Research Article Distributionally Robust Self-Scheduling...

Research ArticleDistributionally Robust Self-Scheduling Optimization with CO

2

Emissions Constraints under Uncertainty of Prices

Minru Bai1 and Zhupei Yang2

1 College of Mathematics and Econometric Hunan University Changsha Hunan 410082 China2 China CITIC Bank Changsha Shuyuan Road Branch Changsha Hunan 410015 China

Correspondence should be addressed to Minru Bai minru-bai163com

Received 7 April 2014 Accepted 18 May 2014 Published 2 June 2014

Academic Editor Nan-Jing Huang

Copyright copy 2014 M Bai and Z Yang This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

As amajor energy-saving industry power industry has implemented energy-saving generation dispatching Apart from security andeconomy low carbon will be the most important target in power dispatch mechanisms In this paper considering a power systemwith many thermal power generators which use different petrochemical fuels (such as coal petroleum and natural gas) to produceelectricity respectively we establish a self-scheduling model based on the forecasted locational marginal prices particularly takinginto account CO

2emission constraint CO

2emission cost and unit heat value of fuels Then we propose a distributionally robust

self-scheduling optimizationmodel under uncertainty in both the distribution form andmoments of the locationalmarginal priceswhere the knowledge of the prices is solely derived from historical data We prove that the proposed robust self-scheduling modelcan be solved to any precision in polynomial time These arguments are confirmed in a practical example on the IEEE 30 bus testsystem

1 Introduction

Generation self-scheduling in a pool-based electricitymarkethas been recently studied in the power systems literature [1ndash3] The self-schedules are required in developing successfulbidding strategies and constructing hourly bidding curvesfor consideration by the independent system operator Inorder to obtain successful generation bids the generationcompanies have to self-schedule their unit by maximizingthe expected profit based on the forecasted location marginalprices and accounting for the network security constraintsFor a price-taker generation and a single time period thegeneration schedule 119909 is obtained from the following deter-ministic self-scheduling problem

max119909isin119883

120585119879119909 minus

119898

sum

119894=1

119862119894(119909119894) (1)

where119909 is an (119898times1) column vector containing the generationschedule 120585 is an (119898times1) column vector of locational marginalprices (LMPs)119883 sub 119877

119898 is the feasible regionwhich is a convex

and compact set and 119862119894(119909119894) is the generation quadratic cost

function

119862119894(119909119894) = 119886119894+ 119887119894119909119894+ 1198881198941199092

119894119894 = 1 119898 (2)

The issue of interest for this work is that since theelectricity prices are of stochastic nature the generationcompany cannot be certain about the revenue Measuringthe underlying risk due to this uncertainty is crucial notonly for assessing profitability but also for generation self-scheduling Stochastic programming can effectively describeself-scheduling problems in uncertain environments Unfor-tunately although the self-scheduling problem is a convexoptimization problem to solve it one must often resort toMonte Carlo approximations which can be computationallychallenging A more challenging difficulty that arises inpractice is the need to commit to a distribution given onlylimited information about the stochastic parameters [4]

In an effort to address these issues robust formulationsfor self-scheduling problems were proposed see [2 3] Jabr[2] considers a generation self-scheduling model based on aworst-case conditional robust profit with partial information

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 356527 7 pageshttpdxdoiorg1011552014356527

2 Journal of Applied Mathematics

on the probability distribution of prices It is assumed that thenominal distribution and a set of possible distributions weregiven Uncertainty of prices is represented by box and ellip-soidal uncertainty sets However in practice true probabilitydistribution of prices cannot be known exactlyTheir solutioncan be misleading when there is ambiguity in the choice ofa distribution for the random prices Recently Delage andYe [4] proposed a distributionally robust optimization modelthat describes uncertainty in both the distribution form andmoments (mean and covariance matrix) By deriving a newform of confidence region for the mean and the covariancematrix of a random vector it was showed how the proposeddistribution set can be well justified when addressing data-driven problems (ie problems where the knowledge of therandom parameters is solely derived from historical data)

On the other hand as a major energy-saving industrypower industry has implemented energy-saving generationdispatching Apart from security and economy low carbonwill be the most important target in power dispatch mech-anisms It becomes a common target for the global powerindustry to build a more safe economic and low-carbonpower system [5] So more attention should be paid to thereduction of CO

2emission in power system operation [5 6]

The main contribution of this paper is twofold Firstby considering a power system with many thermal powergenerators which use different petrochemical fuels (suchas coal petroleum and natural gas) to produce electricityrespectively we establish a self-scheduling model based onthe forecasted locational marginal prices particularly takinginto account CO


2emission cost

and unit heat value of fuels This problem is importantand timely as world leaders and international organizationsdiscuss the roles and responsibilities of each country andsector of economic activity in the path towards a sustainablefuture Second motivated by the work of Delage and Ye [4]we propose a distributionally robust self-scheduling opti-mization model under uncertainty in both the distributionform and moments of the locational marginal prices wherethe knowledge of the prices is solely derived from historicaldataThen we prove that the proposed robust self-schedulingmodel can be solved to any precision in polynomial timeThese arguments are confirmed in a practical example on theIEEE 30 bus test system

2 Robust Self-Scheduling Problemwith Moment Uncertainty and CO

2

Emission Constraint

21 Power Dispatch with 1198621198742Emission Constraint In this

section we describe the power scheduling problemwith CO2

emission constraintGiven is a thermal power system with119872 thermal power

generators 119894 isin 119872 = 1 119898 which use petrochemical fuelsas their fuels such as coal petroleum and natural gas Thequantity of CO

2emission of each power generator 119894 can be

represented as

119864119894= 119865119894ℎ119894 (3)

where 119865119894is the quantity of the fuel which is consumed

to produce electricity and ℎ119894(ge 0) is the amount of CO

2

emissions by per unit of fuel complete combustion Thegenerating capacity of this power generator is

119909119894= 119865119894119901119894120593119894 (4)

where 119901119894(gt 0) is the unit heat value of the fuel and 120593

119894(gt 0) is

the energy conversion efficiencyBy (3) and (4) we get the carbon characteristic function

of power generator 119894

119864119894=

ℎ119894

119901119894

1

120593119894

119909119894 (5)

The objective is themaximization of the expected profit basedon the forecasted locational marginal prices 120585 particularlytaking into account CO

