Effective Restoration Unavailability ($U_R$)

• Effective Restoration Unavailability is a scalar metric defined as the normalized area under the unavailability curve, reflecting mean downtime or missing image details.
• It models restoration using Poisson processes with closed-form solutions under uniform, exponential, or lognormal fits, providing actionable insights for resilience analysis.
• In image super-resolution, U_R is derived from F-score comparisons to assess edge restoration fidelity, offering a rigorous alternative to traditional quality metrics.

The Effective Restoration Unavailability metric, denoted $U_R$, is a scalar measure quantifying the average time or fraction of service during which assets, components, or information remain unrestored after a disruptive event. Its primary applications are in electric power system resilience and objective image super-resolution assessment, where it unifies disparate restoration processes into an interpretable, data-driven indicator of unavailability. The metric expresses either the mean downtime per unit, per event, or the fraction of missing true detail, depending on context, and is typically derived from the area between a normalized recovery curve and the ideal full-restoration line.

1. Formal Definition and General Properties

In electric power systems, $U_R$ is defined as the area under the unavailability curve, normalized per outaged or affected unit, capturing aggregate downtime accrued during an event’s restoration phase. Formally, with $O(t)$ and $R(t)$ as the cumulative counts of outages and restores up to time $t$, and $n$ the total number of units outaged, the metric is

$$U_R = \frac{1}{n} \int_{t_0}^{t_1} [O(t) - R(t)]\,dt$$

where $[t_0, t_1]$ spans from the onset of outages to full restoration. When all outages are assumed instantaneous at $t=0$, this reduces to

$U_R = \int_{0}^{\infty} [1 - r(t)]\,dt$

with $U_R$0 the normalized recovery fraction.

In image super-resolution, $U_R$1 is defined as

$U_R$2

where $U_R$3 is an $U_R$4-score comparing restored and ground-truth image edges, reflecting the fraction of irretrievably missing or spurious image structure (Lyapustin et al., 2022).

Key properties:

• $U_R$5 is homogeneous in units of time for power systems (e.g., hours), or dimensionless in image analysis.
• $U_R$6 implies immediate, perfect restoration; $U_R$7 indicates total unavailability.
• The metric aggregates both the timing and completeness of restoration, subsuming first-restore delay, restoration spread, and overlapping repair processes.

2. Modeling Outage and Restore Processes

The metric’s analytic tractability arises from modeling outages and restores as non-stationary or homogeneous Poisson processes (Dobson, 2023, Dobson et al., 2022). Let $U_R$8 and $U_R$9 be the time-dependent instantaneous rates of outage and restoration, respectively, over their relevant intervals. The cumulative mean outage and restore counts are

$O(t)$0

Assuming rapid outage accumulation (i.e., $O(t)$1 concentrated near $O(t)$2), restoration dominates $O(t)$3.

For restoration phase modeling, three canonical forms are used:

• Uniform (constant-rate): $O(t)$4 on $O(t)$5, giving $O(t)$6
• Exponential: $O(t)$7 with time constant $O(t)$8, yielding $O(t)$9
• Lognormal: $R(t)$0, with $R(t)$1 the lognormal pdf, resulting in $R(t)$2

These forms permit direct closed-form expressions for $R(t)$3 as the first moment (mean) of the fitted restoration time distribution. Thus, $R(t)$4 not only summarizes aggregate downtime but also links directly to probabilistic parameters estimable from historical or utility-collected event data.

3. Calculation Methodologies

Computation of $R(t)$5 depends on available data granularity and context.

Electric Power Systems:

• Integral-based approach: Numerically integrate the sequence $R(t)$6 (outaged minus restored components) over the event time-range, divide by $R(t)$7 to obtain mean downtime per affected component (Dobson et al., 2022, Carrington et al., 2020).
  • For discrete event logs, sum $R(t)$8 over time bins $R(t)$9.
  • Normalize by total outages and possibly by event duration for per-unit, per-time metrics.
• Fitted-distribution approach: Fit restoration times to a parametric model (e.g., lognormal, exponential) and compute the mean as $t$0. For lognormal,

$t$1

where $t$2 counts any simultaneous restores at the first restore time.

Super-Resolution Image Assessment:

• Compute candidate edge gradients for the ground truth and super-resolved outputs.
• Discard the top 15% of gradient magnitudes to reduce noise.
• For each spatial shift, count matched edges where the gradient direction cosine exceeds $t$3.
• Aggregate true positives for the top $t$4 shifts; count false negatives and false positives.
• Compute $t$5-score from these counts, then $t$6 (Lyapustin et al., 2022).

