# Optimal Pressure Based Detection of Compressor Instabilities Using the Hurst Exponent.

INTRODUCTION AND BACKGROUND

Turbo compressors are limited in their operating range at low mass flows by the surge phenomenon. Surge occurs when the compressor can no longer steadily supply the outlet pressure needed to overcome the flow resistances downstream. Typical flow resistances are throttles on a gas stand, or the throttle or intake valves on a turbocharged internal combustion engine. The mass flow and pressure increase through the compressor start oscillating. If the global mass flow becomes negative, the operation conditions is often called deep surge, while mild surge denotes an operation with oscillating, but always positive mass flow through the compression system. Due to the associated noise, uncertainty of the resulting volumetric efficiency, and mechanical stress on the components in the intake system, surge should be avoided on turbocharged combustion engines. It therefore limits the low-end torque in boosted engines.

A common way to model surge is to model the compression system analogously to a mass-spring-damper system. This idea was first introduced by Greitzer [1, 2] for axial compressors, but has been shown to also apply to radial compressors in subsequent studies [3, 4]. In this analogy, the compression system consists of air mass oscillating in a pipe (the mass), connected to a compressible plenum downstream (the spring). The compressor is represented by an actuator disk in the pipe (a damper with negative damping), and at the plenum outlet, there is a flow resistance (a damper with positive damping). A schematic is given in the top right corner of Figure 1. From this analogy, some predictions can be made: The surge frequency should approximately correspond to the eigenfrequency of the system, i.e. the Helmholtz frequency [f.sub.H] in this case. It is given by

[f.sub.H] = a / 2[pi] [square root of [A.sub.c] / [L.sub.c][V.sub.p]] (1)

where a is the speed of sound, [V.sub.p] is the plenum volume, [L.sub.c] is the length of the compression system (including inlet and outlet pipes as well as the flow path through the compressor), and [A.sub.c] is its cross-section area. Further, it can be estimated whether the system is stable or (dynamically) unstable using the criterion

[([partial derivative][psi] / [partial derivative][phi]).sub.C] [less than or equal to] 1 / [B.sup.2] [([partial derivative][psi] / [partial derivative][phi]).sup.-1.sub.T] (2)

B = U / 2 * 2[pi][f.sub.H] * [L.sub.c] (3)

where [phi] = [??] / ([rho]AU) and [psi] = [DELTA]p / (0.5[rho][U.sup.2]) are the mass flow and pressure coefficients, [rho] is the density, U is the blade velocity, and indices C and T denote compressor and throttle, respectively. Since the throttle slope is relatively steep, and B is of the order of magnitude 1, one can expect some oscillations on the positively sloped parts of the compressor speedline. Figure 1 shows the distinction between a stable and a dynamically unstable system in a schematic compressor map.

In order to classify compressor operation as surge, different indicators have been proposed in literature. Many are based on the compressor outlet pressure signal. Subjective noise as experienced by the operator is one of the criteria used in . The standard deviation of the signal was used e.g. in ,  and . It can be calculated based on the raw signal or based on a lowpass filtered signal that removes higher frequency oscillations; further, one can normalize the standard deviation, e.g. with the impeller dynamic pressure 0.5p[U.sup.2], in order to obtain similar results for different speedlines. The height of the peak in the linear frequency spectrum, obtained by a Discrete Fourier Transform (DFT), was proposed in . The signal power in a certain frequency band around the surge frequency - essentially the variance of the signal at these frequencies - has also been used, e.g. in , , and . It was shown that the acoustic sound amplification of the compressor at low frequencies increases strongly as the compressor approaches surge in . Recently, the Hurst exponent has also been proposed as a criterion in . It is an indicator about the long-term memory for statistically similar time series data; more explanation will be given in the subsection Detrended fluctuation analysis in the Method section.

As an alternative to outlet pressure based indicators, the oscillations of the compressor speed can also be used, see e.g. . For a compressor rig with electronic motor instead of a turbine, the authors in  also show that the power input to the electric motor has a steep increase at the onset of oscillations. Due to the backflow of hot gas from the impeller operating close to the surge line, the temperature increase at the inlet is also a commonly used surge indicator [6, 14].

This paper extends the work done in  on the applicability of the Hurst exponent as a surge criterion. The aims are threefold: First, to apply this methodology to a different compression system in order to gain confidence in the method; second, to investigate requirements of the pressure signal with respect to sensor position, sampling time, and sampling frequency; and third, to introduce another, more sensitive surge indicator based on the fractal properties of the signal.

The paper is based on experiments run on a truck-sized turbocharger compressor at the Gas Dynamic and Propulsion Laboratory at the University of Cincinnati. The section Method describes the theoretical background

and the experiments. In its first subsection, the estimation of the Hurst exponent is shortly revisited. Then, the experimental setup is described. Results are presented in section Results and Discussion: First, the applicability of Detrended Fluctuation Analysis (DFA) for this compressor is established. The Hurst exponent as a surge indicator is compared to standard deviation for different pressure sensor locations along one speedline. Sensitivity studies are carried out with respect to the sampling duration and the sampling frequency. An additional Hurst exponent based surge indicator is introduced, which allows distinguishing between large amplitude oscillations and small amplitude oscillations. Finally, the investigation is extended to two other speedlines.

