The Robustness of the Mathematical Model - Literature review Example

Table of Contents 5. Analysis of the Mathematical Model 3 5 Sensitivity Analysis 4 5 1. Noise 5 5 2. Simulation Results 6 5 3. New European Drive Cycle, 1st sub-cycle 7 5.1.4. New European Drive Cycle, 5th sub-cycle 9 5.1.5. Prediction of Fuel Consumption in Artificial Driving Phases 12 5.1.6. Constant Speed 15 5.1.7. Acceleration 17 5.1.8. Deceleration with throttle 19 5.1.9. Gear Change 20 5.2. Conclusion 21 5.3. Recommendations 22 5.4. Bibliography 23 List of Figures Figure 5.1 - Measured and computed fuel consumptions during first sub-cycle of NEDC, (a) without noise, (b) with noise in input for computation 9 Figure 5.2 - Measured and computed fuel consumptions during fifth sub-cycle of NEDC, (a) without noise, (b) with noise in input for computation 11Figure 5.3 - Drive cycle simulated in Test 1, (a) velocity, (b) fuel consumption 13 Figure 5.4 - Drive cycle simulated in Test 2, (a) velocity, (b) fuel consumption 14 Figure 5.5 - Fuel consumption in constant speed phases during Test 1 and Test 2, and its prediction by the mathematical relationship 17 Figure 5.6 - Fuel consumption in acceleration phases during Test 1 and Test 2, and its prediction by the mathematical relationship 18 Figure 5.7 - Fuel consumption in deceleration phases during Test 1 and Test 2, and its prediction by the mathematical relationship 20 Figure 5.8 - Fuel consumption in gear change phases during Test 1 and Test 2, and its prediction by the mathematical relationship 21 Chapter 5 1. Analysis of the Mathematical Model A mathematical model was presented in the previous chapter that was designed to tabulate the fuel consumption by providing various driving input parameters. Testing and validation was carried out in the laboratory and under real life driving conditions in order to decipher how driving input parameters and fuel consumption were connected. The robustness of the mathematical model presented before needs to be verified theoretically to ascertain the validity and reliability under various performance regimes. This chapter will look critically at the mathematical model presented before using sensitivity analysis methods that will target the various inputs to the mathematical model. The contention behind the sensitivity analysis is to vary inputs to the mathematical model as a means of measuring the change in the overall output. This would provide for the amount of variation that could occur in the output of the mathematical model in case that erroneous inputs are received. Firstly, a sensitivity analysis is carried out where perturbations are added to the input data (i.e. velocity, acceleration, and throttle position), and consumption is calculated during a drive cycle using the perturbed data. Then, artificial drive cycles are created, and fuel consumption is measured and calculated during the same drive cycle. In the artificial drive cycles, high and low velocities and accelerations are involved in order to study the limits until the mathematical relationship between fuel consumption and drive cycle parameters can be used. Sensitivity AnalysisSensitivity analysis is carried out in order to determine how any form of uncertainty in the output of a provided mathematical model could be traced back to the various inputs that are being provided. The apportionment of output uncertainty to input uncertainties in their respective contribution levels allows improvements on the mathematical model for bringing about greater reliability and validity (Saltelli et al., 2008). A sensitivity analysis may be performed for a number of reasons. Primarily, sensitivity analysis allows the determination of model robustness when faced with uncertainty in the inputs (Becker et al., 2011). In addition, sensitivity analysis allows gaining a deeper understanding of how various inputs are tied to the final output. As a whole, sensitivity analysis allows for a reduction of uncertainty since the model’s inputs that provide for sizable uncertainty introduction could be focused on. It must be taken to note that sensitivity analysis provides for errors available in mathematical models but this is not its primary purpose. In a similar manner, sensitivity analysis allows simplification of the model. The inputs that have little or no impact on the overall mathematical model, but are still present, and are eliminated in the light of sensitivity analysis results. Comparably, redundancies in the mathematical model can also be eliminated through a comprehensive sensitivity analysis. Another primal function of sensitivity analysis is to allow optimisation of the mathematical model by concentrating on inputs that provide the greatest change in the required direction. In the case of the current research, the sensitivity analysis presented below could be used to perform optimisation if required. NoiseNoise is provided to the mathematical model in order to produce deviations from the expected outcomes of the proposed mathematical model. In order to deal with the sensitivity analysis for the noise, the inputs to the mathematical model (velocity, acceleration and throttle position) are manipulated by providing faulty inputs. In order to deal with the faulty inputs, random numbers are generated and fed to the mathematical model. The noise based sensitivity analysis in this manner occurs when simulations are taking place for gauging fuel consumption against other driving parameters. The choice of noise inputs to the mathematical model holds particular significance. If the noise inputs to the mathematical model are too high or alternatively too low, it would result in significant deviations from the expected output of the mathematical model. The noise on the input should correspond to the uncertainty in the measurement. If the mathematical model provides satisfactory outputs even with such noise in the input, then