Detection of Cavitation in a Centrifugal Pump-as-Turbine Using Time-Domain-Based Analysis of Vibration Signals

Abstract

Pumps-as-Turbines (PATs) are increasingly used in micro-hydropower applications due to their cost competitiveness that is brought about by lower acquisition, design, operation, and maintenance costs. Despite these, limited research exists that investigates PAT failures. Notably, there is a literature gap concerning cavitation in PATs. As such, this study proposes an improvement to the deviation from the normal distribution (DND) technique to facilitate application in PAT cavitation detection. Probability density functions of vibration signals collected during operation at design speed and various cavitation states are developed and the DND computed using two approaches, i.e., the use of baseline data and the original method, for comparison purposes. Normal probability plots are presented to depict suitability of the two approaches in quantifying the DND. Results show higher deviation when using baseline data, hence, improved detection capabilities with amplification of the slope of the trend line under cavitating conditions when using the proposed DND approach. The proposed method also allows for establishing clear alarm limits for the condition monitoring of PATs in practice. Moreover, the proposed method is validated by application at various PAT operating speeds and cavitation states. The proposed method is found to be responsive, reliable, and independent from operating speed.

1. Introduction

The utilization of Pumps-as-Turbines (PATs) has emerged as a cost-effective alternative for energy production in micro- or pico-hydropower applications, including water supply networks, run-of-river hydropower schemes, irrigation, and other pre-existing water infrastructure [1]. This cost competitiveness stems from the lower costs associated with the manufacture of PATs when compared to traditional hydropower turbines for such applications. Moreover, PATs are available in various sizes, accommodating a wide range of heads and flow rates, and demonstrate operational ease following failure-induced downtime.

For critical assets in production facilities, downtime significantly contributes to the total cost, as the loss of production results in no output, and, consequently, lost revenue. Although PATs are generally associated with lower maintenance costs due to the ready availability of spare parts from mass-manufactured pumps [2], the acquisition of spare parts accounts for only a minor portion of the total operation and maintenance cost. According to Hu [3], the cost of spare parts account for 25–30% of the total operating and maintenance support costs.

Despite the claim of lower operation and maintenance cost linked with PATs, there are a very limited number of research works aimed at understanding the failures of these machines. While most failures have been extensively investigated under the umbrella of ‘rotating machinery’, there is a gap in the literature concerning the understanding of cavitation in the machines of this class. This gap requires attention, as while cavitation in pumps is well understood, pumps operating in reverse as turbines do so under flow conditions for which they have not been designed. The susceptibility of PATs to cavitation has also been shown to be larger than for conventional machines [4].

Cavitation involves the change of phase from liquid to gas when the fluid is accelerated through the impeller passages. When the fluid reaches areas of higher pressures downstream, the formed vapor bubbles collapses, leading to performance degradation, violent vibration, noise, and component damage [5]. Some suppression measures for cavitation effects have been studied [6,7,8,9] and for centrifugal pumps measures such as increasing the pressure closer to the cavitation occurrence region, modifying the impeller blade structure and blade impeller parameter optimization have been recommended [6]. These measures are associated with drawbacks such as more complex designs that, in turn, increase manufacturing costs, which could invalidate the cost competitiveness of PATs as an off-the-shelf substitute to conventional bespoke turbines in some applications.

Early detection of cavitation in PATs is crucial to the prevention of damage, ensuring their continued economic viability for micro-hydropower applications. Furthermore, detecting cavitation in PATs is valuable for understanding their maintenance requirements, as cavitation is typically associated with detrimental effects in hydraulic machines. During cavitation, the collapsing vapor bubbles release shockwaves characterized by high-energy waves that can penetrate metallic structures, creating weak points and shortening their operational life. Additionally, these shockwaves may occur at frequencies matching the structural natural frequencies, resulting in excessive structural vibration.

