Electrified Transportation System Performance: Conventional Versus Online Electric Vehicles

Abstract

In recent years, the electrification of ground transportation has emerged as a trend to support energy efficiency and CO₂ emissions reduction targets. The true success, however, of this trend depends on the successful integration of electric vehicles into the infrastructure systems that support them. Left unmanaged, conventional electric vehicles may suffer from delays due to charging or cause destabilizing charging loads on the electrical grid. Online electric vehicles have emerged to remediate the need for stationary charging and its effects. This chapter seeks to objectively compare the systemic impacts of these two electric vehicle concepts on the combined electrical power grid and road transportation system. It applies a recently developed hybrid dynamic system model of the transportation-electricity nexus that holistically incorporates vehicle dispatch, route choice, charging station queues, coordinated charging, and vehicle-to-grid stabilization. It draws upon Axiomatic Design for Large Flexible Engineering System Theory to superimpose a marked petri net model layer on a continuous-time kinematic and electrical state evolution. The results show that online electric vehicles, unlike their conventional vehicle counterparts, are able to avoid charging station queues and thus are able to meet the needs of a greater variety of transportation uses cases including commercial and public fleets. Their impacts on the power system also differ. While conventional electric vehicles are likely to require greater investment to expand power system generation capacity, online electric vehicles are likely to incur greater operating costs to manage their charging loads. The chapter concludes with several directions for future work in the development of intelligent transportation-energy systems which can serve to reduce both costs for both vehicle concepts.

Appendix

The hybrid dynamic system model presented in Sect. 20.2 is based upon graph theory, petri nets, and Axiomatic Design for large flexible systems. As the first two of these subjects are from different disciplines, a brief introduction to their fundamental concepts is provided for the potentially uninitiated reader.

1.1 Graph Theory

Graph theory is a long-established field of mathematics with applications in many fields of science and engineering where artifacts are transported between physical locations [46,47,48]. A number of definitions are introduced for later use in the discussion.

Definition 12

Graph [48]: $G = \{ B,E\}$ consists of a collection of nodes $B$ and collection of edges E. Each edge $e \in E$ is said to join two nodes, which are called its end points. If $e$ joins $b_{1}, b_{2} \in B,$ we write $e = \left\langle {b_{1}, b_{2} } \right\rangle$. Nodes $b_{1}$ and $b_{2}$ in this case are said to be adjacent. Edge $e$ is said to be incident with nodes $b_{1}$ and $b_{2}$, respectively.

Definition 13

Bipartite Graph [48]: Graph G is bipartite if $B(G)$ can be partitioned into two disjoint subsets $B_{1}$ and $B_{2}$ such that each edge $e \in E(G)$ has one end point in $B_{1}$ and the other in $B_{2}$ : $E(G) \subseteq \left\{ {e = \left\langle {b_{1}, b_{2} } \right\rangle |b_{1} \in B_{1} } \right.,$ and $b_{2} \in B_{2} \}$.

Definition 14

Directed Graph (digraph) [48]: D, consists of a collection nodes B, and a collection of arcs E, for which we write $D = (B,E).$ Each arc $e = \left\langle {b_{1}, b_{2} } \right\rangle$, is said to join node $b_{1} \in B$ to another (not necessarily distinct) node $b_{2}$. Vertex $b_{1}$ is called the tail of e, whereas $b_{2}$ is its head.

Definition 15

Incidence In Matrix [48]: $M^{ + }$ of size $\sigma (B) \times \sigma (E)$ is given by:

$$ M^{ + } (i,j) = \left\{ {\begin{array}{*{20}l} 1 \hfill & {{\kern 1pt} if\;b_{y} \;is\;the\;tail\;of\;arc\;e_{j} {\kern 1pt} } \hfill \\ 0 \hfill & {{\kern 1pt} otherwise{\kern 1pt} } \hfill \\ \end{array} } \right. $$

(20.13)

where the operator $\sigma ()$ gives the size of a set.

Definition 16

Incidence Out Matrix [48]: $M^{ - }$ of size $\sigma (B) \times \sigma (E)$ is given by:

$$ M^{ - } (i,j) = \left\{ {\begin{array}{*{20}l} 1 \hfill & {{\kern 1pt} if\;b_{y} \;is\;the\;head\;of\;arc\;e_{j} {\kern 1pt} } \hfill \\ 0 \hfill & {{\kern 1pt} otherwise{\kern 1pt} } \hfill \\ \end{array} } \right. $$

(20.14)

Definition 17

Incidence matrix [48]: M of size $\sigma (B) \times \sigma (E)$ is given by:

$$ M = M^{ + } - M^{ - } $$

(20.15)

Definition 18

Adjacency matrix [48]: A, is binary and of size $\sigma (B) \times \sigma (B)$ and its elements are given by:

$$ A(y_{1}, y_{2} ) = \left\{ {\begin{array}{*{20}l} 1 \hfill & {{\kern 1pt} if\left\langle {b_{{y_{1} }}, \;b_{{y_{2} }} } \right\rangle exists{\kern 1pt} } \hfill \\ 0 \hfill & {{\kern 1pt} otherwise{\kern 1pt} } \hfill \\ \end{array} } \right. $$

(20.16)

1.2 Petri Nets

Petri nets offer a long-established method for modeling and simulating the discrete-event dynamics of a system. Their usage is described by the following definitions.

Definition 19

Marked Petri Net ( Graph ) [49]: A bipartite directed graph represented as a 5-tuple $\mathcal{N} = (B,\mathcal{E},A,W,Q_{B} )$ where:

• B is a finite set of places of size $\sigma (B).$

• $\mathcal{E}$ is a finite set of (instantaneous) transitions/events of size $\sigma (E).$

• $M \subseteq (B \times \mathcal{E}) \cup (\mathcal{E} \times B)$ is a set of arcs of size $\sigma (M)$ from places to transitions and from transitions to places in the graph.

