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Strengthening AC optimal power flow relaxation through offline bound tightening and linear approximations

Xiangxin An, Constance Crozier, Santanu S. Dey

arXiv:2610.32006v1math.OCcs.SYeess.SY

TL;DR

ACOPF is nonconvex, making globally optimal solutions challenging, while the strength of convex relaxations depends on their bounds and cuts. This paper computes reusable bounds and SOC-OA cuts offline with a demand-slack model for related instances. On PGLib instances, fully offline convexification yields a 4.03% average gap versus 3.49% for fully online convexification, with a 272× speedup.

  • Problem

    ACOPF’s nonconvexity makes global optimization challenging, while repeated instance-specific bound tightening is costly when many related cases must be solved.

    Source: §1 · pp. 2–3

  • Method

    The framework computes variable bounds and valid cuts offline over prescribed demand ranges using a demand-slack model, then reuses them for instances with the same topology.

    Source: §1 · p. 3, §5 · p. 14

  • Results

    4.03% average gap for offline bounds and cuts versus 3.49% for fully online convexification, with a 272× speedup; adding online SOC-OA reduces the offline gap to 3.65%.

    Source: §5 · p. 14

  • Takeaways & Limitations

    The reusable bounds and cuts capture most of online FBBT’s gap-closing effect and support repeated scenario evaluations within prescribed demand ranges.

    Source: §5 · p. 14

    Offline FBBT imposes no objective cutoff constraints so its bounds remain representative across instances whose generation cost functions may vary.

    Source: §3.2.3 · p. 8

Abstract

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The alternating-current optimal power flow (ACOPF) problem is fundamental to power system operation and planning, but its nonconvexity makes globally optimal solutions of large-scale instances computationally challenging. Convex relaxations provide tractable lower bounds for assessing solution quality and supporting global optimization, although their effectiveness depends strongly on relaxation strength. This paper proposes an offline convexification and bound tightening framework for constructing strong linear programming (LP) relaxations for families of related ACOPF instances. Rather than repeating the tightening procedure for each instance, we compute variable bounds and second-order cone outer-approximation (SOC-OA) cuts offline using a demand-slack model and reuse them for subsequent instances with the same topology and admissible demand variations. Computational results on PGLib instances show that the fully offline approach achieves an average gap of $4.32\%$, compared with $3.80\%$ for the fully online approach, while achieving an average speedup of approximately $272\times$. The results demonstrate that offline convexification and bound tightening can recover most of the relaxation improvement of online tightening while significantly reducing the computational burden for repeated ACOPF evaluations.

Parameters

The notation identifies generator costs and power outputs, network flows and admittances, demands, and operating limits used in the ACOPF formulation.

Parameters

  • Generator k’s active and reactive power are represented by p_g^k and q_g^k, and its generation cost by C_g^k(·).p_g^k and q_g^k have lower and upper operating limits.Source: Parameters · p. 2
  • The line-admittance terms distinguish the real and imaginary parts of from-side, from-to, to-from, and to-side admittances.The entries identify these respectively as G_ff, G_ft, G_tf, and G_tt.Source: Parameters · p. 2
  • Bus demands are active and reactive power, while generator outputs have lower and upper bounds and ramp-down and ramp-up limits.Bus voltage magnitude bounds and line thermal limits are also specified.Source: Parameters · p. 2

1 Introduction

ACOPF relaxations provide tractable lower bounds, but their strength depends on variable domains, and instance-specific tightening can be costly when many related cases must be solved. The introduction motivates and presents an offline framework that reuses tightening across demand-varying cases.

Motivation

  • Loose variable bounds can weaken convex envelopes and widen relaxation gaps, while tighter bounds can improve relaxation quality.Relaxation gaps measure the difference between the lower bound from a convex relaxation and an AC-feasible upper bound.Source: §1 · p. 2

Bound tightening

  • FBBT can improve LP, SOCP, and QC relaxations, but conventional optimization-based bounds are generally tied to one operating instance.Standard FBBT minimizes and maximizes selected variables over the relaxed feasible region; objective-based tightening adds an objective cutoff, a variant this paper distinguishes from FBBT.Source: §1 · pp. 2–3

Motivation

  • FBBT requires two auxiliary optimization problems per selected variable and may repeat them across iterations, making large-network tightening costly.Existing scalability approaches reduce the cost per instance but do not eliminate repeated tightening across related load instances.Source: §1 · p. 3
  • Repeated instance-specific tightening is especially burdensome when day-ahead operation, contingency analysis, planning, or multi-period optimization requires many similar ACOPF solves.Conventional approaches treat these instances independently, causing redundant computation.Source: §1 · p. 3

