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Leveraging Two-Stage Adaptive Robust Optimization for Power Flexibility Aggregation

Xin Chen, Na Li

arXiv:2005.03768v2eess.SY

TL;DR

Power flexibility aggregation must characterize feasible substation exchanges from many DERs across time while respecting coupled network and device constraints. The paper uses two-stage ARO to compute optimal active-power intervals and elliptical active-reactive regions, with adaptive dispatch ensuring disaggregation feasibility. Numerical tests on a real feeder report greater extracted flexibility than a conservative method and validate feasible disaggregation and accurate elliptical approximation.

  • Problem

    Multi-period aggregation must capture DER and network constraints while achieving both aggregation optimality and disaggregation feasibility; existing heuristic constraints are conservative.

  • Method

    Two-stage ARO models aggregate active power with time-varying feasible intervals and joint active-reactive power with time-variant elliptical regions.

  • Results

    35.39 MWh versus 32.90 MWh shows 2.49 MWh more flexibility from APA than Method 1, while simulations validate disaggregation feasibility and accurate elliptical approximation.

  • Takeaways & Limitations

    The proposed framework guarantees aggregation optimality and disaggregation feasibility for active-power and joint active-reactive flexibility aggregation.

Abstract

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Adaptive robust optimization (ARO) is a well-known technique to deal with the parameter uncertainty in optimization problems. While the ARO framework can actually be borrowed to solve some special problems without uncertain parameters, such as the power flexibility aggregation problem studied in this paper. To effectively harness the significant flexibility from massive distributed energy resources (DERs), power flexibility aggregation is performed for a distribution system to compute the feasible region of the exchanged power at the substation over time. Based on two-stage ARO, this paper proposes a novel method to aggregate system-level multi-period power flexibility, considering heterogeneous DER facilities, network operational constraints, and an unbalanced power flow model. This method is applicable to aggregate only the active (or reactive) power, and the joint active-reactive power domain. Accordingly, two power aggregation models with two-stage optimization are developed: one focuses on aggregating active power and computes its optimal feasible intervals over multiple periods, and the other solves the optimal elliptical feasible regions for the aggregate active-reactive power. By leveraging the ARO technique, the disaggregation feasibility of the obtained feasible regions is guaranteed with optimality. Numerical simulations on a real-world distribution feeder with 126 multi-phase nodes demonstrate the effectiveness of the proposed method.

NOMENCLATURE

The notation defines limits and state variables for voltages, currents, DER outputs, storage, controllable loads, HVAC systems, and substation injections over time.

  • Voltage and current symbols denote upper and lower limits for three-phase nodal magnitudes and distribution-line magnitudes.
  • PV, energy-storage, controllable-load, and HVAC symbols describe phase-level active or reactive power, capacity, state-of-charge, energy, and temperature quantities.
  • HVAC notation includes indoor temperature, comfort-zone bounds, and the discretized time-slot length.
  • Substation vectors collect total three-phase net active and reactive power injections over the time horizon.
  • Superscript-free symbols denote corresponding quantities summed over phases.

C. Notations

This section introduces the distribution-network and DER setting, the multi-period aggregation problem, and the two-stage ARO formulation used to obtain feasible aggregate-power regions.

  • Problem context: Exact aggregate-power feasibility is computationally impractical at scale, motivating approximation methods for heterogeneous DER flexibility.
  • Problem context: Snapshot-based approaches may miss inter-temporal constraints from energy storage and HVAC systems.
  • Problem context: Prior heuristic constraints can ensure disaggregation feasibility but are conservative and produce sub-optimal aggregation solutions.
  • Two-stage ARO: The paper treats the aggregate feasible region as an ARO uncertainty set, with adaptive decisions providing feasible DER dispatch for every aggregate-power realization.
  • Two-stage ARO: The proposed models maximize a parameterized set inscribed in the exact feasible region while considering heterogeneous DERs, unbalanced power flow, and network constraints.
  • Aggregation models: APA computes multi-period active-power intervals, whereas ARPA computes time-variant elliptical regions for the aggregate P-Q domain.
  • Network and DER models: The distribution system is modeled as a multi-phase unbalanced network with wye- or delta-connected devices and phase-specific complex-power injections.
  • Network and DER models: The multi-period model includes dispatchable PV, energy storage, controllable loads, and HVAC systems.

