A comparison of lossless algebraic optimum profiles versus the lossy numerical integration solutions is presented. It is shown that for station to station run times with small time margin then the lossless solutions are a close approximation to the model with detailed losses. However, for higher energy savings longer station to station run time is required. In this case the losses are significant and the more easily calculated lossless soultions introduce error into the optimum traction and regenerative brake profile solutions for minimum energy consumption. It is explained how these solutions are applied to Driving Advice Systems and the significant energy savings that can result.


1.            RECAPITULATION

The first three articles in this series have derived an algebraic solution to optimise station to station tractive effort profiles for a lossless traction drive and regenerative brake system. It is demonstrated how the algebraic equations are solved by application of Newton´s method using first derivative estimates. The solution objectively weights the importance of the station to station run time against the total energy consumed in the process. This provides a set of optimum results ranging from the absolute minimum run time, which inevitably consumes the greatest energy, to longer run times where the energy expended approaches a minimum value.


The fourth article showed that the inclusion of realistic non-linear loss functions for electrical and mechanical train losses, typical of rapid transit systems, prevents the manual derivation of an algebraic solution. In this case, the non-reduced equation set is numerically integrated with a bespoke paradigm, developed to find the minimum run time solution by iterative adjustment of the costing period. Whist this method is computationally much slower than the lossless algebraic approach, when the loss functions were set to zero the same solution resulted. This provided validation for both mathematical and numerical processes, plus confidence that the introduction of realistic losses would yield credible solutions.


To demonstrate the influence of realistic traction drive losses a detailed comparison of the algebraic lossless solutions versus the numerically integrated lossy solutions is presented. The system is assumed to be ideally receptive to regenerative brake. The typical fixed metro train constants used for this comparison are as follows:

Train power to weight ratio = 7.1kW per tonne

Station to station distance = 1Km

Speed limit = 25m/s

Full to weak field ratio (k) = 1.5

Axles that are motored = 50%

Weight on motored axles = 60%

Mechanical losses = Davis formula with typical open (not tunnel) track parameters

Electrical losses = Three region representation of motor iron and copper (<5%) losses

Linear regenerative brake rate = 0.98m/s2

Most of the previous evaluation in earlier articles used 2.5 as the maximum realistic value of k, but in this comparison it is limited to 1.5, both to provide alternative solutions and to particularly suit the design of smaller motors to lower speed transit systems. It is recognised that loss model is not comprehensive, since it excludes supply and return conductor resistive loss. However, a power conversion loss of 2% in both motoring and regenerative brake is added to the loss model.



The first graphical result shows the comparison of consumed energy (EL) against optimal run time (Tra) from station to station for the lossless model versus the lossy case.


The additional energy required in the lossy case is relatively constant, although contributing an increasing proportion of the total energy consumed for heavier weighting of energy reduction in the optimisation procedure, which results in increased station to station run time. It is evident from figure 1 that operating a flat-out service without a coasting phase or time margin is extremely inefficient in energy terms. In practice this would also be more susceptible to disruption due to minor delays.

Comparison of Energy Used against Optimal Station to Station Run Time
Figure 1            Comparison of Energy Used against Optimal Station to Station Run Time


The second graph compares the peak speed attained prior to coasting required by the respective lossless and lossy solutions. Note that the 25kph speed limit is not reached over this distance. The graph indicates that by applying more significance or weighting to saving energy, then the lossy case does not reach the same top speed as the lossless case. The principle reason for this adjustment to the optimum solution is the velocity dependence in the Davis drag resistance formula. Despite this the same distance is covered.

Comparison of Peak Speed against Optimal Station to Station Run Time
Figure 2            Comparison of Peak Speed against Optimal Station to Station Run Time


According to figure 3 the correct distance is covered in the lossy case, because the train accelerates harder in initial phase (region 1) covering more distance than the lossless case in this region


Comparison of Initial Acceleration against Optimal Station to Station Run Time
Figure 3            Comparison of Initial Acceleration against Optimal Station to Station Run Time


Because the power of the motors is fixed, the increased initial acceleration in the lossy case has to be mirrored by a lower base speed and therefore earlier transition into weak field operation.

Comparison of Base Speed against Optimal Station to Station Run Time
Figure 4            Comparison of Base Speed against Optimal Station to Station Run Time


3.            ENERGY SAVING

It is deduced from these results that in accounting for realistic losses, the optimum profile moves towards a lower characteristic motor solution. This is indicated in figure 1 of article 1. However from figures 3 and 4, this effect is only significant when the time margin (latency or make up time allowance) is more than 8%. Timetables include a time margin to allow correction for short delays. The time margin is typically between 5 to 12% of the minimum run time on a metro system. Hence the significance of losses on the choice of optimum profile will vary case by case. For networks with very low time margin the lossless solutions (which are more easily solved) are a close approximation to the lossy solution. Hence, under these conditions, ignoring the losses to determine the optimal profile solution is perfectly acceptable.


The time margin is infrequently used in an energy efficient manner, but mostly just extends station dwell times. Hence, by driving to utilise the available time margin, substantial energy savings are possible. For example, rather than driving flat out from station to station in 73.4s and consuming 30MJ of energy, from figure 1 results it is possible to drive the train optimally in 78s and consume only 18MJ, a 40% energy saving. To achieve this, a time margin of only 6.3% is necessary.  


Hence, time margin availability permits different driving strategies that save energy in comparison with the minimum run time profile. Depending on experience and skill, some drivers may use time margin to apply an energy efficient driving style. Driving Advice Systems (DAS) exist that calculate and continuously update the optimum driving strategy more precisely than a driver can estimate. They are based on train position, route and timetable data, Driving Advice Systems for metro networks are normally limited to issuing coasting recommendations, indicating to the driver when to notch off traction power and then to brake in order to reach the next station just in time. DAS for metro or rapid transit usually only predict one coasting phase before the next stop, predictions determined from solutions to equations of motion and energy derived in this series of articles.


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