ALGEBRAIC OPTIMISATION OF REGENERATIVE RAILWAY SYSTEMS FOR MINIMUM RUN TIME

*ABSTRACT*

This analysis establishes criteria for operating a simple
railway system in a manner that minimises the run time between stations, by
prescribing an appropriate specification of traction motoring and braking
characteristics for a fixed power rating. The use of either, **pure
regenerative braking** (commonly referred to as "regen"),
or **blended mechanical-electric brake** (partially regenerative) is
considered and results permit the degradation in station-station run time to be
established as a consequence of optimally maximising the recovered energy.

*BACKGROUND*

Until recently the wide operating speed range of the DC motor made it an ideal choice for traction purposes. The introduction of AC induction motors in modern rolling stock permits a wider variation in the tractive effort–speed characteristic. However, in practice, the requirements of initial acceleration and speed limits for existing railway networks often result in AC motor performance characteristics similar to DC motored stock. This article outlines the speed-time characteristic of a typical rapid transit vehicle station-station run, and by means of algebraic optimisation, states the exact form of the traction characteristic necessary to minimise the station-station run time. In this analysis mechanical, electrical and gradient losses are ignored, and it also assumes an ideal supply system, capable of supplying and absorbing an unlimited energy.

Traditionally series DC machines had widespread application in traction drives. This was because the ratio of armature to field mmf is better regulated against supply variation than for alternative configurations. A typical supply for a rapid transit system will vary by ±30% because of line impedance and load changes. The unpredictable nature of supply variations makes dynamic modelling of analytic optimisation procedures difficult. These effects, and those of train headway (time between trains) and substation spacing are preferably analysed on network simulators. For this static algebraic study, single vehicle operation on a constant voltage supply is assumed.

Machine control is by a DC chopper (or AC inverter), rather
than switched resistor control, thus permitting electric brake.
The theoretical form of the traditional torque-speed
characteristic for a series machine when motoring is shown in **figure 1**.
The three motoring regions of the characteristic are defined by the control
method:

region 1 Constant Torque - armature voltage control with closed loop regulation of armature current

region 2 Constant Power - field-weakening control with closed loop regulation of armature current.

region 3 Natural Characteristic - operation at weakest field while maintaining a constant armature to field mmf ratio

Figure 1 Three
region Traction curves showing Low and High Motor Characteristic

The complete speed-time characteristic will depend on the type of braking used. Two cases have been considered:

(i) Pure regenerative braking, where all the braking torque is developed by the traction machine.

(ii) Linear braking, where the composite braking characteristic of the blended mechanical-electric brake is linear.

The
speed-time characteristics are shown for both cases in **figure 2**, over a
station-station distance long enough to reach the track speed limit and enter
region 4 where the vehicle is coasting.

Figure 2 Speed-Time
Profiles for Regenerative and Linear constant rate Brake

By assuming a lossless system and a fixed machine power rating, the first problem was to establish optimal values of the region break points for a minimum run time. This is a static optimisation problem since the control input (motor torque) is of a predetermined form, with certain parameters (region break points) permitted to vary. It is the prescribed form of the traction tractive effort (TE) characteristic that allow simplification to a static optimisation problem.

Other authors have tried to solve this problem using dynamic optimisation, by treating it as a feedback position control system. Whilst this can account for gradient and frictional effects, the optimal solutions achieved for simple gradient patterns were totally unrealistic in the power requirements of the vehicle. Solutions demand machines with a power rating many times the conventional value. Both from an economic viewpoint, and the impracticability of mounting a larger motor under a rail car, these results are of academic interest only. Putting a constraint on motor power in the dynamic optimisation procedure is necessary, as well as other constraints on speed and acceleration limits. Unfortunately this then renders the dynamic optimisation insoluble.

In the following sections of this article a run time optimisation procedure is described, and how this result influences the choice of motor characteristic. By way of identification a low-speed, high-torque motor is termed low characteristic, and a high-speed, low-torque motor is defined as high characteristic. The analysis can be applied equally to controlled DC and AC motor applications. The analysis is performed on the basis of total train motored power (for multi-motor stock) being represented by a single motor with a scaled power rating.

