Second-Order-Cone Formulations of Power Flow for Topology Optimization

James Luedkte; Line Roald; Noah Rhodes

REVIEW 4 major objections 5 minor 24 references

Second-Order-Cone Formulations of Power Flow for Topology Optimization

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper shows that an exact reformulation of the voltage-cone constraint in the second-order-cone optimal power shutoff problem cuts solve time from about 13,500 seconds to about 46 seconds on the 57-bus case, while fully linear…

desk verdict An exact single-cone reformulation that is genuinely useful and a plausible negative result, but the AC-deliverability claim is not fully supported and two defining equations appear mis-specified. read the letter →

arxiv 2506.02234 v1 pith:NSWB4TDV submitted 2025-06-02 eess.SY cs.SY

Noah Rhodes , James Luedkte , Line Roald This is my paper

classification eess.SYcs.SY

keywords optimalpowershutoffsecond-orderconeprogrammingtopologyoptimizationflowrelaxationwildfirerisklinearvoltageloaddelivery

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper addresses the optimal power shutoff problem, where a grid operator decides which lines to de-energize to balance wildfire risk against load delivery after large-scale outages. The authors try to make the second-order-cone (SOC) power-flow formulation fast enough to be practical by replacing its nonlinear voltage constraints with a small number of linear cuts. Their first result is a simple exact reformulation of the voltage cone, $(W^R_{ij})^2+(W^I_{ij})^2 \le W^{Fr}_{ij}W^{To}_{ij}$, that both shrinks the formulation and tightens the continuous relaxation, cutting solve time on the 57-bus case from about 13,500 seconds to about 46 seconds. Their second result is negative: the fully linear relaxations they construct overestimate how much load can be delivered, and after re-solving with the SOC power flow they deliver no better load than the much faster DC model. The paper concludes that a good speed-quality trade-off for a fully linear formulation was not found.

What carries the argument

The load-bearing object is the rotated second-order cone inequality $(W^R_{ij})^2+(W^I_{ij})^2 \le W^{Fr}_{ij}W^{To}_{ij}$, labeled Eq. (17). It replaces the earlier product relaxation that used $W_{ii}W_{jj}z_{ij}$, which required three cone constraints and a binary product. The reformulation works because the per-line squared-voltage variables $W^{Fr}_{ij}$ and $W^{To}_{ij}$ are defined as $W_{ii}z_{ij}$ and $W_{jj}z_{ij}$, so when the line is de-energized the right-hand side is zero and the cone is vacuous, and when the line is energized it equals $W_{ii}W_{jj}$. The paper then builds two linear relaxations around this cone: one using McCormick envelopes on the bilinear product $W^{Fr}_{ij}W^{To}_{ij}$, and one using a secant upper bound; both keep tangent cuts for the quadratic terms but relax the right-hand side, and it is that right-hand-side relaxation that the paper identifies as the source of overestimation.

What would settle it

Take the binary line decisions produced by the McCormick and secant formulations on a case like case14 or case57 and evaluate them with a full AC power flow solver instead of the SOC redispatch used in the paper. If the AC-feasible delivered load is still far below the linear solutions' estimates, the overestimation claim is confirmed; if AC delivers substantially more than the SOC redispatch indicates, then the SOC proxy, not the linear relaxation, is the source of the measured gap.

Watch

Extended reading notes

Core claim

The central claim is that the SOC voltage cone in topology-switching power flow can be written exactly as $(W^R_{ij})^2+(W^I_{ij})^2 \le W^{Fr}_{ij}W^{To}_{ij}$ instead of as a product relaxation that needs multiple cone constraints and an explicit binary multiplier. Because $W^{Fr}_{ij}=W_{ii}z_{ij}$ and $W^{To}_{ij}=W_{jj}z_{ij}$, multiplying the original right-hand side by $z_{ij}^2=z_{ij}$ makes the product form exact; the per-line voltage variables already encode the switching state, so the binary product disappears from the constraint. This reformulation, called SOC-OPS-P, is both smaller and tighter and is the paper's main positive result. The paper further claims that replacing the convex cone with tangent-based linear cuts for thermal limits and either McCormick or secant bounds for the voltage product yields fast but optimistic solutions: the linear models report more deliverable load than the SOC redispatch actually achieves, and their final redispatch performance is no better than DC power flow.

