Martin, Bishop, Choudhary & Boies 1 Title: Forecasting ...docs.trb.org/prp/15-4186.pdf · Martin, Bishop, Choudhary & Boies 1 ... 224 20 References: ... 110 distinguish between computer

Martin, Bishop, Choudhary & Boies 1

Title: Forecasting Passenger Fleet Fuel Consumption – A New Methodology to 1 Include Uncertainty Analysis 2

3

Corresponding Author: Niall PD Martin1 ([email protected] | +447597229469) 4 5 Authors: Justin DK Bishop1 6 (Research Associate | [email protected] | +441223746972) 7 Ruchi Chourdhary1 8 (University Lecturer | [email protected] | +441223746972) 9 Adam M Boies1 10 (University Lecturer | [email protected] | +441223746972) 11 12 Addresses: 1Department of Engineering, 13 University of Cambridge, 14 Trumpington Street, 15 Cambridge, CB21PZ 16 17 Text Word Count (inc. Abstract): 5,349 18 Abstract Word Count: 224 19 References: 35 20 Figurers and Tables: 6 21 Total Word Count: 5,469+(6*250) = 6,849 22


ABSTRACT 23

The UK’s light duty vehicle fleet is the largest end user of refined petroleum, accounting 24 for 12.5% of national UK CO2 emissions in 2010. Despite this, current fleet turnover and 25 fuel consumption models do not account for advances in engine technologies, and are 26 unable to account for uncertainties when used to forecast future fleet fuel consumption 27 and CO2 emissions. This paper introduces a methodology for determining new car fleet 28 fuel consumption based on vehicle design metrics, and accounts for uncertainties inherent 29 in such predictions using a Bayesian statistical framework. Sources of uncertainty are also 30 listed and quantified, while the New European Drive Cycle (NEDC) shown to 31 underestimate on-road fuel consumption in Spark and Compression Ignition vehicles by 32 14% and 17%, respectively. A Bayesian fuel consumption model attributes these 33 discrepancies to the NEDC’s misrepresentation of rolling, frictional and aerodynamic 34 resistances, while on-road fuel consumption estimates were derived using inputs for the 35 engine size, vehicle mass, and compression ratio. The likely 2020 on-road fuel 36 consumption of the average SI and CI vehicles was estimated to be 6.5 L/100 km and 5.8 37 L/100 km, respectively, which represents a 21% and 16% reduction over the average 38 2011 NEDC estimate. These forecasts exceed the mandatory fuel equivalent emissions 39 standards set for that year, highlighting the need for more stringent technological 40 developments to ensure vehicle manufacturers meet emissions targets. 41


INTRODUCTION 42

Unprecedented changes in energy-use and emissions are required from the UK’s 43 passenger vehicle fleet for manufacturers to reduce sales weighted emissions to 95 g 44 CO2/km by 2020 [1]. Legislative targets also require the UK government to achieve an 45 80% reduction in national emissions by 2050, of which passenger vehicles were 46 responsible for 12.5% (73.3 MtCO2-eq) in 2010 [2], [3]. Nevertheless, the likelihood of 47 reducing emissions in passenger vehicles is far from certain due to several factors which 48 include ignorance about the future of vehicle designs, variable uncertainty in fuel 49 consumption models and the inadequacy of the models themselves. Our present 50 understanding of the UK’s light-duty vehicle fleet prohibitively restricts our ability to 51 quantify these uncertainties going forward, all of which constrains how legislators make 52 strategic decisions without a full appreciation for the risks involved. 53

A particular source of uncertainty comes from the New European Drive Cycle (NEDC) 54 [4], which is estimated to underrepresent on-road passenger vehicle fuel consumption by 55 approximately 20-25% [5]. Considering that the NEDC standard is used to determine 56 manufacturers’ adherence to legislative standards, this failure has particular repercussions 57 for the 2020 emissions targets that equate to fuel consumption ratings of approximately 58 4.1 L/100 km for Spark-Ignition (SI) vehicles and 3.6 L/100 km for Compression-Ignition 59 (CI) [6]. Indeed, the NEDC testing discrepancies could allow for significant variations of 60 up to 1.0 L/100 km and 0.9 L/100 km per respective vehicle, which must be considered 61 when assessing adherence of the manufacturers to fuel consumption targets using 62 available computer models. 63

Focusing specifically on the UK’s passenger vehicle fleet, deterministic computer design 64 packages are frequently used to quantify the evolutionary vehicle changes required to 65 reduce vehicle fuel consumption [7]. The single point outputs from these models, 66 however, can be misleading to both vehicle designers and policy makers, giving an 67 impression of greater certainty than is warranted. In reality, the underling model structure 68 and input variable are themselves subject to uncertainty. Consequently, the development 69 of stochastic vehicle design simulation packages is necessary to quantify likely fuel 70 consumption that gives consideration to the multitude of modelling uncertainties. 71

