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Appendix 7b F1F2

United Kingdom

Original text. Later amendments are not shown.

Procedure to determine the cumulative positive elevation gain of a trip

1.

INTRODUCTION

This Appendix describes the procedure to determine the cumulative elevation gain of an RDE trip.

2.

SYMBOLS

d(0)

  • distance at the start of a trip [m]

    • d

  • cumulative distance travelled at the discrete way point under consideration [m]

    • d 0

  • cumulative distance travelled until the measurement directly before the respective way point d [m]

    • d 1

  • cumulative distance travelled until the measurement directly after the respective way point d [m]

    • d a

  • reference way point at d(0) [m]

    • d e

  • cumulative distance travelled until the last discrete way point [m]

    • d i

  • instantaneous distance [m]

    • d tot

  • total test distance [m]

    • h(0)

  • vehicle altitude after the screening and principle verification of data quality at the start of a trip [m above sea level]

    • h(t)

  • vehicle altitude after the screening and principle verification of data quality at point t [m above sea level]

    • h(d)

  • vehicle altitude at the way point d [m above sea level]

    • h(t-1)

  • vehicle altitude after the screening and principle verification of data quality at point t-1 [m above sea level]

    • h corr (0)

  • corrected altitude directly before the respective way point d [m above sea level]

    • h corr (1)

  • corrected altitude directly after the respective way point d [m above sea level]

    • h corr (t)

  • corrected instantaneous vehicle altitude at data point t [m above sea level]

    • h corr (t-1)

  • corrected instantaneous vehicle altitude at data point t-1 [m above sea level]

    • h GPS,i

  • instantaneous vehicle altitude measured with GPS [m above sea level]

    • h GPS (t)

  • vehicle altitude measured with GPS at data point t [m above sea level]

    • h int (d)

  • interpolated altitude at the discrete way point under consideration d [m above sea level]

    • h int,sm,1 (d)

  • smoothed interpolated altitude, after the first smoothing run at the discrete way point under consideration d [m above sea level]

    • h map (t)

  • vehicle altitude based on topographic map at data point t [m above sea level]

    • Hz

  • hertz

    • km/h

  • kilometre per hour

    • m

  • metre

    • road grade,1 (d)

  • smoothed road grade at the discrete way point under consideration d after the first smoothing run [m/m]

    • road grade,2 (d)

  • smoothed road grade at the discrete way point under consideration d after the second smoothing run [m/m]

    • sin

  • trigonometric sine function

    • t

  • time passed since test start [s]

    • t 0

  • time passed at the measurement directly located before the respective way point d [s]

    • v i

  • instantaneous vehicle speed [km/h]

    • v(t)

  • vehicle speed of data point t [km/h].

    • 3.

    GENERAL REQUIREMENTS

    The cumulative positive elevation gain of an RDE trip shall be determined based on three parameters: the instantaneous vehicle altitude h GPS,i [m above sea level] as measured with the GPS, the instantaneous vehicle speed v i [km/h] recorded at a frequency of 1 Hz and the corresponding time t [s] that has passed since test start.

    4.

    CALCULATION OF CUMULATIVE POSITIVE ELEVATION GAIN

    4.1.

    General

    The cumulative positive elevation gain of an RDE trip shall be calculated as a three-step procedure, consisting of: (i) the screening and principle verification of data quality; (ii) the correction of instantaneous vehicle altitude data; and (iii) the calculation of the cumulative positive elevation gain.

    4.2.

    Screening and principle verification of data quality

    The instantaneous vehicle speed data shall be checked for completeness. Correcting for missing data is permitted if gaps remain within the requirements specified in Point 7 of Appendix 4; else, the test results shall be voided. The instantaneous altitude data shall be checked for completeness. Data gaps shall be completed by data interpolation. The correctness of interpolated data shall be verified by a topographic map. It is recommended to correct interpolated data if the following condition applies:

    | h GPS (t) – h map (t) | > 40 m

    The altitude correction shall be applied so that:

    h(t) = h map (t)

    where:

    h(t)

  • vehicle altitude after the screening and principle verification of data quality at data point t [m above sea level]

    • h GPS (t)

  • vehicle altitude measured with GPS at data point t [m above sea level]

    • h map (t)

  • vehicle altitude based on topographic map at data point t [m above sea level].

