Distributed Randomized Control for Ancillary Service to the Power Grid

Distributed Randomized Control
for Ancillary Service to the Power Grid
With application to rational pools
LIDS Seminar May 6, 2014
Ana Buˇsi´c and Sean Meyn
Prabir Barooah, Yue Chen, and Jordan Ehren
TREC & D´epartement d’Informatique
INRIA & ENS
Electrical and Computer Engineering
University of Florida
Thanks to NSF, AFOSR, ANR, and DOE / TCIPG

Outline
1 Distributed Control of Loads
2 Design for Intelligent Loads
3 Linearized Dynamics and Passivity
4 One Million Pools
5 Conclusions and Extensions
6 References
0 / 28

Power GridControl WaterPump
Batteries
Coal
GasTurbine
BP
BP
BP C
BP
BP
Voltage
Frequency
Phase
HC
Σ
−
Actuator feedback loop
A
LOAD
Architecture for Distributed Control

Distributed Control of Loads
Power grid as a feedback control system
Massive disturbance rejection problem
One day in Gloucester:
1 / 28

Power grid as a feedback control system
Massive disturbance rejection problem
Two typical weeks in the Paciﬁc Northwest (BPA):
0
2
4
6
8
2
4
6
8
0
0.8
-0.8
1
-1
0
0.8
-0.8
1
-1
0
Sun Mon Tue
October 20-25 October 27 - November 1
Hydro
Wed Thur FriSun Mon Tue Wed Thur Fri
GenerationandLaodGW
GWGW
RegulationGW
Thermal
Wind
Load
Generation
Regulation
1 / 28

Control of Deferrable Loads
Control Goals and Architecture
Batteries
Coal
GasTurbine
BP
BP
BP C
BP
BP
Voltage
Frequency
Phase
HC
Σ
−
A
LOAD
Context: Consider population of similar loads that are deferrable.
Two examples of ﬂexible energy hogs:
Chillers in HVAC systems and residential pool pumps
2 / 28

Control Goals and Architecture
Batteries
Coal
GasTurbine
BP
BP
BP C
BP
BP
Voltage
Frequency
Phase
HC
Σ
−
A
LOAD
Two examples of ﬂexible energy hogs:
Chillers in HVAC systems and residential pool pumps
Control Constraints:
Grid operator demands reliable ancillary service;
Consumers demand reliable service (and more)
2 / 28

One Million Pools in Florida
Pools Service the Grid Today
On Call1: Utility controls residential pool pumps and other loads
1
Florida Power and Light, Florida’s largest utility.
www.fpl.com/residential/energysaving/programs/oncall.shtml
3 / 28

Contract for services: no price signals involved
Used only in times of emergency — Activated only 3-4 times a year
1
3 / 28

Opportunity:
FP&L already has their hand on the switch of nearly one million pools!
1
3 / 28

Opportunity:
Surely pools can provide much more service to the grid
1
3 / 28

Opportunity:
Summary (beyond pools):
For many household loads, contracts with consumers are already in place
1
3 / 28

Opportunity:
Summary (beyond pools):
For many household loads, contracts with consumers are already in place
For other loads, such as plug-in electric vehicles (PEVs),
consumer risk may be too great.
1
3 / 28

Randomized Control Architecture
Constraints: Grid operator demands reliable ancillary service; Consumer demands
reliable service
Control strategy
Requirements:
4 / 28

reliable service
Control strategy
Requirements:
1. Minimal communication: Each load should know the needs of the
grid, and the status of the service it is intended to provide.
4 / 28

reliable service
Control strategy
Requirements:
2. Smooth behavior of the aggregate
=⇒ Randomization
4 / 28

Control strategy
Requirements:
1. Minimal communication: Each load should know the needs of the
grid, and the status of the service it is intended to provide.
2. Smooth behavior of the aggregate
=⇒ Randomization
Need: A practical theory for distributed control based on this architecture
For recent references see thesis of J. Mathieu
4 / 28

