  The power flow equations are fundamental to power system planning, analysis,
and control. However, the inherent non-linearity and non-convexity of these
equations present formidable obstacles in problem-solving processes. To
mitigate these challenges, recent research has proposed adaptive power flow
linearizations that aim to achieve accuracy over wide operating ranges. The
accuracy of these approximations inherently depends on the curvature of the
power flow equations within these ranges, which necessitates looking at
second-order sensitivities. In this paper, we leverage second-order
sensitivities to both analyze and improve power flow approximations. We
evaluate the curvature across broad operational ranges and subsequently utilize
this information to inform the computation of various sampling-based power flow
approximation techniques. Additionally, we leverage second-order sensitivities
to guide the development of rational approximations that yield linear
constraints in optimization problems. This approach is extended to enhance
accuracy beyond the limitations of linear functions across varied operational
scenarios.
