Automobile Exhaust Emission Modal Analysis Model

Calspan Technical Report A UTOMOBILE EX HA US T EMISSION MOD A L ANAL YSIS MODEL Paul Kunselman and H.T. McAdams Galspan Report No. NA-5194-D-3 Calspan Corporation Buffalo, New York 14221 Formerly Cornell Aeronautical Laboratory, Inc. ------- EPA-420-R-73-103 Calspan On November 17, 1972 Cornell Aeronautical Laboratory (CAL) changed it* name to Calspan Corporation and converted to for-profit operations. Calspan is dedicated to carrying on CAL's long-standing tradition of advanced research and development from an independent viewpoint. All of CAL's diverse scientific and engineering programs for government and industry are being continued in the aerosciences, electronics and avionics, computer sciences, transportation and vehicle research, and the environ- mental sciences. Calspan is composed of the same staff, management, and facilities as CAL, which operated since 1946 under federal income tax exemption. A UTOMOBILE EX HA UST EMISSION MODAL ANAL YS/S MODEL Paul Kunselman and H.T. McAdams Calspan Report No. NA-5194-D-3 Prepared For: ENVIRONMENTAL PROTECTION AGENCY DIVISION OF CERTIFICATION AND SURVEILLANCE ANN ARBOR, MICHIGAN JULY 1973 Calspan Corporation Buffalo, New York 1422 ------- TABLE OF CONTENTS 4. COMPUTER IMPLEMENTATION 5. MODEL PERFORMANCE 5.1 Statistical Indicators of Performance 5.2 Performance Results 5.3 Discussion and Evaluation Page No. 1 1. INTRODUCTION 1.1 Modal Analysis of Vehicle Emissions 3 1.2 Composite Analysis of Vehicle Emissions 5 1.3 Report Scope and Preview 7 2. PROBLEM DEFINITION 8 2.1 Input Data 8 2.2 Output Data 9 3. OVERVIEW OF THE MATHEMATICAL MODEL 10 3.1 The Emission Rate Function 11 3.2 Steady-State and Accel/Decel Emission Rate Functions ** 3.3 Determination of the Coefficients (bi, si) 3.4 The composite Emission Rate Function 3.5 Vehicle and Vehicle Group Characterization TABLES FIGURES APPENDIX I - Specification of the 37 Modes and Evaluation of the Average Values of the Basis Functions APPENDIX II - Speed vs. Time Curves for the Surveillance Driving Sequence and first 505 seconds of the Federal Test Procedure APPENDIX III - The Mathematical Model APPENDIX IV - Program Listings APPENDIX V - Vehicle Classification by Discriminant Function Analysis 13 15 15 17 20 20 21 22 6. SUMMARY AND CONCLUSIONS 30 33 47 i ------- LIST OF TABLES Table No. Table 1 Bag Value Statistics for the Surveillance Driving Sequence 2 Bag Value Statistics for the First SOS Seconds of the Federal Test Procedure Driving Sequence 3 Distribution of HC Bap Value Error from the Surveillance Driving Sequence 4 Distribution of CO Bag Value Error from the Surveillance Driving Sequence 5 Distribution of NO Bap Value Error from the x Surveillance Driving Sequence 6 Distribution of HC Bap Value Error from the First 505 Seconds of the Federal Test Procedure Driving Sequence 7 Distribution of 00 Bag Value Error from the First 505 Seconds of the Federal Test Procedure Driving Sequence 8 Distribution of N0x Bap Value Error from the First 505 Seconds of the Federal Test Procedures Driving Sequence 9 Replicate Modal Analyses of HC for 61 Vehicles 10 Replicate Modal Analyses of CO for 61 Vehicles 11 Replicate Modal Analyses of NO^ fro 61 Vehicles 12 Variance Component Analysis for 61 Replicate Tests ii ------- LIST OF FIGURES Mean Steady-State HC Emission Rate Values vs. Speed Mean steady-State CO Emission Rate Values vs. Speed Mean Steady-State NO^ Emission Rate Values vs. Speed Main Program I Main Program II Distribution of HC Bag Error from Surveillance Driving Sequence Distribution of CO Bag Error from Surveillance Driving Sequence Distribution of NO^ Bag Error from Surveillance Driving Sequence Distribution of HC Bag Error from First 505 Seconds Federal Test Procedure Driving Sequence Distribution of CO Bag Error from First 505 Seconds Federal Test Procedure Driving Sequence Distribution of NO^ Bag Error from First 505 Seconds Federal Test Procedure Driving Sequence iii ------- ABSTRACT This report on modal analysis of automobile emissions was prepared for the United States Environmental Protection Agency's Division of Certification and Surveillance, Ann Arbor, Michigan under EPA Contract No. 68-01-0435. The mathematical model and allied computer programs which are described in this report enable an analyst to calculate the amounts of hydrocarbons, carbon monoxide and oxides of nitrogen emitted by individual vehicles or vehicle groups over any specified driving sequence. The model requires as input the amounts of the three pollutants given off by individual vehicles over short duration driving sequences (modes) in which speed is a monotonic function of time. The validity of the model is investigated by using it to predict emissions for individual vehicles over the Surveillance Driving Sequence (SDS) and the first 505 seconds of the hot Federal Test Procedure (FTP) driving sequence. The ability of the model to predict actual vehicle emissions is compared with the reproducibility of the actual vehicle emissions measured in replicated tests. Using this analysis, the model performs extremely well. The model has maximum effectiveness as a predictor of group emission charac- teristics of warmed-up vehicles. It is understood that the mathematical model should be used only within the region of speed and acceleration space which is spanned by the input modal data. It should be noted that because of the specifics of the study design and the data collection process, the same vehicles were tested over the SDS and FTP. Therefore, the test of model performance may have been more favorable than would have been the case if the data had been obtained from two different vehicle fleets. iv ------- TEXT V ------- 1. INTRODUCTION In many geographic regions, the major portion of hydrocarbons (HC), carbon monoxide (CO), and nitrogen oxides (NO^) present in the environment is due to motor vehicle emissions. The impact of motor vehicles on the environ- ment in a given location is a function of many factors. Among these factors are the emission characteristics of individual vehicles, the mix of vehicles in a particular traffic way, the numerical concentration of vehicles per mile or per unit of area, and the driving pattern in which the vehicles are employed. This driving pattern is influenced by the functional use of the traffic artery (i.e., whether inter or intra city, whether it serves industrial or recreational purposes, etc.) as well 9s a number of other items. Other items include the design of the highway (e.g., whether it is designed for high or low speed) and the extent to which it passes through and is limited by population density. It has been well established that emission rates for a particular automobile depend upon the manner in which the vehicle is operated -- that is, emission rates are different for different accelerations or decelerations. In a particular trip taken by an automobile in traveling from Point A to Point B, that automobile will exhibit a particular time profile of acceleration and velocity. The trip may entail a number of starts and stops, as well as a range of speeds determined by traffic conditions, local speed controls, and other factors. As the vehicle travels from A to B, this time profile or "driving sequence", together with the emission characteristics of the vehicle, determines the pollution contribution to the atmosphere. Due to the fact that emissions, expressed in grams per miles, vary along the route, the distribution of the vehicle emissions, as well as the total contribution of pollutants to the atmosphere, can be determined. Indeed, the traffic way can be considered as a line source of pollution, the strength of which depends on vehicle density, vehicle mix, driving sequence, and the emission characteristics of individual automobiles. Finally, the local concentration of pollution along this route is determined by this line source distribution mechanism acting in concert with meteorological transport processes such as wind and diffusion. To assess the impact of vehicular emissions on a particular section of highway requires, therefore, a number of data inputs. These include characterization of both traffic and emission parameters. Traffic parameters 1 ------- include numerical traffic density, traffic composition (makes and models of vehicles), and traffic flow characteristics (speed, starts and stops, rates of acceleration and deceleration). Emission parameters include emission rates for various categories of vehicles where these rates are expressed as functions of driving variables such as speed and acceleration. The required traffic parameters can be readily obtained by monitoring traffic along the route in question. The number of vehicles passing various points along the way per unit of time can be counted, and the total number of vehicles can be broken down into homogeneous groups accord- ing to make, model, age or other factors influencing emissions. Moreover, speeds and accelerations prevailing along the route can be measured by observing a "tagged" automobile or by instrumenting a vehicle and injecting this vehicle as a "probe" into the traffic stream. In contrast to the relatively straightforward approach to traffic parameter assessment, the evaluation of applicable vehicle emission functions proves to be quite difficult unless a means can be found for modeling the infinite multiplicity of driving sequences which can arise. It is to be noted, for exanple, that the EPA Surveillance Driving Sequence is only one of the infinitude of possible sequences. Standard emission tests based on a pre- scribed driving sequence serve the purpose of comparing vehicles according to a standard set of operating conditions and make it possible to implement emission control standards and to check compliance with these standards. However, they are not structured in such a way as to readily provide the ability to predict vehicle emissions over an arbitrary driving sequence. A generalized prediction capability can be accomplished by breaking the standard sequence, or any other available sequence, into segments having specified speeds and accelerations. Then, these segments can be appropriately recombined to form other driving sequences. In this way, one hopes to be able to approx- imate any desired driving sequence by appropriately weighting the various segments according to their time duration in the sequence to be modeled. The segments are referred to as operating "modes" and any analysis based on the use of these modes as a "modal analysis". 2 ------- 1.1 MODAL ANALYSIS OF VEHICLE EMISSIONS In 1971, the Surveillance Driving Sequence (SDS) was developed by EPA to measure vehicle emissions over a variety of steady state and transient driving conditions. The acceleration and deceleration modes represented in the SDS consist of all possible combinations of the following five speeds: 0 mph, 15 mph, 30 mph, 45 n^h, 60 mph. The average acceleration or deceleration rate observed for each mode in the Los Angeles basin is used during operation of 20 of the transient modes. In addition, 6 of the tran- sient modes are repeated using accel/decel rates higher or lower than the average rate in order to determine the affect of accel/decel rate on emissions. These acceleration and decelerations were chosen to represent the full range of accelerations and decelerations observed in the CAPE-10 project.* The concept of modal analysis examined in this report employs as input data the emissions measured for the 37 distinct modes of the SDS. These modes can be characterized by an average speed and an average acceleration. Of the 37 modes, 5 are regarded as "steady state" -- that is, the acceleration is zero. The five modes represent average speeds of 0, 15, 30, 45 and 60 miles per hour. The other 32 modes represent either periods of acceleration or deceleration and are characterized by an average acceleration, which is constant, and an average speed. The importance of the speed constraint can be appre- ciated by noting that if a vehicle accelerates from 0 to 15 miles per hour in t seconds, its emissions response is not the same as when it accelerates from 15 to 30 miles per hour or 45 to 60 miles per hour in the sane time of t seconds. The mathematical model which has been developed to predict vehicle emissions over any specified driving sequence is derived from vehicle data on emissions from the 37 modes of the SDS. One difficulty presented by the use of these discrete modes as inputs to a continuous driving sequence model is that during much of the sequence, the vehicle may be operating at velocities and accelerations not included in the set of five steady state and 32 accel/decel modes. For example, a vehicle traveling at 23 mph is neither in the 15 nph or 30 mph steady state mode. To arrive at a continuous predictive model, one must be able to interpolate or otherwise estimate the appropriate emission rates fbr * "Construction of Chassis Dynamometer Test Cycles", Scott Research Laboratories, Inc., November 18, 1971. 3 ------- all combinations of speed and acceleration encountered in the driving sequence. The primary contribution of this report is the development of a scheme whereby emissions from the 37 discrete modes can be expanded into a continuous function of time. Any driving sequence can be reduced to a speed time profile. Since acceleration is a function of speed change and time, both speed and acceleration can be expressed as continuous functions of time. The emission rate of a vehicle at a given point in time is dependent upon its speed and acceleration. Using these functional relationships, it is possible to integrate the emission rate function according to the time history of speed and acceleration associated with the driving sequence in question. An essential feature of the model presented in this report is a regression function which can, for purposes of visualization, be represented Emission Response Surface For any point (V , a) in the speed-acceleration plane there corresponds an * instantaneous emission rate S (v,a). The surface can be represented by a mathematical equation of the form: £ « f(v,a) in which the function f contains a nunfcer of adjustable constants. These constants can be selected to represent the emission characteristics of a particular automobile or can be selected to represent the mean emission characteristics of a collection of automobiles. This collection need not be homogeneous with regard to make, model, age or other identifying characteristics of automobiles. However, if a comparison of homogeneous sets of vehicles is desired, a characteristic emission function 4 ------- for each set can be derived. In short, the model presented in this report can be applied to individual vehicles or to composite groups of vehicles selected in whatever way is meaningful to pollution assessment. The flex- ibility of the model in this connection rests on the fact that the emission rate function is developed as a linear function of adjustable constants which can be particularized to an individual vehicle. Since the pooling of emissions fbr a composite group of vehicles is itself a linear summing operation, the composite emission function can be derived in a straightfoiward manner as a weighted linear sum of the emission functions for individual vehicles. Determination of the best process for pooling emissions from a collection of vehicles is beyond the scope of this report, but, to provide perspective fbr the use of the modal analysis, it is essential that certain aspects of composite emissions modeling be considered since they affect the modal analysis model. 1.2 COMPOSITE ANALYSIS OF VEHICLE EMISSIONS Computation of the emissions emanating from a particular traffic way in a given period of time is a composite of the emissions produced by all the vehicles which traversed that traffic way during that time. Quite clearly, it is not possible to assess the instantaneous emission rate functions for each automobile. However, if the composition of the vehicle mix is known or can be determined or postulated, one can define what might be called a "pilot mix" or analog of the actual traffic composition. For example, suppose that a fraction p^ of the vehicles belong to Category 1, a fraction P2 of the vehicles belong to Category 2, and so on, and that the number of vehicles traversing the traffic way per unit of time is N. An analog of this mix is n vehicles, where n «n, in which there are p^n vehicles of Category 1, P2n vehicles of Category 2, etc. in the sample. If the constants for the emission-rate function are determined fbr each of the n vehicles in the sample, these constants can be averaged over all vehicles comprising the sample to produce an emission rate function which is "typical" for the mix. This composite emission rate function can then be used to compute a typical or average emission for the specified 5 ------- driving sequence, where by "typical" is meant "representative of the vehicle mix in question." By multiplying this average emission by N, the total number of vehicles traversing the traffic way per unit of time, an estimate of the total emission contribution in the time can be obtained. Note that, by .virtue of the additive nature of the model, the result will be the same as if separate contributions to the composite were computed for each vehicle in the sample by means of its own specific emission-rate function, and then these individual emission outputs were added and scaled up by multiplying by the factor N/n. The approach taken above can be referred to as the method of pro- portional sampling -- that is, the number of vehicles in each category in the sample is proportional to the corresponding number of vehicles in each type in the population. An alternative approach is one in which no attempt is made to produce an analog of the mix in the population but rather an emission- rate function for each category is established independently of all other categories. For example, n^ vehicles of Category 1 would be subjected to modal analysis and an average emission rate function determined for Category 1 vehicles. Similarly, r\2 vehicles of Category 2 would be analyzed to determine an average emission rate function for Category 2 vehicles, and so on. It is presumed that the number of vehicles tested in each category (that is, n^, nj....) would be such that the desired precision is realized; among other things, these nunfcers would depend on the intra-class variability. Then, given that each category of vehicle has been characterized by an emission-rate function, a composite emission-rate function for any mix of vehicles could be computed as a weighted average of the emission-rate functions for the several categories. In view of the flexibility of the modal-analysis model, the definition of homogeneous categories of vehicles is not necessary. Indeed, categories of vehicles can be constructed arbitrarily, so long as these aibitrary categories are useful in the particular problem under study and can be weighted appro- priately in the composite result. Nevertheless, it was considered of interest to examine available modal data to determine if any significant groupings were evident and whether these groupings might influence the application of the emission model. In this connection it was found, by discriminant-ftmction analysis, that vehicles in Denver exhibit somewhat different emission-rate functions from comparable vehicles in other cities (see Appendix 5). The only 6 ------- way in which this observation affects the use of the model, however, is in the choice of input data. In short, to model traffic ways or to compare alternatives in Denver or in high-altitude locations one must use input modal data appropriate to these locations. In all other respects, the application of the emission model would be unaffected. The model is particularly valuable if the analyst wishes to examine alternatives, such as alternative routes or highway designs, before an actual highway is built. By postulating the anticipated mix of vehicles and the anticipated driving sequences, the relative desirability of alternatives can be ranked according to their pollution impact. 1.3 REPORT SCOPE AND PREVIEW In the ensuing sections of this report a methodology will be pre- sented that will enable an analyst to predict the amount of hydrocarbons (HC), carbon monoxide (CO) and oxides of nitrogen (N0x) given off by indivi- dual or specified distributions of light duty vehicles as these vehicles move from point A to point B by some defined speed-time profile. The methodology will be presented in a somewhat classical modeling approach. First, a description of the problem and the pi-oposed model objectives will be given in terms of the data that are available (the iconic model). Secondly, a mathematical model will be developed that parallels the iconic model in its objectives. This model will be amenable to computer implementation. The model's performance will then be analyzed to see if it is able to meet the objectives set for it. The proposed methodology is intended to be flexible enough to accomodate changes in emission parameters which are expected to result from improvements in emission control systems. This flexibility will also allow modification or extensions of the model's objectives as the need for such modifications arise. 7 ------- 2.0 PROBLEM DEFINITION Problem definition can best be understood in terms of the inputs and outputs of the model. Input data consist of vehicle modal emission measurements and speed versus time profiles for specified driving sequences. Output data consist of estimates of emissions for any given driving sequence. 2.1 INPUT Vehicle emission data are given for 1020 individual light duty vehicles that represent variations in model year, manufacturer, geographic location, engine and drive train equipment, accumulated mileage, state of maintenance and attached pollution abatement devices. The individual vehicle characteristics and emission data were obtained by Automotive Environmental Systems, Inc. under EPA Contract No, 68-04-0042.* For each of the 1020 vehicles in the data base, the following emission data are given for the three pollutants (HC, CO and NO^J under consi- deration: (A) Modal Emission Data: The annuat of each polLutant emitted in each of 37 defined speed-time profiles. There are three cases: speed is monotonically increasing and acceleration is constant and positive over time (accel); speed is monotonically decreasing and acceleration is constant and negative over time (decel); speed is constant over time and acceleration is zero (steady-state)j as shown below. Accel Speed Pecel Speed Speed Steady State Time * APTD-1497 "A Study of Emissions from Light Duty Vehicles in Six Cities" March, 1973. ------- The 37 speed-time curves are referred to as modes. There are 32 accel/decel modes and S steady state modes. (See Appendix I for modal specifications.) (B) Driving Sequence Emission Data (1) The total amount of each pollutant emitted during the Surveillance Driving Sequence: The Surveillance Driving Sequence repre- sents a speed-time curve of duration 1054 seconds; it is made up of the 32 accel/decel speed-time curves joined together by the 5 steady state modes. The Surveillance Driving Sequence was performed after the vehicle had performed the Federal Test Procedure Driving Sequence. Therefore, emissions measured over the Surveillance Driving Sequence represent emissions from a warmed-up vehicle. (See Appendix II for the speed-time values in the Surveillance Driving Sequence.) (2) Emissions measured for each vehicle twice using the FTP, once from a cold start and once from a hot start: Emissions were collected from the "transient" and "stabilized" portions of these tests and the data are reported as "cold transient", "cold stabilized", "hot transient" and "hot stabilized" values. Emissions from the Federal Short Cycle were also measured. In the study of model effectiveness, the values used were the amount of each pollutant given off during the FTP "hot transient" driving sequence (see Appendix II for the first 505 seconds of the Federal Test Procedure Driving Sequence). Hot transient data were used since the model has maximum effectiveness as a predictor of emissions for warmed-up vehicles. The use of data from the Federal Short Cycle to measure the effectiveness of the model has been left for future work. NOTE: The total amount of a pollutant emitted by a vehicle as it executes a driving sequence is often referred to as the "bag value". 2.2 OUTPUT DATA Given the modal emission data on an individual vehicle, the basic objective of the model is to develop a method which can predict the emission response of this vehicle over any specified driving sequence. The predictive ability of the model is restricted to accelerations and speeds in the sequence which do not exceed the range of accelerations and speeds spanned by the input modal data. In addition, it is desired to extend these individual vehicle responses so that the emission responses of specified homogeneous groups of vehicles can be predicted. 9 ------- 3.0 OVERVIEW OF THE MATHEMATICAL MODEL In this section a general description of the mathematical model development is given. A more detailed version of the model appears in Appendix III. 3.1 THE EMISSION RATE FUNCTION The mathematical model used to describe the emission response of a vehicle or group of vehicles is built around the concept of an instantaneous emission rate. (The instantaneous emission rate is defined as the rate at which a pollutant is given off at a specific point in time.) If the amount of a pollutant emitted by a vehicle from time ¦ 0 to any time * t is denoted by eft), then the instantaneous emission rate function l(t) is defined as the time rate of change of e(t). (1) §(t) = «[«()! The instantaneous emission rate at a specified time T is the value of the emission rate function evaluated at this time: (2) S(T) .( d^' ¦ ) t . T In the development of this model, it has been assumed that the instantaneous emission rate of a vehicle is a function of its speed, v and acceleration, a. Since speed and acceleration are considered to be time dependent, the emission rate function can be expressed as C3) e(t) = e(v(t) , a(t)) » e (v,a) Inherent in the definition of a driving sequence is a speed-time (and therefore acceleration time) profile. The amount of a pollutant given off by the vehicle over a driving sequence lasting T seconds is then given by integrating the emission rate function over the speed-time curve for the driving sequence of interest: 10 ------- (4) e(T) (v(t),a(t))dt where v(t) and a(t) are the values of speed and acceleration at time ¦ t specified by the driving sequence. In practice the driving sequences are specified by a series of speed-time points along the speed-time curve that are equidistant in time, as shown below. Vr SPEED where The integration in equation (4) is then approximated by the following summation: U-l (5) e(T) where X e(V®i ) A t L=-1 Vi Vi ~ 1 + Vi A ai vi + 1 ' vi At N At » T At this point, it is necessary to determine a suitable functional form of the instantaneous emission rate function in terms of speed and acceleration. 3.2 STEADY STATE AND ACCEL/DECEL* EMISSION RATE FUNCTIONS It is necessary to determine a functional form for the emission rate function for the steady state case and the case of accel/decel. In the steady state case (acceleration equals zero, constant speed) the emission rate function is a function of speed only. This case is presented first. * If a(t) « acceleration at time t, then a(t)decel. a(t)>©=>accel. 11 ------- For each of three pollutants, steady state emission rates averaged over the 1020 vehicles in the data base were plotted against speed. Inspection of these plots (Figures 1, 2 and 3) suggested that the steady state emission rate function eg could be expressed as a quadratic function of speed! (6) si + s2 v ¦* SjV where s^, s2 and s^ are constants. In the case of non-zero acceleration (accel/decel), the assumption is made that the acceleration occurring at a given speed is a perturbation to the steady state emission rate at this speed. This perturbation can be accounted for by letting the coefficients Sj, S2 and Sj become functions of acceleration. If it is assumed that quadratic functions of acceleration represent good approximations to these coefficients, the coefficients can be expressed as follows: 2 a fs 1 * sl(a) * ^11 + * 12 a + ^13 (7)t s2 * s2(a) - «21 + <22a + W2 s3 - s3(a) - q31 + q32a + q33a2 where the qs are constants. The emission rate function used during times of non-zero acceleration can then be written in the form: (8) ®$(v2a) s + b2V + ^3* + b4av + ^5y2 + b6&2 + b7v^a + bg&2v * b92 where the b's are constants and can be expressed in terms of the q's. It is noted that if a ¦ 0 equation (8) reduces to: • 2 (9) e^(v,a s o) s bj + b2v + b5v • » which has the identical form as the equation for es. Thus, e^ could be used to determine emissions for both steady state and non-zero acceleration periods. At this point in the discussion, however, separate functions for steady state and accel/decel emission rates will be retained; the reason for doing so will be given later in this report. 12 ------- The instantaneous emission rate function e for a given vehicle and pollutant is a composite function given by: (10) e(v,a) - h(a)eg(v) + (1 - h(a)) eA(v,a) where h(a) is a weighting function which is bounded by the values 0 and 1 and which is dependent on acceleration. Note that h(a) allows for a smooth, continuous transition from steady state to accel/decel emission rate Amotions or vice versa. The next step is to evaluate the twelve coefficients (b^,i ¦ 1,9; s^,i ¦ 1,3) for each vehicle and pollutant. These coefficients will completely specify the instantaneous emission rate function describing this vehicle's response with respect to the given pollutant. 3.3 DETERMINATION OF THE COEFFICIENTS (b±, s±) The coefficients that specify the instantaneous emission rate function could be determined by a straightforward application of the least squares regression method if values of the instantaneous emission rates were available. However, the data base on vehicle emissions does not contain any instantaneous emission rate observations for accel/decel modes; instead, the observations reported are the total amounts of the pollutants collected over each mode or the average emission rate for the mode (which covers, many speeds). In this light, the following method allows the determination of the coefficients that specify the accel/decel instantaneous emission rate function. (A) Specification of the Accel/Decel Emission Rate Function It can be shown that if the proposed form of the instantaneous accel/decel emission rate function is used to evaluate the functional form of the average emission rate function, the same coefficients that specify the emission rate function also appear in the average emission rate function in a linear fashion (see Appendix III). Now, the values for the average emission rate can be determined for each mode by dividing the amount of pollutant given off in the mode by the time in mode. A standard least squares regression analysis can then be performed on the average emission rate function which will determine the values of the coefficients that specify the instantaneous emission rate function. For exanple, suppose the instantaneous emission rate function is given as: 13 ------- e(v,a) ¦ bj + b£V + b^va Then the average emission rate function over T seconds is defined as: T T 3 4= 5 ®(v'a) dt = T eCT) o Substituting the functional form of the instantaneous emission rate function into the integral gives T s. j C k. + b. v 4- W3 va) dt = e CT)/t T T T K. C t 35 _[_ ^ -J- j" tj. V C^-t + -L ^ ^3 VC^. cl"t T o T o 0 r t t Let V/= ~r [ V jav Jt, and since i Jt = 1, have 6 T0 O eCT) /T = t>» 4- V + b3 oST The total emission e(T) given off in each mode and the time in each mode T are known, v, and Iv can be determined for each mode. TTie coefficients (b^) can therefore be obtained through least squares regression analysis applied to the average emission rate function. For the general model, there are 9 coefficients to determine and 32 accel/decel modes. A least squares regression analysis can be performed on an individual vehicle or on the mean of a group of vehicles. This approach forms a logical bridge from the experimental observations to the specification of the accel/decel emission rate function. (B) Specification of the Steady State Emission Rate Function I* the case of steady state conditions, the speed does not change with time. Thus, the average emission rate is equal to the instantaneous emission rate. Values of the steady state emission rate function are then available from the experimental observations, and the coefficients are evaluated directly using least squares regression techniques. * This expression in only an example, deliberately simplified for illustrative purposes. 14 ------- There is, however, one problem that crops up with the above straight- forward least squares approach. The values of the emission rate vary greatly between speed zero (idle) and a speed of 60mph. As a result, the least squares approach sometimes produces a steady state emission rate function which predicts negative emission rates for certain speeds. In this event, the function is adjusted by means of a constraint on its minimum value. The two lowest emission rates measured experimentally are determined and averaged. Similarly, the speeds corresponding to these two rates are also averaged. The average rate and average speed determined in this way are then taken as the coordinates of the minimum point of the emission rate function. Two of the three coefficients that specify the steady-state emission rate function are thus determined; the third is computed by the least squares method subject to this constraint (see Appendix II for details). 3.4 THE COMPOSITE EMISSION RATE FUNCTION As stated earlier, two separate emission rate functions were desired in order to describe accel/decel and steady state conditions. The two functions are then joined by means of the weighting function as defined by equation (10). The reason for retaining a separate function for steady state conditions when the accel/decel function appears flexible enough to handle the steady state case is that the accel/decel rate function also produces negative emission rates in some cases for steady state speeds; any efforts to modify the coefficients to constrain the function to yield only positive steady state emission rates would produce serious errors when the function is used to evaluate accel/decel emission rates. The composite emission rate function allows the freedom to adjust the coefficients of the steady state emission rate function without disturbing the accuracy of the accel-decel emission rate function. 3.5 VEHICLE AND VEHICLE GROUP CHARACTERIZATION Once the emission rate function for a vehicle and pollutant is specified, it can be used to obtain this vehicle's response, over any given driving sequence, by integrating the rate function over the speed-time curve defined by the driving sequence. Each vehicle is characterized by 36 15 ------- parameters or coefficients; 12 parameters for the specification of each emission rate function describing the HC, CO, and N0x response. The characterization of a group of vehicles can be achieved by- defining the emission rate function for the average vehicle within the group: Let ss k'th coefficient in the emission rate function for the j'th vehicle within the group and ith kind of pollutant. Ng = nunfcer vehicles in the group. "fc ik ~ coefficient in the emission rate function describing the average vehicle's i'th kind of pollutant response. Then, IM, $ »» ik ¦ itH bi* g If* Thus, the group emission rate functions aTe determined by averaging the coefficients which make up the emission rate functions of each vehicle in the group. The emission response of the group over any driving sequence is then deterjnined by multiplying the average vehicle's response by the number of vehicles in the group. The average vehicle's response is obtained by integrating its rate function over the speed-time curve specified by the driving sequence. 16 ------- 4.0 COMPUTER IMPLEMENTATION The computer version of the mathematical model to calculate emissions given off from individual vehicles and vehicle groups over any specified driv- ing sequence is made up of two main programs: Main Program I A main program to compute emissions from individual vehicles over any specified driving sequence. Main Program II A main program to compute emissions from a specified group of vehicles over any specified driving sequence. The main programs are used to read in the speed-time values of the driving sequences, to perform any filtering operations needed to define vehicle groups, to write out calculated emission values, and to call in proper sequence the following set of routines which perform the majority of calculations: Subroutine SETUP Output: Subroutine SETUP determines the basis function factor arrays for the accel/decel and steady state emission rate functions. These arrays are labeled AA and AS, respectively. Input: Speed-time values for the Surveillance Driving Sequence. Utilization: Called once in each main program. Other Subroutines Used: Subroutine INVERS (to calculate the inverse of a matrix). Subroutine EDOT Output: Subroutine EDOT calculates the 36 coefficients specifying the three emission rate functions for an individual vehicle. Input: (i) The amount of each pollutant given off by an individual vehicle in each of 37 modes. (ii) The basis function factor arrays AA and AS determined by subroutine SETUP. 17 ------- Utilization: EDOT is called once for each individual vehicle con- sidered in main programs I and II. Other Subroutines Used: Subroutine PAD. Subroutine PAD Output: Subroutine PAD calculates a set of 3 coefficients that specify the steady state emission rate function for an individual vehicle such that the emission rate function does not produce any negative emission rates. Input: The amount of pollutant given off by an individual vehicle in the five steady state modes. Utilization Called once from subroutine EDOT for each individual vehicle that originally had a steady state emission rate function that pro- duced negative emission rates. Other Subroutines Used: None. Subroutine EDGRP Output: A set of 36 coefficients that specify the emission rate functions for the average vehicle within a group of vehicles. Input: (i) The 36 coefficients specifying the emission rate func- tions of individual vehicles determined by subroutine EDOT. (ii) Sequence indicator INT. Utilization: Called by Main Program II only; once for each vehicle in the group (INT « 1 for first vehicle in the group, INT = 2 for all the following vehicles in the group), and once after all vehicles in the group have been considered (INT = 3). Other Subroutines Used: None. Subroutine ESUM Output: The amounts of the three pollutants (HC, CO, NO^ given off by an individual vehicle or a group average vehicle over any specified driving sequence. 18 ------- Input: (i) The 36 coefficients that specify the emission Tate functions determined by subroutine EDOT (Program I) or subroutine EDGRP (Pro- gram I[). (ii) The velocity-time values for the driving sequence under consideration. Utilization: Called once for each vehicle in Main Program I, called once for the group under consideration in Main Program II. Other Subroutines Used: None. Subroutine INVERS Output: The inverse of any two-dimensional square matrix of dimen- sion less than 20. Input: (i) The matrix whose inverse is desired. (ii) Dimension of the matrix. Utilization: Called once by subroutine SETUP. Other Subroutines Used: None. Flew charts for the main programs (see Figures 4 and 5)) show the calling order for the four main subroutines: SETUP, EDOT, EDGRP and ESUM. In any main program the following calling sequence of the main subrou- tines must be strictly observed: SETUP I EDOT V ESUM Listings of the main programs and subroutines are given in Appendix IV. 19 ------- 5.0 MODEL PERFORMANCE Evaluation of the performance of the model can be approached only by comparing computed and measured quantities. For this purpose, it was found convenient to use the Surveillance Driving Sequence and the first 505 seconds of the Hot Federal Test Procedure driving cycles as measurably quantities and to compare these quantities with the corresponding outputs predicted by the model. Also, it must be appreciated that the degree of agreement between com- puted and observed results will vary from vehicle-to-vehicle and that ultimate evaluation of the validity of the model must take into account this statistical variability. Toward this end, several statistical quantities were employed, as discussed below. 