Research Article Parameter Matching Analysis of …downloads.hindawi.com/journals/jam/2013/615608.pdfResearch Article Parameter Matching Analysis of Hydraulic Hybrid Excavators Based

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 615608 10 pageshttpdxdoiorg1011552013615608

Research ArticleParameter Matching Analysis of Hydraulic Hybrid ExcavatorsBased on Dynamic Programming Algorithm

Wei Shen1 Jihai Jiang1 Xiaoyu Su2 and Hamid Reza Karimi3

1 School of Mechatronics Harbin Institute of Technology Harbin 150080 China2 College of Automation Harbin Engineering University Harbin 150001 China3Department of Engineering Faculty of Engineering and Science University of Agder 4898 Grimstad Norway

Correspondence should be addressed to Wei Shen shenwhiteducn

Received 26 July 2013 Revised 2 September 2013 Accepted 3 September 2013

Academic Editor Baocang Ding

Copyright copy 2013 Wei Shen et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In order to meet the energy saving requirement of the excavator hybrid excavators are becoming the hot spot for researchers Theinitial problem is to match the parameter of each component because the system is tending to be more complicated due to theintroduction of the accumulator In this paper firstly a new architecture is presented which is hydraulic hybrid excavator based oncommon pressure rail combined switched function (HHES) Secondly the general principle of dynamic programming algorithm(DPA) is explainedThen the method by using DPA for parameter matching of HHES is described in detail Furthermore the DPAis translated into the M language for simulation Finally the calculation results are analyzed and the optimal matching group isobtained

1 Introduction

The demand for fuel efficient and low-emission hydraulicexcavators has been increased due to the growing energy cri-sis and environmental deterioration recentlyThe appearanceof hybrid excavator has the immense potential for reducingthe fuel consumption because it can eliminate the throttlingloss theoretically and recover the braking or gravitationalpotential energy Nevertheless the system tends to be morecomplicated by introducing the hydraulic accumulator whichis used as another power source The power flow is alsochanged due to the new power source and the recoveryenergy hence different parameters of the system units canresult in different fuel consumption rate It is important forimproving the system efficiency and reducing the fuel rate ofthe hydraulic hybrid excavator by investigating the parametermatching method which is also a good way to cut down therated engine power and cost

The parameter matching of power transmission systemmakes the parameters of the components in the systemadjust to the working conditions by choosing the param-eters of the components appropriately in the premise that

the system working correctly can guarantee the system inoptimal working condition and then the overall efficiencyof the system is improved the purpose of energy saving isreached [1ndash4] Static matching is the main way in existingmatching methods In this method the maximum valuesin the working process of all actuators are used to choosethe parameters of components However the working char-acteristics of frequent and large-scale power changes whenthe excavator works to some extent lead to oversize ofcomponents However the excavator has characteristic thatmultiple actuators of the excavator act at the same timeso the working conditions and system dynamics under thecondition of composite actions have to be considered tomakevarious components work in high efficiency and reduce thefuel consumption of the engine In the existing optimizationalgorithms the Dynamic Programming Algorithm (DPA)can solve the optimizing problems of any complex systemsin theory so it has been widely used but DPA algorithm ismainly used to solve the optimal trajectory of controlled vari-ables to provide reference for designing suboptimal controller[5ndash9] One of the earliest researchers in this regard is Filipiet al [10] who proposed a design optimization process in

2 Journal of Applied Mathematics

A 4B

6

5 4

4

2

3

1

6

4

6

5

4A

18B14

8 9

7

116

10

5

D DC C

Figure 1 Example of DP

two stages for a parallel hybrid medium truck Then Crossused this algorithm to extend the application in parametermatching [11] In this work the Hydraulic Hybrid Excavatorbased on CPR combined switched function (HHES) is inves-tigated CPR means Common Pressure Rail which is similarwith the electric grid It is divided into two lines includinghigh and low pressure pipelines All of the hydraulic actuatorsare connected with the two lines in parallel it means that it isconvenient to arrange the hydraulic componentsMoreover itnot only eliminates the throttling loss in the theory aspect butcan also recover the braking or gravitational potential energyHence applying this structure on the hydraulic excavator isa promising hydraulic architecture in the aspect of savingenergy However because HHES is a new system there areonly a few relevant research papers published on parametermatching In this paper the optimal control principle basedon DPA is first introduced to the parameters optimizationmatching research of HHES The minimized engine fuelconsumption in typical working condition is treated as theoptimization goal Considering the influence of the factorssuch as the efficiency of components and system dynamicsthe minimum fuel consumption of various componentsparameters matching mode will be excavated most possiblyby choosing a group of optimal parameters and the methodin this paper can guarantee that the fuel consumption ofthe different components parameters can be compared fairlywithout considering the influence of control method