Load-bearing premise

It assumes that a solution that passes the relaxed second-order-cone power-flow check will really be able to deliver that much power on the actual AC grid; if the relaxation is loose in these large-outage topologies, the reported load-delivery numbers could be wrong.

Editorial extensions

If this is right

Any topology-switching SOC power flow model can adopt Eq. (17) to remove the binary product and tighten the relaxation, so the speedup is not limited to the optimal power shutoff problem.
Fully linear relaxations of the voltage cone are not a drop-in replacement for the SOC formulation: they report overly optimistic load delivery and, after redispatch, match rather than beat DC power flow quality.
The secant relaxation is tighter and gives better redispatch than the McCormick relaxation, especially on the worst-case 14-bus network, so it is the better starting point if linear relaxations are pursued further.
Adding more linearization points for the quadratic terms does not improve solution quality; the weak point is the right-hand-side product $W^{Fr}_{ij}W^{To}_{ij}$, not the tangent cuts.
The SOC-OPS-T model, which linearizes only thermal limits, preserves near-identical load delivery with no speedup, suggesting that future gains must come from a better voltage-cone relaxation rather than from thermal-limit cuts.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If Eq. (17) is used inside restoration or topology-control formulations that also switch lines, the same exactness and tightening should carry over, because the argument only uses the definition of the per-line voltage variables.
The paper's diagnosis points to a concrete next step it does not test: a piecewise-linear outer approximation of the bilinear term $W^{Fr}_{ij}W^{To}_{ij}$ might recover quality without full binaries, at the cost of more constraints.
In speed-critical settings, the results suggest starting from the DC solution and repairing it with an SOC or AC check, rather than solving a large linearized conic model; the paper notes this direction but does not develop it.
Because the evaluation uses SOC redispatch rather than full AC redispatch, the quantitative redispatch ratios are upper bounds on what the paper can conclude about true AC load delivery; the qualitative ranking of formulations is likely more robust than the exact percentages.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the exact reformulation in Section II-B is real, the speedup is real, and the negative result about full linearizations is probably right in the large, but not proven as carefully as the abstract implies. The paper deserves serious review, but needs fixes.

The new thing is the voltage-cone rewrite (W_R)^2+(W_I)^2 <= W_Fr*W_To, which is exact when the binary switch is on and forces everything to zero when off. That is a clean observation, and the runtime gain on the 57-bus case (13,530s to 45.75s) is striking. The paper does not oversell it: it notes the reformulation does not help beyond 60 buses. Also good: the negative result about linear relaxations is framed honestly. The authors tried two linearization schemes and report that they overestimate deliverable load and do not beat DC-OPS. That is useful for people deciding where to spend effort.

Where it gets soft: the evaluation metric is SOC redispatch, not AC. The abstract and intro lean on 'AC-feasible', but the tables are ratios against SOC-redispatch denominators. For cells well below 100%, the overestimation conclusion is directionally safe. But the relative rankings among Secant, McCormick, and DC on cases where ratios are close (case39, case60) could shift if checked against AC. That is a real gap, and the paper should either add AC validation or soften the language.

There are also two text errors. Eq. (21a) writes the thermal limit as T_ij z_ij instead of T_ij^2 z_ij. And the 'fully linear' McCormick model in (23a-b) is not linear: it has products of W_Fr, W_To, and z. These might be typos, but they are in the defining equations, so the model is mis-specified in print. Also, no code or data is released, which makes independent checking harder.

The reference list and baselines look fine; the PGLib cases and the DC model from [6] are appropriate. Self-citation is not an issue here, since the prior results are real stepping stones.

Who should read this: anyone working on topology optimization or SOC power-flow relaxations. It is a practical, modest paper. I would send it out for review, but I would want the authors to fix the typos, release code/data, and add an AC redispatch check or at least a sensitivity analysis on a couple of cases.