METHOD

Detrended Fluctuation Analysis

The Hurst exponent can be understood as a scaling relation for statistically self-similar data . Given for example a random walk x(t) and magnifying it by a factor c, i.e. rescaling the time axis by 1/c, the data axis needs to be rescaled by a factor

x(t) [right arrow] [(1/C).sup.H] x(1 / C * t) (4)

where H is the Hurst exponent. H depends on the underlying distribution of the step size of the random walker. If the step sizes are uncorrelated (white noise), the Hurst exponent is H = 0.5; this is another way of saying that the expected displacement of the random walker scales with the square root of length of the walk. For a random walker with a long-term positively autocorrelated step size, e.g. pink noise, H > 0.5 . Turbulent pressure fluctuations exhibit a long-term positive autocorrelation.

The principal idea in applying this methodology to compressor surge is that the pressure signal consists of turbulent pressure fluctuations and signal noise at stable compressor operation, while it is more wave-like near surge, with a frequency corresponding to the eigenfrequency of the compression system, Eq. (1). Thus, one can expect the pressure oscillations at the system natural frequency to be governed by a scaling law with H [greater than or equal to] 0.5 in stable operation. In pure mild or deep surge, on the other hand, H [approximately equal to] 0, because the pressure signal is repeating periodically. Thus, the signal variance is independent of the time scale as long as the time scale is larger than one wave period. In between these two extremes, one can expect a decline of the Hurst exponent from the stable regime H [greater than or equal to] 0.5 downwards.

There are different ways to estimate the Hurst exponent from data. In this paper, DFA is used due to its flexibility. It was first introduced by Peng . Alternatives include the original rescaled range approach , or via a Fourier transform . A step-by-step explanation of how the Hurst exponent can be calculated for some artificial signals is given e.g. in [15, 12]. The main steps will be summarized in the following.

The first step is to calculate a cumulative time series of the signal in question,

[Y.sub.j] = [j.summation over (i=1)][x.sub.i] (5)

The cumulative time series is then divided into windows of length [t.sub.s], and in each window w, the data is detrended,

[y.sub.w] = [Y.sub.w] - [v.sub.w] (6)

where [v.sub.w] is a least square fit of the data in the window. In this case, detrending of order 1 (DFA1) is applied, meaning that the mean and the linear trend in each window are removed. Higher order DFA is also possible, but not used in this analysis. The fluctuation function [F.sup.2] is then the root mean of all window variances after detrending,

[F.sup.2]([t.sub.s]) = [square root of mean(var[y.sub.w])] (7)

and a function of the window length [t.sub.s]. This procedure is performed for many different [t.sub.s], which are logarithmically increasing. The Hurst exponent is then the slope of the fluctuation function [F.sup.2] over the window length [t.sub.s] in a log-log diagram. It is important to note here that this slope is not generally constant for all window lengths, and thus, the Hurst exponent is dependent on the time scale over which it is calculated.

Besides the "standard" Hurst exponent, other orders of the Hurst exponent can be calculated by using a different order root mean instead of the square root mean as given in Eq. (7), see e.g. . The fluctuation function of order q is then defined as

[mathematical expression not reproducible] (8)

The Hurst exponent of order q, [H.sup.q], is the slope of [F.sub.q] in a logarithmic plot against window size [t.sub.s]. For q = 2, Eq. (8) results in Eq. (7), and the "standard" Hurst exponent will henceforth be referred to as [H.sup.2]. For high values of q, the windows with a high variance are weighted heavily in the averaging procedure, and thus dominate the estimate of the Hurst exponent. For negative values of q, windows with small variance are weighted heavily. This method allows treating large amplitude and small amplitude fluctuations differently. A time series is then defined as "monofractal" if the Hurst exponent is independent of its order, [H.sup.q] [not equal to] f(q). If [H.sup.q] = f(q), on the other hand, the time series is called "multifractal" .

Experimental Setup

The turbocharger used in this study is a Honeywell GT40, a truck-sized turbocharger. In the following, a short summary of the relevant features for this investigation will be given; for a detailed description of the turbocharger and the experimental setup, the reader is referred to the works by Guillou  and Gancedo .

The compressor has 10 full blades and an exducer diameter of [D.sup.2] = 88 mm. It inducts air via a bell mouth from the test chamber. Ports in the shroud serve as a passive flow device to stabilize compressor operation near the surge line. The impeller is followed by a vaneless diffuser and a volute. A pipe of length L = 1780 mm, whose diameter is D = 63.5 mm in its first half, and D = 50.3 mm in its second half, connects the compressor outlet to the pneumatic valve downstream. The flow is then guided to an orifice plate, which is instrumented for mass flow measurements.

