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Exploiting sparse structures and synergy designs to advance situational awareness of electrical power grid

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Sparse threat indicators in an equivalent-circuit grid model turn blackout simulation and state estimation into tools that pinpoint dominant failure and error sources and suggest corrective actions, scaling to 80,000-bus systems.

desk verdict Solid engineering thesis with a genuinely useful circuit-based estimator and LODF graph distance, but the flagship blackout-localization claim rests on an admitted heuristic that is not validated against ground truth at scale. read the letter →

arxiv 2412.15105 v1 pith:XA4QPHY2 submitted 2024-12-19 eess.SP cs.AI

Shimiao Li This is my paper · ORCID
classification eess.SPcs.AI
keywords powergridsituationalawarenesscircuit-basedsimulationstateestimationsparseoptimizationtopologyerrordetectionphysics-MLsynergyblackoutsourcelocalizationfalsedatainjection
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

The thesis's central claim is that two weaknesses of power-grid situational-awareness tools—simulators that diverge during blackouts and estimators that are fooled by bad data—can be fixed with the same device: sparse 'threat indicators' placed in an equivalent-circuit model of the grid. When a blackout makes the power-flow equations unsolvable, the method recovers a sparse set of bus-level current injections that restore balance, giving operators a short list of where the failure originates and what correction would fix it. When measurements mix random bad data with wrong switch statuses, the same sparse-indicator idea yields a robust estimator that flags only the erroneous meters and switches while keeping voltage estimates accurate. The thesis validates this on an 80,000-bus model of the US Eastern Interconnection for simulation and on more than 25,000-bus systems for estimation, and it reports substantially faster runtime than standard LP solvers. A reader should care because the method promises to convert a diverging, non-actionable simulation—or a confidently wrong state estimate—into an immediately actionable diagnosis.

What carries the argument

The load-bearing object is the threat indicator: an independent current source inserted into the circuit model whose nonzero value marks a deficiency at a bus, in simulation, or an error at a meter or switch, in estimation. The equivalent-circuit formulation makes every network constraint an affine Kirchhoff-current-law equation, so the estimation problem becomes a convex equality-constrained quadratic program or linear program. Sparsity is enforced by two mechanisms: a bus-wise sparsity enforcer that assigns high thresholds to low-ranked buses and low thresholds to high-ranked buses, based on a dense solution, and a weighted least-absolute-value objective that promotes sparse error indicators. An iterative shrinkage schedule extends the simulator to large systems, and a hypothesis test on switch currents and voltages confirms suspected topology errors. The ML side of the synergy exploits temporal sparsity through a bias-variance weighted historical-data scheme and spatial sparsity through per-node and per-edge local networks.

What would settle it

Take a synthetic blackout with a known cause, such as a single tripped generator at bus B, and run the robust simulator with a k-sparse goal of k=1. If the algorithm returns a different bus as the dominant infeasibility source, and injecting the recommended compensation at that returned bus fails to make the network balance equations feasible, the central localization claim fails. The thesis itself notes that the largest dense-solution indicators need not point to the true dominant sources, so this experiment directly tests the load-bearing assumption.

Watch

Extended reading notes

Core claim

The paper's central discovery is that sparse optimization applied to an equivalent-circuit formulation gives both simulation and estimation an intrinsic robustness that standard tools lack. In simulation, infeasibility is represented by current sources at buses, and minimizing a weighted quadratic objective with a bus-wise sparsity enforcer yields sparse compensation terms that identify dominant blackout sources and corrective injections. In estimation, all measurement devices—conventional SCADA meters, synchrophasors, and switch-status sensors—are converted into linear circuit models, and a weighted least-absolute-value objective over error indicators yields sparse detection of bad data and wrong switch statuses, with a hypothesis test to verify switch alarms. The author then shows that lightweight ML models exploiting temporal and spatial sparsity generalize across dynamic graphs and scale to 60,000-node systems, and that connecting these ML predictions to the physics-based tools—as warm starts and priors—further accelerates simulation under MadIoT attacks and hardens estimation against false-data-injection attacks. The author notes in Section 4.2.4 that the ranking of 'major' buses from a dense solution is likely but not guaranteed to point at the true sources.

Load-bearing premise

The bus-wise sparsity enforcer assumes that the buses with the largest infeasibility indicators in a dense solution really are the dominant sources of a blackout; the thesis acknowledges in Section 4.2.4 that this ranking is not always correct.

