\(Min:\ \ J(P_G,P_D)=P_{Base}[P_G\cdot(a^T\cdot P_G+b) \ \ \ \\ \ \ +C_T^D\cdot(P_D^{\max}-P_D)]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)\)
\[s.t.:\ B\delta=P_G-P_D-P_{tie}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)\]
\[-P_i^{\max}\le H\delta\le P_i^{\max}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)\] \[P_G^{\min}\le P_G^{ }\le P_G^{\max}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(4\right)\] \[P_D^{\min}\le P_D^{ }\le P_D^{\max}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(5\right)\]
In which \(J(P_G,P_D)\) represents total exploitation cost ($/h) , \(P_{Base}\) is base active power (MW), \(C_D^T\) is the transposed of suggested vector of cost load ($/MWh) , a and b are constant coefficients vector in the generator price suggestion function, \(P_G\) and \(P_D\) are vectors of generators output active power and active loads in perunit (P.U.) (these vectors are the output of Optimal Power Flow), \(P_{tie}\) is the output power vector from the studied area to other areas in P.U. , and \(B\) is the linearized Jacobian matrix to P.U.. The term H is the linear matrix of the passing flow from lines to P.U. and \(\delta\) is the buses' voltage angle vector in radian. Objective function of Eq. (1) shows the total exploitation cost. The first part of this equation shows the exploitation cost of generators and the second part shows the deficiency cost of load. Equation (2) is for DC load distribution. The passing power limitation from lines is shown in equation (3). Equations (4) and (5) show generation limits and load limits, respectively. Losses are deleted in this model. Second-order optimization programing method can be used to solve this problem.