2emission constraints and CO

2

emission cost The cost consists of the variable cost ofelectricity production of the thermal generators and the costof CO

2emission So the power scheduling problem with

CO2emission constraints can be modeled as the following

deterministic self-scheduling problem

max119909isin119883

119864[120585119879119909 minus

119898

sum

119894=1

119862119894(119909119894) minus

119898

sum

119894=1

120575 sdot

ℎ119894

119901119894

1

120593119894

119909119894]

st119898

sum

119894=1

ℎ119894

119901119894

1

120593119894

119909119894le 119864

(6)

where119909 is an (119898times1) column vector containing the generationschedule 120585 is an (119898 times 1) column vector of forecastedlocational marginal prices (LMPs) 119883 is the feasible region120575 is the CO

2emission price with the unit poundton 119864 is

the maximum allowable CO2emissions and 119862

119894(119909119894) is the

generator quadratic cost function

119862119894(119909119894) = 119886119894+ 119887119894119909119894+ 1198881198941199092

119894 119894 = 1 119898 (7)

where 119888119894gt 0 for 119894 = 1 119898

Letting119863119894(119909119894) equiv 120575(ℎ

119894119901119894)(1120593119894)119909119894 by (5) problem (6) can

be rewritten as follows

max119909isin119883

119864[120585119879119909 minus

119898

sum

119894=1

119862119894(119909119894) minus

119898

sum

119894=1

119863119894(119909119894)]

st119898

sum

119894=1

119863119894(119909119894)

120575

le 119864

(8)

22 Robust Self-Scheduling ProblemwithMoment Uncertaintyand 119862119874

2Emission Constraints In practice It is often the

case that one has limited information about the locationalmarginal prices 120585 driving the uncertain parameters that areinvolved in the decision making process The data samplesmay be not available or the data samples may be unstable Insuch situations it might instead be safer to rely on estimatesof the mean 120583

0and covariance matrix Σ

0of the random

vector 120585 for example using empirical estimates However webelieve that in such problems it is also rarely the case that

Journal of Applied Mathematics 3

one is entirely confident in these estimates For this reasonmotivated by Delage and Ye [4] we propose representing theuncertainty using two constraints parameterized by 120574

1ge 0

and 1205742ge 1

(E [120585] minus 1205830)119879

Σminus1

0(E [120585] minus 120583

0) le 1205741 (9)

E [(120585 minus 1205830) (120585 minus 120583

0)119879

] ⪯ 1205742Σ0 (10)

where constraint (9) assumes that the mean of price 120585 liesin an ellipsoid of size 120574

1centered at the estimate 120583

0and

constraint (10) forces the centered second-moment matrix of120585 to lie in a positive semidefinite cone defined with a matrixinequality In other words it describes how likely 120585 is tobe close to 120583

0in terms of the correlations expressed in Σ

0

Finally the parameters 1205741and 120574

2provide natural means of

quantifying onersquos confidence in 1205830and Σ

0 respectively

Denote the distributional set as

D1(F 1205830 Σ0 1205741 1205742)

= 119865 isin U | P (120585 isin F) = 1

(E [120585] minus 1205830)119879

Σminus1

0(E [120585] minus 120583

0) le 1205741

E [(120585 minus 1205830) (120585 minus 120583

0)119879

] ⪯ 1205742Σ0

(11)

where U is the set of all probability measures on themeasurable space (R119898B) withB being the Borel 120590-algebraon R119898 and F isin R119898 is any closed convex set known tocontain the support of 119865 The setD

1(F 1205830 Σ0 1205741 1205742) which

will also be referred to in shorthand notation as D1 can be

seen as a generalization of many previously proposed setsNext we will study worst-case expected results over the

choice of a distribution in the distributional set D1 This

leads to solving the distributionally robust self-schedulingoptimization with moment uncertainty of prices (DRSSO)

V = max119909isin119883

(min119865isinD1

E119865[120585119879119909 minus

119898

sum

119894=1

119862119894(119909119894) minus

119898

sum

119894=1

119863119894(119909119894)])

st119898

sum

119894=1

119863119894(119909119894)

120575119894

le 119864

(12)

where E119865[sdot] is the expectation taken with respect to the ran-

dom vector 120585 given that it follows the probability distribution119865 isin D

1 It is easy to see thatDRSSOproblem (12) is equivalent

to the following problem

minus V = min119909isin119883

(max119865isinD1

E119865[minus120585119879119909 +

119898

sum

119894=1

119862119894(119909119894) +

119898

sum

119894=1

119863119894(119909119894)])

st119898

sum

119894=1

119863119894(119909119894)

120575119894

le 119864

(13)

We start by considering the question of solving the innermaximization problem of the problem (13) that uses the setD1

Definition 1 Given any fixed 119909 isin 119883 let Φ(119909 1205741 1205742) be the

optimal value of the moment problem

max119865isinD1

E119865[minus120585119879119909 +

119898

sum

119894=1

119862119894(119909119894) +

119898

sum

119894=1

119863119894(119909119894)] (14)

Applying the duality theory and robust optimizationmethods[7ndash10] by [4 Lemma 1] we can circumvent the difficulty offinding the optimal value of the problem (14)

Lemma 2 For a fixed 119909 isin R119899 suppose that 1205741ge 0 120574

2ge 1

Σ0≻ 0 ThenΦ(119909 120574

1 1205742)must be equal to the optimal value of

the problem (15)

min119876119902119903119905

119903 + 119905

st 119905 ge (1205742Σ0+ 1205830120583119879

0) sdot 119876 + 120583

119879

0119902

+ radic1205741

10038171003817100381710038171003817Σ12

0(119902 + 2119876120583

0)

10038171003817100381710038171003817

119903 ge minus120585119879119909 +

119898

sum

119894=1

119862119894(119909119894)

+

119898

sum

119894=1

119863119894(119909119894) minus 120585119879119876120585 minus 120585

119879119902 forall120585 isin F

119876 ⪰ 0

(15)

where (119860 sdot 119861) refers to the Frobenius inner product betweenmatrices119876 isin R119898times119898 is a symmetric matrix the vector 119902 isin R119898and 119903 119905 isin R In addition ifΦ(119909 120574