This compression system has no compressible plenum between the compressor and the valve downstream that can clearly be distinguished from the outlet pipes. The system shows typical mild surge behavior in certain parts of the operating range. It does not, however, exhibit the classic deep surge cycle of quiescence, instability growth, blowdown, and recovery as described by Fink . Instead, at a low enough mass flow, the (time-averaged) total pressure ratio drops significantly (see also Figure 10 later in the paper), and the oscillation amplitude increases. Guillou (, p. 237 ff.) estimated the instantaneous mass flow to be negative for parts of the surge cycle from the stereo particle image velocimetry data. This operating regime will therefore in the following be denoted as deep surge, and the term surge line denotes the mass flows where the switch between the mild surge like and deep surge behavior occurs.

The turbine that drives the compressor uses electrically heated pressurized air. The maximum air mass flow supplied by the facility is 0.26 kg/s at a gauge pressure of 8.3 bar. The air inlet temperature to the turbine was chosen to be around 200 degree C.

Since the compressor inducts air from the test chamber through a bellmouth, inlet total temperature and pressure are assumed to be equal to the respective test chamber values. The total temperature is measured by a type T thermocouple ca. 300 mm upstream the bellmouth, and the total pressure in the test chamber is measured with an atmospheric barometer. Compressor outlet conditions are measured by a Druck PDCR 130/W/C pressure transducer and a type K thermocouple. The compressor speed is obtained using a center housing speed sensor provided by the turbocharger manufacturer, which detects a machined surface on the turbocharger shaft. For the mass flow, both the pressure drop over the outlet throttle valve or the orifice plate further downstream can be used. The accuracy using the pressure drop over the valve was considered acceptable for this study, cp. , and is thus used.

In addition to the sensors mentioned above for the compressor performance evaluation, static pressure transducers were mounted at different locations in the diffuser and volute. An overview over the position is given in Figure 2. The sensors in the diffuser are called "Dx", where x is an indicator of the circumferential location. They are of type Kulite XTL-123B-19O-65SG and were flush mounted at the back plate at a radius of R = 72.5 mm. Volute sensors are called "Vx", x again being a circumferential location indicator. They are of type Kulite XTEL-160-50G or XTL-123B-19O-65SG and were flush mounted in a radially inwards direction. Table 1 summarizes the pressure sensors used for time-resolved pressure measurements.

Three speedlines are used for this investigation, with corrected impeller speeds of 64 krpm, 80 krpm, and 88 krpm. Each compressor operating point was sampled for 5 seconds after the compressor operation reached steady-state (or revolves around a steady-state point in the case of surge), as determined by the compressor outlet temperature. The sampling frequency for the time-resolved pressure measurements was [f.sub.s] = 32768 Hz.

RESULTS AND DISCUSSION

Applicability of DFA and the Hurst Exponent Criterion

A DFA1 was performed on four different corrected mass flows on the speedline N = 64 krpm in order to establish its applicability for this compressor. The mass flow correction is done to ensure Mach similarity in the axial direction, using Eq. (9),

MF = [square root of [[T.sub.01] / [T.sub.ref] / [p.sub.01] / [p.sub.ref]]] * [MF.sub.phys] (9)

with inlet total temperature [T.sub.01], reference temperature [T.sub.ref] = 298 K, inlet total pressure [p.sub.01], and reference pressure [p.sub.ref] = 100 kPa. The speedline correction, ensuring Mach similarity for the blade velocity, is given by Eq. (10):

N= [square root of [T.sub.ref] / [T.sub.01]] * [N.sub.phys] (10)

The corrected mass flows for this investigation were MF = 0.312 kg/s, MF = 0.161 kg/s, MF = 0.109 kg/s, and MF = 0.075 kg/s, and correspond to the choke, maximum efficiency, maximum Total Pressure Ratio (TPR), and lowest stable mass flows at this compressor speed, cp. also Figure 10 later in this section. The window sizes [t.sub.s] were chosen to be 32 logarithmically spaced values from 1/1024 seconds to 1 second. As a sensor position, the outlet pressure sensor was chosen in order to be able to compare the results to those in . Figure 3 shows the resulting fluctuation functions. At the two higher mass flows, the fluctuation function increases monotonically, with a slightly increasing slope at time scales [t.sub.s] [greater than or equal to] 0.25 s. This means that the pressure signal is self-similar at time scales smaller than this, with a scaling law of approximately H [approximately equal to] 0.6. ..0.7. The increased slope at large time scales means that there is a small long-term trend or oscillations with a very low frequency present in the time series. At MF = 0.109 kg/s, i.e. peak TPR, the fluctuation function starts to deviate from the higher mass flows, and a knee can be observed at [t.sub.s]= 1/18 s. This knee is typical for the onset wave-like oscillations, see e.g.  for an artificially generated sine wave. For time scales larger than the period of this wave, the fluctuation function becomes flatter. The reason is that the signal variance does not increase strongly with increased window length, since the following wave periods are similar to the first one. For the last stable operating point before entering deep surge, MF = 0.075 kg/s, the knee is much more pronounced and at a lower time scale, namely [t.sub.s]= 1/32 s. For comparison, in previous works by Guillou , the surge frequency was estimated to be 43 Hz for this configuration from the outlet geometry, using the Helmholtz resonator analogy described in the Introduction and using Eq. (1). The estimate from the linear frequency spectrum, obtained via a discrete Fourier transform in , came to a result of around 30 Hz, depending also on the operating point.

