In my work on the economics of electric vehicle charging, I start from a simple observation: the way energy flows between a vehicle and the grid is not a technical detail but a design variable that determines who captures value in the system. Under conventional unidirectional electric vehicle charging, energy moves in one direction only, from the grid into the battery, and the charging station operator monetizes the relationship exclusively through a service fee levied per kilowatt-hour delivered. That arrangement was adequate during the early diffusion of electric vehicle charging infrastructure, when the priority was coverage rather than profitability. It is no longer adequate. As fleets scale, three structural weaknesses of unidirectional electric vehicle charging become visible at once: concentrated charging pushes distribution networks toward peak stress, the single-revenue-stream logic leaves operators with thin margins that struggle to amortize construction and maintenance, and the storage embedded in vehicle batteries remains idle for the overwhelming majority of the day.
Bidirectional charging, usually labeled vehicle-to-grid, relaxes the one-way constraint. A bidirectional charger can both deliver energy to the battery and withdraw energy from it, so the battery becomes a distributed storage asset that the operator can arbitrage across time-of-use prices. I treat bidirectional electric vehicle charging as the core object of analysis and ask a question that is operational rather than technological: given a population of commuters with observable behavioral characteristics, what charging policy maximizes the operator’s expected profit, and under what conditions does that profit vanish? Answering this question requires a model that links the physical state of the battery, the stochastic demand of the grid for peak relief, and the contractual floor that drivers impose on their state of charge.
Existing work on the economics of bidirectional electric vehicle charging leans heavily toward the driver’s perspective, estimating whether an individual can earn money by selling energy or by collecting subsidies. That framing is incomplete. The operator is the aggregator that pools thousands of small, dispersed batteries, converts them into a dispatchable block, absorbs metering and settlement costs, and stands between the driver and the wholesale market. Without a credible operator-level profit model, the commercialization narrative of bidirectional electric vehicle charging remains speculative. Engineering studies, meanwhile, have concentrated on converter topologies, communication protocols, and control feasibility, and rarely connect those results to a decision rule that an operator could actually implement. I attempt to close that gap by importing the machinery of stochastic inventory management into the electric vehicle charging domain.
The analogy I exploit is precise. A battery state of charge is an inventory level. Off-peak charging is a replenishment order. Peak-period discharge against random grid demand is a newsvendor-style sale of a perishable asset with zero salvage value at the end of the horizon. The driver’s minimum acceptable state of charge is a service-level constraint, and the daily commute is a deterministic demand that depletes inventory before it can be sold. Once the problem is posed this way, the structure of the optimal electric vehicle charging policy follows from standard dynamic programming arguments, and the resulting rule is remarkably simple to communicate to practitioners.
Modeling the Commuting Cycle
I model a single representative driver who belongs to a homogeneous cluster. Clustering is a deliberate simplification: in practice a station serves a heterogeneous population, but drivers can be grouped by residential community, commuting distance, and stated range anxiety, and within a cluster the parameters can be treated as identical. The driver moves between two anchor locations, home and workplace, and the vehicle is plugged in at both. Two behavioral parameters govern everything that follows. The first is the minimum acceptable state of charge, which I denote by $k$ and which expresses range anxiety as a hard floor on the battery level. The second is the energy consumed by a single one-way commute, denoted $\theta$, which converts physical distance into a deterministic inventory depletion.
The daily timeline proceeds as follows. At the beginning of a period the battery holds $x$. The morning commute consumes $\theta$, leaving $x-\theta$. At the workplace the operator takes control and purchases energy at the off-peak price $p_l$, charging the battery up to a level $y$ that the operator chooses. The vehicle then sits idle until the evening commute, which consumes another $\theta$ and leaves $y-\theta$. At home, during the evening peak, the operator discharges the battery into the grid at price $p_h$, but never below the driver’s floor $k$. The quantity actually sold is therefore limited both by the energy the battery can spare and by the grid’s willingness to absorb it.
Peak-period demand for discharge, which I write as $D$, is random because it depends on seasonal temperature, industrial activity, and commercial load patterns. I let $f(\cdot)$ and $F(\cdot)$ denote its probability density and cumulative distribution functions respectively. Given a charging decision $y$, the energy sold in the peak is $\min\{y-\theta-k, D\}$. This single expression captures the central tension of bidirectional electric vehicle charging: charging too little leaves arbitrage value on the table, while charging too much exposes the operator to unsold energy that earns nothing.