Recognising this necessity, a new Bayesian model was derived to quantify likely SI and 72 CI fuel consumption of UK passenger vehicles. This paper introduces this Cambridge 73 Automotive Research Modelling Application (CARma) and discusses three outcomes of 74 research. 75

1. Hybrid Model Derivation - CARma is formulated from both bottom-up and 76 statistical principals that relate fuel consumption to inputs for engine size, 77 compression ratio, vehicle mass and engine speed. 78

2. Uncertainty Identification – Sources of uncertainty are categorised, and mitigation 79 methods proposed. NEDC fuel consumption data, for example, was used to estimate 80 uncertainties in the coefficients for the rolling resistance, aerodynamic drag, 81 frictional powertrain losses and annual design improvements. These estimates were 82 subsequently calibrated with open-source on-road fuel consumption data. 83

3. Bayesian Model – A Bayesian methodology is introduced to calibrate uncertain 84 parameters, ensuring that combined information from the NEDC and on-road 85 datasets are incorporated into CARma's outputs. Results are presented as stochastic 86 distribution functions, quantifying both expected and likely fuel consumption. 87


Having developed the methodology, CARma was used to quantify the likelihood of the 88 average SI and CI vehicle (available for sale) achieving its 2020 fuel consumption target. 89 Historical vehicle data was used to derive evolutionary projections for mass, engine size 90 and compression ratios out to 2020, allowing for the likely fuel consumption to be 91 determined. 92

BACKGROUND 93

Vehicle Design Models 94

Numerous studies have analysed how best to improve fuel consumption for fleets 95 composed of internal combustion engine (ICE) vehicles [8]–[11]. All works focused on 96 the North American market, where vehicle designs and consumer demands are notably 97 different from those in the UK [12]. The majority of studies also utilise bottom-up 98 strategies to relate vehicle design trade-offs to energy-use [13]–[15]. Nevertheless, these 99 models are unable to quantify uncertainties caused by variability, model inadequacies, 100 and ignorance. Indeed, the Stochastic Transport Emissions Policy (STEP) model [16] was 101 the only identified study to incorporate risk assessment when developing stochastic 2030 102 scenarios for US fuel consumption from inputs of calendar year, vehicle age and vehicle 103 scrappage. The exclusion of technology specific inputs, however, limits STEP’s ability to 104 capture technological trends in its results, while its’ Monte Carlo methodology limits its 105 ability to update results to include new data. 106

Sources of Uncertainty and Mitigation Strategies 107

The categorisation of uncertainties in vehicle simulation models is vital to ensure areas 108 requiring risk mitigation are accurately identified. Several classification systems exist to 109 distinguish between computer model uncertainties [17], [18], yet the Kennedy and 110 O’Hagan scheme [19] is commonly used for statistical models. These uncertainties, and 111 the measures adopted to mitigate them in this paper, are categorised as follows: 112

1. Parameter Uncertainty or Observational Error - Caused by a number of factors 113 including insufficient data availability and NEDC inaccuracies, its influence is 114 reduced by increasing the number of independent model inputs. For this study, open-115 source user-generated data was used to increase the sample size of fuel-consumption 116 data, while Bayesian calibration allowed for improve parameter quantification. 117

2. Model Inadequacy or Parametric Variability - Attributed to over-simplification of 118 systems, leaving unspecified variables, model inadequacy represents the difference 119 between the true fuel consumption and the model estimate. This uncertainty is 120 impossible to eliminate due to the possibility of unknown unknowns. Mitigation 121 entailed dual first-principal and statistical derivation of CARma’s form to ensure its’ 122 twofold validation. 123

3. Variable Uncertainty - Attributed to stochastic variability occurring within similarly 124 defined homogeneous groups. For example, fuel consumption measurements can 125 vary for identical vehicles tested under equivalent drive cycle conditions. Stochastic 126 model inputs were defined for model inputs to account for this uncertainty. 127

4. Ignorance - Attributed to our basic lack of knowledge for the appropriate 128 mathematical model. Ignorance is impossible to eliminate, though reduced with 129 additional research. 130

The validity of the NEDC testing standard for fuel consumption is a particularly 131 influential source of uncertainty, with the results affecting the significance of European 132


emissions standards. Performed with a representative vehicle for each available model 133 over a standard drivecycle, the test advantageously provides a database of repeatable and 134 comparable fuel consumption measurements. Nevertheless, the single drivecycle provides 135 manufactures an opportunity to optimise vehicle energy-use and emissions ratings to 136 NEDC testing conditions. Indeed, a myriad of testing limitations exists that are estimated 137 to cause NEDC to on-road deviances of up to 21% for fuel consumption [20] and 21 ± 138 9% for CO2 emissions [8], as listed below. 139 140 1. Acceleration patterns inaccurately represent on-road driving conditions. For example, 141