5.1 STATISTICAL INDICATORS OF PERFORMANCE Notation 0^ ¦ observed amount of i kind of pollutant given off by j vehicle over a specified driving sequence (observed bag value) ¦ calculated amount of i kind of pollutant given off by j vehicle over a specified driving sequence (calculated bag value) N = number of vehicles in sample (1020) c R.. ¦ bag value error = 0.. - C.. To analyze the performance of the emission rate model in predicting bag values, the following statistics are evaluated: (a) The Mean Bag Error or Bias (FL) for each type of pollutant: Nlc, ^ = 7Nc ^ ^0ij " ) or _Nc 20 ------- (b) The Standard Deviation of the Bag Error (G"R ) for each pollutant: _ i , x V j=t (c) Root Mean Square Deviation of the Bag Error (RMS^) for each pol lutant: / v. ~ ' rms> = 7 Si i- 0" % c Tei /ol ) • too % ¦ m (J at + ir%i / 0c). ,ot •/ The mean, standard deviation, and root mean square deviation of the bag error together provide insight into how the bag errors are distributed. Expressing these statistics in terms of percent of the observed mean gives an indication of how serious the bag error distribution is. 5.2 PERFORMANCE RESULTS The values of the statistics for bag values obtained in the Surveil- lance Driving Sequence and first 505 sec. of the Hot Federal Test Procedure's driving sequence (hot transient) are given in Tables 1 and 2. A visual inspection of the distribution of bag value errors is offered by Tables 3 through 8 and corresponding histograms on Figures 6 through 11. 21 ------- 5.3 DISCUSSION AND EVALUATION To focus attention on the adequacy of the model, it is helpful to condense Table 1 and Table 2 to a somewhat more concise form, as shown below. PERCENT RMS ERROR BETWEEN CALCULATED AND OBSERVED BAG VALUES FOR 1020 VEHICLES Surveillance First 505 Seconds Federal Driving Sequence Test Procedure HC 26.1 32.0 CO 23.9 29.1 NO 27.1 28.0 x The percent RMS error is defined as: J ^ / & j ICO'/, and represents the combined systematip and random errors. It is a particularly meaningful quantity if one assumes that the mean or expected difference between the calculated and observed values should be zero. A? will be noted in Tables 1 and 2, the RMS values are largely dominated by the random error component, as represented by (Tr2. Moreover, these tables, together with the histograms showing the error distributions, suggest that the difference between cpjnputed and observed results cluster rather closely around the average error If and that this average value deviates from zero by only a few percent of the average measured bag values. A logical question arises, however, as to the interpretation which should be put on such terms as "cluster rather closely around the average" or on such quantitative measures of performance as "25% RMS error". Against what criterion are these measures of performance to be judged and is the nodel to be judged satisfactory or unsatisfactory? 22 ------- To answer this question, one must consider the quality of the input data and the manner in which errors in the input propagate into errors in the output. In particular, one must inquire as to the repeatability of emission measurements performed on the same vehicle and ostensibly under identical test conditions. If the results of the Surveillance Driving Sequence or any other specified driving, sequence fail to repeat on replicate tests, this failure can not be traced to the inadequacy of a computational model, because no such model is involved. The acpruement of instantaneous emissions over the driving sequence is a physical, not a mathematical, process of integration, and the vehicle and the measuring instrumentation constitute the only "computer" in the system. Of the 1020 vehicles in the input data set, 61 had been tested twice each. Thus there were available 61 "replicate" measurements from which can be obtained a measure of repeatability of measurements. This measure of repeatability can be easily obtained as follows. Consider a particular vehicle, and compute the mean X^ of the two replicate measurements. Then compute the quantity 0"£ = ( ¥" !k~ Xja. ) -f- (Xz.k — ) N - / where and are the two replicate measurements for the k vehicle. Since N = 2 in this case, the formula for (j~.^ reduces to the simple form fa. (%m- tikf- A 2 Now let us assume that the quantity is one estimate of the variance of replicate determinations and that each of the other 60 pairs of values provide an additional estimate. These 61 estimates can be pooled or averaged to obtain: ^ - t k.-t 23 ------- as a best estimate of the variance of replicate values. Similarly, the quan- tities X, (k * 1, 2, , 61) can be pooled to obtain an estimate of the _ K A _ mean X for the total collection of vehicles, and the quantity be taken as a relative or percent standard deviation characterizing the repeata- bility of measurements. a _ The values are shown below for the Surveillance Driving Se- quence and for the first 505 seconds of the Federal Test Procedure. PERCENT STANDARD DEVIATION BETWEEN REPLICATE BAG VALUES FOR 61 VEHICLES Surveillance First 505 Seconds Federal Driving Sequence Test Procedure HC 68.6 70.6 CO 14.4 26.9 N0x 15.5 15.8 Comparison of these values with those obtained by comparing calculated and measured results suggests that the errors are comparable in the two cases. Consequently, it is concluded that the model is performing quite acceptably and that, indeed, its performance is substantially limited by the variability inherent in the test measurements themselves. Further support for this point of view is found in Tables 9, 10 and 11. Based on the 61 replicates, the quantities >f, These errors are reflected as errors in the determination of the regression coefficients, and these errors in turn determine the error of estimating the instantan- eous emission rate at any point in the (a,v) - space. Procedures are available to trace the error propagation through this rather involved process and to produce 24 ------- "variance maps" in (a,v) - space,* but this type of analysis is beyond the scope of this report. Moreover, even if such a variance surface were available, one must further translate this surface into its effect on the integrated emissions for a particular driving cycle. In view of the relatively large errors in modal input data, however, the 25% to 35% RMS errors obtained for model per- formance do not appear unreasonable. Further insight into this matter can be had by an elementary and straightforward application of analysis of variance as follows. Denote by ijk ^ = 2' ^ a ^ = ^ replicate of the vehicle, where i = 1 denotes measured values and i ® 2 denotes values computed from the model. For each vehicle, therefore, there are four values of total emission for each of the pollutants HC, CO and N0X. These are: Xj^ = Bag value measured for first replicate Xi2k = Ba® vaue measured ^or second replicate X2jk ¦ Bag value computed for first replicate = Bag value computed for second replicate These four values can each be decomposed into components representing the effects of the model, the effects of replication, and the interaction between replica- tions and models. Let us visualize the effect of the model as shown in the sketch below * OUTPUT "Identity" model INPUT INPUT Mi Mc \ » OUTPUT "Computational" model In the "identity model", the measured emissions are subjected to no computation, the bag values being those obtained directly from the measurement process itself. In the "computational model", the modal measurements are used as the basis for generation of an emission-rate surface, and bag values are computed by integra- i H. T. McAdams, "A Computer Method for Hypsometric Analysis of Abrasive Surfaces," Advances in Machine Tool Design and Research, 1968, Pergamon Press, Oxford, 1969, pp. 1149-1171. 25 ------- tion over the appropriate driving sequence. Thus if the data from replicate tests are fed to these two "models" as inputs, the outputs will differ because of the difference in the "transfer functions," Mj and M^,, of the two models. The difference between the outputs of the models can be called the "model effect" and is a measure of the extent to which the computationally inte- grated results fail to agree with the physically integrated results. In pre- vious discussion we have referred to this difference as a measure of the "validity" of the computational model. In the present analysis, however, we wish to examine this difference in relation to the repeatability of the input measurements. An appropriate statistical model for the analysis is y-ijt = + eCik + fijk t i « 1, 2; j > 1,2; k = 1,2,...., 61. where is the output of the it'1 model for the j1 replicate on the k1 vehicle. The convention, of course, is that i ¦ 1 denotes the identity or physical model and i * 2 denotes the computational model. The quantity is the mean of the four output values for each vehicle and can be thought of as the common or reference value against which model and replication effects can be compared. The quantity ^ is a departure from this mean occasioned by the effect of the particular model; since <7^ + " °» 006 of the models will be represented as a negative departure, the other as a positive departure from the mean. The quantity s a departure from the mean occasioned by replication. As far as is concerned, it is assumed that the difference between replicates, in the statistical sense, is the same for the identity model and for that computational model; hence, p ^ represents a pooled estimate incor- porating both expressions of the replication effect. Since ¦ 0, one of the replicates will be represented as a negative departure from the mean, the other as a positive departure from the mean. In reality, there is a distinct possibility that replication will be influenced by the model—that is, it might be anticipated that replicate results emerging from the computational model might be different from replicate results emerging from the identity model, If such is the case, then there is inter- 26 ------- action between replication and models. This interaction is measured by the term ( o( which represents a "correction", in a sense, to the assumption that repeatability of the output does not depend on whether one is considering the identity or the computational model. Decomposition of into components is easily accomplished by the matrix transformation 1/4 1 -1 1 -1 1 1 -1 -1 1 1 1 1 - 1 """ Xllk Ak X12k ik X21k J Y 22k ({{!>) ijk Moreover, and ( J) ^ represent, respectively, the mean squares for models, replication and interaction in a two-way analysis of variance. Each of these mean squares has one degree of freedom. Our analysis is completed by computing these mean squares for each of the 61 vehicles and averaging these values. These results are presented in Table 12 under the heading "mean squares". Viewed directly, however, these mean squares are somewhat misleading, because the statistical expectation of the mean squares for—say, the models effect—is not the variance q associated with models but rather: + r}. „2 where 0"^^ is the interaction variance. Similarly, the expected mean squares for replications is not the variance associated with replications but rather 3 ------- <>2 <7~~ fr" ft " XJ = Teplicates mean squares <¦ J a interaction mean squares / one can extract the variance components ^ , (f^ and • These are displayed in Table 12 under the heading "Variance Components", Though the analysis for HC, CO, and NO^ as well as the analysis for the two driving sequences give different results, the general impression is that the models and replications effects are of comparable magnitude (note, in particular, the results for CO and NO^ for the Surveillance Driving Sequence). In the case of HC, it appears that the replications effect is much larger than the models effect for both the first 505 seconds of the Federal Test Procedure and for the Surveillance Driving Sequence. However, examination of the data for individual vehicles revealed that there was one vehicle for which the bag values replicated so poorly that the case might be considered an outlier. This single vehicle is largely responsible for the large replications Mean square for HC. Special attention must be given to the interaction components. Though there are several examples in which this component is of appreciable magnitude, it can not be concluded that the computational model has poorer repeatability than the identity model, because all of the variance components denote magnitude only, not direction. Indeed, examination of the data for individual vehicles reveals that the interaction effect is about as likely to be negative as posi- tive. Often the interaction is occasioned by the fact that the ranking of the two replicates is reversed when one goes from the identity model to the compu- tational model. For example, in the identity model the first replication might yield a higher bag value than the second, but in the computational model the reverse might be true, yet the magnitude of the difference between the two replicates might be the same in both cases. In conclusion, the computational model performs remarkably well in view of the relatively large errors in the modal emission rates which serve as inputs. Since the model reproduces the measured bag values about as well as a 28 ------- replicate does, it is postulated that the model can predict emissions form a non-standard driving sequence about as well as might be expected from an actual test performed on that driving sequence. Further experience with the model is needed, however, to be more assertive on this point. However, the model has maximum effectiveness as a predictor of vehicle group emission char acteristics, since the input variability of a homogeneous group of vehicles is in general less than the input variability of any individual vehicle. 29 ------- 6.0 SUMMARY AND CONCLUSIONS In the preceeding discussion, a method was presented to calculate the amounts of HC, CO and N0x emitted by individual vehicles and vehicle groups over any specified driving sequence. The method uses, as inputs, the modal emission data on individual warmed-up vehicles. It is to be understood, of course, that the mathematical model should be used only within the region of the speed and acceleration space which is spanned by the input modal data. The model has maximum effectiveness as a predictor of group emissions for warmed-up vehicles. The method, given in terms of a vehicle emissions model, is character- ized by the concept of an instantaneous emission rate. From this concept, the emissions response of individual vehicles and vehicle groups are given in terms of instantaneous emission rate functions. The development of the instantaneous emission rate functions for a vehicle contains two important computational features: the assessment of the coefficients that specify the instantaneous emission rate function and a method to bound the steady state emission rate function so that the function is non-negative (does not produce negative emission rates) on the speed interval (0,60) mph. In a least squares fitting procedure, it is possible to obtain negative predicted values for some points on the speed and acceleration/deceleration surface. Such a possibility is most likejly in extrapolated areas of the surface or areas with very few actual data points. Due to the complexity of the prediction procedure over the range of accel/decel space, the current model does not check for negative emissions for each possible point in the prediction surface of each vehicle. This type of problem did not occur for the set of vehicles considered when appropriate weighting functions were used. However, a test for negative emissions over accel/decel space and an appropriate mathematical correction is planned as a future refinement to the model. The instantaneous emission rate function can be used to characterize an individual vehicle's emission response over any driving sequence as well as to describe a vehicle group's emission response. The latter is accomplished by the determination of the group's average vehicle emission rate function. Further, a means of investigating the homogeneity of hypothesized vehicle groups using linear discriminant function analysis is presented. (Appendix V) 30 ------- The content of the discussion and structure of the model allows fbr the immediate use of the emissions model by an analyst to predict vehicle emission and serves as a base for further research in predicting vehicle emissions and their effect on the environment. 31 ------- TABLES 32 ------- TABLE 1: BAG VALUE STATISTICS FOR THE SURVEILLANCE DRIVING SEQUENCE POLLUTANT 0 R <7R2 Cf — .100% 0 — .100% 0 \ R2 4r2 ' \ r2 2 - .100% 0 HC 53.5 7.2 143.3 12.0 14.0 13.5 22.4 26.1 CO 625.0 43.1 20420.8 143.0 149.3 6.9 22.9 23.9 NO X 48.2 -2.7 163.0 12.8 13.0 -5.6 26.5 27.1 ------- TABLE 2: BAG VALUE STATISTICS FOR THE FIRST 505 SEC OF THE FEDERAL TEST PROCEDURE DRIVING SEQUENCE POLLUTANT 0 R Ci2 Gr — .100% 0 Ci — ~y R2 +(j]f 2 .100% 0 .100% 0 HC 21.0 2.8 37.3 6.1 6.7 13.4 29.0 32.0 00 223.7 9.2 4158.0 64.5 65.1 4.1 28.8 29.1 NO X 17.2 0.5 22.9 4.8 4.8 2.9 27.8 28.0 ------- TABLE 3: DISTRIBUTION OF HC BAG VALUE ERROR (OBSERVED - CALCULATED) FROM THE SURVEILLANCE DRIVING SEQUENCE ERROR (GMS) NUMBER OF VEHICLES -70 to -65 1 -65 to -60 1 -60 to -55 0 -55 to -50 0 -50 to -45 0 -45 to -40 0 -40 to -35 1 -35 to -30 0 -30 to -25 2 -25 to -20 4 -20 to -15 5 -15 to -10 9 -10 to - 5 24 - 5 to - 0 149 0 to 5 309 5 to 10 246 10 to 15 97 15 to 20 80 20 to 25 37 25 to 30 16 30 to 35 18 35 to 40 3 40 to 45 4 45 to 50 5 50 to 55 1 55 to 60 0 60 to 65 2 65 to 70 2 70 to 75 0 75 to 80 2 80 to 85 1 85 to 90 0 90 to 95 0 95 to 100 0 100 to 105 0 105 to 110 0 110 to 115 0 115 to 120 1 TOTAL 1020 35 ------- iE 4 Isi 700 650 ¦600 550 500 450 400 350 300 250 200 150 100 • 50 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 DISTRIBUTION OF CO BAG VALUE ERROR (OBSERVED - CALCULATED) FROM THE SURVEILLANCE DRIVING SEQUENCE NUMBER OF VEHICLES 1 0 0 0 1 1 0 1 4 3 10 22 57 94 170 272 143 92 53 32 27 6 7 7 2 5 2 1 2 1 0 0 0 4 TOTAL 1020 36 ------- TABLE 5: DISTRIBUTION OF NOx BAG VALUE ERROR (OBSERVED - CALCULATED) FROM THE SURVEILLANCE DRIVING SEQUENCE ERROR (CMS) NUMBER OF VEHICLES -65 to -60 1 -60 to -55 0 -55 to -50 0 -50 to -45 2 -45 to -40 1 -40 to -35 3 -35 to -30 8 -30 to -25 13 -25 to -20 26 -20 to -15 81 -15 to -10 98 -10 to -5 203 -5 to 0 298 0 to 5 145 5 to 10 60 10 to 15 44 15 to 20 22 20 to 25 17 25 to 30 14 30 to 35 5 35 to 40 6 40 to 45 1 45 to 50 0 50 to 55 1 55 to 170 1 TOTAL 1020 37 ------- TABLE 6: DISTRIBUTION OF HC BAG VALUE ERROR (OBSERVED - CALCULATED) FROM FIRST 505 SEC, OF THE FEDERAL TEST PROCEDURE DRIVING SEQUENCE (HOT TRANSIENT PORTION) ERROR (GMS) .NUMBER OF VEHICLES -45 to -40 1 -40 to -35 0 -35 to -30 1 -30 to -25 2 -25 to -20 3 -20 to -15 5 -15 to -10 3 -10 to -5 21 -5 to 0 158 0 to 5 592 5 to 10 172 10 to 15 35 15 to 20 11 20 to 25 9 25 to 30 . 1 30 to 35 . . 3 35 to 40 1 40 to 45 1 45 to 50 0 50 to 55 0 55 to 60 . 0 60 to 65 1 38 ------- TABLE 7: DISTRIBUTION OF CO BAG VALUE ERROR (OBSERVED - CALCULATED) FROM FIRST 505 SEC. OF FEDERAL TEST PROCEDURE DRIVING SEQUENCE (HOT TRANSIENT PORTION) ERROR (GMS) NUMBER OF VEHICLES -350 to -300 2 -300 to -250 1 -2S0 to -200 1 -200 to -150 8 -150 to -100 24 -100 to - 50 72 - 50 to 0 279 0 to 50 483 50 to 100 103 100 to 150 30 150 to 200 3 200 to 250 5 250 to 300 0 300 to 350 1 350 to 400 1 400 to 1000 1 TOTAL 1020 39 ------- TABLE 8: DISTRIBUTION OF NOx BAG VALUE ERROR (OBSERVED - CALCULATED) FROM FIRST 505 SEC. OF FEDERAL TEST PROCEDURE DRIVING SEQUENCE (HOT TRANSIENT PORTION) ERROR (GMS) NUMBER OF .VEHICLES -25 to -20 2 -20 to -15 1 -15 to -10 13 -10 to - 5 60 - 5 to 0 468 0 to 5 351 5 to 10 88 10 to 15 24 15 to 20 10 20 to 25 3 40 ------- MODE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 FTP SDS TABLE 9 REPLICATE MDDAL ANALYSES OF HC FOR 61 VEHICLES X (gms/min.) 3.8570 1.6284 2.4522 2.8327 3.5073 1.9178 5.2240 2.7310 4.4227 2.9622 4.9806 3.0453 4.9745 3.2702 1,8480 1.4517 4.2265 2.2881 3. 7102 2.3403 5.7121 3.1666 3.5338 4.2144 2.9225 1.8284 4.2448 3.0080 3.1500 4.5983 2.6819 1.9091 1.2829 1.2182 1.6903 2.5522 3.2911 21.3255 54.4599 C"(gms/min) 2.2342 1.1856 1.2589 2.7032 3.2759 1.0114 3.2673 2.1971 3.7429 1.3290 3.6829 1.3054 4.1268 1.8522 0.8133 0.5117 3.4144 1.5467 3.0242 1.3852 4.1975 1.2412 2.8711 3.7651 1.6542 1.0534 3.3618 0.9896 2.6789 3.8230 1.4452 1.1396 0.3898 0.9104 1.0114 2.3242 2.6043 15.0555 37.3647 -ST .100% X 57.93 72.81 51.34 95.43 93.40 52.74 62.54 80.45 84.63 44.86 73.95 42.87 82.96 56.64 44.01 35.25 80. 79 67.60 81.51 59.19 73.48 39.20 81.25 89.34 56.60 57.61 79.20 32.90 85.04 83.14 53.89 59.69 30.39 74.73 59.84 91.06 79.13 70.60 68.61 41 ------- TABLE 10 REPLICATE MODAL ANALYSES OF CO FOR 61 VEHICLES MODE X (gms/min) A ------- 1 2 3 4 5 6 7 8 9 10 11 12 13 14 IS 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 FTP SDS TABLE 11 REPLICATE MODAL ANALYSIS OF NO^ FOR 61 VEHICLES X (gm/min) 3.5689 0.6435 0.9107 2.8518 5.2695 1.5707 6.9947 2.9842 7.9071 1.8646 7.2462 1.7380 7.0501 1.9580 0.6813 0.3179 4.9960 1.1132 4.8965 0.9471 6.6968 1.3989 2.9016 6.9010 2.0334 0.6391 6.5950 1.3037 2.5598 6.6468 2.1487 0.6203 0.1174 0.1826 1.0192 3.1829 6.3341 18.2670 50.8930 ------- TABLE 12 VARIANCE COMPONENT ANALYSIS FOR 61 REPLICATE TESTS Source First 505 Seconds - Federal Test Procedure Mean Squares Variance Component HC CO NO X HC CO NO X Models 5.74 832.57 3.39 1.54 79.59 1.01 Replications 94.00 784.67 2,64 45.67 55.64 0.63 Tnteraction 2.65 673.39 2.37 2.65 673.39 1.37 Surveillance Driving Sequence Source Models Replications Interaction Mean Squares HC CO NO x 24.95 5560.34 29.76 600.84 4759.36 26.37 9.54 2770.54 5.13 Variance Component HC 7.75 295.65 9.54 CO 1394.90 994.41 2770.54 NO x 12.31 10.61 5.13 44 ------- FIGURES 45 ------- BEE 20x20 TO INCH O* ------- ------- BEE 20x20 TO INCH ------- FIG 4 : MAIN PROGRAM I SUBROUTINE SETUP Subroutine EDOT subrouting ESUtf 49 ------- FXG 5 : MAXN PROGRAM JH READ VELOCITY VS. 77WE ARRAY FOR THE. SURVG/LLANc£ DRIVING SEQUENCE. 1 — READ VELOCITY VS. TIME 4ERAY FOR. WE DRIVING SEQUENCE OVER WHICH EMISSIONS ARE! TO BE CALCULATED. I DETERMINE BASIS FUNCTtOfii FACTOR. ARRA\ l READ IA/ VEHPCLE SPECIFICATIONS. NO TS "77//S VBM/CLE ^ "'¦s, XaI GROUP uhpER CONSIDERATION specified 4Goup filter. READ XA/ VEHICLES MODAL EMISSIONS DATA. D£T£RM/ME EMISSION RATE FUNCTionToR VEHKLE Add vehicles emission rate function COEFFICIENTS TO GROUPS FUAiCTmN SUBROUTING ED6T SUBROUTINE ED&RP, n GROUP? DETERMINE EMISSION rate function for, "avera&e* vehicle Representing group. ?mte&> Cats: AllERACE vehicles EMISS/oN RATE FUNCTION OVER. SPGZ-/Fiei> DRIVING SE$UENC«E. — » DETERMINE TOTAL EMISSIONS &VEN OFF B V QRCUP. SuenoHVA/e ED&RP, tut 3. lUBRMVNE €SUM I WRITE OUT MVUUT HC, CO, UOf &!V£N OFF, 50 ------- ------- BEE 20x20 TO INCH ------- ------- ------- ------- BEE 20x20 TO INCH ------- APPENDICES ------- APPENDIX I ------- APPENDIX I Specification of the 37 Modes and Evaluation of the Average Values of the Basis Functions (I) Modal Specifications: Table 1-1 (II) Evaluation of the Average Values of The Basis Functions The nine functions of speed (v) and acceleration (a) which form the basis functions of the instantaneous emission rate function must be averaged over each mode in order to determine the coefficients which specify the emission rate functions (See Appendix III). To demonstrate how these average values are evaluated for the basis 2 2 2 2 functions: 1.0, v, a, va, v , va , v a , consider the case of determining the average value of the 3rd basis function va over the i'th mode. Let: T. = duration of i'th mode i Vi(t) = speed at time a t in the ilth mode, t £ (.+"} - d Vt 1+0 acceleration at time = t in the i'th mode, It t Ti then by definition: \m. tf viu) cuiAlJt O This integration is evaluated by the following approximation: K* \ Ti u a- S? where: V., * initial speed of mode: ll = average speed over the j'th time interval ¦z. 2 LS— of mode i ------- 4t = average acceleration over the j th time interval of mode i and» N i. At. = TL The averages of the other 9 basis functions are similarity determined. Values for the averages of the basis functions over each mode are given in Table 1-2. ------- Table 1-1 MODAL SPECIFICATIONS MODE DUKATIUNISEC) 0 I STANC£(HI) ¦SPeeO AT LME SEC. INTERVALS I 12. 0 0.06020 0.0 1.8 5. 1 9.1 13.2 30.0 2 " 160 0.07410 30.0 29.6 28.9 ~ 28.1 26.9 6.4 3. 7 1.6 0.3 0. 0 3 80 0.02010 0.0 1.7 4. 6 7.6 10.3 4 ll.Q 0.07050 15.0 15.9 17.3 18. 20.6 5 13.0 0.13600 30.0 30. 7 31. 7 32.9 34.2 44.4 45.0 ~ b 12 . J 0.12680 5.0 44.7 44. 1 43.4 42.2 30.0 7 17.U 0.21630 30.0 30.0 31.0 32. 5 34. 3 51.6 53.6 55.5 57,2 58.6 8 12.0 0.17160 60.0 60.0 59.7 59.1 58. 3 45.L 9 14.0 0.20430 4 5.0 45.6 46.5 47.6 48. 7 58. 5 59.3 60.0 10 30.0 0.33670 600 59.7 59.3 58.8 58. 3 49.2 47.4 45.5 43,4 41. 1 22.1 20.2 18.4 17.0 15.9 U 26.0 0.31360 15.0 16.7 20.0 23.1 26. 1 456 46.8 48.5 50.1 51. 5 58.9 59.3 60.0 12 21.0 0.19730 60.0 59.4 58. 5 57.4 56. 1 31.2 26.6 21.9 17.3 12.8 13 32.0 033130 0.0 15 5.2 8.6 12.0 34.5 36.8 39.0 41.0 43.0 54.5 55.5 56.4 57.2 57.9 14 23.0 0.29940 60.0 597 59.3 58.9 58.3 47.8 45.8 43. 7 41.6. 39. 4 15 9.0 0.05 790 30.0 29.5 28. 7 27.3 25.3 16 B0 0.01730 15.0 14.4 13.3 11.3 8.5 . 17 220 0.17590 0.0 2.4 6.3 10.0 13.5 34.4 36.2 37.8 39.2 40.5 IB 16.0 0.13920 45.0 445 43.9 43.0 41.8 20.7 18.1 16.1 15.1 15.0 19 la.o Q.15280 150 15.9 17.3 18.9 20.7 369 38. 8 40.5 42.6 43.4 _ 20 19.0 0.13040 45.0 44.5 43.7 42,7 M»4_ 17.8 13.8 9.9 6,4 3.5 21 25.0 0.26540 0.0 2.6 7.1 11.5 15.7 41.8 44.2 46.4 48.5 50.3 593 60.0 22 28.0 0.26340 60.0 59.6 59.0 56.31 57.4 43.6 40.8 37.8 34.6 31,3 5.2 3.0 1.3 0.3 0.0 23 15.0 007370 0.0 12 3.5 6.4 9.6 27.8 28.7 29.6 30.0 24 25.0 0.31340 300 30.5 31.4 32,4 33.6 44.6 46.0 47.5 48,9 50.4 59.3 60.0 25 18.0 023620 60 .0 596 59.1 58.4 57.6 41.6 38.9 36.3 34.0 32.1 26 10.0 0.04440 30.0 29. 1 2t. 7 25.4 22.0 2? 38.0 0.40090 0.0 0.9 4.0 6.9 9.0 29.6 31.7 33.7 35.7 37.5 49.6 50.8 51.9 52 ,9 53. 8 58.9 59.3 60.0 2a 35.0 0.32930 60.0 59.7 59.2 58.7 58.1 49.3 47.6 • 45.7 43.6 41.4 20. 1 17.3 14.6 12.0 9.5 29 180 008860 0.0 0.8 2.7 4.9 7.5 25.2 26.5 27.4 28.3 29.0 30 21.0 0.25990 30.0 30.7 31.8 33.1 34.5 48. i 49.8 51.5 53.0 54.6 31 14.0 0.18130 60.0 59.5 58.7 57.6 56. 1 32.1 30.4 30.0 32 13.0 0.05920 30.0 29.4 28.5 27.2 25.4 0.3 0.0 33 60.0 0.0 SP - 0. OFQR 60 SEC 34 60.0 0. 25000 spee o - 15. OFQR 60 SEC 35 60.0 0.5C JCO sPeo » 30. OfOR 60 SEC 36 60.0 0.75000 speep 45. OF OR 60 SEC 37 60.0 l.COOOO SPtfBC - 60. OFOK 60 SEC 17.1 20.b Zi.i, 25.4 23.5 21.2 12.1 22.4 35.6 0.7 36.2 59.8 57.2 50.0 57. 6 38.8 15.2 -ZS9. 52.8 5. 8.7 15.3 44.8 56.5 57.6 37.3 22.7 5.2 jfc.a 41,6 0.2 22.6 .5 ?9.a 13.7 24.2 37.0 38.9 38.3 55.7 51.3 56.9 36 ¦ 4 15.0 31.6 54.0 52. 5.1 18.4 46.5 58.9 56.7 55.4 19.9 2.1 19.9 14.5 25.9 38.5 36. a 40.5 53.8 52.6 56.0 34.0 34,2 55.1 49.9 2.3 21.li „4«. 1 25.7 IB.5 15.0 27.5 39. 9 34.6 42.7 51.7 5 3.9 54.9 31.5 36. 6 56.1 46.9 0.5 24.2 49.6 1.1 19.6 52.0 56.5 271? 12.8 34-. 8 51.7 56.5 30.7 17.6 12.6 39.3 54.7 57.4 39.0 7.2 ID. 1 29.a 36.0 56.0 54.0 42,6 "•2 24.5 45.0 3T.fl 59.3 55,7 33.6 17.3 0.2 0.1 23.4 53.5 55.3 24.3 43.4 35.8 26.5 }},} 60.0 54.5 32.1 15.4 0.0 3 5. 5 27. 1 15. f, 23. 3 41.2 32.6 45.0 49.6 55.2 53.8 29.0 38.9 56.9 43.6 0.0 27.0 _51.0 28. 6 29. 7 12.5 9.4 29.8 42.4 31.0 47.3 * 47.5 56.4 52.4 26.6 41.1 57.7 30.0 43.5 30. 1 49.5 45.9 57.5 50.9 24.3 43.1 58.3 39.8 35.7 29.6 52.3 53.1 30.9 15.0 51.5 30.2 0.0 26.9 54.» 54,0 20,5 44.0 33.1 18.7 44.5 SQti 30.8 29.2 45.0 26.9 32,9 2?. ft 30.3 56.1 52.4 JL71_ 13.4 57.1 50i6 14.a 36.4 58.0 4e.5 JJUL 36.0 53.0 55.1 30.0 12.5 15.3 41.0 55.5 56.7 36.6 5.2 12 8 30.0 37.6 57.2 51.5 37.4 54.3 36.7 55.5 40.2 56.6 41.» 57.6 22.9 19.9 7.4 17.9 42.5 56.3 55.B 34.0 3.4 15.4 39.3 58.4 48.5 16.4 2.9 20.4 44.2 56,9 54.B 31.3 1.9 17.8 41.0 59.3 45. r 12.5 0.3 22.8 45.6 57.5 53.7 28.6 0.8 20.1 42.8 60.0 41.6 B.6 32.1 53.5 49.7 30.0 28.0 30.3 32.4 34.9 21.8 39.2 58. T 46.2 _ tiA 15.9 18.8 21.3 23.5 25.2 26.T 43.1 58.5 53.4 51.5 49.3 46.8 44.3 0.0 25.2 * 47.0 58.1 52.4 25.B 0.2 22.1 44.6 38.0 5.0 27.4 4«.4 58, 5 50.9 22.9 0.0 23.a 46.3 34.7 2. 1 ------- Table 1-2 VALUES OF THE AVERAGES OF THE BASIS FUNCTIONS OVER EACH MODE l 7 ~a 7s 75? To? vN? 1 1.0000 18.0500 £.5000 37.5C0C 42j.5966 7.7567 747.7668 IC4.8537 164C.132C 2 1.0000 16.6625 -1.0750 -28.1250 386.9459 4.<.550 -561.5264 61.9900 11C3.4938 3 1.0000 9.0375 1.8750 14.C625 !Ct>.4d67 4.3150 139.7132 29.2004 255.4498 4 1.0000 23.0727 1.3636 30.6818 556.5805 2.0B00 715.6334 46.5847 1075.^698 5 1.0000 37.6538 1.1536 43.2692 1440.5222 1.4046 1644.0825 52.5375 1986.8048 6 1.0000 38.0500 -1.2500 -46.8750 1476.2113 2.0417 -1760.9423 75.4888 2818.3750 7 1.0000 45.8000 1.7647 79.4118 2192.9562 3.4094 3705.3125 152.8414 7064.6548 8 1.0000 53.00b3 -1.2500 -65.6250 2838.4180 2.0500 -3468.4389 106^5081 5561.0050 9 1.0000 52.5428 1.0714 56.25C0 2783.1422 1.2014 2973.0977 62.9974 3322.1119 TO 1.0000 40.4033 -1.5000 -56.2500 187573556 2.8120 -2362.0152 100.7045 3951.2093 11 I.0000 43.4154 1.7308 64.9038 2067.4378 3.7085 2725.2108 123.4465 4606.5663 12 1.0000 33.8333 -2.8571 -85.7143 1572.5804 10.2076 -3425.2217 282.9109 10063.8078 13 1.0000 36.2375 1.8750 56.2500 1789.5073 4.4462 2248.9990 107.7057 3670.1761 14 1.0000 46.0609 -1.3043 -58.6956 2303.4566 2.1313 -2738.8101 93.5297 4220.4915 1 5 1.0000 23.1778 -1.6667 -37.5000 564.6190 3.5533 -874.3014 78.0627 1761.9989 t6 1.0000 7.8125 -1.8750 -14.06 25 89.8487 4.6950 -139.5406 32.6369 290.9055 17 1.0000 28.8454 £.0455 46.0227 1016.1811 5.2827 1379.3948 96.7243 2467.4704 re itoooo jttszsc -1.8750 -56.2500 1093.6045 4.545c =T8rr.ui5 131.2360 4035.1918 19 1.0000 30.5500 1.6667 50.0000 1026.4346 2.9778 1624.5391 88.6821 2826.1665 20 1.00CO "2477158 -i.3684 -53.2895 861.2722"" 7.193T -1596.6755149.6970 3992.4466 21 1.0000 38.2760 2.4000 72.0000 1791.6623 7.2248 2877.9475 176.2425 6001.1068 Z2 1-OOOC 33.8750 -2.1429 -64.2857 1575.9649 5.7421 -2570.0153 159.2388 5668.85i49 23 1.0000 17.7333 2.0000 30.0000 411.6827 4.8573 598.9142 65.4714 1148.7460 24 170000 45.144C T.200U "54.0000 ZT28.765I 1.5120 2519.8353 67.7954 3133.7277 25 l.OOOO 47.2333 -1.6667 -75.0000 2334.0895 3.4022 -3499.3613 149.2304 6730.2482 26 1.0000 15.9900 -370000-45.0000 369.3C02 11.8100 -895.7151 164.2715 291270867 27 1.0000 38.0053 1.5789 47.3684 1770.6486 3.1421 1094.1427 75.7415 2585.3045 2B 1.0000 33.8686 -1.7143 -51.4286 1575.8225 3.6766 -2056.4151 101.6697 36247S62& 29 1.0000 17.7333 1.6667 25.0000 412.2008 3.3911 499.3674 45.6602 803.7473 —30 1. GOOD 45.2657 I.4ZB6 64.2S57 2X40.4352" 7.1562 2999.7180 96.6297 —4465 .1001 31 1.0000 46.6286 -2.1429 -96.4286 2283.4472 5.8200 -4498.5436 255.5805 11536.7248 32 1.0000 I?.4000 -2.3C77 -34.6154 379.6860 6.8246 -690.4485" 94.9561 16B67S623 33 1.0000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 —3* 1.0000 15.000C 07$ CT70 225.0CC0 C70 0J5 C7Q 070 35 l.OOCO 30.0000 0.0 0.0 900.0000 0.0 0.0 0.0 0.0 56 170000 45.0000 t.O C.F 2C25.0(0C O.C O.C C.C 0.0 37 1.0000 60.0000 0.0 °2_ 3600.0000 0_^0 C.0 O.C. 0.0 ------- APPENDIX II Speed vs. time curves for the Surveillance Driving Sequence nd firs-, 535 seconds ;t the ^¦edacB-l Tast Procedura Driving Saquer.oe. ------- JlRVfULA'-'CF ACCFLERfiTIClh- rf«f SPEhC (SECI (cpm T/.. 10. 11. 12. _!¦• I. 15. 16. 17. 18. 10. "20. 21. 22. 23. 24. _20. 2b. 27. 28. 29. 30. 31. 32. " 33. _ 35. 36. 37. 28. 39. 40. <¦1. 42. S3. 44. <.5. 46. 47 . 4>e. 49. 5n. 51. 52. 53. 54. 5s- 56. 57. C.C 0.0 C.O _0.C C.O 0.0 _ 0.0 C.C" C.O 0.0 1.8 5.1 9.1 13.2 17.1 2C.6 23.A 25.7 27.3 28.6 21.7 3C.0 3C..C 30.C 30.C "30.0 3C.C 3C.0 30.0 3C.C 3C.0 3C.0 3C.C 3C.C 30.C 30.C 30. e 29.6 26.9 28.1 26.9 25.1 23."> 21.2 18.5 15.6 12.5 <».4 _ 3.7 1.6 C.3 C.C C.C _ C.O C.O C.C C £ . 59. to. _ '1 • 2. 63. 64. 65. 6 6, f. 48. 69. _ 71. 72. 74. 75. 76. 77. 78. 79. " eo. pi. _?2. n. e4. 85. 86. 8/. 6ft . 69. "50. _5 i. 2. 53. 54. 15. 56. 17 . S8. ?}. ICO. IC1." 1C2. 1r 3 . 1C4. 105. JC6 ._ 1 C 7 . 1C3. 1C1. lie. 111. 112.. 113. 114. O.C o.p c.c . C. C_ 0.0 0.0 1.7 4. 6 7.6 1C. 3 12.3 13.7 14.6 15.C 15.C 15.C 15.C 15.C 15.C 15.C 15.0 15.C 15.c 15.C IS.C 15.C 15.C 15. C 15.C 15. 9 17. 3 18.9 20.6 22.4 24.2 25.9 27.5 28.8 2S. C 30.0 30.0 3C.C 30.0 30.C 3C.C 30. C 30.C 30.0 3C.C 30. C 30.0 3C. C. 3C. C 30. n 30.0 30. 7 ' 31.7 ri"f si'Ffc, (sro irPH) 115. 32.9 116. 34.;- 117. 35.6 llfl. 37.0 119. ift.5 120. J9.9 121. 41.2 122. 42.4 123. 43.5 1?4. 44.4 125. 45.C 126. 45.0 127. 45.c l?8. 45.C 129. 45.0 nc. 45.c 131. 45.0 132. 45.0 133. 45.0 134. 45.0 135. 45,0 136. /,5.C 137. 45.C 138. 45.0 139. 45.0 14 C. 45. c 141. 44.7 142. 44.1 143. 43.4 144. 42.^ 145. 4C.7 146. 3S.9 147. it. e 148. 14.6 149. J2.6 150. 31.0 151. 3C.1 1'.?. 3C.C 153. M'.C 154. 10.0 ' 155. " 3C.C 156. 3C.0 157. 30.0 ISA. 30.0 15<). 30. C lf-C. 3C.C lfcl. 30.0 162. 3C.C 163. 3C.C 16'.. " 30. : 165. 3C.C 166. 3C.C 16 7. 3 0.C 168. 31.C 16<1. 32,5 17C. 34.3 171. 36.2 Tivr spem isco h) IStzCl (VPh) 4 7.6 786. 15,0 343. 5P.4 48.7 287." 15.0 344. 58.5 50.0 28a. 15.0 345. 57.4 51.3 239. 15.0 34 6. 56.1 52.6 290. 15.0 34 7. 54 .4 53.9 291. 15.0 34fl. 52.4 55.2 29 2. 15.0 349. 49.9 56.4 29 3. 15.0 350. 4('.9 57.5 294. 15.0 351. 43.6 58.5 295. 15.0 352. 39.8 59. J 296. ' ~ I 5 »0 353. 35.7 60.0 207. 15.0 354. 31.2 60.0 298. 15.0 355. 26.6 60 . 0 ~ 299 ."" 15.0" 356. 21 .9 60.C 300. 15.0 357. 1 7. 3 60 .0 301. 15.0 358. 12.8 60. 0 302. 16.7 3Hq. 8.7 60.0 30 3. 20.0 360. 5.1 60. C 304. 23.1 361. 2.3 60.0 305. 2 6.1" " 362 . C . 5 60.C 306. 28,9 363. 0.0 60.0 307. 31,6 364 . 0.0 60.0 " 308. 34.2 365. 0.0 CO.0 309. 36.6 366. C.O 60.0 310. 3 8.9 36 7. 0.0 60 . 0 311. 4 1.1 368. " C.O 60.0 312. 43.\ 369.. c.o 6C.0 313. 45.0 370. r.o 59.7 314. 46 .8 37 1. C.O 59.3 315. 48,5 If2. 0.0 58.8 316. 50.1 373. 0.0 58.3 "317. " 51.5 " " 374 . 1 .5 57.6 318. 52.8 375. 5.2 56.9 319. 54.0 376. 8.6 56.0 320. 55.1 ~ 37 7. 12.0 54.9 321. 56. 1 378. 15.3 53.8 322. 56.9 379. 18.4 52.4 323. 5 7.7 380. ' 21.4 50.9 324. 58,3 381. 24.2 4t>.2 325. 58.9 382. 27.0 47.4 326. 59.3 383. ?9.6 45.5 327. 60.0 384. 32.1 43.4 328 . 60 .0 385. 34.5 41.1 329. 60.0" 396. 36.8 38.8 3 30. 60.0 337. 3-7.0 36.4 331 . 60.0 38 P. 41.0 34.0 332. 60.0 389. 43.0 31.5 333. 60.0 3 70. 44.8 29.0 334 . 60 .0 3<<1 . 46.5 26.6 335. 60.0 392. 48.1" 24.3 336. 60.0 39 3. 4-3.6 77 . 1 33 7. 6 0.0 394 . 5 1.0 20.? 33a. 60.0 395. 52.3 18.4 339. 60.0 3oa. 53.5 17.0 340. 60.0_ 397. 54.5 15.9 341. 60.0 398. 55.5 15.2 342. 60.0 399. 56.4 nxF sprrn riwe speed . 15.0 4 14. 60.0 4 71. 15,0 4 15. 60.0 47 2. .15.0 416. 60.f 473. 15.0 417. 60.0 474. 15.0 418. 60.0 475. 15.0 419. 60.0 476. 15.0 420. 60.0 477. 15.0 471. 59.7 478. 15.0 422. 59.3 479. 15.0 423. 58.9 480. 15.0 424. 58. 3 _ _481 . 15.0 4?5. 57.6 4 82. 15.0 426. 56.7 483. 14.4 427. 55.7 484. 13.3 428. 54.5 485. 11.3 429. 53.1 486. 8.5 430. 51.5 487« 5.2 431. 49.7 48R. 2.1 432. 47.8 489. 0.2 433. 45.8 490. 0.0 434. 43.7" 491. "0.0 435. 41.6 4«2, 0.0 436. 39.4 493. . 0.0 437. 37.3 494. C.O 436. 35.4 495. 0.0 439. 33.6 496. 0.0 440. 32.1 497. ' 0.0 441. 3«.<» 49. 0.0 442. 30.2 493. 0.0 443. 30.C 5CC. 0.0 444. 30.0 5C1• 2.4 445. 30.0 5^2. 6.3 446." 30.0 " 503. lD.0 447. 30.0 504. 13.5 448. 33.0 5C5. 16.8 449. " 30.0 " 506. .19.9 4 50. 30.0 50 7. 22.8 451. 30, 0 5Cfi. 25.5 452. 30. 0 50'J. 2d.O 453. 30.0 510. 30.3 4 54. 30.0 511. 32.4 455. 30.0 512. 34.4 456. 30.0 5\3. 36.2 ------- TIME SPEEC time SPEED TIPE SPEED TIPE SPEEO T I ME tSEC J CStC) C«PH» (SFC» (?«r>H j (SEC) (CPU) (SEC I 51. 37.8 574. 24.5 634. 15.7 694. 5.2 754. 515. 39.2 575. 26.6 635. 19.6 695. 3.0 755. 516. P.5 576. 28.7 636. 23.4 696. I.-3 756. 517. 41.6 577. 3C.8 637. 26.9 697. 0.3 757. 518. 42.6 5/8. 32.9 638. 3C.3 698. 0.0 758. 519. 42.4 579. 34.9 639. 33.4 699. 0.1< 759. 520. 44.0 580. 36.9 640. 36.4 700. 0.0 760. 521. 44.5 581. 38.8 641. 39.2 701. 0.0 761. 522. 45.0 5e2. 4C.5 642. 41.8 702. 0.0_ 762. 523. 45.0 5U3. 42. C 643. 44.2 703. 0.0 763. 524. 45.0 584. 43,4 644. 46.4 704. 0.0 764. 525. 45 .C 585. 44.5 645. 48.5 705. 0.0 765. 526. 45.0" 586. 45iC 646. 5C.3 706. 0.0 766. 527. 45.0 587. 45.0 647. 52.C 707. 0.0 767. 528, 45.0 588. 45.C 64 8. 53.5 708. o.o 76 B. 529. 45.C 5fc9 . 45.0 649. 54.9 709. "" 1.2 769. 530. 45.0 590. 45. e 650. 56.1 710. 3.5 770. 531 . 45.0 I. 45.0 651. 57.1 711. 6.4 771. 532. 45.0 59?. 45.C 652. ie.c 712. 9.6 772. 533. 45.r. 593. 45.r 653. 58.7 713. 12.8 773. 534. 45.0 594. 45.C 654. 59.3 714. 15.9 774. 535. 45.0 595. 45.0 655. 60.0 715. 18.8 775. 536. 45.0 596. 45.C 656. 60.0 7\ 6 . 21.3 776. 537. 45.0 597. 45.0 _ 657. 6C.0 717. 23.5 777. 53B. 44.5 5S 8 • 45.C 658. 6C.C 71 8. 25.2 778. 539. 43.9 599. 45.0 659. 60.0 719. 26.7 779. 540. 43.0 6C0. 45.C 66P. fcC.C 720. 27.8 780. 541. 41.8 6CI. 45. C 661. 6C.0 721. 28.7 781. 542. 4C.2 <-C2. 44.5 662. 60.0 722. 29.6 782. 543. 38.2 6C3. 4 3.7 „ 663. 60.0 723. 30.0 783. 544. 35.8 6C4 . 42.7 664. ec.c 724. 30.0 784. 545. 33.1 6C5. 41.4 665. 60. C 725. 30.0 785. 546. 3C.1 6C6. 39.8 666. 6C. 0 726. 30.P 7Hfi. 547." 26.9 6C7 . 37.8 66 7. er.c 727. 30.0 787. 548. 23.7 fcC°.. 35. 3 668. 6C.C 728 . 30.0 788. 549. 2C.7 6C9 • 32.5 669. 60. 0 729. 30.0 739. 550 . 18.1 610. 29.2 670. 60.0 730. 30.0 790. 551. 16.1 611. 25.6 671. 59.6 731. 30.0 791. 5521_ 15.1 . .612. 21.8 672. _59 • 0 732. 30.0 792. 553. 15.n 613. 17. 8 673. 58.3 733. 30.0 793. 554. 15.0 614. 12.P 674. 57.4 734. 30.0 794. _555. 15.0 615. 9.9 67.5, 56.5 735. _ 30.0 795. 556. 15.0 616. 6.4 676. 55.3 736. 30.0 796. 557. 15.P 617. 3.5 677. 54.C 737. 3C.0 797. 558. 15.0 1.3 678. 52.4 738. 30.0 79P . 559. 15.C 619. C. 1 679. 5C.6 739. 30.5 799. 560. 15.C 620. O.C ten. 48.5 740. 31.4 800. 5fcl. 15,0 621, 0-C 6P 1 . . 46. 2 741. 32.4 . 801 . 562. 15.0 622. 0.0 6P2. 43.6 742. 33.6 802. 563. 15.0 673. O.C 683. 4C.8 74 3. 34.8 803. _ 564. 15.P 624. O.C 6?4. 37.8 744. 36.0 804. 565. 15.0 625. O.C 685. 34.6 745. 37.4 805. 566. 15.0 626. c.c 686. 31.3 746. 38.7 806. _ 567 . 15. C 62 Ij, O.C _ 687. 27.9 74 7. 40.2 8C7. 568. 15.0 628. c.c 688. 24.3 748. 41.6 808. 569. 15.9 629. O.C 689. 2C.8 749. 43.1 8P9. 570. 17f3 630. 0.0 690. _17.3 750. 44.6 810. 571 . 18.9 631. 2.6 691. 14.C 751. 46.0 811. 572. 2C.7 632. 7. 1 692. IC.7 752. 47.5 ei2. 573. 226_ 433. ..J 1,5 . 693. 7,8. 753, 48.9 _813. speec ('•"PHI TIKE SPEEO (SEC) (MPH) TIME SPEED (SEC i (MPH) 50. 4 814 • 29.1 eh',. 60.8 51.7 P15. 27.7 875. 6C.C S3. 0 816. 25.4 876. 60.0 >>4.3 817." 22.0 877. 60.0 55.5 818. 17.6 878. 60.0 56-6 BIT. 12. 5_ 879. 60.0 57.6 820. 7.A 83 C. 60.0 58.5 . 821. 2.0 881. 60.0 •>". 3 822. 0.3 882. 60.0 60.0 823. 0.0 BR3. 6C.0 60.0 824. 0.0 884. 60.0 <.0.0 825. 0.0 _ 885. 60.0 6C.0 876. C.O ~ r.R6. 60.0 60.0 827. 0.0 837. 59.7 60.0 828, 0.0 888. 59.2 60.0 829. 0.0 889. 58.7 60.0 830. 0.0 800. 58.1 60.0 831. 0.0 fl1)!,.. .57.4 0.0 8 3 2. 0.0 892. 56.7~ 6P.0 833. 0.0 893. 55.8 60.n _ 834. 0.5 894. 54.8 "hO.O "" 835. 4/9 895. 53.7 60.0 836. 6.9 816. 52.4 bP.O 837. 9.8 897. 50.9 c.0.0 838. 12.6 89ft. 49.3 60.0 839. 15.3 899. 47.6 59.6 840. _ 17 • 9 900. 45.7 59.1 84 1." 20.4 901. 43.6 5.4 842. 22.8 902. 41.4 57.6 843. _ 25.? 903. _39.0 56.5 044. 27.4 904. 36.6 55.1 845. 29.6 905. 34.0 53. 4 846. 31.7 906. _31.3 51.5 847. £3.7 907. 2R.6 49.3 fa, 8. i 5.7 90S. 25.8 46-8 849. 37.5 909. 22.9 44.3 P50• 39.3 910. 20.1 41.6 ' 851. 41.0 9H. 17.3 3«.9 _ P52,_ .42• 6_ 91?, 14.6 36.3 853. 44.2 913. 12.0 34.0 854. 45.6 914. 95 32.1 855._ 47.0 915. 7..2 30.7 856. 48.4 916. 5.2 30.C 857. 49.6 917. 3.4 30.0 856 50. 8__ 9lg. 1.9. 30.0 859. 51.9 919. 0.8 30.0 860. 52.9 920. 0,2 30.0 861._ 53.8 921. ._.._0,0 30.0 P62. 54.7 922. 0.0 3O.0 863. 55.5 923. 0.0 10.0 _ 86. . 56.3 924,_„.. O.0_. 30.0 865. 56.9 925. 0.0 30.0 866 57.5 »26. 0.0 30.0 867._ 58,1 927. C.O 30.0 868. 58.5 928. C.O 30.0 e69. 58.9 929. 0.0 30.0 _ _870. 59 .3 930-_ 0.0 30.0 87l. 60.0 931- P.C 30.0 872. 60.0 932. 0.8 30.0 873._ 60,0. 933, 2.7 "934. 4.9 994. 60.0 935. 7.5 995. 60.0 936. 10.1 996, 60.0 937". 12.8 997. 60.0 "38. 15.4 998. 60.6 939. 17.8 999. 60.0 940. " 2C.1 1000." 60.0 941. 22.1 1001. 59.5 342. 23.8 1002. _58.7 943. 25.2" * 1003. 57.6 944. 26.5 1004. 56.1 945. 27.4 1005. 54.0 946. 28.3 1006. 51.5 947. 29.0 1007. 48.5 948. 29.6_ 1008. 45.1 949. 30.0 1009. 41.6 950. 30.C 1010. 38.0 951. 30.0_ 1011. 34.7 952. 30.0 1012. 32.i 953. 30.0 1013. 30.4 954. 30.P 1014. 30.0 955. 30.6 1015. 30.0 956. 30.0 1016. 30.0 957. 30.0 _ 1017. 30.0 958. 30.0 1018. 30.0 959. 30.0 1019. 30.0 960._ 30.0 1020. 30.0 9fcl. 30.P 1C21. 30.0 962. 30.0 1022. 30.0 963. _30.0 1023. 30.0 964. 30.0 1024. 30.0 965. 30.7 1025. 30.0 966 . 31.8 1026._ 30,0 967. 33.1 1027. 3C.0 968. 34.5 1028. 30.0 •'fig. 36ifi_102'>. 3C. 9 970. 37.6 1030. 29.4 971. 39.3 1031. 28.5 972. 41.0 1032. _ 27,2 973. 42.8 1033. 25.4 9 74. 44.6 1034. 22.9 ... 975.__ .46.3 .1035. .19.9 976. 43.1 1036. 16.4 9 77. 49.8 1037. 12.5 978. 51.5 103P. 8.6 979. 53.0 1039. 5.0 980. 54.6 1040. 2.1 931. 56.0 .1041. _..0,.3_ 982. 57.2 1042. 0.0 083. 58.4 1 P43 . 0.0 „ °84. 59.3 1044._ 0.0 985. 60.0 1045. 0.0 986. 60.0 1 046 . 0.0 _ 987. 60.0 1047. 0.0 988. 60.0 1048. 0.0 989. 60.0 1049. 0.0 . 990. 60.0 IO5O. .0-0 _ 991. 60. P 1051. 0.0 992. 60.0 1052. 0.0 993. 60.0 .1.053, _ 0,0. 1054. 0.0 ------- FEDERAL TEST PROCEDURE SPtfclMMHH 0.0 0.0 t TUtE(SEC) 5? 