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Aggregation and Disaggregation of Energetic Flexibility from Distributed Energy Resources
Fabian L. Müller, Jácint Szabó, Olle Sundström, John Lygeros
TL;DR
Distributed energy resources offer flexibility, but quantifying and aggregating many small systems is computationally challenging. The paper uses zonotopic feasible-set approximations, efficient aggregation, and economically fair disaggregation, showing that the framework supports aggregate feasibility checks, regulation bids, and lower baseline costs than box approximations.
Problem
Flexibility from distributed small-scale energy resources must be quantified and aggregated to support grid operation, but exact aggregate feasible sets are generally computationally intractable.
Method
The paper approximates individual feasible sets with zonotopes, aggregates them through Minkowski sums, and disaggregates aggregate signals using an economically fair algorithm.
Results
Zonotopic descriptions enable explicit aggregate feasibility checks and regulation-bid computation, while yielding lower baseline costs than axis-aligned box approximations.
Takeaways & Limitations
Zonotopes provide a computationally efficient representation for aggregate control and market decisions involving large populations of flexible systems.
Abstract
from arXiv · showhide
A variety of energy resources has been identified as being flexible in their electric energy consumption or generation. This energetic flexibility can be used for various purposes such as minimizing energy procurement costs or providing ancillary services to power grids. To fully leverage the flexibility available from distributed small-scale resources, their flexibility must be quantified and aggregated. This paper introduces a generic and scalable approach for flexible energy systems to quantitatively describe and price their flexibility based on zonotopic sets. The description proposed allows aggregators to efficiently pool the flexibility of large numbers of systems and to make control and market decisions on the aggregate level. In addition, an algorithm is presented that distributes aggregate-level control decisions among the individual systems of the pool in an economically fair and computationally efficient way. Finally, it is shown how the zonotopic description of flexibility enables an efficient computation of aggregate regulation power bid-curves.
I. INTRODUCTION
Growing renewable penetration increases variability and reduces reliable regulation power, while distributed resources offer additional flexibility that is difficult to quantify and coordinate at scale. The paper proposes zonotopic aggregation and economically fair disaggregation to support control and regulation bids.
- Motivation: Renewable variability and reduced rotational inertia increase power-grid requirements for additional energetic flexibility.Wind and solar introduce intermittency and limited controllability, while replacing traditional generators reduces reliable regulation power.
- Motivation: Distributed resources such as HVAC systems, plug-in vehicles, batteries, and micro-generation can provide flexibility on both demand and supply sides.Their flexibility is defined through feasible electric-power trajectories over a given time horizon.
- Feasible-set formulation: A system’s feasible set consists of the power trajectories it can track over a finite horizon, subject to power, energy, ramp, and state constraints.The resource polytope summarizes these constraints, while PE-systems impose only power and energy constraints.
- Existing approaches: Exact aggregate feasible sets are generally computationally intractable, motivating approximation methods for populations of flexible systems.Outer approximations may include infeasible trajectories, whereas inner approximations remain within the feasible set.
- Contribution: The paper introduces zonotopes as compact approximations that can aggregate efficiently while representing time-variant and asymmetric power, energy, and ramp constraints.Zonotopes are used to support aggregate-level control and market decisions.
- Contribution: An economically fair disaggregation algorithm distributes aggregate control signals among individual systems, and the framework supports regulation-power bid computation.Disaggregation has received limited attention compared with flexibility aggregation.
3) Ramp-rate constraints:
Flexible energy systems are modeled through linear constraints on power, energy, ramp rates, and internal states. Zonotopes then provide centrally symmetric inner approximations that can be aggregated efficiently, though their symmetry can reduce approximation quality.
- 3) Ramp-rate constraints:: Ramp-rate constraints bound the change in power between consecutive time steps.The bound is expressed through the scaled difference (p_k − p_{k−1})/t_s for k = 2, . . . , N.
- State constraints: Internal states such as battery state of charge or heating temperature can be constrained through linear state dynamics.The dynamics include other inputs u_k and power p_k between lower and upper state bounds.
