Formulas

Route & timetable builder

buffer_time

Safety margin added to the timetable, sized by how delay-prone each country's network is.

The formula

$$t_{buffer,l} = t_{drive,l} \times q_{buffer,country(l)}$$

	Symbol	Meaning	Unit	Where it comes from
Input	t_drive,l	Driving time on one country leg (the part of a segment within one country)	min	computed upstream
Input	q_buffer,country(l)	That country's schedule buffer, as a share of driving time	fraction	track_buffer_quota_per
Result	t_buffer,l	Buffer time added for this country leg	min	—

Feeds into: total_time_per_leg

total_time_per_leg

Scheduled time for one country leg: driving, stop braking and acceleration, and buffer.

The formula

$$t_{total,l} = t_{drive,l} + t_{dyn,l} + t_{buffer,l}$$

	Symbol	Meaning	Unit	Where it comes from
Input	t_drive,l	Driving time at cruise speed	min	computed upstream
Input	t_dyn,l	Time lost braking into and accelerating out of stops on this leg	min	stop_dynamics_time_loss
Input	t_buffer,l	Schedule buffer for this leg	min	buffer_time
Result	t_total,l	Scheduled time for the country leg	min	—

Feeds into: auto_stop_added_time, total_time_per_segment

total_time_per_segment

Scheduled time between two neighbouring stops, summed over its country legs.

The formula

$$t_{seg} = \sum_{l \in seg} t_{total,l} + t_{slack,seg}$$

	Symbol	Meaning	Unit	Where it comes from
Input	t_total,l	Scheduled time of each country leg in the segment	min	total_time_per_leg
Input	t_slack,seg	Slack time from night-window stretching (usually zero)	min	computed upstream
Result	t_seg	Scheduled time between the two stops	min	—

Feeds into: arrival_time, total_time

avg_speed

Distance over pure driving time, buffers excluded — shown for orientation only.

The formula

$$\bar{v}_{kmh} = \frac{d_{km}}{t_{drive,h}}$$

	Symbol	Meaning	Unit	Where it comes from
Input	d_km	Distance	km	computed upstream
Input	t_drive,h	Pure driving time	h	computed upstream
Result	v̄_kmh	Average speed	km/h	—

dwell_time_boarding

How long the train waits at a stop where passengers only board.

The formula

$$t_{dwell} = \max(t_{board,comp},\ t_{board,infra})$$

	Symbol	Meaning	Unit	Where it comes from
Input	t_board,comp	Minimum boarding time the train composition needs	min	composition_type_min_boarding_time
Input	t_board,infra	Minimum boarding time the station needs	min	track_min_boarding_time
Result	t_dwell	Waiting time at the stop	min	—

dwell_time_alighting

How long the train waits at a stop where passengers only get off.

The formula

$$t_{dwell} = \max(t_{alight,comp},\ t_{alight,infra})$$

	Symbol	Meaning	Unit	Where it comes from
Input	t_alight,comp	Minimum alighting time the train composition needs	min	composition_type_min_alighting_time
Input	t_alight,infra	Minimum alighting time the station needs	min	track_min_alighting_time
Result	t_dwell	Waiting time at the stop	min	—

dwell_time_both

How long the train waits at a stop where passengers both board and get off.

The formula

$$t_{dwell} = \max(t_{board,comp},\ t_{board,infra},\ t_{alight,comp},\ t_{alight,infra})$$

	Symbol	Meaning	Unit	Where it comes from
Input	t_board,comp	Minimum boarding time the train composition needs	min	composition_type_min_boarding_time
Input	t_board,infra	Minimum boarding time the station needs	min	track_min_boarding_time
Input	t_alight,comp	Minimum alighting time the train composition needs	min	composition_type_min_alighting_time
Input	t_alight,infra	Minimum alighting time the station needs	min	track_min_alighting_time
Result	t_dwell	Waiting time at the stop	min	—

Feeds into: crew_eur, driver_eur, auto_stop_added_time, departure_time

auto_stop_added_time

Extra travel time an additional stop would cost: detour, stop dynamics and the wait itself.