the robustness of the model is acceptable. The faulty noise inputs to the mathematical model were kept between -2% and +2% of the regular input range. The random numbers generated to serve as the faulty input were kept within the 2% range listed above. This made it possible to verify that if faulty inputs corresponding to the measurement error were provided to the mathematical model then the output was still within an acceptable range. The mathematical model was supplied with faulty inputs within the range of operation shown below as a list:allowed to exceed 2 km/hr; Acceleration was varied between ±2%; Throttle position was varied between ±2%. The error limits presented above are the maximum data measurement errors even though both velocity and acceleration are typically measured even more accurately. The absolute throttle position error was discovered to be no greater than ±1.4% which indicates the efficacy of the overall system. Based on these error tabulations, the discrepancy between actual fuel consumption and that predicted by the mathematical model remained low and within recognised limits. These issues are discussed in greater detail under Section 5.1.3 and 5.1.4. Parameter Operating Range (%) Noise Inputs ±2 Velocity ±2 Acceleration ±2 Throttle Position ±2 Simulation Results The New European Drive Cycle (NEDC) was utilised as the basic framework for the purposes of this research. The laboratory testing as well as the real life driving tests conducted relied on NEDC and its various components such as urban driving and extra urba driving. In order for the simulation of the NEDC and validation of the mathematical model, only one sub cycle of the NEDC was simulated. The large amounts of data generated from various sub cycles runs of the NEDC meant that a lot of computing power and time was required for validation and reliability checks of the mathematical model. In order to deal with this problem, various sub cycles of the NEDC were simulated and their validity and reliability against the mathematical model was checked in phases. Moreover, various simulated sub cycles of the NEDC were checked for validity and reliability of the mathematical model using various parameters such as fuel consumption, velocity, acceleration, throttle position etc. For the purposes of this simulation, the first sub cycle of the NEDC was tested for validation and reliability of the mathematical model. New European Drive Cycle, 1st sub-cycleIn order for testing to take place, the requirements of the NEDC such as the idling time, cruising time, acceleration and deceleration were observed as close as possible to the provided guidelines. The results of the NEDC were computed for fuel consumption measurement rather than other parameters for the purposes of this simulation. The measured total consumption of the NEDC first sub cycle testing revealed that fuel consumption stood at 0.3404 g/s. On the other hand, computations of fuel consumption for the first sub cycle of the NEDC using the mathematical model provided a computed fuel consumption of 0.3222 g/s. In order to make the testing more realistic, noise was also added using noise based inputs within the 2% range of inputs specified above. The computed total fuel consumption through the mathematical model with noise revealed a computed fuel consumption of 0.3224 g/s. For evaluation of the mathematical model’s reliability and validity two measures were employed. The first measure relied on the error between the measured total consumption and the computed total consumption of the mathematical model. The second measure relied on the error between the total consumption obtained using the mathematical model and the total consumption obtained by the mathematical model with noise. Results revealed that the error between measured total consumption and computed total consumption was -5.3% while the error between the computed total consumptions without and with noise was +0.1%. The results are summarised in the table shown below. Parameter Fuel Consumption (g/s) Measured total consumption 0.3404 Computed total consumption 0.3222 Computed total consumption with noise 0.3224 (a) (b) Figure 5.1 - Measured and computed fuel consumptions during first sub-cycle of NEDC, (a) without noise, (b) with noise in input for computation The graphs shown above show two different simulation runs for the NEDC first sub cycle that compare measured fuel consumption and computed fuel consumption with and without noise. The plots above make it abundantly clear that the difference between measured fuel consumption and computed fuel consumption is hardly noticeable. This stands to indicate that the validity and reliability of the proposed mathematical model is robust. New European Drive Cycle, 5th sub-cycle The fifth sub cycle of the NEDC was also simulated in order to verify the results produced by the mathematical model. The NEDC cycle was simulated under laboratory conditions as well as under real life driving conditions in order to test the reliability and validity of the mathematical model. The fifth sub cycle was simulated and discussed in this section. The measured total fuel consumption of the fifth sub cycle was tabulated as 0.6905 g/s. On the other hand, the computed total consumption from the mathematical model without noise was tabulated as 0.6761 g/s. In contrast, the computed total consumption from the mathematical model with noise was tabulated as 0.6652 g/s. The errors between various simulations of measurement and computation were tabulated in order to gauge the reliability and validity of the mathematical model. The error between the measured total consumption and the computed total consumption of the mathematical model was -2.1%. In contrast, the error between the computed total consumption of the mathematical model without noise and with noise was tabulated as -1.6%. It must be noticed that the errors between