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Computationally Transparent Power Flexibility Aggregation via Directional Support Evaluation

Mingyu Zhai, Liang Du, Yue Chen

arXiv:2610.30428v1eess.SYcs.SY

TL;DR

Characterizing the flexibility of heterogeneous DER portfolios is difficult because of high dimensionality, diverse operating dynamics, and coupling constraints. This paper uses LP-based directional support evaluations to build feasible inner approximations. The approach is tighter than compact inner-region benchmarks in small-scale tests, matches a dense directional reference with 16–64 directions, and scales approximately linearly in tested large-scale simulations.

  • Problem

    Exact aggregate flexibility characterization is difficult for heterogeneous DER fleets because high-dimensional regions and cross-DER coupling make explicit representations computationally challenging.

    Source: §I · p. 1, B. Aggregate Power Flexibility Region · p. 4

  • Method

    The framework solves LP-based directional support evaluations over a joint model with device and coupling constraints, using the resulting trajectories to form a feasible inner approximation.

    Source: §I · p. 2, C. Proposed Adaptive Directional… · p. 5

  • Results

    The method produces tighter projected regions than box, zonotope, and inner-ball benchmarks, matches the dense directional reference within numerical tolerance using 16–64 directions, and scales approximately linearly in tested simulations up to 2000 DERs.

    Source: C. Comparison with Existing Aggregate… · p. 8, §V · p. 10

  • Takeaways & Limitations

    The results support a scalable and theoretically grounded approach to aggregate flexibility characterization for heterogeneous DER portfolios.

    Source: §V · pp. 9–10

Abstract

from arXiv · show

The growing penetration of distributed energy resources (DERs) creates new opportunities for improving resource adequacy, operational flexibility, and power-system reliability. However, efficiently characterizing the aggregate flexibility of heterogeneous DER portfolios remains challenging because of high dimensionality, diverse operating dynamics, and spatio-temporal coupling constraints. This paper proposes a computationally transparent framework that formulates aggregate flexibility characterization as a sequence of linear programming (LP)-based directional support evaluations. The framework accommodates heterogeneous DER technologies and configurable device-, feeder-, and portfolio-level constraints. The convex hull of the resulting boundary trajectories provides a feasible inner approximation of the coupled aggregate power flexibility region. Theoretical analysis establishes feasibility and non-overestimation guarantees and characterizes the computational dependence on portfolio size, scheduling horizon, and direction budget. In the small-scale case, the proposed method produces tighter projected regions than box, zonotope, and Chebyshev inner-ball benchmarks and matches a dense directional reference within numerical tolerance using 16--64 construction directions. Large-scale simulations with up to 2000 DERs demonstrate approximately linear empirical runtime scaling with the portfolio size and direction budget over the tested ranges. The results establish a scalable and theoretically grounded approach for aggregate DER flexibility characterization.

I. INTRODUCTION

DER growth and climate-linked uncertainty make reliable power-flexibility aggregation important, but heterogeneous portfolios and coupled constraints make exact characterization difficult. The paper proposes transparent, tunable directional support evaluations and complexity analysis to address gaps in scalability and DER generality.

I. INTRODUCTION

  • DER penetration and climate change introduce intertwined uncertainties that amplify power-system reliability challenges, while exact aggregation is computationally intractable in general.The difficulty reflects heterogeneous devices, stochastic behavior, high-dimensional feasible regions, and combinatorial structure.Source: §I · p. 1
  • Existing approaches trade computational efficiency, modeling fidelity, and operational feasibility across inner, outer, directional, and data-driven approximations.Inner approximations preserve feasibility but can underestimate flexibility; outer approximations may include infeasible points, while directional methods can require many samples for tight boundaries.Source: §I · pp. 1–2
  • Data-driven methods use operational data and learning to capture nonlinearities, but may lack explicit feasibility guarantees and be sensitive to data coverage, generalization, and distribution shifts.Prior work includes a classifier framework, inverse optimization for correlated uncertainty, and active-learning calibration of a network-aware approximation.Source: §I · p. 2
  • The literature leaves a gap in transparent, theoretically grounded complexity analysis of scaling with system size, horizon, and constraint coupling.Reported numerical runtimes are case-dependent and offer limited insight into algorithmic efficiency.Source: §I · p. 2
  • Many existing studies focus on conventional DERs, while emerging technologies such as zinc–air and solid-fuel batteries have materially different operating characteristics.The passage identifies DER-agnostic methods that accommodate heterogeneous portfolios without substantial reformulation as a research direction.Source: §I · p. 2
  • The proposed framework uses adaptive LP-based directional support evaluations with configurable device- and system-level constraints, alongside unified complexity analysis.Its feasible boundary trajectories form an inner approximation without explicitly enumerating the high-dimensional region's vertices or facets.Source: §I · p. 2

