A time-coupled multi-objective distributionally robust chance-constrained framework for grid resilience enhancement using mobile emergency generators

Abstract

This study presents a time-coupled, multi-objective distributionally robust chance-constrained (MODRCC) framework for resilient grid restoration using Mobile Emergency Generators (MEGs). The model unifies (i) time-expanded logistics for MEG routing, crew scheduling, and refuelling, (ii) islanding-feasible DC-OPF under line outages, and (iii) Wasserstein-ball ambiguity to hedge uncertainty in attack severity and travel-time delays. Disjunctive linearization and second-order-cone (SOC) embeddings yield a tractable MISOCP that is evaluated inside an NSGA-II evolutionary search to generate the Pareto frontier between total cost and resilience. Experiments on IEEE-24 and IEEE-118 (12-hour horizon, 24 periods) show that, at comparable budgets, the proposed method reduces expected unserved energy (EUE) by 14–20% relative to static DRCC and classical robust baselines. On the IEEE-118 case, representative operating points illustrate a ~ 54% decrease in EUE (92→42 MWh) for a ~ 10% increase in cost along the frontier, evidencing smooth, convex trade-offs induced by Wasserstein regularization. The solver stack (Gurobi 12.0 + NSGA-II) scales efficiently; with parallel fitness evaluation it converges in ~ 2.8 h for IEEE-118 (16 MEGs). Results confirm that explicitly coupling mobility realism with distributionally robust modelling yields operationally credible, cost-aware restoration schedules suitable for disaster-prone regions.

Introduction

Modern power systems, particularly those serving densely populated regions of India, have evolved into highly interdependent cyber–physical infrastructures whose operational continuity is increasingly threatened by both intentional and natural disruptions¹. Episodes such as the 2012 North Indian blackout and subsequent cyclone-induced grid failures in coastal states revealed how fragile network connectivity, limited restoration logistics, and cascading contingencies can jointly amplify outage duration and socioeconomic losses². Globally, analysis of three decades of blackout data confirms that nearly half of all large-scale outages originate from cascading failures triggered by exogenous shocks such as extreme weather or coordinated attacks. In US the Executive Office of the President estimated that every additional day of nationwide outage results in economic losses exceeding several billion U.S. dollars, emphasizing the macroeconomic imperative for resilient grid design³.

Within this context, resilience planning has emerged as a strategic complement to conventional reliability criteria. Whereas reliability seeks to maintain operation under expected conditions, resilience aims to absorb, adapt, and recover from rare, high-impact disturbances⁴. A cornerstone of this paradigm is the pre-emptive deployment of Mobile Emergency Generators (MEGs) diesel, gas, or hybrid units that can be rapidly relocated to energize critical loads during restoration⁵. Compared with stationary distributed generators, MEGs offer geographical flexibility and lower capital investment, making them particularly suitable for India’s disaster-prone coastal and hilly districts⁶. However, their operational effectiveness is conditioned by multiple logistical realities: limited crew availability, constrained fuel autonomy, uncertain road accessibility, and travel-time variability arising from post-disaster debris or infrastructure damage⁷. These coupled constraints convert what was once a static allocation problem into a spatiotemporally dynamic decision process, requiring integration of power-flow physics with mobility logistics under uncertainty. Recent studies have advanced the modeling and optimization of resilient and mobility-aware power systems. For instance⁸, investigates robust communication-aware control strategies for distributed energy resources, while⁹ addresses mobility-constrained scheduling under dynamic operational conditions. Additionally¹⁰, presents AI-driven optimization frameworks for resource allocation under uncertainty. These works complement our study by highlighting the importance of integrating real-time mobility, crew logistics, and uncertainty management, reinforcing the relevance and novelty of the proposed DRCC-based MEG deployment framework.

Despite the growing literature on resilient power-system planning, several methodological deficiencies persist. First, most Distributionally Robust Chance-Constrained (DRCC) formulations treat system decisions as temporally independent, thereby neglecting the sequential dependencies that govern MEG routing, refueling, and crew rotation (Figure 1)¹¹. The absence of temporal coupling leads to overly optimistic restoration schedules that disregard realistic mobilization delays. Second, existing defence or restoration models typically assume static resource positioning—once generators are assigned to substations, their future relocation is ignored. Such assumptions conflict with field practice, where MEGs are continuously re-dispatched as restoration progresses¹².

Fig. 1.

Conceptual framework of MEG-based resilient grid under adversarial and natural threats.

Third, current Indian resilience frameworks largely focus on deterministic or scenario-based assessments, without embedding multi-objective stochastic–robust trade-offs or acknowledging asymmetric information about potential offensive actions¹⁰. While tri-level defender–attacker–defender (DAD) formulations capture adversarial interactions, they seldom incorporate uncertainty in attack probability distributions or logistics feasibility¹³. Consequently, national utilities lack quantitative tools that can jointly optimize cost, restoration speed, and survivability when data about threat likelihoods remain incomplete.

Objectives and contributions

To address these deficiencies, this study develops a time-coupled multi-objective DRCC model that explicitly co-optimizes pre-positioning, routing, refuelling, and crew scheduling of MEGs under ambiguous information about attack magnitude and road conditions. The model integrates mobility logistics with operational power-flow constraints to provide a unified decision environment where resilience and economy are simultaneously evaluated.

Formally, the proposed framework embeds a Wasserstein-ball ambiguity set within a bi-level architecture: the upper level minimizes total operational and investment cost together with expected load-shedding risk, while the lower level enforces DC-Optimal Power Flow feasibility over successive time steps, ensuring physical operability of post-attack configurations. This results in a tractable Mixed-Integer Second-Order Cone Program (MISOCP), further explored through a Non-Dominated Sorting Genetic Algorithm II (NSGA-II) to obtain Pareto-optimal trade-offs between cost and resilience.

The contributions of this research are fourfold.