1. DETERMINATION OF DISTANCE AND RUN TIME EQUATIONS

Breaking down the speed-time characteristic into either five or
seven regions permits time integration of the individual regions to find the
distance travelled in each. The distance and end time for one region was used
as the initial conditions for the next, and by summation the station-station
distance, D_{A}, and run time, Tra, were determined in terms of train
motor characteristic parameters and region speed break points. By ignoring
losses, the motor and brake torque characteristics for a purely regenerative
braking configuration can be identical. This sets identical speed break points
on accelerating and braking, and makes the braking torque, T_{b}, numerically
equal to the motoring torque, T_{m}.

The resulting distance equations for purely regenerative braking:

D_{A} = {M.P_{m}/T_{m}^{2}}[P_{m}/3T_{m}
+ k^{3}P_{m}/6T_{m} – 0.5w_{3} – k^{2}w_{3}/6]
+ M.T_{m}w_{3}^{4}/(6kP_{m}^{2})
+ w_{3}t_{4} …..
equation 1.1

where M = mass of train

P_{m} =
total motor power

k = full to weak field ratio

The corresponding run time equation for regenerative braking:

Tra = M/T_{m}[w_{3}^{3}T_{m}^{2}/(3kP_{m})
+ k^{2}P_{m}/6T_{m} + P_{m}/2T_{m}] + t_{4}
…..
equation 1.2

The distance equations for linear (or blended, constant BE) braking:

D_{A} = {M.P_{m}/(12T_{m}^{2})}[2P_{m}/T_{m}
+ k^{3}P_{m}/T_{m} – 6w_{3} – 2k^{2}w_{3}/6]
– M.T_{m}w_{3}^{4}/(12kP_{m}^{2})
+ M.w_{3}^{2}/2T_{b} + w_{3}t_{4}’

….. equation 1.3

The corresponding run time equation for linear braking:

Tra = M.w_{3}/T_{b} + t_{4}’ …..
equation 1.4

The condition
for coasting shall apply in both cases, that is t_{4} (or t_{4}’)
>= t_{3}.

2. OPTIMISATION OF THE RUN TIME FUNCTION

Elimination of t_{4} from equations 1.1 and 1.2, and t_{4}’
from equations 1.3 and 1.4 respectively, yield an equation for run time in
terms of the station-station distance. Assuming the vehicle reaches its maximum
permitted speed (i.e. the coasting region is entered), then the run time
equation for the regenerative braking case is given by:

Tra = (Mw_{3max}^{3}
T_{m})/6kP_{m}^{2} + M(1+k^{2}/3)P_{m}/T_{m}^{2}
– (M/3w_{3max})(1+k^{3}/2)(P_{m}^{2}/T_{m}^{3})
+ D_{A}/w_{3max } ….. equation
1.5

where w_{3max}
is the station-station track speed limit.

The last term D_{A}/w_{3max} is a constant for
a particular run. The remaining terms in equation 1.5 were defined to
constitute the run time function (RTF).

Performing a grid search minimisation on the RTF with respect
to k and w_{1 }(using the substitution T_{m} = P_{m}/w_{1}
in equation 1.5) gives the following solution,

k
= 2.5 w_{1 }= βw_{3max}
= 0.260w_{3max}

where k = 2.5 was a design choice for the upper boundary of region 2 (i.e. full-field to weak-field ratio).

