State Space Collapse in Resource Allocation for Demand Dispatch - May 2019

Resource Allocation for Demand Dispatch
State Space Collapse & Market Implications
May 3, 2019
Sean Meyn
Department of Electrical and Computer Engineering University of Florida
Inria International Chair Inria, Paris
Based on joint research with Joel Mathias
and Ana Buˇsi´c, Neil Cammardella, & Robert Moye
Thanks to to our sponsors: NSF, DOE, ARPA-E,

Recaps from January
Creation of Virtual Energy Storage

Balancing Reserves for my GRID
Comfort for me and my owner
Recaps from January
Happy Grid, Loads and Customers

Virtual Energy Storage: Recap from January
Goals of our Research
Ancillary services to match supply and demand:
• Balancing Reserves
Sun
0
1
-1
GW
AGC/Secondary control +
2 / 23

• Peak shaving
2 / 23

• Peak shaving
Modified Prices with Demand Dispatch
2 / 23

• Peak shaving
• Ramp service
GW
Forecasted peak: 29,549Forecasted peak: 29,549
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
17
22
27
32
2 / 23

• Peak shaving
• Ramp service
GW
Forecasted peak: 29,549Forecasted peak: 29,549
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
17
22
27
32
Modified Load with Demand Dispatch
2 / 23

• Peak shaving
• Ramp service
• Contingency support
59.915 Hz
60.010 Hz Modified Load with Demand Dispatch
2 / 23

Duck Curves: California in March 2018
12 181 6 24
26
22
18
14
10
6
2
24
20
16
12
8
4
0
Renewables
Thermal
Imports
Nuclear
Hydro
hours
Generation(GW)
Generation at CAISO March 4, 2018
March 4: nearly 50% of demand was served by solar at 1pm
March 5: record solar production, over 10GW at 10am
3 / 23

12 181 6 24
26
22
18
14
10
6
2
24
20
16
12
8
4
0
Renewables
Thermal
Imports
Nuclear
Hydro
hours
Generation(GW)
Implications:
15 GW upward ramp in three hours; 7 GW ramp in just one hour
Morning downward ramp of 10 GW in three hours
10 GW of thermal generation needed for just a few hours
3 / 23

12 181 6 24
26
22
18
14
10
6
2
24
20
16
12
8
4
0
Renewables
Thermal
Imports
Nuclear
Hydro
hours
Generation(GW)
Net Load
3 / 23

12 181 6 24
26
22
18
14
10
6
2
24
20
16
12
8
4
0
Renewables
Thermal
Imports
Nuclear
Hydro
hours
Generation(GW)
Net Load with Demand Dispatch
Net Load
Load ﬂexibility in California can be used to ﬂatten the duck
3 / 23

Virtual Energy Storage
Rational agent wants a hot shower
http://www.onsetcomp.com/learning/application_stories/multi-channel-data-loggers-improve-forensic-analysis-complex-domestic-hot-water-complaints
Θ(t)
G(t)
Ambient
Temperature
Inlet Water
Temperature
3 kW
Power and temperature evolution in a water heater
4 / 23

80
100
120
140
80
100
120
140
0
MWMW
-10
0
10
Tracking Typical Load Response
temp(F)temp(F)
rt≡0Noreg:|rt|≤10MW
LoadOnLoadOn
BPA Reference:
Power Deviation
rt
G(t) (of interest to BA) Spike Train
Θ(t) (of interest to home) Smooth
Mismatch = gift to the control engineer
4 / 23

Power GridControl Flywheels
Batteries
Coal
GasTurbine
BP
BP
BP C
BP
BP
Voltage
Frequency
Phase
HC
Σ
−
Actuator feedback loop
A
LOAD
G(t) (of interest to BA) Spike Train
Θ(t) (of interest to home) Smooth
Mismatch = gift to the control engineer
4 / 23

