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Preference-Oriented Aggregation of Heterogeneous Distributed Energy Resources for Reserve Dispatch

Jingguan Liu, Xiaomeng Ai, Shichang Cui, Xizhen Xue, Shengshi Wang, Jiakun Fang, Wei Yao, Jinyu Wen

arXiv:2609.03252v1eess.SY

TL;DR

Heterogeneous DER aggregation is challenged by incompatible time dimensions, irregular resource profiles, and flexibility loss in conventional approximations. The paper uses matrix-transformation projection and preference-oriented aggregation-dispatch coordination to address these issues. The method captures active flexibility relevant to reserve dispatch and is reported to reduce conservatism for type-heterogeneous DERs.

  • Problem

    Existing aggregation methods struggle with dimensional heterogeneity and diverse, irregular DER profiles, causing flexibility loss and conservative reserve-dispatch representations.

  • Method

    The paper reformulates the Minkowski sum as a matrix-transformation polytope projection and coordinates aggregation with reserve-dispatch preferences through a distributed framework.

  • Results

    The method aggregates DERs across different time dimensions and captures critical active regions, reducing conservatism and identifying more potential reserve flexibility than preference-agnostic methods.

  • Takeaways & Limitations

    Preference-oriented aggregation focuses approximation on flexibility prioritized by optimal reserve dispatch rather than merely maximizing aggregate-set volume.

Abstract

from arXiv · show

Aggregating distributed energy resources (DERs) aims to encode their collective flexibility into a single set for efficient grid dispatch. However, existing aggregation methods are overly conservative for heterogeneous DERs due to two main challenges: 1) dimensional heterogeneity, which complicates the combination of flexibilities across different time dimensions, and 2) type heterogeneity, where diverse and irregular DER profiles hinder accurate approximations, resulting in significant flexibility loss. To resolve these challenges, this paper propose a novel preference-oriented aggregation method for reserve dispatch. For dimensional heterogeneity, we extend existing techniques by reformulating the Minkowski sum as a polytope projection problem using a matrix transformation technique. By unifying DERs in a higher-dimensional space and projecting them back into the aggregate feasible region, the proposed technique effectively aggregates dimensionally heterogeneous DERs. For type heterogeneity, we further develop a distributed aggregation-dispatch coordination framework that incorporates reserve dispatch preferences into aggregation. This framework effectively captures the critical, active aggregate flexibility prioritized in optimal reserve dispatch, thereby significantly reducing the flexibility loss when aggregating type-heterogeneous DERs. Numerical tests validate the effectiveness of our method in addressing both heterogeneities and highlight its promising potential for power systems with high reserve requirements.

A. Abbreviations:

The paper addresses computationally difficult aggregation of heterogeneous DER flexibility by combining matrix-transformation projection with preference-oriented dispatch coordination. Its framework targets dimensional and type heterogeneity while preserving flexibility relevant to reserve dispatch.

  • Background: DER aggregation represents heterogeneous resource flexibilities as one collective set for reserve dispatch.The paper considers ESSs, thermal controllable residents, and deferrable loads, including plug-in EVs.
  • Aggregation challenge: Exact Minkowski-sum aggregation is generally computationally intractable for high-dimensional H-representation DER polytopes.The difficulty increases when DERs have diverse parameters, variables, and time dimensions.
  • Dimensional heterogeneity: Dimensional heterogeneity prevents a single prototype from consistently approximating DERs across different time dimensions.Existing polytope-based methods approximate DERs individually before computing their Minkowski sum.
  • Type heterogeneity: Type-heterogeneous DER profiles can cause significant flexibility loss and overly conservative aggregate sets under volume-maximizing methods.The paper notes that active reserve-dispatch regions may cover only a subset of the submitted aggregate flexibility set.
  • Proposed aggregation: The proposed method reformulates the Minkowski sum as a matrix-transformation polytope projection in a unified higher-dimensional space.The projected result returns to the lower-dimensional aggregate feasible region, with a linear reformulation for computational tractability.
  • Preference-oriented coordination: A distributed aggregation-dispatch framework incorporates optimal dispatch preferences and uses ADMM with parallel-enabled and warm-start strategies.The framework targets critical active aggregate regions rather than simply enlarging aggregate-set volume.

