Network-Constrained Aggregate Flexibility Sets

• Network-constrained aggregate flexibility sets are defined as the collection of feasible multi-period substation power trajectories that respect device limits, network capacities, and inter-temporal dynamics.
• They utilize representations such as constrained zonotopes, ellipsoidal approximations, and polymatroids to enable robust projection and computational tractability under uncertainty.
• These models facilitate implementable disaggregation and hierarchical coordination of distributed energy resources, enhancing grid reliability and dynamic operating envelopes.

Network-constrained aggregate flexibility sets are set-valued representations of what a collection of distributed energy resources, loads, or feeder-level devices can realize at a grid interface when internal network physics and device constraints are respected. In the distribution-systems literature, the object is typically defined as a set of feasible multi-period substation or interconnection power trajectories, often under uncertainty, such that every aggregate trajectory in the set admits at least one feasible disaggregation to individual devices. Equivalent formulations appear as robust feasible projections of device-and-network constraints, as intersections of aggregate resource polytopes with network half-spaces, and as projections of multi-stage optimal power flow feasible sets onto coupling variables (Cui et al., 2020, Taha et al., 2022, Raetsch et al., 4 Nov 2025, Mukhi et al., 19 Sep 2025).

1. Formal definitions and conceptual scope

A canonical formulation models the aggregate flexibility of a feeder by the set of substation injections $p_0 \in \mathbb{R}^T$ that remain feasible for all admissible uncertainty realizations. In "Network-Cognizant Time-Coupled Aggregate Flexibility of Distribution Systems Under Uncertainties" (Cui et al., 2020), this is written as

$$\mathcal{F} = \big\{ p_0 \in \mathbb{R}^T \,\big|\, \forall\, \zeta \in \mathcal{Z},\; \exists\, p \in \mathbb{R}^{nT} \text{ s.t. } W p \le z(\zeta),\; D p + b(\zeta) = p_0 \big\}.$$

Here, $W p \le z(\zeta)$ encodes device-level limits, time-coupled dynamics, and linearized network constraints, while $D p + b(\zeta)=p_0$ maps internal decisions to the substation trajectory.

A second, equivalent viewpoint defines the flexibility set as a projection. In the constrained-zonotope multi-period MS-OPF formulation, the network-constrained aggregate flexibility set is

$$\mathcal{F}_i \doteq \operatorname{proj}_{\mathcal{Z}_i}\mathcal{X}_i = \{ z \in \mathcal{Z}_i \mid \exists\ y\ \text{s.t. }(z,y)\in\mathcal{X}_i\},$$

where $\mathcal{X}_i$ is the full feasible set of the distribution subproblem and $z$ collects coupling variables such as interface active/reactive powers and boundary voltages (Raetsch et al., 4 Nov 2025). In EV aggregation, the same idea appears as an intersection: $U^{\text{net}} := U \cap N,$ where $U$ is the aggregate device-level flexibility set and $N$ is the network admissibility set in the same trajectory space (Taha et al., 2022).

These formulations share three structural features. First, the sets are network-constrained: admissibility is determined by feeder capacities, line flows, voltage magnitudes, or AC/DistFlow relations, not only by device capability. Second, they are often time-coupled: storage state of charge, HVAC thermal states, or EV cumulative-energy constraints couple decisions across periods. Third, they are typically interface-oriented: the set is expressed at the substation, feeder, bus, or TSO–DSO coupling point rather than in the internal device coordinates (Chen et al., 2018, Bandeira et al., 27 Oct 2025).

2. Network and device models shaping the set

The geometry of a network-constrained aggregate flexibility set is inherited from the physical and operational models used underneath. In radial distribution feeders, linearized DistFlow-like and multiphase linear power-flow models are common. For example, the substation real power and bus voltages may be expressed as affine functions of nodal injections,

$$\mathcal{F} = \big\{ p_0 \in \mathbb{R}^T \,\big|\, \forall\, \zeta \in \mathcal{Z},\; \exists\, p \in \mathbb{R}^{nT} \text{ s.t. } W p \le z(\zeta),\; D p + b(\zeta) = p_0 \big\}.$$0

with voltage bounds imposed through corresponding linear relations (Cui et al., 2020). In unbalanced feeders, the same theme appears in fixed-point linearizations of full multiphase power flow,

$$\mathcal{F} = \big\{ p_0 \in \mathbb{R}^T \,\big|\, \forall\, \zeta \in \mathcal{Z},\; \exists\, p \in \mathbb{R}^{nT} \text{ s.t. } W p \le z(\zeta),\; D p + b(\zeta) = p_0 \big\}.$$1

together with phase-wise voltage and current limits (Chen et al., 2018).

