As explained in the second article, the energy used is a measure of operational cost, both run time and energy consumed are important factors in choosing the motoring and braking characteristic. Hence, careful weighting of the two objectives is extremely important. This paper defines the choice of an objective function and mathematical methods to solve for minimum energy consumption and optimum time margin.




The equation for energy consumed (EL) is a function of three variables, β, w1, k1, and also two fixed parameters M and k. Recapping β = w1/w3max, k1 = Tm/Tb, k = full field to weak field ratio and M = effective vehicle mass.


Since the energy required reaching peak speed is independent of machine power, the choice of power rating only influences the energy lost indirectly.  The equation is obviously heavily weighted by the w12term, and since the energy used can not be negative, trivial minimisation of EL is when w1 is zero.


This implies an infinite coasting run, which can not be realised. A method is devised whereby EL can be minimised whilst maintaining a sensible run time. This requires the formulation of a multi-objective minimisation function. The best approach to solve this type of problem is a weighting method, in which the multi-objective problem is converted to a series of single objective problems of the form,


min ( λEL + Tra) subject to a set of constraints                                          .. equation 3.1


The type of solution for a range of λ from zero to infinity is indicated in figure 1. This method finds a point on a portion of the non-inferior solution that is convex to the feasible space, where the gradient equals -λ. It cannot be relied upon to generate points on a concave portion of the solution.


Railway Energy minimisation solutions

Figure 1           Non-Inferior solution to the Run Time / Energy Consumed feasible space


An expression for run time with coasting, for a linear brake, was determined in the course of finding the minimum run time in the first article. This expression includes the power rating (Pm), station-station distance (DA) and the field-weakening ratio (k) as parameters,

Tra = 0.5[(Mw3max3 Tm)/6kPm2 + M(1+k2/3)Pm/Tm2 – (M/3w3max)(1+k3/2)(Pm2/Tm3)]

                     + DA/w3max + Mw3max/2Tb                                                                 .. equation 3.2

Applying substitutions for w3max, Tm and Tb, equation 3.1 reduces to

            Tra = 0.5[Mw12/Pm].[K1/β + 1/(6kβ3) – β/3 + k3β/6 + k2/3 + 1] + DAβ/w1         .. equation 3.3


The objective function comprised equation 3.1 in Megajoules multiplied by the weighting coefficient, plus equation 3.3 in seconds. The dimensional inconsistency of the function to be minimised is not unacceptable, however, large numerical discrepancies in the two components of the function would be.


The speed-time graph was subject to two constraints, the speed limit enforced, w3max, and the adhesion limits.  The speed limit was included in the minimisation program as an inequality constraint function of the form,

            β – w1 / w3max >= 0                                                                                  .. equation 3.4


An adhesion proportion limit, h, was imposed on motoring by a fixed lower bound on w1,

            w1 >= Pm / 2hMg                                                                                    .. equation 3.5


and on braking by another inequality constraint function,

            K1 - Pm / w1 hMg >= 0                                                                              .. equation 3.6

where h = % adhesion figure.


A direct search algorithm due to Powell was tried to minimise the objective function for prescribed values of λ. The constraint equations with weighting multipliers were added to the objective function and a minimum found. By successive reduction of the multipliers, each time starting from the previous solution, an approximation to the final solution was reached. Although this algorithm rapidly converged towards the minimum, it tended not to find the absolute minimum, but remained in a subspace close to the minimum. Powell stated this might happen for some functions because the set of directions generated tend to collapse as the iteration proceeds, and the subsequent search is then limited to a subsection of the total space.


The direct search method was considered unsatisfactory and abandoned in favour of a quasi-Newton gradient technique, which estimated the first partial derivatives by a finite differences algorithm. Although this technique required as many iterations as the direct search method, it was more accurate and computationally less time consuming. This made it a better proposition to tackle the run time / energy lost optimisation problem.


Several sets of solutions were found by keeping all the parameters constant, except the field-weakening ratio, which was changed for each run. When λ = 0 the solution corresponded to the minimum run time case, so the particular solution for λ = 0 and k = 2.5 was identical to the minimum run time solution found algebraically in regeneration_1. This confirmed the accuracy of the numerical solutions, and justified other solutions by this method. Typical rapid transit systems operate on a 5 –10% time margin. Therefore it was considered sufficient to vary over a range which produced optimal solutions with a 0 – 20% time margin Predictably, increasing the run time reduces the energy consumption and the maximum braking rate (10%) is required throughout the range for values of k between 1.5 and 2.5. Figure 2 shows the run time / energy lost trade-off with k as a parameter.


Energy Savings by Driving Strategy
Figure 2           Variation of Station to Station Journey Time against Energy Consumed by using an optimum Driving Strategy


In spite of regeneration the best way to save energy is still to reduce the motoring energy, i.e. the kinetic energy of the train (½Mw32) at the end of region 3. So for longer time margins the optimal solutions predicted a systematic decrease in the value of w3. The solutions follow the same general pattern for all values of k, keeping the time spent in region 3 short for maximum energy saving, then coasting for as long as possible before braking with the limiting deceleration value. The principle of coasting, then braking hard is confirmed as an optimum energy saving strategy, despite from a driving perspective this may not be ideal. Mathematically, it is more important from an energy conservation viewpoint to keep the maximum speed down, and to coast as far as possible, than it is to try for further regeneration by braking more slowly. However, for practical braking characteristics it is better to maximise regenerative braking rather than using blended mechanical braking The diminishing importance of k with run time is further evidence for reduced region 3 operation, although for time margins of 5 -10% its effect is still significant. The speed limit (25m/s in this instance) did not come into effect on these solutions because of the relatively short station-station distance, but it could do when required to satisfy specific track requirements.


The software developed enables a system designer to choose an optimum motoring and braking characteristic for rolling stock for specified proportions of motored axles and train weight. All that needs to be specified is the power rating and weight of the train, the station-station distance, adhesion and speed limits, and a time margin (this may be reviewed after examination of the run time / energy lost curve). The major deficiency is that it fails to account for loss factors in the system. Almost certainly loss terms will not alter the general pattern of solutions, but they will cause some change in the optimal numerical solution, and consequently in the choice of the motor characteristic.


The inclusion of losses into the optimisation problem will be explained in the next part of the series.


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