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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

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Electric Vehicles: The promise of a gasoline-free future 32nd CNLS Annual Conference, Santa Fe, NM 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

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

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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

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

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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, 2011], [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

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

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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 32nd CNLS Annual Conference, Santa Fe, NM Arrival rates are constant – service times are exponentially distributed

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Our model Demand from each queue = (the number of customers receiving service from queue) × (the charging power) A set of M t /GI t /∞ 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 32nd CNLS Annual Conference, Santa Fe, NM

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Why using a queuing model? Important result for a M t /GI t /∞ 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 S c is the stationary-excess service time with 32nd CNLS Annual Conference, Santa Fe, NM

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

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Model elements for each queue Not going to fit the Probability Mass Function (PMF) of the charging rate R i We use the data to fit: M t 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) F S (s; t) We will show an appropriate CDF 32nd CNLS Annual Conference, Santa Fe, NM

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Data used – ICEV and PHEVs Only look at home-based charging 620 real samples of PHEV - UC Davis PH&EV center 10 7 mileage entries - National Household Travel Survey in 2009 – Customers drive ICEVs To convert miles to charge Customer plug in their car with a probability P plug (M) If they plug in : Charge duration = min { max charge duration, energy per mile × traveled miles /rate, the amount of time parked} 32nd CNLS Annual Conference, Santa Fe, NM Miles Charge Conversion

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Estimating the probability of plugging in Data on travelled miles M from NHTS Real charging data from UC Davis We estimate P plug (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

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

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Distribution fitting We found that a lognormal distribution with μ = 5.03 and σ = 0.78 best fits the data Further improvement: clipped lognormal at ≈ 350 minutes 32nd CNLS Annual Conference, Santa Fe, NM

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

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Different degrees of flexibility 32nd CNLS Annual Conference, Santa Fe, NM We separated daytime and nighttime charging events in the UC Davis charging data to verify Fits: Daytime: Exp(1.08) Nighttime: lnN(2.25,0.4 2 )

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Modeling arrivals: is Poisson reasonable? We cannot prove it We construct a test of null hypothesis based on the 2009 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

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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

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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 c j 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

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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 U T CV = Σ The j-th column of the count matrix C, c j can be approximated via K principal components Since V is orthogonal ( α i (1), α i (2), α i (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 ( α i (1), α i (2), α i (N)) produces predictions of λ (t) 32nd CNLS Annual Conference, Santa Fe, NM

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

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Smart load forecasting using AMI In reality, the system is not initialized at N(t 0 ) customers are present in the system at time t 0 If they inform the forecast unit of charge duration upon arrival Smart forecast for t> t 0 : Lower forecast error Deterministic term: customers that were present at t 0 Stochastic term: customers arriving after t> t 0, Poisson distributed 32nd CNLS Annual Conference, Santa Fe, NM

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Numerical Experiment: Normalized MSE of half-hour ahead prediction 32nd CNLS Annual Conference, Santa Fe, NM 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 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

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

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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 a q (k) d q (k) Each M t /GI t /∞ queue is mapped into Q distinct M t /D/∞ queues

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AD conversion: Watts Jobs Each arrival (t i a,S i ) - 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 Arrivals in a specific load class

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

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Mathematical Description 32nd CNLS Annual Conference, Santa Fe, NM Average charging power of these set of queues First Difference OperatorConvolution We can use this synthesis formula to model or design a scheduler that makes scheduling decisions on departures at every epoch d q (l 0 )

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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

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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 32

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

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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 34

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

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

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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 37

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Conclusions 32nd CNLS Annual Conference, Santa Fe, NM 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

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

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