Business-to-Peer Carsharing Systems with Electric Vehicles

Theodoros Mamalis Subhonmesh Bose Lav R. Varshney

Abstract—Carsharing systems are adopting electric vehiclesinto their fleets and may therefore be able to provide not onlytransportation services to drivers/passengers but also energyservices to the power grid through appropriate participationin electricity markets. This paper develops a queuing modelof such carsharing platforms where cars may be deployedfor transportation services at a given price, but also for gridservices during transportation-idle periods through energy pricearbitrage. The model provides a characterization of revenuefrom each of these two streams. The paper further findsoptimal pricing and battery splitting to maximize revenue forthe carsharing platform, via an analysis of the reward structureand an optimization algorithm. Platform revenue is assessed forvarious system parameters under optimal operation.

Index Terms—Electric vehicles, queuing networks, sharingeconomy, transportation, revenue maximization

I. INTRODUCTION

Automobiles have a certain flexibilty in that they can actas both transportation resources and as distributed energyresources. This dual-use flexibility of gasoline engines wasexploited in the early twentieth century by car owners atthe individual level, e.g. through Model T-powered washingmachines [1, Fig. 2]. The resurgence of modern plug-inelectric vehicles (EVs), however, enables leveraging thisflexibility at a large-scale systems level through coupledcontrol of transportation systems and energy grids [2],especially in the context of the sharing economy [3].Indeed on the energy side, EVs can not only serve asboth energy sources and sinks, but also provide voltageand frequency regulation services to the smart grid [4]–[7]. On the transportation side, business-to-peer carsharingplatforms1 are starting to adopt electric vehicles within theirfleets [8]–[10]. This is especially the case for automobilemanufacturer-based platforms including Volkswagen’s WE,Mercedes-Benz’s Car2Go, BMW’s ReachNow, GM’s Maven,and Audi’s Audi-on-Demand.

Carsharing companies can garner additional revenues fromutilizing the battery packs of their vehicle fleet to providegrid services. This paper develops a rigorous queuing-theoreticmodel and computational tools to facilitate the analysisand optimal coordination of EV-based carsharing serviceplatforms. Although queuing models have a long historyin modeling transportation services [11]–[13], including

All authors are with the Department of Electrical and Computer Engineeringat the University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA.Emails: {mamalis2, boses, varshney}@illinois.edu.

Discussions with Tamer Basar are appreciated. This work was supportedin part by a grant from the Siebel Energy Institute, and by the AndreasMentzelopoulos Scholarships for the University of Patras.

1Business-to-peer carsharing platforms provide short-term (e.g., hourly) carrentals, where the customer drives the car provided by the company. On theother hand peer-to-peer ride sharing platforms, such as Lyft and Uber, relyon privately-owned vehicles and their drivers to provide on-demand mobilityto customers.

carsharing platforms [14], we believe this is the firstapplication to settings with EVs. The main goal in developingour queuing-theoretic model for carsharing platforms is to (i)capture the salient features of EV charging processes, and(ii) capture the tradeoffs in providing both transportation andgrid services. Indeed, the inherent time required to charge EVbatteries will impact a transportation service provider’s abilityto deliver rides in a timely manner. Further, the underlyingcost and time requirements of battery charging will affect thepricing decisions for mobility services.

In particular, we first establish a simple queuing modelof EV transportation service provision and recharge, furtherdetermining the revenue garnered. Next we consider thepossibility of using an EV battery to perform price arbitrage inan energy market; this is done in the presence of a stochasticreal-time wholesale market price signal through a dynamicprogramming argument. Finally, we consider the possibilityof (dynamically) splitting the battery into a transportationsegment and a grid services segment to enable energy pricearbitrage when idle from delivering transportation services. Forcertain standard distributional assumptions, we characterizethe candidate optimal transportation prices p upon fixingthe battery capacity B dedicated to provide transportationservices. An efficient algorithm for joint optimization of (p,B)is given, and numerical examples are used to provide insightinto the basic tradeoff.

For ease of presentation and to capture the basic queuing-theoretic insights, we focus on energy services restricted toprice arbitrage against real-time prices in the energy market.We further focus on the setting where the fleet consists of asingle car. Longer presentations of this work will include thesetting with a variety of grid services and larger fleets of cars.

II. MODELING TRANSPORTATION SERVICES

Let customers open A’s application following a Poissonprocess with intensity λ0. When a customer arrives, sheobserves the posted price p (in money/time) for utilizingA’s vehicle and the maximum driving time τmax = B/β−,assuming that the vehicle battery of capacity B depletesat a constant rate β− when driven. She decides to rent avehicle, if (i) the posted price p is lower than her reservationprice, and (ii) her required driving time τ does not exceedthe maximum driving time τmax. We model a customer’sreservation price and driving time as independent randomvariables. Let Fπ denote the complementary cumulative or taildistribution function of the reservation wages, and Fτ denotethe cumulative distribution function of the trip times. It followsthat customers who are willing to pay the posted price and

abide by the driving time restrictions arrive according to aPoisson process with intensity