Where are the rational agents?
Eﬃcient Equilibrium
max
GS
E e−γt
WS(t) dt max
GD
E e−γt
WD(t) dt
The market is eﬃcient if G∗
S
= G∗
D
5 / 23

max
GS
E e−γt
WS(t) dt max
GD
E e−γt
WD(t) dt
S
= G∗
D
Imperfect, but reasonable: WS(t) = P(t)GS(t) − cS(GS(t))
5 / 23

max
GS
E e−γt
WS(t) dt max
GD
E e−γt
WD(t) dt
S
= G∗
D
Not so reasonable: WD(t) = wD(GD(t)) − P(t)GD(t)
What is the “value of power” to consumers?
80
100
120
140
temp(F)
Consumerservice
LoadOn
5 / 23

max
GS
E e−γt
WS(t) dt max
GD
E e−γt
WD(t) dt
S
= G∗
D
Not so reasonable: WD(t) = wD(GD(t)) − P(t)GD(t)
What is the “value of power” to consumers?
This lecture:
Optimal control and market outcomes with rational agents
5 / 23

Irrational Agents and the Power Grid
1 Virtual Energy Storage: Recap from January
2 Social Planner’s Problem (Optimal Control)
3 State Space Collapse
4 Dynamic Competitive Equilibrium
5 Conclusions
6 References

Feed
Forward GRID
Actuation
real-time
feedback
Social Planner’s Problem
Optimal Control

Social Planner’s Problem (Optimal Control)
Resource allocation for feedforward control
Optimizing demand dispatch
Minimize cost of supplying forecast net load
Available resources, costs and constraints:
Generation: energy and ramping each come with cost.
Modeled as convex costs on generation and its derivative
1 / 23

M classes of ﬂexible loads: air conditioning, refrigeration (residential),
refrigeration (commercial), water heating (residential), water heating
(commercial), irrigation.
1 / 23

M classes of ﬂexible loads: air conditioning, refrigeration (residential),
refrigeration (commercial), water heating (residential), water heating
(commercial), irrigation.
The “SoC” (state of charge) is deﬁned for each class
(e.g., temperature of WH).
Cost of deviation from nominal behavior: convex function of SoC
1 / 23

Leaky battery model Hao et. al. [10, 11], UF/Inria [4, 1, 3]
−ui(t) : deviation of average power consumption from nominal.
xi(t) : deviation of average SoC from nominal for class i at time t.
Water heaters: xi(t) ∝ deviation of average temperature from nominal.
2 / 23

Model: d
dt xi(t) = −αixi(t) − ui(t).
2 / 23

Model: d
Why consider average water heater temperature? (over population)
1) xi(t) represents normalized energy storage at time t for class i
2 / 23

Model: d
Why consider average water heater temperature? (over population)
1) xi(t) represents normalized energy storage at time t for class i
2) SoC constraints appear to be suﬃcient for tracking
D(p p0
) +
κ
2
T
k=1
νk, U − rk
2
2 / 23

Model: d
Finite-horizon optimal control problem: cX(x) = ci(xi)
min
g,u
T
0
cg(g(t)) + cd(g (t)) + cX(x(t)) dt
2 / 23

Resource Allocation
The convex formulation
Convex formulation over time-period [0, T ]:
minimize
g, u
T
0
subject to (t) = g(t) +
i
ui(t) (net load)
12 181 6 24
26
22
18
14
10
6
2
24
20
16
12
8
4
0
Renewables
Thermal
Imports
Nuclear
Hydro
hours
Generation(GW)
: Net Load
3 / 23

Resource Allocation
The convex formulation
Convex formulation over time-period [0, T ]:
minimize
g, u
T
0
subject to (t) = g(t) +
i
ui(t) (net load),
d
dt
ui(t) = νi(t), i ∈ {1, ..., M}
with x(0), u(0) ∈ RM given
Cost on ui , ui =⇒ Singular optimal control problem
3 / 23

Maxweight scheduling in a generalized switch:
state space collapse and workload minimization in heavy traffic
Stolyar, 2004
Heavy traffic resource pooling in parallel-server systems.
J. M. Harrison and M. J. L opez.
Heavy traffic analysis of open processing networks
with complete resource pooling: Asymptotic optimality of discrete
review policies complete resource pooling: asymptotic optimality of discrete review policies
B. Ata and S. Kumar.
Dynamic scheduling of a system with two parallel servers
in heavy traffic with complete resource pooling:
Asymptotic optimality of a continuous review threshold policy.
S. L. Bell and R. J. Williams.
State Space Collapse

Control Techniques
FOR
Complex Networks
Sean Meyn

A Langrangian Decomposition
Relax the algebraic constraint
minimize
g, u
T
0
subject to [ (t)= g(t) +
i
ui(t)],
d
dt
xi(t) = −αixi(t) − ui(t) , 1 ≤ i ≤ M
4 / 23

