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Trust-Region Benders for Scalable Multiperiod Expansion Planning for Transmission and Storage

Kevin Wu, Rabab Haider, Pascal Van Hentenryck

arXiv:2610.32862v1eess.SYcs.SY

TL;DR

Multiperiod TEP+Storage planning must coordinate investment timing and location with hourly operations, increasing the scale of an already challenging planning problem. The paper applies trust-region Benders decomposition with per-period stabilization and a relaxation-based warm start; on a 2,000-bus Texas system, it reports a 0.59% final gap and improved solution quality with less computational effort than rolling horizon.

  • Problem

    TEP+Storage must coordinate investment location and timing with hourly operations across periods, while existing high-resolution multiperiod evidence is limited by single-period or rolling-horizon approaches.

    Source: §I · p. 1

  • Method

    The all-years formulation co-optimizes transmission reconductoring and storage across periods using trust-region-stabilized Benders decomposition with per-period constraints and a relaxation-based warm start.

    Source: §VI · p. 7

  • Results

    0.59% final optimality gap on a 2,000-bus synthetic Texas system; compared with rolling horizon, the all-years approach achieves comparable or improved solution quality with substantially reduced computational effort.

    Source: §VI · p. 7

  • Takeaways & Limitations

    Compared with rolling horizon, all-years planning invests more cumulatively in reconductoring, less in storage, and front-loads reconductoring, with lower final objective value and optimality gap.

    Source: C. Investment Result Comparison · p. 7

Abstract

from arXiv · show

Transmission expansion planning (TEP) seeks to identify cost-effective investment strategies that enable power systems to reliably meet future demand, while accounting for the location, timing, and operational impacts of infrastructure upgrades. As demand grows and renewable penetration increases, there is growing interest in jointly modeling energy storage investments to enhance operational flexibility. This work presents a multiperiod formulation that co-optimizes transmission reconductoring and energy storage installation across multiple planning periods. To solve the resulting large-scale problem efficiently, a Benders decomposition with per-period trust-region stabilization and a relaxation-based warm-start procedure is proposed. The approach is demonstrated on a 2,000-bus Texas system, producing an investment plan spanning 2030-2045 with optimality gap below 1.0%, and is compared against a rolling-horizon approach to highlight the benefits of multiperiod coordination.

NOMENCLATURE

The nomenclature indexes buses, branches, days, periods, and hours, while the introduction frames multiperiod TEP+Storage as a large-scale planning problem. The paper extends TR-Benders to coordinate transmission and storage investments across periods and reports a sub-1.0% gap on a 2,000-bus Texas model.

NOMENCLATURE

  • The nomenclature indexes buses, branches, days, planning periods, and hours as N, E, R, Y, and T, respectively.Unless specified otherwise, parameters and operational variables use indices for period y, representative day r, and hour t.Source: NOMENCLATURE · p. 1
  • The variable notation distinguishes reconductoring γ, installed storage σ, state of charge, charging, discharging, generation, flow, and load shedding.The nomenclature also distinguishes generator minimum and maximum output, load, and annualized investment and operating costs.Source: NOMENCLATURE · p. 1

I. INTRODUCTION

  • TEP+Storage co-optimizes investment location, timing, and hourly operation across multiple planning periods, dramatically increasing problem scale.Here, “multiperiod” means planning decisions across multiple investment planning periods.Source: §I · p. 1
  • Prior multiperiod studies used relatively low-dimensional systems, while high-resolution TEP+Storage studies were single-period or used myopic rolling horizons.Aggregated spatial representations cannot capture high-resolution congestion patterns and storage–transmission interactions.Source: §I · pp. 1–2
  • The proposed TR-Benders extension jointly optimizes transmission and storage investments across periods while preserving hourly DC power flow, with a 2030–2045 plan and final optimality gap below 1.0%.The method is demonstrated on a high-resolution 2,000-bus Texas grid model.Source: §I · p. 2

II. FORMULATION

The formulation chooses transmission reconductoring and storage investments over multiple periods, then accounts for their discounted capital and operating expenditures. It represents storage as four-hour lithium-ion batteries and line upgrades as increased capacity on existing branches.