λ := λ0Fπ(p)Fτ (B/β−). (1)

Driving times of customers who ultimately avail A’s servicefollow the same distribution as the driving times of allcustomers truncated at B/β−. The vehicle battery loses β−τamount of energy when driven for τ , where recall that β−

denotes the battery discharge rate. Denote the battery chargingrate at the depot by β+. Then, the car will require τC := β−

β+ τamount of time to charge it back up to its capacity B.Therefore, we model the charging time proportional to thedriving time as Figure 1(a) illustrates. The car (server) isdeemed busy when it is either being driven or charging afterbeing driven. For each ride provided, the car remains busy forτ +τC = τ(1+β−/β+) := τ β amount of time, implying thatthe service time follows a truncated version of the trip timedistribution with mean

1

µ:= βE[τ | τ ≤ B/β−]. (2)

Thus, A’s service has been modeled as an M/G/1 queue,written in Kendall’s notation, with arrival rate λ and servicerate µ. In our current formulation, we allow each arrivingcustomer to wait in the depot if the vehicle has been checkedout by another customer. A more realistic model with renegingor balking, and other behavioral models of customers as in [15]are left to future endeavors.

Too low a price or too large a battery capacity can resultin unstable growth in the passenger queue. Enforcing λ < µensures that the queue remains stable, i.e.,

λ0βFπ(p)

∫ B/β−

0

tfτ (t)dt︸ ︷︷ ︸:=Φτ (B/β−)

< 1, (3)

where fτ is the probability density function of driving times.

A. Computing A’s revenue rate from transport services

Drawing results from the equilibrium analysis of M/G/1queues, we aim to calculate the rate at which A accruesrevenue, assuming the car service is in steady-state. Fromrenewal-reward theory, this rate is given by the ratio ofthe expected revenue A makes from a busy period and theexpected total cycle length (busy + idle period). We calculatethe expected revenue during a busy period as follows. Foreach customer, A is paid pτ for the transportation service,and it pays pretβ+τC = pretβ

−τ for energy. Here, pret (inmoney/energy) denotes the flat retail rate for energy that thedistribution utility (or a retail aggregator) charges A. Overall,A makes (p − pretβ

−)τ from a customer who drives it forτ time. Thus, the expected revenue from the customers in abusy period from transportation service provision is given by

ΠT = E

[NT∑i=1

(p− pretβ−)τi

],

where τ1, . . . , τNT are the driving times of the NT customersin the busy period who seek to use the car service. By Wald’slemma, the above relation simplifies to

ΠT = (p− pretβ−)E[NT ] · E[τ |τ ≤ B/β−]

= (p− pretβ−)λ

µ− λE[τ |τ ≤ B/β−].

The last line in the above equation uses the fact that theexpected length of the busy period for an M/G/1 queue isgiven by (µ−λ)−1. Also, the expected length of the idle periodis λ−1, yielding the following expression for the reward rateRT of A from transportation service provision.

RT =

(1

µ− λ +1

λ

)−1

ΠT

= (p− pretβ−)λ2

µE[τ |τ ≤ B/β−].

Utilizing (1) and (2), we get

RT = λ20β(p− pretβ−)F 2

π (p)Φ2τ (B/β−). (4)

Notice that we do not explicitly model A’s maintenanceand repair costs for its vehicles. One can expect repair coststo be proportional to the driving times, as more time a car isdriven, the more prone it becomes to traffic accidents. Suchcosts can be included in the retail electricity price. Regularmaintenance costs will add a constant to the revenue rate thatwill not affect our conclusions from the model.

III. PRICE ARBITRAGE USING VEHICLE BATTERY DURINGIDLE PERIODS

We aim to find how a vehicle can split its battery capacityfor dual use—a portion set aside to provide transportationservices and utilize the rest to maximize its revenue fromarbitraging against time-varying energy prices. To simplifythe exposition, conceptually consider a vehicle battery withcapacity Btot as a combination of a transportation battery ofcapacity B and a trading battery of capacity B′ = Btot −B.

Conceptually splitting the vehicle battery into two partsensures that a transportation customer always enjoys anonrandom battery capacity for her trip. Allowing the vehiclebattery to provide grid services with the total battery capacitymakes the available battery for transportation a randomquantity, compromising A’s carsharing business. Any residualenergy in the grid battery can be utilized by a transportationcustomer in an emergency. Further, the battery split need notbe constant, but can be varied over time. For example, on dayswith high customer traffic, A might allocate a higher portionof the battery for transportation as opposed to that in a daywith low traffic.

Recall that the car is busy with transportation service whenit is being driven by a customer, or is preparing for it bycharging its transportation battery. It is idle, otherwise. Duringthis idle period, let A receive nonnegative2 energy prices ρ (inmoney/capacity) at regular intervals of length ∆ as shown in

2The nonnegativity assumption on the prices can be relaxed with minormodifications to the results.