II. AGGREGATE FLEXIBILITY FORMULATION

The framework constructs aggregate flexibility from directional support evaluations over a joint convex model, with device-specific operating constraints represented in a unified formulation. It accommodates EVs, conventional and alternative storage, and PV while retaining linear-program structure.

II. AGGREGATE FLEXIBILITY FORMULATION

  • For each sampled direction, a support LP returns a feasible aggregate net-power trajectory; the convex hull of retained trajectories forms an inner representation under the continuous convex model.Sampling stops when validation deficit meets tolerance, improvement saturates, or the direction budget is reached.Source: §II · p. 3

1) EVs:

  • EV flexibility allows vehicle-to-grid operation with bidirectional exchange or vehicle-to-grid operation with controlled charging only.The passage names the latter mode V1G and the bidirectional mode V2G.Source: 1) EVs · p. 3

2) Battery and Alternative Energy Storage Systems:

  • Battery storage is constrained by charging and discharging limits, energy capacity, and energy balance over time.The model tracks charging power, discharging power, and state of charge.Source: 2) Battery and Alternative Energy… · p. 3
  • Zinc–air batteries use a refueling parameter in state-of-charge evolution to represent swap events, with a continuous refueling-activation variable.The passage distinguishes their replenishment behavior from conventional lithium-ion batteries.Source: 2) Battery and Alternative Energy… · p. 4
  • Solid fuel batteries exclude electrical charging and use continuous fuel-activation variables to preserve the convex LP structure.The model imposes additional operational constraints for fuel input.Source: 2) Battery and Alternative Energy… · p. 4

3) Photovoltaic Systems:

  • The PV representation is a modeling choice for evaluating aggregation, rather than a detailed inverter-dynamics model.The model includes available PV power and operational flexibility constraints.Source: 2) Battery and Alternative Energy… · p. 4

2) Battery and Alternative Energy Storage Systems:

  • Technology-specific models are combined into a unified convex representation with device operating limits and intertemporal dynamics.The resulting feasible set contains device trajectories under the continuous convex model.Source: 2) Battery and Alternative Energy… · p. 4

B. Aggregate Power Flexibility Region

The aggregate flexibility region is formed by mapping locally feasible DER decisions to total power while enforcing shared cross-DER constraints. These constraints make the coupled region a projection that generally differs from the uncoupled Minkowski sum.

B. Aggregate Power Flexibility Region

  • For large fleets, exact H- or V-representations become computationally intractable as DER count and polytope dimension grow, and cross-DER coupling prevents a simple Minkowski-sum representation.Without coupling, the aggregate set is the Minkowski sum of individual flexibility sets.Source: B. Aggregate Power Flexibility Region · pp. 4–5
  • The coupled aggregate region projects locally feasible DER decisions satisfying feeder, transformer, and portfolio-level constraints onto aggregate power space.The shared constraints are expressed as Acz ≤ bc.Source: B. Aggregate Power Flexibility Region · pp. 4–5
  • Technology-specific models are assembled from local constraints and output mappings; battery storage includes charging, discharging, and state-of-charge variables and limits.ZAB and SFB models add refueling or fuel-input variables, while PV models include generation and curtailment variables.Source: B. Aggregate Power Flexibility Region · pp. 4–5
  • The aggregate power trajectory is p = Hz, summing the individual DER power trajectories represented by their output mappings.The portfolio-level output matrix H maps the stacked decision vector z to aggregate power.Source: B. Aggregate Power Flexibility Region · p. 5

C. Proposed Adaptive Directional Support Approximation

The method samples aggregate flexibility boundaries with support LPs, then takes their convex hull to form a feasible inner approximation. Adaptive validation directs when to stop sampling.