- Feasible-set representation: Power trajectories satisfying the system constraints form a convex resource polytope P := {p ∈ R^N : Ap ≤ b}.A and b summarize the constraint matrices and limits.
- PE-systems: PE-systems are idealized energy buffers subject only to power and energy constraints, with feasible sets called PE-polytopes.This restriction is used for a class of flexible energy-resource models.
- B. Zonotopic feasible sets: Zonotopes represent feasible sets using a center, generators, and symmetrically bounded scaling factors, enabling efficient aggregation.The paper uses a generator matrix with 2N − 1 generators for PE-polytopes.
- B. Zonotopic feasible sets: For PE-polytopes, the proposed generators can reconstruct every possible facet, but central symmetry introduces additional facets absent from the original polytope.These additional facets influence approximation quality.
III. COMPUTING ZONOTOPIC APPROXIMATIONS
The paper computes inner-approximating zonotopes for resource polytopes by solving optimization problems that enforce containment and maximize a chosen approximation objective. The approach also addresses degeneracy and evaluates approximation quality for plug-in electric vehicles.
- Optimal zonotopic approximations: Zonotopes are optimized as inner approximations of resource polytopes using containment constraints.The containment condition is enforced through linear constraints on the zonotope center, generators, and scaling bounds.
- Optimal zonotopic approximations: The objective Λ measures zonotope coverage, with Λ = 0 for a point and Λ = 1 for perfect polytope approximation.The maximization problem is formulated as a linear program.
- Preventing degeneracy: Degeneracy is addressed by adding constraints requiring the zonotope to contain a full-dimensional object such as a box.Without these constraints, some zonotope facet distances may be zero even when the corresponding polytope extension is positive.
- Approximation results: Approximation quality depends on the generator matrix, polytope shape, objective function, dimension, and number of possible facets.The evaluation considers plug-in electric-vehicle resource polytopes over a 24 h horizon with 2 h sampling.
B. Aggregation of feasible sets
The aggregate feasible set combines individual systems’ feasible trajectories, but general polytopes are computationally difficult to aggregate. Zonotopes provide an efficiently updateable aggregate representation for aggregator control and market operations.
- Aggregation setup: The aggregate feasible set is the Minkowski sum of all individual feasible sets.It contains every collective power trajectory that the population can follow.
- Aggregation of polytopic flexibility: Minkowski sums of general polytopes are computationally challenging, especially in hyperplane representation and higher dimensions.Existing methods rely on special polytope structures, low dimensions, or approximations of the true sum.
- Aggregation of zonotopic flexibility: Zonotopes can be aggregated efficiently because their Minkowski sum is itself a zonotope.The aggregate representation can be updated by adjusting the sums of centers and scaling bounds when systems or feasible sets change.
- Aggregator operation: Aggregators assign feasible reference trajectories and account for system-specific energy prices and flexibility costs.Disaggregation selects individual trajectories that sum to the aggregate trajectory while optimizing the aggregator’s objective.
B. Disaggregation using zonotopes
The zonotopic disaggregation method distributes an aggregate control trajectory among feasible systems by optimizing separable flexibility costs in generator coordinates. Aggregate cost components are constructed from individual piecewise-linear convex costs by ordering their segments by slope.
- Zonotopic disaggregation: The method represents each feasible system as a zonotope parameterized by center, generators, and bounded coefficients.This structure makes aggregate disaggregation scale favorably with the number of systems.
- Cost representation: System costs are rewritten in terms of generator coefficients, separating fixed costs from coefficient-dependent piecewise-linear convex components.The fixed term is independent of the coefficient vector.
- Aggregate cost construction: Each aggregate cost component is formed by concatenating individual line segments in ascending order of slope.The resulting aggregate component is piecewise-linear and convex.
- Aggregate cost construction: The aggregate cost function represents the cheapest disaggregation of an aggregate coefficient vector among the individual systems.The construction is illustrated by combining example cost components from two systems.
- Aggregate cost construction: The ordered segment lists include breakpoint lengths and the originating system index.These lists are prerequisites for the subsequent disaggregation algorithm.
