1. Optimal Decision Making for Electric Vehicles
Providing Electric Grid Frequency Regulation:
A Stochastic Dynamic Programming Approach
Jonathan Donadee
Ph.D. Student, ECE, Carnegie Mellon University
jdonadee@andrew.cmu.edu
Marija Ilic
IEEE Fellow, Professor of ECE and EPP, Carnegie Mellon University
milic@ece.cmu.edu
9th International Conference on Computational Management Science
Imperial College London, UK
April 20th, 2012
Support for this research was provided by
Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology)
through the Carnegie Mellon Portugal Program

2. Outline
Problem Introduction
Deterministic Equivalent Problem Model
Stochastic Dynamic Programming Algorithm
Simulation Results
Conclusions
Questions
2

3. Electrical Grid Frequency Regulation
Power Supply and Demand Balance
 Electrical Grid AC frequency
must be maintained within
± 0.05 Hz
 60Hz (USA)
 50Hz (Europe)
 Demand < Supply, f Regulation is on seconds to minutes timescale
 Demand > Supply, f
 Some generators follow “AGC”
or “Regulation” control signal
on second to minute basis
 Generators bid capacity (MW)
into hourly markets
3
Graphic Source: PJM ISO Training

4. One Day’s AGC Signal
1 Day of PJM AGC Signal
1
0.5
Normalized Signal
0
-0.5
-1
-1.5
0 5 10 15 20 25
Hours
4

5. Motivation
In the future, greater need for fast responding
electrical grid resources
 Compensate renewable resource forecast errors
 Mitigate trend of decreasing system inertia
smaller imbalances causing larger frequency deviations
Integration of Electric Vehicles
 Minimize EV charging cost
Deterministic models and methods are not
well suited for managing uncertain resources
5

6. Electric Vehicle Participation in
Frequency Regulation
 EVs can adjust charge rate to follow AGC signal
 We’ll focus on charging only strategies, no discharging
 Specify preferred charge rate, Pavg
 Specify capacity for regulating, B
 Adjust according to negative of AGC signal scaled by B
0 ≤ 𝐵 ≤ 𝑃𝑎𝑣𝑔
0 ≤ 𝑃𝑎𝑣𝑔 ≤ 𝑃 𝑚𝑎𝑥
0 ≤ 𝐵 ≤ 𝑃 𝑚𝑎𝑥 − 𝑃𝑎𝑣𝑔
6

7. Smart Charging Scenario
 Driver arrives at home and plugs in vehicle
 Inputs time of departure
 Inputs inconvenience cost for not finishing on time ($/hr)
 Smart charger optimizes Pavg, B decisions for each hour
 Smart charger can participate in markets directly, without
delay
7
Picture Source: Ford.com

8. A Stochastic Model is Needed
Ebatt
Emax
t1 t2 t3 tf  State of charge can take
any non-decreasing path
e0
t
Average charge rate to hit Emax at tf
Bound of Possible State of Charge
8

9. A Stochastic Model is Needed
Ebatt Regulation Bid Violated
Emax
t1 t2 t3 tf  State of charge can take
any non-decreasing path
 Regulation contract is
e0 violated if charging finishes
early
t
Average charge rate to hit Emax at tf
Bound of Possible State of Charge
9

10. A Stochastic Model is Needed
Ebatt Regulation Bid Violated
Emax
t1 t2 t3 tf  State of charge can take
any non-decreasing path
 Regulation contract is
e0 Driver violated if charging finishes
Inconvenienced
early
t
Average charge rate to hit Emax at tf  Driver inconvenienced if
Bound of Possible State of Charge
vehicle is not charged on
time
10

11. A Stochastic Model is Needed
Ebatt Regulation Bid Violated
Histogram of July 2011 PJM AGC Signal Energy
18
t1 t2 t3 tf
Emax 16
14
12
Counts
10
8
e0 Driver
6
Inconvenienced
4
2
t
Average charge rate to hit Emax at tf 0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Integrated Hourly Energy
Bound of Possible State of Charge
11