The contract that makes this arrangement incentive compatible deserves emphasis. Ownership of the battery stays with the driver; a limited right of dispatch transfers to the operator during plugged-in hours. In exchange, the driver receives free or heavily discounted electric vehicle charging, and the operator commits to a hard guarantee that the state of charge never falls below $k$. This is a delegated scheduling arrangement, and it explains why the floor $k$ enters the model as a constraint rather than as a preference weight. The driver does not care about the operator’s arbitrage; the driver cares that the car will start in the morning.
| Symbol | Interpretation | Domain or Unit |
|---|---|---|
| $x$ | State of charge at the start of a period | $[k, 100]$ |
| $y$ | Charging target selected by the operator | $[x-\theta, 100]$ |
| $k$ | Driver’s minimum acceptable state of charge | $[0, 100]$ |
| $\theta$ | Energy consumed by a one-way commute | $[0, 100]$ |
| $p_l$ | Off-peak purchase price of energy | currency per kWh |
| $p_h$ | Peak selling price of energy | currency per kWh |
| $D$ | Random peak-period discharge demand | kWh |
| $F, f$ | Distribution and density of $D$ | dimensionless |
| $\gamma$ | Discount factor across periods | $(0, 1]$ |
| $C$ | Usable battery capacity | kWh |
| $S$ | Base-stock level (fixed charging threshold) | kWh |
| $H_t(y)$ | Decision-dependent component of period profit | currency |
| $V_t(x)$ | Value function of the finite-horizon problem | currency |
| $G(S)$ | Long-run average profit per period | currency |
| $a$ | Maximum sellable energy $S-\theta-k$ | kWh |
Finite-Horizon Formulation
I begin with a horizon of $T$ periods, which corresponds to a pilot deployment of bidirectional electric vehicle charging in which the operator commits to a fixed operating window. Let $V_t(x)$ denote the maximum expected profit from period $t$ onward, given that the battery enters period $t$ at level $x$. The operator’s decision variable is the target level $y$, and the Bellman recursion is
$$V_t(x) = \max_{y \ge x-\theta}\Big\{-p_l\big(y-(x-\theta)\big) + \mathbb{E}\big[p_h \min\{y-\theta-k,\, D\}\big] + \gamma\,\mathbb{E}\big[V_{t+1}\big(\max\{y-\theta-D,\, k\}\big)\big]\Big\}.$$
The three terms have clear economic content. The first is the purchase cost incurred at the off-peak price for the energy added to the battery. The second is the expected revenue from selling into the peak, which is truncated by whichever of the available energy and the realized demand is smaller. The third is the continuation value, discounted by $\gamma$, evaluated at the state of charge that remains after discharge. That terminal state is $\max\{y-\theta-D, k\}$ because the operator cannot discharge below the contractual floor even if the grid would happily absorb more energy.
Because the current state $x$ appears in the objective only through the constant $-p_l(x-\theta)$ shifted by a positive sign, I can separate state from decision and write
$$V_t(x) = \max_{y \ge x-\theta}\big\{H_t(y) + p_l(x-\theta)\big\},$$
where the decision-dependent component is
$$H_t(y) = -p_l y + \mathbb{E}\big[p_h \min\{y-\theta-k,\, D\}\big] + \gamma\,\mathbb{E}\big[V_{t+1}\big(\max\{y-\theta-D,\, k\}\big)\big].$$
Expanding the expectations into integrals over the demand distribution yields a form that is convenient for differentiation:
$$H_t(y) = -p_l y + \int_0^{y-\theta-k}\big[p_h s + \gamma V_{t+1}(y-\theta-s)\big] f(s)\,ds + \big[p_h(y-\theta-k) + \gamma V_{t+1}(k)\big]\big[1-F(y-\theta-k)\big].$$
The first integral covers demand realizations small enough that the battery can serve the entire request without breaching the floor; the second term covers realizations large enough that the floor binds and the operator sells exactly $y-\theta-k$. This structure is exactly the structure of a single-period inventory problem with a service constraint, and it is the reason the electric vehicle charging decision admits a threshold rule rather than a state-contingent schedule.
