Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Self-Commitment of Combined Cycle Units
under Electricity Price Uncertainty
University of Liège
Anthony Papavasiliou
Department of Mathematical Engineering
Center for Operations Research and Econometrics
Catholic University of Louvain
September 20th, 2013
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Outline
1
Introduction
2
A Motivating Example
3
Model
4
Solution Methodology
5
Results
6
Conclusions
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Day-Ahead Unit Commitment versus Self-Commitment
Context : this work was motivated by collaboration with
Pacific Gas and Electric, one of the major California utilities
1
Day-ahead unit commitment : bid operating and cost data
to day-ahead market, system operator commits units
Advantage : Hedging real-time uncertainty through
day-ahead forward market
Disadvantage : inefficient scheduling of units in day-ahead
unit commitment (deterministic model, short horizon)
2
Self-commitment : Generators fix day-ahead unit
commitment schedule
Advantage : Generators can commit their units more
efficiently
Disadvantage : Generators assume full risk of real-time
price uncertainty
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Combined Cycle (CC) Units
Multiple combustion turbines (CTs), waste heat can be
used for fueling steam turbines (STs)
CC units increasingly important in balancing due to the
more degrees of control, modularity, flexibility in fuel
Renewable energy integration is increasing the need for
balancing resources, CC units are highly suitable
(Ott, 2010) : increased need for accurate representation of
CC units in operations
Modeling approaches result in ‘heavy’ mixed integer linear
program (Liu, 2009)
Bottom-up modeling of components (CTs and STs) : higher
fidelity, more complexity
Reduced modeling of modes (CTs and STs that are on) :
lower fidelity, less complexity
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Literature Review
(Cohen and Ostrowski, 1996), (Lu and Shahidehpour,
2004), (Lu and Shahidehpour, 2005), (Li and
Shadidehpour, 2005), (Simoglou et al., 2010) : focus on
combined cycle units, no representation of uncertainty
(Cerisola et al., 2009), (Tseng and Zhu, 2009), (Garces
and Conejo, 2010) : focus on uncertainty in
self-commitment, no representation of combined cycle unit
operations
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Objective
Research objective : investigate the influence of risk
aversion and price volatility on decision of CC units to
self-commit versus bidding in the day-ahead market
Features of the model
Modeling : simultaneous representation of uncertainty and
risk aversion in the self-commitment of CC units
Analysis : two-stage decision framework for comparing
self-commitment and day-ahead market
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Setup
~
Separate ownership of conventional and renewable unit
Conventional unit owner and system operator share same
renewable supply forecast
Conventional unit owner is risk neutral
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Market Outcome
To commit in day-ahead (DA) market, u ? , operator solves :
max 100d − 100u − 20p
p + W̄ ≥ d
0 ≤ p ≤ 3u
u ∈ {0, 1}
To clear in real-time (RT) market, operator solves :
max 100d − 20p
p + Wi ≥ d, (λi )
0 ≤ p ≤ 3u ?
λ1 =100$/MWh (W1 =96MW), λ2 =0$/MWh (W2 =104MW)
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Predicting the Market Outcome
Challenging to use bottom-up unit commitment and
economic dispatch models to predict market outcomes
With convex feasible sets and costs, dispatch obtained by
solving profit maximization against market clearing price
No such guarantees in markets with non-convexities,
nevertheless a proxy used in practice (with
λ̄ = 0.5 · 100 + 0.5 · 0 = 50$/MWh) :
max 50p − 20p − 100u
0 ≤ p ≤ 3u
u ∈ {0, 1}
Optimal solution reproduces DA outcome
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Self-Commitment
Conventional unit solves :
max 0.5(100p1 − 20p1 ) + 0.5(0p2 − 20p2 ) − 100u
0 ≤ p1 ≤ 3u
0 ≤ p2 ≤ 3u
u ∈ {0, 1}
Optimal solution is to commit the unit (expected revenue of
0.5 · (3 MW) · (100 $/MWh) = 150$/h, expected fuel cost of
0.5 · (3 MW) · (20$/MWh) = 30$/h, startup cost of 100$)
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Discussion
Two take-aways :
Generators can solve profit maximization against DA price
forecast as proxy of the DA unit commitment
DA market will not necessarily reproduce schedule that is
desirable for generator
The example is not realistic :
Single-period DA model without security constraints
Renewable forecast equal to average supply
Generator does not account for its influence on price
But ... goal is to illuminate difference between
self-scheduling and bidding
Utilties use time series models of RT prices, which ignores
underlying unit commitment and economic dispatch
Divergence of market from generator optimum is
demonstrated in a more realistic example
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Framework
Real-Time
Price
Scenarios
Day-Ahead
Self-Commitment
Commitment
Real-Time
Market
Dispatch
Risked
profit
RT Profit
CVaR
Real-Time
Price
Samples
Day-Ahead
Price
Forecast
Day-Ahead
Market
Commitment
Comm.
