I am reading a book on electricity cost modelling. I understand equation 2.7 below, which indicates that the total cost for an ith plant is a function of fixed cost(FC), fuel cost(FL), plant efficiency (af) and quantity of electricity produced (Q).
Equation 2.9 is a two-step piecewise cost function which describes the existence of two possible ranges of operation, and that producing above a threshold implies that there is an increase in the variable costs. I understand this too.
Equation 2.10 provides a more general functional form that represents a continuous and smooth version of a multiple-step piecewise linear cost function through a quadratic function. However, I do not understand how this was derived. The only difference between equation 2.7 and 2.10 seems to be the squaring of Q in equation 2.10.
My questions are:
- Why was Q squared?
- How was this done? What is the broader concept/principle through which this was done?
- In what situations can this be applied as I build new models in the future?