Dual functional φ∗
Relax constraint with Lagrange multiplier
φ∗
( ) = inf
g,u
T
0
cg(g(t)) + cd(g (t))
+ cX(x(t)) + (t)[ (t) − g(t) − uσ(t)] dt
Subject to:
d
dt
xi(t) = −αixi(t) − ui(t)
5 / 23

Dual functional φ∗
Relax constraint with Lagrange multiplier
φ∗
( ) = inf
g,u
T
0
+ cX(x(t)) + (t)[ (t) − g(t) − uσ(t)] dt
Subject to:
d
dt
xi(t) = −αixi(t) − ui(t)
Separable minimization problem over g, xi
5 / 23

Grid-operator’s minimization problem
inf
g
T
0
Lg(g(t), ˙g(t), t) dt,
Lg(g(t), ˙g(t), t) = cg(g(t)) + cd(˙g(t)) − (t)g(t)
6 / 23

inf
g
T
0
For any C1 function , the optimizer g (t) satisﬁes the
Euler-Lagrange equation:
∂
∂g Lg(g , ˙g , t) − d
dt
∂
∂ ˙g Lg(g , ˙g , t) = 0
6 / 23

inf
g
T
0
For any C1 function , the optimizer g (t) satisﬁes the
Euler-Lagrange equation:
cg (g (t)) − (t) −
d
dt
cd(˙g (t)) = 0
6 / 23

Minimization for each load class
inf
xi
T
0
Li(xi(t), ˙xi(t), t) dt,
Li(xi(t), ˙xi(t), t) = ci xi(t) + αi (t)xi(t) + (t) ˙xi(t)
7 / 23

inf
xi
T
0
For any function that is continuously diﬀerentiable on (0, T ], the
optimizer xi (t) satisﬁes the Euler-Lagrange equation:
∂
∂xi
Li(xi , ˙xi , t) − d
dt
∂
∂ ˙xi
Li(xi , ˙xi , t) = 0
7 / 23

inf
xi
T
0
For any function that is continuously diﬀerentiable on (0, T ], the
optimizer xi (t) satisﬁes the Euler-Lagrange equation:
ci (xi (t)) + αi (t) −
d
dt
(t) = 0
7 / 23

State space collapse
Optimal SoC evolves in a two dimensional subspace
Recall: φ∗
( ) = inf
g,u
T
0
+ cX(x(t)) + (t)[ (t) − g(t) − uσ(t)] dt
8 / 23

Recall: φ∗
( ) = inf
g,u
T
0
+ cX(x(t)) + (t)[ (t) − g(t) − uσ(t)] dt
Assume that a C1 maximizer ∗ exists: φ∗( ∗) ≥ φ∗( ) for all .
8 / 23

Assume that a C1 maximizer ∗ exists: φ∗( ∗) ≥ φ∗( ) for all .
State space collapse:
ci (x∗
i (t)) = −αi
∗
(t) +
d
dt
∗
(t)
Marginal costs evolve on a two-dimensional subspace!
8 / 23

Numerical Experiment: state space collapse
Net load taken from California, March 2018
12 181 6 24
26
22
18
14
10
6
2
24
20
16
12
8
4
0
Renewables
Thermal
Imports
Nuclear
Hydro
hours
Generation(GW)
9 / 23

Net load taken from California, March 2018
Five classes of electric loads:
ACs, residential WHs, commercial WHs, fridges, pools
Time horizon T = 24 hours
Load Parameters
Par. Unit ACs fWHs sWHs RFGs PPs
αi hours-1
0.25 0.04 0.01 0.10 0.004
9 / 23

Given: optimal SoC evolution for ACs, x∗
ac(t) and Pools, x∗
pp(t)
10 / 23

pp(t)
Recover Lagrange multiplier:
∗(t)
d
dt
∗(t)
=
−αac 1
−αpp 1
−1
cac(x∗
ac(t))
cpp(x∗
pp(t))
10 / 23

pp(t)
∗(t)
d
dt
∗(t)
=
−αac 1
−αpp 1
−1
cac(x∗
ac(t))
cpp(x∗
pp(t))
State space collapse ≡ Recover SoC for any other load
10 / 23

pp(t)
∗(t)
d
dt
∗(t)
=
−αac 1
−αpp 1
−1
cac(x∗
ac(t))
cpp(x∗
pp(t))
State space collapse ≡ Recover SoC for any other load
GWh
AC
Pools
Res WH
Res WH (prediction)
24 Hours 24 Hours
Subspaceprediction
-6
-4
-2
0
2
4
6
-3
-2
-1
0
1
2
3
Water heaters: x∗
rwh(t) = (crwh)−1
(−αrwh
∗
(t) + d
dt
∗
(t))
10 / 23