1 1205742) is finite then the set of

optimal solutions to problem (15)must be nonempty

Proof Let ℎ(119909 120585) = minus120585119879119909 + sum

119898

119894=1119862119894(119909119894) + sum

119898

119894=1119863119894(119909119894) Then

ℎ(119909 120585) is 119865-integrable for all 119865 isin D1and the conclusions are

followed by [4 Lemma 1] This completes the proof

The following result shows that the DRSSO problem (13)is a tractable problem

Theorem 3 Assume that the set 119883 is convex and compactThen DRSSO problem (13) is equivalent to the following convexoptimization problem

minus V = min119909119876119902119903119905120578119910120596

119903 + 119905

st 119905 ge (1205742Σ0+ 1205830120583119879

0) sdot 119876 + 120583

119879

0119902

+ radic1205741

10038171003817100381710038171003817Σ12

0(119902 + 2119876120583

0)

10038171003817100381710038171003817

[

[

[

119876

119902

2

+

119909

2

119902119879

2

+

119909119879

2

119903 minus 120578

]

]

]

⪰ 0


119876 ⪰ 0

120578 ge

119898

sum

119894=1

(119886119894+ 119887119894119909119894) +

119898

sum

119894=1

120575119894

ℎ119894

119901119894

1

120593119894

119909119894+ 119910

119910 ge

119898

sum

119894=1

1205962

119894

120596119894= radic119888119894

119909119894

119898

sum

119894=1

ℎ119894

119901119894

1

120593119894

119909119894le 119864

119909 isin 119883

(16)

In addition the DRSSO problem (13) can be solved to anyprecision 120598 in time polynomial in log(1120598) and the size of 119909 and120585

Proof By Lemma 2 the DRSO problem (13) must be equal tothe following problem

minus V = min119909119876119902119903119905

119903 + 119905

st 119905 ge (1205742Σ0+ 1205830120583119879

0) sdot 119876 + 120583

119879

0119902

+ radic1205741

10038171003817100381710038171003817Σ12

0(119902 + 2119876120583

0)

10038171003817100381710038171003817

119903 ge minus120585119879119909 +

119898

sum

119894=1

119862119894(119909119894)

+

119898

sum

119894=1

119863119894(119909119894) minus 120585119879119876120585 minus 120585

119879119902 forall120585 isin F

119862119894(119909119894) = 119886119894+ 119887119894119909119894+ 1198881198941199092

119894 119894 = 1 119898

119863119894(119909119894) = 120575119894

ℎ119894

119901119894

1

120593119894

119909119894 119894 = 1 119898

119876 ⪰ 0

119898

sum

119894=1

ℎ119894

119901119894

1

120593119894

119909119894le 119864

119909 isin 119883

(17)

Let

120578 ge

119898

sum

119894=1

(119886119894+ 119887119894119909119894) +

119898

sum

119894=1

120575119894

ℎ119894

119901119894

1

120593119894

119909119894+ 119910

119910 ge

119898

sum

119894=1

1205962

119894

120596119894= radic119888119894

119909119894

(18)

Then 120578 ge sum119898

119894=1119862119894(119909119894)+sum119898

119894=1119863119894(119909119894) and the inequality (17) can

be replaced by

120585119879119876120585 + 120585

119879119902 + 120585119879119909 + 119903 minus 120578 ge 0 forall120585 isin F (19)

Note that the constraint in (19) can be written as the followingLMI

[

120585

1]

119879

[

[

[

119876

119902 + 119909

2

119902119879+ 119909119879

2

119903 minus 120578

]

]

]

[

120585

1] ge 0 forall120585 isin F

lArrrArr[

[

[

119876

119902

2

+

119909

2

119902119879

2

+

119909119879

2

119903 minus 120578

]

]

]

⪰ 0

(20)

So DRSSO problem (13) is equivalent to the convex optimiza-tion problem (16)

Since 119891(119909 120585) = minus120585119879119909 + sum

119898

119894=1119862119894(119909119894) + sum

119898

119894=1119863119894(119909119894) is

convex in 119909 and concave in 120585 and 119883 is convex and compactthe assumptions in [4] are satisfied So a straightforwardapplication of [4 Proposition 2] shows that DRSSO problem(13) can be solved in polynomial time This completes theproof

If119883 is a convex polyhedron then the convex optimizationproblem (16) is a semidefinite program that can be solved bySeDuMi conic optimizer

3 Numerical Example

We present our simulation results on the IEEE 30 bustest system and get the results by using the SeDuMi conicoptimizer running on an Intel Core i3-2350M (230GHz) PCwith 2GB RAM There are 6 power generations with coalas their fuel in this system Reference [11] gives the networkand load data for this system The generator data is listed inTable 1 A historical data set of 100 samples of prices vector 120585is shown in [2 Table 2] We assume that the generating unitsin Table 1 belong to the same generation company

In implementing our method the distributional set isformulated as D

1(R6 120583

0 Σ0 02 23) where 120583

0and Σ

0are

the empirical estimates of the mean and covariance matrixof prices vector 120585 shown in [2 Table 2]

We choose the parameters 1205741and 1205742based on some simple

statistical analysis of a lot of experiments We convert thecalorific value of different kinds of coals to standard unit ofcoal and let the unit heat value of fuel119901 equal 813 (kW sdoth)kgstandard unit of coal the amount of CO

2emissions by per

unit of fuel complete combustion ℎ equal 262 kgCO2kg

standard unit of coal in uniform and themaximumallowableCO2emissions 119864 equal 150 ton

If we fix 1205742= 12 and let 120574

1vary from 0 to 2 it can be

shown that the profit decreases when 1205741increases from 0 to

12 and the profit is almost invariable after 1205741gt 12 Table 2

illustrates the generation self-scheduling result for 1205741from 0

to 1 and 1205742= 12

The value of 1205741reflects the stability of the electricity price

The electricity price is more instable when the value of 1205741


Table 1 Generator data

Bus number 119909min MW 119909max MW Cost coefficients119886 poundh 119887 poundMWh 119888 poundMW2h

1 50 200 0 200 0003752 20 80 0 175 0017505 15 50 0 100 0062508 10 35 0 325 00083411 10 30 0 300 00250013 12 40 0 300 002500

Table 2 Generation self-scheduling result for 1205741from 0 to 1 (120574

2= 12)