From this investigation into four mass flows at one compressor speed, it can be concluded that the Hurst exponent at time scales from one to several mild surge cycles could be a surge indicator for this compression system. Calculations in the following will be based on time scales from [t.sub.s]= 1.25/32 s (i.e. slightly longer than 1 mild surge period) to [t.sub.s]= 4/32 s with 24 logarithmically spaced window lengths between them, if not explicitly stated otherwise.

Comparison of the Hurst Exponent to Pressure Standard Deviation

Figure 4 shows the standard deviation of selected pressure signals at low mass flows and a corrected speed of N = 64 krpm. Higher mass flows are not shown for readability reasons; the standard deviation is approximately constant there. The sensor locations D1, D4, D-2 in the diffuser, V1, V4, V5 in the volute, as well as the compressor outlet pressure were chosen for this analysis. On the top of the figure, following the dynamic stability estimate described in the Introduction, the compressor operation is distinguished between three regimes: operating on a negative speedline slope, operating on a positive speedline slope, and deep surge. These regimes serve as a simplified version of the dynamic stability criterion; the right hand side in Eq. (2) becomes close to zero for a large enough B value, since the throttle characteristic is very steep. However, it is possible that some operating points in the positively sloped speedline regime are dynamically stable. The lines are interrupted between the last operating point at the positive slope and deep surge, since the standard deviation here is not known and cannot be reasonably interpolated from the measurements. The vertical axis uses a logarithmic spacing to better distinguish the differences between sensor locations and between mass flows in stable operation. One can see that on the negative slope, the standard deviation is around 1 kPa for the sensor locations V4, V5, and [P.sub.out], and around 1.4 kPa for the sensor locations in the diffuser as well as V1. A possible reason could be the blade passing and the resulting jet-wake structure, which is not fully mixed out at the diffuser sensor locations. The sensor V1 is located the beginning of the volute where its cross-section area is very small, and thus the effect of the blades can also be seen here.

At mass flows where the speedline slope turns positive, the standard deviation increases by ca. 0.4 kPa at all locations. Reducing the mass flow further, the standard deviation is approximately constant until the last stable operating point, where another slight increase can be observed. In deep surge, its values are significantly higher and dependent on the measurement location, with the highest standard deviation of ca 10 kPa in the compressor outlet pipe.

The Hurst exponent [H.sup.2] for the same sensors and compressor operating points is shown in Figure 5. On the negatively sloped part of the speedline, it varies between [H.sup.2] = 0.5 and [H.sup.2] = 0.7. There is a slight increase from the mass flow with the peak total pressure ratio to the adjacent lower mass flow, followed by an approximately linear decrease until the last stable operating point. The gradient is similar for all locations except D1 and V1, which have a less steep decline. At the last stable operating point, the [H.sup.2] values are between 0.1 and 0.2 for all pressure signals except D1. The limiting value of [H.sup.2] = 0.2 was also found in  for a smaller turbocharger compressor, using different sensor locations in the outlet pipe.

The different behavior of the sensor positions V1 and D1 is probably related to their locations under the volute tongue. One can explain that by considering the mild surge oscillations present here as a wave in the outlet pipe. This wave is more easily detectable at the outlet pressure position and the compressor pressure sensors that are located circumferentially before the volute exit cone, while it is dampened out at the beginning of the volute.

Comparing the standard deviations and the Hurst exponents, one can conclude that the Hurst exponent has some advantages over the standard deviation for evaluating the proximity of the compressor operation to its surge limit. The advantages are a steady, almost linear decrease as the compressor approaches surge, as well as the [H.sup.2] threshold that could be validated for two different compression systems. A similar comparison (not shown here) for two higher speedlines revealed that while the advantage of an approximately linear decrease towards surge is not fulfilled higher speeds, the values of [H.sup.2] [less than or equal to] 0.1 for deep surge, 0.1 [less than or equal to] [H.sup.2] [less than or equal to] 0.5 at the positive speedline slope operation, and [H.sup.2] [greater than or equal to] 0.5 at the negative speedline slope operation are still fulfilled, see also Figure 10.