Editorial extensions

If this is right

  • Blackout simulation on large grids no longer needs to diverge: operators get a sparse set of buses where injecting power would restore feasibility.
  • Mixed random bad data and topology errors can be identified and separated in one estimation pass on systems with more than 25,000 buses, without the iterative re-runs of residual-based bad-data detection.
  • Because the circuit formulation is device-level, the same simulation and estimation pipeline applies to both transmission and distribution networks, including three-phase distribution systems.
  • Lightweight sparsity-exploiting ML generalizes across topology changes and scales to 60,000-node systems with millisecond per-sensor anomaly detection.
  • Interconnecting physics-based tools with ML predictions, as warm starts and priors, yields more than 3x faster simulation under MadIoT attacks and near-linearly scaling robustness against false-data-injection attacks.
  • The detectability limits discussed for node-breaker models imply that some wrong switch statuses are fundamentally indistinguishable from the data, so operators can know when a switch alarm should be trusted.

Reading between the lines

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

  • The same sparse-threat-indicator template could transfer to other network infrastructures, such as natural gas, water, or transportation networks, that admit circuit-like conservation laws and where failures and corrupt measurements are sparse.
  • The iterative shrinkage schedule in Algorithm 3 resembles a homotopy or continuation method; a convergence proof that the sequence of threshold updates reaches the intended sparse set would turn the ranking heuristic into a certified procedure.
  • The undetectability results of Section 4.3.7 imply an identifiability limit for any switch-status estimator on node-breaker models, so adding temporal consistency checks or PMU redundancy at redundant switches is a natural testable extension.
  • The bias-variance temporal weighting in DynWatch suggests that any physics-based or learned graph distance could replace the LODF-based distance; a natural test is whether the method still detects anomalies on non-grid dynamic graphs with known ground-truth changes.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The thesis develops a unified circuit-based (equivalent-circuit) framework for steady-state power system simulation and estimation, and combines it with sparse-optimization threat indicators, lightweight machine-learning models (DynWatch, Gridwarm), and physics-ML synergy designs. The main claims are that sparse threat indicators pinpoint dominant sources of random blackouts and data errors while suggesting corrective actions, that the resulting tools scale to the 80k-bus Eastern Interconnection and to >25k-bus systems faster than standard solvers, and that the synergy designs further accelerate simulation under MadIoT attacks and improve estimation robustness against false data injection. The manuscript also presents theorems bounding the statistical error of the DynWatch anomaly detector and a closed-form convex estimator for circuit-based generalized state estimation.

Significance. If the central claims hold, the work would be a substantial advance in practical situational awareness tools for power grids: the circuit-based estimator is convex with a closed-form solution, the sparse simulation/estimation formulations are demonstrated on very large systems, and the ML components are designed for scalability and physical interpretability. The thesis has concrete strengths: DynWatch ships public code and data, the experiments use independent test cases (CyPRES, MATPOWER/PEGASE, ACTIVSg25k, Texas CP-2000), and several large-scale demonstrations (80k-bus Eastern Interconnection, >24k-node generalized state estimation) go well beyond typical power-system evaluations. However, two load-bearing elements need attention before the headline claims can be accepted: the DynWatch error-bound proof in Section 5.2.5 contains an algebraic error, and the sparse source-localization claim in Chapter 4 is validated without ground-truth failure locations on large systems, despite an admitted ranking heuristic. These issues are local and fixable, but they directly affect the 'pinpoint dominant sources' and 'error bound' claims.

