Introduction Model Day-ahead formulations Numerical examples Conclusions References
Congestion Management in a Stochastic DispatchModel for Electricity Markets
Endre Bjørndal1, Mette Bjørndal1, Kjetil Midthun2, Golbon
Zakeri3
Workshop on
Optimization and Equilibrium in Energy Economics
UCLA, January 2016
Supported by the Norwegian Research Council1NHH/SNF2SINTEF3University of Auckland 1 / 33
Introduction Model Day-ahead formulations Numerical examples Conclusions References
Congestion management and balancing
Reasons for inadequate congestion handling:
• Congestion within areas not considered (in full)
• «Loop-flow» not included in market clearing
• Inadequate representation of capacity constraints
time
Delivery hour (e.g. 08:00-08:59)
Spot marketwith congestion
avoidance
e.g.
- Nodal pricing
- Zonal pricing
- (Uniform price)
Congestion
alleviation
e.g.
- Counter-purchase
- Re-dispatch auction
- Use of reserves list
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Nordic electricity market
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Nordic electricity market
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Nord Pool Spot - 2015/01/12, 18-19
DK140.33
DK240.33
SE124.34
SE224.34
SE340.33
SE440.33
NO140.33
NO240.33
NO324.34
NO424.34
NO540.33
FI49.58
EE49.58
LV66.03
LT66.03
DEGB92.99
NLPL
BE
RU
FR CZ
BLR
UKR
LU
EIRE
FRE
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Markets associate members of PCR
Markets included in PCR - over 2800 TWh
of yearly consumption
Markets that could join next as
part of an agreed European roadmap
Towards Single European Market:
Next Steps
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Nordic market
time
Delivery hour (e.g. 08:00-08:59)
Elbas(Until 1 hour
Before delivery)
‘Feasible trade’
Elspot(12:00)
Zonal pricing
Pre-delivery markets
Markets and systems for: • Real-time balancing (Regulating power
market, and other ancillary services)
Bidding regulating power market (20:00)
Special regulation
using regulating power list
Real-time balancing
using regulating power and
ancillary services
• Congestion alleviation
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Flexibility costs and uncertainty
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Stochastic market clearing
Literature:
Wong and Fuller (2007); Bouard et al. (2005b,a); Pritchardet al. (2010); Khazaei et al. (2012); Morales et al. (2012);Khazaei et al. (2013, 2014a,b); Morales et al. (2014); Zugnoand Conejo (2013) ...
Our paper:
Energy-only stochastic market clearing as in Pritchard et al.(2010)How should the day-ahead part of the market be modeled?
Eects of day-ahead network ow and balance constraints
Compare to a sequential market clearing model with myopicclearing of the day-ahead part of the market
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Outline
1 Introduction
2 Model
3 Day-ahead formulations
4 Numerical examples
5 Conclusions
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Model
Day-ahead and real-time generation (≥ 0) and load (≤ 0)
quantities:
xi ∈ C 1
i ∀i ∈ I
Xiω ∈ C 2
i (ω, xi ) ∀i ∈ I , ω ∈ Ω
Upregulation X uiω = maxXiω − xi , 0 or downregulation
X diω = maxxi − Xiω, 0 for exible entities.
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Generator cost functions
Gen.
Price
bi
ai
xi
ai + b ixi
X i ω1
Xi ω1
d
bid
ai − aid
X i ω2
Xi ω2
u
biu
aiu − ai
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Load benet curves
Con.
Price
bi
ai
− xi
ai + b ixi
− Xi ω1
Xi ω1
u
biu
aiu − ai
− Xi ω2
Xi ω2
d
bid
ai − aid
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Objective function
Cost of real-time quantity at day-ahead parameters:
ci (Xiω) = aiXiω + 0.5bi (Xiω)2
Flexibility cost:
ci (xi ,Xiω) =(aui − ai )Xuiω + 0.5(bui − bi )(X u
iω)2
+(ai − adi )X diω + 0.5(bdi − bi )(X d
iω)2
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Stochastic market clearing model
minx,f ,X ,F
E
[∑i∈I
(ci (Xi ) + ci (xi ,Xi )
)](1a)
s.t.
xi ∈ C 1i ∀i ∈ I (1b)
Xiω ∈ C 2i (ω, xi ) ∀i ∈ I , ω ∈ Ω (1c)
τn(f ) +∑i∈I (n)
xi = 0 ∀n ∈ N [πn] (1d)
τn(Fω)− τn(f ) +∑i∈I (n)
(Xiω − xi ) = 0 ∀n ∈ N, ω ∈ Ω [pωλnω] (1e)
f ∈ U1 (1f)
Fω ∈ U2 ∀ω ∈ Ω (1g)
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Myopic market clearing model - day-ahead part
minx,f
∑i∈I
ci (xi ) (2a)
s.t.
xi ∈ C 1i ∀i ∈ I (2b)
τn(f ) +∑i∈I (n)
xi = 0 ∀n ∈ N [πn] (2c)
f ∈ U1 (2d)
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Myopic market clearing model - real-time part, scenario ω
minXω,Fω
∑i∈I
(ci (Xiω) + ci (xi ,Xiω)
)(3a)
s.t.
