minimizing forced outage risk in generator bidding
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EUROPEAN TRANSACTIONS ON ELECTRICAL POWEREuro. Trans. Electr. Power 2007; 17:347–357Published online 23 April 2007 in Wiley InterScience
(www.interscience.wiley.com) DOI: 10.1002/etep.156*CoU.SyE-§Fe
Co
Minimizing forced outage risk in generator bidding
Dibyendu Das*,y and Bruce F. Wollenberg§
Department of Electrical and Computer Engineering, University of Minnesota, MN 55455, U.S.A.
SUMMARY
Competition in power markets has exposed the participating companies to physical and financial uncertainties. Arandom outage after acceptance of bids by the ISO forces a generator to buy power from the real-time hourly spotmarket and sell to the ISO at the set day-ahead market-clearing price, incurring losses if the real-time hourly spotmarket is expensive. This paper assesses the financial risk of the generators using risk profiles and VaRs. A riskminimization module is developed which derives optimum bidding strategies of the generator company such thatthe estimated total earning is maximized keeping the Value at Risk (VaR) below a tolerable limit. Copyright #2007 John Wiley & Sons, Ltd.
key words: expected revenue; risk profile; value at risk (VaR)
1. INTRODUCTION
Today, modern companies in every sector use techniques of risk management to be sure of not incurring
large losses. For generator companies, forced outages mean reduced income or even large losses. This
work measures the risk due to forced outages that a generator company faces when bidding in a
competitive market and uses risk minimization technique to minimize the outage risk.
The generators bidding [1] to supply load in a day-ahead market are exposed to risk. Reference [5]
reports on the risk that the bidder is not accepted to supply power. This paper reports on studies that
look at the financial risk associated with forced outages and how that risk varies with change in bids [6]
and location of the generators. The problem investigated is one where the generator’s bid [2] has been
accepted by the ISO, and then it experiences a forced outage and must buy expensive replacement
energy from the real-time hourly spot market to fulfill its commitment. Unit Commitment (UC) is used
as an auction method [3] by the ISO.
Four modules were created for the problem. The first module is the bidding module where a
generator derives its bid function. The second module is the auction module which is a UC run by the
ISO. The third module, the risk assessment module, derives the risk profiles of the generators and
rrespondence to: Dibyendu Das, Department of Electrical and Computer Engineering, University of Minnesota, MN 55455,.A.mails: wollenbe@ece.umn.edu; dibyendu.das@constellation.comllow IEEE.
pyright # 2007 John Wiley & Sons, Ltd.
348 D. DAS AND B. F. WOLLENBERG
calculates generator’s Value at Risk (VaR). The final module is the risk minimization module which
maximizes the profit of the generator keeping its VaR within a tolerable limit.
The paper is organized as follows. Section 2 describes the bidding module and the auction
mechanism. Generator risk assessment and risk minimization are described in Section 3. Section 4
presents the simulations and results and the paper is concluded in Section 5.
2. AUCTION AND BIDDING
This work assumes that the energy market consists of day-ahead market [4] and a real-time spot market.
In the day-ahead market, hourly market clearing is calculated for each hour of the next operating day
based on generation bids submitted to the ISO. The ISO runs a UC as an auction method to schedule the
generators. A transmission constrained UC model [5–6] is included to study the effects of congestion
on generators’ risk. Each generator submits a quadratic bid curve and its operating constraints to
the ISO.
The bidding module of a generator receives feedback of the market-clearing price and load profiles
on a daily basis along with day’s generator schedule. The bid curves [7] are assumed to be continuous
and the incremental bid function of the form
IBiðPÞ ¼ aiPþ bi (1)
where IBi is the incremental bid of generator i and ai and bi are the bidding coefficients. The bidding
coefficients have the following relationship with the marginal cost coefficients:
mi ¼bi
bic
¼ ai
aic
(2)
where aic and bic are the coefficients from generator’s actual cost function. The actual cost function is of
the form:
CiðPÞ ¼1
2aicP
2 þ bicP (3)
The bidding parameter mi represents the markup above or below the marginal cost that a generator i
decides to set its marginal bid. The bid function, Bi(P), derived from the incremental bid is of the form:
BiðPÞ ¼1
2aiP
2 þ biP (4)
Another bid function assumed here derives the function coefficients from normal distribution of
historical market-clearing price. Bids [8] can also be adjusted in response to the load profile or the
previous day-ahead market-clearing price.