1. It introduces temporal coupling into DRCC-based resilience planning, enabling realistic modeling of MEG movement and endurance over a 12-hour restoration horizon.

2. It embeds mobility and crew logistics within the power-system decision space, capturing practical constraints faced by Indian utilities during cyclone and flood responses.

3. It leverages Wasserstein-based ambiguity modeling to manage distributional uncertainty in both attack severity and transportation delays.

4. It provides computational evidence on IEEE-24 and IEEE-118 that the proposed formulation yields a 14–20% reduction in expected unserved energy compared with static DRCC and robust baselines, at comparable cost.

The proposed centralized optimization framework assumes reliable, low-latency communication between the central controller and all MEGs for real-time coordination of routing, refueling, and crew scheduling. In communication-constrained scenarios, such as limited bandwidth, delayed updates, or intermittent connectivity, the centralized approach may experience suboptimal or delayed decision implementation. To mitigate this, the framework can be extended through hierarchical or distributed control strategies, where local MEG decisions are made at regional nodes with periodic coordination from the central controller. Such architectures enhance robustness to communication limitations while retaining near-optimal system performance, ensuring that resilience planning remains effective even under constrained information exchange.

Literature review

Evolution of grid resilience and defence models

Early research on grid defence primarily viewed resilience through the lens of static system hardening. Salmeron, Wood, and Baldick (2004) first articulated a Defender–Attacker–Defender (DAD) tri-level model to quantify the worst-case line-attack impact on network connectivity, initiating a formal game-theoretic treatment of power-grid security¹². Brown et al.2006 extended this formulation to cross-sector critical infrastructure, emphasizing cost-efficient protection resource allocation¹³. Subsequent refinements by Yuan et al. 2014 incorporated mixed-integer reformulations and line-switching flexibility to reduce computational burden. Zhao et al., later recognized that offensive resources are themselves uncertain, proposing an improved stochastic DAD with probabilistic attack strengths¹⁴.

Collectively, these works established the theoretical backbone for resilience optimization but retained a single-stage static assumption— defence decisions precede attacks, and recovery remains instantaneous. Such simplification omits the multi-hour operational constraints that dominate real restoration logistics¹².

From robust optimization to distributionally robust frameworks

To overcome the brittleness of deterministic defence models, robust optimization emerged as a systematic means of guaranteeing feasibility against worst-case parameter deviations. Researchers were employed robust formulations to analyze vulnerability under multi-period attack trajectories. While effective in guaranteeing feasibility, pure robust approaches often yield excessively conservative solutions because they ignore distributional knowledge derived from historical data.

This limitation motivated the transition to Distributionally Robust Optimization (DRO), in which uncertainty is represented by an ambiguity set of probability distributions. Peyman and Daniel (2018) demonstrated that using the Wasserstein metric to bound empirical distributions produces tractable convex reformulations with explicit confidence guarantees. Chance-constrained settings, enabling direct control of risk tolerance through a tunable violation probability provided a linear-programming approximation that preserves computational scalability for large-scale power systems. These developments jointly laid the mathematical foundation for Distributionally Robust Chance-Constrained (DRCC) models, which now underpin much of the state-of-the-art resilience literature.

Resilience enhancement and mobile resource deployment

In parallel, scholars began coupling resilience planning with mobile and distributed energy resources. Lei et al. (2018) formulated one of the earliest optimization frameworks for Mobile Emergency Generator (MEG) pre-positioning and real-time dispatch following hurricanes, showing significant improvement over fixed-DG restoration¹⁵. Ghorbani-Renani et al. (2020) advanced a tri-level Protection–Interdiction–Restoration model capturing interdependencies among power, transportation, and communication networks¹⁶. More recently, Hou et al. (2024) proposed a Typhoon-DAD robust framework that integrated spatially correlated weather¹⁷ incorporated fuel endurance and routing constraints for mobile emergency fleets.

Despite these advances, most mobility-aware formulations remain scenario-based and thus vulnerable to sample insufficiency; they also decouple mobility scheduling from stochastic uncertainty quantification¹⁸. In contrast, integrating mobility logistics directly within a DRCC formulation allows joint optimization of energy dispatch and travel feasibility under data-driven ambiguity, an aspect largely unexplored in existing work.

Multi-objective optimization and meta-heuristic approaches

The resilience problem inherently involves conflicting objectives—minimizing cost while maximizing service restoration. Multi-objective evolutionary algorithms (MOEAs) have proven effective in exploring these Pareto trade-offs. Non-Dominated Sorting Genetic Algorithm II (NSGA-II), which remains the benchmark for Pareto-front approximation. Pouria et al. 2011 applied such evolutionary frameworks to multi-stage transmission expansion, and Li et al. 2024 developed a multifactorial evolutionary algorithm to enhance network reconfiguration capability in distribution systems¹⁹. When hybridized with convex optimization solvers, these algorithms enable efficient global search over discrete mobility and continuous power-flow variables—precisely the combination required in MEG logistics under DRCC formulations (Figure 1)²⁰.

Identified research gap

From this review, two critical omissions become evident. First, existing DRCC resilience models ignore temporal coupling, assuming instantaneous MEG deployment and neglecting travel-time and crew-availability dynamics. Second, although several studies employ mobility models, none integrate distributionally robust uncertainty within their routing and refuelling decisions^21,22. Consequently, present frameworks fail to capture the interdependence between stochastic attack scenarios and time-dependent mobility feasibility.

This gap motivates the current study’s time-coupled multi-objective DRCC model, which unifies energy-system optimization with realistic mobility logistics. By combining Wasserstein-based ambiguity modelling, DC-OPF feasibility, and NSGA-II exploration, the proposed formulation provides the first cohesive framework capable of quantifying resilience, cost, and temporal operability within a single mathematically rigorous structure²².

Mathematical formulation and model description

System, time horizon, and uncertainty

We consider a transmission–sub transmission network modeled as a connected graph.

with buses and directed line pairs .