Referee Report

4 major / 5 minor

Summary. The paper studies topology optimization for power shutoff planning under large-scale outages, comparing a SOC-based optimal power shutoff (OPS) formulation with several linearized alternatives. The authors propose an exact reformulation of the voltage cone as a single rotated SOC constraint linking the line-end voltage variables (Eq. 17), and report that this reformulation reduces solve time dramatically on several PGLib cases. They then develop linear relaxations of the thermal limits and of the voltage cone, using tangent and secant cuts, and evaluate the resulting formulations on PGLib networks from 14 to 118 buses. The paper concludes that the SOC-OPS-P reformulation is a major practical speedup, while the fully linearized models overestimate deliverable load and do not provide a better speed-quality trade-off than DC-OPS.

Significance. The SOC-OPS-P reformulation, if correct, is a valuable and easily reusable modeling improvement for topology-switching SOC power flow, with reported speedups of two to three orders of magnitude on the 57-bus and 60-bus cases (Table I). The paper also makes a useful negative claim: replacing the SOC voltage cone by simple linear cuts does not preserve deliverable load quality. These claims are supported by experiments on public PGLib cases with multiple scenarios. However, the definition of the McCormick-based linear model contains a substantive error, and the evaluation of load deliverability is based on SOC redispatch rather than AC power flow, so the negative conclusions require correction and additional validation before the paper can be accepted.

major comments (4)

[Section III-B, Eq. (23)] The model SOC-OPS-M is claimed to be fully linear, but Eqs. (23a) and (23b) contain the products W_Fr W_To and W_Fr W_To z_ij, which are bilinear and trilinear, respectively. Moreover, the two constraints are identical as printed, and neither resembles a McCormick envelope for the bilinear term W_Fr W_To. Since this is the defining model for the McCormick relaxation, the numerical results for SOC-OPS-M in Tables I-III rest on an undefined model. Please replace Eqs. (23a)-(23b) with the actual McCormick envelopes using the bounds on W_Fr and W_To (or the intended disjunctive formulation), and then re-run or re-report the affected experiments.
[Section III-A, Eq. (21a)] The thermal-limit cut for the reverse direction is written as y_P_ji + y_Q_ji <= T_ij z_ij, but the original thermal constraint (8) has the squared limit T_ij^2 z_ij. The square is missing in (21a). As printed, this constraint is not a valid relaxation of (8): for T_ij > 1 it is tighter than the thermal limit and may cut off feasible flows, while for T_ij < 1 it is too loose. Since the SOC-OPS-T, SOC-OPS-M, and SOC-OPS-S models all include (21a), their feasibility and optimality properties are affected.
[Section IV and Table III] The paper evaluates solution quality by fixing binary decisions and re-solving with SOC-Redispatch from [6], not by solving the AC power flow. The abstract and introduction motivate the work through AC-feasible load delivery and describe the negative result as an overestimation of deliverable load. Because the SOC voltage cone is a relaxation of AC, SOC-Redispatch can report higher deliverable load than AC; consequently, the relative rankings among Secant, McCormick, and DC formulations in Table III are not established for AC deliverability. This is especially important for cells near 100% and for the comparison between Secant and DC. Please either run an AC-feasible redispatch for the fixed topologies or clearly qualify the negative conclusions as statements about SOC-feasible load delivery rather than AC-feasible load delivery.
[Section II-B, Eq. (17)] The paper claims that when z_ij is relaxed to be continuous, the right-hand side W_Fr W_To in (17) is never larger than any of the right-hand-side terms in (15). This claim is false in the continuous relaxation. For example, take z_ij = 0.5, W_ii = W_jj = 1, and V_i^2 = V_j^2 = 0.9, Vbar_i^2 = Vbar_j^2 = 2. Then W_Fr = W_To = 1 is feasible for (11)-(12), giving W_Fr W_To = 1, while W_ii W_jj z_ij = 0.5. Thus (17) is not tighter than (15) in the continuous relaxation. The reformulation is exact on binary feasible sets, which is sufficient for the main speedup claim, but the stronger continuous-relaxation statement should be removed or replaced with a precise proof.

minor comments (5)