2.1. Low bid curve
The bidding parameter mi is set to 1. The generator’s bid to the ISO is:
BiðPÞ ¼1
2aicP
2 þ bicP (5)
Copyright # 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2007; 17:347–357
DOI: 10.1002/etep
MINIMIZING FORCED OUTAGE RISK 349
2.2. High bid curve
The bidding parameter mi is set to a higher value (mi¼ 2) in this case. The generator’s bid to the ISO is:
BiðPÞ ¼ aicP2 þ 2bicP (6)
2.3. Normalized bid curve
The day-ahead market-clearing prices are assumed to follow a normal distribution, then the generators
can estimate market-clearing price for each hour of the day from the normal distribution to derive the
bid function coefficients. For example:
IBiðPÞ P¼Pmin¼ aiPþ bi ¼ MCP90%j (7)
IBiðPÞ P¼Pmax¼ aiPþ bi ¼ MCP95%j (8)
Equations (7) and (8) can be solved for ai and bi of Equation (4).
3. GENERATOR RISK
Random outages of the generators and the transmission lines can occur after the ISO has scheduled the
generators. These outages lead to the risk of revenue losses [9]. The risk module, described here,
simulates this scenario by generating random outages of the generators and accumulating data for
profit-loss plot of each generator.
3.1. Calculation of profit and loss
Acceptance of generator to supply power does not insure its profit. A random outage forces a scheduled
generator to buy power from a real-time hourly spot market. The generator is still paid by the ISO at the
day-ahead hourly market-clearing price. Thus it incurs a loss when the real-time hourly spot market
price is higher than the day-ahead hourly market-clearing price. For a random failure of a generator an
expected profit/loss for time t can be defined as:
Etprofit ¼ ð1 � f Þ½Pt
schedMCPt � CðPtschedÞ� (9)
Etloss ¼ f ½Pt
schedSt � Pt
schedMCPt� (10)
where St is the real-time hourly spot market price at hour t.
In case of binding transmission constraints there is no single market-clearing price. Instead the bus
marginal prices or the locational marginal prices (LMPs) become the market-clearing prices for the
constrained buses. It is assumed that each constrained bus has a separate real-time hourly spot market
(Stj is the real-time hourly spot market price). The expected profit/loss of a generator at bus j for time t
becomes:
Etprofit ¼ ð1 � f Þ½Pt
schedLMPtj � CðPt
sched� (11)
Etloss ¼ ð1 � f Þ½Pt
schedStj � Pt
schedLMPtj� (12)
Copyright # 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2007; 17:347–357
DOI: 10.1002/etep
350 D. DAS AND B. F. WOLLENBERG
3.2. Real-time spot market price
Real-time hourly spot market price can vary randomly from low values during the off-peak periods to
very large values during peak loads. The spot market price is assumed to be a few times higher than the
bus marginal price in most cases but can reach very high value occasionally and can even be lower than
the bus marginal price during off-peak periods. The real-time hourly spot market price is modeled
using:
Stj ¼ LMPtj½1 þ X� (13)
where X is a random variable over (0,1). This random variable is generated as follows:
Let Y1 be a standard Normal random variable and Y2 be a random variable following the standard
Cauchy distribution [10]. Let M¼ (M1,M2)T be a bivariate random variable that takes the value (0, 1)T
with probability 0.9 and the value (1, 0)T with probability 0.1. Let Y¼M1Y1þM2Y2 and let
X ¼ Yj j8t 2 ð5; 20Þ andX ¼ Y8t 2 ð1; 4Þ [ ð21; 24Þ
This ensures that the real-time hourly spot market price is greater than the day-ahead bus marginal
price between the fifth and the twentieth hour but can be lower between the first and the fourth hour and
between the twenty first and the twenty fourth hours, which are assumed to be off-peak periods. Few
random spikes in hourly spot market prices are accounted by Cauchy distribution.