NEDC vehicles are stationary for approximately 20% of the test, which favours stop-142 start technologies. 143

2. Power and weight requirements of auxiliary systems are discounted (i.e. heating, 144 sunroof and audio systems), causing the true vehicle reference mass to be 145 underestimated. 146

3. Air conditioning systems, which are estimated to increase fuel consumption by up to 147 128% [21], are excluded. 148

4. A number of permissible flexibilities exist, including ambient test temperature, tyre 149 specification, running-in periods, laboratory altitude, battery state-of-charge, 150 reference mass, gear change schedule and the test track surface and slope. 151 Cumulatively, flexibilities allow for a maximum deviation of approximately 11% 152 (bandwidth 6 - 16%) [22]. 153

5. Mock et al. [20] note certain modifications are allowed between NEDC and 154 production vehicles, including engine control unit calibration and modification to tire 155 rolling resistance. Consequently, the potential for deviations between NEDC and on-156 road fuel consumption is further increased due to variations in the vehicles 157 themselves. 158

159 Recognising these NEDC limitations, CARma was designed to explicitly account for the 160 uncertainties that cause discrepancies between NEDC and on-road fuel consumption 161 measurements. The model was subsequently used to determine the likelihood of UK 162 passenger vehicles reaching 2020 fuel consumption targets, with uncertainties accounted 163 for in the stochastic outputs. 164


METHODOLOGY 165

Data Collection 166

CARma is designed to quantify the fuel consumption of UK SI and CI passenger vehicles 167 using raw, unnormalised data from two sources - NEDC tests and open-source websites. 168 These data sources were used to develop two separate models that relate physical vehicle 169 characteristics to: 170

1. Rated NEDC fuel consumption in the NEDC Model (NEDC-M); and, 171 2. On-road fuel consumption in the On-Road Model (OR-M). 172

Both models output distributions for unknown parameters that were derived using the 173 Bayesian regression technique discussed in the Bayesian Methodology section of this 174 paper. 175 176 NEDC data was used to determine the drivetrain, engine design and NEDC fuel 177 consumption information of all 35,000 type-approval vehicles made available for sale in 178 the UK since 2000 (CAP, “unpublished data”). For the open-source data, 184,000 179 publically available on-road fuel consumption measurements were collected from 180 European users [23]. Limiting data collection to users logging over 1,500 km of vehicle 181 distance travelled also allowed for increased data integrity, yet the data’s independent to 182 spatial location causes a bias towards continental European drivers whose driving patterns 183 are differ from those of UK drivers (The average vehicle kilometer travelled for a 184 German citizen in 2002, for example, was 13,500 km [24] compared to 14,758 km for the 185 average UK citizen [25]). Nevertheless, the Bayesian model accounts for this bias when 186 estimating distributions for parameter by increasing variance for each estimate. 187 188 Model Selection 189

The Bayesian methodology requires a statistical model of the form shown in Equation 1, 190 where unknown parameters of the ith term are embodied using θi, and known variables are 191 represented using βi. First-principal derivation allowed for the inference of variables in 192 each unknown parameter (θi). Variables selection was also validated using a statistical 193 methodology, ensuring a fundamental understanding of CARma’s mechanical and 194 statistical properties. 195 196 𝑚! = 𝛽!𝜃! + 𝛽!𝜃!…+ 𝛽!𝜃! + 𝑒𝑟𝑟𝑜𝑟 (1) 197 198 Initiating the first-principal derivation, indicated mean effective pressure (imep - a 199 measure of usable work produced) was used to encapsulate both the break mean effective 200 pressure (bmep - a measure of an engine’s ability to produce work) and the frictional 201 mean effective pressure (fmep- an indication of frictional losses within the drivetrain) in 202 Equation 2: 203 204 𝑖𝑚𝑒𝑝 = 𝑏𝑚𝑒𝑝 + 𝑓𝑚𝑒𝑝 (2) 205 206 Imep was subsequently decomposed in Equation 3 to show the total indicated work (𝑊!), 207 normalized with respect to engine size (𝑉!), is dependent on the fuel mass flow rate (𝑚!), 208 lower calorific value (𝑄!"#) and engine efficiency (𝜂!,!) [26]. Likewise, the bmep’s 209 normalized break work (𝑊!) was decomposed in Equation (4 into break power (𝑃!), 210 engine speed (𝑁) and the number of crank revolutions for each power stroke per cylinder 211 (𝑛!). 212


𝑖𝑚𝑒𝑝 = !!!!= !!!!!"!!,!

!! (3) 213

214 𝑏𝑚𝑒𝑝 = !!

!!= !!!!

