How MRT Energy Use Is Optimised Using Mathematics

An MRT train can arrive at the next station at exactly the right time in many different ways.

It can accelerate hard, hold a high speed and brake later.

It can accelerate more gently, reach a lower peak speed, coast for part of the journey and begin braking earlier.

It can even alter its speed profile so that its acceleration overlaps with another train’s regenerative braking.

All three journeys may cover the same distance.

All three may satisfy the same timetable.

They do not necessarily use the same amount of electricity.

The energy problem of an MRT is not merely how much power the train needs. It is when that power is needed, how motion is shaped, and whether energy released by one train can become useful to another.

This is why MRT energy optimisation sits at the intersection of physics, calculus, electrical engineering, timetable design, statistics, data analytics, control theory and operations research.

Singapore provides a particularly useful real example. SMRT’s current decarbonisation information says Phase 1 of its Green Communications-Based Train Control project on the North-South and East-West Lines has delivered around 17 GWh of annual energy savings. SMRT reports an 8 per cent reduction in traction energy from Phase 1 and says the wider project is intended to achieve up to a 15 per cent reduction in traction energy.

These are not savings created by asking passengers to tolerate a dramatically slower railway.

They come from improving the mathematical relationship among train movement, signalling, timetables and energy use.

This article continues the eduKateSG MRT mathematics series. Begin with How MRT Timing Works Using Mathematics, then explore MRT braking, headway, station dwell, delay recovery, and timetable construction.

Now we follow the electricity.

The RFE — Why Does MRT Energy Mathematics Exist?

The simplest objective would be:

use as little electricity as possible

That is incomplete.

A train that never moves uses very little traction energy.

It also fails completely as public transport.

The real objective is to reduce unnecessary energy while preserving the railway’s whole job:

Safety
+ passenger capacity
+ punctuality
+ regular headway
+ comfort
+ accessibility
+ reliability
+ recoverability
+ energy efficiency

So the Reason for Existence is not austerity.

It is intelligent use.

deliver the required transport service
with the least avoidable energy,
without transferring the cost
into slower, less reliable or less accessible journeys

This gives us the governing principle for the article:

A good energy-saving railway does not merely consume less. It performs the same human job more intelligently.

Prompt 1 — Where Does an MRT Railway Use Energy?

When people think about railway energy, they usually imagine the motors that move the train.

That is traction energy.

But a working MRT uses electricity for many other jobs.

Train traction and acceleration
Train air-conditioning, lighting, controls and passenger systems
Station air-conditioning and ventilation
Escalators and lifts
Lighting and platform systems
Signalling and communications
Pumps, fans, control rooms and depots
Maintenance and testing equipment

So total railway energy can be divided conceptually into:

Etotal
=
Etraction
+ Etrain auxiliaries
+ Estations
+ Esignalling
+ Edepots
+ Eother

SMRT’s November 2024 Green CBTC release stated that traction power accounted for about 50 per cent of energy across the SMRT Trains network at that time. This is a reminder that train movement is a major energy job, but not the only one.

LTA also describes energy-saving station systems such as energy-recovering lifts, dual-speed escalators, hybrid cooling and smarter air-conditioning. SMRT has separately used data-driven facilities management to optimise station HVAC while preserving commuter comfort.

This creates two related optimisation problems:

How can trains move with less traction energy?

How can the whole railway operate with less total energy?

This article concentrates on the first, while keeping the second visible.

Power is not energy

These two words are often confused.

Power is the rate at which energy is transferred or used.

P = dE/dt

Power is measured in watts, kilowatts or megawatts.

Energy is accumulated power over time.

E = ∫P(t)dt

Energy is measured in joules or, for electrical systems, often kilowatt-hours and megawatt-hours.

A train may draw very high power for a short acceleration but use less total energy than another train drawing moderate power for much longer.

That means railway engineers care about both:

total energy consumed
and
maximum power demanded

Reducing total kilowatt-hours lowers consumption.

Reducing simultaneous megawatt peaks can reduce stress on the traction-power system and change the infrastructure needed to support the service.

Energy asks “how much altogether?” Power asks “how hard right now?”

Prompt 2 — How Does Train Motion Become an Energy Equation?

A train moves because traction force acts against resistance, gradient and the need to change speed.

