Research Article A DCT-Based Driving Cycle Generation...

Research ArticleA DCT-Based Driving Cycle Generation Method andIts Application for Electric Vehicles

Cheng Lin Li Zhao Xingqun Cheng and Wenwei Wang

National Engineering Laboratory for Electric Vehicles Collaborative Innovation Center of Electric Vehicles in BeijingBeijing Institute of Technology Beijing 100081 China

Correspondence should be addressed to Li Zhao zhaolibiteducn

Received 2 November 2014 Revised 7 May 2015 Accepted 20 May 2015

Academic Editor Oleg V Gendelman

Copyright copy 2015 Cheng Lin 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

Nowadays many widely used driving cycle (DC) representing and generating methods are designed for traditional vehicles withinternal combustion engines (ICE) The real-world driving is viewed as a sequence of acceleration cruise deceleration and idlemodes The emission and fuel consumption in each period should be taken into account carefully However for electric vehicles(EVs) most of them are powered by low or zero-emission renewable energy sources The working status and energy managementalgorithms of them are very different from traditional vehicles To facilitate the EVdesign we proposed a novel DC representing andconstruction method to generate DCs for EVs The whole driving route is divided into several length-fixed segments and each ofthese segments is converted into a frequency sequence After doing that we can adjust the frequency and amplitude of the generateddriving cycle directly The experiment results showed that the proposed method was effective and convenient

1 Introduction

Recently the rising cost of gasoline and the increasinglystringent fuel-efficiency standards make electric vehicles aworthwhile alternative to vehicles with internal combustionengine [1ndash5] As an important research tool driving cycleis widely used in the areas of traffic management vehicularpollutionmeasurement and fuel consumption estimation Ingeneral DC gives a speed-time profile of driving behaviorin a specific area or city [6ndash8] For example the US FederalTest Procedure (FTP) the Urban Dynamometer DrivingSchedule for Heavy-Duty Vehicles (UDDSHDV) and theNew European Driving Cycle (NEDC) were developed fordifferent countries or cities In most of these DCs the real-world driving is viewed as a sequence of acceleration cruisedeceleration and idle modes This is suitable for designingan internal combustion (IC) power train in which theemission and fuel consumption in each period should betaken into account carefully However the electric vehicle orsome of hybrid EVs are powered by low or zero-emissionrenewable energy sources and the energy consumed in theidle period is very low or close to zero Additionally most ofthe electric vehicles have an energy-recovery system and the

running efficiencies of them are largely decided by matchingthe electromotor batteries driving cycles and so on Thetraditional DC generation and representing methods areno longer suitable for EVs or HEVs to predict the realworking condition and to indicate the residual energy ortravel distance An adapted and representative driving cycleand a new DC constructing method are needed to facilitatethe design of them

However without a special DCs generation method forelectric vehicles a large number of studies on EV design andenergy management algorithm have to use the traditionaldriving cycles as their objective functions Won and Langaripresented a fuzzy logic based control system and evaluated itvia computational simulation under the FTP75 (Federal TestProcedure) urban drive cycle [9] Lin et al used dynamicprogramming (DP) to find the optimal control actionsand defined a cost function over the Urban DynamometerDriving Schedule for Heavy-Duty Vehicles DC [10] Rodatzet al proposed an original concept to convert the electricalpower flow into equivalent hydrogen cost [11] And thenthey tested the proposed consumptionminimization strategyon the New European Driving Cycle (NEDC) Delprat etal developed a global optimization algorithm to derive a

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 178902 13 pageshttpdxdoiorg1011552015178902

2 Mathematical Problems in Engineering

real-time control strategy and tested the algorithm on theNEDC and a highway DC [12] Kim and Peng developeda combined power management and design optimizationframework and tested it for three different DCs [13] Assaniset al described a methodology for integrating vehicle andengine simulations and tested it for the SAE J1711 DC [14]Gao and Porandla used simulated annealing and geneticalgorithm to optimize the design of a parallel hybrid electricvehicle [15] A combined FTP75Highway Fuel Economy Test(HWFET) DC was used to optimize a parallel hybrid electricpower train design Zhang et al proposed an optimizationmethod which has the capability of providing a set of trade-off optimal solutions among the fuel economy and variousemissions and tested it with a combined UDDSHWFETDC [16] Liu et al developed a real-coded adaptive basedhybrid genetic algorithm and applied it to the optimal sizingof a series hybrid electric vehicle [17] The performance ofthe proposed algorithm was verified with the City-Hwy testprocedure in the ADVISOR environment

Additionally a system design or controller configurationmay be best for a given cycle but not robust for another cycleThe parameters optimized for a special DC are not necessarilyoptimal for others or for the entire driving profile [18ndash20]Sciarretta and Guzzella believed that the control and designparameters that perform well under a given conditions maylead to poor behavior under different conditions [21] Fugelet al showed that the fuel consumption obtained on standardDCs does not sufficiently correlate with the consumptionobtained in real conditions [22] For electric buses andspecial purpose vehicles (such as electric sanitation vehicleelectric garbage truck and electric tank car) the working andrunning routes are determined and the traditional drivingcycles can not be used for the special driving missionprofiles An adapted and representative driving cycle has tobe constructed to facilitate the design of them

So in this paper we introduce a new definition of drivingcycle (distance-based driving cycle DBDC) and presenta novel methodology to generate it With the proposedmethodology a representative DBDC can be easily pro-duced and the stochastic parameters can be determined byobserved data or be manually adjustedThe generated DBDCrepresents the stochastic nature of the observed data Theproposed methodology gives the following advantages thatare important for EVs or HEVs optimization

(1) A novel representing method for driving cyclemdashDBDCmdashwas proposed with which a driving cyclecan be presented by a frequency sequence Thecomponents which appeared in this sequence areorthogonal and can be adjusted independently Itis more advantageous than many traditional rep-resenting methods in which many parameters aredependent on each other

(2) A new DC generation method was proposed withwhich the original DBDC array can be compressedinto several frequency coefficients and these coeffi-cients can be determined from observed data or bemanually adjustedThe generated driving profiles canrepresent the stochastic nature of the observed data

(3) The proposed method is very useful for electricvehicle design and the independent frequency coef-ficients are very useful for testing the compatibility ofelectromotor batteries power train and so on

The rest of this paper is organized as follows We firstreview the related work on the discrete cosine transform(DCT) Secondly we represent the DCT-based approach forconstructing distance-based drive cycle Then we describeseveral experiments in which different DBDC-constructionmethods were compared And with these experiments wedemonstrate how to use the methodology to build effectiveand efficient DBDCs Finally we conclude this paper byhighlighting the key contributions of this work

2 Related Work

21 Discrete Cosine Transform (DCT) A discrete cosinetransform is a finite sequence of data points in terms of asum of cosine functions oscillating at different frequencies Itwas first defined by Ahmed et al in 1974 and widely appliedin science and engineering especially in lossy compressionof audios (eg MP3) and images (eg JPEG) [23] It is verysimilar to the discrete Fourier transform (DFT) but usingonly real numbers There are 8 standard DCT variants andonly 4 of them are commonly used The main feature of it isthe property of energy compaction most of the informationin a given signal is concentrated in a few low-frequencycomponents

Formally the discrete cosine transform is a linear invert-ible function 119891 119877119873 rarr 119877119873 (where 119877 denotes the set of realnumbers) or equivalently an invertible119873times119873 square matrixThere are several variants of the DCT with slightly modifieddefinitions and the four commonly used forms are DCT-I DCT-II DCT-III and DCT-IV In each of them 119873 realnumbers 119909

0 119909

119873minus1are transformed into 119873 real numbers

1198830 119883

119873minus1according to the formulas below

DCT-I

119883119896=12(1199090 + (minus1)

119896 119909119873minus1) +

119873minus2sum119899=1

119909119899cos [ 120587

119873 minus 1119899119896]

119896 = 0 1 119873 minus 1

(1)

DCT-II

119883119896=119873minus1sum119899=0

119909119899cos [ 120587

119873(119899+

12) 119896] 119896 = 0 1 119873 minus 1 (2)

DCT-III

119883119896=121199090 +119873minus1sum119899=1

119909119899cos [ 120587

119873119899(119896+

12)]

119896 = 0 1 119873 minus 1

(3)

Mathematical Problems in Engineering 3

DCT-IV

119883119896=119873minus1sum119899=0

119909119899cos [ 120587

119873(119899+

12)(119896+

12)]

119896 = 0 1 119873 minus 1

(4)

22 Distance-Based Drive Cycle (DBDC) Driving cycle is theset of data points representing vehicle speed versus timewhich can provide a wide range of applications and themost important usages of it are vehicle fuel consumptionestimate and pollutants estimate [24 25]Many driving cycleshave been developed and most of them are designed forvehicleswith traditional internal combustion engines insteadof electric vehicles (EVs) and hybrid EVs (HEVs) [26ndash29]For an internal combustion engine the energy consumed instop-and-go scenario is very vast However for EV no electricenergy will be consumed in the idle period The researchemphasis of EV or HEV is not the fuel consumption orpollution but the electric energy saving and the performanceof the given vehicle The drawback of the traditional DCgenerating methods is that they pay too much attentionto characteristics in time domain However the frequencycharacteristics of DCs have the same importance on energysaving vehicle performance pollution and so on For anEV or HEV designer state-of-charge (SOC) estimation andenergy recycle are significant issues How to calculate theresidual energy and estimate the rest of the journey are thekey points they concerned So in this section we introducea new defined driving cycle (distance-based driving cycleDBDC) to describe the relationship between vehicle speedand distanceThe difference of the two definitions can be seenfrom Figure 1 in which no idle phase was presented in thegiven DBDC (In Figure 1(a) the driving cycle was describedby speed-time In Figure 1(b) the driving cycle was describedby speed-distance)