!!! (4) 215

216 Additional inference of vehicle efficiency allowed for the incorporation of the 217 compression ratio (𝑟!) into Equation 3, where 𝐴 and 𝛾 were used as coefficients specific 218 to the constant-volume (i.e. SI) and constant-pressure (i.e. CI) idealized heat addition 219 processes [26]. This relationship is represented in Equation 5, where the compression 220 ratio variables are incorporated into a simplified compression ratio term (𝜂!,! = 𝑓 𝑟! =221 𝑆𝑟!). The imep derivation was subsequently substituted into Equation 2, yielding the 222 relationship presented in Equation 6. 223 224 𝑖𝑚𝑒𝑝 = !!!!"#!!,!

!!= !!!!"#

!!1− !

!!!!!= !!!!"#

!!𝑆𝑟! (5) 225

226 !!!!"#!!

𝑆𝑟! = 𝑏𝑚𝑒𝑝 + 𝑓𝑚𝑒𝑝⟹ 𝑚! =!!

!!"#!!!𝑏𝑚𝑒𝑝 + 𝑓𝑚𝑒𝑝 (6) 227

228 Similarly, the substitution of the break power with road-loaded power under constant 229 velocity in Equation 4 allowed for the inclusion of additional vehicle metrics in Equation 230 7 [26]. These included vehicle mass (𝑀!), the coefficient of rolling resistance (𝐶!), 231 acceleration due to gravity (𝑔), vehicle speed (𝑆!), air density (𝜌), the coefficient of drag 232 (𝐶!) and vehicle frontal area (𝐴!). 233 234 𝑏𝑚𝑒𝑝 = !!!!

!!!= !!

!!!𝐶!𝑀!𝑔𝑆! +

!!𝐶!𝐴!𝑆!! (7) 235

236 Finally, Equations 6 and 7 were combined and compared against the required Bayesian 237 statistical form in Equation 1. Variables for which data is not available are represented by 238 the unknown 𝛽! parameter leaving the known variables (𝑀! [ln(kg)], 𝑉! [ln(cc)], 𝑆𝑟!, 239 𝑁 [ln kg ], 𝑌𝑒𝑎𝑟 [ln(year)], as shown in bold in Equation 8) to be used in the 𝜃 240 parameters. The vehicle model year and error terms were also included to represent 241 annual effects and model inaccuracies, while the natural logarithm of each variable was 242 used to ensure model linearity. The resulting model gives fuel consumption as a result of 243 four θ parameters, 244 245 𝑚! = 𝜃!

!" 𝑴𝐯 ln(𝑺𝒓𝐜) !" (𝑵)

!!

+ 𝜃!!

ln(𝑺𝒓𝐜) !" (𝑵)!!

+ 𝜃!!" 𝑽𝐝 ln(𝑺𝒓𝐜) !!

+ 𝜃! 𝑙𝑛 𝒀𝒆𝒂𝒓!!

+ 𝑒!""#"

(8) 246

where 𝜃! represents !!!!!!!!!"#!"##$%&

, 𝜃! represents !!!!!!!!!!

!!!"#!"#$

and 𝜃! represents !"#$!!"#

!"#$%#&'

. 247

248 The statistical derivation of parameters was performed using a Variance Information 249 Factors (VIF) stepwise selection process that eliminated multicollinearity amongst 250 explanatory variables [27]. VIF was calculated using the coefficient of determination (R2) 251 in Equation 9, with the jth explanatory variable regressed against all remaining variables 252 to compare variability. 253 254


𝑉𝐼𝐹! = !

!!!!! (9) 255

256 Using all available explanatory variables (engine size, stroke, bore, cylinder numbers, 257 rated power, rated torque, acceleration time, engine speeds at maximum power and 258 torque, vehicle mass, compression ratios and capital costs), stepwise selection was 259 performed with a VIF threshold of 10 (i.e. VIF ≥ 10 indicates variables are not 260 independent) [27]. The results showed engine size (𝑉!), vehicle mass (𝑀!), compression 261 ratio (𝑆𝑟!) and engine speeds at maximum rated power and torque (𝑁) to be the most 262 appropriate variables on which to derive the linear statistical model, with the highest VIF 263 factor of 7.6. A derivation was subsequently undertaken using Mallow’s Cp selection 264 criterion [28] to determine the combination of variables that achieve highest statistical 265 significance. Defined as the model whose Cp value is closest to the number of parameters 266 in the model (including the intercept), the results showed that all explanatory variables 267 were required for use in CARma, thus justifying the first-principal derivation in Equation 268 8. 269 270 Finally, the inclusion of additional vehicle characteristic variables could be considered to 271 calibrate a greater number of model parameters, yet the accuracy of the calibration 272 reduces as the Bayesian algorithm searches over a greater parameter space. A note of 273 caution also surrounds the interpretation of CARma’s parameter estimates, since the 274 calibration of just four parameters causes uncertainties about other (uncalibrated) 275 parameters to be “lumped” into the developed parameter estimates. The chosen 276 calibration parameters, therefore, are in some sense “pseudo-variables”, and partially 277 cease to correspond to any physically meaningful quantity. Though this approach is 278 useful when developing fuel consumption forecasts from inherently uncertain input data 279 (as is the intended function of this model), uncertainties due to ignorance are also 280 partially lumped into the calibration parameters and caution is required when interpreting 281 a physical meaning from parameter estimates. 282 283 Bayesian Methodology 284

