Power Flexibility Aggregation Insights

• Power Flexibility Aggregation is defined as the aggregate feasible set of power trajectories from distributed loads, characterized by the Minkowski sum of individual convex polytopes.
• The method employs a polytopic projection approximation with variable substitutions and robust linear programming to reduce exponential complexity to a tractable, scalable formulation.
• The approach enables decentralized scheduling and reliable disaggregation, achieving significant cost reductions and operational efficiency in grid flexibility markets.

The Power Flexibility Aggregation Problem addresses the analytic quantification, tractable modeling, and practical deployment of the aggregate active power flexibility available from collections of distributed, controllable loads. This flexibility is instrumental for grid operators and aggregators seeking to participate in demand response, arbitrage, and ancillary services markets, while accurately reflecting the physical and operational constraints of potentially heterogeneous load populations such as plug-in electric vehicles, deferrable loads, or distributed storage. The main technical challenge arises from faithfully characterizing the aggregate feasible set, typically defined as the Minkowski sum of individual polytopes, while overcoming the formidable exponential growth in description complexity for heterogeneous, high-dimensional systems.

1. Analytic Characterization of Aggregate Flexibility

The aggregate flexibility of a collection of deferrable loads is defined as the set of aggregate power trajectories that can be feasibly decomposed into admissible individual charging profiles. Each individual load’s admissible charging trajectories is a convex polytope, e.g.,

$$\mathcal{P}^i = \Big\{ u^i \in \mathbb{R}^m : u_t^i \in [0, p^i] \,\, \forall\, t \in \mathcal{A}^i,\, u_t^i = 0 \,\, \forall\, t \notin \mathcal{A}^i,\, \sum_t u_t^i \in [\underline{E}^i,\,\overline{E}^i] \Big\}$$

where $\mathcal{A}^i$ represents the allowed time indices for charging. The aggregate admissible set is the Minkowski sum: $$\mathcal{P} = \bigoplus_{i=1}^N \mathcal{P}^i = \left\{ u \in \mathbb{R}^m : u = \sum_{i=1}^N u^i,\,\, u^i \in \mathcal{P}^i \right\}$$ A fundamental result gives an exact analytic characterization: an aggregate profile $u$ is feasible if and only if

$$\min \left\{\sum_{i \in \alpha} \overline{E}^i - \sum_{t \in \beta} u_t,\;\;\; \sum_{t \in \beta^c} u_t - \sum_{i \in \alpha^c} \underline{E}^i \right\} \geq -\sum_{i \in \alpha^c} |\beta \cap \mathcal{A}^i| p^i$$

holds for all $\alpha \subseteq \{1, ..., N\}$, $\beta \subseteq \{1, ..., m\}$. However, the exact representation involves $O(2^{mN})$ inequality constraints in the general heterogeneous case, rendering it impractical for nontrivial system sizes. In the homogeneous case, the representation collapses to majorization constraints with $O(N + m)$ inequalities.

2. Polytopic Projection Approximation

Given the intractable complexity of the exact analytic representation, the aggregate flexibility is approximated using a polytopic projection approach. Geometrically, the individual flexibility sets are polytopes in $\mathbb{R}^m$, and their aggregation via Minkowski sum can equivalently be viewed as projecting a higher-dimensional polytope: $$\tilde{\mathcal{P}} = \{ (u, \tilde{u}) \in \mathbb{R}^{m + \tilde{m}} : u = \sum_{i=1}^N u^i,\,\, u^i \in \mathcal{P}^i \}$$ onto the $u$-subspace: $$\mathcal{P} = \operatorname{Proj}_u({\tilde{\mathcal{P}}})$$ This interpretation enables one to target a tractable inner approximation by seeking the largest homothet of a nominal polytope $\mathbb{B}$ (such as a “virtual battery” polytope) that is contained in the projection: $$\lambda \mathbb{B} + \mu \subseteq \operatorname{Proj}_u({\tilde{\mathcal{P}}})$$ After reformulation with variable substitutions $s = 1/\lambda$, $r = -\mu/\lambda$, the inclusion becomes amenable to constraint generation and efficient computation.

