Model Predictive Black Start (MPBS)

• MPBS is a predictive restoration framework that uses finite-horizon optimization with receding control to determine the first-step actions after blackout.
• It integrates inverter-based resource coordination, synchronization, load pickup, and dynamic security constraints to enable safe, bottom-up grid recovery.
• Direct MPBS models screen for transient issues like transformer inrush and frequency nadir violations, with reported validations exceeding 90% accuracy in key scenarios.

Model Predictive Black Start (MPBS) denotes a class of black-start and restoration methods in which post-blackout recovery is posed as a finite-horizon predictive optimization problem and only the first restoration action is implemented before the horizon shifts and the problem is solved again. In the literature represented here, direct MPBS formulations appear most clearly in distribution-system restoration with inverter-based resources, where bottom-up black start, switching, load pickup, synchronization, and dynamic security are optimized over a receding horizon (Bai et al., 16 Jul 2025, Zheng et al., 18 Aug 2025). Closely related transmission-level work also uses an $N$-step receding or rolling horizon with explicit frequency-nadir constraints and coordinated energy storage support, although its control model remains a restoration planner rather than a fully measurement-driven MPC implementation (Zou et al., 13 May 2026). Around these direct formulations lies a broader body of precursor work on generator startup sequencing, sectionalization, line recovery, converter black start, and decentralized restoration; these papers are often structurally important for MPBS, but several explicitly are not themselves MPBS because they omit receding-horizon execution, closed-loop feedback, or explicit predictive optimization (Chopra et al., 2022, Zhao et al., 2023, Zeng et al., 2024).

1. Scope and classification

Within the current literature, MPBS is best distinguished from three neighboring categories: direct receding-horizon black-start frameworks, predictive but non-receding restoration optimizations, and lower-layer or benchmark control architectures. The distinction is substantive because many black-start papers optimize over time yet do not implement horizon shifting or first-step-only execution.

Class	Representative papers	Characterization
Direct MPBS	(Bai et al., 16 Jul 2025, Zheng et al., 18 Aug 2025, Zou et al., 13 May 2026)	Receding or rolling horizon with first-step implementation and explicit security models
Predictive but not MPBS	(Chopra et al., 2022, Zhao et al., 2023, Maharjan et al., 2024, Bai et al., 18 Aug 2025)	Finite-horizon, stochastic, or time-indexed restoration without full receding-horizon control
Lower-layer or benchmark black start	(Zeng et al., 2024, Lee et al., 24 Nov 2025, Nguyen et al., 2021)	Device- or feeder-level black-start feasibility and control, not predictive supervisors

This classification matters because the literature repeatedly warns against conflating optimization-based restoration with MPBS. “Parallel Power System Restoration” formulates integrated sectionalization and generator startup sequencing, but it is “not” an MPC controller and is best interpreted as an open-loop deterministic predictive schedule (Chopra et al., 2022). “Power System Recovery Coordinated with (Non-)Black-Start Generators” is similarly a multi-period MILP with binary line recovery and non-black-start generator energization, but the model is solved over a fixed horizon rather than in receding-horizon form (Zhao et al., 2023). “DERs-Aided Blackstart and Load Restoration Framework for Distribution Systems Considering Synchronization and Frequency Security Constraints” is highly relevant to MPBS because it already contains time-indexed switching, synchronization, and frequency-security constraints, yet it explicitly remains a precursor rather than an MPC implementation (Maharjan et al., 2024).

A further distinction concerns black-start context. Distribution-level MPBS is shaped by grid-forming inverters, cold-load pickup, bus-block energization, synchronization of microgrids, and increasingly transformer inrush and protection constraints (Bai et al., 16 Jul 2025, Zheng et al., 18 Aug 2025). Transmission-level predictive restoration, by contrast, emphasizes generator startup sequencing, cranking power, line energization, inertia, governor response, and system frequency nadir (Zou et al., 13 May 2026, Chopra et al., 2022).