The formula

$$\Delta t_{cand} = \left(\sum_{l \in reroute(a, cand, b)} t_{total,l}\right) - t_{total,(a,b)} + t_{dwell,cand}$$

	Symbol	Meaning	Unit	Where it comes from
Input	Σ t_total (reroute)	Travel time of the leg rerouted via the candidate stop	min	total_time_per_leg
Input	t_total,(a,b)	Travel time of the original leg without the stop	min	total_time_per_leg
Input	t_dwell,cand	Waiting time at the candidate stop	min	dwell_time_both
Result	Δt_cand	Added travel time if the stop is included	min	—

stop_dynamics_time_loss

Time lost braking into and accelerating out of every stop, which routing alone ignores.

The formula

$$\Delta t_{leg} = \underbrace{\frac{v}{2\,a_{dec}}}_{braking} + \underbrace{t_{acc}(v) - \frac{d_{acc}(v)}{v}}_{acceleration},\quad t_{acc},d_{acc}\ \text{from}\ F(u)=\min\!\left(F_{loco},\ \frac{P_{loco}}{u}\right),\ m = m_{coaches} + m_{loco}$$

	Symbol	Meaning	Unit	Where it comes from
Input	v	Cruise speed on the line before and after the stop	km/h	computed upstream
Input	a_dec	Comfortable service braking deceleration	m/s²	TRACTION_BRAKE_DECELERATION_MS2
Input	F_loco	Locomotive pulling force from standstill	kN	TRACTION_LOCO_TRACTIVE_EFFORT_KN
Input	P_loco	Locomotive power	kW	TRACTION_LOCO_POWER_KW
Input	m_coaches	Weight of all coaches of the composition	t	coach_type_weight_gross_t
Input	m_loco	Weight of the assumed standard locomotive	t	loco_type_weight_t
Result	Δt_leg	Time lost per stop compared to passing at constant speed	min	—

Feeds into: total_time_per_leg

arrival_time

When the train reaches a stop.

The formula

$$t_{arr,i} = t_{dep,i-1} + t_{seg,i-1}$$

	Symbol	Meaning	Unit	Where it comes from
Input	t_dep,i-1	Departure time at the previous stop	hh:mm	departure_time
Input	t_seg,i-1	Scheduled travel time of the segment in between	min	total_time_per_segment
Result	t_arr,i	Arrival time at the stop	hh:mm	—

Feeds into: departure_time

departure_time

When the train leaves an intermediate stop.

The formula

$$t_{dep,i} = t_{arr,i} + t_{dwell,i}$$

	Symbol	Meaning	Unit	Where it comes from
Input	t_arr,i	Arrival time at the stop	hh:mm	arrival_time
Input	t_dwell,i	Waiting time at the stop	min	dwell_time_both
Result	t_dep,i	Departure time at the stop	hh:mm	—

Feeds into: arrival_time

total_distance

Total trip distance across all country legs.

The formula

$$d_{total} = \sum_{seg} \sum_{l \in seg} d_{m,l}$$

	Symbol	Meaning	Unit	Where it comes from
Input	d_m,l	Distance of each country leg	m	computed upstream
Result	d_total	Total trip distance	m	—

total_driving_time

Pure driving time across all country legs, without buffers or waiting times.

The formula

$$t_{drive,total} = \sum_{seg} \sum_{l \in seg} t_{drive,l}$$

	Symbol	Meaning	Unit	Where it comes from
Input	t_drive,l	Driving time of each country leg	min	computed upstream
Result	t_drive,total	Total driving time of the trip	min	—

total_time

Total scheduled trip time, including buffers, waiting times and stop dynamics.

The formula

$$t_{total} = \sum_{seg} t_{seg}$$

	Symbol	Meaning	Unit	Where it comes from
Input	t_seg	Scheduled time of each segment	min	total_time_per_segment
Result	t_total	Total scheduled trip time	min	—

Energy model

energy_per_leg

Electricity used on one country leg: traction, rolling and air resistance, and on-board power.