    • 4.3.

    Correction of instantaneous vehicle altitude data

    The altitude h(0) at the start of a trip at d(0) shall be obtained by GPS and verified for correctness with information from a topographic map. The deviation shall not be larger than 40 m. Any instantaneous altitude data h(t) shall be corrected if the following condition applies:

    | h(t) – h ( t – 1)| > ( v(t) /3,6 * sin 45°)

    The altitude correction shall be applied so that:

    h corr (t) = h corr ( t -1)

    where:

    h(t)

  • vehicle altitude after the screening and principle verification of data quality at data point t [m above sea level]

    • h(t-1)

  • vehicle altitude after the screening and principle verification of data quality at data point t-1 [m above sea level]

    • v(t)

  • vehicle speed of data point t [km/h]

    • h corr (t)

  • corrected instantaneous vehicle altitude at data point t [m above sea level]

    • h corr (t-1)

  • corrected instantaneous vehicle altitude at data point t-1 [m above sea level].

    • Upon the completion of the correction procedure, a valid set of altitude data is established. This data set shall be used for the final calculation of the cumulative positive elevation gain as described in point 4.4.

    4.4.

    Final calculation of the cumulative positive elevation gain

    4.4.1.

    Establishment of a uniform spatial resolution

    The total distance d tot [m] covered by a trip shall be determined as sum of the instantaneous distances d i . The instantaneous distance d i shall be determined as:

    d i = v i 3,6

    Where:

    d i

  • instantaneous distance [m]

    • v i

  • instantaneous vehicle speed [km/h]

    • The cumulative elevation gain shall be calculated from data of a constant spatial resolution of 1 m starting with the first measurement at the start of a trip d(0) . The discrete data points at a resolution of 1 m are referred to as way points, characterised by a specific distance value d (e.g., 0, 1, 2, 3 m…) and their corresponding altitude h(d) [m above sea level].

    The altitude of each discrete way point d shall be calculated through interpolation of the instantaneous altitude h corr (t) as:

    h int d = h corr 0 + h corr 1 − h corr 0 d 1 − d 0 × d − d 0

    Where:

    h int (d)

  • interpolated altitude at the discrete way point under consideration d [m above sea level]

    • h corr (0)

  • corrected altitude directly before the respective way point d [m above sea level]

    • hcorr(1)

  • corrected altitude directly after the respective way point d [m above sea level]

    • d

  • cumulative distance travelled until the discrete way point under consideration d [m]

    • d 0

  • cumulative distance travelled until the measurement located directly before the respective way point d [m]

    • d 1

  • cumulative distance travelled until the measurement located directly after the respective way point d [m].

    • 4.4.2.

    Additional data smoothing

    The altitude data obtained for each discrete way point shall be smoothed by applying a two-step procedure; d a and d e denote the first and last data point respectively (Figure 1). The first smoothing run shall be applied as follows:

    road grade ,1 d = h int d + 200 m − h int d a d + 200 m for d ≤ 200 m

    road grade ,1 d = h int d + 200 m − h int d − 200 m d + 200 m − d − 200 m for 200 m < d < ( d e – 200 m )

    road grade ,1 d = h int d e − h int d − 200 m d e − d − 200 m for d ≥ ( d e – 200 m )

    h int,sm ,1 ( d ) = h int,sm ,1 ( d – 1 m ) + road grade ,1 ( d ), d = d a + 1 to d e

    h int,sm ,1 ( d a ) = h int ( d a ) + road grade ,1 ( d a )

    Where:

    road grade,1 (d)

  • smoothed road grade at the discrete way point under consideration after the first smoothing run [m/m]

    • h int (d)

  • interpolated altitude at the discrete way point under consideration d [m above sea level]

    • h int,sm,1 (d)

  • smoothed interpolated altitude, after the first smoothing run at the discrete way point under consideration d [m above sea level]

    • d

  • cumulative distance travelled at the discrete way point under consideration [m]

    • d a

  • reference way point at a distance of zero metres [m]

    • d e

  • cumulative distance travelled until the last discrete way point [m].

