PaperlayerView exact revision on arXiv

Source-linked AI summary

Learning to Solve Two-Stage Stochastic Unit Commitment Problems with Quality Guarantees

Andrea Fusco, Andrea Lodi, Lavanya Marla

arXiv:2609.18859v1eess.SYcs.SYmath.OC

TL;DR

Large scenario sets make extensive-form stochastic unit commitment computationally difficult under day-ahead time constraints. The paper combines a convex ICNN surrogate with Neural-Benders correction, reporting zero optimality gap and speedups up to 214× over the Extensive Form in integer recourse.

  • Problem

    Large scenario sets make stochastic unit-commitment MILPs grow rapidly, potentially exceeding commercial solvers’ day-ahead time budgets.

    Source: §I · p. 2

  • Method

    An ICNN learns a convex recourse-cost surrogate, while a Neural-Benders loop refines first-stage solutions using exact-subproblem cuts to certify quality independently of surrogate accuracy.

    Source: C. Neural-Benders Correction Loop · pp. 6–7, §VI · p. 10

  • Results

    Across IEEE case9–case118 benchmarks with continuous and integer recourse, the method achieves zero optimality gap on the K-scenario instance and speedups up to 214× over EF in the integer setting.

    Source: §VI · p. 10

  • Takeaways & Limitations

    On the tested benchmark instances, solve times are stable across scenario realizations and compatible with day-ahead time windows.

    Source: §VI · p. 10

    Evaluation covers IEEE SUC benchmarks case9–case118 under continuous and integer recourse; the reported scope is the tested benchmark instances.

    Source: §VI · p. 10

Abstract

from arXiv · show

Two-stage stochastic Mixed-Integer Linear Programs are a canonical modeling tool to optimize power system operations under uncertainty, yet their extensive-form counterparts scale linearly with the number of scenarios and quickly become computationally prohibitive under day-ahead time constraints. We propose an Input Convex Neural Network architecture to learn a convex surrogate of the second-stage value function, enabling fast first-stage optimization while preserving convexity by construction. We couple the surrogate with a Neural-Benders correction loop that refines the first-stage solution a posteriori, recovering the exact optimum whether the network overestimates or underestimates the recourse cost, and certifying solution quality independently of surrogate accuracy. We evaluate the method on IEEE Stochastic Unit Commitment benchmarks (case9-case118) under both continuous and integer recourse. The proposed approach achieves solutions with zero optimality gap on the $K$-scenario instance across all test cases and scenario dimensions $K \in \{10, 50, 100\}$, with speedups up to $214\times$ over the Extensive Form in the integer setting. Solve times are stable across scenario realizations and compatible with day-ahead time windows on the tested benchmark instances.

Indices and Sets

The notation defines the sets and parameters for the unit-commitment model, the neural surrogate, and the Neural-Benders correction loop. It also names the decision variables and quantities used to track bounds and convergence.

Indices and Sets

  • G, T, B, E, and S index generators, periods, buses, lines, and uncertainty scenarios, with Gs and Gf denoting slow- and fast-start generators.The notation also identifies bus-level generator and line sets and the sending and receiving buses of each line.Source: Indices and Sets · p. 1

SUC parameters and variables

  • The SUC notation covers generator costs and operating limits, network parameters, scenario-dependent net load, and commitment, dispatch, load-shedding, flow, and voltage-angle variables.The parameters include start-up and no-load costs, minimum up/down times, generation costs and limits, ramp rates, line susceptance and limits, and net load.Source: Indices and Sets · p. 1

Recourse and correction-loop notation

  • Q(x, ξs) is scenario recourse cost, Eξ[Q(x, ξ)] is expected recourse cost, and Q̂(x, z) is the ICNN surrogate; the remaining symbols track correction-loop cuts, bounds, slack, and convergence.These include scenario duals and Farkas rays, the cut-based lower bound θ(x), sample upper bound Q(x(i)), solution gap, and convergence tolerance.Source: Indices and Sets · pp. 1–2

Scenario and ICNN notation

  • K is the scenario count, Ψ1 and Ψ2 encode and aggregate scenarios, z is the aggregated embedding, and L and h(l) describe ICNN depth and activations.The notation also specifies the first-stage dimension nx, embedding dimension d2, and layer weight matrices W(l).Source: Indices and Sets · p. 1

I. INTRODUCTION

The introduction frames scenario growth and integer recourse as barriers to scalable, certified stochastic unit commitment. It motivates an ICNN surrogate and a Neural-Benders loop as a way to address those barriers.