The formula

$$E_{kWh,l} = A \cdot m_t + d_{km,l} \left( B + C \cdot m_t + (K_m m_t + K_L L_m) \cdot \bar{v}^2_{kmh,l} \right) + (P_{loco} + P_{hotel} \cdot n_{coach}) \cdot t_{h,l}$$

	Symbol	Meaning	Unit	Where it comes from
Input	m_t	Coach train weight at 80% load, locomotive excluded	t	train_weight
Input	L_m	Coach rake length, locomotive excluded	m	computed upstream
Input	n_coach	Number of coaches	1	computed upstream
Input	d_km,l	Distance of the country leg	km	computed upstream
Input	v̄_kmh,l	Average speed on the leg	km/h	avg_speed
Input	t_h,l	Driving time on the leg, dynamics included	h	computed upstream
Input	A, B, C, K_m, K_L, P_loco, P_hotel	Calibrated coefficients, fleet-wide. Fitted against DB Trassenfinder technical runs; the values live in models/energy/calibrated_coefficients.py and are regenerated by calib/02_energy_calibration.ipynb	see models/energy/calibrated_coefficients.py	computed upstream
Result	E_kWh,l	Electricity used on the country leg	kWh	—

Feeds into: energy_eur, energy_per_km, total_energy

energy_per_km

Energy use per kilometre on a country leg, for display and comparison between countries.

The formula

$$e_{kWh/km,l} = \frac{E_{kWh,l}}{d_{km,l}}$$

	Symbol	Meaning	Unit	Where it comes from
Input	E_kWh,l	Electricity used on the country leg	kWh	energy_per_leg
Input	d_km,l	Distance of the country leg	km	computed upstream
Result	e_kWh/km,l	Energy use per kilometre	kWh/km	—

total_energy

Total electricity used across the whole trip.

The formula

$$E_{total} = \sum_{seg} \sum_{l \in seg} E_{kWh,l}$$

	Symbol	Meaning	Unit	Where it comes from
Input	E_kWh,l	Electricity used on each country leg	kWh	energy_per_leg
Result	E_total	Total trip energy use	kWh	—

avg_speed

Average speed on a country leg; feeds the air-resistance term of the energy model.

The formula

$$\bar{v}_{kmh,l} = \frac{d_{km,l}}{t_{drive,h,l}}$$

	Symbol	Meaning	Unit	Where it comes from
Input	d_km,l	Distance of the country leg	km	computed upstream
Input	t_drive,h,l	Driving time on the leg	h	computed upstream
Result	v̄_kmh,l	Average speed on the leg	km/h	—

Feeds into: energy_per_leg

train_weight

Train weight as the energy model uses it: coaches at 80% load, locomotive excluded.

The formula

$$m_t = \sum_{coach} m_{coach,t}$$

	Symbol	Meaning	Unit	Where it comes from
Input	m_coach,t	Weight of each coach	t	coach_type_weight_gross_t
Result	m_t	Coach train weight, locomotive excluded	t	—

Feeds into: energy_per_leg

Cost allocation to accommodation classes

Cost share by accommodation class — class_main_allocation

How costs shared by the whole train are split between accommodation classes.

The formula

$$s_{c} = (1-f_{svc})\left(X \frac{L_c}{L_{rev}} + (1-X)\frac{W_c}{W_{rev}}\right) + f_{svc}\frac{P_c}{P}$$

	Symbol	Meaning	Unit	Where it comes from
Input	L_c	Length of the class's sections in the train	m	section_length_m
Input	L_rev	Length of all passenger space in the train	m	coach_type_length_wo_service_m
Input	W_c	Weight of the class's sections in the train	t	section_weight_t
Input	W_rev	Weight of all passenger space in the train	t	coach_type_weight_wo_service_t
Input	X	Weighting between length and weight (0.7 = 70% length)	fraction	composition_type_length_cost_prop
Input	f_svc	Share of the train that is service area (dining etc.)	fraction	computed upstream
Input	P_c	Places of this class on the train	places	coach_type_class_places
Input	P	All places on the train	places	coach_type_class_places
Result	s_c	Share of a shared cost carried by the class	fraction	—

Feeds into: per_sold_place_km_by_class

Cost per sold place-km, by class — per_sold_place_km_by_class

A class's cost per sold place-kilometre; empty berths make the sold ones dearer.

