This document summarizes the process which utilized EPA's laboratory engine and chassis dynamometer test data to generate a full engine fuel consumption map for the 2015 Ford 2.7L EcoBoost V6 engine from the 2015 Ford F150 tested on Tier 3 fuel. The generated full engine map defines the complete operating boundaries of the engine which are needed for ALPHA modeling including idle, WOT, minimum torque, and maximum speed.
SUGGESTED CITATION:
2015 Ford 2.7L EcoBoost V6 Engine Tier 3 Fuel - ALPHA Map Package. Version 2019-11. Ann Arbor MI: US EPA National Vehicle and Fuel Emissions Laboratory, National Center for Advanced Technology, 2019.
Engine Physical Characteristics
Set physical characteristics based on published information titled: 2015 FORD F-150 TECHNICAL SPECIFICATIONS. (n.d.). Retrieved from https://media.ford.com/content/dam/fordmedia/North%20America/US/2015_Specs/2015_F150_Specs.pdf
engine = [];
engine.name = '2015 Ford 2.7L EcoBoost V6 Engine Tier 3 Fuel';
engine.combustion_type = enum_engine_combustion_type.spark_ignition;
engine.displacement_L = 2.694;
engine.num_cylinders = 6;
engine.compression_ratio = 10.0;
engine.bore_mm = 83;
engine.stroke_mm = 83;
Test Data
Import EPA engine dynamometer test data from "2015 Ford 2.7L EcoBoost V6 Engine LEV III Fuel Cell 9 - Test Data Package. Version 2019-11. Ann Arbor, MI: US EPA, National Vehicle and Fuel Emissions Laboratory, National Center for Advanced Technology, 2019." and from "2015 Ford 2.7L EcoBoost V6 Engine Tier 3 Fuel HD1 - Test Data Package. Version 2019-11. Ann Arbor, MI: US EPA, National Vehicle and Fuel Emissions Laboratory, National Center for Advanced Technology, 2019." The Cell 9 data set includes detailed steady-state operation measurements at mid and low engine torque. The HD1 data set was run on a higher capacity engine dynamometer and includes transient measurements at high engine torque. These two data sets are combined to generate the full engine map that defines the complete operating boundaries of the engine including, idle, WOT, minimum torque, and maximum speed.
test_data = readtable2('4- 2015 Ford 2.7L EcoBoost V6 Engine LEV III Fuel Cell 9 - Test Data.xlsx','Sheet','Steady State','VariableNamesRange',4,'VariableUnitsRange',5,'DataRange',6); ss_test_data.name = 'SS Test Data - Cell 9'; ss_test_data.speed_rpm = test_data.Speed; ss_test_data.torque_Nm = test_data.Torque; ss_test_data.fuel_gps = test_data.FuelMeterFlow; ss_test_data.manifold_press_kPaA = test_data.IntakeManifoldPress; test_data = readtable2('4- 2015 Ford 2.7L EcoBoost V6 Engine Tier 3 Fuel HD1 - Test Data.xlsx','Sheet','High Load Initial','VariableNamesRange',4,'VariableUnitsRange',5,'DataRange',6); hl_test_data.name = 'HL Test Data - Cell HD1'; hl_test_data.speed_rpm = test_data.Speed; hl_test_data.torque_Nm = test_data.Torque; hl_test_data.fuel_gps = test_data.InjectorFuelFlow; hl_test_data.manifold_press_kPaA = test_data.IntakeManifoldPress;
Additional Fueling Data
Add estimated points from vehicle chassis testing for improved loaded idle fuel consumption estimates; Published report pending. The test data did not contain engine torque measurements, so they have been estimated from transmission and accessory information. Additionally these test were conducted with Tier 2 fuel, so the quantities have been adjusted based on the fuel net heating value.
est_idle_data.name = 'Estimated Idle Data';
est_idle_data.speed_rpm = [ 625, 580, 600];
est_idle_data.torque_Nm = [ 12, 30, 80];
est_idle_data.fuel_gps = [ 0.248, 0.200, 0.45];
Maximum Torque (WOT) Curve Data
Use "2015 FORD F-150 TECHNICAL SPECIFICATIONS." (n.d.). Retrieved from https://media.ford.com/content/dam/fordmedia/North%20America/US/2015_Specs/2015_F150_Specs.pdf
Peak Torque: 375 ft-lbs @ 3000 RPM
Peak Power: 325 hp @ 5750 RPM
Additional points added to produce an approximate curve:
WOT_data{1}.speed_rpm = [2000, 3000, 4250, 5750];
WOT_data{1}.torque_Nm = [340, 375, 360, 297] * convert.ftlbs2Nm;
Minimum Torque (Zero Accelerator Pedal) Curve Data
Use zero pedal test points provided in the Engine Test Data packages referenced above; no additional inputs are necessary.