56 SPfcEU(MPH) I5.B 17.7 6 7 3 9 10 11 12 13 ~T4 15 15 17 TB 19 ~7cr 21 22' 23 ~24 25 uvtr o.o o.o o.o o.o 0.0 —5T- 58 59 60 —6T- 62 i$.a 21.6 "23.7" 24.2 T^.fc 24.9 25.0 24.6 24.5 24.7 24.8 24.7 T1ME(SEC» 110 ~ 111 TIT SPtEDIMPHI 31 .i jl • b 113 114 115 116 117 -3772~ 32.4 32.2 31.7 28.6 25.3 22.0 18.7 15.4 12.1 8.8 5.5 - 2.2' 0.0 TlHt(SfcC) lf.5 166 TE7 SPtED(MPH» 6.5 9.9 168 169 170 TT1 172 " US 174 —13.2 16.5 T978 22. i ~2T3 25.8 2b.4 25.7 TIME(St C1 22 0 221 7TI SPIED(MPHI 50.0 50.6 22 3 224 22 5 226 227 22 B 22 9 230 231 732 233 2 34 235 """"5T.Tr" 51.5 52 .2 53.2 54.1 54 .6 54.9 55.0 54.9 54.6 "TT.6 54.8 55.1 55.5 "55.7 56.1 "55T3" 56 . 6 56.? 56.7 5r."5 56.5 56 .5" 56.5 0.0" 0.0 0.0 0.0 0.0 0.0 TT 64 ~E5~ 66 ET 68 0.0" 0.0 0.0 0.0 0.0 0.0 ~ET~ 70 TT 72 TT 74 ~2E~ 27 29 31 0.0 3.0 5.9 8.6 n. 5 14.3 "75" 76 -rr 78 79" 60 ~32~ 33 16.9 17.3 TB n " 20.7 ZI.7 22.4 ~BT~ 82 83 84 T5" 86 34 35 37 2Z.5 22.1 Zl.5^ 20.9 "87" 88 ~3B~ 39 41 ~52 43 20.4 19.8 By 90 ~9T~ 92 17.0 14.9 14.9 15.2 15.5 16.0 ~7T~ 94 96 ~ ------- TIMUStU SPttDI MPH) TIMEISECI SPEED(MPH) TIHttStCI SPfctOIHPH) TIMEISEC) SPEEDtHPHJ TIME1SEC 1 SPEEOIMPH) 28.0 25.5 2 75 276 53.0 "" 53. 5 330 331 8.0 4.7 385 386 33.5 31.4 440 441 0.0 0.0 95 496 ill 278 54.0 54. V 332 333 1.4 0.0 3« T 388 <:v.u 25.7 42 443 ¦ trm 0.0 ~ 49T 498 22.5 19.8 279 280 ^>5.4 5b. b 334 335 0 .0 0.0 389 390 a .u 20.3 444 445 0.0 0.0 499 500 — 13.2 261 282 56.0 56.0 336 337 0 .0 0.0 391 392 1 f .5 14.5 44b 47 O.C 0.0 iCl 502 10.3 7.2 2U3 284 5b • y 5b. c 33H 339 0.0 0.0 ifi 394 14 . u 8.7 448 449 3.3 6.6 50 3 504 4.0 1.0 28b 28b 54.5 53.6 340 341 0.0 0.0 396 5.* 2.1 450 451 9.9 13.2 ZB7 288 52.5 51.5 342 343 0.0 0.0 3V 1 398 0.0 0.0 45Z 453 16 ."5 14.8 2BV 290 SI . 5 51.5 345 0 .0 0.0 400 0.0 3 455 23.1 26.4 2VI 292 51.1 50.1 36 347 0.0 1.0 402 0.0 96 457 2/.8 29. 1 2V3 294 50.0 50.1 30 349 • 3 7.6 404 2.6 5.9 458 459 31.15 ' 33.0 2V5 29b 50 .U 49.6 350 351. 10. V 14.2 406 9.4 12.5 460 461 33.6 34.8 297 298 49.5 49.5 352 353 1 /. s 20.0 408 19.1 0 463 35.X 35.6 299 300 4 V. 5 49. 1 354 355 a.» 23.7 410 25.0 HI 465 36 ;t~ 36.6 iiUl 302 48.6 48.1 i!>6 357 26.6 412 27.5 6b 67 36 « i 36.2 303 ' 304 47 . 2 46.1 358 359 . 1 30.0 414 30.0 469 36 .u 35.7 305 306 45.0 43. a 361) 361 30.8 31.6 416 30.0 471 36 b 36.0 307 308 4 2.6 41.5 362 363 32.1 32.£ f 418 29.3 14 473 35 .U 35.5 309 310 40.3 38.5 364 365 33 .6 34.5 420 26 0 475 35. 35.2 311 312 3/.U 35. i 366 367 3* .e 34.9 41 422 21.7 477 35.4 35.2 313 31 33.8 32.5 368 369 34. 8 34.5 1 42 J r424 13.1 479 39 .4 35.2 3I5 316 31.5 30.6 370 371 34.7 35.5 425 426 8.5 481 35.^ 35.0 317 318 ' 30.5 30.0 372 373 36.0 36.0 42 1 428 3.4: 1.9 48Z 483 35.1 35.2 319 320 29.0 27.5 374 375 "36 .U 36.0 429 430 u .u 0.0 484 485 35.5 35.2 3;rl 322 — 24.a 21.5 376 377 36. (J 36.0 431 432 0.0 0.0 48b 487 35.0 35.0 323 32* 20.1 19.1 J7B 37«J 36 n 3b .4 434 0.0 489 m U 34.8 32 5 326 la . a 17.0 IS.* 12.5 3UU 381 3sr i«i 36.5 3o . 4 36.0 ' ' 35.1 U5 4j6 "TJT 438 tJ . u 0.0 491- 92 493 3.b 34.5 33.S 32.0 32T 32& 0.0 •S2* 10.ft ?w 3.l - -wif - 0.0 49 ».l ------- APPENDIX III THE MATHEMATICAL MDDEL The mathematical model presents a method to calculate the amount of a particular pollutant given off by a vehicle or vehicle group over any specified driving sequence given the modal emissions data on the individual vehicles. Notation e(t) = amount of a pollutant given off by a vehicle from time = 0 to time = t. e(t) = — = instantaneous emission rate function. dt eA(t) « accel/decel instantaneous emission rate function. eg(t) = steady state instantaneous emission rate function. E^ = amount of a pollutant given off by a vehicle in the j'th mode. Tj = duration of jth mode, (in seconds) v(t) = speed at time = t for any driving sequence (t) a ——= acceleration at time = t in the j'th mode dv. (t) a(t) ¦ ^rr^- = acceleration at time = t in any driving sequence. V ) fa j£ - average value of the emission rate function for the ------- (A) Functional Form of the Steady State and Accel/Decel Emission Rate Functions Based on Figures 1, 2, 3 (Appendix VI) where average steady state emission rates over the 1020 vehicles in the data base are plotted versus speed, the assumption is made that the steady state emission rate function eg can be represented as a quadratic function of speed: CD &s<-V) = 5, + S. V/ +- S3 V2- where , S2, and aTe constants. The functional form of an emission rate function that will be appli- cable during periods of acceleration/deceleration is obtained by assuming the constant coefficients (S^) become functions of the acceleration and that this functional dependence can be sufficiently represented by quadratic functions of acce 1 e rat ion. The re fo re: S, = S, ca.) - $,t +-ql3. a- -h OL- (2) Sj. = Sa. = +~ 4LL ^ 3 ~ S3 Ca) = c^3i e^3JL ^ ^ qJL. where the q's are constants and (3) <£^(v,a.) - Si Cct) +- S^ca.) ir -h S3 taj Ul C4) eA 4 ( + q.n.c,+ a?) c t- Qt Cu -h a2-) - U + C$3i t- %-gi, a, f ^33 oJ-) ¦ tr- Upon multiplying and redefining the constant coefficients we have the following expression for the accel/decel emission rate function (5) <2/j (\Jta) = t>, f- 1/ + Ai U$ + bcf ------- ^ The Instantaneous Emission Rate Function Although the accel/decel emission rate function is functionally identical to the steady-state rate function when the acceleration is zero, both functions will be retained to allow independent adjustment of the coefficients (b. and S^) specifying the accel/decel and steady state emission rate functions. The two functions can be joined by an acceleration dependent weighting function h(a) to form the instantaneous emission rate function: <6) e where h(a) = = /WaJ e%L\f) 4- C f- hca.)) dfitV,£L) — _L- (X, -h I or JL- CL + I S & > O j k IM ^2. <& < 0 a, >/ c^ ^ 6 (u) )eA ------- In practice 1 and °^-2 have been arbitrarily set to -1.2 sec and 1.0 mph/sec respectively. (C) Specification of The Emission Rate Function Given the functional form of the emission rate function, the coefficients (Ik, S.,) must be determined in order to specify the emission rate function which characterizes a vehicle's emission response. (1) Determination of The Accel/Pecel Emission Rate Function Coefficients fb^) The only modal emission observations available are the total amount of each pollutant given off in each mode of the SDS; there are no measures of the instantaneous emission rate given for accel/decel modes. Therefore, the following procedure is used to determine the coefficients (b^) that specify the instaneous emission rate function for a given vehicle and pollutant: Ut: " , h * A- X. /7 - uV , t 4~q - where the "A. , l - tjf basis functions of the accel/decel emission rate function. Then using equation (5) we have: _! , , (6) eA IV/D - 2- i =/ For an accel/decel mode the weighting function h(a) ~ 0, then: (7) = e (v,a) Now consider the average emission rate over the k' th mode: 5 (8) ik 0 ------- by equation (7) (9) = = ±- f Tit = Ek /Tk and Tt J* (11) -f- - 7- [&A(v,a)Jt = -L j ^ 7 Tk U 0 ' Since the b. are constants: (12) Now, '/Tie I 1~uav~ is just the average value of the i'th basis function Gl /7i ' '/Tk X ( I l/T f k Jt ' /T" I over the K'th mode, which can be easly evaluated knowing the speed as function of time for the k'th mode , and acceleration as a function of time, CLf, Lb\ T* Let' ^Lk ' I di IfL ) O (see Appendix I for values of ) Thus, ^ (13) M~~ ~ / ^ ^Lk- i-t and since E^/T^ is known for all modes and the f^ can be evaluated then the b^ can be determined using standard least squares regression techniques. ------- The result of the least squares regression method can best be given by defining the following matrices and elements: Ek Y, Y(k) = •?— (response] k =1, 32 Tfc (32 accel/decel modes) X, X(i,k) = fik i - 1,9 B, B(i) = bi and using the convention y' = Transpose of Y Y= Inverse of Y we have by equation (13): (14) Y = BX The method of least squares then gives: (15) B = (X ' X)_1 X 'Y Let: A^ s (X ' X) 1 X7 (A^ basis function factor array) Then, (16) B = A^Y and the i'th coefficient is given by (b^): 21 (16) b. = y Aa (i,k) Y(k) i = 1, 9 i. - i In summary, the average emission rate, which can be evaluated by using experimental observables, is used to determine the coefficients that specify the instantaneous accel/decel emission rate function. (2) Determination of the Steady State Emission Rate Function Coefficients (S^) In the case of steady state emissions the average emission rate is equal to the instantaneous emission rate because the speed is constant in time. ------- Let: U -- 1.0 , Lb -- Vs. t = Vt then, by equation (1): 3_ (17) e£ ^ Ut. L - I Consider the K'th steady state mode ^ ^ 1 C (18) \t <£CVJ,aOrt- T^= Effe _ I ( ( S, Ufe ) Jt TV rfe j since both S. and f.. are now constants ilk _ 09) f-j = ^ ^/"ft "" ^ Vl l O t The coefficients can now be determined using standard least squares repression techniques, Consider the situation in which the have been determined by the above procedure and the specified emission rate function produces negative emission rates as shown below. (3"->observed pts e6 + Steady state Emission rate ^3 [ 0 Negative Emission Rates VL '.it. Speed (v) <£> V. Regression Curve t V 4 V 4- S/VJ') ------- When this happens (each steady state emission rate function is tested for this possibility) a new set of coefficients (S^) is calculated in a manner that will guarantee the steady state emission rate function will be positive for all values of speed between 0 and 60 mph. The procedure to determine the new set of coefficients is described below. The minimum of the regression curve representing the steady state emission rate function is forced throught the point given by the average of the 2 lowest emission rates and the average of their speeds. In our example above this point (e ,v ) is given by e= et4 Jz j V = V 3 / Requiring the minimum of the regression curve (steady state emission rate _ » function) to go through (e,v) specifys two of the three coefficients. Since e is a quadratic function of speed, the coordinates of the minimum ate given bv: i. ^ SvSjL - SC _ Therefore y J - - ^ ^3 e = 4 ^ Solving for S and S2 in terms of Sj, v and e we obtain: ^ ^ •3 , - + "v ** substituting these values into equation (1) (20) ^ (vj -- e -h U2- S3 + C~ Z.V ) V + S3 V1 Regrouping: (21) £ v l/ + The only coefficient left to determine is now S^ which is obtained by the standard least squares method. ------- This method will remove any negative emission rates over the 0 to 60 mph speed interval and still retain the general trend of the observed data. (D) Characterization of Individual Vehicle and Vehicle Group Emission Responses Assuming input modal data is available, each vehicle's emission response can be characterized by the 36 coefficients that specify the thTee emission rate functions for HC, CO and NO^ (12 coefficients per function). A vehicle group can be similarly characterized by determining the coefficients for the average vehicle in the group. Let: b... = k'th coefficient of the emission rate function for the j th vehicle in group sample and i th kind of pollutant Ng = number of vehicle in sample representing group ^ik = ^'th coefficient of the emission rate function for the average vehicle for i th kind of pollutant The coefficients describing the average vehicle's emission rate function is thus given by: ^ (22) ^ = -L L ^ (e) Determination of Individual Vehicle and Vehicle Group Emissions Over a Specified Driving Sequence The amount of a pollutant e(T) given off by a vehicle i\ undergoing a driving sequence from time = 0 to time = T is obtained by integrating the instan- taneous emission rate function describing this vehicle's response with respect to pollutant under consideration over the sequence: (23) T e U) -- j e ] (ku) ------- _N C24) eCT) ^ 4 ) ¦- L i-Wuu)) U. -W Vi. + ^aL . = 1 + \/c + 14 Q.I ^t]\J- t- t* ------- APPENDIX IV Program Listings Main Program I Main Program II Subroutine SETUP Subroutine EDOT Subroutine PAD Subroutine ESUM Subroutine EDGRP Subroutine INVERS ------- LEVEL 21.6 ( MAY 12 ) OS/36 0 FORTRAN H COMPILER OPTIONS - NAME = IN,~PT = 02»L1N£CNT=60,SIZE = OOOOK, SQUR.CL.fcBCDIC.NQLl ST « NQDHC K « L n A 0 .M AP »NHi niT.irt.XRPf t »~»~~«»»»** MAIN PROGRAM I »*****•»»~•****#** C MAIM PROGRAM I QE-IEBHIHES JIHE-AMDUNT QF EMISSIONS GIVEN OFF BY C INDIVIDUAL VEHICLES OVER A DRIVING SEOUENCE SPfCJFJED BY ARR'WT. C VTM(1)«>VEL0C1TY VS. TIKfcllN ONE SECOND INTERVALS! OF THE SURVEIL- -Z -LANCE DRIVING SEQUENCE .tf THI I ) = VF I nr. T TV 1 HPH) AT TIME II-l)Cfr C (REAL ) C ... _ C WT < I )=>VELOCITY VS. TI ME 1 IN ONE SECOND INTERVALS) OF ANY DRIVING C SEQUENCE _QV_ER HHI CH_ FWT^SIDMS ABF Tfl RF C ALCl II ATF D .VVT < 11 =VFI f)C - c -rrv at rr«E AMOUNT QF I'TH EMH1TTANT GIVEN OFF IN i'TH KQOfc . C DSlI)=OISTANCE(MILES!TRAVELED IN I	l7<4.26l .UTAH M 1 .261 .QSrSTT DIMENSION YTM(1055), VYT 12000 I . C 13 , 37 I ,C < 3-> REAL»8 At9.321. fcStl. SI.»tai3.12l QATA OS /.06O2,.071,.02O1t.OTD5t.1360,.126£,.2l63,.L116 C..2 C<»3. ¦ 3U7.. 3136 ¦- 197^.-3313.. .0^79.. CI 71. ^17i«.. 1392.. IS2B C. 13Ch .2654, .263NXl)-GT.9<.Q)G0TQ3111 ISM 0015 3000 CONTINUE -ISN 0Q16 JU1 tQNIiMUi C r- REAP IN nR IVING SEQUENCE OVER WHICH EHHISSIQNS ARE I&J1E-. CALCULATED C r TN THIS FXAMPLF tfVTg> FIRST 505 SFC. OF FTP C ISN 0Q17 _ DO 15QQ 1=1.1QQ ISN OOIS NXl = t U-l )16 11 XStf-0012 .NX2_=NX1+U? ISN 0020 READ t 5 »100)(VVT{K)»K=NX1»NX2) TCN 0021 IF I VVTlNXl ).r.T.99.0>r.nTniS65 ISM 0023 1500 CONTiNUe ISN . QQ2 ------- ISN 0027 500 _ c FORMAT(lHlJ c c PUT MODAL EMISSIONS DATA INTO ARRAY AMTC c IN THIS EXAMPLE THE MODAL EMSSIONS DATA IS READ OFF A DISK FILE ISN ISN 0023 0 029 L NCART=0 READIOO^SJITAB ISN 0030 ISN QQ3L 00 2000 I Y=5 7t71 IREC=1Y—56 ISN 0032 _ ISM 0013 JSTART=ITABlIREC 111 ISN ISN 0034 0035 DO 2001 J=JSTART,JEND READC9'Jl( ( IDAT t L • K t . L = 1. 41 . 1 R DAT 1L. K J . L= 1 . 1? 1 1 .K= 1 .261 ISN ISN 0036 0017 DO 2002 K=l,2f> IF I 1DAT< 1 . K1 .FQ.-?1GC]T(J2QQ1 ISN 0039 isn OG-m NCART=NCART1 DO 1C00 IR=1.37 ISN ISN 0041 0042 D0«1 .0 IF I IP—Lb.321DDDS tIK) ISN 0044 15N 0045 ca ioai ic=if3 IW=I IIF4-I l3>+10+IC ISN ISN 0046 0047 1001 AMTCIICtIB J = 3OAT CIH,K100 CONTINUE ISN 0C43 1000 c CONTINUE c c DETERMINE INDIVIDUAL VEHICLE EMISSION RATE FUNCTION COEFFICIENTS ISN 0049 c CALL EDOT c c. TNTEflRATP THT VFHTCIFS FMISSION RATF FUNCTION DVFR THE c c DRIVING SEQUENCE. ISN 0050 c CALL ESUK(VVT.506.&AD.CI1).CC2),C<3I.DIST> C c c c WRITE OUT EMISSION RESULTS.... ISN ISN. 0051 0052 501 WRITE(6,5011NCART, (C(LI,L = 1(3> FORMAT!1H .»CAR= • . I10.2X.•HC=•.F10.2.2X . 'C0 = «.F10.2. 2X.1 NOX= .F10. ISN. 0053 1 2002 -2, ) CONTINUE ISN ISN 0054 0055 2001 2000 CONTINUE CONTINUE ISN m 0054 0057 1234 STOP END ------- LEVEL 2 l.S> C «4r U I OS/36 0 FO® T° 4N H C1KP] L 1-R C __.IL JP1IUNS - N4MF= MAIN,IJPT=02,LINECN7=.SDSI7E = 00CQK, i.JUjl££XS1C iNQL.1 ST.tNGD^CK .IQA&iBAgiaCLU.LtltuMEf.. ******************** wiiit PMUfiAi ri ******** ****** C c c L C _L_ MAIN PROGRAM II DETERMINES THE AMOUNT OP EMISSIONS GIVEN OFF BY .A.. GJiQULEL-Qf -YliilLLiS SPiClFILD a_Y_TfcL£ FILTER OYER. 7HE .DRIVING SLt>. SPECIFIED BY !kR«A> ' VVT". VTM(I )=>VLLnCITV VS. TIHEIIN ONE SECOND INTERVALS) OF THE SURVFIL- -±M4££_ DRIVING. iLiiUENCEjVIMA 11 = ¥£LD£ 1IY IHPJ41 AT 11M£. .11-11SEC (R E A L* 4 ) C C c c _£L_ WT I I » = >VFLOtTTY VS. TIMFtlN ONE SFCOND INTERVALS] OF ANY DRIVING 5taU£ML£ OVER WHICH nrtMISSIQKS A££ TO BE CALCULATED.VVTIJ)=YEtaC- -IT Y AT TIHt (1-1) SEC. AMOUNT UF I 'TH EHMITTANT GIVEN OFF IN J'TH MODE. C DS< Il=BISTANCt(MILESJTRAVELED IN I 1 TH MODE .NOTE , STEADY STATE MODES C ARt 60 iL: Ih DURATION. ..... C c 4 + +: ******* *** ************ ***************** **** ISM 0002 ISN QQQ3 31 MENS ION I TAB(20,2)iIDAT<•,26>,RDAT< 12 1 f26 I ,DS(37) DIMENSION VTMtl0 5 5).VVT{20001.AHTC(3.37).C(3) ISN 000-k -iSRjaaa. ISN JJQD c c c ISM MAI ISN OOOS 3SN 0QQ9 REAL»6 AM9,32J,AS(3,51,aU>(3,L2) _QAT A _HS /.0fet2..07ti. .CZC1. .070?. .13feu. - Uefi . .ilfc31. 1716 C ,.2 043, . 336 7,. 31 3to,.1973, .3313, .294-, .0579,.0173,. 1759,.13921528 t. ..13Q0137,.3134. .2362. .Q444. C t«1313 «.0 592 f.0000».2 500,.5000,.75C0,1.000/ DEFINE J1LL^99<75.3256.U.NI) — K6AD IN. SUR.VL ILLANCL DRIVING SECLlNCt . rjO 300C 1=1 jISO — NX I = I (I-1)16>»1 NX2=N)U+15 ISN 0010 READ<5,100)(VTH(K),K=NXl,NX2t ISN 0011 100 FORMATf 16FS..D I JSN 0012 IF< VTMCgXl) .GT. 9<3 .0JGDTD3111 ISN 0014 3'iQQ CQNTINi't ISN 0015 3111 CONTINUE C C c BEAD IN DRIVING SEUUFNCE OVER WHICH FHMISSIONS ARE TH BE CALCULATED IN "THIS" E X AMPLE VVT-> FIRST 505 S EC. OF FTP ISN 0016 L5N 0^17 to 1500 1=1,100 U-ll»16)l ISN 00 13 NX2 =NX 1 + 15 ISN 0013 REA D(5 .100 i t YVLUl »=NXLiN Xi J ISN 0020 1FIVVT (NX1>.GT.9<5.0IGDTQ1555 ISN 002Z 15LC iOMIN''- . ..... ISN 0D23 1555 LUNTINJE C SET UP iiASIS FUNCTION FACTUR ARRAYS A A t A S . C ------- ISN 00^4 £ CALL SFTUPtVTM.AA.AS) C £ re ad in individual vehicle modal emission data FILTER VEHICILES FOR GROUP RFPRfSfNTATIVE c c IN THIS FXAMPLF THF DATA 1$ BEING R FAD TFF A DISC FILF AMR TWF c c INI VISUAL VEHICLE MODAL EMISSION DATA IS PUT INTO ARRAY AMTC IN THIS EXfMPLE THE GROUP IS. DENVER. PRE F MIS SI ON mNTRtiL. I$N 0025 c NVIG0 ISN ISN 0026 0027 REA019P«75)ITAB DO 2000 I Y»57>71 ISN ISN 0026 0029 IREC«=XY-56 JSTART = ITAR1fREC.l ) ISN ISN 0030 0031 JENDITAB(IREC,2i DO 2001 J.JSTART.JFNC ISN ISN 0032 0333 READ!99 * J1 ((IDAT ------- C INTFGRATF this GROUP'S EMISSinN 'ATE FUNCTION CIVE ^ THE: DRIVING £ - ..... ISN 0U61 CALL tSL',<(yVT.5J6,bAL'.L(l] ,w(£J ,C (3>.01S JJ C ___.£, DETERMINE TDAIAL LMlSSIQfi-rXMJJLTIPLY. ECh .EMISSION AMUUN7 BY NO, C VEHICLFS IN GROUP VN=NVIG DQ 5.Q3U. 1 = 1.1.3 .. ._ . .... 5030 CI I)=C{I>VN u Xfi.iI.E_ QlLl EMISSION .RESULTS. .,_. ...._ WRITF<6,500) »C(L 1 ,1=1,It 500 FORMAT I 1H . ' HC= ' . F 1C . 2 . 2X . ' Ct1= ' . F 10. 2 »2 X . ' NOX = '.x£l.Q^2 I .... STOP LNC ..... - - .. ISN 00 62 ISN QQ63 ISN 0064 ISN Q0&5 ISN aflt>6. ISN 0J67 ISN Q.QM ------- LEVEL 21.6 ( HAY 7? ) Oi)/3fcO FORTRAN H C3MPIL cR OPTIONS - NAM£= MAIN,OPT=02,LINECNT=60,SIZE=OOOOK, SOURCF. .EBCDIC.NOLIST. NO DECK.LOAD,MAP.NOEDIT«ID«XR£F ISN 0u02 c SUb10UTIN E SfcTUP < VTM,AA,AS I ******************************************************** ****** c £- c c ACCEL/DECEL AND STEADY STATE EMISSION RATE FUNCTION DETERMINATIONS •ARRAYS 'AA' AND 'AS» RESPECTIVELY. GIVEN THE VELOCITY VS. TIME c .£ HISTORY OF THE SURVEILLANCE DRIVING SEQUENCEVELOCITY VS. TIME(IN ONE SEC. INTERVALS) OF THE SURVEILL- ANCE DRIVING SEQUENCE. VTM(I)"VELOCITY AT TIME I.(REAL4) c c MVT <1)=TIME I•TH ACCEL/DECEL MODE STARTS IN THE SURVEILLANCE c ... c DRIVING SEQUENCE.(REAL4) c c TM(I)=TIME(SEC 1 IN ITH ACCEL/DECEL MODE.(REAL4) c c AA=> BASIS FUNCTION FACTOR ARRAY FOR ACCEL/DECEL.(REAL»81 c c AS=>BASIS FUNCTION FACTOR ARRAY FOR STEADY STATE.(REAL8 > ISN 0003 c DIMENSION VTMf1055 1 .MVT(321.TM(321 ISN ISN 0004 0005 REALMS XI32,9),SV ,SA,TMD,CJ 9,9),AA<9,32),SUM,AS<3,5» DATA MVT/11..38..64..87.,113.,141..167..199..227..257..302..343.. C374.,421.,59.,483.,501.,538.,569.,60 2.,631.,671.,709.,739.,780., ISN 0006 DATA TM/12.,16.,8.,11.,13.,12.,17.,12.,14.,30.,26.,21.,32.,23.,9., C8..22..16..18..19.,25..28..15.,25..18.,10.,38..35..1«..21..14..13. ISN 0007 C/ N0BSA=32 ISN ISN 0008 0009 N0BSS=5 NBFA=9 ISN 0010 c N8FS=3 c c **** CALCULATE AA *********** c c CALCULATE BASIS FUNCTION ARRAY X ISN ISN 0011 0012 DO 1000 IM=1,N0BSA NTSMVT(IM) ISN ISN 0013 0014 NTM=TM(IM) NTF=NTSNTM-1 ISN ISN 0015 .0.016.. DO 999 IKS2,NBFA 999 X(IM.IK)=0.0D0 ISN Ii_N 0017 0013 X(IM,1)=1.0D0 DO 001 IT-NTS.NTF ISN ISN 0019 0020 KT= IT 1 SV=lVTM(IT)»VTMtKT) 1/2.0 ISN 0021 ISN 0022 SA=VTM(KT)-VTM(IT) X(IM,21=X(IM.2J+SV ISN ISN ISN ISN 0023 0 024 X(IM,3)=X(IM,3)SA X (IM•4)=X(IM•4)~(SVSAI 0025 002* X(IM,5)=X(IM,5IMSV2) X(IM.6»=X(IM,6IMSA2) ISN ...ISM 0027 Mie X(IM,7)=X(IM,7) + ( (SV*2)S A) X ( I M« 8 ) = X ( IM , C ) ~ ( 1 S_A*2 J S V ) ------- ISN 0029 X~((SV*2>ISA*^() ... ISN OOiJ 1Q.CLL XQNTINUL ISN 0031 DO 1002 IK.=2,NBFA ISN 0032 IMD = TMC1M) ISN 003J XX(KrJ)> .. ISN. IQQ5.XQNTINUL ISN 0042 C(I , J) = SUM ISN OOW 1003 CONTINUE £ C C c SET UP A A AflRAY TSN 0046 ISN Q047 DO 1006 I = 1»NBf A DQ 1007 J=1.N0BSA ISN ISN 0048 0049 SUM=0.0DO 00 lOOfl K=1.NBFA ISN ISN 00 50 0051 inofi SUM=SUH(C(I»K)X(J,K)J f.flN T T Nil F ISN 0052 AA(I,J)=SUN ISN 0054 1006 c CONTINUE c c **** CALCULATE AS ARRAY ******* c c CALCULATE BASIS FUNCTIONS ISN ISN 0055 00 56 DO 2000 1=1,N0BSS XI=I-1 ISN ISN 0057 0058 VXI15 .0 XXIK,J)l 2003 CON TIN Ufc ISN ISN 008 QQfeS 2Q02 C(I,J)=SUM CCNTIN'Jl . ------- ISN J07G ISH 0J/1. ISN 0072 ISN OQXL- c ./.ALL lMVIftSI Cj.3» 91.. DO 2004 1=1»NBFS DO 2005 J=1.N0BSS ISM 0 374 ISN 0075 ISN 0076 JSH JiZTL- ISN 0078 fSft 007? ISN LSfL 0080 £as.L ISN 0082 SUM=0.000 2Q_.2.QQfe- K=li.NfiFi SUM = SUM* (C (I ,K)H(JiUI 2Q06 CONTINUE AS ------- LFVFL 21.6 ( MAY 72 ) 0S/36G FORTRAN H COMPILER OPTIONS - NAMt= MA IN,0PT=02,LINtCN1=foO,SIZ£=OOOOK, 5QU.R.C 11EtJCDICi_NQUI STiNUSKKiLQAUj^ap.NOEDI T . IQ.XREF ISN 0)02 SUBROUTINE EQOT(AMTC,AA,AS,BAD) c *****************************************+****+»*******» C SUBROUTINE £ DOT COMPUTES THE COEFFICIENTS THAT SPECIFV AN AUTO'S c INST ANTANEOUS EM.liS.ION RATE FUNCTIONS FOR HC .CO ..NQXl ARRAY • B AO • I « C GIVEN THE AMOUNT OF EACH EM1TTANT GIVFN OFF BY THE AUTO IN 32 A/D £ MODES ANL 5 STEADY STATE MODES (ARRAY 'AMTC'l.AND THE BASIS C FUNCTION FACTOR ARRAYStAA,AS I. c .. . . C AM TC =AMOUNT(GNS) OF THE I'TN EMITTANT GIVEN OFF BY THIS AUTO C- IN THE J»TH MDDE.I = 1=>HC,1=2=>C0.I=2=>NQX,J=L.3713Z A/D MODES C 5 STEADY STATE MOOES ).( RE AL<.) — c C BAD(I,J I=J¦TH COEFFICIENT OF THIS AUTO'S INSTANTANEOUS FMISSION C RATE FUNCTION FQ.R THE IlTH KIND. Of EMITTANT1= 1=>HC, I = 2.=>C Q* C I=3=>N0X.(REAL8) C C AA=>BASIS FI £ AA=BASIS FUNCTION FACTOR ARRAY FOR ACEL/DECELtCALCULATED BY SUBROU C -TINE SETUP). C _ _... C AS=9ASIS FUNCTION FACTOR ARRAY FOR STEADY STATEHC. IC = ?=>C Q. I = 3=>NQX __ C c ISN 0011 DO 1100 1 = 1.32 ..._ ISN 0012 Al = AMTC(1C,1) ISN OOli A2=TM< 1) _ . ..._ ISN 0014 YA(I>=A1/A2 ISN Q315 1100 CONTINUE C C CALCULATE COEFFICIENTS THAT SPECIFY A/D EMISSION RATE FUNCTIONS C ISN. Ofllfc DO 12.U0 I=1.,NRFA ISN 0317 SUM=O.ODO ISN 0018 DO 12i>C J=l.NO£>iA ISN 0019 SUM = SUM +(AA(I,J)YA(J)) -ISN.OOJM 1250 CONTINUE ------- ISN ISN 0021 00 22 1 ?0C1 BAD I IC, ~ I ) = SUH CONTINUE C r. f.ALCULATF flflSFRVFO AVFRAGF EMISSION RATFS OVER 5 SS MfinFS ISN Q;i?3 C nn 2000 1=33.37 ISN ISN 0024 0075 IP«I—32 A1=AMTC(IC.I1 ISN TSN 0026 on?7 A2=TM(I) YSIIPIxAl/A? ISN 0028 2000 r. CONTINUE c c CALCULATE COEFFICIENTS THAT SPECIFY SS EMISSION RATE FUNCTIONS ISN ISN 0029 0010 DO 2001 11,NBFS SIIMkO.ODO ISN ISN 0031 Q01? 00 2100 JxltNOBSS SIIM-SUMfAS( I.J)YS C J ft 1 ISN ISN 0033 0034 2100 CONTINUE B(I1=SUM ISN 0035 2001 C CONTINUE C C CHECK ON EXISTANCE OF NEGATTVE EMISSION RATES ISN ISN 0036 0037 L00P«0 IF(B(3 1.EQ.O.ODOIGOT02151 ISN ISN 0039 0040 XO(B(21*21-(4.ODOB(3 JB(11) IF 1XO. LT.0.000)GOTO?153 ISN ISN 0042 0043 XODSQRT( (BU)*2I-(4.0D0BJ3)B(1»)) X1 1 —B 12) ~X0I/I2.0D0BI31 1 ISN ISN 0044 0045 X2=(-BI2>—XO)/(2.0D0B(31» IF < m.GT.O.OnO.AND.Xl.LT.tO.OOOI.flR. IX2.GT.0.000.AND.X2-LT.60-OnO ISN 0047 C)) L00P=1 G0T02153 ISN ISN 0048 0049 2151 X0-B(1)/B(2) IF(XO.GT.0.ODO.AND.XO.LT.60.0001L00P«2 ISN 0051 2153 C IF(LOOP.EQ.O)GOT02154 C c IF L00P0>N0 NEGATIVE tHISSIONS FOR VELOCITYS BETWEEN 0,60 c c IF L00P=1 OR 2=> NEGATIVE EMISSION RATES BETWEEN 0.60MPH. c c CALL SUBROUTINE PAD TO FIND COEFFICIENTS WHICH DO NOT PRODUCE NEGATIVE EMISSION RATES. ISN 0053 c CALL PAD(YS.B) ISN 0054 c 2154 BADlIC.lOJrBU) ISN ISN 0055 0056 BAD(IC,11)=B<2) BAD(IC•12)=B{3) ISN 0057 ISN 0058 1000 CONTINUE RFIURN ISN 0059 END ------- LEVEL 21.6 ( MAY 72 I GS/360 FORTRAN H CQMPIL ER ISN 0002 OPTIONS - NAME = MA IN,0PT = C2 , LINECNT=60,S1ZE = 000OK, — -- SQURC fc . £ B CO I C . NOL I ST. NDDF CK«LOAD»MAP«NDt:DIT.ID»XRFF SUBROUTINE PAOtZ.BTI *************_**** **************************************** SUBRQUIIML PAD COMPUTES A SET OF COEFFICIENTS EMISSION RATE FUNCTION FOR STEADY STATE CONDITIONS THAT IS _NQ_N_ NEGATIVE BETWEEN VELQCTY 0 AND feO MPH. ..JUL. - * ************** ******************************************* ISN JJ03 ISN 0Q04 ISN 0 305 HiALc Hi). IPCS) . 71 = Z I IJ L2 = Z12J ISN 0006 ISM aoai isn oooe ISN QJ10 ISN 0011 11=1 12=2 IF(Z1.LT.Z2]GOTOI Z1=IL£J Z2=Z(1) ISN 0013 12= 1 ISN 0014 I Dfl 2 1=3.5 ISN 0015 ISN 0017 IF(2(IJ.GT.Z2IG0T02 ISN 0 018 12=1 ISN 0019 IF{Z1.LT.Z21G0T0? ISN 0021 Cl = Zl ISN 0022 Z1 = Z 2 ISN 0023 Z2 = C 1 ISN 0024 IX= I 1 ISN 0025 11 = 12 ISN 0026 12= IX ISN 0027 2 CONTINUE ISN 0028 B=(ZH-Z2I/2.CD0 ISN 0029 Vl = Il ISN 0030 V?=I? ISN 0031 Vl=m-1.0>15.0 ISN 0032 V2=(V2-1.0>15.0 ISN 0033 A=(V1V2J/2.0 ISN 0034 nr> 4 1 = 1.5 ISN 0035 4 ZP(I)=Z(I\|-B ISN <3036 SUM 1=0.000 ISN 0037 SUM2=O.ODC ISN 003B DQ 5 1=1.5 ISN 0039 Vl = I ISN 0040 vi= rvi-i.o)i5.o ISN 0041 V=V1 ISN 0042 X=(V2)(-2.0D0%AV1»A*2) ISN 0043 SUM1 = SUM1~{ZPCI> X1 ISN 0044 SUM2=SUM2»(X*2> ISN 0045 ISN 004b ISN 0047 isn o)4a ISN 004<» ISN 0050 CONTINUE BTU)=SUM1/SJ«2 BTI I 1=B+(BT<3)tA*21) BTl2)=-2«lfDQ31(3)A RETURN END ------- LEVEL 21.iS ( HAY 72 ) OS/360 FORTRAN M COMPILfcR OPTIONS ISN QJ02 NAME = MAIN,QPT=0 2,LINECNT=60,SUE = OOOOK, SnURCF.EBCDIC. NQLI ST.NOD ECK.LOAD«MAP »NQEDIT»ID «XREF suaROUTINE tSUMIVVT,NT,BAD.AHC.ACQ.ANOX.DIST) ********** 4* + * + +******************** ******************. _5UaMHJIlM£ £SilM—CALCULATES . AND THE DISTANCE TRAVELED BY THE AUTO THAT HAS THE INSTANTANEOUS EMISSION RATE FUNCTION SPECIFIED BY THE ARRAY 'BAD'. AND A DRIVING CYCLE SPECIFIED BY THE ARRAY 'VVT'. VVT(I) = >VEL0CITY VS. TIME HISTORY(OR IVING CYCLEI IN ONE SECOND NTOMAXIMUH NUMBER SECONDS IN DRIVING CYCLE+1 SECOND ^AQIJjXI^XI'TH COEFFICIENT OF THIS AUTO'S INSTANTANEOUS EHHIS&IOK RATE FUNCTION FOR THE I'TH KIND OF EMITTANT,I«1=>HC, C C AHC=AMTIGMS) HC GIVEN DFF BY THIS AUTO IN GOING THRU DRIVING CYCLE c c REAL»* c c ACO-AHT(GMS) CO GIVEN OFF BY THIS AUTO IN GOING THRU DRIVING REAM* CYCLE c c ANOX=AMT£GMSI NO* GIVEN OFF BY THIS AUTD IN GOING THRU DRIVE CYCLE c c REAL** c c 01STDISTANCE(MILES)IN SPECIFIED DRIVING CYCLE,REAL* ISN 0003 c DIMENSION VVTINT 1 ISN 000* ISN 0305 REAL8 BA0(3,12).AMTI3),X<12),DIS,AMIN,AMAX,AI,A2,HOA,SOA AMAX=1.0D0 ISN 0006 I SN 0007 AMIN=-1.2000 A1=-1.0DO/AMIN ISN 0008 c A2 = — 1.ODD/AMAX c c CLEAR AMT ARRAY ISN 0009 ISN 0010 1000 DO 1000 1=1,3 AMT(11=0.000 c c INTEGRATE AUTO'S EMISSION RATE FUNCTION OVER DRIVING CYCLE t<;n noil c DIS =0 a ODO ISN 0012 kw 0013 NTT =NT-1 00 3000 IT=1.NTT ISN 001 ISN 0015 KT= I T1 X(11=1.000 ISN 0016 I SN 0017 X(2 1=D0LE((VVT(IT)+VVT(KT)1/2.01 X( 3 ) =D8 Lfc I VVT 1 K.T) -VVT (IT)) ISN 0018 I cm Q019 X(4)-X(2IX(3) X(S1=Xl2)2 ISN 0020 ISN 0021 X( 6)=X(312 x(7l»(X<2>*2]XO> ISN 0022 ISN 0023 X(8)= ------- 1SN 00?4 ISN QQ25 ISN 0026 JSN mil ISN 0029 ISN 3031 ISN 0033 ISN 0035 X(10)=X(1) XLL1J=X121 X(12)=X(5) IF ( X(3 1 «Gt.AKAXl HQA=0.GDQ _ IF .j-T.AMAXlHQA = ( A2Xi^) ) +1.000 1F(X(3).LE.0.0D0.AND.X13).GT.AMINIHOA=(Al X C 3)J+1.0D0 DQ 2999IC-1«3 ISN 0036 ISN Q&3I ISN 0038 ISN 0040 ISN 0041 ISN QQ42- 2998 2999 DO 2998 IE= 1» 12 jQA=1.0D0-HGA 1F tIE.GT.91S0A=H0A CQNTINUE CONTINUE ISN 0043 ISN ISN 0045 ISN 0046 ISN 004T ISN 0048 DIS=D1S+X< 2) ^000 CONTINUE AHC =AMT(1) ACD=AMT 12 i AN0X=AMT(3) DIS=QIS/3600.QDQ ISN 0049 ISN QQSO DIST=DIS DQ 4000 ICK=1.3 ISN 0051 ISN .0,053- ISN 0054 ISN 0055 4000 IF(AMT ------- LFVEL 2L.6 ( HAY 72 f QS/360 FORTRAN H C9MPILLR OPTIONS - NAME = MA IN,OPT=02.LINECNT»60 . S I ZE=OOOOK, SClltRf-F . PB CD 1C . NflL 1ST . IMHnF CK • 1 fun . M AP . NflF fll T . T ft . XRF F ISN 0002 r. SUBROUTINE EDGRPINSTANT ANEDUS EMISSION RATE FUNCTIONS ARRAY.(RE AL8} c c SET INT = 1 FOR THE FIRST AUTO IN THE GROUP c c SET INT=2 FOR THE REST OF THE AUTOS IN THE GROUP c SET 1KT = 3 AFTEH ILL THE AUTOS IN THE GROUP HAVE BEEN CONSIDERED 1 Shi QQ03 u BFALM BOI1.1?I.B(3.12l.r.TM U n ISN ISN 0004 0006 IF( INT.GT.DGOT02000 nn 1000 1=1.3 ISN 0007 00 1001 J=l,12 ft(l.JJ=G.ODO ISN ISN 0009 nil in 1001 1000 CONTINUE rnNTlNlIF ISN ISN 0011 001? ?000 CTN=O.ODO TFi TNT.EQ.31G0TO4133 ISN isn 0014 nm s CTN-CTN+l.ODO nn ?ooi 1=1.3 ISN 0016 ISN 0017 DO 2002 J=l,12 RiI.JlsBII.Jl4-BADII.JI ISN I SN 0 J13 0019 2002 7001 CONTINUE mNTiNUF ...... ISN 0020 IF(INT.NE.3)60T04000 rm 3000 1=1.3 ISN ISN 0023 0024 00 3001 j«=lt 12 A AO t I - J i=BH . Jl/CTN ISN 0325 ISN O02J. 3001 tooo CONTINUE rnNTINUF _ ISN ASiL 0027 0Q2.8 4000 RETURN END - ------- LEVEL 21.6 t MAY 72 ) OS/'SfcO FORTP4N M COMPILLR OPTIONS - NAME = MAIN,0PT = 02 ,LINECN1=0,SIZE = OOOOK, - SOL'RCL.EeClUC.NQLIST.rtJDECK.LJA^.ilAP.NCtSIT.l;;, Htf ISN 0003 SUBROUTINE INVERStA,N,N1) C.» . _ . * c****************************** INVERSE ********************************** C ~ * c* * C» P.LMARKS : THt R2UIIN£.-L0WPUT_£S The INVERSE QF MATRIXA; DIMENSION CO. * C* WHERE H IS THt DIMENSION OF A. * C* . . . * C+ THE INVERSE IS RETURNED IN MATRIX A. * C* .. ... . .. * C« N IS RETURNED AS -I WHEN A IS SINGULAR » C* _ _ __ _ _ * ISN 0003 ISN QQQ4 ISN 0005 C*************************************************************************** c ..... ._ _ . ... ... DIMENSION A(N1 .MI) .S(10,20) DaJBLt PRECISION S.BUFF.OVH.fPY.A . . K= N+N INITIALIZATION . ISN £KLQi. ISN 0007 isn norm ISN 0009 ISN Ml? -Qa_5Ql 0=Jj10_ D3 502 J = 1, ------- I SN 0o41 ISN QQJtZ- 49 FPY = S (M ,L ) IF ( FPY . BO. G.U(J Jin ic is- ISN 0044 50 00 70 J= 11K -QQ45 ELUf S-LUOi ISN 0046 = S(HtJ) - BUFF _ ISii QQ <»7 1X1 CflliTIMUt ISN 0048 7b JIN = M+l TSN Q.149 IF t JIN . GT.N ) GQ TQ ,149 ISN 0051 XSN-QQ52... ISN 0053 1SHQQ3A 100 M = M1 .XZil. GO TP 49 CONTINUE _X5lQ cast IHU-fc . DIAGONALIZAT ION OF S lan ISN 0056 L = I t<;m 0157 H s 1-1 ISN isn 00 58 0059 350 FPY = S(M,U IF i FPY . EO. 0-00 1 GQ TCI 375 ISN 0061 351 00 370 J=1,K ISN 0062 BUFF = FPY • S(I.J) ISN 0063 S(M,J) = SIM,J) - BUFF I SN 0064 370 CANTINUF ISN 0065 375 IF < K .LE. 1 I GO TO 3 84 I SN 0067 380 M = M-l ISN 0068 GO TO 3 50 ISN 0069 384 CONTINUE ISN 0070 385 f. CONTINUE STORE INVERSE IN A ISN 0071 390 DO 402 Il=lN ISN 0072 LL = 1 1 ISN 0071 395 00 400 Jl=lN ISN or>74 KK = N + J1 ISN 0075 396 A(I 1,J1t = S< LL,KK) ISN 0O76 400 CONTINUE ISN 0077 402 CONTINUE ISN 00 7B RETURN C NO INVERSE ISN 0079 900 WRI TE( 6 ¦7000] ISN 0080 7000 FORMAT(1H0» N0 INVERSE') ISN nnfii N=-l ISN 0082 RETURN LSJ1 00fl3 END ... ------- APPENDIX V VEHICLE CLASSIFICATION BY DISCRIMINANT FUNCTION ANALYSIS The implementation of the emission-rate function model is assessing the impact of a collection of vehicles on the environment requires that the input data be appropriately chosen according to the problem under consideration. A matter of central concern in this regard is whether available data can be employed in a setting different from that in which it originated. In particular, such questions as geographic location and time frame are of interest. For example, can surveillance data obtained in one location be employed as a basis for modeling emissions in a different location, and can data collected in the past be used as a basis for modeling present or future problems? In a previous report, it was noted that a distinction exists between the emissions of vehicles in Denver as compared with five other cities of lower altitudes. Also, as would be expected, the implementation of emission controls clearly affected the emissions of vehicles in both geographic categories. The concept of discriminant function analysis was employed to crystallize and quantify these differences. The distinctions alluded to above were based on an analysis of total emissions of HC, CO and NO^ over the surveillance driving sequerce. Though it is presumed that these differences among categories of vehicles vrould persist in other driving sequences, it was considered of interest to investigate this assumption in a systematic way. Toward this end a discriminant function approach was undertaken. The logic of this approach is as follows. Each driving cycle consists of a series of accel/decel and steady-state modes. Conceivably some of these modes might be more sensititve to geographic or time differences than others. This would result in a situation where driving sequences consisting of combina- tions of these modes could themselves differ in this regard. A linear discriminant function is a weighted combination of the modes constructed in such a way that differences between vehicle categories, if they exist at all, will be most clearly delineated. ------- Suppose that only Denver versus non-Denver vehicles are considered and the emissions for a particular pollutant for the various modes are denoted X^, X^, X^. Then the linear discriminant function is defined as where a^, , ...» a^ are coefficients selected according to an optimization logic. For each vehicle, one can compute a value of F. If one groups together all F values for Denver vehicles and all F values for non-Denver vehicles, one can compute a mean value of F for each group and a variance or standard deviation for each group. The two groups will separate clearly, with minimal overlap of the two distributions of F values, if the difference between the mean values for the groups is large and the dispersion or scatter of F values within each group is small. The coefficients a^, ^ , • ••, are chosen so as to make as large as possible the ratio of the dispersion of the group means to the pooled dispersion of the individual vehicles within groups. If only two groups are involved, it is possible to confute only one discriminant function for each vehicle. However, if there are G groups, one can compute G-l such functions, provided G-l does not exceed the number of modes. For example, consider the following groups of vehicles GROUP 1 - Non-Denver, pre-emission control GROUP 2 - Denver, pre-emission control GROUP 3 - Non-Denver, emission control GROUP 4 - Denver, emission control In this case three discriminant functions can be computed as follows: F\ — &.vYi 4- - - - - CX.in X i"7 Fi_ c Y3-1 )(. v V-Cv Kv 4- _ — — V- 3 7 These functions employ three distinct sets of weighting coefficients [ci] the efficacy of which can be appreciated best by a geometric argument. ------- For each vehicle, one can compute F , , and F^. Consider these three values as the coordinates of a point in three-dimensional space. If each vehicle is plotted as such a point in 3-space, then there will result a collection of points which, hopefully, will tend to cluster into four distinct "clouds", these clouds being associated with the four groups of vehicles under study. If the separation of these groups is complete, one can partition the space into disjoint compartments and there will be no overlap or spillover from one compartment to the other. In reality, this will not. be the case, and there will be a certain amount of "mixing" or "confusion" if one attempts to assign a particular vehicle to its correct group merely by looking at the F-value coordinates of that vehicle. Quantification of the degree of correct and incorrect classification is therefore required in order to assess how "good" such a classification matrix is. To appreciate the nature of this quantification, consider first the case in which there are only two categories--say, Denver and non-Denver-- and only a single discriminant function F. Divide all vehicles into two sets on the basis of their known membership—that is, consider all Denver vehicles as a group and all non-Denver vehicles as a group. There are 169 vehicles in the Denver group, 851 in the non-Denver group. For the Denver group, compute F for each of the 169 vehicles and display these results as a histogram. Similarly, compute F for each of the 859 vehicles in the non-Denver group and display these results as a histogram. The results are shown in Figures 5-1, 5-2, and 5-3 for HC, CO and N0x respectively. Quite clearly, a certain amount of overlap is evident, so that if one attempts to assign class membership on the basis of the F value only, a certain number of both the Denver and non-Denver vehicles will be misclassified. A technique which can be used to determine the quantitative separation of groups is the construction of a "classification matrix" for the groups. The concept, as illustrated below, represents classifying automobiles as coming from City A or City B on the basis of a particular test value. CITY A CITY B CITY A CITY B 83 17 42 58 CLASSIFICATION MATRIX FOR TWO CITIES ------- The classification matrix should be interpreted in the following way. Assume a technique which takes a test value for an automobile from City A and, on the basis of probabilities for this test value, classifies this automobile as coming from either City A or City B, whichever has the highest probability for the given test value. If al1 the automobiles from City A are subjected to this probability test, one will find that some are correctly classified as coming from City A while others are incorrectly classified as coming from City B. The same test can be performed on the automobiles from City B. Since the number of automobiles from each city is known, one can convert the number of automobiles correctly or incorrectly classified into percent of automobiles from each city. The diagonal elements of the matrix represent the percent of automobiles coming from City A and City B which were correctly classified as coming from City A and City B. The off-diagonal elements give the percent of automobiles incorrectly classified. Therefore, 83 percent of the automobiles from City A were correctly classified as coming from City A, while 17 percent were incorrectly classified as coming from City B. Likewise, 58 percent of the automobiles from City B were classified as coming from City B while 42 percent were classified as coming from City A. In this case, the probability for classifying automobiles was more heavily weighted in favor of City A. Classification matrices for Denver versus non-Denver vehicles are shown in Table 5-1 and the corresponding histograms on which they are based are shown in Figures 5-1, 5-2 and 5-3. It can be seen that the separation of the two groups is somewhat better for CO and NO^ than for HC. Also, in the case of HC, it is more likely that Denver vehicles will be misclassified as non-Denver vehicles than that non-Denver vehicles will be classified as Denver vehicles. Thus once the probabilities for a set of groups have been determined for values of a given test the classification matrix technique can be used to give an easy-to-interpret quantitative measure of the separation between groups. ------- Tf the number of groups used in the classification is increased, the size of the classification matrix increases accordingly, with the diagonal elements indicating the percent correct classification. In this more elaborate situation, a vehicle can be miscl assi fi ed in more than one way, and the nature of the misc.1 assi fication can provide insight into which categories of vehicles are most "alike". Just as the number of groups (cities) in the classification matrix can be increased, the number of discriminant functions to be used as the basis of classification can also be increased. In this case one calculates, for a given automobile, the probability of occurrence of each of the discriminant- function values associated with the vehicle. For reasons discussed later, it can be assumed that the several discriminant functions are statistical]y independent. Therefore, these probabilities can be multiplied together and it can be assumed that the product is the probability of the joint occurrence of the particular discriminant function values encountered for the vehicle in question. Then compare the probability products calculated for all the groups and assign the automobile to the group with the highest probability product. In the case of Denver and non-Denver vehicles before and after the advent of emission controls, there are four categories and three F values, Ry virtue of the theory on which discriminant function analysis rests, the three F values are statistically uncorrelated. Also, by virtue of the fact that each of the F values is a linear combination of the outcomes of 37 random variables, it can be presumed that the Central Limit Theorem of statistics will cause the distribution of the F values to tend toward a Gaussian distribut on. Under these conditions, the assumption of independence is believed justified. The classification matrices for HC, CO and N0x according vo Denver and non-Denver, pre-control and control eras are. presented in Table 5-2. The type of inferences which can be drawn from these tables is exemplified by noting the classification matrix for N0x. For example, note that non-Denver, pre-emission controls (category ]) is misclassi fied more frequently as non-Denver emission controls (category 3) than as either of the Denver categories (categories 2 and 4"). Similarly, the Denver pre-emission control is confused more with Denver post-emission controls than with the other two categories, and the Denver emission ------- TABLE 5-1 CLASSIFICATION MATRICES GROUP 1 = NON-DENVER GROUP 2 = DENVER HC I 2 1) 82 17 2) 42 57 CO I 2 2) B4 15 2) H 88 NO x I I 1) 88 11 2) 11 88 ------- TABLE 5-2 CLASSIFICATION MATRICES GROUP 1 = NON-DENVER PRE-EMISSION CONTROL GROUP 2 = DENVER PRE-EMISSION CONTROL GROUP 3 = NON-DENVER EMISSION CONTROL GROUP 4 = DENVER EMISSION CONTROL HC 2 3. £ 1) 47 13 23 14 2) 19 48 3 28 3) 16 3 58 21 4) 5 15 11 68 CO £ 2_ 3 £ 1) 56 8 25 9 2) 12 72 0 15 3) 15 1 79 3 4) 12 11 4 72 NO X 1 2_ 3 4 1) 58 10 20 9 2) 7 73 2 17 3) 27 2 66 3 4) 5 23 2 68 ------- K*E lO X lO TO THE CENTIMETER 46 1512 18 X 25 CM. MADE IK U.S.A. < KEUFFEL & ESSER CO, ------- ------- U.C 10 X IO TO THE CENTIMETER 46 1512 «• 16 X 25 CM. naDE in u. S.a. • KEUFFEL « ESSER CO. -------

