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How suboptimal is my stochastic network controller allowed to be? Completion certificates with application to power grids hosting AI data centers

Michael Chertkov

arXiv:2610.03275v1eess.SYcs.SYmath.DSphysics.soc-ph

TL;DR

The paper asks how to assess implementable controllers when large-network optima are out of reach and power grids face abrupt, correlated AI-data-center load changes. It constructs HJB lower bounds through control completion and tests the resulting certificates on stochastic swing dynamics. On the IEEE 118-bus system, repricing the same total authority sharply narrows the certified gap under heterogeneous forcing.

  • Problem

    For large networks, the optimal feedback is out of reach, leaving the question of how far an implementable controller is from optimal.

    Source: Context and significance · pp. 1–2

  • Method

    The paper uses control completion to make a path-integral surrogate linearly solvable, whose value supplies an automatic HJB-subsolution lower bound.

    Source: §9 · p. 29

  • Results

    On the IEEE 118-bus system after severe load loss, the certified gap is 2.8% under homogeneous forcing and, under heterogeneous forcing, 39% with uniform prices versus 1.2% with the same authority priced in proportion to noise.

    Source: §9 · p. 29

  • Takeaways & Limitations

    The geometry provides shadow values to rank sparse reinforcement and indicates that pricing authority in proportion to local noise variance can reduce completion burden.

    Source: §9 · pp. 29–30

    The deflated value’s domain-wide enclosure remains out of reach with generic derivative bounds, crossing probabilities are empirical, and reinforcement experiments do not compare against unrestricted optimal investment.

    Source: §8.5 · pp. 28–29

Abstract

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Power grids are beginning to host AI data centers whose demand can change abruptly and in a correlated way, and operators and planners must decide whether existing controllers can absorb the resulting transients and where new flexibility is worth installing. We cast this as a question in stochastic control: how far from optimal is an implementable controller? For controlled diffusions with affine, state-independent actuation, additive and possibly degenerate noise and quadratic control cost, any Hamilton-Jacobi-Bellman (HJB) subsolution bounds the optimal cost from below and simulation bounds the deployed cost from above, so their gap certifies the permissible suboptimality. We construct subsolutions from path-integral control by completing the control geometry: enlarging the control Gramian until it matches the physical noise makes the problem linearly solvable, and its Feynman-Kac value is an automatic lower bound whose HJB residual is exactly the energy of the fictitious control. A dual noise-deflation construction can be tighter but requires a curvature condition. The geometry yields planning rules: price control authority in proportion to local noise variance, and use shadow values to guide sparse reinforcement. For nonlinear stochastic swing dynamics of the IEEE 118-bus system after a severe load loss, a simple generator controller is certified within 2.8% of optimal under homogeneous forcing; under heterogeneous forcing the gap is 39% with uniform prices and 1.2% once the same total authority is repriced, before any hardware is added.

Context and significance

The paper asks how much better an implementable controller could be and offers a computable certificate for that gap. It connects path-integral control to power-grid planning under abrupt, correlated data-center load changes.

Context and significance

  • The certificate compares a deployed controller’s simulated cost with an HJB-subsolution lower bound to show how much better any controller could do.The paper frames a small gap as evidence that redesign cannot recover much, while a large gap quantifies what is at stake.Source: Context and significance · pp. 1–2
  • Path-integral control recasts certain optimal-control problems as free energies of path ensembles, and completing the physical problem into that form supplies a subsolution automatically.The value is a log-partition function, control is an exponential tilt of uncontrolled paths, and regret is relative entropy.Source: Context and significance · p. 2
  • In transmission networks, noise enters at every bus while fast active-power control is concentrated at a few, making network structure central to the control geometry.The swing equations are nonlinear; the relevant structure includes the network Laplacian, inertia, and actuator placement.Source: Context and significance · p. 2
  • AI training loads can change abruptly and in correlated ways, while protective ride-through can remove a large block of load at once.The paper notes reported operational challenges involving large-load reductions and associated frequency and voltage excursions.Source: Context and significance · p. 2
  • On the IEEE 118-bus system after a severe load loss, the same total generator authority yields a 39% gap with uniform prices and 1.2% when priced in proportion to noise.The repricing result is presented as actionable before new hardware is purchased.Source: Context and significance · p. 2

1 Introduction

The introduction frames a certificate for implementable controllers as the answer to an intractable large-network optimum. It previews completion and deflation, their design implications, and the power-grid application.