1205741

Bus 1 MW Bus 2 MW Bus 5 MW Bus 8 MW Bus 11 MW Bus 13 MW Profit pound0 13500 3537 1645 3500 3000 3867 2295202 10336 3344 1620 3500 3000 3677 1887804 9142 3245 1608 3500 3000 3594 1741406 8287 3162 1598 3500 3000 3529 1637108 7613 3088 1590 3500 3000 3473 155411 7055 3020 1582 3500 3000 3423 14844

is higher If generations produce too much electricity theywould face more risks The decision makers would produceless electricity when the value of 120574

1is high And due to the

basic power needs the generating capacity would tend to bestable if the value of 120574

1is too high Figure 1 shows the result

intuitivelyHowever if we fix 120574

1and let 120574

2vary from 1 to 10 numerical

results show that the profit is almost invariable This impliesthat 1205742is not sensitive to the model

Now we study three kinds of fuels with different unit heatvalues The unit heat value 119901 of natural gas coal and oil isshown in [12 Table 1] And we can get ℎ the quantity ofCO2released by the unit fuel burnt fully by some simple

calculations The relevant data is listed in Table 3 From thistable we can know that the unit heat value of natural gasis higher than that of coking coal and the quantity of CO

2

released by the unit natural gas is lower than that of it Let120575 = 02 poundton 119864 = 1350 ton We assume the 6 buses all useone fuel as their power fuel the three results are shown inTable 4

The results show that using natural gas gets more eco-nomic benefits This is because the unit natural gas canproduce more quantity of heat and release fewer CO

2 And

the result of using oil is between natural gas and coking coalIn addition we can find that the quantity of CO

2emission

of coking coal and the quantity of CO2emission of oil have

reached the maximum allowable value respectivelyFor different power generators we let their efficiency

increase in turns Numerical results show that the profitsincrease as their efficiency increases

We consider the carbon pricersquos effect in the followingFigure 2 illustrates the effect of carbon price 120575 on thegeneration self-schedulingTheupper part of the figure showsthat the profit and CO

2emission decrease when the carbon

price 120575 increases with the limit of 119864 And the lower part of thefigure shows the result without the limit of 119864 When 120575 is very

Table 3 Relevant data of fuels

Power fuel ℎ119894 kg(CO2)kg 119901

119894 Kwhkg

Coking coal 304 790Natural gas 218 1081Oil 306 1161

0 05 1 15 2140150160170180190200210220230

f(x

ksi)

1205741

Figure 1 Relation between profit and 1205741

small the CO2emission is so large because there is no limit of

119864That is119864makes a difference on reducing carbon emissionsWhen 120575 increases the profit decreases So government shouldconsider the effect of carbon price when they set the carbonprice

We let the carbon price 120575 equal 02 poundton and 06 poundtonrespectively and 119864 range from 120 ton to 180 ton Figure 3illustrates this result Due to the emission restrictiondecreases the object 119891 and CO

2emission go up And they

tend to be stable along with the value of 119864 becoming largerBy the comparison between the upper and the lower part


Table 4 The result of using one kind of fuel (120575 = 02 poundton 119864 = 1350 ton)

Bus number 1 MW 2 MW 5 MW 8 MW 11 MW 13 MW Profit pound CO2 t

Natural gas 9924 3118 1547 3500 3000 3531 17412 1191Oil 6842 2536 1500 3500 3000 3110 15489 1350Coking coal 5000 2000 1500 2488 1555 1490 11384 1350

50

100

150

200

Profi

t

Profit

130

132

134

136

0 01 02 03 04 05 06 07 08120575 (E = 135)

CO2

CO2

(a)

Profi

t

406080

100120140160180200

120140160180200

0 01 02 03 04 05 06 07 08

Profit

120575 (without E)

CO2

CO2

(b)

Figure 2 Profit and CO2emission with increasing carbon price 120575

120 130 140 150 160 170 180140

145

150

155

160

Profi

t

100

120

140

160

180

CO2

E (120575 = 02)

ProfitCO2

(a)

120 130 140 150 160 170 180

Profi

t

93

94

95

96

110

120

130

140

CO2

E (120575 = 06)

ProfitCO2

(b)

Figure 3 Profit and CO2emission with increasing 119864

of Figure 3 the quality of CO2emission tends to reach an

invariable value earlier as 120575 = 06 poundton This shows that asuitable carbon tax is a good means to reduce carbon dioxideemissions

Figure 4 demonstrates the change of generation self-schedule when 119864 increases At first when 119864 increases somegeneration schedule 119909 increases But when the 119864 achieves apoint the effect of restriction disappears and the generationschedules are invariable

4 Conclusion

This paper studies worst-case profit self-schedules of price-taker generators with the constraints of CO

2emission in

pool-based electricity markets A distributionally robustself-scheduling optimization model describes uncertainty ofprices in both distribution form and moments (mean andcovariance matrix) where the knowledge of the prices issolely derived from historical data It is proved that theproposed robust self-scheduling model can be solved toany precision in polynomial time These arguments areconfirmed in a practical example on the IEEE 30 bus test

120 125 130 135 140 145 150 155 160 165 17010152025303540455055

Bus 1Bus 2Bus 5

Bus 8Bus 11Bus 13

E

x

Figure 4 Generation self-schedule 119909 with increasing 119864

system Numerical results show that parameter 1205741of mean

is sensitive to the solution and parameter 1205742of covariance


is not sensitive to the solution Through the comparisonbetween the different fuels and energy conversion efficiencieswe find that the power stations should choose the fuels withlower ℎ and higher 119901 as their power fuels And the energyconversion efficiency is also very important to the powerstations we should try to improve itThrough the comparisonbetween the carbon prices 120575 and 119864 it is showed that themaximum allowable CO

2emissions makes a difference in

reducing carbon emissions and a reasonable carbon tax is agood means to reduce carbon dioxide emissions

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported by the Hunan Provincial NaturalScience Foundation of China (no 14JJ2053)

References

[1] H Yamin S Al-Agtash and M Shahidehpour ldquoSecurity-con-strained optimal generation scheduling for GENCOsrdquo IEEETransactions on Power Systems vol 19 no 3 pp 1365ndash13722004

[2] R A Jabr ldquoGeneration self-scheduling with partial informationon the probability distribution of pricesrdquo IET GenerationTransmission amp Distribution vol 4 no 2 pp 138ndash149 2010

[3] R A Jabr ldquoRobust self-scheduling under price uncertaintyusing conditional value-at-riskrdquo IEEE Transactions on PowerSystems vol 20 no 4 pp 1852ndash1858 2005

