Economic Dispatch

We have three power plants each producing electricity to satisfy the demand of $s=3000$. Each of the power plants has an upper and lower capacity limit $u_i$ and $l_i$ and a quadratic model of the cost associated with its power output $a_i+b_i{P}_i+c_i{P}_i^2$. Numerical values for the capacity limits and cost coefficients are provided in the table bellow.

How much power should each power plant provide, to minimize the overall cost?

Mathematical Model

Always ON

Assuming all power plants must be powered on at all times, the problem can be modeled as

$$\begin{array}{rl}\underset{x_{1:3}}{\min }& \sum _{i=1}^3{a}_i+b_i{x}_i+c_i{x}_i^2\\ \text{s.t.}& l_i\le x_i\le u_i,\forall i\in \{1,\dots ,3\}\\ & \sum _{i=1}^3{x}_i=s\end{array}$$

ON/OFF

Lets now assume that we may also power on/off each power plant based on the demand. This necessitates the inclusion of additional binary decision variables $z_i$ in the problem

$$\begin{array}{rl}\underset{x_{1:3},z_{1:3}}{\min }& \sum _{i=1}^3{z}_i{a}_i+b_i{x}_i+c_i{x}_i^2\\ \text{s.t.}& z_i{l}_i\le x_i\le z_i{u}_i,\forall i\in \{1,\dots ,3\}\\ & \sum _{i=1}^3{x}_i=s\\ & z_i\in \{0,1\}.\end{array}$$

The problem remains quadratic but, due to the binary decision variables $z_i$, is now classified as a MIQP.