1 Introduction

  • For large networks, an implementable controller’s suboptimality can be bounded by subtracting an HJB-subsolution lower bound from its simulated cost.The paper’s setting has affine, state-independent actuation, additive noise that may be degenerate, quadratic control cost, and otherwise general nonlinear, nonseparable drift and state costs.Source: §1 · pp. 2–3

1.1 Contributions

  • Control completion enlarges the control Gramian to Q/λ when λG ⪯ Q, making the surrogate linearly solvable and yielding an automatic lower bound.The completed value’s physical HJB residual is the energy of its fictitious control; the least inflation gives the tightest bound in this family.Source: §1.1 · p. 3
  • Noise deflation instead shrinks Q to λG and can give a tighter lower bound, but only when the value has the required upward curvature in deleted-noise directions.The two constructions are not generally ordered.Source: §1.1 · p. 3
  • For fixed total generator authority, pricing it in proportion to local noise variance minimizes completion burden; shadow values guide sparse reinforcement.A closed-form robust allocation covers multiple noise scenarios, while exact solvability for two scenarios requires proportional covariances.Source: §1.1 · p. 3
  • The Monte Carlo estimator of the completed value is biased upward by approximately λ/(2 ESS), while the distribution-free finite-sample lower confidence bound is vacuous at practical small matching temperatures.The paper distinguishes what is and is not numerically certified.Source: §1.1 · p. 4
  • In lossless stochastic swing dynamics with all-bus noise and generator-only control, completion adds virtual authority at load buses, and the matching temperature is set by the worst-matched channel.The application tests simple generator controllers on IEEE 39- and 118-bus systems after severe load loss.Source: §1.1 · p. 4

1.2 Relation to existing work

  • Unlike BOTC [1], which hardens matched tasks to obtain upper bounds, this paper changes control–noise geometry to obtain lower bounds; unactuated noisy directions create a BOTC support mismatch.The comparison is illustrated as moves among noise variance and control authority in Figure 1.Source: §1.2 · p. 4, Figure 1 · p. 6

1.3 How to read the paper

  • The paper’s roadmap separates control theory, design rules, power-system applications, claim-status accounting, and curvature verification for deflation.Its modelling scope is stated in Section 7.1, and further figure descriptions appear in Section D.Source: §1.3 · p. 4

2 Performance identities and subsolution certificates

This section establishes performance identities that turn HJB subsolutions into lower bounds and simulated policy costs into upper bounds. Their difference certifies deployed-controller regret.

2 Performance identities and subsolution certificates

  • The performance identity measures terminal and HJB defects for candidate cost-to-go functions rather than requiring an exact HJB solution.The identity follows from Dynkin’s formula and completing the square, under the standing regularity and integrability assumptions.Source: §2 · p. 7
  • A classical subsolution must have nonnegative physical HJB residual and terminal value no greater than the terminal cost.The sign convention ensures a subsolution can only understate cost.Source: §2 · p. 7
  • For every admissible deployed policy, a subsolution J bounds the optimal value below, while the policy cost bounds it above, so their difference upper-bounds regret.The certificate gap is the policy cost minus J.Source: §2 · p. 7
  • When the optimum is classical, the identity prices regret exactly, and a small feedback error has no first-order effect on performance.The feedback-error identity is evaluated under the deployed closed loop, not as a fixed norm shared by all policies.Source: §2 · pp. 7–8
  • A candidate with residual at least −δ can be shifted downward by δ(T−t) to obtain a subsolution, degrading the certificate by exactly that amount.The terminal condition is preserved by the shift.Source: §2 · p. 8
  • Stopped-domain certificates can bound exit-time questions, but the paper uses a smooth, unstopped performance value for its main objective.A path-space relative-entropy reading additionally requires the drift change to lie in the diffusion range.Source: §2 · p. 8

3 Two linearly solvable relaxations

The section builds two matched, linearly solvable surrogates for bounding the physical control problem. Completion gives an unconditional lower bound under its regularity assumptions; deflation can be tighter but requires curvature.