3. Robust Optimization Formulation and Computational Strategies

The approximation task is cast as a robust optimization problem: $$\max \lambda \quad \text{subject to} \quad \lambda \mathbb{B} + \mu \subseteq \operatorname{Proj}_u(\tilde{\mathcal{P}})$$ Rewriting via variable substitution allows the problem to be handled as: $$\min s \quad \text{subject to} \quad \mathbb{B} \subseteq s \operatorname{Proj}_u(\tilde{\mathcal{P}}) + r$$ Leveraging the linearity of both the projection and translation operators, the problem is further reformulated using Farkas’s lemma. The affine adjustable optimization—where variables representing individual charging profiles are taken as affine functions of $u$—leads to a linear programming (LP) formulation: $$\begin{aligned} \min_{s} \qquad & s \ \text{s.t.} \qquad & G F = B [I; W] \ & G H \leq B [r; -V] + s c \ & G \geq 0 \end{aligned}$$ where $F$, $H$, $c$, $B$ arise from the facet representations of $\mathbb{B}$ and $\tilde{\mathcal{P}}$.

To manage scalability, a multi-stage divide-and-conquer aggregation is proposed: small groups of loads are aggregated first, each yielding an intermediate virtual battery model (dilation and translation), then groups are recursively merged using the same approach. Key properties, such as the additive structure for homothetic Minkowski sums, ensure mathematical consistency and allow parallelization.

4. Scheduling Policy and Aggregate-to-Individual Disaggregation

The projection-based approach yields not only the aggregate feasible set but also an explicit disaggregation (scheduling) policy for the individual loads. For $K$ groups, suppose the aggregated flexibility is modeled as a “sufficient battery” $\mathbb{B}_s = \lambda \mathbb{B} + \mu$ with parameters

$$\lambda = \sum_k \lambda_k \quad (\lambda_k = 1/s_k),\qquad \mu = \sum_k \mu_k \quad (\mu_k = -r_k/s_k)$$

Given a feasible aggregate power profile $u \in \mathbb{B}_s$, each group receives

$$\tilde{u}_k = W_k \cdot \left(\frac{\lambda_k}{\lambda} (u - \mu) + V_k\right)$$

This ensures that feasible aggregate behaviors can always be mapped to individual load schedules that respect local constraints, thus guaranteeing implementability even for highly heterogeneous populations.

5. Application to Energy Arbitrage and Market Participation

To demonstrate operational relevance, the approach is applied to an energy arbitrage problem for a population of 1000 plug-in electric vehicles (PEVs). The LP-based aggregate flexibility model serves as a surrogate for the combined fleet: $$\min \pi^T z \qquad \text{s.t.} \qquad z \in \mathbb{B}_s$$ where $\pi$ are day-ahead LMPs and $z$ is the scheduled aggregate power trajectory. The derived scheduler reduces total energy costs by approximately 20% compared to immediate charging. Furthermore, the computed flexibility region accurately reflects underlying variability, with largest charging flexibility during low-price periods. The approach thus provides both economic efficiency and operational reliability in leveraging distributed flexible loads for energy services.

6. Trade-offs, Computational Scalability, and Deployment

Direct use of the exact analytic representation is rendered infeasible due to its exponential growth in constraint number. The polytopic projection-and-scaling approach achieves marked reductions in constraint count (from exponential to polynomial in $N$ and $m$), supporting tractable convex optimization and decomposable parallel implementation. The method also naturally accommodates load heterogeneity and general time-varying availability.

The main trade-off is between model fidelity and tractability: outer or inner approximations will necessarily exclude some feasible points, but strong numerical results indicate the maximum-homothet approach closely matches the true feasible set for practical populations (as shown for 1000 PEVs). The multi-stage aggregation supports application to systems with tens of thousands of loads without centralized orchestration, enabling integration with distributed decision-making architectures.

7. Significance and Implications for Grid Flexibility Markets

This polytopic projection approximation framework provides a rigorous, practical, and computation-ready means to capture, communicate, and schedule the aggregate power flexibility of large, heterogeneous portfolios of deferrable loads. By guaranteeing that every aggregate schedule in the computed set can be mapped to feasible individual-level actions—and by enabling large-scale parallel scheduling algorithms—this approach is foundational for system operators, aggregators, and market participants seeking to unlock flexibility for renewable integration, demand response, and grid services while ensuring individual feasibility and privacy constraints are satisfied.