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Fair Dynamic Operating Envelopes using Distributed Multi-Period Optimal Power Flow and Jain Index for Active Distribution Networks

Pedro Salomão Quessongo, Daniel Gebbran, Clodomiro Unsihuay-Vila

arXiv:2608.23444v1eess.SYcs.CE

TL;DR

The paper addresses the lack of a multi-period DOE workflow that separates technical feasibility from cumulative fairness under storage coupling. It proposes a two-stage fair-DOE framework with scenario-dependent BESS recourse, regional ADMM, and AC validation; on the IEEE 33-bus feeder, fairness increased curtailment while improving cumulative fairness without validated voltage or thermal violations.

  • Problem

    Existing approaches loosely couple technical DOE computation, fairness, storage-coupled multi-period operation, and distributed coordination without an explicit budget separating equity from efficiency.

    Source: §I · p. 1, A. Literature Review · p. 2

  • Method

    The framework computes technical envelopes first, redistributes capacity through budgeted cumulative fairness, uses fair DOEs as first-stage decisions with BESS recourse, and solves regional LinDistFlow DOPF using ADMM with AC validation.

    Source: §VI · p. 9

  • Results

    5.7216 MWh versus 2.1097 MWh of technical-stage curtailment accompanied Γ = 11.20%, Jain indices close to unity, and no AC voltage or thermal violations with ΔV AC below 0.01 p.u.

    Source: §VI · p. 9, E. Discussion of Technical Implications · p. 9

  • Takeaways & Limitations

    Co-designing multi-period fairness, which increases curtailment, with battery storage, which alleviates curtailment impact, offers an alternative to single-period DOE design.

    Source: §VI · p. 9

Abstract

from arXiv · show

Dynamic operating envelopes (DOEs) are increasingly used to publish time-varying export limits that keep distribution networks within operational limits. Purely technical DOE allocation, however, can systematically privilege electrically favorable prosumers, while embedding fairness directly into a single-period optimal power flow (OPF) objective mixes network feasibility, equity and efficiency in a way that obscures the cost of fairness. This paper proposes a two-stage, multi-period framework that addresses both of these. Initially, a technical distributed OPF computes network-feasible export envelopes. The subsequent stage then applies a dynamic aggregate export budget and redistributes capacity through cumulative proportional fairness, limiting the additional curtailment by an admissible efficiency budget. The resulting fair DOEs are treated as first-stage decisions, while battery storage provides scenario-dependent recourse under demand and renewable uncertainty. The operational problem is solved by a calibrated regional alternating direction method of multipliers (ADMM) on a lossless LinDistFlow model and independently validated using AC power flow. On the IEEE 33-bus feeder over a 24-hour horizon, the technical benchmark yields 2.1097 MWh of renewable curtailment, whereas the fairness-constrained allocation increases curtailment to 5.7216 MWh but caps the maximum cumulative curtailment ratio at 11.20% and raises Jain fairness indices close to unity, with AC voltage deviations below 0.01 p.u. and no voltage or thermal violations under the adopted 0.90-1.05 p.u. limits. Results show that considering both storage (which alleviates curtailment impact) and multi-period fairness (which increases curtailment) is an interesting approach for modern DOE design, which in turn requires a multi-period, co-designed approach.

I. INTRODUCTION

The paper addresses the need to preserve network integrity, reduce renewable curtailment, and allocate export capacity fairly as DERs make distribution systems active. It proposes a multi-period, two-stage DOE workflow that separates technical feasibility from budgeted cumulative fairness, with storage recourse and distributed solution methods.

  • DER penetration creates bidirectional distribution-network flows and requires tools balancing feeder integrity, renewable curtailment, and fair prosumer export allocation.Source: §I · p. 1
  • DOEs publish time-varying export limits, but technical-only allocation favors electrically favorable locations and distributes curtailment unevenly.Source: §I · p. 1
  • Storage couples operating intervals through state of charge, making independently fair single-period allocations insufficient for cumulative curtailment control.Source: §I · p. 1
  • Existing literature loosely couples network modeling, DOE allocation, fairness, and distributed multi-period optimization without an explicit cumulative budget separating technical feasibility from equity.Source: A. Literature Review · p. 2
  • The framework first computes network-feasible technical envelopes, then redistributes capacity through cumulative proportional fairness under an admissible curtailment-efficiency limit.Source: B. Contributions · p. 2
  • Fair DOEs become first-stage decisions, BESS supplies scenario-dependent recourse, and calibrated regional ADMM solves the LinDistFlow DOPF with independent AC validation.Source: B. Contributions · p. 2

II. MATHEMATICAL FORMULATION

The formulation models a radial feeder over buses, branches, time periods, prosumers, BESS units, and uncertainty scenarios. Its objective combines import, curtailment, storage-cycling, and fairness terms, while a LinDistFlow network model enforces operational limits before AC validation.