[4] EDelage andYYe ldquoDistributionally robust optimization undermoment uncertainty with application to data-driven problemsrdquoOperations Research vol 58 no 3 pp 595ndash612 2010

[5] Q Chen C Kang Q Xia and D S Kirschen ldquoMechanism andmodelling approach to low-carbon power dispatchrdquo Automa-tion of Electric Power Systems vol 34 no 12 pp 18ndash23 2010

[6] C Li Y Liu Y Cao Y Tan C Xue and S Tang ldquoConsistencyevaluation of low-carbon generation dispatching and energy-saving generation dispatchingrdquo Proceedings of the Chinese Soci-ety of Electrical Engineering vol 31 no 31 pp 94ndash101 2011

[7] A Ben-Tal L El Ghaoui and A Nemirovski Robust Optimiza-tion Princeton University Press 2009

[8] M Sim Robust Optimization Operations Research 2004[9] F J Fabozzi P N Kolm D A Pachamanova and S M Focardi

Robust Portfolio Optimization and Management John Wiley ampSons 2007

[10] D Bertsimas D B Brown and C Caramanis ldquoTheory andapplications of robust optimizationrdquo SIAM Review vol 53 no3 pp 464ndash501 2011

[11] O Alsac and B Stott ldquoOptimal load flow with steady-statesecurityrdquo IEEE Transactions on Power Apparatus and Systemsvol 93 no 3 pp 745ndash751 1974

[12] T Hongqing ldquoResearch on energy transformation rate in coalchemical plantrdquo Chemical Engineering Design Communicationsvol 37 no 4 pp 5ndash13 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


on the probability distribution of prices It is assumed that thenominal distribution and a set of possible distributions weregiven Uncertainty of prices is represented by box and ellip-soidal uncertainty sets However in practice true probabilitydistribution of prices cannot be known exactlyTheir solutioncan be misleading when there is ambiguity in the choice ofa distribution for the random prices Recently Delage andYe [4] proposed a distributionally robust optimization modelthat describes uncertainty in both the distribution form andmoments (mean and covariance matrix) By deriving a newform of confidence region for the mean and the covariancematrix of a random vector it was showed how the proposeddistribution set can be well justified when addressing data-driven problems (ie problems where the knowledge of therandom parameters is solely derived from historical data)

On the other hand as a major energy-saving industrypower industry has implemented energy-saving generationdispatching Apart from security and economy low carbonwill be the most important target in power dispatch mech-anisms It becomes a common target for the global powerindustry to build a more safe economic and low-carbonpower system [5] So more attention should be paid to thereduction of CO

2emission in power system operation [5 6]

The main contribution of this paper is twofold Firstby considering a power system with many thermal powergenerators which use different petrochemical fuels (suchas coal petroleum and natural gas) to produce electricityrespectively we establish a self-scheduling model based onthe forecasted locational marginal prices particularly takinginto account CO


2emission cost

and unit heat value of fuels This problem is importantand timely as world leaders and international organizationsdiscuss the roles and responsibilities of each country andsector of economic activity in the path towards a sustainablefuture Second motivated by the work of Delage and Ye [4]we propose a distributionally robust self-scheduling opti-mization model under uncertainty in both the distributionform and moments of the locational marginal prices wherethe knowledge of the prices is solely derived from historicaldataThen we prove that the proposed robust self-schedulingmodel can be solved to any precision in polynomial timeThese arguments are confirmed in a practical example on theIEEE 30 bus test system

2 Robust Self-Scheduling Problemwith Moment Uncertainty and CO

2

Emission Constraint

21 Power Dispatch with 1198621198742Emission Constraint In this

section we describe the power scheduling problemwith CO2

emission constraintGiven is a thermal power system with119872 thermal power

generators 119894 isin 119872 = 1 119898 which use petrochemical fuelsas their fuels such as coal petroleum and natural gas Thequantity of CO

2emission of each power generator 119894 can be

represented as

119864119894= 119865119894ℎ119894 (3)

where 119865119894is the quantity of the fuel which is consumed

to produce electricity and ℎ119894(ge 0) is the amount of CO

2

emissions by per unit of fuel complete combustion Thegenerating capacity of this power generator is

119909119894= 119865119894119901119894120593119894 (4)

where 119901119894(gt 0) is the unit heat value of the fuel and 120593

119894(gt 0) is

the energy conversion efficiencyBy (3) and (4) we get the carbon characteristic function

of power generator 119894

119864119894=

ℎ119894

119901119894

1

120593119894

119909119894 (5)

The objective is themaximization of the expected profit basedon the forecasted locational marginal prices 120585 particularlytaking into account CO

2emission constraints and CO

2

emission cost The cost consists of the variable cost ofelectricity production of the thermal generators and the costof CO

2emission So the power scheduling problem with

CO2emission constraints can be modeled as the following

deterministic self-scheduling problem

max119909isin119883

119864[120585119879119909 minus

119898

sum

119894=1

119862119894(119909119894) minus

119898

sum

119894=1

120575 sdot

ℎ119894

119901119894

1

120593119894

119909119894]

st119898

sum

119894=1

ℎ119894

119901119894

1

120593119894

119909119894le 119864

(6)

where119909 is an (119898times1) column vector containing the generationschedule 120585 is an (119898 times 1) column vector of forecastedlocational marginal prices (LMPs) 119883 is the feasible region120575 is the CO

2emission price with the unit poundton 119864 is

the maximum allowable CO2emissions and 119862

119894(119909119894) is the

generator quadratic cost function

119862119894(119909119894) = 119886119894+ 119887119894119909119894+ 1198881198941199092

119894 119894 = 1 119898 (7)

where 119888119894gt 0 for 119894 = 1 119898

Letting119863119894(119909119894) equiv 120575(ℎ

119894119901119894)(1120593119894)119909119894 by (5) problem (6) can

be rewritten as follows

max119909isin119883

119864[120585119879119909 minus

119898

sum

119894=1

119862119894(119909119894) minus

119898

sum

119894=1

119863119894(119909119894)]

st119898

sum

119894=1

119863119894(119909119894)

120575

le 119864

(8)

22 Robust Self-Scheduling ProblemwithMoment Uncertaintyand 119862119874

2Emission Constraints In practice It is often the

case that one has limited information about the locationalmarginal prices 120585 driving the uncertain parameters that areinvolved in the decision making process The data samplesmay be not available or the data samples may be unstable Insuch situations it might instead be safer to rely on estimatesof the mean 120583