    • The second smoothing run shall be applied as follows:

    road grade ,2 d = h int,sm ,1 d + 200 m − h int,sm ,1 d a d + 200 m for d ≤ 200 m

    road grade ,2 d = h int,sm ,1 d + 200 m − h int,sm ,1 d − 200 m d + 200 m − d − 200 m for 200 m < d < ( d e – 200 m )

    road grade ,2 d = h int,sm ,1 d e − h int,sm ,1 d − 200 m d e − d − 200 m for d ≥ ( d e – 200 m )

    Where:

    road grade,2 (d)

  • smoothed road grade at the discrete way point under consideration after the second smoothing run [m/m]

    • h int,sm,1 (d)

  • smoothed interpolated altitude, after the first smoothing run at the discrete way point under consideration d [m above sea level]

    • d

  • cumulative distance travelled at the discrete way point under consideration [m]

    • d a

  • reference way point at a distance of zero metres [m]

    • d e

  • cumulative distance travelled until the last discrete way point [m].

    • Figure 1

    Illustration of the procedure to smooth the interpolated altitude signals

    4.4.3.

    Calculation of the final result

    The positive cumulative elevation gain of a trip shall be calculated by integrating all positive interpolated and smoothed road grades, i.e. road grade,2 (d) . The result should be normalised by the total test distance d tot and expressed in meters of cumulative elevation gain per 100 kilometres of distance.

    5.

    NUMERICAL EXAMPLE

    Tables 1 and 2 show the steps performed in order to calculate the positive elevation gain on the basis of data recorded during an on-road test performed with PEMS. For the sake of brevity an extract of 800 m and 160 s is presented here.

    5.1.

    Screening and principle verification of data quality

    The screening and principle verification of data quality consists of two steps. First, the completeness of vehicle speed data is checked. No data gaps related to vehicle speed are detected in the present data sample (see Table 1). Second, the altitude data are checked for completeness; in the data sample, altitude data related to seconds 2 and 3 are missing. The gaps are filled by interpolating the GPS signal. In addition, the GPS altitude is verified by a topographic map; this verification includes the altitude h(0) at the start of the trip. Altitude data related to seconds 112-114 are corrected on the basis of the topographic map to satisfy the following condition:

    h GPS (t) – h map (t) < – 40 m

    As result of the applied data verification, the data in the fifth column h(t) are obtained.

    5.2.

    Correction of instantaneous vehicle altitude data

    As a next step, the altitude data h(t) of seconds 1 to 4, 111 to 112 and 159 to 160 are corrected assuming the altitude values of seconds 0, 110 and 158 respectively since the following condition applies:

    | h(t) – h ( t – 1)| > ( v(t) /3,6 * sin 45°)

    As result of the applied data correction, the data in the sixth column h corr (t) are obtained. The effect of the applied verification and correction steps on the altitude data is depicted in Figure 2.

    5.3.

    Calculation of the cumulative positive elevation gain

    5.3.1.

    Establishment of a uniform spatial resolution

    The instantaneous distance d i is calculated by dividing the instantaneous vehicle speed measured in km/h by 3,6 (Column 7 in Table 1). Recalculating the altitude data to obtain a uniform spatial resolution of 1 m yields the discrete way points d (Column 1 in Table 2) and their corresponding altitude values h int (d) (Column 7 in Table 2). The altitude of each discrete way point d is calculated through interpolation of the measured instantaneous altitude h corr as:

    h int 0 = 120,3 + 120,3 − 120,3 0,1 − 0,0 × 0 − 0 = 120,3000

    h int 520 = 132,5 + 132,6 − 132,5 523,6 − 519,9 × 520 − 519,9 = 132,5027

    5.3.2.