Xiω ∈ C 2i (ω, xi ) ∀i ∈ I (3b)
τn(Fω)− τn(f ) +∑i∈I (n)
(Xiω − xi ) = 0 ∀n ∈ N [pωλnω] (3c)
Fω ∈ U2 (3d)
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Day-ahead constraints
We assume that U2 represents the network constraints in a DC
load ow model without losses
What should U1 represent?
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Alternative day-ahead network representations
1 Nodal model, i.e., U1 = U2
2 Zonal model
No loop owAggregate ow capacities set by system operator(s)
3 Unconstrained ow, i.e., U1 = R|L|
4 Unconstrained ow and balance
min[v stochnodal , vstochzonal ] ≥ v stochbal ≥ v stochunc
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Example 1
ω = 1
130
10
20
10
3−3020
ω = 2
10
20
260
40
3−6020
P(1) = P(2) = 0.5
Real-time quantities Xω are given above
All cost parameters equal zero, except au1
= au2
= 1 and
au3
= 0.25
All lines have identical impedances
Capacity of line (2,3) is 40
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Example 1 - stochastic model
min 0.5 · 1 ·([30− x1]+ + [0− x1]+ + [0− x3]+ + [60− x3]+
)+ 0.5 · 0.25 ·
([−30− x3]+ + [−60− x3]+
)s.t.
x1 + x2 + x3 = 0
− 40 ≤ x2 − x33
≤ 40
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Example 1 - optimal day-ahead schedules
x1
−50
0
50 x2−50
0
50
x3
−100
0
100
(30,0,−30)
(30,0,−30)
(0,60,−60)(30,0,−30)
(0,60,−60)
(30,0,−30)
(0,60,−60)
xbal = (30,60,−90)
(30,0,−30)
(0,60,−60)
xbal = (30,60,−90)
(30,0,−30)
(0,60,−60)
xbal = (30,60,−90)xnodal = (30,45,−75)
(30,0,−30)
(0,60,−60)
xbal = (30,60,−90)xnodal = (30,45,−75)
(30,0,−30)
(0,60,−60)
xbal = (30,60,−90)
xunc = (30,60,−30)
xnodal = (30,45,−75)
(30,0,−30)
(0,60,−60)
xbal = (30,60,−90)
xunc = (30,60,−30)
xnodal = (30,45,−75)
(30,0,−30)
(0,60,−60)
xbal = (30,60,−90)
xunc = (30,60,−30)
xnodal = (30,45,−75)
(30,0,−30)
(0,60,−60)
xbal = (30,60,−90)
xunc = (30,60,−30)
xnodal = (30,45,−75)
(30,0,−30)
(0,60,−60)
xbal = (30,60,−90)
xunc = (30,60,−30)
xnodal = (30,45,−75)
(30,0,−30)
v stochunc = 0
v stochbal = 0.5 · (−30− (−90)) · 0.25+ 0.5 · (−60− (−90)) · 0.25 = 11.25
v stochnodal = 0.5 · (−30− (−75)) · 0.25
+ 0.5 ·[(60− 45) · 1
+ (−60− (−75)) · 0.25]= 15
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Example 2
Node 2: Nuclear + Thermal
Eur
os/M
Wh
MWh/h
000
1000
010
000
1500
0
221010
150150
Node 1: Wind (scen. 2) − Consumption
Eur
os/M
Wh
MWh/h
00
7000
0
150150
b = 0.01
1
2
32000
3000
3000 Node 3: Hydro
Eur
os/M
Wh
MWh/h
000
1500
0
150150150
b = 0.01
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Wind scenarios
p1 = 0.2
p2 = 0.5
p3 = 0.3
Wind = 0
Wind = 7000
Wind = 15000
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Cost and benet parameters
Entity Node Intercept (a) Slope (b) Flexible? Flex. cost up Flex. cost down
Wind 1 0 0 Partly - -Load 1 150 0.01 Yes bu = 30b -Nucl. 2 2 0 No - -Therm. 2 10 0 Yes au − a = 6 -Hydro 3 0 0.01 Yes bu = 10b -
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Example 2 - optimal values, stochastic model
Model Value (1000 es)
Wait-and-see 956.620
Unconstrained 952.500
Balanced 950.808
Nodal 950.542
Zonal (cap1,2,3 = 3000) 938.986
Zonal (cap1,2,3 = 5000) 950.808
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Example 2 - optimal schedules, stochastic model
Nodal model
Real-time adj.Entity Day-head Low Medium High
Wind 0 7000 13800Nucl. 1200Therm. 42 -42 438 -42Hydro 3517 83 -877 -3517Load -4758 -42 -6562 -10242
Total 0 0 0 0
Unconstrained model
Real-time adj.Entity Day-head Low Medium High
Wind 0 7000 14000Nucl. 1000Therm. 800 -800 -800Hydro 4000 -1600 -4000Load -5000 -6200 -10000
Total 800 -800 -800 -800
Power ow - nodal model
1
2000
−4758
2
758
1242
32758 3517
1
2000 (0)
−4800 (−42)