Fuel Properties
Load fuel properties described in the Engine Test Data packages referenced above. The two tests used different Lev III / Tier 3 fuels as indicated by the Fuel Analysis report numbers. The Cell 9 data using the Lev III fuel was adjusted based on the heating value.
engine.fuel = class_REVS_fuel('FTAG 25402');
disp(engine.fuel);
class_REVS_fuel with properties:
id: 'FTAG 25402'
description: 'Tier 3 Cert'
density_kgpL_15C: 0.74506
energy_density_MJpkg: 41.791
carbon_weight_fraction: 82.67
research_octane_number: 97.8
motor_octane_number: 88.4
alcohol_pct_vol: 9.8
gCO2pgal: 8.5434e+05
energy_density_BTUplbm: 17967
specific_gravity: 0.74578
alternate_fuel = class_REVS_fuel('FTAG 24670');
disp(engine.fuel);
class_REVS_fuel with properties:
id: 'FTAG 25402'
description: 'Tier 3 Cert'
density_kgpL_15C: 0.74506
energy_density_MJpkg: 41.791
carbon_weight_fraction: 82.67
research_octane_number: 97.8
motor_octane_number: 88.4
alcohol_pct_vol: 9.8
gCO2pgal: 8.5434e+05
energy_density_BTUplbm: 17967
specific_gravity: 0.74578
tier_2_fuel = class_REVS_fuel('FTAG 25278');
disp(engine.fuel);
class_REVS_fuel with properties:
id: 'FTAG 25402'
description: 'Tier 3 Cert'
density_kgpL_15C: 0.74506
energy_density_MJpkg: 41.791
carbon_weight_fraction: 82.67
research_octane_number: 97.8
motor_octane_number: 88.4
alcohol_pct_vol: 9.8
gCO2pgal: 8.5434e+05
energy_density_BTUplbm: 17967
specific_gravity: 0.74578
ss_test_data.fuel_gps = ss_test_data.fuel_gps .* engine.fuel.energy_density_MJpkg / alternate_fuel.energy_density_MJpkg; est_idle_data.fuel_gps = est_idle_data.fuel_gps .* engine.fuel.energy_density_MJpkg / tier_2_fuel.energy_density_MJpkg;
Idle Speed
Set estimated idle speed using preliminary EPA chassis dynamometer test data for the 2015 Ford F150; EPA published report pending.
engine.idle_speed_radps = 700 * convert.rpm2radps ;
Engine Build for Simulation in ALPHA
Construct the ALPHA engine, combining all of the above inputs. Fit the fueling data onto a grid for simple interpolation and establish points that represent the maximum and minimum engine torque versus speed. Confirm the quality of fit between the input data and the resulting engine description by examining the resulting plots.
engine = REVS_build_engine(engine, {ss_test_data, hl_test_data, est_idle_data}, 'plot_bsfc', 'plot_efficiency');
Idle Fuel Verification
Confirm appropriate fuel consumption near idle conditions. The conditions below represent the idle observed during chassis testing using estimates for accessory load and with engine torque. The speeds and fuel flow agree with measurements recorded on this engine however; in general, idle speeds and fueling depend on the drivetrain the engine is coupled to and may change depending on the application.
% Idle @ 580 RPM in drive measured fuel rate ~0.200 g/sec engine.interp_fuel_gps( 580 * convert.rpm2radps, 30); % Idle @ 625 RPM in neutral measured fuel rate ~0.248 g/sec engine.interp_fuel_gps( 625 * convert.rpm2radps, 12);
0.21051
0.22759
Fuel Map Linearity
Confirm expected trends in output fuel map, specifically in regions where test data is unavailable.
REVS_plot_engine_fuel_linearity(engine)
Model Calibration Options
Calibrate additional properties of the ALPHA engine model which can be set to match observed behaviors during the validation process.
