Spatial and Temporal Model of Electric
Vehicle Charging Demand
Presented by: Hao Liang
Broadband Communications
Research (BBCR) Lab
2012.5.31
Smart Grid Research Group
Reference: S. Bae and A. Kwasinski, “Spatial and temporal model of electric vehicle charging demand,”
IEEE Transactions on Smart Grid (SI on Transportation Electrification and Vehicle-to-Grid Applications),
vol. 3, no. 1, pp. 394-403, Mar. 2012.
Outline
• Introduction
• Highway Model Description
• Model Formulations
• Numerical Example and Discussions
• Conclusions
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Introduction
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Transportation Electrification
Plug-in Electric Vehicle (PEV)
BMW
Electric Mini Cooper
Nissan
Leaf
Tesla
Model S
Plug-in Hybrid Electric Vehicle (PHEV)
Toyota
Prius
Chevrolet
Volt
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Benefits of Transportation Electrification
• Reduce gasoline consumption
• Decrease greenhouse gas emission
• Reduce energy bill of vehicle owner ?
• Increase profit of vehicle manufacturer ?
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Challenges in Transportation Electrification
• Stress on the power system during the peak
time (temporal changing nature)
• Limitations in distribution transformers, which
are aggravated by the uneven PEV and PHEV
penetration favoring high-income areas, e.g.,
downtown vs. rural (spatial changing nature)
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Main Contributions of the Paper
• Present a mathematical model of rapid charging station’s electricity
demand which may vary both spatially and temporally
• The arrival rate of discharged electric vehicles at a specific charging
station is anticipated by the fluid traffic model (modified based on
highway Poisson-arrival-location model (PALM))
• EV charging demand is predicted by the M/M/s queueing theory
(Poisson vehicle arrival, exponential vehicle charging time, s identical
charging pumps)
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Highway Model Description
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Traffic Model for Semi-infinite, One-way, Single-lane Freeway
• x – Distance along the highway from the spatial origin which is the
beginning point of the highway
• v(x, t) – Velocity field of each vehicle
• Charging stations are located on each exit or entrance
• Vehicle arrives at each entrance with the Poisson distribution
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Extensions of the Traffic Model
Multiple-lane Highway Model
– Combine basic highway models
in which vehicles have different
velocities
Bidirectional Highway Model
– Eastbound ve(x, t) ≥ 0
– Westbound vw(x, t) ≤ 0
Elaborate Highway Network Model
– Superimpose groups of multiple-lane
and bidirectional highways
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Model Formulations
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Assumptions
• The battery for an already charged vehicle entering the highway has a
full state-of-charge (e.g., due to the night-time charging at home)
• Fully charged batteries can last for the entire range of the trip. Hence,
the user of an EV that enters the highway fully charged may exit the
highway not because the batteries are discharged but rather because
he/she may require to rest
=> This study focuses on the discharged EV’s user who forgets to
charge it at night, thus requiring visiting a charging station on a
highway (consider the highway as a prison or jail)
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Definitions of Variables
– The number of discharged EVs remaining in the interval
(0, x] at time t
– The number of discharged EVs that have already passed
through the position x before time t
– The number of discharged EVs that have entered the
highway along the interval (0, x] before time t
– The number of discharged EVs that have exited the
highway along the interval (0, x] before time t
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Representations of Variables
Discharged EVs leaving the system (i.e.,
) are divided into:
1) Permanently depart from the highway and recharge at their final
destinations (e.g., arrive home)
2) Temporarily leave the highway in order to recharge their batteries at the
highway exit charging station, and will return to the highway after recharging
their batteries
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Deterministic Fluid Dynamic Model
Conservation Equation (Foundation)
Density of Discharged Vehicles, veh/km (Will show: this is the only unknown variable)
Traffic Flow of Discharged Vehicles, veh/min
Densities of Discharged Vehicles Entering or Leaving the Highway
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Deterministic Fluid Dynamic Model (Cont’d)
In traffic theory, traffic flow can be defined as the multiplication
of a traffic density by a vehicle’s velocity
Known (or Measurable)
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Deterministic Fluid Dynamic Model (Cont’d)
– All discharged EVs actually arriving at the highway in
the interval (0, x] before time t. Equivalent to
– All discharged EVs permanently departing from the
highway in the interval (0, x] before time t
– All discharged EVs temporarily leaving the highway in
order to recharge their batteries in the interval (0, x]
before time t
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Deterministic Fluid Dynamic Model (Cont’d)
These rate densities can be identified with the actual arrival rate (i.e., i(t)) and the
permanent departure rate (i.e., βi(t)) of discharged EVs at the ith highway entrance/exit
and at time t typically measured in the number of vehicles per minute
Known (or Measurable)
The condition which discharged vehicles can only arrive at and depart from the highway
through entrances/exits:
Dirac Delta
Function
Distance from the Spatial Origin to the
ith highway entrance/exit
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Deterministic Fluid Dynamic Model (Cont’d)
Assume that discharged EVs will return to the highway immediately after
finishing to recharge their batteries
Average Charging
Power per Vehicle
1/60
Charging Completion Rate per Minute
Temporarily Departing Rate per Minute
Average Recharged
SOC per Vehicle
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Deterministic Fluid Dynamic Model (Cont’d)
By substituting the equations into the conservation equation, we have
An ordinary differential equation (ODE) which can be solved with numerical
methods without many difficulties, given certain boundary condition
The arrival rate of discharged EVs at the ith highway charging station
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EVs’ Charging Demand by the M/M/s Queueing Theory
• The queueing system is stable if and only if the occupation rate (ρ) of
charging pumps is less than 1
• The minimum number of charging pumps
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EVs’ Charging Demand by the M/M/s Queueing Theory
(Cont’d)
• The expected number of busy charging pumps
• The power demand of charging station
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Stochastic Model
• The purpose of the stochastic model presented here is to identify
the expected value of the stochastic EV charging demand
Analogous to the deterministic model
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Numerical Example and Discussions
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Basic Highway Model for a Numerical Example (1/2)
• Charging power: 70 kW (level 3 charging station )
• Battery capacity: 8.6 kWh
• Average charge per vehicle at the highway charging station is 4 kWh
which is about 50% of the battery capacity (3.4 min to recharge)
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Basic Highway Model for a Numerical Example (2/2)
• Velocity fields of vehicles on the highway is 1 km/min for all
x ≥ 0 when t ≤ 40 or t > 55 min. During the time interval (40, 55]
min corresponding to rush hour, given by
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Simulated Mean Density of Discharged Vehicles
Non-Rush Hour
Rush Hour
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Simulated Mean Traffic Flow of Discharged Vehicles
Non-Rush Hour
Rush Hour
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Expected Number of Charging Pumps in Service and
Expected Charging Demand
Non-Rush Hour
Rush Hour
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Application: Sizing the Energy Storage System
Off-Peak Time
Peak Time
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Conclusions
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• Highway EV model is based on the fluid traffic model
• EV charging demand is calculated with the arrival rate of
discharged EVs by the M/M/s queueing theory
• Application I: Sizing the energy storage system
• Application II: Distribution system planning
- Traditionally, the planning focuses on local demand
- With EVs, demand may move from another utility into the
area of the utility under consideration => coordination among
neighboring utilities is necessary
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Thank you!
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