Identification and quantification analysis of nonlinear dynamics properties of combustion instability in a diesel engine.

Identification and quantification analysis of nonlinear dynamics properties of combustion instability in a diesel engine Li-Ping Yang, Shun-Liang Ding, Grzegorz Litak, En-Zhe Song, and Xiu-Zhen Ma Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 25, 013105 (2015); doi: 10.1063/1.4899056 View online: http://dx.doi.org/10.1063/1.4899056 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/25/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Reduction of NOx emission on NiCrAl-Titanium Oxide coated direct injection diesel engine fuelled with radish (Raphanus sativus) biodiesel J. Renewable Sustainable Energy 5, 063121 (2013); 10.1063/1.4843915 Characterization of complexities in combustion instability in a lean premixed gas-turbine model combustor Chaos 22, 043128 (2012); 10.1063/1.4766589 Analysis on performance, emission and combustion characteristics of diesel engine fueled with methyl–ethyl esters J. Renewable Sustainable Energy 4, 063116 (2012); 10.1063/1.4767911 Experimental investigation effects of blend hazelnut oil on compression ignition engine performance characteristics and emission J. Renewable Sustainable Energy 4, 042701 (2012); 10.1063/1.4737921 Effect of palm methyl ester-diesel blends performance and emission of a single-cylinder direct-injection diesel engine AIP Conf. Proc. 1440, 562 (2012); 10.1063/1.4704263

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CHAOS 25, 013105 (2015)

Identification and quantification analysis of nonlinear dynamics properties of combustion instability in a diesel engine Li-Ping Yang,1,a) Shun-Liang Ding,1 Grzegorz Litak,2 En-Zhe Song,1 and Xiu-Zhen Ma1 1

Institute of Power and Energy Engineering, Harbin Engineering University, No. 145-1, Nantong Street, Nangang District, Harbin 150001, China 2 Faculty of Mechanical Engineering, Lublin University of Technology, Nadbystrzycka 36, 20-618 Lublin, Poland

(Received 26 April 2014; accepted 10 October 2014; published online 6 January 2015) The cycling combustion instabilities in a diesel engine have been analyzed based on chaos theory. The objective was to investigate the dynamical characteristics of combustion in diesel engine. In this study, experiments were performed under the entire operating range of a diesel engine (the engine speed was changed from 600 to 1400 rpm and the engine load rate was from 0% to 100%), and acquired real-time series of in-cylinder combustion pressure using a piezoelectric transducer installed on the cylinder head. Several methods were applied to identify and quantitatively analyze the combustion process complexity in the diesel engine including delay-coordinate embedding, recurrence plot (RP), Recurrence Quantification Analysis, correlation dimension (CD), and the largest Lyapunov exponent (LLE) estimation. The results show that the combustion process exhibits some determinism. If LLE is positive, then the combustion system has a fractal dimension and CD is no more than 1.6 and within the diesel engine operating range. We have concluded that the combustion system of diesel engine is a low-dimensional chaotic system and the maximum values of CD and LLE occur at the lowest engine speed and load. This means that combustion system is more complex and sensitive to initial conditions and that poor combustion quality leads to the C 2015 AIP Publishing LLC. decrease of fuel economy and the increase of exhaust emissions. V [http://dx.doi.org/10.1063/1.4899056] Although the durations of each engine cycles are same when engine speed is held constant, the cyclic combustion of diesel engines is separated by the intake and exhaust and the indicated pressure exhibits fluctuations reducing the output power. It is expected that better understanding of the harmful cycle-to-cycle variations could help to eliminate them and improve the engine efficiency. Using experimental time series, we identify and quantitatively analyzed the instabilities of combustion process in the diesel engine using chaos theory and nonlinear signal analysis techniques. In particular, the delay-coordinate reconstruction, recurrence plot (RP), and recurrence quantification analysis (RQA) are used to describe the dynamical response of the combustion system, to reconstruct the attractor and to identify the deterministic nature of combustion process. The correlation dimension (CD) and Lyapunov exponent calculation are applied to quantify the complexity of the combustion system and the sensitivity to initial conditions. The obtained results show that the combustion system of diesel engine is a lowdimensional chaotic system. Interestingly, as the engine speed and load are decreased, the structure of the attractors of combustion system becomes more complex with increasing combustion instabilities.

a)

Author to whom correspondence should be addressed. Electronic mail: [email protected].

1054-1500/2015/25(1)/013105/13/$30.00

I. INTRODUCTION

Under constant nominal operating conditions, reciprocating internal combustion engines would exhibit substantial cycle-to-cycle variations of in-cylinder combustion pressure. Any deviation in the pressure level or its time development can reduce power output, the efficiency, and the reliability of the engine, and simultaneously can increase engine exhaust gas emissions and noise.1 Cycle-to-cycle variations have been observed and studied for more than a century,2 and research on this phenomenon has continued.3–43 Earlier studies on cycle-to-cycle variations were focused mainly on spark ignition (SI) engines,3–6,19–43 while the corresponding analyses for diesel engines were investigated less intensively because of the lower strength cyclic pressure variations in diesel engines. However, with the worldwide energy crisis and environmental pollution problems getting more and more serious, the requirements for diesel engines to save energy and to control emissions have become increasingly strict. Thus, a few studies have directed attentions also to the combustion instability in diesel engines.7–18 In general, cycle-to-cycle variations have been described as either stochastic or deterministic in their nature.19,20 Recently, researchers have analyzed the nonlinear properties of cycle-to-cycle variations based on nonlinear dynamics and chaos theory.21–40 Daily pointed out that combustion cycle-to-cycle variations are an inherent consequence of nonlinear kinetics in the combustion process. Highly chaotic behavior occurred when the burn time occupies an excessive fraction of the cycle time.21 Chaos analysis was conducted by these measurements and the phase,