B. Aggregate Flexibility Set

The paper represents DER flexibility aggregation as a Minkowski-sum problem, then addresses dimensional heterogeneity by lifting DER sets into a common high-dimensional space and projecting them into aggregate power space. This global matrix-transformation approach avoids dimensional collapse and yields a less conservative inner approximation.

  • Aggregation challenge: Exact aggregation is generally computationally intractable because Minkowski sums of high-dimensional H-representation polytopes are NP-hard.The difficulty is especially pronounced for DERs with diverse parameters, variables, and time dimensions.
  • Feasibility: The resulting aggregate set remains an inner approximation, preserving feasible points for subsequent reserve-dispatch disaggregation.The paper states that any profile within the aggregate set has a feasible disaggregation strategy satisfying DER operational constraints, while detailed disaggregation strategies remain future work.
  • Proposed aggregation: The proposed method lifts heterogeneous DER flexibility sets into a high-dimensional polytope and projects the result onto the VPP’s lower-dimensional aggregate set.This provides a uniform high-dimensional representation while avoiding direct summation of inconsistent individual approximations.
  • Dimensional heterogeneity: Existing polytope methods approximate each DER with a common base set, but dimensional incompatibility can substantially underestimate available flexibility.A lower-dimensional DER may be reduced to a lower-dimensional subset, such as collapsing a three-dimensional feasible region to one dimension.
  • Proposed aggregation: Matrix transformation replaces direct cross-dimensional summation with a structured projection that requires compatibility with the union, rather than the intersection, of DER dimensions.The less restrictive condition avoids dimensional collapse and supports more accurate inner approximations for highly heterogeneous DERs.
  • Relationship to prior methods: The method generalizes existing polytope-based approaches and is reported to address dimensional heterogeneity more effectively than prior approaches.Existing methods arise as a special case of the proposed formulation under a particular transformation setting.

B. Tractable Reformulation

The containment constraints underlying the aggregate-set approximation are reformulated into a tractable linear representation. This enables commercial-solver computation, although formulation size still grows with the number of DERs.

  • Linear reformulation: Farkas’ Lemma is used to convert the difficult polytope-containment constraint into solvable linear constraints.Proposition 3 provides the reformulation in constraints (14.a)–(14.b).
  • Computational tractability: The reformulated constraints provide a tractable linear representation of the original complex containment condition.The paper reports satisfactory computational efficiency in numerical simulations.
  • Limitation: The number of continuous variables and constraints scales with the number of DERs, increasing overall problem size for larger inputs.This growth is identified as a concern for large-scale implementation.
  • Scalability: Existing grouping strategies can further reduce computational complexity, and the linear-programming formulation allows commercial solvers such as GUROBI to handle sizable instances.The paper gives partitioning large DER populations into groups as an example of this extension.
  • Preference coordination: After linear reformulation, preference-oriented aggregation uses updated reserve-dispatch preference signals to guide VPP aggregation iteratively until convergence.The coordination parameters remain fixed after convergence.

IV. PREFERENCE-ORIENTED AGGREGATION METHOD

The method incorporates reserve-dispatch preferences into aggregate-set optimization to capture critical active flexibility for heterogeneous DERs. A distributed coordination framework exchanges preference signals between the power system and VPPs while avoiding centralized coordination.

  • Motivation: Type-heterogeneous DER profiles can cause conventional inner approximations to sacrifice substantial flexibility because their feasible sets are diverse and irregular.The method therefore focuses on critical active aggregate regions rather than maximizing aggregate-set volume alone.
  • Motivation: Reserve-dispatch preferences vary with grid conditions and scenarios, making the critical active aggregate flexibility difficult to predict in advance.Centralized coordination is also impractical because of computational complexity and VPP privacy concerns.
  • Preference signals: Lagrange multipliers associated with the boundary equality constraint are interpreted as preference signals that guide VPP aggregate-set optimization.The aggregate-set parameters used by VPPs and the power-system dispatch subproblem are represented separately but linked by the constraint.
  • Distributed coordination: The distributed framework lets the power system share dispatch preferences with VPPs, which locally compute aggregate sets aligned with system requirements while preserving privacy.The framework addresses unpredictability, computational burden, and privacy through distributed coordination.