Vibration analysis is widely used in damage detection [10,11,12], making it the most reliable technique, particularly when coupled with a diagnostic decision making system [12]. Another important diagnostic signal for cavitation is the acoustic signal [6,10,13], which measures the noise generated by the machine as it operates. Both structure-borne and fluid-borne noise can be used for detection, but measurement of cavitation noise is difficult due to signal attenuation during its propagation [10]. Zhu et al. [6] note that the widespread use of the acoustic emission signals in cavitation detection is limited by its reliance on precise and expensive supporting facilities. It is, however, possible to obtain and use the spectral content of high frequency signals and modulating frequencies.

Frequency domain analysis of vibration signals by means of Fourier transform is the dominant technique in vibration-analysis-based fault detection and diagnosis, even though it provides average information that ignores the time-varying nature of fault signals. As a result of continued research and development in the sensing and data acquisition techniques, a myriad of signal processing techniques has emerged over time. These include techniques based on the Hilbert transform, time frequency methods such as short-time Fourier transform, wavelets, and empirical mode decomposition, and have proved their worth in structural health monitoring applications. Nagarajaiah and Basu [14] reviewed these techniques in detail and proposed a method of identifying multi-degree of freedom linear time-invariant and linear time-variant systems based on such time frequency techniques.

While the use of frequency spectrums obtained from transformed time waveforms is prevalent in fault detection, the adoption of time domain vibration response for fault detection has been limited. Nonetheless, methods derived from time domain vibration signals such as PeakVue^®, root mean square (RMS), kurtosis, and pseudo fractal-based techniques have been developed for fault detection purposes and, sometimes, time-domain-based methods dependent on probability density functions (PDF) have been presented. Hadjileontiadis, Douka, and Trochidis [15] proposed a kurtosis crack detector based on the property of kurtosis to identify deviations from Gaussianity of vibration signals in the time domain. The method was compared to the RMS value method, which measures the power content in the vibration signal, and it was proven to show superior performance.

Furthermore, Pakrashi, O’Connor, and Basu [16] compared the performance of the kurtosis-based crack detector to wavelet-based and pseudo fractal-based techniques, and found it to be inferior because of its lack of flexibility and its fragility against noise. In addition, it was found that the kurtosis-based technique is more susceptible to events of non-detection and false alarm. Based on the comparison of the actual PDF with the normal/Gaussian PDF of vibration signals, Rzeszucinski et al. [17] proposed the deviation from the normal distribution (DND) technique. Asnaashari and Sinha [18] proposed utilizing the DND of vibration signals to detect cracks on beamlike structures. Moreover, Rzeszucinski et al. [19] proposed some condition indicators for gearbox fault detection and monitoring based on the amplitude of the Gaussian PDF.

Boylan and Cho [20] present a comprehensive demonstration and analysis of how various statistical properties within a data set can influence the shape and corresponding properties in a normal probability plot (NPP). Umesh and Ganguli [21] used the NPP of the PDF to show the difference between the dispersions of damaged and undamaged cases when studying the effect of uncertainty on the structural health monitoring of a smart composite plate with matric crack. In Sharma and Parey [22], a fault detection method was proposed using entropy measurement and a modified PDF.

In this paper, time domain vibration signals are used to propose a PDF-based cavitation detection method. The approach entails modification of the DND technique by computing the deviation of the actual PDFs for various operating conditions from the normal PDF of the baseline data. Results indicate better performance when using the proposed method in comparison to the original DND method. The paper is structured as follows: theoretical background to the PDF, NPP, and the DND method are discussed in Section 2. This is followed by the experimental setup and methodology in Section 3 and Section 4. Section 5, Section 6 and Section 7 present the findings and discussions, validation of the technique, and implications to practical applicability, respectively.

2. Theoretical Background

2.1. Probability Density Functions

Vibration signals obtained from rotating machines can be classed as either deterministic or random. Deterministic signals can be predicted through a mathematical function whereas random signals cannot be predicted at any future instant. When obtained from a cavitating machine, vibration signals can be either deterministic, characterized by a change in the vibration energy of a specific frequency band, or random in nature, indicated by a change in the overall vibration levels [10,23,24]. It is usually convenient to utilize the concept of probability density functions to gain further understanding of signals. PDFs are defined by the value of instantaneous amplitude within a certain interval divided by the size of that interval [25].