2. Predictive optimization structure

Direct MPBS formulations in the current literature share a common control pattern: forecast over a finite horizon, solve a mixed-integer restoration problem, implement only the first step, update the network state, and repeat. In “Grid Edge Intelligence-Assisted Model Predictive Framework for Black Start of Distribution Systems with Inverter-Based Resources,” the common prediction horizon is

$$\{t+k\Delta t : k \in \mathcal{N}_k\}, \qquad \mathcal{N}_k = \{1,2,\dots,N\},$$

with a 3-hour horizon, a 15-minute time step, and thus $N=12$ in the case study (Zheng et al., 18 Aug 2025). The utility-side objective is

$$\max \sum_{k \in \mathcal{N}_k} \sum_{i \in \mathcal{N}_i} \sum_{p \in \mathcal{N}_p} P^D_{i,p,k} \Delta t,$$

so restored active power is maximized over the horizon subject to switching, power-flow, energy, synchronization, and frequency constraints (Zheng et al., 18 Aug 2025). The same paper adds a second predictive layer at the grid edge: each Grid Edge Intelligence (GEI) device computes multi-period upper and lower flexibility bounds and receives only aggregate dispatch signals, so the utility coordinates behind-the-meter DERs without requiring detailed asset information (Zheng et al., 18 Aug 2025).

A closely related receding-horizon structure appears in “Frequency Nadir-Constrained Power System Restoration Planning with Energy Storage.” There the restoration plan is built as a sequence of discrete actions every $t_a$ minutes; the modified IEEE 9-bus case uses actions every 2 minutes, and the algorithm solves an $N$-step rolling-horizon MILP, commits only the first step, updates the restoration state, and repeats (Zou et al., 13 May 2026). Its objective maximizes weighted occupancy of restored generators, loads, lines, and ESSs over the horizon, with weights satisfying $w_{\rm g}\gg w_{\rm d}\gg w_{\rm l},w_{\rm s}$, thereby prioritizing early generator recovery and then load recovery (Zou et al., 13 May 2026). The distinctive feature is that the planner embeds a predictive frequency-security model rather than a static pickup rule.

A direct distribution-system MPBS architecture is formalized in “Model Predictive Black Start for Dynamic Formation of DER-Led Microgrids with Inrush Current Impacts.” The controller receives short-term forecasts of DER output and transmission-grid availability, solves a restoration optimization over a control horizon $\mathcal{T}_c$, submits the first-step switching action to an inrush-feasibility module, modifies the decision if transformer inrush would threaten protection devices, executes the first step, and then rolls the horizon forward (Bai et al., 16 Jul 2025). This is an MPC structure in the conventional sense: optimization over a horizon, feasibility screening, first-step implementation, and iterative replanning.

By contrast, several mathematically explicit restoration models remain outside direct MPBS. “Parallel Power System Restoration” minimizes restoration time $RT$ with

$\min RT$

under assignment, capacity-balance, and connectivity constraints, but solves a deterministic planning problem over a fixed horizon (Chopra et al., 2022). Its generator startup sequencing subproblem is NP-hard, which is significant for MPBS because it identifies a hard combinatorial core that any receding-horizon controller must confront (Chopra et al., 2022).

3. Distribution-system realizations

The most developed MPBS realizations in the present literature are distribution-level and DER-led. “Grid Edge Intelligence-Assisted Model Predictive Framework for Black Start of Distribution Systems with Inverter-Based Resources” considers bottom-up restoration after upstream transmission-grid outage, with two grid-forming BESSs, grid-following PVs, bus blocks, energizing switches, synchronizing switches, and GEI-enabled households (Zheng et al., 18 Aug 2025). The case study uses a modified IEEE 123-bus distribution system with 88 load nodes, 8 GFL PVs, 2 GFM BESS units, and 11 bus blocks. The restoration sequence begins at 9:00, all bus blocks are energized by 10:00, the two islands synchronize at 10:15, the transmission grid recovers at 11:45, and TG-DS synchronization occurs at 12:00 (Zheng et al., 18 Aug 2025). GEI penetration is varied from 15% to 100%; the total restored load-hours increase from 5.36 MWh at 15% GEI to 6.44 MWh at 100% GEI, an improvement of about 20.1% (Zheng et al., 18 Aug 2025).

