research article energy losses and voltage stability study...

Research ArticleEnergy Losses and Voltage Stability Study inDistribution Network with Distributed Generation

Hongwei Ren1 Congying Han1 Tiande Guo1 and Wei Pei2

1 School of Mathematical Sciences University of Chinese Academy of Sciences Beijing 100049 China2 Institute of Electrical Engineering Chinese Academy of Sciences Beijing 100190 China

Correspondence should be addressed to Congying Han hancyucasaccn

Received 13 January 2014 Revised 19 July 2014 Accepted 30 July 2014 Published 21 August 2014

Academic Editor Hongjie Jia

Copyright copy 2014 Hongwei Ren et al 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

With the distributed generation technology widely applied some system problems such as overvoltages and undervoltages aregradually remarkable which are caused by distributed generations like wind energy system (WES) and photovoltaic system (PVS)because of their probabilistic output power which relied on natural conditions Since the impacts ofWES and PVS are important inthe distribution system voltage quality we study these in this paper using newmodels with the probability density function of nodevoltage and the cumulative distribution function of total losses We apply these models to solve the IEEE33 distribution systemto be chosen in IEEE standard database We compare our method with the Monte Carlo simulation method in three differentcases respectively In the three cases these results not only can provide the important reference information for the next stageoptimization design system reliability and safety analysis but also can reduce amount of calculation

1 Introduction

Electric power systems have been originally designed basedon the unidirectional power flow Nevertheless in the lastyears the conception of distributed generation (DG) such aswind power generation and solar power generation has ledto new consideration on the distribution networks (DN) [1]As the ratio of the distributed generation in power systemexpanded to study the effect of distributed generation onthe system steady run is more and more important Becausewind power and solar power are with stochastic volatility thepenetration of DG may impact the operation of DN in bothbeneficial and detrimental ways [2ndash9] The positive impactsof DG may possibly be voltage support power loss reduc-tion support of ancillary services and improved reliabilitywhereas negative ones included protection coordinationdynamic stability and islanding Numerous researchers havedealt with the issue of size and site ofDG intoDNs A group ofarticles optimize sizing andor siting of DG units in order toobtain maximum benefits such as maximum loss reductionor reliability andminimum cost [10ndash18] However the above-mentioned papers use power flow analysis either for a certainloading condition or for a few specific scenarios (eg seasonal

loadings) based on measured data or default test cases [3ndash5 9 10 14ndash21] And some have not considered stochasticvolatility of distributed generationThis paper ismainly aboutenergy losses and voltage stability assessment in distributionnetwork with distributed generation considering stochasticvolatility

With the development of science and technology and theraised awareness of environmental protection every countryis becoming more and more interested in the renewableenergy sources specifically because they are reproducibleand nonpolluting These technologies include hydro- wind[9] solar [10] biomass [11] and tidal technology Amongthese renewable energies wind and solar technology haveevolved very rapidly over the past decade and the reduction ofcapital costs the improvement of reliability and the efficiencyhave made the wind and solar power be able to competewith conventional power generation [12] The renewable DGtechnologies like wind and solar have special characteristicsdue to their main source of energy Obviously the primaryenergy source of a wind turbine is wind The wind speed isnot a constant quantity during the operation of wind turbineand is highly dependent on climate condition of the areawhere wind turbine is installed The solar technology is also

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 939482 7 pageshttpdxdoiorg1011552014939482

2 Journal of Applied Mathematics

0

02

04

06

08

1

Mem

bers

hip

degr

ee

120572

120572

min 120572L120572U 120572max

A120572

Predicted value of A

Univers of discourse

Figure 1 Fuzzy trapezoidal number

dependent on the climate and geographical location It is thereason that they exhibit uncertainty and variability in theiroutput [13] Somemethods are proposed tomodel the impactof these uncertainties on distribution network performancethe general and powerful tool is based on Monte Carlosimulation for simulating the uncertainties but this methodneeds too large amount of calculation especially the powerflow calculation Hence this paper proposed a tool to reducethe amount of calculation about uncertainties for distributionnetwork

This paper is organized as follows in Section 2 prelimi-nary theory the fuzzy theory Monte Carlo simulation andthe half invariant method used for the stochastic analysisare presented The model is given in Section 3 The pro-posed solution algorithm and simulation result are presentedin Section 4 Some general conclusions are presented inSection 5

2 Preliminary Theory

The value of load in each bus and DG generation arecontrollable with decisions of their owners In this section weprepare the preliminary theory used in this paper One partis about fuzzy mathematics theory the other is Monte Carloand half invariants method

21 Fuzzy MathematicsTheory In fuzzy mathematics theoryamembership function is defined which describes howmucheach element belongs to a fuzzy set 120583

