The document discusses several issues related to dynamic power systems including: 1) Political mandates for renewable energy are changing rapidly and coupled generators and consumers can lead to instability. 2) Power systems are complex interconnected networks that require reliable operation while market power poses risks. 3) Balancing supply and demand is challenging with intermittent renewable resources like wind. 4) Power flow is subject to physical constraints that create friction in the system.

1. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Dynamic Models and Dynamic Markets
for Electric Power Networks
Sean P. Meyn
Joint work with: In-Koo Cho, Matias Negrete-Pincetic, Gui Wang,
Anupama Kowli, and Ehsan Shafieeporfaard
Coordinated Science Laboratory
and the Department of Electrical and Computer Engineering
University of Illinois at Urbana-Champaign, USA
May 25, 2010
1 / 66

2. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Outline
1 Issues and Approaches
2 Reliability by Design
3 Dynamic Markets and their Equilibria
4 Who Commands the Wind?
5 Research Frontiers
6 References
2 / 66

3. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
1,400
1,200 Tasmania 1,000
1,000
Demand
800
0
600
ia
400 Victor
Prices
Price (Aus $/MWh)
Volume (MW)
200 - 1,000
0 - 1,500
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
09:00
10:00
11:00
12:00
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14:00
15:00
16:00
17:00
18:00
19:00
20:00
21:00
22:00
23:00
24:00
nia
Tasma
10,000 19,000
9,000
Victoria Demand
8,000
7,000
6,000
10,000
5,000
Prices
4,000
3,000
2,000
1,000 1,000
0 - 1,000
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
09:00
10:00
11:00
12:00
13:00
14:00
15:00
16:00
17:00
18:00
19:00
20:00
21:00
22:00
23:00
24:00
Australia - January 16, 2007
Issues and Approaches
3 / 66

4. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Issues: Political Mandates come, and go...
Renewable Portfolio Standards
www.dsireusa.org / February 2010
VT: (1) RE meets any increase ME: 30% x 2000
WA: 15% x 2020* New RE: 10% x 2017
MN: 25% x 2025 in retail sales x 2012;
MT: 15% x 2015 (Xcel: 30% x 2020) (2) 20% RE & CHP x 2017 NH: 23.8% x 2025
OR: 25% x 2025 (large utilities)* ND: 10% x 2015 MI: 10% + 1,100 MW MA: 15% x 2020
5% - 10% x 2025 (smaller utilities) x 2015* + 1% annual increase
(Class I RE)
SD: 10% x 2015 WI: Varies by utility; NY: 29% x 2015
10% x 2015 goal RI: 16% x 2020
NV: 25% x 2025* CT: 23% x 2020
IA: 105 MW
W O
OH: 25% x 2025†
CO: 20% by 2020 (IOUs) PA: 18% x 2020†
10% by 2020 (co-ops & large munis)*
IL: 25% x 2025 WV: 25% x 2025*† NJ: 22.5% x 2021
CA: 33% x 2020 UT: 20% by 2025* KS: 20% x 2020 VA: 15% x 2025* MD: 20% x 2022
MO: 15% x 2021
DC DE: 20% x 2019*
AZ: 15% x 2025
NC: 12.5% x 2021 (IOUs) DC: 20% x 2020
0
10% x 2018 (co-ops & munis)
NM: 20% x 2020 (IOUs)
10% x 2020 (co-ops)
TX: 5,880 MW x 2015
HI: 40% x 2030
State renewable portfolio standard Minimum solar or customer-sited requirement 29 states +
State renewable portfolio goal
6
Solar water heating eligible *
†
Extra credit for solar or customer-sited renewables
Includes non-renewable alternative resources
DC have an RPS
(6 states have goals)
4 / 66

5. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Issues: Dynamics Coupled generators and consumers
4600 MW
4400
4200
4000
Mass-spring model
0 20 40 60 80 Time in Seconds
Instability in the North-West U.S.
5 / 66

6. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Issues: Complexity
Interconnected
network of
ENTSO-E
European Network of Transmission
System Operators for Electricity
6 / 66

7. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Issues: Market Power Risk to consumers
California, July 2000
Spinning reserve prices PX prices $/MWh
Enron traders openly discussed manipulating
California’s power market during profanity-laced 70
250
telephone conversations in which they merrily
60
gloated about ripping o “those poor 200
grandmothers” during the state’s energy crunch 50
in 2000-01 ... [AP by Kristen Hays, 06/03/04] 150 40
30
100
NRON As Je rey Skilling, the rst but sadly not the last guy
to package up a big sample of nothing and sell it to a greedy 20
public, Samuel West oozes self-belief; his boss, Ken Lay (Tim 50
Piggott-Smith) and stooge, Andy Fastow (Tom Goodman-Hill) 10
end up smeared with it. But Lucy Prebble's play is rather more
than a simple tale ...
simple tale -Time Out, London 2010 0
Weds Thurs Fri Sat Sun Mon Tues Weds
“Ripping off those poor grandmothers”
7 / 66

8. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Issues: Risks to suppliers
Balancing Authority Load (MW)
7000 January 16-22, 2010
6000
5000
Wind generation (MW)
1000
0
Sun Mon Tues Weds Thurs Fri Sat
Who will buy my power if the wind is blowing?
8 / 66

9. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Issues: Friction
1,400 Power flows according to
1,200 Tasmania
Kirchoff’s Laws, and subject
1,000
1,000
Demand
800
600
0 to constraints ...
400
Prices
Price (Aus $/MWh)
Volume (MW)
200 - 1,000
0 - 1,500
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
09:00
10:00
11:00
12:00
13:00
14:00
15:00
16:00
17:00
18:00
19:00
20:00
21:00
22:00
23:00
10,000 24:00 19,000
9,000
Victoria Demand
8,000
7,000
6,000
10,000
5,000
Prices
4,000
3,000
2,000
1,000 1,000
0 - 1,000
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
09:00
10:00
11:00
12:00
13:00
14:00
15:00
16:00
17:00
18:00
19:00
20:00
21:00
22:00
23:00
24:00
Australia - January 16, 2007
9 / 66

10. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Issues: Friction
1,400 Power flows according to
1,200 Tasmania
Kirchoff’s Laws, and subject
1,000
1,000
Demand
800
600
0 to constraints ...
400
Prices ... Mechanical and
Price (Aus $/MWh)
Volume (MW)
200 - 1,000
0 - 1,500 thermal constraints, ramp
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
09:00
10:00
11:00
12:00
13:00
14:00
15:00
16:00
17:00
18:00
19:00
20:00
21:00
22:00
23:00
10,000 24:00 19,000 constraints, security
9,000
Victoria
8,000
7,000
Demand
constraints, ...
6,000
10,000
5,000
Prices
4,000
3,000
2,000
1,000 1,000
0 - 1,000
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
09:00
10:00
11:00
12:00
13:00
14:00
15:00
16:00
17:00
18:00
19:00
20:00
21:00
22:00
23:00
24:00
Australia - January 16, 2007
9 / 66

11. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Issues: Friction
1,400 Power flows according to
1,200 Tasmania
Kirchoff’s Laws, and subject
1,000
1,000
Demand
800
600
0 to constraints ...
400
Prices ... Mechanical and
Price (Aus $/MWh)
Volume (MW)
200 - 1,000
0 - 1,500 thermal constraints, ramp
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
09:00
10:00
11:00
12:00
13:00
14:00
15:00
16:00
17:00
18:00
19:00
20:00
21:00
22:00
23:00
10,000 24:00 19,000 constraints, security
9,000
Victoria
8,000
7,000
Demand
constraints, ...
6,000
10,000
5,000
Prices
4,000
3,000
2,000
1,000 1,000
0 - 1,000 Market and physical system
00:00
01:00
02:00
03:00
04:00
05:00
06:00
07:00
08:00
09:00
10:00
11:00
12:00
13:00
14:00
15:00
16:00
17:00
18:00
19:00
20:00
21:00
22:00
23:00
24:00
Australia - January 16, 2007
are tightly coupled
9 / 66

12. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Issues: Uncertainty
32
Available Resources Forecast (GW)
30
28
California ISO
www.caiso.com
26
24
22 Day Ahead Demand Forecast
20
Actual Demand (GW)
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Hour
CAISO Mission:
• Operate the grid reliably and efficiently
• Provide fair and open transmission access
• Promote environmental stewardship
• Facilitate effective markets ...
10 / 66

13. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Control solution: Model based
Simplest model that leads to useful policy
Listed in order of complexity:
1. Generation is modeled via overall ramp-rate:
G(t1 )−G(t0 )
t1 −t0 ≤ ζ +, 0 ≤ t0 < t1 < ∞
We may also impose lower bounds.
11 / 66

14. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Control solution: Model based
Simplest model that leads to useful policy
Listed in order of complexity:
1. Generation is modeled via overall ramp-rate:
G(t1 )−G(t0 )
t1 −t0 ≤ ζ +, 0 ≤ t0 < t1 < ∞
We may also impose lower bounds.
2. Distinguish primary and ancillary service:
Gp (t1 )−Gp (t0 ) Ga (t1 )−Ga (t0 )
t1 −t0 ≤ ζ p+ , t1 −t0 ≤ ζ a+
11 / 66

15. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Control solution: Model based
Simplest model that leads to useful policy
Listed in order of complexity:
1. Generation is modeled via overall ramp-rate:
G(t1 )−G(t0 )
t1 −t0 ≤ ζ +, 0 ≤ t0 < t1 < ∞
We may also impose lower bounds.
2. Distinguish primary and ancillary service:
Gp (t1 )−Gp (t0 ) Ga (t1 )−Ga (t0 )
t1 −t0 ≤ ζ p+ , t1 −t0 ≤ ζ a+
3. Introduce transmission constraints:
Power flow on transmission lines = linear function of nodal
power values
F (t) = ∆Z(t), ∆ = distribution factor matrix
11 / 66

16. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Control solution: Model based
Simplest model that leads to useful policy
Listed in order of complexity:
1. Generation is modeled via overall ramp-rate:
G(t1 )−G(t0 )
t1 −t0 ≤ ζ +, 0 ≤ t0 < t1 < ∞
We may also impose lower bounds.
2. Distinguish primary and ancillary service:
Gp (t1 )−Gp (t0 ) Ga (t1 )−Ga (t0 )
t1 −t0 ≤ ζ p+ , t1 −t0 ≤ ζ a+
3. Introduce transmission constraints:
Power flow on transmission lines = linear function of nodal
power values
F (t) = ∆Z(t), ∆ = distribution factor matrix
F (t) ∈ F := {x ∈ Rℓt : −f + ≤ x ≤ f + }
11 / 66

17. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Market Analysis: Perfect competition
A beautiful world...
In this lecture,
Dynamic market equilibria
under the most ideal circumstances:
12 / 66

18. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Market Analysis: Perfect competition
A beautiful world...
In this lecture,
Dynamic market equilibria
under the most ideal circumstances:
Price manipulation is excluded
12 / 66

19. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Market Analysis: Perfect competition
A beautiful world...
In this lecture,
Dynamic market equilibria
under the most ideal circumstances:
Price manipulation is excluded
No “ENRON games” – no “market power”
12 / 66

20. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Market Analysis: Perfect competition
A beautiful world...
In this lecture,
Dynamic market equilibria
under the most ideal circumstances:
Price manipulation is excluded
No “ENRON games” – no “market power”
Externalities are disregarded
No government mandates
12 / 66

21. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Market Analysis: Perfect competition
A beautiful world...
In this lecture,
Dynamic market equilibria
under the most ideal circumstances:
Price manipulation is excluded
No “ENRON games” – no “market power”
Externalities are disregarded
No government mandates
Consumers own available wind generation resources
12 / 66

22. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
32
Available Resources Forecast (GW)
30
28
California ISO
www.caiso.com
26
24
22 Day Ahead Demand Forecast
20
Actual Demand (GW)
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Hour
Reliability by Design
13 / 66

23. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Reliability: How to achieve the CAISO Mission?
32
Available Resources Forecast (GW)
30
28
California ISO
www.caiso.com
26
24
22 Day Ahead Demand Forecast
20
Actual Demand (GW)
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Hour
Begin with,
• Operate the grid reliably and efficiently
14 / 66

24. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Reliability: How to achieve the CAISO Mission?
32
Available Resources Forecast (GW)
30
28
California ISO
www.caiso.com
26
24
22 Day Ahead Demand Forecast
20
Actual Demand (GW)
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Hour
Begin with,
• Operate the grid reliably and efficiently
These can wait:
• Provide fair and open transmission access
• Promote environmental stewardship
• Facilitate effective markets ...
14 / 66

25. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
A diffusion model
Dynamic model for reserves
R(t) = Available power − Demand = G(t) − D(t)
15 / 66

26. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
A diffusion model
Dynamic model for reserves
R(t) = Available power − Demand = G(t) − D(t)
D(t) = Actual demand − Forecast demand
G(t): Deviation in on-line capacity from day-ahead market
15 / 66

27. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
A diffusion model
Dynamic model for reserves
R(t) = Available power − Demand = G(t) − D(t)
D(t) = Actual demand − Forecast demand
G(t): Deviation in on-line capacity from day-ahead market
Volatility
Deviation in demand D is modeled as a driftless Brownian motion
with instantaneous variance σ 2 (imposed for computation)
15 / 66

28. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
A diffusion model
Dynamic model for reserves
R(t) = Available power − Demand = G(t) − D(t)
D(t) = Actual demand − Forecast demand
G(t): Deviation in on-line capacity from day-ahead market
Volatility
Deviation in demand D is modeled as a driftless Brownian motion
with instantaneous variance σ 2 (imposed for computation)
Friction
Generation cannot increase instantaneously:
G(t′ ) − G(t)
≤ζ, for all t ≥ 0, and t′ > t.
t′ − t
15 / 66

29. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Social planner’s objective
Discounted welfare: For given initial generation and demand,
K(g, d) = E e−γt WS (t) + WD (t) + WE (t) dt .
Welfare functions:
Supplier, WS (t): payments - cost
16 / 66

30. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Social planner’s objective
Discounted welfare: For given initial generation and demand,
K(g, d) = E e−γt WS (t) + WD (t) + WE (t) dt .
Welfare functions:
Supplier, WS (t): payments - cost
Consumer, WD (t): utility - payments - cost of b/o
16 / 66

31. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Social planner’s objective
Discounted welfare: For given initial generation and demand,
K(g, d) = E e−γt WS (t) + WD (t) + WE (t) dt .
Welfare functions:
Supplier, WS (t): payments - cost
Consumer, WD (t): utility - payments - cost of b/o
External, WE (t): environmental, political
16 / 66

32. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Social planner’s objective
Discounted welfare: For given initial generation and demand,
K(g, d) = E e−γt WS (t) + WD (t) + WE (t) dt .
Welfare functions:
Supplier, WS (t): payments - cost
Consumer, WD (t): utility - payments - cost of b/o
External, WE (t): environmental, political
Goal: Maximize the discounted social welfare
16 / 66

33. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Social planner’s objective
Piecewise linear welfare
Linear costs and values
WS (t) := P (t)G(t) − cG(t)
WD (t) := v min(D(t), G(t))
− cbo max(0, −R(t)) − P (t)G(t)
17 / 66

34. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Social planner’s objective
Piecewise linear welfare
Linear costs and values
WS (t) := P (t)G(t) − cG(t)
WD (t) := v min(D(t), G(t))
− cbo max(0, −R(t)) − P (t)G(t)
=⇒ discounted welfare is a function of reserves
K(g, d) = (v − c)d + E e−γt w(R(t)) dt
w(R(t)) = v min(0, R(t)) + cbo min(R(t), 0) − c(R(t))
17 / 66

35. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Social planner’s objective
Piecewise linear welfare
Discounted welfare is a function of reserves
K(g, d) = (v − c)d + E e−γt w(R(t)) dt
w(R(t)) = v min(0, R(t)) + cbo min(R(t), 0) − c(R(t))
Threshold policy
1 cbo + v
Optimal threshold: r∗ =
¯ log
θ+ c
1
where θ+ > 0 solves, 2 σ 2 θ 2 − ζθ − γ = 0
18 / 66

36. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Social planner’s objective
Piecewise linear welfare
Threshold policy
1 cbo + v
Optimal threshold: r∗ =
¯ log
θ+ c
1
where θ+ > 0 solves, 2 σ 2 θ 2 − ζθ − γ = 0
1 2
1σ
Average cost case: → 2 ζ , as γ ↓ 0.
θ+
19 / 66

37. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Social planner’s objective
Reserve Dynamics
Optimal reserves = reflected Brownian motion:
Reserves
Normalized demand
r∗
0
1 cbo +v
Optimal threshold r ∗ =
¯ θ+ log c ,
1 2 2
where θ+ > 0 solves, 2 σ θ − ζθ − γ =0
20 / 66

38. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Primary and Ancillary Service
Multiple thresholds
R(t) = Available power − Demand = Gp (t) + Ga (t) − D(t)
21 / 66

39. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Primary and Ancillary Service
Multiple thresholds
R(t) = Available power − Demand = Gp (t) + Ga (t) − D(t)
Social Planner’s Problem solved using an affine policy:
21 / 66

40. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Primary and Ancillary Service
Multiple thresholds
R(t) = Available power − Demand = Gp (t) + Ga (t) − D(t)
Social Planner’s Problem solved using an affine policy:
The cheaper resource Gp (t) ramps up when R(t) < r p
¯
21 / 66

41. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Primary and Ancillary Service
Multiple thresholds
R(t) = Available power − Demand = Gp (t) + Ga (t) − D(t)
Social Planner’s Problem solved using an affine policy:
The cheaper resource Gp (t) ramps up when R(t) < r p
¯
Ancillary service Ga (t) ramps up when R(t) < r a
¯
21 / 66

42. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Primary and Ancillary Service
Multiple thresholds
R(t) = Available power − Demand = Gp (t) + Ga (t) − D(t)
Social Planner’s Problem solved using an affine policy:
The cheaper resource Gp (t) ramps up when R(t) < r p
¯
Ancillary service Ga (t) ramps up when R(t) < r a
¯
Ga
(R(t), Ga (t)) constrained
by affine policy
R
r∗ r∗
a p
0
Trajectory of the two-dimensional model under an affine policy
21 / 66

43. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Primary and Ancillary Service
Multiple thresholds
R(t) = Available power − Demand = Gp (t) + Ga (t) − D(t)
Social Planner’s Problem solved using an affine policy:
The cheaper resource Gp (t) ramps up when R(t) < r p
¯
Ancillary service Ga (t) ramps up when R(t) < r a
¯
Ga 2
σD = 6 2
σD = 18 2
σD = 24
40 40 40
(R(t), G (t)) constrained
a
30 30 30
by affine policy
20 20 20
10 10 10
R
- 20 - 10 0 10 20 30 - 20 - 10 0 10 20 30 - 20 - 10 0 10 20 30
p p∗
CRW model q CBM model q
qa q a∗
Solidarity between optimal policies for non-Brownian demand
22 / 66

44. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Reliability by Design in Networks
Three-Node Example
E1 Gp
1
D1 Dallas
D2 Austin
Line 1 Line 2 D3 Houston
Ga
2 Ga
3
Line 3
E2 E3
F (t) = ∆Z(t)
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45. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Reliability by Design in Networks
Three-Node Example
E1 Gp
1
D1 Dallas
D2 Austin
Line 1 Line 2 D3 Houston
Ga
2 Ga
3
Line 3
E2 E3
F (t) = ∆Z(t)
Power flow on transmission lines F = (F1 , F2 , F3 )T
Nodal power Z = (Gp − E1 , Ga − E2 , Ga − E3 )T
1 2 3
23 / 66

46. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Reliability by Design in Networks
Three-Node Example
Power flow on transmission lines F = (F1 , F2 , F3 )T
Nodal power Z = (Gp − E1 , Ga − E2 , Ga − E3 )T
1 2 3
Distribution factor matrix
 
1 −1 0
1
∆= 3 2 1 0
−1 −2 0
24 / 66

47. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Reliability by Design in Networks
Three-Node Example
Power flow on transmission lines F = (F1 , F2 , F3 )T
Nodal power Z = (Gp − E1 , Ga − E2 , Ga − E3 )T
1 2 3
Distribution factor matrix
 
1 −1 0
1
∆= 3 2 1 0
−1 −2 0
Constraints: F (t) ∈ F := {x ∈ Rℓt : −f + ≤ x ≤ f + }
24 / 66

48. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Reliability by Design in Networks
Extension to general topology
Primary and ancillary capacity processes at each node are
subject to ramp constraints
Deviation in demand is a multi-dimensional stochastic process
– Brownian motion for computation.
Cost in social planner’s problem,
cp gi + ca gi + cbo (ei − di )−
i
p
i
a
i
i
25 / 66

49. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Reliability by Design in Networks
Extension to general topology
Primary and ancillary capacity processes at each node are
subject to ramp constraints
Deviation in demand is a multi-dimensional stochastic process
– Brownian motion for computation.
Cost in social planner’s problem,
cp gi + ca gi + cbo (ei − di )−
i
p
i
a
i
i
Optimization problem is now intractable!
25 / 66

50. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Reliability by Design in Networks
Relaxations
Consider aggregate reserves and ancillary generalization:
RA (t) = Ri (t) , Ga (t) =
A Ga (t).
i
26 / 66

51. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Reliability by Design in Networks
Relaxations
Consider aggregate reserves and ancillary generalization:
RA (t) = Ri (t) , Ga (t) =
A Ga (t).
i
For given
a
Aggregate state xA = (rA , gA )T ∈ XA := R × R+
Demand vector d ∈ R ℓn
Effective cost: Cost of cheapest state vector consistent with xA :
26 / 66

52. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Reliability by Design in Networks
Relaxations
Consider aggregate reserves and ancillary generalization:
RA (t) = Ri (t) , Ga (t) =
A Ga (t).
i
For given
a
Aggregate state xA = (rA , gA )T ∈ XA := R × R+
Demand vector d ∈ R ℓn
Effective cost: Cost of cheapest state vector consistent with xA :
min (cp gi + ca gi + cbo ri− )
i
p
i
a
i
i
rA = a a
subject to i (ei − di ) gA = i gi
p a
0 = i (gi + gi − ei ) 0 = e−d−q
f = ∆p ∈ F, e, q, g ∈ R , g ∈ Rℓn
p
+
ℓn a
26 / 66

53. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Reliability by Design in Networks
Relaxations and Effective Cost
Ga
A Level sets of
effective cost
40
c(xA , d)
30
20
10
RA
0
- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 50
r a
r p
27 / 66

54. Issues and Approaches
Reliability by Design
Dynamic Markets and their Equilibria
Who Commands the Wind?
Research Frontiers
References
Reliability by Design in Networks
Affine Policy for Relaxation
Ga
A Level sets of
effective cost
40
c(xA , d)
30
Monotone region
20
Constraint region
under affine policy
10
RA
0
- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 50
r a
r p
Affine policy is an affine enlargement of “monotone region” in
which Ga ramps at maximum rate.
A
28 / 66