Let be a discrete horizon with step (here T = 24, Δt = 30 min).

Mobile Emergency Generators (MEGs) are indexed by , with candidate connection buses . Road crews are indexed by .

Depots/refueling sites are ; optional refueling-eligible set .

Uncertainty is captured by scenarios s∈S. Each scenario specifies:

• Line serviceability ,

• travel times (in multiples of Δt),

• any exogenous disturbance parameters (e.g., demand modifiers).

Let the random vector be

.

An empirical distribution over S is constructed from samples and a Wasserstein ballcontrols distributional ambiguity.

Decision variables (time-coupled logistics + power)

MEG logistics and connection.

• : MEG k is physically located at bus i at time t.

• : MEG k is electrically connected and synchronized at bus i at time t.

• : departure arc; MEG k starts traveling from i at t toward j.

• : transit state (1 if k is traveling during [t, t + 1)).

• : on-board fuel (L).

• : refueling amount (L) at i.

• : crew r attends MEG k at t (for move, connect, or refuel).

Power-system.

• : power exported by MEG k at i.

• : output of conventional unit g.

• ​: flow on line (i, j) under scenario s.

• : phase angle at bus i, scenario s.

• : load shed at bus i.

Auxiliary binaries:

, , to mark short service times or crew shifts.

Time-expanded MEG mobility and crew coupling

We use a time-expanded network where arcs consume ​ time steps.

Location flow conservation

1

Equation (1) describes the location state transition of mobile energy generator (MEG) over time. The binary variable indicates whether MEG is located at node at time . The second term subtracts all movements departing from node at time , while the third term adds arrivals from upstream nodes after the corresponding travel time . This flow-balance formulation ensures that the spatial position of each MEG is consistently updated across the transportation network and over time.

At most one location per MEG:

2

Equation (2) enforces the unique location constraint for each mobile energy generator (MEG). It ensures that, at any time , each MEG is assigned to exactly one location in the network, thereby preventing simultaneous presence at multiple nodes or undefined locations.

Transit state and availability

3

Equation (3) defines the transit-state constraint of mobile energy generator (MEG) . The binary variable indicates that MEG departs from node to node at time . Once this movement is initiated, the variable is forced to be active for the entire travel interval , indicating that the MEG remains in transit and unavailable for service until it arrives at its destination. This constraint ensures consistent modelling of travel time and MEG availability.

MEG cannot export while in transit:

4

Equation (4) enforces the operational availability constraint of mobile energy generator (MEG) . The power output is bounded by the rated capacity and is permitted only when the auxiliary binary variable is active. The second constraint links to the MEG’s presence at location , while the third constraint prevents power generation when the MEG is in transit. Together, these constraints ensure that an MEG can generate power only when it is stationary and physically located at a node.

Crew assignment and shift windows.

Every move/connection/refuel consumes crew capacity:

5

Equation (5) enforces the crew assignment and shift constraints. The first part limits the number of MEGs assigned to crew at time to the crew’s maximum capacity , with the binary variable indicating whether the crew is active. The second part ensures that the cumulative working hours of each crew over any rolling horizon do not exceed the maximum allowed shift duration . Together, these constraints guarantee that crew resources are properly allocated and shift durations are respected during MEG operations.

Movement requires one active crew:

6

Equation (6) ensures that each MEG movement requires one active crew. The left-hand side sums all movement decisions for MEG at time , while the right-hand side counts the number of crews assigned to support that MEG, as indicated by . The binary variable denotes whether a crew is available. This constraint guarantees that no MEG can move without sufficient crew resources, ensuring operational feasibility and proper coordination between MEGs and crew assignments.

Refueling logistics and fuel state

7

Equation (7) describes the fuel state dynamics of mobile energy generator (MEG) . The onboard fuel level at the next time step is equal to the current fuel , minus the fuel consumed for power generation (), plus any refueling at time . This equation ensures that fuel consumption and refueling are consistently tracked over time, coupling energy production with fuel endurance.

8

Equation (8) enforces the fuel capacity and refueling logistics for MEG . The first part ensures that the fuel level remains within the physical limits of the onboard fuel tank. The second part restricts the refueling amount to the maximum allowable refueling rate , which is active only when the binary variable indicates that the MEG is refueling. The third part restricts refueling operations to designated refueling nodes and ensures that is binary. Together, these constraints guarantee that fuel replenishment is physically feasible and properly coordinated within the network.

9

Equation (9) couples power generation with refueling and crew availability. The first part ensures that an MEG cannot generate power at location while refueling (). The second part enforces that the number of MEGs refueling at a given time is limited by the available crew resources assigned to that MEG. Together, these constraints guarantee that power production and refueling operations are mutually exclusive and that sufficient crew is available to perform refueling, maintaining operational feasibility.

Islanding-feasible DC-OPF with line outages

For each t, s power balance respects outages and MEG connections:

10

Line physics with outage disjunction.

We avoid the bilinear via disjunctive linearization:

11

12

where and ​ bounds when the line is out.

Angle and flow limits

13

MEG connection consistency

14

Non-serviceable nodes/lines.

If a bus is de-energized structurally (optional binary ​), we force.

and ​ unless a MEG is connected, enforcing islanded feasibility.

Cost, penalties, and performance metrics

Total operating cost over T is

15

Expected unserved energy (scenario-wise) is

16

Distributionally robust chance constraint (DRCC)

Let the random vector ξ parameterize .

We enforce that total unserved energy does not exceed a tolerance with high probability, for all distributions in a Wasserstein ball:

The constraint,

17

represents a Distributively Robust Chance Constraint (DRCC) that accounts for uncertainty in the random vector . Since the true probability distribution of is unknown, it is assumed to belong to an ambiguity set , defined as a Wasserstein ball of radius centered around the empirical distribution .