4. Statistical Estimation and Practical Implementation

For restoration events in critical infrastructure, estimation of $t$7 leverages utility records or large-scale historical event logs:

• Extract event windows ($t$8), count $t$9.
• Compute restoration delays $n$0 or fits to restoration curves.
• Fit parameters ($n$1, $n$2 for lognormal; $n$3 for exponential) via method-of-moments or maximum likelihood.
• Estimate $n$4 using analytic closed forms.
• To reduce variability, use geometric mean (median fit) or quantile-based metrics for robust reporting; the paper recommends $n$5 or $n$6 for routine planning (Dobson et al., 2022).
• For image benchmarks, all steps are deterministic and computationally efficient (batchable gradient convolutions, vectorized edge matching).

Typical empirical results for North American power transmission events: $n$7 (lognormal fit) $n$8 16.3 h overall, with substantial increase for weather-driven (e.g., hurricane) events ($n$9 h), and lower for non-weather events ($$U_R = \frac{1}{n} \int_{t_0}^{t_1} [O(t) - R(t)]\,dt$$0 h) (Dobson et al., 2022). For super-resolution, $$U_R = \frac{1}{n} \int_{t_0}^{t_1} [O(t) - R(t)]\,dt$$1 strongly correlates (PLCC = $$U_R = \frac{1}{n} \int_{t_0}^{t_1} [O(t) - R(t)]\,dt$$2, SRCC = $$U_R = \frac{1}{n} \int_{t_0}^{t_1} [O(t) - R(t)]\,dt$$3) with human judgment of restoration quality, outperforming PSNR and SSIM (Lyapustin et al., 2022).

5. Interpretational Context and Use Cases

$$U_R = \frac{1}{n} \int_{t_0}^{t_1} [O(t) - R(t)]\,dt$$4 condenses multiple dynamic and stochastic factors—outage intensity, restoration timing, crew resource allocation, and event severity—into a standardized, cross-comparable scalar. In infrastructure, it supports:

• Benchmarking utility or regional system restoration performance.
• “What-if” analysis for resource planning (e.g., reducing restore delays or increasing crew counts).
• Regulatory or resilience targets (e.g., U_R < threshold for given event size).
• Integration into risk models or cost-benefit frameworks, translating restoration dynamics into effective downtime and societal impact metrics (Carrington et al., 2020).

In image analysis, $$U_R = \frac{1}{n} \int_{t_0}^{t_1} [O(t) - R(t)]\,dt$$5 objectively quantifies both missing and hallucinated structure in super-resolved output, focusing on restoration “fidelity” (true edges restored) rather than only visual quality, and corrects for spurious detail addition typical in many generative image models (Lyapustin et al., 2022).

6. Assumptions, Variability, and Limitations

• Independence: Restoration actions (and, optionally, outages) are modeled as mutually independent, disregarding queueing or spatial dependencies.
• Phase simplification: Restoration is typically modeled as a unimodal distribution (lognormal, exponential), which may not fully capture multi-phase or strongly multimodal restore processes.
• First-restore delay: $$U_R = \frac{1}{n} \int_{t_0}^{t_1} [O(t) - R(t)]\,dt$$6 is sensitive to the "dead time" before initial restoration activity, especially in small $$U_R = \frac{1}{n} \int_{t_0}^{t_1} [O(t) - R(t)]\,dt$$7 events.
• Tail behavior: As $$U_R = \frac{1}{n} \int_{t_0}^{t_1} [O(t) - R(t)]\,dt$$8 is the mean of the fitted restoration time distribution, it is affected by heavy tails; for consistent reporting (especially when last-few restores dominate), practitioners may prefer geometric means or quantiles for volatility reduction (Dobson et al., 2022).
• Image analysis context: $$U_R = \frac{1}{n} \int_{t_0}^{t_1} [O(t) - R(t)]\,dt$$9’s parametrization (gradient norm thresholds, spatial shifts, cosine cutoffs) may need tuning across datasets; edge-matching concentration (top $[t_0, t_1]$0 shifts) is critical for reproducibility (Lyapustin et al., 2022).

A plausible implication is that $[t_0, t_1]$1’s tail sensitivity may limit its utility for operational tracking where last-restore times are dominated by extreme outliers. Selecting alternative central-tendency metrics (e.g., $[t_0, t_1]$2) is preferable for stability in small or variable event populations (Dobson et al., 2022).

7. Summary Table: $[t_0, t_1]$3 Formulations Across Domains

Application Domain	$[t_0, t_1]$4 Computation	Interpretation
Power System Resilience	$[t_0, t_1]$5 or analytic fit (lognormal)	Mean downtime per affected unit (hours)
Transmission Events	$[t_0, t_1]$6 (lognormal fit)	Mean restoration time, weather/eventwise
Image Super-Resolution	$[t_0, t_1]$7 from matching edge pixels	Fraction of unavailable true detail

$[t_0, t_1]$8 thus serves as a unifying metric for restoration efficacy across infrastructure and digital domains, providing rigorous, closed-form insight into the effective speed and completeness of recovery processes (Dobson, 2023, Dobson et al., 2022, Carrington et al., 2020, Lyapustin et al., 2022).