2 Basic Principle

21 Dynamic Program Algorithm Principle DPA algorithm isan effective computing method combined with sorting deci-sion method and optimization principle In 1953 Americanmathematician Robert Bellman proposed the optimizationprinciple in his writing ldquoAn optimal policy has the propertythat whatever the initial state and optimal first decision maybe the remaining decisions constitute an optimal policy withregard to the state resulting from the first decisionrdquo [12]According to this theory the sorting decision can be appliedin a complicated system and ldquooptimization procedurerdquo isused at each level so as to achieve the overall optimizationgoal

Now the basic principle of sorting decision is simplyillustrated by Figure 1 For Figure 1 numbers close to theconnecting lines between two points are the distance of two

points The red lines in Figure 1 show the trajectory between119860 and 119861

119869lowast

119860119861= 119869

119860119863+ 119869

lowast

119863119861 (1)

where 119869119860119863

constitutes the initial control and 119869lowast119863119861

representsthe shortest distance from 119863 to 119861 So we can calculate everypossible route and compare to get the shortest distanceHowever if the number of the points is large it tends to beimpossible to get the suitable result through the calculationprocess

119869lowast

119863119861= min (119869

119863119864119861 119869

119863119865119861 119869

119863119866119861 119869

119863119864119861 119869

119863119864119865119861 119869

119863119866119865119861) (2)

so we can get 119869lowast119860119861= 18

The application of optimization algorithm can reduce thenumber of trajectories to be considered as shown in Figure 1Taking the reverse calculation from point 119861 as an example ifoptimal path passes state point119862 the optimal path between119862and119861 is from the above node to119861 (the required time is 2+5 =7) instead of the path from the below node to 119861 (the requiredtime is 6 + 6 = 12) then the minimum cost and optimalpath from this point to terminal point are determined Byrepeating the calculation process to all stated points theminimum costs and optimal paths for all state points can becalculated and the optimal path of the whole process can beobtained until the calculation of point 119860 is finished Becauseof the iteration method used in DPA the main applicationbackground is for discrete system For continuous system itshould be converted into discrete system and the optimalsolution can be solved after discretization

For a given system the system dynamics can be describedas

119896= 119891 (119883

119896 119906

119896 119889

119896) (3)

where 119883 is the state vector 119906 is the control vector 119889 is thedisturbance vector and the subscript 119896 is the time instantGenerally to simplify the problem the system dynamics canbe described in a discrete domain in other words differentialequations are replaced by difference equations

119883119896+1= 119883

119896+ 119891 (119883

119896 119906

119896 119889

119896) (4)

Generalizing the principle of optimal control to discrete timesystems results in [11]

119862lowast

119896119873(119883

119896 119906

119896) = 119869

119896119896+1(119883

119896 119906

119896 119889

119896) + 119869

lowast

119896+1119873(119883 (119896 + 1)) (5)

where 119862lowast

119896119873is the minimum cost of operation from k to N for

a specific state 119909(119896) and control 119906(119896) The minimum cost ofoperation for all combinations of control is calculated from

119869lowast

119896119873(119883

119896) = min

119906(119896)

[119862lowast

119896119873(119883

119896 119906

119896)] (6)

22 Hydraulic Hybrid Excavator Based on CPR CombinedSwitched Function In CPR the constant pressure variablepump and hydraulic accumulator constitute the high pressureline and the low pressure line is connecting the oil tankdirectly Multiple different loads connect in parallel between