Calspan
Technical Report
A UTOMOBILE EX HA US T EMISSION MOD A L ANAL YSIS MODEL
Paul Kunselman and H.T. McAdams
Galspan Report No. NA-5194-D-3
Calspan Corporation
Buffalo, New York 14221
Formerly Cornell Aeronautical Laboratory, Inc.

-------
EPA-420-R-73-103
Calspan
On November 17, 1972 Cornell Aeronautical Laboratory (CAL) changed it* name to Calspan Corporation and converted to
for-profit operations. Calspan is dedicated to carrying on CAL's long-standing tradition of advanced research and development
from an independent viewpoint. All of CAL's diverse scientific and engineering programs for government and industry are being
continued in the aerosciences, electronics and avionics, computer sciences, transportation and vehicle research, and the environ-
mental sciences. Calspan is composed of the same staff, management, and facilities as CAL, which operated since 1946 under
federal income tax exemption.
A UTOMOBILE EX HA UST EMISSION MODAL ANAL YS/S MODEL
Paul Kunselman and H.T. McAdams
Calspan Report No. NA-5194-D-3
Prepared For:
ENVIRONMENTAL PROTECTION AGENCY
DIVISION OF CERTIFICATION AND SURVEILLANCE
ANN ARBOR, MICHIGAN
JULY 1973
Calspan Corporation
Buffalo, New York 1422

-------
TABLE OF CONTENTS
4.	COMPUTER IMPLEMENTATION
5.	MODEL PERFORMANCE
5.1	Statistical Indicators of Performance
5.2	Performance Results
5.3	Discussion and Evaluation
Page No.
1
1.	INTRODUCTION
1.1	Modal Analysis of Vehicle Emissions	3
1.2	Composite Analysis of Vehicle Emissions	5
1.3	Report Scope and Preview	7
2.	PROBLEM DEFINITION	8
2.1	Input Data	8
2.2	Output Data	9
3.	OVERVIEW OF THE MATHEMATICAL MODEL	10
3.1	The Emission Rate Function	11
3.2	Steady-State and Accel/Decel Emission Rate Functions	**
3.3	Determination of the Coefficients (bi, si)
3.4	The composite Emission Rate Function
3.5	Vehicle and Vehicle Group Characterization
TABLES
FIGURES
APPENDIX I - Specification of the 37 Modes and Evaluation of the
Average Values of the Basis Functions
APPENDIX II - Speed vs. Time Curves for the Surveillance
Driving Sequence and first 505 seconds of the Federal Test Procedure
APPENDIX III - The Mathematical Model
APPENDIX IV - Program Listings
APPENDIX V - Vehicle Classification by Discriminant Function Analysis
13
15
15
17
20
20
21
22
6. SUMMARY AND CONCLUSIONS	30
33
47
i

-------
LIST OF TABLES
Table No.	Table
1	Bag Value Statistics for the Surveillance
Driving Sequence
2	Bag Value Statistics for the First SOS Seconds
of the Federal Test Procedure Driving Sequence
3	Distribution of HC Bap Value Error from the
Surveillance Driving Sequence
4	Distribution of CO Bag Value Error from the
Surveillance Driving Sequence
5	Distribution of NO Bap Value Error from the
x
Surveillance Driving Sequence
6	Distribution of HC Bap Value Error from the
First 505 Seconds of the Federal Test Procedure
Driving Sequence
7	Distribution of 00 Bag Value Error from the
First 505 Seconds of the Federal Test Procedure
Driving Sequence
8	Distribution of N0x Bap Value Error from the
First 505 Seconds of the Federal Test Procedures
Driving Sequence
9	Replicate Modal Analyses of HC for 61 Vehicles
10	Replicate Modal Analyses of CO for 61 Vehicles
11	Replicate Modal Analyses of NO^ fro 61 Vehicles
12	Variance Component Analysis for 61 Replicate Tests
ii