119860(119909)is a membership

function that takes values in the interval [0 1] For eachelement 119909 isin 119860 119860 is a fuzzy set of universe of discourse 119880In this paper fuzzy trapezoidal membership (FTM) with anotation 119860 = (119886min 119886119871 119886119880 119886max) is used as shown in Figure 1

In engineering problems the question is knowing theuncertain input variables 119909

119894 how to give the membership

function of 119910The 120572-cut method [16] answers this question inthe following way for a given fuzzy set119860 defined on universeof discourse that is 119880 the crisp set 119860120572 is defined as all

elements of 119880 which have membership degree to 119860 greaterthan or equal to 120572 as calculated in the following equation

119860120572= 119909 isin 119880 | 120583

119860 (119909) ⩾ 120572 (1)

The120572-cut of each input variable that is119909120572119894 is calculated using

(1) then the 120572-cut of 119910 that is 119884120572 is calculated as follows

119884120572= (119910120572 119910120572) (2)

119910120572= min119891 (119909) (3)

119910120572= max119891 (119909) (4)

119909 isin (119909120572 119909120572) (5)

This means for each 120572-cut (3) and (4) are solved The upperbound of 119910120572 is obtained by (4) and the lower bound of 119910

120572is

obtained by (3)The defuzzification is amathematical processfor converting a fuzzy number into a crisp one [16] In [19] thecentroidmethod is used for defuzzification of fuzzy numbersThe defuzzified value of a given fuzzy quantity that is 119860lowast iscalculated as follows

119860lowast=int120583119860 (119909) 119909 119889119909

int120583119860 (119909) 119889119909

(6)

Transforming fuzzy variables into random variables is com-monly used in engineering This paper adopts a conversionmethod defined as follows

119891 (119909) =120583119860 (119909)

int 120583119860 (119909) 119889119909

(7)

This method not only retained the distribution informationof fuzzy variables membership functions but also met thecompleteness and the nonnegativity of the probability densityfunction

22 Monte Carlo Simulation and Half Invariants (Cumu-lants) Method Themain concept of Monte Carlo simulation(MCS) method is described as follows suppose a multivari-able function namely 119910 119910 = 119891(119885) where 119885 = (119885

1 119885

119898)

in which 1198851to 119885119898

are random variables with their ownprobability distribution function (PDF) In [21] the MCSacts as follows first of all it will generate a value that is119885Θ

119894 for each input variable 119885

119894using its own PDF and form

119885Θ= (119885Θ

1 119885

Θ

119898) and then calculate the value of 119910Θ using

119910Θ= 119891(119885

Θ) This process will be repeated for a number of

iterations The trend of the output that is 119910 will determineits PDF

Some of uncertain input parameters follow from PDFsuch as the value of wind which follows a Weibull PDF [20]MCS is a powerful tool for analyzing the uncertainties whichfollow any PDF But MCS calculation is too big hence thispaper presents a new methodmdashhalf invariants (cumulants)method instead of Monte Carlo simulation method

Just as expectation and variance half invariant is also anumerical characteristic of random variable119883 which can becalculated by the moment of the random variable119883

Journal of Applied Mathematics 3

If 119884 is a linear function of random variables 1198831and 119883

2

with their own PDF the problem is knowing the PDFs ofall variables 119883

1and 119883

2 what would be the PDF of 119884 The

half invariants (cumulants) method applied half invariantproperties and Gram-Charlier series theorem first calculatethe half invariant of random variable 119884 using the halfinvariant of random variables 119883

1and 119883

2 then get the PDF

of random variable 119884 using Gram-Charlier series theoremThis can avoid complicated convolution operation and a largenumber of Monte Carlo simulations

Gram-Charlier Series Theorem Suppose 119883 is a randomvariable then the PDF of119883 can be expressed as follows

PDF (119909) = 120593 (119909) + 1198881120593(1)

(119909) + 1198882120593(2)

(119909) + sdot sdot sdot (8)

where 120593(119909) is the PDF of the standard normal distribution120593(119894)(119909) is the 119894th derivative and 119888

119894is the coefficient which can

be calculated by the moment of the random variable119883

3 Half Invariants Modeling

The assumptions for modeling the two types of uncertaintiesconstraints and the objective functions are described asfollows

31 Uncertainty Modeling

311 Load It is assumed that the values of load in eachbus are controllable with decisions of their owners In thispaper we assume that the distribution network operators(DNO) can just describe them with a membership functionas follows

119878119863

ℎ119894= 119878119863

119894119891times DLF

ℎtimes (120585min 120585119871 120585119880 120585max) (9)

where 119878119863119894119891

is the apparent forecasted value of peak load in bus119894 andDLF

ℎis the demand level factor at demand level ℎwhich

takes values between 0 and 1 Finally 119878119863ℎ119894is the fuzzy value of

demand in bus 119894 and demand level ℎDG generation pattern the amount of energy which a

controllable DG unit injects into the network is uncertainand usually it depends on the decisions of DG owner so theDNO cannot have a PDF of it if there is not much historicdata about it The output power of a controllable DG unit ismodeled using a membership function as follows