II. FORMULATION

  • The two-stage model chooses investment timing and location in its first stage, then determines hourly operations in its second stage.The stages minimize capital expenditures and operating costs, respectively.Source: §II · p. 2

A. Investment Decisions and Variables

  • Transmission upgrades reconductor existing lines, with each binary branch-period decision adding Δc_ij MW of capacity.The text notes reconductoring is frequently favored over new line construction due to cost-effectiveness.Source: A. Investment Decisions and Variables · p. 2
  • Storage investment represents cumulative installed energy capacity at each node, modeled as four-hour lithium-ion batteries with charge and discharge power capped at one-quarter of capacity.The text motivates lithium-ion batteries by their technological maturity and widespread adoption.Source: A. Investment Decisions and Variables · p. 2
  • The full investment vector x concatenates transmission-upgrade decisions γ and storage decisions σ across planning periods, while x_y contains the decisions for period y.γ_y and σ_y collect decisions across branches and nodes, respectively.Source: A. Investment Decisions and Variables · p. 2

B. Objective

  • The objective minimizes capital and operating expenditures across all planning periods, discounting period costs by d_y = (1+α)^−p(y).The elapsed years p(y) are zero in the first period and increase with each planning period.Source: B. Objective · p. 2
  • Capital expenditure is linear in cumulative reconductoring and storage decisions, with both investment variables initialized to zero in baseline period y = 0.The investment costs for reconductoring and storage are annualized.Source: B. Objective · p. 2
  • Operating cost includes generation, storage charging and discharging, and load-shedding penalties, and is annualized across representative days and summed over planning periods.The period cost uses representative-day probabilities and a 365-day annual scaling.Source: B. Objective · p. 2

C. Constraints

The operational model uses PTDF-based DC power flow, balances injections, enforces equipment limits, and models storage within representative days. Its storage boundary conditions preserve inter-day independence, while the model size grows with periods and representative days.

C. Constraints

  • PTDF matrix Φ maps net nodal injections to transmission line flows in the DC power-flow model.Reconductoring increases thermal limits while leaving electrical parameters largely unchanged, permitting the fixed-parameter PTDF representation.Source: C. Constraints · p. 3
  • Operational constraints enforce system-wide power balance, thermal line limits, and generator dispatch limits, while allowing renewable curtailment through scenario-dependent upper bounds.Renewables have zero minimum output; conventional generation remains within operating limits.Source: C. Constraints · p. 3
  • Storage constraints model state-of-charge dynamics with efficiency losses, charging and discharging limits, and zero state of charge at each representative day's start and end.This preserves inter-day independence but limits inter-day energy shifting.Source: C. Constraints · p. 3
  • Transmission and storage investments are constrained to be irreversible through monotonicity constraints.Source: C. Constraints · p. 3
  • Model size grows linearly with the product of planning periods and representative days, Y × R, motivating scalable solution strategies.Source: C. Constraints · p. 3

III. METHODOLOGY

The method decomposes multiperiod planning into a master investment problem and operational subproblems, adding Benders cuts from subproblem dual information. It stabilizes this process with per-period trust regions and a relaxation-based warm start.

III. METHODOLOGY

  • Baseline multicut Benders decomposition separates investment decisions in a master problem from operational subproblems and adds a cut from each subproblem's dual information at each iteration.The baseline can suffer oscillating master iterates and dual degeneracy across investment configurations.Source: §III · p. 3
  • The proposed extension adds per-period trust-region constraints and a relaxation-based warm start to the all-years multiperiod formulation.Trust-region and warm-start techniques had previously been applied to single-period expansion or simplified multiperiod settings [8],.Source: §III · p. 3

A. AY TR-Benders Master Problem (MP)

  • The master problem chooses reconductoring and storage investments for each planning period, with auxiliary θ_y,r variables lower-bounding operating costs for each period and representative day.Benders cuts refine θ and the load-shedding proxy ρ across iterations.Source: A. AY TR-Benders Master Problem (MP) · p. 3
  • The Benders master uses a proxy variable ρ to enforce zero load shedding in the final solution, avoiding a penalty parameter that would need tuning as representative days increase.Load-shed slack variables remain in both formulations to ensure recourse feasibility.Source: A. AY TR-Benders Master Problem (MP) · p. 4

B. Benders Subproblem (SP)

The subproblems test feasibility before optimizing operating cost, while per-period trust regions stabilize investment decisions around the best-known trajectory. Stabilization also reduces the burden of subproblem solves and supports lazy constraint generation.

B. Benders Subproblem (SP)

  • Each subproblem first minimizes load shedding; if it is nonzero, the algorithm generates a feasibility cut, otherwise it minimizes dispatch cost and generates an optimality cut.The two phases produce cuts for the load-shed and operational-cost underestimators, respectively.Source: B. Benders Subproblem (SP) · p. 4
  • Lazy constraint generation imposes the PTDF flow constraints in subproblems while preserving an equivalent, more tractable formulation.The implementation details are provided in Appendix B.Source: B. Benders Subproblem (SP) · p. 4