2) Energy Storage Devices:

The energy-storage model represents charging and discharging through signed power, enforces state-of-charge dynamics and limits, and requires final SOC recovery. The broader DER formulation uses approximations while retaining compatibility with the ARO aggregation framework.

  • Signed storage power represents discharging as positive and charging as negative, with storage efficiency modeling energy loss over time.
  • SOC constraints bound the storage state and require the final SOC to recover its initial value for sustainability.
  • The system formulation also includes indoor-temperature dynamics, linearized multi-phase power flow, and voltage and line-thermal network constraints.
  • The DER models use common approximations for a precision-efficiency trade-off, while more realistic models incorporating charging and discharging power loss can be adopted.
  • Equation (5b) prevents simultaneous charging and discharging through a complementarity constraint, whose non-convexity can be handled by penalty reformulation.
  • As long as DER constraints remain convex, they can be incorporated into the aggregation method while retaining the ARO framework and CCG solution algorithm.

C. Comprehensive System Model

The comprehensive model stacks multi-period DER and network constraints into a compact formulation, then interprets aggregation as a two-stage ARO projection problem. The resulting feasible set is designed to be both large and exactly disaggregable.

  • C. Comprehensive System Model: The compact system model stacks multi-period DER equations and the linearized network model into variables x and aggregate powers p0 and q0.
  • C. Comprehensive System Model: The formulation includes apparent-power capacity constraints for PV units and energy-storage devices, alongside remaining DER and network constraints.
  • A. Power Aggregation Modelling via ARO: The aggregation problem seeks an inner approximation D of the exact aggregate-power region that maximizes volume while preserving disaggregation feasibility.
  • A. Power Aggregation Modelling via ARO: ARO treats D as the uncertainty set, aggregate power as the uncertainty variable, and DER dispatch x(p0,q0) as an adaptive response.
  • A. Power Aggregation Modelling via ARO: The ARO constraint enforces D ⊆ Proj(F), so every aggregate-power point in D has a feasible DER dispatch within the operational region.
  • A. Power Aggregation Modelling via ARO: Two concrete models compute multi-period feasible intervals for active or reactive power and elliptical regions for the joint active-reactive domain.

B. Active Power Aggregation Model

The active power aggregation model characterizes time-decoupled feasible intervals at the substation using a two-stage ARO formulation. It maximizes aggregate flexibility while guaranteeing disaggregation feasibility for all represented power trajectories.

  • The model depicts aggregate active power flexibility with time-decoupled feasible intervals over multiple periods.
  • The aggregate power trajectory is represented as a linear combination of lower and upper bounds using ξ_t ∈ [0,1].This change of variables replaces the original feasible-region condition with a fixed box uncertainty set.
  • The APA model obtains the optimal feasible region with maximal flexibility.
  • The first-stage decision maximizes total flexibility, while the second-stage adaptive dispatch responds after ξ is revealed.The minimax formulation requires a feasible DER dispatch for every worst-case scenario in U1.
  • Adaptive robust constraints guarantee disaggregation feasibility exactly, avoiding conservative heuristic constraints that can produce sub-optimal solutions.The proposed formulation also applies to joint active-reactive power aggregation, unlike the prior heuristic constraints described here.

C. Active-Reactive Power Aggregation Model

The active-reactive power aggregation model targets the coupled P-Q flexibility of a distribution system. It optimizes time-varying elliptical feasible regions through a two-stage ARO formulation with exact disaggregation feasibility.

