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Efficient Quantification and Representation of Aggregate Flexibility in Electric Vehicles

Nanda Kishor Panda, Simon H. Tindemans

Abstract

Aggregation is crucial to the effective use of flexibility, especially in the case of electric vehicles (EVs) because of their limited individual battery sizes and large aggregate impact. This research proposes a novel method to quantify and represent the aggregate charging flexibility of EV fleets within a fixed flexibility request window. These windows can be chosen based on relevant network operator needs, such as evening congestion periods. The proposed representation is independent of the number of assets but scales only with the number of discrete time steps in the chosen window. The representation involves $2T$ parameters, with T being the number of consecutive time steps in the window. The feasibility of aggregate power signals can be checked using $2T$ constraints and optimized using $2(2^T-1)$ constraints, both exactly capturing the flexibility region. Using a request window eliminates uncertainty related to EV arrival and departure times outside the window. We present the necessary theoretical framework for our proposed methods and outline steps for transitioning between representations. Additionally, we compare the computational efficiency of the proposed method with the common direct aggregation method, where individual EV constraints are concatenated.

Efficient Quantification and Representation of Aggregate Flexibility in Electric Vehicles

Abstract

Aggregation is crucial to the effective use of flexibility, especially in the case of electric vehicles (EVs) because of their limited individual battery sizes and large aggregate impact. This research proposes a novel method to quantify and represent the aggregate charging flexibility of EV fleets within a fixed flexibility request window. These windows can be chosen based on relevant network operator needs, such as evening congestion periods. The proposed representation is independent of the number of assets but scales only with the number of discrete time steps in the chosen window. The representation involves parameters, with T being the number of consecutive time steps in the window. The feasibility of aggregate power signals can be checked using constraints and optimized using constraints, both exactly capturing the flexibility region. Using a request window eliminates uncertainty related to EV arrival and departure times outside the window. We present the necessary theoretical framework for our proposed methods and outline steps for transitioning between representations. Additionally, we compare the computational efficiency of the proposed method with the common direct aggregation method, where individual EV constraints are concatenated.
Paper Structure (13 sections, 7 theorems, 26 equations, 4 figures)

This paper contains 13 sections, 7 theorems, 26 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem definition
  3. Notation
  4. Single EV flexibility
  5. Aggregate flexibility
  6. Motivating example
  7. Definition
  8. Polytope representation
  9. Aggregation of UL-flexibility
  10. Representing EV flexibility
  11. Numerical examples
  12. Computational efficiency
  13. Conclusion

Key Result

Lemma 1

The UL-flexibility for vectors $\vec{u}, \vec{l}$ can be expressed as Here, $\vec{\tilde{p}}$ is the component-wise descending permutation of $\vec{p}$, $\mathbbm{1}^L \in \mathbb{R}^{T\times T}$ is the lower triangular matrix where all lower triangular elements are 1 and $R\in \mathbb{R}^{T\times T}$ is the order reversing matrix, where the 1 elements reside on the a

Figures (4)

  • Figure 1: Different approaches for aggregating flexibility in EVs. Computational complexity is indicated by the number of parameters (P), variables (V) and constraints (C), where N is the number of vehicles and T is the number of time steps
  • Figure 2: Graphical check of feasibility using ordered UL-flexibility. The UL values plotted here correspond to EV 1 as explained in section \ref{['section: motivating example']}. $\Vec{u}$ and $\Vec{l}$ for the EV is represented by and respectively. The feasibility of the reference signals ( ) is checked by comparing the $\sum \textrm{descending}(\Vec{p})$ ( ) and $\sum \textrm{ascending}(\Vec{p})$ ( ) by their respective upper and lower limits.
  • Figure 3: Graphical representation of 3-d polytope representing the feasible flexibility of EV 1, EV 2 and EV 1 $\oplus$ EV 2 respectively as described in Section \ref{['section: motivating example']}.
  • Figure 4: Comparison of computational resources used by the UL-flexibility approach with direct aggregation for different time steps and quantities of EVs. The results of the base case are shown using dashed lines.

Theorems & Definitions (19)

  • Definition 1: Feasible EV flexibility
  • Definition 2: Polytope representation of feasible EV flexibility
  • Definition 3: Minkowski summation
  • Definition 4: Convex/concave vectors
  • Definition 5: Zero-extended vectors
  • Definition 6: UL-flexibility
  • Lemma 1: Ordered UL representation
  • proof
  • Lemma 2: Properties of UL-flexibility
  • proof
  • ...and 9 more