A simplified longitudinal force equation is:

meff dv/dt
=
Ftraction
− Fbrake
− R(v)
− mg sin θ

where:

m_eff is effective moving mass,
F_traction is motor force,
F_brake is braking force,
R(v) represents resistance,
θ represents track gradient.

Mechanical traction power is approximately:

Pmechanical = Ftraction v

If motors and electrical conversion have an overall efficiency η:

Pelectrical ≈ Pmechanical/η

Electrical traction energy over a journey is then approximately:

Etraction = ∫Pelectrical(t)dt

This integral is the area under the electrical power-time graph.

Every acceleration pulse adds positive area.

Coasting can reduce positive traction power towards zero.

Regenerative braking can make train traction power negative from the perspective of the electrical network: the train is returning power rather than drawing it.

Kinetic energy

A moving train stores kinetic energy:

Ek = ½mv²

Velocity is squared.

Double speed and kinetic energy becomes four times as large for the same mass.

Consider a deliberately hypothetical 220,000 kg train travelling at 20 m/s, or 72 km/h.

Ek = ½(220,000)(20²)
   = 44,000,000 J
   = 44 MJ

Since one kilowatt-hour is 3.6 MJ:

44 MJ / 3.6
≈ 12.2 kWh

This is only the train’s idealised translational kinetic energy at that instant. It is not the total grid energy used for the journey. Real electrical losses, rotating components, auxiliaries, resistance, gradients and braking efficiency make the full accounting more complex.

But the example reveals the scale of energy repeatedly built up and removed between stations.

Potential energy and gradient

When a train climbs, it gains gravitational potential energy:

Ep = mgh

For the same hypothetical 220,000 kg train climbing 10 metres:

Ep = 220,000 × 9.81 × 10
   ≈ 21.6 MJ
   ≈ 6.0 kWh

Going downhill can return part of that energy through reduced traction demand or regenerative braking, but not perfectly. Efficiency losses and operating constraints remain.

This is why line topography belongs in the energy model.

SMRT’s Green CBTC description explicitly says line topography and train speed at different locations are integrated into the software when coasting locations are determined.

Resistance

A common generic representation of train resistance is:

R(v) = A + Bv + Cv²

The exact constants depend on the train and conditions.

The v² term means resistance can grow strongly at higher speed.

Therefore a slightly higher cruising speed can require disproportionately more power.

The fastest physically permitted journey is not automatically the lowest-energy journey.

Prompt 3 — Why Is the Fastest Speed Profile Often Not the Most Efficient?

Imagine two speed profiles that both reach the next station on time.

Profile A — sprint and brake

maximum acceleration
→ high cruise speed
→ late braking
→ stop

Profile B — shape the journey

appropriate acceleration
→ shorter cruise
→ coast
→ controlled braking
→ stop

If the timetable contains enough running time, Profile B may use less positive traction energy.

Why?

It may avoid building unnecessary kinetic energy.
It may spend less time at speeds where resistance is high.
It may use momentum during coasting instead of continued motor power.
It may create a braking event better aligned with electrical receptivity elsewhere.

But Profile B may take slightly longer.

This creates a Pareto trade-off:

lower journey time ↔ lower energy

A solution is Pareto-efficient when no objective can be improved without worsening another.

One speed profile may be faster but more energy-intensive.

Another may save more energy but use too much timetable margin.

The useful operating point lies between them.

A simplified objective can be written:

minimise J
=
wE Egrid
+ wT (Tarrival − Ttarget)²
+ wJ ∫j(t)²dt
+ wH headway penalty

subject to:

v(x) ≤ vmax(x)
a_min ≤ a(t) ≤ a_max
|j(t)| ≤ jmax
safe separation maintained
arrival inside timetable window

The weights describe the priorities assigned to energy, punctuality, comfort and regularity.

Research on energy-efficient train control often finds optimal or near-optimal trajectories composed of regimes such as maximum or moderated acceleration, cruising, coasting and braking. The hard mathematical question is where the switching points should occur.

Energy optimisation is often the art of deciding when to stop applying power, not merely how much power the train can produce.

Prompt 4 — How Does Coasting Save Energy?

Coasting means the train continues moving while traction power is not being applied and the brakes are not being used.

