Multifractal analysis vehicle’s in-use speed profile for application in driving cycles Multifraktalna analiza eksploatacyjnego profilu prędkości

Each methodology for creating driving tests requires an analysis of the nature of recorded in-use driving speed signals. Most of the actual time signals, including the time series recorded by the on-board diagnostic system (OBD) and describing the way the manner of vehicle movement in actual road conditions, show non-stationarity and non-linearity as well as statistical multiscalarity. In practice, it means that the analysis of recorded time series requires the use of non-linear analysis methods [1, 7]. The methods of multifractal formalism using local power measures, such as Holder signal regularity exponents or probabilistic indicators, are a good way of modelling the dynamics of such systems [5, 8]. The Holder singularity exponent, determined at each point where the function is defined, reflects the level of amplitude fluctuation in the vicinity of this point. The scaling factor of the probabilistic meas-Andrzej


Introduction
Tests of the performance of automotive drive systems find application in the phase of concept development (basic cognitive research), in the design phase (as part of prototype testing), in the production phase (during quality control), at the admission stage (type approval tests) and in the maintenance phase (diagnostic tests). This goal is attained through various types of examination at engine test stands, which differ with regard to the stipulated working conditions. This work undertakes to address the problem of analysing the in-use vehicle speed profile with the view to applying it to driving tests. The Vehicle Driving Cycle (VDC) is a time series of data representing vehicle speed, which is intended to reflect a vehicle's movement in real road conditions and be used in evaluation of a vehicle or engine in terms of its economy and emissions.
Many years of research have led to several hundred practical driving cycles in various countries and regions. Most of the current tests used to determine emissions of motor vehicles are developed in accordance with the principle of accurate simulation in the time domain. The most popular tests are ones like FTP-75 (Federal Test Procedure), NEDC (New European Driving Cycle) or JC08 (Japanese cycle). NEDC is used in Europe, the FTP 75 cycle is used in the United States and JC08 in Japan. NEDC is a synthetic cycle of the theoretical driving profile, while the other two come from actual usage data [3,9].
More reliable results for fuel economy during the simulation of vehicle operating conditions at engine test stands are to be provided by the WLTC test (Worldwide Harmonised Light duty Test Cycle), which is a result of the world-wide harmonised light vehicle testing procedure. The cycle is based on actual driving profiles derived from statistical research [18].
Each methodology for creating driving tests requires an analysis of the nature of recorded in-use driving speed signals. Most of the actual time signals, including the time series recorded by the on-board diagnostic system (OBD) and describing the way the manner of vehicle movement in actual road conditions, show non-stationarity and non-linearity as well as statistical multiscalarity. In practice, it means that the analysis of recorded time series requires the use of non-linear analysis methods [1,7]. The methods of multifractal formalism using local power measures, such as Holder signal regularity exponents or probabilistic indicators, are a good way of modelling the dynamics of such systems [5,8].
The Holder singularity exponent, determined at each point where the function is defined, reflects the level of amplitude fluctuation in the vicinity of this point. The scaling factor of the probabilistic meas-
Słowa kluczowe: analiza multifraktalna, testy jezdne, rzeczywiste warunki drogowe. ure distribution function, based on the values of the signal amplitude, allows for segmentation with considerations for the level of entropy. The statistical distribution represented by the histogram of singularity exponents is a picture of the multifractality of the signal. There are two ways to determine the multifractal spectrum of a singularity exponent. The direct method relies on the approximation of the histogram for determined Holder exponents. The indirect method in which the fractal dimension is usually assumed to be the capacitive dimension of the curve being a graph of the signal considered, is based on the algorithm of detrended fluctuation analysis of time series [6].
The article presents the following issues. Chapter 2 signals the idea of the applied multifractal analysis method. The results of this analysis for the driving cycle of the harmonised light vehicle test procedure (WLTC) are presented in Chapter 3. The tests carried out in actual urban driving conditions are discussed in Chapter 4. Chapter 5 constitutes the concluding section.

