  Despite strong connections through shared application areas, research efforts
on power market optimization (e.g., unit commitment) and power network
optimization (e.g., optimal power flow) remain largely independent. A notable
illustration of this is the treatment of power generation cost functions, where
nonlinear network optimization has largely used polynomial representations and
market optimization has adopted piecewise linear encodings. This work combines
state-of-the-art results from both lines of research to understand the best
mathematical formulations of the nonlinear AC optimal power flow problem with
piecewise linear generation cost functions. An extensive numerical analysis of
non-convex models, linear approximations, and convex relaxations across
fifty-four realistic test cases illustrates that nonlinear optimization methods
are surprisingly sensitive to the mathematical formulation of piecewise linear
functions. The results indicate that a poor formulation choice can slow down
algorithm performance by a factor of ten, increasing the runtime from seconds
to minutes. These results provide valuable insights into the best formulations
of nonlinear optimal power flow problems with piecewise linear cost functions,
a important step towards building a new generation of energy markets that
incorporate the nonlinear AC power flow model.