Before deriving the optimal policy I record the simplifications that make the analysis tractable. Drivers within a community are treated as homogeneous in $k$ and $\theta$, so the operator faces one representative problem per cluster. Commuting follows a strict two-point pattern, and only weekday round trips deplete the battery; discretionary trips are excluded. Conversion losses and battery degradation are suppressed, with the understanding that any compensation paid to drivers for degradation can be folded into the effective purchase price $p_l$, which raises the hurdle that arbitrage must clear. These assumptions do not drive the qualitative conclusions; they only keep the notation readable.
Optimality of the Base-Stock Charging Policy
The natural candidate policy is a base-stock rule: charge the battery up to a fixed threshold $S$ whenever the current level falls short of it, and do nothing otherwise. I now establish that this candidate is in fact optimal for the finite-horizon problem, which means the operator’s electric vehicle charging decision can be reduced to the selection of a single number.
Consider the terminal period, where no continuation value exists because unsold energy has zero salvage value. The problem collapses to a pure newsvendor formulation:
$$H_T(y) = -p_l y + \mathbb{E}\big[p_h \min\{y-\theta-k,\, D\}\big] = -p_l y + p_h\Big[\int_0^{y-\theta-k} s f(s)\,ds + \int_{y-\theta-k}^{+\infty}(y-\theta-k) f(s)\,ds\Big].$$
The second derivative is strictly negative whenever the density is positive,
$$\frac{d^2 H_T(y)}{dy^2} = -p_h\, f(y-\theta-k) < 0,$$
so $H_T$ is strictly concave and possesses a unique maximizer characterized by the first-order condition. Differentiating once and setting the result to zero gives the critical fractile
$$S_T^{*} = F^{-1}\!\left(\frac{p_h – p_l}{p_h}\right) + \theta + k.$$
The interpretation of this expression is instructive for anyone designing electric vehicle charging tariffs. The first term is the standard newsvendor quantile, the ratio of the margin on a sold unit to the price obtained per unit sold. The remaining terms are pure behavioral adjustments: the operator must buy enough energy to cover two commute legs and must respect the driver’s floor. The optimal target therefore rises one-for-one with the driver’s range anxiety and with commuting distance. The resulting policy is
$$y^{*}(x) = \begin{cases} S_T^{*}, & x-\theta \le S_T^{*}, \\[4pt] x-\theta, & x-\theta > S_T^{*}, \end{cases}$$
which is precisely the base-stock structure familiar from inventory theory. What matters for the operator is that $S_T^{*}$ does not depend on the entering state $x$, so the electric vehicle charging rule can be pre-programmed and audited without observing the vehicle’s history.
The multi-period case requires an induction. The inductive step is contained in the following result, which I state and then prove in outline.
Lemma 1. Define $S^{*} = F^{-1}\!\left(\dfrac{p_h – p_l}{p_h – \gamma p_l}\right) + \theta + k$. Suppose that $V_{t+1}(x)$ is concave and satisfies $V’_{t+1}(x) \le p_l$ for all $x$. Then: (i) $H_t(y)$ is concave and attains its maximum at $S^{*}$; (ii) the optimal policy in period $t$ is a base-stock policy with level $S^{*}$; (iii) $V_t(x)$ is concave and satisfies $V’_t(x) \le p_l$.
To see (i), write $H_t(y) = -p_l y + \mathbb{E}_D[g_t(y;D)]$ with
$$g_t(y;d) = p_h \min\{y-\theta-k,\, d\} + \gamma V_{t+1}\big(\max\{y-\theta-d,\, k\}\big),$$
which is piecewise defined by
$$g_t(y;d) = \begin{cases} p_h (y-\theta-k) + \gamma V_{t+1}(k), & y-\theta-k \le d, \\[4pt] p_h d + \gamma V_{t+1}(y-\theta-d), & y-\theta-k > d. \end{cases}$$
Differentiating with respect to $y$ gives
$$\frac{\partial g_t}{\partial y} = \begin{cases} p_h, & y-\theta-k \le d, \\[4pt] \gamma\, \dfrac{\partial V_{t+1}(y-\theta-d)}{\partial y}, & y-\theta-k > d. \end{cases}$$
By the inductive hypothesis the second branch is bounded above by $\gamma p_l$, which is no larger than $p_h$ because the peak price exceeds the off-peak price and $\gamma \le 1$. Hence the derivative is non-increasing in $y$, so $g_t$ is concave for every realization $d$. Expectation preserves concavity and the linear cost term is concave, so $H_t$ is concave. Setting the derivative to zero produces the level $S^{*}$ quoted above. The proof of (ii) follows from concavity: if $x-\theta \le S^{*}$ the operator charges to $S^{*}$, and if $x-\theta > S^{*}$ the derivative of $H_t$ at $x-\theta$ is already non-positive, so the operator charges nothing. For (iii), substituting the policy back yields
$$V_t(x) = \begin{cases} H_t(S^{*}) + p_l(x-\theta), & x-\theta \le S^{*}, \\[4pt] H_t(x-\theta) + p_l(x-\theta), & x-\theta > S^{*}, \end{cases}$$
whose derivative is $p_l$ in the first region and $H’_t(x-\theta) + p_l \le p_l$ in the second, which establishes the claim.