Journal of Applied Mathematics 3

12057531205752A

B

T

E

AB

T

AB

T

Swing

LP

HP

Boom

Arm Bucket

Travel 1 Travel 2

1205732

1205731

1205751

u1

Figure 2 Hydraulic hybrid excavator based on CPR combinedswitched function

the two lines The rotating loads can be controlled byregulating the displacement of hydraulic pumpmotor whilethe linear loads are actuated by hydraulic transformer becausethe hydraulic cylinders are hard to change displacementnormally [13ndash17] Since the system includes secondary com-ponents and accumulators energy can be recovered whenthe actuator brakes or falls and then is stored in the accu-mulator Hence the excavator possesses two kinds of powersource The low fuel consumption can be obtained by usingadopted appropriate control strategy In this configurationthe former three fixed displacement motors which are usedfor swinging and driving respectively should be replacedby three hydraulic pumpmotors [18 19] However the keycomponent is not popular and expensive We propose a newarchitecture which uses on-off valves to switch the hydraulictransformer control and Figure 2 shows the schematic Thereason for this modification is the working condition ofexcavators because the travel part and the arm cylinder orthe bucket cylinder are not working at the same time Sothe fixed displacement motors which are used for travelingin the original nonhybrid excavator can remain There aretravel 1 and arm cylinder in Group 1 and Group 2 includestravel 2 and bucket cylinder Moreover two sets of valvesin which there are four on-off valves are used to switch thehydraulic transformer control motor or cylinder Hence notonly the energy-saving characteristic is remained but alsothe cost can be reduced because of the manipulation of thefixed displacement motor instead of variable displacementpumpmotor Furthermore it is easier tomodify based on theexisting manufacture process

3 Application of DPA for HydraulicHybrid Excavators

The purpose of this paper is to calculate the componentparameter configuration that minimizes the fuel consump-tion in typical working condition of the excavator by DPAalgorithm and a 5 ton LS-control prototype is used asresearch object and the existing components in proto-type should be changed as less as possible to reduce thereform cost The main components of the entire hydraulic

Table 1 Parameter names and their ranges

Parameter name Unit RangeHydraulic accumulator 119881

0Volume (L) 10 16 25 40

Precharge pressure 1198750

Pressure (bar) 50 100 150 200 250Swing pumpmotor 119881

2Volume (L) 28 40 71

system include constant pressure variable pump hydraulicaccumulator and hydraulic transformer and the actuatorscontain boom hydraulic cylinder bucket hydraulic cylinderarm hydraulic cylinder swing motor and travel motors Byusing switch control principle the actuators except for thequantitative swing motor are reserved and the quantitativeswing motor is replaced by variable pumpmotor Becauseof the limitation of current technical level the hydraulictransformers have not been applied widely and the displace-ment of hydraulic transformer is not a choice In additionthe main pump of original system also has the function ofelectronically controlling variables so it has been in useThus Table 1 shows that the components parameters need tobe optimized matching in the entire system

Installation space of a 5-ton excavator is limited sothe optional maximum volume of the hydraulic accumu-lator is determined as 40 L The decision of swing motormainly refers to the existing parameters of the hydraulicpumpmotor We need to know the relevant data of cir-culatory working condition when using DPA algorithm Inaddition we need to determine the state and controlledvariables of the system and the dynamic state equation alsoneeds to be established

31 Working Cycle The standard working cycle is used forcalculation This cycle represents an excavator digging a loadof dirt rotating and releasing the load into a truck or ontoa pile and then returning to its initial position It shouldbe noticed that the travel part is not considered in thispaper This process is divided into four parts Figure 3 showsthe velocity of each actuator respectively [20] During thebeginning part the boom cylinder and the swing keep theposition basically but the arm cylinder and bucket cylindermove out to dig Then the boom cylinder extends and theswing rotates to lift the dirt and prepare for dumping Nextthe bucket cylinder retracts to dump the dirt Finally theswing rotates back and the boom cylinder retracts to go backto the initial status

32 State Variables and Controls of the System The criticalstate variables of the system can be selected by (7) andTable 2shows the symbol and the meaning

119883 = [119899eng 119899119904 1199011 bm 1199012 bm 1199011 119860 119901

2 119860 119901

1 bk 1199012 bk 119901ℎ

Vbm V119860 Vbk] (7)

According to the DPA principle if all the state variablesin the state matrix we establish are unknown then it isdifficult to realize the optimization process because the calcu-lation amount will increase rapidly [11] Hence according to

4 Journal of Applied Mathematics

0 1 2 3 4 5 6 7 8 9Time (s)

BoomArmBucket

Dig DumpLift and turn Return and go down03

02

01

0

minus01

minus02

minus03

minus04

v(m

s)

(a)

1 2 3 4 5 6 7 8 9Time (s)

Dig DumpLift and turn Return and go down86420

minus2

minus4

minus6

minus8

Spee

d (r

pm)

Swing

0

(b)

