Optimal Pricing to Manage Electric Vehicles in
Coupled Power and Transportation Networks

We model the transportation network through a connected directional graph $\mathcal{G}=(\mathcal{V},\mathcal{A})$, where $\mathcal{V}$ and $\mathcal{A}$ respectively denote the set of nodes and arcs of the graph.

Traveling on the transportation network is associated with a cost for the user since he/she values the time spent en route. The time to travel between an origin and a destination node is comprised of the time spent on arcs that connect these two nodes on $\mathcal{G}$. Here we adopt the popular Beckmann model for the cost of traveling an arc, i.e., a road section [33]. Accordingly, we assume that the travel time for each arc $a\in\mathcal{A}$ only depends on the rate of EVs per time unit that travel on the arc, which we refer to as the arc flow and denote by $\boldsymbol{\lambda}=[\lambda_{a}]_{a\in\mathcal{A}}$. The time it takes to travel arc $a$ is then represented by a latency function $\tau_{a}(\lambda_{a})$, which is convex, continuous, non-negative, and increasing in $\boldsymbol{\lambda}$. The congestion cost that a user incurs for traveling arc $a$ is given by:

where $\gamma$ is the cost of one unit of time spent en route. Hence, the cost to travel link $a$ is:

where $\theta_{a}$ corresponds to any tolls that the driver should pay to the ITSO, if any such toll is enforced for link $a$.

Moreover, traveling each arc $a\in\mathcal{A}$ requires a certain amount of energy $e_{a}$ (see Fig. 2). Energy needs to be received from the power grid and stored in the EV battery. The cost that the user incurs to receive battery charge is modeled next.

A subset of nodes on the transportation network $\mathcal{N}\subseteq\mathcal{V}$ are equipped with battery charging facilities and, hence, the EV drivers have the choice of charging their batteries at these locations. Naturally, to be able to provide charging services, the nodes $\mathcal{N}$ are also a subset of the nodes $\mathcal{B}$ that constitute the power grid graph $\mathcal{R}=(\mathcal{B},\mathcal{F})$. Each node $v\in\mathcal{B}$ has an associated price for electricity $p_{v}$. Consequently, if the EV driver chooses to charge at location $v$, he/she will pay:

where $e_{v}\in\mathcal{E}_{v}$ is the battery charge amount received at $v$ chosen from a finite set $\mathcal{E}_{v}$, and $\theta_{v}$ corresponds to a one-time plug-in fee for the charging station at $v$. Moreover, if $v$ is not the origin of the trip, the driver needs to spend some extra time en route in order to receive charge at $v$. This is due to any congestion at the fast charging stations (FCS) plus the time it takes to receive charge. Hence, an extra inconvenience cost is incurred. If the charging rate at FCS $v$ is $\rho_{v}$, and the rate of EVs being plugged in for charge at this location is denoted by $\lambda_{v}$, this cost is equal to:

where $\tau_{v}(\lambda_{v})$ denotes the wait time due to congestion, and is a soft cost model to capture limited station capacity.

The EV driver’s goal would be to find the least cost path that connects the origin and the destination on the transportation graph $\mathcal{G}$ while making sure that the EV battery never runs out of charge¹¹1Note that with the cost structure we have defined, the EV driver will reach the destination with minimum-possible leftover charge. An extension of the analytical framework to include the value of leftover charge at the destination in the optimization is trivial and has been removed for brevity of notation. We refer the reader to our conference paper [32] for this extension..

Here we show that we can recast the EV driver’s route and charge problem as a resource-constrained shortest path problem on a new extended transportation multigraph $G^{e}(\mathcal{S},\mathcal{L})$. This definition will help us study and control the aggregate effect of individual EVs on power and transportation systems.

To complete a trip, the driver incurs two forms of costs: the cost associated with arcs and the cost associated with the charging decisions taken at nodes. However, we can observe charging is very similar in nature to traveling: 1) it takes a certain amount of time (due to both charging rate limitations and congestion); 2) it has a cost; and 3) it changes the energy level of the battery. Acknowledging this similarity, we transform the EV driver’s decision problem to a shortest path problem by associating charging decisions made at the nodes $v$ of the transportation graph to a set of new virtual arcs to be added at each node $v\in\mathcal{N}$ where charging is possible. At each origin and destination node where charging is possible, the following transformation would capture all decisions:

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  The decision of how much to charge: a set of virtual arcs $\mathcal{L}_{v}$ added at node $v$ are each associated with a specific choice of how much to charge, i.e., $e_{v}\in\mathcal{E}_{v}$. Hence, the energy gained by traveling each new virtual arc is set to be one such member of $\mathcal{E}_{v}$ (red arcs in Fig. 4). Equivalently, we can say that the energy required to travel the virtual arc is negative. Travel time is $\frac{e_{v}}{\rho_{v}}$.