Sensitivity to Sampling Duration and Frequency

As a next step, the sensitivity of this criterion to the sampling duration is investigated. Each signal of a total duration of 5 seconds was divided into 10 parts of 0.5 seconds duration each, and the Hurst exponent is calculated for each part; these ten samples are then analyzed independently, which allows to estimate the expected variance in the Hurst exponent for short sampling times. This size was chosen due to the time scales for the DFA as established in the section on applicability: Since the largest window size is [t.sub.s,max] = 1/8 s, and one should average the variance (Eq. (7)) over at least four windows, see , 0.5 seconds is the minimum signal length. If a shorter signal should be used, the algorithm can also be tweaked by limiting the window size to e.g. [t.sub.s,max] = 1/16 s, which would correspond to approximately two mild surge cycles.

Figure 6 shows the resulting error bars for the sensors V4, V5, and [P.sub.out]. The ends of the error bars mark the maximum and minimum of the 10 estimates for each operating point, while the lines are drawn using the average of the estimates. At high mass flows and a negative speedline slope, the difference between maximum and minimum [H.sup.2] value is around 0.3 for all sensors; however, since the absolute minimum found in this operating regime is [H.sup.2] = 0.4, this variation is still acceptable to avoid false positives in surge detection. Please note that only two of these mass flows are shown in Figure 6 for readability reasons. More problematic than the high mass flows are the operating points MF = 0.095 kg/s and MF = 0.087 kg/s. At these points, the differences between the [H.sup.2] values are large enough to generate some false positives. Close to the surge line, the [H.sup.2] values become consistently small, so that this criterion can again give an accurate prediction. In deep surge, the pressure oscillations of the surge cycles are so similar that the uncertainty in the estimate of the Hurst exponent becomes very small.

In order to estimate how much of this uncertainty in the [H.sup.2] estimate is due to the short signal lengths and how much is due to changes in the underlying signal, this result is compared to a short-term Fourier transform of the signal. Again, it was split into 10 parts, and the power spectrum was calculated. The power spectrum was then integrated in the interval from f = 20 Hz to f = 40 Hz, to isolate only fluctuations with the systems natural frequency, and converted to a sound pressure level. The equation for the signal power SP is thus

SP = 10 * [log.sub.10] (1 / [(20 [micro]Pa).sup.2][[integral].sup.40 Hz.sub.20 Hz][S.sub.pp]df) [dB] (11)

where [S.sub.pp] is the power spectral density of the respective pressure signal. Note that although the reference pressure of 20 [mu]Pa was chosen analogous to the calculation of sound pressure levels, the sensors here also capture hydrodynamic pressure oscillations, and thus the levels do not only reflect acoustic waves. The results are shown in Figure 7. One can see that the power of the pressure oscillations increases non-linearly towards the surge line, although the decibel is already a logarithmic scale itself. The difference in SP between the sensor positions is approximately independent of the compressor operating point. An exception is the last operating point before entering deep surge, where the SP values of the different sensors are closer together. Comparing the values of the signal power SP with the Hurst exponent [H.sup.2] in Figure 6, one can see that the [H.sup.2] error bars in one operating point are large at mass flows of MF = 0.8.. .0.1 kg/s. They are much smaller for the SP values at the same compressor mass flows. During operation very close to deep surge, on the other hand, the [H.sup.2] error bars are very small, while they are large for the SP values. This observation is valid for all sensor positions. A strict surge criterion based on the Hurst exponent with short sampling times would thus give earlier warnings compared to the signal power, at the drawback of more false positives. Another difference is that the lowest possible value for [H.sup.2] is 0 per definition, while there is no theoretical limit to SP. The SP value itself is also dependent on the chosen reference pressure, which shows up as a constant addition (due to the logarithm) in Eq. (11).

In the following, the sensitivity of the Hurst exponent estimate to the sensor sampling rate will be investigated. The data was programmatically resampled, applying a low-pass Finite Impulse Response (FIR) filter to avoid aliasing. The new sampling rates were chosen as 1/32, 1/64, and 1/128 of the original sampling rate of 32768 Hz. Again, the speedlines N = 64 krpm was used for this analysis. Results for the outlet pressure sensor [P.sub.out] are shown in Figure 8. It can be seen that the lowest sampling rate of [f.sub.s] = 256 Hz results in significantly lower estimates of [H.sup.2] at high mass flows. This is mainly due to the lowpass filter, which takes away large parts of the signal oscillations and noise at high frequency which lead to the higher Hurst exponent. For comparison, an estimate of the Hurst exponent of data resampled with [f.sub.s] = 256 Hz, without a filter, is also shown. This estimate gives a high enough estimate at stable operation, but does not react quickly to the onset of oscillations near surge. Resampling the signal with [f.sub.s]= 512 Hz, the estimated Hurst exponents are the same in near surge operation, and they are consistently high enough in stable operation. For sampling frequencies of [f.sub.s] = 1024 Hz and higher, there is no significant difference in between the [H.sup.2] values of the resampled signal and the original signal at all mass flows. Please note that the results with respect to the necessary sampling frequency should be used carefully when applying this methodology to another compression system. Both the systems natural frequency and the sensor mounting, e.g. flush mounting versus indicator passage, will likely influence the necessary sampling frequency.