Hence, the optimum motoring and braking characteristic for minimum run time has k = maximum design value = 2.5, β = 0.260, and a run time given by:

Tra _{opt}
= D_{A}/w_{3max }+ αMw_{3max}^{2}/P_{m} …..
equation 1.6

where α = 0.412 α being the constant multiplier of the minimised RTF

For the linear brake case the
elimination of t_{4 }gives,

Tra = 0.5[(Mw_{3max}^{3}
T_{m})/6kP_{m}^{2} + M(1+k^{2}/3)P_{m}/T_{m}^{2}
– (M/3w_{3max})(1+k^{3}/2)(P_{m}^{2}/T_{m}^{3})]

+
D_{A}/w_{3max} + Mw_{3max}/2T_{b} …..
equation 1.7

The term in square brackets is just the RTF, which again is
minimised when k = 2.5, w_{1 }= 0.26w_{3max} and α =
0.412. Hence, the equation for the optimised run time for a linear brake
simplifies to:

Tra _{opt}
= D_{A}/w_{3max }+ 0.5Mw_{3max}(αw_{3max}/P_{m
}+ 1/T_{b}) …..
equation 1.8

The first term enclosed by brackets
is inversely proportional to motoring torque, and the second term to inversely
proportional to braking torque. So for the case of equal time spent motoring
and braking (equivalent to the purely regenerative brake case) then T_{b}
= P_{m}/(αw_{3max}) = βP_{m}/(αw_{1})
= 0.631P_{m}/w_{1}.

3. MINIMISATION OF STATION-STATION RUN TIME

The optimum run time equations 1.6 and 1.8 establish the
minimum run time for an optimised motor characteristic (k and β). From
inspection of these equations it is apparent that a minimum run time is
dependent on the speed limit enforced. For the regenerative braking case the
value of w_{3max} that minimises Tra was obtained by differentiating
equation 1.6 with respect to w_{3max} and equating to zero.

Hence, Tra_{min}
= 1.89α^{1/3}[M.D_{A}^{2}/P_{m}]^{1/3 and} this occurs when w_{3max}
= [D_{A}P_{m}/(2αM)]^{1/3} ….. equation 1.9

Significantly this value of w_{3max}
results in t_{4} - t_{3} = 0 when substituted back into the
equations. This is proof of the intuitive precept that minimum run time is
achieved using an optimised motor characteristic without a coasting period.
This result requires that the track speed limit is at least [D_{A}P_{m}/(2αM)]^{1/3 }.

When using a linear blended brake, the optimum run time
equation is a function of w_{3max} and T_{b}. Tra was minimised
by partial differentiation with respect to both w_{3max} and T_{b}.

Hence, Tra_{min}
= 1.5α^{1/3}[M.D_{A}^{2}/P_{m}]^{1/3 and} this occurs when w_{3max}
= [D_{A}P_{m}/(αM)]^{1/3} and T_{b} » T_{m}.

This result is of only academic interest
since an infinite braking torque not realisable, and practical adhesion limits
ensure it is not even approachable. However, it does demonstrate that the run
time will be minimised when T_{b} is maximised, and that the physical constraint
on T_{b} is ultimately the adhesion limit.

The value of w_{3max} that
minimises Tra for a finite and constant value of T_{b} is:

(a) w_{3max}
= [P_{m}/(3αT_{b})]{cosθ – 0.5) for T_{b} < [M.P_{m}^{2}/(54D_{A}^{2})]^{1/3}

where θ = 1/3 . cos^{-1}[1-(108D_{A}α^{2}T_{b}^{3})/(M.P_{m}^{2})] for 0 < T_{b} < [M.P_{m}^{2}/(108D_{A}^{2})]^{1/3}

and θ = 1/3 . cos^{-1}[(108D_{A}α^{2}T_{b}^{3})/(M.P_{m}^{2})-1] for [M.P_{m}^{2}/(108D_{A}^{2})]^{1/3} < T_{b} < [M.P_{m}^{2}/(54D_{A}^{2})]^{1/3}

(b) w_{3max} = [P_{m}/(3αT_{b})]{coshθ
– 0.5) for T_{b}
> [M.P_{m}^{2}/(54D_{A}^{2})]^{1/3}

where θ = 1/3 . cosh^{-1}[(108D_{A}α^{2}T_{b}^{3})/(M.P_{m}^{2})-1]

