Appendix 7b

United Kingdom

Procedure to determine the cumulative positive elevation gain of a PEMS trip

INTRODUCTION

This Appendix describes the procedure to determine the cumulative elevation gain of a PEMS trip.

SYMBOLS, PARAMETERS AND UNITS

d(0)

distance at the start of a trip [m]

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

d₀

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

d₁

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 and 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]

hertz

km/h

kilometer per hour

meter

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

time passed since test start [s]

t₀

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 at a data point t [km/h]

GENERAL REQUIREMENTS

The cumulative positive elevation gain of a 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.

CALCULATION OF CUMULATIVE POSITIVE ELEVATION GAIN

4.1.

General

The cumulative positive elevation gain of a 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)| > 40m

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 \times sin45°)

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 calculation of the cumulative positive elevation gain as described in Point 13.4.

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} = \frac{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, characterized 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) + \frac{h_{corr} (1) - h_{corr} (0)}{d_{1} - d_{0}} \times (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]

h_corr(1)

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

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

d₀

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

d₁

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) = \frac{h_{int} (d + 200m) - h_{int} (d_{a})}{(d + 200m)} for d \leq 200m

{road}_{grade,1} (d) = \frac{h_{int} (d + 200m) - h_{int} (d - 200m)}{(d + 200m) - (d - 200m)} for 200m < d < (d_{e} - 200m)

{road}_{grade,1} (d) = \frac{h_{int} (d_{e}) - h_{int} (d - 200m)}{d_{e} - (d - 200m)} for d \geq (d_{e} - 200m)

h_{int,sm,1} (d) = h_{int,sm,1} (d - 1m) + {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]

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

d_a

reference way point at a distance of zero meters [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) = \frac{h_{int,sm,1} (d + 200m) - h_{int,sm,1} (d_{a})}{(d + 200m)} for d \leq 200m

{road}_{grade,2} (d) = \frac{h_{int,sm,1} (d + 200m) - h_{int,sm,1} (d - 200m)}{(d + 200m) - (d - 200m)} for 200m < d < (d_{e} - 200m)

{road}_{grade,2} (d) = \frac{h_{int,sm,1} (d_{e}) - h_{int,sm,1} (d - 200m)}{d_{e} - (d - 200m)} for d \geq (d_{e} - 200m)

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]

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

d_a

reference way point at a distance of zero meters [m]

d_e

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

F14.4.3.

Calculation of the final result

The positive cumulative elevation gain of a total trip shall be calculated by integrating all positive interpolated and smoothed road grades, i.e. road _grade,2(d). The result should be normalized by the total test distance dtot and expressed in metres of cumulative elevation gain per one hundred kilometres of distance.

The positive cumulative elevation gain of the urban part of a trip shall then be calculated based on the vehicle speed over each discrete way point:

v_w = 1 / (t_w,i – t_{w,i – 1}) · 60² / 1 000

Where:

v_w - waypoint vehicle speed [km/h]

All datasets with v_w =< 60 km/h belong to the urban part of the trip.

Integrate all of the positive interpolated and smoothed road grades that correspond to urban datasets.

Integrate the number of 1m waypoints which correspond to urban datasets and divide by 1 000 to calculate urban test distance d _urban [km].

The positive cumulative elevation gain of the urban part of trip shall then be calculated by dividing the urban elevation gain by the urban test distance, and expressed in metres of cumulative elevation gain per one hundred kilometres of distance.

NUMERICAL EXAMPLE

Tables 1 and 2 show how 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 800m and 160s 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) < - 40m

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 for the altitude data in these time periods the following condition applies:

|h (t) - h (t - 1)| > (v (t) ∕ 3,6 \times sin45°)

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 + \frac{120,3 - 120,3}{0,1 - 0,0} \times (0 - 0) = 120,3000

h_{int} (520) = 132,5 + \frac{132,6 - 132,5}{523,6 - 519,9} \times (520 - 519,9) = 132,5027

5.3.2.

Additional data smoothing

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

{road}_{grade,1} (0) = \frac{h_{int} (200m) - h_{int} (0)}{(0 + 200m)} = \frac{120,9682 - 120,3000}{200} = 0,0033

chosen to demonstrate the smoothing for d ≤ 200m

{road}_{grade,1} (320) = \frac{h_{int} (520) - h_{int} (120)}{(520) - (120)} = \frac{132,5027 - 121,0}{400} = 0,0288

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

{road}_{grade,1} (720) = \frac{h_{int} (799) - h_{int} (520)}{799 - (520)} = \frac{121,2000 - 132,5027}{279} = - 0,0405

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

The smoothed and interpolated altitude is calculated as:

h_{int,sm,1} (0) = h_{int} (0) + {road}_{grade,1} (0) = 120,3 + 0,0033 \approx 120,3033m

h_{int,sm,1} (799) = h_{int,sm,1} (798) + {road}_{grade,1} (799) = 121,2550 - 0,0220 = 121,2330m

Second smoothing run:

{road}_{grade,2} (0) = \frac{h_{int,sm,1} (200) - h_{int,sm,1} (0)}{(200)} = \frac{119,9618 - 120,3033}{(200)} = - 0,0017

chosen to demonstrate the smoothing for d ≤ 200m

{road}_{grade,2} (320) = \frac{h_{int,sm,1} (520) - h_{int,sm,1} (120)}{(520) - (120)} = \frac{123,6809 - 120,1843}{400} = 0,0087

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

{road}_{grade,2} (720) = \frac{h_{int,sm,1} (799) - h_{int,sm,1} (520)}{799 - (520)} = \frac{121,2330 - 123,6809}{279} = - 0,0088

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

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. values in the column road_grade,2(d) in Table 2. For the entire data set the total covered distance was $d_{tot} = 139,7 km$ and all positive interpolated and smoothed road grades were of 516m. Therefore the positive cumulative elevation gain reached 516*100/139,7=370m/100km.

Table 1

Correction of instantaneous vehicle altitude data

Timet [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]
— denotes data gaps

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]	t₀[s]	d₀[m]	d₁[m]	h₀[m]	h₁[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

Table 3

Calculation of the positive elevation gain

d[m]	t₀[s]	d₀[m]	d₁[m]	h₀[m]	h₁[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: Substituted by Commission Regulation (EU) 2018/1832 of 5 November 2018 amending Directive 2007/46/EC of the European Parliament and of the Council, Commission Regulation (EC) No 692/2008 and Commission Regulation (EU) 2017/1151 for the purpose of improving the emission type approval tests and procedures for light passenger and commercial vehicles, including those for in-service conformity and real-driving emissions and introducing devices for monitoring the consumption of fuel and electric energy (Text with EEA relevance).