A summary of CARma’s methodology is presented in Figure 1. Heterogeneous clustering 285 was performed by fuel type to reduce the variability of the data caused by categorical 286 dichotomies. Two methodologies were also adopted to account for different uncertainties, 287 with the Bayesian Model allowing for the quantification of both parameter uncertainty 288 and model inadequacy. Likewise, stochastic projections were developed using Holt 289 exponential smoothing forecasts that account for variable uncertainties [29], thus ensuring 290 the mitigation of all possible sources of uncertainty (excluding model ignorance, which 291 can only be reduced with a cumulative increase in scientific knowledge over time). 292 Finally, variable and parameter distributions from both models were combines using 293 Monte Carlo sampling to yield the final stochastic estimates for SI and CI fuel 294 consumption. 295

296


FIGURE 1: CARma’s structure. 297 298

For the Bayesian Model, Bayes’ Theorem was used to compute updated posterior 299 distributions for the uncertain parameters, given the priors and likelihood function (i.e. 300 statistical form). This process is represented using the Bayesian formulation in Equation 301 10: 302

𝑝(𝜃|𝐷) ∝ 𝑝(𝐷|𝜃) ∙ 𝑝(𝜃) (10) 303 304 where, θ represents the vector of uncertain parameters and p(θ) represents their prior 305 distributions. Likewise, the likelihood function is symbolised using p(D|θ), while p(θ|D) 306 represents the posterior distributions of the parameters. The data (conditional on the 307 predefined parameters in the statistical model) is represented as D. This Bayesian 308 methodology ensures all available knowledge is incorporated into previously held 309 estimates for unknown parameters and provides the means of evaluating the effectiveness 310 of the NEDC standard. 311

Three specific components are encapsulated in the Bayesian methodology, the first of 312 which are the prior probability distributions. Prior distributions were specified for each of 313 the unknown parameters to reflect preceding knowledge, independent to the data at hand. 314 For the NEDC-M, no preceding estimates were available for the unknown parameters and 315 vague priors were thus chosen. For the OR-M, the priors were specified as the posterior 316 probability distributions of the NEDC-M. These posteriors are the second component of 317 the Bayesian methodology and characterize the updated parameter estimates, given the 318 available data. Finally, the Bayesian methodology requires the specification of a 319 likelihood function to determine the conditional distribution of the new data given the 320 model parameters, which was represented using a statistical relationship of model 321 variables. All results were developed using 500,000 Markov Chain Monte Carlo iterations 322 in the Bayesian OpenBUGS software platform [30]. 323


324 The probabilistic posterior distributions from the Bayesian regression represent the first-325 order uncertainty (i.e. the random variation around an average value) for each parameter 326 within a sub-group of vehicles. Combining these posteriors with single value inputs for 327 the vehicle mass, engine size and compression ratio allows for the stochastic estimation 328 of fuel consumption, accounting for model inadequacy and data uncertainty. However, 329 the additional specification of input variables as probability distribution functions 330 incorporates second-order uncertainties into CARma, which stems from a lack of 331 knowledge about the values of the input parameters themselves. These distributions were 332 produced using the Holt exponential smoothing method [29], where the weighted average 333 of past observations was used to forecast expected values out to 2020. Weights were 334 chosen to decline exponentially over time so that recent observations contribute to the 335 forecasted estimate more than earlier observations. This technique is widely used for the 336 development of national statistical forecasts [31] and provides the means of projecting 337 future vehicle mass, engine size and compression ratios in CARma. 338


RESULTS 339

NEDC Discrepancy and Model Validation 340

A comparison between the 35,000 NEDC and 184,000 on-road fuel consumption 341 measurements in Table 1 shows mean on-road fuel consumption to be 16.1% and 12.5% 342 higher for SI and CI vehicles, respectively. On average, NEDC tests underestimate actual 343 fuel consumption by 0.96 L/100 km for SI vehicles and 0.98 L/100 km for CI. Higher 344 standard deviations for on-road data are attributed its open-source nature, where there is a 345 larger variation in drive cycles and user driving styles. 346

TABLE 1: Comparison of NEDC and on-road fuel consumption for UK model years 347 2000-2011. 348