Example: One Million Pools in Florida
How Pools Can Help Regulate The Grid
1,5KW 400V
Needs of a single pool
Filtration system circulates and cleans: Average pool pump uses
1.3kW and runs 6-12 hours per day, 7 days per week
5 / 28

1,5KW 400V
Pool owners are oblivious, until they see frogs and algae
5 / 28

1,5KW 400V
Pool owners do not trust anyone: Privacy is a big concern
5 / 28

1,5KW 400V
Pool owners do not trust anyone: Privacy is a big concern
Randomized control strategy is needed.
5 / 28

yt
Control @ Utility
Gain
One Million Pools
Disturbance to be rejected
Proportion of pools on
desired
ζt
d
dt
µt = µtDζt yt = µt, U
Intelligent Appliances

General Model
Controlled Markovian Dynamics
Assumptions
1 Continuous-time model on ﬁnite state space X
2 Controlled Markovian generator Dζ(x, y) : x, y ∈ X, ζ ∈ R.
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General Model
Assumptions
D = D0 models nominal behavior of load.
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General Model
Assumptions
3 Smooth and Lipschitz: For a constant ¯b,
d
dζ
Dζ(x, y) ≤ ¯b, for x, y ∈ X, ζ ∈ R.
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General Model
Assumptions
3 Smooth and Lipschitz: For a constant ¯b,
d
dζ
Dζ(x, y) ≤ ¯b, for x, y ∈ X, ζ ∈ R.
4 Utility function U : X → R
represents state-dependent power consumption
6 / 28

General Model
Assumptions
Each load is subject to common controlled Markovian dynamics.
For the ith load,
P{Xi
(t + s) = x+
| Xi
(t) = x−
} ≈ sDζt (x−
, x+
) + O(s2
)
6 / 28

General Model
Mean Field Model
Aggregate model: N loads running independently,
each under the command ζ.
Empirical Distributions:
µN
t (x) =
1
N
N
i=1
I{Xi
(t) = x}, x ∈ X
7 / 28

General Model
Mean Field Model
µN
t (x) =
1
N
N
i=1
I{Xi
(t) = x}, x ∈ X
Limiting model:
d
dt
µt(x ) =
x∈X
µt(x)Dζt (x, x ), yt :=
x
µt(x)U(x)
via Law of Large Numbers for martingales
7 / 28

General Model
Mean Field Model
µN
t (x) =
1
N
N
i=1
I{Xi
(t) = x}, x ∈ X
Mean-ﬁeld model:
d
dt
µt = µtDζt , yt = µt(U)
ζt = ft(µ0, . . . , µt) by design
7 / 28

General Model
Mean Field Model
Mean-ﬁeld model:
d
dt
yt
Control @ Utility
Gain
N Loads
desired
ζt
µt+1 = µtPζt
yt = µt, U
Questions: 1. How to control this complex nonlinear system? CDC 2013
8 / 28

General Model
Mean Field Model
Mean-ﬁeld model:
d
dt
yt
Control @ Utility
Gain
N Loads
desired
ζt
µt+1 = µtPζt
yt = µt, U
2. How to design {Dζ} so that control is easy?
8 / 28

General Model
Mean Field Model
Mean-ﬁeld model:
d
dt
yt
Control @ Utility
Gain
N Loads
desired
ζt
µt+1 = µtPζt
yt = µt, U
2. How to design {Dζ} so that control is easy? focus here
8 / 28

Design for Intelligent Loads
Mean Field Model
Design via “Bellman-Shannon” Optimal Control
Goal: Construct a family of generators {Dζ : ζ ∈ R}.
9 / 28

