Robust Aggregation of Electric Vehicle Flexiblity

TL;DR

The paper addresses how to characterize and bound the aggregate flexibility of populations of EVs with uncertain charging needs. It uses generalized polymatroids to obtain an exact representation of aggregate flexibility as a $g$-polymatroid and then constructs distributionally robust sets $A^\beta$ around a known charging-requirement distribution using a Wasserstein ambiguity set and measure-concentration guarantees. The result is a tractable convex reformulation that yields tight, finite-sample guarantees even for small populations. The framework supports robust bidding into flexibility markets and local networks, enabling higher-confidence participation while controlling conservatism, with $F(\Xi_N)$ denoting the aggregate flexibility and $A^\beta$ providing probabilistic feasibility guarantees.

Abstract

We address the problem of characterizing the aggregate flexibility in populations of electric vehicles (EVs) with uncertain charging requirements. Extending upon prior results that provide exact characterizations of aggregate flexibility in populations of electric vehicle (EVs), we adapt the framework to encompass more general charging requirements. In doing so we give a characterization of the exact aggregate flexibility as a generalized polymatroid. Furthermore, this paper advances these aggregation methodologies to address the case in which charging requirements are uncertain. In this extended framework, requirements are instead sampled from a specified distribution. In particular, we construct robust aggregate flexibility sets, sets of aggregate charging profiles over which we can provide probabilistic guarantees that actual realized populations will be able to track. By leveraging measure concentration results that establish powerful finite sample guarantees, we are able to give tight bounds on these robust flexibility sets, even in low sample regimes that are well suited for aggregating small populations of EVs. We detail explicit methods of calculating these sets. Finally, we provide numerical results that validate our results and case studies that demonstrate the applicability of the theory developed herein.

Figures (7)

• Figure 1: A schematic of the work presented in this paper. We are provided with a distribution over charging requirements $\mathbb{P}$. From this distribution, a population of $N$ charging requirements is obtained by drawing $N$ independent samples from $\mathbb{P}$. Each of the charging requirements generate their own individual flexibility sets $F(\hat{\xi}_i)$. The aggregate flexibility set for the population is the Minkowski sum of these, $F(\hat{\Xi}_i) = \sum_i^N F(\hat{\xi}_i)$, characterizing this is the subject of \ref{['sec:aggregation']}. This aggregate flexibility set is a random object and so we would like define robust sets $A^\beta$, in which we have varying confidence that $F(\hat{\Xi}_i)$ will be contained in. We derive these robust sets in \ref{['sec:agg_uncertainty']} and provide a tractable reformulation for their computation in \ref{['sec:reformulation']}.
• Figure 2: Polyhedra generated by the paramodular pair $(p,b)$. The red and blue lines are the submodular and supermodular base polyhedra. The red and blue shaded regions are the submodular and supermodular polyhedra, $S(b)$ and $S'(p)$. The g-polymatroid, $Q(p, b)$, is generated by the intersection of $S(b)$ and $S'(p)$.
• Figure 3: A cube $F'(\xi)$ (red shaded region), and a plank $K(\underline{e}, \overline{e}$) (blue shaded region), intersecting to form the g-polymatroid $Q(p',b')$ (region outlined in blue).
• Figure 4: A visualization of the proof of \ref{['thm:robust_flex']}. The robust set is the intersection of the aggregate flexibility sets generated by populations that lie in the ambiguity set. Here, we show the aggregate flexibility sets associated with two sample populations, $\hat{\Xi}_N^1$ and $\hat{\Xi}_N^2$. The aggregate flexibility sets generated by them are, $F(\hat{\Xi}_N^1)$ and $F(\hat{\Xi}_N^2)$, and their intersection, $A^\beta$. The robust set is defined by the maximum of the supermodular functions (in this case $p_{\hat{\Xi}_N^2}$) and minimum of the submodular functions (in this case $b_{\hat{\Xi}_N^1}$). For clarity we only show the sub- and supermodular functions for $\mathcal{A} = \{1\}$.
• Figure 5: Empirical results showing the probability of not being able to satisfy all aggregate charging profiles within the robust set $A^\beta$, for different values of $\varepsilon$ and $N$.

Theorems & Definitions (18)

• definition 1
• definition 2
• definition 3
• definition 4: Submodular functions
• definition 5: Paramodularity
• definition 6: Generalized polymatroids
• definition 7: Plank
• lemma 1
• theorem 1: Plank Intersection theorem
• corollary 1