• $W:A \to \{ 0,1\}$ is the weighting function on arcs.

• $Q_{B}$ is a marking (or discrete state) vector of size $\sigma (B) \times 1 \in {\mathbb{N}}^{\sigma (B)}$.

As with graphs, three incidence matrices are defined for petri nets.

Definition 20

Petri Net Incidence In Matrix [49]: $M_{{\mathcal{N}}}^{ + }$ of size $\sigma (B) \times \sigma (\mathcal{E})$ is given by:

$$ M_{{\mathcal{N}}}^{ + } (i,j) = \left\{ {\begin{array}{*{20}l} 1 \hfill & {if\;b_{y} \;is\;the\;tail\;of\;event\; {\epsilon }_{j} } \hfill \\ 0 \hfill & {otherwise{\kern 1pt} } \hfill \\ \end{array} } \right. $$

(20.17)

Definition 21

Petri Net Incidence Out Matrix [49]: $M_{{\mathcal{N}}}^{ - }$ of size $\sigma (B) \times \sigma (\mathcal{E})$ is given by:

$$ M_{{\mathcal{N}}}^{ - } (i,j) = \left\{ {\begin{array}{*{20}l} 1 \hfill & {if\;b_{y} \;is\;the\;head\;of\;event\; {\epsilon }_{j} {\kern 1pt} } \hfill \\ 0 \hfill & {otherwise{\kern 1pt} } \hfill \\ \end{array} } \right. $$

(20.18)

Definition 22

Petri Net Incidence matrix [49]: $M_{{\mathcal{N}}}$ of size $\sigma (B) \times \sigma (\mathcal{E})$ is given by:

$$ M_{{\mathcal{N}}} = M_{{\mathcal{N}}}^{ + } - M_{{\mathcal{N}}}^{ - } $$

(20.19)

Definition 23

Petri Net (Discrete-Event) Dynamics [49]: Given a binary firing vector $U_{Dk}$ of size $\sigma (\mathcal{E}) \times 1$ and a petri net incidence matrix $M_{{\mathcal{N}}}$ of size $\sigma (B) \times \sigma (\mathcal{E}),$ the evolution of the marking vector $Q_{B}$ is given by the state transition function $\varPhi (Q_{B}, U_{Dk} ):$

$$ Q_{B} [k + 1] = {\varPhi }(Q_{B}, U_{Dk} ) = Q_{B} [k] + M_{{\mathcal{N}}} U_{k} $$

(20.20)

While marked petri nets are sufficient for discrete-event dynamics, they do assume events of infinitesimal duration. In the case of transportation systems, it is necessary to associate a duration to each of these events which may be either deterministic [50] or stochastic [101]. The former is defined as follows:

Definition 24

Timed Petri Net ( Graph ) [50]: A 6-tuple $\mathcal{N}_{T} = (B,\mathcal{E},A,W,Q,D)$ where $(B,\mathcal{E},A,W,Q)$ is a marked petri net where $Q$ is marking vector of size $[\sigma (B) + \sigma (\mathcal{E})] \times 1 \in {\mathbb{N}}^{{[\sigma (B) + \sigma (\mathcal{E})]}}$ that includes marking of events $Q_{{\mathcal{E}}}$ in addition to the marking of places $Q_{B}$. $Q = [Q_{B} ;Q_{{\mathcal{E}}} ].$ D is a duration vector of size $\sigma (\mathcal{E}) \times 1$ representing the finite duration required to fire the event.

The following petri net dynamics are used in the context of this work.

Definition 25

Timed Petri Net (Discrete-Event) Dynamics: The evolution of the marking vector $Q = [Q_{B} ;Q_{{\mathcal{E}}} ]$ is given by the state transition function $Q[k + 1] = {\varPhi }_{T} \left( {Q[k],U_{k}^{ + }, U_{k}^{ - } } \right).$

$$ Q_{B} [k + 1] = Q_{B} [k] + M_{{\mathcal{N}}}^{ + } U_{k}^{ + } - M_{{\mathcal{N}}}^{ - } U_{k}^{ - } $$

(20.21)

$$ Q_{{\mathcal{E}}} [k + 1] = Q_{{\mathcal{E}}} [k] - U_{k}^{ + } + U_{k}^{ - } $$

(20.22)

where $U_{k}^{ - }$ is the $k^{th}$ input firing vector and $U_{k}^{ + }$ is $k^{th}$ output firing vector.

The input firing vector $U_{Dk}^{ - }$ is taken as exogenous, while the output firing vector $U_{Dk}^{ + }$ is calculated from the event durations D by means of a scheduled event list.

The state transition function in Definition 25 has a minor modification from the one commonly used elsewhere in the petri net literature [50]. Normally, tokens remain in place until the event duration has passed. Here, the tokens are taken from the place marking vector and appear instead in the transition marking vector. They reappear in the marking vector after the event duration has passed. This modification is made so that the petri net dynamics more closely represent the physical reality as described in Sect. 20.2. The rules of timed petri net operation, including when transitions are enabled, remain otherwise the same.

Definition 26

Scheduled Event List [49]: A tuple $S = (u_{vk}, t_{k} )$ consisting of all elements $u_{vk}$ in firing vectors $U_{k}$ and their associated times $t_{k}$. For every element, $u_{vk}^{ - } \in U_{k}^{ - }$, there exists another element $u_{vk}^{ + } \in U_{k}^{ + }$ which occurs at $t = t_{k} + d_{v}$.

The output firing vectors $U_{Dk}^{ + }$ are then calculated from their elements for all the unique times $t = t_{k} + d_{v}$.