A. Aggregation Problem of VPPs

The VPP aggregation problem optimizes approximate aggregate-set shape parameters under inner-approximation constraints and a Lagrangian penalty linking VPP aggregation with reserve dispatch. The reserve-dispatch model is a two-stage stochastic optimization over day-ahead capacity and scenario-based regulation.

  • VPP aggregation formulation: VPPs optimize approximate aggregate-set shape parameters based on received preference signals while satisfying inner-approximation constraints.The Lagrangian penalty term links the VPP parameters to corresponding parameters used by the power-system dispatch problem.
  • VPP aggregation formulation: Separable VPP objectives and constraints make the aggregation variables orthogonal and allow the Lagrangian penalty to be reformulated into independent terms.The reformulation uses the squared differences between each VPP's parameter vectors and their dispatch-side counterparts.
  • Reserve dispatch: The grid reserve-dispatch model is a two-stage stochastic optimization with day-ahead reserve-capacity decisions followed by scenario-based intra-day regulation.The model includes VPPs and thermal units in both reserve provision and deployment.
  • Reserve dispatch: The objective minimizes total operational cost and the Lagrangian penalty, while constraints enforce power balance, transmission limits, generation limits, and operational bounds.Second-stage constraints also address wind curtailment and load shedding limits.

C. ADMM Solution Algorithm

The proposed coordination problem is solved by ADMM, alternating between VPP aggregation, grid reserve dispatch, and preference-signal updates until residual-based stopping criteria are met. The method is parallelizable and, in the toy example, captures dimensionally heterogeneous flexibility while preserving optimal dispatch points with a smaller aggregate region.

  • Algorithm procedure: ADMM alternates between parallel VPP aggregation, reserve-dispatch optimization, and updates of the Lagrange-multiplier preference signals.The procedure repeats until primal and dual residuals satisfy the convergence threshold.
  • Algorithm procedure: The orthogonality of VPP parameter vectors decomposes the aggregation step into independent subproblems that can be solved concurrently.Only boundary aggregation parameters and preference signals are shared, supporting privacy protection.
  • Convergence: ADMM convergence is guaranteed for the stated convex aggregation and reserve-dispatch subproblems, whereas nonconvex dispatch applications require additional modifications.This convergence guarantee is scoped to the convex formulation used in the framework.
  • Toy-example results: The toy example shows that the matrix-transformation cases approximate the aggregate flexibility of ESSs represented with different dimensions.The exact aggregate set requires enumerating extreme points, whose number grows exponentially with period dimension, while the inner approximation uses fewer parameters.
  • Toy-example results: 65.78% of Case 2's volume in Case 3 still captures Case 1's optimal operating points and yields the same optimal reserve-dispatch results.The preference-oriented set focuses on critical regions rather than inactive feasible areas.

B. Method Validation in Small-Scale System

Validation studies show that preference-oriented aggregation improves reserve dispatch quality and flexibility utilization for heterogeneous DERs while retaining computational practicality.

  • System setup: Ten stochastic net-load scenarios are generated for a 24-hour, 1-hour-granularity modified 6-bus system with an 80 MW wind farm.The system has three thermal units and 210 MW total load.
  • Computational efficiency: 12 iterations and 66 seconds suffice for warm-started Case 3b, compared with 19 iterations and 105 seconds for Case 3a, with the same final objective value.Each preference-oriented iteration takes approximately 5.5 seconds, and aggregating 45 heterogeneous DER clusters takes 2.1 seconds.
  • Reserve dispatch comparison: 93.07% of expected DER reserve deployment is achieved by preference-oriented Case 3, versus 78.67% for preference-agnostic Case 2 relative to unaggregated Case 1.Case 3 incurs only a 6.30% reserve-dispatch-cost increase, whereas Case 2 incurs a 21.32% increase.
  • Reserve dispatch comparison: 29.6% lower overall DER reserve capacity in Case 2 than Case 3 reflects its inability to anticipate increased reserve demand during 17:00–18:00.Case 3 also has smaller reserve-deployment deviation and exploits more potential DER regulation flexibility.
  • Practical-scale validation: 18.1% higher expected DER reserve deployment and 19.4% lower total reserve-dispatch cost are obtained by Case 3 than Case 2 in the practical-scale system.Unaggregated Case 1 does not achieve optimal dispatch within one day of CPU time, while Cases 2 and 3 solve within reasonable time.
  • Method validation: The method combines matrix-transformation aggregation across time dimensions with preference-oriented aggregation of critical active regions for heterogeneous DER reserve dispatch.The paper reports applicability to practical-scale systems, while noting future work on computational efficiency, disaggregation, and base-set configuration.