Theorem 1. With terminal condition $V_{T+1}(x)=0$, the optimal electric vehicle charging levels are
$$S^{*} = \begin{cases} F^{-1}\!\left(\dfrac{p_h – p_l}{p_h – \gamma p_l}\right) + \theta + k, & t \le T-1, \\[10pt] F^{-1}\!\left(\dfrac{p_h – p_l}{p_h}\right) + \theta + k, & t = T. \end{cases}$$
The proof runs backward. The terminal period is the newsvendor case already solved. For period $T-1$, substituting $V_T$ into the recursion and differentiating yields
$$\frac{d H_{T-1}(y)}{dy} = p_h – p_l + (\gamma p_l – p_h)\, F(y-\theta-k) = 0,$$
which produces $S^{*}_{T-1} = F^{-1}\!\left(\frac{p_h-p_l}{p_h-\gamma p_l}\right) + \theta + k$. Since the parameter tuple $(p_h, p_l, \gamma, \theta, k)$ is stationary, Lemma 1 propagates the same expression to every earlier period.
Two features of Theorem 1 deserve comment. First, the terminal quantile is smaller than the interior quantile, because $p_h – \gamma p_l < p_h$ makes the ratio $(p_h-p_l)/(p_h-\gamma p_l)$ larger than $(p_h-p_l)/p_h$; equivalently, the interior periods value leftover energy through the continuation term and therefore charge more aggressively. This terminal effect is a familiar artifact of finite-horizon inventory models and it matters for operators running fixed-length pilots, who should expect a gradual drawdown of the charging target as the end of the commitment approaches. Second, the base level responds monotonically to the behavioral parameters: both $k$ and $\theta$ shift $S^{*}$ upward one-for-one, while their effect on profit is negative, a tension I examine quantitatively later.
| Parameter | Direction of effect on $S^{*}$ | Direction of effect on operator profit | Mechanism |
|---|---|---|---|
| Peak price $p_h$ | Increases | Increases | Higher margin per unit sold |
| Off-peak price $p_l$ | Decreases | Decreases | Higher replenishment cost |
| Discount factor $\gamma$ | Increases | Increases | Leftover energy retains value |
| Commuting energy $\theta$ | Increases | Decreases | Two legs must be financed per cycle |
| Floor $k$ | Increases | Decreases | Sellable energy shrinks |
| Demand scale $D_{max}$ | Increases | Increases | More peak absorption capacity |
Infinite-Horizon Operation
When bidirectional electric vehicle charging is treated as a durable business line rather than a time-boxed pilot, the horizon extends indefinitely and the terminal effect disappears. In that regime the initial state of charge becomes irrelevant to long-run performance, and the appropriate objective is the maximization of average profit per period. The problem maps onto a lost-sales inventory system with zero fixed ordering cost and a discount factor of one, for which base-stock policies are known to be optimal. Because the periods decouple once the terminal effect is removed, the optimal electric vehicle charging level is constant across periods and equals the maximizer of the single-period expected profit.