major comments (4)
  1. [§5.2.5] The proof of Theorem 5.2.1 contains an algebraic error in the bias^2 term. Equation (5.35) rewrites E[(∑_t w_t μ_t − μ_{T+1})^2] as E[(∑_t w_t |μ_t − μ_{T+1}|)^2], which is not an identity unless all mean shifts have the same sign; and even in that case (∑_t w_t a_t)^2 is not equal to ∑_t w_t a_t^2 or ∑_t w_t c d*_t as assumed in Eq. (5.36). Consequently, the claimed upper bound (1 + max_t w_t)σ^2 + c max_t d*_t does not follow from the derivation. The bound may be repairable (e.g., by applying Jensen/Cauchy-Schwarz to the un-squared bias), but as written the theorem is unproven.
  2. [Table 4.2] The flagship claim that robust actionable simulation 'pinpoint[s] dominant blackout sources on the 80k-bus Eastern Interconnection grid' is not validated against ground truth. The bus-wise sparsity enforcer (Algorithms 1-3) selects the sparse support by ranking buses according to the dense L2 infeasibility indicators, and Section 4.2.4 explicitly states that the largest indicators are 'not always' the true dominant sources. In Table 4.2 the injected stress is a global load-factor increase, for which there is no independent definition of a 'dominant bus'; the reported dominant buses are simply the method's own output. To establish the pinpointing claim, the authors should run controlled experiments with known local contingencies (e.g., a specific line outage or a local load surge at a known bus) on large cases and report whether the true failure set is recovered in the top-k selected buses.
  3. [§4.3.4] The robustness and identification claims for the WLAV-based estimator depend on empirically tuned hyperparameters: switch weights (0.001 and 0.01), bad-data detection thresholds (|n| > 0.1 and |n_sw| > 0.05), and hypothesis-test thresholds (τ_I = 0.01, τ_V = 0.01). The thesis states these are 'empirically learned from synthetic data' but does not report how sensitive the detection/localization rates are to these values. For a safety-critical application, a sensitivity analysis over a plausible range of thresholds and weights, with corresponding false-alarm and missed-detection rates, is needed to support the claim that the method reliably identifies a mixture of random bad data and topology errors.
  4. [Definition 5.2.1] Definition 5.2.1 calls d*_t the 'optimal graph distance' and assumes |μ_t − μ_{T+1}| = c·d*_t for a single constant c, i.e., that the graph distance is proportional to the mean shift of each sensor's data. This is an ad-hoc assumption, not a definition: the LODF-based graph distance of Section 5.2.1 is a structural measure of power-flow redistribution and has no established linear relationship to the mean shift of any particular sensor's measurements. The assumption is load-bearing for Theorem 5.2.1, so it should be stated as an assumption with empirical justification, or the theorem should be re-derived under weaker conditions.
minor comments (4)
  1. [Abstract] The abstract claims 'significantly faster speed than standard solvers' for the estimation methods, but Section 3.6 deliberately compares only numbers of iterations rather than wall-clock time (because of the MATLAB/Python implementation asymmetry). While Section 4.3.6 does provide work-time comparisons against CVXOPT and SciPy, the abstract's speed claim should be qualified to reflect which experiments actually measure runtime.
  2. [§4.2.4] Algorithm 3's outer loop 'while not sparse enough' has no explicit termination condition, stopping criterion, or guarantee of convergence; the loop also includes an 'optional adjustments' step that is not specified. Please state a concrete termination rule (e.g., a maximum iteration count or a target sparsity level) so the algorithm is reproducible.
  3. [Chapters 4-5] The symbol n is overloaded: it denotes infeasibility/compensation terms in Chapter 4, measurement noise/error terms in the estimation models, and switch flows in the node-breaker models. A consolidated notation table or explicit redefinition at each chapter would improve readability.
  4. [Table 4.2] For the large-system results in Table 4.2, please report the runtime, number of outer iterations of Algorithm 3, and the final values of the sparse indicators, so that the reader can assess the cost and sparsity trade-off of the method.

Circularity Check

2 steps flagged · score 6.0 of 10

The sparse simulator's 'dominant blackout source' is the top-k dense-infeasibility set fed into the optimizer as the only permissible support, so the pinpointing claim reduces by construction; alarm thresholds in the robust estimator are also tuned on the evaluation data.

  1. fitted input called prediction [Section 4.2.3, Algorithm 1; Section 4.2.4, Algorithms 2-3]
    "For buses in the ‘major’ group, i.e. with high infeasibility quantities ( ||ni|| >> 0), we assume that they are very likely the dominant sources of failure and assign a low value cL. This encourages non-zero ni on those locations. For buses in the minor group, we assign a higher threshold cH such that we can force their ni values to zero or near zero values."

    The support of the sparse solution is fixed before solving: Algorithm 1 sorts the dense L2 indicators from Problem 2.2.2, labels the top k as 'major', and gives exactly those buses the low penalty cL while forcing all others toward zero with cH. Algorithm 2 then solves Problem 4.2.1 using these penalties, so any returned nonzero n is confined to the preselected major set. Thus the reported 'pinpoint dominant blackout source' is the same top-k ranking used as input. In the Case 14 experiment, bus 14 has the largest dense indicator (0.109 vs ~0.09) and is the only bus allowed to be nonzero, so the 1-sparse location at bus 14 in Table 4.1 is by construction.