    Additional data smoothing

    In Table 2, the first and last discrete way points are: d a = 0 m and d e = 799 m, respectively. The altitude data of each discrete way point is smoothed by applying a two-step procedure. The first smoothing run consists of:

    road grade ,1 0 = h int 200 m − h int 0 0 + 200 m = 120,9682 − 120,3000 200 = 0,0033

    chosen to demonstrate the smoothing for d ≤ 200 m

    road grade ,1 320 = h int 520 − h int 120 520 − 120 = 132,5027 − 121,9808 400 = 0,0288

    chosen to demonstrate the smoothing for 200 m < d < (599 m )

    road grade ,1 720 = h int 799 − h int 520 799 − 520 = 121,2000 − 132,5027 279 = − 0,0405

    chosen to demonstrate the smoothing for d ≥ (599 m )

    The smoothed and interpolated altitude is calculated as:

    h int,sm ,1 (0) = h int (0) + road grade ,1 (0) = 120,3 + 0,0033 ≈ 120,3033 m

    h int,sm ,1 (799) = h int,sm ,1 (798) + road grade ,1 (799) = 121,2550 – 0,0220 = 121,2330 m

    Second smoothing run:

    road grade ,2 0 = h int,sm ,1 200 − h int,sm ,1 0 200 = 119,9618 − 120,3033 200 = − 0,0017

    chosen to demonstrate the smoothing for d ≤ 200 m

    road grade ,2 320 = h int,sm ,1 520 − h int,sm ,1 120 520 − 120 = 123,6809 − 120,1843 400 = 0,0087

    chosen to demonstrate the smoothing for 200 m < d < (599 m )

    road grade ,2 720 = h int,sm ,1 799 − h int,sm ,1 520 799 − 520 = 121,2330 − 123,6809 279 = − 0,0088

    chosen to demonstrate the smoothing for d ≥ (599 m )

    5.3.3.

    Calculation of the final result

    The positive cumulative elevation gain of a trip is calculated by integrating all positive interpolated and smoothed road grades, i.e. road grade,2 (d) . For the presented example the total covered distance was d tot = 139,7 km and all positive interpolated and smoothed road grades were of 516 m. Therefore a positive cumulative elevation gain of 516 × 100/139,7 = 370 m/100 km was achieved.

    Table 1

    Correction of instantaneous vehicle altitude data

    - denotes data gaps

    Time t [s] v(t) [km/h] h GPS (t) [m] h map (t) [m] h(t) [m] h corr (t) [m] d i [m] Cum. d [m]
    0 0,00 122,7 129,0 122,7 122,7 0,0 0,0
    1 0,00 122,8 129,0 122,8 122,7 0,0 0,0
    2 0,00 - 129,1 123,6 122,7 0,0 0,0
    3 0,00 - 129,2 124,3 122,7 0,0 0,0
    4 0,00 125,1 129,0 125,1 122,7 0,0 0,0
    … … … … … … … …
    18 0,00 120,2 129,4 120,2 120,2 0,0 0,0
    19 0,32 120,2 129,4 120,2 120,2 0,1 0,1
    … … … … … … … …
    37 24,31 120,9 132,7 120,9 120,9 6,8 117,9
    38 28,18 121,2 133,0 121,2 121,2 7,8 125,7
    … … … … … … … …
    46 13,52 121,4 131,9 121,4 121,4 3,8 193,4
    47 38,48 120,7 131,5 120,7 120,7 10,7 204,1
    … … … … … … … …
    56 42,67 119,8 125,2 119,8 119,8 11,9 308,4
    57 41,70 119,7 124,8 119,7 119,7 11,6 320,0
    … … … … … … … …
    110 10,95 125,2 132,2 125,2 125,2 3,0 509,0
    111 11,75 100,8 132,3 100,8 125,2 3,3 512,2
    112 13,52 0,0 132,4 132,4 125,2 3,8 516,0
    113 14,01 0,0 132,5 132,5 132,5 3,9 519,9
    114 13,36 24,30 132,6 132,6 132,6 3,7 523,6
    … … … … … … …
    149 39,93 123,6 129,6 123,6 123,6 11,1 719,2
    150 39,61 123,4 129,5 123,4 123,4 11,0 730,2
    … … … … … … …
    157 14,81 121,3 126,1 121,3 121,3 4,1 792,1
    158 14,19 121,2 126,2 121,2 121,2 3,9 796,1
    159 10,00 128,5 126,1 128,5 121,2 2,8 798,8
    160 4,10 130,6 126,0 130,6 121,2 1,2 800,0