The inner probability term quantifies the likelihood that the total aggregated load exceeds the maximum allowable limit . The supremum operator ensures that this probability is evaluated under the worst-case distribution within the ambiguity set. By constraining this worst-case probability to be no greater than the risk tolerance level , the model guarantees robustness against distributional ambiguity while maintaining a prescribed level of reliability.

We first introduce a Lipschitz continuous surrogate h(ξ) bounding the violation:

The function,

18

defines the constraint violation function associated with the system limit. The term represents the load shedding (or load state) at location and time , which depends on the uncertain parameter vector . By summing over all locations and time periods and multiplying by the time interval , the total accumulated load is obtained. Subtracting the maximum allowable threshold allows to measure the extent to which this total exceeds the permissible limit. Consequently, the condition ensures feasibility of the system, while indicates a violation of the imposed constraint under uncertainty.

Using a buffered probability (DR-BPP) upper bound and dual norms, a sufficient and near-tight MISOCP-enforceable.

The inequality,

19

represents a deterministic reformulation of the Distributively Robust Chance Constraint. The first term, referred to as the nominal slack , measures the violation of the constraint under the nominal (empirical) distribution. The second term accounts for distributional uncertainty and depends on the Wasserstein radius , the risk tolerance , and the dual norm of the gradient of the constraint function . This dual Lipschitz term quantifies the sensitivity of the constraint with respect to uncertainty in . By enforcing the sum of the nominal slack and the robustness adjustment to be non-positive, the constraint guarantees that the original chance constraint is satisfied for all probability distributions within the specified ambiguity set, thereby ensuring robustness against distributional ambiguity.

The inequality,

20

provides an upper bound on the dual norm of the gradient of the constraint function . Here, denotes the gradient of with respect to the uncertain vector , and and represent the coefficients associated with the linear dependence of on . Bounding the dual norm by the auxiliary variable allows the sensitivity of the constraint to uncertainty to be explicitly controlled.

To ensure computational tractability, the bound is expressed in second-order cone (SOC) form, which preserves convexity and enables the problem to be efficiently solved using standard conic optimization solvers. This SOC reformulation is a key step in converting the original Distributively Robust Chance Constraint into a deterministic and tractable optimization constraint.

If one prefers a moment-free DRCC with indicator via order-statistics (worst-γ samples) under Wasserstein balls, a mixed-integer conic reformulation is also obtainable by selecting the γ∣S∣ most adversarial scenarios (using binary selectors) and imposing their violation to be non-positive; we omit for brevity.

Upper-level multi-objective program

We minimize (expected) operating cost and a distributionally robust risk functional:

21

subject to logistics (1)–(9), DC-OPF (10)–(14), bounds (13), and the DRCC SOC (19)–(20).

Here x, y compactly denote all continuous and binary decisions. A scalarization used internally for evaluation is.

22

s.t. constraints (1)–(14),(19)–(20).

Deterministic MISOCP reformulation details

All nonconvexities arise from (i) outage disjunctions, (ii) logistics–power couplings, and (iii) DRCC.

• Outages. Disjunctive DC-OPF is linearized by (11)–(12) with tight .

• MEG power–binary couplings. Use perspective/McCormick:

23

Time-of-travel. Constraint (3) encodes a reservation of future slots; equivalently, one can pre-build a time-expanded arc set and write

24

eliminating (3); both are linear. Fuel SOC. If fuel-to-power rate is uncertain, (7) can be replaced by to embed bounds in SOC. DRCC SOC. The term (ψ/γ) u in (19) is enforced by a single SOC

25

• with v stacking line- and travel-sensitivities and H a diagonal or block-diagonal bound matrix.

The resulting problem is a Mixed-Integer Second-Order Cone Program (MISOCP) that modern conic solvers (Gurobi/CPLEX/MOSEK) handle efficiently.

Hybrid solution: NSGA-II + MISOCP

Because of discrete logistics and multi-objective trade-offs, we embed the MISOCP in a non-dominated sorting genetic algorithm:

Algorithm 1 (MODRCC–NSGA-II)

1. Initialize population ​ with diversified (MEG placements, routes, crew and refuel patterns).

2. For each candidate : solve MISOCP (22)–(25) to obtain (F₁, F₂) and feasibility.

3. Apply non-dominated sorting, crowding distance; form elites.

4. Crossover/mutation on the logistics chromosome (routes, refuels, crew shifts) with feasibility repair (respecting (1)–(9)).

5. Terminate when Pareto hypervolume improvement <ϵ or (e.g., 100).

6. Return Pareto set P^∗ (each point is a 12-h feasible restoration schedule).

This design exploits the conic solver for exact inner evaluations while retaining global exploration over combinatorial logistics.

The proposed model integrates three key layers: (i) time-expanded MEG logistics that capture realistic routing, refueling, and crew operations; (ii) an islanding-feasible DC-OPF ensuring power-flow stability under line outages; and (iii) a Wasserstein DRCC that safeguards resilience under uncertain attacks and mobility delays. By applying disjunctive linearization and SOC embeddings, the formulation becomes a tractable MISOCP, and when combined with NSGA-II, it efficiently produces Pareto-optimal strategies balancing cost and resilience.

Solution methodology and algorithmic implementation

Overview of the computational framework

The solution architecture combines the analytical robustness of Mixed-Integer Second-Order Cone Programming (MISOCP) with the exploratory adaptability of evolutionary²³ multi-objective search.

The optimization proceeds in two tightly coupled layers:

1. Inner Layer – Deterministic Conic Optimization:

   For each candidate MEG deployment and routing configuration, the deterministic MISOCP subproblem derived in Sect. 3.5 is solved to obtain the optimal operational schedule, line flows, and resilience indicators (cost and load shedding).

2. Outer Layer – Evolutionary Search:

   The Non-Dominated Sorting Genetic Algorithm II (NSGA-II) explores the combinatorial space of mobility and crew decisions. It continuously refines the Pareto front representing the trade-off between total operational cost and resilience performance.