Self-commitment
vs
Market
Real-Time
Market
RT
Dispatch Profit
DA Profit
A. Papavasiliou
max
Profit
CVaR
Risked
profit
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Framework Details
Two stages of decision-making :
1
2
3
Generators commit units
Generators observe RT prices
Generators dispatch their units with fixed commitment
DA market determines secured DA profit
RT profits computed by running RT dispatch, distribution of
profits is transformed through CVaR risk criterion
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
State Diagram
Off
A1
1x1 A3 2x1 A10 3x1
A7
A5
A11
A9
A12
State diagram corresponding to CC unit with 3 CTs
Operating constraints within each state
Costs incurred for operating within each state and
transitioning between states
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Heat Rate Curve
MMBtu/
Mwh
ugxt=1
HR3
HR1
HR2
MWh
BP1
BP2
BP3
BP4
pgx1t
pgx2t
pgx3t
Non-linear heat rate curve for each state
Heat rate curve needs to be increasing, in order to obtain
convex cost function
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Conditional Value at Risk
% of
outcomes
20
10
Profit ($)
-1000
3000
8000
Conditional value at risk (CVaR) averages the a% worst
outcomes
Example : CVaR0.04 = −1000,
4
CVaR0.14 = 14
· (−1000) + 10
14 · 0 = −286
(Rockafellar and Uryasev, 2002) propose a linear program
for computing CVaR
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Structure of Self-Commitment Problem
Self-commitment problem has the following form :
1X
min c T w + ζ +
πs (Q(w, λs ) − ζ)+
a
(1)
s∈S
w ∈ W,
(2)
Second-stage cost for a given realization is given by
(P2s ) : Q(w, λs ) = min λTs z
(3)
Aw + Bz = h
(4)
z≥0
(5)
Sole source of uncertainty is the coefficient vector of the
objective function
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Convexity of Value Function
Proposition
The value function V (w, ζ) =
convex function of (w, ζ).
P
s∈S
πs (Q(w, λs ) − ζ)+ is a
Démonstration.
According to theorem 2, paragraph 3.1 of (Birge, 2010), we
have that Q(w, λs ) is a convex function of w. We get convexity
of the value function from the fact that the non-negative sum of
convex functions, Q(wt , λs ) − ζt , is convex ; the composition of
convex functions, (Q(wt , λs ) − ζt )+ , is convex ; and expectation
operator preserves convexity.
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Subgradient of Value Function
Proposition
The subgradient of V (w, ζ) at (w, ζ) is given by
X
−σsT A
∂V (w, ζ) =
πs 1s
−1
(6)
s∈S
where 1s = 1Q(w,λs )≥ζ and σs are the dual optimal multipliers of
the coupling constraints in Eq. (4).
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Proof : Case 1
Démonstration.
Show
(Q(w 0 , λs ) − ζ 0 )+ ≥ 1s [(Q(w, λs ) − ζ)+ − σsT A(w 0 − w) − (ζ 0 − ζ)]
Suppose Q(w, λs ) ≥ ζ. For any (w 0 , ζ 0 ) 6= (w, ζ) we have that
1s [(Q(w, λs ) − ζ)+ − σsT A(w 0 − w) − (ζ 0 − ζ)] =
Q(w, λs ) − σsT A(w 0 − w) − ζ 0
≤ Q(w 0 , λs ) − ζ 0
≤ (Q(w 0 , λs ) − ζ 0 )+
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Proof : Case 2
Démonstration.