-------
LIST OF FIGURES
Mean Steady-State HC Emission Rate Values
vs. Speed
Mean steady-State CO Emission Rate Values
vs. Speed
Mean Steady-State NO^ Emission Rate Values
vs. Speed
Main Program I
Main Program II
Distribution of HC Bag Error from Surveillance
Driving Sequence
Distribution of CO Bag Error from Surveillance
Driving Sequence
Distribution of NO^ Bag Error from Surveillance
Driving Sequence
Distribution of HC Bag Error from First 505
Seconds Federal Test Procedure Driving Sequence
Distribution of CO Bag Error from First 505
Seconds Federal Test Procedure Driving Sequence
Distribution of NO^ Bag Error from First 505
Seconds Federal Test Procedure Driving Sequence
iii

-------
ABSTRACT
This report on modal analysis of automobile emissions was prepared
for the United States Environmental Protection Agency's Division of
Certification and Surveillance, Ann Arbor, Michigan under EPA Contract
No. 68-01-0435.
The mathematical model and allied computer programs which are
described in this report enable an analyst to calculate the amounts of
hydrocarbons, carbon monoxide and oxides of nitrogen emitted by individual
vehicles or vehicle groups over any specified driving sequence. The model
requires as input the amounts of the three pollutants given off by individual
vehicles over short duration driving sequences (modes) in which speed is
a monotonic function of time.
The validity of the model is investigated by using it to predict
emissions for individual vehicles over the Surveillance Driving Sequence (SDS)
and the first 505 seconds of the hot Federal Test Procedure (FTP) driving
sequence. The ability of the model to predict actual vehicle emissions is
compared with the reproducibility of the actual vehicle emissions measured
in replicated tests. Using this analysis, the model performs extremely well.
The model has maximum effectiveness as a predictor of group emission charac-
teristics of warmed-up vehicles. It is understood that the mathematical
model should be used only within the region of speed and acceleration space
which is spanned by the input modal data.
It should be noted that because of the specifics of the study design
and the data collection process, the same vehicles were tested over the SDS
and FTP. Therefore, the test of model performance may have been more favorable
than would have been the case if the data had been obtained from two different
vehicle fleets.
iv

-------
TEXT
V

-------
1.	INTRODUCTION
In many geographic regions, the major portion of hydrocarbons (HC),
carbon monoxide (CO), and nitrogen oxides (NO^) present in the environment is
due to motor vehicle emissions. The impact of motor vehicles on the environ-
ment in a given location is a function of many factors. Among these factors
are the emission characteristics of individual vehicles, the mix of vehicles
in a particular traffic way, the numerical concentration of vehicles per mile
or per unit of area, and the driving pattern in which the vehicles are employed.
This driving pattern is influenced by the functional use of the traffic artery
(i.e., whether inter or intra city, whether it serves industrial or recreational
purposes, etc.) as well 9s a number of other items. Other items include the
design of the highway (e.g., whether it is designed for high or low speed) and
the extent to which it passes through and is limited by population density.
It has been well established that emission rates for a particular
automobile depend upon the manner in which the vehicle is operated -- that is,
emission rates are different for different accelerations or decelerations. In
a particular trip taken by an automobile in traveling from Point A to Point B,
that automobile will exhibit a particular time profile of acceleration and
velocity. The trip may entail a number of starts and stops, as well as a range
of speeds determined by traffic conditions, local speed controls, and other
factors. As the vehicle travels from A to B, this time profile or "driving
sequence", together with the emission characteristics of the vehicle, determines
the pollution contribution to the atmosphere. Due to the fact that emissions,
expressed in grams per miles, vary along the route, the distribution of the
vehicle emissions, as well as the total contribution of pollutants to the
atmosphere, can be determined. Indeed, the traffic way can be considered as
a line source of pollution, the strength of which depends on vehicle density,
vehicle mix, driving sequence, and the emission characteristics of individual
automobiles. Finally, the local concentration of pollution along this route
is determined by this line source distribution mechanism acting in concert with
meteorological transport processes such as wind and diffusion.
To assess the impact of vehicular emissions on a particular section
of highway requires, therefore, a number of data inputs. These include
characterization of both traffic and emission parameters. Traffic parameters
1

-------
include numerical traffic density, traffic composition (makes	and models
of vehicles), and traffic flow characteristics (speed, starts	and stops,
rates of acceleration and deceleration). Emission parameters	include
emission rates for various categories of vehicles where these	rates are
expressed as functions of driving variables such as speed and acceleration.
The required traffic parameters can be readily obtained by
monitoring traffic along the route in question. The number of vehicles
passing various points along the way per unit of time can be counted, and the
total number of vehicles can be broken down into homogeneous groups accord-
ing to make, model, age or other factors influencing emissions. Moreover,
speeds and accelerations prevailing along the route can be measured by
observing a "tagged" automobile or by instrumenting a vehicle and injecting
this vehicle as a "probe" into the traffic stream.
In contrast to the relatively straightforward approach to traffic
parameter assessment, the evaluation of applicable vehicle emission functions
proves to be quite difficult unless a means can be found for modeling the
infinite multiplicity of driving sequences which can arise. It is to be noted,
for exanple, that the EPA Surveillance Driving Sequence is only one of the
infinitude of possible sequences. Standard emission tests based on a pre-
scribed driving sequence serve the purpose of comparing vehicles according to
a standard set of operating conditions and make it possible to implement
emission control standards and to check compliance with these standards.
However, they are not structured in such a way as to readily provide the
ability to predict vehicle emissions over an arbitrary driving sequence. A
generalized prediction capability can be accomplished by breaking the standard
sequence, or any other available sequence, into segments having specified
speeds and accelerations. Then, these segments can be appropriately recombined
to form other driving sequences. In this way, one hopes to be able to approx-
imate any desired driving sequence by appropriately weighting the various
segments according to their time duration in the sequence to be modeled. The
segments are referred to as operating "modes" and any analysis based on the
use of these modes as a "modal analysis".
2

-------
1.1	MODAL ANALYSIS OF VEHICLE EMISSIONS
In 1971, the Surveillance Driving Sequence (SDS) was developed by
EPA to measure vehicle emissions over a variety of steady state and transient
driving conditions. The acceleration and deceleration modes represented in
the SDS consist of all possible combinations of the following five speeds:
0 mph, 15 mph, 30 mph, 45 n^h, 60 mph. The average acceleration or
deceleration rate observed for each mode in the Los Angeles basin is used
during operation of 20 of the transient modes. In addition, 6 of the tran-
sient modes are repeated using accel/decel rates higher or lower than the
average rate in order to determine the affect of accel/decel rate on emissions.
These acceleration and decelerations were chosen to represent the full range
of accelerations and decelerations observed in the CAPE-10 project.*
The concept of modal analysis examined in this report employs as
input data the emissions measured for the 37 distinct modes of the SDS. These
modes can be characterized by an average speed and an average acceleration.
Of the 37 modes, 5 are regarded as "steady state" -- that is, the acceleration
is zero. The five modes represent average speeds of 0, 15, 30, 45 and 60 miles
per hour. The other 32 modes represent either periods of acceleration or
deceleration and are characterized by an average acceleration, which is constant,
and an average speed. The importance of the speed constraint can be appre-
ciated by noting that if a vehicle accelerates from 0 to 15 miles per hour in
t seconds, its emissions response is not the same as when it accelerates from
15 to 30 miles per hour or 45 to 60 miles per hour in the sane time of t
seconds.
The mathematical model which has been developed to predict vehicle
emissions over any specified driving sequence is derived from vehicle data on
emissions from the 37 modes of the SDS. One difficulty presented by the use
of these discrete modes as inputs to a continuous driving sequence model is
that during much of the sequence, the vehicle may be operating at velocities
and accelerations not included in the set of five steady state and 32 accel/decel
modes. For example, a vehicle traveling at 23 mph is neither in the 15 nph or
30 mph steady state mode. To arrive at a continuous predictive model, one must
be able to interpolate or otherwise estimate the appropriate emission rates fbr
* "Construction of Chassis Dynamometer Test Cycles", Scott Research Laboratories,
Inc., November 18, 1971.
3

-------
all combinations of speed and acceleration encountered in the driving
sequence.
The primary contribution of this report is the development of a
scheme whereby emissions from the 37 discrete modes can be expanded into a
continuous function of time. Any driving sequence can be reduced to a
speed time profile. Since acceleration is a function of speed change and
time, both speed and acceleration can be expressed as continuous functions
of time. The emission rate of a vehicle at a given point in time is dependent
upon its speed and acceleration. Using these functional relationships, it is
possible to integrate the emission rate function according to the time history
of speed and acceleration associated with the driving sequence in question.
An essential feature of the model presented in this report is a
regression function which can, for purposes of visualization, be represented
Emission Response Surface
For any point (V , a) in the speed-acceleration plane there corresponds an
*
instantaneous emission rate S (v,a). The surface can be represented by a
mathematical equation of the form: £ « f(v,a) in which the function f contains
a nunfcer of adjustable constants. These constants can be selected to represent
the emission characteristics of a particular automobile or can be selected to
represent the mean emission characteristics of a collection of automobiles.
This collection need not be homogeneous with regard to make, model, age or
other identifying characteristics of automobiles. However, if a comparison
of homogeneous sets of vehicles is desired, a characteristic emission function
4

-------
for each set can be derived. In short, the model presented in this report
can be applied to individual vehicles or to composite groups of vehicles
selected in whatever way is meaningful to pollution assessment. The flex-
ibility of the model in this connection rests on the fact that the emission
rate function is developed as a linear function of adjustable constants
which can be particularized to an individual vehicle. Since the pooling of
emissions fbr a composite group of vehicles is itself a linear summing
operation, the composite emission function can be derived in a straightfoiward
manner as a weighted linear sum of the emission functions for individual
vehicles. Determination of the best process for pooling emissions from a
collection of vehicles is beyond the scope of this report, but, to provide
perspective fbr the use of the modal analysis, it is essential that
certain aspects of composite emissions modeling be considered since they
affect the modal analysis model.
1.2	COMPOSITE ANALYSIS OF VEHICLE EMISSIONS
Computation of the emissions emanating from a particular traffic way
in a given period of time is a composite of the emissions produced by all the
vehicles which traversed that traffic way during that time. Quite clearly, it
is not possible to assess the instantaneous emission rate functions for each
automobile. However, if the composition of the vehicle mix is known or can be
determined or postulated, one can define what might be called a "pilot mix" or
analog of the actual traffic composition. For example, suppose that a fraction
p^ of the vehicles belong to Category 1, a fraction P2 of the vehicles belong to
Category 2, and so on, and that the number of vehicles traversing the traffic way
per unit of time is N. An analog of this mix is n vehicles, where n «n, in which
there are p^n vehicles of Category 1, P2n vehicles of Category 2, etc. in the
sample. If the constants for the emission-rate function are determined fbr
each of the n vehicles in the sample, these constants can be averaged over
all vehicles comprising the sample to produce an emission rate function
which is "typical" for the mix. This composite emission rate function can
then be used to compute a typical or average emission for the specified
5

-------
driving sequence, where by "typical" is meant "representative of the
vehicle mix in question." By multiplying this average emission by N, the
total number of vehicles traversing the traffic way per unit of time, an
estimate of the total emission contribution in the time can be obtained.
Note that, by .virtue of the additive nature of the model, the result will
be the same as if separate contributions to the composite were computed
for each vehicle in the sample by means of its own specific emission-rate
function, and then these individual emission outputs were added and scaled
up by multiplying by the factor N/n.
The approach taken above can be referred to as the method of pro-
portional sampling -- that is, the number of vehicles in each category in the
sample is proportional to the corresponding number of vehicles in each type
in the population. An alternative approach is one in which no attempt is
made to produce an analog of the mix in the population but rather an emission-
rate function for each category is established independently of all other
categories. For example, n^ vehicles of Category 1 would be subjected to
modal analysis and an average emission rate function determined for Category 1
vehicles. Similarly, r\2 vehicles of Category 2 would be analyzed to determine
an average emission rate function for Category 2 vehicles, and so on. It is
presumed that the number of vehicles tested in each category (that is, n^,
nj....) would be such that the desired precision is realized; among other
things, these nunfcers would depend on the intra-class variability. Then, given
that each category of vehicle has been characterized by an emission-rate
function, a composite emission-rate function for any mix of vehicles could be
computed as a weighted average of the emission-rate functions for the several
categories.
In view of the flexibility of the modal-analysis model, the definition
of homogeneous categories of vehicles is not necessary. Indeed, categories of
vehicles can be constructed arbitrarily, so long as these aibitrary categories
are useful in the particular problem under study and can be weighted appro-
priately in the composite result. Nevertheless, it was considered of interest
to examine available modal data to determine if any significant groupings
were evident and whether these groupings might influence the application of
the emission model. In this connection it was found, by discriminant-ftmction
analysis, that vehicles in Denver exhibit somewhat different emission-rate
functions from comparable vehicles in other cities (see Appendix 5). The only
6

-------
way in which this observation affects the use of the model, however, is
in the choice of input data. In short, to model traffic ways or to compare
alternatives in Denver or in high-altitude locations one must use input
modal data appropriate to these locations. In all other respects, the
application of the emission model would be unaffected.
The model is particularly valuable if the analyst wishes to examine
alternatives, such as alternative routes or highway designs, before an actual
highway is built. By postulating the anticipated mix of vehicles and the
anticipated driving sequences, the relative desirability of alternatives can
be ranked according to their pollution impact.
1.3	REPORT SCOPE AND PREVIEW
In the ensuing sections of this report a methodology will be pre-
sented that will enable an analyst to predict the amount of hydrocarbons
(HC), carbon monoxide (CO) and oxides of nitrogen (N0x) given off by indivi-
dual or specified distributions of light duty vehicles as these vehicles
move from point A to point B by some defined speed-time profile.
The methodology will be presented in a somewhat classical modeling
approach. First, a description of the problem and the pi-oposed model
objectives will be given in terms of the data that are available (the iconic
model). Secondly, a mathematical model will be developed that parallels the
iconic model in its objectives. This model will be amenable to computer
implementation. The model's performance will then be analyzed to see if it
is able to meet the objectives set for it.
The proposed methodology is intended to be flexible enough to
accomodate changes in emission parameters which are expected to result from
improvements in emission control systems. This flexibility will also allow
modification or extensions of the model's objectives as the need for such
modifications arise.
7

-------
2.0	PROBLEM DEFINITION
Problem definition can best be understood in terms of the inputs
and outputs of the model. Input data consist of vehicle modal emission
measurements and speed versus time profiles for specified driving sequences.
Output data consist of estimates of emissions for any given driving sequence.
2.1	INPUT
Vehicle emission data are given for 1020 individual light duty
vehicles that represent variations in model year, manufacturer, geographic
location, engine and drive train equipment, accumulated mileage, state of
maintenance and attached pollution abatement devices. The individual vehicle
characteristics and emission data were obtained by Automotive Environmental
Systems, Inc. under EPA Contract No, 68-04-0042.*
For each of the 1020 vehicles in the data base, the following
emission data are given for the three pollutants (HC, CO and NO^J under consi-
deration:
(A) Modal Emission Data: The annuat of each polLutant emitted in
each of 37 defined speed-time profiles. There are three cases: speed is
monotonically increasing and acceleration is constant and positive over time
(accel); speed is monotonically decreasing and acceleration is constant and
negative over time (decel); speed is constant over time and acceleration is
zero (steady-state)j as shown below.
Accel Speed
Pecel Speed
Speed
Steady State
Time
* APTD-1497 "A Study of Emissions from Light Duty Vehicles in Six Cities"
March, 1973.

-------
The 37 speed-time curves are referred to as modes. There are 32 accel/decel
modes and S steady state modes. (See Appendix I for modal specifications.)
(B) Driving Sequence Emission Data
(1)	The total amount of each pollutant emitted during
the Surveillance Driving Sequence: The Surveillance Driving Sequence repre-
sents a speed-time curve of duration 1054 seconds; it is made up of the
32 accel/decel speed-time curves joined together by the 5 steady state
modes. The Surveillance Driving Sequence was performed after the vehicle
had performed the Federal Test Procedure Driving Sequence. Therefore,
emissions measured over the Surveillance Driving Sequence represent emissions
from a warmed-up vehicle. (See Appendix II for the speed-time values in the
Surveillance Driving Sequence.)
(2)	Emissions measured for each vehicle twice using the
FTP, once from a cold start and once from a hot start: Emissions were
collected from the "transient" and "stabilized" portions of these tests and
the data are reported as "cold transient", "cold stabilized", "hot transient"
and "hot stabilized" values. Emissions from the Federal Short Cycle were
also measured. In the study of model effectiveness, the values used were the
amount of each pollutant given off during the FTP "hot transient" driving
sequence (see Appendix II for the first 505 seconds of the Federal Test
Procedure Driving Sequence). Hot transient data were used since the model has
maximum effectiveness as a predictor of emissions for warmed-up vehicles. The
use of data from the Federal Short Cycle to measure the effectiveness of the
model has been left for future work.
NOTE: The total amount of a pollutant emitted by a vehicle as it executes
a driving sequence is often referred to as the "bag value".
2.2	OUTPUT DATA
Given the modal emission data on an individual vehicle, the basic
objective of the model is to develop a method which can predict the emission
response of this vehicle over any specified driving sequence. The predictive
ability of the model is restricted to accelerations and speeds in the sequence
which do not exceed the range of accelerations and speeds spanned by the input
modal data. In addition, it is desired to extend these individual vehicle
responses so that the emission responses of specified homogeneous groups of
vehicles can be predicted.
9

-------
3.0	OVERVIEW OF THE MATHEMATICAL MODEL
In this section a general description of the mathematical model
development is given. A more detailed version of the model appears in
Appendix III.
3.1	THE EMISSION RATE FUNCTION
The mathematical model used to describe the emission response of a
vehicle or group of vehicles is built around the concept of an instantaneous
emission rate. (The instantaneous emission rate is defined as the rate at
which a pollutant is given off at a specific point in time.)
If the amount of a pollutant emitted by a vehicle from time ¦ 0
to any time * t is denoted by eft), then the instantaneous emission rate
function l(t) is defined as the time rate of change of e(t).
(1)	§(t) = «*[«(*)!
The instantaneous emission rate at a specified time T is the value of the
emission rate function evaluated at this time:
(2)	S(T) .( d^' ¦ ) t . T
In the development of this model, it has been assumed that the instantaneous
emission rate of a vehicle is a function of its speed, v and acceleration, a.
Since speed and acceleration are considered to be time dependent, the emission
rate function can be expressed as
C3)	e(t) = e(v(t) , a(t)) » e (v,a)
Inherent in the definition of a driving sequence is a speed-time
(and therefore acceleration time) profile. The amount of a pollutant given
off by the vehicle over a driving sequence lasting T seconds is then given
by integrating the emission rate function over the speed-time curve for the
driving sequence of interest:
10

-------
(4)
e(T)

(v(t),a(t))dt
where v(t) and a(t) are the values of speed and acceleration at time ¦ t
specified by the driving sequence.
In practice the driving sequences are specified by a series of
speed-time points along the speed-time curve that are equidistant in time,
as shown below.
Vr
SPEED
where
The integration in equation (4) is then approximated by the following summation:
U-l
(5)	e(T)
where
X e(V®i
) A t
L=-1
Vi
Vi ~ 1 + Vi
A
ai
vi + 1 ' vi
At
N At » T
At this point, it is necessary to determine a suitable functional form
of the instantaneous emission rate function in terms of speed and acceleration.
3.2	STEADY STATE AND ACCEL/DECEL* EMISSION RATE FUNCTIONS
It is necessary to determine a functional form for the emission rate
function for the steady state case and the case of accel/decel. In the steady
state case (acceleration equals zero, constant speed) the emission rate
function is a function of speed only. This case is presented first.
* If a(t) « acceleration at time t, then a(t)decel. a(t)>©=*>accel.
11

-------
For each of three pollutants, steady state emission rates averaged
over the 1020 vehicles in the data base were plotted against speed.
Inspection of these plots (Figures 1, 2 and 3) suggested that the steady
state emission rate function eg could be expressed as a quadratic function
of speed!
(6)	* si
+ s2 v ¦* SjV
where s^, s2 and s^ are constants.
In the case of non-zero acceleration (accel/decel), the assumption
is made that the acceleration occurring at a given speed is a perturbation
to the steady state emission rate at this speed. This perturbation can be
accounted for by letting the coefficients Sj, S2 and Sj become functions of
acceleration. If it is assumed that quadratic functions of acceleration represent
good approximations to these coefficients, the coefficients can be expressed
as follows:
2
a
fs 1 * sl(a) * ^11 + * 12 a + ^13
(7)t	s2 * s2(a) - «21 + <*22a + W2
s3 - s3(a) - q31 + q32a + q33a2
where the q*s are constants. The emission rate function used during times of
non-zero acceleration can then be written in the form:
(8)	®$(v2a) s + b2V + ^3* + b4av + ^5y2 + b6&2 + b7v^a + bg&2v * b9*2
where the b's are constants and can be expressed in terms of the q's. It is
noted that if a ¦ 0 equation (8) reduces to:
•	2
(9)	e^(v,a s o) s bj + b2v + b5v
• »
which has the identical form as the equation for es. Thus, e^ could be used
to determine emissions for both steady state and non-zero acceleration periods.
At this point in the discussion, however, separate functions for steady state
and accel/decel emission rates will be retained; the reason for doing so will
be given later in this report.
12

-------
The instantaneous emission rate function e for a given vehicle
and pollutant is a composite function given by:
(10) e(v,a) - h(a)eg(v) + (1 - h(a)) eA(v,a)
where h(a) is a weighting function which is bounded by the values 0 and 1
and which is dependent on acceleration. Note that h(a) allows for a smooth,
continuous transition from steady state to accel/decel emission rate Amotions
or vice versa.
The next step is to evaluate the twelve coefficients (b^,i ¦ 1,9;
s^,i ¦ 1,3) for each vehicle and pollutant. These coefficients will completely
specify the instantaneous emission rate function describing this vehicle's
response with respect to the given pollutant.
3.3	DETERMINATION OF THE COEFFICIENTS (b±, s±)
The coefficients that specify the instantaneous emission rate function
could be determined by a straightforward application of the least squares
regression method if values of the instantaneous emission rates were available.
However, the data base on vehicle emissions does not contain any instantaneous
emission rate observations for accel/decel modes; instead, the observations
reported are the total amounts of the pollutants collected over each mode or
the average emission rate for the mode (which covers, many speeds). In this
light, the following method allows the determination of the coefficients that
specify the accel/decel instantaneous emission rate function.
(A) Specification of the Accel/Decel Emission Rate Function
It can be shown that if the proposed form of the instantaneous
accel/decel emission rate function is used to evaluate the functional form
of the average emission rate function, the same coefficients that specify
the emission rate function also appear in the average emission rate function
in a linear fashion (see Appendix III). Now, the values for the average
emission rate can be determined for each mode by dividing the amount of
pollutant given off in the mode by the time in mode. A standard least
squares regression analysis can then be performed on the average emission
rate function which will determine the values of the coefficients that
specify the instantaneous emission rate function. For exanple, suppose
the instantaneous emission rate function is given as:
13

-------
e(v,a) ¦ bj + b£V + b^va
Then the average emission rate function  over T seconds is
defined as:	T
 T 3 4= 5 ®(v'a) dt = T eCT)
o
Substituting the functional form of the instantaneous emission rate function
into the integral gives
T s. j C k. + b*. v 4- W3 va) dt = e CT)/t
T	T	T
K. C t	35 _[_ ^	-J- j" tj. V C^-t + -L ^ ^3 VC^. cl"t
T o	T o	0
r	t	t
Let V/= ~r [ V	jav Jt, and since i Jt = 1, have
6	T0	O
eCT) /T = t>» 4- V + b3 oST
The total emission e(T) given off in each mode and the time in each mode T are
known, v, and Iv can be determined for each mode. TTie coefficients (b^) can
therefore be obtained through least squares regression analysis applied to
the average emission rate function.
For the general model, there are 9 coefficients to determine and 32
accel/decel modes. A least squares regression analysis can be performed on
an individual vehicle or on the mean of a group of vehicles. This approach
forms a logical bridge from the experimental observations to the specification
of the accel/decel emission rate function.
(B) Specification of the Steady State Emission Rate Function
I* the case of steady state conditions, the speed does not change
with time. Thus, the average emission rate is equal to the instantaneous
emission rate. Values of the steady state emission rate function are then
available from the experimental observations, and the coefficients are evaluated
directly using least squares regression techniques.
* This expression in only an example, deliberately simplified for illustrative
purposes.
14

-------
There is, however, one problem that crops up with the above straight-
forward least squares approach. The values of the emission rate vary greatly
between speed zero (idle) and a speed of 60mph. As a result, the least
squares approach sometimes produces a steady state emission rate function which
predicts negative emission rates for certain speeds. In this event, the function
is adjusted by means of a constraint on its minimum value. The two lowest emission
rates measured experimentally are determined and averaged. Similarly, the speeds
corresponding to these two rates are also averaged. The average rate and average
speed determined in this way are then taken as the coordinates of the minimum
point of the emission rate function. Two of the three coefficients that specify
the steady-state emission rate function are thus determined; the third is computed
by the least squares method subject to this constraint (see Appendix II for details).
3.4	THE COMPOSITE EMISSION RATE FUNCTION
As stated earlier, two separate emission rate functions were desired
in order to describe accel/decel and steady state conditions. The two functions
are then joined by means of the weighting function as defined by equation (10).
The reason for retaining a separate function for steady state conditions when
the accel/decel function appears flexible enough to handle the steady state case
is that the accel/decel rate function also produces negative emission rates in
some cases for steady state speeds; any efforts to modify the coefficients to
constrain the function to yield only positive steady state emission rates would
produce serious errors when the function is used to evaluate accel/decel
emission rates. The composite emission rate function allows the freedom to
adjust the coefficients of the steady state emission rate function without
disturbing the accuracy of the accel-decel emission rate function.
3.5	VEHICLE AND VEHICLE GROUP CHARACTERIZATION
Once the emission rate function for a vehicle and pollutant is
specified, it can be used to obtain this vehicle's response, over any given
driving sequence, by integrating the rate function over the speed-time curve
defined by the driving sequence. Each vehicle is characterized by 36
15

-------
parameters or coefficients; 12 parameters for the specification of each
emission rate function describing the HC, CO, and N0x response.
The characterization of a group of vehicles can be achieved by-
defining the emission rate function for the average vehicle within the group:
Let	ss k'th coefficient in the emission rate function for the j'th
vehicle within the group and i*th kind of pollutant.
Ng = nunfcer vehicles in the group.
"fc ik ~	coefficient in the emission rate function describing the
average vehicle's i'th kind of pollutant response.
Then,
IM,
$
»» * ik ¦ itH bi*
g
If*
Thus, the group emission rate functions aTe determined by averaging
the coefficients which make up the emission rate functions of each vehicle
in the group. The emission response of the group over any driving sequence
is then deterjnined by multiplying the average vehicle's response by the number
of vehicles in the group. The average vehicle's response is obtained by
integrating its rate function over the speed-time curve specified by the
driving sequence.
16

-------
4.0	COMPUTER IMPLEMENTATION
The computer version of the mathematical model to calculate emissions
given off from individual vehicles and vehicle groups over any specified driv-
ing sequence is made up of two main programs:
Main Program I
A main program to compute emissions from individual vehicles over
any specified driving sequence.
Main Program II
A main program to compute emissions from a specified group of vehicles
over any specified driving sequence.
The main programs are used to read in the speed-time values of the
driving sequences, to perform any filtering operations needed to define vehicle
groups, to write out calculated emission values, and to call in proper sequence
the following set of routines which perform the majority of calculations:
Subroutine SETUP
Output: Subroutine SETUP determines the basis function factor arrays
for the accel/decel and steady state emission rate functions. These arrays are
labeled AA and AS, respectively.
Input: Speed-time values for the Surveillance Driving Sequence.
Utilization: Called once in each main program.
Other Subroutines Used: Subroutine INVERS (to calculate the inverse
of a matrix).
Subroutine EDOT
Output: Subroutine EDOT calculates the 36 coefficients specifying
the three emission rate functions for an individual vehicle.
Input: (i) The amount of each pollutant given off by an individual
vehicle in each of 37 modes.
(ii) The basis function factor arrays AA and AS determined
by subroutine SETUP.
17

-------
Utilization: EDOT is called once for each individual vehicle con-
sidered in main programs I and II.
Other Subroutines Used: Subroutine PAD.
Subroutine PAD
Output: Subroutine PAD calculates a set of 3 coefficients that
specify the steady state emission rate function for an individual vehicle such
that the emission rate function does not produce any negative emission rates.
Input: The amount of pollutant given off by an individual vehicle
in the five steady state modes.
Utilization Called once from subroutine EDOT for each individual
vehicle that originally had a steady state emission rate function that pro-
duced negative emission rates.
Other Subroutines Used: None.
Subroutine EDGRP
Output: A set of 36 coefficients that specify the emission rate
functions for the average vehicle within a group of vehicles.
Input: (i) The 36 coefficients specifying the emission rate func-
tions of individual vehicles determined by subroutine EDOT.
(ii) Sequence indicator INT.
Utilization: Called by Main Program II only; once for each vehicle
in the group (INT « 1 for first vehicle in the group, INT = 2 for all the
following vehicles in the group), and once after all vehicles in the group
have been considered (INT = 3).
Other Subroutines Used: None.
Subroutine ESUM
Output: The amounts of the three pollutants (HC, CO, NO^ given off
by an individual vehicle or a group average vehicle over any specified driving
sequence.
18

-------
Input: (i) The 36 coefficients that specify the emission Tate
functions determined by subroutine EDOT (Program I) or subroutine EDGRP (Pro-
gram I[).
(ii) The velocity-time values for the driving sequence under
consideration.
Utilization: Called once for each vehicle in Main Program I, called
once for the group under consideration in Main Program II.
Other Subroutines Used: None.
Subroutine INVERS
Output: The inverse of any two-dimensional square matrix of dimen-
sion less than 20.
Input: (i) The matrix whose inverse is desired.
(ii) Dimension of the matrix.
Utilization: Called once by subroutine SETUP.
Other Subroutines Used: None.
Flew charts for the main programs (see Figures 4 and 5))
show the calling order for the four main subroutines: SETUP, EDOT, EDGRP and
ESUM. In any main program the following calling sequence of the main subrou-
tines must be strictly observed:
SETUP
I
EDOT
V
ESUM
Listings of the main programs and subroutines are given in Appendix IV.
19

-------
5.0	MODEL PERFORMANCE
Evaluation of the performance of the model can be approached only by
comparing computed and measured quantities. For this purpose, it was found
convenient to use the Surveillance Driving Sequence and the first 505 seconds
of the Hot Federal Test Procedure driving cycles as measurably quantities and
to compare these quantities with the corresponding outputs predicted by the
model. Also, it must be appreciated that the degree of agreement between com-
puted and observed results will vary from vehicle-to-vehicle and that ultimate
evaluation of the validity of the model must take into account this statistical
variability. Toward this end, several statistical quantities were employed,
as discussed below.
5.1	STATISTICAL INDICATORS OF PERFORMANCE
Notation
0^ ¦ observed amount of i kind of pollutant given off by j
vehicle over a specified driving sequence (observed bag value)
¦ calculated amount of i kind of pollutant given off by j
vehicle over a specified driving sequence (calculated bag value)
N = number of vehicles in sample (1020)
c
R.. ¦ bag value error = 0.. - C..
To analyze the performance of the emission rate model in predicting
bag values, the following statistics are evaluated:
(a) The Mean Bag Error or Bias (FL) for each type of pollutant:
			Nlc,
^ = 7Nc ^ ^0ij " )
or
		_Nc
20

-------
(b)	The Standard Deviation of the Bag Error (G"R ) for each pollutant:
_	i	,
x
V	j=t
(c)	Root Mean Square Deviation of the Bag Error (RMS^) for each
pol lutant: /	v.	~ '
rms> = 7 Si i- 0" %
c Tei /ol ) • too %
¦ m
(J at + ir%i / 0c). ,ot •/
The mean, standard deviation, and root mean square deviation of the
bag error together provide insight into how the bag errors are distributed.
Expressing these statistics in terms of percent of the observed mean gives an
indication of how serious the bag error distribution is.
5.2	PERFORMANCE RESULTS
The values of the statistics for bag values obtained in the Surveil-
lance Driving Sequence and first 505 sec. of the Hot Federal Test Procedure's
driving sequence (hot transient) are given in Tables 1 and 2.
A visual inspection of the distribution of bag value errors is offered
by Tables 3 through 8 and corresponding histograms on Figures 6 through 11.
21

-------
5.3	DISCUSSION AND EVALUATION
To focus attention on the adequacy of the model, it is helpful to
condense Table 1 and Table 2 to a somewhat more concise form, as shown below.
PERCENT RMS ERROR
BETWEEN CALCULATED AND OBSERVED BAG VALUES
FOR 1020 VEHICLES
Surveillance	First 505 Seconds Federal
Driving Sequence	Test Procedure
HC	26.1	32.0
CO	23.9	29.1
NO	27.1	28.0
x
The percent RMS error is defined as: J ^	/ & j
ICO'/,
and represents the combined systematip and random errors. It is a particularly
meaningful quantity if one assumes that the mean or expected difference between
the calculated and observed values should be zero. A? will be noted in Tables
1 and 2, the RMS values are largely dominated by the random error component, as
represented by (Tr2. Moreover, these tables, together with the histograms
showing the error distributions, suggest that the difference between cpjnputed
and observed results cluster rather closely around the average error If and that
this average value deviates from zero by only a few percent of the average
measured bag values.
A logical question arises, however, as to the interpretation which
should be put on such terms as "cluster rather closely around the average" or
on such quantitative measures of performance as "25% RMS error". Against what
criterion are these measures of performance to be judged and is the nodel to be
judged satisfactory or unsatisfactory?
22

-------
To answer this question, one must consider the quality of the input
data and the manner in which errors in the input propagate into errors in the
output. In particular, one must inquire as to the repeatability of emission
measurements performed on the same vehicle and ostensibly under identical test
conditions. If the results of the Surveillance Driving Sequence or any other
specified driving, sequence fail to repeat on replicate tests, this failure can
not be traced to the inadequacy of a computational model, because no such model
is involved. The acpruement of instantaneous emissions over the driving
sequence is a physical, not a mathematical, process of integration, and the
vehicle and the measuring instrumentation constitute the only "computer" in
the system.
Of the 1020 vehicles in the input data set, 61 had been tested twice
each. Thus there were available 61 "replicate" measurements from which can be
obtained a measure of repeatability of measurements.
This measure of repeatability can be easily obtained as follows.
Consider a particular vehicle, and compute the mean X^ of the two replicate
measurements. Then compute the quantity
0"£ =	( ¥" !k~ Xja. ) -f- (Xz.k —	)
N - /
where and are the two replicate measurements for the k vehicle. Since
N = 2 in this case, the formula for (j~.^ reduces to the simple form
fa. (%m- tikf-
A 2
Now let us assume that the quantity	is one estimate of the variance of
replicate determinations and that each of the other 60 pairs of values provide
an additional estimate. These 61 estimates can be pooled or averaged to obtain:
^ -  t
k.-t
23

-------
as a best estimate of the variance of replicate values. Similarly, the quan-
tities X, (k * 1, 2,	, 61) can be pooled to obtain an estimate of the
_ K	A _
mean X for the total collection of vehicles, and the quantity	be
taken as a relative or percent standard deviation characterizing the repeata-
bility of measurements.
a _
The values	are shown below for the Surveillance Driving Se-
quence and for the first 505 seconds of the Federal Test Procedure.
PERCENT STANDARD DEVIATION BETWEEN
REPLICATE BAG VALUES FOR 61 VEHICLES
Surveillance First 505 Seconds Federal
Driving Sequence		Test Procedure	
HC	68.6	70.6
CO	14.4	26.9
N0x	15.5	15.8
Comparison of these values with those obtained by comparing calculated and
measured results suggests that the errors are comparable in the two cases.
Consequently, it is concluded that the model is performing quite acceptably
and that, indeed, its performance is substantially limited by the variability
inherent in the test measurements themselves.
Further support for this point of view is found in Tables 9, 10
and 11. Based on the 61 replicates, the quantities >f,  These errors are
reflected as errors in the determination of the regression coefficients,
and these errors in turn determine the error of estimating the instantan-
eous emission rate at any point in the (a,v) - space. Procedures are available to
trace the error propagation through this rather involved process and to produce
24

-------
"variance maps" in (a,v) - space,* but this type of analysis is beyond the scope
of this report. Moreover, even if such a variance surface were available, one
must further translate this surface into its effect on the integrated emissions
for a particular driving cycle. In view of the relatively large errors in
modal input data, however, the 25% to 35% RMS errors obtained for model per-
formance do not appear unreasonable.
Further insight into this matter can be had by an elementary and
straightforward application of analysis of variance as follows. Denote by
*ijk ^ = 2' ^ a	^ =			 ^	replicate of the
vehicle, where i = 1 denotes measured values and i ® 2 denotes values computed
from the model. For each vehicle, therefore, there are four values of total
emission for each of the pollutants HC, CO and N0X. These are:
Xj^ = Bag value measured for first replicate
Xi2k = Ba® va*ue measured ^or second replicate
X2jk ¦ Bag value computed for first replicate
= Bag value computed for second replicate
These four values can each be decomposed into components representing the effects
of the model, the effects of replication, and the interaction between replica-
tions and models.
Let us visualize the effect of the model as shown in the sketch below
	* OUTPUT "Identity" model
INPUT
INPUT
Mi
Mc \	» OUTPUT "Computational" model
In the "identity model", the measured emissions are subjected to no computation,
the bag values being those obtained directly from the measurement process itself.
In the "computational model", the modal measurements are used as the basis for
generation of an emission-rate surface, and bag values are computed by integra-
i
H. T. McAdams, "A Computer Method for Hypsometric Analysis of Abrasive
Surfaces," Advances in Machine Tool Design and Research, 1968, Pergamon Press,
Oxford, 1969, pp. 1149-1171.
25

-------
tion over the appropriate driving sequence. Thus if the data from replicate
tests are fed to these two "models" as inputs, the outputs will differ because
of the difference in the "transfer functions," Mj and M^,, of the two models.
The difference between the outputs of the models can be called the "model
effect" and is a measure of the extent to which the computationally inte-
grated results fail to agree with the physically integrated results. In pre-
vious discussion we have referred to this difference as a measure of the
"validity" of the computational model. In the present analysis, however, we
wish to examine this difference in relation to the repeatability of the input
measurements.
An appropriate statistical model for the analysis is
y-ijt =	+ eCik + fijk t
i « 1, 2; j > 1,2; k = 1,2,...., 61.
where is the output of the it'1 model for the j**1 replicate on the k**1
vehicle. The convention, of course, is that i ¦ 1 denotes the identity or
physical model and i * 2 denotes the computational model. The quantity
is the mean of the four output values for each vehicle and can be thought of
as the common or reference value against which model and replication effects
can be compared. The quantity ^ is a departure from this mean occasioned
by the effect of the particular model; since <7^ + " °» 006 of the models
will be represented as a negative departure, the other as a positive departure
from the mean. The quantity *s a departure from the mean occasioned by
replication. As far as is concerned, it is assumed that the difference
between replicates, in the statistical sense, is the same for the identity model
and for that computational model; hence, p ^ represents a pooled estimate incor-
porating both expressions of the replication effect. Since	¦ 0, one
of the replicates will be represented as a negative departure from the mean, the
other as a positive departure from the mean.
In reality, there is a distinct possibility that replication will be
influenced by the model—that is, it might be anticipated that replicate results
emerging from the computational model might be different from replicate results
emerging from the identity model, If such is the case, then there is inter-
26

-------
action between replication and models. This interaction is measured by the
term ( o(	which represents a "correction", in a sense, to the assumption
that repeatability of the output does not depend on whether one is considering
the identity or the computational model.
Decomposition of into components is easily accomplished by the
matrix transformation
1/4
1
-1
1
-1
1
1
-1
-1
1
1
1
1

- 1 """



Xllk

Ak

X12k

ik

X21k


J
Y
22k

(*{{!>) ijk
Moreover,	and ( J) ^ represent, respectively, the mean squares
for models, replication and interaction in a two-way analysis of variance. Each
of these mean squares has one degree of freedom.
Our analysis is completed by computing these mean squares for each of
the 61 vehicles and averaging these values. These results are presented in
Table 12 under the heading "mean squares". Viewed directly, however, these mean
squares are somewhat misleading, because the statistical expectation of the mean
squares for—say, the models effect—is not the variance q associated with
models but rather:
+ r}.