This nested structure exploits the mathematical tractability of the lower level and the global search power of NSGA-II at the upper level, enabling convergence toward high-quality non-dominated solutions without linearization of discrete routing variables²⁴.

Algorithmic flow

The complete computational pipeline is illustrated conceptually in Fig. 2 (Flowchart of the Time-Coupled MODRCC–NSGA-II Framework).

Fig. 2.

Flowchart of the time-Coupled MODRCC – NSGA -II framework showing the interaction between evolutionary multi – objective search deterministic MISOCP evaluation under distributionally robust chance constraints.

The key procedural steps are summarized below.

1. Initialization:

   Generate an initial population of MEG configurations (x, y) through random feasible sampling that respects location and crew-availability constraints.

2. Fitness Evaluation:

   For each configuration, solve the deterministic MISOCP using Gurobi 12. The solver outputs.

   

   where C_op​ denotes total cost and R_shed​ total expected unserved energy.

3. Non-Dominated Sorting:

   Rank individuals into Pareto fronts based on dominance relationships. Each solution receives a rank and a crowding distance that balance convergence and diversity.

4. Selection, Crossover, and Mutation:

   Apply tournament selection (size = 2), single-point crossover (probability = 0.8), and bit-flip mutation (probability = 0.1) on binary variables representing routing and allocation.

5. Population Update:

   Combine parent and offspring populations, perform fast non-dominated sorting, and select the top N_p​ individuals for the next generation.

6. Termination:

   Repeat Steps 2–5 until either (a) the maximum number of generations N_gen=100 N is reached or (b) the Pareto front variation falls below 1%. The final non-dominated archive constitutes the decision support set.

Complexity and convergence analysis

Here N_p​ denote the population size, N_gen​ the number of generations, and T_MISOCP​ the average solution time of a single deterministic conic subproblem. The theoretical sequential complexity of the hybrid solver is

.

where ​ increases approximately polynomial with the number of buses and MEGs owing to the convex second-order cone constraints. In practice, fitness evaluations are executed in parallel across W = 32 worker threads, reducing the effective wall-clock complexity to

.

Empirical profiling on the IEEE-118 network (118 buses, 16 MEGs, 24 periods) with N_p=36, N_gen=70, and median T_MISOCP=86.7s yielded convergence after ≈ 70 generations with a total wall-clock time of about 2.8 h on an Intel Xeon Silver 4314 @ 2.4 GHz (32 threads, 64 GB RAM). Smaller IEEE-24 runs (6 MEGs, 24 buses) converged within ≈ 25 min. These results confirm that the proposed MODRCC–NSGA-II hybrid maintains a favourable trade-off between computational efficiency and solution quality, scaling effectively to real-world contingency-planning dimensions.

Solver integration and numerical stability

The deterministic MISOCPs are solved using Gurobi 12.0 with barrier and SOCP relaxation modes. Integer cuts from NSGA-II outputs are directly translated into binary constraints through a MATLAB–Python interface employing gurobipy and Py-MEX²⁵. To prevent oscillation in dual feasibility, warm-starting is used with previous iteration values of voltage angles and line flows. Constraint tolerance was set to 10^{− 6} and parallel branch-and-bound was enabled with thread count = 16. All subproblems achieved optimality gaps below 0.3%, ensuring reliable evaluation of Pareto fitness functions.

Flow of the hybrid solver

Each Pareto solution corresponds to a fully feasible 12-hour restoration schedule, including MEG locations, crew shifts, refueling windows, and power-flow decisions that satisfy DC-OPF and logistics simultaneously.

Advantages of the hybridization

The proposed framework is highly effective, combining mathematical precision with practical efficiency and scalability. It guarantees that all solutions are not only optimal but also physically viable, while efficiently exploring millions of potential combinations without getting bogged down. The system scales exceptionally well, maintaining fast performance even as the problem grows larger. Crucially, it delivers clear, interpretable results that visually present the trade-offs between competing goals, such as cost versus risk, to directly support strategic decision-making.

Validation and reproducibility

Benchmark runs were validated by comparing deterministic (ψ = 0) results against baseline DRCC and stochastic optimization models. Relative differences in unserved energy and cost were within ± 3%, confirming numerical consistency. All model files, parameter sheets, and NSGA-II seeds were version-controlled via MATLAB Live Link and Python 3.11, ensuring full reproducibility.

Results and discussion

Experimental setup

The developed multi-objective distributionally robust chance-constrained (MODRCC) framework was evaluated on the IEEE-24 Reliability Test System for validation and the IEEE-118 test network for large-scale performance assessment. The simulation horizon spans 12 h, discretized into 24 half-hour periods, representing an extended restoration window following a large-scale outage. Sixteen Mobile Emergency Generators (MEGs), each rated at 1.5 MW, were deployed across pre-defined staging depots. Randomized attack scenarios and road-blockage patterns were generated using Monte Carlo sampling of 500 realizations. The ambiguity set was built with Wasserstein radius ψ = 0.003 and risk tolerance γ = 0.1. All computations were executed on MATLAB R2023b with Gurobi 12.0 as the MISOCP solver.

Convergence characteristics

Figure 3 illustrates the convergence behaviour of the MODRCC–NSGA-II hybrid over 100 generations. The mean Pareto hypervolume stabilized after approximately 70 generations, indicating a balance between cost and resilience performance. Compared to baseline NSGA-II (without deterministic subproblem refinement), the proposed hybrid achieved a 27% faster convergence and a 15% improvement in Pareto diversity due to MISOCP-informed evaluation. To quantify computational efficiency, Table 1 lists average run times per subproblem and per generation.

Fig. 3.

Convergence characteristics of MODRCC-NSGA-II (IEEE-118).

Table 1.

Convergence and computational efficiency for MODRCC–NSGA-II.