[Eq. (15)] Equations (15b) and (15c) are identical as printed, with the same right-hand side W_ii W_jj z_ij. This appears to be a typo; please clarify the intended three constraints or remove the duplicate.
[Section V-C and Figure 2] The text in Section V-C describes a case study on the 73-bus system, but Figure 2 is titled 'SOC-OPS-M on the 89 bus system'. Please align the text and the figure.
[Section VI] The discussion states that SOC-OPS-T reduces the number of SOC constraints from 3|B| to |B|, but the relevant count is per line, so it should read 3|L| to |L| (or the sentence should be reworded accordingly).
[Section III-C, Eq. (24)] The variable y_WSum is introduced as an upper bound for the squared term [1/2(W_To - W_Fr)]^2, but the cut in Eq. (24) is written directly in terms of W_To - W_Fr. Please clarify the scaling of the linearization points l and the relation between l and the argument of the square, to avoid dimensional ambiguity.
[Introduction and Section IV] The introduction says the goal is to find good binary decisions and then solve an AC power flow continuous problem, but the evaluation in Section IV uses SOC-Redispatch. Please align the terminology so that the evaluation metric is consistently described as SOC-based, or change the implementation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SOC-OPS-P reformulation is derived algebraically from the model equations, and the linear relaxations are evaluated against external PGLib cases and a published DC baseline.

full rationale

The central derivation is self-contained. Equation (17) is obtained by multiplying W_ii W_jj z_ij by z_ij and substituting the defining relations W_Fr_ij = W_ii z_ij and W_To_ij = W_jj z_ij, with constraints (11)-(12) enforcing W_Fr_ij = W_ii and W_To_ij = W_jj when z_ij=1 and zero when z_ij=0; this is an algebraic equivalence, not a fitted or imported result. The McCormick and secant linearizations are standard relaxation constructions applied to the paper's own conic constraint, with no parameter tuned to make the negative conclusion come out. The computational evaluation uses PGLib benchmark cases and compares against the published SOC-OPS and DC-OPS baselines. The SOC-Redispatch step from [6] is a proxy for AC deliverability that is not directly validated here, but that is a correctness or external-validity limitation, not a circular reduction, because no quantity predicted by the paper is defined in terms of that evaluation metric. Moreover, in the cells where the linear models' predicted load exceeds the SOC-redispatch value, the conclusion that the linear model overestimates deliverable load is logically sound, since SOC feasibility is a relaxation of AC feasibility for the fixed topology. Self-citations such as [6] supply the starting model and redispatch routine, but the speedup and overestimation findings do not reduce to those citations: the reformulation is shown from the stated equations, and the overestimation is measured by solving a distinct fixed-binary SOC problem. No self-definitional, fitted-input, imported-uniqueness, or renamed-result pattern is present. Score 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to target results; alpha and the linearization points are user choices, not estimated quantities. No new physical entities are introduced. The main assumptions are the SOC-as-AC proxy and the standard switching and big-M modeling inherited from prior work.

free parameters (3)

alpha (load/risk trade-off weight) = not reported
Eq. (1) balances load delivery against wildfire risk; the case study never states alpha, so the objective used in Tables I-III is not fully specified.
load weights w_d = not reported (presumably 1)
Weights in the load-delivery term of Eq. (1) are not specified; if they are not 1, objective values and comparisons change.
linearization point set L per quadratic term = 5 to 15 cuts; exact point locations not given
Cuts (19), (20), (22), (24) depend on the chosen linearization points. The paper varies only the number of cuts on one case and never defines how the points are selected.

assumptions (4)

domain assumption The SOC power flow relaxation is a valid relaxation of AC power flow and a reliable proxy for AC-feasible load delivery in large-outage scenarios.
Evaluation in Section V.B uses SOC-Redispatch from [6], not the nonlinear AC equations; the abstract claims 'AC feasible' while the paper verifies only SOC feasibility.
domain assumption The binary energization variables enforce zero power flow on de-energized components through big-M bounds (10)-(13).
The equivalence of (17) to (15) relies on W_Fr = W_ii z and W_To = W_jj z holding exactly, which is only true when z is binary and the big-M constraints are tight.
domain assumption Voltage angle limits (14) and the McCormick shunt relaxation (16) from the prior model are valid and carried over.
These constraints are taken unchanged from [6]; the paper does not re-derive them.
standard math The secant upper bound (25) is a valid outer approximation of the squared term s^2 over the interval [ell, u], including when z is fractional.
For convex x^2, the chord lies above the graph; the z-scaling used in (25) preserves this for z in [0,1].