349 350

351 FIGURE 2: Comparison between measured and actual fuel consumption, using 10-352

fold cross validation of CARma, with R2 = 0.858 353 354 The form of the statistical model in Equation 8 was validated using 10-fold cross-355 validation to compare model estimates against separate test data [32]. For this process, 356 NEDC data was partitioned into 10 equal subsamples, each of which were randomly split 357 into two groups - 90% for model training and 10% for model testing. The 10 accuracy 358 assessments were combined to give a measure of the predictive performance of the model 359 using the mean squared error and sum of squares values, estimate at 0.0106 and 27.5, 360 respectively. The results from the 10-fold cross validation are depicted in Figure 2, where 361

Mean SD Mean SD Mean SD[L/100km] [L/100km] [L/100km]5.95 1.22 6.9 1.48 0.96 17.84 1.76 8.82 2.01 0.98 1.287.02 1.81 7.99 2.04 0.97 1.16

Rated Fuel Consumption

On-Road Fuel Consumption

Discrepancy


modelled CARma estimates are shown to compare favourably against collected fuel 362 consumption values. The statistical model form was also validated using linear 363 regression, where the coefficient of determination was 0.858 for all vehicles and the 364 residual standard error was 0.0921. 365 366 Bayesian Calibration 367

Results from the Bayesian calibration process are shown in Figure 3 as prior and posterior 368 distributions for three of the unknown model parameters (θ1, θ 2, θ 3). These distributions 369 are the primary outcomes from the calibration process and help to determine how the 370 input variables influences fuel consumption under NEDC and on-road driving conditions, 371 where larger absolute magnitudes indicate a greater influence of the known variables on 372 fuel consumption. 373

374 FIGURE 3: SI and CI prior (red dashed line for NEDC-M) and posterior density 375 distributions (blue solid line for OR-M) for θ1 [𝐥𝐧(𝐤𝐠) 𝐥𝐧(𝐫𝐩𝐦)], θ2 [𝟏 𝐥𝐧(𝐫𝐩𝐦)], 376 and θ3 [ln(cc)]. 377 378 A comparison between the SI and CI distributions for the first set of unknown parameters 379 (θ1) shows that the NEDC-M underestimates the influence of the rolling resistance (Cr) by 380 an average of 12% for SI vehicles and 14% for CI. These distortions can be achieved by 381 over-inflating tires [33], reducing frictional losses [34], wheel realignment and break 382 adjustment, all of which are free parameters set by manufacturers during the NEDC tests 383 [22]. Similar trends for θ3 are noted in Figure 3 c and f, where the NEDC tests 384 overestimate frictional powertrain losses. This is particularly apparent for SI vehicles, 385 whose mean on-road parameter is 26% lower than the rated NEDC prior compared to 386 18% for CI. Overestimation of frictional losses may be attributed to a higher number of 387 trips running under low engine load conditions in the on-road dataset. Considering that 388 50% of European trips are known to be less than 3 km in length [35], the results may 389 highlight an overrepresentation of the extra-urban drivecycle in the combined NEDC 390 estimates for fuel consumption. 391


The second group of unknown parameters (θ2), which is dominated by the coefficient of 392 drag term, achieved the closest approximation between NEDC-M and OR-M parameter 393 estimates. This aerodynamic drag term is underestimated in NEDC tests by an average of 394 2% for SI vehicles but overestimate by 2% for CI. The slight discrepancy may be caused 395 by differences in the average speeds at which SI and CI vehicles are driven on-road, or 396 deviations between the average coefficients of drag for SI and CI vehicles. Uncertainties 397 about the parameters are noted to have reduced following the calibrations process, as 398 shown by the reductions in distribution variance of 18.4% for SI and 32.6% for CI. 399 Indeed, uncertainties reduced for all estimates following Bayesian calibration. 400 401 Mean error terms (not shown) reduced from 265.7 to 243.9 for SI and 293.4 to 283.9 for 402 CI with little (<2%) change in variance. The influence of the vehicle model year 403 parameter (θ4) on SI and CI fuel consumption also reduced from -34.9 to -32.0, and -38.5 404 to -34.0, respectively, between the NEDC-M and OR-M models. This trend is attributed 405 to the increased year-on-year optimization of vehicle designs to the NEDC standard; a 406 practice that allows vehicle manufactures to maximize adherence to legislative emissions 407 standards. Such annual changes in vehicle design, however, may have a limited influence 408 on realistic fuel-consumption and further undermine the accuracy of the NEDC results. 409 410 Finally, the complete unnormalised Bayesian formula for both NEDC-M and OR-M 411 models are shown in Equations 11 and 12, where mean values and units of the priors and 412 posteriors are presented for each θ parameter. 413 414 𝑚!,!"#$ [

!!""

km] = 𝛽! 18.06!!

!"(!")!"(!"#)

− 𝛽! 155.3!!

!!"(!"#)

+ 𝛽! 0.194!![!"(!!)]

− 𝛽! 34.93!! !"(!"#$)

+ 265.7!""#"

415

(11) 416 417 𝑚!,!" [

!!""

km] = 𝛽! 20.41!!