The actual PDF of a dataset is usually unknown; therefore, an estimate of the general PDF must be obtained. This can be carried out through either a parametric or a non-parametric approach. Parametrically, the data are assumed to be sampled from a known distribution such as the Gaussian distribution, whereas in a non-parametric approach, PDFs are estimated by adapting the data via means such as the use of histograms and use of estimators such as naïve, kernel, and nearest neighbour method [26]. Al-Hashmi [25] reported that the PDFs of vibration data captured from a cavitating pump exhibit high count around the mean amplitude, spreading out symmetrically as amplitude increases in both positive and negative directions, indicating closer resemblance to the Gaussian distribution. It is as such that we assume a Gaussian PDF for our study case as the PDFs have similar characteristics. The Gaussian distribution is expressed as per Equation (1) [25]:

$$f_n\left(x\right)=\mathrm{}\frac{1}{\sigma \sqrt{2\pi }}{e}^{\left(\frac{-(x-\mu {)}^2}{2{\sigma }^2}\right)}$$

(1)

and the true density function of the data was obtained by substituting its mean ($\mu$) and standard deviation ($\sigma$) into the expression.

In this study, the vibration signals captured at various operating conditions were used to estimate the PDFs using a histogram estimator. Equation (2) was used to define the histogram:

$$f_a\left(x\right)=\mathrm{}\frac{1}{M}\times \frac{(number of x_1 in same bin as x)}{(width of bin containing x)}$$

(2)

where $M$ represents total number of data points and $x$ represents individual data points in the signal.

2.2. Normal Probability Plot

The normal probability plot (NPP) graphically assesses the normality of a dataset. The relative cumulative frequencies of the data are demonstrated by plotting the data on a straight line and comparing the quantiles of the observed data with that of the theoretical distribution, which is Gaussian in the current study. The shape of the curve in the NPP and the way it deviates from the reference line provides information about the characteristics of the actual distribution in terms of the first four sample moments, i.e., mean, variance, skewness and kurtosis [20].

If $[x_1,x_2,\dots ,x_n$] represents an ordered sample of data and $F\left(X\right)=(x-\mu /\sigma )$ represents its cumulative distribution function with $\mu$ being the mean and $\sigma$ the standard deviation, the abscissa of the NPP is the sample ordered statistics $x_i$, and the ordinate is the inverse of the normal cumulative distribution function (CDF) $Z_i=F^{-1}\left(p_i\right)$ where $p_i$ is the cumulative probability associated with each rank-ordered data. The cumulative probability was obtained using Equation (3) as proposed by Blom [27]:

$$p_i=\frac{i-0.375}{n+0.25}$$

(3)

where $i$ represents the rank order number and $n$ represents the number of points.

2.3. Deviation from Normal Distribution

The DND technique as proposed by Rzeszucinski et al. [17] is based on the comparison of the deviation of vibration responses from their corresponding normal distributions. The idea here is that these deviations are different between varying structural health conditions, thus being able to differentiate between healthy and faulty conditions. If the reviewed dataset follows a Gaussian distribution, most of the data points will fall along a straight line in an NPP. However, non-linearity is indicated by deviations from the straight line and these can be quantified as the DND. The area between the actual and Gaussian distribution was computed according to Equation (4) [18]:

$$DND=\mathrm{}\sum _{i=1}^n\left(\frac{\left|Z_{a,i}-Z_{n,i}\right|+|Z_{a,i+1}-Z_{n,i+1}|}{2}\right)\left(\Delta x\right)$$

(4)

where $n$ represents the number of data points in a signal, $Z_{a,i}$ and $Z_{n,i}$ represent values for the actual and Gaussian distribution, respectively, at $i\mathrm{th}$ data point, and $\Delta x$ is the spacing between data points.

