Generator Startup Sequence (GSUS)

• Generator Startup Sequence (GSUS) is a control framework that defines the transition from inactivity to productive operation using parameterized trajectories and dynamic constraints.
• It applies in diverse contexts—from hydro turbines and space nuclear reactors to grid black-start and GPU initialization—each tailored with specific optimization models.
• Active-learning, mixed-integer, and continuous optimization techniques in GSUS reduce startup fatigue, cut recovery times, and balance energy and operational constraints.

Searching arXiv for the cited GSUS-related papers to ground the article in current records. Search query: "(Mai et al., 2024) Active Learning-Based Optimization of Hydroelectric Turbine Startup to Minimize Fatigue Damage" Generator Startup Sequence (GSUS) denotes a family of startup-control or startup-ordering formulations used to transition a generator, generating unit, or generator-like subsystem from rest, outage, or initialization to productive operation under dynamic and operational constraints. In the recent literature, the term is not tied to a single canonical model. In hydroelectric operation, GSUS is a four-parameter guide-vane trajectory optimized to reduce strain-cycle amplitude during startup of a Francis turbine prototype (Mai et al., 2024). In space nuclear power systems, it is a ten-parameter piecewise-linear startup schedule for a helium-xenon Closed Brayton Cycle reactor-generator system (Li et al., 2024). In black-start restoration, it is a mixed-integer optimization problem over startup times, energization paths, and resource constraints for thermal generators, fuel cells, and batteries (Lu et al., 19 Jul 2025), while closely related work uses the term generator startup sequencing (GSS) for the corresponding integer programming problem in parallel power system restoration (Chopra et al., 2022). A distinct computational usage also appears in GPU stochastic simulation, where GSUS denotes the initialization sequence of the Mersenne Twister for Graphics Processors (MTGP) (Passerat-Palmbach et al., 2015).

1. Terminological scope and conceptual core

Across the cited work, GSUS always concerns the structured progression from an inactive state to an operational state, but the object being started and the mathematical machinery vary substantially. In hydropower and space nuclear systems, GSUS is a parameterized control trajectory over continuous plant dynamics. In blackout restoration, it is a scheduling and network-energization problem over discrete time. In GPU random-number generation, it is an initialization protocol for a parameterized pseudo-random number generator (Mai et al., 2024, Li et al., 2024, Lu et al., 19 Jul 2025, Passerat-Palmbach et al., 2015).

A common source of ambiguity is terminological rather than methodological. The power-system restoration literature often uses generator startup sequencing or GSS rather than GSUS, although the underlying concern remains the same: determine feasible and efficient startup orders subject to cranking, ramping, and network constraints (Chopra et al., 2022). The GPU usage is more divergent: there, “generator” refers to a pseudo-random number generator, not an electromechanical or thermodynamic power asset (Passerat-Palmbach et al., 2015).

This suggests that GSUS is best understood as a domain-dependent startup formalism rather than a single algorithm. The recurring ingredients are a startup parametrization, a dynamic or time-indexed feasibility model, an objective that penalizes undesirable transients or delays, and an optimization layer that chooses a startup policy under limited resources or strict operating constraints.

2. Hydroelectric GSUS as a four-parameter startup law

For the Francis turbine prototype, the GSUS is fully defined by four scalar tuning parameters,

$$\theta = \{ r_o, o_{\mathrm{ini}}, \omega_{\mathrm{trigger}}, o_{\mathrm{trigger}} \},$$

with ranges specified as follows: $r_o \in [1\%/s,10\%/s]$, $o_{\mathrm{ini}} \in [0,0.34]$, $\omega_{\mathrm{trigger}} \in [0,0.95]$, and $o_{\mathrm{trigger}} \in [0,0.21]$ (Mai et al., 2024). Here $r_o$ is the constant opening-rate of the guide-vane setpoint $u(t)$, $o_{\mathrm{ini}}$ is the first plateau level, $\omega_{\mathrm{trigger}}$ is the normalized rotational-speed threshold at which the setpoint jumps, and $o_{\mathrm{trigger}}$ is the second plateau level.