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waveform, Poincare, and FFT plots were presented by Chew et al.22 These results showed conclusively that the pressure fluctuation in cylinder is governed by a broadband chaos. Wagner et al.24,25 observed the transition from fairly stochastic behavior to more deterministic as the equivalence ratio is decreased from stoichiometric to very lean mixture conditions. The transition to nonlinear deterministic behavior was realized via a period-doubling bifurcation sequence. Finney et al.26 adopted the technique of symbolic time series analysis to analyze temporal patterns in dynamic measurements of engine combustion variables, and showed their utility in detecting deterministic features of data amidst high level noise. Green et al.27 indicated that the behavior of a spark-ignition lean burn engine was inconsistent with a linear Gaussian random process and was more appropriately described as a nonlinear dynamical process effected by noise. Time irreversibility of related heat release time series has been shown in Refs. 3, 4, and 27. Daw et al.28 proposed a physically oriented model and a cycle-resolved dynamic model to explain characteristics of combustion instabilities in spark ignition engines. The physically oriented model could analyze the interaction between stochastic, small-scale fluctuations in engine parameters, and nonlinear deterministic coupling in successive engine cycles, and the model could rapidly simulate thousands of engine cycles.28–30 Sen et al.31–34 adopted the methods of statistics and continuous wavelet transform to analyze the effect of exhaust gas recirculation, equivalence ratio, spark advance angle, and compression ratio on combustion dynamics in a spark ignition engine. Based on nonlinear deterministic mechanism, Davis et al.35 exploited a recognition system of the combustion instability in order to control cycle-to-cycle variations in lean fueled spark ignition engines. Wendeker et al.36 examined the combustion process in a spark ignition engine using nonlinear dynamics theory. More recently, Curto-Risso et al.37 also applied nonlinear time series methods to study combustion fluctuations in a spark-ignition engine. Though the RP, RQA, and Recurrence Networks are widely used in the field of economy, physiology, neuroscience, earth sciences, astrophysics, engineering, etc.,44–49 the corresponding analyses are less used in the field of internal combustion engine. Litak et al.38–40 estimated the deterministic nature and the noise level of cyclic combustion variations in a spark ignition engine by RPs and RQA. Sen et al.41 adopted the methods of RPs to analyze the mean indicated pressure cycle-tocycle variations in a diesel engine. Longwic et al.42 analyzed the peak pressure cycle-to-cycle variations in a diesel engine. From what we discussed above, we can easily find that most research on the dynamics characteristics of combustion system are focused on the spark ignition engine. In the field of diesel engines, the studies on nonlinear dynamic problems of combustion process are still scarce.15,18 This paper is focused on this direction. In this research, we adopted phase space embedding theory, RP, RQA, and the estimation of chaotic characteristics parameters to identify and quantitatively analyze dynamic properties of combustion instability in diesel engine under the entire engine operating conditions.

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II. FACILITIES AND EXPERIMENT

All the experimental data analyzed were obtained from an in-line, two cylinder, water-cooled, direct injection marine diesel engine. The test dynamometer is an eddy current dynamometer with automatic control. It can work at the constant speed control and constant torque control modes. During the test process, other feedback controls were canceled except for engine speed. Air-to-fuel ratio was monitored using an ETAS LA4 Lambda Meter. The multi-cycle cylinder pressure data were captured by a high-speed data acquisition system. This system is composed of a Kistler 6125c piezoelectric pressure sensor, Kistler 5011 charge amplifier, AVL365C crank angle encoder, AVL 621 combustion analyzer, and computer. The pressure sensor is mounted on the cylinder head of the first cylinder. This system provided the direct measurements of the combustion process in an internal combustion engine. The direct interaction between the pressure sensor and in-cylinder gas enabled us to effectively gain combustion information without additional signal interference. Each voltage signal produced by piezoelectric pressure sensor corresponds to a cylinder pressure after experimental calibration. The natural frequency of the piezoelectric transducer is 160 kHz. The highest sampling resolution of the data acquisition system was 0.1 of crankshaft angle ( CA). However, the higher sampling resolution meant a larger data file. In our experiment, a sampling resolution of 1 CA was chosen. An engine cycle is composed of two rotations; therefore, each cycle contains 720 sample points. Such sampling resolution was adequately precise to describe each combustion process. Tests were conducted under the entire engine operating conditions. Namely, the engine speed was from 600 to 1400 rpm (corresponding sample frequency is from 3.6 to 8.4 kHz) and the engine load rate (LR) was from 0% to 100%. The cylinder pressure data file of a single experimental test contained 2000 cycles of engine work. The operating conditions of the diesel engine are shown in Fig. 1 and the schematic of experimental stand is presented in Fig. 2. First, we acquired the in-cylinder pressure time series including 2000 cycles for each operation condition. Although the in-cylinder pressure is only one parameter, it is

FIG. 1. Operation conditions of diesel engine.


