minimizing forced outage risk in generator bidding

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
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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: [email protected]; [email protected] 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

schedÞ� (11)

Etloss ¼ ð1 � f Þ½Pt

schedStj � Pt

schedLMPtj� (12)


DOI: 10.1002/etep


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.


DOI: 10.1002/etep


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.


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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.


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


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


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


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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 $)


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)



VaR $22 368 $19 985 $12 819Expected revenue $44 650 000 $30 250 000 $25 300 000


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)



VaR $10 435 $14 435 $21 430Expected Revenue $3 457 000 $6 330 000 $13 423 000


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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 $)


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.


DOI: 10.1002/etep



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 t
B(P) b
id function
Etloss e
xpected loss at time t
Etprofit e
xpected profit at time t
f p
robability of failure of a generator
MCP d
ay-ahead market-clearing price ($/MW hour)
LMPtj b
us j’s locational marginal price at time t
St r
eal-time hourly spot market price at time t ($/MW hour)
Stj b
us j’s real-time hourly spot market price at time t ($/MW hour)
Pmin m
inimum generation capacity
Pmax m
aximum generation capacity
MCP90% v
alue of estimated MCP from the normal distribution with a probability of 90%
6.2. Abbreviations

MW
mega watt
FOR
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.


DOI: 10.1002/etep

minimizing forced outage risk in generator bidding

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