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Cite this review

Pith. "Pith review of Second-Order-Cone Formulations of Power Flow for Topology Optimization." pith.science (2026). https://pith.science/paper/NSWB4TDV

@misc{pith2026250602234,
  author       = {Pith},
  title        = {Pith review of: Second-Order-Cone Formulations of Power Flow for Topology Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NSWB4TDV}},
  note         = {Machine review of arXiv:2506.02234}
}

read the original abstract

Optimization problems that involve topology optimization in scenarios with large scale outages, such as post-disaster restoration or public safety power shutoff planning, are very challenging to solve. Using simple power flow representations such as DC power flow or network flow models results in low quality solutions which requires significantly higher-than-predicted load shed to become AC feasible. Recent work has shown that formulations based on the Second Order Cone (SOC) power flow formulation find very high quality solutions with low load shed, but the computational burden of these formulations remains a significant challenge. With the aim of reducing computational time while maintaining high solution quality, this work explores formulations which replace the conic constraints with a small number of linear cuts. The goal of this approach is not to find an exact power flow solution, but rather to identify good binary decisions, where the power flow can be resolved after the binary variables are fixed. We find that a simple reformulation of the Second Order Cone Optimal Power Shutoff problem can greatly improve the solution speed, but that a full linearization of the SOC voltage cone equation results in an overestimation of the amount of power that can be delivered to loads.

Figures

Figures reproduced from arXiv: 2506.02234 by the authors.

**Figure 1.** Comparison of SOC-OPS and SOC-OPS-P solve time. The SOC-OPS-P problem is faster than the SOC-OPS problem, especially on the 57 and 60 bus cases. The 73, 89, 118 cases generally terminate at the time limit. V. RESULTS We explore three key results from this work. First, we examine the computational speed improvements of the presented formulations in comparison to the SOC and DC power flow formulations in [3]. Next we… view at source ↗

**Figure 2.** Redispatch Performance and Solve Time of SOC-OPS-M on the 89 bus system as a the number of linearization points are varied. Points represent individual scenarios, and the line is the average value. the two linear relaxations, the McCormick relaxation performs worse on all networks than the Secant relaxation, a trade-off for the faster runtime shown in Table I. The Secant relaxation has worse performance than the DC-… view at source ↗

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Reference graph

Works this paper leans on

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Pith tools

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Long solution times or low solution quality: On trade-offs in choosing a power flow formulation for the optimal power shutoff problem,

E. Haag, N. Rhodes, and L. Roald, “Long solution times or low solution quality: On trade-offs in choosing a power flow formulation for the optimal power shutoff problem,”Electric Power Systems Research, vol. 234, p. 110713, 2024

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Risk analysis and management in power outage and restoration: A literature survey,

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work page 2014

[3] [2]

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N. Rhodes, L. Ntaimo, and L. Roald, “Balancing wildfire risk and power outages through optimized power shut-offs,”IEEE Trans. on Power Syst., vol. 36, no. 4, pp. 3118–3128, 2020

work page 2020

[5] [4]

Relax- ations of ac maximal load delivery for severe contingency analysis,

C. Coffrin, R. Bent, B. Tasseff, K. Sundar, and S. Backhaus, “Relax- ations of ac maximal load delivery for severe contingency analysis,” IEEE Trans. on Power Syst., vol. 34, no. 2, pp. 1450–1458, March 2019

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[6] [5]

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P. Van Hentenryck and C. Coffrin, “Transmission system repair and restoration,”Mathematical Programming, vol. 151, pp. 347–373, 2015

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Powermodel- srestoration. jl: An open-source framework for exploring power network restoration algorithms,

N. Rhodes, D. M. Fobes, C. Coffrin, and L. Roald, “Powermodel- srestoration. jl: An open-source framework for exploring power network restoration algorithms,” in2021 Power Syst. Comput. Conf., 2021

work page 2021

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A. Kody, A. West, and D. K. Molzahn, “Sharing the load: Considering fairness in de-energization scheduling to mitigate wildfire ignition risk using rolling optimization,” in2022 IEEE 61st Conference on Decision and Control (CDC). IEEE, 2022, pp. 5705–5712

work page 2022