If Q(w, λs ) < ζ we have
1s [(Q(w, λs ) − ζ)+ − σsT A(w 0 − w) − (ζ 0 − ζ)] = 0
≤ (Q(w 0 , λs ) − ζ 0 )+
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Benders Decomposition
Formulate relaxation of first-stage problem :
1
(P1) : min c T w + ζ + θ
a
w
θ ≥ Dl
+ dl, 1 ≤ l ≤ k
ζ
w ∈ W , θ ≥ 0, ζ ≥ ζLB
(7)
(8)
(9)
ζLB is lower bound on the value at risk, and prevents
unboundedness of (P1)
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Benders Steps
Step 0 : Set k = 1. Initialize θ̂1 = −∞, and (ŵ 1 , ζ̂ 1 ). Go to
step 2.
Step 1 : Solve (P1). Set (ŵ k , ζ̂ k ) equal to the optimal
first-stage solution. Go to step 2.
Step 2 : For all s ∈ S, solve (P2s ) using ŵ k as input. Set
σ̂sk equal to the dual optimal multipliers of the coupling
constraints in Eq. (4). Set 1ks = 1(σ̂k )T (h−Aŵ k )≥ζ̂ k . Go to step
s
3.
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Benders Steps (II)
Step 3 : Set
Dk
=
X
=
X
πs 1ks (−(σ̂sk )T A, −1)
(10)
πs 1ks (σ̂sk )T h
(11)
s∈S
d
k
s∈S
If θ̂k =
X
πs 1ks ((σ̂sk )T (h − Aŵ k ) − ζ̂ k ) then exit with
s∈S
(ŵ k , ζ̂ k ) as the optimal solution. Otherwise, set k = k + 1
and go to step 1.
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Data Sources and Simulation Setup
3 × 1 unit ; UTgx = DTgx = 4 hours ; UTg = DTg = 6 hours
Heat rate curve is shown in Table 1
48-hour horizon
Spring and Summer, Friday-Saturday and
Saturday-Sunday
Fit RT price time series model to 2007 RT electricity price
data from the San Francisco (SF) bus
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Bids versus Self-Commitment
a = 100%
75%
50%
25%
SpFriSat
S-C
M
51,500
25,159
22,474
9,273
5,347
1,495
0
938
SpSatSun
S-C
M
25,504
0
-1,274
0
0
0
0
0
SuFriSat
S-C
M
994,490
1,097,800
864,210
1,001,900
775,060
981,100
657,960
981,100
SuSatSun
S-C
M
33,200
0
18,167
0
6,624
0
0
0
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Observations : Spring Friday-Saturday
Self-commitment produces better outcome for all but the
most risk averse case (a = 25%)
Unit commitment schedules :
Hours
1-6
7-9
10 - 24
25 - 48
Self-commit
Market
0
0
301
0
301
301
301
0
Market model does not recognize the profit opportunities
that arise from high prices
In the most risk averse case, self-scheduling will shut the
unit down
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Observations : Spring Saturday-Sunday
For a = 100%, self-commitment performs better
For a = 75%, self-commitment results in negative risked
profit
For a = 50% and a = 25%, both approaches keep the unit
off for both days
If both models produce the same unit commitment
schedule, self-commitment cannot dominate bidding in the
market
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Observations : Summer
Friday-Saturday : optimal schedule for both approaches is
keeping the unit at 1 × 1
Hedge of DA market makes market preferable
Saturday-Sunday : like Spring Friday-Saturday,
self-commitment keeps unit on, market keeps unit off
For a = 25%, self-commitment also shuts the unit down
Incentive for generators to self-commit is enhanced by less
risk aversion because increased risk aversion reduces
differences in schedules
In most instances the market is outperformed by
self-commitment : this undermines the depth and very
purpose of the DA market