Let $S$ denote the base level and define $a \equiv S-\theta-k$, the maximum energy that can be offered in a single evening. Writing $D$ and $D’$ for two independent draws from the peak demand distribution, the long-run average profit is
$$G(S) = -p_l\,\mathbb{E}\big[\min\{D’ + 2\theta,\, S-k+\theta\}\big] + p_h\,\mathbb{E}\big[\min\{D,\, a\}\big].$$
The first term is the expected energy purchased overnight to restore the battery after two commute legs plus whatever was sold; the second is the expected revenue from the evening discharge. Introducing the shorthand $m(a) \equiv \mathbb{E}[D \cdot \mathbf{1}\{D \le a\}] = \int_0^a y\, dF(y)$ and $\bar{F}(a) = 1-F(a)$, the two expectations admit the closed forms
$$\mathbb{E}\big[\min\{D, a\}\big] = m(a) + a\bar{F}(a), \qquad \mathbb{E}\big[\min\{D’ + 2\theta,\, S-k+\theta\}\big] = m(a) + 2\theta F(a) + (S-k+\theta)\bar{F}(a).$$
Substituting these identities collapses the objective to
$$G(S) = (p_h – p_l)\big[m(a) + a\bar{F}(a)\big] – 2\theta p_l,$$
whose derivatives are
$$\frac{dG}{dS} = (p_h – p_l)\,\bar{F}(S-\theta-k), \qquad \frac{d^2 G}{dS^2} = -(p_h – p_l)\, f(S-\theta-k) < 0.$$
The second derivative confirms strict concavity in the region where the density is positive, so the long-run problem is well behaved and admits a clean characterization.
Proposition 1. Under the long-run average profit objective, the optimal electric vehicle charging level is $S^{*} = \min\{C,\, \theta + k + D_{max}\}$.
The reasoning is a boundary argument. If the price spread is strictly positive, then $\bar{F}(a) > 0$ whenever $a$ lies below the upper support of demand, so the derivative is strictly positive and profit increases monotonically in $S$; the optimum is therefore pushed to the largest feasible level, the battery capacity $C$. If demand has a finite upper bound $D_{max}$, then once $a \ge D_{max}$ the survival function vanishes, the profit function becomes flat at $G \equiv (p_h-p_l)\mathbb{E}[D] – 2\theta p_l$, and any level satisfying $S \ge \theta + k + D_{max}$ is equally good, so the smallest such level is selected. If the two prices coincide, the derivative is identically zero and bidirectional electric vehicle charging generates no arbitrage value at all, which is the degenerate case an operator must avoid by site selection.
The contrast with the finite-horizon result is the central economic message of the long-run analysis. With no terminal penalty, there is no over-stocking risk in the relevant range, and the only ceiling on the charging target comes from the physical capacity of the battery and the absorptive capacity of the peak market. Growth in profit therefore does not come from fine-tuning the threshold; it comes from widening the corridor of energy that can be traded, whether by sourcing vehicles with larger usable capacity, by contracting with drivers who accept lower floors, or by locating in networks where peak demand is deep.
Behavioral Parameters and Critical Thresholds
The model implicitly defines a free-charging contract: the operator absorbs the entire off-peak energy bill and guarantees the floor $k$, while the driver supplies plug-in time and dispatch rights. Whether that contract is profitable depends entirely on who signs it. Drivers with long commutes or high range anxiety consume the operator’s margin before any energy reaches the peak market, whereas drivers with short commutes and low floors are cheap to serve but may find free charging an insufficient inducement once battery degradation is priced in. I therefore derive the boundary conditions that separate profitable from unprofitable contracts.
Fixing the optimal base level at $S^{*}$ and writing $a^{*} = S^{*}-\theta-k = \min\{C-\theta-k,\, D_{max}\}$, the maximal profit as a function of the behavioral parameters is
$$G_{max}(k, \theta) = (p_h – p_l)\big[m(a^{*}) + a^{*}\bar{F}(a^{*})\big] – 2\theta p_l.$$
Differentiating with respect to the floor gives
$$\frac{\partial G_{max}}{\partial k} = \begin{cases} 0, & k \le C – \theta – D_{max}, \\[6pt] -(p_h – p_l)\,\bar{F}(a^{*}), & k > C – \theta – D_{max}. \end{cases}$$
There is a plateau in which the binding constraint is the absorptive capacity of the grid rather than the battery, and beyond that plateau profit declines strictly in $k$. Since the function is continuous and strictly decreasing on the declining branch, and since it eventually becomes negative, there is a unique crossing point.
Proposition 2. Under the long-run average profit objective, there exists a unique critical floor $k_0$ satisfying $G_{max}(k_0, \theta) = 0$. For $k > k_0$ the operator cannot earn a positive return from the contract.