Definition 1 (drive cycle DC) DC is the set of data pointsrepresenting vehicle speed versus time

Definition 2 (distance-based drive cycle DBDC) DBDC isthe set of data points representing vehicle speed versusdistance travelled from the start point in a given route

3 Generating DBDCs with DiscreteCosine Transform

31 Splitting the Whole Route into Length-Fixed SegmentsNowadays a lot of DC generation methodologies classifydriving scenarios into several categories for example stop-and-go urban suburban rural and highway And thenthey build DCs according to different scenarios [30ndash32]However the classification is intuitive and there is not a clearboundary or criterion existing for it Particularly for electricspecial purpose vehicles their driving mission profiles arevery special An adapted and representative driving cycle hasto be built to facilitate the design of them In fact manyfactors such as location topography driving characteristicsenvironment and traffic give their effects to the driving

Spee

d

Time

(a) Drive cycle

Distance

Spee

d

(b) Distance-based drive cycle

Figure 1 Drive cycle and distance-based drive cycle

profiles The complexities of the relationship between themmake it impossible to construct an accurate driving profilewith limited parameters Researchers listed some importantparameters and modeled the stochastic characteristics of adriving profile with these parameters But the accuracy ofthem is not easy to be controlled and the parameters aredifficult to be extended

In this section we separate the whole route into seg-ments by distance Each segment presents a length-fixedroute with which distance-based driving cycles can be freelyconstructed For each segment it is very easy to convertthe representing manner of them from time domain to fre-quency domain After changing the number of the frequencycomponent coefficients of the sequence is fixed (the lengthsof the segments are fixed) So we can easily calculate theprobabilistic characteristics of it and rebuild a new drivingcycle for EVs As the frequency components are orthogonalwe can independently adjust the coefficients to facilitate theEV or HEV design

32 Changing the Segments into Frequency Domain Havingdivided the whole route into segments and expressed thesegments in speed-distance manner the discrete cosinetransform can be used to change the segments into frequencycomponents Let 119909(119899) be the sequential data points represent-ing vehicle speed versus distance let 119910(119896) be the coefficientsof different frequency components let119873 be the total numberof the data points 119909(119899) and DBDCs of the route-segmentscan be changed into a series of components with differentfrequency coefficients

119910 (119896) = 119908 (119896)119873minus1sum119899=0

119909 (119899) cos(120587 (2119899 + 1) 1198962119873

)

119896 = 0 1 119873 minus 1

(5)

where

119908 (119896) =

1radic119873

119896 = 0

radic2119873

1 le 119896 le 119873 minus 1(6)


=+ + + + middot middot middot

Figure 2 Discrete cosine transformation

119910 return the unitary discrete cosine transform of 119909 119909 and119910 are the same size The series is indexed from 119899 = 0 and119896 = 0

In fact these frequency components are very useful fordescribing the internal and external parameters disturbingthe constructed driving cycles Generally speaking the low-frequency components are decided by terrain topology trafficcongestionThe medium-frequency components are decidedby traffic flow driving characteristics and the high-frequencycomponents are decided by road condition rapid drivingmaneuvers spontaneous reactions and so forth We can seethis in Figure 2

33 Calculating Probability Functions After changing theDBDC into frequency domain a probability function (orprobability density function) should be assigned to eachcoefficient of these frequency components whichwill be usedto generate random driving cycles These probability func-tions are derived by interpolating data from driving profileanalysis

In this section we implement one hundred times oftests with electric sanitation vehicles (ZLJ5071TSL) whichbelong to Beijing Sanitation Group Corporation In the samework route we recorded 100 groups of speed-distance dataand divided them into length-fixed segments After doingthat we changed these segments into frequency domainand calculated the distributions of the frequency componentcoefficients For each of these coefficients both of the largestvalue and smallest value of themweremarkedWedivided theinterval of them into 10 parts and calculated the probabilitydistributions for each of them Having done this we use thecurve fitting functionCftool (Matlab) to create the probabilitysurface to fit the inputs and outputs The frequency of thecoefficient values and the surface were shown in Figure 3from which we can see that with the increasing of the serialnumber of coefficient the values and the ranges of it keptgetting down For example the ranges of 119910(0) 119910(10) 119910(50)and 119910(100) are 360 315 90 and 45 This is decided by theenergy-compaction property of DCT So we can present aroute-segment with very few coefficients

34 Reconstructing DBDC Sequence Based on real statisticaldistributions of parameters it is possible to generate a drivingcycle that is statistically representative of real driving patternsIn this section we use the inverse discrete cosine transform toreconstruct a sequenceThe discrete cosine transform (DCT)coefficients were produced with the probability functionsobtained in Section 33 Based on the frequency coefficients119910(119896) we can calculate the segment sequence 119909(119899) and thenreconstruct DBDC sequence by connecting different seg-ments The terminal sequence presents the driving cycle ina speed-distance manner

Let 119910(119896) be the coefficients of different frequency com-ponents let 119909(119899) be the sequential data points representingvehicle speed versus distance and let119873 be the total number ofthe data points 119909(119899) then the DBDCs of the route-segmentscan be obtained by the formula below

119909 (119899) =119873minus1sum119896=0

119908 (119896) 119910 (119896) cos(120587 (2119899 + 1) 1198962119873

)

119899 = 0 1 119873 minus 1

(7)

where

119908 (119896) =

1radic119873

119896 = 0

radic2119873

1 le 119896 le 119873 minus 1(8)

and119873 = length(119909) which is the same as length(119910)The seriesis indexed from 119899 = 0 and 119896 = 0

4 Algorithm Analysis

41 Orthogonality Analysis

Definition 3 (record interval 119872) The record interval 119872 isa fixed distance between two successive speed records Ingeneral the distance119872 is one meter

Definition 4 (DBDC sequence 119909(119899)) The DBDC sequence119909(119899) 119899 isin [0119873minus1] is a set of sequential speed-distance pairs119899 is the serial number the value of 119909(119899) is the speed at the 119899thpoint 119909(119899) can be used to represent the vehicle speed versusdistance traveled from the start point in a given route

Definition 5 (DBDC frequency coefficient sequence 119910(119896))The DBDC frequency coefficient sequence 119910(119896) 119896 isin [0119873 minus1] is the set of frequency coefficients 119896 is the serial number119910(119896) is the value of the 119896th coefficient

Theorem 6 Let 119909(119899) 119899 isin [0119873 minus 1] be a DBDC sequenceLet 119910(119896) 119896 isin [0119873 minus 1] be the coefficient sequence of 119909(119899) Ifthe coefficient sequence 119910(119896) was obtained from sequence 119909(119899)by the discrete cosine transform DCT-II then the coefficientsof 119910(119896) are orthogonal

Proof According to the definition of DCT-II

119910 (0) = radic 1119873

119873minus1sum119899=0

119909 (119899)

119910 (119896) = radic2119873

119873minus1sum119899=0

119909 (119899) cos(120587 (2119899 + 1) 1198962119873

)

(9)


Freq

uenc

y

10

20

30

y(0)

440 640280 320 480360 400 520 560 600

Freq

uenc

y

10

20

30

y(1)

120 295minus20 15 15550 85 190 225 260

120 295minus20 15 15550 85 190 225 260

y(10)

Freq

uenc

y10

20

0

Freq

uenc

y

10

20

30

y(2)

40 240minus120 minus80 80minus40 0 120 160 2000

Freq

uenc

y

10

20

30

0

y(50)

30 80minus10 0 4010 20 50 60 70

Freq

uenc

y

minus35 minus30 minus25 minus20 minus15 minus10 minus5 0 5 10

y(100)

0

25

50

Figure 3 Statistical distributions of 119910(0) 119910(1) 119910(2) 119910(10) 119910(50) and 119910(100)

The base vector of them is radic1119873radic2119873 cos(120587(2119899 +1)1198962119873)

According to the Chebyshev polynomials

1198790 (119901) = radic1119873

119879119896(119911119899) = radic

2119873

cos [119896 arccos (119911119899)]

(10)

119879119896(119911119899) is a polynomial represented by parameters 119896 and

119911119899 The119873th polynomial of it is

119879119873(119911119899) = radic

2119873

cos [119873 arccos (119911119899)] (11)

If 119879119873(119911119899) = 0 then

119911119899= cos (2119899 + 1) 120587

2119873 (12)


30

15

Spee

d (k

mh

)

0 3 6 9

Distance (km)

(a) Distance-based driving cycle

minus100

050

900

Valu

es y(0) y(100) y(11040)

Frequency coefficients y(n)

(b) The DCT coefficients

08

09

1

Mea

n sq

uare

erro

r

E[132] = 95

0 2000 4000 6000 8000 10000

(c) The relative energy of 119864[119888]

Figure 4 The energy-compaction property of the DCT

Bringing it in 119879119873(119911119899) we get

119879119896= radic

2119873

cos 119896 arccos [cos (2119899 + 1) 1205872119873

]

= cos (2119899 + 1) 1198961205872119873

(13)

Obviously it is the same with the base vector Since theChebyshev polynomial is orthogonal polynomial the DBDCtransformation is orthogonal too

42 Energy Analysis In fact the key feature of DCT isthe property of energy compaction in which most of theinformation in a given signal can be concentrated in a smallnumber of low-frequency components DCT is closely relatedto the discrete Fourier transform (DFT) but by using onlyreal numbers [23] We can accurately reconstruct a sequencewith only a small number of DCT coefficients In this waythe DBDC sequence (119909(119899) 119899 isin [1119873]) can be expressed byreduced frequency components (119910(119896) 119896 isin [1 119870lowast]) where119870lowast ≪ 119873

In Figure 4 we plot the speed-distance picture of anelectric sanitation vehicle Totally 11041 data points areobtained and each of them is a speed-distance pair We didnot split the route into segments and the whole route waschanged from time domain into frequency domain Aftertransforming the obtained coefficients sequence 119910(119899) wasshown in Figure 4(b) from which we can see that most of