The startup unfolds in four phases. In Ramp-up, $r_o \in [1\%/s,10\%/s]$0 increases linearly at slope $r_o \in [1\%/s,10\%/s]$1 from $r_o \in [1\%/s,10\%/s]$2 until $r_o \in [1\%/s,10\%/s]$3. In the 1st plateau, $r_o \in [1\%/s,10\%/s]$4 until $r_o \in [1\%/s,10\%/s]$5. In the 2nd plateau, $r_o \in [1\%/s,10\%/s]$6 until $r_o \in [1\%/s,10\%/s]$7, the synchronous speed. In Feedback control, a PID governor closes the speed error to zero (Mai et al., 2024).

The fatigue-damage objective is defined from the measured strain trajectory $r_o \in [1\%/s,10\%/s]$8 over time indices $r_o \in [1\%/s,10\%/s]$9, containing $o_{\mathrm{ini}} \in [0,0.34]$0. The largest cycle amplitude is

$o_{\mathrm{ini}} \in [0,0.34]$1

The optimal parameters solve

$o_{\mathrm{ini}} \in [0,0.34]$2

subject to the synchronization-time constraint

$o_{\mathrm{ini}} \in [0,0.34]$3

This formulation makes the startup sequence a constrained stochastic optimization problem rather than a deterministic tuning exercise (Mai et al., 2024).

The operational motivation is explicit. Hydro-generating units play a crucial role in integrating intermittent renewable energy sources, and the resulting increase in transient events such as startups imposes significant stresses on turbines, increasing fatigue and reducing operational lifespan. The GSUS formulation therefore targets fatigue mitigation under a limited budget of prototype measurements (Mai et al., 2024).

3. Active learning, virtual sensing, and black-box optimization in hydropower

The hydroelectric GSUS methodology nests a black-box optimizer inside an active-learning outer loop. Initialization selects $o_{\mathrm{ini}} \in [0,0.34]$4 initial $o_{\mathrm{ini}} \in [0,0.34]$5 values, such as “fast,” “slow,” and “standard,” runs those startups on the real turbine, gathers measured trajectories, and builds a dataset of enveloped trajectories. During the active-learning phase, an ensemble of $o_{\mathrm{ini}} \in [0,0.34]$6 neural nets $o_{\mathrm{ini}} \in [0,0.34]$7 is trained on the current dataset, and an “optimistic” strain cost is defined as

$o_{\mathrm{ini}} \in [0,0.34]$8

where $o_{\mathrm{ini}} \in [0,0.34]$9 are aleatoric moments predicted by the virtual sensor and $\omega_{\mathrm{trigger}} \in [0,0.95]$0 is the ensemble standard deviation, interpreted as epistemic uncertainty. NOMAD is then used to solve

$\omega_{\mathrm{trigger}} \in [0,0.95]$1

for $\omega_{\mathrm{trigger}} \in [0,0.95]$2 inner-loop simulator calls, after which the chosen startup is executed on the turbine and the dataset is updated. In the final optimization phase, the epistemic term is removed and the “risk-averse” strain cost becomes

$\omega_{\mathrm{trigger}} \in [0,0.95]$3

The overall black-box cost is

$\omega_{\mathrm{trigger}} \in [0,0.95]$4

with $\omega_{\mathrm{trigger}} \in [0,0.95]$5 set so that $\omega_{\mathrm{trigger}} \in [0,0.95]$6 (Mai et al., 2024).

The simulation model couples turbine and governor dynamics. The rotational dynamics obey

$\omega_{\mathrm{trigger}} \in [0,0.95]$7

where $\omega_{\mathrm{trigger}} \in [0,0.95]$8 is obtained via a fast, bilinear surrogate of a precomputed torque map based on steady-state SIMSEN runs. More generally, the governor and turbine form a coupled system of $\omega_{\mathrm{trigger}} \in [0,0.95]$9 nonlinear ODEs,

$o_{\mathrm{trigger}} \in [0,0.21]$0

with state vector $o_{\mathrm{trigger}} \in [0,0.21]$1. A classical 4th-order Runge–Kutta integrator with $o_{\mathrm{trigger}} \in [0,0.21]$2 yields the dynamic trajectory $o_{\mathrm{trigger}} \in [0,0.21]$3, which is then passed pointwise through the virtual sensor $o_{\mathrm{trigger}} \in [0,0.21]$4 to produce

$o_{\mathrm{trigger}} \in [0,0.21]$5

The synchronization-time limit $o_{\mathrm{trigger}} \in [0,0.21]$6 is enforced via early stopping and the penalty term $o_{\mathrm{trigger}} \in [0,0.21]$7 (Mai et al., 2024).