„2
where 0"^^ is the interaction variance. Similarly, the expected mean squares for
replications is not the variance	associated with replications but rather
3 
-------
<>2 <7~~ fr"
ft " XJ = Teplicates mean squares
<*¦ J	a interaction mean squares
/
one can extract the variance components	^ , (f^ and • These are
displayed in Table 12 under the heading "Variance Components",
Though the analysis for HC, CO, and NO^ as well as the analysis for
the two driving sequences give different results, the general impression is
that the models and replications effects are of comparable magnitude (note, in
particular, the results for CO and NO^ for the Surveillance Driving Sequence).
In the case of HC, it appears that the replications effect is much larger than
the models effect for both the first 505 seconds of the Federal Test Procedure
and for the Surveillance Driving Sequence. However, examination of the data
for individual vehicles revealed that there was one vehicle for which the bag
values replicated so poorly that the case might be considered an outlier. This
single vehicle is largely responsible for the large replications Mean square
for HC.
Special attention must be given to the interaction components. Though
there are several examples in which this component is of appreciable magnitude,
it can not be concluded that the computational model has poorer repeatability
than the identity model, because all of the variance components denote magnitude
only, not direction. Indeed, examination of the data for individual vehicles
reveals that the interaction effect is about as likely to be negative as posi-
tive. Often the interaction is occasioned by the fact that the ranking of the
two replicates is reversed when one goes from the identity model to the compu-
tational model. For example, in the identity model the first replication might
yield a higher bag value than the second, but in the computational model the
reverse might be true, yet the magnitude of the difference between the two
replicates might be the same in both cases.
In conclusion, the computational model performs remarkably well in
view of the relatively large errors in the modal emission rates which serve as
inputs. Since the model reproduces the measured bag values about as well as a
28
-------
replicate does, it is postulated that the model can predict emissions form a
non-standard driving sequence about as well as might be expected from an
actual test performed on that driving sequence. Further experience with the
model is needed, however, to be more assertive on this point. However, the
model has maximum effectiveness as a predictor of vehicle group emission char
acteristics, since the input variability of a homogeneous group of vehicles
is in general less than the input variability of any individual vehicle.
29
-------
6.0
SUMMARY AND CONCLUSIONS
In the preceeding discussion, a method was presented to calculate
the amounts of HC, CO and N0x emitted by individual vehicles and vehicle groups
over any specified driving sequence. The method uses, as inputs, the modal
emission data on individual warmed-up vehicles. It is to be understood, of
course, that the mathematical model should be used only within the region of
the speed and acceleration space which is spanned by the input modal data.
The model has maximum effectiveness as a predictor of group emissions for
warmed-up vehicles.
The method, given in terms of a vehicle emissions model, is character-
ized by the concept of an instantaneous emission rate. From this concept, the
emissions response of individual vehicles and vehicle groups are given in terms
of instantaneous emission rate functions. The development of the instantaneous
emission rate functions for a vehicle contains two important computational
features: the assessment of the coefficients that specify the instantaneous
emission rate function and a method to bound the steady state emission rate
function so that the function is non-negative (does not produce negative
emission rates) on the speed interval (0,60) mph.
In a least squares fitting procedure, it is possible to obtain negative
predicted values for some points on the speed and acceleration/deceleration
surface. Such a possibility is most likejly in extrapolated areas of the surface
or areas with very few actual data points. Due to the complexity of the
prediction procedure over the range of accel/decel space, the current model
does not check for negative emissions for each possible point in the prediction
surface of each vehicle. This type of problem did not occur for the set of
vehicles considered when appropriate weighting functions were used. However,
a test for negative emissions over accel/decel space and an appropriate
mathematical correction is planned as a future refinement to the model.
The instantaneous emission rate function can be used to characterize
an individual vehicle's emission response over any driving sequence as well
as to describe a vehicle group's emission response. The latter is accomplished
by the determination of the group's average vehicle emission rate function.
Further, a means of investigating the homogeneity of hypothesized vehicle
groups using linear discriminant function analysis is presented. (Appendix V)
30
-------
The content of the discussion and structure of the model allows fbr
the immediate use of the emissions model by an analyst to predict vehicle
emission and serves as a base for further research in predicting vehicle
emissions and their effect on the environment.
31
-------
TABLES
32
-------
TABLE 1: BAG VALUE STATISTICS FOR THE SURVEILLANCE DRIVING SEQUENCE
POLLUTANT
0
R
<7R2
Cf

— .100%
0
— .100%
0
\ R2 4r2 '
\ r2 2
-	 .100%
0
HC
53.5
7.2
143.3
12.0
14.0
13.5
22.4
26.1
CO
625.0
43.1
20420.8
143.0
149.3
6.9
22.9
23.9
NO
X
48.2
-2.7
163.0
12.8
13.0
-5.6
26.5
27.1
-------
TABLE 2: BAG VALUE STATISTICS FOR THE FIRST 505 SEC* OF THE
FEDERAL TEST PROCEDURE DRIVING SEQUENCE
POLLUTANT
0
R
Ci2
Gr

— .100%
0
Ci
—
~y R2 +(j]f 2
	 .100%
0
		 .100%
0
HC
21.0
2.8
37.3
6.1
6.7
13.4
29.0
32.0
00
223.7
9.2
4158.0
64.5
65.1
4.1
28.8
29.1
NO
X
17.2
0.5
22.9
4.8
4.8
2.9
27.8
28.0
-------
TABLE 3: DISTRIBUTION OF HC BAG VALUE ERROR (OBSERVED - CALCULATED)
FROM THE SURVEILLANCE DRIVING SEQUENCE
ERROR (GMS)	NUMBER OF VEHICLES
-70 to -65			1
-65 to -60			1
-60 to -55			0
-55 to -50			0
-50 to -45			0
-45 to -40			0
-40 to -35			1
-35 to -30			0
-30 to -25			2
-25 to -20			4
-20 to -15			5
-15 to -10			9
-10 to - 5				24
- 5 to - 0			149
0 to 5			309
5 to 10			246
10 to 15			97
15 to 20			80
20 to 25			37
25 to 30			16
30 to 35			18
35 to 40			3
40 to 45			4
45 to 50			5
50 to 55			1
55 to 60			0
60 to 65			2
65 to 70			2
70 to 75			0
75 to 80			2
80 to 85			1
85 to 90			0
90 to 95					0
95 to 100			0
100 to 105			0
105 to 110			0
110 to 115				0
115 to 120			1
TOTAL	1020
35
-------
iE 4
Isi
700
650
¦600
550
500
450
400
350
300
250
200
150
100
• 50
0
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
DISTRIBUTION OF CO BAG VALUE ERROR (OBSERVED - CALCULATED)
FROM THE SURVEILLANCE DRIVING SEQUENCE
NUMBER OF VEHICLES
1
0
0
0
1
1
0
1
4
3
10
22
57
94
170
272
143
92
53
32
27
6
7
7
2
5
2
1
2
1
0
0
0
4
TOTAL	1020
36
-------
TABLE 5: DISTRIBUTION OF NOx BAG VALUE ERROR (OBSERVED - CALCULATED)
FROM THE SURVEILLANCE DRIVING SEQUENCE
ERROR (CMS)	NUMBER OF VEHICLES
-65 to -60			1
-60 to -55			0
-55 to -50			0
-50 to -45			2
-45 to -40			1
-40 to -35			3
-35 to -30			8
-30 to -25			13
-25 to -20			26
-20 to -15			81
-15 to -10			98
-10 to -5			203
-5 to 0			298
0 to 5			145
5 to 10			60
10 to 15			44
15 to 20			22
20 to 25			17
25 to 30			14
30 to 35			5
35 to 40			6
40 to 45			1
45 to 50			0
50 to 55			1
55 to 170			1
TOTAL	1020
37
-------
TABLE 6: DISTRIBUTION OF HC BAG VALUE ERROR (OBSERVED - CALCULATED)
FROM FIRST 505 SEC, OF THE FEDERAL TEST PROCEDURE
DRIVING SEQUENCE (HOT TRANSIENT PORTION)
ERROR	(GMS)	.NUMBER OF VEHICLES
-45 to	-40			1
-40 to	-35			0
-35 to	-30					1
-30 to	-25			2
-25 to	-20			3
-20 to	-15			5
-15 to	-10					3
-10 to	-5			21
-5 to	0			158
0 to	5			592
5 to	10			172
10 to	15					35
15 to	20					11
20 to	25			9
25 to	30	. 			1
30 to	35	. 			 .	3
35 to	40				1
40 to	45			1
45 to	50			0
50 to	55			0
55 to	60	. 			0
60 to	65			1
38
-------
TABLE 7: DISTRIBUTION OF CO BAG VALUE ERROR (OBSERVED - CALCULATED)
FROM FIRST 505 SEC. OF FEDERAL TEST PROCEDURE
DRIVING SEQUENCE (HOT TRANSIENT PORTION)
ERROR (GMS)	NUMBER OF VEHICLES
-350 to -300 		2
-300 to -250 		1
-2S0 to -200 		1
-200 to -150 		8
-150 to -100 		24
-100 to - 50 		72
- 50 to 0 		279
0 to 50 		483
50 to 100 		103
100 to 150 		30
150 to 200 		3
200 to 250 		5
250 to 300 		0
300 to 350 		1
350 to 400 		1
400 to 1000 		1
TOTAL	1020
39
-------
TABLE 8:
DISTRIBUTION OF NOx BAG VALUE ERROR (OBSERVED - CALCULATED)
FROM FIRST 505 SEC. OF FEDERAL TEST PROCEDURE DRIVING
SEQUENCE (HOT TRANSIENT PORTION)
ERROR (GMS)	NUMBER OF .VEHICLES
-25 to -20				2
-20 to -15			1
-15 to -10			13
-10 to - 5			60
- 5 to 0					468
0 to 5				351
5 to 10			88
10 to 15			24
15 to 20			10
20 to 25			3
40
-------
MODE
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
FTP
SDS
TABLE 9
REPLICATE MDDAL ANALYSES OF HC FOR 61 VEHICLES
X (gms/min.)
3.8570
1.6284
2.4522
2.8327
3.5073
1.9178
5.2240
2.7310
4.4227
2.9622
4.9806
3.0453
4.9745
3.2702
1,8480
1.4517
4.2265
2.2881
3. 7102
2.3403
5.7121
3.1666
3.5338
4.2144
2.9225
1.8284
4.2448
3.0080
3.1500
4.5983
2.6819
1.9091
1.2829
1.2182
1.6903
2.5522
3.2911
21.3255
54.4599
C"(gms/min)
2.2342
1.1856
1.2589
2.7032
3.2759
1.0114
3.2673
2.1971
3.7429
1.3290
3.6829
1.3054
4.1268
1.8522
0.8133
0.5117
3.4144
1.5467
3.0242
1.3852
4.1975
1.2412
2.8711
3.7651
1.6542
1.0534
3.3618
0.9896
2.6789
3.8230
1.4452
1.1396
0.3898
0.9104
1.0114
2.3242
2.6043
15.0555
37.3647
-ST .100%
X
57.93
72.81
51.34
95.43
93.40
52.74
62.54
80.45
84.63
44.86
73.95
42.87
82.96
56.64
44.01
35.25
80. 79
67.60
81.51
59.19
73.48
39.20
81.25
89.34
56.60
57.61
79.20
32.90
85.04
83.14
53.89
59.69
30.39
74.73
59.84
91.06
79.13
70.60
68.61
41
-------
TABLE 10
REPLICATE MODAL ANALYSES OF CO FOR 61 VEHICLES
MODE
X (gms/min)
A
-------
1
2
3
4
5
6
7
8
9
10
11
12
13
14
IS
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
FTP
SDS
TABLE 11
REPLICATE MODAL ANALYSIS OF NO^ FOR 61 VEHICLES
X (gm/min)
3.5689
0.6435
0.9107
2.8518
5.2695
1.5707
6.9947
2.9842
7.9071
1.8646
7.2462
1.7380
7.0501
1.9580
0.6813
0.3179
4.9960
1.1132
4.8965
0.9471
6.6968
1.3989
2.9016
6.9010
2.0334
0.6391
6.5950
1.3037
2.5598
6.6468
2.1487
0.6203
0.1174
0.1826
1.0192
3.1829
6.3341
18.2670
50.8930
-------
TABLE 12
VARIANCE COMPONENT ANALYSIS FOR 61 REPLICATE TESTS
Source
First 505 Seconds - Federal Test Procedure
Mean Squares
Variance Component

HC
CO
NO
X
HC
CO
NO
X
Models
5.74
832.57
3.39
1.54
79.59
1.01
Replications
94.00
784.67
2,64
45.67
55.64
0.63
Tnteraction
2.65
673.39
2.37
2.65
673.39
1.37
Surveillance Driving Sequence
Source
Models
Replications
Interaction
Mean Squares
HC	CO NO
x
24.95 5560.34 29.76
600.84 4759.36 26.37
9.54 2770.54 5.13
Variance Component
HC
7.75
295.65
9.54
CO
1394.90
994.41
2770.54
NO
x
12.31
10.61
5.13
44
-------
FIGURES
45
-------
BEE 20x20 TO INCH

O*
-------
-------
BEE 20x20 TO INCH
-------
FIG 4 : MAIN PROGRAM I
SUBROUTINE SETUP
Subroutine EDOT
subrouting ESUtf
49
-------
FXG 5 : MAXN PROGRAM JH
READ VELOCITY VS. 77WE ARRAY FOR THE.
SURVG/LLANc£ DRIVING SEQUENCE.
1	—
READ VELOCITY VS. TIME 4ERAY FOR. WE
DRIVING SEQUENCE OVER WHICH EMISSIONS
ARE! TO BE CALCULATED.	
I
DETERMINE BASIS FUNCTtOfii FACTOR. ARRA\
l
READ IA/ VEHPCLE SPECIFICATIONS.
NO
TS "77//S VBM/CLE
^	"'¦s,
XaI GROUP uhpER
CONSIDERATION
specified 4Goup filter.
READ XA/ VEHICLES MODAL EMISSIONS DATA.
D£T£RM/ME EMISSION RATE FUNCTionToR
VEHKLE				
Add vehicles emission rate function
COEFFICIENTS TO GROUPS FUAiCTmN
SUBROUTING ED6T
SUBROUTINE ED&RP,
n GROUP?
DETERMINE EMISSION rate function for,
"avera&e* vehicle Representing group.
?mte&> Cats: AllERACE vehicles EMISS/oN
RATE FUNCTION OVER. SPGZ-/Fiei>
DRIVING SE$UENC«E.
	 — »		
DETERMINE TOTAL EMISSIONS &VEN OFF
B V QRCUP. 	
SuenoHVA/e ED&RP,
tut *3.
lUBRMVNE €SUM
I
WRITE OUT MVUUT HC, CO, UOf &!V£N OFF,
50
-------
-------
BEE 20x20 TO INCH
-------
-------
-------
-------
BEE 20x20 TO INCH
-------
APPENDICES
-------
APPENDIX I
-------
APPENDIX I
Specification of the 37 Modes and Evaluation of the Average Values
of the Basis Functions
(I)	Modal Specifications: Table 1-1
(II)	Evaluation of the Average Values of The Basis Functions
The nine functions of speed (v) and acceleration (a) which form the
basis functions of the instantaneous emission rate function must be averaged over
each mode in order to determine the coefficients which specify the emission rate
functions (See Appendix III).
To demonstrate how these average values are evaluated for the basis
2 2 2 2
functions: 1.0, v, a, va, v , va , v a , consider the case of determining the
average value of the 3rd basis function va over the i'th mode.
Let:	T. = duration of i'th mode
i
Vi(t) = speed at time a t in the ilth mode, t £
(.+"} - d Vt 1+0	acceleration at time = t in the i'th mode,
It	t * Ti
then by definition:
\m. tf viu) cuiAlJt
O
This integration is evaluated by the following approximation:
K* \
Ti u a- 	S?	
where: V., * initial speed of mode:
ll
= average speed over the j'th time interval
¦z.
	2	LS—	of mode i
-------
4t
= average acceleration over the j th time interval of
mode i
and» N i. At. = TL
The averages of the other 9 basis functions are similarity determined. Values
for the averages of the basis functions over each mode are given in Table 1-2.
-------
Table 1-1
MODAL SPECIFICATIONS
MODE DUKATIUNISEC) 0 I STANC£(HI)
¦SPeeO	AT LME SEC. INTERVALS
I
12. 0
0.06020
0.0
1.8
5. 1
9.1
13.2



30.0




2 "
16*0
	 0.07410
30.0
29.6
28.9 ~
28.1
26.9



6.4
3. 7
1.6
0.3
0. 0
3
8*0
0.02010
0.0
1.7
4. 6
7.6
10.3
4
ll.Q
0.07050
15.0
15.9
17.3
18.
20.6
5
13.0
0.13600
30.0
30. 7
31. 7
32.9
34.2



44.4
45.0



~ b
12 . J
0.12680
*5.0
44.7
44. 1
43.4
42.2



30.0




7
17.U
0.21630
30.0
30.0
31.0
32. 5
34. 3



51.6
53.6
55.5
57,2
58.6
8
12.0
0.17160
60.0
60.0
59.7
59.1
58. 3



45.L




9
14.0
0.20430
4 5.0
45.6
46.5
47.6
48. 7



58. 5
59.3
60.0


10
30.0
0.33670
60*0
59.7
59.3
58.8
58. 3



49.2
47.4
45.5
43,4
41. 1



22.1
20.2
18.4
17.0
15.9
U	
26.0
0.31360
15.0
16.7
20.0
23.1
26. 1



45*6
46.8
48.5
50.1
51. 5



58.9
59.3
60.0


12
21.0
0.19730
60.0
59.4
58. 5
57.4
56. 1



31.2
26.6
21.9
17.3
12.8
13
32.0
0*33130
0.0
1*5
5.2
8.6
12.0



34.5
36.8
39.0
41.0
43.0



54.5
55.5
56.4
57.2
57.9
14
23.0
0.29940
60.0
59*7
59.3
58.9
58.3



47.8
45.8
43. 7
41.6.
39. 4
15
9.0
0.05 790
30.0
29.5
28. 7
27.3
25.3
16
B*0
0.01730
15.0
14.4
13.3
11.3
8.5
. 17
22*0
0.17590
0.0
2.4
6.3
10.0
13.5



34.4
36.2
37.8
39.2
40.5
IB
16.0
0.13920
45.0
44*5
43.9
43.0
41.8



20.7
18.1
16.1
15.1
15.0
19
la.o
Q.15280
15*0
15.9
17.3
18.9
20.7



36*9
38. 8
40.5
42.6
43.4
_ 20
19.0
0.13040
45.0
44.5
43.7
42,7
M»4_



17.8
13.8
9.9
6,4
3.5
21
25.0
0.26540
0.0
2.6
7.1
11.5
15.7



41.8
44.2
46.4
48.5
50.3



59*3
60.0



22
28.0
0.26340
60.0
59.6
59.0
56.31
57.4



43.6
40.8
37.8
34.6
31,3



5.2
3.0
1.3
0.3
0.0
23
15.0
0*07370
0.0
1*2
3.5
6.4
9.6



27.8
28.7
29.6
30.0

24
25.0
0.31340
30*0
30.5
31.4
32,4
33.6


44.6
46.0
47.5
48,9
50.4



59.3
60.0



25
18.0
0*23620
60 .0
59*6
59.1
58.4
57.6



41.6
38.9
36.3
34.0
32.1
26
10.0
0.04440
30.0
29. 1
2t. 7
25.4
22.0
2?
38.0
0.40090
0.0
0.9
4.0
6.9
9.0


29.6
31.7
33.7
35.7
37.5



49.6
50.8
51.9
52 ,9
53. 8



58.9
59.3
60.0


2a
35.0
0.32930
60.0
59.7
59.2
58.7
58.1



49.3
47.6
• 45.7
43.6
41.4



20. 1
17.3
14.6
12.0
9.5
29
18*0
0*08860
0.0
0.8
2.7
4.9
7.5



25.2
26.5
27.4
28.3
29.0
30
21.0
0.25990
30.0
30.7
31.8
33.1
34.5



48. i
49.8
51.5
53.0
54.6
31
14.0
0.18130
60.0
59.5
58.7
57.6
56. 1



32.1
30.4
30.0


32
13.0
0.05920
30.0
29.4
28.5
27.2
25.4



0.3
0.0



33
60.0
0.0
SP*
- 0.
OFQR 60
SEC

34
60.0
0. 25000
spee o
- 15.
OFQR 60
SEC

35
60.0
0.5C JCO
sP*eo
» 30.
OfOR 60
SEC

36
60.0
0.75000
speep
* 45.
OF OR 60
SEC

37
60.0
l.COOOO
SPtfBC
- 60.
OFOK 60
SEC

17.1
20.b Zi.i,
25.4 23.5 21.2
12.1
22.4
35.6
*0.7
36.2
59.8
57.2
50.0
57. 6
38.8
15.2
-ZS*9.
52.8
5*.*
8.7
15.3
44.8
56.5
57.6
37.3
22.7
5.2
jfc.a
41,6
*0.2
22.6
**.5
?9.a
13.7
24.2
37.0
38.9
38.3
55.7
51.3
56.9
36 ¦ 4
15.0
31.6
54.0
52. *
5.1
18.4
46.5
58.9
56.7
55.4
19.9
2.1
19.9
14.5
25.9
38.5
36. a
40.5
53.8
52.6
56.0
34.0
34,2
55.1
49.9
2.3
21.li
„4«. 1
25.7
IB.5
15.0
27.5
39. 9
34.6
42.7
51.7
5 3.9
54.9
31.5
36. 6
56.1
46.9
0.5
24.2
49.6
1.1
19.6
52.0
56.5
271?
12.8
34-. 8
51.7
56.5
30.7
17.6
12.6
39.3
54.7
57.4
39.0
7.2
ID. 1
29.a
36.0
56.0
54.0
42,6
"•2
24.5
45.0
3T.fl
59.3
55,7
33.6
17.3
0.2
0.1
23.4
53.5
55.3
24.3
43.4
35.8
26.5
}},}
60.0
54.5
32.1
15.4
0.0
3 5. 5
27. 1
15. f,
23. 3
41.2
32.6
45.0
49.6
55.2
53.8
29.0
38.9
56.9
43.6
0.0
27.0
_51.0
28. 6 29. 7
12.5 9.4
29.8
42.4
31.0
47.3
*
47.5
56.4
52.4
26.6
41.1
57.7
30.0
43.5
30. 1
49.5
45.9
57.5
50.9
24.3
43.1
58.3
39.8 35.7
29.6
52.3
53.1
30.9
15.0
51.5
30.2
0.0
26.9
54.»
54,0
20,5
44.0
33.1
18.7
44.5
SQti
30.8
29.2
45.0
26.9
32,9
2?. ft
30.3
56.1
52.4
JL7*1_
13.4
57.1
50i6
14.a
36.4
58.0
4e.5
JJUL
36.0
53.0
55.1
30.0
12.5
15.3
41.0
55.5
56.7
36.6
5.2
12* 8
30.0
37.6
57.2
51.5
37.4
54.3
36.7
55.5
40.2
56.6
41.»
57.6
22.9 19.9
7.4
17.9
42.5
56.3
55.B
34.0
3.4
15.4
39.3
58.4
48.5
16.4
2.9
20.4
44.2
56,9
54.B
31.3
1.9
17.8
41.0
59.3
45. r
12.5
0.3
22.8
45.6
57.5
53.7
28.6
0.8
20.1
42.8
60.0
41.6
B.6
32.1
53.5
49.7
30.0
28.0 30.3 	32.4
34.9
21.8
39.2
58. T
46.2
_ tiA
15.9 18.8 21.3 23.5 25.2 26.T
43.1
58.5
53.4 51.5 49.3 46.8 44.3
0.0
25.2 *
47.0
58.1
52.4
25.B
0.2
22.1
44.6
38.0
5.0
27.4
4«.4
58, 5
50.9
22.9
0.0
23.a
46.3
34.7
2. 1
-------
Table 1-2
VALUES OF THE AVERAGES OF THE BASIS FUNCTIONS OVER EACH MODE
l 7 	~a	7s	75?	To?	vN?
	1 1.0000	18.0500 £.5000	37.5C0C	42j.5966 7.7567	747.7668	IC4.8537	164C.132C
2	1.0000	16.6625 -1.0750	-28.1250	386.9459 4.<.550	-561.5264	61.9900	11C3.4938
	3	1.0000	9.0375	1.8750	14.C625 !Ct>.4d67	4.3150	139.7132	29.2004	255.4498
4	1.0000	23.0727 1.3636	30.6818	556.5805 2.0B00	715.6334	46.5847	1075.^698
5	1.0000	37.6538 1.1536	43.2692	1440.5222 1.4046	1644.0825	52.5375	1986.8048
6	1.0000	38.0500 -1.2500	-46.8750	1476.2113 2.0417	-1760.9423	75.4888	2818.3750
7	1.0000	45.8000 1.7647	79.4118	2192.9562 3.4094	3705.3125	152.8414	7064.6548
8	1.0000	53.00b3 -1.2500	-65.6250	2838.4180 2.0500	-3468.4389	106^5081	5561.0050
	9	1.0000	52.5428	1.0714	56.25C0 2783.1422	1.2014	2973.0977	62.9974	3322.1119
TO 1.0000	40.4033 -1.5000	-56.2500	187573556 2.8120	-2362.0152	100.7045	3951.2093
11	I.0000	43.4154 1.7308	64.9038	2067.4378 3.7085	2725.2108	123.4465	4606.5663
12	1.0000	33.8333 -2.8571	-85.7143	1572.5804 10.2076	-3425.2217	282.9109	10063.8078
13	1.0000	36.2375 1.8750	56.2500	1789.5073 4.4462	2248.9990	107.7057 3670.1761
14	1.0000	46.0609 -1.3043	-58.6956	2303.4566 2.1313	-2738.8101	93.5297	4220.4915
1	5	1.0000	23.1778	-1.6667	-37.5000 564.6190	3.5533	-874.3014	78.0627 1761.9989
t6 1.0000	7.8125 -1.8750	-14.06 25	89.8487 4.6950	-139.5406	32.6369 290.9055
17 1.0000	28.8454 £.0455	46.0227	1016.1811 5.2827	1379.3948	96.7243 2467.4704
re itoooo	jttszsc -1.8750	-56.2500	1093.6045 4.545c	=T8rr.ui5	131.2360 4035.1918
19	1.0000	30.5500 1.6667	50.0000	1026.4346 2.9778	1624.5391	88.6821 2826.1665
20	1.00CO	"2477158 -i.3684	-53.2895	861.2722"" 7.193T -1596.6755149.6970	3992.4466
21	1.0000	38.2760 2.4000	72.0000 1791.6623	7.2248	2877.9475	176.2425	6001.1068
Z2 1-OOOC	33.8750 -2.1429	-64.2857 1575.9649	5.7421	-2570.0153	159.2388	5668.85i49
23	1.0000	17.7333 2.0000	30.0000 411.6827	4.8573	598.9142	65.4714	1148.7460
24	170000	45.144C T.200U	"54.0000 ZT28.765I	1.5120	2519.8353	67.7954	3133.7277
25	l.OOOO	47.2333 -1.6667	-75.0000 2334.0895	3.4022	-3499.3613	149.2304	6730.2482
26	1.0000	15.9900 -370000-45.0000 369.3C02	11.8100	-895.7151	164.2715	291270867
27	1.0000	38.0053 1.5789	47.3684 1770.6486	3.1421	1094.1427	75.7415	2585.3045
2B	1.0000	33.8686	-1.7143	-51.4286	1575.8225	3.6766	-2056.4151	101.6697	36247S62&
29 1.0000	17.7333 1.6667	25.0000	412.2008 3.3911	499.3674	45.6602	803.7473
—30	1. GOOD	45.2657 I.4ZB6	64.2S57 2X40.4352"	7.1562	2999.7180	96.6297 —4465 .1001
31	1.0000	46.6286 -2.1429	-96.4286	2283.4472 5.8200	-4498.5436	255.5805	11536.7248
32	1.0000	I?.4000 -2.3C77	-34.6154	379.6860 6.8246	-690.4485"	94.9561	16B67S623
33	1.0000	0.0 0.0	0.0	0.0 0.0	0.0	0.0	0.0
—3*	1.0000	15.000C	07$	CT70	225.0CC0	C70	0J5	C7Q	070	
35 l.OOCO	30.0000 0.0	0.0	900.0000 0.0	0.0	0.0	0.0
56 170000	45.0000 t.O	C.F	2C25.0(0C O.C	O.C	C.C	0.0
37 1.0000 60.0000	0.0	°*2_ 3600.0000 	0_^0	C.0 	O.C.	0.0
-------
APPENDIX II
Speed vs. time curves
for the Surveillance Driving Sequence
nd firs-, 535 seconds ;t the ^¦edacB-l
Tast Procedura Driving Saquer.oe.
-------
JlRVfULA'-'CF ACCFLERfiTIClh-
rf«f SPEhC
(SECI (cpm
T/..
10.
11.
12.
_!¦*•
I*.
15.
16.
17.
18.
10.
"20.
21.
22.
23.
24.
_20.
2b.
27.
28.
29.
30.
31.
32.	"
33.
_
35.
36.
37.
28.
39.
40.
<¦1.
42.
S3.
44.
<.5.
46.
47 .
4>e.
49.
5n.
51.
52.
53.
54.
5s-
56.
57.
C.C
0.0
C.O
_0.C
C.O
0.0
_ 0.0
C.C"
C.O
0.0
1.8
5.1
9.1
13.2
17.1
2C.6
23.A
25.7
27.3
28.6
21.7
3C.0
3C..C
30.C
30.C
"30.0
3C.C
3C.0
30.0
3C.C
3C.0
3C.0
3C.C
3C.C
30.C
30.C
30. e
29.6
26.9
28.1
26.9
25.1
23.">
21.2
18.5
15.6
12.5
<».4
_
3.7
1.6
C.3
C.C
C.C
_ C.O
C.O
C.C
C £ .
59.
to.
_ '1 •
*2.
63.
64.
65.
6 6,
f.
48.
69.
_
71.
72.
74.
75.
76.
77.
78.
79.
" eo.
pi.
_?2.
n.
e4.
85.
86.
8/.
6ft .
69.
"50.
_5 i.
*2.
53.
54.
15.
56.
17 .
S8.
?}.
ICO.
IC1."
1C2.
1r 3 .
1C4.
105.
JC6 ._
1 C 7 .
1C3.
1C1.
lie.
111.
112..
113.
114.
O.C
o.p
c.c
. C. C_
0.0
0.0
1.7
4. 6
7.6
1C. 3
12.3
13.7
14.6
15.C
15.C
15.C
15.C
15.C
15.C
15.C
15.0
15.C
15.c
15.C
IS.C
15.C
15.C
15. C
15.C
15. 9
17. 3
18.9
20.6
22.4
24.2
25.9
27.5
28.8
2S. C
30.0
30.0
3C.C
30.0
30.C
3C.C
30. C
30.C
30.0
3C.C
30. C
30.0
3C. C.
3C. C
30. n
30.0
30. 7 '
31.7
ri"f	si'Ffc,
(sro	irPH)
115.	32.9
116.	34.;-
117.	35.6
llfl.	37.0
119.	ift.5
120.	J9.9
121.	41.2
122.	42.4
123.	43.5
1?4.	44.4
125.	45.C
126.	45.0
127.	45.c
l?8.	45.C
129.	45.0
nc.	45.c
131.	45.0
132.	45.0
133.	45.0
134.	45.0
135.	45,0
136.	/,5.C
137.	45.C
138.	45.0
139.	45.0
14 C.	45. c
141.	44.7
142.	44.1
143.	43.4
144.	42.^
145.	4C.7
146.	3S.9
147.	it. e
148.	14.6
149.	J2.6
150.	31.0
151.	3C.1
1'.?.	3C.C
153.	M'.C
154.	10.0
' 155. " 3C.C
156.	3C.0
157.	30.0
ISA.	30.0
15<).	30. C
lf-C.	3C.C
lfcl.	30.0
162.	3C.C
163.	3C.C
16'.. "	30. :
165.	3C.C
166.	3C.C
16 7.	3 0.C
168.	31.C
16<1.	32,5
17C.	34.3
171.	36.2
Tivr spem
isco h)
IStzCl
(VPh)
4 7.6
786.
15,0
343.
5P.4
48.7
287."
15.0
344.
58.5
50.0
28a.
15.0
345.
57.4
51.3
239.
15.0
34 6.
56.1
52.6
290.
15.0
34 7.
54 .4
53.9
291.
15.0
34fl.
52.4
55.2
29 2.
15.0
349.
49.9
56.4
29 3.
15.0
350.
4('.9
57.5
294.
15.0
351.
43.6
58.5
295.
15.0
352.
39.8
59. J
296. '
~ I 5 »0
353.
35.7
60.0
207.
15.0
354.
31.2
60.0
298.
15.0
355.
26.6
60 . 0 ~
299 .""
15.0"
356.
21 .9
60.C
300.
15.0
357.
1 7. 3
60 .0
301.
15.0
358.
12.8
60. 0
302.
16.7
3Hq.
8.7
60.0
30 3.
20.0
360.
5.1
60. C
304.
23.1
361.
2.3
60.0
305.
2 6.1" "
362 .
C . 5
60.C
306.
28,9
363.
0.0
60.0
307.
31,6
364 .
0.0
60.0 "
308.
34.2
365.
0.0
CO.0
309.
36.6
366.
C.O
60.0
310.
3 8.9
36 7.
0.0
60 . 0
311.
4 1.1
368. "
C.O
60.0
312.
43.\
369..
c.o
6C.0
313.
45.0
370.
r.o
59.7
314.
46 .8
37 1.
C.O
59.3
315.
48,5
If2.
0.0
58.8
316.
50.1
373.
0.0
58.3
"317. "
51.5 "
" 374 .
1 .5
57.6
318.
52.8
375.
5.2
56.9
319.
54.0
376.
8.6
56.0
320.
55.1 ~
37 7.
12.0
54.9
321.
56. 1
378.
15.3
53.8
322.
56.9
379.
18.4
52.4
323.
5 7.7
380. '
21.4
50.9
324.
58,3
381.
24.2
4t>.2
325.
58.9
382.
27.0
47.4
326.
59.3
383.
?9.6
45.5
327.
60.0
384.
32.1
43.4
328 .
60 .0
385.
34.5
41.1
329.
60.0"
396.
36.8
38.8
3 30.
60.0
337.
3-7.0
36.4
331 .
60.0
38 P.
41.0
34.0
332.
60.0
389.
43.0
31.5
333.
60.0
3 70.
44.8
29.0
334 .
60 .0
3<<1 .
46.5
26.6
335.
60.0
392.
48.1"
24.3
336.
60.0
39 3.
4-3.6
77 . 1
33 7.
6 0.0
394 .
5 1.0
20.?
33a.
60.0
395.
52.3
18.4
339.
60.0
3oa.
53.5
17.0
340.
60.0_
397.
54.5
15.9
341.
60.0
398.
55.5
15.2
342.
60.0
399.
56.4
nxF	sprrn	riwe speed
.	15.0
4 14.	60.0	4 71.	15,0
4 15.	60.0	47 2.	.15.0
416.	60.f	473.	15.0
417.	60.0	474.	15.0
418.	60.0	475.	15.0
419.	60.0	476.	15.0
420.	60.0	477.	15.0
471.	59.7	478.	15.0
422.	59.3	479.	15.0
423.	58.9	480.	15.0
424.	58. 3 _ _481 .	15.0
4?5.	57.6	4 82.	15.0
426.	56.7	483.	14.4
427.	55.7	484.	13.3
428.	54.5	485.	11.3
429.	53.1	486.	8.5
430.	51.5	487«	5.2
431.	49.7	48R.	2.1
432.	47.8	489.	0.2
433.	45.8	490.	0.0
434.	43.7"	491.	"0.0
435.	41.6	4«2,	0.0
436.	39.4	 493.	. 0.0
437.	37.3	494.	C.O
436.	35.4	495.	0.0
439.	33.6	496.	0.0
440.	32.1	497.	' 0.0
441.	3«.<»	49*.	0.0
442.	30.2	493.	0.0
443.	30.C	5CC.	0.0
444.	30.0	5C1•	2.4
445.	30.0	5^2.	6.3
446."	30.0 "	503.	lD.0
447.	30.0	504.	13.5
448.	33.0	5C5.	16.8
449.	"	30.0 "	506.	.19.9
4 50.	30.0	50 7.	22.8
451.	30, 0	5Cfi.	25.5
452.	30. 0	50'J.	2d.O
453.	30.0	510.	30.3
4 54.	30.0	511.	32.4
455.	30.0	512.	34.4
456.	30.0	5\3.	36.2
-------
TIME
SPEEC
time
SPEED
TIPE
SPEED
TIPE
SPEEO
T I ME
tSEC J