In formulations that retain more of the AC structure, the flexibility set is derived from branch-flow or AC power-flow constraints. For radial grids with active and reactive flexibility, feasible PQ regions are defined through nodal balances, nonlinear branch-flow equations, voltage bounds, and apparent-power limits, then sampled to construct nodal and region-level PQ capability curves (Hinneck, 4 Apr 2026). In multi-period constrained-zonotope aggregation, DistFlow equations with voltage and current box constraints are linearized over all time steps, yielding a bounded convex polytope before projection (Raetsch et al., 4 Nov 2025).

Device dynamics determine the time-coupled dimension of the set. Storage typically follows equations such as

$$\mathcal{F} = \big\{ p_0 \in \mathbb{R}^T \,\big|\, \forall\, \zeta \in \mathcal{Z},\; \exists\, p \in \mathbb{R}^{nT} \text{ s.t. } W p \le z(\zeta),\; D p + b(\zeta) = p_0 \big\}.$$2

with power-rate and state-of-charge bounds; HVAC loads obey thermal recursions with indoor-temperature bands; EVs are modeled through cumulative-energy inequalities $$\mathcal{F} = \big\{ p_0 \in \mathbb{R}^T \,\big|\, \forall\, \zeta \in \mathcal{Z},\; \exists\, p \in \mathbb{R}^{nT} \text{ s.t. } W p \le z(\zeta),\; D p + b(\zeta) = p_0 \big\}.$$3 coupled with pointwise charging-power bounds (Cui et al., 2020, Taha et al., 2022, Panda et al., 2023). This structure makes the aggregate set fundamentally trajectory-based rather than a static $$\mathcal{F} = \big\{ p_0 \in \mathbb{R}^T \,\big|\, \forall\, \zeta \in \mathcal{Z},\; \exists\, p \in \mathbb{R}^{nT} \text{ s.t. } W p \le z(\zeta),\; D p + b(\zeta) = p_0 \big\}.$$4–$$\mathcal{F} = \big\{ p_0 \in \mathbb{R}^T \,\big|\, \forall\, \zeta \in \mathcal{Z},\; \exists\, p \in \mathbb{R}^{nT} \text{ s.t. } W p \le z(\zeta),\; D p + b(\zeta) = p_0 \big\}.$$5 capability chart.

Uncertainty enters either through explicit robust sets or through bounded exogenous disturbances. In the robust feeder formulation, load uncertainty is modeled by multiplicative perturbations with $$\mathcal{F} = \big\{ p_0 \in \mathbb{R}^T \,\big|\, \forall\, \zeta \in \mathcal{Z},\; \exists\, p \in \mathbb{R}^{nT} \text{ s.t. } W p \le z(\zeta),\; D p + b(\zeta) = p_0 \big\}.$$6, so the aggregate flexibility set contains only trajectories implementable for all $$\mathcal{F} = \big\{ p_0 \in \mathbb{R}^T \,\big|\, \forall\, \zeta \in \mathcal{Z},\; \exists\, p \in \mathbb{R}^{nT} \text{ s.t. } W p \le z(\zeta),\; D p + b(\zeta) = p_0 \big\}.$$7 in the prescribed uncertainty set (Cui et al., 2020). A plausible implication is that the same set-theoretic machinery can absorb broader uncertainty descriptions whenever those descriptions preserve tractable robust counterparts.

3. Set representations and approximation paradigms

Recent work uses several non-equivalent geometric representations for network-constrained aggregate flexibility. They differ in exactness, tractability, and the kinds of coupling they preserve.

Representation	Source	Characteristic
Ellipsoidal inner approximation	(Cui et al., 2020)	Robust maximum-volume ellipsoid inside the feasible region
Hyperbox / interval band	(Chen et al., 2018)	Cartesian product of per-period bounds
AH-polytope	(Taha et al., 2022)	Affine image of a base H-polytope
Constrained zonotope	(Raetsch et al., 4 Nov 2025)	Projection-friendly representation of convex polytopes
Generalized polymatroid	(Mukhi et al., 19 Sep 2025)	Exact EV aggregate set under box-type feeder limits
UL-flexibility polytope	(Panda et al., 2023)	Exact $$\mathcal{F} = \big\{ p_0 \in \mathbb{R}^T \,\big|\, \forall\, \zeta \in \mathcal{Z},\; \exists\, p \in \mathbb{R}^{nT} \text{ s.t. } W p \le z(\zeta),\; D p + b(\zeta) = p_0 \big\}.$$8-parameter aggregate EV representation