The reported results are unusually data-efficient. The total measured startups were 7, decomposed as 5 initial, 2 active-learning, and 1 final optimization startup. Under a 90 s constraint, the final $o_{\mathrm{trigger}} \in [0,0.21]$8 reduced the maximum strain-cycle amplitude by 42% relative to the standard parameters. Under a tighter 60 s constraint, a second validation still achieved a 26% reduction. Convergence was attained with only two active-learning iterations before the final optimization loop, and runtime per active-learning iteration, including signal processing, retraining, and a 200-call NOMAD solve, was $o_{\mathrm{trigger}} \in [0,0.21]$9 minutes on a standard laptop (Mai et al., 2024).

The same paper explicitly states a generalization pattern to other generator types, including Kaplan and Pelton units: redefine the control-sequence parametrization $r_o$0, build or adapt a simulator, instrument blades or use high-fidelity FEA/CFD to train virtual sensors, and plug these components into the same active learning / black-box framework (Mai et al., 2024).

4. GSUS in Closed Brayton Cycle space nuclear power systems

In the NuHeXSys framework for a helium-xenon gas-cooled Closed Brayton Cycle, GSUS is divided into five sequential phases (Li et al., 2024). Phase 1 – Zero-Power Heat-Up raises core power from thermal-leakage levels to $r_o$1, taking the core average temperature from $r_o$2 to $r_o$3. Phase 2 – Turbine Spin-Up & First Reactivity Ramp launches the turbine/alternator system at core $r_o$4, accelerates the shaft to an intermediate speed $r_o$5 of rated, deploys the cooler, and ramps reactivity from $r_o$6 to $r_o$7, bringing power into the tens of kW. Phase 3 – Low-Power Steady State holds reactivity at $r_o$8 and shaft speed at $r_o$9, with core outlet temperature stabilizing near $u(t)$0 and net electric output near $u(t)$1. Phase 4 – Final Spin-Up & Second Reactivity Ramp increases reactivity from $u(t)$2 to zero insertion margin while ramping $u(t)$3 from $u(t)$4 to rated speed $u(t)$5, causing reactor power to climb linearly toward full power near $u(t)$6. Phase 5 – Full-Power Steady State fixes rated set-points and extracts full electric power near $u(t)$7 at designed efficiency near 25%, with $u(t)$8 and $u(t)$9 (Li et al., 2024).

The startup schedule is parameterized by ten variables,

$o_{\mathrm{ini}}$0

These control the timing and magnitude of the reactivity ramps, the two-segment shaft-speed set-point $o_{\mathrm{ini}}$1, and the cooler background temperature drop from $o_{\mathrm{ini}}$2 to $o_{\mathrm{ini}}$3 via radiator deployment (Li et al., 2024).

The governing dynamics combine reactor thermal-hydraulics, point kinetics, shaft dynamics, and non-ideal gas closure. In lumped form,

$o_{\mathrm{ini}}$4

The neutron dynamics are

$o_{\mathrm{ini}}$5

The shaft dynamics are

$o_{\mathrm{ini}}$6

with

$o_{\mathrm{ini}}$7

The working fluid uses a virial-based equation of state,

$o_{\mathrm{ini}}$8

with corresponding corrections for $o_{\mathrm{ini}}$9, $\omega_{\mathrm{trigger}}$0, and $\omega_{\mathrm{trigger}}$1 (Li et al., 2024).

Optimization is cast as a two-objective NSGA-II problem with

$\omega_{\mathrm{trigger}}$2

subject to safety bounds $\omega_{\mathrm{trigger}}$3, $\omega_{\mathrm{trigger}}$4, $\omega_{\mathrm{trigger}}$5, temporal ordering constraints, and box limits $\omega_{\mathrm{trigger}}$6. The NSGA-II settings are population size 100, 20 generations, crossover probability 0.9 with simulated-binary crossover, mutation probability $\omega_{\mathrm{trigger}}$7 with polynomial mutation, and binary tournament selection using Pareto rank and crowding distance (Li et al., 2024).