When calculating the original DND, the values of Z_(n,i) and Z_(n,i+1) are drawn from the same dataset used to compute the actual distribution, thus comparing it to its theoretical Gaussian distribution, which may diverge from the Gaussian distribution of vibration data for healthy operating conditions. To ensure that the DND reflects the machine’s healthy state, we generate the Gaussian distribution from the baseline data and employ it to compute the DND for subsequent operational conditions, clearly delineating the machine’s healthy condition from other operating points. In this context, the baseline data pertain to the data collected when the machine is operated at its optimal efficiency point for the designated operating speed. The underlying concept is that the actual distribution for each operational condition varies depending on the machine’s cavitation status, leading to fluctuations in the DND. Under the conditions of ‘no cavitation’, DND values are expected to align with or approximate those of the baseline condition, while cavitating conditions are anticipated to yield distinctly different values from the baseline.

3. Materials and Methods

3.1. Experimental Setup

The experimental setup consisted of a KSB ETN 050-32-200 (KSB Limited, Loughborough, UK) volute type centrifugal pump operating in reverse as a turbine and driving a standard 5.5 kW ABB 3GBA 132 210-ADD (ABB Limited, Cork, Ireland) induction motor operated as a generator. This was connected to the grid via a 5.5 kW Mitsubishi Electric FR-A741-5.5K (Mitsubishi Electric, Dublin, Ireland) regenerative variable speed drive (VSD)/inverter. The purpose of VSD was to provide loading to the PAT. By setting the desired operational speed on the VSD, the generator speed is controlled to match the grid frequency requirements, ensuring efficient energy conversion. The water flowing through the system was supplied from a 2 m³ water reservoir through a 9.2 kW Ebara 3D 50–200 (Ebara Pumps Europe, Vicenza, Italy) centrifugal pump operating at 2900 rpm. Figure 1 shows the schematic of the experimental test rig with an insert picture showing the test PAT.

The PAT consisted of six rearward facing blades on an enclosed impeller with outer diameter of 209 mm, inlet diameter of 63 mm, and rated speed, flow rate, and efficiency of 1520 rpm, 0.006 m³/s, and 51.5%, respectively. A TUF 2000M ultrasonic flow sensor (Dalian Taijia Technology, Dalian, China) with an accuracy error of less than 1% was used to obtain flow measurements and flow through the system was varied by adjusting a manually operated control valve at the outlet end of the supply pump. To validate its measurements, the flow sensor measurements were compared with the computed flow readings manually obtained using a measuring beaker and a stopwatch. The readings were found to be in agreement. At the outlet of the PAT a viewing pane was installed to allow observation of the cavitation condition of the machine (see Figure 1).

The instrumentation and software schematic for the vibration data acquisition is displayed in Figure 2. The operating speed of the test rig was set through the VSD connected to the induction motor. Vibration measurements were then measured with 100 ± 10% mV/g sensitivity PCB ICP type accelerometers (PCB Piezotronics, Stevenage, UK) with non-linearity of less than 1%, frequency range of 0.5–10,000 (±5%) Hz, and resonant frequency greater than 50 kHz. The accelerometers were factory calibrated and for validation purposes their outputs were compared when attached to a same vibrator mounted frame, in the same measuring direction and found to be yielding similar results. Three single axis accelerometers were stud-mounted at the drive end bearing housing of the PAT in mutually perpendicular directions. A single triaxial accelerometer was adhesively mounted at the PAT outlet flange. The accelerometers were connected via a signal conditioner unit to a 16-bit NI 6210 analogue to digital data acquisition system (National Instruments, Newbury, UK). An NI DAQExpress Version 5.0.0 data logging software was used to store the digitized vibration signal on a personal computer for further analysis.

3.2. Methods

The supply pump was started to provide flow through the system while driving the PAT, which was free spinning with no load connected. After constant operation for some time, the flow through the system was reduced to allow easy connection of the load without tripping. The VSD was set to 1500 rpm of generator speed and provided the load to the PAT. The system was allowed to run for some further time to allow synchronization of generator speed with the flow through the system.