12. A Stochastic Model is Needed
Ebatt Regulation Bid Violated
Histogram of July 2011 PJM AGC Signal Energy
18
t1 t2 t3 tf
Emax 16
14
12
Counts
10
8
e0 Driver
6
Inconvenienced
4
2
t
Average charge rate to hit Emax at tf 0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
Integrated Hourly Energy
Bound of Possible State of Charge
 Providing regulation makes future battery state of charge uncertain
 Literature ignores effect of regulation or optimizes considering expected value
 Stochastic Model needed to:
 value risk of regulation contract violation (pro-rated by time)
 value risk of inconveniencing EV driver
 Optimize choice of average charge rates and regulation contracts size under
uncertainty 12

13. Dynamic Programming Solution
 Solve many hour long optimization problems in a backwards
recursion
𝑉ℎ 𝐸 𝑖,ℎ = min 𝔼 𝜔 𝐽ℎ 𝐸 𝑖,ℎ , 𝑃𝑖,ℎ , 𝐵 𝑖,ℎ , 𝑅ℎ𝜔 + 𝑉ℎ+1 𝑒 𝑡 𝜔𝑓
𝑃 𝑖,ℎ ,𝐵 𝑖,ℎ
 Minimize Expected Future Cost given current time and state
of charge
 Find a single decision to minimize average cost over all future
outcomes 𝜔 𝑖,ℎ ∈ Ω ℎ
 𝑉ℎ 𝐸 𝑖,ℎ is a Stochastic Deterministic Equivalent Problem
13

14. DEP and Optimal Value Function
Solving for V5(16.8) Using 30 Sample Regulation Signals
0.25
Optimal Value Function Cost ($)
0.2
0.15
0.1 𝑉6
0.05 Energy at Pavg
0
Hour 6
Path Bounds
(from Regulation Bid)
Time Energy in sample ω
24
22
20
Hour 5 18
16
Energy (kWh)
14

15. Sample Path Generation
Each DEP uses 30, hour long, AGC signals, 𝑅ℎ𝜔
 Sample historical data using crude monte carlo
 𝜔 𝑖,ℎ ∈ Ω ℎ
Integrate signal over 5 minutes
 Becomes normalized energy for discrete state
equations
𝜔
 𝑅 𝐻 is a correlated 12 dimensional vector
 Assume AGC independent across hours
15

16. State Equations
Energy Actually
𝝎 𝝎 𝝎 Consumed
𝒆 𝒕𝝎 = 𝒆 𝒕−𝟏 + ∆𝒕 ∙ 𝑷 − 𝑩 ∙ ∆𝒕 ∙ 𝑹 𝒕−𝟏 − 𝒔 𝒕−𝟏 , 𝒕 ≥ 𝟐, ∀𝝎 ≠
𝑃 ∙ ∆𝑡
𝑒 𝑡 𝜔 = 𝐸 𝑖,ℎ , 𝑡 = 1, ∀𝜔
𝜔
𝑒 𝑡 𝜔 ≥ 𝑒 𝑡−1 , ∀𝑡, ∀𝜔 24
Example State Dynamics
23
𝜔
𝑒 𝑡 ≤ 𝐸 𝑚𝑎𝑥 , ∀𝑡, ∀𝜔 22
Energy (kWh)
21
𝑠 𝑡 𝜔 ≥ 0, ∀𝑡, ∀𝜔 20
19
18
17
16
T0 1Hr
Time
16

17. Regulation Contract Risk
 If the battery reaches full state of charge, cannot provide
regulation
 Payment is pro-rated, time-based
 Regulation contract violation indicator, 𝑢
 Allows penalty to be a function of time, not energy
 𝑢 is a binary variable Ebatt
𝑢1 = 0 𝑢2 = 0 𝑢3 = 1
Emax
𝑢 𝑡𝜔 ∙ 𝑃 𝑚𝑎𝑥 ∙ ∆𝑡 ≥ 𝑠 𝑡 𝜔 , 𝑡 ≠ 𝑡 𝑓 , ∀𝜔
𝜔
𝑒 𝑡+1 ≥ 𝑢 𝑡𝜔 ∙ 𝐸 𝑚𝑎𝑥 , 𝑡 ≠ 𝑡 𝑓 , , ∀𝜔
𝜔
𝑢 𝑡𝜔 ≥ 𝑢 𝑡−1 , 𝑡 ≠ 𝑡 𝑓 , ∀𝜔 e0
t
Average charge rate to hit Emax at tf
Bound of Possible State of Charge
17