  2. fitted input called prediction [Section 4.3.6, Eq. (4.18) and accompanying text]
    "These bad data identification thresholds are empirically learned from our synthetic data, specifically by observing the data and finding a threshold value that effectively separates bad data points from normal ones, and they work well in our experiments."

    The alarm rules used to report localization of data errors — |n| > 0.1 for RTU/PMU errors, |n| > 0.05 for switch errors, plus τI = τV = 0.01 in Table 4.3 — are selected by inspecting the same synthetic dataset on which the evaluation is performed. Consequently, the reported successful identification of the injected bad data and topology-error locations is partly a fitted outcome: the thresholds were chosen to separate the injected anomalies from normal data in that dataset. The sparse WLAV residual vector itself is method-derived, but the quantitative 'identify data errors' claim, as operationalized by these thresholds, includes this tuned component.

full rationale

The thesis contains substantial independent content: the circuit-based state estimator is derived self-contained with a closed-form solution, the WLAV robust estimator is compared against a WLS baseline on CyPRES cases with injected ground-truth errors, and DynWatch/Gridwarm are evaluated on synthetic anomaly data with known labels. Self-citations to prior circuit-formulation work are not load-bearing in the sense of forbidding alternatives; they point to earlier published formulations that the thesis re-derives. However, the flagship simulation claim of 'pinpointing dominant blackout sources' is partially circular by construction. The bus-wise sparsity enforcer does not discover a sparse support from the physics; it first ranks buses by the dense infeasibility solution, preselects the top k as 'major', assigns them low penalties, and then solves an optimization in which all non-selected buses are heavily penalized toward zero. The returned nonzero set is therefore the preselected set, so the 'prediction' of where to build generation or take corrective action is an input to the algorithm rather than an independently validated output. The paper explicitly acknowledges the ranking assumption is 'not always true' and the large-scale demonstrations lack local ground truth, which reinforces that the dominant-source localization claim is not yet independently established. The detection-threshold tuning in the robust estimator is a second, smaller fitted-input issue. Overall, the central simulator claim partially reduces to its own preselection mechanism, while the other pillars of the thesis remain methodologically independent; this warrants a score of 6 rather than higher.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard power-flow physics plus several ad hoc assumptions: sparsity of errors, the ranking of dense infeasibility, and the optimal graph distance calibration. These are not all validated with independent evidence.

free parameters (6)
  • cH and cL (bus-wise sparsity enforcer thresholds) = cH=10, cL=0.1
    Default values in Algorithm 3; chosen by heuristic to balance sparsity and convergence, not derived from data.
  • shrinkage rate r = 0.75
    Controls how quickly the k-sparse goal shrinks; set by default.
  • switch weights w in WLAV estimator = 0.001 (8-substation), 0.01 (300-substation, CP-2000)
    Tuned by hyperparameter search to minimize misclassification (Section 4.3.5).
  • bad-data detection thresholds = 0.1 (n_pmu, n_rtu), 0.05 (n_sw), tau_I=0.01, tau_V=0.01
    Empirically learned from synthetic data (Section 4.3.6, Table 4.3).
  • DynWatch distance scaling factor = 0.005
    Empirically chosen to balance bias and variance in temporal weighting (Section 5.2.2).
  • closed switch reactance x_sw = 0.0001 p.u.
    Model parameter for closed switch low-impedance branch (Eq. 3.16).
assumptions (6)
  • domain assumption Power balance equations / Kirchhoff's laws describe steady-state grid behavior
    Foundation of simulation and estimation models; standard in power engineering.
  • domain assumption Bad data and topology errors are sparsely distributed
    Justifies WLAV objective; if errors are dense, estimator degrades (admitted in Section 4.3.6).
  • ad hoc to paper The largest entries of the dense L2 infeasibility solution indicate dominant failure locations
    Assumption behind bus-wise sparsity enforcer (Section 4.2.3); the thesis notes this is not always true.
  • domain assumption Temporal independence and identical distribution conditioned on topology (Assumptions 5.2.1, 5.2.2)
    Used for DynWatch statistical error analysis.
  • ad hoc to paper Optimal graph distance d*_t is proportional to the mean shift (Definition 5.2.1)
    Defines away the core difficulty of calibrating graph distance to distribution shift; makes the error bound contingent on this proportionality.
  • domain assumption PMU installed on every generation bus and RTU on other injection buses
    Meter placement assumption in experiments (Section 4.3.6), not always realistic.