I. INTRODUCTION

  • Large scenario sets make stochastic unit-commitment MILPs grow rapidly, potentially exceeding commercial solvers’ day-ahead time budgets.Renewable forecast uncertainty motivates stochastic optimization, while deterministic UC can yield infeasible or highly suboptimal schedules.Source: §I · p. 2

A. Literature Review

  • Extensive-form MILP size grows linearly with K, while Benders iterations solve K subproblems and classical dual cuts rely on continuous recourse.Integer recourse makes the value function generally nonconvex and invalidates those dual-based cuts without substantial extensions.Source: A. Literature Review · p. 2
  • Scenario reduction and learned embeddings approximate the uncertainty distribution, but the reduced distribution carries error without a bound on first-stage suboptimality.This limits what those approaches can certify about the resulting first-stage solution.Source: A. Literature Review · p. 2
  • Neur2SP embeds a learned recourse surrogate in the first-stage MILP, but its nonconvex MLP approximation can create spurious basins and provides no a posteriori certificate.End-to-end learned dispatch can also struggle to enforce hard physical constraints.Source: A. Literature Review · p. 2

B. Contribution

  • The proposed ICNN matches the convex, piecewise-linear recourse structure in continuous settings, avoids spurious basins, and yields a tighter LP relaxation bound per master iteration.The continuous expected recourse cost is convex in the first-stage decision.Source: B. Contribution · p. 3
  • The Neural-Benders loop uses exact-subproblem cuts and scenario evaluation to certify solutions independently of surrogate accuracy, extending the approach to integer recourse.The authors report 1–13 loop iterations across benchmarks versus up to 100 uncertified iterations for classical Integer L-Shaped; the extension targets settings where classical decompositions are intractable.Source: B. Contribution · p. 3

II. BACKGROUND

The background defines the two-stage stochastic-program structure and applies it to unit commitment. First-stage commitment decisions precede uncertainty realization, while second-stage dispatch responds to scenario-specific net load.

A. Two-Stage Stochastic Programming

  • A two-stage stochastic program chooses x subject to deterministic constraints before uncertainty ξ is realized, then selects recourse y to satisfy operational requirements.The recourse value Q(x, ξ) is the optimal value of the second-stage problem, whose data include cost vectors, technology links, recourse constraints, and uncertainty-dependent right-hand sides.Source: A. Two-Stage Stochastic Programming · p. 3
  • With continuous recourse, Q(x, ξ) is convex and piecewise linear in x, supporting Benders cuts; integer recourse generally makes it nonconvex and discontinuous, invalidating standard dual-based cuts.The discussion identifies this distinction as the structural basis for classical decomposition’s applicability and limits.Source: A. Two-Stage Stochastic Programming · p. 3

B. Stochastic Unit Commitment Problem

  • In SUC, scenario-indexed net load represents demand and renewable-generation variability across buses and scheduling periods.The model’s sets cover generators, periods, buses, and transmission lines.Source: B. Stochastic Unit Commitment Problem · p. 3

1) First Stage (Unit Commitment):

  • The first-stage unit-commitment decision x contains binary on/off, start-up, and shut-down variables, with commitment logic and minimum operating-time constraints defining feasibility.The first-stage cost includes start-up and no-load costs.Source: 1) First Stage (Unit Commitment) · p. 3

2) Second Stage (Economic Dispatch):

  • Given a commitment schedule and scenario, second-stage economic dispatch minimizes cost using generation, power-flow, and voltage-angle decisions while enforcing power balance and operating limits.Load shedding is penalized by a value-of-lost-load coefficient set orders of magnitude above marginal generation costs; constraints also condition generation on commitment and limit inter-temporal output changes.Source: 2) Second Stage (Economic Dispatch) · pp. 3–4
  • Continuous recourse uses real-valued dispatch variables and yields a convex recourse function, whereas fast-start activation or transmission switching adds integer decisions and generally makes it nonconvex.The integer variant therefore falls outside standard decomposition methods described for continuous recourse.Source: 2) Second Stage (Economic Dispatch) · p. 4

C. SUC with Integer Recourse: Fast-Start Unit Commitment

The integer-recourse SUC variant separates generators into slow units committed before uncertainty and fast-start units activated after load is observed. Its first-stage decision governs only the slow generators.

C. SUC with Integer Recourse: Fast-Start Unit Commitment

  • Slow-generator commitment is fixed before uncertainty is revealed, while fast-start generators may be activated after the load realization is observed.The benchmark SUC is extended to include these two generator classes.Source: C. SUC with Integer Recourse:… · p. 4

1) First Stage:

  • The first-stage decision x contains only slow-generator commitment, start-up, and shut-down decisions, with commitment logic and minimum operating-time constraints applying to that subset.The slow/fast partition is based on marginal-cost rank; the fast-start subset comprises the highest-cost units according to percentile parameter pf.Source: 1) First Stage · p. 4, C. SUC with Integer Recourse:… · p. 4

2) Second Stage:

The methodology combines scenario aggregation, a convex ICNN surrogate, and a Neural-Benders correction loop. Its scenario representation is fixed-dimensional and permutation-invariant, while the second-stage model includes scenario-specific fast-unit commitment.