CStC)
C«PH»
(SFC»
(?«r>H j
(SEC)
(CPU)
(SEC I
51*.
37.8
574.
24.5
634.
15.7
694.
5.2
754.
515.
39.2
575.
26.6
635.
19.6
695.
3.0
755.
516.
*P.5
576.
28.7
636.
23.4
696.
I.-3
756.
517.
41.6
577.
3C.8
637.
26.9
697.
0.3
757.
518.
42.6
5/8.
32.9
638.
3C.3
698.
0.0
758.
519.
42.4
579.
34.9
639.
33.4
699.
0.1<
759.
520.
44.0
580.
36.9
640.
36.4
700.
0.0
760.
521.
44.5
581.
38.8
641.
39.2
701.
0.0
761.
522.
45.0
5e2.
4C.5
642.
41.8
702.
	0.0_
762.
523.
45.0
5U3.
42. C
643.
44.2
703.
0.0
763.
524.
45.0
584.
43,4
644.
46.4
704.
0.0
764.
525.
45 .C
585.
44.5
645.
48.5
705.
0.0
765.
526.
45.0"
586.
45iC
646.
5C.3
706.
0.0
766.
527.
45.0
587.
45.0
647.
52.C
707.
0.0
767.
528,
45.0
588.
45.C
64 8.
53.5
708.
o.o
	76 B.
529.
45.C
5fc9 .
45.0
649.
54.9
709.
"" 1.2
769.
530.
45.0
590.
45. e
650.
56.1
710.
3.5
770.
531 .
45.0
I.
45.0
651.
57.1
711.
6.4
771.
532.
45.0
59?.
45.C
652.
ie.c
712.
9.6
772.
533.
45.r.
593.
45.r
653.
58.7
713.
12.8
773.
534.
45.0
594.
45.C
654.
59.3
714.
15.9
774.
535.
45.0
595.
45.0
655.
60.0
715.
18.8
775.
536.
45.0
596.
45.C
656.
60.0
7\ 6 .
21.3
776.
537.
45.0
597.
45.0
_ 657.
6C.0
717.
23.5
777.
53B.
44.5
5S 8 •
45.C
658.
6C.C
71 8.
25.2
778.
539.
43.9
599.
45.0
659.
60.0
719.
26.7
779.
540.
43.0
6C0.
45.C
66P.
fcC.C
720.
27.8
780.
541.
41.8
6CI.
45. C
661.
6C.0
721.
28.7
781.
542.
4C.2
<-C2.
44.5
662.
60.0
722.
29.6
782.
543.
38.2
6C3.
4 3.7
„ 663.
60.0
723.
30.0
783.
544.
35.8
6C4 .
42.7
664.
ec.c
724.
30.0
784.
545.
33.1
6C5.
41.4
665.
60. C
725.
30.0
785.
546.
3C.1
6C6.
39.8
	666.
6C. 0
726.
30.P
7Hfi.
547."
26.9
6C7 .
37.8
66 7.
er.c
727.
30.0
787.
548.
23.7
fcC°..
35. 3
668.
6C.C
728 .
30.0
788.
	549.
2C.7
6C9 •
32.5
669.
60. 0
729.
30.0
739.
550 .
18.1
610.
29.2
670.
60.0
730.
30.0
790.
551.
16.1
611.
25.6
671.
59.6
731.
30.0
791.
	5521_
15.1
. .612.
21.8
672.
_59 • 0
732.
30.0
	792.
553.
15.n
613.
17. 8
673.
58.3
733.
30.0
793.
554.
15.0
614.
12.P
674.
57.4
734.
30.0
794.
_555.
15.0
615.
9.9
67.5,
56.5
735.
_ 30.0
	 795.
556.
15.0
616.
6.4
676.
55.3
736.
30.0
796.
557.
15.P
617.
3.5
677.
54.C
737.
3C.0
797.
558.
15.0

1.3
678.
52.4
738.
30.0
	79P .
559.
15.C
619.
C. 1
679.
5C.6
739.
30.5
799.
560.
15.C
620.
O.C
ten.
48.5
740.
31.4
800.
	5fcl.
15,0
621,
0-C	
	6P 1 .
. 46. 2
741.
32.4 .
	801 .
562.
15.0
622.
0.0
6P2.
43.6
742.
33.6
802.
563.
15.0
673.
O.C
683.
4C.8
74 3.
34.8
803.
_ 564.
15.P
624.
O.C
	6?4.
37.8
744.
36.0
804.
565.
15.0
625.
O.C
685.
34.6
745.
37.4
805.
566.
15.0
626.
c.c
686.
31.3
746.
38.7
806.
_ 567 .
15. C
62 Ij,	O.C
_ 687.
27.9
74 7.
40.2
8C7.
568.
15.0
628.
c.c
688.
24.3
748.
41.6
808.
569.
15.9
629.
O.C
689.
2C.8
749.
43.1
8P9.
570.
17f3
630.
0.0
690*.
_17.3
750.
44.6
810.
571 .
18.9
631.
2.6
691.
14.C
751.
46.0
811.
572.
2C.7
632.
7. 1
692.
IC.7
752.
47.5
ei2.
573.
22*6_
	433.
..J 1,5
. 693.
7,8.
753,
48.9
_813.
speec
('•"PHI
TIKE SPEEO
(SEC) (MPH)
TIME SPEED
(SEC i (MPH)
50. 4 814 • 29.1 eh',. 60.8
51.7 P15. 27.7 875. 6C.C
S3. 0	816. 25.4 876.	60.0
>>4.3 817." 22.0 877. 60.0
55.5	818. 17.6 878. 60.0
56-6	BIT.	12. 5_	 879. 60.0
57.6	820. 7.A 83 C. 60.0
58.5	. 821. 2.0 881. 60.0
•>". 3	822.	0.3	882. 60.0
60.0 823. 0.0 BR3. 6C.0
60.0 824. 0.0 884. 60.0
<.0.0	825. 0.0 _ 885. 60.0
6C.0 876. C.O ~ r.R6. 60.0
60.0 827. 0.0 837. 59.7
60.0	 828,	0.0 888. 59.2
60.0 829. 0.0 889. 58.7
60.0 830. 0.0 800. 58.1
60.0	831. 	0.0	fl1)!,.. .57.4
*0.0 8 3 2. 0.0 892. 56.7~
6P.0 833. 0.0 893. 55.8
60.n _ 834.	0.5	894. 54.8
"hO.O "" 835. * 4/9 895. 53.7
60.0 836. 6.9 816. 52.4
bP.O	837. 9.8	897.	50.9
c.0.0 838. 12.6 89ft. 49.3
60.0	839. 15.3 899. 47.6
59.6		 840. _ 17 • 9	900. 45.7
59.1	84 1." 20.4 901. 43.6
5*.4 842. 22.8 902. 41.4
57.6 843. _ 25.?	 903. _39.0
56.5 044. 27.4 904. 36.6
55.1 845. 29.6 905. 34.0
53.	4	846. 31.7	906. _31.3
51.5	847. £3.7 907. 2R.6
49.3 fa, 8. i 5.7 90S. 25.8
46-8 	849. 37.5	 909. 22.9
44.3 P50• 39.3 910. 20.1
41.6	' 851. 41.0 9H. 17.3
3«.9 _ P52,_ .42• 6_	91?,	14.6
36.3 853. 44.2 913. 12.0
34.0	854. 45.6 914. 9*5
32.1		855._ 47.0	915.	7..2
30.7	856. 48.4 916. 5.2
30.C 857. 49.6 917. 3.4
30.0	856 50. 8__	9lg.	1.9.
30.0 859. 51.9 919. 0.8
30.0 860. 52.9 920. 0,2
30.0	861._ 53.8	921. ._.._0,0
30.0 P62. 54.7 922. 0.0
3O.0 863. 55.5 923. 0.0
10.0 _ 86*. . 56.3	924,_„.. O.0_.
30.0 865. 56.9 925. 0.0
30.0 866* 57.5 »26. 0.0
30.0	 867._ 58,1	927. C.O
30.0 868. 58.5 928. C.O
30.0 e69. 58.9 929. 0.0
30.0 _ _870.	59 .3	930-_ 0.0
30.0 87l. 60.0 931- P.C
30.0 872. 60.0 932. 0.8
30.0	 873._ 60,0.	933,	2.7
"934.	4.9 994. 60.0
935.	7.5 995. 60.0
936.	10.1 	996,	60.0
937".	12.8 997. 60.0
"38.	15.4 998. 60.6
939.	17.8	 999. 60.0
940.	" 2C.1 1000." 60.0
941.	22.1 1001. 59.5
342. 23.8	1002. _58.7
943.	25.2" * 1003. 57.6
944.	26.5 1004. 56.1
945.	27.4 1005. 54.0
946.	28.3 1006. 51.5
947.	29.0 1007. 48.5
948.	29.6_ 1008. 45.1
949.	30.0 1009. 41.6
950.	30.C 1010. 38.0
951.	30.0_ 1011.	 34.7
952.	30.0 1012. 32.i
953.	30.0 1013. 30.4
954.	30.P 1014. 30.0
955.	30.6 1015. 30.0
956.	30.0 1016. 30.0
957.	30.0 _ 1017.	30.0
958.	30.0 1018. 30.0
959.	30.0 1019. 30.0
960._	30.0 1020. 30.0
9fcl.	30.P 1C21. 30.0
962.	30.0 1022. 30.0
963.	_30.0 1023. 30.0
964.	30.0 1024. 30.0
965.	30.7 1025. 30.0
966	.	31.8 1026._ 30,0
967.	33.1 1027. 3C.0
968.	34.5 1028. 30.0
•'fig. 36ifi_102'>.	3C. 9
970.	37.6 1030. 29.4
971.	39.3 1031. 28.5
972.	41.0	1032. _ 27,2
973.
42.8
1033.
25.4
9 74.
44.6
1034.
22.9
... 975.__
.46.3	
.1035.
.19.9
976.
43.1
1036.
16.4
9 77.
49.8
1037.
12.5
978.
51.5
103P.
8.6
979.
53.0
1039.
5.0
980.
54.6
1040.
2.1
931.
56.0	
.1041.
_..0,.3_
982.
57.2
1042.
0.0
083.
58.4
1 P43 .
0.0
„ °84.
59.3
1044._
	0.0
985.
60.0
1045.
0.0
986.
60.0
1 046 .
0.0
_ 987.
60.0
1047.
0.0
988.
60.0
1048.
0.0
989.
60.0
1049.
0.0
. 990.
60.0
IO5O.
	.0-0 _
991.
60. P
1051.
0.0
992.
60.0
1052.
0.0
993.
60.0
.1.053, _
	0,0.


1054.
0.0
-------
FEDERAL TEST PROCEDURE
SPtfclMMHH
0.0
0.0
t TUtE(SEC)
5?
56
SPfcEU(MPH)
	I5.B
17.7
6
7
3
9
10
11
12
13
~T4
15
15
17
TB
19
~7cr
21
22'
23
~24
25
uvtr
o.o
o.o
o.o
o.o
0.0
—5T-
58
59
60
—6T-
62
i$.a
21.6
"23.7"
24.2
T^.fc
24.9
25.0	
24.6
24.5	
24.7
24.8
24.7
T1ME(SEC»
110	~
111
	TIT
SPtEDIMPHI
31 .i
jl • b
113
114
115
116
117
-3772~
32.4
32.2
31.7
28.6
25.3
22.0
18.7
15.4
12.1
8.8
5.5
- 2.2'
0.0
TlHt(SfcC)
lf.5
166
	TE7
SPtED(MPH»
	6.5
9.9
168
169
170
TT1
172
" US
174
—13.2
16.5
T978
22. i
~2*T3
25.8
2b.4
25.7
TIME(St C1
22 0
221
7TI	
SPIED(MPHI
50.0
50.6
22 3
224
22 5
226
227
22 B
22 9
230
231
732
233
2 34
235
""""5T.Tr"
51.5
52 .2
53.2
54.1
54 .6
54.9
55.0
54.9
54.6
"TT.6
54.8
55.1
55.5
"55.7
56.1
"55T3"
56 . 6
56.?
56.7
5r."5
56.5
56 .5"
56.5
0.0"
0.0
0.0
0.0
0.0
0.0
TT
64
~E5~
66
ET
68
0.0"
0.0
0.0
0.0
0.0
0.0
~ET~
70
TT
72
TT
74
~2E~
27
29
31
0.0
3.0
5.9
8.6
n. 5
14.3
"75"
76
-rr
78
79"
60
~32~
33
16.9
17.3
TB n "
20.7
ZI.7
22.4
~BT~
82
83
84
T5"
86
34
35
37
2Z.5
22.1
Zl.5^
20.9
"87"
88
~3B~
39
41
~52
43
20.4
19.8
By
90
~9T~
92
17.0
14.9
14.9
15.2
15.5
16.0
~7T~
94
96
~
-------
TIMUStU
SPttDI MPH)
TIMEISECI
SPEED(MPH)
TIHttStCI
SPfctOIHPH)
TIMEISEC)
SPEEDtHPHJ
TIME1SEC 1
SPEEOIMPH)
	28.0
25.5
2 75
276
53.0 ""
53. 5
330
331
8.0
4.7
385
386
33.5
31.4
440
441
0.0
0.0
*95
496
ill
278
54.0
54. V
332
333
1.4
0.0
3« T
388
<:v.u
25.7
4*2
443
¦ trm	
0.0
~ 49T
498
22.5
19.8
279
280
^>5.4
5b. b
334
335
0 .0
0.0
389
390
a .u
20.3
444
445
0.0
0.0
499
500
	—
13.2
261
282
56.0
56.0
336
337
0 .0
0.0
391
392
1 f .5
14.5
44b
4*7
O.C
0.0
iCl
502
10.3
7.2
2U3
284
5b • y
5b. c
33H
339
0.0
0.0
ifi
394
14 . u
8.7
448
449
3.3
6.6
50	3	
504
4.0
1.0
28b
28b
54.5
53.6
340
341
0.0
0.0
396
5.*
2.1
450
451
9.9
13.2


ZB7
288
52.5
51.5
342
343
0.0
0.0
3V 1
398
0.0
0.0
45Z
453
16 ."5	
14.8


2BV
290
SI . 5
51.5
345
0 .0
0.0
400
0.0
*3*
455
23.1
26.4


2VI
292
51.1
50.1
3*6
347
0.0
1.0
402
0.0
*96
457
2/.8
29. 1


2V3
294
50.0
50.1
3*0
349
*• 3
7.6
404
2.6
5.9
458
459
31.15	'
33.0


2V5
29b
50 .U
49.6
350
351.
10. V
14.2
406
9.4
12.5
460
461
33.6
34.8


297
298
49.5
49.5
352
353
1 /. s
20.0
408
19.1
*0*
463
35.X
35.6


299
300
4 V. 5
49. 1
354
355
a.»
23.7
410
25.0
HI
465
36 ;t~
36.6


iiUl
302
48.6
48.1
i!>6
357
26.6
412
27.5
*6b
*67
36 « i
36.2


	303 '
304
47 . 2
46.1
358
359
. 1
30.0
414
30.0
469
36 .u
35.7


305
306
45.0
43. a
361)
361
30.8
31.6
416
30.0
471
36 *b
36.0


	307	
308
4 2.6
41.5
362
363
32.1
32.£
f
418
29.3
*14
473
35 .U
35.5


309
310
40.3
38.5
364
365
33 .6
34.5
420
26 *0
475
35.*
35.2


311
312
3/.U
35. i
366
367
3* .e
34.9
*41
422
21.7
477
35.4
35.2


313
31*
33.8
32.5
368
369
34. 8
34.5
1 42 J
r424
13.1
479
39 .4
35.2


	 3I5
316
31.5
30.6
370
371
34.7
35.5
425
426
8.5
481
35.^
35.0


317
318
' 30.5
30.0
372
373
36.0
36.0
42 1
428
3.4:
1.9
48Z
483
35.1
35.2


319
320
29.0
27.5
374
375
"36 .U
36.0
429
430
u .u
0.0
484
485
35.5
35.2


	3;rl	
322
— 24.a
21.5
376
377
36. (J
36.0
431
432
0.0
0.0
48b
487
35.0
35.0


323
32*
20.1
19.1
J7B
37«J
36 n
3b .4
434
0.0
489
m U
34.8


32 5
326
la . a
17.0
IS.*
12.5
3UU
381
3sr
i«i
36.5
3o . 4
	 36.0 ' '
35.1
U5
4j6
"TJT
438
tJ . u
0.0
491-
*92
493
3*.b
34.5
	33.S	
32.0


32T
32&
0.0


•S2*	
10.ft
?w
3*.l	
-	-wif -
	0.0
49*
».l


-------
APPENDIX III
THE MATHEMATICAL MDDEL
The mathematical model presents a method to calculate the amount of
a particular pollutant given off by a vehicle or vehicle group over any specified
driving sequence given the modal emissions data on the individual vehicles.
Notation
e(t) = amount of a pollutant given off by a vehicle from time = 0
to time = t.
e(t) =	— = instantaneous emission rate function.
dt
eA(t) « accel/decel instantaneous emission rate function.
eg(t) = steady state instantaneous emission rate function.
E^ = amount of a pollutant given off by a vehicle in the j'th mode.
Tj = duration of jth mode, (in seconds)
v(t) = speed at time = t for any driving sequence
(t) a ——= acceleration at time = t in the j'th mode
dv. (t)
a(t) ¦ ^rr^- = acceleration at time = t in any driving sequence.
V
) fa j£ - average value of the emission rate function for the
-------
(A) Functional Form of the Steady State and Accel/Decel Emission Rate Functions
Based on Figures 1, 2, 3 (Appendix VI) where average steady state
emission rates over the 1020 vehicles in the data base are plotted versus
speed, the assumption is made that the steady state emission rate function
eg can be represented as a quadratic function of speed:
CD	&s<-V) = 5, + S*. V/ +- S3 V2-
where , S2, and aTe constants.
The functional form of an emission rate function that will be appli-
cable during periods of acceleration/deceleration is obtained by assuming the
constant coefficients (S^) become functions of the acceleration and that this
functional dependence can be sufficiently represented by quadratic functions of
acce 1 e rat ion. The re fo re:
S, = S, ca.) - $,t +-ql3. a- -h	OL*-
(2)
Sj. = Sa.	=	+~	4LL
^ 3 ~ S3 Ca) = c^3i	e^3JL ^ ^	qJL.
where the q's are constants
and
(3)	<£^(v,a.) - Si Cct) +- S^ca.) ir -h S3 taj Ul
C4) eA	4 ( + q.n.c*,+ a?) c t- Qt* Cu -h a2-) - U
+ C$3i t- %-gi, a, f ^33 oJ-) ¦ tr*-
Upon multiplying and redefining the constant coefficients we have the following
expression for the accel/decel emission rate function
(5) <2/j (\Jta) = t>, f- 1/ + Ai	U$	+ bcf
-------
^ The Instantaneous Emission Rate Function
Although the accel/decel emission rate function is functionally
identical to the steady-state rate function when the acceleration is zero,
both functions will be retained to allow independent adjustment of the
coefficients (b. and S^) specifying the accel/decel and steady state emission
rate functions.
The two functions can be joined by an acceleration dependent
weighting function h(a) to form the instantaneous emission rate function:
<6) e
where h(a) =
= /WaJ e%L\f) 4- C f- hca.)) dfitV,£L)
— _L- (X, -h I

or
JL- CL + I S
& >
O j
k IM
^2. <& < 0
a, >/ c*^
^ 6 (u) )eA
-------
In practice 1 and °^-2 have been arbitrarily set to -1.2 sec and
1.0 mph/sec respectively.
(C) Specification of The Emission Rate Function
Given the functional form of the emission rate function, the
coefficients (Ik, S.,) must be determined in order to specify the emission
rate function which characterizes a vehicle's emission response.
(1) Determination of The Accel/Pecel Emission Rate Function
Coefficients fb^)
The only modal emission observations available are the total
amount of each pollutant given off in each mode of the SDS; there are no
measures of the instantaneous emission rate given for accel/decel modes.
Therefore, the following procedure is used to determine the coefficients
(b^) that specify the instaneous emission rate function for a given vehicle
and pollutant:
Ut:	" , h * A-
X.
/7 - uV ,	t 4~q -
where the	"A. , l - tjf basis functions of the accel/decel emission
rate function.
Then using equation (5) we have:
_! , ,
(6)	eA IV/D - 2-
i =/
For an accel/decel mode the weighting function h(a) ~ 0, then:
(7)	= e (v,a)
Now consider the average emission rate over the k' th mode:
5
(8)
ik 0
-------
by equation (7)
(9)  =  = ±- f Tit = Ek /Tk
and	Tt	J*
(11)	-f*- - 7- [&A(v,a)Jt = -L j	^
7* Tk	U 0	'
Since the b. are constants:
(12)
Now, '/Tie I 1~uav~	is just the average value of the i'th basis function
Gl /7i ' '/Tk X ( I
l/T f k Jt
' /T" I
over the K'th mode, which can be easly evaluated knowing the speed as function
of time for the k'th mode	, and acceleration as a function of time, CLf, Lb\
T*
Let' ^Lk ' I di
IfL )
O
(see Appendix I for values of	)
Thus,	^
(13)	M~~ ~ / ^ ^Lk-
i-t
and since E^/T^ is known for all modes and the f^ can be evaluated then the
b^ can be determined using standard least squares regression techniques.
-------
The result of the least squares regression method can best be
given by defining the following matrices and elements:
Ek
Y, Y(k) = •?— (response] k =1, 32
Tfc
(32 accel/decel modes)
X, X(i,k) = fik	i - 1,9
B, B(i) = bi
and using the convention
y' = Transpose of Y
Y= Inverse of Y
we have by equation (13):
(14)	Y = BX
The method of least squares then gives:
(15)	B = (X ' X)_1 X 'Y
Let: A^ s (X ' X) 1 X7	(A^ basis function factor array)
Then, (16) B = A^Y
and the i'th coefficient is given by (b^):
21
(16)	b. = y Aa (i,k) Y(k)	i = 1, 9
i. - i
In summary, the average emission rate, which can be evaluated by
using experimental observables, is used to determine the coefficients that specify
the instantaneous accel/decel emission rate function.
(2) Determination of the Steady State Emission Rate Function
Coefficients (S^)
In the case of steady state emissions the average emission rate is
equal to the instantaneous emission rate because the speed is constant in time.
-------
Let: U -- 1.0 , Lb -- Vs. t	= Vt
then, by equation (1):
3_
(17)	e£	^ Ut.
L - I
Consider the K'th steady state mode ^	^ 1 C
(18)	\t
<£CVJ,aOrt- T^= Effe _ I ( ( S, Ufe ) Jt
TV rfe j
since both S. and f.. are now constants
ilk	_
09) f-j =	^ ^/"ft "" ^ Vl
l	O	t
The coefficients can now be determined using standard least squares repression
techniques,
Consider the situation in which the have been determined by the
above procedure and the specified emission rate function produces negative
emission rates as shown below.
(3"->observed pts
e6 +
Steady state
Emission rate
^3 [	
0

Negative Emission
Rates
VL
'.it.
Speed (v)
<£>
V.
	 Regression Curve
t V 4 V 4- S/VJ')
-------
When this happens (each steady state emission rate function is tested for
this possibility) a new set of coefficients (S^) is calculated in a manner
that will guarantee the steady state emission rate function will be positive
for all values of speed between 0 and 60 mph. The procedure to determine
the new set of coefficients is described below.
The minimum of the regression curve representing the steady state
emission rate function is forced throught the point given by the average of
the 2 lowest emission rates and the average of their speeds. In our
example above this point (e ,v ) is given by
e= et4 Jz j V =	V 3 /
Requiring the minimum of the regression curve (steady state emission rate
_ 	 »
function) to go through (e,v) specifys two of the three coefficients. Since e
is a quadratic function of speed, the coordinates of the minimum ate given bv:
i.
^ SvSjL - SC	_
Therefore
y J	-	-	^ ^3
e = 4	^
Solving for S and S2 in terms of Sj, v and e we obtain:
^ ^
•3 , - + "v **
substituting these values into equation (1)
(20)	^ (vj -- e -h U2- S3 + C~ Z.V ) V + S3 V1
Regrouping:
(21)		£ v l/ +
The only coefficient left to determine is now S^ which is obtained by
the standard least squares method.
-------
This method will remove any negative emission rates over the 0 to 60
mph speed interval and still retain the general trend of the observed data.
(D) Characterization of Individual Vehicle and Vehicle Group Emission Responses
Assuming input modal data is available, each vehicle's emission response
can be characterized by the 36 coefficients that specify the thTee emission rate
functions for HC, CO and NO^ (12 coefficients per function).
A vehicle group can be similarly characterized by determining the
coefficients for the average vehicle in the group.
Let: b... = k'th coefficient of the emission rate function for the j th
vehicle in group sample and i th kind of pollutant
Ng = number of vehicle in sample representing group
^ik = ^'th coefficient of the emission rate function for the
average vehicle for i th kind of pollutant
The coefficients describing the average vehicle's emission rate function is thus
given by:	^
(22)	^ = -L L ^
(e) Determination of Individual Vehicle and Vehicle Group Emissions Over a
Specified Driving Sequence
The amount of a pollutant e(T) given off by a vehicle i\ undergoing a
driving sequence from time = 0 to time = T is obtained by integrating the instan-
taneous emission rate function describing this vehicle's response with respect to
pollutant under consideration over the sequence:
(23)
T
e U) -- j e	] (ku) 
-------
_N
C24) eCT) ^	4 ) ¦*- L i-Wuu)) U. *-W Vi. + ^aL
*. = 1
+	\/c + 1*4 Q.I ^t]\J- t- t*
-------
APPENDIX IV
Program Listings
Main Program I
Main Program II
Subroutine SETUP
Subroutine EDOT
Subroutine PAD
Subroutine ESUM
Subroutine EDGRP
Subroutine INVERS
-------
LEVEL 21.6 ( MAY 12 )
OS/36 0 FORTRAN H
COMPILER OPTIONS - NAME = IN,~PT = 02»L1N£CNT=60,SIZE = OOOOK,
					SQUR.CL.fcBCDIC.NQLl ST « NQDHC K « L n A 0 .M AP »NHi niT.irt.XRPf
t	»~»*~~«*»**»**»**** MAIN PROGRAM I »********•*»»*~*•******#****
C MAIM PROGRAM I QE-IEBHIHES JIHE-AMDUNT QF EMISSIONS GIVEN OFF BY
C	INDIVIDUAL VEHICLES OVER A DRIVING SEOUENCE SPfCJFJED BY ARR'WT.
C VTM(1)«>VEL0C1TY VS. TIKfcllN ONE SECOND INTERVALS! OF THE SURVEIL-
-Z	-LANCE DRIVING SEQUENCE .tf THI I ) = VF I nr. T TV 1 HPH) AT TIME II-l)Cfr 	
C	(REAL )
C	 	...	_ 					
C WT < I )=>VELOCITY VS. TI ME 1 IN ONE SECOND INTERVALS) OF ANY DRIVING
C	SEQUENCE _QV_ER HHI CH_ FWT^SIDMS ABF Tfl RF C ALCl II ATF D .VVT < 11 =VFI f)C -
c -rrv at rr«E  AMOUNT QF I'TH EMH1TTANT GIVEN OFF IN i'TH KQOfc .
C DSlI)=OISTANCE(MILES!TRAVELED IN I

l7<4.26l .UTAH M 1 .261 .QSrSTT DIMENSION YTM(1055), VYT 12000 I . C 13 , 37 I ,C < 3-> REAL»8 A*t9.321. fcStl. SI.»tai3.12l QATA OS /.06O2,.07*1,.02O1t.OTD5t.1360,.126£,.2l63,.L116 C..2 C<»3. ¦ 3U7.. 3136 ¦- 197^.-3313.. .0^79.. CI 71. ^17i«.. 1392.. IS2B C*. 13Ch .2654, .263NXl)-GT.9<*.Q)G0TQ3111 ISM 0015 3000 CONTINUE -ISN 0Q16 JU1 tQNIiMUi C r- REAP IN nR IVING SEQUENCE OVER WHICH EHHISSIQNS ARE I&J1E-. CALCULATED C r TN THIS FXAMPLF tfVTg> FIRST 505 SFC. OF FTP C ISN 0Q17 _ DO 15QQ 1=1.1QQ ISN OOIS NXl = t U-l )*16 1*1 XStf-0012 .NX2_=NX1+U? ISN 0020 READ t 5 »100)(VVT{K)»K=NX1»NX2) TCN 0021 IF I VVTlNXl ).r.T.99.0>r.nTniS65 ISM 0023 1500 CONTiNUe ISN . QQ2

-------
ISN
0027
500
_ c
FORMAT(lHlJ



c
c
PUT MODAL EMISSIONS DATA INTO ARRAY AMTC



c
IN THIS EXAMPLE THE MODAL EMSSIONS DATA IS READ OFF
A DISK FILE
ISN
ISN
0023
0 029
	L	
NCART=0
READIOO^SJITAB

ISN 0030
ISN QQ3L

00 2000 I Y=5 7t71
IREC=1Y—56

ISN 0032
_ ISM 0013

JSTART=ITABlIREC 111

ISN
ISN
0034
0035

DO 2001 J=JSTART,JEND
READC9'Jl( ( IDAT t L • K t . L = 1. 41 . 1 R DAT 1L. K J . L= 1 . 1? 1 1 .K= 1
.261
ISN
ISN
0036
0017

DO 2002 K=l,2f>
IF I 1DAT< 1 . K1 .FQ.-?1GC]T(J2QQ1

ISN 0039
isn OG-m

NCART=NCART*1
DO 1C00 IR=1.37

ISN
ISN
0041
0042

D0«1 .0
IF I IP—Lb.321DD*DS tIK)

ISN 0044
15N 0045

ca ioai ic=if3
IW=I IIF4-I l*3>+10+IC

ISN
ISN
0046
0047
1001
AMTCIICtIB J = 3OAT CIH,K1*00
CONTINUE

ISN
0C43
1000
c
CONTINUE



c
c
DETERMINE INDIVIDUAL VEHICLE EMISSION RATE FUNCTION
COEFFICIENTS
ISN
0049
c
CALL EDOT



c
c.
TNTEflRATP THT VFHTCIFS FMISSION RATF FUNCTION DVFR
THE


c
c
DRIVING SEQUENCE.

ISN
0050
c
CALL ESUK(VVT.506.&AD.CI1).CC2),C<3I.DIST>

C
c


c
c
WRITE OUT EMISSION RESULTS....