various measured and computed simulation results has gone up from before indicating that the mathematical model is robust although the error has increased. The results are summarised in the table shown below. Parameter Fuel Consumption (g/s) Measured total consumption 0.6905 Computed total consumption 0.6761 Computed total consumption with noise 0.6652 (a) (b) Figure 5.2 - Measured and computed fuel consumptions during fifth sub-cycle of NEDC, (a) without noise, (b) with noise in input for computation Prediction of Fuel Consumption in Artificial Driving Phases As discussed before, the mathematical model was created based on the NEDC framework. The sensitivity tests conducted before relied on the NEDC cycle too and it could be expected that their results would provide little aberration from the mathematical model. In order to touch upon the claims made in Chapter 4 before that the proposed mathematical model could be applied to any other drive cycles, a number of different tests were conducted on artificially designed drive cycles. Two different drive cycles were utilised in order to test the reliability and validity of the proposed mathematical model. The drive cycles created for testing consisted of both constant speed as well as gear change phases. Furthermore, the first artificial drive cycle created for testing purposes consisted of high acceleration phases (with acceleration varying between 4 km/h/s and 5 km/h/s) and medium deceleration phases (with deceleration varying between -2 km/h/s and -3 km/h/s). On the other hand, the second artificial drive cycle created for testing purposes consisted of low acceleration phases (below 1 km/h/s) and high deceleration (varying around -5 km/h/s). The combination of acceleration and deceleration in both tests was meant to cover all available possibilities i.e. low to high acceleration as well as low to high deceleration. The drive cycle testing results are shown in the plots below as Figure 5.3 for the first artificial drive cycle and Figure 5.4 for the second artificial drive cycle.The velocity as well as the resulting fuel consumption for both driving cycles was plotted as shown in Figs. 5.3 and 5.4. The plot characteristics are discussed below for explanation purposes. (a) (b) Figure 5.3 - Drive cycle simulated in Test 1, (a) velocity, (b) fuel consumption (a) (b) Figure 5.4 - Drive cycle simulated in Test 2, (a) velocity, (b) fuel consumption As shown in Figure 5.3, the velocity was varied in phases for testing purposes. The velocity was allowed to increase followed by constant velocity phases for time periods of 25 seconds. This signifies that the acceleration was varied in constant increments between the start of testing (t = 0 seconds) and the end of testing (t = 225 seconds). The jumps in velocity also signify the presence of gear changes of 25 seconds each from the start of the testing to 150 seconds after which a phase of deceleration and decreasing velocity is witnessed. In the second artificial driving cycle, the velocity was allowed to increase throughout the acceleration phases with constant acceleration. Velocity was allowed to increase between 50 seconds and 325 seconds after which constant deceleration and velocity decrease is available. If the fuel consumption for both phases is compared, it becomes clear that for the first artificial driving cycle the fuel consumption tends to show sharp increases with every gear change. On the other hand, the fuel consumption for the second artificial driving cycle tends to show a constant increase without such high local maxima as in the first cycle. On the other hand, the deceleration phases in both aritificial drive cycles tend to display constant decreases. It could also be inferred that fuel consumption in the first artificial drive cycle would be higher due to the presence of local maxima, when compared to the second artificial drive cycle. Constant SpeedFirst, the constant speed regimes are evaluated to validate the efficacy of the mathematical model. The data points from the two artificial driving tests were plotted against the fuel consumption in order to validate the relationship between fuel consumption and constant speed. Regression was performed in order to discern the relationship between fuel consumption and constant speed. Based on the previously proposed mathematical model from Chapter 4, the equation for constant speed could be expressed as shown in the equation below: (4.8) An observation of the plot below (Figure 5.5) clearly shows that the plot has a quadratic nature. The examination of equation 4.8 above also affirms this observation given the presence of quadratic elements. The derived relationship provides a satisfactory prediction (errors below 20% for most cases), except for the highest velocity where the consumption is significantly underestimated. This is delineated in Figure 5.5 by the outlier points (last two points of the artificial driving test cycles) that are not covered by the regression relationship. In the first test, there is a considerable discrepancy between measured and calculated consumption; however, this may be due to the unexpected behaviour of measured data. Consumption hardly increases with velocity up to 80 km/h, and then it jumps up to a very high value (2.02 g/s). This would indicate that the acceleration and deceleration parameters enunciated before for the first artificial driving cycle are not as valid and reliable as those for the second artificial driving cycle. Figure 5.5 - Fuel consumption in constant speed phases during Test 1 and Test 2, and its prediction by the mathematical relationship Acceleration The acceleration regimes are also evaluated to validate the efficacy of the mathematical model. The data points from the two artificial driving tests were plotted against the fuel consumption in order to validate the relationship between fuel consumption and