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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**

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

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**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

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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, ], [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**

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

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

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**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

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**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

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**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

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**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

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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 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

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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 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

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**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

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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

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**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

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**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

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**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

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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 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

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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 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

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**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

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**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

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**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

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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 aq(k) dq(k) Each Mt/GIt/∞ queue is mapped into Q distinct Mt/D/∞ queues

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**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

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**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

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**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

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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

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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)

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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

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**Mathematical formulation**

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

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**Linear Programming Approximation**

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

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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

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**Thank you Any questions?**

32nd CNLS Annual Conference, Santa Fe, NM

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