‘Appendix 7a

Verification of overall trip dynamics

1.   INTRODUCTION

This Appendix describes the calculation procedures to verify the overall trip dynamics, to determine the overall excess or absence of dynamics during urban, rural and motorway driving.

2.   SYMBOLS

RPA	relative positive acceleration
‘acceleration resolution ares ’	minimum acceleration > 0 measured in m/s2
T4253H	compound data smoother
‘positive acceleration apos ’	acceleration [m/s2] greater than 0,1 m/s2

Index (i) refers to the time step

Index (j) refers to the time step of positive acceleration datasets

Index (k) refers to the category (t = total, u = urban, r = rural, m = motorway)

Δ	—	difference
>	—	larger
≥	—	larger or equal
%	—	per cent
<	—	smaller
≤	—	smaller or equal
a	—	acceleration [m/s2]
ai	—	acceleration in time step i [m/s2]
apos	—	positive acceleration greater than 0,1 m/s2 [m/s2]
apos,i,k	—	positive acceleration greater than 0,1 m/s2 in time step i considering the urban, rural and motorway shares [m/s2]
ares	—	acceleration resolution [m/s2]
di	—	distance covered in time step i [m]
di,k	—	distance covered in time step i considering the urban, rural and motorway shares [m]
Mk	—	number of samples for urban, rural and motorway shares with positive acceleration greater than 0,1 m/s2
Nk	—	total number of samples for the urban, rural and motorway shares and the complete trip
RPAk	—	relative positive acceleration for urban, rural and motorway shares [m/s2 or kWs/(kg × km)]
tk	—	duration of the urban, rural and motorway shares and the complete trip [s]
v	—	vehicle speed [km/h]
vi	—	actual vehicle speed in time step i [km/h]
vi,k	—	actual vehicle speed in time step i considering the urban, rural and motorway shares [km/h]
(v · a)i	—	actual vehicle speed per acceleration in time step i [m2/s3 or W/kg]
(v · apos)j,k	—	actual vehicle speed per positive acceleration greater than 0,1 m/s2 in time step j considering the urban, rural and motorway shares [m2/s3 or W/kg].
(v · apos)k _[95]	—	95th percentile of the product of vehicle speed per positive acceleration greater than 0,1 m/s2 for urban, rural and motorway shares [m2/s3 or W/kg]
	—	average vehicle speed for urban, rural and motorway shares [km/h]

3.   TRIP INDICATORS

3.1.   Calculations

3.1.1.   Data pre-processing

Dynamic parameters like acceleration, v · apos or RPA shall be determined with a speed signal of an accuracy of 0,1 % above 3 km/h and a sampling frequency of 1 Hz. This accuracy requirement is generally fulfilled by wheel (rotational) speed signals.

The speed trace shall be checked for faulty or implausible sections. The vehicle speed trace of such sections is characterised by steps, jumps, terraced speed traces or missing values. Short faulty sections shall be corrected, for example by data interpolation or benchmarking against a secondary speed signal. Alternatively, short trips containing faulty sections could be excluded from the subsequent data analysis. In a second step the acceleration values shall be ranked in ascending order, in order to determine the acceleration resolution ares = (minimum acceleration value > 0).

If ares ≤ 0,01 m/s 2, the vehicle speed measurement is accurate enough.

If 0,01 < ares ≤ rmax m/s2, smoothing by using a T4253 Hanning filter.

If ares > rmax m/s2, the trip is invalid.

The T4253 Hanning filter performs the following calculations: The smoother starts with a running median of 4, which is centred by a running median of 2. It then re-smoothes these values by applying a running median of 5, a running median of 3, and Hanning (running weighted averages). Residuals are computed by subtracting the smoothed series from the original series. This whole process is then repeated on the computed residuals. Finally, the smoothed residuals are computed by subtracting the smoothed values obtained the first time through the process.

The correct speed trace builds the basis for further calculations and binning as described in paragraph 3.1.2.

3.1.2.   Calculation of distance, acceleration and v · a

The following calculations shall be performed over the whole time-based speed trace (1 Hz resolution) from second 1 to second tt (last second).