Driving timeCharging time

Busy period

Busy period

time

Idle period

Driving timeCharging time= ⌧

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Charging time

Driving time

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Idle period⇡ T�<latexit sha1_base64="Bu7cXHkqGL6SsdramVHh3TUE+qU=">AAAB+XicbZDLSgMxFIYz9VbHS0ddugmWgqsyI4Iui7pwWaE36Awlk2ba0EwSkoxYhj6JK0FB3Poornwb03YW2vpD4OM/53BO/lgyqo3vfzuljc2t7Z3yrru3f3BY8Y6OO1pkCpM2FkyoXow0YZSTtqGGkZ5UBKUxI914cjuvdx+J0lTwlplKEqVoxGlCMTLWGniVEEmpxBNshXeEGTTwqn7dXwiuQ1BAFRRqDryvcChwlhJuMENa9wNfmihHylDMyMythZkmEuEJGpG+RY5SoqN8cfkM1qwzhIlQ9nEDF677ayJHqdbTNLadKTJjvVqbm//V+plJrqOccpkZwvFyUZIxaAScxwCHVBFs2NQCworaYyEeI4WwsWG5NoVg9c/r0LmoB5YfLquNmyKPMjgFZ+AcBOAKNMA9aII2wCADz+AVvDm58+K8Ox/L1pJTzJyAP3I+fwDfgZMc</latexit><latexit sha1_base64="Bu7cXHkqGL6SsdramVHh3TUE+qU=">AAAB+XicbZDLSgMxFIYz9VbHS0ddugmWgqsyI4Iui7pwWaE36Awlk2ba0EwSkoxYhj6JK0FB3Poornwb03YW2vpD4OM/53BO/lgyqo3vfzuljc2t7Z3yrru3f3BY8Y6OO1pkCpM2FkyoXow0YZSTtqGGkZ5UBKUxI914cjuvdx+J0lTwlplKEqVoxGlCMTLWGniVEEmpxBNshXeEGTTwqn7dXwiuQ1BAFRRqDryvcChwlhJuMENa9wNfmihHylDMyMythZkmEuEJGpG+RY5SoqN8cfkM1qwzhIlQ9nEDF677ayJHqdbTNLadKTJjvVqbm//V+plJrqOccpkZwvFyUZIxaAScxwCHVBFs2NQCworaYyEeI4WwsWG5NoVg9c/r0LmoB5YfLquNmyKPMjgFZ+AcBOAKNMA9aII2wCADz+AVvDm58+K8Ox/L1pJTzJyAP3I+fwDfgZMc</latexit><latexit sha1_base64="Bu7cXHkqGL6SsdramVHh3TUE+qU=">AAAB+XicbZDLSgMxFIYz9VbHS0ddugmWgqsyI4Iui7pwWaE36Awlk2ba0EwSkoxYhj6JK0FB3Poornwb03YW2vpD4OM/53BO/lgyqo3vfzuljc2t7Z3yrru3f3BY8Y6OO1pkCpM2FkyoXow0YZSTtqGGkZ5UBKUxI914cjuvdx+J0lTwlplKEqVoxGlCMTLWGniVEEmpxBNshXeEGTTwqn7dXwiuQ1BAFRRqDryvcChwlhJuMENa9wNfmihHylDMyMythZkmEuEJGpG+RY5SoqN8cfkM1qwzhIlQ9nEDF677ayJHqdbTNLadKTJjvVqbm//V+plJrqOccpkZwvFyUZIxaAScxwCHVBFs2NQCworaYyEeI4WwsWG5NoVg9c/r0LmoB5YfLquNmyKPMjgFZ+AcBOAKNMA9aII2wCADz+AVvDm58+K8Ox/L1pJTzJyAP3I+fwDfgZMc</latexit><latexit sha1_base64="Bu7cXHkqGL6SsdramVHh3TUE+qU=">AAAB+XicbZDLSgMxFIYz9VbHS0ddugmWgqsyI4Iui7pwWaE36Awlk2ba0EwSkoxYhj6JK0FB3Poornwb03YW2vpD4OM/53BO/lgyqo3vfzuljc2t7Z3yrru3f3BY8Y6OO1pkCpM2FkyoXow0YZSTtqGGkZ5UBKUxI914cjuvdx+J0lTwlplKEqVoxGlCMTLWGniVEEmpxBNshXeEGTTwqn7dXwiuQ1BAFRRqDryvcChwlhJuMENa9wNfmihHylDMyMythZkmEuEJGpG+RY5SoqN8cfkM1qwzhIlQ9nEDF677ayJHqdbTNLadKTJjvVqbm//V+plJrqOccpkZwvFyUZIxaAScxwCHVBFs2NQCworaYyEeI4WwsWG5NoVg9c/r0LmoB5YfLquNmyKPMjgFZ+AcBOAKNMA9aII2wCADz+AVvDm58+K8Ox/L1pJTzJyAP3I+fwDfgZMc</latexit>