ISN
ISN.
0051
0052
501
WRITE(6,5011NCART, (C(LI,L = 1(3>
FORMAT!1H .»CAR= • . I10.2X.•HC=•.F10.2.2X . 'C0 = «.F10.2.
2X.1 NOX= *.F10.
ISN.
0053
1
2002
-2, )
CONTINUE

ISN
ISN
0054
0055
2001
2000
CONTINUE
CONTINUE

ISN
m
0054
0057
1234
STOP
END

-------
LEVEL 2 l.S> C «4r U I
OS/36 0 FO® T° 4N H
C1KP] L 1-R
C
__.IL
JP1IUNS - N4MF= MAIN,IJPT=02,LINECN7=.SD*SI7E = 00CQK,
	i.JUjl££XS1C iNQL.1 ST.tNGD^CK .IQA&iBAgiaCLU.LtltuMEf..
**********************
wiiit PMUfiAi ri
************ **** ********
C
c
c
L
C
_L_
MAIN PROGRAM II DETERMINES THE AMOUNT OP EMISSIONS GIVEN OFF BY
.A.. GJiQULEL-Qf -YliilLLiS SPiClFILD a_Y_TfcL£ FILTER OYER. 7HE .DRIVING SLt>.
SPECIFIED BY !kR«A> ' VVT".
VTM(I )=>VLLnCITV VS. TIHEIIN ONE SECOND INTERVALS) OF THE SURVFIL-
-±M4££_ DRIVING. iLiiUENCEjVIMA 11 = ¥£LD£ 1IY IHPJ41 AT 11M£. .11-11SEC
(R E A L* 4 )
C
C
c
c
_£L_
WT I I » = >VFLOtTTY VS. TIMFtlN ONE SFCOND INTERVALS] OF ANY DRIVING
5taU£ML£ OVER WHICH nrtMISSIQKS A££ TO BE CALCULATED.VVTIJ)=YEtaC-
-IT Y AT TIHt (1-1) SEC.  AMOUNT UF I 'TH EHMITTANT GIVEN OFF IN J'TH MODE.
C	DS< Il=BISTANCt(MILESJTRAVELED IN I 1 TH MODE .NOTE , STEADY STATE MODES
C		ARt 60 iL: Ih DURATION.		 		 ..... 	
C
c
*4* + +*: ********** ******* **************** ********************* ******
ISM 0002
ISN QQQ3
31 MENS ION I TAB(20,2)iIDAT<*•,26>,RDAT< 12 1 f26 I ,DS(37)
DIMENSION VTMtl0 5 5).VVT{20001.AHTC(3.37).C(3)	
ISN 000-k
-iSRjaaa.
ISN JJQD
c
c
c
ISM MAI
ISN OOOS
3SN 0QQ9
REAL»6 AM9,32J,AS(3,51,aU>(3,L2)
_QAT A _HS 	/.0fet2..07ti. .CZC1. .070?. .13feu. - Uefi . .ilfc31. 1716
C ,.2 043, . 336 7,. 31 3to,.1973, .3313, .2*94-, .0579,.0173,. 1759,.13921528
t. ..13Q0137,.3134. .2362. .Q444.
C t«1313 «.0 592 f.0000».2 500,.5000,.75C0,1.000/
DEFINE J1LL^99<75.3256.U.NI)		—	
K6AD IN. SUR.VL ILLANCL DRIVING SECLlNCt
. rjO 300C 1=1 jISO	 	 — 	
NX I = I (I-1)*16>»1
NX2=N)U+15	
ISN 0010	READ<5,100)(VTH(K),K=NXl,NX2t
ISN 0011	100 FORMATf 16FS..D I	
JSN 0012	IF< VTMCgXl) .GT. 9<3 .0JGDTD3111
ISN 0014	3'iQQ CQNTINi't	
ISN 0015	3111 CONTINUE
C
C
c
BEAD IN DRIVING SEUUFNCE OVER WHICH FHMISSIONS ARE TH BE CALCULATED
IN "THIS" E X AMPLE VVT-> FIRST 505 S EC. OF FTP
ISN 0016
L5N 0^17
to 1500 1=1,100
U-ll»16)*l
ISN 00 13	NX2 =NX 1 + 15
ISN 0013	REA D(5 .100 i t YVLUl »*=NXLiN Xi J
ISN 0020	1FIVVT (NX1>.GT.9<5.0IGDTQ1555
ISN 002Z 	15LC iOMIN''- .	..... 	
ISN 0D23	1555 LUNTINJE
C	SET UP iiASIS FUNCTION FACTUR ARRAYS A A t A S .
C

-------
ISN
00^4
£
CALL SFTUPtVTM.AA.AS)


C
£
re ad in individual vehicle modal emission data
FILTER VEHICILES FOR GROUP RFPRfSfNTATIVE*


c
c
IN THIS FXAMPLF THF DATA 1$ BEING R FAD TFF A DISC FILF AMR TWF


c
c
INI VISUAL VEHICLE MODAL EMISSION DATA IS PUT INTO ARRAY AMTC
IN THIS EXfMPLE THE GROUP IS. DENVER. PRE F MIS SI ON mNTRtiL.
I$N
0025
c
NVIG*0
ISN
ISN
0026
0027

REA019P«75)ITAB
DO 2000 I Y»57>71
ISN
ISN
0026
0029

IREC«=XY-56
JSTART = ITAR1fREC.l )
ISN
ISN
0030
0031

JEND*ITAB(IREC,2i
DO 2001 J.JSTART.JFNC
ISN
ISN
0032
0333

READ!99 * J1 ((IDAT
-------
C	INTFGRATF this GROUP'S EMISSinN 'ATE FUNCTION CIVE ^ THE: DRIVING
	£				 - .....
ISN 0U61		CALL tSL',<(yVT.5J6,bAL'.L(l] ,w(£J ,C (3>.01S JJ
C
	___.£,	DETERMINE TDAIAL LMlSSIQfi-rXMJJLTIPLY. E*Ch .EMISSION AMUUN7 BY NO,
C	VEHICLFS IN GROUP
VN=NVIG
	 DQ 5.Q3U. 1 = 1.1.3 		.. ._	 . .... 		
5030 CI I)=C{I>*VN
u	Xfi.iI.E_ QlLl EMISSION .RESULTS. .,_. 	...._
WRITF<6,500) »C(L 1 ,1=1,It
500 FORMAT I 1H . ' HC= ' . F 1C . 2 . 2X . ' Ct1= ' . F 10. 2 »2 X . ' NOX = '.x£l.Q^2 I	.... 	
STOP
	LNC ..... 				 -	 - 	 .. 	
ISN 00 62
ISN QQ63
ISN 0064
ISN Q0&5
ISN aflt>6.
ISN 0J67
ISN Q.QM
-------
LEVEL 21.6 ( HAY 7? )
Oi)/3fcO FORTRAN H

C3MPIL
cR OPTIONS - NAM£= MAIN,OPT=02,LINECNT=60,SIZE=OOOOK,
SOURCF. .EBCDIC.NOLIST. NO DECK.LOAD,MAP.NOEDIT«ID«XR£F
ISN
0u02
c
SUb10UTIN E SfcTUP < VTM,AA,AS I
********************************************************** ********


c
	£-
c
c
ACCEL/DECEL AND STEADY STATE EMISSION RATE FUNCTION DETERMINATIONS
•ARRAYS 'AA' AND 'AS» RESPECTIVELY. GIVEN THE VELOCITY VS. TIME


c
.£
HISTORY OF THE SURVEILLANCE DRIVING SEQUENCEVELOCITY VS. TIME(IN ONE SEC. INTERVALS) OF THE SURVEILL-
ANCE DRIVING SEQUENCE. VTM(I)"VELOCITY AT TIME I.(REAL*4)


c
c
MVT <1)=TIME I•TH ACCEL/DECEL MODE STARTS IN THE SURVEILLANCE


c
...	 c
DRIVING SEQUENCE.(REAL*4)


c
c
TM(I)=TIME(SEC 1 IN I*TH ACCEL/DECEL MODE.(REAL*4)


c
c
AA=> BASIS FUNCTION FACTOR ARRAY FOR ACCEL/DECEL.(REAL»81


c
c
AS=>BASIS FUNCTION FACTOR ARRAY FOR STEADY STATE.(REAL*8 >
ISN
0003
c
DIMENSION VTMf1055 1 .MVT(321.TM(321
ISN
ISN
0004
0005

REALMS XI32,9),SV ,SA,TMD,CJ 9,9),AA<9,32),SUM,AS<3,5»
DATA MVT/11..38..64..87.,113.,141..167..199..227..257..302..343..
C374.,421.,*59.,483.,501.,538.,569.,60 2.,631.,671.,709.,739.,780.,
ISN
0006

DATA TM/12.,16.,8.,11.,13.,12.,17.,12.,14.,30.,26.,21.,32.,23.,9.,
C8..22..16..18..19.,25..28..15.,25..18.,10.,38..35..1«..21..14..13.
ISN
0007

C/
N0BSA=32
ISN
ISN
0008
0009

N0BSS=5
NBFA=9
ISN
0010
c
N8FS=3


c
c
****** CALCULATE AA *************


c
c
CALCULATE BASIS FUNCTION ARRAY X
ISN
ISN
0011
0012

DO 1000 IM=1,N0BSA
NTS*MVT(IM)
ISN
ISN
0013
0014

NTM=TM(IM)
NTF=NTS*NTM-1
ISN
ISN
0015
.0.016..

DO 999 IKS2,NBFA
999 X(IM.IK)=0.0D0
ISN
	Ii_N
0017
0013

X(IM,1)=1.0D0
DO *001 IT-NTS.NTF
ISN
ISN
0019
0020

KT= IT* 1
SV=lVTM(IT)»VTMtKT) 1/2.0
ISN 0021
ISN 0022

SA=VTM(KT)-VTM(IT)
X(IM,21=X(IM.2J+SV
ISN
	ISN
ISN
ISN
0023
0 024

X(IM,3)=X(IM,3)*SA
X (IM•4)=X(IM•4)~(SV*SAI
0025
002*

X(IM,5)=X(IM,5IMSV**2)
X(IM.6»=X(IM,6IMSA**2)
ISN
...ISM
0027
Mie

X(IM,7)=X(IM,7) + ( (SV**2)*S A)
X ( I M« 8 ) = X ( IM , C ) ~ ( 1 S_A**2 J *S V ) 	
-------
ISN 0029	X~((SV**2>*ISA**^()
... ISN OOiJ	1Q.CLL XQNTINUL	
ISN 0031	DO 1002 IK.=2,NBFA
ISN 0032		IMD = TMC1M)
ISN 003J	X*X(KrJ)>
.. ISN. 	IQQ5.XQNTINUL	
ISN 0042	C(I , J) = SUM
ISN OOW	1003 CONTINUE
	£				
C


C
c
SET UP A A AflRAY

TSN 0046
ISN Q047

DO 1006 I = 1»NBf A
DQ 1007 J=1.N0BSA

ISN
ISN
0048
0049

SUM=0.0DO
00 lOOfl K=1.NBFA

ISN
ISN
00 50
0051
inofi
SUM=SUH*(C(I»K)*X(J,K)J
f.flN T T Nil F

ISN
0052

AA(I,J)=SUN

ISN
0054
1006
c
CONTINUE



c
c
****** CALCULATE AS ARRAY
*********


c
c
CALCULATE BASIS FUNCTIONS

ISN
ISN
0055
00 56

DO 2000 1=1,N0BSS
XI=I-1

ISN
ISN
0057
0058

V*XI*15 .0
X*XIK,J)l
2003 CON TIN Ufc
	
ISN
ISN
00*8
QQfeS
2Q02
C(I,J)=SUM
CCNTIN'Jl .

-------
ISN J07G
ISH 0J/1.
ISN 0072
ISN OQXL-
c
./.ALL lMVIftSI Cj.3» 91..
DO 2004 1=1»NBFS
DO 2005 J=1.N0BSS
ISM 0 374
ISN 0075
ISN 0076
JSH JiZTL-
ISN 0078
fSft 007?
ISN
	LSfL
0080
£as.L
ISN 0082
SUM=0.000
2Q_.2.QQfe- K=li.NfiFi
SUM = SUM* (C (I ,K)H(JiUI
2Q06 CONTINUE	
AS
-------
LFVFL 21.6 ( MAY 72 )
0S/36G FORTRAN H
COMPILER OPTIONS - NAMt= MA IN,0PT=02,LINtCN1=foO,SIZ£=OOOOK,
			5QU.R.C 11EtJCDICi_NQUI STiNUSKKiLQAUj^ap.NOEDI T . IQ.XREF
ISN 0)02	SUBROUTINE EQOT(AMTC,AA,AS,BAD)
c *******************************************+********+**»********»*
C	SUBROUTINE £ DOT COMPUTES THE COEFFICIENTS THAT SPECIFV AN AUTO'S
c	INST ANTANEOUS EM.liS.ION RATE FUNCTIONS FOR HC .CO ..NQXl ARRAY • B AO • I «
C	GIVEN THE AMOUNT OF EACH EM1TTANT GIVFN OFF BY THE AUTO IN 32 A/D
	 			£	MODES ANL 5 STEADY STATE MODES (ARRAY 'AMTC'l.AND THE BASIS
C	FUNCTION FACTOR ARRAYStAA,AS I.
c ..		 .		.
C	AM TC =AMOUNT(GNS) OF THE I'TN EMITTANT GIVEN OFF BY THIS AUTO
	C- IN THE J»TH MDDE.I = 1=>HC,1=2=>C0.I=2=>NQX,J=L.3713Z A/D MODES
C	5 STEADY STATE MOOES ).( RE AL*<.)
—		c			 		
C	BAD(I,J I=J¦TH COEFFICIENT OF THIS AUTO'S INSTANTANEOUS FMISSION
	C	RATE FUNCTION FQ.R THE IlTH KIND. Of EMITTANT*1= 1=>HC, I = 2.=>C Q*
C	I=3=>N0X.(REAL*8)
	C 	
C	AA=>BASIS FI
	£	AA=BASIS FUNCTION FACTOR ARRAY FOR ACEL/DECELtCALCULATED BY SUBROU
C -TINE SETUP).
	 	C 		_ 							 _... 			
C	AS=9ASIS FUNCTION FACTOR ARRAY FOR STEADY STATEHC. IC = ?=>C Q. I = 3=>NQX 			 __		
C
c
ISN 0011		DO 1100 1 = 1.32			 ..._ 	
ISN 0012	Al = AMTC(1C,1)
ISN OOli	A2=TM< 1) _				 . 	 ..._ 	
ISN 0014	YA(I>=A1/A2
ISN Q315	1100 CONTINUE			
C
	C	CALCULATE COEFFICIENTS THAT SPECIFY A/D EMISSION RATE FUNCTIONS
C
ISN. Ofllfc	DO 12.U0 I=1.,NRFA
ISN 0317	SUM=O.ODO
ISN 0018	DO 12i>C J=l.NO£>iA	 	
ISN 0019	SUM = SUM +(AA(I,J)*YA(J))
-ISN.OOJM	1250 CONTINUE
-------
ISN
ISN
0021
00 22
1 ?0C1
BAD I IC, ~ I ) = SUH
CONTINUE


C
r.
f.ALCULATF flflSFRVFO AVFRAGF EMISSION RATFS OVER 5 SS MfinFS
ISN
Q;i?3
C
nn 2000 1=33.37
ISN
ISN
0024
0075

IP«I—32
A1=AMTC(IC.I1
ISN
TSN
0026
on?7

A2=TM(I)
YSIIPIxAl/A?
ISN
0028
2000
r.
CONTINUE


c
c
CALCULATE COEFFICIENTS THAT SPECIFY SS EMISSION RATE FUNCTIONS
ISN
ISN
0029
0010

DO 2001 1*1,NBFS
SIIMkO.ODO
ISN
ISN
0031
Q01?

00 2100 JxltNOBSS
SIIM-SUM*fAS( I.J)*YS C J ft 1
ISN
ISN
0033
0034
2100
CONTINUE
B(I1=SUM
ISN
0035
2001
C
CONTINUE


C
C
CHECK ON EXISTANCE OF NEGATTVE EMISSION RATES
ISN
ISN
0036
0037

L00P«0
IF(B(3 1.EQ.O.ODOIGOT02151
ISN
ISN
0039
0040

XO*(B(21**21-(4.ODO*B(3 J*B(11)
IF 1XO. LT.0.000)GOTO?153
ISN
ISN
0042
0043

XO*DSQRT( (BU)**2I-(4.0D0*BJ3)*B(1»))
X1 * 1 —B 12) ~X0I/I2.0D0*BI31 1
ISN
ISN
0044
0045

X2=(-BI2>—XO)/(2.0D0*B(31»
IF < m.GT.O.OnO.AND.Xl.LT.tO.OOOI.flR. IX2.GT.0.000.AND.X2-LT.60-OnO
ISN
0047
C)) L00P=1
G0T02153
ISN
ISN
0048
0049
2151
X0*-B(1)/B(2)
IF(XO.GT.0.ODO.AND.XO.LT.60.0001L00P«2
ISN
0051
2153
C
IF(LOOP.EQ.O)GOT02154


C
c
IF L00P*0*>N0 NEGATIVE tHISSIONS FOR VELOCITYS BETWEEN 0,60


c
c
IF L00P=1 OR 2=> NEGATIVE EMISSION RATES BETWEEN 0.60MPH.


c
c
CALL SUBROUTINE PAD TO FIND COEFFICIENTS WHICH DO NOT PRODUCE
NEGATIVE EMISSION RATES.
ISN
0053
c
CALL PAD(YS.B)
ISN
0054
c
2154
BADlIC.lOJrBU)
ISN
ISN
0055
0056

BAD(IC,11)=B<2)
BAD(IC•12)=B{3)
ISN 0057
ISN 0058
1000
CONTINUE
RFIURN
ISN
0059

END
-------
LEVEL 21.6 ( MAY 72 I
GS/360 FORTRAN H
CQMPIL ER
ISN 0002
OPTIONS - NAME = MA IN,0PT = C2 , LINECNT=60,S1ZE = 000OK,
—	--	SQURC fc . £ B CO I C . NOL I ST. NDDF CK«LOAD»MAP«NDt:DIT.ID»XRFF	
SUBROUTINE PAOtZ.BTI
****************_******** ******************************************
SUBRQUIIML PAD COMPUTES A SET OF COEFFICIENTS
EMISSION RATE FUNCTION FOR STEADY STATE CONDITIONS THAT IS
_NQ_N_ NEGATIVE BETWEEN VELQCTY 0 AND feO MPH.	
..JUL.
- * ** ****************** *********************************************
ISN JJ03
ISN 0Q04
	ISN 0 305
HiAL*c Hi). IPCS)	.
71 = Z I IJ
L2 = Z12J	
ISN 0006
ISM aoai
isn oooe
ISN QJ10
ISN 0011
11=1
12=2
IF(Z1.LT.Z2]GOTOI
Z1=IL£J	
Z2=Z(1)
ISN 0013
12= 1

ISN 0014
I Dfl 2 1=3.5

ISN 0015
ISN 0017
IF(2(IJ.GT.Z2IG0T02

ISN 0 018
12=1

ISN 0019
IF{Z1.LT.Z21G0T0?

ISN 0021
Cl = Zl

ISN 0022
Z1 = Z 2

ISN 0023
Z2 = C 1

ISN 0024
IX= I 1

ISN 0025
11 = 12

ISN 0026
12= IX

ISN 0027
2 CONTINUE

ISN 0028
B=(ZH-Z2I/2.CD0

ISN 0029
Vl = Il

ISN 0030
V?=I?

ISN 0031
Vl=m-1.0>*15.0

ISN 0032
V2=(V2-1.0>*15.0

ISN 0033
A=(V1*V2J/2.0

ISN 0034
nr> 4 1 = 1.5

ISN 0035
4 ZP(I)=Z(I|-B

ISN <3036
SUM 1=0.000

ISN 0037
SUM2=O.ODC

ISN 003B
DQ 5 1=1.5

ISN 0039
Vl = I

ISN 0040
vi= rvi-i.o)*i5.o

ISN 0041
V=V1

ISN 0042
X=(V**2)*(-2.0D0%A*V1*»A**2)

ISN 0043
SUM1 = SUM1~{ZPCI> *X1

ISN 0044
SUM2=SUM2»(X**2>

ISN	0045
ISN 004b
ISN 0047
isn	o)4a
ISN	004<»
	ISN 0050
CONTINUE
BTU)=SUM1/SJ«2	
BTI I 1=B+(BT<3)*tA**21)
BTl2)=-2«lfDQ*31(3)*A	
RETURN
END	
-------
LEVEL 21.iS ( HAY 72 )
OS/360 FORTRAN M
COMPILfcR OPTIONS
ISN QJ02
NAME = MAIN,QPT=0 2,LINECNT=60,SUE = OOOOK,
		 SnURCF.EBCDIC. NQLI ST.NOD ECK.LOAD«MAP »NQEDIT»ID «XREF			
suaROUTINE tSUMIVVT,NT,BAD.AHC.ACQ.ANOX.DIST)
*********** 4* + * + +********************** **********************.**
_5UaMHJIlM£ £SilM—CALCULATES .
AND THE DISTANCE TRAVELED BY THE AUTO THAT HAS THE INSTANTANEOUS
EMISSION RATE FUNCTION SPECIFIED BY THE ARRAY 'BAD'. AND A DRIVING
CYCLE SPECIFIED BY THE ARRAY 'VVT'.
VVT(I) = >VEL0CITY VS. TIME HISTORY(OR IVING CYCLEI IN ONE SECOND
NTOMAXIMUH NUMBER SECONDS IN DRIVING CYCLE+1 SECOND
^AQIJjXI^XI'TH COEFFICIENT OF THIS AUTO'S INSTANTANEOUS EHHIS&IOK
RATE FUNCTION FOR THE I'TH KIND OF EMITTANT,I«1=>HC,

C
C
AHC=AMTIGMS) HC GIVEN DFF BY THIS AUTO IN GOING THRU DRIVING
CYCLE

c
c
REAL»*


c
c
ACO-AHT(GMS) CO GIVEN OFF BY THIS AUTO IN GOING THRU DRIVING
REAM*
CYCLE

c
c
ANOX=AMT£GMSI NO* GIVEN OFF BY THIS AUTD IN GOING THRU DRIVE
CYCLE

c
c
REAL**


c
c
01ST*DISTANCE(MILES)IN SPECIFIED DRIVING CYCLE,REAL**

ISN 0003
c
DIMENSION VVTINT 1

ISN 000*
ISN 0305

REAL*8 BA0(3,12).AMTI3),X<12),DIS,AMIN,AMAX,AI,A2,HOA,SOA
AMAX=1.0D0

ISN 0006
I SN 0007

AMIN=-1.2000
A1=-1.0DO/AMIN

ISN 0008
c
A2 = — 1.ODD/AMAX


c
c
CLEAR AMT ARRAY

ISN 0009
ISN 0010
1000
DO 1000 1=1,3
AMT(11=0.000


c
c
INTEGRATE AUTO'S EMISSION RATE FUNCTION OVER DRIVING CYCLE

t<;n noil
c
DIS =0 a ODO

ISN 0012
kw 0013

NTT =NT-1
00 3000 IT=1.NTT

ISN 001*
ISN 0015

KT= I T*1
X(11=1.000

ISN 0016
I SN 0017

X(2 1=D0LE((VVT(IT)+VVT(KT)1/2.01
X( 3 ) =D8 Lfc I VVT 1 K.T) -VVT (IT))	

ISN 0018
I cm Q019

X(4)-X(2I*X(3)
X(S1=Xl2)**2

ISN 0020
ISN 0021

X( 6)=X(31**2
x(7l»(X<2>**2]*XO>

ISN 0022
ISN 0023

X(8)=
-------
1SN 00?4
ISN QQ25
ISN 0026
JSN mil
ISN 0029
ISN 3031
ISN 0033
ISN 0035
X(10)=X(1)
XLL1J=X121
X(12)=X(5)
IF ( X(3 1 «Gt.AKAXl HQA=0.GDQ	 	 _
IF  .j-T.AMAXlHQA = ( A2*Xi^) ) +1.000
1F(X(3).LE.0.0D0.AND.X13).GT.AMINIHOA=(Al* X C 3)J+1.0D0
DQ 2999IC-1«3	
ISN 0036
ISN Q&3I
ISN 0038
ISN 0040
ISN 0041
ISN QQ42-
2998
2999
DO 2998 IE= 1» 12
jQA=1.0D0-HGA	
1F tIE.GT.91S0A=H0A
CQNTINUE
CONTINUE
ISN 0043
	ISN
ISN 0045
ISN 0046
ISN 004T
ISN 0048
DIS=D1S+X< 2)
^000 CONTINUE	
AHC =AMT(1)
		ACD=AMT 12 i	
AN0X=AMT(3)
DIS=QIS/3600.QDQ
ISN 0049
ISN QQSO
DIST=DIS
DQ 4000 ICK=1.3
ISN 0051
ISN .0,053-
ISN 0054
ISN 0055
4000 IF(AMT
-------
LFVEL
2L.6 (
HAY 72
f QS/360 FORTRAN H

C9MPILLR OPTIONS - NAME = MA IN,OPT=02.LINECNT»60 . S I ZE=OOOOK,
SClltRf-F . PB CD 1C . NflL 1ST . IMHnF CK • 1 fun . M AP . NflF fll T . T ft . XRF F
ISN
0002
r.
SUBROUTINE EDGRPINSTANT ANEDUS EMISSION RATE FUNCTIONS ARRAY.(RE AL*8}


c
c
SET INT = 1 FOR THE FIRST AUTO IN THE GROUP


c
c
SET INT=2 FOR THE REST OF THE AUTOS IN THE GROUP


c
SET 1KT = 3 AFTEH ILL THE AUTOS IN THE GROUP HAVE BEEN CONSIDERED
1 Shi
QQ03
u
BFALM B*OI1.1?I.B(3.12l.r.TM
U
n
ISN
ISN
0004
0006

IF( INT.GT.DGOT02000
nn 1000 1=1.3
ISN
0007

00 1001 J=l,12
ft(l.JJ=G.ODO
ISN
ISN
0009
nil in
1001
1000
CONTINUE
rnNTlNlIF
ISN
ISN
0011
001?
?000
CTN=O.ODO
TFi TNT.EQ.31G0TO4133
ISN
isn
0014
nm s

CTN-CTN+l.ODO
nn ?ooi 1=1.3
ISN 0016
ISN 0017

DO 2002 J=l,12
RiI.JlsBII.Jl4-BADII.JI
ISN
I SN
0 J13
0019
2002
7001
CONTINUE
mNTiNUF ......
ISN
0020

IF(INT.NE.3)60T04000
rm 3000 1=1.3
ISN
ISN
0023
0024

00 3001 j«=lt 12
A AO t I - J i=BH . Jl/CTN
ISN 0325
ISN O02J.
3001
tooo
CONTINUE
rnNTINUF _
ISN
ASiL
0027
0Q2.8
4000
RETURN
END 	-
-------
LEVEL 21.6 t MAY 72 )
OS/'SfcO FORTP4N M
COMPILLR OPTIONS - NAME = MAIN,0PT = 02 ,LINECN1=*0,SIZE = OOOOK,
			 - 	SOL'RCL.EeClUC.NQLIST.rtJDECK.LJA^.ilAP.NCtSIT.l;;, *Htf
ISN 0003	SUBROUTINE INVERStA,N,N1)
C.» . _ .	*
c******************************** INVERSE ************************************
	C ~	*
c*	*
	C» P.LMARKS : THt R2UIIN£.-L0WPUT_£S The INVERSE QF MATRIXA; DIMENSION CO.	*
C* WHERE H IS THt DIMENSION OF A.	*
C* . .	. *
C+ THE INVERSE IS RETURNED IN MATRIX A.	*
C* 		 .. ... . ..	*
C« N IS RETURNED AS -I WHEN A IS SINGULAR	»
	C*			 _ 			 	 	 _ __	_ _ *
ISN 0003
ISN QQQ4
ISN 0005
C***************************************************************************** **
c .....		 ._ 	_	. ...		 ... 						 		
DIMENSION A(N1 .MI) .S(10,20)
	DaJBLt PRECISION S.BUFF.OVH.fPY.A	 	 . . 	
K= N+N
INITIALIZATION
. ISN £KLQi.
ISN 0007
isn norm
ISN 0009
ISN Ml?
-Qa_5Ql 0=Jj10_
D3 502 J = 1,
-------
I SN 0o41
ISN QQJtZ-
49 FPY = S (M ,L )
IF ( FPY . BO.
G.U(J
Jin	ic	is-
ISN 0044	50 00 70 J= 11K
-QQ45	ELUf	S-LUOi	
ISN 0046	= S(HtJ) - BUFF
_ ISii QQ <»7	1X1 CflliTIMUt	
ISN 0048	7b JIN = M+l
TSN Q.149 	IF t JIN . GT.N ) GQ TQ
,149
ISN 0051
XSN-QQ52...
ISN 0053
1SHQQ3A
100 M = M*1
.XZil. GO TP 49
CONTINUE
_X5lQ cast IHU-fc .
DIAGONALIZAT ION OF S
lan
ISN
0056

L = I

t<;m
0157

H s 1-1

ISN
isn
00 58
0059
350
FPY = S(M,U
IF i FPY . EO. 0-00 1 GQ TCI 375

ISN
0061
351
00 370 J=1,K

ISN
0062

BUFF = FPY • S(I.J)

ISN
0063

S(M,J) = SIM,J) - BUFF

I SN
0064
370
CANTINUF

ISN
0065
375
IF < K .LE. 1 I GO TO 3 84

I SN
0067
380
M = M-l

ISN
0068

GO TO 3 50

ISN
0069
384
CONTINUE

ISN
0070
385
f.
CONTINUE
STORE INVERSE IN A
ISN
0071
390
DO 402 Il=l*N

ISN
0072

LL = 1 1

ISN
0071
395
00 400 Jl=l*N

ISN
or>74

KK = N + J1

ISN
0075
396
A(I 1,J1t = S< LL,KK)

ISN
0O76
400
CONTINUE

ISN
0077
402
CONTINUE

ISN
00 7B

RETURN



C

NO INVERSE
ISN
0079
900
WRI TE( 6 ¦7000]

ISN
0080
7000
FORMAT(1H0» *N0 INVERSE')

ISN
nnfii

N=-l

ISN
0082

RETURN

	LSJ1
00fl3

END ...

-------
APPENDIX V
VEHICLE CLASSIFICATION BY DISCRIMINANT FUNCTION ANALYSIS
The implementation of the emission-rate function model is assessing
the impact of a collection of vehicles on the environment requires that the input
data be appropriately chosen according to the problem under consideration. A
matter of central concern in this regard is whether available data can be
employed in a setting different from that in which it originated. In particular,
such questions as geographic location and time frame are of interest. For
example, can surveillance data obtained in one location be employed as a basis
for modeling emissions in a different location, and can data collected in the
past be used as a basis for modeling present or future problems?
In a previous report, it was noted that a distinction exists between
the emissions of vehicles in Denver as compared with five other cities of lower
altitudes. Also, as would be expected, the implementation of emission controls
clearly affected the emissions of vehicles in both geographic categories. The
concept of discriminant function analysis was employed to crystallize and quantify
these differences.
The distinctions alluded to above were based on an analysis of total
emissions of HC, CO and NO^ over the surveillance driving sequerce. Though it
is presumed that these differences among categories of vehicles vrould persist
in other driving sequences, it was considered of interest to investigate this
assumption in a systematic way. Toward this end a discriminant function approach
was undertaken.
The logic of this approach is as follows. Each driving cycle consists
of a series of accel/decel and steady-state modes. Conceivably some of these
modes might be more sensititve to geographic or time differences than others.
This would result in a situation where driving sequences consisting of combina-
tions of these modes could themselves differ in this regard. A linear discriminant
function is a weighted combination of the modes constructed in such a way that
differences between vehicle categories, if they exist at all, will be most
clearly delineated.
-------
Suppose that only Denver versus non-Denver vehicles are considered
and the emissions for a particular pollutant for the various modes are denoted
X^, X^,	X^. Then the linear discriminant function is defined as
where a^, , ...» a^ are coefficients selected according to an optimization
logic. For each vehicle, one can compute a value of F. If one groups together
all F values for Denver vehicles and all F values for non-Denver vehicles,
one can compute a mean value of F for each group and a variance or standard
deviation for each group. The two groups will separate clearly, with minimal
overlap of the two distributions of F values, if the difference between the
mean values for the groups is large and the dispersion or scatter of F values
within each group is small. The coefficients a^, ^ , • ••, are chosen so
as to make as large as possible the ratio of the dispersion of the group means
to the pooled dispersion of the individual vehicles within groups.
If only two groups are involved, it is possible to confute only one
discriminant function for each vehicle. However, if there are G groups, one
can compute G-l such functions, provided G-l does not exceed the number of modes.
For example, consider the following groups of vehicles
GROUP 1	- Non-Denver, pre-emission control
GROUP 2	- Denver, pre-emission control
GROUP 3	- Non-Denver, emission control
GROUP 4	- Denver, emission control
In this case three discriminant functions can be computed as follows:
F\ — &.vYi	4- - - - - CX.in X i"7
Fi_ c				Y3-1
)(. v V-Cv Kv 4- _ — —	V- 3 7
These functions employ three distinct sets of weighting coefficients	[ci]
the efficacy of which can be appreciated best by a geometric argument.
-------
For each vehicle, one can compute F , , and F^. Consider these
three values as the coordinates of a point in three-dimensional space. If
each vehicle is plotted as such a point in 3-space, then there will result a
collection of points which, hopefully, will tend to cluster into four distinct
"clouds", these clouds being associated with the four groups of vehicles
under study. If the separation of these groups is complete, one can partition
the space into disjoint compartments and there will be no overlap or spillover
from one compartment to the other. In reality, this will not. be the case, and
there will be a certain amount of "mixing" or "confusion" if one attempts to
assign a particular vehicle to its correct group merely by looking at the F-value
coordinates of that vehicle. Quantification of the degree of correct and
incorrect classification is therefore required in order to assess how "good"
such a classification matrix is.
To appreciate the nature of this quantification, consider first
the case in which there are only two categories--say, Denver and non-Denver--
and only a single discriminant function F. Divide all vehicles into two sets
on the basis of their known membership—that is, consider all Denver vehicles
as a group and all non-Denver vehicles as a group. There are 169 vehicles
in the Denver group, 851 in the non-Denver group. For the Denver group,
compute F for each of the 169 vehicles and display these results as a histogram.
Similarly, compute F for each of the 859 vehicles in the non-Denver group and
display these results as a histogram. The results are shown in Figures 5-1,
5-2, and 5-3 for HC, CO and N0x respectively. Quite clearly, a certain
amount of overlap is evident, so that if one attempts to assign class
membership on the basis of the F value only, a certain number of both the Denver
and non-Denver vehicles will be misclassified.
A technique which can be used to determine the quantitative separation
of groups is the construction of a "classification matrix" for the groups. The
concept, as illustrated below, represents classifying automobiles as coming from
City A or City B on the basis of a particular test value.
CITY A
CITY B
CITY A CITY B
83
17
42
58
CLASSIFICATION MATRIX FOR TWO CITIES
-------
The classification matrix should be interpreted in the following way.
Assume a technique which takes a test value for an automobile from City A
and, on the basis of probabilities for this test value, classifies this
automobile as coming from either City A or City B, whichever has the highest
probability for the given test value. If al1 the automobiles from City A
are subjected to this probability test, one will find that some are correctly
classified as coming from City A while others are incorrectly classified
as coming from City B. The same test can be performed on the automobiles from
City B. Since the number of automobiles from each city is known, one can
convert the number of automobiles correctly or incorrectly classified into
percent of automobiles from each city. The diagonal elements of the matrix
represent the percent of automobiles coming from City A and City B which were
correctly classified as coming from City A and City B. The off-diagonal elements
give the percent of automobiles incorrectly classified. Therefore, 83 percent
of the automobiles from City A were correctly classified as coming from City A,
while 17 percent were incorrectly classified as coming from City B. Likewise,
58 percent of the automobiles from City B were classified as coming from
City B while 42 percent were classified as coming from City A. In this case,
the probability for classifying automobiles was more heavily weighted in favor
of City A.
Classification matrices for Denver versus non-Denver vehicles are
shown in Table 5-1 and the corresponding histograms on which they are based are
shown in Figures 5-1, 5-2 and 5-3. It can be seen that the separation of the
two groups is somewhat better for CO and NO^ than for HC. Also, in the case of
HC, it is more likely that Denver vehicles will be misclassified as non-Denver
vehicles than that non-Denver vehicles will be classified as Denver vehicles.
Thus once the probabilities for a set of groups have been determined for
values of a given test the classification matrix technique can be used to give
an easy-to-interpret quantitative measure of the separation between groups.
-------
Tf the number of groups used in the classification is increased,
the size of the classification matrix increases accordingly, with the diagonal
elements indicating the percent correct classification. In this more elaborate
situation, a vehicle can be miscl assi fi ed in more than one way, and the nature
of the misc.1 assi fication can provide insight into which categories of vehicles
are most "alike".
Just as the number of groups (cities) in the classification matrix
can be increased, the number of discriminant functions to be used as the basis
of classification can also be increased. In this case one calculates, for
a given automobile, the probability of occurrence of each of the discriminant-
function values associated with the vehicle. For reasons discussed later, it
can be assumed that the several discriminant functions are statistical]y
independent. Therefore, these probabilities can be multiplied together and it
can be assumed that the product is the probability of the joint occurrence of the
particular discriminant function values encountered for the vehicle in question.
Then compare the probability products calculated for all the groups and assign
the automobile to the group with the highest probability product.
In the case of Denver and non-Denver vehicles before and after the
advent of emission controls, there are four categories and three F values, Ry
virtue of the theory on which discriminant function analysis rests, the three F
values are statistically uncorrelated. Also, by virtue of the fact that each of
the F values is a linear combination of the outcomes of 37 random variables,
it can be presumed that the Central Limit Theorem of statistics will cause the
distribution of the F values to tend toward a Gaussian distribut on. Under these
conditions, the assumption of independence is believed justified.
The classification matrices for HC, CO and N0x according vo Denver and
non-Denver, pre-control and control eras are. presented in Table 5-2.
The type of inferences which can be drawn from these tables is exemplified by
noting the classification matrix for N0x. For example, note that non-Denver,
pre-emission controls (category ]) is misclassi fied more frequently as non-Denver
emission controls (category 3) than as either of the Denver categories (categories
2 and 4"). Similarly, the Denver pre-emission control is confused more with Denver
post-emission controls than with the other two categories, and the Denver emission
-------
TABLE 5-1
CLASSIFICATION MATRICES
GROUP 1 = NON-DENVER
GROUP 2 = DENVER
HC
I 2
1)	82 17
2)	42 57
CO
I 2
2) B4 15
2) H 88
NO
x
I I
1)	88 11
2)	11 88
-------
TABLE 5-2
CLASSIFICATION MATRICES
GROUP 1 = NON-DENVER PRE-EMISSION CONTROL
GROUP 2 = DENVER PRE-EMISSION CONTROL
GROUP 3 = NON-DENVER EMISSION CONTROL
GROUP 4 = DENVER EMISSION CONTROL
HC


2
3.
£
1)
47
13
23
14
2)
19
48
3
28
3)
16
3
58
21
4)
5
15
11
68


CO



£
2_
3
£
1)
56
8
25
9
2)
12
72
0
15
3)
15
1
79
3
4)
12
11
4
72


NO




X



1
2_
3
4
1)
58
10
20
9
2)
7
73
2
17
3)
27
2
66
3
4)
5
23
2
68
-------
K*E
lO X lO TO THE CENTIMETER 46 1512
18 X 25 CM.	MADE IK U.S.A. <
KEUFFEL & ESSER CO,
-------
-------
U.C 10 X IO TO THE CENTIMETER 46 1512
«•	16 X 25 CM.	naDE in u. S.a. •
KEUFFEL « ESSER CO.
-------