Simulation & Data processing

The processes that constitute the simulation work and subsequent data processing are illustrated within Figure 4-6 and discussed below.

FIGURE 4-6–SUMMARY OF THE DATA PROCESSING STEPS.FOLLOWING SIMULATION WORK, THE SIGNALS FOR VELOCITY AND POWER DEMAND ARE RECORDED.AN INITIAL PART OF THE SIGNAL IS DELETED TO AVOID THE STANDSTILL PORTION OF EACH SIMULATION. THE PERIOD, I.E. LAP LENGTH OF EACH CIRCUIT IS DETERMINED THROUGH AN UNBIASED AUTO-CORRELATION.THE COLLECTED SIGNAL IS SPLIT INTO INDIVIDUAL LAPS, FROM WHICH A MEAN LAP IS CALCULATED.

A simulation was carried out for each circuit. The data extracted from simulations are the vehicle speed and battery power profiles at the power source terminals as illustrated within Figure 4-7a and Figure 4-7b, respectively, on the example of the Silverstone circuit simulation. Each simulation starts with the vehicle at standstill and 1800 seconds of driving are recorded for each racing-circuit. This is done to ensure that each recorded power demand profile contains at least

Simulation

79 one complete lap without a start from standstill. The sampling frequency was set to 10 Hz, which is the maximum sampling frequency of the available battery testing equipment, discussed in more detail in Chapter 6. For real-life data, sampling at higher frequencies would be useful to analyse the effect of traction limiting scenarios such as tyre slipping, free spinning, or locked up wheels, on the power profiles. However, this level of fidelity cannot be achieved with the current models.

FIGURE 4-7– A) VEHICLE SPEED RECORDED FROM THE SILVERSTONE SIMULATION; B) ASSOCIATED BATTERY DUTY CYCLE PROFILE FROM THE SILVERSTONE SIMULATION

Both, the vehicle speed and the battery power profile follow a periodic pattern, with almost identical profiles for every lap, as would be expected for a racing driver. As such, a typical duty cycle for each circuit can be approximated as a mean lap from the simulations. An initial step in this process is to determine the period, thus lap duration, of each profile. To reduce the effect of the standing start on each mean lap, the initial portion of each power profile is eliminated. This initial starting point can be defined at the the first point at which charging of the power source exceeds a minimum threshold, or after a specific amount of time has passed. Within this work, a minimum charging threshold value of 10kW was chosen arbitrarily. From this reduced profile, the

80 number of individual periods and their duration can be determined through an unbiased auto-correlation function [218].

Assuming the recorded data is a signal 𝑥(𝑛) containing 𝑁 data points as described in equation (34).

𝑥(𝑛) = [𝑥₀, 𝑥₁, … 𝑥_𝑁] (34)

The unbiased auto-correlation (𝑅𝑥𝑥) of 𝑥(𝑛) is the unbiased correlation of 𝑥(𝑛) with a shifted copy of itself as a function of the shift (𝑚) as described in equation (35), where 𝑥^∗ is the complex conjugate of 𝑥.

𝑅𝑥,𝑥,𝑢𝑛𝑏𝑖𝑎𝑠𝑒𝑑(𝑚) =∑𝑁−𝑚−1𝑥𝑛+𝑚. 𝑥𝑛∗ 𝑛=0

𝑁 − |𝑚| (35)

𝑚 = [0,1, … 𝑁 − 1] (36)

The calculated correlation function is unique for each power profile. The peak values for 𝑅 have a different amplitude for each individual circuit. To ease the processing of multiple profiles within MATLAB®, the values of each correlation function are normalised through equation (37) to provide values between a maximum and minimum limit of 1 and -1, respectively.

𝑅_{𝑁𝑜𝑟𝑚}= 𝑅

max(𝑅) (37)

A plot of 𝑅_{𝑁𝑜𝑟𝑚} vs the lag 𝑚 for the Silverstone profile is shown in Figure 4-8.

81 FIGURE 4-8–NORMALISED UNBIASED AUTOCORRELATION VS SAMPLE SHIFT.THE RED CROSSES MARK THE LOCATION OF THE HIGHEST PEAKS, INDICATING COMPLETION OF A FULL LAP. THE NUMBER OF SAMPLE POINTS BETWEEN PEAKS IS EQUAL TO THE SAMPLES CONTAINED WITHIN EACH LAP.

The number of samples for each period, thus lap, is equal to the number of samples between the highest peaks marked by a red cross. These are determined through use of the MATLAB®

“findpeaks” function. The number of peaks and their location indices (𝑙𝑜𝑐 = [𝑙𝑜𝑐0, 𝑙𝑜𝑐₁… 𝑙𝑜𝑐_𝐼]) with respect to the sample shift are stored. The definition for a single lap profile (𝑥_𝑖(𝑛_𝑖)), is described in equation (38) and equation (39), where 𝑖 denotes the lap number, and 𝐼 is the number of peaks in Figure 4-8.

𝑥_𝑖(𝑛) = [𝑥_{𝑙𝑜𝑐𝑖}, 𝑥_{𝑙𝑜𝑐0+1}, … 𝑥_{𝑙𝑜𝑐(𝑖+1)−1}] (38)

𝑖 = [1,2, … 𝐼 − 1] (39)

The resulting power profiles for the laps are shown in Figure 4-9a. As the data stems from simulations, the individual laps vary only by one sample, i.e. 0.1 s. For the Silverstone simulation, the shortest period contains 1631 samples, and the longest period 1632 samples. The shorter period length was used for every lap, and the last data sample for longer laps was omitted, such that all laps have the same length. A typical lap profile (𝑃(𝑡)) is then approximated as the mean lap over the periods displayed in Figure 4-9b.

82 FIGURE 4-9– A)LAP PROFILES EXTRACTED FROM ORIGINAL RECORDING FOR THE SILVERSTONE SIMULATION; B)

MEAN LAP CALCULATED FROM THE LAPS SHOWN IN SUBFIGURE A).

This approximation is only justifiable as each simulation returns lap profiles, which are almost indistinguishable from one another. For real world data, where more variation is expected, an intermediate data processing step may be required. For larger differences between individual periods, each lap profile could be scaled and interpolated onto the same time vector, as suggested within [219], and a mean lap calculated.

4.5.1 Normalised duty cycles

Normalisation of the mean lap power profiles with equation (40) allows decoupling of the power profile from battery and system parameters with the resulting profiles containing values between -100% and 100%. This processing step allows the scaling of the profiles to a subsystem or cell level, as it is independent of the internal design features of the power management system.

𝑃_{𝑁𝑜𝑟𝑚}(𝑡) = 𝑃_{𝐶𝑦𝑐𝑙𝑒}(𝑡)

max(𝑃𝑆𝑦𝑠𝑡𝑒𝑚) (40)

83 𝑃_{𝑁𝑜𝑟𝑚}(𝑡) is the normalised power profile, referred to as a duty cycle within this work, 𝑃_{𝐶𝑦𝑐𝑙𝑒}(𝑡) is the power demand profile resulting from the simulation, and max (𝑃_{𝑆𝑦𝑠𝑡𝑒𝑚}) is the peak power that the power source can supply to the electric machines. These resulting duty cycles, although dimensionless, are still fully dependent on the specific properties of the vehicle and driver models, and race-circuit selection. As such, they relay information about their intensity as a function of the system’s peak capability, without eliminating crucial information about differences between individual profiles. Furthermore it enables direct comparison with the battery testing profiles within the ISO 12405-2, IEC 62620-1 and Battery Test Manual [34–36], that aim to represent the best practice approach for LIB testing.