18. Driver Inconvenience Risk
 When not fully charged by
unplug time
 Charge at 𝑃 𝑚𝑎𝑥 until battery is Ebatt 𝜔
𝑇
full Emax
t1 t2 t3 tf
𝐸 𝑚𝑎𝑥 −𝑒 𝑡 𝜔
𝑓
 𝑇𝜔= time late on
𝑃 𝑚𝑎𝑥
sample path ω
e0 Driver
 𝐿 Driver’s inconvenience cost Inconvenienced
($/hr)
t
 𝐿 ∙ 𝑇 𝜔 + 𝑐 𝐻+1 (𝐸 𝑚𝑎𝑥 − 𝑒 𝑡 𝜔𝑓 ) Average charge rate to hit Emax at tf
Bound of Possible State of Charge
18

19. Objective Function
(if final decision)
𝑉 𝐻 𝐸 𝑖,𝐻 = min 𝑃 𝑖,𝐻 ,𝐵 𝑖,𝐻 𝑐 𝐻 𝑃𝑖,𝐻 − 𝑟 𝐻 𝐵 𝑖,𝐻 + 𝜃
𝑡 𝑓 −1
1
𝜃= 𝑐𝐻 −𝐵 𝑖,𝐻 ∙ 𝑅 𝑡𝜔 ∙ ∆𝑡 − 𝑠 𝑡 𝜔 + 𝑄 ∙ ∆𝑡 ∙ 𝐵 𝑖,𝐻 ∙ 𝑢 𝑡𝜔 + 𝐿 ∙ 𝑇 𝜔 + 𝑐 𝐻+1 (𝐸 𝑚𝑎𝑥 − 𝑒 𝑡 𝜔𝑓 )
𝑁
𝜔∈Ω 𝑁 𝑡=1 𝑡
19

20. Objective Function
(if final decision)
Baseline
Energy Cost
𝑉 𝐻 𝐸 𝑖,𝐻 = min 𝑃 𝑖,𝐻 ,𝐵 𝑖,𝐻 𝑐 𝐻 𝑃𝑖,𝐻 − 𝑟 𝐻 𝐵 𝑖,𝐻 + 𝜃
𝑡 𝑓 −1
1
𝜃= 𝑐𝐻 −𝐵 𝑖,𝐻 ∙ 𝑅 𝑡𝜔 ∙ ∆𝑡 − 𝑠 𝑡 𝜔 + 𝑄 ∙ ∆𝑡 ∙ 𝐵 𝑖,𝐻 ∙ 𝑢 𝑡𝜔 + 𝐿 ∙ 𝑇 𝜔 + 𝑐 𝐻+1 (𝐸 𝑚𝑎𝑥 − 𝑒 𝑡 𝜔𝑓 )
𝑁
𝜔∈Ω 𝑁 𝑡=1 𝑡
20

21. Objective Function
(if final decision)
Baseline Regulation Contract
Energy Cost Revenue
𝑉 𝐻 𝐸 𝑖,𝐻 = min 𝑃 𝑖,𝐻 ,𝐵 𝑖,𝐻 𝑐 𝐻 𝑃𝑖,𝐻 − 𝑟 𝐻 𝐵 𝑖,𝐻 + 𝜃
𝑡 𝑓 −1
1
𝜃= 𝑐𝐻 −𝐵 𝑖,𝐻 ∙ 𝑅 𝑡𝜔 ∙ ∆𝑡 − 𝑠 𝑡 𝜔 + 𝑄 ∙ ∆𝑡 ∙ 𝐵 𝑖,𝐻 ∙ 𝑢 𝑡𝜔 + 𝐿 ∙ 𝑇 𝜔 + 𝑐 𝐻+1 (𝐸 𝑚𝑎𝑥 − 𝑒 𝑡 𝜔𝑓 )
𝑁
𝜔∈Ω 𝑁 𝑡=1 𝑡
21