II. MATHEMATICAL FORMULATION

  • The formulation defines buses, branches, time periods, prosumers, BESS units, scenarios, and the substation slack bus as the main index sets.Source: §II · p. 2

A. Objective Function

  • The operational objective combines import cost, renewable-curtailment cost, BESS cycling cost, and fairness-related penalties.Source: A. Objective Function · p. 2
  • The fairness term represents avoiding systematic discrimination against electrically remote prosumers, while an efficiency budget limits excessive curtailment.Source: A. Objective Function · pp. 2–3

B. Network and Branch Flow Model

C. Technical and Fair Dynamic Operating Envelopes

The DOE design first maximizes network-feasible technical export capacity, then applies a dynamic aggregate budget and cumulative fairness redistribution. An admissible curtailment budget explicitly bounds the efficiency cost of fair allocation.

D. BESS and Scenario Recourse

The BESS model links charging and discharging decisions across time through state-of-charge dynamics, while scenarios represent uncertainty in demand and renewable availability.

PVP PV

The framework fixes fair DOEs across scenarios while using battery storage as scenario-dependent recourse. Regional ADMM coordinates LinDistFlow subproblems, whose converged dispatch is independently checked with AC power flow.

  • Fair DOEs remain scenario-independent first-stage decisions, while BESS powers and states of charge adapt to each uncertainty scenario.Source: PVP PV · p. 4
  • The feeder is decomposed into electrical regions with local copies of interface active power, reactive power, and squared voltages.Source: E. Distributed ADMM Coordination · p. 4
  • Each regional ADMM subproblem minimizes its local objective contribution subject to regional LinDistFlow and device constraints.Source: E. Distributed ADMM Coordination · p. 4
  • Consensus and scaled-dual updates coordinate shared interface variables across regions, with convergence assessed using primal and dual residuals.Source: E. Distributed ADMM Coordination · p. 4
  • The final dispatch is fixed and validated by AC power flow using corresponding AC and LinDistFlow/DOPF voltages and substation imports.Source: E. Distributed ADMM Coordination · p. 4

F. Fairness and Validation Indicators

Fairness is evaluated from cumulative renewable acceptance and curtailment across prosumers, using Jain and Gini indicators alongside the proposed DOE workflow.

III. PROPOSED ALGORITHM

The proposed algorithm first computes technical envelopes and then allocates fair, budgeted DOEs; scenario-based regional ADMM dispatches storage recourse and adds AC validation.

III. PROPOSED ALGORITHM

  • The two-stage workflow obtains technical envelopes from network limits, then redistributes capacity through cumulative proportional fairness under a dynamic export budget.Source: §III · p. 5

IV. CASE STUDY

  • The IEEE 33-bus radial feeder is partitioned into nine electrical regions for distributed coordination.Source: §IV · p. 5
  • The 24-hour case study includes fifteen PV units, eight wind units, and eight BESS units providing temporal recourse.Source: §IV · p. 5
  • The dynamic DOE budget uses βt = 0.70 from hours 10 to 16, coinciding with peak PV availability.Source: §IV · p. 5

III. PROPOSED ALGORITHM

  • For each scenario, demand, PV, and wind are scaled, fair DOEs are fixed, BESS variables provide recourse, and regional subproblems run in parallel under ADMM.Source: §IV · p. 5
  • After convergence, the dispatch is fixed, AC power flow is run for every period, and curtailment, cost, fairness, and convergence metrics are computed.Source: §IV · p. 5

IV. CASE STUDY

  • The cases incrementally compare baselines, technical and budgeted DOEs, fairness rules, uncertainty scenarios, and AC validation.Source: §IV · p. 5, Table I · p. 5

V. RESULTS AND DISCUSSION

The reported operating-condition results use six critical uncertainty cases, labeled consistently across the scenario table and result figures.

  • Six uncertainty cases ω1–ω6 represent nominal, high PV/low load, high wind, peak/low RES, night peak, and mixed congestion conditions.Source: §V · p. 6
  • The six cases were selected as operating conditions of interest with critical conditions for the quantitative results.Source: §V · p. 6

A. Technical and Fair DOE Allocation

The technical DOE provides network-feasible export capacity, while the fairness stage redistributes capacity over time under an aggregate budget, increasing curtailment to improve cumulative equity. Storage then shifts renewable energy across the day while maintaining acceptable voltage performance.