Mean Field Model
Design: Consider ﬁrst the ﬁnite-horizon control problem:
Choose distribution pζ to maximize
1
T
ζEpζ
T
t=0
U(Xt) − D(pζ p0)
D denotes relative entropy.
p0 denotes nominal Markovian model.
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Mean Field Model
Design: Consider first the finite-horizon control problem:
Choose distribution pζ to maximize
1
T
ζEpζ
T
t=0
U(Xt) − D(pζ p0)
D denotes relative entropy.
p0 denotes nominal Markovian model.
Explicit solution for finite T:
p∗
ζ(xT
0 ) ∝ exp ζ
T
t=0
U(xt) dt p0(xT
0 )
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Mean Field Model
p∗
ζ(xT
0 ) ∝ exp ζ
T
t=0
U(xt) dt p0(xT
0 )
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Mean Field Model
p∗
ζ(xT
0 ) ∝ exp ζ
T
t=0
U(xt) dt p0(xT
0 )
Markovian, but not time-homogeneous.
As T → ∞, we obtain generator Dζ
9 / 28

Mean Field Model
p∗
ζ(xT
0 ) ∝ exp ζ
T
t=0
U(xt) dt p0(xT
0 )
Simple construction via eigenvector problem:
9 / 28

Mean Field Model
p∗
ζ(xT
0 ) ∝ exp ζ
T
t=0
U(xt) dt p0(xT
0 )
Dζ(x, y) =
v(y)
v(x)
ζU(x) − Λ + D(x, y)
9 / 28

Mean Field Model
p∗
ζ(xT
0 ) ∝ exp ζ
T
t=0
U(xt) dt p0(xT
0 )
Dζ(x, y) =
v(y)
v(x)
ζU(x) − Λ + D(x, y)
where Dv = Λv, D(x, y) = ζU(x) + D(x, y)
Extension/reinterpretation of [Todorov 2007] ⊕ Kontoyiannis & M 200X
9 / 28

d
dt
µt = µtDζt
yt = µt, U
d
dt
Φt = AΦt + Bζt
γt = CΦt
Linearized Dynamics

Linearized Dynamics and Passivity
Mean Field Model
Linearized Dynamics
Mean-ﬁeld model: d
dt µt = µtDζt , yt = µt(U)
Linear state space model:
Φt+1 = AΦt + Bζt
γt = CΦt
10 / 28

Mean Field Model
Linearized Dynamics
Φt+1 = AΦt + Bζt
γt = CΦt
Interpretations: |ζt| is small, and π denotes invariant measure for D.
10 / 28

Mean Field Model
Linearized Dynamics
Φt+1 = AΦt + Bζt
γt = CΦt
• Φt ∈ R|X|, a column vector with
Φt(x) ≈ µt(x) − π(x), x ∈ X
10 / 28

Mean Field Model
Linearized Dynamics
Φt+1 = AΦt + Bζt
γt = CΦt
Φt(x) ≈ µt(x) − π(x), x ∈ X
• γt ≈ yt − y0; deviation from nominal steady-state
10 / 28

Mean Field Model
Linearized Dynamics
Φt+1 = AΦt + Bζt
γt = CΦt
Φt(x) ≈ µt(x) − π(x), x ∈ X
• A = DT
, Ci = U(xi), and input dynamics linearized:
10 / 28

Mean Field Model
Linearized Dynamics
Φt+1 = AΦt + Bζt
γt = CΦt
Φt(x) ≈ µt(x) − π(x), x ∈ X
• A = DT
, Ci = U(xi), and input dynamics linearized:
BT
=
d
dζ
πDζ
ζ=0
10 / 28

Linearized Dynamics
Transfer Function
Φt+1 = AΦt + Bζt
γt = CΦt A = DT
, Ci = U(xi
), BT
=
d
dζ
πDζ
ζ=0
11 / 28

Linearized Dynamics
Transfer Function
Φt+1 = AΦt + Bζt
γt = CΦt A = DT
, Ci = U(xi
), BT
=
d
dζ
πDζ
ζ=0
Transfer Function:
G(s) = C[Is − A]−1
B = C[Is − DT
]−1
B hmmmm...
11 / 28

Linearized Dynamics
Transfer Function
Φt+1 = AΦt + Bζt
γt = CΦt A = DT
, Ci = U(xi
), BT
=
d
dζ
πDζ
ζ=0
Transfer Function:
G(s) = C[Is − A]−1
B = C[Is − DT
]−1
B hmmmm...
Resolvent Matrix:
Rs =
∞
0
e−st
etD
dt = [Is − D]−1
11 / 28

