dissertation