The complete solution for w_{3max}
is continuous, but when expressed analytically it has three distinct
regions. Substituting the expressions
for w_{3max} into the condition for coasting again gives a coasting
time of zero. Hence, the minimum run time for a linear brake is given by,

Tra_{min}
= 3D_{A}αT_{b}/[P_{m}(cosθ
– 0.5)] + M.P_{m}(cosθ – 0.5)/(6αT_{b}^{2})
+ M.P_{m}{(cosθ – 0.5)^{2}}/(18αT_{b}^{2})

for T_{b} < [M.P_{m}^{2}/(54D_{A}^{2})]^{1/3}
….. equation
1.10a

Tra_{min}
= 3D_{A}αT_{b}/[P_{m}(coshθ
– 0.5)] + M.P_{m}(coshθ – 0.5)/(6αT_{b}^{2})
+ M.P_{m}{(coshθ – 0.5)^{2}}/(18αT_{b}^{2})

for T_{b} > [M.P_{m}^{2}/(54D_{A}^{2})]^{1/3}
….. equation
1.10b

The equation is more complex than its equivalent case for pure regenerative braking because of the use of a fixed blended braking rate.

4. COMPARISON WITH ESTABLISHED TRACTION PRACTICE

Thus far the emphasis is on minimising run time. This is an important factor, but needs to be considered economically before shorter run times are proposed. The cost penalty is due to the extra energy that would inevitably need supplying to achieve a minimum run time. In future articles a technique for examining the trade-off between run time and energy will be discussed, but for now comparison is drawn with existing traction practice.

It is remembered that the system was simplified to obtain analytical solutions. The following effects were not included in the analysis; mechanical losses, gradient variations, electrical losses, supply variations, non-ideal receptivity conditions for regenerated energy. None of these effects individually should have a major influence on the choice of motor characteristic for a minimum run-time, but together they may necessitate some alteration in the optimum characteristic. The important feature of this analysis of a lossless system was the optimising of the motor characteristic (embodied in the RTF) for either electric brake or blended mechanical-electric braking. It suggests the use of a fairly low characteristic machine, irrespective of speed limits, station-station distance, train mass and power rating. Maximum permissible field-weakening (2.5:1) was required in region 2, and a base speed to maximum speed ratio of 0.26:1 (or 1:3.85) is necessary to optimise the motoring characteristic for minimum run-time.

A base speed to full speed ratio of 1:4 is typical of rapid
transit practice with DC chopper controlled motors. Historically, a low
traction characteristic was used to minimise the loss due to resistor heating
in region 1. Experience has approved this choice of characteristic from a
run-time consideration, although the heating
loss argument no longer applies with modern motor
controllers that dispense with starting resistors. A graph of α and β
against k (figure 3) confirms the requirement to make k as large as possible.
Particularly for AC drives limiting values greater than 2.5 are possible.
Another feature of **figure 3** is that a lower characteristic machine is required
for minimum run-time when k is larger. This discourages excessive operation in
region 3. For DC traction motor design
these results emphasise the need to make the field-weakening region as large as
practical. Analogously for AC motor design it infers that the onset of pullout
torque limitation should be as close to the speed limit as possible.

Figure 3 Grid Search
Minimisation solution to the Run Time Function

In **figure 4** a minimum station-station run time characteristic
is shown. Once a speed limit of w_{3max} is enforced a (global) minimum
run time is no longer possible, and a coasting period results (coasting period
is at constant speed for a no loss system). An optimum motoring characteristic
is required to reach the speed limit, necessitating a lower characteristic
motor than for a minimum run time characteristic. So when ‘make-up
time’ or a time margin is included in a run that permits coasting, the
optimum characteristic is slightly lower than the case for a minimum run time.
A number of operational factors also influence the choice of motor
characteristic. Principally these include headways, supply power constraints
and track sections, all of which influence the receptivity of the system to
regenerated energy. Evidently, energy consideration is an important factor in
the choice of characteristic, and this is the subject of a later article.

Figure 4
Optimum Speed Profile when a Speed Limit (w_{3max}) prevents
Minimum Run Time Solution

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