Once synchronized, the flow through the system was adjusted via a manually operated control valve to nominal conditions and allowed to run for few minutes to allow the system to stabilize at a constant flow condition. Vibration data, flow, and pressure measurements were collected at a sampling frequency of 12.8 kHz for a total sample time of 60 seconds. These data were then saved as the baseline data, as they represented the machine operation at nominal conditions.

The flow through the system was then adjusted to the eight other operating conditions, listed in

Table 1. Summary of the PAT operating conditions investigated.

Operating Point	Flow Rate (L/s)	Cavitation Number, σ	Code *	Observed Operating Condition
1	7.69	0.22	OP1	No cavitation
2	6.79	0.22	OP2 †	No cavitation
3	6.91	0.24	OP3	No cavitation
4	6.67	0.26	OP4	No cavitation
5	6.13	0.29	OP5	No cavitation
6	5.48	0.34	OP6	Inception cavitation
7	5.03	0.39	OP7	Inception cavitation
8	4.18	0.46	OP8	Full cavitation
9	2.94	0.59	OP9	Full cavitation

* Same nomenclature is used in figures. ^† Baseline operating condition.

. After adjusting the flow to a new operating point, the system was allowed to run for some time to stabilize before measurements can be collected. Table 1 shows the summary of the studied operating conditions and the observed cavitation status of each condition. The experiments were repeated four times to observe the variability of results and ensure reproducibility before final measurements could be taken for analysis. No major variations were observed and the experiments were then deemed reproduceable.

As the control valve was being throttled, flow conditions through the PAT were observed via a clear viewing pane. Three operating conditions could be identified, i.e., no cavitation, inception cavitation, and full cavitation. Figure 3 shows the appearance of the viewing pane under both inception and full cavitating conditions. For no cavitation conditions, only clear water could be observed.

At inception cavitation, the flow develops a thin clear sheet-like cavity that is indicative of the start of the cavitation process in the PAT. This cavity develops from the centre of the impeller extending outwards into the outlet pipe. On the contrary, full cavitation conditions are identified by the formation of a thicker whitish vapor-filled vortex core at the centre of the flow that extends the full length of the viewing pane. The formation of this vapor-filled vortex core is a result of a swirling velocity component at non-optimal conditions that reaches vaporization pressure.

4. Results and Discussions

4.1. Probability Density Functions

The study computed the PDFs of the captured vibration data to qualitatively describe existing variations as the machine condition is being varied. Figure 4a shows the Gaussian PDF of all studied operating conditions computed using their mean and standard deviations in Equation (1), whereas Figure 4b shows their actual PDFs obtained through the histogram estimator defined in Equation (2). An in-depth observation of the PDFs indicates that the studied operating conditions can be grouped into three categories. The PDFs for operating conditions OP1 to OP4 form the first group, OP5 and OP6 form the second group, and OP7 to OP9 form the third group. These groups are clearly visible in the tails of the Gaussian PDF plot.

Operating point OP5 falls under the inception cavitation conditions, contrary to the observed flow regime through the viewing pane. It is likely that inception cavitation occurs before it can be visible in the viewing pane, especially since the viewing pane is a little downstream of the impeller centre where the cavitation process initiates. This presents a possible source of error in the experiment, as there is over reliance on the observer’s judgement to confirm the cavitation status of the machine. The same applies to operating condition OP7, which, according to the data, falls under the full cavitation regime as opposed to inception cavitation per Table 1. The boundary between observed flow under ‘inception cavitation’ and ‘fully developed cavitation’ can be difficult to judge, which might have resulted in the machine operating conditions being incorrectly described based on the experimenter’s judgement. Nonetheless, the groupings observed in the PDF plots provide an immediate correction of this experimental error.