At the FCS, these transformations capture all decisions:

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  The decision to stop or skip stopping at a charge station en route: the driver can either take a charging station entrance arc (labeled “stop” in Fig 3) and plug in their EV at the station, or skip stopping at the station via a bypass arc with zero travel time and energy requirements (green arcs in Fig. 3). Charging station entrance arcs can be congested;

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  The decision of how much to charge if stopping at $v$: the charging station entrance arc is connected in series to a set of virtual arcs $\mathcal{L}_{v}$ capturing the choice of amount of charge $e_{v}\in\mathcal{E}_{v}$ (blue arcs in Fig. 3).

The flow on the charging station entrance arc will capture the wait time to be plugged in at the station. The set of all entrance and bypass arcs for all charging stations is $\mathcal{C}$. The new extended transportation graph with these virtual arcs would then have the following set of arcs:

Consequently, the transformed problem seeks a shortest path on this extended graph from the origin to the destination, with the cost of traveling each arc being the sum of its travel time cost and all monetary charges such as the electricity bill or tolls. The travel time costs on $G^{e}(\mathcal{S},\mathcal{L})$ is given by:

All other monetary costs can be captured as:

and hence, each driver selfishly optimizes their trip plan by solving the Energy-aware Shortest Path Problem (ESPP):

where $\mathcal{K}$ is now the set of energy-feasible paths that connect the origin and the destination on the extended graph $G^{e}(\mathcal{S},\mathcal{L})$. Energy-feasibility of a path ensures that the battery will never run out of charge en route. Next, we define energy-feasibility mathematically.

These two constraints ensure that the EV never runs out of charge en route if taking path $k$, and that the battery charge state never exceeds battery capacity. A vector of dimension zero is simply empty. Note that with this definition, we can determine whether a path is energy-feasible independently of the network congestion mirrored through $\boldsymbol{\lambda}_{\mathbf{a}_{k}}$. Hence, for system-level control of $\boldsymbol{\lambda}_{\mathbf{a}_{k}}$, the set of energy-feasible paths can be calculated offline.

Now, imagine that every EV owner in the society solved (7) to plan their trips. These users share two infrastructures: the transportation network, and the power grid. Hence, collectively, EVs give rise to a traffic and load pattern, determining which roads and grid buses are congested and hence, will have longer travel times and higher electricity prices. Through this interaction, individuals affect each other’s cost, leading to the system-level problem that we are interested in.

We assume that drivers belong to a finite number of classes $q\in\mathcal{Q}$. Vehicles in the same class share the same origin and destination. Vehicles could include both EVs as well as ICEVs. A given class q would contain either EVs or ICEVs but not both. Drivers of the same class $q$ are represented by a set of feasible paths $\mathcal{K}_{q}$, each of which allows them to to finish their trip on the extended graph. For EVs, this is equivalent to the set of energy-feasible paths given in Definition II.2 and can be enumerated offline for each class. For ICEVs, we can assume these paths simply include transportation arcs in $\mathcal{A}$ that connect the origin and destination, and do not enter charging stations. Clearly, any other path selection method that considers more realistic constraints can also be applied. We leave the study of optimal clustering mechanisms that assign heterogeneous users to a finite of number of classes to future work.

Each customer directly affects the flow of the arcs that constitute his/her path. To model this effect, we define:

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  $m_{q}$ as the travel demand rate (flow) of EVs in cluster $q$. This demand is taken as deterministic and given;

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  $f_{q}^{k}$ as the rate of cluster $q$ EVs that choose path $k\in\mathcal{K}_{q}$. We define $\mathbf{f}_{q}=[f_{q}^{k}]_{k\in\mathcal{K}_{q}}$.

Naturally, since every driver has to pick one path, the following conservation rule holds:

Given the path decisions of all EV drivers, i.e., the $f_{q}^{k}$’s, the flow of EVs on arc $a$ is given by $\lambda_{a}=\sum_{q\in\mathcal{Q},k\in\mathcal{K}_{q}}\delta_{a}^{k}f_{q}^{k}$, where $\delta_{a}^{k}$ is an arc-path incidence indicator (1 if arc $a$ is on path $k$ and 0 otherwise). This is written in matrix form as:

where $\boldsymbol{\lambda}=[\lambda_{a}]_{a\in{\mathcal{L}}}$ denotes the vector of network flows and $\mathbf{A}_{q}$ is a $|\mathcal{L}|\times|\mathcal{K}_{q}|$ matrix such that $[\mathbf{A}_{q}]_{a,k}=\delta_{a}^{k}$.