Figure 3 The velocity of the actuator during the working cycle

Table 2 Meanings of the state variables

Symbol Meaning Unit119899eng Engine speed rpm1198992

Swing speed rpm1199011 bm Boom cylinder bore side Pa1199011 119860

Arm cylinder bore side Pa1199011 bk Bucket cylinder bore side Pa119901ℎ

Pressure of high pressure pipe line in CPR PaVbm Boom cylinder speed msV119860

Arm cylinder speed msVbk Bucket cylinder speed ms

the known working conditions state variables can be dividedinto two categories namely state variables decided by work-ing conditions and the optimal state trajectory calculated byDPA algorithm Because there is no coupling relationshipbetween the engine of HHE and the key state variables insystem the rotating speed of engine and the pressure of highpressure pipe line are selected to be the state variables foroptimization Some state variables are limited by workingcondition requirements other state variables such as thepressure between two chambers of actuators and the resultanttorques (or resultant forces) of actuators calculated by thepressure between two chambers can also be regarded asknown in the calculation process

119883 = [119899eng 119901ℎ] (8)

In addition the critical control of the system is

119862 = [1199061 120573

1 120573

2 120575

1 120575

2 120575

3] (9)

The controls can also be divided into two parts one is beingdecided by the working cycle and the other is the optimizingtrajectory In order to finish the working cycle the torqueand force requirement should be met For instance 120573

2would

be decided during each step after the state variable 119901ℎis

confirmed by the next equation

1205732=

2120587

119901ℎsdot 119881

2

(119872119903+ sign (119899

2) sdot1003816100381610038161003816119872119897

1003816100381610038161003816) (10)

where119872119903is the requirement torque of the swing and119872

119897is

the torque lossHence the free controls are chosen as

119883 = [1199061 120573

1] (11)

33 Discretization of the System After the state and con-trolled variables are determined we need to ensure thescope of the state and controlled variables and perform themesh generation The rotating speed range of the engine isdetermined by the inherent curve of the original engineand the maximum value of the high pressure pipe line isdefined by the allowable maximum pressure 350 bar of thecomponents The interval of the engine rotating speed is100 rpm and the interval of the pressure in high pressure pipeline is 5Mpa both the range of the controlled variables 120583

1 120573

1

being from 0 to 100 The grids are shown in Figure 4 [21]Generally speaking the more dense grids the more

accurate results but the calculated amount will be greatlyincreased The purpose of this paper is to obtain minimumfuel consumption in the same cycle Dynamic performancesof the variable displacement mechanism in pump have notbeen considered so it ismore reasonable to choose the similartime interval with the variable displacement mechanismsince the frequency of the variable displacement mechanismis 5Hz and dt is chosen as 02 s

34 Optimizing Object The fewest fuel consumption rate ofthe engine is the optimization objective for the hydraulic

Journal of Applied Mathematics 5

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

middot middot middot

1300

1400

2700

2600

350345

State Control

0

10

90

100

0 5 95 100

u1

1205731

middot middot middot

middot middot middot

ph min (bar)

Nen

g(r

pm)

ph min

Figure 4 Discretization of the system

hybrid excavator deterministic dynamic programming sim-ulation

119869119888=

119873

sum

119896=1

119891(119896) sdot Δ119905 =

119873

sum

119896=1

119865 (119872eng 119899119890) sdot Δ119905 (12)

Due to the big difference among the different componentsespecially for the excavator which is used widely we considerthe cost combined with the object of optimal fuel consump-tion by using weight factor method

119865 (1198810 119881

2 119901

0 119901max) = 1205721 sdot

119869119888minus 119869min

119869max minus 119869min+ 120572

2sdot119862119888minus 119862min

119862max minus 119862min

(13)

where 119862 represents the additional cost for different compo-nents

35 Equations of System Dynamics

351 Engine Dynamics The engine dynamics is a compli-cated process It is difficult to state the detailed procedureby using mathematical analysis especially how to modela model is not the object of this work Hence one effectivemethod which is based on the experience data is adopted Itmeans that the main torque types such as friction torque andloss are obtained from the lookup table which is calculatedfrom the exact speed and torque For theHHEC the only loadtorque of the engine is the torque of the main pump and thefriction torque

119899eng =1

119869eng[119906

1sdot 119872

119882119874119879minus119872

119901minus119872loss minus119872119891

] (14)

where 119872119901= ((119901

ℎsdot 119881

1)(2 sdot 120587))120573

1is the torque of the main

pump 119872119882119874119879

represents the maximum torque for differentengine speed119872loss is the loss torque which is a lookup tableby using the experimental dates and 119872

119891is the friction

torque

A discrete difference equation is required by using DPAso the continuous differential equations are approximated as

Δ119899eng

=Δ119905

119869eng[119906

1sdot 119872

119882119874119879minus119901ℎsdot 119881

1

2 sdot 1205871205731minus119891

119872loss(119899eng 119901ℎ 1205731)minus119872119891

]

