The quadratic function shows a good fit with the measured heat rate vs. active power production (H-P curve) of fossil-based thermal units. The cost function is obtained by multiplying the heat rate by the fuel cost per unit heat rate (usually, $/MBtu) which is a constant for the same fuel. Consequently, a quadratic cost function is obtained. There are many other mathematical representations of the cost function, but the quadratic function is popularly used.

If we are talking about thermal units, the cost function depends on the type of the fuel used. Generally speaking, we search for the curve of best fit. Probably, one can get better fit with polynomials with higher degrees, but the optimization problem would be more complicated. Thus, we usually suffice with the quadratic function. Besides, in case we use the piecewise linear approximation of the cost function, then there is no need for the better fit in the first place.

One can use linearized approximation but it will not be accurate. One can use higher order polynomials but matter will get worsened from the point of view of solutions. Quadratic function offers us a very good approximation resulting in fairly ttaccurate results. Further it does not complicate the solution process much and leads acceptably accurate solutions.