Spatial and Temporal Model of Electric Vehicle Charging Demand

1. Spatial and Temporal Model of Electric Vehicle Charging Demand Presented by: Hao Liang 2012.5.31 Broadband Communications Research (BBCR) Lab 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.

2. Outline • Introduction • Highway Model Description • Model Formulations • Numerical Example and Discussions • Conclusions Broadband Communications Research (BBCR) Lab Smart Grid Research Group

3. Introduction Broadband Communications Research (BBCR) Lab Smart Grid Research Group

4. 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 Broadband Communications Research (BBCR) Lab Smart Grid Research Group

5. Benefits of Transportation Electrification • Reduce gasoline consumption • Decrease greenhouse gas emission • Reduce energy bill of vehicle owner ? • Increase profit of vehicle manufacturer ? Broadband Communications Research (BBCR) Lab Smart Grid Research Group

6. 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) Broadband Communications Research (BBCR) Lab Smart Grid Research Group

7. 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) Broadband Communications Research (BBCR) Lab Smart Grid Research Group

8. Highway Model Description Broadband Communications Research (BBCR) Lab Smart Grid Research Group

9. 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 Broadband Communications Research (BBCR) Lab Smart Grid Research Group

10. 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 Broadband Communications Research (BBCR) Lab Smart Grid Research Group

11. Model Formulations Broadband Communications Research (BBCR) Lab Smart Grid Research Group

12. 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) Broadband Communications Research (BBCR) Lab Smart Grid Research Group

13. 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 highwayalong the interval (0, x] before time t Broadband Communications Research (BBCR) Lab Smart Grid Research Group

14. 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 leavethe highway in order to recharge their batteries at the highway exit charging station, and will return to the highway after recharging their batteries Broadband Communications Research (BBCR) Lab Smart Grid Research Group

15. 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 Broadband Communications Research (BBCR) Lab Smart Grid Research Group

16. 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) Broadband Communications Research (BBCR) Lab Smart Grid Research Group

17. 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 Broadband Communications Research (BBCR) Lab Smart Grid Research Group

18. 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 The condition which discharged vehicles can only arrive at and depart from the highway through entrances/exits: Known (or Measurable) Dirac Delta Function Distance from the Spatial Origin to the ith highway entrance/exit Broadband Communications Research (BBCR) Lab Smart Grid Research Group

19. 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 Average Recharged SOC per Vehicle Temporarily Departing Rate per Minute Broadband Communications Research (BBCR) Lab Smart Grid Research Group

20. 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 Broadband Communications Research (BBCR) Lab Smart Grid Research Group

21. 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 Broadband Communications Research (BBCR) Lab Smart Grid Research Group

22. 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 Broadband Communications Research (BBCR) Lab Smart Grid Research Group

23. 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 Broadband Communications Research (BBCR) Lab Smart Grid Research Group

24. Numerical Example and Discussions Broadband Communications Research (BBCR) Lab Smart Grid Research Group

25. 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) Broadband Communications Research (BBCR) Lab Smart Grid Research Group

26. 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 Broadband Communications Research (BBCR) Lab Smart Grid Research Group

27. Simulated Mean Density of Discharged Vehicles Rush Hour Non-Rush Hour Broadband Communications Research (BBCR) Lab Smart Grid Research Group

28. Simulated Mean Traffic Flow of Discharged Vehicles Rush Hour Non-Rush Hour Broadband Communications Research (BBCR) Lab Smart Grid Research Group

29. Expected Number of Charging Pumps in Service and Expected Charging Demand Rush Hour Non-Rush Hour Broadband Communications Research (BBCR) Lab Smart Grid Research Group

30. Application: Sizing the Energy Storage System Off-Peak Time Peak Time Broadband Communications Research (BBCR) Lab Smart Grid Research Group

31. Conclusions Broadband Communications Research (BBCR) Lab Smart Grid Research Group

32. 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 Broadband Communications Research (BBCR) Lab Smart Grid Research Group

33. Thank you! Broadband Communications Research (BBCR) Lab Smart Grid Research Group