The same framework explicitly embeds three analytical frequency-security metrics for VSG-controlled GFM BESSs. The quasi-steady-state frequency deviation is

$$\{t+k\Delta t : k \in \mathcal{N}_k\}, \qquad \mathcal{N}_k = \{1,2,\dots,N\},$$0

the RoCoF proxy is

$$\{t+k\Delta t : k \in \mathcal{N}_k\}, \qquad \mathcal{N}_k = \{1,2,\dots,N\},$$1

and the nadir approximation is

$$\{t+k\Delta t : k \in \mathcal{N}_k\}, \qquad \mathcal{N}_k = \{1,2,\dots,N\},$$2

all constrained within prescribed bounds during load pickup and synchronization (Zheng et al., 18 Aug 2025). This gives MPBS a tractable surrogate for inverter-frequency dynamics without embedding EMT simulation in the optimization.

“Model Predictive Black Start for Dynamic Formation of DER-Led Microgrids with Inrush Current Impacts” extends the distribution-level MPBS idea by making transformer inrush feasibility part of the receding-horizon loop (Bai et al., 16 Jul 2025). The distribution system is partitioned into bus blocks connected by ESWs and SSWs; GFMI-based BESSs form initial microgrids, PV output and TG availability are forecast over the horizon, and a mixed-integer restoration schedule is repeatedly recomputed (Bai et al., 16 Jul 2025). What differentiates this framework is that each candidate first-step ESW action is screened for no-load distribution-transformer inrush current and consequent fuse or recloser misoperation. If the action is infeasible, the framework triggers emergency-operation-inspired voltage reduction and switch blocking, then re-optimizes (Bai et al., 16 Jul 2025).

A precursor with many of the same structural ingredients is “DERs-Aided Blackstart and Load Restoration Framework for Distribution Systems Considering Synchronization and Frequency Security Constraints.” It uses a modified unbalanced IEEE-123-bus feeder, bus-block restoration, synchronization switches, BES-based GFMIs, GFLI PV delay, cold-load pickup, and dynamic radiality, but it is not formulated as receding-horizon MPC (Maharjan et al., 2024). The paper’s synchronization-aware optimization improved customer-hours restored by 0.22% to 2.5% over a benchmark in the 2-GFMI case and by 4.86% to 11.46% in the 4-GFMI case, while reducing restoration time by 15 minutes and 30–45 minutes respectively as TG recovery was delayed (Maharjan et al., 2024). This suggests that synchronization logic and dynamic root-count evolution are core ingredients for distribution-level MPBS even when the supervisory controller is not yet receding-horizon.

A related stochastic planning framework is “Stochastic Black Start Resource Allocation to Enable Dynamic Formation of Networked Microgrids and DER-aided Restoration,” which allocates GFMI-based BESS black-start resources and smart switches in a first stage and optimizes scenario-dependent restoration of a modified IEEE 123-node feeder in a second stage (Bai et al., 18 Aug 2025). The paper is not MPBS, but its synchronizing dynamic microgrids (SDMG-BS) formulation shows how scenario-based uncertainty in PV, load, and TG outage duration can be coupled with frequency-security constraints and dynamic microgrid formation (Bai et al., 18 Aug 2025). A plausible implication is that such two-stage stochastic allocation can serve as an offline planning layer above a real-time MPBS supervisor.