Linearized Dynamics
Transfer Function
Φt+1 = AΦt + Bζt
γt = CΦt A = DT
, Ci = U(xi
), BT
=
d
dζ
πDζ
ζ=0
Transfer Function:
G(s) = C[Is − A]−1
B = C[Is − DT
]−1
B hmmmm...
= CRT
sB TF for L-MFM Resolvent for one load
Resolvent Matrix:
Rs =
∞
0
e−st
etD
dt = [Is − D]−1
11 / 28

Linearized Dynamics
Passive Pools
Theorem: Reversibility =⇒ Passivity
12 / 28

Linearized Dynamics
Passive Pools
Suppose that the nominal model is reversible.
Then its linearization satisﬁes,
Re G(jω) = PSDY (ω), ω ∈ R ,
where
G(s) = C[Is − A]−1
B for s ∈ C.
PSDY (ω) =
∞
−∞
e−jω
Eπ[U(X0)U(Xt)] dt
12 / 28

Linearized Dynamics
Passive Pools
Suppose that the nominal model is reversible.
Then its linearization satisﬁes,
Re G(jω) = PSDY (ω), ω ∈ R ,
where
G(s) = C[Is − A]−1
B for s ∈ C.
PSDY (ω) =
∞
−∞
e−jω
Eπ[U(X0)U(Xt)] dt
Implication for control: G(s) is positive real
12 / 28

Linearized Dynamics
Example Without Passivity
Example: Eight state model Utility function U(xi) = i.
1 2
3 4
5 6
7 8
a c
c
a
a a
a a
a a
b b b b
Not reversible
13 / 28

Linearized Dynamics
Example: Eight state model Utility function U(xi) = i.
Generator is the 8 × 8 matrix,
D =







−a a 0 0 0 0 0 0
a −(a + b + c) 0 b c 0 0 0
b 0 −(a + b) a 0 0 0 0
0 0 a −a 0 0 0 0
0 0 0 0 −a a 0 0
0 0 0 0 a −(a + b) 0 b
0 0 0 c b 0 −(a + b + c) a
0 0 0 0 0 0 a −a







13 / 28

Linearized Dynamics
1 2
3 4
5 6
7 8
a c
c
a
a a
a a
a a
b b b b
Example: Eight state model a = c = 10, b = 1
−2
−1
0
1
2
Real Axis
−1 0 1 2
ImaginaryAxis
−25 −20 −15 −10 −5 0 5 10
−15
−10
−5
0
5
10
15
Real Part
ImaginaryPart
Pole-zero Plot
G(jω), ω >0
Non-minimum
phase zero
Nyquist Plot
G(s) = C[Is − A]−1B not positive real
13 / 28

yt
Control @ Utility
Gain
One Million Pools
Proportion of pools on
desired
ζt
µt+1 = µtPζt
yt = µt, U
One Million Pools

One Million Pools
A Single Pool
Control Architecture
Transition diagram for a single pool:
1 2 T−1 T
. . .
T
On
Off
12T−1
...
Numerics using discrete-time model
14 / 28

One Million Pools
A Single Pool
1 2 T−1 T
. . .
T
On
Off
12T−1
...
Utility:
U(x) = I Pool is on
x
∝ (Power consumption)
x
14 / 28

One Million Pools
A Single Pool
1 2 T−1 T
. . .
T
On
Off
12T−1
...
Utility:
U(x) = I Pool is on
x
∝ (Power consumption)
x
Controlled dynamics: As ζ increases, probability of turning on increases:
0 2412
0
0.5
1
zζ
=-4
ζ=-2
ζ = 4
ζ = 2
ζ = 0
(ζ)
14 / 28