Table 1 Parameters of ZLJ5071TSL electric sanitation vehicle

Vehicle type ZLJ5071TSL sanitation EVChassis type BJ1071VDE0A-1Max mass 7495 kgDriving battery 384V200Ah Li-ionElectromotor rated power 60110 kWFan rated power 28 kW2900 rpmMax velocity 80 kmhMax travel distance (running) 150 kmMax travel distance (working) 50 kmCleaning width 3mGarbage bin 5m3

Gradient coefficient (120579) 0Drag coefficient (119888

119889) 033

Tire rolling resistance (120583) 0015Front area (119860 front) 25m2

Density of air (120588air) 12 kgm3

the energy of the sequence is concentrated into the low-frequency coefficients ahead

For further indicating the energy-compaction propertywe calculated the relative energies between the DBDCsequence 119909(119899) and the new obtained sequence 119909

119888(119899) pro-

duced by cutting down the high-frequency components from119910(119888 + 1) to 119910(119873 minus 1) Let 119864[119888] be the relative energy then

119864[119888] =

sum119888

119899=01003816100381610038161003816119909119888 (119899)

1003816100381610038161003816

sum119873minus1119899=0 |119909 (119899)|

(14)

From Figure 4(c) we can see that 95 energy of theDBDC sequence 119909(119899) can be expressed by 119 (13211041)frequency components of sequence 119910(119899)

5 Experimental Verification

51 DBDC Generation In this section one hundred hours(800 km) of distance-based driving cycle data were acquiredin the city Beijing in February and March of 2014 Theelectric sanitation vehicles used for test are ZLJ5071TSL theequipment number of which is 400 We obtained these datawith twenty electric sanitation vehicles the parameters ofwhich were listed in Table 1

The probability functions of 150 coefficients were extrap-olated from the test data of ZLJ5071TSL using a VC++ scriptThe whole DBDC consists of several route-segments Each ofthemwas constructedwith themethod above Figure 5 showsthe generated DBDC snippets For further investigating andcomparing the recorded data and the created driving profilewe showed the speed-acceleration frequency distributionsof the two DBDCs in Figure 6 in which the frequencycoefficients were normalized There are two frequency peaks(speed = 35 acceleration = 0ms2 speed = 15 acceleration= 0ms2) which appeared in Figure 6(a) Similarly twofrequency peaks appeared in Figure 6(b) too We can see thatthe frequency distributions of the two kinds of driving dataare very similar


Distance (km)0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

15

45

30

0

(a) Produced data

Distance (km)0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

15

45

30

0

(b) Recorded data

Figure 5 The generated and recorded DBDC snippets

52 Frequency Coefficient Analysis In fact one of the keysuperiorities of the proposed DBDC-construction methodis the convenience on reconstructing the frequency andamplitude of the DBDC sequences Having changed thedriving cycle into frequency domain we can reconstruct anew DBDC sequence easily by combining some of the coeffi-cients which appeared in different DBDCs For example weshowed original DBDC sequence 119884 and some new sequences(1198841 1198842 1198848) evolved from 119884 in Figure 7 Most of the newbuilding sequences have only partly frequency componentsFor sequence 1198841 the sequence was constructed with low-frequency components It leads to a smooth profile and onlystructure information is left For sequence 1198847 the sequenceis constructed with high-frequency components So onlydetails are left In fact the low-frequency components aremainly decided by terrain topology or traffic congestion andthe high-frequency components are mainly decided by driv-ing maneuvers or spontaneous reaction Thus we can freelyexamine the influences coming from terrain information ordriving maneuvers by reconstructing the driving cycles withdifferent coefficients

53 Power Demand Analysis We can further examine thephysical impact of a driving profile by comparing the fre-quency features of the power demand Firstly we converta DBDC into a duty cycle Then the internal and externalforces are calculated with a vehicle dynamic model Let 119898be the vehicle mass 119886 the acceleration 119892 the gravitationalacceleration 120579 the road angle 120583 the tire rolling resistance120588air the density of air V the velocity 119888

119889the drag coefficient

119860 front the front area of the vehicle 119865acc the external force foracceleration 119865grav the projected normal force 119865roll the rollingresistance and119865drag the aerodynamic drag forceThus we canpresent the dynamic model like this

119865acc = 119898times119886

119865grav = 119898times119892times sin 120579

Table 2 Parameters of the dynamic model

Max mass 7495 kgGradient coefficient (120579) 0Drag coefficient (119888

119889) 033



119865roll = 120583times119898times119892times cos 120579

119865drag = 120588air times V2 times 119888119889times119860 front2

(15)

In Figure 8 we showed the frequency spectra of severalchanged driving profiles in power frequency distribution plotfor the given electric sanitation vehicle ZLJ5071TSL Thespecifications of it were shown in Table 2The driving profileswe used are sequences 119884 1198841 1198846 and 1198848 We obtained themby adjusting the frequency coefficients of 119884 The specificprocedures and methods are shown in Figure 7

From Figure 8 we can see that the changes in frequencyor amplitude for a given DBDC sequence will lead tolarge changes on power distribution And the changes arealways useful for an electric vehicle designer For examplein Figure 8 (1198841) and (1198846) if we leave only low-frequencycomponents (1198841) or high-frequency components (1198846) ofthe given sequence 119884 the range of the power demandwill become very narrow And the low-frequency compo-nents carry more power demand information If we enlargeonly partial components (1198848) and keep other componentsunchanged the distribution of the power demandwill changegreatly So for an electric vehicle designer it is a special pointof view to analyze the effects of terrain topology drivingcharacteristics road condition and driving maneuvers byreconstructing DBDC with suitable coefficients

54 Comparisons For many traditional driving cycle devel-opingmethodologies selecting test route collecting real dataand constructing driving cycle are three major componentsGenerally speaking the developed driving cycle provides aspeed-time profile that is representative for specific drivingcharacteristics And the driving cycle is usually used fortesting vehicles determining the emission factors and build-ing a platform for governmentrsquos emissions control testingTo clearly show the features of the proposed driving cycledeveloping method we compare it with the traditionalmethods in three aspects driving cycle expression mannerconstruction method and error

541 Comparison-1 (Driving Cycle Expression Manner) Tocompare the two different driving cycle expression manners(DC and DBDC) two separate trips of the same route areshown In Figure 9 two different expression manners areexhibited In (a) the driving cycle was displayed by distanceand it provides the spatial correspondence In (b) the drivingcycle was illustrated in time and it showed the temporal


Speed (kmh)

020

40

0

15

30

450

025

05

075

1Fr

eque

ncy

inde

x

Acceleration (ms)minus20

minus40

(a) Recorded data

Speed (kmh)0

2040

0

15

30

450

025

05

075

1

Freq

uenc

y in

dex


minus40

(b) Generated data

Figure 6 Speed-acceleration frequency distribution histograms for recorded data and generated data

Y = y(0) + y(1) + middot middot middot + y(2066)

Y1 = y(0) + y(1) + middot middot middot + y(10)



Y4 = y(201) + y(202) + middot middot middot + y(500)Y5 = y(501) + y(502) + middot middot middot + y(1000)


Y8 = y(0) + y(1) + middot middot middot + y(1001)998400(1000) + middot middot middot + y998400(2066)+y

Figure 7 Different frequency coefficients

correspondence The trip of the upper driving cycle spannedfor 3912 s and the one below spanned for 3624 s If we changethem into frequency domain the low-frequency coefficientsof them might be available for analyzing the global variationtendency It contains the traffic signal location gradingthe time of the day and so forth From Figure 9(b) wecan observe that the two driving cycles can not reflectthe correlation of the trips (at the same time points 600 sor 1750 s the speeds or accelerations have no correlation)However in (a) the two driving cycles show better corre-spondence (at the same locations 30 km or 99 km both ofthe two speedaccelerations decreased to zero) By comparingthe distinct speed or accelerations at certain points whichappeared in the same route the distinct landmarks (egthe traffic lights and stop signs) and road conditions canbe estimated Thus we can say that the distance-basedrepresenting method (used in this paper) is more suitablethan traditional representing method to develop driving

cycles for electric buses and special purpose vehicles whoseworking or running routes are fixed

542 Comparison-2 (Driving Cycle ConstructionMethod) Infact the most difficult task for developing a driving cycle isto consolidate the huge amount of driving data records intoa representative driving cycle In general random selectionapproach is a commonly usedmethod for constructing candi-date driving cycles with whichmanymicrotrips (bounded byidling periods) are randomly selected to form a driving cycle[1 2 4ndash7] However the random selection approach is onlysuitable for building DCs in a certain region or district Forelectric buses and special purpose vehicles (such as electricsanitation vehicle electric garbage truck and electric tankcar) the working and running routes of them are fixed andsequential Constructing candidate driving cycles with thetraditional random selection method will take the risk oflosing most of the geography information of the routes and


200

150

100

50

0

minus8

0

8 minus15

0

15

30

45

(Y)

Freq

uenc

y

120575power120575t (kws)Power

demand (k

w)

1000

800

600

400

200

0

0

4 minus5

0

5

10

15

(Y6)

Freq

uenc

y

120575power120575t (kws) Power dem

and (kw)

300

200

100

0

minus8

0

8 minus15

0

15

30

45

(Y8)

Freq

uenc

y

120575power120575t (kws)Power d

emand (kw)

300

200

100

0

minus8

0

8 minus15

0

15

30

45

(Y1)

