1. Stochastic Models and Architecture for Predicting and Scheduling Electric Vehicle Charging Demand
Presented by: Anna Scaglione
University of California Davis
Coauthors: Mahnoosh Alizadeh, Jamie Davis,
Kenneth Kurani, Bob Thomas (Cornell)
32nd CNLS Annual Conference, Santa Fe, NM

2. Electric Vehicles: The promise of a gasoline-free future
The number of electric vehicles (PHEV or EV) is expected to rise considerably during the next two decades
Need: Stochastic models of what we can schedule
California rules give push to 'green' vehicles in the near future
Green Jobs: The Future of the Plug In Electric Vehicle Market Is Looking Bright
Electric Car Greener Alternative to Gas-
Powered Vehicles
Stackable Robot Cars: The Future of Green Mass Transit
32nd CNLS Annual Conference, Santa Fe, NM

3. From “Watts” to “Jobs” Generators $$$  Load following Arrivals 
Load injection
Service = (Charging)
Generators $$$ 
Load following
Arrivals  Jobs
Service requests
Scheduled Service
 follow generation
Generators $ 
Load following
$ Renewables
32nd CNLS Annual Conference, Santa Fe, NM

4. Overview of the talk Background
Part 1: A new stochastic model for EV and PHEV charging demand
Model tuned on real data for PHEV
Test: Smart Load Forecasting capability of model
Communication protocol for submetering
Part 2: Digital Direct Load Scheduling: scheduling of EVs for minimum average delay
Numerical experiments on DDLS
Market integration of EVs as deferrable load
32nd CNLS Annual Conference, Santa Fe, NM

5. Part 1: a scalable mathematical model for uncontrolled EV charging demand
32nd CNLS Annual Conference, Santa Fe, NM

6. Some previous work
Focus: grid impact of a large-scale integration of EVs
See e.g., [Green et al, 2011]’s review, [Wu, Aliprantis and Gkritza, ], [Taylor et al., 2009], [Clement-Nyns et al.,2010]…
Method: Traces of car travel data are used to create an ensemble of “realistic” charges and load injections
derive statistical features (e.g., mean and variance) of EV charging demand based on their proposed arrival and charge pattern
Not an analytical stochastic model for the charging demand useful to assess if “schedulable”
32nd CNLS Annual Conference, Santa Fe, NM

7. Parameters in our model
Load injections of each car is ≈ rectangular
The rate Ri of charge is height, charge time Si is width
N(t): number of cars present in the system
Si: service time for cars present in the system
tia: arrival time of car
What we model:
(tia,Si,Ri, χia) = (arrival time, charge duration, rate) + laxity (slack time or max delay to schedule)
32nd CNLS Annual Conference, Santa Fe, NM

8. Basic idea and more related work
Queueing theory can be used to capture N(t)
[Li and Zhang, 2012]  M/M/c and M/M/c/k/N
[Vlachogiannis 2009]  M/M/∞
[Garcia-Valle and Vlachogiannis, 2009]  M/M/∞
[Alizadeh, Scaglione and Wang,2010]
M/M/c or M/M/∞ queues  fail to capture the temporal variations in the arrival rate and the lognormal distribution of the charge duration
Arrival rates are constant – service times are exponentially distributed
32nd CNLS Annual Conference, Santa Fe, NM

9. for a specific charging level
Our model
A set of Mt/GIt/∞ queues – each queue models demand from either EVs or PHEVs and
for a specific charging level
∞ Servers  once plugged the device is ON = served
The total EV charging demand = Σ of demand from queues
Demand from each queue = (the number of customers receiving service from queue) × (the charging power)
32nd CNLS Annual Conference, Santa Fe, NM

10. Why using a queuing model?
Important result for a Mt/GIt/∞ system
(Proved by Palm and Khinchin)
The number of customers receiving service at t = N(t)
If we initialize the system at t= -∞, N(t) is Poisson distributed with mean
Sc is the stationary-excess service time with
32nd CNLS Annual Conference, Santa Fe, NM

11. New aspects of our model
Arrival rates are non-homogenous with rate λ(t) random process itself (Periodic ARMA process)
Each arrival has a random quadruplet:
(tia,Si,Ri, χia) = (arrival time, charge duration, rate) + laxity (slack time)
Separate arrivals at different rates R into different queues
Unscheduled load
Set of all
possible rates
32nd CNLS Annual Conference, Santa Fe, NM