Monofractal and Multifractal Signals

As described in the section Method, it is possible to estimate different orders q of the Hurst exponent [H.sup.q] using Eq. (8). The orders are able to distinguish between large amplitude oscillations, which are weighted strongly at high orders, and small amplitude oscillations, which are weighted strongly at negative orders. The assumption made here is that the mild surge-like pressure oscillations with the compression system natural frequency will be large compared to oscillations due to turbulence and signal noise. Therefore, higher order [H.sup.q] should be closer to zero once surge-like fluctuations appear, while negative order H (q) should behave similarly for all operating points. A common way to investigate for monofractality and multifractality is the Renyi scaling exponent [tau], see , as given by

[tau] = [H.sup.q]*q-1 (12)

Plotting [tau] versus order q, a linear function means that the signal is monofractal (since [H.sup.q] [not equal to] f(q)), while a concave function means that the signal is multifractal. Results are shown in Figure 9. The sensor position for this analysis used was [P.sub.out], and the corrected speed was N = 64 krpm . Different corrected mass flows are distinguished by the line color, with red colors representing high mass flows and blue colors representing low mass flows. One can see that high mass flows have an approximately monofractal outlet pressure signal. Lower mass flows in mild surge tend to have a concave shaped [tau] function, where the small amplitude oscillations scale similar to high mass flows, while large amplitude fluctuations scale differently. These observations support the assumption made above. It should be noted here that this observation holds true for most mass flows, but there are some operating points where a reduction in mass flow does not lead to a more concave shaped [tau] function. The same phenomenon was also found for the general Hurst exponent, see Figure 5, and the signal power in the surge frequency range, see Figure 7. Another interesting observation is that at deep surge, curves become almost linear again, indicating that even very small amplitude oscillations are governed by surge.

Extension to Other Speedlines

Both the [H.sup.2] criterion analysis and the monofractal - multifractal analysis are now extended to two other speedlines, N = 80 krpm and N = 88 krpm. Figure 10 shows all three speedlines in a compressor map. The contour levels drawn are the Hurst exponents [H.sup.2] of the outlet pressure, in levels from 0.2 (dark red) to 0.8 (dark blue) with a step size of 0.1. The marker of each measurement point is a circle in the case of monofractal outlet pressure, and a cross in the case of multifractal outlet pressure. The operating points in deep surge can easily be detected by the preceding sharp drop in total pressure ratio.

Considering the lowest speedline N = 64 krpm first, which was extensively investigated before, it is noteworthy that the outlet pressure starts showing multifractal characteristics at MF = 0.172 kg/s, where the speedline slope is still positive and the system is thus theoretically stable. With the exception of three operating points on this speedline at 0.099 kg/s, 0.124 kg/s, and 0.161 kg/s , the rule elsewhere is that the high mass flows have a monofractal outlet pressure signal and the lower ones a multifractal outlet pressure signal. At the next highest speedline, N = 80 krpm, the same rule can be observed, in this case without exceptions. Here also, the multifractal characteristics can be observed where the speedline slope is still negative. The highest speedline, N = 88 krpm, follows this rule with one exception at slightly higher mass flows than the peak total pressure ratio. At this speedline, there exists also a second change in slope from negative to positive at MF = 0.185 kg/s, giving the speedline a "camelback"-like shape. In the region of this local maximum of the speedline, the [H.sup.2] values are around 0.5, indicating stable operation. The monofractal-multifractal distinction, on the other hand, consistently detects mild surge type oscillations at the system natural frequency with very low amplitude. The operating points in deep surge are marked as multifractal, since [tau] has a small concave curvature as can be seen from Figure 9. This effect is very small, however, so that one could also reasonably classify them as monofractal.

CONCLUSIONS

In this paper, the stable operating range of a compressor was investigated using stability criteria based on the Hurst exponent. The compressor is a truck-sized turbocharger compressor with ported shroud and a vaneless diffuser, operated on a cold gas stand.

The Hurst exponent of different pressure signals, for time scales ranging from one to several surge cycles, was estimated using detrended fluctuation analysis. It was shown for one speedline that the Hurst exponent declines almost monotonically with an approximately constant slope as the compressor is throttled towards surge. This characteristic can have advantages compared to the pressure standard deviation, where the increase towards surge is exponential. It should be repeated here that for the higher speedlines, this advantage still persists, but it is less pronounced. For all speedlines, it was found that deep surge is characterized by values [H.sup.2] [less than or equal to] 0.1, while theoretically stable operation results in [H.sup.2] [greater than or equal to] 0.5. Comparing the [H.sup.2] surge threshold for this compressor with a small passenger car sized turbocharger which was investigated in , one can conclude that the values are very similar.

Different sensor locations in the stator were compared in order to assess their suitability for a surge detection algorithm based on the Hurst exponent. The result was that sensor locations at the compressor outlet and those that are circumferentially located before the volute tongue (in the compressor rotational direction) are better suited than those after the tongue. The difference between a sensor in the diffuser and one in the volute, on the other hand, was less pronounced.