C. Per-period Stabilization

  • The master problem uses per-period l1 trust regions around the incumbent trajectory with the best known upper bound to mitigate oscillatory iterates and dual degeneracy.These issues arise when multiple transmission–storage configurations have similar operating costs.Source: C. Per-period Stabilization · p. 4
  • Trust-region stabilization restricts subproblem searches to nearby investments, producing more consistent congestion patterns and fewer newly violated flow constraints for lazy generation.The passage states that this improves convergence and accelerates subproblem solves.Source: C. Per-period Stabilization · p. 4
  • Valid per-period minimum-storage inequalities ensure load feasibility and strengthen the master problem.Their derivation and proofs are provided in Appendix C.Source: C. Per-period Stabilization · p. 4

D. Benders Warm-start (WS)

The warm-start builds an intertemporally monotone investment trajectory from sequential continuous relaxations across periods and representative days. TR-Benders separately maintains a valid global lower bound and stops when its relative optimality gap meets the specified tolerance.

D. Benders Warm-start (WS)

  • Sequential continuous relaxations across periods aggregate each investment component by its maximum across representative days, round up to admissible granularity, and enforce monotonic investments in later periods.The resulting trajectory is initialized consistently with worst-case operational requirements across representative days.Source: D. Benders Warm-start (WS) · p. 4
  • The warm-start trajectory initializes Benders decomposition, and candidate investment sets inferred from relaxed final-period solutions further restrict decisions to reduce degeneracy.The candidate-set restriction is based on relaxed solutions in the final planning period.Source: D. Benders Warm-start (WS) · p. 4

E. Convergence of TR-Benders

  • Because trust-region constraints make the master optimum only a local lower bound, the algorithm constructs a separate valid global lower bound.The global lower-bound construction is presented in Appendix D.Source: E. Convergence of TR-Benders · p. 5
  • When all subproblems are solved without load shedding, the feasible solution updates the upper bound; the algorithm terminates once the relative gap between that bound and the global lower bound meets tolerance.The practical level-set constraint requires the master objective to be no greater than the upper bound minus a small numerical tolerance.Source: E. Convergence of TR-Benders · p. 5

IV. CASE STUDY

The case study evaluates all-years multiperiod planning on a 2,000-bus Texas system using representative operating days and specified investment assumptions. AY reaches a sub-1% optimality gap within 64 hours, while the study also compares it with rolling-horizon planning.

A. Network Data

  • The ACTIVSg2000 system contains 2,000 buses, 3,206 transmission lines, 423 conventional generators, and 175 renewable generators.The case applies 1.5% annual load growth and a 0.005 PTDF cutoff.Source: A. Network Data · p. 5
  • The 18 representative days each span 24 hours and are selected for profile diversity using a max–min criterion, with equal probabilities.The planning periods are 2030, 2035, 2040, and 2045; operations are evaluated at each period's start year.Source: A. Network Data · p. 5, §IV · p. 5

B. Investment Parameters

  • Reconductoring doubles line thermal limits, while storage investments use discrete 250 MWh units with up to 12 units per node and 90.25% round-trip efficiency.Storage is assigned a 15-year lifetime and line upgrades an effective 20-year lifetime for the planning horizon.Source: B. Investment Parameters · p. 5

C. Comparison with Rolling Horizon (RH)

  • Rolling horizon solves single-period models sequentially without anticipating future-period projections, whereas AY jointly optimizes investments across all periods after one warm-start.The rolling-horizon procedure alternates a warm-start and TR-Benders solve across periods.Source: C. Comparison with Rolling Horizon (RH) · p. 5

D. Computational Setup and Hyperparameters

  • The study uses a 144-hour time limit for AY and sequential 24/24/24/72-hour limits for rolling horizon; storage trust-region radius is fixed at 1.The transmission radius starts at 1 and is updated dynamically; warm-start load-shed penalty is $50,000/MWh.Source: D. Computational Setup and… · p. 5

A. All-Years Multiperiod (AY) Computational Performance

  • AY-Final achieved a total cost of $28.25B with a final optimality gap of 0.59%, and AY reached under 1% optimality gap within 64 hours.The full run took 148.2 hours including the 4.2-hour warm-start; AY-WS produced $29.14B with a 3.63% gap.Source: A. All-Years Multiperiod (AY)… · p. 6
  • The August 11, 2045 representative day is the most computationally challenging, with severe congestion and substantial storage utilization driving the longest subproblem solves and iteration duration.Across AY iterations, the master averages 0.43 minutes per solve, compared with 25.1 minutes for subproblems; master solves account for 1.7% of runtime.Source: A. All-Years Multiperiod (AY)… · p. 6

B. Comparison with Rolling Horizon (RH) Formulation

AY reaches a 0.59% final optimality gap and achieves a gap below RH-Final’s 0.81% in less than half its runtime. Its coordinated plan favors earlier reconductoring and less cumulative storage investment, although the study’s uniform future projections may limit the comparison with RH.