22. Objective Function
(if final decision)
Baseline Regulation Contract
Energy Cost Revenue
𝑉 𝐻 𝐸 𝑖,𝐻 = min 𝑃 𝑖,𝐻 ,𝐵 𝑖,𝐻 𝑐 𝐻 𝑃𝑖,𝐻 − 𝑟 𝐻 𝐵 𝑖,𝐻 + 𝜃
𝑡 𝑓 −1
1
𝜃= 𝑐𝐻 −𝐵 𝑖,𝐻 ∙ 𝑅 𝑡𝜔 ∙ ∆𝑡 − 𝑠 𝑡 𝜔 + 𝑄 ∙ ∆𝑡 ∙ 𝐵 𝑖,𝐻 ∙ 𝑢 𝑡𝜔 + 𝐿 ∙ 𝑇 𝜔 + 𝑐 𝐻+1 (𝐸 𝑚𝑎𝑥 − 𝑒 𝑡 𝜔𝑓 )
𝑁
𝜔∈Ω 𝑁 𝑡=1 𝑡
Energy Cost
Adjustement
22

23. Objective Function
(if final decision)
Baseline Regulation Contract
Energy Cost Revenue
𝑉 𝐻 𝐸 𝑖,𝐻 = min 𝑃 𝑖,𝐻 ,𝐵 𝑖,𝐻 𝑐 𝐻 𝑃𝑖,𝐻 − 𝑟 𝐻 𝐵 𝑖,𝐻 + 𝜃
𝑡 𝑓 −1
1
𝜃= 𝑐𝐻 −𝐵 𝑖,𝐻 ∙ 𝑅 𝑡𝜔 ∙ ∆𝑡 − 𝑠 𝑡 𝜔 + 𝑄 ∙ ∆𝑡 ∙ 𝐵 𝑖,𝐻 ∙ 𝑢 𝑡𝜔 + 𝐿 ∙ 𝑇 𝜔 + 𝑐 𝐻+1 (𝐸 𝑚𝑎𝑥 − 𝑒 𝑡 𝜔𝑓 )
𝑁
𝜔∈Ω 𝑁 𝑡=1 𝑡
Energy Cost Contract
Adjustement Violation Cost
23

24. Objective Function
(if final decision)
Baseline Regulation Contract
Energy Cost Revenue
𝑉 𝐻 𝐸 𝑖,𝐻 = min 𝑃 𝑖,𝐻 ,𝐵 𝑖,𝐻 𝑐 𝐻 𝑃𝑖,𝐻 − 𝑟 𝐻 𝐵 𝑖,𝐻 + 𝜃
𝑡 𝑓 −1
1
𝜃= 𝑐𝐻 −𝐵 𝑖,𝐻 ∙ 𝑅 𝑡𝜔 ∙ ∆𝑡 − 𝑠 𝑡 𝜔 + 𝑄 ∙ ∆𝑡 ∙ 𝐵 𝑖,𝐻 ∙ 𝑢 𝑡𝜔 + 𝐿 ∙ 𝑇 𝜔 + 𝑐 𝐻+1 (𝐸 𝑚𝑎𝑥 − 𝑒 𝑡 𝜔𝑓 )
𝑁
𝜔∈Ω 𝑁 𝑡=1 𝑡
Energy Cost Contract Driver
Adjustement Violation Cost Inconvenience Cost
24

25. Objective Function
(if final decision)
Baseline Regulation Contract
Energy Cost Revenue
𝑉ℎ 𝐸 𝑖,𝐻 = min 𝑃 𝑖,𝐻 ,𝐵 𝑖,𝐻 𝑐 𝐻 𝑃𝑖,𝐻 − 𝑟 𝐻 𝐵 𝑖,𝐻 + 𝜃
Future Energy
Purchases
𝑡 𝑓 −1
1
𝜃= 𝑐𝐻 −𝐵 𝑖,𝐻 ∙ 𝑅 𝑡𝜔 ∙ ∆𝑡 − 𝑠 𝑡 𝜔 + 𝑄 ∙ ∆𝑡 ∙ 𝐵 𝑖,𝐻 ∙ 𝑢 𝑡𝜔 + 𝐿 ∙ 𝑇 𝜔 + 𝑐 𝐻+1 (𝐸 𝑚𝑎𝑥 − 𝑒 𝑡 𝜔𝑓 )
𝑁
𝜔∈Ω 𝑁 𝑡=1 𝑡
Energy Cost Contract Driver
Adjustement Violation Cost Inconvenience Cost
25