0and covariance matrix Σ

0of the random

vector 120585 for example using empirical estimates However webelieve that in such problems it is also rarely the case that



1ge 0

and 1205742ge 1

(E [120585] minus 1205830)119879

Σminus1

0(E [120585] minus 120583

0) le 1205741 (9)

E [(120585 minus 1205830) (120585 minus 120583

0)119879

] ⪯ 1205742Σ0 (10)



0and



0




0 respectively


D1(F 1205830 Σ0 1205741 1205742)

= 119865 isin U | P (120585 isin F) = 1

(E [120585] minus 1205830)119879

Σminus1

0(E [120585] minus 120583

0) le 1205741

E [(120585 minus 1205830) (120585 minus 120583

0)119879

] ⪯ 1205742Σ0

(11)


1(F 1205830 Σ0 1205741 1205742) which





V = max119909isin119883

(min119865isinD1

E119865[120585119879119909 minus

119898

sum

119894=1

119862119894(119909119894) minus

119898

sum

119894=1

119863119894(119909119894)])

st119898

sum

119894=1

119863119894(119909119894)

120575119894

le 119864

(12)






(max119865isinD1

E119865[minus120585119879119909 +

119898

sum

119894=1

119862119894(119909119894) +

119898

sum

119894=1

119863119894(119909119894)])

st119898

sum

119894=1

119863119894(119909119894)

120575119894

le 119864

(13)




max119865isinD1

E119865[minus120585119879119909 +

119898

sum

119894=1

119862119894(119909119894) +

119898

sum

119894=1

119863119894(119909119894)] (14)



2ge 1

Σ0≻ 0 ThenΦ(119909 120574


the problem (15)

min119876119902119903119905

119903 + 119905

st 119905 ge (1205742Σ0+ 1205830120583119879

0) sdot 119876 + 120583

119879

0119902

+ radic1205741

10038171003817100381710038171003817Σ12

0(119902 + 2119876120583

0)

10038171003817100381710038171003817

119903 ge minus120585119879119909 +

119898

sum

119894=1

119862119894(119909119894)

+

119898

sum

119894=1

119863119894(119909119894) minus 120585119879119876120585 minus 120585

119879119902 forall120585 isin F

119876 ⪰ 0

(15)





119898

119894=1119862119894(119909119894) + sum

119898

119894=1119863119894(119909119894) Then





minus V = min119909119876119902119903119905120578119910120596

119903 + 119905

st 119905 ge (1205742Σ0+ 1205830120583119879

0) sdot 119876 + 120583

119879

0119902

+ radic1205741

10038171003817100381710038171003817Σ12

0(119902 + 2119876120583

0)

10038171003817100381710038171003817

[

[

[

119876

119902

2

+

119909

2

119902119879

2

+

119909119879

2

119903 minus 120578

]

]

]

⪰ 0


119876 ⪰ 0

120578 ge

119898

sum

119894=1

(119886119894+ 119887119894119909119894) +

119898

sum

119894=1

120575119894

ℎ119894

119901119894

1

120593119894

119909119894+ 119910

119910 ge

119898

sum

119894=1

1205962

119894

120596119894= radic119888119894

119909119894

119898

sum

119894=1

ℎ119894

119901119894

1

120593119894

119909119894le 119864

119909 isin 119883

(16)



minus V = min119909119876119902119903119905

119903 + 119905

st 119905 ge (1205742Σ0+ 1205830120583119879

0) sdot 119876 + 120583

119879

0119902

+ radic1205741

10038171003817100381710038171003817Σ12

0(119902 + 2119876120583

0)

10038171003817100381710038171003817

119903 ge minus120585119879119909 +

119898

sum

119894=1

119862119894(119909119894)

+

119898

sum

119894=1

119863119894(119909119894) minus 120585119879119876120585 minus 120585

119879119902 forall120585 isin F

119862119894(119909119894) = 119886119894+ 119887119894119909119894+ 1198881198941199092

119894 119894 = 1 119898

119863119894(119909119894) = 120575119894

ℎ119894

119901119894

1

120593119894

119909119894 119894 = 1 119898

119876 ⪰ 0

119898

sum

119894=1

ℎ119894

119901119894

1

120593119894

119909119894le 119864

119909 isin 119883

(17)

Let

120578 ge

119898

sum

119894=1

(119886119894+ 119887119894119909119894) +

119898

sum

119894=1

120575119894

ℎ119894

119901119894

1

120593119894

119909119894+ 119910

119910 ge

119898

sum

119894=1

1205962

119894

120596119894= radic119888119894

119909119894

(18)

Then 120578 ge sum119898

119894=1119862119894(119909119894)+sum119898


be replaced by

120585119879119876120585 + 120585



[

120585

1]

119879

[

[

[

119876

119902 + 119909

2

119902119879+ 119909119879

2

119903 minus 120578

]

]

]

[

120585


lArrrArr[

[

[

119876

119902

2

+

119909

2

119902119879

2

+

119909119879

2

119903 minus 120578

]

]

]

⪰ 0

(20)


Since 119891(119909 120585) = minus120585119879119909 + sum

119898

119894=1119862119894(119909119894) + sum

119898

119894=1119863119894(119909119894) is



3 Numerical Example



1(R6 120583

0 Σ0 02 23) where 120583

0and Σ

0are




2emissions by per



If we fix 1205742= 12 and let 120574





to 1 and 1205742= 12






1 50 200 0 200 0003752 20 80 0 175 0017505 15 50 0 100 0062508 10 35 0 325 00083411 10 30 0 300 00250013 12 40 0 300 002500


2= 12)

1205741







1and let 120574





2



2 And


2emission









119894 Kwhkg


0 05 1 15 2140150160170180190200210220230

f(x

ksi)

1205741











50

100

150

200

Profi

t

Profit

130

132

134

136

0 01 02 03 04 05 06 07 08120575 (E = 135)

CO2

CO2

(a)

Profi

t

406080

100120140160180200

120140160180200

0 01 02 03 04 05 06 07 08

Profit

120575 (without E)

CO2

CO2

(b)


120 130 140 150 160 170 180140

145

150

155

160

Profi

t

100

120

140

160

180

CO2

E (120575 = 02)

ProfitCO2

(a)

