Derivation and generation of path-based valid inequalities for transmission expansion planning

Abstract

This paper seeks to solve the long-term transmission expansion planning problem in power systems more effectively by reducing the solution search space and the computational effort. The proposed methodology finds and adds cutting planes based on structural insights about bus angle differences along paths. Two lemmas and a theorem are proposed which formally establish the validity of these cutting planes onto the underlying mathematical formulations. These path-based bus angle difference constraints, which tighten the relaxed feasible region, are used in combination with branch-and-bound to find lower bounds on the optimal investment of the transmission expansion planning problem. This work also creates an algorithm that automates the process of finding and applying the most effective valid inequalities, resulting in significantly reduced testing and computational time. The algorithm is implemented in Python, using Gurobi to add constraints and solve the exact DCOPF-based transmission expansion problem. This paper uses two different-sized systems to illustrate the effectiveness of the proposed framework: the GOC 500-bus system and a modified Polish 2383-bus system.

Appendix

In order to generalize Theorem 1, we introduce new definitions. Given a path $\rho _k$, a line path $\ell _k$ is a sequence of exactly one line per corridor $(i,j) \in \rho _k$. The $k^{th}$ line in corridor (i, j) will be denoted (i, j, k) for the purposes of a line path. For example, in a network with 3 lines per corridor, the simple path $\rho =(1,2), (2,3)$ might have line paths $\ell _1 = (1,2,1), (2,3,3)$, $\ell _2 = (1,2,3), (2,3,3)$, or $\ell _3 = (1,2,2), (2,3,2)$. That is, $\ell _1$ is comprised of the first line from corridor (1, 2) and the third line from corridor (2, 3). In this basic case, there are 9 possible such line paths corresponding to the path $\rho$. Additionally, an established line path is a line path composed entirely of existing lines, hence it corresponds to a path composed of only established paths. Let $\mathcal {C}_\ell$ be the set of all line paths. Let $N_{e}(\ell _k)$ denote the number of candidate lines in the line path $\ell _k$, when $N_e$ is applied as a function to a line path instead of a path. Let $\mathbb {I}_{ijk}$ represent the indicator function for candidate lines (i.e., $\mathbb {I}_{ijk} = 1$ means that line (i, j, k) is a candidate line). Given the above definitions and notations, Theorem 2 follows immediately from Theorem 1.

Theorem 2

The following are valid inequalities for TEP, for all line paths $\ell _k \in \mathcal {C}_\ell$:

$$\begin{aligned} \vert \theta _n - \theta _m \vert \le CR(\ell _k) + \left( \overline{CR(\ell )} - CR(\ell _k) \right) \left( N_e \left( \ell _k \right) - \sum _{(i,j,r) \in \ell _k} \mathbb {I}_{ijr} y_{ij,r} \right) . \end{aligned}$$

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Furthermore, let $\mathcal {C}^0 \subseteq \mathcal {C}$ denote the set of paths comprised solely of established corridors, with $\ell _k^0$ denoting an element of this set. Additionally, let $\underline{CR(\ell ^0)} = \min \{ CR(\ell _k^0) \}$. If $\mathcal {C}^0$ is nonempty, then the above inequalities can be strengthened as follows:

$$\begin{aligned} \vert \theta _n - \theta _m \vert \le CR(\ell _k) + \left( \underline{CR(\ell ^0)} - CR(\ell _k) \right) \left( N_e \left( \ell _k \right) - \sum _{(i,j,r) \in \ell } \mathbb {I}_{ijr} y_{ij,r} \right) . \end{aligned}$$

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