  • Joint P-Q aggregation is needed because active and reactive power are coupled in network and DER operational constraints.Reactive power flexibility can support voltage security, reduce network loss, and support transmission-system operation.
  • The ARPA model optimally aggregates both active and reactive power using time-varying elliptical feasible regions.The ellipse is parameterized by its center and a positive semidefinite matrix describing rotation and stretching.
  • The ellipse area is represented by det(Y_t), so the objective maximizes total aggregate P-Q flexibility over all time periods.
  • The ARPA model is a two-stage semidefinite program whose adaptive robust constraints guarantee disaggregation feasibility exactly.Other parameterized convex sets, including polygons, can also represent aggregate P-Q flexibility under the proposed method.
  • ARPA and APA address distinct applications: ARPA aggregates joint P-Q flexibility, whereas APA aggregates active power alone.APA is intended for settings where active power is traded or dispatched, while ARPA supports aggregation of reactive power as well.

D. Practical Application and Power Disaggregation

The aggregated feasible region can be reported to the transmission operator for system-level scheduling, after which the distribution system disaggregates the assigned trajectory among DERs. The resulting disaggregation problem remains feasible for any trajectory inside the obtained region.

  • A distribution system can provide its optimal feasible region for transmission-level scheduling as a virtual power plant.The region is used similarly to a conventional generator’s capability curve.
  • Each distribution system reports its optimal feasible intervals, and the transmission operator uses them with generator capability curves to determine dispatch.
  • After receiving a regulation trajectory inside the feasible region, the distribution system solves a power-disaggregation problem to determine the DER dispatch.The dispatch tracks the assigned trajectory while satisfying network and DER operational constraints.
  • The disaggregation problem is feasible for any regulation trajectory contained in the proposed aggregate feasible region.
  • The operational cost model can include storage degradation, HVAC discomfort, photovoltaic curtailment, and electricity-purchasing costs.

IV. SOLUTION ALGORITHM

The solution algorithm uses column-and-constraint generation to solve the two-stage ARO models. Its sub-problems are reformulated into tractable mixed-integer second-order cone programs, while circular uncertainty is conservatively linearized.

  • Column-and-constraint generation decomposes each two-stage ARO model into a master problem and a sub-problem solved iteratively.
  • The master problem enumerates uncertainty scenarios and assigns adaptive decisions, providing an upper bound that approaches the original optimum as scenarios are added.
  • The sub-problem searches for a worst-case scenario lacking a feasible adaptive dispatch and adds it to the master problem when found.This process tests whether the candidate feasible interval guarantees disaggregation feasibility.
  • The active-power sub-problem becomes an MISOCP, with integer-variable dimension equal to the number of time periods rather than network or DER size.
  • The circular uncertainty set for ARPA is approximated by rotated square constraints; using more squares improves accuracy while preserving robustness.The outer approximation ensures solutions feasible for the approximation remain feasible for the original circular uncertainty set.
  • The ARPA sub-problem uses the same reformulation approach and is also a MISOCP.

D. Column and Constraint Generation Algorithm

The APA model is solved with a column-and-constraint generation algorithm that iteratively refines the master problem using subproblem-generated constraints until convergence.

  • Algorithm 1: The CCG algorithm alternates between solving the master problem and adding new variables and constraints generated from the subproblem.Initialization sets K = 1 and a tolerance ε; iterations continue while |fS| ≥ ε.
  • Algorithm properties: CCG is guaranteed to generate the optimal solution within finitely many iterations of order O(L), where L is the number of uncertainty-set extreme points.The master and subproblems can be solved with optimizers such as IBM CPLEX and Gurobi.
  • Algorithm 1: The algorithm terminates when the convergence criterion is met and outputs the final feasible intervals for aggregate active power.The final output is represented as (p∨*0, p∧*0).
  • Overall workflow: The proposed aggregation method and its solution algorithm are organized into the overall structure shown in Figure 3.