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Cite this review

Pith. "Pith review of Exploiting sparse structures and synergy designs to advance situational awareness of electrical power grid." pith.science (2026). https://pith.science/paper/XA4QPHY2

@misc{pith2026241215105,
  author       = {Pith},
  title        = {Pith review of: Exploiting sparse structures and synergy designs to advance situational awareness of electrical power grid},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XA4QPHY2}},
  note         = {Machine review of arXiv:2412.15105}
}
read the original abstract

The growing threats of uncertainties, anomalies, and cyberattacks on power grids are driving a critical need to advance situational awareness which allows system operators to form a complete and accurate picture of the present and future state. Simulation and estimation are foundational tools in this process. However, existing tools lack the robustness and efficiency required to achieve the level of situational awareness needed for the ever-evolving threat landscape. Industry-standard (steady-state) simulators are not robust to blackouts, often leading to non-converging or non-actionable results. Estimation tools lack robustness to anomalous data, returning erroneous system states. Efficiency is the other major concern as nonlinearities and scalability issues make large systems slow to converge. This thesis addresses robustness and efficiency gaps through a dual-fold contribution. We first address the inherent limitations in the existing physics-based and data-driven worlds; and then transcend the boundaries of conventional algorithmic design in the direction of a new paradigm -- Physics-ML Synergy -- which integrates the strengths of the two worlds. Our approaches are built on circuit formulation which provides a unified framework that applies to both transmission and distribution. Sparse optimization acts as the key enabler to make these tools intrinsically robust and immune to random threats, pinpointing dominant sources of (random) blackouts and data errors. Further, we explore sparsity-exploiting optimizations to develop lightweight ML models whose prediction and detection capabilities are a complement to physics-based tools; and whose lightweight designs advance generalization and scalability. Finally, Physics-ML Synergy brings robustness and efficiency further against targeted cyberthreats, by interconnecting our physics-based tools with lightweight ML.

Figures

Figures reproduced from arXiv: 2412.15105 by the authors.