Ellipsoidal inner approximations arise when the exact projected feasible set is intractable and one seeks a compact, smooth surrogate. The robust formulation in (Cui et al., 2020) computes

$$\mathcal{F} = \big\{ p_0 \in \mathbb{R}^T \,\big|\, \forall\, \zeta \in \mathcal{Z},\; \exists\, p \in \mathbb{R}^{nT} \text{ s.t. } W p \le z(\zeta),\; D p + b(\zeta) = p_0 \big\}.$$9

as a robust maximum-volume ellipsoid contained in the true network-aware region. Because $W p \le z(\zeta)$0 encodes correlations across time, ellipsoids capture inter-temporal coupling more effectively than axis-aligned boxes.

Hyperboxes remain important because of simplicity and explicit disaggregation guarantees. In unbalanced distribution systems, the approximate feasible region is

$W p \le z(\zeta)$1

and any trajectory in this band is guaranteed to admit a feasible disaggregation under the paper’s joint ordering constraints (Chen et al., 2018).

Polyhedral approaches are more varied. AH-polytopes represent aggregate flexibility as affine transformations of a common template polytope and are especially prominent for EV fleets (Taha et al., 2022). Generalized polymatroids provide an exact description for aggregate EV charging under power and cumulative-energy bounds, and retain tractability after intersection with feeder-capacity boxes (Mukhi et al., 19 Sep 2025). UL-flexibility gives an exact $W p \le z(\zeta)$2-parameter representation over a request window, with feasibility checked by $W p \le z(\zeta)$3 constraints and optimization represented by $W p \le z(\zeta)$4 inequalities (Panda et al., 2023). Multi-battery models approximate heterogeneous EV populations by a linear combination of a small number of base batteries and are designed to minimize a conservative approximation of the Hausdorff distance while remaining an inner approximation (Taha et al., 2023).

Constrained zonotopes occupy a distinctive position because linear maps and projections are cheap. In the multi-period grid-aware formulation, the feasible set of a linearized MS-OPF is converted into a constrained zonotope and then projected onto coupling variables by a sparse matrix, yielding an exact representation of the convex approximation used in the underlying model (Raetsch et al., 4 Nov 2025).

4. Computational tractability and algorithmic formulations

The central computational difficulty is that exact set projection and exact Minkowski-sum construction are often intractable. Computing the maximum-volume ellipsoid of a projected polytope is NP-hard even without uncertainty, and computing the Minkowski sum of H-polytopes is likewise NP-hard in general (Cui et al., 2020, Taha et al., 2022). Direct polytope projection in multi-period network models becomes computationally expensive as the number of buses, constraints, and time steps grows (Raetsch et al., 4 Nov 2025).

The literature therefore relies on structured approximations and special-purpose reformulations. In robust feeder aggregation, the intractable adaptive robust problem is converted into tractable convex programs by restricting second-stage disaggregation policies. Quadratic policies yield an SDP approximation using an approximate S-lemma, whereas affine policies yield an exact SOC reformulation (Cui et al., 2020). In AH-polytope aggregation, containment of affine images inside individual H-polytopes is encoded by nonnegative multipliers in a Farkas-like linear system, leading to LPs whose size grows polynomially in the number of EVs and time periods (Taha et al., 2022).

For generalized polymatroids, linear optimization over the aggregate set is handled by greedy algorithms using only evaluations of the submodular and supermodular set functions, rather than explicit enumeration of exponentially many inequalities (Mukhi et al., 19 Sep 2025). For constrained zonotopes, the heavy computation is shifted offline: a bounding zonotope is intersected with all half-spaces of the linearized MS-OPF, and subsequent projection is a linear map (Raetsch et al., 4 Nov 2025).

Numerical comparisons show the importance of representation choice. On a real 126-node feeder without load uncertainty, the ellipsoid with quadratic policy had volume approximately $W p \le z(\zeta)$5, the ellipsoid with affine policy approximately $W p \le z(\zeta)$6, and the hyperbox approximately $W p \le z(\zeta)$7, illustrating the conservatism of time-decoupled boxes when inter-temporal coupling is active (Cui et al., 2020). In the constrained-zonotope study, for a 15-bus grid with $W p \le z(\zeta)$8 time steps, offline construction took about $W p \le z(\zeta)$9 s while online projection remained about $D p + b(\zeta)=p_0$0 s, which the paper treats as acceptable pre-computation for real-time use (Raetsch et al., 4 Nov 2025).