Irrational Agents
Dynamic Competitive Equilibrium

Dynamic competitive equilibrium and real-time prices
CE model with M + 1 players:
Single supplier of generation g
M consumer classes with power deviation −ui
11 / 23

Social Planner’s Problem:
max
g,ui
T
0
US(g(t), g (t)) +
M
i=1
UDi (ui(t)) dt
subject to supply equals demand, g(t) = l(t) − i ui(t) (physics)
11 / 23

max
g,ui
T
0
US(g(t), g (t)) +
M
i=1
UDi (ui(t)) dt
Utility functions ≡ negative of cost:
UDi (ui) = −ci xi), US(g, g ) = −cg(g) − cd(−g )
11 / 23

max
g,ui
T
0
US(g(t), g (t)) +
M
i=1
UDi (ui(t)) dt
Dynamic competitive equilibrium:
u∗
i = arg max
ui
T
0
UDi (ui(t)) − p∗
(t)ui(t) dt
g∗
= arg max
g
T
0
US(g(t), g (t)) + p∗
(t)g(t) dt
p∗ = ∗ is the equilibrium price
see also Chen & Low 2011 and Cruise and Zachary 2019 (?)
11 / 23

Price, marginal cost, and marginal value
Averages of prices, marginal cost, and marginal value
avg
=
1
T
T
0
∗
(t) dt
MCavg
g =
1
T
T
0
cg(g∗
(t)) dt
MVavg
i = −
1
T
T
0
ci (x∗
i (t)) dt
12 / 23

Return to Euler-Lagrange equations:
∗
(t) = cg (g∗
(t)) −
d
dt
cd(˙g∗
(t))
−αi
∗
(t) = ci (x∗
i (t)) −
d
dt
∗
(t)
Price is not equal to marginal cost or value
12 / 23

Return to Euler-Lagrange equations:
∗
(t) = cg (g∗
(t)) −
d
dt
cd(˙g∗
(t))
−αi
∗
(t) = ci (x∗
i (t)) −
d
dt
∗
(t)
Integrating:
Avg price is approximately the scaled average marginal value:
avg
=
1
αi
MVavg
i + ed
i /T , 1 ≤ i ≤ M
Avg price is approximately the average marginal cost:
avg
= MCavg
g + eg
/T
12 / 23

Balancing California in 2018
Optimal load trajectories
Optimal SoC trajectories for a 24 hour time horizon:
-5
0
5
GWhNormalized
AC
Res WH
Comm WH
Refrigerators
Pools
State of Charge
Price
-1
0
1
24 Hours
Net Load 22
18
14
10
20
16
12
8
Load(GW)
13 / 23

Balancing California in 2018
Optimal load trajectories
Optimal SoC trajectories for a 24 hour time horizon:
-5
0
5
GWhNormalized
AC
Res WH
Comm WH
Refrigerators
Pools
State of Charge
Price
-1
0
1
24 Hours
Net Load 22
18
14
10
20
16
12
8
Load(GW)
Prices to devices success story! ∗(t) ∼ (t)!! (approximately)
13 / 23

Prices for Load Reduction
∗
(t) ∼ (t) Prices to devices success story?
0 1 2 3 4
-2
-1
0
1
2
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
SoC(GWh)
20
25
30
power(GW)
hrs
-5
0
5
power(GW)NormalizedOptimalPrice
Net Load piece-wise constant:
AC fWH sWH FR PPu∗
i (t) = neg load
similar to ?Price estimate
14 / 23

Prices for Load Reduction
The solution to the E-L equations tells a diﬀerent story
0 1 2 3 4
-2
-1
0
1
2
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
SoC(GWh)
20
25
30
power(GW)
hrs
-5
0
5
power(GW)NormalizedOptimalPrice
Net Load piece-wise constant:
AC fWH sWH FR PPu∗
i (t) = neg load
Competitive equilibrium price
is smooth, even when net load is not
14 / 23