119875dgℎ119894

= 119862dg119894119891

times (120577min 120577119871 120577119880 120577max) (10)

where119862dg119894119891

is the capacity of DG unit installed in bus 119894 and119875dgℎ119894

is the active power of a DG unit in bus 119894 in demand level ℎPhotovoltaic generation pattern the amount of solar radi-

ation that reaches the ground besides on the daily and yearlyapparent motion of the sun depends on the geographicallocation (latitude and altitude) and on the climatic conditions(eg cloud cover) The generation schedule of a photovoltaicgeneration pattern highly depends on the irradiance in the

site The variation of irradiance that is 119864 can be modeledusing a beta PDF as follows

PDF (119864) =Γ (119901 + 119902)

Γ (119901) Γ (119902)(

119864

119864max)

119901minus1

(1 minus119864

119864max)

119902minus1

(11)

where 119864 and 119864max are the actual light intensity andmaximumlight intensity and 119901 and 119902 are the shape of the betadistribution parameters

The generated power of the photovoltaic generation isdetermined as follows

119875sloarℎ119894

= 119864119860120574 (12)

where119860 is the area of the solar panels and 120574 is the photoelec-tric conversion efficiency

Wind turbine generation pattern the generation scheduleof a wind turbine highly depends on the wind speed in thesite The variation of wind speed that is V can be modeledusing a Weibull PDF [22] and its characteristic functionwhich relates the wind speed and the output of a wind turbine[23] as follows

PDF (V) = (120573

120572)(

V120572)

120573minus1

exp(minus( V120572)

120573

) (13)

where 120573 is the shape parameters and 120572 is the scale parametersof the Weibull PDF of wind speed in the zone under studyThe generated power of the wind turbine is determined usingits characteristics as follows

119875windℎ119894

=

119901wind119894119903

V119888in lt V lt VrateV minus V119888inV119888out minus V119888in

119901wind119894119903

Vrate lt V lt V119888out

0 else

(14)

where 119875wind119894119903

is the rated power of wind turbine installed inbus 119894 119875wind

ℎ119894is the generated power of wind turbine in bus 119894

and demand level ℎ V119888out is the cut out speed V119888

in is the cut inspeed and Vrate is the rated speed of the wind turbine

32 Active Losses Thepower flow equationsmust be satisfiedin each demand level ℎ and at each bus 119894 as follows

119875netℎ119894

= 119881ℎ119894sum(119884119894119895119881ℎ119894(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895))

119876netℎ119894

= 119881ℎ119894sum(119884119894119895119881ℎ119894(119866119894119895cos 120579119894119895minus 119861119894119895sin 120579119894119895))

(15)

where 119875netℎ119894

and 119876netℎ119894

are the net active and reactive powerinjected to the network in bus 119894 at level ℎ The above equationcan be written in the matrix form as 119882 = 119891(119883) The powerflow equations at each branch are given as follows

119875ℎ119894119895

= 119881119894119881119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895) + 1199051198941198951198661198941198951198812

119894

119876ℎ119894119895

= minus119881119894119881119895(119866119894119895cos 120579119894119895minus 119861119894119895sin 120579119894119895) + (119861

119894119895minus 1198871198941198950)1198661198941198951198812

119894

(16)


Table 1 Data used in this paper

Parameters Unit Value119888 878119881min pu 095V119888in ms 3Vrate ms 13120585119871

0925120585max 115120577119871

09120577max 1119896 175119881max pu 105V119888out ms 25120585min 0850120585119880

1075120577min 0120577119880

1120591ℎ

h 365

The above equation can be written in the matrix form as 119885 =

119892(119883) Make the first order Taylor expansion as follows

119882 = 119891 (1198830) + 119869

10038161003816100381610038161198830Δ119883 119885 = 119892 (119883

0) + 119866

10038161003816100381610038161198830Δ119883 (17)

Then we get the following linear relationship

Δ119883 = 1198780sdot Δ119882 Δ119885 = 119866

0sdot ΔX = 119866

0sdot 1198780sdot Δ119882 = 119879

0Δ119882

(18)

The total active loss of the network in each demand level isequal to the sum of all active power injected to each bus asfollows

Lossℎ= sum(119875

ℎ119894119895+ 119875ℎ119895119894

) sdot 120591ℎLossℎ= 119867 (119883) (19)

By the Taylor expansion and linearization the total active lossof the network is equal to the sum of all active power injectedto each bus that is

Lossnet = sum Lossℎ (20)

where Lossℎ= 119867(119883

0) + ΔLoss

ℎ= Loss

ℎ0+ ΔLoss

ℎ

4 Algorithm and Simulation Results

This paper supposed that the injection power at each bus isindependent and made power flow equation linearizationThe uncertainties are neither random variables nor fuzzyvariables Transforming fuzzy variables into random vari-ables with the method is proposed ahead

