Nomenclature

$$U, P$$

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

$$K$$

$$N,L$$

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.