DC Formulation


\(U, P\)

set of generating units(nodes) and firms(portfolios).


set of loads.


set of power system nodes and lines.

\(c_u, k_u\)

Variable cost ($/MWh) and capacity(MW) of unit \(u\)


Fixed load (demand) at node \(j\).

\(\hat{c}_u, \hat{g}_u\)

Offered price-quantity pair of unit \(u (0\leq \hat{c}_u \leq \hat{C}_u).\)

\(g_u, \pi_u\)

Dispatch (MW) and profit ($/h) of unit \(u\).

\(x_{uk}, y_{uk}\)

Binaries which select the price-quantity strategy of each unit \(u\).


Set of equations which determine optimal economic dispatch.


in the candidate strategy combination and in the alternative in which portfolio \(p\) chooses strategy \(s\).


active power injection from node i to j, more specific it is the amount of power measured at node j.


\(\operatorname{Im} (1/Y_{ij})\) imaginary part of inverse admittance matrix (or, alternatively \(1/x_{ij}\), where \(x_{ij}\) is line reactance).


is the coefficient for loss modeling which is equal to \(\frac{R^{(i,j)}\cdot 4}{(V_i + V_j)^2}\)


resistance of line \((i,j)\).


nominal voltage at bus \(i\).

The Reformulation Technique Basics

The first case is \[XY = 0\] which is then represented as \[Xb + Y(1-b) = 0,\] which could be replaced by \[\begin{aligned} 0\leq X&\leq Mb\\ 0\leq Y&\leq M(1-b)\\ M&\rightarrow \infty.\end{aligned}\]

The second case is the \(XY\) as a term but when \[X = \sum_i a_i x_i + a_0, x_i\in \{0,1\}\] representation is allowed. Then \(XY\) can be expressed as \[\begin{aligned} XY &= \sum_i a_i z_i + a_0 Y, \\ z_i &= x_iY.\end{aligned}\]

The last expression could in turn be expressed as a pair \[\begin{aligned} z_i&\leq Y\\ 0\leq z_i &\leq x_i\cdot M.\end{aligned}\] However, this will work only for a certain type of the functional that will ensure that the maximum \(z_i\) value is always preferable. This inequalities could be tightened \[\begin{aligned} Y \leq &z_i + (1 - x_i) \cdot M \leq M \\ 0\leq &z_i \leq Y\end{aligned}\] which ensures that \(z_i = Y\) when \(x_i = 1\) and \(z_i = 0\) otherwise.