Data-Driven Distributionally Robust Electric Vehicle Balancing for Mobility-on-Demand Systems under Demand and Supply Uncertainties Sihong He Lynn Pepin Guang Wang Desheng Zhang Fei Miao

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Data-Driven Distributionally Robust Electric Vehicle Balancing for

Mobility-on-Demand Systems under Demand and Supply Uncertainties

Sihong He Lynn Pepin Guang Wang Desheng Zhang Fei Miao

Abstract— As electric vehicle (EV) technologies become

mature, EV has been rapidly adopted in modern trans-

portation systems, and is expected to provide future au-

tonomous mobility-on-demand (AMoD) service with eco-

nomic and societal beneﬁts. However, EVs require frequent

recharges due to their limited and unpredictable cruising

ranges, and they have to be managed efﬁciently given the

dynamic charging process. It is urgent and challenging

to investigate a computationally efﬁcient algorithm that

provide EV AMoD system performance guarantees under

model uncertainties, instead of using heuristic demand

or charging models. To accomplish this goal, this work

designs a data-driven distributionally robust optimization

approach for vehicle supply-demand ratio and charging

station utilization balancing, while minimizing the worst-

case expected cost considering both passenger mobility

demand uncertainties and EV supply uncertainties. We

then derive an equivalent computationally tractable form

for solving the distributionally robust problem in a com-

putationally efﬁcient way under ellipsoid uncertainty sets

constructed from data. Based on E-taxi system data of

Shenzhen city, we show that the average total balancing

cost is reduced by 14.49%, the average unfairness of

supply-demand ratio and utilization is reduced by 15.78%

and 34.51% respectively with the distributionally robust

vehicle balancing method, compared with solutions which

do not consider model uncertainties.

I. INTRODUCTION

There are over 5 million EVs by December 2018

globally, and this ﬁgure is predicted to increase to 100-

125 million by 2030 [3]. Compared to conventional gas

vehicles, EV ﬂeets have prolonged charging time and

concentrated mobility patterns due to current charging

technologies and limited charging infrastructures, espe-

cially for commercial EV ﬂeets, e.g., e-taxi, future au-

tonomous mobility-on-demand (AMoD) systems, given

their long daily travel distances [14].

Researchers have been focusing on models and algo-

rithms to study EVs [16], [14]. There have also been

This work has been published in International Conference on

Intelligent Robots and Systems (IROS 2020). Sihong He, Lynn Pepin,

and Fei Miao are with the Department of Computer Science and

Engineering, University of Connecticut, Storrs Mansﬁeld, CT, USA

06268. Email: {sihong.he, lynn.pepin, fei.miao}@uconn.edu. This

work is also partially supported by NSF SAS-1849238 and CPS-

1932223. Guang Wang and Desheng Zhang are with the Department

of Computer Science, Rutgers University, Piscataway, NJ, USA 08901.

Email: {gw255, desheng.zhang}@cs.rutgers.edu.

focusing on how to choose the optimal locations for

charging stations and how to assign charging points

to EVs in each station to minimize the charging time

of EVs considering various constraints, e.g., demand,

costs, and charging compatibility [6], [15]. However,

high costs of charging infrastructures and land resources

make it impractical to deploy abundant charging stations

and points at the early promotion stage [15]. Even when

there is enough charging infrastructure for all EVs in

theory, the uncontrolled and decentralized charging and

mobility behaviors of some EV ﬂeets, e.g., e-taxi, cause

long waiting times when the demand for charging points

greatly exceeds the availability [12].

The above mentioned EV management issues have

posed key optimization and scheduling algorithm chal-

lenges for world-wide EV adoption of AMoD. The

interaction between AMoD systems and power networks

through EVs based on the vehicles’ charging require-

ments, battery depreciation, and power transmission

constraints have been investigated, and the economic

and societal value of EV AMoD has been analyzed [11].

To improve the performance of general AMoD systems,

mobility demand based vehicle balancing methods have

been proposed with various system design objectives,

such as reducing the number of vehicles needed to serve

all passengers [19], [4], reducing customers’ waiting

time [13], or taxis’ total idle distance [8]. However, the

limited knowledge we have about charging patterns [14]

affect the performance of vehicle balancing strategies,

and make real-time decisions under demand model un-

certainties still a challenging and unsolved task.

The contributions of this work are as follows:

•We are the ﬁrst to consider both future demand

uncertainties and EV supply uncertainties predicted

based on charging activity data in designing a

system-level vehicle balancing algorithm. While

model predictive control algorithms [19], [4] have

been designed considering AMoD system demand

uncertainties in the literature, the supply side un-

certainties for EV AMoD is not well studied yet.

•We design a distributionally robust optimization

approach to balance EVs across a city for min-

imum total idle distance and balanced charging

arXiv:2210.10887v1 [math.OC] 19 Oct 2022

station utilization with respect to the worst-case

expected cost. The approach considers probability

distribution uncertainties of the passenger mobility

demand and the EV supply caused by the challenge

of charging process prediction [15], [12].