⇢0<latexit sha1_base64="AFPCKUS1jXiNTs3LgvClV4SH0fo=">AAAB7nicbZDLSgMxFIbPeK3jrerSTbAUXJUZERRXBTcuK9gLtEPJpJk2NpOEJCOUoe/gSlAQt76PK9/GtJ2Ftv4Q+PjPOeScP1acGRsE397a+sbm1nZpx9/d2z84LB8dt4zMNKFNIrnUnRgbypmgTcsspx2lKU5jTtvx+HZWbz9RbZgUD3aiaJTioWAJI9g6q9XTI9kP+uVKUAvmQqsQFlCBQo1++as3kCRLqbCEY2O6YaBslGNtGeF06ld7maEKkzEe0q5DgVNqony+7hRVnTNAidTuCYvmrv9rIsepMZM0dp0ptiOzXJuZ/9W6mU2uo5wJlVkqyOKjJOPISjS7HQ2YpsTyiQNMNHPLIjLCGhPrEvJdCuHyzavQuqiFju8vK/WbIo8SnMIZnEMIV1CHO2hAEwg8wjO8wpunvBfv3ftYtK55xcwJ/JH3+QOqe48N</latexit><latexit sha1_base64="AFPCKUS1jXiNTs3LgvClV4SH0fo=">AAAB7nicbZDLSgMxFIbPeK3jrerSTbAUXJUZERRXBTcuK9gLtEPJpJk2NpOEJCOUoe/gSlAQt76PK9/GtJ2Ftv4Q+PjPOeScP1acGRsE397a+sbm1nZpx9/d2z84LB8dt4zMNKFNIrnUnRgbypmgTcsspx2lKU5jTtvx+HZWbz9RbZgUD3aiaJTioWAJI9g6q9XTI9kP+uVKUAvmQqsQFlCBQo1++as3kCRLqbCEY2O6YaBslGNtGeF06ld7maEKkzEe0q5DgVNqony+7hRVnTNAidTuCYvmrv9rIsepMZM0dp0ptiOzXJuZ/9W6mU2uo5wJlVkqyOKjJOPISjS7HQ2YpsTyiQNMNHPLIjLCGhPrEvJdCuHyzavQuqiFju8vK/WbIo8SnMIZnEMIV1CHO2hAEwg8wjO8wpunvBfv3ftYtK55xcwJ/JH3+QOqe48N</latexit><latexit sha1_base64="AFPCKUS1jXiNTs3LgvClV4SH0fo=">AAAB7nicbZDLSgMxFIbPeK3jrerSTbAUXJUZERRXBTcuK9gLtEPJpJk2NpOEJCOUoe/gSlAQt76PK9/GtJ2Ftv4Q+PjPOeScP1acGRsE397a+sbm1nZpx9/d2z84LB8dt4zMNKFNIrnUnRgbypmgTcsspx2lKU5jTtvx+HZWbz9RbZgUD3aiaJTioWAJI9g6q9XTI9kP+uVKUAvmQqsQFlCBQo1++as3kCRLqbCEY2O6YaBslGNtGeF06ld7maEKkzEe0q5DgVNqony+7hRVnTNAidTuCYvmrv9rIsepMZM0dp0ptiOzXJuZ/9W6mU2uo5wJlVkqyOKjJOPISjS7HQ2YpsTyiQNMNHPLIjLCGhPrEvJdCuHyzavQuqiFju8vK/WbIo8SnMIZnEMIV1CHO2hAEwg8wjO8wpunvBfv3ftYtK55xcwJ/JH3+QOqe48N</latexit><latexit sha1_base64="AFPCKUS1jXiNTs3LgvClV4SH0fo=">AAAB7nicbZDLSgMxFIbPeK3jrerSTbAUXJUZERRXBTcuK9gLtEPJpJk2NpOEJCOUoe/gSlAQt76PK9/GtJ2Ftv4Q+PjPOeScP1acGRsE397a+sbm1nZpx9/d2z84LB8dt4zMNKFNIrnUnRgbypmgTcsspx2lKU5jTtvx+HZWbz9RbZgUD3aiaJTioWAJI9g6q9XTI9kP+uVKUAvmQqsQFlCBQo1++as3kCRLqbCEY2O6YaBslGNtGeF06ld7maEKkzEe0q5DgVNqony+7hRVnTNAidTuCYvmrv9rIsepMZM0dp0ptiOzXJuZ/9W6mU2uo5wJlVkqyOKjJOPISjS7HQ2YpsTyiQNMNHPLIjLCGhPrEvJdCuHyzavQuqiFju8vK/WbIo8SnMIZnEMIV1CHO2hAEwg8wjO8wpunvBfv3ftYtK55xcwJ/JH3+QOqe48N</latexit>

