A Scalable Formulation for Look-Ahead Security-Constrained Optimal Power Flow

Lamia Varawala, Mohammad Reza Hesamzadeh, György Dán, Ross Baldick

Abstract: We consider the look-ahead security-const- rained optimal power flow (LASCOPF) problem under transmission line and generator contingencies. We first formulate LASCOPF under the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N-1$</tex-math></inline-formula> contingency criterion ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{LASCOPF}_1$</tex-math></inline-formula> ) using the dc power flow model. We observe that the number of decision variables in the comprehensive formulation increases quadratically with the number of look-ahead intervals, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> , making the problem infeasible to solve for large <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> . To overcome this, we propose the reduced LASCOPF problem ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{LASCOPF-r}_1$</tex-math></inline-formula> ) in which the number of decision variables increases only linearly with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$T$</tex-math></inline-formula> . Thereafter, we prove that, barring borderline cases, if <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{LASCOPF}_1$</tex-math></inline-formula> is feasible then the optimal solutions of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{LASCOPF}_1$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{LASCOPF-r}_1$</tex-math></inline-formula> are equivalent. We then extend our results to the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N-k$</tex-math></inline-formula> contingency criterion ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{LASCOPF-ru}_k$</tex-math></inline-formula> ) for any collection of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> contingencies, and we prove that the ordering of the contingencies does not affect the optimal solution. We then illustrate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{LASCOPF}_1$</tex-math></inline-formula> on a simple 2-bus 2-generator system. We show the numerical benefits of the proposed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text{LASCOPF-r}_1$</tex-math></inline-formula> formulation on the IEEE 118-bus, the IEEE 300-bus, and the 2383-bus Polish systems.