26. Objective Function
(not final decision)
Baseline Regulation Contract
Energy Cost Revenue
𝑉ℎ 𝐸 𝑖,ℎ = min 𝑃 𝑖,ℎ ,𝐵 𝑖,ℎ 𝑐ℎ 𝑃𝑖,ℎ − 𝑟ℎ 𝐵 𝑖,ℎ + 𝜃
Future Optimal Value Function
𝑡 𝑓 −1
+ 𝑉ℎ+1 𝑒 𝑡 𝜔𝑓
1
𝜃= 𝑐ℎ −𝐵 𝑖,ℎ ∙ 𝑅 𝑡𝜔 ∙ ∆𝑡 − 𝑠 𝑡 𝜔 + 𝑄 ∙ ∆𝑡 ∙ 𝐵 𝑖,ℎ ∙ 𝑢 𝑡𝜔 + 𝐿 ∙ 𝑇 𝜔 + 𝑐ℎ+1 (𝐸 𝑚𝑎𝑥 − 𝑒 𝑡 𝜔𝑓 )
𝑁
𝜔∈Ω 𝑁 𝑡=1 𝑡
Energy Cost Contract
Adjustement Violation Cost
26

27. Exact Linearization of Contract
Violation Cost
𝑄 ∙ ∆𝑡 ∙ 𝐵 ∙ 𝑡 𝑢 𝑡𝜔 is nonlinear
Replace with 𝑄 ∙ ∆𝑡 ∙ 𝑡 𝑥 𝑡𝜔
Add constraints
𝑥 𝑡𝜔 ≤ 𝐵
𝜔
𝑃 𝑚𝑎𝑥 𝜔
𝑥𝑡 ≤ 𝑢𝑡
2
𝜔
𝑃 𝑚𝑎𝑥
𝑥𝑡 ≥ 𝐵 − 1 − 𝑢 𝑡𝜔
2
 When 𝑢 = 0, 𝑥=0
 When 𝑢 = 1, 𝑥= 𝐵
27

28. Stochastic Dynamic Programming
𝑉5 𝐸5
Unplug Time
E5
Ebatt
Emax
E1
h1 h2 h3 h4 h5 Time (Hours)
28

29. Stochastic Dynamic Programming
𝑉5 𝐸5
Unplug Time
E5
Ebatt
Emax
E1
h1 h2 h3 h4 h5 Time (Hours)
29

30. Stochastic Dynamic Programming
𝑉5 𝐸5
Unplug Time
E5
Ebatt
Emax
E1
h1 h2 h3 h4 h5 Time (Hours)
30

31. Stochastic Dynamic Programming
𝑉5 𝐸5
Unplug Time
E5
Ebatt
Emax
E1
h1 h2 h3 h4 h5 Time (Hours)
31

32. Stochastic Dynamic Programming
𝑉5 𝐸5
Unplug Time
E5
Ebatt
Emax
E1
h1 h2 h3 h4 h5 Time (Hours)
32

33. Stochastic Dynamic Programming
𝑉5 𝐸5
Unplug Time
E5
Ebatt
Emax
E1
h1 h2 h3 h4 h5 Time (Hours)
33

34. Stochastic Dynamic Programming
𝑉5 𝐸5
Unplug Time
E5
Ebatt
Emax
E1
h1 h2 h3 h4 h5 Time (Hours)
34

35. Stochastic Dynamic Programming
𝑉5 𝐸5
Unplug Time
E5
Ebatt
Emax
E1
h1 h2 h3 h4 h5 Time (Hours)
35

36. Future Cost - 𝑉 𝟓 𝑒 𝑡 𝜔𝑓
Find state, cost points on the convex hull of all
points
 Andrews Monotone Chain Algorithm
 Basically compares slopes
𝑉5 𝐸5
E5 36

37. Future Cost - 𝑉 𝟓 𝑒 𝑡 𝜔𝑓
Find state, cost points on the convex hull of all
points
 Andrews Monotone Chain Algorithm
 Basically compares slopes
𝑉5 𝐸5
Not on the hull
On the hull
E5 37

38. Future Cost - 𝑉 𝟓 𝑒 𝑡 𝜔𝑓
 Create inequalities from points on the convex hull
 Add new inequality constraints to DEP 𝑉4 𝐸 𝑖,4
𝑉ℎ+1 𝑒 𝑡 𝜔𝑓 ≥ 𝐼𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 𝑗 − 𝑆𝑙𝑜𝑝𝑒 𝑗 ∗ 𝑒 𝑡 𝜔𝑓 , ∀𝑗
𝑉5 𝐸5
Cut j
E5 38