Technical allocation

Fair allocation

  • 11.20% is the maximum cumulative curtailment ratio after fairness redistribution, with total fair curtailment of 5.7216 MWh.The admissible curtailment budget is 20.0130 MWh, so the fair solution remains within the efficiency allowance.Source: A. Technical and Fair DOE Allocation · p. 6
  • Fairness binds during restrictive periods: fair and proportional aggregates are near 2.5 MW, about 1.1 MW below the technical DOE, while they coincide outside the budget-binding window.This produces additional midday curtailment relative to the technical formulation.Source: A. Technical and Fair DOE Allocation · p. 6
  • At hour 14, final fair envelopes redistribute export capacity across prosumer buses, including near-zero fair export at buses 29 and 31.The cumulative criterion uses prior curtailment history, unlike a snapshot OPF.Source: A. Technical and Fair DOE Allocation · p. 6

Voltage performance

  • Nominal voltages remain above the 0.90 p.u. lower limit, although several remote buses fall below 0.95 p.u. after hour 21.The result supports co-designing DOE limits and multi-period storage when stricter statutory bands are required.Source: B. Nominal Dispatch and Voltage… · p. 7

C. ADMM Convergence under Operating Conditions ω1–ω6

Regional ADMM coordinates the six operating conditions through normalized primal and dual residual tests. All conditions converge within 208–234 iterations, while stressed renewable scenarios produce the largest curtailment and fairness remains generally high.

Convergence

  • The primal residual decays largely monotonically, while the dual residual is more oscillatory before remaining below the unit threshold.The residuals monitor agreement on interface active power, reactive power, and voltage copies.Source: C. ADMM Convergence under Operating… · p. 7, Figure 7 · p. 8
  • 208–234 iterations are required for all six operating conditions, with ω1 converging in 211 and ω2 in 234 iterations.None of the conditions fails to meet both convergence tolerances.Source: C. ADMM Convergence under Operating… · p. 7

Operating conditions

  • 17.528 MWh (25.47%) is the highest renewable curtailment, occurring under ω2 high PV/low load; ω3 and ω6 follow with 12.177 MWh and 9.898 MWh.These stressed conditions require connection-point envelopes to bind to preserve feeder integrity.Source: C. ADMM Convergence under Operating… · p. 7

Fairness

  • JFI stays above 0.98 where curtailment is material, reaching near 0.998 for ω1 and ω6, while ω3 has a higher curtailment Gini coefficient of 0.4216.High-wind congestion still concentrates some curtailment on electrically weaker buses despite high JFI.Source: C. ADMM Convergence under Operating… · p. 7

Operating conditions

  • ω4 and ω5 require limited emergency demand response of 0.0807 and 0.0137 MWh and are not strictly feasible without that recourse.Near-zero curtailment makes relative Gini values unstable in these conditions.Source: C. ADMM Convergence under Operating… · p. 8

D. Independent AC Validation

Independent AC power-flow validation confirms that the distributed fair-DOE schedules remain physically feasible under nonlinear network physics. The validation also clarifies that this check is separate from optimization and does not provide a comparable AC-OPF runtime.

Validation results

  • AC and LinDistFlow substation exchanges track closely, with AC values slightly higher because losses are represented.Nominal validation compares the fixed DOPF schedule against nonlinear AC power flow.Source: D. Independent AC Validation · p. 8
  • The corrected substation power-deviation metric stays near zero after accounting for AC losses.Voltage and power deviations are evaluated after the nonlinear AC re-evaluation.Source: D. Independent AC Validation · p. 9
  • Maximum AC voltage deviation remains below approximately 0.01 p.u., with no voltage or thermal violations under the 0.90–1.05 p.u. limits.Minimum AC voltages remain above 0.90 p.u. and maximum voltages remain near 1.00 p.u.Source: D. Independent AC Validation · pp. 8–9

Validation scope

  • The AC validation holds the DER/BESS schedule fixed and is not a separate AC OPF solve, so its runtime is not comparable with LinDistFlow DOPF timing.A dedicated AC-validation wall-clock time was not measured in the submitted runs.Source: D. Independent AC Validation · p. 9

Pipeline implication

  • The two-stage fair-DOE pipeline separates technical envelope computation, regional ADMM coordination, and independent AC feasibility checking.This combination addresses the need to avoid direct DER micro-management while retaining network-physics verification.Source: E. Discussion of Technical Implications · p. 9, §VI · p. 9

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