Network	Buses	MEGs	Median subproblem time (s/worker)	Population size (Np)	Generations (Ngen)	Total wall-clock time (h)
IEEE-24	24	6	6.3	24	45	0.42
IEEE-118	118	16	86.7	36	70	2.8

Pareto front analysis

The proposed MODRCC formulation generated a smooth and continuous Pareto front between total operational cost and expected unserved energy (EUE). Figure 4 illustrates the Pareto surface for the IEEE-118 system, displaying the trade-off envelope obtained after 70 generations of the hybrid NSGA-II + MISOCP optimization. The front exhibits pronounced convexity, indicating well-balanced transitions between cost-oriented and resilience-oriented regimes—an effect attributed to the Wasserstein regularization, which penalizes abrupt probability-mass shifts within the ambiguity set and therefore ensures gradual risk transitions.

Fig. 4.

Pareto front: cost vs. resilience.

Three representative solutions characterize the front:

• Extreme Point A (Cost-Optimal): ₹45.3 million total cost, EUE = 92 MWh.

• Extreme Point B (Balanced): ₹47.1 million total cost, EUE = 61 MWh.

• Extreme Point C (Resilience-Optimal): ₹49.8 million total cost, EUE = 42 MWh.

The evolution from A to C demonstrates the intrinsic efficiency of risk-aware mobility planning: for roughly a 10% increase in total cost, the EUE drops by around 54% (92 → 42 MWh). This within-front improvement highlights how the MODRCC’s probabilistic constraints shift resource allocation toward high-risk load pockets while avoiding unnecessary redundancy elsewhere.

When compared against baseline models at a similar budget, the MODRCC reduces expected unserved energy by 14–20% relative to static DRCC and classical robust optimization approaches (ψ = 0). This advantage originates from the framework’s ability to explicitly capture correlated uncertainties in attack severity and travel-time delays via Wasserstein ambiguity sets. Unlike scenario-based methods that rely on finite samples, the distributionally robust representation guarantees resilience over an entire neighbourhood of probability distributions, translating into more stable restoration performance across unseen contingencies.

The convex shape of the Pareto front is analytically meaningful: it implies that the marginal cost of resilience increases smoothly rather than discontinuously, confirming that the multi-objective formulation is well-posed and not dominated by solver artifacts. Such curvature parallels the nonlinear trade-offs reported in multi-stage defence–restoration models, where coordinated resource mobility yields diminishing returns beyond a certain investment threshold. Hence, the MODRCC frontier serves as a practical decision-support curve for planners, enabling selection of operating points that balance fiscal limits and resilience targets without over-committing emergency assets.

Spatial deployment and routing outcomes

Figure 5 visualizes the spatiotemporal routing pattern of MEGs over the 12-hour horizon for one representative scenario. MEGs originating from Depots D1–D3 were dynamically repositioned toward the most affected load pockets during hours 4–10, achieving faster restoration of critical substations (buses 47, 61, 79). Crew fatigue limits and refueling constraints were strictly satisfied. The average travel delay across all units was 0.56 h, corresponding to 6–8% of total response time. This modest delay, relative to the mobility benefit, highlights the effectiveness of time-coupled routing optimization. Table 2 summarizes the dispatch and routing metrics for the three representative Pareto-optimal strategies.

Fig. 5.

The spatiotemporal routing pattern of MEGs over the 12-hour horizon.

Table 2.

Key logistics and operational metrics under three Pareto-optimal strategies.

Scenario	Total energy restored (MWh)	Mean travel delay (h)	Fuel utilization (%)	Crew shifts used
Cost-optimal	827	0.41	74.2	2
Balanced	861	0.56	81.3	3
Resilience-Optimal	882	0.63	88.1	4

Sensitivity to risk parameters (ψ, γ)

To examine model robustness, sensitivity analyses were conducted by varying Wasserstein radius ψ and violation probability γ. Results are shown in Fig. 6; Table 3.

Fig. 6.

Sensitivity of cost and EUE to ψ and γ.

Table 3.

Sensitivity of cost and resilience to risk parameters.

ψ	γ	Expected cost (₹ million)	EUE (MWh)	ΔEUE vs. base (%)
0.001	0.05	46.2	51	–17.3
0.003	0.10	47.1	61	Base
0.005	0.15	47.6	67	+ 9.8

A smaller ψ implies tighter distributional ambiguity (less uncertainty tolerance), improving resilience at a slight increase in cost. Conversely, relaxing γ beyond 0.1 noticeably increases unserved energy, confirming the importance of calibrated risk thresholds.

Overall, ψ = 0.003 and γ = 0.1 offered the best balance between computational tractability and resilience quality.

Comparative benchmarking

To benchmark performance, three existing models were reimplemented under the same test conditions:

1. RO Model — Robust Optimization without distributional ambiguity.

2. DAD Model — Classical tri-level Defender–Attacker–Defender.

3. Scenario-Based Mobility Model — MEG routing with deterministic travel times.

The proposed MODRCC consistently outperforms existing models by reducing EUE by 14–20% (Table 4)at comparable or lower operational cost, demonstrating the benefit of integrating mobility and chance-constrained robustness.

Table 4.

Comparative performance benchmarking with baseline models.

Model	Cost (₹ million)	EUE (MWh)	Avg. travel delay (h)	Computation time (h)
DAD	44.9	118	–	1.7
RO	46.5	71	–	2.1
Mobility (Lei et al.)	47.8	65	0.71	3.5
Proposed MODRCC	47.1	61	0.56	2.8

Positioning and status-reporting uncertainty

In field operations, crew dispatch relies on the MEG’s reported position and availability status, which can be affected by GNSS errors, multipath, intermittent connectivity, and delayed telemetry. To reflect this practical issue without overcomplicating the core restoration model, we evaluate the sensitivity of crew-feasibility and restoration performance to two dispatch frictions: (i) positioning error and (ii) status-reporting delay. Specifically, we model the reported location of MEG k at time t as ​, which may differ from the true node with probability ρ (mislocalization rate). When mislocalized, crews may be dispatched to causing an additional search/relocation delay before service begins. Likewise, this model delayed status updates via a reporting latency δ, such that crew decisions at time t are based on MEG state information from t − δ. These two effects reduce the effective availability of crews and increase the time-to-service, which can propagate into delayed refuelling/connection actions and higher unserved energy.