Note: Default values are used unless properties are specified below, which are derived from the composite set of EPA testing and model validation activities.
Accelerator Pedal Map
Apply default pedal mapping for passenger cars, which will linearly increase engine power with accelerator pedal position.
engine.pedal_map_type = enum_engine_pedal_map_type.max_engine_power;
Deceleration Fuel Cut-Off (DFCO) Transition Adjustment
Calibrate properties for additional fueling after a DFCO event that aligns with test data.
This sample test data shows a return to fueling after DFCO, and the additional fuel used relative to the steady state map. Chassis dynamometer test data was used to calibrate the values below.
engine.DFCO_min_duration_secs = 2.0; % DFCO defueled duration before triggering additional fueling engine.DFCO_refuel_multiplier = [1.0, 1.3, 1.0]; % DFCO refuel multiplier profile engine.DFCO_refuel_multiplier_time_secs = [0.0, 0.1, 4.0]; % DFCO refuel multiplier profile time [sec]
File Description
Generate and write the created ALPHA engine definition into a file for later simulation.
write_REVS_engine( 'engine_2015_Ford_EcoBoost_2L7_Tier3.m', engine);
Engine Build: engine_2015_Ford_EcoBoost_2L7_Tier3.m
% ALPHA ENGINE DEFINITION % Generated 26-Nov-2019 14:03:35 % Constructor engine = class_REVS_engine(); engine.name = '2015 Ford 2.7L EcoBoost V6 Engine Tier 3 Fuel'; engine.source_filename = mfilename;engine.matrix_vintage = enum_matrix_vintage.present; % Physical Description engine.displacement_L = 2.694; engine.num_cylinders = 6; engine.combustion_type = enum_engine_combustion_type.spark_ignition; engine.compression_ratio = 10; engine.bore_mm = 83; engine.stroke_mm = 83; % Maximum Torque Curve engine.full_throttle_speed_radps = [ 0.00000000000000000 ; 105.03391438501875 ; 157.18435243460931 ; 183.36429121452426 ; 209.54422999443918 ; 235.72416877435413 ; 315.54389323222813 ; 367.31035884637993 ; 420.10075095503510 ; 473.46419283476172 ; 525.51863777549261 ; 575.68638179481752 ; 604.47070088455837 ; 633.25501997429933 ]; engine.full_throttle_torque_Nm = [ 0.00000000000000000 ; 232.08000000000001 ; 363.56999999999999 ; 421.47000000000003 ; 457.49000000000001 ; 454.13000000000000 ; 479.25499999999988 ; 461.78888888888895 ; 455.93333333333328 ; 433.23000000000002 ; 418.07083333333327 ; 382.85000000000002 ; 328.15714285714290 ; 0.00000000000000000 ]; engine.naturally_aspirated_speed_radps = [ 0.00000000000000000 ; 157.19744240399928 ; 209.55170997694773 ; 261.90410755426905 ; 366.68967613113944 ; 471.48192756884436 ; 525.30807553883392 ; 633.25501997429933 ]; engine.naturally_aspirated_torque_Nm = [ 180.81847488547547 ; 171.09394843590599 ; 162.67011541367296 ; 174.15161620570851 ; 181.38153555259967 ; 214.54295007430514 ; 209.89137951473592 ; 166.71947068814549 ]; % Minimum Torque Curve engine.closed_throttle_speed_radps = [ 0.00000000000000000 ; 131.00441365469436 ; 288.08404633418399 ; 314.26398511409894 ; 471.34361779358858 ; 633.25501997429933 ]; engine.closed_throttle_torque_Nm = [ -9.8423753846153854 ; -10.720000000000001 ; -20.280000000000001 ; -27.460000000000001 ; -13.000000000000000 ; -14.084676676923078 ]; % Fuel Map engine.fuel_map_speed_radps = [ 0.00000000000000000 ; 31.459559767197788 ; 62.919119534395577 ; 104.87159876458327 ; 131.00441365469436 ; 157.19744240399928 ; 183.37045120011953 ; 209.55170997694773 ; 235.73032875994937 ; 261.90410755426905 ; 288.08712632698155 ; 314.31052722748552 ; 366.68967613113944 ; 419.09090531649349 ; 471.48192756884436 ; 498.39500155383917 ; 525.30807553883392 ; 575.92736789795401 ; 604.59119393612673 ; 