Freq

uenc

y

120575power120575t (kws) Power demand (kw)

minus4

Figure 8 Frequency distribution of different power demands

the obtained driving cycles can not reflect the sequentialfeatures of them This can be seen in Figure 10 where threeDBDCs were shown In (a) the DBDC-1 was constructedwith the random selection method In (b) the DBDC-2 wasconstructed with the proposed method In (c) the DBDC-3 was obtained with the primary data by deleting the idlingperiods The sequence of the microtrips reflects specifictraffic-conditions that appeared in the route so does DBDC-2 which was constructed with the proposed method Thedifference of the two driving cycles DBDC-3 and DBDC-2is that DBDC-2 implied not only the sequence informationof the microtrips but also the statistical information comingfrom all of the recorded data while DBDC-3 is just onetime of running However in (a) since the random selectionmethod is adopted the order of the microtrips is randomly

decided Even though the values of the assessment parameters(appeared in Section 543) in the two driving cycles DBDC-1and DBDC-3 are very similar the obtained driving cycle cannot be used for evaluating the special EVs Why

The reason lies in the characteristics of EVs We cansee this from Figure 11 where a driving cycle and theassociated duty cycle are shown In the figure the rotatespeed RPM is used to illustrate the transient and dynamicperformances involved in the power train systemThe batterypack information including the capacity state-of-charge andenergy profiles is also illustrated From it we can find thatduring the running process the SOC or residual capacity ofthe battery pack keeps decreasing For each of the microtripswhich appeared in the DBDC sequence the correspondingSOC or residual capacity is different If a demand of peak


Distance (km)0 2 4 6 8

0 202 4 6 8 10 12 14 16 18

2010 12 14 16 18

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

0

(a)

Time (s)

0 4000400 800 1200 1600 2000 2400 2800 3200 3600

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

00 4000400 800 1200 1600 2000 2400 2800 3200 3600

(b)

Figure 9 The different expression manners (a) Plots are illustrated in distance (b) Plots are illustrated in time


15

45

30

02010 12 14 16 18

Spee

d (k

mh

)

(a) Driving cycle DBDC-1 (constructed with the random selectionmethod)

0 2 4 6 8 2010 12 14 16 18Distance (km)

15

45

30

0Spee

d (k

mh

)

(b) Driving cycle DBDC-2 (constructed with the proposed method)

Spee

d (k

mh

)

0 202 4 6 8 10 12 14 16 18Distance (km)

15

45

30

0

(c) Driving cycle DBDC-3 (constructed with the primary records)

Figure 10 The generated DBDCs (with different driving cycleconstruction methods)

power appeared in a microtrip close to the start point wherethe residual capacity of the pack is sufficient the demand willbe fulfilled However if it appears in a microtrip close to theend where no enough power can be supplied the demandwill not be fulfilled Thus the sequence of microtrips is a keyfactor for electric vehicles whose working routes are fixedand sequential It should not be randomly constructed

Additionally the energy-recovery characteristic of EVis another reason for us to discard the traditional random

selection method For an EV the effectiveness of energy-recovery system changes with the battery pack status Atdifferent positions (in a route) the residual capacity of thebattery pack is different Ignoring the sequence of microtripsand estimating SOC or residual capacity for an EV withenergy-recovery systemwill lead to large errors For exampleif the battery pack is fully charged at the start point a longdistance of downhill will give no help for charging

543 Comparison-3 (Errors) For most of the driving-cyclegeneration methods after establishing a candidate cycle theassessment parameters will be calculated and compared withthe target statistics (the assessment criteria and parametersare derived and determined in the period of data collection)If each of the assessment parameters is different from thetarget mean values by less than a given value 120598 the candidatecycle will be accepted Otherwise the next candidate cyclewill be sampled and this exercise will repeat In Table 3we compare the errors of the five candidate driving cycles(constructed with the traditional method) with driving cycleDBDC-2 (constructed with the proposed method) Theassessment parameters employed in the comparison processare as follows

(1) Average running speed (1198811)(2) Average acceleration of all acceleration phases (119886)(3) Average deceleration of all deceleration phases (119889)(4) Distance proportions of driving modes-acceleration

(Pa)(5) Distance proportions of driving modes-deceleration

(Pd)(6) Average number of acceleration-deceleration changes

(119873)(7) Root mean square acceleration (RMS)(8) Positive acceleration kinetic energy-PKE (ms2)

By comparing the values of the eight parameters wecan see that the error of DBDC-2 is less than that of the


0 2 4 6 8200

400

300

Volta

ge (V

)

2010 12 14 16 18

0 2 4 6 8

Spee

d (R

PMs)

4

2

02010 12 14 16 18

2010 12 14 16 18

Pow

er (k

W)

0 2 4 6 8Distance (km)

minus60

0

minus30

minus90

Curr

ent (

A)

0 2 4 6 8 2010 12 14 16 18

minus200

0

minus100

0 2 4 6 8

SOC

()

70

90

80

602010 12 14 16 18

Ener

gy (k

Wh)


50

0

25

752010 12 14 16 18

30

90

60

0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

0

Capa

city

(Ah)

0 2 4 6 8

0

2010 12 14 16 18

minus10

minus30

minus20

Figure 11 An example DBDC and the associated parameters

Table 3 Assessment criteria and the values of them in candidate driving cycles and DBDC-2

DC 1198811 (ms) 119886 (ms2) 119889 (ms2) Pa () Pd () 119873 (time) RMS (ms2) PKE (ms2)Average 571 045 048 3862 3566 125 054 016Candidate-1 592 044 049 4051 3342 121 053 015Candidate-2 545 045 049 3965 3506 114 056 017Candidate-3 561 043 047 3961 3382 129 052 015Candidate-4 597 044 048 3972 3412 119 054 016Candidate-5 555 045 047 3821 3642 132 057 017DBDC-2 572 045 048 3888 3561 127 055 016

five candidate driving cycles What is the reason After allboth the average values of the eight parameters and the fivecandidate driving cycles are obtained with the recorded datain time domain while DBDC-2 is constructed in frequencydomainThe reason lies in the two different driving cycle con-struction methods For the traditional construction methodthe first step is to calculate the assessment measures for thecollected data And then a candidate cycle will be developedby randomly selecting the microtrips from the microtrippool Finally if the error between the candidate cycle and the

assessment measure is very small the candidate cycle will bereserved Otherwise a new construction process will beginSince each of the microtrips can only be selected as a wholesection the probability for constructing a driving cycle whosetest statistics are very close to the target statistics is very small

On the contrary for the proposed cycle constructionmethod the frequency coefficients of the route-segments arefirstly calculated And then the probability functions of thesecoefficients are used to generate a randomdriving cycle Sincethe driving cycle is directly constructed with the obtained


probability functions and at the same time the probabilityfunctions are also the evaluation criterions in frequencydomain thus the error between the assessment criteria withthe obtained driving cycle will be very small In fact theerror calculated in time domain and frequency domain isconsistent Thus the test statistics of DBDC-2 are more closeto the target statistics than other five candidate driving cycles

6 Conclusion

For EVs the major purpose for building driving cycle or dutycycle is to develop a method for assessing predicting andtesting the performance andmatching degree of batteries andvehiclesThe road condition and driving habit take importantroles in constructing a complex system to characterize orpredict the longshort term effects of battery performanceand life In fact many pieces of battery pack informationincluding capacity state-of-charge current voltage andstate-of-health have intensively relationships with frequencycharacteristics of DCs For example peak power and thefrequency of duty pulses are primary factors affecting batteryperformance and life the effectiveness of the energy-recoverysystem has great correlations on the high-frequency compo-nents of DCs So developing a novel DC expressing and con-structing method to facilitate EV design is important for us

Conflict of Interests

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

Acknowledgments

The authors are grateful to the anonymous referees for theirinvaluable suggestions to improve the paper This work wassupported by National Science-Technology Support PlanProjects ofChina (2014BAG02B02) International SampTCoop-eration Program of China (ISTCP 2015DFG81930) NationalNatural Science Foundation of China (51175043 51205022)and China Postdoctoral Science Foundation funded project(2013M530534)

References

[1] R Kumar B K Durai W Saleh and C Boswell ldquoComparisonand evaluation of emissions for different driving cycles ofmotorcycles a noterdquo Transportation Research Part D Transportand Environment vol 16 no 1 pp 61ndash64 2011

[2] S Stockar V Marano M Canova G Rizzoni and L GuzzellaldquoEnergy-optimal control of plug-in hybrid electric vehiclesfor real-world driving cyclesrdquo IEEE Transactions on VehicularTechnology vol 60 no 7 pp 2949ndash2962 2011

[3] T Austin F DiGenova T Carlson R Joy K Gianolini and JLee ldquoCharacterization of driving patterns and emissions fromlight-duty vehicles in Californiardquo Tech Rep Sierra ResearchInc Sacramento Calif USA 1993

[4] J Lin and D A Niemeier ldquoEstimating regional air quality vehi-cle emission inventories constructing robust driving cyclesrdquoTransportation Science vol 37 no 3 pp 330ndash346 2003

[5] M Kuhler and D Karstens ldquoImproved driving cycle for testingautomotive exhaust emissionsrdquo SAE Technical Paper 7806501978

[6] J Wu C-H Zhang and N-X Cui ldquoFuzzy energy managementstrategy for a hybrid electric vehicle based on driving cyclerecognitionrdquo International Journal of Automotive Technologyvol 13 no 7 pp 1159ndash1167 2012

[7] S H Kamble T VMathew andG K Sharma ldquoDevelopment ofreal-world driving cycle case study of Pune Indiardquo Transporta-tion Research Part D Transport and Environment vol 14 no 2pp 132ndash140 2009

[8] D Jung J Bang S Choi H Choi and K Min ldquoOptimizationalgorithm for diesel engine operating parameters based on avehicle driving test cyclerdquo Journal of Mechanical Science andTechnology vol 27 no 7 pp 2171ndash2179 2013

[9] J-S Won and R Langari ldquoFuzzy torque distribution control fora parallel hybrid vehiclerdquo Expert Systems vol 19 no 1 pp 4ndash102002