39. Stochastic Dynamic Programming
Optimal Value Function Cost ($) 0.25
0.2
𝑉ℎ+1 𝑒 𝑡 𝜔𝑓
0.15
0.1
0.05
𝑒1𝑓
𝑡
0
Hour 6
5
Time
24
22
20
Hour 5
4 18
16
Energy (kWh)
39

40. Stochastic Dynamic Programming
Repeat backwards recursion until the current
state is reached
Unplug Time
Ebatt
Emax
E1
h1 h2 h3 h4 h5 Time (Hours)
40

41. Stochastic Dynamic Programming
Repeat backwards recursion until the current
state is reached
Unplug Time
Ebatt
Emax
E1
h1 h2 h3 h4 h5 Time (Hours)
41

42. Stochastic Dynamic Programming
Repeat backwards recursion until the current
state is reached
Unplug Time
Ebatt
Emax
E1
h1 h2 h3 h4 h5 Time (Hours)
42

43. Stochastic Dynamic Programming
Repeat backwards recursion until the current
state is reached
Unplug Time
Ebatt
Emax
E1
h1 h2 h3 h4 h5 Time (Hours)
43

44. Implementation
 Calculate Optimal Value
Functions, Vh 26
Simulation Results
 At initial state, time 24
 Solve one DEP for P1, B1
Battery State of Charge (kWh)
22
 Implement decision and 20 Energy Bounds
wait 1hr, see what happens (from Regulation Bid)
18
 Given new state, 16 Energy at Pavg
 Optimize decision, Actual Energy
14
implement, wait State
 At unplug time 12
1 2 3 4 5 6 7 8
 If not full, charge at Pmax Simulation Timestep (hr)
 Else, Done!
44

45. Forward Simulation for Comparison
Simulate 150 different, 7 hour long
realizations of AGC Signal
Each trial uses the same
 Optimal Value Functions
 Initial state
 Deterministic prices
 Set of samples in DEPs
Compare with an expected value formulation
 1 sample, using expected value of AGC signal
45

46. Results- 150 forward Simulations
Histogram of Expected Value Formulation Costs
$20/hr Inconvenience cost 120
Stochastic Expected Value 100
Model Model 80
μ $ 0.23 $ 0.44
Counts
60
Σ2 5.0 E-4 0.35 40
Late trials 0% 29% 20
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Cost($)
$200/hr Inconvenience cost
Histogram of Stochastic Formulation Costs
35
Stochastic Expected Value
Model Model 30
Μ $ 0.23 $ 2.29 25
Σ2 5.0 E-4 35.58 20
Counts
Late trials 0% 29% 15
10
5
0
0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
Cost($)
46

47. Observations
 Expected Value formulation often inconveniences
driver, while Stochastic formulation is robust
 For final decision, P,B are chosen such that driver is not
inconvenienced on any sample path
 Cost of uncharged energy ÷ 30 > All hourly Energy Prices
 Vast majority of DEP solutions are on the CH
 good approximation of 𝑉ℎ
 If Charging, regulation contract size, B is on upper
bound , but decisions are dependent
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48. Future Work
 Method Improvement/Evaluation
 Bias Estimation and Correction
 Number of AGC Samples
 Number of Discretizations
 Parallelize
 Model Expansion
 Investigate AGC signal properties
 Uncertain Prices- ARIMA or GARCH
 Form CH in 4 dimensions with QuickHull
 Fleet Aggregation
 Apply method to other technologies (flywheels)
 Integrate into broader Smart Distribution Network model
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49. Conclusions
Stochastic models are necessary for demand
side frequency regulation
We have accurately modeled risks of
providing frequency regulation
Our method is tractable and parallelizable
49

50. Thank you!
Histogram of Stochastic Formulation Costs
35
30
25
20
Counts
15
10
5
0
0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3
Cost($)
Simulation Results
26
24
Battery State of Charge (kWh)
22
20
18
16
14
12
1 2 3 4 5 6 7 8
Simulation Timestep (hr)
Support for this research was provided by
Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology)
through the Carnegie Mellon Portugal Program
50