La intertextualidad

Shortest Path Stochastic Dynamic Programming (SP-SDP) uses a terminal state to define the end of a journey in order to avoid using a discount factor. This has a number of benefits because it allows the optimisation to be aware of when and how likely the drive-cycle is to finish and allows additional costs to be applied to the end of the cycle such as based the terminal battery State of Charge (SoC). For a Charge-Depleting (CD) algorithm, this allows the battery depletion to be managed effectively over the duty cycle and therefore maximise the benefits of being able to charge the vehicle whilst it is parked. For a Charge-Sustaining (CS) algorithm, it allows more fluctuation in the SoC during the cycle, without losing the ability to constrain the final SoC.

The probability of the drive-cycle ending can be calculated in a similar manner to the transitions between other states. The logged data have already been separated into individ-ual trips during the data processing, see Section 4.3.2. Therefore, the probability of the trip finishing can be calculated using the number of times each state was the final state divided by the number of times the state was entered in total.

The definition of the trips in Section 4.3.2 means that the terminal state can only be entered when the vehicle is stationary. As a result, the final acceleration can only be zero or negative. The probability of transitioning to the terminal state for the Loughborough mail room duty cycle can be seen in Figure 4.14. It can be seen that there is a very low overall probability of approximately 0.0012 when the previous acceleration was 0ms⁻²and no possibility at any other time. The transition to the terminal state does not form part of the main probability matrix in order to allow testing of both SP-SDP and infinite horizon algorithms.

-3 -2 -1 0 1 2 3

Acceleration / ms^-2 0

0.2 0.4 0.6 0.8 1 1.2 1.4

Probability

×10^-3 Terminal State Transition

Figure 4.14: Terminal State Transition Probability - Loughborough Mail Room Cycle

4.5 Validation

Some validation of the Markov chain can be achieved using the Monte Carlo method. The vehicle is initialised at zero speed and acceleration and then the transition to the next state is randomly chosen based on the probabilities in the Markov Chain. This process contin-ues until the vehicle transitions into the terminal state. This will generate a random trip based on the transitional probabilities contained in the Markov Chain, which can then be compared to the original logged data, see Figure 4.15. By running a large number of such simulations, the distribution of various parameters can be compared and SAFD analysis can be used to calculate a numerical assessment of correlation. A high correlation will show that the Markov chain effectively models the behaviour of the driver and the vehicle for the assessed duty cycle.

0 200 400 600 800 1000 1200

0 5 10 15

Speed /ms-1

Logged Trip

0 200 400 600 800 1000 1200

Time /s 0

5 10 15

Speed /ms-1

Randomly Generated Trip

Figure 4.15: Randomly Generated Drive-cycle Comparison

Figure 4.15 shows a comparison between a logged trip and one of a similar length gener-ated using the Markov Chain. It can be seen that the trip genergener-ated using the Markov Chain is significantly different to that from the logged data although the generalised behaviour of the vehicle is quite similar. Both trips show similar top speeds of 10ms⁻¹and 11ms⁻¹, and are characterised by a number of journeys at approximately 7ms⁻¹average speed separated by periods where the vehicle is stationary. One significant difference, however, is that the speeds in the logged data are continuous whereas in the randomly generated trip they are not. This is due to the discretization of the data for development of the Markov Chain.

The Monte Carlo method has been used to randomly generate 10,000 trips for a number of the duty cycles logged. This allows the distribution of journey lengths, modal speeds, maximum speeds and peak accelerations to be compared to that of the original logged data. The results of the Loughborough University mail room duty cycle using the rounding method are shown in Figure 4.16. It can be seen that the distribution of journey lengths are quite similar, although the Markov model has produced a number of trips which are longer in length than the original data. As a result, a lower proportion of trips are between 1 and 2km than in the logged data, but this is still the most common length. The peak modal speeds are identical, although the logged data show a greater variation either side of this peak. Both the logged data and randomly generated cycles have a modal maximum speed of approximately 9ms⁻¹, although the proportion of trips with a maximum speed of between 10 and 14ms⁻¹ is higher in the randomly generated cycles. The logged data show a peak acceleration of 1ms⁻²in 40% of the journeys and 2ms⁻²in the rest. In comparison only 10%

of the randomly generated cycles have a peak acceleration of 1ms⁻¹, with approximately 90% reaching 2ms⁻¹. This is perhaps due to the increase in average journey length, meaning that more trips are likely to hit higher maximum speeds.

Figure 4.16: Statistical Comparison for Loughborough Mail Room Cycle using Rounding Discretization Method

Figure 4.17: Statistical Comparison for Loughborough Mail Room Cycle using Interpolation Discretization Method

In comparison, the results for the same data using the interpolation discretization method are shown in Figure 4.17. In this case, the estimation of journey length and modal speed is slightly improved and the peak acceleration seen is identical to the rounding method. How-ever, it is clear that the interpolation method tends to overestimate the maximum speed.

This is likely due to the issue mentioned in Section 4.4.2, leading to a higher usage of ex-trapolated data. In the Monte Carlo simulation, as in the SDP optimisation, any errors will be cumulative. Another point to note is that the interpolation method may result in impos-sible accelerations under some circumstances. For example, if the maximum acceleration of the vehicle was 2.1ms⁻²at a certain speed, the rounding method would give a maximum acceleration of 2ms⁻², however the interpolation method would result in some weighting being given to 2ms⁻² and some to 3ms⁻². Although it is possible to have a similar effect for the rounding method at 2.6ms⁻², for example, this would occur as often, and would result in a lower over-estimation of the maximum acceleration by the model. Therefore, these results suggest that the interpolation method is not adequate to substitute for higher resolution grid spacing.

Figure 4.18: Statistical Comparison for Loughborough Mail Room Cycle using Gaussian Discretization Method

Finally, the results for the Gaussian discretization method are shown in Figure 4.18. It can be seen that the maximum speeds reached in the generated trips are much lower than for the other two methods and closer to the original data, although they are approximately 1ms⁻¹ lower than the original data on average. The estimation of modal speed slightly different, with the approximately 50% of cycles averaging 7ms⁻¹, and only approximately 30% having a modal speed of 6ms⁻¹, which is again closer to the original data than the other two methods, although no trips had a modal speed of below 6ms⁻¹as was seen in the logged data. The peak acceleration in approximately 60% of the trips was 1ms⁻² and in the rest, it was 2ms⁻². This is the opposite of the distribution seen in the original data and is subject to approximately the same magnitude of error when compared to the other methods. The journey length, however, is vastly underestimated by the Gaussian distribution method with approximately 90% of trips lasting less than 2km and no trips exceeding 4km.