A sensitivity analysis with respect to sampling duration and sampling frequency was performed for the compressor outlet pressure signal. It was shown that a sampling duration of 500 ms could result in some false positives, if a warning level of around [H.sup.2] = 0.3 is chosen. This was different from a surge criterion based on the short term Fourier transform, which was less likely to result in false positives, at the cost of giving false negatives. A sampling frequency of [f.sub.s][greater than or equal to] 512 Hz was shown to be necessary for obtaining [H.sup.2] estimates that are only very weakly dependent on the sampling frequency.

Finally, a second stability indicator based on a distinction between monofractal and multifractal outlet pressures was introduced. This indicator was found to be able to detect very small mild surge-like oscillations in some operating points on the negatively sloped section of the compressor speedlines. Theoretical lumped parameter models like Fink's  and 1D models like Bozza and de Belli's  predict a stable operation in this regime. The monofractal-multifractal distinction indicator could have some potential to provide an earlier warning compared to these common theoretical models as well as the [H.sup.2] value presented here.

The method of using the Hurst exponent as a surge criterion has now been used on two compression systems with a turbocharger compressor, a light duty one in  and a heavy duty one in this paper. Both had a relatively small outlet volume. It would be interesting to do a similar investigation on different compression systems, e.g. with a clearly distinguishable compressible plenum so that it exhibits typical deep surge behavior with instability growth and a blowdown. Testing the algorithm on a turbocharger on an actual engine could reveal possibilities and drawbacks as a real-time surge detection method. In particular, it is likely that the pressure oscillations produced by the opening and closing of the engine intake valves will result in Hurst exponent close to zero even during stable compressor operation.

From a more methodological perspective, comparing different algorithms to estimate the Hurst exponent, see  and , and their applicability to surge detection could be a worthwhile endeavor. Furthermore, this concept is closely related to other time series analysis methods from chaos theory. Surge and rotating stall detection on axial compressors using chaotic time series methods has been performed by Bright et al. [21, 22]. An extension of this work to turbocharger compressors, and comparison with the Hurst exponent criterion, could reveal additional insights into surge generating mechanisms.

REFERENCES

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[2.] Greitzer, E., "Surge and Rotating Stall in Axial Compressors II: Experimental Results and Comparison with Theory," Journal of Engineering for Gas Turbines and Power, vol. 2, pp. 199-211, 1976. doi:10.1115/1.3446139

[3.] Fink, D., Cumpsty, N. and Greitzer, E., "Surge Dynamics in a Free-Spool Centrifugal Compressor System," Journal of Turbomachinery, vol. 114, pp. 321-332, 1992. doi:10.1115/1.2929146

[4.] Hansen, K., Joergensen, P. and Larsen, P., "Experimental and Theoretical Study of Surge in a Small Centrifugal Compressor," Journal of Fluids Engineering, vol. 103(3), pp. 391-395, 1981. doi:10.1115/1.3240796

[5.] van den Braembussche, R., Centrifugal Compressor Analysis and Design, Von Karman Insitute for Fluid Dynamics, 2013.

[6.] Andersen, J., Lindstrom, F., and Westin, F., "Surge Definitions for Radial Compressors in Automotive Turbochargers," SAE Int. J. Engines 1(1):218-231, 2009, doi:10.4271/2008-01-0296.

[7.] Gancedo, M., "Effect of Self Recirculation Casing Treatment on the Performance of a Turbocharger Centrifugal Compressor" PhD thesis, School of Aerospace Systems, College of Engineering and Applied Science, University of Cincinatti, OH, USA, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1439305804

[8.] Guillou, E., "Flow Characterization and Dynamic Analysis of a Radial Compressor with Passive Method of Surge Control," PhD thesis, School of Aerospace Systems, College of Engineering and Applied Science, University of Cincinatti, OH, USA, 2011. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1321371782

[9.] Galindo, J., Serrano, J. R., Guardiola, C. and Cervello, C., "Surge Limit Definition in a Specific Test Bench for the Characterization of Automotive Turbochargers," Experimental Thermal and Fluid Science, vol. 30, pp. 449-462, 2005. doi:10.1016/j.expthermflusci.2005.06.002

[10.] Dehner, R., Figurella, N., Selamet, A., Keller, P. et al., "Instabilities at the Low-Flow Range of a Turbocharger Compressor," SAE Int. J. Engines 6(2):1356-1367, 2013, doi:10.4271/2013-01-1886.

[11.] Kerres, B., Cronhjort, A. and Mihaescu, M., "Experimental investigation of upstream installation effects on the turbocharger compressor map," Proceedings of the IMechE 12th International Conference on Turbochargers and Turbocharging, London, UK, 17-18 May, 2016.