3 Two linearly solvable relaxations

  • Exact linear solvability holds when Q = λG, and matching requires the physical control range to lie within the noise range.When the range condition holds, admissible temperatures satisfy 0 < λ ≤ λ⋆; the physical problem is generally unmatched.Source: §3 · pp. 8–9

3.1 Control completion: an automatic lower bound

  • Control completion sets the enlarged Gramian to Q/λ, so the Feynman–Kac value solves a linearized surrogate and is a subsolution of the physical problem.Its physical HJB residual is exactly the energy of the fictitious control, and certification follows from the linear equation without requiring an optimality argument for the completed problem.Source: §3.1 · pp. 9–10
  • The completion certificate gap splits into fictitious-authority energy and deployed-policy mismatch, while λ⋆ gives the tightest lower bound within the scalar completion family.The completion burden and whitened noise-direction deficit are geometry diagnostics, not bounds on the value gap.Source: §3.1 · p. 10
  • The completed value’s smoothness can follow from a parabolic Hörmander condition even with degenerate noise, subject also to global integrability for Dynkin’s formula.The paper notes this covers noise acting only on mechanical-system velocities.Source: §3.1 · p. 11

3.2 Noise deflation: a conditional lower bound

  • Noise deflation keeps G and shrinks diffusion to λG/2; its value is a subsolution when curvature is nonnegative along deleted-noise directions.For affine drift and convex running and terminal costs, the condition is automatic; for nonlinear drift, convex costs alone do not suffice.Source: §3.2 · p. 11
  • Completion and deflation are not ordered in general: deflation can be tighter but is conditional, whereas completion is unconditional and is tighter in the larger test system.There, completion’s bound is tighter by a factor between three and ten.Source: §3.2 · p. 12

4 From certificates to design

The completion geometry supports both investment–performance comparisons and planning rules for allocating or repricing control authority. It also supplies shadow values for sparse reinforcement and a common way to price implementation defects.

4.1 The price of linearisability

  • The linearly solvable restriction yields an exactly computable investment–performance frontier as decreasing λ installs more authority and lowers J + λ.The price of linearisability Π is not computed because its unrestricted planning value requires solving the unrestricted planning problem.Source: §4.1 · p. 13

4.2 Shadow prices of authority

  • At a completed design, the value and gradient provide shadow prices that rank sparse reinforcements without requiring the full completion to be built.The paper contrasts this ranking with local-noise ranking in Section 7.Source: §4.2 · p. 13

4.3 Pricing existing authority

  • With fixed total authority, noise-proportional pricing maximizes the matching temperature and minimizes completion burden by equalizing r_iq_i across controlled channels.After optimal pricing, the remaining completion burden reflects uncontrolled relative to controlled noise variance; a robust allocation also minimizes worst-case burden across scenarios.Source: §4.3 · pp. 13–14
  • A fixed architecture can exactly match two noise scenarios only when their covariances are proportional; scenarios with redistributed noise require completion or a compromise.Repricing changes the physical control problem, so a loose certificate under original prices does not establish that mispricing caused the gap.Source: §4.3 · p. 14

4.4 Implementation defects are priced in the same currency

  • Implementation defects including hold, lag, quantisation, saturation, dropout, and event-triggered sampling can be priced analogously to the control mismatch.A 0.2 s hold or lag cost about 2% of the objective in the 39-bus study; the summary cautions that performance certificates do not certify safety probabilities.Source: §4.4 · p. 15
  • Under heterogeneous forcing, repricing the same total generator authority reduced the certified gap from 39% with uniform prices to 1.2%; homogeneous forcing yielded a 2.8% gap.The engineering summary also reports 5.3% under heterogeneous and 4.8% under homogeneous forcing with one fixed pricing across scenarios.Source: §4.4 · p. 15

5 Computing a certificate, and what is certified

The section distinguishes what is mathematically bounded from what is estimated: deployed cost is simulated, the completed value is an exact lower value estimated numerically, and the deflated value also needs residual verification. At low matching temperatures, available distribution-free confidence bounds are vacuous, while deflated-value verification remains a separate domain-wide challenge.