41 Algorithm

Step 1 Input feeder data ℎ = 1

Step 2 Read load and DG data at ℎ level

Step 3 Run power flow with Newton-Raphson at ℎ level

19 20 21 22

23 24 25

26 27 28 29 30 31 32 33

35

34

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Figure 2 IEEE33 distribution network

Step 4 Calculate the half invariance of the injection powerflow of the generator and load at ℎ level

Step 5 Calculate the half invariance of Δ119882

Step 6 Calculate the half invariance of Δ119883 according to 119878ℎ0

Step 7 Calculate the PDF of variate Δ119883 according to Gram-Charlier Series theorem


Step 9 Calculate the half invariance of ΔLoss according to119872ℎ0

Step 10 Calculate the PDF of ΔLossℎaccording to Gram-

Charlier series theorem at ℎ level and store the data

Step 11 According Step 7 calculate the probability of out-of-limit voltage at ℎ level and store the data

Step 12 ℎ = ℎ + 1 if ℎ ⩽ 24 then turn to Step 2 else turn toStep 13

Step 13 According to the data stored in Step 10 fit the PDFof ΔLoss of one year

Step 14 According to the data stored in Step 11 estimate thetime of out-of-limit voltage of one year

Step 15 End

42 Simulation Results The proposed methodology isapplied to a IEEE33 distribution network which is shown inFigure 2 and joined with two distributed generations (ienode34 and node35)

The first one is the Monte Carlo simulation method andthe second is the method proposed by us in this paper Thetwo methods respectively are used in three cases Case 1

is not joined with any distributed generation (node34 andnode35 are out of work) case 2 is joined with two windturbine generations (node34 and node35) while case 3 isjoined with one wind turbine generation (node34) and onephotovoltaic generation (node35) The simulation results areas shown in Figures 4 and 5 Simulation parameters are givenin Table 1 It is assumed that there are 24 demand levels in


Table 2 Moment of bus voltage in case 1

Node The 1st order cumulant The 2nd order cumulant The 3rd order cumulant The 4th order cumulant5 10178 03181 times 10

minus6 0 010 09893 03482 times 10

minus5 0 018 09786 08142 times 10

minus5 0 022 09756 08796 times 10

minus5 0 0



minus501928 times 10

minus10minus01012 times 10

minus16

10 10097 06032 times 10minus4

01074 times 10minus8

minus01404 times 10minus15

18 09676 01542 times 10minus5



22 09458 01696 times 10minus5



1 3 5 7 9 11 13 15 17 19 21 230

01

02

03

04

05

06

07

08

09

1

(hours)

Dem

and

leve

l fac

tors

(DLF

)

Figure 3 The variations of DLFℎin each demand level

each year with equal duration of 120591ℎ= 365 hThe variations of

demand level factors are depicted in Figure 3Some cumulants are given in Tables 1 2 and 3 The first

order cumulant is expected of node voltage the second ordercumulant is the variance Figure 4 is the probability densityfunction of the voltage amplitude of node33 From this figurewe can know that in case 1 the random fluctuations ofvoltage are similar to normal distribution probability of out-of-limit voltage is almost zero When adding distributed gen-eration on one hand line voltage had improved significantlyand load node voltage rises On the other hand because of therandomness of the wind power and photovoltaic power thenode voltage fluctuation and the probability of out-of-limitvoltage significantly increased By comparing with case 2 andcase 3 we can also find that for wind and solar hybrid powersystems due to its complementarity its impact on systemvoltage fluctuation is relatively smaller compared with singlewind power and the probability of out-of-limit voltage isreduced obviously The experimental results show that theproposed fuzzy variables are effective to describe load insteadof normal variable (Table 4)

09 095 1 105 110

005

01

015

02

025

03

035

04

045

05

Voltage amplitude

Prob

abili

ty d

ensit

y

Case 1Case 2Case 3

Figure 4 Voltage probability density function of the 33rd node

Figure 5 is the distribution of network loss the calculationresults of two methods are almost the same but calculatedamount of method 2 is far less thanmethod 1 In other wordsthe method of this paper is effective

5 Conclusions

A method combining half invariants and fuzzy mathematicstheory is proposed for evaluation of active losses in thedistribution network and the time of out-of-limit voltage Bycomparing the simulation it can be found that the proposedmethod can completely replace the Monte Carlo simulationmethod and moreover reduce a large amount of calculationThemodel considers probabilistic presentation of wind speedusing a Weibull PDF and probabilistic description of loadsusing normal distribution

On the other hand as the reduction of calculation thismethod can not only conveniently be used to calculate the






minus17

10 79887 04582 times 10minus4



18 69693 01124 times 10minus3



22 69876 01685 times 10minus3



2500 3000 35000

01

02

03

04

05

06

07

08

09

1

Network loss (MWh)