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FIG. 2. Schematic of the experimental stand. 1-diesel engine; 2-eddy current dynamometer; 3-computer; 4-combustion analyzer; 5-charge amplifier; 6-piezoelectric transducer; 7-terminal box; 8-light source; 9-grating disc; 10-encoder; 11-Lambda Meter; and 12-oxygen sensor.

easily obtained and it is the result of comprehensive effects and reflects of all the parameters of combustion system such as the intake air pressure, the injected diesel fuel pressure, and the structure of combustion chamber. The instantaneous values of which are so hard to obtain. The indicated pressure of 10 engine combustion cycles of each engine load at speed of 600 rpm is showed in Fig. 3. For each engine cycle, it is

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composed of four primary processes: intake process, compression process, combustion process, and exhaust process. We notice that the combustion process appears only once for each two crankshaft rotations, the combustion process is separated by the intake and exhaust processes, and consequently, combustion in a diesel engine is intermittent. When engine speed is held constant, the total time of each cycle and the pressure profile of intake and compression process are approximately the same, the differences between combustion processes lead to the cycle-to-cycle variations in the diesel engine, especially when the engine load rate is less than 50%. It leads to decreased engine fuel economics and increased exhaust emissions rapidly. As the engine fuel consumptions and emissions regulations become more and more strict, it is very urgent to study combustion fluctuations, and their source identification. The understanding and possible elimination of combustion instability in diesel engine is also important in the context of new fuels.43 III. THE NONLINEAR ANALYSIS A. Phase space reconstruction

Cycle-to-cycle combustion variations can be defined in terms of variations in the cylinder pressure between different

FIG. 3. The in-cylinder pressure time series for each load rate. The operation condition was chosen: load rate LR ¼ 0%, 10%, 25%, 50%, 75%, and 100% from (a) to (f), engine speed n ¼ 600 rpm.


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xðiÞ i ¼ 1; 2:::; N:

(1)

Made a set of m dimensions vectors Xi ¼ fxði þ sÞ; xði þ 2sÞ; :::; xði þ ðm 1ÞsÞg; i ¼ 1; 2; ::::; N ðm 1Þs;

(2)

where N is the sampling number, m is the embedding dimension, and s is the delay time. B. Delay time determination

In the process of phase space reconstruction, one of the most important problems is how to choose the appropriate phase space reconstruction parameters including delay time s and embedding dimension m. In theory, s can be any value when the scalar of time series is infinite and there is no noise in the data. In fact, however, the measurements of time series are always limited and include inevitable noise in test, and s directly affects the calculation accuracy of CD, Lyapunov exponent, and so on. When s is too small, the correlation between the components is very strong, and reconstruction attractors are compressed near the main diagonal in the embedded space. When the s is too big, the delay vectors of each component are almost irrelevant, and the structure of the system attractor is complicated. In our study, the autocorrelation function method and the mutual information function method were adopted to comprehensively determine the optimal s, and then optimal values of s were verified through the visualization of the data in a two-dimensional embedding space. The autocorrelation function approach is the most common way to determine the delay time.51,52 The autocorrelation function is defined as P xði þ sÞxðiÞ : (3) CðsÞ ¼ N

FIG. 4. Autocorrelation function of in-cylinder pressure time series when engine speed n ¼ 600 rpm and engine load rate LR ¼ 0%.

cycles, or in terms of variations in the details of the burning process. However, pressure-related quantities are easiest to determine, these parameters include the peak pressure pmax, the crank angle at which this peak pressure occurs hpmax, the maximum rate of pressure rise (dp/dh)max, the crank angle at which (dp/dh)max occurs, and indicated mean effective pressure (IMEP).1 These variables directly derive from the engine combustion cycle and give partial information of the whole combustion process, so these parameters from incylinder pressure cannot directly reflect the correlation of pressure signals in the same cycle and effects between cycles. However, in our study, more attention was given to the effect of the prior cycle on each subsequent cycle and the correlation between cyclic pressure fluctuations. We adopted the phase space reconstruction theory to display the internal characteristics of combustion system attractors and their evolution rules. Based on Takens’ theory, the evolution of any component (indicated by a variable) in the nonlinear system is determined by the interaction with other components. Namely, the information of the relevant higher dimension embedding space components is implied in the time history of any component.50 Thus, the scalar cylinder pressure time series of diesel engine contain the effects of all other variables on cycle-to-cycle combustion variations, reconstructing cylinder pressure time series in appropriately high dimension phase space is an effective means to understand the nonlinear dynamics of diesel engine combustion system. We have used the delay coordinate reconstruction method to reconstruct the attractor of combustion system

An example of the autocorrelation function for incylinder combustion pressure data is shown in Fig. 4. One can note that auto-correlation function C(s) is quasi-periodic, and only part of the first period is shown in Fig. 4. The corresponding value s was selected when the autocorrelation function of the time series decreased 0 or 1/e (see Table I). For example, when engine speed n ¼ 600 rpm, engine load rate LR ¼ 0%, s equaled 82 and 42, respectively.