V. NUMERICAL SIMULATION

The numerical study evaluates active-power aggregation on a real unbalanced feeder using measured load and solar profiles, comparing APA with a prior method across storage capacities.

  • Test system: The test feeder contains 126 multi-phase buses, 366 single-phase connections, 33 PV units, 28 ES devices, and 5 HVAC systems.The study uses a 12 kV substation, voltage limits of 1.02 p.u. and 0.98 p.u., real load and irradiance profiles, and 30-minute intervals.
  • Input profiles: The simulation covers total available PV power and uncontrollable loads from 9:00 to 16:00 at 30-minute granularity.Figure 4 presents the corresponding time-series inputs.
  • Active-power comparison: APA is compared with Method 1 using the feasible intervals of aggregate active power and aggregate flexibility under different ES capacities.The comparison is illustrated in Figure 5 and summarized in Table I.
  • Active-power comparison: 35.39 MWh versus 32.90 MWh: APA extracts 2.49 MWh more aggregate flexibility than Method 1.The paper attributes the difference to Method 1’s conservative ES and HVAC constraints, whereas APA guarantees disaggregation feasibility through ARO.

B. Implementation of Power Disaggregation

The study tests whether aggregate flexibility points can be disaggregated into feasible DER dispatches for active power and approximates active-reactive flexibility with time-varying elliptical regions.

  • Active-power disaggregation: Up to 3000 randomly generated regulation trajectories are tested, and the disaggregation problem is feasible for every trajectory.The trajectories follow a uniform distribution independently at each time, and an optimal DER dispatch is obtained for each case.
  • Active-power disaggregation: Figure 6 compares the optimal aggregate active-power interval in blue with a randomly generated regulation trajectory in red.The figure illustrates one of the tested trajectories.
  • Active-power disaggregation: Figure 7 shows the optimal PV, ES, and HVAC dispatch used to track the regulation trajectory from Figure 6.The plotted PV, ES, and HVAC outputs are summed over the corresponding DER types.
  • Active-reactive aggregation: ARPA computes time-varying elliptical feasible regions for the aggregate active-reactive domain from 9:00 to 14:00 at one-hour granularity.The red point denotes the center and the blue areas denote feasible aggregate P-Q regions.
  • Active-reactive aggregation: The computed ellipses lie within the tested feasible P-Q points and cover most of the green feasible area.Feasible points are tested at 0.5 MW(MVar) resolution at four separate times; larger regions at 12pm and 14pm are associated with higher available PV generation.
  • Computational efficiency: The APA master and subproblem take 9.25s and 213.9s on average, while ARPA takes 145.2s and 571.6s.CCG for APA usually converges within one or two iterations, and the paper considers the computational time acceptable for hours-ahead/day-ahead scheduling.

APPENDIX A LINEAR MULTI-PHASE POWER FLOW MODEL

The appendix derives a linear multi-phase power-flow model from complex power injections, nodal voltages, and a fixed-point voltage equation for use in the aggregation formulation.

  • Model formulation: Three-phase complex power injections are collected into wye- and delta-connected vectors, which are combined with DER and network variables.The formulation defines sY := pY + qY and s∆:= p∆+q∆ and constructs the associated variable and matrix representations.
  • Voltage model: The complex nodal voltage vector satisfies a fixed-point equation, which is linearized to obtain model (29) for voltage magnitudes.The derivation uses a given operational point and the real part and complex-conjugate operators.
  • Voltage model: Model (29) can be interpreted as a linear interpolation between a given operational point and the zero-power-injection operating point.
  • Model formulation: Kirchhoff’s laws provide the matrices, vectors, and scalars used in the multi-phase power-flow representation.The appendix refers to reference for their detailed derivation.
  • Subproblem reformulation: The subproblem reformulates ||yl|| ≤ sl using yl = Elx and constructs a Lagrangian dual representation with nonnegative dual variables.The resulting dual form follows from the stated Lagrangian conditions, including σl ≥ 0.
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