C. Subgradient-based disaggregation algorithm
The paper reformulates zonotope-based disaggregation so its optimization size is independent of the population size, then solves it with a projected subgradient method and assigns the aggregate solution to individual systems.
- Zonotope structure makes the disaggregation problem’s decision variables and constraints independent of the population size J.
- The convex disaggregation problem has a piece-wise linear objective and can be formulated as a linear program through an epigraph reformulation.
- The proposed algorithm initializes a feasible aggregate variable, alternates subgradient steps with projections, and assigns the best aggregate solution to individual systems.
2) Subgradient step:
The subgradient step updates the aggregate decision using a diminishing step size, projects it back into the feasible set, and retains the best objective value until termination.
- 2) Subgradient step:: The update is β(k + 1) = β(k) − α(k)∇(k), with α(k) = a/k and a > 0.The cited convergence statement gives limk→∞ T(β(k)) = T*; setting a = (β̄(agg)g)/g is reported as effective.
- 3) Projection:: After each subgradient update, Euclidean projection Q(·) restores feasibility with respect to the disaggregation constraints.
- Because the subgradient method is not a descent method, the algorithm stores the best objective value found and stops using a termination criterion.
D. Performance of subgradient-based disaggregation
The subgradient approach addresses the population-dependent size and memory demands of the original LP formulation, while zonotopes also make aggregate-feasibility checks explicit and inexpensive.
- The performance comparison uses uniformly sampled PEV populations, random linear flexibility costs, MATLAB, and CPLEX on a desktop computer.
- The original disaggregation problem grows with both population size J and time steps N, requiring NJ variables and J(4N − 2) inequalities for PE systems.Memory is reported as exhausted for practical problem sizes, including a weekly secondary-reserve formulation with N = 672.
- 96-time-step instances take 34.0 s to 3811.1 s with LP-based disaggregation, versus 0.17 s to 0.48 s with the subgradient method.
- For zonotopes, aggregate feasibility is checked by validating q inequalities rather than solving the computationally expensive polytopic disaggregation problem.
B. The cost of offering regulation power
The regulation-power formulation reserves an axis-aligned cube around the aggregate baseline and finds the cheapest feasible baseline for a specified symmetric reserve level.
- Offering r̄ units of symmetric constant regulation requires every time-step deviation of r̄ from the baseline to remain feasible.In power space, this is represented by an axis-aligned cube of edge length 2r̄ centered at the aggregate baseline.
- For a fixed reserve r̄ and expected wholesale price v̂, the aggregator solves for the cheapest baseline.
- Each system reserves a portion η of its available flexibility to maintain r̄ = ηr̄max regulation power.
- Zonotopic approximations are suboptimal relative to polytopes but yield significantly lower baseline costs than axis-aligned boxes.The results characterize zonotopes as a compromise between accurate, complex polytopes and simple, inaccurate boxes.
- At 20 kW reserve, baseline costs are −59.5 EUR for polytopes, −58.6 EUR for zonotopes, and −25.8 EUR for boxes.
APPENDIX A FACETS OF PE-ZONOTOPES
The appendix characterizes zonotope facets through structured normal vectors and shows that these facets can reconstruct every possible facet of a PE-polytope.
- Zonotopes can reconstruct every possible facet of a PE-polytope.
- Every facet of a full-dimensional zonotope has a normal vector formed by summing consecutive unit vectors.The consecutive index interval satisfies 1 ≤ j ≤ k ≤ N.
- The proof partitions the coordinates into the maximizing interval and its complement, then combines points from the corresponding projected facets.The combination γ(x, y) preserves membership in the facet through compatible generator coefficients.
- The constructed combinations show that the facet decomposes across the coordinate partition and that each projected set consists of points maximizing the restricted center in the projected zonotope.
- The generator vectors listed for the supporting hyperplane have rank N − 1, establishing that the hyperplane is a facet.
- The maximum number of facets of a zonotope is N^2 + N, and this bound is attained if the stated construction is realized.