Aggregate Flexibility of Thermostatically Controlled Loads using Generalized Polymatroids

TL;DR

The paper addresses aggregating the flexible demand of heterogeneous TCL populations with dissipative dynamics by extending generalized polymatroid methods. It introduces a base polytope $\mathcal{B}_i(\underline{y},\overline{y})$ that is itself a g-polymatroid and computes maximal inner approximations $\mathcal{B}(\underline{y}^*,\overline{y}^*)$ to each device's flexibility, enabling tractable aggregation via $\mathcal{Q}(p_\mathcal{N}^*,b_\mathcal{N}^*)$ with $p_\mathcal{N}^* = \sum_i p_i^T$ and $b_\mathcal{N}^* = \sum_i b_i^T$. The authors derive recursive expressions for $p_i^t(A)$ and $b_i^t(A)$ to construct the aggregate g-polymatroid and prove containment $\mathcal{Q}(p_\mathcal{N}^*,b_\mathcal{N}^*) \subseteq \mathcal{F}_\mathcal{N}$, with numerical results showing tighter approximations and improved generation-signal tracking over baselines. This yields a scalable, near-exact framework for representing and exploiting TCL flexibility in hierarchical grid operation for heterogeneous, dissipative devices.

Abstract

Leveraging populations of thermostatically controlled loads could provide vast storage capacity to the grid. To realize this potential, their flexibility must be accurately aggregated and represented to the system operator as a single, controllable virtual device. Mathematically this is computed by calculating the Minkowski sum of the individual flexibility of each of the devices. Previous work showed how to exactly characterize the flexibility of lossless storage devices as generalized polymatroids-a family of polytope that enable an efficient computation of the Minkowski sum. In this paper we build on these results to encompass devices with dissipative storage dynamics. In doing so we are able to provide tractable methods of accurately characterizing the flexibility in populations consisting of a variety of heterogeneous devices. Numerical results demonstrate that the proposed characterizations are tight.

Figures (4)

• Figure 1: The individual flexibility set of a TCL $\mathcal{F}_i$ (blue shaded region) and its maximal inner approximation as a g-polymatroid $\mathcal{Q}(p_i, b_i)$ (green shaded region).
• Figure 2: $\overline{y}^*(t)$ is found by minimizing $\sum_s^tu(t)$ s.t. $u \in \overline{\mathcal{H}} \cap \mathcal{F}^{t}_{ i}$
• Figure 3: Approximation quality of the different methods as a function of the time horizon length. The proposed method consistently achieves lower approximation error compared to benchmark approaches.
• Figure 4: Aggregate charging profile during generation signal tracking. The proposed method follows the signal more accurately than the two benchmark approaches.

Theorems & Definitions (12)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Lemma 1
• Lemma 2
• Theorem 2
• proof
• Lemma 3
• proof