Multifractal formalism and the regularity measure of a time series
Multifractal formalism has found a number of applications in the analysis of real signals, in particular in performance diagnostic tests of motor vehicles [4,[11][12][13][14][15][16][17]. The conducted research indicates various procedures for identifying the multifractal nature of actual time series.
This chapter presents an approach based on the exponents of signal singularities and multifractal spectrum. The point exponent of Holder singularity function ( ) f x in point 0 x denotes the number h defined as the supremum of all exponents satisfying, for a certain C> 0, the condition: shows that the exponent 1 h > describes the regularity of a function more accurately than its subsequent derivatives. If the signal representation is a time series , 1, 2, , where ( ) h i is the singularity exponent of time series at point .
where ( ) H q is a generalised Hurst exponent, and the parameter q allows for decomposing the measure with regard to its value. Legendre's transform of generalised large-scale exponent τ q qH q ( ) = ( )−1 allows for obtaining a multifractal spectrum: is Holder singularity exponent.
The multifractal spectrum constitutes a histogram of exponents reflecting the level of signal amplitude fluctuation. Moreover, the parallelism of multifractal formalism and statistical thermodynamics indicates that Holder singularity exponent and the multifractal spectrum can be interpreted as the energy and entropy of the studied process respectively.
The description of the system's dynamic properties based on the multifractal spectrum of the time series is permitted by:

Simulation research of the WLTC test
A number of studies have confirmed that the driving tests used for many years are not representative for the actual operating conditions of vehicles. As a result, the emission and fuel consumption of vehicles are underestimated. Striving for a more dynamic, harmonised test cycle, a new WLTC (Worldwide harmonised Light duty Test Cycle) has been developed. The synthesis was made on the basis of data on traffic parameters in Europe, India, Japan, Korea and the USA, taking into account situations related to driving in urban, countryside and motorway traffic. Compared to NEDC, the test cycle is longer, significantly more dynamic and features a lot more acceleration and deceleration cycles, shorter stops and higher average and maximum speed values. Due to the introduced changes, the WLTC driving cycle will provide much more accurate conditions for calculating fuel consumption and exhaust emissions [10]. Its first application will apply to vehicle models introduced to the market for the first time since September 2017. The WLTC driving cycle has been divided into four parts, corresponding to different driving speeds: low, medium, high and extra high. If the maximum speed does not exceed 135 km/h, then part of extra high speed should be replaced with part of the low speed. The WLTC time series for 3b class cars, with the power to weight ratio of PWr> 34, is shown in Fig.1, and its basic parameters in Table  1. The drive dynamics in the case of these vehicles is determined by the driver's behaviour and the intensity of the road traffic, not by the car's technical parameters.
Singularity spectra for the studied WLTC driving test shown in Fig. 2, confirm its multifractal nature. The level of multifractality is: , while the dimensional range of segments with the highest and lowest probability Δ 0,92 f =  . The most common singularity exponents refer to those fragments of recorded time series which describe the greatest variability. In turn, the lowest probability of recording is shown by the drive periods of the highest regularity. Most of the points are concentrated around the dimensions corresponding to the singularities with the largest and smallest fluctuation of vehicles speed time series. sciENcE aNd tEchNology

Research and analysis of empirical data
The article presents the results of vehicle traffic tests in actual road conditions, represented by urban driving in a large agglomeration, between 9 am and 1 pm for five working days. The conducted analysis was based on vehicle's speed time series with a sampling period equal to 1s. The applied measuring device, shown in Fig. 3 is capable of sampling the speed once every second. The range of speeds available for recording is 0 -255 km/h, with a resolution of 1 km/h while the maximum recordable research distance is 16,000 km. The device has the ability to record a number of other operating parameters, such as engine speed, vehicle acceleration, pressure in the inlet manifold, and throttle position and allows communication with the engine governing unit using the following communication protocols: J1850-41,6, J1850-10,4, ISO9141, KWP2000 (ISO 14230), CAN (Control Area Network ISO 11898).
The measuring route consisted of a 12kilometre-long road section and ran between Plac Wilsona and the Galeria Mokotów in Warsaw. The map presenting the measurement route, developed with the use of Google maps, is shown in Fig. 4.  Multifractal spectra shown in Fig.6 -Fig.7 reflect the decomposition of the actual vehicle speed time series with reference to the abundance of fragments with a specific dynamics of variation.

Conclusions
The conducted analysis confirmed the multifractal nature of vehicle speed fluctuations both within WLTC driving cycle of the universal harmonised test procedure for light vehicles as well as real road conditions of urban driving along the measurement route, being subject of experimental research in Warsaw. Multifractal spectra are characterised by a similar shape and similar values of the level of multifractality. The asymmetric course of the spectra indicates the analogous properties for multifractals obtained as a result of the implementation of the generalised process of the binomial multiplicative cascade [2]. The obtained results indicate the possibility of using the multiplicative cascade process for the synthesis of real operating conditions and short-term forecasting of traffic conditions, which is a critical aspect in the development of intelligent transport systems. This step requires further research including driving in conditions outside urban centres and on motorways.