DC Formulation

Nomenclature

\(U, P\)

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

\(K\)

set of loads.

\(N,L\)

set of power system nodes and lines.

\(c_u, k_u\)

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

\(d_j\)

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 \(u\).

\(E^0\)

Set of equations which determine optimal economic dispatch.

\(E^{(p,s)}\)

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

\(p_{ij}\)

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

\(B_{ij}\)

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

\(k^{(i,j)}\)

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

\(R^{(i,j)}\)

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

\(V_i\)

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.