The first group is characterized by broader PDFs with low peaks. This coincides with the non-cavitating condition of the PAT. In the second group of PDFs, the peaks increase and the PDF narrows, as compared to the first group. This matches with the inception cavitation condition. When fully cavitated, the PDFs becomes narrower with higher peaks. In Figure 4b, the actual PDFs for all operating conditions consists of longer left tails of the distribution, which is indicative of negative skewness with the preponderance of values lying on the right of the mean.

From this analysis, it is evident that the magnitude of the vibration acceleration is decreasing as the cavitation status worsens. This can imply less vibration and reduced chances of wear and tear with cavitation, consequently meaning cavitation detection in this case is counterintuitive. However, analysis of PDFs of vibration data captured at other PAT operating speeds and varying cavitation status showed different trends (indicated later on). At varying operating speeds, both above and below the design speed, a combination of increasing and decreasing trends of vibration amplitudes was observed, hence, invalidating the inference that monitoring cavitation in PATs defies logic. As such, this prompts for a cavitation detection technique that is resistant to the speed dependency of vibration amplitudes but sensitive to changes in the PAT cavitation state. The prior described variation between the three groups of investigated PDFs might be useful for condition monitoring purposes, but it becomes difficult to quantitatively evaluate the differences in a PDF plot. As such, there exists a need for a more quantifiable means of segregation between the different operating conditions for fault detection.

4.2. Normal Probability Plots

To facilitate understanding of how the data deviate from the normal distribution, the normal probability plot was used. Two approaches were adopted in creating the NPP, (1) the reflection of distribution of the data relative to their own Gaussian distribution, and (2) the reflection of the distribution of the data relative to the Gaussian distribution of the baseline data. For brevity, only one case of each operating condition is shown alongside the baseline condition.

Figure 5 shows the NPPs obtained by comparing the actual distribution of data for each operating condition with its own Gaussian distribution. Comparing the plots for each operating condition does not show any significant differences that can be the basis for condition monitoring. This indicates that comparing the actual distribution of data with its normal distribution in an NPP will not yield reasonable differences suitable for monitoring in a condition monitoring setup, hence, proving the need for alternative approaches.

In Figure 6, the NPP was obtained by comparing the actual distribution of data with the normal distribution of the baseline data. When the PAT is not under cavitating conditions (Figure 6a), the data mostly lie on the normal distribution, diverting only at the extreme ends. The actual distributions of both the baseline data and the ‘no cavitation’ condition match exactly. When cavitation initiates, see Figure 6b, there is an evident variation in the data, as indicated by the actual distribution line visibly crossing the normal distribution line three times, creating a larger area between the two curves. The extreme ends of the actual distribution show that the vibration acceleration tends to increase beyond that of the baseline, which is indicative of the increased effects of outliers in the data.

Under fully cavitating conditions (see Figure 6c), the NPP resembles that of inception cavitation, with the area between the baseline normal and the actual distribution of the full cavitation data significantly increasing. Further comparison of the actual distributions shows that the full cavitation condition curve further crosses the baseline data curve at extreme ends, signalling a reduction in vibration acceleration amplitude. This reduction in amplitude must be because of the effects of the developed vortex rope in the dynamics of the machine. Alatorre-Frenk [28] noted that when a PAT goes through inception cavitation, the effects of the vortex rope in the dynamics of the machine increases, which then normalizes because of the reduction in the skin friction due to the formation of vapor at the structural surface.

These observed differences in distributions from various operating conditions also provide a qualitative differentiation of cavitation and it is important to quantify this for condition monitoring purposes. The quantification of the areas observed between the actual distribution and the normal distribution of the baseline data provides the basis of the proposed time-based detection method for cavitation presented in this paper.