Cum

ulat

ive d

istrib

utio

n

Method 1Method 2

Figure 5 Cumulative distribution function of total losses

network loss and the time of out-of-limit voltage and to assessthe effects of a distributed generation to the distribution net-work but also be used as the distributed power optimizationindex when considering size and site

Conflict of Interests

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

Acknowledgments

This research was funded by Knowledge Innovation Projectof the Chinese Academy of Sciences (KGCX2-RW-329) andthe Chinese National Natural Science Foundation (7127120411331012 and 11101420)

References

[1] T Ackermann G Andersson and L Soder ldquoDistributed gener-ation a definitionrdquo Electric Power Systems Research vol 57 no3 pp 195ndash204 2001

[2] J A P Lopes N Hatziargyriou J Mutale P Djapic and NJenkins ldquoIntegrating distributed generation into electric powersystems a review of drivers challenges and opportunitiesrdquoElectric Power Systems Research vol 77 no 9 pp 1189ndash12032007

[3] J Mutale G Strbac S Curcic and N Jenkins ldquoAllocation oflosses in distribution systems with embedded generationrdquo IEEProceedings Generation Transmission andDistribution vol 147no 1 pp 7ndash14 2000

[4] P Chiradeja and R Ramakumar ldquoAn approach to quantify thetechnical benefits of distributed generationrdquo IEEE Transactionson Energy Conversion vol 19 no 4 pp 764ndash773 2004

[5] P Chiradeja ldquoBenefit of distributed generation a line lossreduction analysisrdquo in Proceeding of the IEEEPES TransmissionandDistributionConference andExhibition Asia andPacific pp1ndash5 Dalian China August 2005

[6] M Thomson and D G Infield ldquoImpact of widespread pho-tovoltaics generation on distribution systemsrdquo IET RenewablePower Generation vol 1 no 1 pp 33ndash40 2007

[7] M Thomson and D G Infield ldquoNetwork power-flow analysisfor a high penetration of distributed generationrdquo IEEE Transac-tions on Power Systems vol 22 no 3 pp 1157ndash1162 2007

[8] R AWalling R Saint R C Dugan J Burke and L A KojovicldquoSummary of distributed resources impact on power deliverysystemsrdquo IEEE Transactions on Power Delivery vol 23 no 3pp 1636ndash1644 2008

[9] J Deuse S Grenard D Benintendi P J Agrell and P BogetoftldquoUse of system charge smethodology and norm models fordistribution system including DERrdquo in Proceedings of theInternational Conference on Electricity Distribution vol 7 p 762007

[10] S Kotamarty S Khushalani and N Schulz ldquoImpact of dis-tributed generation on distribution contingency analysisrdquo Elec-tric Power Systems Research vol 78 no 9 pp 1537ndash1545 2008

[11] S A Papathanassiou ldquoA technical evaluation framework for theconnection of DG to the distribution networkrdquo Electric PowerSystems Research vol 77 no 1 pp 24ndash34 2007

[12] G Pepermans J Driesen D Haeseldonckx R Belmans andWDrsquohaeseleer ldquoDistributed generation definition benefits andissuesrdquo Energy Policy vol 33 no 6 pp 787ndash798 2005

[13] P P Barker and R W De Mello ldquoDetermining the impact ofdistributed generation on power systems 1 Radial distributionsystemsrdquo in Proceedings of the Power Engineering Society Sum-mer Meeting pp 1645ndash1656 July 2000

[14] G P Harrison and A R Wallace ldquoOptimal power flow eval-uation of distribution network capacity for the connection ofdistributed generationrdquo IEE Proceedings C Generation Trans-mission amp Distribution vol 152 no 1 pp 115ndash122 2005

[15] C Wang and M H Nehrir ldquoAnalytical approaches for optimalplacement of distributed generation sources in power systemsrdquoIEEE Transactions on Power Systems vol 19 no 4 pp 2068ndash2076 2004

[16] H Zhang and D Liu Fuzzy Modeling and Fuzzy ControlBirkhaauser Boston Mass USA 2006

[17] M A Matos and E M Gouveia ldquoThe fuzzy power flowrevisitedrdquo IEEE Transactions on Power Systems vol 23 no 1pp 213ndash218 2008


[18] E M Gouveia and M A Matos ldquoSymmetric AC fuzzy powerflow modelrdquo European Journal of Operational Research vol 197no 3 pp 1012ndash1018 2009

[19] T Ross Ed Fuzzy Logic with Engineering Applications JohnWiley amp Sons New York NY USA 2004

[20] C Kongnam S Nuchprayoon S Premrudeepreechacharn andS Uatrongjit ldquoDecision analysis on generation capacity of awind parkrdquo Renewable and Sustainable Energy Reviews vol 13no 8 pp 2126ndash2133 2009