An analogous argument applies to commuting energy. Holding $k$ fixed,
$$\frac{\partial G_{max}}{\partial \theta} = \begin{cases} -2p_l, & \theta \le C – k – D_{max}, \\[6pt] -(p_h – p_l)\,\bar{F}(C-k-\theta) – 2p_l, & \theta > C – k – D_{max}, \end{cases}$$
which is strictly negative throughout, so the profit function decreases monotonically in commuting energy and crosses zero exactly once.
Proposition 3. Under the long-run average profit objective, there exists a unique critical commuting energy $\theta_0$ satisfying $G_{max}(k, \theta_0) = 0$. For $\theta > \theta_0$ the operator cannot earn a positive return from the contract.
Closed forms become available under a uniform demand assumption. Let $D \sim U[0, D_{max}]$, so that $m(a) = a^2/(2D_{max})$ and $\bar{F}(a) = 1 – a/D_{max}$. The profit function becomes
$$G(S) = (p_h – p_l)\left(a – \frac{a^2}{2D_{max}}\right) – 2\theta p_l,$$
and setting $G = 0$ produces a quadratic in $a^{*}$ whose smaller root is
$$a^{*} = D_{max}\left(1 – \sqrt{1 – \frac{4\theta p_l}{(p_h – p_l)D_{max}}}\right).$$
When the term under the radical is small, a first-order expansion yields the clean approximation $a^{*} \approx \dfrac{2\theta p_l}{p_h – p_l}$, and therefore
$$k_0 \approx C – \theta – \frac{2\theta p_l}{p_h – p_l} > C – \theta – D_{max}.$$
This expression is worth memorizing because it decomposes the break-even floor into three interpretable pieces: the capacity available, the energy consumed by commuting, and the energy that must be reserved to fund the purchase cost of the round trip. A driver whose floor exceeds the residual is simply too expensive to serve under a free-charging contract, no matter how the operator schedules the electric vehicle charging.
| Price spread $p_h – p_l$ | Commuting energy $\theta$ | Break-even floor $k_0$ (exact) | Break-even floor $k_0$ (approximation) | Break-even commuting energy $\theta_0$ |
|---|---|---|---|---|
| 0.36 | 5.0 | 41.2 | 40.9 | 12.4 |
| 0.56 | 5.0 | 48.9 | 49.3 | 17.6 |
| 0.76 | 5.0 | 52.1 | 52.1 | 20.7 |
| 0.56 | 8.0 | 44.6 | 45.1 | 15.2 |
| 0.56 | 12.0 | 37.6 | 38.3 | 12.5 |
The table makes the policy trade-off visible. Raising the price spread widens the set of acceptable drivers along both dimensions, while increasing commuting energy shrinks it along both. An operator who cannot influence prices can still influence the composition of the driver pool, and the critical thresholds tell the operator exactly where to draw the line.
Numerical Assessment
To translate the analytical results into magnitudes, I built a simulation of a fleet of ten vehicles operating under both unidirectional and bidirectional electric vehicle charging. Under the unidirectional regime the operator collects only a service fee and the driver bears the energy cost; under the bidirectional regime the operator supplies energy free of charge and captures the arbitrage. Peak demand for the fleet is drawn from a uniform distribution on $[100, 450]$ kWh, so that both the regime in which demand falls short of available energy and the regime in which it exceeds available energy are represented. Allocation across vehicles follows an equal-sharing rule when demand is scarce and a full-discharge rule when demand is abundant. All results are obtained from ten thousand Monte Carlo replications.
| Parameter | Value | Basis |
|---|---|---|
| Usable battery capacity per vehicle | 60 kWh | Mainstream passenger electric vehicle configuration |
| Driver floor $k$ | 12 kWh | Twenty percent state of charge, common safety convention |
| One-way commuting energy $\theta$ | 5 kWh | Measured urban commuting consumption |
| Charging service fee | 0.80 per kWh | Average public charging tariff |
| Off-peak price $p_l$ | 0.32 per kWh | Industrial and commercial valley tariff |
| Peak price $p_h$ | 0.88 per kWh | Industrial and commercial peak tariff |
| Conventional peak-shaving cost | 1.20 per kWh | Gas turbine regulation cost |
| Emission factor of displaced generation | 0.80 kg CO2 per kWh | Regional grid accounting guideline |
| Carbon price | 0.08 per kg CO2 | Average allowance settlement price |
The comparison across the two operating modes is summarized below. The finite-horizon column reports a five-day pilot, while the infinite-horizon column reports steady-state daily averages.