[10] C-C Lin H Peng J W Grizzle and J-M Kang ldquoPower man-agement strategy for a parallel hybrid electric truckrdquo IEEETransactions on Control Systems Technology vol 11 no 6 pp839ndash849 2003

[11] P Rodatz G Paganelli A Sciarretta and L Guzzella ldquoOptimalpower management of an experimental fuel cellsupercapa-citor-powered hybrid vehiclerdquoControl Engineering Practice vol13 no 1 pp 41ndash53 2005

[12] S Delprat T Guerra and J Rimaux ldquoOptimal control of aparallel powertrain from global optimization to real time con-trol strategyrdquo in Proceedings of the IEEE 55th Vehicular Technol-ogy Conference (VTC 02) vol 4 pp 2082ndash2088 2002

[13] M-J Kim and H Peng ldquoPower management and designoptimization of fuel cellbattery hybrid vehiclesrdquo Journal ofPower Sources vol 165 no 2 pp 819ndash832 2007

[14] D Assanis G Delagrammatikas R Fellini et al ldquoAn opti-mization approach to hybrid electric propulsion system designrdquoMechanics of Structures andMachines vol 27 no 4 pp 393ndash4211999

[15] W Gao and S K Porandla ldquoDesign optimization of a parallelhybrid electric powertrainrdquo in Proceedings of the IEEE VehiclePower and Propulsion Conference (VPPC rsquo05) pp 530ndash535September 2005

[16] B Zhang Z Chen C Mi and Y L Murphey ldquoMulti-objectiveparameter optimization of a series hybrid electric vehicle usingevolutionary algorithmsrdquo in Proceedings of the 5th IEEE VehiclePower and Propulsion Conference (VPPC rsquo09) pp 921ndash925September 2009

[17] X Liu Y Wu and J Duan ldquoOptimal sizing of a series hybridelectric vehicle using a hybrid genetic algorithmrdquo in Proceed-ings of the IEEE International Conference on Automation andLogistics pp 1125ndash1129 Jinan China August 2007

[18] S Campanari G Manzolini and F Garcia de la IglesialdquoEnergy analysis of electric vehicles using batteries or fuel cellsthrough well-to-wheel driving cycle simulationsrdquo Journal ofPower Sources vol 186 no 2 pp 464ndash477 2009

[19] J Lin and D A Niemeier ldquoRegional driving characteristicsregional driving cyclesrdquo Transportation Research Part D Trans-port and Environment vol 8 no 5 pp 361ndash381 2003

[20] S Shahidinejad E Bibeau and S Filizadeh ldquoStatistical develop-ment of a duty cycle for plug-in vehicles in a North Americanurban setting using fleet informationrdquo IEEE Transactions onVehicular Technology vol 59 no 8 pp 3710ndash3719 2010


[21] A Sciarretta and L Guzzella ldquoControl of hybrid electric vehi-clesrdquo IEEE Control Systems Magazine vol 27 no 2 pp 60ndash702007

[22] M Fugel T Kassel and F Kucukay ldquoSimulation eines parallelenhybridantriebs im kundeneinsatzrdquo in VDI-Berichte vol 2030pp 125ndash149 2008

[23] N Ahmed T Natarajan and K R Rao ldquoDiscrete cosine trans-formrdquo IEEE Transactions on Computers vol 23 no 1 pp 90ndash931974

[24] A Ravey N Watrin B Blunier D Bouquain and A MiraouildquoEnergy-source-sizingmethodology for hybrid fuel cell vehiclesbased on statistical description of driving cyclesrdquo IEEE Transac-tions on Vehicular Technology vol 60 no 9 pp 4164ndash4174 2011

[25] B Y Liaw and M Dubarry ldquoFrom driving cycle analysis tounderstanding battery performance in real-life electric hybridvehicle operationrdquo Journal of Power Sources vol 174 no 1 pp76ndash88 2007

[26] K Brundell-Freij and E Ericsson ldquoInfluence of street characte-ristics driver category and car performance on urban drivingpatternsrdquo Transportation Research Part D Transport and Envi-ronment vol 10 no 3 pp 213ndash229 2005

[27] J-H Tsai H-L Chiang Y-C Hsu B-J Peng and R-F HungldquoDevelopment of a local real world driving cycle formotorcyclesfor emission factor measurementsrdquo Atmospheric Environmentvol 39 no 35 pp 6631ndash6641 2005

[28] B H Wang Y G Luo and J W Zhang ldquoSimulation of citybus performance based on actual urban driving cycle in ChinardquoInternational Journal of Automotive Technology vol 9 no 4 pp501ndash507 2008

[29] V Schwarzer and R Ghorbani ldquoNew opportunities for large-scale design optimization of electric vehicles using GPU tech-nologyrdquo in Proceedings of the 7th IEEE Vehicle Power andPropulsion Conference (VPPC rsquo11) pp 1ndash6 September 2011

[30] P H Nguyen E Hoang and M Gabsi ldquoPerformance synthesisof permanent-magnet synchronous machines during the driv-ing cycle of a hybrid electric vehiclerdquo IEEE Transactions onVehicular Technology vol 60 no 5 pp 1991ndash1998 2011

[31] T-K Lee B Adornato and Z S Filipi ldquoSynthesis of real-world driving cycles and their use for estimating PHEV energyconsumption and charging opportunities case study for mid-westUSrdquo IEEE Transactions on Vehicular Technology vol 60no 9 pp 4153ndash4163 2011

[32] E Tazelaar J Bruinsma B Veenhuizen and P van den BoschldquoDriving cycle characterization and generation for design andcontrol of fuel cell busesrdquoWorld Electric Vehicle Journal vol 3pp 1ndash8 2009

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


real-time control strategy and tested the algorithm on theNEDC and a highway DC [12] Kim and Peng developeda combined power management and design optimizationframework and tested it for three different DCs [13] Assaniset al described a methodology for integrating vehicle andengine simulations and tested it for the SAE J1711 DC [14]Gao and Porandla used simulated annealing and geneticalgorithm to optimize the design of a parallel hybrid electricvehicle [15] A combined FTP75Highway Fuel Economy Test(HWFET) DC was used to optimize a parallel hybrid electricpower train design Zhang et al proposed an optimizationmethod which has the capability of providing a set of trade-off optimal solutions among the fuel economy and variousemissions and tested it with a combined UDDSHWFETDC [16] Liu et al developed a real-coded adaptive basedhybrid genetic algorithm and applied it to the optimal sizingof a series hybrid electric vehicle [17] The performance ofthe proposed algorithm was verified with the City-Hwy testprocedure in the ADVISOR environment

Additionally a system design or controller configurationmay be best for a given cycle but not robust for another cycleThe parameters optimized for a special DC are not necessarilyoptimal for others or for the entire driving profile [18ndash20]Sciarretta and Guzzella believed that the control and designparameters that perform well under a given conditions maylead to poor behavior under different conditions [21] Fugelet al showed that the fuel consumption obtained on standardDCs does not sufficiently correlate with the consumptionobtained in real conditions [22] For electric buses andspecial purpose vehicles (such as electric sanitation vehicleelectric garbage truck and electric tank car) the working andrunning routes are determined and the traditional drivingcycles can not be used for the special driving missionprofiles An adapted and representative driving cycle has tobe constructed to facilitate the design of them

So in this paper we introduce a new definition of drivingcycle (distance-based driving cycle DBDC) and presenta novel methodology to generate it With the proposedmethodology a representative DBDC can be easily pro-duced and the stochastic parameters can be determined byobserved data or be manually adjustedThe generated DBDCrepresents the stochastic nature of the observed data Theproposed methodology gives the following advantages thatare important for EVs or HEVs optimization

(1) A novel representing method for driving cyclemdashDBDCmdashwas proposed with which a driving cyclecan be presented by a frequency sequence Thecomponents which appeared in this sequence areorthogonal and can be adjusted independently Itis more advantageous than many traditional rep-resenting methods in which many parameters aredependent on each other

(2) A new DC generation method was proposed withwhich the original DBDC array can be compressedinto several frequency coefficients and these coeffi-cients can be determined from observed data or bemanually adjustedThe generated driving profiles canrepresent the stochastic nature of the observed data

(3) The proposed method is very useful for electricvehicle design and the independent frequency coef-ficients are very useful for testing the compatibility ofelectromotor batteries power train and so on

The rest of this paper is organized as follows We firstreview the related work on the discrete cosine transform(DCT) Secondly we represent the DCT-based approach forconstructing distance-based drive cycle Then we describeseveral experiments in which different DBDC-constructionmethods were compared And with these experiments wedemonstrate how to use the methodology to build effectiveand efficient DBDCs Finally we conclude this paper byhighlighting the key contributions of this work

2 Related Work

21 Discrete Cosine Transform (DCT) A discrete cosinetransform is a finite sequence of data points in terms of asum of cosine functions oscillating at different frequencies Itwas first defined by Ahmed et al in 1974 and widely appliedin science and engineering especially in lossy compressionof audios (eg MP3) and images (eg JPEG) [23] It is verysimilar to the discrete Fourier transform (DFT) but usingonly real numbers There are 8 standard DCT variants andonly 4 of them are commonly used The main feature of it isthe property of energy compaction most of the informationin a given signal is concentrated in a few low-frequencycomponents

Formally the discrete cosine transform is a linear invert-ible function 119891 119877119873 rarr 119877119873 (where 119877 denotes the set of realnumbers) or equivalently an invertible119873times119873 square matrixThere are several variants of the DCT with slightly modifieddefinitions and the four commonly used forms are DCT-I DCT-II DCT-III and DCT-IV In each of them 119873 realnumbers 119909

0 119909

119873minus1are transformed into 119873 real numbers

1198830 119883

119873minus1according to the formulas below

DCT-I

119883119896=12(1199090 + (minus1)

119896 119909119873minus1) +

119873minus2sum119899=1

119909119899cos [ 120587

119873 minus 1119899119896]