The train does not stop immediately because it already has momentum.

During coasting, the simplified force equation becomes:

m dv/dt = −R(v) − mg sin θ

On level track, speed gradually falls because of resistance.

Downhill, gradient may partly offset resistance.

Uphill, speed falls faster.

The coasting decision therefore depends on:

current speed,
distance remaining,
track gradient,
speed restrictions,
required arrival time,
braking point,
the train ahead,
and available recovery margin.

SMRT’s public Green CBTC explanation says predefined coasting locations are selected using line topography and train speed, with the aim of saving energy with negligible impact on travel time. Automatic Train Operation controls the acceleration, braking and coasting actions.

A simple coasting thought experiment

Suppose a train has 500 metres before its braking region.

Option A maintains 20 m/s using traction for the whole 500 metres.

Journey time for this section is:

tA = 500/20
   = 25 s

Option B begins coasting and averages 18.5 m/s across the same distance.

tB = 500/18.5
   ≈ 27.0 s

Coasting adds about two seconds in this deliberately simple example.

But positive traction energy over that portion can fall substantially because the motors are no longer supplying the force needed to hold 20 m/s.

If the timetable already contains two seconds of usable margin, the passenger may notice no late arrival at all.

This is stored timetable flexibility being converted into energy savings.

The switching-point problem

Coast too early and the train becomes slow.

Coast too late and little energy is saved.

So the optimiser chooses a switching position x_c or switching time t_c:

traction for x < xc
coasting for xc ≤ x < xb
braking for x ≥ xb

where x_b is the braking transition.

The optimum depends on the full route and timetable, not one local equation.

This is why train trajectory optimisation is a genuine optimal-control problem.

Coasting looks like doing nothing. Mathematically, it is a carefully timed use of energy already stored in motion.

Prompt 5 — How Does Regenerative Braking Turn a Train into a Generator?

During acceleration, electrical energy becomes mechanical motion.

electrical energy
→ motor
→ traction force
→ kinetic energy

During regenerative braking, the process can partly reverse.

kinetic energy
→ traction motor acting as generator
→ electrical energy
→ traction power system

LTA states that regenerative braking systems on Singapore trains recoup about 30 per cent of the energy they use and reduce wear on braking components.

LTA also notes that the Circle Line can reuse energy from one braking train to power another, while Downtown Line trains use regenerative brakes that recover kinetic energy for other uses.

This gives us the central regenerative-energy equation:

Erecovered
=
ηmotor
× ηconversion
× ηnetwork
× ρreceptivity
× Eavailable

where:

η terms represent conversion and transmission efficiencies,
ρ_receptivity represents how much regenerated energy the network can use or store at that moment,
E_available is the kinetic and potential energy available for recovery.

The word receptivity is crucial.

A train can generate braking energy only if the electrical system has somewhere useful for that energy to go.

In a general rail system, possible destinations include:

another train that is accelerating,
other electrical loads on the traction network,
an energy-storage device,
or a reversible grid interface where such infrastructure exists.

If there is no receptive load or storage, some potentially regenerative energy may need to be dissipated through other braking means.

This is a general engineering principle; the exact architecture and operating rules vary by railway.

A hypothetical recovery calculation

Return to our hypothetical train with:

Ek = 12.2 kWh

Suppose 75 per cent remains electrically recoverable after motor and conversion losses, and the network is able to use 80 per cent of that regenerated electricity at that moment.

Euseful
= 12.2 × 0.75 × 0.80
≈ 7.3 kWh

This is not a claim about an actual Singapore train.

It illustrates the chain:

available motion energy
→ conversion efficiency
→ network receptivity
→ useful recovered energy

Improving any one link can increase useful recovery.

Why regenerative braking does not create free energy

The train first needed energy to accelerate.

Some energy was lost to resistance, electrical conversion and auxiliary loads.

Braking recovers only part of what remains.

Erecovered < Etraction supplied

Regeneration reduces waste.

It does not violate conservation of energy.

Regenerative braking is not free motion. It is a partial return of energy that would otherwise be lost.

Prompt 6 — How Can One Train’s Timing Change Another Train’s Energy Use?

Now the problem leaves the individual train.

Suppose Train A is braking and returning electrical power.

Train B is accelerating and demanding power.