⇢1<latexit sha1_base64="qx3hurWYxX5sA48LWbeVi3S9m60=">AAAB7nicbZDLSgMxFIbPeK3jrerSTbAUXJUZERRXBTcuK9gLtEPJpJk2NpOEJCOUoe/gSlAQt76PK9/GtJ2Ftv4Q+PjPOeScP1acGRsE397a+sbm1nZpx9/d2z84LB8dt4zMNKFNIrnUnRgbypmgTcsspx2lKU5jTtvx+HZWbz9RbZgUD3aiaJTioWAJI9g6q9XTI9kP++VKUAvmQqsQFlCBQo1++as3kCRLqbCEY2O6YaBslGNtGeF06ld7maEKkzEe0q5DgVNqony+7hRVnTNAidTuCYvmrv9rIsepMZM0dp0ptiOzXJuZ/9W6mU2uo5wJlVkqyOKjJOPISjS7HQ2YpsTyiQNMNHPLIjLCGhPrEvJdCuHyzavQuqiFju8vK/WbIo8SnMIZnEMIV1CHO2hAEwg8wjO8wpunvBfv3ftYtK55xcwJ/JH3+QOsAI8O</latexit><latexit sha1_base64="qx3hurWYxX5sA48LWbeVi3S9m60=">AAAB7nicbZDLSgMxFIbPeK3jrerSTbAUXJUZERRXBTcuK9gLtEPJpJk2NpOEJCOUoe/gSlAQt76PK9/GtJ2Ftv4Q+PjPOeScP1acGRsE397a+sbm1nZpx9/d2z84LB8dt4zMNKFNIrnUnRgbypmgTcsspx2lKU5jTtvx+HZWbz9RbZgUD3aiaJTioWAJI9g6q9XTI9kP++VKUAvmQqsQFlCBQo1++as3kCRLqbCEY2O6YaBslGNtGeF06ld7maEKkzEe0q5DgVNqony+7hRVnTNAidTuCYvmrv9rIsepMZM0dp0ptiOzXJuZ/9W6mU2uo5wJlVkqyOKjJOPISjS7HQ2YpsTyiQNMNHPLIjLCGhPrEvJdCuHyzavQuqiFju8vK/WbIo8SnMIZnEMIV1CHO2hAEwg8wjO8wpunvBfv3ftYtK55xcwJ/JH3+QOsAI8O</latexit><latexit sha1_base64="qx3hurWYxX5sA48LWbeVi3S9m60=">AAAB7nicbZDLSgMxFIbPeK3jrerSTbAUXJUZERRXBTcuK9gLtEPJpJk2NpOEJCOUoe/gSlAQt76PK9/GtJ2Ftv4Q+PjPOeScP1acGRsE397a+sbm1nZpx9/d2z84LB8dt4zMNKFNIrnUnRgbypmgTcsspx2lKU5jTtvx+HZWbz9RbZgUD3aiaJTioWAJI9g6q9XTI9kP++VKUAvmQqsQFlCBQo1++as3kCRLqbCEY2O6YaBslGNtGeF06ld7maEKkzEe0q5DgVNqony+7hRVnTNAidTuCYvmrv9rIsepMZM0dp0ptiOzXJuZ/9W6mU2uo5wJlVkqyOKjJOPISjS7HQ2YpsTyiQNMNHPLIjLCGhPrEvJdCuHyzavQuqiFju8vK/WbIo8SnMIZnEMIV1CHO2hAEwg8wjO8wpunvBfv3ftYtK55xcwJ/JH3+QOsAI8O</latexit><latexit sha1_base64="qx3hurWYxX5sA48LWbeVi3S9m60=">AAAB7nicbZDLSgMxFIbPeK3jrerSTbAUXJUZERRXBTcuK9gLtEPJpJk2NpOEJCOUoe/gSlAQt76PK9/GtJ2Ftv4Q+PjPOeScP1acGRsE397a+sbm1nZpx9/d2z84LB8dt4zMNKFNIrnUnRgbypmgTcsspx2lKU5jTtvx+HZWbz9RbZgUD3aiaJTioWAJI9g6q9XTI9kP++VKUAvmQqsQFlCBQo1++as3kCRLqbCEY2O6YaBslGNtGeF06ld7maEKkzEe0q5DgVNqony+7hRVnTNAidTuCYvmrv9rIsepMZM0dp0ptiOzXJuZ/9W6mU2uo5wJlVkqyOKjJOPISjS7HQ2YpsTyiQNMNHPLIjLCGhPrEvJdCuHyzavQuqiFju8vK/WbIo8SnMIZnEMIV1CHO2hAEwg8wjO8wpunvBfv3ftYtK55xcwJ/JH3+QOsAI8O</latexit> ⇢T�1

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= pricing interval

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time

(b)

Fig. 1: (a) The relation between driving times, charging times, and busy times. (b) The price process during the idle period.

Figure 1(b). These prices may be the locational marginal pricesfrom a real-time wholesale market,3 or from an emergentretail market. Notice that ρ is different from the flat retailenergy price pret that A pays to charge its transportationbattery. A participates as a regular consumer of a distributionutility that offers it a flat retail rate to charge its transportationbattery. And during an idle period, A participates as an owner-operator of a distributed energy resource (the vehicle battery)or one who relinquishes control over its trading battery to anaggregator of such resources.