B. Comparison with Rolling Horizon (RH) Formulation

  • Below 0.81% optimality gap within 72 hours, AY beats RH-Final’s 0.81% gap at 147.7 hours, though the final gaps are comparable.Reported runtimes include warm-start; final solves stopped at imposed time limits, not proven optimality.Source: B. Comparison with Rolling Horizon… · p. 6, Figure 1 · p. 6
  • Uniform future load and generation projections do not represent spatially and temporally heterogeneous shifts that might mitigate RH’s myopic limitations.The authors give an example of a new load at one node followed by a new generator at another node in a later year.Source: B. Comparison with Rolling Horizon… · p. 6

C. Investment Result Comparison

  • AY invests more cumulatively in transmission reconductoring and less in storage than RH, with earlier reconductoring reducing later storage needs.The authors relate this pattern to the multiperiod perspective, which accounts for increasing future system requirements.Source: C. Investment Result Comparison · p. 7, Table II · p. 7, Table III · p. 7
  • The RH and AY plans have almost identical total period costs in 2030 despite differing line configurations, indicating degeneracy among similar-cost investment plans.Incremental first-stage investments are reported undiscounted; total period costs are discounted and include operational costs.Source: C. Investment Result Comparison · pp. 6–7, Table III · p. 7

VI. CONCLUSION

  • AY jointly optimizes transmission and storage investments across planning periods using an all-years formulation and trust-region-stabilized Benders decomposition.The method adds per-period trust-region constraints and a relaxation-based warm-start procedure.Source: §VI · p. 7
  • On a 2,000-bus synthetic Texas system, AY achieves a final optimality gap of 0.59%.The conclusion reports comparable or improved solution quality versus RH with substantially reduced computational effort.Source: §VI · p. 7
  • The authors identify low master-problem computational burden as favorable for longer planning horizons and model extensions.Future work includes enhanced stabilization, more heterogeneous projections, linking storage state-of-charge across representative days, and additional parallel procedures for valid inequalities.Source: §VI · p. 7

APPENDIX

The appendix outlines the AY TR-Benders algorithm and PTDF lazy constraint generation, then gives valid minimum-storage-capacity inequalities and a relaxation-based storage warm start.

B. PTDF Lazy Constraint Generation

  • PTDF lazy constraint generation adds flow and thermal-limit constraints for the largest violations, with generated constraints retained across TR-Benders iterations.The procedure is used in subproblem solves for both Phase I and Phase II.Source: B. PTDF Lazy Constraint Generation · p. 7

C. Valid Inequalities for Minimum Required Storage Capacity

  • The minimum-storage-capacity inequality uses a feasibility relaxation enforcing zero load shedding, neglecting operating costs, and fixing transmission upgrades at their maximum values.The resulting minimum storage requirement is imposed in the master problem; the relaxation-based warm start gives 0.0, 27.75, 96.00, and 165.00 GWh for 2030, 2035, 2040, and 2045, respectively.Source: C. Valid Inequalities for Minimum… · pp. 7–8

A. Multiperiod All-Years TR-Benders Algorithm

  • The AY TR-Benders algorithm initializes and updates a master problem, solves period-day subproblems in parallel, adds feasibility or optimality cuts, and updates bounds and trust regions.The procedure stops when the relative optimality gap meets the tolerance or the time limit is reached.Source: C. Valid Inequalities for Minimum… · p. 8

D. Global Lower Bound for Optimality Assessment

The section constructs a valid global lower bound from period-wise investment and operating-cost bounds, and describes PTDF lazy constraint generation in the subproblem solves.

D. Global Lower Bound for Optimality Assessment

  • The first-stage lower bound uses minimum investment costs that ensure zero load shedding across representative days, while the second-stage bound assumes maximal feasible investments to yield optimistic operating costs.The discounted sum of the period-wise bounds gives a valid global bound.Source: D. Global Lower Bound for Optimality… · p. 8
  • PTDF lazy constraint generation identifies violated flow limits and adds PTDF flow and thermal-limit constraints for the largest violations.Generated constraints persist across iterations, and the procedure is used for Phase I and Phase II subproblem solves.Source: D. Global Lower Bound for Optimality… · p. 8
  • $28.08 B is the valid global lower bound for the multiperiod problem.The period-wise discounted lower bounds are summed; Table IV presents their per-period decomposition.Source: D. Global Lower Bound for Optimality… · p. 8, Table IV · p. 8

E. Case Study Parameters

The case study parameters summarize generation and load projections relative to 2022 and investment-cost configuration; the study uses 18 equally weighted representative days.

E. Case Study Parameters

F. Representative days

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