5 Computing a certificate, and what is certified

  • The deployed cost J_u is estimated by plain Monte Carlo, J+ is an exact lower value estimated numerically, and J− additionally requires residual verification.The three ingredients therefore have distinct certification status.Source: §5 · p. 15

5.1 Estimating the completed value

  • The J+ estimator overstates the lower bound, with approximate bias λ/(2 ESS), and the bias is small when λ/ESS is small relative to the certificate gap.The estimator can miss probability mass carried by weights never sampled, so a rigorous finite-sample statement requires weight control.Source: §5.1 · p. 16
  • At λ⋆≈1.76 × 10−4 and J+≈0.025, the distribution-free bound gives d J+LCB ≈6×10−4 for N=2048 and δ=0.05, which the authors call useless.The weight-range term is about e142; a useful low-temperature rigorous bound would need a pathwise action lower bound close to its typical value, which is unavailable.Source: §5.1 · p. 16

5.2 Verifying the deflated value

  • A confidence interval for the deflated value at one state does not establish the required residual sign across a domain; the nonlinear application has only partial verification.A certificate needs an analytic sign theorem, a validated domain-wide enclosure, or a uniform negative bound.Source: §5.2 · p. 16

5.3 Relation to finite-dimensional searches

  • Finite-dimensional searches use polynomial or moment relaxations, whereas these constructions produce one candidate by solving a surrogate exactly and use Monte Carlo integration instead of a state-dimension semidefinite program.The subsolution inequalities are linear before minimizing over control and become convex but nonlinear in the reduced value function afterward.Source: §5.3 · p. 17

6 Stochastic swing networks

The application specializes the certificate framework to stochastic swing dynamics with noise at every bus and fast active-power control only at generator buses. Its control–noise geometry yields matching and planning interpretations, while the performance objective treats synchrony safety separately.

6 Stochastic swing networks

  • The stochastic swing-network application models a lossless fixed-voltage grid after a large load loss, with random injections at every bus and fast active-power control only at generator buses.The underactuated noise directions are the setting for which completion was designed, and the network structure makes the framework's quantities explicit.Source: §6 · p. 17, §6.1 · p. 17

6.2 Performance objective and separate safety

  • The objective rewards concentration near the post-event equilibrium over a finite horizon, while synchrony safety is reported separately as the probability of a first line-angle crossing of π/2.The smooth objective remains finite beyond 90°; the authors note that regularity on the whole state space is assumed rather than proved.Source: §6.2 · pp. 17–18

6.3 Geometry of generator-only actuation

  • In the matching geometry, inertia cancels, and λG ⪯ Q reduces to comparing control authority with noise variance at each generator bus.The equivalent power rating is an authority proxy, not a device rating; energy capacity, state of charge, ramp limits, and degradation are not modeled.Source: §6.3 · pp. 18–19
  • For homogeneous noise, the least completion adds virtual authority only at load buses; with heterogeneous noise, one weak-noise generator sets the matching temperature for all.Adding an actuator on a quiet bus can lower λ⋆ and raise completion burden I even while improving the physical problem.Source: §6.3 · p. 18
  • For deflation, rotor-angle curvature is irrelevant; the curvature condition concerns only the value function in deleted frequency directions.The deleted diffusion comprises load-bus noise plus any unmatched generator surplus.Source: §6.3 · p. 19

7 Numerical study

The numerical study applies completion certificates to load-loss transients on IEEE 39- and 118-bus systems, comparing controller architectures, noise profiles, and authority allocations. It shows that repricing fixed generator authority can sharply reduce the certified gap under heterogeneous forcing, while sparse reinforcement yields small absolute benefits in the studied setting.

7.1 Modelling scope and protocol

  • The studies use stressed lossless, fixed-voltage models with synthetic load-bus inertia, independent white noise in homogeneous or activity-scaled profiles, and simple controllers; governor, voltage, loss, protection, and actuator-energy-limit dynamics are absent.The activity-scaled noise profile is a stylized proxy, not fitted to data.Source: §7.1 · p. 19

7.3 The 118-bus system: scaling

  • 2.8%: the Case118 H3 scaled-LQR controller’s certified gap under homogeneous forcing; under heterogeneous forcing, the gap depends strongly on pricing existing authority.The activity-specific price achieves 1.2% under activity-scaled noise, whereas uniform pricing gives 39.1%.Source: Table 1 · p. 20, Pricing existing authority · p. 24