A vehicle is connected to the grid when it charges itstransportation battery. We insist in our model that it does notarbitrage energy to accrue revenue during that period. Thatis, we separate the times when the car provides or preparesfor transportation services, and when it takes part in pricearbitrage. This separation facilitates easy auditing.

A. Optimal control for price arbitrage

We now formulate the question of expected revenuemaximization from price arbitrage with a battery of capacityB′ over an idle period as a discrete-time stochastic controlproblem. Suppose there are T intervals of length ∆ within anidle period, where recall that ∆ is the time interval betweenconsecutive price changes. Let ρρρ := (ρ0, . . . , ρT−1) denotethe stochastic price process against which A maximizes itsexpected revenue from arbitrage. Starting from a state ofcharge z0 ∈ [0, B′], the trading battery state at interval tprogresses as

zt+1 := zt + ut

for t = 0, . . . , T −1. Here, ut stands for the energy transactedat time t. A positive u indicates charging the trading battery,and a negative one indicates discharging.

We seek a control policy γγγ := {γ0, . . . , γT−1}, whereγt maps the available information at time t to the storagecontrol action uγt . The relevant information for control designcomprises the state at that time and the history of prices tillthat time. A policy is deemed admissible, denoted γγγ ∈M(B′),if the induced control actions respect the capacity constraints

3While A needs a sufficiently large aggregate battery capacity to participatein a wholesale market, our analysis on the utilization of one car battery revealsthe dependency of the revenue rate on said battery capacity and informs futurework with multiple vehicles.

of the trading battery, i.e.,

0 ≤ z0 +

t∑k=0

uγk ≤ B′

almost surely for t = 0, . . . , T − 1. The optimal expectedrevenue over an idle period of length T is then given by

J∗(z0) := maxγγγ∈M(B′)

E

[T−1∑k=0

ρk(−uγk)

], (5)

where E stands for the expectation computed with respect tothe distribution on the prices. In our next result, we providea closed-form expression for the optimal revenue J∗(z0). Theproof in the Appendix relies on a dynamic programmingbased argument, adopted from [16]. We use the notationz+ = max{z, 0} in stating the result.

Proposition 1. The maximum expected revenue from pricearbitrage with a trading battery of capacity B′ is given by

J∗(z0) = z0E[ρ0] +B′T−2∑j=0

E[(ρj+1|j − ρj

)+], (6)

where ρt+1|t = E [ρt+1|ρ≤t] is the one-step look-ahead priceforecast, given the history of prices till time t.

The derivation of the above result proves that the optimalstorage control policy has a threshold structure. It prescribesto charge the trading battery completely when the price isexpected to go up in the next time step, and to fully dischargeit, otherwise.

Our model of storage operation for price arbitrage neglectsthree important considerations. First, we do not modelroundtrip efficiency losses. The results can be extended toconsider such losses, but are not modeled to maintain clarityof exposition. Second, we do not model the effect of batterydegradation from storage cycling. Vehicle batteries havelimited cycle life and replacement costs can be significant,e.g., see [17]. We aim to address this modeling limitation infuture work. Finally, we do not consider ramping limitationson the battery’s charging and discharging abilities. One wayto account for ramping limitations is to constrain the split ofthe battery in a way that the trading battery size is chargeablewithin ∆ time.

B. Computing A’s revenue rate from price arbitrage

With the aid of Proposition 1, we now derive the revenuerate from energy trading that A garners. To simplify thecalculation, assume that the price process ρρρ is stationaryMarkov4. Then, the distribution of ρt :=

(ρt+1|t − ρt

)+becomes independent of t. Denoting its expectation by 〈ρ〉,the expected revenue during an idle period becomes

ΠE ≈ B′ · E[T − 2]〈ρ〉 ≈ B′

λ∆〈ρ〉,

where ∆ denotes the length of the pricing interval. We maketwo approximations in deriving ΠE . First, we ignore thecontribution of the initial state of charge at the start of anidle period to the revenue from arbitrage. Second, the numberof pricing intervals T − 2 has been approximated by T whoseexpectation is given by (λ∆)−1. If the number of pricingintervals are sufficiently high within an idle period, theseapproximations are accurate.

The expected revenue in the above relation yields thefollowing revenue rate from energy trading using renewal-reward theory.

RE =

(1

µ− λ +1

λ

)−1

ΠE

=1

∆B′〈ρ〉

[1− λ0βFπ(p)Φτ (B/β−)

].

When splitting the vehicle battery for transportation and pricearbitrage, a larger trading battery size increases the arbitragerevenue in each idle period. Also, it leaves lesser capacity fortransportation, leading to a decrease in the incoming traffic ofcustomers seeking transport, thereby increasing the idle time.The transportation price also has a similar effect on the revenuerate from energy trading in that it impacts the rate of arrivingcustomers that in turn affects the lengths of the idle periods.

4The published version is missing the Markov assumption.

Fig. 2: Variation of aggregate revenue rate with transportation priceand battery allocated for transportation. For this experiment, we choseλ0 = 1, Btot = 8, β− = 1, β+ = 7, pret = 3

10, and 〈ρ〉

∆= 1

20.