•We derive an equivalent form of convex opti-

mization problem for the proposed distributionally

robust optimization problem to provide system-

level performance guarantee in a computationally

tractable way under model uncertainties. Based

on real data of Shenzhen city, we show that the

average total balancing cost is reduced by 14.49%,

the average unfairness of supply-demand ratio and

utilization is reduced by 15.78% and 34.51%, re-

spectively, with the proposed method, compared

with solutions which do not consider model un-

certainties.

The rest of the paper is organized as follows. The dis-

tributionally robust EV balancing problem is presented

in Section II. An equivalent computationally tractable

form is derived in Section III. We show performance

improvement in experiments based on real data in Sec-

tion IV. Concluding remarks are provided in Section V.

II. PROBLEM FORMULATION

In this section, we formulate a distributionally robust

optimization problem to balance EVs across a city with

minimum total idle distance and balanced charging sta-

tion utilization. Both passenger mobility demand and EV

supply uncertainties are considered. The region every

empty EV will go is updated in a receding horizon

control process. At each time step, the EV status is up-

dated to the dispatch center ﬁrst, then the dispatch center

calculates a vehicle balancing decision by solving the

proposed distributionally robust optimization problem,

and sent solutions to EVs. The goal is to dispatch vacant

EVs to different regions to pick up current and predicted

passengers if the EVs have enough energy, or to charging

stations if the EVs are short of energy, while minimize

the cost of dispatching for the following τtime steps.

Local dispatchers that match individual EV with one or

several passengers (for carpool) is out the scope of this

work.

A. EV States and Corresponding Actions

We assume there are three possible states for one EV:

vacant, occupied, and low-battery. Vacant means there

are no passengers in this EV, and it has enough energy to

ﬁnish the next trip. The controller dispatches vacant EVs

according to current and predicted passengers demand.

When a vacant EV picks up one or more passengers,

it turns to occupied, and the controller has no actions

for it until it becomes vacant again. An occupied EV

will be ﬁnishing current order in a time period and will

become a vacant EV once it drops off its passengers.

One occupied EV can only become a vacant EV when

it ﬁnishes the current order. When a vacant EV can

not ﬁnish the next trip with the remaining battery, this

EV becomes a low-battery EV and will go to regions

assigned by the controller where it can ﬁnd a charging

station. Before a low-battery EV gets fully charged, it

stays in the low-battery status until it leaves the charging

station and becomes vacant. A low-battery can only

transfer to a vacant EV or stay in current state.

B. Problem Description

We assume that one day is divided into Ktime

intervals, and we use k= 1,2, ..., K to denote time

index. We assume the entire city is divided into N

regions and we use nto denote region index, where

n= 1,2, ..., N. At time k, the system-level controller

makes vacant and low-battery EVs to go to other regions

or stay in the same region for picking up passengers

or charging, respectively. After one EV arrives at its

dispatched region, a local-level controller assign the EV

to pick up passengers or to charge according to the EV’s

battery status.

During time k, there are rk

ipredicted total amount

of passengers demand and ck

ipredicted total number

of EVs ﬁnish charging (new supply of EVs) in re-

gion i, where i= 1,2, ..., N, k = 1,2, ..., K. Let

demand vector rk= [rk

1, rk

2, ..., rk

N]Tand supply vec-

tor ck= [ck

1, ck

2, ..., ck

N]T∈RNbe random vectors

instead of deterministic vectors. And assuming they

are independent. To model the spatial and temporal

relations of deman(supply) during every τconsecutive

time interval, we deﬁne concatenation of demand as

r= [r1, r2, ..., rτ], and concatenation of supply as

c= [c1, c2, ..., cτ]. We use F∗

rand F∗

cto denote the

unknown true probability distributions of r, c ∈RNτ

respectively, i.e. r∼F∗

rand c∼F∗

We use non-negative matrices Xkand Ykas the

decision matrices at time kwhere Xk, Y k∈RN×N

+and

ij (yk

ij )is the total amount of vacant(low-battery) EVs

will be dispatched from region ito region jat the begin-

ning of time k. Minimizing the expected allocating cost

given true probability distributions of demand vector

and supply vector is deﬁned as the following stochastic

programming problem:

min.

X1:τ,Y 1:τ

Er∼F∗

r,c∼F∗

cJ(X1:τ, Y 1:τ, r, c)

s.t. X1:τ, Y 1:τ∈ D,

(1)

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Data-DrivenDistributionallyRobustElectricVehicleBalancingforMobility-on-DemandSystemsunderDemandandSupplyUncertaintiesSihongHeLynnPepinGuangWangDeshengZhangFeiMiaoAbstractAselectricvehicle(EV)technologiesbecomemature,EVhasbeenrapidlyadoptedinmoderntrans-portationsystems,andisexpectedtoprovidefuture...

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Data-Driven Distributionally Robust Electric Vehicle Balancing for Mobility-on-Demand Systems under Demand and Supply Uncertainties Sihong He Lynn Pepin Guang Wang Desheng Zhang Fei Miao

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