2) Second Stage:

  • The second stage is a mixed-integer dispatch problem with scenario-specific binary commitment decisions for fast-start generators, subject to power balance, network-flow, ramping, and generation constraints.Source: 2) Second Stage · p. 4
  • Load-shedding variables ensure nodal balance can always be satisfied, giving the formulation complete recourse by construction.Source: 2) Second Stage · p. 4
  • Integer recourse makes the value function generally non-convex and discontinuous, so the correction loop uses combinatorial cuts rather than LP duality-based cuts,.Source: 2) Second Stage · p. 4

III. METHODOLOGY

  • The methodology combines permutation-invariant scenario aggregation, an ICNN surrogate convex in the first-stage decision, and a Neural-Benders loop that certifies solution quality a posteriori.Source: §III · p. 4

A. Scenario Encoding and Aggregation

  • A probability-weighted mean pools independently encoded scenarios into a fixed-dimensional representation, and the encoding networks run offline because scenarios are known at solve time.Source: A. Scenario Encoding and Aggregation · p. 4

B. Input Convex Neural Networks Architecture

The ICNN takes the first-stage decision and an aggregated scenario representation as inputs, with architectural constraints ensuring convexity in the decision. This structure matches continuous-recourse properties and supports a convex approximation in the integer-recourse setting, where a separate correction loop provides certification.

B. Input Convex Neural Networks Architecture

  • Nonnegative passthrough and output weights make the scalar ICNN prediction convex in the first-stage decision for fixed scenario representation z.Source: B. Input Convex Neural Networks… · p. 5
  • In continuous recourse, convex piecewise-linear value functions match the ICNN function class, while convexity avoids spurious basins and yields tighter LP bounds per master iteration.Source: B. Input Convex Neural Networks… · p. 5
  • For integer recourse, the convex surrogate may overestimate or underestimate the value function, so the Neural-Benders loop supplies an a posteriori gap guarantee independent of its local accuracy.Source: B. Input Convex Neural Networks… · p. 5
  • ReLU activations admit an exact MILP reformulation, and the non-negativity constraints are enforced during offline training without online optimization overhead.Source: B. Input Convex Neural Networks… · p. 5

C. Neural-Benders Correction Loop

The Neural-Benders loop corrects errors in the learned recourse surrogate and certifies the first-stage solution using exact subproblem information. Its cuts and penalized slack allow refinement whether the surrogate underestimates or overestimates recourse cost.

C. Neural-Benders Correction Loop

  • The loop addresses both surrogate error directions: cuts correct underestimation, while penalized slack lets the solver explore overestimated regions and obtain corrective cuts.Without that exploration, an overestimated region may be avoided and its bias remain undetected.Source: C. Neural-Benders Correction Loop · pp. 5–6
  • For continuous recourse, feasible LP subproblems yield dual optimality cuts, while infeasible subproblems can yield Farkas feasibility cuts.The feasibility cuts exclude first-stage decisions that make a scenario infeasible.Source: C. Neural-Benders Correction Loop · p. 6
  • For integer recourse, complete recourse avoids feasibility cuts, but MILP subproblems invalidate classical LP-duality cuts, so the method uses the Integer L-Shaped Method.A global lower bound is obtained once by relaxing first- and second-stage binary variables to the unit interval; combinatorial cuts then encode exact recourse at a binary point and remain globally valid.Source: C. Neural-Benders Correction Loop · p. 6
  • 1–13 iterations certify convergence across the benchmark, versus up to 100 uncertified iterations for the classical Integer L-Shaped method.Each iteration solves all K second-stage subproblems; the iteration-count advantage reduces total subproblem solves.Source: C. Neural-Benders Correction Loop · p. 6
  • The certified lower bound comes from accumulated exact-subproblem cuts, not the penalized hybrid objective or the ICNN surrogate.Convergence is declared when the relative gap is at most 10^-3; when progress stagnates, the penalty is halved after three nonimproving iterations, down to 10^-3.Source: C. Neural-Benders Correction Loop · pp. 6–7

IV. CASE STUDIES

The case studies evaluate both problem classes on IEEE transmission test systems from 9 to 118 buses, using 10, 50, or 100 scenarios.

IV. CASE STUDIES

  • The evaluation covers two problem classes on IEEE transmission test systems from 9 to 118 buses, with K ∈ {10, 50, 100} scenarios.The benchmarks are presented as controlled and reproducible tests of methodological soundness and scalability.Source: §IV · p. 7

A. Stochastic UC on IEEE Transmission Test Systems

The continuous-recourse case studies use IEEE transmission systems with scenario uncertainty represented by spatially correlated net-load variation over a 24-hour load shape.