Trip times and reservation wages were assumed to be exponentiallydistributed with means 1, and 5

2, respectively. The dashed line plots

p∗(B) in (7) for each B ∈ [BL, Btot].

IV. SPLITTING BATTERY CAPACITY FOR DUAL USE

Having computed the revenue rate from transportation andenergy trading for price arbitrage, we now derive how A cansplit its battery capacity to maximize its aggregate revenue rateRtot := RT+RE . Recall that we concluded Section III with theobservation that the transportation price p affects not only therevenue from transportation, but also the revenue from pricearbitrage. Hence, A seeks to jointly optimize its transportationprice and the battery split for the two services.

maximizep,B

λ20β(p− pretβ−)F 2

π (p)Φ2τ (B/β−)

+〈ρ〉∆

(Btot −B)[1− λ0βFπ(p)Φτ (B/β−)

],

subject to λ0βFπ(p)Φτ (B/β−) < 1,

p ≥ 0, 0 ≤ B ≤ Btot.

The stability constraint defines an open set. To avoid technicaldifficulties in optimizing over such an open set, we take itsclosure with the understanding that the parameters are suitablyperturbed to make the closed set feasible.

Optimizing the aggregate revenue rate Rtot can bechallenging, owing to its nonlinear non-concave variation inthe parameters p and B as Figure 2 illustrates. In what follows,we make additional assumptions on the trip times and identifystructural properties of Rtot that facilitate the development ofan algorithm towards solving the above optimization problem.

A. The case with exponentially distributed reservation prices

We characterize the candidate optimal transportation pricesupon fixing the battery capacity B dedicated to providetransportation services.

Proposition 2. Suppose the reservation prices fortransportation customers are exponentially distributedwith mean 〈π〉. For a given B ∈ (0, Btot], the maximum ofRtot occurs at p∗ = +∞ or at

p∗(B) = max

{p0 − 〈π〉W

(− p1

〈π〉ep0〈π〉

), pstable

}, (7)

if 1 + p0/〈π〉 + log(p1/〈π〉) ≥ 0, where W is the principalbranch of the Lambert-W function, and

p0 := pretβ− +

1

2〈π〉, p1 :=

(Btot −B)〈ρ〉2λ0∆Φτ (B/β−)

,

pstable := F−1(

[λ0βΦτ (B/β−)]−1).

Proposition 2 is crucial to our design of an algorithm tocompute the optimal battery split B∗ and the optimal price p∗.Several remarks on the result are in order before presentingits proof and the algorithm design. The ensuing discussionignores the stability constraints for the ease of exposition.

First, the Lambert-W function is a solution to the implicitequation W(xex) = x. For x ∈ [−1/e, 0), there are twosolutions to that equation, both of which are negative. Theprincipal branch selects the one with the smallest absolutevalue; see [18] for details. As a result, the candidate optimizerp∗(B) in (7) is no less than p0.

Second, notice that as B sweeps from the total batterycapacity to zero, p1 increases from zero to∞. Said differently,A charges its passengers more as the size of the tradingbattery Btot −B increases, to both compensate from reducedpassenger car-requests and to garner higher energy-tradingrevenue from increased idle times. Further, beyond a certainsize of the trading battery, p1 becomes large enough to makethe argument inside the Lambert-W function less than −1/e.Then p∗(B) in (7) no longer defines a candidate optimizer.In other words, when the trading battery is large enough,marginally increasing the trading battery will marginallyincrease the price-arbitrage profit, but this increase is unable tocover A’s subsequent marginal loss in transportation revenue,and p∗ =∞ becomes the only candidate optimizer for such Bvalues. Therefore, p∗(B) in (7) is a candidate optimizer onlyfor B ∈ [BL, Btot], where BL is implicitly defined by

BL +1

〈ρ〉2λ0∆〈π〉e−(1+p0/〈π〉)φτ (BL/β−) = Btot. (8)

Third, p∗ = ∞ corresponds to the case that no customeravails A’s transportation service, effectively reducing itsvehicle to a static battery. It readily follows that A willsimultaneously reduce B to zero, thus maximizing its revenuefrom energy trading.

Fourth, setting B = Btot amounts to fully utilizingthe vehicle battery for transportation. In that case, theonly candidate optimal transportation price becomes p0 thatdepends both on the retail energy price and the meanreservation price. Higher the energy retail price to charge itsvehicles, the more A charges customers to compensate. Higherthe customers’ mean reservation price, the more A chargesthem to exploit it.

Proof of Proposition 2. We ignore the queue stabilityconstraint in the rest of the proof. The derivative of Rtot withrespect to p is given by

∂Rtot

∂p= λ0βΦτ (B/β−)Fπ(p)fπ(p)

(p1e

p〈π〉 + p0 − p

),

where recall that fπ is the probability density function of thereservation prices. Therefore, the derivative is positive at zeroand for large p. The candidate maximizers of the aggregaterevenue are infinity and the roots of the derivative. These rootsare the solutions of p1e

p〈π〉 +p0−p = 0, which can be written

as1

〈π〉 (p0 − p)e1〈π〉 (p0−p) = − p1

〈π〉ep0〈π〉 .