Effective service time penalty

Replace your crew service time (for connect/refuel/move) by an effective service time:

where is the base service time, is the additional time due to searching/relocating when location is wrong, and is the expected delay due to stale information (e.g., waiting for confirmation before starting service). Operationally, this can be implemented by tightening the crew shift window ​ or increasing the minimum crew-hours consumed per action.

Sensitivity of personnel scheduling to localization and reporting delays

We conducted a sensitivity study by varying the mislocalization rate , the additional localization-induced delay minutes, and reporting latency time steps (each step = 30 min). For each setting, we re-evaluated the selected Pareto solutions and measured (i) crew-feasibility violations (number of time steps where required crew could not be assigned), (ii) MEG service start delay, and (iii) the resulting changes in EUE and total cost. As expected, increasing ρ and δ reduces the effectiveness of personnel scheduling by delaying connect/refuel actions and increasing travel/idle time; this translates into a measurable increase in EUE and a mild increase in cost due to more conservative schedules and/or additional crew utilization. Importantly, the degradation is graceful: moderate errors ( step) cause limited performance loss, while larger errors (ρ = 10%, δ = 2 steps) noticeably shift the Pareto front upward in EUE, indicating that reliable telemetry is a key enabling requirement for centralized restoration coordination.

Table 5 quantifies the sensitivity of personnel scheduling effectiveness to MEG positioning errors and delayed status reporting. While moderate telemetry imperfections cause only marginal degradation, larger localization errors and reporting delays significantly increase crew infeasibility and expected unserved energy, highlighting the importance of reliable real-time communication for centralized coordination.

Table 5.

Impact of positioning error and reporting delay on crew effectiveness and restoration outcomes.

Mislocalization rate ρ	Δtloc (min)	Reporting delay δ (steps)	Avg. service start delay (min)	Crew-feasibility violations (%)	ΔEUE (%) vs. baseline	ΔCost (%) vs. baseline
0.00	0	0	0	0.0	0.0	0.0
0.05	15	1	18	2.1	+ 3.4	+ 1.2
0.05	30	1	27	3.6	+ 5.9	+ 1.9
0.10	15	2	34	6.8	+ 9.7	+ 2.8
0.10	30	2	46	9.4	+ 14.6	+ 3.7

Communication signaling overhead and network requirements

The proposed centralized coordination requires periodic status reporting from Mobile Emergency Generators (MEGs) and crews, together with downlink control commands for routing, connection, and dispatch. The signaling load is modeled as the combination of uplink telemetry, downlink control, and acknowledgement messages.

Let K be the number of MEGs, RRR the number of crews, T the number of time steps with duration Δt, and ​ the reporting interval. Event-triggered updates (e.g., rerouting, refueling) occur with probability p_e per step. Telemetry, control, and acknowledgement payloads are denoted by S_u​, S_d​, and S_a​, respectively.

The expected total number of messages over the horizon is approximated as:

and the total data volume as:

yielding an average required throughput:

This provides a conservative sizing estimate for disaster-resilient communication technologies such as DSRC/ITS-G5, LTE/5G, private LTE, or satellite backhaul. In practice, latency and link availability are more critical than bandwidth.

Even for IEEE-118, the required average throughput is only a few kbps, indicating that bandwidth is not a limiting factor. The dominant challenges are latency, reliability, and intermittent connectivity in post-disaster conditions (Table 6). The framework therefore remains feasible under DSRC or LTE/5G-class links and can be further strengthened using adaptive reporting, event-triggered updates, and hierarchical coordination.

Table 6.

Estimated signaling overhead and implied network requirements (12-h horizon, 24 steps) .

Network	MEGs (K)	Crews (R)	Su​ (B)	Sd(B)	Sa(B)	Nmsg​	Volume (MB)	(kbps)	Latency target
IEEE-24	6	6	256	192	64	~ 17,500	~ 4.2	~ 0.8	< 200 ms
IEEE-118	16	8	256	192	64	~ 46,500	~ 11.1	~ 2.1	< 200 ms

Discussion

The improvement observed when switching from a classical robust model to a distributionally robust chance-constrained (DRCC) formulation (EUE ↓ 14–20%) stems from the model’s ability to bound risk using data-driven ambiguity sets. In conventional robust optimization, uncertainty is treated deterministically, which forces all constraints to be satisfied under the worst-case scenario. This guarantees feasibility but creates over-conservatism, leading to excessive reserve scheduling and inflated cost. By contrast, the Wasserstein-based DRCC framework defines a probabilistic confidence region around empirical sample. This approach leverages the actual dispersion of contingency data rather than assuming a single extreme realization. Hence, the optimizer learns to allocate MEGs preferentially toward high-risk clusters while avoiding redundant redundancy in low-probability regions. This balance explains why cost increases by only ~ 3% while resilience improves drastically²⁶.

The inclusion of travel-time, crew-shift, and refueling constraints generated an average delay of 0.56 h per MEG, yet overall outage restoration accelerated compared with static-allocation models.

This apparently paradoxical outcome can be explained by the temporal coupling mechanism: once mobility is modeled explicitly, early-period MEG dispatches are dynamically re-routed to restore critical buses first, producing a cascading acceleration in later-period recovery.

This phenomenon parallels the findings, that mobility scheduling can reduce restoration time by up to 30% when prioritized by load importance. However, their work neglected stochastic travel-time variability; our DRCC formulation internalizes this variability via chance constraints, ensuring that these gains persist even under road disruptions. In essence, temporal awareness replaces brute-force redundancy with intelligent anticipation, making resilience both faster and leaner²⁷.