7.2 The 39-bus system: mechanism

  • In Case39, homogeneous noise gives every actuator architecture the same completion, while activity-scaled noise produces a weakest-channel effect and a burden that rises from about 62.4 for H0 to 744.5 for H3.Across the Case39 hierarchy, the completion gap tracks burden more closely than the whitened deficit; repricing activity-scaled authority lowers burden from 744.5 to 38.0 without adding authority.Source: §7.2 · p. 21, Figure 3 · p. 22

Pricing existing authority

  • 39.1% to 1.2%: for Case118 under activity-scaled noise, activity-specific pricing cuts the certified gap at fixed total generator authority, while deployed cost barely changes.The gap is 11.6% when activity-specific pricing is mismatched to homogeneous noise; robust pricing gives 4.8% and 5.3% in the two scenarios.Source: Pricing existing authority · p. 24, Figure 5 · p. 25
  • Sparse Case118 load-side reinforcement under activity-scaled noise has a small absolute effect: even all 64 load directions improve deployed cost by only 0.09%, with crossing frequency unchanged at 0.167.At matched added authority, shadow-value placement beats local-noise placement for k = 5, 10, 20, and 40 devices.Source: Pricing existing authority · p. 25, Figure 7 · p. 26

8 Discussion

The discussion distinguishes exact lower-bound identities from numerically estimated or conditional claims, and relates the approach to other value-bounding methods. The grid application highlights noise-aligned pricing, while the limitations identify where certification remains incomplete.

8.1 What is certified, and how

  • The completed value is an exact lower bound, but the deflated value is valid only if a domain-wide curvature condition holds, which remains unverified.The Monte Carlo estimate of the completed value does not change its status as an exact lower bound.Source: §8.1 · p. 27

8.2 Relation to Bound-Optimized Task Choice

  • Unlike BOTC’s hard-task upper bounds, completion and deflation approach the physical problem from the lower spectral edge; BOTC also requires noise in the directly actuated subspace.The application’s load-bus noise has no corresponding actuator, so a literal BOTC application would require an extension of its task class.Source: §8.2 · p. 27

8.3 Relation to HJB bounds and sum-of-squares

  • Rather than approximate the physical HJB equation, the method changes control–noise geometry to obtain a linearly solvable surrogate whose value is evaluated by Feynman–Kac.The resulting physical-equation residual identifies actuator directions, prices, and candidate locations in the network application.Source: §8.3 · p. 27

8.4 What the application adds

  • In the swing network, heterogeneous forcing makes a single quiet channel set the matching temperature, while pricing authority in proportion to noise removes much of the apparent underactuation.The application makes the role of noise, network structure, and actuator placement concrete; it asks not only how much authority exists, but how that authority is priced relative to noise.Source: §8.4 · p. 28

8.5 Limitations

  • A domain-wide curvature enclosure for deflation remains out of reach, crossing probabilities are empirical, and reinforcement experiments do not price linearisability against unrestricted optimal investment.The sparse curves are costs for a fixed controller family, while the exact-PIC curves are solvable reference values; the modelling limitations of Section 7.1 also apply.Source: §8.5 · pp. 28–29

9 Conclusion and Path Forward

The conclusion presents HJB subsolutions and deployed-policy simulation as a way to certify controller suboptimality, with completion supplying an automatic lower bound. In the 118-bus application, noise-aligned repricing sharply narrows the certified gap under heterogeneous forcing, while the authors identify further design and certification work.

Conclusion

  • An HJB subsolution bounds optimal cost from below and deployed-policy simulation bounds it from above, so their difference certifies permissible suboptimality without computing the nonlinear stochastic optimum.Completing the control geometry to match physical noise gives an automatic subsolution; its physical-HJB residual is the energy of the fictitious control.Source: §9 · p. 29
  • 2.8% of optimal under homogeneous forcing; under heterogeneous forcing, the gap falls from 39% with uniform prices to 1.2% when the same total authority is priced in proportion to noise.These results are for a simple generator controller on the IEEE 118-bus system after a severe load loss; the repricing occurs before hardware is added.Source: §9 · p. 29