4. Dynamic security and device-level feasibility

MPBS departs from earlier heuristic black-start scheduling chiefly by embedding explicit dynamic-security and device-feasibility models. In the transmission-level receding-horizon planner of (Zou et al., 13 May 2026), the essential frequency-security relation is an affine disturbance budget: $$\{t+k\Delta t : k \in \mathcal{N}_k\}, \qquad \mathcal{N}_k = \{1,2,\dots,N\},$$3 where $$\{t+k\Delta t : k \in \mathcal{N}_k\}, \qquad \mathcal{N}_k = \{1,2,\dots,N\},$$4 is the maximum tolerable disturbance without ESS support and $$\{t+k\Delta t : k \in \mathcal{N}_k\}, \qquad \mathcal{N}_k = \{1,2,\dots,N\},$$5 captures the marginal benefit of ESS setpoint changes (Zou et al., 13 May 2026). This relation is derived from a reduced-order aggregate-frequency model with online inertia, available primary frequency response, turbine-governor dynamics, and ESS first-order response. The paper shows that a frequency-unconstrained restoration plan can produce frequency excursions exceeding $$\{t+k\Delta t : k \in \mathcal{N}_k\}, \qquad \mathcal{N}_k = \{1,2,\dots,N\},$$6 Hz, a 5% heuristic cap can still produce excursions as large as $$\{t+k\Delta t : k \in \mathcal{N}_k\}, \qquad \mathcal{N}_k = \{1,2,\dots,N\},$$7 Hz, and the nadir-constrained receding-horizon plan respects a 1 Hz nadir limit in MATLAB and PSS/E validation (Zou et al., 13 May 2026). With a 50 MWh / 10 MW ESS at bus 5, restoration time drops from 94 minutes to 50 minutes while maintaining frequency security, although the paper notes slight limit exceedance caused by linearization of the ESS-dependent nadir bound (Zou et al., 13 May 2026).

In the distribution-level MPBS of (Bai et al., 16 Jul 2025), dynamic feasibility is protection-centered. The transformer-saturation indicator is

$$\{t+k\Delta t : k \in \mathcal{N}_k\}, \qquad \mathcal{N}_k = \{1,2,\dots,N\},$$8

and, when saturation occurs, peak inrush current is estimated by

$$\{t+k\Delta t : k \in \mathcal{N}_k\}, \qquad \mathcal{N}_k = \{1,2,\dots,N\},$$9

The steady saturated current is computed from a dynamic microgrid Thevenin equivalent,

$N=12$0

so the inrush estimate changes as the microgrid expands and source impedance changes (Bai et al., 16 Jul 2025). The inrush module then aggregates transformer inrush over laterals and microgrid feeder heads and compares those currents against fuse minimum 2-cycle melting currents and recloser 2-cycle rapid-act currents (Bai et al., 16 Jul 2025). The analytical model was validated against PowerFactory EMT simulation with peak-current estimation accuracies of 93.29%, 94.36%, 95.17%, and 92.70% at switching angles $N=12$1, $N=12$2, $N=12$3, and $N=12$4 (Bai et al., 16 Jul 2025).

Lower-layer converter behavior is another major constraint source for MPBS. “Black Start Operation of Grid-Forming Converters Based on Generalized Three-phase Droop Control Under Unbalanced Conditions” studies bottom-up restoration after a complete blackout in the IEEE 13-bus feeder using two grid-forming voltage-source converters and generalized three-phase droop control under unbalanced conditions (Zeng et al., 2024). The per-phase droop equations are

$N=12$5

$N=12$6

with phase-balancing gain $N=12$7 trading voltage balance against power balance and per-phase current limiting

$N=12$8

during asymmetrical faults or motor starting (Zeng et al., 2024). The paper explicitly is not MPC, but it provides dynamics and constraints that a converter-dominated MPBS must respect.

At the transformer-energization timescale, “Accelerated Transformer Energization Sequence for Inverter Based Resources in Black-Start Procedures with Active Flux Trajectory Manipulation in the Stationary Reference Frame” isolates the subproblem of safe transformer energization by a GFM inverter with LC filter (Lee et al., 24 Nov 2025). Hard energization induces flux offset

$N=12$9

whereas the ultra-fast soft-magnetization method completes startup in

$$\max \sum_{k \in \mathcal{N}_k} \sum_{i \in \mathcal{N}_i} \sum_{p \in \mathcal{N}_p} P^D_{i,p,k} \Delta t,$$0

the Archimedean spiral method completes in

$$\max \sum_{k \in \mathcal{N}_k} \sum_{i \in \mathcal{N}_i} \sum_{p \in \mathcal{N}_p} P^D_{i,p,k} \Delta t,$$1

and a demagnetization-plus-soft-start sequence completes in about 60 ms (Lee et al., 24 Nov 2025). The paper is not MPBS, but it shows that transformer energization itself is a constrained transient-trajectory problem, not merely a discrete switching action.