If these events overlap appropriately on an electrically connected section, part of A’s regenerative output can support B’s traction demand.

Let train power P_i(t) be positive during traction and negative during regeneration.

Net traction-network power is conceptually:

Pnet(t)
=
Σ Pi(t)
+ Ploss(t)
+ Pother(t)

Suppose at one moment:

Train A regenerates:  PA = −2 MW
Train B accelerates:  PB = +3 MW

Ignoring losses and other loads for the illustration:

Pnet = −2 + 3
     = +1 MW

The external supply needs to provide only the net difference in this idealised instant.

If the two events do not overlap, Train B may demand 3 MW while Train A’s regenerative event occurs at a time when less of it can be used.

Therefore timetable offsets and speed profiles can influence both:

total grid energy
and
peak grid power

The overlap-time problem

Let A’s regenerative interval be:

[ta1, ta2]

and B’s acceleration interval be:

[tb1, tb2]

Their overlap length is:

Toverlap
=
max(0,
min(ta2,tb2) − max(ta1,tb1))

A larger overlap does not automatically guarantee maximum useful energy because power magnitudes, network losses and electrical connectivity also matter.

But it is an important scheduling variable.

Research on metro energy optimisation explicitly adjusts timetables and speed profiles to increase regenerative-energy utilisation by coordinating acceleration and braking events.

Why every train should not accelerate together

Suppose four trains each demand 3 MW at the same instant.

Ppeak = 4 × 3
      = 12 MW

If their acceleration events can be staggered without harming headway or journey time, the maximum simultaneous demand may be lower.

This is peak-shaving through scheduling.

But staggering cannot be done carelessly.

It must still respect:

passenger frequency,
safe separation,
junction order,
terminal slots,
arrival times,
and delay recovery.

The electrical optimum and the passenger optimum must be solved together.

Once trains share a power network, energy efficiency becomes a choreography problem.

Prompt 7 — How Do Timetables, Signalling and Data Analytics Optimise Energy?

A train’s speed profile is constrained by the timetable.

The timetable is constrained by headway and passenger demand.

Signalling constrains where and how fast the train may move.

The power system records the electrical result.

Therefore energy optimisation cannot be isolated from operation.

timetable
→ permitted running time
→ available speed profiles
→ acceleration/coasting/braking
→ power curve
→ measured energy

SMRT’s Green CBTC programme is a real example of this integration.

SMRT says the system analyses train movement and energy-use profiles to recommend improvements in timetables and train operations. Automatic Train Operation controls acceleration, braking and coasting. Earlier project descriptions said revenue-service operational data would be used to improve the algorithms and recommend suitable changes.

The current reported result is significant:

Phase 1 produced an 8 per cent traction-energy reduction on NSEWL.
SMRT currently reports around 17 GWh in annual energy savings.
The overall project is set to achieve up to a 15 per cent reduction in traction energy.

This is data analytics operating on a physical railway.

The prediction loop

Suppose a model predicts energy for journey j:

Êj

The railway measures actual energy:

Ej

Prediction error is:

ej = Ej − Êj

If the model systematically underestimates energy on an uphill, crowded or high-speed section, the residuals contain information.

They may indicate:

resistance parameters need recalibration,
passenger-load assumptions are wrong,
the actual speed profile differs from the planned profile,
regenerative receptivity is lower than expected,
or operational conditions have changed.

The model is then improved and tested again.

predict
→ operate
→ measure
→ compare
→ recalibrate
→ optimise again

This is the World Return.

Why the optimisation must be robust

A speed profile that saves energy only when every parameter is perfect may fail in real operation.

Train mass varies with passenger load.

Dwell varies.

Resistance parameters and operating conditions vary.

So robust optimisation studies scenarios:

light load
normal load
heavy load
small delay
lower receptivity
changed running condition

and looks for a trajectory that performs well across them rather than beautifully in only one ideal case.

A chance-style constraint might be written:

P(arrival on time) ≥ 1 − ε

while still minimising expected or worst-case energy.

The best energy curve is not merely the one that wins a simulation. It is the one that survives the railway.

Prompt 8 — What Does Whole-Railway Energy Optimisation Look Like?