The distance increment per data sample shall be calculated as follows:

di = vi /3,6, i = 1 to Nt

Where:

di is the distance covered in time step i [m]

v i is the actual vehicle speed in time step i [km/h]

N t is the total number of samples

The acceleration shall be calculated as follows:

ai = (v i + 1 – v i – 1)/(2 · 3,6), i = 1 to Nt

Where:

ai is the acceleration in time step i [m/s2]. For i = 1: vi – 1 = 0, for i = Nt : vi + 1 = 0.

The product of vehicle speed per acceleration shall be calculated as follows:

(v · a)i = vi · ai /3,6, i = 1 to Nt

Where:

(v · a)i is the product of the actual vehicle speed per acceleration in time step i [m2/s3 or W/kg].

3.1.3.   Binning of the results

After the calculation of ai and (v · a)i , the values vi , di , ai and (v · a)i shall be ranked in ascending order of the vehicle speed.

All datasets with vi ≤ 60 km/h belong to the ‘urban’ speed bin, all datasets with 60 km/h < vi ≤ 90 km/h belong to the ‘rural’ speed bin and all datasets with vi > 90 km/h belong to the ‘motorway’ speed bin.

The number of datasets with acceleration values ai > 0,1 m/s2 shall be bigger or equal to 150 in each speed bin.

For each speed bin the average vehicle speed

shall be calculated as follows:

, i = 1 to Nk,k = u,r,m

Where:

Nk is the total number of samples of the urban, rural, and motorway shares.

3.1.4.   Calculation of v · apos_[95] per speed bin

The 95th percentile of the v · apos values shall be calculated as follows:

The (v · a)i,k values in each speed bin shall be ranked in ascending order for all datasets with ai,k ≥ 0,1 m/s 2 and the total number of these samples Mk shall be determined.

Percentile values are then assigned to the (v · apos)j,k values with ai,k ≥ 0,1 m/s 2 as follows:

The lowest v · apos value gets the percentile 1/Mk , the second lowest 2/Mk , the third lowest 3/Mk and the highest value Mk/Mk =100 %.

(v · apos)k _[95] is the (v · apos)j,k value, with j/Mk = 95 %. If j/Mk = 95 % cannot be met, (v · apos)k _[95] shall be calculated by linear interpolation between consecutive samples j and j + 1 with j/Mk < 95 % and (j + 1)/Mk > 95 %.

The relative positive acceleration per speed bin shall be calculated as follows:

RPAk = Σ j (Δt · (v · apos ) j,k )/Σ idi,k , j = 1 to Mk,i = 1 to Nk,k = u,r,m

Where:

RPAk is the relative positive acceleration for urban, rural and motorway shares in [m/s2 or kWs/(kg*km)]

Δt	time difference equal to 1 second
Mk	the sample number for urban, rural and motorway shares with positive acceleration
Nk	the total sample number for urban, rural and motorway shares.

4.   VERIFICATION OF TRIP VALIDITY

4.1.1.   Verification of v*apos_[95] per speed bin (with v in [km/h])

If

and

is fulfilled, the trip is invalid.

If

and

is fulfilled, the trip is invalid.

4.1.2.   Verification of RPA per speed bin

If

and

is fulfilled, the trip is invalid.

If

and RPAk < 0,025 is fulfilled, the trip is invalid.

‘Appendix 7b

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]
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]
da	—	reference way point at d(0) [m]
de	—	cumulative distance travelled until the last discrete way point [m]
di	—	instantaneous distance [m]
dtot	—	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]
hcorr(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]
hcorr(t)	—	corrected instantaneous vehicle altitude at data point t [m above sea level]
hcorr(t-1)	—	corrected instantaneous vehicle altitude at data point t-1 [m above sea level]
hGPS,i	—	instantaneous vehicle altitude measured with GPS [m above sea level]
hGPS(t)	—	vehicle altitude measured with GPS at data point t [m above sea level]
hint(d)	—	interpolated altitude at the discrete way point under consideration d [m above sea level]
hint,sm,1(d)	—	smoothed interpolated altitude, after the first smoothing run at the discrete way point under consideration d [m above sea level]
hmap(t)	—	vehicle altitude based on topographic map at data point t [m above sea level]
Hz	—	hertz
km/h	—	kilometre per hour
m	—	metre
roadgrade,1(d)	—	smoothed road grade at the discrete way point under consideration d after the first smoothing run [m/m]
roadgrade,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]
t0	—	time passed at the measurement directly located before the respective way point d [s]
vi	—	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 hGPS,i [m above sea level] as measured with the GPS, the instantaneous vehicle speed vi [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:

|hGPS(t) – hmap(t)| > 40 m

The altitude correction shall be applied so that:

h(t) = hmap(t)

where:

h(t)	—	vehicle altitude after the screening and principle verification of data quality at data point t [m above sea level]
hGPS(t)	—	vehicle altitude measured with GPS at data point t [m above sea level]
hmap(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 * sin45°)