(15)

352 Pressure of the High Pressure Pipe The pressure build-up equation describes the change of pressure in the systemwith respect to time

Because all of the high pressure sides of componentsin CPR are connected together every component flow rateshould be considered to calculate the pressure changeIn detail the high pressure pipe contains a main pumphydraulic accumulator and the actuators which are depictedin Figure 5Thedirection of the flow rate is defined by positiveif coming from the component to the high pressure pipeand negative for the opposite direction Then the pressureis calculated by the following equation and it is noticed thatagain travel motors are omitted in the part

h =1198761minus sum

3

119894=1119876HT 119894

minus 1198762+ sum

3

119894=11198761198602 119894minus 119876

119871

(1120573119890) [sum

3

119894=1119881119894 119886+119881

119898 119886+sum

3

119894=1119860

1198941sdot (119867

119894 sk minus 119897119894)]+119862accu

(16)

where 119894 represents the index of each actuator such as bucketarm and boom cylinderssum3

119894=1119881119894 119886

is the total capacity whichincludes each 119860 port of the HTs every cylinder volume ofthe rod side and the pipe line volume the initial volume ofthe motorpump is represented as 119881

119898 119886119867

119894 sk is the stroke ofeach cylinder and 119897

119894is the displacement of every cylinder119876

119901

represents the output flow rate of the main pump 1198761198602 119894

isthe flow rate of the rod side of each cylinder 119876

2is the flow

rate which goes into the motorpump and119876119871is the total flow

rate of leakage 119876HT 119894is the flow rate that goes into the HT

respectively

6 Journal of Applied Mathematics

Q1

Q2

QHT bkQHT A

1205753

QA1 bkQA1 A

QA2 bkQA2 AQA2 bm

A

T

E

Swing

Boom

AB

T

Arm

AB

T

Bucket

1205732QA1 bm

12057521205751

Caph

1205731

Ch QHT bm

Figure 5 Schematic of the PHP

Also in the previous equation 119862accu is defined as thecapacity of the accumulator which is the function [22]

119862accu =119881119886

119896(

119875pre

119875ℎ

119896+1

)

1119896

(17)

Again the discrete difference equation is as follows

Δ119901ℎ=

Δ119905 sdot (119876119901minus sum

3

119894=1119876HT 119894

minus 1198762+ sum

3

119894=11198761198602 119894minus 119876

119871)

(1120573119890)[sum

3

119894=1119881119894 119886+119881

119898 119886+sum

3

119894=1119860

1198941sdot (119867

119894 sk minus 119897119894)]+119862accu

(18)

where 1198761198602 119894

is confirmed by the working cycle which equalsthe velocity times to the area of the rod side for each cylinderHowever the way to calculate 119876HT 119894

should be pointed outThe SHT of boom cylinder is chosen to show the processThemethod for the other two HTs is the same

The boom cylinder is controlled by regulating the portplate angle of the HT in HHEC Firstly we define thetransformer ratio and the next equation is [23]

120582 =119901119861

119901119860

= (minus sin 1205722sdot sin 120575 minus

119901119879

119901119860

sdot sin120574

2sdot sin(120575 + 120572

2+120574

2))

times (sin120573

2sdot sin(120575 minus 120572

2minus120573

2))

minus1

=119865net bm + 119901ℎ sdot 1198602

119901ℎsdot 119860

1

(19)

where 119865net bm = 1199011 bm sdot 1198601 bm minus 1199012 bm sdot 1198602 bm means the netforce of the boom cylinder because all of the pressure and thearea are known according to the cycle data

Moreover the flow rate of 119860 and 119861 can be obtained by

119876HT bm = 119902119860 =120596HT sdot 119881HT2120587

sdot sin 1205722sdot sin 120575 + 119871

119894119898(119901

119860minus 119901

119861)

+ 119871119894119898(119901

119860minus 119901

119879) + 119871

119890119898119901119860

1198761198601 bm = 119902119861 =

120596HT sdot 119881HT2120587

sdot sin120573

2sdot sin(120575 minus 120572

2minus120573

2)

minus 119871119894119898(119901

119861minus 119901

119860) minus 119871

119894119898(119901

119861minus 119901

119879) minus 119871

119890119898119901119861

(20)