3.3. Risk minimization
A cumulative profit/loss distribution is generated for each generator using risk assessment process, as
shown in Figure 1. The simulation is run for each hour over 1 year. The cumulative distribution is the
risk profile (risk profile is defined as the distribution pattern of probabilities of certain outcomes) which
will give us the probabilities of profit and loss. An important aspect of risk profile is VaR. VaR [11–12]
is the maximum expected loss over a given period of time at a given confidence level. In this experiment
VaR is a measure of losses due to different factors like outages and bid functions.
Risk assessment process calculates VaR and expected revenue of the generators for different
scenarios like change in bidding functions and transmission capacity.
VarRiskProfiles
AuctionModule
BiddingModule
Generator Risk Module
SpotMarket
Bi(P)
GenScheduleand MCP
Pi
ForcastedSpot Price
MCP
Figure 1. Risk assessment process.
Copyright # 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2007; 17:347–357
DOI: 10.1002/etep
MINIMIZING FORCED OUTAGE RISK 351
Risk minimization process gives an optimized mixed bidding strategy using expected revenues and
VaRs for different bidding functions. The optimization is as follows:
maxPNs
i¼1
Ripi
such that
XNs
i¼1
Vipi � Vtol (14)
0 � pi � 1 (15)
where Ri is the expected revenue for ith bidding function, pi is the fraction of time the generator should
adopt bidding function i, Ns is the number of bidding functions, Vi is the VaR calculated for bidding
function i, and Vtol is the maximum VaR the generator is willing to accept. The optimized mixed bidding
strategy maximizes the expected revenue of the generator while reducing the risk to a tolerable limit.
4. SIMULATIONS
The simulation setup consists of two buses as shown in Figure 2. Each bus consists of five generators
and a load. Most of the generators at Bus 1 are cheaper than the generators at Bus 2.
Two different cases are investigated to see the effects of generation capacity, transmission line
capacity, and generator locations on the risk profiles of the generators. In all the cases, it is assumed that
each generating company has one generator.
Ayearly load profile is created for the simulation with seasonal, weekly, and daily variations. Load 2
is higher than Load 1 most of the time. The same load profile is used for different cases.
4.1. Case 1: different bid function—same transmission capacity
In this experiment, Generator 1 bids Low Bid function in the first scenario, High Bid function in the
second scenario, and Normalized Bid function in the third scenario keeping transmission line capacity
(400 MW) the same in all three scenarios. The forced outage rate of Generator 1 is 15%.
Load 1 Load 2
Transmission Line
Bus 1 Bus 2
10G
6G1G
5G
Figure 2. Two bus test case.
Copyright # 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2007; 17:347–357
DOI: 10.1002/etep
-20 -15 -10 -5 0 5 10 15 20 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Expected Profit/Loss of the generators (10 3 $)
Cum
ulat
ive
prob
ablit
y
Expected Profit/Loss distribution of generator 1
Gen1: Low Bid Gen1:High Bid Gen1:Normalized Bid
Figure 3. Effect of bidding functions on risk profile of generator 1.
352 D. DAS AND B. F. WOLLENBERG
As the generator increases its bid from low to high it supplies less power (Ptsched) to the day-ahead
market. It also increases the bus marginal price. According to Equations (11) and (12), profit and loss of
a generator depends on Ptsched and bus marginal price. This leads to a change in risk profile for different
bidding functions. The changes in risk profiles will be evident from the expected revenue and VaR of
the generator for different bidding functions (Table I).