VII. APPENDIX

The appendix specifies operational constraints for ESSs and TCRs, including power, energy, timing, and thermal-dynamics relationships.

  • ESS model: ESS constraints bound charging and discharging power during each DER’s available start and end periods.The model also constrains storage energy and represents charging and discharging efficiencies.
  • ESS model: ESS storage energy is constrained between its minimum and maximum values over the available operating horizon.The appendix identifies storage energy, charging power, discharging power, and availability periods as model quantities.
  • TCR model: TCR constraints represent thermal inertia through discrete-time power and energy relationships over the available operating periods.The TCR model includes a conversion coefficient from active power to temperature and an ambient-parameter impact factor.

3) Deferrable Loads:

The appendix models deferrable loads through availability, power, residual-energy, and energy-requirement constraints, then derives full-dimensional DER polytopes by eliminating auxiliary variables.

  • Deferrable-load model: Deferrable-load constraints bound electric power input and residual energy between specified start and end periods.The model includes nonnegative power and energy relationships over the load’s available timeframe.
  • Deferrable-load model: The deferrable-load energy state evolves with input power, dissipation, and conversion parameters across discrete periods.A terminal requirement enforces the required residual energy by the end of the available interval.
  • Coefficient-matrix derivation: QR decomposition and Gaussian elimination remove equality-constrained variables before the remaining inequalities are transformed into a full-dimensional DER set.The resulting representation retains only linear inequalities in the relevant power variables.
  • Coefficient-matrix derivation: Fourier–Motzkin elimination combined with redundancy identification removes extraneous energy variables and redundant constraints until the original DER set becomes full-dimensional and non-redundant.The process iteratively eliminates variables while pruning redundant inequalities.
  • Computational consideration: Independent DER models permit parallel elimination, limiting the effect of Fourier–Motzkin elimination complexity on the number of DERs.The paper notes that Fourier–Motzkin elimination can otherwise have exponential complexity because redundant constraints accumulate.

C. Proof of Proposition 3

The proposition proof establishes linear containment conditions for the aggregate approximation by combining transformed aggregate variables, auxiliary variables, and a base-set approximation.

  • Aggregate representation: The aggregate constraints combine transformed aggregate power with auxiliary variables through linear inequalities and equality relations.These relations define the transformed aggregate representation used in the containment proof.
  • Containment proof: A conservative approximation is obtained by dropping a constraint in the minimization, yielding a tractable sufficient condition for containment.The proof then invokes the polytope containment relationship between the relevant sets.
  • Containment proof: Strong duality converts the containment condition into a dual minimization form involving the aggregate and base-set representations.The resulting comparison establishes that the right-hand-side set dominates the left-hand-side set for the relevant coefficient vectors.
  • Existence proof: A full-dimensional base set provides a unit-box construction that supports the existence argument for the aggregate representation.The proof uses a nonnegative epsilon and a dimensional unit box associated with the base set.

D. Parameters in Case Studies

The case studies specify parameter probability distributions for ESS, plug-in EV, and TCR clusters, alongside load, wind, and real-system representations.

  • Parameter probability distributions are provided for ESS clusters.
  • Parameter probability distributions are provided for plug-in EV clusters.
  • Parameter probability distributions are provided for TCR clusters.
  • The parameter description includes distribution statistics and maximum indoor temperature.
  • The case studies also include load-demand and wind-power profiles and a topological diagram of the real power system case.
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