A. Stochastic UC on IEEE Transmission Test Systems

  • The six IEEE cases span 9 to 118 buses, with generator and network parameters taken directly from MATPOWER files without modification.The systems are used in their stochastic unit commitment variant.Source: A. Stochastic UC on IEEE Transmission… · p. 7
  • Each scenario scales base bus loads by a per-period multiplier drawn independently from U[0.7, 1.0] and applied uniformly across buses.The shared period multiplier yields spatially correlated scenarios, described as consistent with correlated day-ahead UC forecast errors.Source: A. Stochastic UC on IEEE Transmission… · p. 7

B. IEEE Transmission Benchmarks with Integer Recourse

The integer-recourse benchmark defers commitment of the most expensive generators to the second stage. The study uses structured data generation, a shared surrogate-training protocol, and comparisons against exact and surrogate-based alternatives.

B. IEEE Transmission Benchmarks with Integer Recourse

  • Integer recourse uses the same systems, parameters, and scenarios as the continuous case, with the 20% most expensive generators assigned to fast-start recourse at p_f = 0.8.Their commitment is deferred to the second stage as binary variables.Source: B. IEEE Transmission Benchmarks with… · p. 7

C. Data Generation

  • Training labels are expected recourse costs evaluated over a finite scenario bank, and feasible commitment samples are structured perturbations around reference UC solutions.Infeasible samples are retained and labeled using dual information; integer-recourse labeling solves MILPs rather than LPs.Source: C. Data Generation · p. 7

D. Training Setup

  • Neur2SP and ICNN share a training protocol that minimizes MSE on normalized targets, with hyperparameters selected over 50 random-search trials by validation MAE.A per-case scaler standardizes recourse costs; offline labeling and search range from a few hours on small cases to about 48 hours for case118.Source: D. Training Setup · pp. 7–8

E. Methodology Comparison

  • Five methods are compared: Extensive Form, L-Shaped or Integer L-Shaped, Neur2SP, ICNN without refinement, and the full Neural-Benders method with an a posteriori certificate.The decomposition baselines are capped at 100 iterations, with nonconverged cases marked in the tables.Source: E. Methodology Comparison · p. 8

V. NUMERICAL RESULTS AND DISCUSSION

  • Reported experimental figures average 30 random-seed runs on an Apple M1 with Pyomo and Gurobi 12.0.The seeds vary scenario sampling.Source: §V · p. 8

A. Neural Approximation Quality and Surrogate Performance

Across the IEEE benchmarks, ICNN+ certifies zero optimality gap, while surrogate accuracy varies most on case30. Refinement preserves certification, and solve times remain practical on the tested instances.

A. Neural Approximation Quality and Surrogate Performance

  • On validation data, median relative absolute prediction errors stay below 0.2% except case30, where both surrogates have higher errors.For case30, errors reach 6.0% (ICNN) and 4.7% (Neur2SP) under continuous recourse, and 0.8% and 0.7% under integer recourse; excluding case30, their errors are comparable.Source: A. Neural Approximation Quality and… · p. 8

B. Computational Performance and Solution Quality

  • Surrogate methods have stable solve times across random seeds, unlike EF, whose case118, K = 100 integer runs ranged from approximately 2,300 s to 4,300 s.EF timings are averages over 30 runs.Source: B. Computational Performance and… · p. 9
  • Larger, deeper networks can improve approximation but create larger MILP embeddings, potentially eroding speedups; compact ICNN+ delegates improvement to refinement while retaining certification.This makes network-complexity control important as recourse functions become more complex.Source: B. Computational Performance and… · pp. 9–10
  • ICNN+ corrects surrogate gaps to zero and certifies solution quality independently of surrogate accuracy, with up to approximately 13 Benders iterations on case30 at K = 100 in continuous recourse.LS fails to converge in some continuous-recourse cases, while i-LS reaches its 100-iteration limit without convergence across the integer-recourse benchmark.Source: B. Computational Performance and… · pp. 9–10
  • ICNN+ completes case118 at K = 100 in under 20 s with a certified zero optimality gap on the SAA instance.The reported solve time remains compatible with day-ahead time windows.Source: B. Computational Performance and… · p. 10

VI. CONCLUSION

  • Across IEEE benchmark cases case9–case118, experiments under continuous and integer recourse certify zero optimality gap on every tested K-scenario SAA instance, with integer-setting speedups up to 214× over EF.Solve times are stable across scenario realizations and compatible with day-ahead time windows on the tested benchmark instances.Source: §VI · p. 10
Loading 2609.18859v1…