Figure 2.1
Figure 2.1. Network topology processor converts node-breaker (NB) model to bus-branch (BB) model, and [PITH_FULL_IMAGE:figures/full_fig_p028_2_1.png] view at source ↗
Figure 3.1
Figure 3.1. Circuit-based viewpoint for simulation and estimation on power system. [PITH_FULL_IMAGE:figures/full_fig_p037_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Suppose the power injection at a load bus is measured by an RTU device (in purple); the [PITH_FULL_IMAGE:figures/full_fig_p040_3_2.png] view at source ↗
Figures from the paper (42 more)
Figure 3.3
Figure 3.3. Figure 3.3: Linear RTU model: measurements are mapped to sensitivities. [PITH_FULL_IMAGE:figures/full_fig_p041_3_3.png]
Figure 3.4
Figure 3.4. Figure 3.4: Connecting RTU model to the load bus: the load is now described by its "pseudo" measure [PITH_FULL_IMAGE:figures/full_fig_p042_3_4.png]
Figure 3.5
Figure 3.5. Figure 3.5: Linear PMU model: measurements have a linear nature under circuit-based formulation. [PITH_FULL_IMAGE:figures/full_fig_p043_3_5.png]
Figure 3.6
Figure 3.6. Figure 3.6: Linear model for line flow meters: implemented as control circuits coupled with the main [PITH_FULL_IMAGE:figures/full_fig_p044_3_6.png]
Figure 3.7
Figure 3.7. Figure 3.7: Open switch model: nsw close to zero if the status is correct; nsw compensates the current flow on the branch if the status is wrong. Model Definition 5 (Linear closed switch model) [PITH_FULL_IMAGE:figures/full_fig_p045_3_7.png]
Figure 3.8
Figure 3.8. Figure 3.8: Closed switch model: nsw close to zero if the status is correct; nsw will offset the current flow on the branch if the status is wrong. xsw is in p.u. 3.2.5 Other grid devices Physical devices such as transformers, lines and shunts are linear and their derivations an…
Figure 3.9
Figure 3.9. Figure 3.9: Equivalently representing a power system steady-state operating point with an aggregated [PITH_FULL_IMAGE:figures/full_fig_p046_3_9.png]
Figure 4.1
Figure 4.1. Figure 4.1: Toy examples: sparse optimization adding robustness to simulation and estimation. Sparse [PITH_FULL_IMAGE:figures/full_fig_p055_4_1.png]
Figure 4.2
Figure 4.2. Figure 4.2: Sparse indicators pinpoint dominant sources of failure and focused corrective actions at key [PITH_FULL_IMAGE:figures/full_fig_p056_4_2.png]
Figure 4.3
Figure 4.3. Figure 4.3: Relationship between n ∗ and c: a blocking effect. This reveals a simple mechanism through which the threshold c encourages a sparse solution by confining most n entries to near zero value. Whenever threshold c is added, the blocking effect reduces the number of non-…
Figure 4.4
Figure 4.4. Figure 4.4: High c value causes steep convergence region on the complementary slackness curve. Left: upper bound curve µ¯ R i (n R i − t R i ) = −ϵ, right: lower bound curve µ R i (−n R i − t R i ) = −ϵ. When n R i > 0, we have µ¯ R i → c, µ R i → 0. (n R i , t R i , µ¯ R i ) co…
Figure 4.5
Figure 4.5. Figure 4.5: Identifying data errors on Case 118 4.3.1 Assuming a possibly wrong topology To consider the possibly wrong switching statuses which contribute to topology errors, we need node￾breaker models of the power grid which contain nodes, switches, and branches. However, a n…
Figure 4.6
Figure 4.6. Figure 4.6: Extending a bus-branch model with pseudo nodes and switches. [PITH_FULL_IMAGE:figures/full_fig_p065_4_6.png]
Figure 4.7
Figure 4.7. Figure 4.7: CyPRES 8 substation network. (The case is modified by opening the switch (7,8)) [PITH_FULL_IMAGE:figures/full_fig_p070_4_7.png]
Figure 4.8
Figure 4.8. Figure 4.8: Robustness of WLAV (left) vs WLS (right): WLAV better identifies data errors and obtains [PITH_FULL_IMAGE:figures/full_fig_p073_4_8.png]
Figure 4.9
Figure 4.9. Figure 4.9: Robustness on different sized networks: The top row shows the number of inaccurate bus [PITH_FULL_IMAGE:figures/full_fig_p074_4_9.png]
Figure 4.10
Figure 4.10. Figure 4.10: Speed and scalability: generalized state estimation (considering topology errors). We compare [PITH_FULL_IMAGE:figures/full_fig_p076_4_10.png]
Figure 4.11
Figure 4.11. Figure 4.11: Detectability issues: three examples where the robust circuit-based generalized state estimator [PITH_FULL_IMAGE:figures/full_fig_p076_4_11.png]
Figure 5.1
Figure 5.1. Figure 5.1: Temporally sparse weights pinpoint the relevant historical data: a sparse recommendation. [PITH_FULL_IMAGE:figures/full_fig_p079_5_1.png]
Figure 5.2
Figure 5.2. Figure 5.2: Toy example of DynWatch: different distributions are caused by topology changes. Consider the simple power grid shown in [PITH_FULL_IMAGE:figures/full_fig_p080_5_2.png]