The smooth convex Pareto front (Fig. 4) reflects a fundamental property of the combined cost–risk objective: both terms share partially correlated sensitivities to MEG allocation.

At low-cost regimes, additional MEGs are under-utilized, resulting in marginal resilience gain.

As cost weight (α) relaxes, the optimizer shifts toward high-risk nodes, producing nonlinear improvements in unserved energy reduction. Such Pareto curvature has also been reported in multi-stage defence–restoration models by Ghorbani-Renani et al. (2022), validating that our observed trade-off is not numerical artifact but a structural feature of resilience optimization²⁸. The flattening of the Pareto tail at higher cost implies diminishing marginal resilience returns, a principle that can inform policy budgeting for emergency infrastructure investment. The sensitivity results (Table 5) reveal two important dynamics; one is reducing the ambiguity radius tightens the confidence set around observed scenarios²⁹. The optimizer thus concentrates on plausible near-empirical outcomes rather than extreme hypotheticals, improving load-restoration reliability. It shows that decreasing ψ shrinks dual penalties in the conic reformulation, thereby tightening feasible space³⁰. Second one is increasing allowable violation probability effectively relaxes the risk constraint, letting rare high-loss events exceed target thresholds.

As shown in chance-constrained under-frequency load-shedding, small γ values yield better balance between cost and security. Our γ = 0.1 optimum is consistent with their reported 0.08–0.12 operationally stable range. Together these trends confirm that the model’s risk–cost sensitivity is mathematically consistent with existing literature, reinforcing confidence in parameter selection. Spatially, MEGs exhibited a distinct pattern of radial-to-central migration³¹ during the 6–10 h window. Early dispatches favoured peripheral feeders with isolated loads, while later redeployments targeted central substations supplying urban corridors. This pattern arises because the lower-level DC-OPF penalizes voltage-angle deviations, making central nodes more effective for restoring network synchronism once peripheral islands have power. The same physical rationale underpins the results of Wan et al. (2024). who observed similar load-pocket prioritization in pre-disaster allocation models³². Additionally, fuel consumption peaked between hours 8–10 when multiple MEGs converged near high-load buses. Because refueling delay constraints prevented over-commitment, the algorithm naturally scheduled rolling replenishment cycles — a behaviour absent in deterministic mobility models. This self-regulating dynamic demonstrates the realism introduced by coupling energy and logistics variables within one optimization layer³³.

Compared with the classical DAD formulation and the robust counterpart of the MODRCC consistently produced lower expected unserved energy at marginally higher but more stable costs³⁴. This outcome validates the theoretical assertion of Mehrotra that distributional robustness yields risk-calibrated solutions, achieving superior expected performance without over-conservatism³⁵. Furthermore, unlike the scenario-based mobility model of, which depends on predefined route probabilities, our method derives routes endogenously within the optimization, producing adaptive reallocation when new attack patterns arise. The 14–20% improvement in resilience thus originates not from heuristic tuning but from the joint mathematical coupling of (i) Wasserstein ambiguity and (ii) temporal mobility constraints—a novel interaction not present in prior works. From a system-operation standpoint, the results imply that mobility-aware robust dispatch can replace static pre-positioning strategies currently used in many regional utilities. For a similar budget, planners can achieve shorter recovery times and higher served energy by pre-training MEG deployment schedules on distributionally robust policies³⁶. The Pareto frontier effectively acts as a decision-support curve for planners to balance expenditure versus expected resilience gain.

Limitations and future work

Despite the contributions of this study, several limitations should be acknowledged. The current model assumes accurate real-time MEG positioning, perfect communication, and deterministic travel times, which may not fully reflect practical post-disaster conditions. Similarly, the model assumes that personnel can be dispatched instantly to service MEGs, without delays caused by inaccurate location reporting or status updates.

Future research could address these limitations and extend the framework in several directions. Potential extensions include incorporating probabilistic location uncertainty, communication delays, and predictive or hierarchical crew dispatch strategies; moving from DC-OPF to AC-feasible relaxations with voltage and reactive power limits; co-optimizing MEGs with other mobile assets such as storage units, repair crews, or sectionalizing switches; incorporating learning-based priors for road accessibility and correlated hazards; and adding equity or critical-service weights to prioritize hospitals and lifeline loads. Finally, field-pilot calibration with Indian utility partners could validate travel-time assumptions and crew productivity parameters. These extensions would enhance the robustness, realism, and operational applicability of MEG deployment planning under uncertain, post-disaster conditions.

Conclusion

We proposed a MODRCC framework that co-optimizes MEG pre-positioning, routing, crew shifts, and refueling while enforcing islanding-feasible DC-OPF under line outages. The Wasserstein DRCC treatment provides risk-calibrated protection against ambiguity in both interdiction severity and mobility delays, and the MISOCP + NSGA-II hybrid efficiently explores discrete–continuous trade-offs. Across IEEE-24 and IEEE-118, the approach consistently achieves 14–20% lower EUE than robust/static DRCC baselines at similar cost, and exhibits a convex Pareto frontier that enables transparent budget–resilience decisions for planners.

Author contributions

**D. Ashokaraju: Conceptualization, Methodology, Investigation, Data curation.**ML Ramamoorthy: Formal analysis, Validation, Investigation, Writing - review & editing.**Deepa Simon: Funding acquisition, Writing - review & editing.**N Ashok: Funding, Supervision**Abhijit Bhowmik: Conceptualization, Methodology.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial or not-for-profit sectors.

Data availability

All the data required are available within the manuscript.

Declarations

Ethical statement

This study did not involve human or animal subjects. All experiments followed institutional safety and waste management guidelines. Ethical research practices were maintained, ensuring accurate data collection, analysis and reporting.

Competing interests

The authors declare no competing interests.