Path Forward

  • Future work includes matrix- or direction-dependent matching, a structural proof of the deflation curvature condition, and a joint performance and first-exit certificate.For applications, the authors call for measured and correlated fluctuations, multiple contingencies, and explicit actuator energy, ramp, and saturation limits.Source: §9 · p. 29
  • The planning prices are in weighted cost units rather than dollars; calibrating those weights to economic costs could give shadow prices a monetary interpretation.The paper also cautions that repricing changes certifiability more than performance, so these rules concern tight certification rather than which designs perform best.Source: §9 · pp. 29–30
  • Source-side buffering can reduce noise instead of increasing authority, but reducing noise at the worst-matched actuated channel can lower the matching temperature and loosen the scalar certificate.Under heterogeneous forcing, shadow-value rankings outperform ranking devices by local noise variance because value depends on noise projected onto the value gradient.Source: §9 · p. 30

A Verifying the curvature condition for the deflated value

The appendix investigates the curvature condition needed for the deflated value to be a lower bound. It establishes short positive terminal layers and positive finite-set screens, but not a domain-wide enclosure.

A Verifying the curvature condition for the deflated value

  • The deflated value is a lower bound only when its deleted-noise curvature condition holds throughout the domain; the appendix records partial verification, not a whole-domain certificate.The generic derivative bounds are too conservative to extend the check across the domain.Source: A Verifying the curvature condition… · p. 30

A.1 Terminal layer

  • The coupled potential-and-kinetic energy budget verifies positive terminal-curvature layers of about 1.43 ms and 1.13 ms for case39, and 2.9 ms and 1.6 ms for case118.The values correspond respectively to homogeneous and activity-scaled forcing; against a 3 s horizon, this is a proof of concept, not a certificate.Source: §A.1 · p. 31

A.2 Curvature screens

  • Curvature estimates were positive on tested finite sets, including 34 representative case39 states and 24 case118 directional-curvature traces, but do not enclose the continuum.The smallest reported two-standard-error lower values for case39 were about 6.45 × 10^-3 under homogeneous and 4.50 × 10^-3 under activity-scaled forcing; the smallest case118 trace was about 9.95 × 10^-3.Source: §A.2 · p. 31

A.3 Flow variations in a mechanical metric

The appendix propagates flow sensitivities in a local mechanical metric to investigate curvature bounds, but generic third-derivative closure cannot produce a useful domain-wide enclosure. It separately describes a supermartingale route to first-exit probability bounds, which the paper has not constructed for its event.

A.3 Flow variations in a mechanical metric

  • Local first-, second-, and third-variation majorants propagate sensitivities while trajectories remain within a cohesive cell, using the local mechanical metric.For a representative case39 transient, a 0.10° cell gives a first-variation majorant near 20.5, versus about 4.35 × 10^74 for a single 84° sector bound.Source: §A.3 · p. 31

A.4 Why a domain-wide enclosure is out of reach

  • A domain-wide curvature enclosure remains out of reach: generic third-derivative bounds force case118 cell radii to about 9 × 10^-12 and 2 × 10^-14 degrees for the two H3 settings.The authors say finer cells will not fix the problem; a qualitatively sharper argument is needed.Source: §A.4 · p. 31, Figure 9 · p. 32

B First-exit probability as a separate certificate

  • First-exit probabilities can be bounded by a nonnegative supermartingale certificate, but no such certificate was constructed for the studied event, so reported crossing probabilities are empirical.This stopped-domain certificate is distinct from the smooth performance value used in the main text.Source: B First-exit probability as a separate… · p. 32

C Numerical protocol and reproducibility

The numerical protocol uses frozen, disjoint tuning and evaluation seeds, common random numbers for paired comparisons, and fine time steps to address stiffness. Across the 39-bus results, controller performance and safety differ, while certificate tightness depends on noise geometry and architecture.

