A leader-followers model of power transmission capacity expansion in a market driven environment

Abstract

We introduce a model for analyzing the upgrade of the national transmission grid that explicitly accounts for responses given by the power producers in terms of generation unit expansion. The problem is modeled as a bilevel program with a mixed integer structure in both upper and lower level. The upper level is defined by the transmission company problem which has to decide on how to upgrade the network. The lower level models the reactions of both power producers, who take a decision on new facilities and power output, and Market Operator, which strikes a new balance between demand and supply, providing new Locational Marginal Prices. We illustrate our methodology by means of an example based on the Garver’s 6-bus Network.

Appendix

Proof

of Proposition 1 The proposition can be proved by making considerations on the structure of the Karush–Kuhn–Tucker conditions for the modified MO problem. We report the complementarity conditions of such system

$$\begin{aligned} \begin{aligned}&\left( \pi _{zt}- \sigma ^{E}_{ikt} - b_{ikt}\right) q_{ikt}=0&\qquad i \in I, t \in T, z \in Z, k \in K^E_{iz} \\&\left( \pi _{zt}- \sigma ^{C}_{ikt} -b_{ikt}\right) q_{ikt}=0&\qquad i \in I, t \in T, z \in Z, k \in K^C_{iz} \\&\left( q_{ikt} - \varGamma ^E_{ik} \right) \sigma ^{E}_{ikt}=0&\qquad i \in I, t \in T, z \in Z, k \in K^E_{iz} \\&\left( q_{ikt} - \varGamma ^C_{ik}Y_{ik} \right) \sigma ^{C}_{ikt}=0&\qquad i \in I, t \in T, z \in Z, k \in K^C_{iz} \\&\left( \overline{TR}_{lt}-TR_{lt}\right) \eta ^+_{lt}=0&\qquad l \in L^E \cup L^C, t \in T \\&\left( TR_{lt}-\underline{TR}_{lt}\right) \eta ^-_{lt}=0&\qquad l \in L^E \cup L^C, t \in T \\&\left( \overline{\theta }_z-\theta _{zt}\right) \kappa ^+_{zt}=0&\qquad z \in Z, t \in T \\&\left( \theta _{zt}-\underline{\theta }_{z}\right) \kappa ^-_{zt}=0&\qquad z \in Z, t \in T \end{aligned} \end{aligned}$$

(105)

Let us first prove that the margin of every GenCo is non negative. From the first two equations of the complementarity system (105) one has

$$\begin{aligned} \pi _{zt}q_{ikt} = \sigma ^{E}_{ikt}q_{ikt} + b_{ikt}q_{ikt} \end{aligned}$$

$$\begin{aligned} \pi _{zt}q_{ikt} = \sigma ^{C}_{ikts}q_{ikt} + b_{ikt}q_{ikt} \end{aligned}$$

with $\sigma ^{E}_{ikt}q_{ikt}, \sigma ^{C}_{ikt}q_{ikt} \ge 0$.

This implies that

$$\begin{aligned} \pi _{zt}q_{ikt} \ge b_{ikt}q_{ikt} \end{aligned}$$

and since by assumption $b_{ikt} \ge c_{ik}$, one has that

$$\begin{aligned} \pi _{zt}q_{ikt} \ge c_{ik}q_{ikt} \end{aligned}$$

therefore the margin is non negative, which means that each GenCo is willing to increase its production until either the maximum accepted bid or a capacity bound is reached.

Now we show the main claim of the proposition. The power produced by each GenCo cannot be such that $\tilde{q}_{ikt} > q_{ikt}$ and it can be $\tilde{q}_{ikt} < q_{ikt}$ if and only if $\tilde{q}_{ikt}=\varGamma ^E_{ik}$ for existing generation units or $\tilde{q}_{ikt}=\varGamma ^C_{ik}Y_{ik}$ for candidate generation units. But then it would be $q_{ikt} > \varGamma ^E_{ik}$ or $q_{ikt} > \varGamma ^C_{ik}Y_{ik}$ which is not feasible for the modified MO Problem. Therefore, once the MO solves the aforementioned problem it must be $\tilde{q}_{ikt}=q_{ikt}$ for each GenCo. $\square$