| Stakeholder | Metric | Unidirectional, five-day pilot | Bidirectional, five-day pilot | Unidirectional, steady state (daily) | Bidirectional, steady state (daily) |
|---|---|---|---|---|---|
| Operator | Profit | 400.00 | 722.91 | 80.00 | 121.74 |
| Grid | Peak-shaving cost avoided | 0.00 | 1611.84 | 0.00 | 329.43 |
| Society | Carbon value | 0.00 | 85.96 | 0.00 | 17.57 |
Three conclusions follow. First, the operator’s profit under bidirectional electric vehicle charging exceeds the service-fee benchmark by a wide margin even though charging is provided free to drivers, because the arbitrage margin is captured in full. Second, the value created for the grid is substantial: the fleet substitutes for combustion-based peaking capacity and thereby avoids a cost that the unidirectional regime cannot address at all. Third, the carbon value, while smaller in absolute terms, is non-trivial and would grow with the carbon price. The bidirectional regime thus produces a triple dividend, and the electric vehicle charging infrastructure becomes a grid asset rather than merely a load.
The sensitivity of steady-state profit to driver characteristics is displayed below, computed per vehicle per day from the closed-form expression with $D_{max} = 45$ kWh per vehicle, $C = 60$ kWh, $p_l = 0.32$, and $p_h = 0.88$.
| Floor $k$ (kWh) | Profit at $\theta = 5$ | Profit at $\theta = 10$ | Profit at $\theta = 15$ | Optimal base level $S^{*}$ at $\theta = 5$ |
|---|---|---|---|---|
| 12 | 9.38 | 5.90 | 2.10 | 60.0 |
| 20 | 8.78 | 4.80 | 0.51 | 60.0 |
| 30 | 6.91 | 2.31 | −0.89 | 48.0 |
| 40 | 3.80 | −0.58 | −2.60 | 48.0 |
| 50 | −0.56 | −3.20 | −4.80 | 48.0 |
Negative entries mark contracts that should be declined. The pattern confirms Propositions 2 and 3 empirically: profit declines monotonically in both behavioral parameters, and the crossing point arrives sooner for drivers with longer commutes. It is also worth noting that the base level saturates at capacity for low floors and short commutes, which means the operator is physically constrained rather than economically constrained in exactly the segment that is most attractive to serve.
Finally, I examine the price spread, which is the variable over which the operator has the least direct control but the greatest strategic interest.
| Price spread (per kWh) | Daily profit per vehicle | Relative to service fee benchmark | Interpretation |
|---|---|---|---|
| 0.20 | 2.91 | Below | Arbitrage cannot cover acquisition cost |
| 0.30 | 4.62 | Below | Marginal viability |
| 0.40 | 6.39 | Approaching parity | Break-even against service fee near 0.44 |
| 0.56 | 9.38 | Above | Comfortable margin at base parameters |
| 0.80 | 13.11 | Well above | Strong incentive to expand the fleet |
Empirically, many electricity markets exhibit spreads above 0.70 per kWh, and a smaller number exceed 1.00. For an operator deciding where to deploy bidirectional electric vehicle charging, the spread is therefore the first screening variable, and the analysis suggests that the technology is economically attractive across a large share of jurisdictions without any subsidy.
Managerial Implications
The results support a set of concrete operating rules, which I summarize in the table below together with the analytical result that justifies each.
| Decision | Recommended rule | Supporting result |
|---|---|---|
| Charging target selection | Charge to a fixed base level; never to a state-contingent schedule | Theorem 1 and Proposition 1 |
| Terminal period handling | Lower the target as the end of a fixed contract approaches | Terminal effect in Theorem 1 |
| Site selection | Prioritize high price spread and predictable, deep peak demand | Monotonicity of $G$ in $p_h – p_l$ and $D_{max}$ |
| Driver screening | Decline contracts with $k > k_0$ or $\theta > \theta_0$ | Propositions 2 and 3 |
| Incentive design | Offer revenue sharing or guaranteed minimum charge to lower effective $k$ | Monotone decline of profit in $k$ |
| Fleet composition | Prefer vehicles with larger usable capacity when demand is deep | Capacity ceiling in Proposition 1 |
| Cost accounting | Fold degradation compensation into the effective purchase price | Model assumption on $p_l$ |
On site selection, the logic is straightforward. The base level is capped either by battery capacity or by the depth of the peak market, so a location with shallow peak demand wastes the storage that drivers bring. Residential and commercial districts with regular, forecastable evening peaks are the natural candidates, and the electric vehicle charging infrastructure there can be sized with confidence because demand variability is bounded.
