As demonstrated in the first article, the motor characteristic significantly influences the total energy expended. Since the energy used is a measure of operational cost, both run time and energy consumed are important in choosing the motoring characteristic. Both factors are contradictory, since less energy consumption (or lost) generally requires a longer run time. Hence, careful weighting of the two objectives is extremely important. After determination of the energy lost function, starting with the lossless case, this paper explains the choice of objective functions.



The use of inverter or chopper control technology naturally permits regenerative braking capability. For a purely regenerative brake, the energy lost is zero in the idealised case. This is a trivial and unrealistic solution in practice. Normally only a proportion of the train axles are motored, and as braking is applied to all axles, some braking needs to be mechanical, such that the blended mechanical / electrical brake results in linear braking. For this energy study, typical electrical multiple unit (EMU) values of tare train weight and power are used with an assumed station-station distance of 1km.

Using the nomenclature defined in the first paper, for sensible values of Tb it is normal to assume for Tb > [M.Pm2/(54DA2]1/3 for low characteristic machines. The optimum value of w3max to minimise Tra was shown to be,

w3max = [Pm/(3αTb)]{coshθ - 0.5)  for Tb > [M.Pm2/(54DA2)]1/3                      .. equation 2.1

Where   θ = 1/3.cosh-1[(108DAα2Tb3)/(M.Pm2) - 1]                                       

By substituting Pm = Tm .w1 and equating w1 = βw3max

             Then Tm = 3αTb / [β{coshθ - 0.5)]

= 4.75Tb / {coshθ - 0.5) for the minimum run time solution               .. equation 2.2

By allowing a 10% adhesion limit on braking (i.e. Tb = 0.1Mg), Tb is substituted in equation 2.1, and θ numerically evaluated. By substituting θ in equation 2.2, this gives Tm = 1.2Tb.

For the typical 50% axles motored, this would correspond to a 24% motoring adhesion requirement to satisfy the optimum run time criteria. In most circumstances this demands good condition rails, not satisfied by a wet track conditions. However, it emphasises the importance of practical operating constraints on the motor characteristic. The ratio of motoring to braking torque is an important parameter in the energy study, and was defined as K1.

Energy recovery

Figure 1           Typical Blended Regenerative and Friction Brake Effort Profiles

For a train with 50% motored axles, approximately 60% of the train weight will be upon those axles. So assuming ideal receptivity of regenerated energy, a maximum of 60% of the total brake can be provided electrically, whilst keeping the same braking force on each axle. Figure 1 shows what proportion of the power supplied can be recovered at different braking speeds, and how the electrically braked axles need augmenting with mechanical braking above a critical speed (w1K1 / 0.6). By integrating the torque-speed braking characteristic of figure 1, a power-speed curve was established (see figure.2). The significance of the brake power-speed graph is that it is similar to the power-time curve for a linear brake characteristic, apart from a dimensional (time-speed) scaling factor on the speed axis. Hence, the shaded areas in figure 2 (a), (b) & (c) are indicative of the energy lost (not recovered) by the mechanical braking component of overall braking strategy.

Energy lost function

Figure 2           Alternative Power–Speed Distributions of Braking Energy

An expression for energy consumption was found for each of the three possible cases by subtracting the total energy regenerated when braking from the total energy required by the system to stop the train. This could include coast­ing when necessary, since this is a region of zero energy transfer in the lossless case. The energy expended is the train kinetic energy at brake application minus the recovered electrical energy. The resulting function was termed the energy lost equation,

EL = 0.5M.w12 / β2 - regenerated energy function                                  .. equation 2.3

Where the appropriate “regenerated energy function” is chosen from figure 2 according to the specific values of K1, β, and the field-weakening ratio, K, which is a fixed parameter for a particular design. It is clearly desirable to minimise EL, but without compromising the minimisation of the run time function.

The third article in this series will explain how this multiple objective dilemma is transformed into a series of numerically soluble single objective problems.

(back to top)     Copyright Terms     (article 1)     (article 3)     (article 4)     (article 5)    


Cecube homeHome