ApplyingW on both sides, we infer that the above relation hasat most two solutions. A candidate maximizer of Rtot is thesmallest root, given by (7). Further, W is only well-definedover negative arguments in [−1/e, 0), leveraging which, theinequality identifying a potential optimizer in (7) follows. �

B. Algorithm to optimize transportation price and battery split

Proposition 2 allows us to design the following algorithmto optimize the transportation price and the battery split fordual use. For convenience, we use the notation Rtot(p,B) to

make explicit the dependency of the aggregate revenue rate onp and B.• Compute BL in (8) using a bisection search over [0, Btot].• Maximize Rtot(p

∗(B), B) over [BL, Btot] using projectedgradient descent with diminishing step sizes, starting from(BL +Btot)/2. Call the optimizers p∗,1, B∗,1.

• If Rtot(∞, 0) > Rtot(p∗,1, B∗,1), then return p∗ =

∞, B∗ = 0. Otherwise, return p∗,1, B∗,1.We cannot guarantee that the gradient method converges tothe true optimizer in general. It does so for the numericalexperiments described next.

C. Variation of optimal solution with problem parametersFigure 3 illustrates the variation of the optimal revenue rate

R∗tot and the transportation battery capacity B∗ at optimalitywith various problem parameters. As one expects, A investsall its battery for grid services for low customer traffic(low λ0). The part dedicated to transportation increases withthe customer traffic upto a point, after which transportationbecomes more attractive than grid services. Similarly, thelonger the expected trip times 〈τ〉 the higher the revenuethe platform accrues from transportation. Therefore, whentrip times become high enough the battery is allocated fullyto carsharing services. Next, recall that 〈ρ〉/∆ equals theexpected gain from energy arbitrage with unit battery capacity,and its effect on B∗ is exactly opposite to that of λ0

and 〈τ〉. As 〈ρ〉/∆ increases, grid services become morerewarding, and thus our algorithm favors a battery split againsttransportation services, reducing it to zero when the gridprofits become high enough.

V. CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS

This paper defines a framework to analyze carsharingservices that can utilize the vehicle battery for grid serviceswhen idle. The framework conceptually splits the batterycapacity into two parts—one solely for transportation servicesand the other for grid services. To keep the analysis concrete,the focus is on using the battery for price arbitrage against real-time prices during A’s idle periods. Leveraging equilibriumanalysis of queues, we characterize the revenue rate ofsuch a platform as a function of the price it charges itstransport customers and the battery split. We further providean algorithm to compute the optimal prices and battery splitfor exponentially distributed trip times, and use it to studythe dependency of the optimal revenue rate and the resultingbattery split on various system parameters.

We aim to extend our analysis to the case where thecarsharing company A commands a fleet with more thanone vehicle. This work has ignored the possibility thatA maintains more than one charging depot across a cityresulting in a queuing network—an interesting considerationfor future work. Spatio-temporal variations in demand oftenlead carsharing systems to have excess cars in one location,and a paucity in another. We aim to extend our analysis,accounting for rebalancing costs among depots throughappropriate incentive mechanisms, e.g., in [19]. This workonly considers vehicle batteries to garner revenues from price

Fig. 3: Variation of the optimal revenue rate and battery split with 〈ρ〉/∆, 〈τ〉 and λ0. Other parameters are the same as used for Figure 2.

arbitrage; we will consider alternate grid services that the‘grid’ battery might provide. Finally, we wish to utilize ourframework and data from carsharing services to analyze howprofitable such dual service provision will be in practice.

APPENDIX

Consider the optimal value functions

J∗T−1(z, ρ≤T−1) :=maximumu

− ρT−1u,

subject to 0 ≤ z + u ≤ B′,(9)

J∗t (z, ρ≤t)

:= maximumu

− ρtu+ E[J∗t+1(z + u, ρ≤t+1) | ρ≤t

],

subject to 0 ≤ z + u ≤ B′.(10)

for t = 0, . . . , T−2. By [20, Proposition 1.3.1], the parametricoptimizers of the above optimization problems identify theoptimal policy, and the required optimal cost is given by

J∗(z0) = E [J∗0 (z0, ρ0)] .

Since ρT−1 ≥ 0, the optimizer of (9) is given by u∗ = −zthat yields

J∗T−1(z, ρ≤T−1) = ρT−1z, γ∗T−1(z, ρ≤T−1) = −z. (11)

Next, we utilize (10) and backward induction to prove

J∗t (z, ρ≤t) = ρtz +B′T−2∑j=t

E[(ρj+1|j − ρj

)+ | ρ≤t] ,γ∗t (z, ρ≤t) =

{B′ − zt, if ρt ≤ ρt+1|t,

−zt, otherwise

(12)

for each z ∈ [0, B′], price sequence ρρρ and t = 0, . . . , T − 2.The rest follows from (12) with t = 0. Details are omitted forspace constraints.

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