The formula

$$c_{c} = \frac{s_{c} \cdot C}{pkm^{sold}_{c}}$$

	Symbol	Meaning	Unit	Where it comes from
Input	s_c	Share of the cost carried by the class	fraction	class_main_allocation
Input	C	The cost being normalised	€/year	computed upstream
Input	pkm_sold,c	Sold place-kilometres of the class per year	place-km/year	set by you
Result	c_c	Cost per sold place-kilometre of the class	€/place-km	—

Upstream derivations

Roster efficiency (Dienstplanwirkungsgrad) — roster_efficiency_driver

The share of paid staff hours that is actually productive.

The formula

$$\eta = \eta_{ref} \cdot \frac{t_{train,h}}{t_{train,h} + t_{relief} \cdot (n_{duty} - 1)}, \quad n_{duty} = \left\lceil \frac{t_{basis,h}}{t_{duty,max}} \right\rceil$$

	Symbol	Meaning	Unit	Where it comes from
Input	eta_ref	Efficiency when the trip fits a single shift (operator_crew_roster_eff_ref for onboard staff)	–	operator_driver_roster_eff_ref
Input	t_train,h	Time the staff member is on the train	h	computed upstream
Input	t_basis,h	Hours measured against the shift cap — driving time for drivers, time on train for onboard staff	h	computed upstream
Input	t_duty,max	Longest permitted shift (operator_crew_max_duty_h for onboard staff)	h	operator_driver_max_duty_h
Input	t_relief	Unproductive hours added per crew handover	h	operator_relief_allowance_h
Result	eta	Productive share of paid hours for this trip	–	—

Feeds into: crew_eur, driver_eur

Night share of a country run (track access) — tac_night_share

How much of a country run is charged at that country's night track rate.

The formula

$$\nu_c = \begin{cases} 0 & \text{no night tariff} \\ 1 & \text{widening applies} \\ \dfrac{|[t_{in}, t_{out}) \cap B_{night,c}|}{t_{out} - t_{in}} & \text{otherwise}\end{cases}$$

	Symbol	Meaning	Unit	Where it comes from
Input	t_in, t_out	When the train enters and leaves the country on this segment	min	computed upstream
Input	B_night,c	The country's night tariff band (band end: track_tac_night_band_end)	time of day	track_tac_night_band_start
Result	nu_c	Share of the country run priced at the night rate	–	—

Feeds into: tac_eur

Rush-hour share of a country run (track access) — tac_peak_share

How much of a country run falls inside a rush-hour surcharge window.

The formula

$$\pi_c = w \cdot \frac{\sum_{j} |[t_{in}, t_{out}) \cap B_{peak,c,j}|}{t_{out} - t_{in}}, \quad w = \tfrac{5}{7} \text{ if weekdays only, else } 1$$

	Symbol	Meaning	Unit	Where it comes from
Input	t_in, t_out	When the train enters and leaves the country on this segment	min	computed upstream
Input	B_peak,c,j	The country's two daily peak bands (track_tac_peak_band1_* and track_tac_peak_band2_*)	time of day	track_tac_peak_band1_start
Input	w	Weekday blend, applied where the bands run Monday to Friday only	–	WEEKDAY_BLEND
Result	pi_c	Share of the country run falling in rush hour	–	—

Feeds into: tac_eur

Night share of a country run (electricity) — energy_night_share

How much of a country leg's electricity is billed at the night rate.

The formula

$$\nu^{E}_{c} = \frac{|[t_{in},t_{out}] \cap B^{E}_{c}|}{t_{out}-t_{in}}$$

	Symbol	Meaning	Unit	Where it comes from
Input	t_in, t_out	When the train enters and leaves the country on this segment	min	computed upstream
Input	B^E_c	The country's electricity night band (track_energy_night_band_start and _end)	time of day	track_energy_night_band_start
Result	nu^E_c	Share of the country leg billed at the night rate	–	—

Feeds into: energy_eur

Generic aggregation

Generic level total — total_eur

The sum of the items directly below this one in the breakdown.

The formula

$$x_{total} = \sum_i x_i$$

	Symbol	Meaning	Unit	Where it comes from
Input	x_i	The individual items on that level	€/year	computed upstream
Result	x_total	Sum of the level's items	€/year	—

Found a mistake?

This documentation describes a model that is still being built, and some of it will be wrong. If a number looks off, a source is missing, or an explanation does not hold up, tell us — it reaches the working group with the page you were reading.