By now, we have several energy layers.

individual train trajectory
+
regenerative interaction among trains
+
traction-network power peaks
+
timetable and headway
+
station and train auxiliary loads
+
reliability and recovery
=
whole-railway optimisation

A conceptual objective might be:

minimise J
=
w1 Egrid
+ w2 Ppeak
+ w3 passenger waiting
+ w4 journey-time deviation
+ w5 headway variance
+ w6 discomfort
+ w7 delay risk
+ w8 asset and maintenance cost

subject to:

safe movement authority
speed and braking limits
required station stops
passenger dwell requirements
headway constraints
junction and terminal capacity
available trains
power-system limits
recovery requirements

This is a multi-objective, multi-train, network-constrained optimisation problem.

Energy per train is not the same as energy per passenger

Suppose one train journey uses E_train and carries N passengers.

Energy per passenger is:

epp = Etrain/N

A fuller transport intensity measure can include distance:

I = Etrain/(N × L)

measured as energy per passenger-kilometre.

A lightly occupied train may use less total energy than a crowded train because it has less mass, yet use more energy per passenger.

This creates another important distinction:

vehicle efficiency
≠ transport-system efficiency

Running too few trains can increase crowding and waiting.

Running too many nearly empty trains can waste energy.

The timetable must match service supply to passenger demand.

Auxiliary energy changes the optimum

Train air-conditioning, lighting and control systems use energy even while the train is coasting or stationary.

Let auxiliary power be P_aux(t).

Total onboard electrical energy is:

Eonboard
=
∫[Ptraction,grid(t) + Paux(t)]dt

If an energy-saving trajectory makes the journey much longer, auxiliary systems operate for longer.

So excessively slow operation can give back part of the traction saving through longer auxiliary use and lower service productivity.

Once again, the optimum is not an extreme.

Reliability has an energy cost

A disruption can force trains into stop-start movement, lower-speed operation, extra dwell, altered turnbacks or extended service recovery.

These can consume additional energy and reduce regenerative coordination.

Conversely, an overly aggressive energy-saving profile with no recovery margin may make the timetable fragile.

So reliability and energy are coupled:

efficient normal trajectory
+
adequate recovery capability
=
lower whole-day cost

The lowest-energy perfect journey may not create the lowest-energy operating day.

A Complete Worked Energy Journey

Let us build one deliberately simplified station-to-station model.

This is not an actual Singapore MRT train or operating profile.

Assume:

distance L = 1,200 m
train mass m = 220,000 kg
maximum speed = 20 m/s
acceleration = 1.0 m/s²
braking magnitude = 0.8 m/s²
level track for the base case

Step 1 — Acceleration

Time to reach 20 m/s:

ta = v/a
   = 20 s

Distance:

sa = v²/(2a)
   = 400/2
   = 200 m

Idealised kinetic energy at the end:

Ek = ½mv²
   = 44 MJ
   = 12.2 kWh

The grid must supply more than 12.2 kWh because of resistance and conversion losses.

Step 2 — Choose a speed strategy

Braking distance at 0.8 m/s² is:

sb = v²/(2b)
   = 400/1.6
   = 250 m

Acceleration and braking use:

200 + 250 = 450 m

Leaving:

1,200 − 450 = 750 m

between those regions.

Profile A maintains 20 m/s for all 750 metres:

tcruise = 750/20
        = 37.5 s

Profile B cruises for 300 metres and coasts for 450 metres, averaging 18.5 m/s during coasting.

tcruise = 300/20
        = 15 s

tcoast = 450/18.5
       ≈ 24.3 s

Profile B uses about 1.8 additional seconds across the middle section compared with 37.5 seconds of continuous cruising.

But it avoids traction power over most of the 450-metre coast.

Step 3 — Braking recovery

If useful regenerative recovery equals 7.3 kWh in our earlier hypothetical chain, net traction energy associated with building and removing that kinetic energy is much lower than if all 12.2 kWh were discarded as heat.

But resistance losses and auxiliary energy remain.

Step 4 — Add auxiliaries

Suppose train auxiliary power is an illustrative 120 kW during the 80-second station-to-station journey.

Eaux = 120 kW × (80/3600 h)
     ≈ 2.67 kWh

If the journey is lengthened by ten seconds:

extra Eaux
= 120 × (10/3600)
≈ 0.33 kWh

This is why slower is not automatically better.

