Kernel Structure Design for Data-Driven Probabilistic Load Flow Studies

A Framework for Analytical Power Flow Solution Using Gaussian Process Learning

“…By constraining the injection space to the feasible power flow region, we understand that voltage is a smooth function of injections. Therefore, smooth kernels such as square exponential kernel, is extensively used for power flow learning [15]- [17], [20]. It is important to note that using a kernel function k : R 2|B| → R will imply that GP learning via log-likelihood maximization is done over a 2|B| dimensional injection space.…”

“…It is important to note that using a kernel function k : R 2|B| → R will imply that GP learning via log-likelihood maximization is done over a 2|B| dimensional injection space. Although, [20] works with one dimensional sub-kernel (k r in equation ( 3) of [20]), the effective kernel dimension is much larger [20].Note that the additive GP idea considers that a high-dimensional function can be partitioned in low-dimensional sub-functions [18].…”

“…By observing the power flow equations, we know that such correlations exist via admittance matrix and quadratic voltage multiplication terms in power flow [28]. Authors in [20] attempted to solve this issue by searching a kernel among higher-order kernels, making kernel design expensive. Thus, our objective of kernel design is to design an additive kernel using i) sub-kernels working over a small subset of injection vector, ii) capture injection correlation effect on voltage, and iii) determine its structure without solving a large optimization problem.…”

“…Each sub-kernel operates on a specific subset of input variables or features, and their combination yields the final kernel, allowing for a fast and flexible representation of the data. In [20], authors apply this idea of additive GP to improve the learning performance for probabilistic power flow. The kernel design problem is formulated as a bi-level optimization problem, which is NP-hard in nature.…”

“…However, AL (and BO) suffers from curse-of-dimensionality as search space grows exponentially with dimensions of input, which is the number of injections. Standard GP [17] and higher-order additive kernel GP [20] are hence not suitable for AL. Based on aforementioned limitations, it is clear that for large-scale power flow approximations, we require a kernel design that can be expressed using lower-dimensional input sets, exhibits lower sample complexity, and yet captures the correlated injection effects.…”