An Efficient Method for Quantifying the Aggregate Flexibility of Plug-in Electric Vehicle Populations

Abstract

Plug-in electric vehicles (EVs) are widely recognized as being highly flexible electric loads that can be pooled and controlled via aggregators to provide low-cost energy and ancillary services to wholesale electricity markets. To participate in these markets, an aggregator must encode the aggregate flexibility of the population of EVs under their command as a single polytope that is compliant with existing market rules. To this end, we investigate the problem of characterizing the aggregate flexibility set of a heterogeneous population of EVs whose individual flexibility sets are given as convex polytopes in half-space representation. As the exact computation of the aggregate flexibility set -- the Minkowski sum of the individual flexibility sets -- is known to be intractable, we study the problem of computing maximum-volume inner approximations to the aggregate flexibility set by optimizing over affine transformations of a given convex polytope in half-space representation. We show how to conservatively approximate these set containment problems as linear programs that scale polynomially with the number and dimension of the individual flexibility sets. The inner approximation methods provided in this paper generalize and improve upon existing methods from the literature. We illustrate the improvement in approximation accuracy and performance achievable by our methods with numerical experiments.

Figures (7)

• Figure 1: Example of an individual EV flexibility set. The power and net energy profile constraints are depicted as solid black lines. Three different feasible power profiles and net-energy profiles are depicted. In this example, we take $\delta=1/2$ hour, $T=24$, and associate the initial period $t=0$ with the 6:00-6:30 PM time interval. The EV charging parameters used in this example are: $a_i=0$ (6:00 PM arrival), $d_i=23$ (6:00 AM departure), $u_i^{\rm max}=-u_i^{\rm min} = 10$ kW, $x_i^{\rm max} = 60$ kWh, $x_i^{\rm init} = 20$ kWh, and $x_i^{\rm fin} = 50$ kWh.
• Figure 2: Illustration of the inner approximation method proposed in this paper. Depicted are: the base set $\mathbb{U}_0$; the affine transformations $\gamma_i+\Gamma_i\mathbb{U}_0$ (red) that inner approximate the individual flexibility sets $\mathbb{U}_i$ for $i=1,\dots,N$; and the corresponding affine transformation $(\sum_{i\in{\mathcal{N}}}\gamma_i)+(\sum_{i\in{\mathcal{N}}}\Gamma_i)\mathbb{U}_0$ (green) that inner approximates the aggregate flexibility set $\mathbb{U}$.
• Figure 3: Comparison of inner approximation methods. (a), (b) Two individual flexibility sets $\mathbb{U}_i = \{x \mid Hx \leq h_i\}$ ($i= 1,2$) with randomly sampled right-hand side vectors and (c) their sum $\mathbb{U} = \mathbb{U}_1 + \mathbb{U}_2$ are depicted as black solid lines.
• Figure 4: Suboptimality gap distributions associated with each approximation method, as applied to the peak power minimization \ref{['eq:power_min']} and electricity cost minimization \ref{['eq:cost_min']} problems. The whiskers delimit the interdecile range, the box delimits the interquartile range, and the red line represents the median of each distribution.
• Figure 5: Inner approximations of the aggregate flexibility set computed from data sampled on May 3, 2019. Depicted are the homothet-based inner approximation (solid orange lines), the structure-preserving inner approximation (dotted blue lines), and the outer approximation $N\mathbb{U}_0$ (dashed black lines).

Theorems & Definitions (11)

• Example 1: Constructing EV flexibility sets
• Remark 1: Net-energy representation
• Remark 2: Lossy charging dynamics
• Definition 1: Structure-preserving transformations
• Lemma 1: AH-polytope in H-polytope
• proof
• Theorem 2: AH-polytope in Sum of H-polytopes
• proof
• Remark 3: Heterogeneity in EV charging dynamics
• Remark 4: Comparison to the method in zhao2017geometric