120 130 140 150 160 170 180

Profi

t

93

94

95

96

110

120

130

140

CO2

E (120575 = 06)

ProfitCO2

(b)





4 Conclusion


2emission in


120 125 130 135 140 145 150 155 160 165 17010152025303540455055

Bus 1Bus 2Bus 5

Bus 8Bus 11Bus 13

E

x










Acknowledgment


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1ge 0

and 1205742ge 1

(E [120585] minus 1205830)119879

Σminus1

0(E [120585] minus 120583

0) le 1205741 (9)

E [(120585 minus 1205830) (120585 minus 120583

0)119879

] ⪯ 1205742Σ0 (10)



0and



0




0 respectively


D1(F 1205830 Σ0 1205741 1205742)

= 119865 isin U | P (120585 isin F) = 1

(E [120585] minus 1205830)119879

Σminus1

0(E [120585] minus 120583

0) le 1205741

E [(120585 minus 1205830) (120585 minus 120583

0)119879

] ⪯ 1205742Σ0

(11)


1(F 1205830 Σ0 1205741 1205742) which





V = max119909isin119883

(min119865isinD1

E119865[120585119879119909 minus

119898

sum

119894=1

119862119894(119909119894) minus

119898

sum

119894=1

119863119894(119909119894)])

st119898

sum

119894=1

119863119894(119909119894)

120575119894

le 119864

(12)






(max119865isinD1

E119865[minus120585119879119909 +

119898

sum

119894=1

119862119894(119909119894) +

119898

sum

119894=1

119863119894(119909119894)])

st119898

sum

119894=1

119863119894(119909119894)

120575119894

le 119864

(13)




max119865isinD1

E119865[minus120585119879119909 +

119898

sum

119894=1

119862119894(119909119894) +

119898

sum

119894=1

119863119894(119909119894)] (14)



2ge 1

Σ0≻ 0 ThenΦ(119909 120574


the problem (15)

min119876119902119903119905

119903 + 119905

st 119905 ge (1205742Σ0+ 1205830120583119879

0) sdot 119876 + 120583

119879

0119902

+ radic1205741

10038171003817100381710038171003817Σ12

0(119902 + 2119876120583

0)

10038171003817100381710038171003817

119903 ge minus120585119879119909 +

119898

sum

119894=1

119862119894(119909119894)

+

119898

sum

119894=1

119863119894(119909119894) minus 120585119879119876120585 minus 120585

119879119902 forall120585 isin F

119876 ⪰ 0

(15)





119898

119894=1119862119894(119909119894) + sum

119898

119894=1119863119894(119909119894) Then





minus V = min119909119876119902119903119905120578119910120596

119903 + 119905

st 119905 ge (1205742Σ0+ 1205830120583119879

0) sdot 119876 + 120583

119879

0119902

+ radic1205741

10038171003817100381710038171003817Σ12

0(119902 + 2119876120583

0)

10038171003817100381710038171003817

[

[

[

119876

119902

2

+

119909

2

119902119879

2

+

119909119879

2

119903 minus 120578

]

]

]

⪰ 0


119876 ⪰ 0

120578 ge

119898

sum

119894=1

(119886119894+ 119887119894119909119894) +

119898

sum

119894=1

120575119894

ℎ119894

119901119894

1

120593119894

119909119894+ 119910

119910 ge

119898

sum

119894=1

1205962

119894

120596119894= radic119888119894

119909119894

119898

sum

119894=1

ℎ119894

119901119894

1

120593119894

119909119894le 119864

119909 isin 119883

(16)



minus V = min119909119876119902119903119905

119903 + 119905

st 119905 ge (1205742Σ0+ 1205830120583119879

0) sdot 119876 + 120583

119879

0119902

+ radic1205741

10038171003817100381710038171003817Σ12

0(119902 + 2119876120583

0)

10038171003817100381710038171003817

119903 ge minus120585119879119909 +

119898

sum

119894=1

119862119894(119909119894)

+

119898

sum

119894=1

119863119894(119909119894) minus 120585119879119876120585 minus 120585

119879119902 forall120585 isin F

119862119894(119909119894) = 119886119894+ 119887119894119909119894+ 1198881198941199092

119894 119894 = 1 119898

119863119894(119909119894) = 120575119894

ℎ119894

119901119894

1

120593119894

119909119894 119894 = 1 119898

119876 ⪰ 0

119898

sum

119894=1

ℎ119894

119901119894

1

120593119894

119909119894le 119864

119909 isin 119883

(17)

Let

120578 ge

119898

sum

119894=1

(119886119894+ 119887119894119909119894) +

119898

sum

119894=1

120575119894

ℎ119894

119901119894

1

120593119894

119909119894+ 119910

119910 ge

119898

sum

119894=1

1205962

119894

120596119894= radic119888119894

119909119894

(18)

Then 120578 ge sum119898

119894=1119862119894(119909119894)+sum119898


be replaced by

120585119879119876120585 + 120585



[

120585

1]

119879

[

[

[

119876

119902 + 119909

2

119902119879+ 119909119879

2

119903 minus 120578

]

]

]

[

120585


lArrrArr[

[

[

119876

119902

2

+

119909

2

119902119879

2

+

119909119879

2

119903 minus 120578

]

]

]

⪰ 0

(20)


Since 119891(119909 120585) = minus120585119879119909 + sum

119898

119894=1119862119894(119909119894) + sum

119898

119894=1119863119894(119909119894) is



3 Numerical Example



1(R6 120583

0 Σ0 02 23) where 120583

0and Σ

0are




2emissions by per



If we fix 1205742= 12 and let 120574





to 1 and 1205742= 12






1 50 200 0 200 0003752 20 80 0 175 0017505 15 50 0 100 0062508 10 35 0 325 00083411 10 30 0 300 00250013 12 40 0 300 002500


2= 12)

1205741







1and let 120574





2



2 And


2emission









119894 Kwhkg


0 05 1 15 2140150160170180190200210220230

f(x

ksi)

1205741











50

100

150

200

Profi

t

Profit

130

132

134

136

0 01 02 03 04 05 06 07 08120575 (E = 135)

CO2

CO2

(a)

Profi

t

406080

100120140160180200

120140160180200

0 01 02 03 04 05 06 07 08

Profit

120575 (without E)

CO2

CO2

(b)


120 130 140 150 160 170 180140

145

150

155

160

Profi

t

100

120

140

160

180

CO2

E (120575 = 02)