One Million Pools
Mean Field Pool Model
Linearization: Minimum Phase Pole-Zero Plot
−1 0 1
−1
0
1
Real PartImaginaryPart
Transfer function
Normalized Frequency (×π rad/sample)1/24 hrs
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-180
0
20
Phase(degrees)Magnitude(dB)
40
-90
0
15 / 28

One Million Pools
Stochastic simulation — Filtered regulation signal
−400
−200
0
200
400
MW
July 18 July 20July 19 July 21 July 23July 22 July 24
BPA Balancing Reserves Deployed (July 2013)
∗transmission.bpa.gov/Business/Operations/Wind/reserves.aspx
16 / 28

One Million Pools
Stochastic simulation — Filtered regulation signal
−400
−200
0
200
400
MW
July 18 July 20July 19 July 21 July 23July 22 July 24
Filtered BPA reserve signal that can be tracked by pools
BPA Balancing Reserves Deployed (July 2013)
∗transmission.bpa.gov/Business/Operations/Wind/reserves.aspx
16 / 28

One Million Pools
Stochastic simulation using N = 105
pools PI control
Reference (from Bonneville Power Authority)
−300
−200
−100
0
100
200
300
TrackingBPARegulationSignal
(MW)
17 / 28

One Million Pools
pools PI control
Output deviation y
−300
−200
−100
0
100
200
300
(MW)
17 / 28

One Million Pools
pools PI control
Output deviation y
−300
−200
−100
0
100
200
300
-2
-1
-3
2
3
0
1
Input ζ
ζ
(MW)
17 / 28

One Million Pools
pools PI control
Two scenarios, using two diﬀerent reference signals:
0 20 40 60 80 100 120 140 160
t/hour
0 20 40 60 80 100 120 140 160 0 20 40 60 80 100 120 140 160
t/hour
0 20 40 60 80 100 120 140 160
ReferenceOutputdeviation(MW)
−300
−200
−100
0
100
200
300
−300
−200
−100
0
100
200
300
−600
−400
−200
0
200
400
600
−400
−200
0
200
400
600
800
-2
-1
-3
2
3
0
1
−3
−2
−1
0
1
2
3
−5
0
5
−4
−2
0
2
4
6
Input
Input
Input
Input
12 Hour Cleaning Cycle 8 Hour Cleaning Cycle
TrackingBPARegulationSignalBPARegulationSignal-Doubled
(a) (b)
(c) (d)
18 / 28

One Million Pools
pools PI control
The impact of exceeding capacity
0 20 40 60 80 100 120 140 1600 20 40 60 80 100 120 140 160
0 20 40 60 80 100 120 140 160
t/hour
0 20 40 60 80 100 120 140 160
−1000
−500
0
500
1000
−1000
−500
0
500
1000
−1000
−500
0
500
1000
−1000
−500
0
500
1000
−20
−10
0
10
20
−20
−10
0
10
20
−10
0
10
−10
0
10
Input
Input
Input
Input
12 Hour Cleaning Cycle 8 Hour Cleaning Cycle
RegulationexceedscapacityIntegratorwind-upRecovery
(a) (b)
(c) (d)
19 / 28

Batteries
Coal
GasTurbine
BP
BP
BP C
BP
BP
Voltage
Frequency
Phase
HC
Σ
−
A
LOAD
Conclusions and Extensions

Conclusions
Recap
QUIZ: Why intelligence at the loads?
20 / 28

Conclusions
Recap
To simplify control at the grid level
20 / 28

Conclusions
Recap
A particular approach to distributed control is proposed
20 / 28

Conclusions
Recap
The grid level control problem is simple because:
Mean ﬁeld model is simple, and a good approximation of ﬁnite system
20 / 28

Conclusions
Recap
The grid level control problem is simple because:
Mean ﬁeld model is simple, and a good approximation of ﬁnite system
LTI approximation is positive real, which implies minimum phase
20 / 28

Conclusions
Are the customers happy?
No time for details, but no ...
21 / 28

Conclusions
QoS without local control
−100 −80 −60 −40 −20 0 20 40 60 80 100
0
0.01
0.02
0.03
0.04
0.05
fractionofpools
Histogram of pool health
Gaussian distribution
21 / 28