  • For each unit direction, a support LP returns a feasible aggregate boundary trajectory.Construction directions are sampled in batches by normalizing Gaussian vectors.Source: C. Proposed Adaptive Directional… · p. 5
  • A separate validation set measures the maximum normalized support deficit, with true validation supports computed once and cached.The algorithm stops when the deficit meets tolerance, the direction budget is reached, or improvement saturates for the specified patience.Source: C. Proposed Adaptive Directional… · pp. 5–6
  • The convex hull of retained support points is a feasible inner approximation of the coupled aggregate flexibility region.Each retained point is feasible, and convex combinations preserve feasibility.Source: C. Proposed Adaptive Directional… · p. 5

III. THEORETICAL ANALYSIS OF COMPUTATIONAL COMPLEXITY

The complexity analysis relates the sequence of LP solves to solver iterations and sparse Newton-system costs. Under stated locality and bounded-width assumptions, each Newton step scales linearly with portfolio size and horizon.

  • Each sampled projection direction requires an LP solve, and total computational burden depends on model dimensions and the selected solver.The number of decision variables and equality and inequality constraints are identified as important cost factors.Source: §III · p. 6
  • Newton-system solve costs depend on constraint-matrix sparsity and factorization fill-in, while the proposed LP matrix is described as highly sparse.Its constraints involve limited subsets of variables, including time-local power balances and mostly diagonal or block-diagonal bounds.Source: §III · pp. 6–7
  • Under a short-step path-following scheme with a logarithmic barrier, the analysis bounds Newton iterations needed to reach a prescribed duality-gap tolerance.Iteration count depends on the barrier parameter and solution accuracy.Source: §III · p. 6
  • O(NT) Newton-step cost follows when device interactions are local in time, cross-DER constraints remain sparse, and elimination width is uniformly bounded.For fixed scheduling horizon, the per-iteration cost is O(N).Source: §III · p. 7

IV. NUMERICAL SIMULATION

The numerical evaluation uses small- and large-scale cases to examine accuracy, validity, computability, and applicability with heterogeneous DER models.

  • The evaluation comprises a small case of 20 DERs over six time steps and a large-scale case extending to 2000 DERs.The small case illustrates accuracy and validity; the large case examines computability and applicability.Source: §IV · p. 7
  • The large-scale case includes sensitivity analysis to assess the proposed approximation method's performance.DERs are modeled as heterogeneous to reflect real-world conditions and demonstrate generality.Source: §IV · p. 7
  • The small-scale case is designed to illustrate the method's accuracy and validity.Source: §IV · p. 7

A. Small-scale Simulation Case

The simulation setup specifies a heterogeneous 20-device portfolio for the small case and scales the evaluation to 2000 DERs in the large case.

A. Small-scale Simulation Case

  • The small-scale portfolio has 20 DERs—8 EVs, 5 LIBs, 3 ZABs, 2 SFBs, and 2 PVs—over six one-hour time steps.The setup specifies device power, efficiency, energy, and ramp limits, alongside 25 kW grid-power and 40 kWh aggregate-energy limits.Source: A. Small-scale Simulation Case · p. 7

B. A Large-scale Simulation Case

  • The large-scale case evaluates up to 2000 DERs over six one-hour time steps, with direction budgets from 1 to 256.Runtime experiments vary portfolio size from 20 to 2000 and assess model size, construction time, per-direction solve time, and total runtime.Source: B. A Large-scale Simulation Case · p. 7

C. Comparison with Existing Aggregate Models

The proposed directional approximation matches the 180-direction reference in the evaluated two-dimensional projections using 16–64 construction directions, while the simpler models capture less projected area or overestimate the coupled region.

