Adaptive Restoration for IEGDS

• The paper introduces a POMDP-based framework with Bayesian updates to optimize joint inspection and repair decisions under uncertainty.
• It leverages an advanced Belief Tree Search and Value-Prioritized Flow Allocation to minimize outage costs, achieving near-optimal performance.
• The framework integrates gas crew dispatch with rolling-horizon MILP for power scheduling, ensuring scalable, real-time adaptive restoration of coupled systems.

An adaptive restoration framework for Integrated Electricity-Gas Distribution Systems (IEGDS) is a real-time, model-based strategy for the recovery of interconnected power and gas infrastructures after extreme disruptive events that induce component failures with partial observability. Such frameworks unify damage diagnosis (typically via field inspection), optimal repair crew routing, system constraint management, and incremental restoration decisions under uncertainty regarding system state and evolving contingencies. These approaches resolve the fundamental challenge that post-disaster states, particularly the location and severity of faults, can only be partially ascertained initially and become gradually revealed as field inspection progresses. The following sections detail the core methodologies, mathematical models, algorithmic strategies, and empirical performance established in contemporary research.

1. Problem Formulation: POMDP-Based Joint Inspection and Repair

The restoration of IEGDS under partial observability is formally modeled as a Partially Observable Markov Decision Process (POMDP), encapsulating the coupled evolution of the physical system and field crew states, observations, and operational costs. The POMDP tuple is $$\langle \mathcal{S},\mathcal{A},\Omega,T,O,C,b_0\rangle$$:

• State Space $\mathcal{S}$: Comprises the binary operational status for each electricity line and gas pipeline ($z^L_{\ell,t}, z^W_{m,t}\in\{0,1\}$) and the local state of each repair crew (target assignment, activity indicator, remaining work/travel time).
• Action Space $\mathcal{A}$: Specifies, at each decision epoch, the next target (line or pipeline) dispatched to each crew.
• Observation Space $\Omega$: At each time, consists of reported inspection results, as well as indirect evidence such as generator gas shortages and customer outage notifications.
• Transition Function $T$: Encodes deterministic crew activity and repair completion, with component status updated only upon successful repair.
• Observation Function $O$: Yields perfect information on the inspected component, none otherwise.
• Cost Function $C$: Aggregates instantaneous power and gas outage costs across all downstream customers.
• Belief Update: Bayesian filtering over system states, refined as new observations accrue.

This POMDP governs both the scheduling of inspection and repair, and the cascade of dynamic system behavior under evolving damage and intervention (Li et al., 6 Jan 2026).

2. Advanced Belief Tree Search (BTS) Algorithm

Solving the high-dimensional, real-time POMDP for IEGDS is computationally intractable using classical exact methods. Instead, an advanced Belief Tree Search (BTS) algorithm is deployed to select optimal crew dispatches under current belief:

• Scenario Sampling: Draws samples from the calibrated belief distribution over component failures to instantiate $M_s$ full-system scenarios.
• Tree Expansion: Alternates action (crew routing choices) with observation (resulting component inspection outcomes) nodes, branching at each step.
• Action Selection: Implements an upper confidence bound (UCB) mechanism, controlling depth $d$ and progressive widening for tractability.
• Simulation/Rollout: Completes rollouts with a base policy that maximizes expected load gain per repair time, employing the Value-Prioritized Flow Allocation (VPFA) approximation for fast estimation of system-wide outage impact.
• Back-propagation and Pruning: Updates expected costs and action counters, converging to the action with minimum posterior outage cost.
• Root Action Selection: Chooses action $a^* = \arg\min_a Q(a)$ at the root.

This method achieves real-time feasibility by reducing each scenario to $O(d)$ expansions plus a linear-time flow approximation, enabling update times from sub-minute (small systems) to a few minutes (large systems, e.g. $M_s=1000$, $d=2$). The complexity scales linearly with $M_s \times d$ and is substantially more efficient than stochastic programming approaches, which rapidly become intractable with increasing scenario count (Li et al., 6 Jan 2026).

3. Real-Time Adaptive Decision-Making Framework

Operationally, the IEGDS adaptive restoration framework integrates BTS-driven gas crew dispatch with rolling-horizon mixed-integer linear programming (MILP) for power system scheduling:

• Continuous Event-Triggered Loop: The BTS dispatcher updates decisions whenever a crew task completes, a new outage is reported, or fresh observations are received.
• Online Belief Updates: Bayesian posteriors for uncertain component states are updated from field and user reports, feeding subsequent scenario sampling.
• Field Crew Dispatch and Feedback: Decisions are relayed to field crews, whose real-time positional and task completion feedback further closes the decision loop.
• Power System Scheduling: The evolving availability of gas infrastructure is reflected in synchronous rolling-horizon MILP subproblems for power crew dispatch and power flow, ensuring coupling between energy domains.

This framework operates as a unified, real-time architecture for both gas and electricity restoration, with continuous adaptive replanning as new information arrives (Li et al., 6 Jan 2026).

4. Empirical Performance and Scalability

Large-scale case studies substantiate the adaptive restoration framework's efficacy:

System	Outage Cost (BTS)	Hindsight Optimum	Two-Stage SP	Heuristics	Update Time
13-bus, 7-gas	$76,602$	$75,984$ (-0.8%)	$88,548$	$99,034$+	$<$1m ($M_s=500$)
123-bus, 20-gas	$61,682$	$59,071$ (-4.2%)	--	$91,462$+	$<$5m ($M_s=1000$)

The BTS approach closely tracks the full-information optimum, consistently reducing total outage cost by $15-48\%$ relative to stochastic programming and heuristic baselines. Sensitivity studies reveal that convergence and decision stability are achieved for $M_s \geq 300$ (small systems) and $M_s \geq 800$ (large systems) with $d \geq 2$ tree depth. The VPFA approximation ensures each rollout is linear in the number of lines and pipelines, preserving scalability (Li et al., 6 Jan 2026).

5. Comparative Methodology: Adaptive MILP Formulations

Complementary approaches recast restoration as a cost-constrained, reward-maximizing multiple traveling salesman problem on doubly weighted graphs (CCRM-mTSP-DW):

• Graph Reduction: Represents damaged lines and repair jobs as nodes with associated repair times and rewards, and possible crew tours as paths constrained by electrical continuity.
• MILP Formulation: Incorporates assignment, tour continuity, time budget, and precedence constraints, and is solved within a rolling-horizon scheme.
• Real-Time Adaptivity: Embeds periodic re-optimization, warm-starting, stochastic chance constraints for uncertain durations, and online heuristic adjustments to accommodate new events or crew dynamics.

This approach is shown to deliver exact or near-exact restoration plans at moderate scales; embedding the MILP in a rolling-horizon with online updates yields a flexible, adaptive framework for IEGDS (Wei et al., 2024).

6. Theoretical and Practical Significance

The research establishes that adaptive restoration frameworks based on dynamic belief models and real-time optimization offer both near-optimality and operational scalability even under deep uncertainty and adversarially incomplete information. Central practical contributions include:

• Real-time performance on commodity hardware for large-scale systems
• Robustness to delayed or partial observations
• Significant reductions in customer outage costs and restoration times
• Seamless integration of gas and electricity system constraints
• Flexibility to augment or adjust as field conditions evolve

Further, by leveraging advanced belief search, scenario sampling, and tractable flow approximations, these frameworks circumvent the exponential blow-up inherent to direct POMDP or stochastic programming solutions, setting a new standard for real-world post-disaster infrastructure resiliency (Li et al., 6 Jan 2026, Wei et al., 2024).