The best compromise solution reduced startup time by 1260 s, from a baseline of approximately 9780 s to approximately 8520 s, corresponding to a 12.9% reduction. External energy draw was reduced by approximately 17%, and the turbine inlet temperature threshold of $\omega_{\mathrm{trigger}}$8 was reached 1980 s sooner. The same source attributes these improvements to smoother cooler deployment, an optimized spin-up waveform, and smoother reactivity ramps that kept hotspot factor excursions within 1.02–1.05 instead of 1.08–1.12 (Li et al., 2024).

5. Black-start restoration, generator startup sequencing, and network-constrained GSUS

In hydrogen-integrated renewable grids with fuel cells and battery energy storage systems, GSUS is formulated as a single-stage mixed-integer linear program whose core goal is to bring all offline generators back online as quickly as possible while respecting generator cranking needs, ramping limits, network energization dependencies, and the distinct properties of fuel cells and batteries (Lu et al., 19 Jul 2025). The objective is

$\omega_{\mathrm{trigger}}$9

which combines weighted generator startup times with a reward for early energization of critical buses. The formulation includes generator cranking-time bounds, ramping limits, transmission-path energization constraints $o_{\mathrm{trigger}}$0, fuel-cell ON/OFF and output linearization using auxiliary variables $o_{\mathrm{trigger}}$1, and battery discharge with state-of-charge constraints (Lu et al., 19 Jul 2025).

The special treatment of black-start resources is central. Fuel cells have effectively infinite energy through pipeline hydrogen and are constrained by cranking time $o_{\mathrm{trigger}}$2, minimum up-time $o_{\mathrm{trigger}}$3, and ramping slope $o_{\mathrm{trigger}}$4. Batteries instead have limited stored energy $o_{\mathrm{trigger}}$5, minimum and maximum discharge power, minimum state of charge, and initial $o_{\mathrm{trigger}}$6. The full model is solved as a large-scale MILP using commercial solvers such as Gurobi or CPLEX after McCormick-type linearization of bilinearities (Lu et al., 19 Jul 2025).

On the modified IEEE-39 “New England” system, the case study uses 10 thermal units, the bus 30 generator as the sole classical black-start unit, and two additional black-start resources at buses 6 and 16, modeled either as fuel cells of 50 MW each with unlimited duration or batteries rated 50 MW/50 MWh with $o_{\mathrm{trigger}}$7. With 1 h time steps and all generator cranking times set to 1 h, the reported average startup times over 10 units are 182.2 min with no ESS, 100 min with fuel cells, and 102.2 min with batteries. Fuel-cell capacity, battery size, initial SOC, and resource location are all sensitivity dimensions, with high-degree buses 6 and 16 yielding the fastest GSUS and low SOC values causing markedly slower restoration (Lu et al., 19 Jul 2025).

Closely related work on generator startup sequencing embeds the startup-order problem into Parallel Power System Restoration. There, the post-blackout network is $o_{\mathrm{trigger}}$8, $o_{\mathrm{trigger}}$9 is the set of black-start generator buses, $r_o \in [1\%/s,10\%/s]$00 is the set of non-black-start generator buses, and time is discretized into periods. For a single island with one BS unit and a copper-plate assumption, the ILP minimizes restoration time $r_o \in [1\%/s,10\%/s]$01 subject to each unit starting exactly once, nonnegative total on-line capacity in every period,

$r_o \in [1\%/s,10\%/s]$02

and the bottleneck definition of $r_o \in [1\%/s,10\%/s]$03 (Chopra et al., 2022). The paper shows that the GSS problem is NP-hard via a reduction from PARTITION. It then integrates startup sequencing with sectionalization by introducing bus-assignment variables $r_o \in [1\%/s,10\%/s]$04, startup variables $r_o \in [1\%/s,10\%/s]$05, edge variables $r_o \in [1\%/s,10\%/s]$06, and commodity-flow variables $r_o \in [1\%/s,10\%/s]$07 to enforce island connectivity (Chopra et al., 2022).