TABLE I. Delay time determined by autocorrelation function. n ¼ 600 rpm

n ¼ 800 rpm

n ¼ 1000 rpm

n ¼ 1200 rpm

n ¼ 1400 rpm

s

s

s

s

s

Load rate

% 0 10 25 50 75 100

0 82 83 84 85 87 89

1/e

0

1/e

0

1/e

0

1/e

0

1/e

42 42 43 43 43 43

82 82 83 86 88 91

42 42 42 43 43 44

83 83 84 86 88 92

43 43 43 44 44 45

83 84 84 87 89 93

44 44 44 44 45 46

84 85 86 88 90 92

45 45 45 45 46 47


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We compared the structures of combustion system attractor when delay time s was equal to 1, 42 (at which C(s) ¼ 1/e), and 82 (at which C(s) ¼ 0), respectively (see Fig. 5). The results showed that the combustion system attractor is unfolded fully when s ¼ 42, therefore, we determined the optimal value s at which C(s) ¼ 1/e. The complete information about the estimated delays s is included in Table I. In all the examined cases, the delay (measured in terms of the revolution angle) was fairly similar with small increasing tendency for larger loading and higher engine speed. The other method to calculate s is the mutual information function.52,53 In general, the sampling lag value based on the auto-correlation function is not the same as the value from the mutual information function. The mutual information function is suitable to analyze the topological property of the nonlinear system. The original concept of mutual information is based on Shannon’s information theory, which gives a measure of the general independence of two variables.54 The mutual information function is defined as I ðs Þ ¼

N X P xðiÞ; xði þ sÞ i¼1

( ) P xðiÞ; xði þ sÞ ; log2 P½ xðiÞP½ xði þ sÞ

(4)

FIG. 6. Mutual information function of in-cylinder pressure time series when engine speed n ¼ 600 rpm and engine load rate LR ¼ 0%. (Including the inset of multi-period mutual information function.)

where P½xðiÞ; xði þ sÞ is the joint probability of xðiÞ and xði þ sÞ, namely, the interrelated level of the system. When xðiÞ is completely irrelevant to xði þ sÞ, P½xðiÞ; xði þ sÞ ¼ P½xðiÞP½xði þ sÞ. Because I(0) ¼ 1, the optimal value s was selected when mutual information function I(s) was reduced to 1/e (see Fig. 6). Comparing the results obtained, using both the autocorrelation function method and the mutual information method, we found there was a small difference between the values of s at which C(s) ¼ 1/e and I(s) ¼ 1/e. The problem with autocorrelation function method is that it is only based on linear statistic, and this method does not account for any nonlinear dynamical correlation. Therefore, the final values of s we determined after comprehensive consideration are shown in Table II. C. Embedding dimension

Embedding dimension is another important parameter for phase space reconstruction. Based on embedding theory,50 the chaotic attractor can be unfolded as long as the embedding dimension m > 2d þ 1, d is the dimension of the attractor defined by the orbits. However, we must select an appropriate embedding dimension m in the process of phase space reconstruction using experimental time series. If the embedding dimension m is too large, it needs a large amount of computer resources to calculate the chaotic characteristic parameters including the Lyapunov exponent, fractal dimension, entropy, etc. If m is too small, the influence of noise TABLE II. The final values of s.

FIG. 5. The delayed return map of in-cylinder pressure with the effect of delay time s on the in-cylinder pressure attractor structure of combustion system when engine speed n ¼ 600 rpm and engine load rate LR ¼ 0%.

Load rate %

n ¼ 600 rpm s

n ¼ 800 rpm s

n ¼ 1000 rpm s

n ¼ 1200 rpm s

n ¼ 1400 rpm s

0 10 25 50 75 100

37 42 50 49 52 55

39 43 49 52 53 58

43 47 49 53 56 60

50 50 50 54 58 62

51 50 49 54 60 62


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will be enlarged, and it is unable to demonstrate the complex structure of non-linear system. The false nearest neighbors (FNN) approach, which is used in this paper, is one of the most common techniques to estimate the embedding dimension.55 One can calculate the minimal embedding dimension which corresponds to the minimum number of false neighbors "

R2 ðiÞ R2m ðiÞ fm ðiÞ ¼ mþ1 2 Rm ðiÞ ¼

#1=2

jxði þ mnÞ xNN ði þ mnÞj 10%; Rm ðiÞ

(5)

where xi is the in-cylinder pressure time series, ss is the time interval, s is the delay time, n is equal to s/ss, Rm is the distance between two false neighbors points when embedding dimension is m, and Rmþ1 is the distance between two false neighbors points when embedding dimension is m þ 1. Percentages of FNN are showed in Fig. 7. The test condition is: engine speed is 600 rpm and engine load rate increases from 0% to 100%. As the embedding dimension increased, the number of false neighbor points decreased. In theory, the embedding dimension can be determined when the percentage of FNN is decreased to 0. However, in our experiment, there were some noise disturbances if the percentage of FNN is decreased to 0 which meant the embedding dimension was too large. Therefore, values of embedding dimension are between m ¼ 14 and m ¼ 16 for different engine operating conditions, and the attractor can be completely unfolded. D. RP

HðsÞ ¼

1 0

s0 s < 0:

(7)