5. Transmission-level and precursor formulations

A large portion of the MPBS-adjacent literature concerns bulk-system restoration planning rather than direct receding-horizon control. “Parallel Power System Restoration” formulates restoration as coupled sectionalization and generator startup sequencing. For a single black-start source, its generator startup sequencing ILP is

$$\max \sum_{k \in \mathcal{N}_k} \sum_{i \in \mathcal{N}_i} \sum_{p \in \mathcal{N}_p} P^D_{i,p,k} \Delta t,$$2

subject to each NBS generator starting exactly once and total available capacity remaining nonnegative at every period (Chopra et al., 2022). The paper proves that GSS is NP-hard and then integrates startup scheduling with connected BS-rooted island formation through assignment variables, line-island variables, and single-commodity flow constraints (Chopra et al., 2022). The contribution to MPBS is foundational rather than direct: it shows that topology and startup decisions should be optimized jointly, and it supplies exact ILP models, lower bounds, upper bounds via random spanning trees and local search, and heuristic scalability to systems above 1000 buses (Chopra et al., 2022).

“Power System Recovery Coordinated with (Non-)Black-Start Generators” supplies a different precursor. It formulates a DCOPF-based MILP in which non-black-start generators are loads before startup and generators after startup (Zhao et al., 2023). The key mixed-integer transition is

$$\max \sum_{k \in \mathcal{N}_k} \sum_{i \in \mathcal{N}_i} \sum_{p \in \mathcal{N}_p} P^D_{i,p,k} \Delta t,$$3

with

$$\max \sum_{k \in \mathcal{N}_k} \sum_{i \in \mathcal{N}_i} \sum_{p \in \mathcal{N}_p} P^D_{i,p,k} \Delta t,$$4

so an NBS unit consumes fixed cranking power until energized, then becomes dispatchable (Zhao et al., 2023). Binary line recovery variables and cumulative line states couple startup feasibility to transmission-path recovery. This is directly relevant to MPBS because it provides state-like variables for line availability and NBS status, but the paper itself is a finite-horizon open-loop restoration MILP rather than a receding-horizon controller (Zhao et al., 2023).

“Frequency Nadir-Constrained Power System Restoration Planning with Energy Storage” moves closest to transmission-level MPBS proper because it uses rolling-horizon replanning, generator startup phases, ESS state of charge, line energization, and predictive dynamic-security coefficients in a single subsystem already sectionalized around one black-start unit (Zou et al., 13 May 2026). Yet even here the controller state is only partially feedback-driven: future dynamic-security coefficients are frozen each iteration under a simplified assumption so that the overall problem remains MILP rather than MINLP (Zou et al., 13 May 2026).

Resource-model papers broaden the scope of what MPBS can coordinate. “A Black Start Strategy for Hydrogen-integrated Renewable Grids with Energy Storage Systems” gives a time-indexed restoration optimization for fuel cells and batteries as black-start resources, including FC startup, cranking, ramping, and full-output phases, together with bus and line energization (Lu et al., 19 Jul 2025). On the IEEE 39-bus system, adding two 50 MW fuel cells reduced average generator startup time from 182.2 minutes to 100.0 minutes, while adding two 50 MW batteries reduced it to 102.2 minutes (Lu et al., 19 Jul 2025). “Control and Simulation of a Grid-Forming Inverter for Hybrid PV-Battery Plants in Power System Black Start” contributes a dynamic EMT resource model for a grid-forming PV–battery plant at bus 1 of a modified IEEE 9-bus system, with primary droop, secondary PI restoration, staged load pickup, and auxiliary-load support during line and transformer energization (Nguyen et al., 2021). “Load Restoration Methodology Considering Renewable Energies and Combined Heat and Power Systems” provides an earlier heuristic precursor in which a Greedy algorithm ranks loads by the compound-load score

$$\max \sum_{k \in \mathcal{N}_k} \sum_{i \in \mathcal{N}_i} \sum_{p \in \mathcal{N}_p} P^D_{i,p,k} \Delta t,$$5

then validates each pickup with frequency-sensitive dynamic power-flow constraints (Alwan et al., 2018). These are not MPBS, but they supply objective terms, constraints, and resource models that later predictive black-start formulations can absorb.