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Impact of Price Volatility
Increase price spread by 50%
a = 100%
75%
50%
25%
SpFriSat
S-C
M
95,776
40,754
50,280
15,947
24,130
2,706
-825
938
SpSatSun
S-C
M
66,333
0
23,936
0
0
0
0
0
SuFriSat
S-C
M
1,026,700
1,165,200
834,410
1,019,000
704,080
981,120
535,690
981,100
SuSatSun
S-C
M
59,327
0
36,538
0
18,844
0
-1,241
0
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Observations
Increased price volatility increases incentive to
self-commit, with exception of summer Friday-Saturday
Intuition : higher price volatility implies higher risk, DA
market should be more desirable
But... higher volatility increases value of explicitly
accounting for uncertainty
For summer Friday-Saturday, both approaches have same
schedule so market is better option
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Observations for Spring Friday-Saturday
10
Frequency (%)
Frequency (%)
60
40
20
0
0
50
100
Profit (1,000 $)
0
−50
150
Frequency (%)
Frequency (%)
0
100
Profit (1,000$)
200
250
10
60
40
20
0
5
0
50
100
150
Profit (1,000 $)
200
250
5
0
−50
0
100
200
Profit (1,000 $)
300
400
Spring Friday-Saturday : DAM Ref. (UL), S-C Ref. (UR),
DAM Vol. (LL), S-C Vol. (LR)
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Run Time
SpringFriSat
SpringSatSun
SummerFriSat
SummerSatSun
Seconds
Cuts
Seconds
Cuts
Seconds
Cuts
Seconds
Cuts
a = 100%
9
6
118
12
5
4
11
5
75%
56
34
149
42
15
11
38
18
50%
87
41
16
10
10
14
47
20
25%
54
34
11
18
17
11
30
14
CPLEX 12.5.0.0 on Macbook (2.4 GHz Intel Core i5, 8GB
1333 MHz DDR3)
Running times range between 5 seconds and 2.5 minutes
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Conclusions
DA markets vs Self-Commitment : Limited time horizon
and deterministic formulation of DA market model can
undermine the depth and purpose of the DA market
Impact of risk aversion : Incentive to self-commit is
enhanced by less risk aversion
Impact of price volatility : Prince volatility can increase
the incentive of self-commitment instead of mitigating it.
This is driven by the positive bias of the profit distribution
due to the dominant role of high-price periods on the profit
distribution.
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Introduction
A Motivating Example
Model
Solution Methodology
Results
Conclusions
Thank you
Questions ?
Contact : [email protected]
http ://perso.uclouvain.be/anthony.papavasiliou/public_html/
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Market Model
Self-Commitment Model
Evaluation Model
Results
Objective function
Minimize
Market cost :
X
λ̄t bt
t∈T
X
Fuel costs :
HRx,m+1 · F · pxmt
x∈X ,m∈1...M−1,t∈T
X
Variable O&M :
VOMx pxt
x∈X ,t∈T
Transition costs :
X
TCa vat
a∈A,t∈T
Fixed operating costs :
X
(OCx + F · HRx1 · BPx1 )uxt
x∈X ,t∈T
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Market Model
Self-Commitment Model
Evaluation Model
Results
Non-Linear Heat Rate Curve
Total power output in a certain state :
pxt = uxt BPx1 +
M−1
X
pxmt , x ∈ X , t ∈ T
(12)
m=1
Limits on the production of each segment :
pxmt ≤ (BPx,m+1 − BPxm )uxt ,
x ∈ X , 1 ≤ m ≤ M − 1, t ∈ T
A. Papavasiliou
(13)
Self-Commitment of CC Units under Price Uncertainty
Market Model
Self-Commitment Model
Evaluation Model
Results
Ramp Rates
Ramp rate limits for transitions from state to state and
within a state.