Results show that the Normalized bid function is most risk averse. But as Generator 1 bids the
Normalized bid function, it is scheduled less but does not affect the bus marginal price as much as it
does with high bid. Thus its expected revenue decreases. For this reason, a generator must trade-off
between the decrease in VaR and expected revenue to find a suitable bidding strategy. The optimized
mixed bidding strategy for a tolerable VaR (1.05 times the minimum of VaRs) is found out for this case
(Table II).
The effect of bidding functions on the bus marginal prices for a fixed transmission capacity is shown
below (Figure 4).
The bus marginal price at Bus 1 is greater for higher bid of Generator 1. Higher bid of Generator 1
increases the bus marginal price of Bus 2 for low load periods, when the transmission constraint is not
binding. For high load periods, the bus marginal price at Bus 2 is determined by the bids of generators at
that bus and hence they are same for different bids of Generator 1.
Table I. Comparison of expected revenue and VaR of generator1.
Low bid High bid Normalized
VaR $25 383 $20 244 $18 044Expected revenue $16 772 000 $35 936 000 $35 416 000
Copyright # 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2007; 17:347–357
DOI: 10.1002/etep
Figure 4. Effect of bidding functions on bus marginal prices.
Table II. Optimization results for generator 1.
Bid functions Percentage of time adopted
Low bid 0%High bid 41%Normalized 59%Mixed strategy expected revenue $35 629 000Mixed strategy tolerable VaR $18 946
MINIMIZING FORCED OUTAGE RISK 353
4.2. Case 2: same bid function—different transmission capacity
Generator 1 bids High Bid function for different transmission capacities. Ptsched decreases with a
decrease in transmission capacity as Generator 1 supplies less to Bus 2 with low transmission capacity.
This affects the generator’s profit and loss in Equations (11) and (12). Generator 1’s expected revenue
and VaR decreases with decrease in transmission capacity. The effect of transmission capacity on risk
profile of Generator 1 is shown below in Figure 5.
The effect of transmission line capacity on expected revenue and VaR of the generator is shown in
Table III.
Expected revenue and VaR of a generator at the cheaper bus (Bus 1 in this case) decreases with
decreasing transmission capacity. The effect of variation of transmission capacity on the risk profile of
an expensive generator at Bus 2 is exactly the opposite. Generators at Bus 2 increase their generation to
Copyright # 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2007; 17:347–357
DOI: 10.1002/etep
-20 -15 -10 -5 0 5 10 15 20 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Expected Profit/Loss of the generators (10 3 $)
Expected Profit/Loss distribution of generator 1
TieLine: 500 MW TieLine: 300 MW TieLine: 0 MW
Cum
ulat
ive
prob
ablit
y
Figure 5. Effect of transmission capacity on risk profile of generator 1.
Table III. Comparison of expected revenue and VaR of generator 1.
Transmissioncapacity (500 MW)
Transmissioncapacity (300 MW)
Transmissioncapacity (0 MW)
VaR $22 368 $19 985 $12 819Expected revenue $44 650 000 $30 250 000 $25 300 000
354 D. DAS AND B. F. WOLLENBERG
meet the load at that bus as transmission capacity decreases. Increased generation at Bus 2 due to
reduced transmission capacity increases the expected revenue and VaR of generators at Bus 2 as shown
in Table IV. The effect of transmission capacity on risk profile of Generator 9 is shown in Figure 6.
The effect of transmission capacity on the bus marginal prices is shown below in Figure 7.
The bus marginal prices at Bus 2 increases with decrease in transmission capacity, as more and more
of the expensive generators at Bus 2 are scheduled. Conversely, the bus marginal prices at Bus 1
decreases with decrease in transmission capacity as generators are scheduled less.
Table IV. Comparison of expected revenue and VaR of generator 9.