ProfitCO2

(a)

120 130 140 150 160 170 180

Profi

t

93

94

95

96

110

120

130

140

CO2

E (120575 = 06)

ProfitCO2

(b)





4 Conclusion


2emission in


120 125 130 135 140 145 150 155 160 165 17010152025303540455055

Bus 1Bus 2Bus 5

Bus 8Bus 11Bus 13

E

x










Acknowledgment


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119876 ⪰ 0

120578 ge

119898

sum

119894=1

(119886119894+ 119887119894119909119894) +

119898

sum

119894=1

120575119894

ℎ119894

119901119894

1

120593119894

119909119894+ 119910

119910 ge

119898

sum

119894=1

1205962

119894

120596119894= radic119888119894

119909119894

119898

sum

119894=1

ℎ119894

119901119894

1

120593119894

119909119894le 119864

119909 isin 119883

(16)



minus V = min119909119876119902119903119905

119903 + 119905

st 119905 ge (1205742Σ0+ 1205830120583119879

0) sdot 119876 + 120583

119879

0119902

+ radic1205741

10038171003817100381710038171003817Σ12

0(119902 + 2119876120583

0)

10038171003817100381710038171003817

119903 ge minus120585119879119909 +

119898

sum

119894=1

119862119894(119909119894)

+

119898

sum

119894=1

119863119894(119909119894) minus 120585119879119876120585 minus 120585

119879119902 forall120585 isin F

119862119894(119909119894) = 119886119894+ 119887119894119909119894+ 1198881198941199092

119894 119894 = 1 119898

119863119894(119909119894) = 120575119894

ℎ119894

119901119894

1

120593119894

119909119894 119894 = 1 119898

119876 ⪰ 0

119898

sum

119894=1

ℎ119894

119901119894

1

120593119894

119909119894le 119864

119909 isin 119883

(17)

Let

120578 ge

119898

sum

119894=1

(119886119894+ 119887119894119909119894) +

119898

sum

119894=1

120575119894

ℎ119894

119901119894

1

120593119894

119909119894+ 119910

119910 ge

119898

sum

119894=1

1205962

119894

120596119894= radic119888119894

119909119894

(18)

Then 120578 ge sum119898

119894=1119862119894(119909119894)+sum119898


be replaced by

120585119879119876120585 + 120585



[

120585

1]

119879

[

[

[

119876

119902 + 119909

2

119902119879+ 119909119879

2

119903 minus 120578

]

]

]

[

120585


lArrrArr[

[

[

119876

119902

2

+

119909

2

119902119879

2

+

119909119879

2

119903 minus 120578

]

]

]

⪰ 0

(20)


Since 119891(119909 120585) = minus120585119879119909 + sum

119898

119894=1119862119894(119909119894) + sum

119898

119894=1119863119894(119909119894) is



3 Numerical Example



1(R6 120583

0 Σ0 02 23) where 120583

0and Σ

0are




2emissions by per



If we fix 1205742= 12 and let 120574





to 1 and 1205742= 12






1 50 200 0 200 0003752 20 80 0 175 0017505 15 50 0 100 0062508 10 35 0 325 00083411 10 30 0 300 00250013 12 40 0 300 002500


2= 12)

1205741







1and let 120574





2



2 And


2emission









119894 Kwhkg


0 05 1 15 2140150160170180190200210220230

f(x

ksi)

1205741











50

100

150

200

Profi

t

Profit

130

132

134

136

0 01 02 03 04 05 06 07 08120575 (E = 135)

CO2

CO2

(a)

Profi

t

406080

100120140160180200

120140160180200

0 01 02 03 04 05 06 07 08

Profit

120575 (without E)

CO2

CO2

(b)


120 130 140 150 160 170 180140

145

150

155

160

Profi

t

100

120

140

160

180

CO2

E (120575 = 02)

ProfitCO2

(a)

120 130 140 150 160 170 180

Profi

t

93

94

95

96

110

120

130

140

CO2

E (120575 = 06)

ProfitCO2

(b)





4 Conclusion


2emission in


120 125 130 135 140 145 150 155 160 165 17010152025303540455055

Bus 1Bus 2Bus 5

Bus 8Bus 11Bus 13

E

x










Acknowledgment


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1 50 200 0 200 0003752 20 80 0 175 0017505 15 50 0 100 0062508 10 35 0 325 00083411 10 30 0 300 00250013 12 40 0 300 002500


2= 12)

1205741







1and let 120574





2



2 And


2emission









119894 Kwhkg


0 05 1 15 2140150160170180190200210220230

f(x

ksi)

1205741











50

100

150

200

Profi

t

Profit

130

132

134

136

0 01 02 03 04 05 06 07 08120575 (E = 135)

CO2

CO2

(a)

Profi

t

406080

100120140160180200

120140160180200

0 01 02 03 04 05 06 07 08

Profit

120575 (without E)

CO2

CO2

(b)


120 130 140 150 160 170 180140

145

150

155

160

Profi

t

100

120

140

160

180

CO2

E (120575 = 02)

ProfitCO2

(a)

120 130 140 150 160 170 180

Profi

t

93

94

95

96

110

120

130

140

CO2

E (120575 = 06)

ProfitCO2

(b)





4 Conclusion


2emission in


120 125 130 135 140 145 150 155 160 165 17010152025303540455055

Bus 1Bus 2Bus 5

Bus 8Bus 11Bus 13

E

x










Acknowledgment


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra













50

100

150

200

Profi

t

Profit

130

132

134

136

0 01 02 03 04 05 06 07 08120575 (E = 135)

CO2

CO2

(a)

Profi

t

406080

100120140160180200

120140160180200

0 01 02 03 04 05 06 07 08

Profit

120575 (without E)

CO2

CO2

(b)


120 130 140 150 160 170 180140

145

150

155

160

Profi

t

100

120

140

160

180

CO2

E (120575 = 02)

ProfitCO2

(a)

120 130 140 150 160 170 180

Profi

t

93

94

95

96

110

120

130

140

CO2

E (120575 = 06)

ProfitCO2

(b)





4 Conclusion


2emission in


120 125 130 135 140 145 150 155 160 165 17010152025303540455055

Bus 1Bus 2Bus 5

Bus 8Bus 11Bus 13

E

x










Acknowledgment


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Acknowledgment


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Distributionally Robust Self-Scheduling...

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