After considering the leakage coefficient in total

1198761198601 bm

119876119867119879 bm

=119902119861

119902119860

=sin (120575 minus (1205722) minus (1205732))

sin 120575= minus120582 (21)

where1198761198601 bm equals the velocity times to the area of the bore

side for boom cylinder and it is also the known data

36 Programming Figure 6 shows thewhole flow chart of theprogram [21] The program can be divided into three loopsin which the inner is the control loop and the middle is thestate loop the outside ones are the district layers which aredivided by district time dt Then every state in per layershould be calculated by using all of the controls through thedynamic equations During the calculation the control valuesresult in the result which exceeds the state domain that shouldbe abandoned and the calculation should go on by usingthe next control values For those accepted controls the fuelconsumption for that state and the controls should be addedAfter comparing all the controls in that state the minimumone is stored The middle loop includes the same cycle foreach state

Figure 7 shows the process in detail119873 represents the stepThe calculation begins from the end In fact the dynamicprogramming is one type of iterative algorithms It beginsfrom the end hence the initial value must be given In thiswork the initial value 119869 and 119906 are set to 0 Some states areunavailable which are represented by red rectangles Theblack cycles represent the minimum fuel consumption valuescorresponding to those states respectively And the bluetriangle means the optimal value in the step All of the fuelconsumption values (matrix 119869) in each step should be used as

Journal of Applied Mathematics 7

Load cycle dates

Meshing the states and controls

N stepsS stagesI controls

The initialvalues

Calculation from the endI = I + 1

Load the states defined by

cycle

Calculate cycle defined controls

Assign free controls

Calculate projected state

Calculate cost for current state

and controls

Is the state admissible

Is the cost minimum for current state

Store minimum cost

Move to next

control

j = C

Save minimum cost to end of

cycle

i = S

Move to next state

I = N

End

Start

Figure 6 Flow chart of the program

the initial value for calculating in the next step For examplethe matrix 119869

1is used for 119873 minus 1 step It should be noticed

that when calculating the new state by using the controls thevalues may not fit well in the mesh grid Hence the bilinearinterpolation algorithm is introduced to calculate the fuelconsumption for 119869

119909119910[24]

119869119909119910= [119869 (119901

119898 119899

119898minus1) minus 119869 (119901

119898 119899

119898minus1)] sdot 119901

119909

+ [119869 (119901119898minus1 119899

119898) minus 119869 (119901

119898minus1 119899

119898minus1)] sdot 119899

119910

+ [119869 (119901119898 119899

119898) + 119869 (119901

119898minus1 119899

119898minus1) minus 119869 (119901

119898 119899

119898minus1)

minus119869 (119901119898minus1 119899

119898)]

sdot 119901119909sdot 119899

119910+ 119869 (119901

119898minus1 119899

119898minus1)

(22)

where 119869(119901119898 119899

119898minus1)119869(119901

119898minus1 119899

119898minus1) 119869(119901

119898 119899

119898) and 119869(119901

119898minus1 119899

119898)

are the fuel consumption which are coming from the formerresults

4 Simulation Results

There are 60 combinations of the three parameters in totalHence the simulation runs 60 times for each group ofparameters It takes about 5 hours once by using a single corecomputer In order to eliminate the influence of the initialstate 5 cycles are input into the simulation but only themiddle three are used to compare the fuel consumption

Figure 8 shows the relationship among 1198810 119881

2 and 119875

0

It can be found the general tendency with the incrementof 119881

0 the fuel consumption decreases However the fuel

consumption reduces slowly after 1198810approaches 40 L 119881

2is

not independent from the other parameters but it is coupledwith 119881

0and 119901

0 In general the fuel consumption reduces

with the increment of 1198812 and it shows the similar tendency

with 1198810 that is the fuel consumption reduces slowly after

1198810approaches 40 L Furthermore the precharge pressure is

a key variable to impact the fuel consumption The optimalpressure value locates from 100 bar to 150 bar normallyaccording to the simulation results

In order to state it in detail different fuel consumptionvalues corresponding to different precharge pressure values ofthe 16 L accumulator are plotted in Figure 9 which shows thatthe minimum fuel consumption appears in 150 bar All of thesimulation results show the similar trend This is because theenergy storage reaches the maximum around this pressurelevel

119864 = minus int

119881119891

119881119894

119901119889119881 =11990101198810

119899 minus 1[(119881

1198810

)

1minus119899

minus 1]

=11990101198810

119899 minus 1[(1199010

119901)

(1minus119899)119899

minus 1]

(23)

To get the precharge pressure which results in the maximumenergy the derivation of E is calculated as follows

