Automating Verification of Cooperation, Control, and Design in Traffic Applications - Formal Methods and Hybrid Real-Time Systems - page 119

2.2 Outer Losses

The outer losses are summarized in the resistance force F e based on air resistance
and roll resistance.

· ·

sin φ

F e = F air + F r + m g (7)

The term m sin φ is the loss due to the lateral slope angle φ of the rail. The

roll resistance is described by the formula

· g ·

F r = m · c r (8)

where m is the total mass of the train and c r the velocity independent roll
resistance coecient. The air resistance is described by

F air = c air v 2 + c air

2

· · v (9)

1

The coecient c ai 1 depends on the density of air, the cross section of the train
and such things, the coecient c ai 2 describes aerodynamic phenomenon which
cannot be described as functions of v 2 .

2.3 Brake

The main goal of this brake model is to bring up the train model in a non-moving
state and not to study the brake behaviour in detail. For this reason the brake
model is very simple. The brake model consists of two kinds of brake systems:
an eddy current brake and an emergency brake. An eddy current brake consists
of an electromagnetic shoe where the electromagnetic force is controlled by the
brake current. The change of the magnetic field caused by the speed difference
between the brake and the adjacent rail induces an eddy current in the rail. This
eddy current leads to a resistance force which depends on the current and the
speed difference. In high speed region ( v 20 m / s ) the resistance force is nearly

linear to the speed difference. The resistance force tends to zero if the speed
difference becomes zero. Therefore we need an additional brake mechanism for
the low speed region. These two brake systems work as the service brake for
the train model. For the emergency case there exists an additional brake called
emergency brake. This brake is typically an electromagnetic rail brake. Both
brake types, the eddy current brake and the electromagnetic rail brake, work
directly between the train and the rail and do not depend on friction between
the wheels and the train. This simplifies the brake model. The brake force F b is
modelled by

≥

F b =( I b · v +( v offset − v )) · sb sc + eb c (10)

where I b is the brake current to control the service brake, v is the present velocity
of the train and v offset is a constant to ensure brake force if the present velocity
is close to zero. The constant sb sc is a scaling factor for sucient brake force.
The emergency brake is modelled as a constant and denoted by eb c . Deceleration
is controlled through setting a proper brake current using a PI controller. The