633.25501997429933 ]; engine.fuel_map_torque_Nm = [ -28.833000000000002 ; -12.955833333333334 ; 6.3006018518518516 ; 25.557037037037038 ; 46.987154882154876 ; 68.417272727272717 ; 96.838750000000005 ; 105.69100000000000 ; 118.74500000000000 ; 148.19499999999999 ; 181.41750000000002 ; 203.42916666666667 ; 228.28970588235293 ; 254.16499999999999 ; 283.63850000000002 ; 301.06270588235293 ; 318.48691176470584 ; 340.56046296296290 ; 368.60109907120739 ; 395.56000000000000 ; 410.11434228650137 ; 439.20826719576723 ; 457.02361111111111 ; 479.25499999999988 ; 503.21774999999991 ]; engine.fuel_map_gps = [ 0.12960299435860118 0.18578887157746796 0.23766523881917367 0.29476071033436235 0.32429986799551874 0.35349088842988724 0.38715058846399419 0.42133399536505695 0.45136444026731348 0.49030472282356458 0.54911494576774289 0.61529473557432124 0.76735621774867513 0.91958653153100034 1.0225394790806557 0.99695861425011723 0.97419778485681396 0.95791618192981665 0.95519507748192289 0.95370723361928988 ; 0.10234046853934604 0.16944031654390440 0.23452459495430372 0.30212132876504988 0.34037775592539116 0.37216534167205478 0.41629905355519159 0.46564242957426544 0.50937451202464079 0.55600924687884212 0.60618164371625338 0.65090411422959815 0.80652410057556967 0.99750369081221302 1.1999401121908029 1.2900231025712552 1.3586413740746850 1.4426054397857055 1.4843938247808628 1.5255516320239848 ; 0.072092100944542270 0.14832922389961847 0.22895608819555846 0.30696810211876441 0.35749879743237384 0.40692379536670847 0.45961864228099109 0.51929071841755692 0.59408978183445760 0.66778625101034239 0.72608309654783831 0.81571713397590084 1.0138515488097439 1.2196954629067864 1.5262416225736044 1.6881613672246727 1.8323491702799899 2.0299220242710514 2.1253432090433373 2.2179367995671040 ; 0.051253094327114973 0.13003875255428538 0.20584690567712793 0.32701699819301699 0.41144554579248405 0.49075820795896979 0.58104329486567596 0.67529615536226673 0.78235399275728768 0.88403634509125906 1.0050050299099211 1.1637707488178450 1.4071485231076355 1.6170383674573809 1.9683823440847930 2.1554334197320997 2.3348906241937506 2.6211030166233313 2.7663617939260061 2.9080835682245878 ; 0.042038316994002671 0.14822129295582592 0.25589235417747308 0.43180843739588526 0.56404715129595118 0.68988315746167928 0.82177202257947068 0.93548172470202573 1.0556219571071717 1.1860379603908677 1.3409319401832318 1.5091902573281655 1.8242991645584112 2.1443616690691631 2.5433904085535026 2.7422709586176439 2.9383846906557212 3.2932873681375536 3.4862917416994974 3.6771739482467782 ; 0.055511400176665032 0.20939400288315480 0.37011269343212622 0.58243099599780934 0.73352042405531903 0.87621609690987712 1.0085205281026037 1.1534068513559332 1.3241320746994445 1.4910634735091268 1.6688683115230338 1.8734512518166682 2.2611205186418304 2.6927704701348163 3.1562796481431268 3.3659245189505227 3.5776667782960678 3.9903864066566492 4.2251179749293772 4.4596331306729198 ; 0.12306437724176281 0.32064077132721891 0.51785294016285943 0.76722609401994513 0.94478532934323545 1.1192399599908522 1.3012763614527671 1.5002798671595803 1.7077120941877502 1.9236179062169720 2.1479506991623003 2.3936228357451625 2.8766094705928871 3.3956706966765524 3.9303518036068530 4.1962820852236700 4.4615752940295046 4.9684703463015616 5.2594333844842955 5.5513374282541772 ; 0.15069071966984551 0.35874356528816992 0.56526252244659592 0.83401362134892421 1.0214407605303550 1.2114002949832003 1.4084874095840225 1.6199938691948133 1.8384134953154767 2.0671496428386145 2.3078149073363057 2.5643836375363458 3.0732810975979410 3.6143744509725284 