acceleration. Regression was performed in order to discern the relationship between fuel consumption and acceleration. Based on the previously proposed mathematical model from Chapter 4, the equation for acceleration could be expressed as shown below: (4.10) An observation of the plot below (Figure 5.6) clearly shows that the plot has an exponential nature. The examination of equation 4.10 above also affirms this observation given the presence of exponential elements. The derived relationship provides a satisfactory prediction (errors below 20% in most cases). This indicates that for most data points the mathematical model enunciated before tends to hold true. However, for the highest velocities the consumption is underestimated as the error goes up to 30% to 40%. This signals that for both artificial driving cycles the highest velocities tend to provide unsatisfactory results since the error becomes significantly large. There is need to further investigate this part of the mathematical model to verify if this observation arises out of mistaken data points during testing or if the relationship provided above does not adequately cover this testing regime. Figure 5.6 - Fuel consumption in acceleration phases during Test 1 and Test 2, and its prediction by the mathematical relationship Deceleration with throttle The deceleration regimes are evaluated in this section. The data points from the two artificial driving tests were plotted against the fuel consumption in order to validate the relationship between fuel consumption and deceleration. Regression was performed in order to discern the relationship between fuel consumption and deceleration. Based on the previously proposed mathematical model from Chapter 4, the equation for deceleration could be expressed as shown in the equation below: (4.12) An observation of the plot below (Figure 5.7) clearly shows that the plot has a linear nature. The examination of equation 4.12 above also affirms this observation given the presence of linear elements. Deceleration may happen with or without throttle. In case of deceleration without throttle, the fuel consumption is 0 g/s if a gear in engaged; otherwise, when the gear is in neutral, the consumption is approximately the same as during idling, which is a special case of constant speed. Therefore, the phases of deceleration with throttle are studied in both the first and second artificial driving cycles. There is a significant scatter in the data; therefore, the prediction provided by the relationship is considerably less accurate, than in the constant speed and acceleration phases. However, it must be taken to note that predicted values follow the same tendency as the measured data. Figure 5.7 - Fuel consumption in deceleration phases during Test 1 and Test 2, and its prediction by the mathematical relationship Gear Change Finally, the gear change regime is evaluated. The data points from the two artificial driving tests were plotted against the fuel consumption in order to validate the relationship between fuel consumption and gear change. Regression was performed in order to discern the relationship between fuel consumption and gear change. Based on the previously proposed mathematical model from Chapter 4, the equation for gear change could be expressed as shown in the equation below: (4.14) An observation of the plot below (Figure 5.8) clearly shows that the plot has a linear nature. The examination of equation 4.14 above also affirms this observation given the presence of linear elements.There is a scatter in the data during gear change phases although it seems less significant as during the deceleration phases. The prediction by the relationship is satisfactory in most of the cases during testing for the second artificial driving cycle; however, there is a consistent overestimation during testing for the first artificial driving cycle. Similarly to the deceleration phases, predicted values follow the same tendency as the measured data. Figure 5.8 - Fuel consumption in gear change phases during Test 1 and Test 2, and its prediction by the mathematical relationshipThe mathematical relationship is reliable during constant speed and acceleration phases, although there is a significant underestimation of consumption for the highest velocities (above 100 km/h). The relationship is inaccurate in some cases during deceleration and gear change phases; however, it follows the measured tendency even during these phases. This means that the prediction at a given time instance may not be close, but the relationship is applicable to predict the average consumption during a drive cycle. Recommendations Based on the observations of the tested data and the proposed mathematical model above, it could be seen that there is further need to investigate certain regimes where the proposed mathematical model does not hold as true. More importantly more research is required for regimes where the velocity is high. Although the measured data and computed data are far apart but there is a tendency for both data sets to follow each other. This implies that the constants used in the mathematical relationships above would require revision. It may also be the case that the mathematical relationships listed above would require revision so as to derive differing mathematical relationships in differing operating regimes. There is also a need for further research into deceleration and gear change phases of any kinds of driving cycles. This is expected to provide greater reliability and validity to the existing mathematical relationship; thus, making it more robust for the purposes of all kinds of driving cycles. Bibliography Becker, W. et al., 2011. Bayesian sensitivity analysis of a model of the aortic valve. Journal of Biomechanics, 44(8), pp.1499-506. Saltelli, A. et al., 2008. Global Sensitivity Analysis: The Primer. 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