4.3. Deviation from Normal Distribution

The concept of DND was utilized to quantify the observed differences in the NPP obtained by comparing the actual distributions of various operating conditions with the Gaussian of the baseline data. In the prior sections, it was demonstrated that the NPP obtained by computing the actual distributions of each operating conditions with its own Gaussian provides little variation between the operating conditions. As such, the DND was computed using the two methods, the original method of using actual distribution with its Gaussian PDF and the proposed method of comparing the actual PDF to the Gaussian PDF of the baseline data. In this study, instead of applying the DND method for detection of structural defects, we investigate the application of the DND technique in detecting fluid disturbances in the form of cavitation in a PAT. Unlike structural defects, fluid-induced disturbances are inherently non-linear, which means their characteristics vary highly with time. It is assumed that since measurements are taken on structural elements of the machine, the changes in the captured vibration data are indicative of the changes taking place within the flowing fluid that is transmitted to the structure. This is deemed adequate to characterize cavitation occurrence inside the PAT.

Figure 7 shows a trend plot of the DND as the operating condition is being varied from non-cavitating to fully cavitating conditions. The solid line across the figure indicates the separation between observation of cavitating and non-cavitating conditions. The DND value obtained from the proposed method increases to higher values after inception cavitation into fully cavitating conditions. The gradual increase in the plot from non-cavitating to cavitating conditions depicts the capability of trending the DND obtained from the comparison with the normal distribution of a baseline data in detecting changes in the fluid flow regime. As such, this is an indication of strong detection capabilities for cavitation in a PAT.

For the original method, the DND plot decreases continuously from operating point OP2. This consistent drop in the value of the DND would make it difficult to use for condition monitoring purposes. It is evident in the plot that the original method fails to provide a consistent variation between the PAT’s cavitation status. There is variation in the DND values for the same condition, i.e., no-cavitation condition consistently increases when using this approach, indicating some possible unreliability issues associated with this method when used in detecting cavitation in PATs. Moreover, this consistent increase in the magnitude change from one operating point to the next has no respect for the change in operating condition, as its also observed into full cavitating condition.

5. Validation of Detection Capabilities

Further experimental tests were conducted to show the effectiveness of the proposed method in detecting cavitation in a PAT. Further measurements were obtained with the PAT operating at different operating speeds and various flow rate conditions.

Table 2. Investigated operating conditions at 1350 rpm.

Operating Point	Flow Rate (L/s)	Cavitation Number, σ	Code *	Observed Operating Conditions
1	8.14	0.22	OP1	No cavitation
2	6.91	0.23	OP2 †	No cavitation
3	7.06	0.25	OP3	No cavitation
4	6.76	0.27	OP4	No cavitation
5	6.18	0.30	OP5	Inception cavitation
6	5.60	0.36	OP6	Full cavitation
7	5.17	0.41	OP7	Full cavitation
8	4.51	0.49	OP8	Full cavitation
9	3.11	0.66	OP9	Full cavitation

* Same nomenclature is used in figures. ^† Baseline operating condition.

indicates the investigated test conditions for the operating speed of 1350 rpm. For operating points 1–4, the machine was observed as not cavitating, at operating point 5 the machine was observed to be under inception cavitation, and for operating points 6–9, the machine was in full cavitation state.

Table 3. Investigated operating conditions at 1800 rpm.

Operating Point	Flow Rate (L/s)	Cavitation Number, σ	Code *	Observed Operating Conditions
1	7.15	0.22	OP1	No cavitation
2	6.52	0.22	OP2 †	No cavitation
3	6.56	0.24	OP3	No cavitation
4	6.00	0.26	OP4	No cavitation
5	5.50	0.29	OP5	Inception cavitation
6	4.71	0.33	OP6	Inception cavitation
7	4.00	0.37	OP7	Full cavitation
8	3.02	0.43	OP8	Full cavitation
9	1.28	0.51	OP9	Full cavitation

* Same nomenclature is used in figures. ^† Baseline operating condition.

summarizes further test conditions when the operating speed was increased to 1800 rpm and cavitation conditions were observed. At operating points 1–4, the PAT was not cavitating, for operating points 5–6, the machine cavitation state changed to inception cavitation, whereas at operating points 7–9, the machine was under full cavitation conditions.