    Table 2

    Calculation of road grade

    d [m] t 0 [s] d 0 [m] d 1 [m] h 0 [m] h 1 [m] h int (d) [m] road grade,1 (d) [m/m] h int,sm,1 (d) [m] road grade,2 (d) [m/m]
    0 18 0,0 0,1 120,3 120,4 120,3 0,0035 120,3 – 0,0015
    … … … … … … … … … …
    120 37 117,9 125,7 120,9 121,2 121,0 – 0,0019 120,2 0,0035
    … … … … … … … … … …
    200 46 193,4 204,1 121,4 120,7 121,0 – 0,0040 120,0 0,0051
    … … … … … … … … … …
    320 56 308,4 320,0 119,8 119,7 119,7 0,0288 121,4 0,0088
    … … … … … … … … … …
    520 113 519,9 523,6 132,5 132,6 132,5 0,0097 123,7 0,0037
    … … … … … … … … … …
    720 149 719,2 730,2 123,6 123,4 123,6 – 0,0405 122,9 – 0,0086
    … … … … … … … … … …
    798 158 796,1 798,8 121,2 121,2 121,2 – 0,0219 121,3 – 0,0151
    799 159 798,8 800,0 121,2 121,2 121,2 – 0,0220 121,3 – 0,0152

    Figure 2

    The effect of data verification and correction — The altitude profile measured by GPS h GPS (t) , the altitude profile provided by the topographic map h map (t) , the altitude profile obtained after the screening and principle verification of data quality h(t) and the correction hcorr(t) of data listed in Table 1

    Figure 3

    Comparison between the corrected altitude profile h corr (t) and the smoothed and interpolated altitude h int,sm,1

    Table 2

    Calculation of the positive elevation gain

    d [m] t 0 [s] d 0 [m] d 1 [m] h 0 [m] h 1 [m] h int (d) [m] road grade,1 (d) [m/m] h int,sm,1 (d) [m] road grade,2 (d) [m/m]
    0 18 0,0 0,1 120,3 120,4 120,3 0,0035 120,3 – 0,0015
    … … … … … … … … … …
    120 37 117,9 125,7 120,9 121,2 121,0 – 0,0019 120,2 0,0035
    … … … … … … … … … …
    200 46 193,4 204,1 121,4 120,7 121,0 – 0,0040 120,0 0,0051
    … … … … … … … … … …
    320 56 308,4 320,0 119,8 119,7 119,7 0,0288 121,4 0,0088
    … … … … … … … … … …
    520 113 519,9 523,6 132,5 132,6 132,5 0,0097 123,7 0,0037
    … … … … … … … … … …
    720 149 719,2 730,2 123,6 123,4 123,6 – 0,0405 122,9 – 0,0086
    … … … … … … … … … …
    798 158 796,1 798,8 121,2 121,2 121,2 – 0,0219 121,3 – 0,0151
    799 159 798,8 800,0 121,2 121,2 121,2 – 0,0220 121,3 – 0,0152

    Annotations

    Textual amendments

    F1

    Inserted by Commission Regulation (EU) 2016/427 of 10 March 2016 amending Regulation (EC) No 692/2008 as regards emissions from light passenger and commercial vehicles (Euro 6) (Text with EEA relevance).

    F2

    Inserted by Commission Regulation (EU) 2016/646 of 20 April 2016 amending Regulation (EC) No 692/2008 as regards emissions from light passenger and commercial vehicles (Euro 6) (Text with EEA relevance).