The RP is an array of dots in a square. The values 0 and 1 in this matrix can be visualized by the black dot and white dot on the RP. Such a RP exhibits characteristic patterns, which is caused by typical dynamical behavior. Therefore, we compared the RP characteristics of random process, sine function, chaos system, and in-cylinder pressure time series from diesel engine, each signal includes 3600 points. The signals of function and dynamic system are showed in Figs. 8(a)–8(d), and the RPs which correspond to the function and the dynamic systems mentioned above are shown in Figs. 9(a)–9(d). For the white noise signal, which is a random signal with a constant power spectrum density, the RP shows a uniform distribution in reconstructed state space, as presented in Fig. 9(a). The RP of sine function is shown in Fig. 9(b), the RP of sine function is formed by the main diagonal, and their parallel lines and all points are included in the lines. The main diagonal indicates that a pattern is identical with itself. There are no vertical or horizontal lines. The distance reveals the period length of the oscillation. The chaotic signal of Lorenz system with (a, r, b) ¼ (16, 45.92, 4) is shown in Fig. 8(c), its RP has a rich and complex characteristic structure (see Fig. 9(c)), and this figure shows that the system goes into oscillations superimposed on the chaotic motion. The RP of the in-cylinder pressure time series in this study is given in Fig. 9(d). The RP has the checkerboard texture, which provides evidence of the determinism (DET) in the diesel engine combustion system, and that the combustion system dynamics was inconsistent with random process or periodic motion while it was more similar to chaotic processes.

The RP is an effective tool for the analysis of the dynamical system; it is a straightforward visualization of the recurrence matrix.39,56 The RP can be applied to time series data in order to bring out temporal correlations. The system dynamics can be appropriately presented by a reconstruction of the phase space trajectory. The RP is defined as Ri;j ¼ Hðe kXi Xj kÞ;

(6)

where e is a predefined threshold, Xi and Xj are vectors in the state space, and H(s) is the Heaviside function defined as

FIG. 7. Percentage of false nearest neighbors versus embedding dimension m. The dashed line corresponds to 10% of FNN.

FIG. 8. The signals for different functions or systems: the white noise (a), the sine function (b), Lorenz system (c), and in-cylinder combustion pressure of the diesel engine (d).


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CD is defined as CD ¼ lim e!0

ln CðeÞ : ln e

(9)

G. Largest Lyapunov exponent (LLE)

FIG. 9. The RPs for different functions or systems: the RP of white noise (a), the RP of sine function (b), the RP of Lorenz system (c), and the RP of combustion system of diesel engine (d). For each RP, its embedding dimension m is equal to 2, and s ¼ 2, 92, 19, and 37, respectively.

E. RQA

Although RPs are a useful tool to analyze the characteristics of dynamical system by means of visualization, what yielded to the shortage that the user had to detect and interpret the patterns and structures revealed by the RPs. Sometimes, low printer resolutions of RPs further worsened this issue. In order to overcome this disadvantage of the method, RQA was developed by Zbilut and Webber57 and extended by Marwan et al.44,45 to provide statistical measures of RPs. RQA provides an effective means to quantify laminar, divergent, and nonlinear transition dynamics in complex systems. Several measures of complexity which quantify the small scale structures in RPs have been proposed. In our study, several RQA measures were applied: recurrence rate (RR), DET, entropy (RNTR), the length of the longest diagonal line found in the RP (Lmax), and others. F. CD

The CD provides information about the minimum number of independent variables to describe the state of dynamical system.58 We calculated the correlation sum C(e), it represents the probability of any two arbitrary vector points that are separated by a distance less than or equal to e on the trajectories in the state space. The CD of combustion system is calculated by the correlation sum C(e). The correlation sum C(e) is given by N X N X 2 H e kXi Xj k ; C ðeÞ ¼ N ð N 1Þ i¼1 j¼iþ1

i; j ¼ 1; :::; N

(8)

where e is a predefined threshold, Xi and Xj are vectors in the state space, N is the number of vectors, and k.k is a norm (e.g., the Euclidean norm).

Lyapunov exponent is used to estimate the chaotic properties of dynamical system. It can quantify sensitivity of the system to initial conditions. For a chaotic system, Lyapunov exponent will be positive and means that initially neighboring trajectories diverge exponentially, the system is sensitively dependent to initial conditions. For a regular deterministic signal (quasi-periodic or periodic), the Lyapunov exponent is negative and means that the initially neighboring trajectories are converged on a stable orbit. An m-dimensional system has m Lyapunov exponents, among them the LLE, to characterize the chaotic system; and in our work, we have used the algorithm proposed by Rosenstein et al.59 to calculate LLE. After reconstructing phase space, this algorithm searches for nearest neighbors Xj^ of the particular reference point Xj on trajectory of the attractor. The initial distance from the jth point to its nearest neighbor is defined by dj ð0Þ ¼ min jjXj Xj^jj; Xj^

(10)

^ is greater than the where k.k is the Euclidean norm, jj jj mean period of the time series. We assume the jth pair of nearest neighbors diverge approximately at a rate given by the LLE. The distance dj ðiÞ between the jth pair of the nearest neighbors after i discrete-time steps is defined as dj ðiÞ Cj ek1 ðiDtÞ ;

(11)

where Cj ¼ dj ð0Þ. Taking the logarithm of both sides of Eq. (11), we obtain ln dj ðiÞ ln Cj þ k1 ði DtÞ:

(12)

Equation (12) represents a set of approximately parallel lines (for j ¼ 1, 2,…,M), each with a slope is roughly proportional to the k1 . The LLE is easily and accurately calculated using a least-square fit to the “average” line defined by yðiÞ ¼