119896 = 0 1 119873 minus 1

(1)

DCT-II

119883119896=119873minus1sum119899=0

119909119899cos [ 120587

119873(119899+

12) 119896] 119896 = 0 1 119873 minus 1 (2)

DCT-III

119883119896=121199090 +119873minus1sum119899=1

119909119899cos [ 120587

119873119899(119896+

12)]

119896 = 0 1 119873 minus 1

(3)


DCT-IV

119883119896=119873minus1sum119899=0

119909119899cos [ 120587

119873(119899+

12)(119896+

12)]

119896 = 0 1 119873 minus 1

(4)






Spee

d

Time

(a) Drive cycle

Distance

Spee

d






119910 (119896) = 119908 (119896)119873minus1sum119899=0

119909 (119899) cos(120587 (2119899 + 1) 1198962119873

)

119896 = 0 1 119873 minus 1

(5)

where

119908 (119896) =

1radic119873

119896 = 0

radic2119873

1 le 119896 le 119873 minus 1(6)










119909 (119899) =119873minus1sum119896=0

119908 (119896) 119910 (119896) cos(120587 (2119899 + 1) 1198962119873

)

119899 = 0 1 119873 minus 1

(7)

where

119908 (119896) =

1radic119873

119896 = 0

radic2119873

1 le 119896 le 119873 minus 1(8)









119910 (0) = radic 1119873

119873minus1sum119899=0

119909 (119899)

119910 (119896) = radic2119873

119873minus1sum119899=0

119909 (119899) cos(120587 (2119899 + 1) 1198962119873

)

(9)


Freq

uenc

y

10

20

30

y(0)

440 640280 320 480360 400 520 560 600

Freq

uenc

y

10

20

30

y(1)

120 295minus20 15 15550 85 190 225 260

120 295minus20 15 15550 85 190 225 260

y(10)

Freq

uenc

y10

20

0

Freq

uenc

y

10

20

30

y(2)


Freq

uenc

y

10

20

30

0

y(50)

30 80minus10 0 4010 20 50 60 70

Freq

uenc

y


y(100)

0

25

50




1198790 (119901) = radic1119873

119879119896(119911119899) = radic

2119873

cos [119896 arccos (119911119899)]

(10)



119879119873(119911119899) = radic

2119873

cos [119873 arccos (119911119899)] (11)

If 119879119873(119911119899) = 0 then

119911119899= cos (2119899 + 1) 120587

2119873 (12)


30

15

Spee

d (k

mh

)

0 3 6 9

Distance (km)


minus100

050

900

Valu

es y(0) y(100) y(11040)



08

09

1

Mea

n sq

uare

erro

r

E[132] = 95

0 2000 4000 6000 8000 10000




119879119896= radic

2119873

cos 119896 arccos [cos (2119899 + 1) 1205872119873

]

= cos (2119899 + 1) 1198961205872119873

(13)







119889) 033





119888(119899) pro-


119864[119888] =

sum119888

119899=01003816100381610038161003816119909119888 (119899)

1003816100381610038161003816

sum119873minus1119899=0 |119909 (119899)|

(14)






Distance (km)0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

15

45

30

0

(a) Produced data

Distance (km)0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

15

45

30

0

(b) Recorded data






119865acc = 119898times119886




119889) 033





(15)






Speed (kmh)

020

40

0

15

30

450

025

05

075

1Fr

eque

ncy

inde

x


minus40

(a) Recorded data

Speed (kmh)0

2040

0

15

30

450

025

05

075

1

Freq

uenc

y in

dex


minus40

(b) Generated data














200

150

100

50

0

minus8

0

8 minus15

0

15

30

45

(Y)

Freq

uenc

y


demand (k

w)

1000

800

600

400

200

0

0

4 minus5

0

5

10

15

(Y6)

Freq

uenc

y


and (kw)

300

200

100

0

minus8

0

8 minus15

0

15

30

45

(Y8)

Freq

uenc

y


emand (kw)

300

200

100

0

minus8

0

8 minus15

0

15

30

45

(Y1)

Freq

uenc

y


minus4







0 202 4 6 8 10 12 14 16 18

2010 12 14 16 18

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

0

(a)

Time (s)

0 4000400 800 1200 1600 2000 2400 2800 3200 3600

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

00 4000400 800 1200 1600 2000 2400 2800 3200 3600

(b)



15

45

30

02010 12 14 16 18

Spee

d (k

mh

)


0 2 4 6 8 2010 12 14 16 18Distance (km)

15

45

30

0Spee

d (k

mh

)


Spee

d (k

mh

)

0 202 4 6 8 10 12 14 16 18Distance (km)

15

45

30

0













0 2 4 6 8200

400

300

Volta

ge (V

)

2010 12 14 16 18

0 2 4 6 8

Spee

d (R

PMs)

4

2

02010 12 14 16 18

2010 12 14 16 18

Pow

er (k

W)


minus60

0

minus30

minus90

Curr

ent (

A)

0 2 4 6 8 2010 12 14 16 18

minus200

0

minus100

0 2 4 6 8

SOC

()

70

90

80

602010 12 14 16 18

Ener

gy (k

Wh)


50

0

25

752010 12 14 16 18

30

90

60

0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

0

Capa

city

(Ah)

0 2 4 6 8

0

2010 12 14 16 18

minus10

minus30

minus20









6 Conclusion




Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










DCT-IV

119883119896=119873minus1sum119899=0

119909119899cos [ 120587

119873(119899+

12)(119896+

12)]

119896 = 0 1 119873 minus 1

(4)






Spee

d

Time

(a) Drive cycle

Distance

Spee

d






119910 (119896) = 119908 (119896)119873minus1sum119899=0

119909 (119899) cos(120587 (2119899 + 1) 1198962119873

)

119896 = 0 1 119873 minus 1

(5)

where

119908 (119896) =

1radic119873

119896 = 0

radic2119873

1 le 119896 le 119873 minus 1(6)










119909 (119899) =119873minus1sum119896=0

119908 (119896) 119910 (119896) cos(120587 (2119899 + 1) 1198962119873

)

119899 = 0 1 119873 minus 1

(7)

where

119908 (119896) =

1radic119873

119896 = 0

radic2119873

1 le 119896 le 119873 minus 1(8)









119910 (0) = radic 1119873

119873minus1sum119899=0

119909 (119899)

119910 (119896) = radic2119873

119873minus1sum119899=0

119909 (119899) cos(120587 (2119899 + 1) 1198962119873

)

(9)


Freq

uenc

y

10

20

30

y(0)

440 640280 320 480360 400 520 560 600

Freq

uenc

y

10

20

30

y(1)

120 295minus20 15 15550 85 190 225 260

120 295minus20 15 15550 85 190 225 260

y(10)

Freq

uenc

y10

20

0

Freq

uenc

y

10

20

30

y(2)


Freq

uenc

y

10

20

30

0

y(50)

30 80minus10 0 4010 20 50 60 70

Freq

uenc

y


y(100)

0

25

50




1198790 (119901) = radic1119873

119879119896(119911119899) = radic

2119873

cos [119896 arccos (119911119899)]

(10)



119879119873(119911119899) = radic

2119873

cos [119873 arccos (119911119899)] (11)

If 119879119873(119911119899) = 0 then

119911119899= cos (2119899 + 1) 120587

2119873 (12)


30

15

Spee

d (k

mh

)

0 3 6 9

Distance (km)


minus100

050

900

Valu

es y(0) y(100) y(11040)



08

09

1

Mea

n sq

uare

erro

r

E[132] = 95

0 2000 4000 6000 8000 10000




119879119896= radic

2119873

cos 119896 arccos [cos (2119899 + 1) 1205872119873

]

= cos (2119899 + 1) 1198961205872119873

(13)







119889) 033





119888(119899) pro-


119864[119888] =

sum119888

119899=01003816100381610038161003816119909119888 (119899)

1003816100381610038161003816

sum119873minus1119899=0 |119909 (119899)|

(14)






Distance (km)0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

15

45

30

0

(a) Produced data

Distance (km)0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

15

45

30

0

(b) Recorded data






119865acc = 119898times119886




119889) 033





(15)






Speed (kmh)

020

40

0

15

30

450

025

05

075

1Fr

eque

ncy

inde

x


minus40

(a) Recorded data

Speed (kmh)0

2040

0

15

30

450

025

05

075

1

Freq

uenc

y in

dex


minus40

(b) Generated data














200

150

100

50

0

minus8

0

8 minus15

0

15

30

45

(Y)

Freq

uenc

y


demand (k

w)

1000

800

600

400

200

0

0

4 minus5

0

5

10

15

(Y6)

Freq

uenc

y


and (kw)

300

200

100

0

minus8

0

8 minus15

0

15

30

45

(Y8)

Freq

uenc

y


emand (kw)

300

200

100

0

minus8

0

8 minus15

0

15

30

45

(Y1)

Freq

uenc

y


minus4







0 202 4 6 8 10 12 14 16 18

2010 12 14 16 18

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

0

(a)

Time (s)

0 4000400 800 1200 1600 2000 2400 2800 3200 3600

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

00 4000400 800 1200 1600 2000 2400 2800 3200 3600

(b)



15

45

30

02010 12 14 16 18

Spee

d (k

mh

)


0 2 4 6 8 2010 12 14 16 18Distance (km)

15

45

30

0Spee

d (k

mh

)


Spee

d (k

mh

)

0 202 4 6 8 10 12 14 16 18Distance (km)

15

45

30

0













0 2 4 6 8200

400

300

Volta

ge (V

)

2010 12 14 16 18

0 2 4 6 8

Spee

d (R

PMs)

4

2

02010 12 14 16 18

2010 12 14 16 18

Pow

er (k

W)


minus60

0

minus30

minus90

Curr

ent (

A)

0 2 4 6 8 2010 12 14 16 18

minus200

0

minus100

0 2 4 6 8

SOC

()