The flow on the virtual arcs of the extended graph leads to a power load. We denote the charging demand at each node $v\in\mathcal{B}$ of the grid as a vector $\mathbf{d}=[d_{v}]_{v\in\mathcal{B}}$, given by:

where $\mathbf{M}$ is a $|\mathcal{B}|\times|\mathcal{L}|$ matrix given by:

Let $\boldsymbol{s}(\boldsymbol{\lambda})=[s_{a}(\lambda_{a})]_{a\in\mathcal{L}}$. If an ITSO is in charge of determining the optimal path and charge schedule for each EV such that the aggregate cost is minimized, it can solve a modified version of the classic static traffic assignment problem [39] on the extended graph, which we refer to as the charge and traffic assignment problem (CTAP):

where $\mathbf{p}=[p_{v}]_{v\in\mathcal{B}}$.

To serve the charging demand of EVs, a set of generators are located at different nodes of the power network $\mathcal{R}=(\mathcal{B},\mathcal{F})$. For brevity, let us assume that a single merged generator is located at each node of the grid. Assuming that the generation at each node is denoted by a vector $\mathbf{g}=[g_{v}]_{v\in\mathcal{B}}$ and the baseload (any load that does not serve EVs) by a vector $\mathbf{u}=[u_{v}]_{v\in\mathcal{B}}$, there are three constraints that define a feasible generation mix $\mathbf{g}$ in the power grid. First of all, $g_{v}$ must be within the capacity range of the generator at node $v$, i.e., $\mathbf{g}^{\min}\preceq\mathbf{g}\preceq\mathbf{g}^{\max}$. Second, the demand/supply balance requirement of the power grid should be met, i.e.,

Third and last, the transmission line flow constraints of the grid under the DC approximation [40] translate into:

where the matrix $\mathbf{H}$ is the power transfer distribution matrix of the grid, explicitly defined in [40], and $\mathbf{c}=[c_{f}]_{f\in\mathcal{F}}$ is a vector containing arc (line) flow limits (in both directions).

In most power grids, one such feasible generation mix $\mathbf{g}$ is picked by an IPSO to serve demand. We assume that at least one feasible generation mix always exists for every possible load profile. The IPSO’s objective is to pick the cheapest feasible generation mix. Let us denote the cost of generating $g_{v}$ units of energy at node $v\in\mathcal{B}$ as a strongly convex and continuous function $c_{v}(g_{v})$, and the vector of generation costs as $\boldsymbol{c}(\mathbf{g})=[c_{v}(g_{v})]_{v\in\mathcal{B}}$. Given a demand $\mathbf{d}$ from EVs, the IPSO solves an economic dispatch problem to decide the optimal generation dispatch $\mathbf{g}*$ [41]:

Note that the optimal traffic and generation schedules determined through (12) and (15) minimize the total cost to society. However, they do not necessarily minimize the cost of each individual entity that is involved, e.g., the EV drivers or the generators. Hence, one cannot merely ask these selfish users to stick to the socially optimal schedule. An economic mechanism is necessary to align selfish behavior with socially optimal resource consumption behavior. The use of pricing mechanisms is a way of achieving this goal in a distributed and incentive-compatible fashion. We highlight pricing mechanisms that can be used for (12) and (15) next.

To incentivize profit-maximizing generators to produce at an output level $g_{v}$, we apply the principle of marginal cost pricing. The principle states that the electricity price at node $v$, i.e., $p_{v}$, must satisfy:

Let us introduce Lagrange multipliers $\gamma$ and $\boldsymbol{\mu}$ respectively for the balance and line flow constraints in (15). Writing the KKT stationarity condition for (15) then leads to:

commonly referred to as Locational Marginal Prices (LMP) in the power system literature. The reader should note that this is the same price vector $\mathbf{p}$ that is fed into the ITSO optimization (12) and would affect the charging demand at different nodes of the grid, i.e., $\mathbf{d}$ in (15), which would in turn affect the price $\mathbf{p}$ again. This feedback loop highlights the coupling between smart power and transportation systems that we are interested in, further studied in Section V.

In the transportation network, if every user solves an ESPP given in (7) and no tolls are imposed by the ITSO ($\theta_{a}=0,\forall a\in\mathcal{A}\cup\mathcal{C}$), the aggregate flow would be determined based on a state of user-equilibrium. This user equilibrium flow is most likely not equivalent to the social optimal flow in (12). To mathematically characterize this equilibrium, define an auxilary modified cost function $\boldsymbol{s}^{e}(\boldsymbol{\lambda})=[s^{e}_{a}(\lambda_{a})]_{a\in\mathcal{L}}$ for the extended transportation graph’s arcs as:

with $\boldsymbol{s}(\boldsymbol{\lambda})$ and $\mathbf{M}$ given by (5) and (11). Moreover, let $\boldsymbol{w}^{e}(\boldsymbol{\lambda})=[\int_{x=0}^{\lambda_{a}}s^{e}_{a}(x)dx]_{a\in\mathcal{L}}$. Then, according to the well-known Wardrop’s first principle [42], the user equilibrium flow would be the solution of the optimization problem:

So how can the ITSO get the individual drivers to follow the socially optimal flow calculated in (12)? We present the answer in Theorem IV.1.