Conclusions
−100 −80 −60 −40 −20 0 20 40 60 80 100
0
0.01
0.02
0.03
0.04
0.05fractionofpools
There will be “rare events” in which the pool is under- or over-cleaned.
21 / 28

Conclusions
−100 −80 −60 −40 −20 0 20 40 60 80 100
0
0.01
0.02
0.03
0.04
0.05fractionofpools
Proposed approach: Additional layer of control at each load, so that hard
constraints on performance can be assured.
21 / 28

Conclusions
−100 −80 −60 −40 −20 0 20 40 60 80 100
0
0.01
0.02
0.03
0.04
0.05fractionofpools
This will create some modeling error at grid level.
Preliminary experiments on the pool model:
No loss of performance at grid level.
21 / 28

Conclusions
The customers are happy
−100 −80 −60 −40 −20 0 20 40 60 80 100
0
0.01
0.02
0.03
0.04
0.05
fractionofpools
22 / 28

Conclusions
The customers are happy
QoS without local control QoS with local control
−100 −80 −60 −40 −20 0 20 40 60 80 100
0
0.01
0.02
0.03
0.04
0.05
fractionofpools
Conditional Gaussian distribution
−100 −80 −60 −40 −20 0 20 40 60 80 100
Load opts-out when its QoS is outside of prescribed bounds
Analysis: LQG approximation for individual load – CDC 2014
22 / 28

Conclusions
Many issues skipped because they are topics of current research
1 Information at the load: Should these loads simply act as frequency
regulators? Measurement of local frequency deviation will suﬃce.
23 / 28

Conclusions
Or, more information may be valuable: Two measurements at each
load, the BA command, and local frequency measurements.
23 / 28

Conclusions
2 Information at the BA: Does this macro control view suﬃce?
Power Grid
Actuation
Control
WaterPump
Batteries
HVAC
Coal
GasTurbine
BP
BP
BP
BP
BP
Baseline
Generation
Disturbances
from nature
Measurements:
Voltage
Frequency
Phase
HC
Σ
−
23 / 28

Conclusions
2 Information at the BA: Does this macro control view suﬃce?
Power Grid
Actuation
Control
WaterPump
Batteries
HVAC
Coal
GasTurbine
BP
BP
BP
BP
BP
Baseline
Generation
Disturbances
from nature
Measurements:
Voltage
Frequency
Phase
HC
Σ
−
Does the grid operator need to know the real-time power
consumption of each population of loads?
23 / 28

Conclusions
3 How can we engage consumers?
24 / 28

Conclusions
The formulation of contracts with customers requires a better
understanding of the value of ancillary service, as well as consumer
preferences.
24 / 28

Conclusions
The formulation of contracts with customers requires a better
understanding of the value of ancillary service, as well as consumer
preferences.
We will not use real time
prices, even via aggregator,
if we want responsive, reliable
ancillary service
−300
−200
−100
0
100
200
300
-2
-1
-3
2
3
0
1
Input
(a)
24 / 28

Conclusions
Thank You!
25 / 28

Control Techniques
FOR
Complex Networks
Sean Meyn
Pre-publication version for on-line viewing. Monograph available for purchase at your favorite retailer
More information available at http://www.cambridge.org/us/catalogue/catalogue.asp?isbn=9780521884419
Markov Chains
and
Stochastic Stability
S. P. Meyn and R. L. Tweedie
August 2008 Pre-publication version for on-line viewing. Monograph to appear Februrary 2009
π(f)<∞
∆V (x) ≤ −f(x) + bIC(x)
Pn
(x, · ) − π f → 0
sup
C
Ex[SτC(f)]<∞
References