119889119864

1198891199010

=1198810

119899 minus 1[1

119899(1199010

119901)

(1minus119899)119899

minus 1] = 0 (24)

119901

1199010

= 119899119899(119899minus1)

(25)

It means that if the maximum pressure and the parameter119899 are decided then the optimal precharge pressure can beobtained from (25) Hence the same accumulator under theoptimal precharge pressure can store the maximum energyand then the fuel consumption can be reduced

In general large components have low fuel consump-tion under the same condition Because in this algorithmthe minimum engine fuel consumption is taken as theoptimizing objective so every state pursues the highestefficiency However for key components such as the axialpiston type component the efficiency gets lower with thepressure increasing then when we want to achieve the sametorque large components can reach the purpose of efficiencyimprovement in smaller pressure conditions however weneed to take into account the price growth of completemachine After the comprehensive comparison a set ofparameters we choose are 119881

2= 40mLr 119875

0= 15Mpa and

1198810= 16 L

8 Journal of Applied Mathematics

Dynamic calculation

N1

Stages

Steps

(pmminus1 nm)

(pmminus1 nmminus1)

(pm nm)

(px ny)

(pm nmminus1)

u2

u1

nengneng

phph

J1

Δt

middot middot middot

middot middot middot

N minus 1

Figure 7 The detailed calculation process of the program

10 15 20 25 30 35 40

50100

150200

2500018

002

0022

0024

V1 = 45mLrV2 = 28mLr

Cos

t (g)

Minimum pressure (bar) V0 (L)

(a)

0016001700180019

00200210022

10 15 20 25 30 35 40

50100

150200

250

V1 = 45mLrV2 = 40mLr

Cos

t (g)

Minimum pressure (bar) V0 (L)

(b)

0016001700180019

00200210022

10 15 20 25 30 35 40

50100

150200

250

V1 = 45mLrV2 = 71mLr

Cos

t (g)

Minimum pressure (bar) V0 (L)

(c)

Figure 8 Fuel consumption with the changing accumulator volume and precharge pressure

Journal of Applied Mathematics 9

50 100 150 200 250001

0015

002

0025

Minimum pressure (bar)

Cos

t (g)

V0 = 16L

Figure 9 Fuel consumption with the different 1199010under the same

1198810= 16 L

5 Conclusion

Optimal parametermatching results forHHESwere analyzedwith the aim of reducing the fuel consumption and modi-fication cost Firstly a new architecture HHES is presentedwhich not only keeps the advantages of the hydraulic hybridexcavator but also reduces the modification cost Then theDPA was applied in the matching process successfully Theresults show that the fuel consumption reduces with theincrement of the 119881

0 And the similar tendency is obtained

for the swing pumpmotor However it is coupled with 1198810

and 1199010 The precharge pressure shows the independent rela-

tionship for the fuel consumption among other parametersThe optimal value is located around 10sim15Mpa under theconditions that themaximumpressure is 35Mpa and 119899 is 125By combining the cost factor the optimal group is obtainedwhich is 119881

2= 40mLr 119875

0= 15Mpa and 119881

0= 16 L The

future work will focus on the optimal trajectory of the statevariable based on the dynamic programming result firstlyThen design the suboptimal control strategy according to theoptimal trajectory and test it in the real excavator

Conflict of Interests

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

Acknowledgments

The authors acknowledge the contribution of National Nat-ural Science Foundation of China (50875054 and 51275123)andOpen fund of State Key Laboratory of Fluid Power Trans-mission and Control Zhejiang University (GZKF-2008003)

References

[1] Q Xiao and Q Wang ldquoParameter matching method for hybridpower system of hydraulic excavatorrdquoChina Journal of Highwayand Transport vol 21 no 1 pp 121ndash126 2008

[2] T Li D Zhao H Kang and Z Zhang ldquoParameter matchingof parallel hybrid power loadersrdquo Journal of Jilin University vol42 no 4 pp 916ndash921 2012

[3] H Li H Liu H Gao and P Shi ldquoReliable fuzzy control foractive suspension systems with actuator delay and faultrdquo IEEETransactions on Fuzzy Systems vol 20 no 2 pp 342ndash357 2012

[4] H Li J Yu C Hilton and H Liu ldquoAdaptive sliding modecontrol for nonlinear active suspension systems using T-S fuzzymodelrdquo IEEE Transactions on Industrial Electronics vol 60 no8 pp 3328ndash3338 2013

[5] S M Aceves J R Smith L J Perkins et al ldquoOptimization ofCNG series hybrid concept vehiclerdquo SAE Paper 960234 SAEWarrendale Pa USA 1996