Contribution

  • Aquino et al. use demand flexibility to screen redundant constraints; this work instead uses it for reusable bounds and SOC outer-approximation cuts.The proposed framework targets ACOPF instances sharing a topology and varying demands.Source: §1 · p. 3
  • The framework tightens lifted voltage-related variables with FBBT and adds arctangent envelopes and SOC-OA cuts to strengthen an LP relaxation while retaining linear structure.It computes bounds and cuts offline over prescribed demand variations for reuse, and reports relaxation strength comparable to bounds generated from the nominal model.Source: §1 · p. 3
  • The generated bounds and cuts are globally valid over the targeted demand family and can be reused across scenarios; the framework can also extend to mixed-integer ACUC and ACOTS variants.The contribution separates expensive strengthening from subsequent online solution.Source: §1 · p. 3

2 Problem formulation

The formulation represents AC power flows with voltage-derived lifted variables, then relaxes nonlinear relationships to obtain conic and linear approximations. The LP uses McCormick envelopes, whose strength benefits from tighter variable bounds.

2 Problem formulation

  • Lifted variables compactly represent line flows that are nonlinear functions of voltage magnitudes and angle differences.The network is described by buses and transmission lines.Source: §2 · p. 3

2.1 Exact model

  • The exact single-period ACOPF minimizes total generator cost subject to power-flow equations, bus balances, generator and voltage limits, and line thermal limits.The lifted-variable relationships and voltage-angle differences enforce exact nonlinear relationships.Source: §2.1 · p. 4
  • The exact model includes power-flow constraints in both directions on each transmission line.These directional constraints are part of the branch-flow representation.Source: §2.1 · p. 4

2.2 SOCP relaxation

  • The SOCP relaxation replaces a nonlinear equality with a rotated second-order cone constraint and omits the voltage angle-difference constraint.It retains the linear flow, balance, generator, voltage, and thermal constraints.Source: §2.2 · p. 4

2.3 LP relaxation

  • The LP relaxation replaces quadratic and bilinear terms with McCormick-envelope approximations and auxiliary variables.These replacements include terms in the thermal limits and the lifted-variable relationship.Source: §2.3 · pp. 4–5
  • Tighter variable bounds strengthen McCormick envelopes, making the LP relaxation especially useful alongside bound tightening.The LP is a computationally efficient approximation of the ACOPF feasible region.Source: §2.3 · p. 5

3 Offline convexification and bound tightening Framework

The framework performs expensive bound tightening and cut generation offline over a prescribed demand range, then reuses the resulting bounds and cuts for eligible instances with the same topology. Its workflow bypasses online tightening and cut generation for each new demand instance.

Offline convexification and bound tightening Framework

  • The framework carries out bound tightening and cut generation once offline over prescribed demand ranges, storing bounds and valid cuts for future instances with the same topology.It is designed for related instances that differ mainly in demand profiles and generator cost functions.Source: §3 · p. 5
  • Historical demand profiles are aggregated into a demand-slack model, on which FBBT tightens bounds and SOC-OA generates linear inequalities tangent to SOC constraints.These outputs are reused in the online phase.Source: §3 · p. 5
  • For a new instance within the prescribed demand range and with the same topology, the online phase strengthens an initial LP using precomputed bounds and cuts, bypassing online tightening and cut generation.The overall workflow is identified as Figure 1.Source: §3 · p. 5, Figure 1 · p. 5

3.1 Linear approximation enhancement tools

The framework combines arctangent envelopes, used during offline bound tightening, with SOC outer-approximation cuts generated afterward. The cuts come from conic geometry and can be reused across instances while the associated line remains in service.

3.1 Linear approximation enhancement tools

  • The framework uses arctangent envelopes during offline bound tightening and generates SOC outer-approximation cuts afterward to strengthen the LP relaxation.The two tools serve different phases of the algorithm.Source: §3.1 · p. 5

3.1.1 Arctangent envelopes (AT)

  • Arctangent envelopes outer-approximate the arctangent constraint with planes and depend only on the bounds of c and s, not load demand.They are constructed over the bounded c-s region.Source: §3.1.1 · p. 6

3.1.2 SOC outer-approximation (SOC-OA)

  • SOC-OA iteratively adds linear cuts for violated conic constraints, progressively tightening the LP while keeping the cuts valid across instances when the line remains in service.The cuts approximate SOC constraints without imposing the conic constraints explicitly.Source: §3.1.2 · p. 6

3.2 Offline bound tightening

The offline bound-tightening framework focuses on lifted c and s variables, whose bounds affect the strength of McCormick and arctangent envelopes. It uses a demand-slack model to compute reusable bounds for related instances and adds parallel subproblems and selective envelope updates to manage computation.