The computational results show a strong dependence on formulation and decomposition strategy. The full MILP solves the IEEE-30 case to optimality in 2 s, while the IEEE-118 case is solved to optimality at $r_o \in [1\%/s,10\%/s]$08 in 302 s total using a bounding approach and tree + local search + BFS warm-start. For larger systems, including PEG-1354 and RTE-1888, randomized sectionalization plus local search yields high-quality feasible solutions within 10 min, with reported best restoration times close to the lower bounds (Chopra et al., 2022).

6. Runtime-constrained control and non-power-system extensions

A further formalization appears in economic nonlinear model predictive control for microgrids with generator up and downtime constraints. Here the startup sequence is represented by binary on/off variables $r_o \in [1\%/s,10\%/s]$09, switch indicators $r_o \in [1\%/s,10\%/s]$10, and integer counters $r_o \in [1\%/s,10\%/s]$11 that encode consecutive run-time in the current mode (Gutekunst et al., 26 May 2026). The logical flip law is

$r_o \in [1\%/s,10\%/s]$12

and the paper also notes the equivalent startup definition

$r_o \in [1\%/s,10\%/s]$13

Minimum up- and down-time constraints are enforced through the counter bounds

$r_o \in [1\%/s,10\%/s]$14

together with logical implications requiring $r_o \in [1\%/s,10\%/s]$15 when $r_o \in [1\%/s,10\%/s]$16 and $r_o \in [1\%/s,10\%/s]$17 when $r_o \in [1\%/s,10\%/s]$18 (Gutekunst et al., 26 May 2026).

The economic stage cost explicitly includes startup cost,

$r_o \in [1\%/s,10\%/s]$19

while AC power flow is replaced by a Quadratic Convex relaxation, giving a mixed-integer quadratically constrained program. The full periodic reference problem includes QC-relaxed AC flow, power balance, battery and generator dynamics, mode-dependent bounds, a bound on the number of startups in a 24 h window, periodicity constraints, and integer constraints on the $r_o \in [1\%/s,10\%/s]$20-counters. The paper proves practical recursive feasibility under added time-coupled constraints and asymptotic stability about the periodic reference under strict periodic dissipativity. In the 6-bus microgrid case study, the system has 2 identical diesel generators, 1 battery, and 1 PV; $r_o \in [1\%/s,10\%/s]$21, horizon $r_o \in [1\%/s,10\%/s]$22, minimum up/down times are 2 h, the maximum number of startups is 2 per day, and the resulting MIQCP can be solved in approximately seconds with CPLEX (Gutekunst et al., 26 May 2026).

A distinct, non-electromechanical use of GSUS appears in GPU-enabled stochastic simulation with MTGP. There the startup sequence consists of two phases: an offline phase that uses the Dynamic Creator algorithm to generate many parameterized statuses and validate them with TestU01 BigCrush, and an online phase in which the host allocates device arrays with cudaMalloc, copies precomputed parameterized statuses and the common seed status with cudaMemcpy, and launches an initialization kernel that loads each block’s parameter set into shared memory, seeds the $r_o \in [1\%/s,10\%/s]$23-word state, and performs one twist pass before generation begins (Passerat-Palmbach et al., 2015). The underlying recurrence is the MTGP twist transformation,

$r_o \in [1\%/s,10\%/s]$24

followed by tempering, while seeding uses the standard multiply-xor initialization

$r_o \in [1\%/s,10\%/s]$25

with $r_o \in [1\%/s,10\%/s]$26 (Passerat-Palmbach et al., 2015). The best-practice guidance is explicit: use one parameterized status per CUDA block, keep only BigCrush-approved parameter sets, prefer longer periods such as $r_o \in [1\%/s,10\%/s]$27 or $r_o \in [1\%/s,10\%/s]$28 for critical applications if memory permits, avoid arbitrary runtime reseeding, and do not reuse a parameter set across two blocks (Passerat-Palmbach et al., 2015).

Taken together, these formulations show that GSUS is not a narrow term tied to one device class. It denotes a technically precise startup abstraction whose implementation may be continuous-time and fatigue-aware, kinetics-driven and multi-objective, network-constrained and mixed-integer, runtime-constrained in NMPC, or initialization-centric in high-performance stochastic computing. The unifying idea is structured startup under explicit dynamic, combinatorial, or statistical constraints, with optimization used to trade off speed, stress, energy, feasibility, and reliability across very different engineering settings.