2
800 (42)
1200 (−42)
32800 (42) 3600 (83)
1
2000 (0)
−4320 (438)
2
320 (−438)
1680 (438)
32320 (−438) 2640 (−877)
1
800 (−1200)
−1200 (3558)
2
400 (1158)
1200 (−42)
3400 (−2358) 0 (−3517)
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Example 2 - cost and benet eects, stochastic model
Nodal model
Flex. costs (−c)Entity E[−c] Low Medium High E[−c − c]
Wind 0.000 0.000Nucl. -2.400 -2.400Therm. -2.400 -2.630 -3.715Hydro -30.384 -0.313 -30.447Load 987.104 987.104
Total 951.920 -0.313 -2.630 -0.000 950.542
Unconstrained model
Flex. costs (−c)E[−c] Low Medium High E[−c − c]
0.000 0.000-2.000 -2.000-4.000 -4.000-30.400 -30.400988.900 988.900
952.500 0.000 0.000 0.000 952.500
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Myopic model
Infeasibility issue, e.g. due to scheduling of non-exible nuclear
in day-ahead market
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Myopic / nodal model - eect of day-ahead wind capacity
Wind capacity
Tota
l sur
plus
−46
53.9
950.
3
0 15000
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Myopic / balanced model - eect of day-ahead wind capacity
Wind capacity
Tota
l sur
plus
−47
95.9
−28
63.9
0 13600
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
Conclusions and further research
Too restrictive constraints in the day-ahead stage of a
stochastic market clearing model may hinder exibility and
yield sub-optimal solutions
Examples of such constraints are DC load ow capacitites,
European-style zonal capacities, and even nodal balance
constraints
Further research:
PricingInvestigate relevance for deterministic (sequential) marketclearing models
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Introduction Model Day-ahead formulations Numerical examples Conclusions References
References
Bouard, F., Galiana, F. D., and Conejo, A. J. (2005a). Market-clearing with stochastic security-part i:formulation. Power Systems, IEEE Transactions on, 20(4):18181826.
Bouard, F., Galiana, F. D., and Conejo, A. J. (2005b). Market-clearing with stochastic security-part ii:Case studies. Power Systems, IEEE Transactions on, 20(4):18271835.
Khazaei, J., Zakeri, G., and Oren, S. (2012). Single and multi-settlement approaches to market clearingmechanisms under demand uncertainty.
Khazaei, J., Zakeri, G., and Oren, S. (2013). Deterministic vs. stochastic settlement approaches tomarket clearing mechanisms under demand uncertainty.
Khazaei, J., Zakeri, G., and Oren, S. S. (2014a). Market clearing mechanisms under uncertainty.
Khazaei, J., Zakeri, G., and Pritchard, G. (2014b). The eects of stochastic market clearing on the costof wind integration: a case of new zealand electricity market. Energy Systems, 5(4):657675.
Morales, J., Conejo, A., Liu, K., and Zhong, J. (2012). Pricing electricity in pools with wind producers.Power Systems, IEEE Transactions on, 27(3):13661376.
Morales, J. M., Zugno, M., Pineda, S., and Pinson, P. (2014). Electricity market clearing with improvedscheduling of stochastic production. European Journal of Operational Research, 235(3):765 774.
Pritchard, G., Zakeri, G., and Philpott, A. (2010). A single-settlement, energy-only electric power marketfor unpredictable and intermittent participants. Operations Research, 58(4-part-2):12101219.
Wong, S. and Fuller, J. D. (2007). Pricing energy and reserves using stochastic optimization in analternative electricity market. Power Systems, IEEE Transactions on, 22(2):631638.
Zugno, M. and Conejo, A. J. (2013). A robust optimization approach to energy and reserve dispatch inelectricity markets. Technical report, Technical University of Denmark.
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