1 hln dj ðiÞi; Dt

(13)

where hi denotes the average over all values of j. IV. RESULTS AND DISCUSSIONS

The phase space reconstruction plots of in-cylinder pressure time series for different speed (speed is from 600 to 1400 rpm) and load (load rate is from 0% to 100%) are illustrated in Fig. 10. We can note that the motion trajectories of combustion system attractors are limited in the finite range of phase space for all operation conditions. When engine speed is held constant, the attractor of combustion system has twist, folded, and loose geometry structure at a lower


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FIG. 10. Phase space reconstruction of in-cylinder combustion pressure data for different engine speeds and loads: (a1)–(a4) correspond to the load from 0% to 100% at a speed of 600 rpm, (b1)–(b4) correspond to the load from 0% to 100% at a speed of 1000 rpm and (c1)–(c4) correspond to the load from 0% to 100% at a speed of 1400 rpm.

load. The cyclic combustion process exhibits stronger randomness and the combustion process variations make the structure of combustion system attractor more complex. With an increased load rate, the scale of combustion system attractor is enlarged and the trajectories of combustion system attractor gradually close to each other. We have observed the quasi-periodic characteristics of combustion system in a diesel engine. When engine load is held constant, as the engine speed is increased, the periodicity of combustion process is enhanced. We also noticed that the engine speed has a larger effect on combustion instability of diesel engine under a lower load than under a higher load. Therefore, for improved fuel economy and emission, it is more meaningful to study the deterministic characteristics under lower speed and load conditions. The characteristics of RPs of different in-cylinder pressure time series under different engine operation conditions are shown in Fig. 11. By visual inspection, there are significant differences in RP structures for different engine speeds and load rates. In addition, the engine speed has a larger effect on the structures of RPs. At a lower engine speed, when the engine speed was held constant, the over-all structures of all the RPs were the same. They were composed of square blocks, main diagonal, and many short lines paralleled to the main diagonal. These diagonals connected square blocks making the RPs look like checkerboard (see Figs. 11(a1)–11(a4)). The diagonal lines in the RP probably indicate the existence of some determinism in the system. At same time, we notice that the diagonal lines are interrupted

by square blocks. For a chaotic oscillator, diagonal lines are interrupted due to the divergence of nearby trajectories. In order to clearly display the internal texture and the differences of square blocks, the small-scale structures of the square blocks were illustrated under different loads at an engine speed of 600 rpm (see Fig. 12). We observed that there were many black stripes and white stripes in the square blocks, and the stripes were horizontal and vertical and crossed each other. Moreover, at a lower load rate, the texture of square blocks in the RPs were vague. However, as the engine load rate increased, the new clear checkerboard structures formed. With an increased engine speed and load rate, the structures of RPs at a higher speed are obviously different from structures of RPs at a lower engine speed. The square blocks gradually disappeared while short lines paralleled to the main diagonal occurred and the length of the lines increased compared to the conditions of lower engine speeds and smaller load rates. Meanwhile, less single, isolated recurrence points were discovered. In order to reveal the dynamical characteristics of combustion instability in diesel engine, the RQA was adopted. It enabled to perform quantitative analysis on the experimental data. We estimated the RQA measures based on the recurrence density, diagonal lines and vertical lines: RR, DET, Lmax, entropy (ENTR), TREND, laminarity (LAM), the maximal length of the vertical lines in the RP (Vmax), and trapping time (TT). The used threshold e and RQA measures for different engine operation conditions are shown in Table III. The results show that the RR, DET, Lmax, ENT, TREND, LAM, Vmax, and TT exhibit larger changes and have the


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FIG. 11. The RPs of in-cylinder combustion pressure data for different engine speeds and loads: (a1)–(a4) correspond to the load from 0% to 100% at speed of 600 rpm, (b1)–(b4) correspond to the load from 0% to 100% at speed of 1000 rpm and (c1)–(c4) correspond to the load from 0% to 100% at speed of 1400 rpm.

FIG. 12. The RPs for different loads at a speed of 600 rpm: 0% load (a), 25% load (b), 50% load (c), and 100% load (d). For each engine load, the embedding dimension m of the RP is equal to 2 and the delay time s ¼ 37, 50, 49, and 55, respectively.


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TABLE III. Used threshold e and RQA measures for different engine operation conditions. n (rpm) 600

1000

1400

Load rate (%)

e

RR (%)

DET (%)

Lmax

ENTR

TREND

LAM (%)

Vmax

TT

0 25 50 100 0 25 50 100 0 25 50 100

0.050 0.051 0.060 0.070 0.048 0.056 0.059 0.068 0.048 0.058 0.062 0.067

0.825 1.720 3.523 5.202 0.779 1.205 1.692 1.948 0.358 1.175 1.306 1.617

11.336 12.769 45.505 59.649 22.341 54.418 61.823 72.135 21.352 53.649 60.171 74.604

6.0 7.0 11.0 18.0 7.0 16.0 23.0 35.0 6.0 13.0 28.0 37.0

0.409 0.427 1.267 1.735 0.703 1.656 1.879 2.410 0.712 1.527 1.702 2.436

0.020 0.023 0.070 0.075 0.021 0.043 0.086 0.115 0.029 0.049 0.055 0.092

20.515 22.003 58.762 71.591 35.757 66.971 73.693 80.982 34.889 65.901 70.671 82.344