Let us look at the scenario that would happen if no corrective action is taken in regards to how the grid is operated today and hence, smart transportation and energy systems are operated separately. In this disjoint model, the IPSO ignores the fact that the load due to EV charge requests can move from one grid bus to another in response to posted prices. Instead, LMPs are designed assuming that charge events will happen exactly as in the last period (this could be the previous day, the average of the previous month, etc.). On the other hand, the ITSO ignores the effect of EV charge requests on electricity prices, and takes electricity prices as a given when designing road and FCS congestion tolls.

Claim: Under this greedy pricing scheme, the congestion and electricity prices $\boldsymbol{\theta}$ and $\mathbf{p}$ could oscillate indefinitely.

We substantiate this claim through a numerical example in Section VI. This, along with the loss of welfare experienced when our infrastructure is operated at a suboptimal state, motivates us to look into schemes which can allow the ITSO and IPSO to operate their networks optimally and reliably.

In theory, the above algorithm can eliminate the need for the existence of an ex-ante²²2The term ex-ante refers to actions that are adjusted as a result of forecasting user behavior and not actual observations, while ex-post refers to actions that are based on actual observations rather than forecasts. ITSO collaboration for calculating electricity prices. Instead, imagine that the IPSO can actually post electricity prices according to (26) and gradually find the optimal market clearing LMP by observing the charging demand of the actual transportation system³³3Note that this is only possible if the time-scale at which the network flow $\boldsymbol{\lambda}$ reaches its new equilibrium in response to new posted prices ${\bf p}$ and tolls $\boldsymbol{\theta}$ is much smaller that the time-scale at which electricity costs or travel demands change.. When dealing with unknown demand functions in commodity pricing, this is referred to as the trial-and-error approach, see, e.g., [44], for prior use of such approaches in toll design.

Implementing this approach has two requirements:

1) The IPSO should be willing to move away from the greedy pricing scheme in order to eventually maximize societal welfare (even though the extra welfare generated might not be easily quantifiable and the operating point might not correspond to minimum generation costs);

2) More importantly, primal feasibility is most likely violated when using Lagrangian relaxation to handle coupling constraints in (22). This means that when posting prices according to (26), the IPSO should expect the balance and flow constraints to be violated in order for the algorithm to converge, and plan accordingly. In power grids, any unpredictable violation of reliable system operation is referred to as a contingency (a threat to the security of the system) and is handled through generation reserves.

Note that the dispatched reserve generation $\mathbf{y}$ affects the line flows and hence the flow constraint becomes:

Here we will use bounds on primal infeasibility to determine the reserve capacity $\mathbf{r}$ that needs to be procured by the IPSO in this type of ex-post LMP adjustment. For simplicity, we consider a constant step size rule such that $\alpha_{k}=\alpha$ for all $k$. Note that in this scheme, after each price adjustment iteration $\boldsymbol{p}^{(k)}$, the approximate primal solutions, which are the last iterate $\boldsymbol{\lambda}^{(k)}$ and $\boldsymbol{g}^{(k)}$, are actually implemented. Assume that the IPSO knows that during the $k$-th iteration,

where $d^{(k)}=\mathbf{M}\boldsymbol{\lambda}^{(k)}$. For dual first order algorithms such as dual gradient and dual fast gradient methods (any of which can be employed by the IPSO for price update), such bounds were recently provided by [45]. For example, for dual gradient methods, one possible bound is given by:

where $\alpha=\alpha_{k}\leq 1/L_{d},\forall k$ and $L_{d}$ is the Lipschitz constant for the dual problem of (22). We now need to show that the bound in (30) is well-defined.

To calculate (30), the IPSO has to access to the set of potentially optimal energy and congestion prices $\gamma$ and $\boldsymbol{\mu}$. Without access to travel patterns, an estimate of the upper bound to $\|(\gamma^{\star},\bm{\mu}^{\star})\|_{2}$ can be evaluated using many methods. For example, we can calculate these bounds by a) performing Monte-Carlo simulations by setting ${\bf d}$ to different values between its upper and lower bounds given the limited capacity of charging stations (charging stations operating anywhere until full capacity) [46]; b) studying critical load levels as suggested in [47]; c) solving a Mathematical Program with Equilibrium Constraints (MPEC) [48]. We chose to solve an MPEC to find this bound; see Appendix A.

Given (29), the IPSO needs to ensure that (27) and (28) hold by appropriately choosing a reserve capacity $\mathbf{r}_{k}$ to be procured for each future $k$ to ensure system security.