Step 5 — Add the timetable

Suppose scheduled running time is 82 seconds.

If Profile B arrives in 80 seconds, it is feasible and retains two seconds of recovery margin.

If crowding causes the train to leave the previous station five seconds late, the system may use a faster permitted profile and recover part of the delay, sacrificing some energy saving to protect regularity.

The control priority can therefore change with state.

on time
→ favour efficient coasting

slightly late
→ use part of performance margin

safety constraint ahead
→ safety dominates both

This is not one fixed speed curve.

It is a family of feasible curves selected according to the railway’s condition.

The Five Energy Deletion Tests

We can understand the system by removing one layer at a time.

Remove the timetable

The train can save energy by travelling slowly, but passenger service loses its required time and frequency.

Remove coasting

The train may continue applying traction even when momentum could perform part of the journey.

Remove regenerative braking

More kinetic energy is dissipated rather than returned for possible reuse.

Remove network coordination

A braking train may generate energy at a time when fewer receptive loads exist, while several other trains accelerate together and create a high power peak.

Remove recovery margin

The most economical normal speed profile may leave the railway unable to recover ordinary delay without damaging headway.

The deletion tests reveal that energy efficiency is not one component.

It is an agreement among physics, electricity, timetable and control.

The Energy Paradoxes

Paradox 1 — Doing nothing can save energy

During coasting, the traction motors are not pushing the train.

Yet the train continues to perform useful transport work using motion already created.

Paradox 2 — Braking can power acceleration

One train is slowing while another is using part of the returned electrical energy to speed up.

Paradox 3 — A slightly slower journey can protect punctuality

An energy-efficient curve may retain performance margin that can later be used for delay recovery.

Paradox 4 — The lowest-energy train may not create the lowest-energy railway

Optimising each train independently can produce simultaneous power peaks or poor regenerative overlap.

Paradox 5 — A fuller train can be more efficient

Total train energy may increase with passenger mass, but energy per passenger can decrease because one movement serves more people.

Why 17 GWh Matters Mathematically

A gigawatt-hour is one million kilowatt-hours.

1 GWh = 1,000,000 kWh

So SMRT’s current figure of around 17 GWh in annual Green CBTC Phase 1 savings means:

17 GWh
= 17,000 MWh
= 17,000,000 kWh

The 2024 project announcement described more than 15 million kWh in annual additional savings at Phase 1 completion; SMRT’s current decarbonisation page now reports the figure at around 17 GWh.

The lesson is not merely that one algorithm produced one large number.

The lesson is multiplication.

small improvement per train movement
× many interstation runs
× many trains
× many operating hours
× many days
= large annual saving

Infrastructure often changes the world through repetition.

How the Mathematics Grows from School to Research

Primary Mathematics

speed, distance and time,
unit conversion,
rates,
multiplication across repeated journeys.

Secondary Mathematics and Science

force and acceleration,
kinetic and potential energy,
quadratic relationships,
graphs and areas,
efficiency percentages.

Junior College

calculus,
integration of power,
differential equations,
probability and statistics,
constrained optimisation.

University and Research

optimal control,
dynamic programming,
mixed-integer optimisation,
power-system modelling,
robust optimisation,
model predictive control,
multi-agent simulation,
machine learning and data-driven calibration.

The railway does not keep these subjects in separate classrooms.

It makes them solve one journey together.

The World Return — When the Meter Answers the Model

The optimiser proposes a trajectory.

The train operates it.

The energy meter answers.

Perhaps the train consumed less than before.

Perhaps it saved energy but arrived too late.

Perhaps regeneration was lower because no nearby train was receptive.

Perhaps a gradient or passenger load was modelled poorly.

The next version must learn from that return.

model
→ speed profile
→ train movement
→ measured power
→ measured energy
→ timetable outcome
→ comparison
→ revised model

This is why operational data is so powerful.

It closes the loop between an equation and a real train full of people.

RFE Return — What Is an Energy-Efficient MRT?

It is not an MRT that moves as slowly as possible.

It is not an MRT that keeps stations uncomfortable to save electricity.

It is not an MRT that sacrifices accessibility, frequency or reliability.

It is a railway that understands where energy is doing useful work and where it is being wasted.