3.2.1 Base algorithm

  • FBBT tightens selected c_ij and s_ij bounds by solving lower- and upper-bound auxiliary problems, then repeats until changes fall below tolerance or the iteration limit is reached.The method can require many optimization problems and must otherwise be rerun when operating conditions change.Source: §3.2.1 · p. 7

3.2.2 Demand slack model

  • The demand-slack model represents historical demand ranges for buses on a fixed topology, capturing load realizations within the prescribed active- and reactive-demand ranges.The range parameters are estimated from historical demand data.Source: §3.2.2 · pp. 7–8

3.2.3 Offline bound tightening algorithm

  • Offline FBBT computes reusable c and s bounds on the demand-slack model; parallel subproblems, updated McCormick envelopes, and selectively refreshed arctangent envelopes support the iterations.Objective cutoff constraints are omitted so bounds remain representative across varying generation cost functions.Source: §3.2.3 · pp. 8–9

3.3 Offline reusable SOC cut generation

SOC-OA cuts are generated offline on the demand-slack model using the offline tightened bounds. The stored cuts are reused for subsequent instances, avoiding online repetition of FBBT and cut generation.

3.3 Offline reusable SOC cut generation

  • Starting from the McCormick relaxation with offline bounds, the procedure adds arctangent envelopes, iteratively cuts violated SOC and rotated-SOC constraints, and stops when violations fall within tolerance.The resulting cuts are stored for future instances with the same topology.Source: §3.3 · p. 9
  • Stored SOC-OA cuts and offline bounds are applied to subsequent instances, while arctangent envelopes are regenerated online from each instance’s available c-s bounds.This avoids repeating FBBT and SOC cut generation online.Source: §3.3 · p. 9

4 Computational Results

Across PGLib instances, reusing offline bounds and cuts retains most of the fully online relaxation strength while sharply reducing runtime. Bound tightening yields large early reductions, and the demand-slack model supports reusable bounds across admissible demand variations.

4.1 Data and parameters settings

  • The tests use PGLib instances primarily below 1000 buses, with 100 perturbed-load scenarios per original instance and active demands sampled within ±10% of nominal.FBBT is capped at 10 iterations, with arctangent envelopes added at iteration two; SOC-OA is capped at 20 iterations.Source: §4.1 · pp. 9–10

4.2.1 Bounds evolution

  • The c–s bounding region shrinks by 55.26% after the first iteration and 67.59% after the second, then stabilizes near 68.14% after the third.The relaxation gap falls from 8.36% in the first iteration to 7.18% in the final round; the procedure terminates after nine iterations when improvement falls below its threshold.Source: §4.2.1 · pp. 10–11

4.3.1 Offline bounds and cuts generation comparison

  • The demand-slack model increases average c–s bounding-area reduction from 80.09% for the nominal model to 83.78%, while average runtime rises from 2489.76 to 4593.75 seconds.For case588-sdet and case793-goc, numerical issues stop nominal-model FBBT early; the slack model completes ten iterations and achieves 90.30% and 91.10% area reductions, respectively.Source: §4.3.1 · p. 11, Table 2 · p. 12

4.3.2 Comparison between offline and online convexification

  • 4.32% average gap and 9.42 seconds case-shifted SGM time for Offline Bounds + Offline Cuts compare with 3.80% and 2566.38 seconds for the fully online strategy, a roughly 272× speedup.The hybrid strategy records a 3.96% average gap and 23.62 seconds case-average SGM time, more than 108× faster than fully online.Source: Table 3 · p. 12, §4.3.2 · p. 13
  • Fully online FBBT accounts for 2665.88 seconds, or 99.42%, of its 2681.55-second average runtime, identifying repeated instance-specific tightening as the main bottleneck.Offline Bounds + Offline Cuts takes 8.58 seconds on average, while Offline Bounds + Online SOC-OA takes 18.61 seconds.Source: §4.3.2 · p. 14

5 Conclusion

The framework precomputes reusable bounds and valid cuts over prescribed demand ranges for related ACOPF instances. On PGLib, offline bounds and cuts achieve a 4.03% average gap versus 3.49% fully online, with a 272× speedup.

5 Conclusion

  • The framework combines bound tightening, arctangent-envelope enhancement, and SOC outer-approximation cut generation in an offline convexification approach for stronger ACOPF linear relaxations.A demand slack model spans prescribed load ranges; resulting bounds and valid cuts are reusable for future scenarios whose demands fall within those ranges.Source: §5 · p. 14
  • A 4.03% average gap with offline bounds and cuts is close to the fully online 3.49% gap, while delivering a 272× speedup; online SOC outer approximation reduces the gap to 3.65%.The demand slack model produces tightened bounds representative of the feasible c-s region across perturbed demand instances.Source: §5 · p. 14
  • Reusable bounds and cuts over prescribed demand ranges make the framework suited to repeated ACOPF evaluations, including multi-period planning and contingency analysis on the same network structure.The paper concludes that offline bounds capture most of online FBBT's gap-closing effect, while offline cuts further strengthen the LP relaxation.Source: §5 · p. 14
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