70

90

80

602010 12 14 16 18

Ener

gy (k

Wh)


50

0

25

752010 12 14 16 18

30

90

60

0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

0

Capa

city

(Ah)

0 2 4 6 8

0

2010 12 14 16 18

minus10

minus30

minus20









6 Conclusion




Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















119909 (119899) =119873minus1sum119896=0

119908 (119896) 119910 (119896) cos(120587 (2119899 + 1) 1198962119873

)

119899 = 0 1 119873 minus 1

(7)

where

119908 (119896) =

1radic119873

119896 = 0

radic2119873

1 le 119896 le 119873 minus 1(8)









119910 (0) = radic 1119873

119873minus1sum119899=0

119909 (119899)

119910 (119896) = radic2119873

119873minus1sum119899=0

119909 (119899) cos(120587 (2119899 + 1) 1198962119873

)

(9)


Freq

uenc

y

10

20

30

y(0)

440 640280 320 480360 400 520 560 600

Freq

uenc

y

10

20

30

y(1)

120 295minus20 15 15550 85 190 225 260

120 295minus20 15 15550 85 190 225 260

y(10)

Freq

uenc

y10

20

0

Freq

uenc

y

10

20

30

y(2)


Freq

uenc

y

10

20

30

0

y(50)

30 80minus10 0 4010 20 50 60 70

Freq

uenc

y


y(100)

0

25

50




1198790 (119901) = radic1119873

119879119896(119911119899) = radic

2119873

cos [119896 arccos (119911119899)]

(10)



119879119873(119911119899) = radic

2119873

cos [119873 arccos (119911119899)] (11)

If 119879119873(119911119899) = 0 then

119911119899= cos (2119899 + 1) 120587

2119873 (12)


30

15

Spee

d (k

mh

)

0 3 6 9

Distance (km)


minus100

050

900

Valu

es y(0) y(100) y(11040)



08

09

1

Mea

n sq

uare

erro

r

E[132] = 95

0 2000 4000 6000 8000 10000




119879119896= radic

2119873

cos 119896 arccos [cos (2119899 + 1) 1205872119873

]

= cos (2119899 + 1) 1198961205872119873

(13)







119889) 033





119888(119899) pro-


119864[119888] =

sum119888

119899=01003816100381610038161003816119909119888 (119899)

1003816100381610038161003816

sum119873minus1119899=0 |119909 (119899)|

(14)






Distance (km)0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

15

45

30

0

(a) Produced data

Distance (km)0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

15

45

30

0

(b) Recorded data






119865acc = 119898times119886




119889) 033





(15)






Speed (kmh)

020

40

0

15

30

450

025

05

075

1Fr

eque

ncy

inde

x


minus40

(a) Recorded data

Speed (kmh)0

2040

0

15

30

450

025

05

075

1

Freq

uenc

y in

dex


minus40

(b) Generated data














200

150

100

50

0

minus8

0

8 minus15

0

15

30

45

(Y)

Freq

uenc

y


demand (k

w)

1000

800

600

400

200

0

0

4 minus5

0

5

10

15

(Y6)

Freq

uenc

y


and (kw)

300

200

100

0

minus8

0

8 minus15

0

15

30

45

(Y8)

Freq

uenc

y


emand (kw)

300

200

100

0

minus8

0

8 minus15

0

15

30

45

(Y1)

Freq

uenc

y


minus4







0 202 4 6 8 10 12 14 16 18

2010 12 14 16 18

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

0

(a)

Time (s)

0 4000400 800 1200 1600 2000 2400 2800 3200 3600

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

00 4000400 800 1200 1600 2000 2400 2800 3200 3600

(b)



15

45

30

02010 12 14 16 18

Spee

d (k

mh

)


0 2 4 6 8 2010 12 14 16 18Distance (km)

15

45

30

0Spee

d (k

mh

)


Spee

d (k

mh

)

0 202 4 6 8 10 12 14 16 18Distance (km)

15

45

30

0













0 2 4 6 8200

400

300

Volta

ge (V

)

2010 12 14 16 18

0 2 4 6 8

Spee

d (R

PMs)

4

2

02010 12 14 16 18

2010 12 14 16 18

Pow

er (k

W)


minus60

0

minus30

minus90

Curr

ent (

A)

0 2 4 6 8 2010 12 14 16 18

minus200

0

minus100

0 2 4 6 8

SOC

()

70

90

80

602010 12 14 16 18

Ener

gy (k

Wh)


50

0

25

752010 12 14 16 18

30

90

60

0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

0

Capa

city

(Ah)

0 2 4 6 8

0

2010 12 14 16 18

minus10

minus30

minus20









6 Conclusion




Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Freq

uenc

y

10

20

30

y(0)

440 640280 320 480360 400 520 560 600

Freq

uenc

y

10

20

30

y(1)

120 295minus20 15 15550 85 190 225 260

120 295minus20 15 15550 85 190 225 260

y(10)

Freq

uenc

y10

20

0

Freq

uenc

y

10

20

30

y(2)


Freq

uenc

y

10

20

30

0

y(50)

30 80minus10 0 4010 20 50 60 70

Freq

uenc

y


y(100)

0

25

50




1198790 (119901) = radic1119873

119879119896(119911119899) = radic

2119873

cos [119896 arccos (119911119899)]

(10)



119879119873(119911119899) = radic

2119873

cos [119873 arccos (119911119899)] (11)

If 119879119873(119911119899) = 0 then

119911119899= cos (2119899 + 1) 120587

2119873 (12)


30

15

Spee

d (k

mh

)

0 3 6 9

Distance (km)


minus100

050

900

Valu

es y(0) y(100) y(11040)



08

09

1

Mea

n sq

uare

erro

r

E[132] = 95

0 2000 4000 6000 8000 10000




119879119896= radic

2119873

cos 119896 arccos [cos (2119899 + 1) 1205872119873

]

= cos (2119899 + 1) 1198961205872119873

(13)







119889) 033





119888(119899) pro-


119864[119888] =

sum119888

119899=01003816100381610038161003816119909119888 (119899)

1003816100381610038161003816

sum119873minus1119899=0 |119909 (119899)|

(14)






Distance (km)0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

15

45

30

0

(a) Produced data

Distance (km)0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

15

45

30

0

(b) Recorded data






119865acc = 119898times119886




119889) 033





(15)






Speed (kmh)

020

40

0

15

30

450

025

05

075

1Fr

eque

ncy

inde

x


minus40

(a) Recorded data

Speed (kmh)0

2040

0

15

30

450

025

05

075

1

Freq

uenc

y in

dex


minus40

(b) Generated data














200

150

100

50

0

minus8

0

8 minus15

0

15

30

45

(Y)

Freq

uenc

y


demand (k

w)

1000

800

600

400

200

0

0

4 minus5

0

5

10

15

(Y6)

Freq

uenc

y


and (kw)

300

200

100

0

minus8

0

8 minus15

0

15

30

45

(Y8)

Freq

uenc

y


emand (kw)

300

200

100

0

minus8

0

8 minus15

0

15

30

45

(Y1)

Freq

uenc

y


minus4







0 202 4 6 8 10 12 14 16 18

2010 12 14 16 18

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

0

(a)

Time (s)

0 4000400 800 1200 1600 2000 2400 2800 3200 3600

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

00 4000400 800 1200 1600 2000 2400 2800 3200 3600

(b)



15

45

30

02010 12 14 16 18

Spee

d (k

mh

)


0 2 4 6 8 2010 12 14 16 18Distance (km)

15

45

30

0Spee

d (k

mh

)


Spee

d (k

mh

)

0 202 4 6 8 10 12 14 16 18Distance (km)

15

45

30

0













0 2 4 6 8200

400

300

Volta

ge (V

)

2010 12 14 16 18

0 2 4 6 8

Spee

d (R

PMs)

4

2

02010 12 14 16 18

2010 12 14 16 18

Pow

er (k

W)


minus60

0

minus30

minus90

Curr

ent (

A)

0 2 4 6 8 2010 12 14 16 18

minus200

0

minus100

0 2 4 6 8

SOC

()

70

90

80

602010 12 14 16 18

Ener

gy (k

Wh)


50

0

25

752010 12 14 16 18

30

90

60

0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

0

Capa

city

(Ah)

0 2 4 6 8

0

2010 12 14 16 18

minus10

minus30

minus20









6 Conclusion




Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










30

15

Spee

d (k

mh

)

0 3 6 9

Distance (km)


minus100

050

900

Valu

es y(0) y(100) y(11040)



08

09

1

Mea

n sq

uare

erro

r

E[132] = 95

0 2000 4000 6000 8000 10000




119879119896= radic

2119873

cos 119896 arccos [cos (2119899 + 1) 1205872119873

]

= cos (2119899 + 1) 1198961205872119873

(13)







119889) 033





119888(119899) pro-


119864[119888] =

sum119888

119899=01003816100381610038161003816119909119888 (119899)

1003816100381610038161003816

sum119873minus1119899=0 |119909 (119899)|

(14)






Distance (km)0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

15

45

30

0

(a) Produced data

Distance (km)0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

15

45

30

0

(b) Recorded data






119865acc = 119898times119886




119889) 033





(15)






Speed (kmh)

020

40

0

15

30

450

025

05

075

1Fr

eque

ncy

inde

x


minus40

(a) Recorded data

Speed (kmh)0

2040

0

15

30

450

025

05

075

1

Freq

uenc

y in

dex


minus40

(b) Generated data














200

150

100

50

0

minus8

0

8 minus15

0

15

30

45

(Y)

Freq

uenc

y


demand (k

w)

1000

800

600

400

200

0

0

4 minus5

0

5

10

15

(Y6)

Freq

uenc

y


and (kw)

300

200

100

0

minus8

0

8 minus15

0

15

30

45

(Y8)