C. Comparison with Existing Aggregate Models

D. Direction-Selection Sensitivity

Increasing the direction budget generally reduces support deficit, though improvements can plateau; for the small case, the deficit reaches numerical tolerance at 64 directions with no measurable gain at 96.

D. Direction-Selection Sensitivity

  • Construction directions are added in batches of eight, and a separate set of 180 validation directions evaluates the maximum normalized support deficit.The validation supports are cached and reused across direction budgets.Source: D. Direction-Selection Sensitivity · p. 8
  • The mean support deficit falls from 1.56 × 10−1 at M = 16 to 4.64 × 10−3 at M = 24, then reaches numerical tolerance at M = 64.Increasing the budget from 64 to 96 yields no measurable improvement.Source: D. Direction-Selection Sensitivity · p. 8
  • Intermediate plateaus occur when sampled directions return vertices already identified; improvement requires exposing a new facet or extreme point.Thus, flat segments followed by abrupt decreases are expected for this support-function-based approximation and do not necessarily indicate algorithmic stagnation.Source: D. Direction-Selection Sensitivity · pp. 8–9
  • With ϵtol = 0.02, the five projections terminate after 16–64 directions, averaging 30.4 support LPs per projection.For this small case, M = 64 is the empirical accuracy elbow; Mmax = 96 is retained as a conservative upper budget.Source: D. Direction-Selection Sensitivity · p. 9

E. Large-Scale Scalability Analysis

The large-scale experiments show runtime growing approximately linearly with portfolio size and direction budget over the tested ranges. For fixed direction budget, grid-constrained runtimes increased from 368.852 s at 1,000 DERs to 854.071 s at 2,000 DERs.

E. Large-Scale Scalability Analysis

  • 368.852 s at N = 1000, 635.823 s at N = 1500, and 854.071 s at N = 2000; grid-constrained runtime grows approximately linearly at M = 128.The reported trend is consistent with growth in support-LP dimensions.Source: E. Large-Scale Scalability Analysis · p. 9, Figure 4 · p. 9
  • At N = 2000, total runtime increases approximately linearly with M, while median per-direction solve time remains nearly constant for moderate and large M.Runtime growth is primarily caused by the increasing number of support evaluations.Source: E. Large-Scale Scalability Analysis · p. 9
  • The method remains tractable for portfolios up to 2000 DERs; runtime dependence on N is governed by support-LP size and sparsity.Among coupled formulations, the grid-constrained case was faster in the reported experiments, possibly because tighter bounds facilitate solver presolve; this advantage is instance-dependent.Source: E. Large-Scale Scalability Analysis · p. 9

V. CONCLUSION

The conclusion presents a unified linear framework for heterogeneous DER flexibility aggregation and an adaptive directional method for constructing feasible inner approximations. Theoretical guarantees and small- and large-scale results support its scalability and accuracy within the reported settings.

V. CONCLUSION

  • The framework represents conventional batteries, EVs, ZABs, SFBs, and PV systems through a unified linear formulation with feeder- and portfolio-level coupling constraints.It also accommodates device-level operating characteristics.Source: §V · p. 9
  • Adaptive directional support constructs feasible inner approximations without explicitly enumerating flexibility-region facets or vertices.The convex hull of the resulting boundary trajectories provides the approximation.Source: §V · pp. 9–10
  • The approximation has feasibility and non-overestimation guarantees, and small-scale projected regions are tighter than box, zonotope, and inner-ball benchmarks.With 16–64 construction directions, it matches the dense directional reference within numerical tolerance.Source: §V · p. 10
  • Large-scale simulations with up to 2000 DERs show approximately linear empirical scaling with portfolio size and direction budget over the tested ranges.The conclusion identifies uncertainty-aware models, adaptive high-dimensional direction selection, parallel computation, and nonlinear distribution-network constraints as future work.Source: §V · p. 10, Figure 5 · p. 10

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