12. Model elements for each queue
Not going to fit the Probability Mass Function (PMF) of the charging rate Ri
We use the data to fit:
Mt  Arrivals follow a time-dependent Poisson process
Non-homogeneous random arrival rate λ(t)
We will show that the Poisson model fits the real data
We will provide a tunable stochastic model for λ(t) itself
Each arrival  duration of charge S
Cumulative Distribution Function (CDF) FS(s; t)
We will show an appropriate CDF
32nd CNLS Annual Conference, Santa Fe, NM

13. Data used – ICEV and PHEVs
Only look at home-based charging
620 real samples of PHEV - UC Davis PH&EV center
107 mileage entries - National Household Travel Survey in 2009 – Customers drive ICEVs
To convert miles to charge
Customer plug in their car with a probability Pplug(M)
If they plug in :
Miles  Charge Conversion
Charge duration = min { max charge duration, energy per mile × traveled miles /rate, the amount of time parked}
32nd CNLS Annual Conference, Santa Fe, NM

14. Estimating the probability of plugging in
Data on travelled miles M from NHTS
Real charging data from UC Davis
We estimate Pplug(M) so that the charge distribution derived from NHTS matches the real-world data
For PHEVs:
Can be easily extended to EVs ( we don’t have data)
32nd CNLS Annual Conference, Santa Fe, NM

15. PDF of charge duration for PHEVs
Assumption:
Vehicle
type
Kwh/mile
(uniform)
Sedans
Vans
SUVs
Trucks
Remark! The miles to charge conversion without Pplug produces a very different PDF
32nd CNLS Annual Conference, Santa Fe, NM

16. Distribution fitting
We found that a lognormal distribution with μ = and σ = 0.78 best fits the data
Further improvement: clipped lognormal at ≈ 350 minutes
32nd CNLS Annual Conference, Santa Fe, NM

17. Laxity of PHEV charging requests
Time between arrival and departure – charge duration
Two distinct peaks
Daytime slack times
are shorter
Night hours
32nd CNLS Annual Conference, Santa Fe, NM

18. Different degrees of flexibility
We separated daytime and nighttime charging events in the UC Davis charging data to verify
Fits:
Daytime:
Exp(1.08)
Nighttime:
lnN(2.25,0.42)
32nd CNLS Annual Conference, Santa Fe, NM

19. Modeling arrivals: is Poisson reasonable?
We cannot prove it
We construct a test of null hypothesis based on the National Household Travel Survey (NHTS) data
Null hypothesis: arrivals follow a Poisson process with a piecewise constant rate (30 min intervals)
Standard test of uniformity: one sample Kolmogorov- Smirnov test
The null hypothesis is never rejected from NHTS data
32nd CNLS Annual Conference, Santa Fe, NM

20. Quantile-Quantile plot
Q-Q plot comparing conditionally normalized EV arrival times between 13:30-14:00 on a Monday to U(0,1)
32nd CNLS Annual Conference, Santa Fe, NM

21. Modeling and forecasting λ(t)
λ(t) ≈ piecewise constant between samples
The rate profiles are unobservable  count profiles c(t)
We consider vectors of count profiles cj for each day j
Since the variance of a Poisson RV varies with its mean  we apply a variance-stabilizing transformation  a slightly modified version of the Anscombe square-root transform
We store the count profiles of N days in a matrix C
32nd CNLS Annual Conference, Santa Fe, NM

22. Forecasting future arrival rates from historical data stored in C
C has very large dimensions  Multivariate time series may not be the best way for forecasting
We apply an SVD  UTCV = Σ
The j-th column of the count matrix C, cj can be approximated via K principal components
Since V is orthogonal  (αi(1), αi(2), αi(N))’s orthogonal to each other for different i’s
 Reasonable to apply univariate time series to each  train a Periodic ARMA model for each (αi(1), αi(2), αi(N)) produces predictions of λ(t)
32nd CNLS Annual Conference, Santa Fe, NM

23. Principal component analysis
We found that the first 6 principal components account for 96% of the variance in the NHTS database
Weekends
variations
The most
significant pc
(weekday variations)
NHTS arrivals seem different from PHEV arrivals
Need more data
32nd CNLS Annual Conference, Santa Fe, NM