4.1727914052879118 4.4563368744582688 4.7404416977912742 5.2831796131889002 5.5949619377821502 5.9078772298667923 ; 0.19449199970968081 0.41707056871130166 0.63808937576599201 0.93728157946604018 1.1393321734035076 1.3520012929914502 1.5688235047526218 1.7986339515683154 2.0341445195510324 2.2813665852533500 2.5441291938870521 2.8169200726850714 3.3645302685382648 3.9374517348085929 4.5307245394414544 4.8408214175822017 5.1535651765394750 5.7526386512504883 6.0971580764433337 6.4430589609130982 ; 0.31749397449923655 0.57228947910198313 0.82984633120666429 1.1818485607865423 1.4005346617231174 1.6580404817249679 1.9146683698948168 2.1895645647621618 2.4675433790174393 2.7568327605535989 3.0619715577059701 3.3792512425536865 4.0239378122400549 4.6785270615025167 5.3459569726799883 5.7211126656360962 6.1064600474119484 6.8622157881971715 7.3003697331847306 7.7408668204264046 ; 0.51588195315697605 0.79852027368657397 1.0870141119451342 1.5013871739436675 1.7958674789276265 2.0593537655530190 2.3271402073861682 2.6148487848443160 2.9185525878932244 3.2432670115222164 3.5974618719203604 3.9704172771420252 4.7355018753002280 5.5051392784165678 6.3033078834835798 6.7710807389854386 7.2641835123591774 8.2746302826816613 8.8705902421046314 9.4715787166288266 ; 0.66752135084553732 0.97391526783861215 1.2859081959852878 1.7297100715103113 2.0275496402613404 2.3145555251695566 2.6056327826276311 2.9142407964868151 3.2429667030107279 3.5904433934760096 3.9689322449495990 4.3630164691810709 5.1799735029706993 5.9850705276184657 6.9310079692805546 7.4931558939828316 8.0894036071174327 9.3177937072842631 10.040433718977830 10.768398183680118 ; 0.83736918614831202 1.1826915483491358 1.5267242745624379 1.9967779476826824 2.3380034926406763 2.6073942564005490 2.9170090094090435 3.2664394712755240 3.6404392142785387 4.0285730683628813 4.4257906239955354 4.8277044346804745 5.6451242522372160 6.5225082215033412 7.6391973959275674 8.3394476292515911 9.0840284711121040 10.596270094729174 11.470567308445439 12.346929458742492 ; 0.97178728724956343 1.3874344256825284 1.8007256723167788 2.3390639038926073 2.6588690622534448 2.9520167848917103 3.2895188968870399 3.6707375596638645 4.0840979464019895 4.5050046830711201 4.9295194743698856 5.3555222126161031 6.2348639166046960 7.1610051648724546 8.4311840700728524 9.3009924689703318 10.225488626555931 12.059121729668735 13.082228961300986 14.095253571251428 ; 1.0153568624793528 1.5319745239505043 2.0475805323531580 2.7178502049164992 3.0937750532251549 3.4199313658835093 3.7503047320748437 4.1349285064177961 4.5760614906991162 5.0217154413112146 5.4841467866838345 5.9768647679540674 6.9710301583682828 8.1089793475967316 9.5727300458473348 10.579337139389164 11.677247763114378 13.877736485896886 15.040454491296813 16.170133907373049 ; 0.97403576542443171 1.5497226152307100 2.1228691025882824 2.8665301787405375 3.2918765475182830 3.6693512730519311 4.0396630727682776 4.4393215077509263 4.8746729553179824 5.3523903057730102 5.8752550026390677 6.4068158372658353 7.4699611793414702 8.7528560448719102 10.327712132896773 11.393498410160836 12.552428390121049 14.950632456406526 16.193915702919107 17.432083876965244 ; 0.88960285619415869 1.5250712849851087 2.1555963235033713 2.9694424991926280 3.4442363527843902 3.8988317802514421 4.3429188762289783 4.7822488138689323 5.2208157983564965 5.7255904746329387 6.3485516460456148 6.8977725636527483 8.0346039672543679 9.4687753074152194 11.197669774111079 12.270060427322417 13.398890648859949 15.945830631850571 17.305864585017122 18.744115159957943 ; 0.70523490382829535 1.4232161603140874 2.1343199606676091 