TransmissionCapacity (500 MW)
TransmissionCapacity (300 MW)
TransmissionCapacity (0 MW)
VaR $10 435 $14 435 $21 430Expected Revenue $3 457 000 $6 330 000 $13 423 000
Copyright # 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2007; 17:347–357
DOI: 10.1002/etep
-20 -15 -10 -5 0 5 10 15 20 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Expected Profit/Loss of the generator (10 3 $)
Expected Profit/Loss distribution of generator 9
TieLine: 500 MW TieLine: 300 MW TieLine: 0 MW
Cum
ulat
ive
prob
ablit
y
Figure 6. Effect of transmission capacity on risk profile of generator 9.
Figure 7. Effect of transmission capacity on bus marginal prices.
Copyright # 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2007; 17:347–357
DOI: 10.1002/etep
MINIMIZING FORCED OUTAGE RISK 355
356 D. DAS AND B. F. WOLLENBERG
5. CONCLUSIONS AND FUTURE WORK
This paper presents the simulation of risk profiles of generators for different scenarios. Effects of
transmission capacity, bidding functions, and generator location are understood. Prediction of expected
revenue and VaR for different scenarios will help the generating companies to act accordingly and
reduce their risk to a tolerable limit under such circumstances. Knowledge of effects of transmission
capacity and generator locations on generator risk will specifically help in new investment decisions.
Future work will include outage of transmission lines and their effect on risk profiles of the
generators. Inclusion of GENCOs, owning more than one generator will be helpful in generalizing the
results further.
6. LIST OF SYMBOLS AND ABBREVIATIONS
6.1. Symbols
t ti
Copyright #
me
Ptsched p
ower (MW) scheduled to be supplied by a generator at time tB(P) b
id functionEtloss e
xpected loss at time tEtprofit e
xpected profit at time tf p
robability of failure of a generatorMCP d
ay-ahead market-clearing price ($/MW hour)LMPtj b
us j’s locational marginal price at time tSt r
eal-time hourly spot market price at time t ($/MW hour)Stj b
us j’s real-time hourly spot market price at time t ($/MW hour)Pmin m
inimum generation capacityPmax m
aximum generation capacityMCP90% v
alue of estimated MCP from the normal distribution with a probability of 90%6.2. Abbreviations
MW
mega wattFOR
forced outage rate (percent of a total time)ACKNOWLEDGEMENTS
The authors thank Snigdhansu Chatterjee of School of Statistics, University of Minnesota, for his advice andencouragement during the research reported in this paper.
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AUTHORS’ BIOGRAPHIES
Dibyendu Das received his Bachelors degree in Energy Engineering from Indian Institute of Technology,Kharagpur India in 2000. He received a Masters degree and a Ph.D. in Electrical Engineering from the Universityof Minnesota, Twin Cities Minnesota in 2002 and 2004. Dr. Das worked at LCG Consulting in Los Altos Californiafrom 2004 to 2006. He is currently working at Constellation Energy Group in Baltimore Maryland.
Bruce F. Wollenberg received his Bachelors degree in Electrical Engineering and Masters degree in ElectricPower Engineering from Rensselaer Polytechnic Institute, Troy New York in 1964 and 1966 respectively. Hereceived a Ph.D. in Systems Engineering from the University of Pennsylvania, Philadelphia Pennsylvania in 1974.Dr. Wollenberg worked at Leeds and Northrup Company in Philadelphia, Pennsylvania from 1966 to 1974, heworked at Power Technologies Incorporated in Schenectady New York from 1974 to 1984, and he worked atControl Data Corporation’s Energy Management Systems Division in Plymouth Minnesota from 1984 to 1989. In1989, he was appointed to a professorship at the University of Minnesota in Minneapolis Minnesota. His currentinterests are in the development of new deregulated electric power market configurations using mechanism design,in new algorithms to accurately allocate losses and other ancillary service-related quantities to transactions madeon a transmission system, and control systems which use small-area transmission models and network com-munications to solve large transmission system problems.
Copyright # 2007 John Wiley & Sons, Ltd. Euro. Trans. Electr. Power 2007; 17:347–357
DOI: 10.1002/etep
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