The altitude correction shall be applied so that:

hcorr(t) = hcorr (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]
hcorr(t)	—	corrected instantaneous vehicle altitude at data point t [m above sea level]
hcorr(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 dtot [m] covered by a trip shall be determined as sum of the instantaneous distances di . The instantaneous distance di shall be determined as:

Where:

di	—	instantaneous distance [m]
vi	—	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 hcorr(t) as:

Where:

hint(d)	—	interpolated altitude at the discrete way point under consideration d [m above sea level]
hcorr(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]
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; da and de denote the first and last data point respectively (Figure 1). The first smoothing run shall be applied as follows:

for d ≤ 200 m

for 200 m < d < (de – 200 m)

for d ≥ (de – 200 m)

h int,sm,1(d) = h int,sm,1(d – 1 m) + road grade,1(d), d = da + 1 to de

h int,sm,1(da ) = hint (da ) + road grade,1(da )

Where:

roadgrade,1(d)	—	smoothed road grade at the discrete way point under consideration after the first smoothing run [m/m]
hint(d)	—	interpolated altitude at the discrete way point under consideration d [m above sea level]
hint,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]
da	—	reference way point at a distance of zero metres [m]
de	—	cumulative distance travelled until the last discrete way point [m].

The second smoothing run shall be applied as follows:

for d ≤ 200 m

for 200 m < d < (de – 200 m)

for d ≥ (de – 200 m)

Where:

roadgrade,2(d)	—	smoothed road grade at the discrete way point under consideration after the second smoothing run [m/m]
hint,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]
da	—	reference way point at a distance of zero metres [m]
de	—	cumulative distance travelled until the last discrete way point [m].

Figure 1

Illustration of the procedure to smooth the interpolated altitude signals

hint (d – 200m)

or

hint,sm,1 (d – 200m)

hint

or

hint,sm,1

[m above the sea level]

roadgrade,1 (d)

or

roadgrade,2 (d)

hint (d)

or

hint,sm,1 (d)

hint (d + 200m)

or

hint,sm,1 (d + 200m)

d [m]

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. roadgrade,2(d). The result should be normalised by the total test distance dtot 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:

hGPS(t) – hmap(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 * sin45°)

As result of the applied data correction, the data in the sixth column hcorr(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 di 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 hint(d) (Column 7 in Table 2). The altitude of each discrete way point d is calculated through interpolation of the measured instantaneous altitude hcorr as:

5.3.2.   Additional data smoothing

In Table 2, the first and last discrete way points are: da  = 0 m and de  = 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:

chosen to demonstrate the smoothing for d ≤ 200 m

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

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

The smoothed and interpolated altitude is calculated as:

h int,sm,1(0) = hint (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:

chosen to demonstrate the smoothing for d ≤ 200 m

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

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. roadgrade,2(d). For the presented example the total covered distance was dtot  = 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

Time t [s]	v(t) [km/h]	hGPS(t) [m]	hmap(t) [m]	h(t) [m]	hcorr(t) [m]	di [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
- denotes data gaps

Table 2

Calculation of road grade

d [m]	t0 [s]	d0 [m]	d1 [m]	h0 [m]	h1 [m]	hint(d) [m]	roadgrade,1(d) [m/m]	hint,sm,1(d) [m]	roadgrade,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 hGPS(t), the altitude profile provided by the topographic map hmap(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

hmap(t)

hcorr(t)

h(t)

hGPS(t)

Time [s]

Altitude [m above sea level]

Figure 3

Comparison between the corrected altitude profile hcorr(t) and the smoothed and interpolated altitude hint,sm,1

hint,sm,1

hcorr(t)

Time [s]

Altitude [m above sea level]

Table 2

Calculation of the positive elevation gain

d [m]	t0 [s]	d0 [m]	d1 [m]	h0 [m]	h1 [m]	hint(d) [m]	roadgrade,1(d) [m/m]	hint,sm,1(d) [m]	roadgrade,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