[Paper Review] A Mixed-Integer QCQP Formulation of AC Power System Upgrade Planning Problems

Sandro Merkli|arXiv (Cornell University)|Oct 25, 2017

Electric Power System Optimization2 references3 citations

TL;DR

This paper presents a mixed-integer quadratically constrained quadratic program (QCQP) formulation for AC power system upgrade planning, transforming complex power flow and security constraints into a structured QCQP framework. By reformulating voltage, current, and power flow constraints using real and imaginary voltage components and line power flows, the method enables efficient mixed-integer optimization while preserving AC power system physics and enabling scalable solution via standard solvers.

ABSTRACT

This document describes the detailed reformulation of a power system upgrade planning problem into a more generic quadratically constrained quadratic problem (QCQP). The problem is one of deciding what lines to upgrade in an existing power system in order to enable it to handle specific load situations. The reformulation presented here is based on other formulations already used in the field, such as the ones presented in [1] and [2]. This document was created as a reference for publications by the author that refer to the planning problem but avoid outlining reformulation details.

Motivation & Objective

• To develop a generic, mathematically rigorous reformulation of AC power system upgrade planning as a mixed-integer QCQP.
• To enable the use of standard optimization solvers on power system planning problems by transforming non-convex, non-linear constraints into QCQP-compatible forms.
• To reduce computational complexity by avoiding exponential constraint generation through per-line upgrade case distinction instead of per-bus.
• To preserve the physical accuracy of AC power flow models while enabling mixed-integer decision variables for line upgrades.
• To provide a reusable, modular formulation for future research on transmission expansion and security-constrained planning.

Proposed method

• Reformulates the original mixed-integer non-convex problem into a standard QCQP form using real and imaginary parts of voltage variables (v_r, v_q) and line power flows (l_r, l_q).
• Expresses voltage magnitude constraints as quadratic forms in z, with Q_j matrices having 1s at (j,j) and (N+j,N+j), enabling direct inclusion in QCQP structure.
• Transforms current limits into quadratic constraints by squaring both sides and expressing the resulting inequality as z^T Q_jl z + m_jl^T a ≤ u_jl, where coefficients depend on upgrade options.
• Models line power flows using big-M constraints with binary indicators for each upgrade option, ensuring correct power flow equations are enforced only when the corresponding upgrade is selected.
• Introduces per-line case distinction (rather than per-bus) to avoid exponential growth in constraints, reducing complexity to linear in upgrade options per line.
• Uses auxiliary variables and bilinear terms in z and a to represent power flow expressions, with coefficients derived from Y_upg and I_max parameters under different upgrade scenarios.

Experimental results

Research questions

• RQ1 Can AC power system upgrade planning be reformulated as a mixed-integer QCQP while preserving the physical accuracy of AC power flow equations?
• RQ2 How can voltage and current limits be expressed as quadratic constraints in a mixed-integer optimization framework?
• RQ3 What is the most computationally efficient way to model line power flows under multiple upgrade options without exponential constraint growth?
• RQ4 Can big-M constraints be effectively used to select correct power flow equations based on binary upgrade decisions?
• RQ5 How can the reformulation maintain non-convexity while enabling solution via standard QCQP solvers?

Key findings

• The reformulation successfully transforms the original non-convex, mixed-integer, non-linear power system planning problem into a standard mixed-integer QCQP form.
• Voltage constraints are expressed as quadratic inequalities in the form z^T Q_j z ∈ [v_min,j², v_max,j²], directly compatible with QCQP solvers.
• Current limits are reformulated as quadratic constraints in z with linear terms in a, enabling efficient handling via QCQP solvers.
• Line power flow constraints are modeled using big-M techniques with per-upgrade-case constraints, reducing the number of constraints from exponential to linear in upgrade options per line.
• The method avoids per-bus case distinction, significantly reducing the number of constraints compared to alternative formulations.
• The final formulation maintains the full AC power flow physics while enabling scalable solution through standard optimization tools.