4.0 7.0 12.0 27.0 6.0 21.0 23.0 33.0 7.0 17.0 21.0 32.0

2.206 2.236 2.771 3.391 2.396 3.276 3.569 4.084 2.485 3.055 3.292 4.750

same change trend as the engine load rate increased at the different engine speeds. For instance, when the engine speed is held at 600 rpm, the engine load increased from 0% to 100%, the RR increased from 0.825% to 5.202%, DET increased from 11.336% to 59.649%, Lmax increased from 6.0 to 18, ENTR increased from 0.409 to 1.735, and TREND increased from 0.020 to 0.075. The RR quantifies the percentage of recurrent points that fell within the specified radius. The DET, Lmax, ENTR, and TREND are the RQA measures based on the diagonal line, the length of diagonal line means a segment of the trajectory is rather close to another segment of the trajectory at a different time, therefore, these diagonal lines are related to the divergence of the trajectory segments. The deterministic processes cause longer diagonals, less single, and isolated recurrence points; while the processes with uncorrelated or weakly correlated, stochastic or chaotic behavior cause none or very short diagonals. The shorter and intermittent diagonal lines at a lower engine speed while the longer diagonals and less isolated recurrence points can be revealed at a higher engine speed and the load rate is observed in Fig. 11. Such structural characteristics of RPs probably means that the combustion processes of the diesel engine at lower engine speeds look like the stochastic or chaotic process, while the longer diagonal lines in the RP at a higher engine speed and a larger load rate indicate the existence of some determinism in the combustion system in a diesel engine. The ENTR reflects the complexity of the RP in respect of the diagonal lines. Meanwhile, the vertical structures based measures LAM, Vmax, and TT are also computed. LAM, Vmax, and TT are quite different in their amplitudes in Table III. For example, when the engine speed is held at 600 rpm, the engine load increased from 0% to 100%, the LAM increased from 20.515% to 71.591%, Vmax increased from 4 to 27, and TT from 2.206 to 3.391. LAM represents the occurrence of laminar states in the system and it measures the percentage of recurrent points comprising vertical line structures. Vmax is the analogue to the standard measure Lmax. Finally, TT can be used to estimate the mean time that the system will abide at a specific state or how long the state will be trapped. We can notice that the higher value of RR means the trajectories of combustion system often visit the same phase space regions when the engine operates at higher engine speed, the

vertical line length Vmax is more powerful in discriminating the complexity of the combustion process at a lower speed and a lower load rate than the diagonal line length Lmax (see Table III, Figs. 11(a1)–11(a4) and Figs. 11(b1)–11(b3)). We also found the regions of intermittency which represent states with short laminar behavior and cause vertically and horizontally spread black areas in the RPs. The plots of CD versus e, when engine speed is equal to 600 rpm and the load rate is equal to 0%, are presented in Fig. 13. The plots of lnC(e) versus lne can be observed in the upper right in Fig. 13. We calculated the slope of each curve corresponding to different embedding dimensions m and obtained the plots of CD versus e. For each embedding dimension m, there is a linear portion in the middle third of the horizontal range of the plot of lnC(e) versus lne. The slope of the linear portion increased with increased embedding dimension m. When the slope reaches saturation the slope no longer increased as embedding dimension m increased. We determined the CD (which corresponds to a plateau at the bottom in Fig. 13) of the combustion system. By this means, we obtained the MAP of CD for different engine speeds and loads (see Fig. 14). We note that the combustion system of diesel engine possessed the characteristics of fractal dimension, and the largest CD is no more than 1.6. When engine speed is held

FIG. 13. Plots of CD versus e for different embedding dimensions (see Eq. (9)) when the engine speed is equal to 600 rpm and the load rate is equal to 0%. The inset illustrates lnC(e) versus lne. The red line in the main plot corresponds to minimum saturated embedding dimension m ¼ 14.


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FIG. 14. MAP of CD plotted against the load rate and engine speed n, where the engine speed ranges from 600 to 1400 rpm and the engine load rate ranges from 0% to 100%.

constant, the CD decreased with increasing the load. It means that the combustion system at the higher load is less complex than at the lower load. When the load is held constant, CD first decreased and then increased with the increased speed. CD can be used to quantify the number of independent variables to describe a dynamical system, and we conclude that the combustion process of diesel engine is due to a low dimensional deterministic process. The LLE computation method for combustion system of diesel engine is presented in Fig. 15. In this figure, there are strong fluctuations of curves of yi versus i, but the overall variation tendency of each curve shows the distinct linear increase. The linear fit is showed by the dashed line. The slope of the dashed line gives the estimated value of the LLE. The LLEs for whole operation range of diesel engine were calculated, as shown in Fig. 16. The LLEs increased from 1.94 105 to 7.2 104 when the engine speed rose from 600 to 1400 rpm and the load rate rose from 0% to 100%. This implies that the initially neighboring trajectories of combustion system diverge exponentially, which means that the combustion system is sensitive to its initial conditions and is chaotic. When the engine speed is held constant, the LLE decreased with increasing the engine load. However, the effects of the engine load on the LLEs became

FIG. 15. Plots of yi versus i (see Eq. (13)) for different embedding dimensions when the engine speed is equal to 600 rpm and the load rate is equal to 0%.