pxt − px,t−1 ≤ (2 − ux,t−1 − uxt )BPx1 +
(1 + ux,t−1 − uxst )Rx+ , x ∈ X , 2 ≤ t ≤ N
(14)
px,t−1 − pxst ≤ (2 − ux,t−1 − uxst )BPx1
+(1 + ux,t−1 − uxt )Rx− , x ∈ X , 2 ≤ t ≤ N
A. Papavasiliou
(15)
Self-Commitment of CC Units under Price Uncertainty
Market Model
Self-Commitment Model
Evaluation Model
Results
States and State Transitions
No more than one state transition per period
X
vat ≤ 1, t ∈ T
(16)
a∈A
Transition dynamics
X
uxt = ux,t−1 +
X
vat −
a∈A:T (a)=x
vat ,
a∈A:F (a)=x
x ∈ X, 2 ≤ t ≤ N
(17)
Exactly one state per period
X
uxt = 1, t ∈ T
(18)
x∈X
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Market Model
Self-Commitment Model
Evaluation Model
Results
Minimum Up/Down Times Per State
t
X
X
vaτ ≤ uxt , x ∈ X , UTx ≤ t ≤ N
(19)
τ =t−UTx +1 a∈A:T (a)=x
t+DT
Xx
X
vaτ ≤ 1 − uxt , 1 ≤ t ≤ N − DTx
(20)
τ =t+1 a∈A:T (a)=x
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Market Model
Self-Commitment Model
Evaluation Model
Results
Overall Minimum Up/Down Times
On/startup binary variables
X
ut =
uxt , t ∈ T
(21)
x∈X −{0 Off 0 }
X
vt =
vat , t ∈ T
(22)
a∈A:F (a)=0 Off 0
Overall minimum up/down times
t
X
vτ ≤ ut , UT ≤ t ≤ N
(23)
vτ ≤ ut , 1 ≤ t ≤ N − DT
(24)
τ =t−UT +1
t+DT
X
τ =t+1
A. Papavasiliou
(25)
Self-Commitment of CC Units under Price Uncertainty
Market Model
Self-Commitment Model
Evaluation Model
Results
Boundary Conditions
Minimum up / down time constraints per state
t
X
X
vaτ +
τ =1 a∈A:T (a)=x
N
X
N
X
X
0
vaτ
≤ uxt , x ∈ X ,
τ =N−UTx +t+1 a∈A:T (a)=x
X
0
vaτ
+
1 ≤ t ≤ UTx − 1
X
DTxX
+t−N
τ =t+1 a∈A:T (a)=x
τ =1
(26)
vaτ ≤ 1 − uxt , x ∈ X ,
a∈A:T (a)=x
N − DTx + 1 ≤ t ≤ N
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
(27)
Market Model
Self-Commitment Model
Evaluation Model
Results
Boundary Conditions (II)
Overall Minimum up / down time constraints
t
X
vτ +
τ =1
N
X
τ =t+1
N
X
vτ0
vτ0 ≤ ut , 1 ≤ t ≤ UTx − 1
(28)
τ =N−UTx +t+1
+
DTxX
+t−N
vτ ≤ 1 − ut , N − DTx + 1 ≤ t ≤ N(29)
τ =1
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Market Model
Self-Commitment Model
Evaluation Model
Results
Bounds and Integrality
vat ≤ 1, a ∈ A, t ∈ T
(30)
uxt ∈ {0, 1}, ut , vt , vat , pxt , pxmt ≥ 0, x ∈ X , a ∈ A, t ∈ T (31)
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Market Model
Self-Commitment Model
Evaluation Model
Results
Modeling CVaR
Consider risk-averse generator with RT payoff Q(w, λs ),
where w are first-stage decisions. Theorem 10 of
(Rockafellar and Uryasev, 2002) guarantees that CVaRa is
the optimal objective function value of the following
optimization :
min ζ +
ζ
1X
πs (Q(w, ξs ) − ζ)+ ,
a
(32)
s∈S
In addition, ζ is the VaR of the random payoff Q(w, λs )
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Market Model
Self-Commitment Model
Evaluation Model
Results
Self-Commitment Model
First-stage problem :
X
min ζ +
(OCx + F · HRx1 · BPx1 )uxt
x∈X ,t∈T
+
X
TCa vat +
a∈A,t∈T
1X
πs (Q(w, λs ) − ζ)+
a
(33)
s∈S
s.t.(16), (17), (18), (19), (20), (21), (22), (23),
(24), (30), (31), (26), (27), (28), (29)
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Market Model
Self-Commitment Model
Evaluation Model
Results
Self-Commitment Model (II)
Second-stage problem :
Q(w, λs ) =
X
min λst bt
t∈T
X
+
HRx,m+1 · F · pxmt
x∈X ,m∈1...M−1,t∈T
+
X
VOMx pxt
(34)
x∈X ,t∈T
s.t.(12), (13), (14), (15).