C Numerical protocol and reproducibility

  • The numerical protocol freezes disjoint tuning and evaluation seeds, uses common random numbers for paired policy comparisons, and employs fine time steps because the reduced model is stiff.Case39 production runs use Δt ≤ 5 × 10^-3 s; case118 uses Δt = 2.5 × 10^-3 s. The paper provides code, frozen inputs, cached tables, and notebooks in a public repository.Source: C Numerical protocol and reproducibility · p. 33

D Additional figures

  • The additional figures cover case39 controller, relaxation, and planning results, plus formulation and numerical checks; J+ is the exact lower bound, whereas J− is conditional on a curvature condition.Figures 10–18 concern the 39-bus system, Figures 19–20 formulation and numerics, and Figure 21 the 118-bus system.Source: D Additional figures · p. 33, Figure 10 · p. 34, Figure 12 · p. 35

D.1 The 39-bus system: deployed controllers

  • At σrms = 0.2, homogeneous-noise crossing probabilities are 0.129, 0.098, and 0.094 for static balancing, droop, and H3 LQR, respectively.The policies are nearly indistinguishable on smooth cost, but weak controllers remove about a quarter of high-noise boundary crossings; equal-RMS noise profiles also produce different safety statistics.Source: §D.1 · pp. 33–34
  • Beyond the initial estimator singularity, path-integral feedback estimates settle near the H3 LQR scale of 0.01 for both noise profiles, supporting the gain selected by the cost scan.At t = 0, the projections are 0.35 for homogeneous noise and 0.18 for activity-scaled noise.Source: §D.1 · p. 34, Figure 11 · p. 35

D.2 The 39-bus system: the two relaxations across the hierarchy

  • For homogeneous noise, deployed cost falls from about 0.096 for H0/H1 to 0.091 for H2/H3, while the deleted-subspace rank falls from 38 to 29.With activity-scaled noise, deployed cost falls from 0.084 to 0.079, but deleted rank remains 38 for every architecture.Source: §D.2 · p. 34
  • For H3, the homogeneous-noise certificate gap grows from 0.0040 at σrms = 0.05 to 0.0313 at 0.3, while activity-scaled noise makes H3 loosest and H0 tightest.The difference between the completed and deflated values grows with noise in both cases, and neither relaxation dominates the other in general.Source: §D.2 · p. 36

D.3 The 39-bus system: planning

Planning results show that noise-proportional pricing can tighten certificates without adding authority, while scenario-specific and robust allocations trade off matching a known noise profile against flexibility across profiles. The numerical and 118-bus results also characterize reinforcement, integration-step sensitivity, and controller performance.

D.3 The 39-bus system: planning

  • Under activity-scaled noise, five completion-frontier points reduce J+ from near 0.053 to about 0.038, while requiring roughly twice the proxy authority of the homogeneous-noise curve.The frontier is monotone with diminishing returns, and exact completion is useful as a reference but expensive as a construction target.Source: §D.3 · p. 37
  • Under activity-scaled noise, repricing existing authority changes the completion burden from about 744 to 38 and J+ from 0.053 to 0.077, without adding authority.Under homogeneous noise, the two metrics coincide at burden 145 and J+ = 0.085.Source: §D.3 · pp. 37–38
  • At the same total authority, the robust allocation equalizes completion burden across scenarios at 169, while scenario-specific allocations give the tightest certificate when the noise profile is known.Each scenario-specific allocation performs best for its own noise shape and noticeably worse for the other.Source: §D.3 · p. 38, Figure 17 · p. 39
  • With five load-side devices under homogeneous noise, shadow-proxy placement lowers deployed cost to 0.08887 ± 0.00047, versus 0.09025 ± 0.00048 for noise-based placement.The advantage shrinks as devices are added and vanishes at full completion; under activity-scaled noise, the two rankings remain within about one standard error at every budget.Source: Figure 18 · p. 39, §D.3 · p. 40

D.4 Formulation and numerics

  • At Δt = 0.02 s, case39 reports a crossing near 90.4° and cost about 0.0968; for Δt ≤ 5 × 10^-3 s, these settle at 81.1° and about 0.0909–0.0911.The smooth-barrier formulation keeps the path-integral estimate usable near the cohesion boundary, unlike an absorbing failure cost, whose effective sample fraction collapses.Source: §D.4 · pp. 40–41

D.5 The 118-bus system: deployed controllers

  • Under activity-scaled noise, H3 reduces cost relative to static balancing by 1.64%, 1.37%, and 1.29% at σrms = 0.10, 0.15, and 0.20, respectively.The H2 reductions are only 0.12–0.28%; the difference between actuator ranks exceeds that between droop and scaled LQR in this weak-feedback regime.Source: §D.5 · pp. 41–42
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