Prices to devices optimal response
Recall from January Slightly diﬀerent formulation: modeled as desired load reduction
TotalPower(GW)
0
5
10
15
20
hrs
P nominal
P desired
P delivered
1 2 3 4 5
Example: Aggregator has contracts with consumers
7 million residential ACs
700,000 water heaters
700,000 commercial water heaters
17 million refrigerators
All the pools in California
Promises strict bounds on QoS for each customer
15 / 23

Recall from January Slightly diﬀerent formulation: modeled as desired load reduction
TotalPower(GW)
Temperature,cycling,energy
0
5
10
15
20
hrs
P nominal
P desired
P delivered
1 2 3 4 5
hrs1 2 3 4 5
ACs
Small WHs
Commercial WHs
Refrigerators
Pools
QoS
15 / 23

Recall from January
TotalPower(GW)
0
5
10
15
20
hrs
P nominal
P desired
P delivered
1 2 3 4 5
Balancing authority desires power reduction over 2 hours
Sends PRICE SIGNAL: 10% increase
Aggregator optimizes subject to QoS constraints
15 / 23

Recall from January
TotalPower(GW)Power(GW)
0
5
10
15
20
hrs
0
2
4
6
P nominal
P desired
P delivered
ACs FWHs
SWHs Fridges
Pools
1 2 3 4 5
Price event: 10% increase
15 / 23

Recall from January
TotalPower(GW)Power(GW)
0
5
10
15
20
hrs
0
2
4
6
P nominal
P desired
P delivered
ACs FWHs
SWHs Fridges
Pools
1 2 3 4 5
Price event: 10% increase
Intuitive CE price
will blow up the grid
15 / 23

Power GridControl WaterPump
Batteries
Coal
GasTurbine
BP
BP
BP C
BP
BP
Voltage
Frequency
Phase
HC
Σ
−
Actuator feedback loop
A
LOAD
Conclusions

Conclusions Summary and Spring Homework Due
Conclusions
Summaries and clariﬁcations
Θ(t)
G(t)
Ambient
Temperature
Inlet Water
Temperature
3 kW
Proposal: VOLL ?
16 / 23

Conclusions
Θ(t)
G(t)
Ambient
Temperature
Inlet Water
Temperature
3 kW
Proposal: VOLL VOLQoS
16 / 23

Conclusions
VOLL VOLQoS
Please remember: ρ∗ is NOT a price. What is it?
16 / 23

Conclusions
VOLL VOLQoS
ρ∗ is a Lagrange multiplier — as part of a control solution.
It is computed based on forecasts of renewable generation,
and energy needs over the day.
It will be part of an MPC control architecture
16 / 23

Conclusions
VOLL VOLQoS
The CE interpretation is superﬁcial
16 / 23

Conclusions
VOLL VOLQoS
We are excited about the model reduction conclusion:
M = 105 load classes
HJB equation dimension = two
16 / 23

Conclusions
Questions answered this semester?
History (of power economics)
How did we get here?
Why are spot prices seen as the control solution?
Can someone ﬁnd an economic justiﬁcation?
Fixed cost > 50% of total cost is nothing new! [13, 12]
17 / 23

Conclusions
Can we validate the claims that PJM FP&L?
17 / 23

Conclusions
Market design: Let’s create a theoretical foundation for zero
marginal cost resources such as batteries, wind, and Demand
Dispatch
A working solution requires a CEO model, combined with stable public
policy to enable long-term planning
17 / 23

Conclusions
Market design: Let’s create a theoretical foundation for zero
marginal cost resources such as batteries, wind, and Demand
Dispatch
A working solution requires a CEO model, combined with stable public
policy to enable long-term planning
Control architectures
If our goal is smoothing net-load and congestion control, what is
essentially diﬀerent between bits vs. watts?
Our work and research@Vermont suggests the gap isn’t always wide
What questions arise when we look seriously at distribution along with
transmission?
17 / 23

Conclusions Thank You
Thank You
18 / 23

Conclusions Backup question: which is of these is a dictatorship?
19 / 23

References
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
References
20 / 23