[21] M H Kalos and P A Whitlock Monte Carlo Methods Wiley-VCH Verlag GmbH amp Co KGaA 2004

[22] G Boyle Renewable Energy Oxford University Press 2004[23] M Jafarian andAM Ranjbar ldquoFuzzymodeling techniques and

artificial neural networks to estimate annual energy output of awind turbinerdquo Renewable Energy vol 35 no 9 pp 2008ndash20142010

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


0

02

04

06

08

1

Mem

bers

hip

degr

ee

120572

120572

min 120572L120572U 120572max

A120572

Predicted value of A

Univers of discourse

Figure 1 Fuzzy trapezoidal number

dependent on the climate and geographical location It is thereason that they exhibit uncertainty and variability in theiroutput [13] Somemethods are proposed tomodel the impactof these uncertainties on distribution network performancethe general and powerful tool is based on Monte Carlosimulation for simulating the uncertainties but this methodneeds too large amount of calculation especially the powerflow calculation Hence this paper proposed a tool to reducethe amount of calculation about uncertainties for distributionnetwork

This paper is organized as follows in Section 2 prelimi-nary theory the fuzzy theory Monte Carlo simulation andthe half invariant method used for the stochastic analysisare presented The model is given in Section 3 The pro-posed solution algorithm and simulation result are presentedin Section 4 Some general conclusions are presented inSection 5

2 Preliminary Theory

The value of load in each bus and DG generation arecontrollable with decisions of their owners In this section weprepare the preliminary theory used in this paper One partis about fuzzy mathematics theory the other is Monte Carloand half invariants method

21 Fuzzy MathematicsTheory In fuzzy mathematics theoryamembership function is defined which describes howmucheach element belongs to a fuzzy set 120583

119860(119909)is a membership

function that takes values in the interval [0 1] For eachelement 119909 isin 119860 119860 is a fuzzy set of universe of discourse 119880In this paper fuzzy trapezoidal membership (FTM) with anotation 119860 = (119886min 119886119871 119886119880 119886max) is used as shown in Figure 1

In engineering problems the question is knowing theuncertain input variables 119909

119894 how to give the membership

function of 119910The 120572-cut method [16] answers this question inthe following way for a given fuzzy set119860 defined on universeof discourse that is 119880 the crisp set 119860120572 is defined as all

elements of 119880 which have membership degree to 119860 greaterthan or equal to 120572 as calculated in the following equation

119860120572= 119909 isin 119880 | 120583

119860 (119909) ⩾ 120572 (1)

The120572-cut of each input variable that is119909120572119894 is calculated using

(1) then the 120572-cut of 119910 that is 119884120572 is calculated as follows

119884120572= (119910120572 119910120572) (2)

119910120572= min119891 (119909) (3)

119910120572= max119891 (119909) (4)

119909 isin (119909120572 119909120572) (5)

This means for each 120572-cut (3) and (4) are solved The upperbound of 119910120572 is obtained by (4) and the lower bound of 119910

120572is

obtained by (3)The defuzzification is amathematical processfor converting a fuzzy number into a crisp one [16] In [19] thecentroidmethod is used for defuzzification of fuzzy numbersThe defuzzified value of a given fuzzy quantity that is 119860lowast iscalculated as follows

119860lowast=int120583119860 (119909) 119909 119889119909

int120583119860 (119909) 119889119909

(6)

Transforming fuzzy variables into random variables is com-monly used in engineering This paper adopts a conversionmethod defined as follows

119891 (119909) =120583119860 (119909)

int 120583119860 (119909) 119889119909

(7)

This method not only retained the distribution informationof fuzzy variables membership functions but also met thecompleteness and the nonnegativity of the probability densityfunction

22 Monte Carlo Simulation and Half Invariants (Cumu-lants) Method Themain concept of Monte Carlo simulation(MCS) method is described as follows suppose a multivari-able function namely 119910 119910 = 119891(119885) where 119885 = (119885

1 119885

119898)

in which 1198851to 119885119898

are random variables with their ownprobability distribution function (PDF) In [21] the MCSacts as follows first of all it will generate a value that is119885Θ

119894 for each input variable 119885

119894using its own PDF and form

119885Θ= (119885Θ

1 119885

Θ

119898) and then calculate the value of 119910Θ using

119910Θ= 119891(119885

Θ) This process will be repeated for a number of

iterations The trend of the output that is 119910 will determineits PDF

Some of uncertain input parameters follow from PDFsuch as the value of wind which follows a Weibull PDF [20]MCS is a powerful tool for analyzing the uncertainties whichfollow any PDF But MCS calculation is too big hence thispaper presents a new methodmdashhalf invariants (cumulants)method instead of Monte Carlo simulation method