!"(!")!"(!"#)

− 𝛽! 158.9!!

!!"(!"#)

+ 𝛽! 0.143!![!"(!!)]

− 𝛽! 32.00!! !"(!"#$)

+ 243.9!""#"

418

(12) 419 420 Holt Projections 421

Projections for the engine size, vehicle mass and the compression ratio of SI and CI 422 vehicles were derived from an analysis of NEDC rated data from 2001 to 2011. Historical 423 annual averages were used as inputs into the Holt exponential smoothing model, where 424 known data is represented in Figure 4 using the red regression line and the Holt model is 425 represented using the blue. This methodology allows an accurate representation of 426 historical and irregular trends, while the second-order uncertainties are shown to increase 427 from 2011 to 2020 using a 95% normal predictive interval about mean forecasted values. 428 429


430 FIGURE 4: Holt forecasts (dashed blue line) and historical data (solid red line) for 431 (a-b) compression ratio, (c-d) mass and (e-f) engine size of the average CI (left) and 432 SI (right) passenger vehicle available for sale in the UK from 2011 to 2020. 433 434 The historical CAP data shows evolutionary changes in UK passenger vehicles designs 435 that have helped improve fuel efficiencies over time. For CI vehicles, a 6.55% reduction 436 in the average engine size from 2001 to 2011 helped to reduce fuel consumption. The 437 weight of CI vehicles modestly increased by 2.43% from 2001 to 2011, though a 438 reduction from 1571 kg to 1553 kg is noted from 2009 onwards. For SI vehicles, average 439 compression ratios increased from 10.2 to 10.6 across the decade as manufacturers 440 endeavoured to increase fuel conversion efficiencies and reductions in the mass and 441 engine size of SI vehicles (-1.09% and 5.73%, respectively) also helped to improved 442 efficiencies. These trends were projected to continue using the Holt forecasts. 443 444 The largest forecasted changes, relative to 2011 data, are for the compression ratio of CI 445 vehicles and the engine size of SI vehicles that are correspondingly projected to decrease 446 by 19.08% (16.56 to 13.40) and 18.39% (2007 kg to 1638 kg). Both trends are consistent 447 with historical data and allow for the continued improvement of SI and CI environmental 448 efficiencies. Modest changes in SI compression ratio (+2.64%) and mass (-3.59%) are 449 also projected out to 2020, while the forecasts for CI mass and engine size are shown to 450 be more significant, at -4.96% and -5.37%, respectively. 451


Case Study 452

Having determined first and second-order uncertainties for the unknown model 453 parameters and input variables, estimates were combined using Monte Carlo factorial 454 sampling to forecast the likely fuel consumption of the average SI and CI vehicle 455 available for sale in 2020. These results are presented in Figure 5 as probability 456 distribution functions, where the ordinate specifies the relative probability of the estimate, 457 and the variance in fuel consumption is shown on the abscissa. Likely 2020 estimates for 458 SI fuel consumption were 6.50 L/100 km, with a 50% probability consumption between 459 5.75 and 7.17 L/100 km. Similarly, the expected fuel consumption of the average CI 460 vehicle was 5.81 L/100 km, where the 50% confidence interval (C-Inv) was between 4.95 461 and 6.54 L/100 km. 462

463 FIGURE 5: CARma forecasts for the likely on-road fuel consumption of the average 464 SI (left) and CI (right) vehicle available for sale in 2020 (top), and temporal 465 projections for parameter inputs and fuel consumption (bottom). 466 467 Comparing the projected 2020 on-road fuel consumption to 2011 averages (8.25 L/100 468 km for SI and 6.94 L/100 km for CI; NEDC ratings of 7.16 and 5.47 L/100 km, 469 respectively), shows likely reductions in the fuel consumption of 21.2% and 16.3% for 470 the respective SI and CI vehicle. These reductions are lower than those achieved from 471 2001 to 2011 (estimated using CAP data to be 22.0% for SI vehicles and 18.7% for CI), 472 and indicate that the potential to improve fuel economy from evolutionary technological 473 developments alone is diminishing. Furthermore, a 50% chance exists that the reductions 474 in fuel consumption will be between 13.1% and 30.3% for SI vehicles and 5.8% and 475 28.7% for CI, when confidence intervals are considered. 476 477 Forecasts for on-road consumption were shown to exceed their fuel equivalent targets by 478 an average of 58.5% for CI vehicles and 61.4% for SI. The 2020 goals, however, are 479 based on NEDC estimates, and a direct comparison to CARma forecasts (which account 480 for on-road uncertainties) is not entirely appropriate. Consequently, the NEDC-M 481


distributions were used to quantify likely 2020 fuel consumption based on NEDC data 482 alone, from which estimates of 5.80 and 5.50 L/100 km were derived for the 483 corresponding SI and CI vehicle. Even with this consideration, however, the average CI 484 and SI vehicles were still expected to exceed its respective target by 52.8% and 41.5%. 485 Additional technologies changes beyond evolutionary design developments are therefore 486 required to ensure vehicle manufacturers are able to achieve the required reductions in 487 fuel consumption and emissions. 488