Chaos 25, 013105 (2015)

FIG. 16. MAP of LLE when the engine speed is from 600 to 1400 rpm and the engine load rate is from 0% to 100%.

smaller under the higher engine speed. When the engine load is held constant, the estimated LLE decreased with increasing engine speed. For more specific interpretation of these results in terms of the engine transition to the chaotic combustion regime small threshold value to results in Fig. 16 should be applied. However, by noting that in our case LLE increased by a larger amount (over 30 times), we are legitimated to conclude that in the limits of fairly low loading and higher speeds combustion is chaotic. To sum up, there are some potential reasons to explain the non-linear dynamic phenomenon of the combustion instability in diesel engine we discussed above. The combustion process of diesel engine is a complex process with many factors that can affect the combustion quality of a diesel engine: aerodynamics in the cylinder during combustion, the different amount of fuel, air, and recycled exhaust gases supplied to the cylinder, the mixture quality of in-cylinder gas and fuel, and the structure of combustion system, to name a few. In our study, in order to minimize the effect of intake flow, the test was conducted under a constant engine speed, and the load rate was changed by controlling fuel injection. Therefore, the injection quality worsened when the engine operated at a lower load rate, which affected the atomization and vaporization of the diesel spray. In addition, the combustion duration will be longer, the probability of misfire or partial combustion increased, and the intermittent heat released make combustion process in diesel engine more complex. Therefore, we observed the structure of intermittency and laminarity in RPs. The combustion cyclic variation is the inherent result of nonlinear combustion dynamics, and chaotic behavior will be revealed when the burn time occupies an excessive fraction of the cycle time.21 The value of CD and LLE is the largest at the lowest speed and load rate and presented in Fig. 16. The larger value of CD means higher complexity of the combustion process in a diesel engine and positive LLE means that the initially neighboring trajectories of combustion system diverge exponentially, and the combustion process exhibited inherent chaotic nature. Furthermore, when the quantity of fuel oil and air supplied into cylinder was constant, with increased engine speed, the quality of fuel oil injection was improved, and the interaction between fuel injection and in-cylinder gas flow was


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enhanced. The improved atomization and vaporization of injection spray can accelerate the combustion speed and improve the stability of combustion in diesel, hence, the complexity of combustion system and sensibility to the change of initial conditions decreased. V. CONCLUSIONS

By adopting embedding theory, the RP technique, the RQA, and estimation of chaotic system indicators, we tested the deterministic dynamics of combustion instabilities in a diesel engine and quantitatively analyzed the effect of engine operation conditions on the complexity of the cycle-to-cycle combustion fluctuations. Experiments covering the whole operating range of the engine have been performed. For all the speeds and loads considered, in-cylinder pressure time series have been reconstructed into a two-dimensional phase space. The trajectories of the combustion system attractor have twist, folded, and loose geometry structure at a lower load rate. Cycle-to-cycle combustion process exhibits stronger randomness, while the evolution process of combustion system trends to quasi-periodic at higher speeds and load rates. The trajectories of the attractor in a combustion system are limited in finite range in the way they are twisted and folded. The RP analysis presented that the checkerboard structure of RPs and the diagonal lines in the RPs at lower engine speeds probably indicated the existence of some determinism in the system under consideration. We observed intermittency and laminarity in RPs. With an increased engine speed and load rate, the checkerboard structure gradually disappeared while short lines parallel to the main diagonal occurred and the length of the lines increased compared to the conditions of a lower engine speed and a smaller load rate. At the same time, less single, isolated recurrence points were discovered. The measures of the RQA based on the recurrence density, diagonal lines, and vertical lines were computed. The results show that under constant speed, the RR, DET, Lmax, ENT, TREND, LAM, Vmax, and TT exhibited larger changes and have the same overall trend. The DET, Lmax, ENTR, and TREND showed that the combustion processes of a diesel engine at a lower engine speed look like a stochastic or chaotic process, while at a higher engine speed and a larger load rate indicate the existence of some determinism in combustion system in a diesel engine. The LAM, Vmax, and TT described intermittency which represented states with short laminar behavior and caused vertically and horizontally spread black areas in the RPs. The CD and the LLE have been calculated and the results showed that the CD of the combustion system is no more than 1.6 and the LLE is positive within the operating range of engine, which prove that the combustion system of diesel engine is a low-dimensional chaotic system. It means that the cycle-to-cycle combustion fluctuations became more complex and sensitive to smaller variations of initial conditions with the decreased engine speed and load. The increased complexity at lower speeds and loads may be explained as follows. As the values of engine speed and load

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decreased, the effect of cycle-to-cycle variation of fuel injection quantity on combustion system are enlarged. Meanwhile, the poor quality of fuel injection leads to an inadequate atomization, vaporization, and mixture of fuel oil and air than at higher engine speeds and loads. In addition, the probability of misfire or partial combustion increases and the intermittent heat release makes the combustion process in a diesel engine more complex. In the above analysis, we noticed a fairly large deference in the combustion system response. Thus, in the first approximation, we neglected a low level of external noise influences which were always present in the experimental data. In conclusion, the increased complexity at lower speeds and loads caused combustion instabilities, which loads made the fuel economy and exhaust emissions of diesel engine worse. It is worth noting that understanding the conditions of combustion fluctuations could be important to improve the combustion control procedure. ACKNOWLEDGMENTS

This work was supported by National Natural Science Foundation of China (5130-6041), Natural Science Foundation of Heilongjiang Province of China (QC2013C057), and Fundamental Research Funds for the Central Universities (002030020803). G.L. was also supported by the Polish National Science Center under Grant No. 2012/ 05/B/ST8/00077. 1

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