5. Disaggregation, coordination, and operational use

A defining requirement of network-constrained aggregate flexibility sets is implementability: an aggregate point is useful only if it can be mapped to feasible internal actions. Several constructions make this explicit. In the robust feeder ellipsoid, for any $D p + b(\zeta)=p_0$1 in the ellipsoid and any admissible uncertainty realization, the affine second-stage policy reconstructs a feasible DER vector $D p + b(\zeta)=p_0$2 (Cui et al., 2020). In the unbalanced-feeder box model, any trajectory within the interval band is implemented by a time-varying convex interpolation between upper and lower DER trajectories (Chen et al., 2018). In AH-polytope EV aggregation, if $D p + b(\zeta)=p_0$3, then

$D p + b(\zeta)=p_0$4

provides a closed-form affine disaggregation map when $D p + b(\zeta)=p_0$5 is invertible (Taha et al., 2022).

The same principle appears in exact polymatroidal EV aggregation. After optimizing over the network-constrained aggregate set $D p + b(\zeta)=p_0$6, the selected aggregate trajectory is decomposed through a Frank–Wolfe and Carathéodory vertex decomposition, together with the property that vertices of a Minkowski sum decompose into sums of vertices of the summand polytopes (Mukhi et al., 19 Sep 2025). Multi-battery aggregation similarly returns affine maps that recover individual EV schedules from a chosen aggregate trajectory (Taha et al., 2023).

At the operational interface, these sets support hierarchical coordination. At the TSO–DSO boundary they appear as feasible substation power trajectories or feasible operational regions. For multiple interconnection points, the flexibility set becomes a high-dimensional polyhedron in the active/reactive power coordinates of all interfaces, rather than a single $D p + b(\zeta)=p_0$7–$D p + b(\zeta)=p_0$8 chart (Bandeira et al., 27 Oct 2025). Dynamic operating envelopes perform a closely related role at the grid edge: the DOE construction in (Jalilian et al., 18 Apr 2026) allocates customer-facing flexibility sets whose Cartesian product is guaranteed to lie inside the network-feasible polytope under partial coordination, fairness constraints, and bounded uncertainty.

Empirically, coordination enlarges usable flexibility. In the DOE framework, coordinating 30% of customers increased the achievable aggregate active-power injection range by approximately 25% relative to the non-coordinated baseline (Jalilian et al., 18 Apr 2026). This suggests that network-constrained aggregate flexibility is not only a representation problem but also a coordination problem: the geometry of the feasible set depends on which internal decisions can be jointly orchestrated.

6. Limitations, misconceptions, and research directions

A common misconception is that aggregate flexibility is simply the Minkowski sum of device capabilities. The literature repeatedly rejects that view. Once voltage limits, thermal limits, feeder capacities, and topology are enforced, the feasible aggregate set is generally strictly smaller and structurally different from a network-agnostic sum (Cui et al., 2020, Raetsch et al., 4 Nov 2025). Another misconception is that time-wise bounds are sufficient; they are not when storage, thermal, or charging deadlines induce inter-temporal coupling (Chen et al., 2018, Panda et al., 2023).

The dominant limitations are also consistent across papers. Many formulations rely on linearized power flow, fixed power factor, or simplified network constraints such as feeder-capacity boxes (Cui et al., 2020, Mukhi et al., 19 Sep 2025). Exact AC models are more faithful but much harder to project or robustify (Hinneck, 4 Apr 2026). Uncertainty models are often norm-bounded or box-like, and richer temporal or spatial correlations remain less developed. Some methods provide only inner approximations, so conservatism is intrinsic even when the representation is computationally elegant (Cui et al., 2020, Taha et al., 2023).

Current directions point toward broader grid awareness. Constrained-zonotope projection extends aggregation to up to 96 time steps on a 15-bus grid while preserving exactness for the convexified MS-OPF (Raetsch et al., 4 Nov 2025). Polymatroidal EV aggregation incorporates node-level network limits while preserving exact polyhedral structure (Mukhi et al., 19 Sep 2025). AC-constrained flexibility assessment under distribution system reconfiguration shows that topology can significantly influence and improve operational flexibility; in the reported 95-bus MV case, moving from an unfavorable to an optimal topology increased normalized PQ capability area by 173% (Hinneck, 4 Apr 2026). This suggests that future network-constrained aggregate flexibility sets will increasingly couple set representation, topology control, and hierarchical coordination rather than treating them as separate layers.