CONCLUSIONS 489

European vehicle manufacturers must reduce their fleet average passenger vehicle 490 emissions to 95 gCO2/km by 2020, which equates to approximately 4.1 L/100 km for SI 491 vehicles and 3.6 L/100 km for CI. Models are frequently employed to quantify how 492 changes in vehicle design might be used to achieve these goals yet such models are 493 fraught with uncertainties that collectively obscure the interpretation of results. With little 494 analysis of risk in previously-developed fuel consumption packages, this paper presents a 495 methodology to incorporate uncertainties into a UK specific passenger vehicle model. 496 The Cambridge Automotive Research Modelling Application (CARma) was designed to 497 account for uncertainties about data, model form, and the inherent unpredictability when 498 testing future vehicle designs. To illustrate the effectiveness of the model, 2020 forecasts 499 for SI and CI fuel consumption were presented, whereby the potential of vehicle 500 manufacturers to reduce fuel consumption using evolutionary design modifications was 501 determined. 502 503 Before outlining the structure of the CARma model, discrepancies between NEDC and 504 on-road fuel consumption were first established using open source, online data. Results 505 showed average deviations of 14% for SI vehicles (SD = 0.1648) and 17% for CI (SD = 506 0.1683), with the NEDC underestimating the fuel consumption of both technologies. 507 These factors were subsequently used to update 2020 targets to represent to on-road 508 driving conditions, equating to 4.66 and 4.19 L/100 km for SI and CI respectively. 509 510 Having derived the statistical form of a fuel consumption model, a Bayesian methodology 511 was introduced to estimate unknown parameters using both NEDC and on-road data. This 512 allows for updated stochastic estimates for model coefficients to be developed that 513 directly account for the uncertainties in the NEDC testing standard. A comparison 514 between the prior and posterior coefficients revealed the NEDC tests to overestimate the 515 influence of frictional losses (themselves a function of engine speed) in SI and CI 516 vehicles. Similarly, NEDC tests were shown to accurately account aerodynamic losses, 517 yet the influences of rolling resistance were underestimated for SI and CI cars. 518 519 Recognising that derived distributions fail to account for the inherent uncertainty caused 520 by the development of future projections, Holt exponential smoothing was used to project 521 the average mass, engine size, and compression ratio of SI and CI vehicles with a 95% 522 prediction interval about the mean. Reductions were forecast for all SI vehicle variables 523 out to 2020, with engine size expected to fall from 2020 cc to 1783 cc, weight from 1368 524 kg to 1305 kg and compression ratio from 10.63 to 9.75. Similar reductions were 525 projected for the average CI vehicle, as engine size was forecast to fall from 2029 cc to 526 1909 cc, weight from 1541 kg to 1397 kg and compression ratio from 16.55 to 13.63. 527 528 The effectiveness of Bayesian calibration model was demonstrated using a case study of 529 the average UK SI and CI vehicle from 2001 to 2011. Holt projections were combined 530 with posterior parameters estimates from which forecasts for likely 2020 fuel 531 consumption were established. For the average SI vehicle, this was estimated to be 6.50 532 L/100 km, with a 50% likelihood between 5.75 and 7.17 L/100 km. Likewise, the most 533 likely estimate for the average CI vehicle was 5.81 L/100 km, with a 50% likelihood 534 between 4.95 to 6.54 L/100 km. Both scenarios exceeded their respective on-road fuel 535 equivalent targets by 35% (SI) and 39% (CI), indicating that evolutionary developments 536 alone are unlikely to allow for the required reductions to be achieved. Nevertheless, this 537 study excludes sale-weighted data and simply reviews the average available SI and CI 538


vehicle. Consequently, the fleet averaged targets may still be achieved with the sale of 539 smaller, lighter vehicles that compensate for the average technologies. 540 541 Finally, variability in the results highlights an underlying need to incorporate uncertainty 542 when forecasting scenarios. A true understanding of possible outcomes from policy and 543 vehicle design changes provides a better appreciation of the risks involved, as 544 demonstrated by the case study. The CARma model applies clustering to handle 545 heterogeneity of SI and CI vehicles, whilst a Monte Carlo simulation was used to quantify 546 the uncertainties about future estimates for the input parameters. As CARma is designed 547 to utilise open-source fuel consumption data, the collection of additional information over 548 time introduces the possibility of extending the model to account for other technologies if 549 and when they become available. Work is presently underway to allow CARma to be 550 used in both fleet-wide and single vehicle projection studies, while the inclusion of fuel 551 consumption ratings from additional testing standards can also be considered to improve 552 the accuracy of parameter estimates. 553


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