Just as expectation and variance half invariant is also anumerical characteristic of random variable119883 which can becalculated by the moment of the random variable119883



2


1and 119883



1and 119883

2 then get the PDF



PDF (119909) = 120593 (119909) + 1198881120593(1)

(119909) + 1198882120593(2)









119878119863

ℎ119894= 119878119863

119894119891times DLF

ℎtimes (120585min 120585119871 120585119880 120585max) (9)

where 119878119863119894119891






119875dgℎ119894

= 119862dg119894119891

times (120577min 120577119871 120577119880 120577max) (10)

where119862dg119894119891





PDF (119864) =Γ (119901 + 119902)

Γ (119901) Γ (119902)(

119864

119864max)

119901minus1

(1 minus119864

119864max)

119902minus1

(11)



119875sloarℎ119894

= 119864119860120574 (12)



PDF (V) = (120573

120572)(

V120572)

120573minus1

exp(minus( V120572)

120573

) (13)


119875windℎ119894

=

119901wind119894119903


119901wind119894119903


0 else

(14)

where 119875wind119894119903






119875netℎ119894

= 119881ℎ119894sum(119884119894119895119881ℎ119894(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895))

119876netℎ119894


(15)


and 119876netℎ119894


119875ℎ119894119895

= 119881119894119881119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895) + 1199051198941198951198661198941198951198812

119894

119876ℎ119894119895


119894119895minus 1198871198941198950)1198661198941198951198812

119894

(16)




0925120585max 115120577119871


1075120577min 0120577119880

1120591ℎ

h 365



119882 = 119891 (1198830) + 119869

10038161003816100381610038161198830Δ119883 119885 = 119892 (119883

0) + 119866

10038161003816100381610038161198830Δ119883 (17)


Δ119883 = 1198780sdot Δ119882 Δ119885 = 119866

0sdot ΔX = 119866

0sdot 1198780sdot Δ119882 = 119879

0Δ119882

(18)


Lossℎ= sum(119875

ℎ119894119895+ 119875ℎ119895119894

) sdot 120591ℎLossℎ= 119867 (119883) (19)



where Lossℎ= 119867(119883

0) + ΔLoss

ℎ= Loss

ℎ0+ ΔLoss

ℎ



41 Algorithm




19 20 21 22

23 24 25

26 27 28 29 30 31 32 33

35

34

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18














Step 15 End







minus6 0 010 09893 03482 times 10

minus5 0 018 09786 08142 times 10

minus5 0 022 09756 08796 times 10

minus5 0 0





minus16

10 10097 06032 times 10minus4



18 09676 01542 times 10minus5



22 09458 01696 times 10minus5



1 3 5 7 9 11 13 15 17 19 21 230

01

02

03

04

05

06

07

08

09

1

(hours)

Dem

and

leve

l fac

tors

(DLF

)





09 095 1 105 110

005

01

015

02

025

03

035

04

045

05

Voltage amplitude

Prob

abili

ty d

ensit

y

Case 1Case 2Case 3



5 Conclusions








minus17

10 79887 04582 times 10minus4



18 69693 01124 times 10minus3



22 69876 01685 times 10minus3



2500 3000 35000

01

02

03

04

05

06

07

08

09

1

Network loss (MWh)

Cum

ulat

ive d

istrib

utio

n

Method 1Method 2





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2


1and 119883



1and 119883

2 then get the PDF



PDF (119909) = 120593 (119909) + 1198881120593(1)

(119909) + 1198882120593(2)









119878119863

ℎ119894= 119878119863

119894119891times DLF

ℎtimes (120585min 120585119871 120585119880 120585max) (9)

where 119878119863119894119891






119875dgℎ119894

= 119862dg119894119891

times (120577min 120577119871 120577119880 120577max) (10)

where119862dg119894119891





PDF (119864) =Γ (119901 + 119902)

Γ (119901) Γ (119902)(

119864

119864max)

119901minus1

(1 minus119864

119864max)

119902minus1

(11)



119875sloarℎ119894

= 119864119860120574 (12)



PDF (V) = (120573

120572)(

V120572)

120573minus1

exp(minus( V120572)

120573

) (13)


119875windℎ119894

=

119901wind119894119903


119901wind119894119903


0 else

(14)

where 119875wind119894119903






119875netℎ119894

= 119881ℎ119894sum(119884119894119895119881ℎ119894(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895))

119876netℎ119894


(15)


and 119876netℎ119894


119875ℎ119894119895

= 119881119894119881119895(119866119894119895cos 120579119894119895+ 119861119894119895sin 120579119894119895) + 1199051198941198951198661198941198951198812

119894

119876ℎ119894119895


119894119895minus 1198871198941198950)1198661198941198951198812

119894

(16)




0925120585max 115120577119871


1075120577min 0120577119880

1120591ℎ

h 365



119882 = 119891 (1198830) + 119869

10038161003816100381610038161198830Δ119883 119885 = 119892 (119883

0) + 119866

10038161003816100381610038161198830Δ119883 (17)