3.0543743701805246 3.6088003391905961 4.1618009824298623 4.7284094261520027 5.3146644025079421 5.8161943227118416 6.3999905535756820 7.0898051565628935 7.5481984948459715 8.8281173604377816 10.466638355236743 12.729221225642508 13.554432964517664 14.399378460569016 16.764001793061553 18.658205700923268 20.545221723420269 ; 0.30727253257582754 1.1585843836523699 2.0059538443782077 3.1205034860656125 3.8035922493665484 4.4843081374845823 5.1717111818901387 5.9297239699126907 6.5833767677413091 7.1925610206974424 7.8571374827615665 8.5964379693454145 10.016451563974412 11.734167309734087 14.581985205353684 15.323992922642384 16.106753783610767 19.186522902559556 21.069068424829769 22.960806002761338 ; 0.00000000000000000 0.81513257464348265 1.7813845521090175 3.1308765235505378 3.9699988909847828 4.8053469074034885 5.6246127962547892 6.4289579003379869 7.1822135408892152 7.9635294784277031 8.7139820316586807 9.6862195578225023 11.106615289294625 12.591164940139254 15.715996567910658 16.944975395542173 18.237156135846561 21.891327259520807 23.560259304375595 25.241847723792944 ; 0.00000000000000000 0.76291473409776023 1.7017163677734655 3.1095168669433240 4.0314858482131424 4.9554647394884999 5.8671956570487742 6.7192042078739771 7.6126808415998601 8.4818422637660404 9.3477930957607835 10.330476937113360 11.746119453162505 13.265330770859487 16.366567290463991 17.767036078030817 19.261907461973141 22.870644021256087 24.661845512681271 26.389058410906507 ; 0.00000000000000000 0.63755394625497808 1.6791878427402231 3.0665355272650845 4.0798301418083049 5.1434103427626026 6.1891819544611808 7.2013774368402563 8.3070071922176894 9.3959459291515426 10.372480114536847 11.455761486750923 13.321465190708766 15.194164315102379 17.606986469138871 19.234770036227527 20.958225344349803 24.485380294430406 26.447835587077069 28.385105844877309 ; 0.00000000000000000 0.54892234846614241 1.6531657130660795 3.1249309995696430 4.0838838678255982 5.2459999957832020 6.3886271690715795 7.5371146008601047 8.8041127114414284 10.142569394493497 11.411044361602828 12.593926037386090 14.242508160655639 15.976764820970969 18.616868399737331 20.246236344000504 21.969459326303916 25.437247665113627 27.430501709271972 29.423619860300498 ; 0.00000000000000000 0.42998411656855207 1.6111884603937356 3.1896622184456116 4.1757205211878468 5.4242898704456346 6.7853086809651559 8.1981567314196475 9.6884502991838914 11.241247381920545 12.800434898255116 14.268590296592524 15.724066233324534 17.400589757339290 19.999894063744673 21.572041919620712 23.234869312116050 26.620149900351937 28.606587066302612 30.607385181146750 ; 0.00000000000000000 0.29576016206254946 1.5586722013597920 3.2552804393903507 4.3295421674866077 5.6826867596832047 7.3198709997118776 9.0176083649358674 10.748207842059934 12.460358298190373 14.079736628249201 15.524482637252303 17.461671967712487 19.161724766528099 21.562465675096135 23.036655720396890 24.611920504084850 27.895450925683988 29.848213274233398 31.821167326566947 ]; % Fuel Properties engine.fuel = class_REVS_fuel('FTAG 25402'); % Idle Speed engine.idle_speed_radps = class_REVS_dynamic_lookup; engine.idle_speed_radps.axis_1.signal = 'veh_spd_mps'; engine.idle_speed_radps.axis_1.breakpoints = [ 0.00000000000000000 10.000000000000000 ]; engine.idle_speed_radps.table = [ 73.303828583761842 ; 73.303828583761842 ]; % Pedal Calibration engine.pedal_map_type = enum_engine_pedal_map_type.max_engine_power; % Calibration Adjustment Factors engine.DFCO_min_duration_secs = 2; engine.DFCO_refuel_multiplier_time_secs = [ 0.00000000000000000 0.10000000000000001 4.0000000000000000 ];