References
References: Demand Response
S. Meyn, P. Barooah, A. Buˇsić, and J. Ehren. Ancillary service to the grid from deferrable
loads: the case for intelligent pool pumps in Florida (Invited). In Proceedings of the 52nd
IEEE Conf. on Decision and Control, 2013.
A. Buˇsić and S. Meyn. Passive dynamics in mean field control. ArXiv e-prints:
arXiv:1402.4618. Submitted to the 53rd IEEE Conf. on Decision and Control (Invited),
2014.
S. Meyn, Y. Chen, and A. Buˇsić. Individual risk in mean-field control models for
decentralized control, with application to automated demand response. Submitted to the
53rd IEEE Conf. on Decision and Control (Invited), 2014.
J. L. Mathieu. Modeling, Analysis, and Control of Demand Response Resources. PhD
thesis, Berkeley, 2012.
D. Callaway and I. Hiskens, Achieving controllability of electric loads. Proceedings of the
IEEE, 99(1):184–199, 2011.
S. Koch, J. Mathieu, and D. Callaway, Modeling and control of aggregated heterogeneous
thermostatically controlled loads for ancillary services, in Proc. PSCC, 2011, 1–7.
H. Hao, A. Kowli, Y. Lin, P. Barooah, and S. Meyn Ancillary Service for the Grid Via
Control of Commercial Building HVAC Systems. ACC 2013
(much more on our websites)
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References
References: Markov Models
I. Kontoyiannis and S. P. Meyn. Spectral theory and limit theorems for geometrically
ergodic Markov processes. Ann. Appl. Probab., 13:304–362, 2003.
I. Kontoyiannis and S. P. Meyn. Large deviations asymptotics and the spectral theory of
multiplicatively regular Markov processes. Electron. J. Probab., 10(3):61–123 (electronic),
2005.
E. Todorov. Linearly-solvable Markov decision problems. In B. Schölkopf, J. Platt, and
T. Hoffman, editors, Advances in Neural Information Processing Systems, (19) 1369–1376.
MIT Press, Cambridge, MA, 2007.
M. Huang, P. E. Caines, and R. P. Malhame. Large-population cost-coupled LQG problems
with nonuniform agents: Individual-mass behavior and decentralized ε-Nash equilibria.
IEEE Trans. Automat. Control, 52(9):1560–1571, 2007.
H. Yin, P. Mehta, S. Meyn, and U. Shanbhag. Synchronization of coupled oscillators is a
game. IEEE Transactions on Automatic Control, 57(4):920–935, 2012.
P. Guan, M. Raginsky, and R. Willett. Online Markov decision processes with
Kullback-Leibler control cost. In American Control Conference (ACC), 2012, 1388–1393,
2012.
V.S.Borkar and R.Sundaresan Asympotics of the invariant measure in mean field models
with jumps. Stochastic Systems, 2(2):322-380, 2012.
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References
References: Economics
G. Wang, M. Negrete-Pincetic, A. Kowli, E. Shafieepoorfard, S. Meyn, and U. Shanbhag.
Dynamic competitive equilibria in electricity markets. In A. Chakrabortty and M. Illic,
editors, Control and Optimization Theory for Electric Smart Grids. Springer, 2011.
G. Wang, A. Kowli, M. Negrete-Pincetic, E. Shafieepoorfard, and S. Meyn. A control
theorist’s perspective on dynamic competitive equilibria in electricity markets. In Proc.
18th World Congress of the International Federation of Automatic Control (IFAC), Milano,
Italy, 2011.
S. Meyn, M. Negrete-Pincetic, G. Wang, A. Kowli, and E. Shafieepoorfard. The value of
volatile resources in electricity markets. In Proc. of the 10th IEEE Conf. on Dec. and
Control, Atlanta, GA, 2010.
I.-K. Cho and S. P. Meyn. Efficiency and marginal cost pricing in dynamic competitive
markets with friction. Theoretical Economics, 5(2):215–239, 2010.
U.S. Energy Information Administration. Smart grid legislative and regulatory policies and
case studies. December 12 2011.
http://www.eia.gov/analysis/studies/electricity/pdf/smartggrid.pdf
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Distributed Randomized Control for Ancillary Service to the Power Grid

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