On driver screening, the temptation is to maximize enrollment, since each additional vehicle adds dispatchable capacity. The model says otherwise. Every contract carries a floor and a commuting burden, and contracts that exceed the critical thresholds destroy value on every cycle. An operator that indiscriminately enrolls high-floor, long-commute drivers will find that a profitable portfolio is diluted into a loss-making one. The correct posture is selective enrollment supplemented by differentiated incentives: revenue sharing, guaranteed minimum charge, or dynamic tariffs that reward drivers for accepting a lower floor or for plugging in on days when peak demand is expected to be high.
On battery degradation, the model is deliberately silent on the physical mechanism but explicit on the accounting fix. Frequent bidirectional cycling imposes a cost on the asset owner, and if that cost is not compensated, participation will erode. The clean treatment is to raise the effective purchase price $p_l$ by the per-kilowatt-hour degradation compensation, which uniformly compresses the profit margin and shifts every threshold inward. An operator that ignores degradation will overestimate the size of the addressable pool.
Extensions and Robustness
Several extensions follow naturally and do not disturb the qualitative conclusions. Heterogeneous fleets can be handled by partitioning drivers into clusters indexed by $(k, \theta)$ and solving the base-stock problem cluster by cluster; because the optimal level is additive in $\theta$ and $k$, the aggregate policy is a menu of thresholds rather than a single number. Stochastic commuting distance can be accommodated by treating $\theta$ as a random variable with its own distribution, which effectively convolves the demand distribution and preserves concavity. Battery capacity that degrades over time can be modeled as a deterministic downward drift in $C$, which tightens the ceiling in Proposition 1 and causes the optimal base level to decline slowly over the asset’s life.
The most consequential extension concerns market design. My formulation assumes the operator is a price taker in both the off-peak and peak markets. If the operator’s fleet is large enough to move prices, the arbitrage margin becomes endogenous and the optimal electric vehicle charging volume is determined jointly with the market-clearing price. In that setting the base-stock structure survives, but the critical fractile must be computed against a residual demand curve rather than an exogenous price. This is the natural next step for a fleet-scale analysis.
Concluding Remarks
I set out to determine how an operator of bidirectional electric vehicle charging should schedule energy flows when drivers impose a hard floor on their state of charge and the grid presents a random appetite for peak relief. The answer is a base-stock policy: charge to a fixed threshold determined by the price spread, the demand distribution, the commuting burden, and the driver’s floor, and do nothing when the battery already exceeds that threshold. The policy is optimal in both finite and infinite horizons, though the finite horizon carries a terminal effect that lowers the target in the final period.
The long-run analysis sharpens the message. Because there is no salvage-value penalty in steady state, the operator should push the base level until it hits either battery capacity or the upper support of peak demand. Profit growth therefore comes from widening the tradable corridor rather than from refining the threshold, which reframes the strategic question from scheduling to segmentation. The critical thresholds $k_0$ and $\theta_0$ operationalize that reframing: they identify precisely which drivers can be served profitably and which cannot, and they give the operator a quantitative basis for incentives aimed at relaxing the binding constraint.
For the broader energy system, the numerical exercise indicates that bidirectional electric vehicle charging delivers value on three fronts simultaneously. The operator earns more than under a service-fee model, the grid avoids peaking costs, and emissions fall as cleaner generation displaces combustion-based regulation. None of these benefits requires a technological breakthrough; they require a contractual architecture in which dispatch rights are exchanged for free electric vehicle charging, and a scheduling rule simple enough to be implemented at scale. The base-stock policy is exactly such a rule. It requires only the price spread, an estimate of the demand distribution, and two behavioral parameters per driver cluster, all of which are observable in operational data. That simplicity is what makes the transition from pilot to platform credible, and it is the reason I regard the inventory-theoretic framing of electric vehicle charging as more than a modeling convenience.