A. Papavasiliou
(35)
Self-Commitment of CC Units under Price Uncertainty
Market Model
Self-Commitment Model
Evaluation Model
Results
Market-Based Dispatch
Given DA market unit commitment, generators incur
following cost :
X
X
?
C DA = min
λ̄t bt? +
HRx,m+1 · F · pxmt
t∈T
+
X
x∈X ,m∈1...M−1,t∈T
?
VOMx pxt
x∈X ,t∈T
+
X
?
(OCx + F · HRx1 · BPx1 )uxt
x∈X ,t∈T
+
X
?
TCa vat
(36)
a∈A,t∈T
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Market Model
Self-Commitment Model
Evaluation Model
Results
Real-Time Dispatch
To compute RT profit opportunity of generators we solve :
X
X
λst bt +
HRx,m+1 · F · pxmt
CsRT = min
t∈T
+
X
x∈X ,m∈1...M−1,t∈T
VOMx pxt
x∈X ,t∈T
+
X
(OCx + F · HRx1 · BPx1 )uxt
x∈X ,t∈T
+
X
TCa vat
(37)
a∈A,t∈T
s.t.(12), (13), (14), (15)
?
?
uxt = uxt
, vat = vat
, ut = ut? , vt = vt?
A. Papavasiliou
(38)
Self-Commitment of CC Units under Price Uncertainty
Market Model
Self-Commitment Model
Evaluation Model
Results
Hedging of DA Market
Generators will respond to RT prices if they can improve
their hedged position :
CsM = min(CsRT , C DA )
A. Papavasiliou
(39)
Self-Commitment of CC Units under Price Uncertainty
Market Model
Self-Commitment Model
Evaluation Model
Results
Self-Commitment-Based Dispatch
Cost of self-commitment is computed as :
X
X
λst bt +
HRx,m+1 · F · pxmt
CsSC = min
t∈T
+
X
x∈X ,m∈1...M−1,t∈T
VOMx pxt
x∈X ,t∈T
+
X
(OCx + F · HRx1 · BPx1 )uxt
x∈X ,t∈T
+
X
TCa vat
(40)
a∈A,t∈T
s.t.(12), (13), (14), (15)
?
?
uxt = uxt
, vat = vat
, ut = ut? , vt = vt?
A. Papavasiliou
(41)
Self-Commitment of CC Units under Price Uncertainty
Market Model
Self-Commitment Model
Evaluation Model
Results
To Bid or not to Bid ?
1
Compute distribution of costs over set of price samples, O
2
Apply the CVaRa operator :
RM
= CVaRa (C M )
SC
SC
R
3
= CVaRa (C
(42)
)
(43)
Generator self-commits if R SC ≤ R M , bids in DA market
otherwise
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty
Market Model
Self-Commitment Model
Evaluation Model
Results
Heat Rate Curve
Seg. 1
Seg. 2
Seg. 3
Seg. 4
Seg. 5
Seg. 6
BP1×1,?
BP2×1,?
BP3×1,?
15
316
624.5
72.2
373.2
710.14
129.4
430.4
795.78
186.6
487.6
881.42
243.8
544.8
967.06
301
602
1052.7
HR1×1,?
HR2×1,?
HR3×1,?
9.04
9.25
8.87
8.55
8.75
8.39
8.88
9.08
8.71
9.21
9.42
9.04
9.54
9.76
9.36
9.87
10.09
9.68
Breakpoints are in MW and heat rates are in MMBtu/MWh
A. Papavasiliou
Self-Commitment of CC Units under Price Uncertainty