6. Limitations and distinctions

A recurring misconception is that any time-indexed black-start optimization is MPBS. The literature here draws a stricter boundary. Direct MPBS requires receding or rolling horizon execution with first-step implementation and state update (Bai et al., 16 Jul 2025, Zheng et al., 18 Aug 2025, Zou et al., 13 May 2026). Deterministic fixed-horizon planning models, even when highly predictive and multi-period, remain distinct (Chopra et al., 2022, Zhao et al., 2023). Synchronization-aware distribution restoration with frequency-security constraints and stochastic black-start resource allocation are similarly close precursors rather than full MPBS implementations (Maharjan et al., 2024, Bai et al., 18 Aug 2025).

A second misconception is that black-start feasibility can be protected by static heuristics alone. The transmission-level nadir study shows that unconstrained restoration can produce frequency deviations exceeding $$\max \sum_{k \in \mathcal{N}_k} \sum_{i \in \mathcal{N}_i} \sum_{p \in \mathcal{N}_p} P^D_{i,p,k} \Delta t,$$6 Hz, while a 5% heuristic still permits excursions as large as $$\max \sum_{k \in \mathcal{N}_k} \sum_{i \in \mathcal{N}_i} \sum_{p \in \mathcal{N}_p} P^D_{i,p,k} \Delta t,$$7 Hz; the predictive nadir constraint is materially tighter (Zou et al., 13 May 2026). The DER-led MPBS with inrush feasibility makes the same point at the protection layer: a restoration action that is acceptable in power-flow terms can still be infeasible because no-load transformer inrush may melt fuses or operate reclosers (Bai et al., 16 Jul 2025).

A third misconception is that converter-based black start can be abstracted entirely at the scheduling layer. The converter literature indicates otherwise. Distribution-level GFM restoration must represent limited converter current capability, per-phase current saturation, unbalanced loading, motor inrush, and transformer energization transients (Zeng et al., 2024, Lee et al., 24 Nov 2025). This suggests that practical MPBS will remain hierarchical: a supervisory predictive optimizer will choose sequencing, synchronization, and dispatch, while lower-layer GFM controls, current limiters, and energization trajectories enforce fast electromagnetic feasibility. This hierarchical interpretation is explicit in several supporting papers even when those papers are not themselves predictive controllers (Zeng et al., 2024, Nguyen et al., 2021).

Current MPBS formulations also have visible limitations. The GEI-assisted framework uses deterministic forecasts, requires blackout-resilient communications between utility and edge devices, and centralizes envelope-based coordination over many households without reporting scalability limits (Zheng et al., 18 Aug 2025). The transmission-level nadir-constrained planner uses a reduced-order aggregate-frequency model, constrains nadir but not RoCoF explicitly, and exhibits slight limit exceedance in the ESS case because the nonlinear nadir bound is linearized around zero ESS action (Zou et al., 13 May 2026). The inrush-aware distribution MPBS relies on analytical no-load transformer models and a worst-case switching angle rather than full EMT simulation inside the controller, although the reported validation accuracy exceeds 90% (Bai et al., 16 Jul 2025). A plausible implication is that future MPBS research will continue to combine tractable predictive optimization with progressively richer transient-feasibility filters rather than attempting to internalize full EMT models.

Finally, predictive control in power systems should not be confused with black-start control. “A Model Predictive Approach to Preventing Cascading Failures of Power Systems” is a model-predictive emergency load-shedding method for blackout prevention, not a black-start paper (Zhai et al., 2017). Its relevance to MPBS is methodological: prediction-informed sequential optimization under evolving topology. Its operational problem, however, is cascading-failure mitigation rather than restoration (Zhai et al., 2017). That distinction reinforces the defining feature of MPBS: it is prediction-based restoration after blackout, not predictive protection before or during collapse.