References
Selected References I
[1] J. Mathias, R. Moye, and S. Meyn. State space collapse in resource allocation for
demand dispatch. In IEEE Conference on Decision and Control (submitted), 2019.
[2] N. L. Chen, and S. H. Low, Optimal demand response based on utility maximization in
power networks. IEEE Power and Energy Society General Meeting, July 2011, pp. 1–8.
[3] N. Cammardella, A. Buˇsić, Y. Ji, and S. Meyn. Kullback-Leibler-Quadratic optimal
control of flexible power demand. In IEEE Conference on Decision and Control
(submitted), 2019.
[4] N. Cammardella, J. Mathias, M. Kiener, A. Buˇsić, and S. Meyn. Balancing California’s
grid without batteries. In IEEE Conf. on Decision and Control (CDC), pages 7314–7321,
Dec 2018.
[5] Y. Chen, M. U. Hashmi, J. Mathias, A. Buˇsić, and S. Meyn. Distributed control design
for balancing the grid using flexible loads. In S. Meyn, T. Samad, I. Hiskens, and
J. Stoustrup, editors, Energy Markets and Responsive Grids: Modeling, Control, and
Optimization, pages 383–411. Springer, New York, NY, 2018.
[6] J. Mathias, A. Buˇsić, and S. Meyn. Demand dispatch with heterogeneous intelligent
loads. In Proc. 50th Annual Hawaii International Conference on System Sciences
(HICSS), and arXiv 1610.00813, 2017.
21 / 23

References
Selected References II
[7] J. Mathias, R. Kaddah, A. Buˇsić, and S. Meyn. Smart fridge / dumb grid? Demand
Dispatch for the power grid of 2020. In Proc. 49th Annual Hawaii International
Conference on System Sciences (HICSS), pages 2498–2507, Jan 2016.
[8] Y. Chen, U. Hashmi, J. Mathias, A. Buˇsić, and S. Meyn. Distributed Control Design for
Balancing the Grid Using Flexible Loads. In IMA volume on the control of energy
markets and grids Springer, 2018.
[9] S. Meyn, P. Barooah, A. Buˇsić, Y. Chen, and J. Ehren. Ancillary service to the grid using
intelligent deferrable loads. IEEE Trans. Automat. Control, 60(11):2847–2862, Nov 2015.
[10] H. Hao, B. Sanandaji, K. Poolla, and T. Vincent. A generalized battery model of a
collection of thermostatically controlled loads for providing ancillary service. In 51st
Annual Allerton Conference on Communication, Control, and Computing, pages 551–558,
Oct 2013.
[11] H. Hao, B. M. Sanandaji, K. Poolla, and T. L. Vincent. Aggregate flexibility of
thermostatically controlled loads. IEEE Trans. on Power Systems, 30(1):189–198, Jan
2015.
[12] H. Lo, S. Blumsack, P. Hines, and S. Meyn. Electricity rates for the zero marginal cost
grid. The Electricity Journal, 32(3):39 – 43, 2019.
22 / 23

References
Selected References III
[13] R. Moye and S. Meyn. Redesign of U.S. electricity capacity markets. In IMA volume on
the control of energy markets and grids. Springer, 2018.
[14] R. Moye and S. Meyn. The use of marginal energy costs in the design of U.S. capacity
markets. In Proc. 51st Annual Hawaii International Conference on System Sciences
(HICSS), 2018.
[15] R. Moye and S. Meyn. Scarcity pricing in U.S. wholesale electricity markets. In Proc.
52nd Annual Hawaii International Conference on System Sciences (HICSS) (submitted),
2018.
[16] M. Negrete-Pincetic. Intelligence by design in an entropic power grid. PhD thesis, UIUC,
Urbana, IL, 2012.
[17] G. Wang, M. Negrete-Pincetic, A. Kowli, E. Shaﬁeepoorfard, S. Meyn, and U. V.
Shanbhag. Dynamic competitive equilibria in electricity markets. In A. Chakrabortty and
M. Illic, editors, Control and Optimization Methods for Electric Smart Grids, pages
35–62. Springer, 2012.
[18] R. A¨ıd, D. Possama¨ı, and N. Touzi. Electricity demand response and optimal contract
theory. SIAM News, 2017.
[19] Coase, R.H. The marginal cost controversy. Econometrica 13(51), 169–182 (1946)
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