Freq

uenc

y


emand (kw)

300

200

100

0

minus8

0

8 minus15

0

15

30

45

(Y1)

Freq

uenc

y


minus4







0 202 4 6 8 10 12 14 16 18

2010 12 14 16 18

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

0

(a)

Time (s)

0 4000400 800 1200 1600 2000 2400 2800 3200 3600

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

00 4000400 800 1200 1600 2000 2400 2800 3200 3600

(b)



15

45

30

02010 12 14 16 18

Spee

d (k

mh

)


0 2 4 6 8 2010 12 14 16 18Distance (km)

15

45

30

0Spee

d (k

mh

)


Spee

d (k

mh

)

0 202 4 6 8 10 12 14 16 18Distance (km)

15

45

30

0













0 2 4 6 8200

400

300

Volta

ge (V

)

2010 12 14 16 18

0 2 4 6 8

Spee

d (R

PMs)

4

2

02010 12 14 16 18

2010 12 14 16 18

Pow

er (k

W)


minus60

0

minus30

minus90

Curr

ent (

A)

0 2 4 6 8 2010 12 14 16 18

minus200

0

minus100

0 2 4 6 8

SOC

()

70

90

80

602010 12 14 16 18

Ener

gy (k

Wh)


50

0

25

752010 12 14 16 18

30

90

60

0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

0

Capa

city

(Ah)

0 2 4 6 8

0

2010 12 14 16 18

minus10

minus30

minus20









6 Conclusion




Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Distance (km)0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

15

45

30

0

(a) Produced data

Distance (km)0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

15

45

30

0

(b) Recorded data






119865acc = 119898times119886




119889) 033





(15)






Speed (kmh)

020

40

0

15

30

450

025

05

075

1Fr

eque

ncy

inde

x


minus40

(a) Recorded data

Speed (kmh)0

2040

0

15

30

450

025

05

075

1

Freq

uenc

y in

dex


minus40

(b) Generated data














200

150

100

50

0

minus8

0

8 minus15

0

15

30

45

(Y)

Freq

uenc

y


demand (k

w)

1000

800

600

400

200

0

0

4 minus5

0

5

10

15

(Y6)

Freq

uenc

y


and (kw)

300

200

100

0

minus8

0

8 minus15

0

15

30

45

(Y8)

Freq

uenc

y


emand (kw)

300

200

100

0

minus8

0

8 minus15

0

15

30

45

(Y1)

Freq

uenc

y


minus4







0 202 4 6 8 10 12 14 16 18

2010 12 14 16 18

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

0

(a)

Time (s)

0 4000400 800 1200 1600 2000 2400 2800 3200 3600

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

00 4000400 800 1200 1600 2000 2400 2800 3200 3600

(b)



15

45

30

02010 12 14 16 18

Spee

d (k

mh

)


0 2 4 6 8 2010 12 14 16 18Distance (km)

15

45

30

0Spee

d (k

mh

)


Spee

d (k

mh

)

0 202 4 6 8 10 12 14 16 18Distance (km)

15

45

30

0













0 2 4 6 8200

400

300

Volta

ge (V

)

2010 12 14 16 18

0 2 4 6 8

Spee

d (R

PMs)

4

2

02010 12 14 16 18

2010 12 14 16 18

Pow

er (k

W)


minus60

0

minus30

minus90

Curr

ent (

A)

0 2 4 6 8 2010 12 14 16 18

minus200

0

minus100

0 2 4 6 8

SOC

()

70

90

80

602010 12 14 16 18

Ener

gy (k

Wh)


50

0

25

752010 12 14 16 18

30

90

60

0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

0

Capa

city

(Ah)

0 2 4 6 8

0

2010 12 14 16 18

minus10

minus30

minus20









6 Conclusion




Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Speed (kmh)

020

40

0

15

30

450

025

05

075

1Fr

eque

ncy

inde

x


minus40

(a) Recorded data

Speed (kmh)0

2040

0

15

30

450

025

05

075

1

Freq

uenc

y in

dex


minus40

(b) Generated data














200

150

100

50

0

minus8

0

8 minus15

0

15

30

45

(Y)

Freq

uenc

y


demand (k

w)

1000

800

600

400

200

0

0

4 minus5

0

5

10

15

(Y6)

Freq

uenc

y


and (kw)

300

200

100

0

minus8

0

8 minus15

0

15

30

45

(Y8)

Freq

uenc

y


emand (kw)

300

200

100

0

minus8

0

8 minus15

0

15

30

45

(Y1)

Freq

uenc

y


minus4







0 202 4 6 8 10 12 14 16 18

2010 12 14 16 18

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

0

(a)

Time (s)

0 4000400 800 1200 1600 2000 2400 2800 3200 3600

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

00 4000400 800 1200 1600 2000 2400 2800 3200 3600

(b)



15

45

30

02010 12 14 16 18

Spee

d (k

mh

)


0 2 4 6 8 2010 12 14 16 18Distance (km)

15

45

30

0Spee

d (k

mh

)


Spee

d (k

mh

)

0 202 4 6 8 10 12 14 16 18Distance (km)

15

45

30

0













0 2 4 6 8200

400

300

Volta

ge (V

)

2010 12 14 16 18

0 2 4 6 8

Spee

d (R

PMs)

4

2

02010 12 14 16 18

2010 12 14 16 18

Pow

er (k

W)


minus60

0

minus30

minus90

Curr

ent (

A)

0 2 4 6 8 2010 12 14 16 18

minus200

0

minus100

0 2 4 6 8

SOC

()

70

90

80

602010 12 14 16 18

Ener

gy (k

Wh)


50

0

25

752010 12 14 16 18

30

90

60

0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

0

Capa

city

(Ah)

0 2 4 6 8

0

2010 12 14 16 18

minus10

minus30

minus20









6 Conclusion




Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










200

150

100

50

0

minus8

0

8 minus15

0

15

30

45

(Y)

Freq

uenc

y


demand (k

w)

1000

800

600

400

200

0

0

4 minus5

0

5

10

15

(Y6)

Freq

uenc

y


and (kw)

300

200

100

0

minus8

0

8 minus15

0

15

30

45

(Y8)

Freq

uenc

y


emand (kw)

300

200

100

0

minus8

0

8 minus15

0

15

30

45

(Y1)

Freq

uenc

y


minus4







0 202 4 6 8 10 12 14 16 18

2010 12 14 16 18

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

0

(a)

Time (s)

0 4000400 800 1200 1600 2000 2400 2800 3200 3600

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

00 4000400 800 1200 1600 2000 2400 2800 3200 3600

(b)



15

45

30

02010 12 14 16 18

Spee

d (k

mh

)


0 2 4 6 8 2010 12 14 16 18Distance (km)

15

45

30

0Spee

d (k

mh

)


Spee

d (k

mh

)

0 202 4 6 8 10 12 14 16 18Distance (km)

15

45

30

0













0 2 4 6 8200

400

300

Volta

ge (V

)

2010 12 14 16 18

0 2 4 6 8

Spee

d (R

PMs)

4

2

02010 12 14 16 18

2010 12 14 16 18

Pow

er (k

W)


minus60

0

minus30

minus90

Curr

ent (

A)

0 2 4 6 8 2010 12 14 16 18

minus200

0

minus100

0 2 4 6 8

SOC

()

70

90

80

602010 12 14 16 18

Ener

gy (k

Wh)


50

0

25

752010 12 14 16 18

30

90

60

0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

0

Capa

city

(Ah)

0 2 4 6 8

0

2010 12 14 16 18

minus10

minus30

minus20









6 Conclusion




Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











0 202 4 6 8 10 12 14 16 18

2010 12 14 16 18

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

0

(a)

Time (s)

0 4000400 800 1200 1600 2000 2400 2800 3200 3600

15

45

30

0Spee

d (k

mh

)Sp

eed

(km

h)

15

45

30

00 4000400 800 1200 1600 2000 2400 2800 3200 3600

(b)



15

45

30

02010 12 14 16 18

Spee

d (k

mh

)


0 2 4 6 8 2010 12 14 16 18Distance (km)

15

45

30

0Spee

d (k

mh

)


Spee

d (k

mh

)

0 202 4 6 8 10 12 14 16 18Distance (km)

15

45

30

0













0 2 4 6 8200

400

300

Volta

ge (V

)

2010 12 14 16 18

0 2 4 6 8

Spee

d (R

PMs)

4

2

02010 12 14 16 18

2010 12 14 16 18

Pow

er (k

W)


minus60

0

minus30

minus90

Curr

ent (

A)

0 2 4 6 8 2010 12 14 16 18

minus200

0

minus100

0 2 4 6 8

SOC

()

70

90

80

602010 12 14 16 18

Ener

gy (k

Wh)


50

0

25

752010 12 14 16 18

30

90

60

0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

0

Capa

city

(Ah)

0 2 4 6 8

0

2010 12 14 16 18

minus10

minus30

minus20









6 Conclusion




Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 2 4 6 8200

400

300

Volta

ge (V

)

2010 12 14 16 18

0 2 4 6 8

Spee

d (R

PMs)

4

2

02010 12 14 16 18

2010 12 14 16 18

Pow

er (k

W)


minus60

0

minus30

minus90

Curr

ent (

A)

0 2 4 6 8 2010 12 14 16 18

minus200

0

minus100

0 2 4 6 8

SOC

()

70

90

80

602010 12 14 16 18

Ener

gy (k

Wh)


50

0

25

752010 12 14 16 18

30

90

60

0 202 4 6 8 10 12 14 16 18

Spee

d (k

mh

)

0

Capa

city

(Ah)

0 2 4 6 8

0

2010 12 14 16 18

minus10

minus30

minus20









6 Conclusion




Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











6 Conclusion




Acknowledgments


References









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article A DCT-Based Driving Cycle Generation...

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Transcript of Research Article A DCT-Based Driving Cycle Generation...