[12.] Kerres, B., Nair, V., Cronhjort, A., and Mihaescu, M., "Analysis of the Turbocharger Compressor Surge Margin Using a Hurst-Exponent-based Criterion," SAE Int. J. Engines 9(3):1795-1806, 2016, doi:10.4271/2016-01-1027.

[13.] Kabral, R., Rammal, H., and Abom, M., "Acoustical Methods for Investigating Turbocharger Flow Instabilities," SAE Technical Paper 2013-01-1879, 2013, doi:10.4271/2013-01-1879.

[14.] Bergmann, M. and Parma, P., "The E-drive: An isentropic compressor performance test rig," Proceedings of the 12th International Conference on Turbochargers and Turbocharging, May 17-18, 2016, London, UK, 2016.

[15.] Kantelhardt, J., "Fractal and Multifractal Time Series", in Mathematics of Complexity and Dynamical Systems, Meyers, R., Ed., Springer, New York, 2011, pp. 463-487.

[16.] Peng, C.-K., Buldyrev, S. V., Havlin, S., Simons, M., et al., "Mosaic organization of DNA nucleotides," Physical Review E, vol. 49, pp. 16851689, Feb 1994. doi:10.1103/PhysRevE.49.1685

[17.] Hurst, H., "Long-Term Storage Capacity of Reservoirs," Transactions of the American Society of Civil Engineers, vol. 116 (1), pp. 770-799, 1951.

[18.] Heneghan, C. and McDarby, G., "Establishing the Relation between Detrended Fluctuation Analysis and Power Spectral Density Analysis for Stochastic Processes," Physical Review E, vol. 62 (5), 2000. doi:10.1103/PhysRevE.62.6103

[19.] Gancedo, M., Guillou, E., Gutmark, E., and Mohamed, A., "Dynamic Features and their Propagation in a Centrifugal Compressor Housing with Ported Shroud," SAE Technical Paper 2012-01-0706, 2012, doi:10.4271/2012-01-0706.

[20.] Bozza, F. and De Bellis, V., "Steady and Unsteady Modeling of Turbocharger Compressors for Automotive Engines," SAE Technical Paper 2010-01-1536, 2010, doi:10.4271/2010-01-1536.

[21.] Bright, M., Qammar, H., Weigl, H. and Paduano, J., "Stall Precursor Identification in High-Speed Compressor Stages Using Chaotic Time Series Analysis Methods," Journal of Turbomachinery, vol. 119, pp. 491-499, 1997. doi:10.1115/1.2841148

[22.] Bright, M., Qammar, H. and Wang, L., "Investigation of Pre-stall Mode and Pip Inception in High-Speed Compressors Through the Use of Correlation Integral," Journal of Turbomachinery, vol. 121 (4), pp. 743-750, 1999. doi:10.1115/1.2836728

CONTACT INFORMATION

Bertrand Kerres, PhD Student

KTH Royal Institute of Technology School of Industrial Engineering and Management Department of Machine Design Brinellvagen 83, SE-100 44 Stockholm, Sweden

Phone: +46-8-790 95 07

kerres@kth.se

www.kth.se

ACKNOWLEDGMENTS

The authors would like to acknowledge Dr. Erwann Guillou (Honeywell Turbo Technologies), who carried out parts of the experiments on which this investigation is based. Bertrand Kerres and Mihai Mihaescu would also like to acknowledge the financial support provided by Scania, Volvo Group Truck Technology, Volvo Car Group, BorgWarner TurboSystems, and the Swedish Energy Agency through the Competence Center for Gas Exchange (CCGEx).

ABBREVIATIONS

Latin letters

a - Speed of sound [m/s]

A - Cross-section area

B - Greitzer B parameter

D - Inner diameter

f - Frequency [Hz]

[F.sup.2] - Fluctuation function

[H.sup.q] - Hurst exponent of order q

L - Length

MF - Corrected mass flow [kg/s]

N - Impeller corrected speed [rpm]

SP - Signal power

p - Pressure

S - Power spectral density

T - Temperature

[t.sub.s] - Time scale

U - Impeller tip speed [m/s]

V - Volume

Greek letters

[rho] - Density

[phi] - Mass flow parameter

[psi] - Pressure parameter

Indices

1 - Static - before compressor

C - Compressor

H - Helmholtz resonator model

p - Based on pressure signal or plenum

ref - Reference value

s - sampling

T - Throttle

Bertrand Kerres and Mihai Mihaescu

KTH Royal Institute of Technology

Matthieu Gancedo and Ephraim Gutmark

University of Cincinnati

doi:10.4271/2017-01-1040
```Table 1. Pressure sensors used for time-resolved pressure measurements.

Location                   Sensor name               Range

Pout                       Drack PDCR 130/W/C        0- 344.74 kPa rel.
D-2, D-1, D0, D1, D2, D3,  Kulite XTL-123B-190-65SG  0-448.16 kPa diff.
D4, D5, V1, V4, V5
V2, V3                     Kulite XTEL-190-50G       0- 344.74 kPa diff.
```