24. Smart load forecasting using AMI
In reality, the system is not initialized at
N(t0) customers are present in the system at time t0
If they inform the forecast unit of charge duration upon arrival
Smart forecast for t> t0 :
Lower forecast error
Stochastic term:
customers arriving after
t> t0 , Poisson distributed
Deterministic term:
customers that were
present at t0
32nd CNLS Annual Conference, Santa Fe, NM

25. Numerical Experiment: Normalized MSE of half-hour ahead prediction
A substation serving a base load of 400 kWs (without vehicles)
400 PHEVS
Arrivals based on NHTS
Charging durations randomly generated from the clipped lognormal
Classical aggregate load predictor: ARMA
The classical predictor is applied in one scenario to the base load and in the other to the total load
32nd CNLS Annual Conference, Santa Fe, NM

26. Part 2: Digital Direct Load Scheduling
32nd CNLS Annual Conference, Santa Fe, NM

27. Scheduling departures
Many methods in the literature for EV scheduling…
Our idea: quantize energy requests to bundle the associated transactions into queues
Arrival and Departure processes
aq(k)
dq(k)
Each Mt/GIt/∞ queue is mapped into Q distinct Mt/D/∞ queues

28. AD conversion: Watts  Jobs
Each arrival (tia,Si) - for simplicity only one rate
Digital signature
Quantize into codes
Quantize time, record the # of loads arriving with a certain code
Load Synthesis
Arrivals in a specific
load class

29. Queuing model to schedule the load
Decide a departure processes dq(t) from each queue
Model complexity depends on Q not on actual number of EV
2 hour charge
3 hour charge
4 hour charge
32nd CNLS Annual Conference, Santa Fe, NM

30. Mathematical Description
Average charging power of these set of queues
First Difference Operator
Convolution
We can use this synthesis formula to model or design a scheduler that makes scheduling decisions on departures at every epoch  dq(l0)
32nd CNLS Annual Conference, Santa Fe, NM

31. EV/PHEV model for the wholesale market
In most present work: demand from deferrable loads = filling a tank by a deadline
see [Lambert et. al, 2006] and [Lamadrid et. al, 2011]
DDLS is a concrete yet simple enough mapping between individual energy requests and aggregate load
The model predicts the aggregate flexibility offered by a populations of EVs in the wholesale market

32. DDLS online scheduling by an aggregator
DDLS avoids the curse of dimensionality of taking individual appliance deadlines into account
Low telemetry and complexity cost
Instead of hard deadlines we use average QoS level
This is one case where the analogy with the Internet works

33. Objective and Costs
Cost of deviation from most economic power profile (power purchased on the day ahead B + locally owned renewables R + load we cannot schedule LT)
Inconvenience cost (experienced by the community, not individuals)
queues weighted differently (different QoS)

34. Real-time Scheduling: Model Predictive Control
A sequential decision maker that determines the schedules for the EVs over a sliding finite horizon (N∆)
Objective: minimize increment in accumulated cost over horizon
Uncertainty: arrival of smart loads, traditional load, renewables, price
The scheduler has:
Predictions of local marginal prices (LMP) for deviating from the day ahead bid at the particular load injection bus
The statistics of both smart and traditional loads
Predictions of available local (green) generation
Output: a decision matrix with N-1 dummy decision vectors to account for the future

35. Mathematical formulation?
05:07:36
Schedule decisions: Model Predictive
Day ahead bid
Renewable
Power cost
Delay cost
Example of typical cost :

36. Linear Programming Approximation
Certainty Equivalent Controller for time  LP
Where:

37. Numerical Results 18k Electric Vehicles 0 ≤ Charge time ≤ 8 hours
Optimization is run every 15 minutes
Charge code quantization step = 15 minutes
Arrival process is Poisson with constant rate λ =3 arrivals/each 15 minutes for each queue (32 queues)
Solver: Certainty equivalent controller that uses LP to schedule the Electric Vehicles
Look-ahead horizon = 8 hours
For fairness, the number of scheduled appliances is equal in the two profiles and no arriving appliances is delayed beyond t = 50 h

38. Conclusions
We presented a mathematical model for uncontrolled EV charging demand useful for scalable simulation, telemetry and forecasting
Parameters of the model were optimized based on actual customer behavior
Based on this model, we presented a best-effort scheduling scheme that can be used to exploit the aggregate flexibility offered by EV charging demand to opportunistically use renewables
32nd CNLS Annual Conference, Santa Fe, NM

39. Thank you Any questions?
32nd CNLS Annual Conference, Santa Fe, NM