[6] B Wu C C Lin Z Filipi H Peng and D Assanis ldquoOptimalpower management for a hydraulic hybrid delivery truckrdquoVehicle System Dynamics vol 42 no 1-2 pp 23ndash40 2004

[7] A brahma Y Guezennec and G Rizzoni ldquoOptimal energymanagement in series hybrid electric vehiclesrdquo Proceedings ofAmerican Control Conference vol 1 no 6 pp 60ndash64 2000

[8] C Lin Z Filipi L Louca H Peng D Assanis and J SteinldquoModelling and control of a medium-duty hybrid electrictruckrdquo International Journal of Heavy Vehicle Systems vol 11 no3-4 pp 349ndash370 2004

[9] B Wu C Lin Z Filipi et al ldquoOptimization of power man-agement strategies for a hydraulic hybrid medium truckrdquo inProceedings of the 2002 Advanced Vehicle Control ConferenceHiroshima Japan 2002

[10] Z Filipi L Louca B Daran et al ldquoCombined optimisationof design and power management of the hydraulic hybridpropulsion system for the 6 times 6 medium truckrdquo InternationalJournal of Heavy Vehicle Systems vol 11 no 3-4 pp 372ndash4022004

[11] M Cross and Z Filipi ldquoSeries hydraulic hybrid propulsion fora light truck optimizing the thermostatic power managementrdquoin Proceedings of the 8th International Conference on Engines forAutomobile SAE Technical Paper 2007-24-0080 Capris Italy2007

[12] D Feng Study on power management strategies for a serieshydraulic hybrid vehicle [Doctor thesis] University of ElectronicScience and Technology of China Chengdou China 2009

[13] T J Kim Optimal control an effective method for designinghydraulic hybrid vehicles [MS thesis] Purdue University WestLafayette Ind USA 2011

[14] W Shen and J Jiang ldquoAnalysis and development of the hydraulicsecondary regulation system based on the CPRrdquo in Proceedingsof theInternational Conference on Fluid Power andMechatronics(FPM rsquo11) pp 117ndash122 August 2011

[15] P A J Achten F Zhao and G E M Vael ldquoTransformingfuture hydraulics a new design of a hydraulic transformerrdquo inProceedings of the 5th Scandinavian International Conference onFluid Power Linkoping Sweden May 1997

[16] W Shen and J Jiang ldquoAnalysis of energy recovery efficiency ofhydraulic hybrid excavatorrdquo Journal of South China Universityof Technology vol 40 no 1 pp 82ndash87 2012

[17] K K Ahn T H Ho and Q T Dinh ldquoA study on energy savingpotential of hydraulic on energy saving potential of hydrauliccontrol system using switching type closed loop constantpressure systemrdquo in Proceedings of the 7th JFPS InternationalSymposium on Fluid Power pp 317ndash322 2008

[18] W Shen J Jiang and H R Karimi ldquoObserver-based robustcontrol for hydraulic velocity control systemrdquo MathematicalProblems in Engineering vol 2013 Article ID 689132 9 pages2013

10 Journal of Applied Mathematics

[19] W Shen and J Jiang ldquoStudy on control performance and energyrecovery efficiency of hydraulic hybrid excavatorrdquo ICIC ExpressLetters vol 5 no 12 pp 4439ndash4444 2011

[20] J Zimmerman and M Ivantysynova ldquoThe effect of systempressure level on the energy consumption of displacementcontrolled actuator systemsrdquo in Proceedings of the 5th FPNI PhDSymposium pp 77ndash92 Krakow Poland 2008

[21] J Zimmerman Toward optimal multi-actuator displacementcontrolled mobile hydraulic system [PhD thesis] School ofMechanical Engineering Purdue University West LafayetteInd USA 2012

[22] J Zimmerman R Hippalgaonkar andM Ivantysynova ldquoOpti-mal control for the series-paralleld displacement controlledhydraulic hybrid excavatorrdquo in Proceedings of the ASME 2011Dynamic Systems and Control Conference vol October 2011 pp1ndash8

[23] P A J Achten G E M Vael and F Zhao ldquoThe Innas hydraulictransformermdashthe key to the hydrostatic common pressure railrdquoSAE Paper 2000-01-2561 SAE Warrendale Pa USA

[24] L Chen and C Gao ldquoFast discrete bilinear interpolationalgorithmrdquo Computer Engineering and Design vol 28 no 15pp 3787ndash3790 2007

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