Useful energy:
move passengers safely and reliably
maintain comfort and essential systems
preserve recoverability

Avoidable energy:
unnecessary acceleration
unnecessary high-speed resistance
poor coasting decisions
unused regenerative energy
uncoordinated power peaks
inefficient auxiliary operation

The job of optimisation is to reduce the second category without damaging the first.

The RFE is not to make the railway consume the least electricity. It is to make every useful kilowatt-hour carry as much public value as possible.

Conclusion — The Train Has a Power Curve We Never See

A passenger feels acceleration.

The power system sees a rising electrical load.

A passenger feels the train coast.

The optimiser sees useful motion continuing with little or no positive traction power.

A passenger feels braking.

The electrical network may see a generator returning power.

A second train accelerates.

Its demand can overlap with the first train’s return.

The timetable shifts one event by a few seconds.

Peak power changes.

Regenerative utilisation changes.

Headway must still remain safe and regular.

The train must still arrive on time.

This is why energy optimisation is not a small environmental feature attached to an MRT.

It reaches into the movement of every train.

Force
→ acceleration
→ velocity
→ kinetic energy
→ coasting
→ braking
→ regeneration
→ network receptivity
→ timetable coordination
→ measured energy
→ revised operation

Singapore’s Green CBTC results show what this can mean at scale: an 8 per cent Phase 1 traction-energy reduction and around 17 GWh of annual savings currently reported by SMRT, with the broader project intended to reach up to a 15 per cent reduction.

The achievement is not one train using one clever equation once.

It is a railway making better second-by-second decisions, then repeating them millions of times.

The passenger sees the next station.

The physicist sees energy changing form.

The electrical engineer sees power flowing in both directions.

The operations researcher sees a multi-train optimisation.

The controller sees a trajectory.

The city receives a quieter result:

The train arrives, and less energy was needed to make ordinary life work.

Key Equations

P = dE/dt
Power is the rate of energy transfer

E = ∫P(t)dt
Energy is accumulated power

m dv/dt = Ftraction − Fbrake − R(v) − mg sin θ
Simplified train force balance

Pmechanical = Ftraction v
Mechanical traction power

Pelectrical ≈ Pmechanical/η
Electrical input with efficiency

Ek = ½mv²
Kinetic energy

Ep = mgh
Gravitational potential energy

R(v) = A + Bv + Cv²
Generic train-resistance model

Erecovered = ηmotor ηconversion ηnetwork ρreceptivity Eavailable
Conceptual regenerative-energy chain

Pnet(t) = ΣPi(t) + Ploss(t) + Pother(t)
Multi-train network power

Toverlap = max(0, min(ta2,tb2) − max(ta1,tb1))
Overlap of braking and acceleration intervals

epp = Etrain/N
Energy per passenger

I = Etrain/(N × L)
Energy intensity per passenger-kilometre

Eonboard = ∫[Ptraction,grid(t) + Paux(t)]dt
Traction plus auxiliary energy

ej = Ej − Êj
Observed minus predicted energy

J = weighted energy + peak power + time + comfort
    + regularity + reliability costs
Multi-objective operating optimisation

Reader-safety note: The numerical examples and equations above are educational abstractions. They do not reproduce proprietary Singapore MRT speed profiles, power-system configurations, control algorithms, internal timetables or safety procedures. Actual rail operation uses train-specific, line-specific and safety-validated engineering models.

Continue the MRT Mathematics Series

How MRT Timing Works Using Mathematics — the system-wide pillar.
How MRT Braking Works Using Mathematics — stopping distance and regenerative braking.
How MRT Headway Works Using Mathematics — how train spacing becomes capacity.
How MRT Station Dwell Time Works Using Mathematics — queues and doorway bottlenecks.
How MRT Delays Propagate and Recover Using Mathematics — instability, control and resilience.
How an MRT Timetable Is Built Using Mathematics — demand, fleet cycles and operating constraints.

The next natural leg is How MRT Passenger Capacity Is Calculated Using Mathematics: train capacity, load factors, bottleneck sections, passengers per hour per direction, platform queues and why capacity is never just the number printed for one train.

How MRT Energy Use Is Optimised Using Mathematics: The Hidden Power Curve Behind Every Journey