Δ119883 = 1198780sdot Δ119882 Δ119885 = 119866

0sdot ΔX = 119866

0sdot 1198780sdot Δ119882 = 119879

0Δ119882

(18)


Lossℎ= sum(119875

ℎ119894119895+ 119875ℎ119895119894

) sdot 120591ℎLossℎ= 119867 (119883) (19)



where Lossℎ= 119867(119883

0) + ΔLoss

ℎ= Loss

ℎ0+ ΔLoss

ℎ



41 Algorithm




19 20 21 22

23 24 25

26 27 28 29 30 31 32 33

35

34

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18














Step 15 End







minus6 0 010 09893 03482 times 10

minus5 0 018 09786 08142 times 10

minus5 0 022 09756 08796 times 10

minus5 0 0





minus16

10 10097 06032 times 10minus4



18 09676 01542 times 10minus5



22 09458 01696 times 10minus5



1 3 5 7 9 11 13 15 17 19 21 230

01

02

03

04

05

06

07

08

09

1

(hours)

Dem

and

leve

l fac

tors

(DLF

)





09 095 1 105 110

005

01

015

02

025

03

035

04

045

05

Voltage amplitude

Prob

abili

ty d

ensit

y

Case 1Case 2Case 3



5 Conclusions








minus17

10 79887 04582 times 10minus4



18 69693 01124 times 10minus3



22 69876 01685 times 10minus3



2500 3000 35000

01

02

03

04

05

06

07

08

09

1

Network loss (MWh)

Cum

ulat

ive d

istrib

utio

n

Method 1Method 2





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0925120585max 115120577119871


1075120577min 0120577119880

1120591ℎ

h 365



119882 = 119891 (1198830) + 119869

10038161003816100381610038161198830Δ119883 119885 = 119892 (119883

0) + 119866

10038161003816100381610038161198830Δ119883 (17)


Δ119883 = 1198780sdot Δ119882 Δ119885 = 119866

0sdot ΔX = 119866

0sdot 1198780sdot Δ119882 = 119879

0Δ119882

(18)


Lossℎ= sum(119875

ℎ119894119895+ 119875ℎ119895119894

) sdot 120591ℎLossℎ= 119867 (119883) (19)



where Lossℎ= 119867(119883

0) + ΔLoss

ℎ= Loss

ℎ0+ ΔLoss

ℎ



41 Algorithm




19 20 21 22

23 24 25

26 27 28 29 30 31 32 33

35

34

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18














Step 15 End







minus6 0 010 09893 03482 times 10

minus5 0 018 09786 08142 times 10

minus5 0 022 09756 08796 times 10

minus5 0 0





minus16

10 10097 06032 times 10minus4



18 09676 01542 times 10minus5



22 09458 01696 times 10minus5



1 3 5 7 9 11 13 15 17 19 21 230

01

02

03

04

05

06

07

08

09

1

(hours)

Dem

and

leve

l fac

tors

(DLF

)





09 095 1 105 110

005

01

015

02

025

03

035

04

045

05

Voltage amplitude

Prob

abili

ty d

ensit

y

Case 1Case 2Case 3



5 Conclusions








minus17

10 79887 04582 times 10minus4



18 69693 01124 times 10minus3



22 69876 01685 times 10minus3



2500 3000 35000

01

02

03

04

05

06

07

08

09

1

Network loss (MWh)

Cum

ulat

ive d

istrib

utio

n

Method 1Method 2





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












minus6 0 010 09893 03482 times 10

minus5 0 018 09786 08142 times 10

minus5 0 022 09756 08796 times 10

minus5 0 0





minus16

10 10097 06032 times 10minus4



18 09676 01542 times 10minus5



22 09458 01696 times 10minus5



1 3 5 7 9 11 13 15 17 19 21 230

01

02

03

04

05

06

07

08

09

1

(hours)

Dem

and

leve

l fac

tors

(DLF

)





09 095 1 105 110

005

01

015

02

025

03

035

04

045

05

Voltage amplitude

Prob

abili

ty d

ensit

y

Case 1Case 2Case 3



5 Conclusions








minus17

10 79887 04582 times 10minus4



18 69693 01124 times 10minus3



22 69876 01685 times 10minus3



2500 3000 35000

01

02

03

04

05

06

07

08

09

1

Network loss (MWh)

Cum

ulat

ive d

istrib

utio

n

Method 1Method 2





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














minus17

10 79887 04582 times 10minus4



18 69693 01124 times 10minus3



22 69876 01685 times 10minus3



2500 3000 35000

01

02

03

04

05

06

07

08

09

1

Network loss (MWh)

Cum

ulat

ive d

istrib

utio

n

Method 1Method 2





Acknowledgments


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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