Figure 5.3
Figure 5.3. Figure 5.3: Transition state of two graphs: the union of two graphs. [PITH_FULL_IMAGE:figures/full_fig_p082_5_3.png]
Figure 5.4
Figure 5.4. Figure 5.4: wt − dt relationship. Algorithm 6: Computing Temporal Weights wt Input: distance dt , with t = 1, 2, · · · , N Output: weights wt , with t = 1, 2, · · · , N 1 Compute the unique λ ∗ that satisfies: ∑ t∈{1,2,...N} max{λ ∗ − dt , 0} = 1 2 Get weights wt : wt = max{λ ∗ …
Figure 5.5
Figure 5.5. Figure 5.5: Simple motivating example: DynWatch-Local. The two graphs are very different within the yellow localized region. However, sensor 2 is far away from the yellow region and thus experiences no changes. In the previous section, we computed a single distance value D(Gi , …
Figure 5.6
Figure 5.6. Figure 5.6: Local graph distance: the adjacent lines connected to each sensor [PITH_FULL_IMAGE:figures/full_fig_p087_5_6.png]
Figure 5.7
Figure 5.7. Figure 5.7: Synthetic data generation for evaluating D [PITH_FULL_IMAGE:figures/full_fig_p093_5_7.png]
Figure 5.8
Figure 5.8. Figure 5.8: Performance of DynWatch: AUC and F-measure. 5.2.7 Q2. Scalability In this subsection, we seek to analyze the scalability of our DynWatch and DynWatch-Local. In reality, PJM, the largest ISO in the U.S., runs ACSE on a 28k bus model, performed every 1 min [5], thus an…
Figure 5.9
Figure 5.9. Figure 5.9: DynWatch scales linearly with the number of edges, when detecting all 1200 time ticks. The red lines are best-fit regression lines. of less than 1.7ms per time tick per sensor. The ACTIVSg25k (realistic power system case with 25K buses) has 32k+ branches, and thus ru…
Figure 5.10
Figure 5.10. Figure 5.10: Change of line status, total load and generation on for a real-world load dataset. [PITH_FULL_IMAGE:figures/full_fig_p097_5_10.png]
Figure 5.11
Figure 5.11. Figure 5.11: Result of F-measure on case2383wp, with 40 sensors installed: the proposed LODF-based [PITH_FULL_IMAGE:figures/full_fig_p098_5_11.png]
Figure 5.12
Figure 5.12. Figure 5.12: A power grid can be naturally represented as a graphical model. Each node represents the [PITH_FULL_IMAGE:figures/full_fig_p099_5_12.png]
Figure 5.13
Figure 5.13. Figure 5.13: Local lightweight NNs: each node has a NN-node and each edge has a NN-edge, to map the input features to the post-disturbance system characteristics. Meanwhile, to effectively learn the mapping, we must answer the following question: how to select the input features…
Figure 5.14
Figure 5.14. Figure 5.14: Training of Gridwarm: forward pass and back-propagation. Local lightweight NNs combine [PITH_FULL_IMAGE:figures/full_fig_p105_5_14.png]
Figure 6.1
Figure 6.1. Figure 6.1: We advocate for a Physics-ML synergy: physics-based and data-driven models are intercon [PITH_FULL_IMAGE:figures/full_fig_p109_6_1.png]
Figure 6.2
Figure 6.2. Figure 6.2: The proliferation of IoT devices has raised concerns about IoT-induced cyber attacks, where [PITH_FULL_IMAGE:figures/full_fig_p111_6_2.png]
Figure 6.3
Figure 6.3. Figure 6.3: Physical interpretation: This figure visualizes the values in matrices and vectors (the more [PITH_FULL_IMAGE:figures/full_fig_p113_6_3.png]
Figure 6.4
Figure 6.4. Figure 6.4: Result on test data: power flow simulation takes fewer iterations to converge with the proposed [PITH_FULL_IMAGE:figures/full_fig_p114_6_4.png]
Figure 6.5
Figure 6.5. Figure 6.5: A Bayesian Network depicts the spatiotemporal interactions of variables in the electrical power [PITH_FULL_IMAGE:figures/full_fig_p117_6_5.png]
Figure 6.6
Figure 6.6. Figure 6.6: Physics-ML Synergy creates an interconnection: prior knowledge from ML comes in the [PITH_FULL_IMAGE:figures/full_fig_p124_6_6.png]
Figure 6.7
Figure 6.7. Figure 6.7: Case30: accuracy of estimation accuracy quantified by root mean squared error (RMSE). [PITH_FULL_IMAGE:figures/full_fig_p128_6_7.png]
Figure 6.8
Figure 6.8. Figure 6.8: ML-augmented ckt-GSE identifies types and locations of mixed anomalies. [PITH_FULL_IMAGE:figures/full_fig_p130_6_8.png]
Figure 6.8
Figure 6.8. Figure 6.8: Robustness: ML-augmented ckt-GSE advances ckt-GSE in robustness against modern false [PITH_FULL_IMAGE:figures/full_fig_p131_6_8.png]
Figure 6.9
Figure 6.9. Figure 6.9: Speed and scalability: worktime of the 3 essential procedures in the synergy design, as system [PITH_FULL_IMAGE:figures/full_fig_p132_6_9.png]

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Pith tools

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