Gaussian PDFs for these operating speeds were computed and the results are shown in Figure 8a,b for 1350 rpm and 1800 rpm, respectively. As discussed in Section 4.1, the PDFs in Figure 8 shows a different trend of vibration acceleration to the one discussed. For these operating speeds, vibration acceleration for cavitating conditions depict a wider range, thus, wider PDF plots with lower peaks in comparison to narrower PDF plots with higher peaks for non-cavitating conditions, indicative of narrow acceleration ranges with majority of points lying close to the mean acceleration value. Furthermore, it is evident from PDF plots that it is difficult to describe the relationship between rotational speed and vibration acceleration. This is indicated by acceleration amplitude’s peak-to-peak values for non-cavitating conditions increasing from 2 g at 1350 rpm to 5 g at 1500 rpm, followed by 4 g at 1800 rpm. The same random characteristic is observed at other values of rotational speeds not presented here. A similar characteristic is observed for cavitating conditions.

Based on this analysis and the observed trend of data captured at various other speeds not presented in this paper, it is clear that cavitation occurrence alters vibration response differently at various operating speeds. At some operating speeds, there is a resultant increase in vibration levels, whereas at other speeds, vibration levels of the PAT decrease. This variable response at various speeds can be considered evidence for a requirement of a cavitation detection system in PATs that is robust and independent from operational speed.

Similar to the analysis conducted in the prior section, the DND value changed trends as the machine cavitation status varied from ‘no cavitation’ to ‘full cavitation’. Figure 9 shows the obtained trend plot when the machine was operated at 1350 rpm. Similarly, the plot compares the proposed method of computing the DND to its original version. It is evident from the plot that the proposed method shows a higher magnitude change when the cavitation state changes from no cavitation to full cavitation. There is an observed magnitude increment in the value of DND during cavitation as compared to the no cavitations state. Moreover, the comparison of the proposed method with the original version indicates that the magnitude of the change observed when using the original version is very small and difficult to discern when compared with the proposed new method, especially when the cavitation state changes from no cavitation to full cavitation. This can be a very important factor when the DND is being trended for condition monitoring purposes, which could yield false negatives.

Similarly, when the PAT was operated at a slightly higher rotational speed of 1800 rpm (Figure 10) and cavitation monitored using the two methods, it was observed that the proposed method again performed better, as the value of the DND was amplified when moving from ‘no cavitation’ condition to full cavitation state. For the original method, the observed increase in the value of the DND was minimal, hence limiting the detection capabilities of the method when used for condition monitoring purposes. The performance of the proposed method in detecting the occurrence of cavitation inside a PAT is quite promising and with careful selection of limits, the method can be of valuable importance in avoiding failures caused by cavitation.

6. Implications and Practical Applicability

The experiments were conducted in as close as possible conditions to PAT installations in the field. No additional suction or vacuum systems were included that are typical of laboratory studies of this nature, but are not representative of real-world applications. As such, it is evident that in real world operational settings, PATs will be exposed to cavitating conditions that can be detrimental to their operation, especially in cases where there is a wide variation in flow conditions and/or frequent start-up or shut-down operations. Application of the proposed DND approach to detect cavitation occurrence can be helpful in averting such effects, particularly where there is lack of expert understanding of the cavitation phenomenon in the generation mode of PATs [9].

Deployment of the technique in a typical industrial scenario would entail computation of the DND from captured vibration data and trending the value over time. This can be accomplished at the data capture point via edge computing as it does not require complex lines of codes to compute the DND and large data storage space. The potential limitations associated with field deployment include the reliance on baseline data and the possibility of baseline drift due to varying operating conditions. The effects of the changes in operating speed have been investigated here and the robustness of the technique was demonstrated under varying speed conditions. Variation in operating conditions could also be a result of changes in head–flow conditions and maybe the accumulation of material on rotating parts of the impeller due to the type of application such as waste water treatment. Since the technique was proven efficient in detecting cavitation when flow rate was varied, it is, therefore, assumed that similar changes will not result in many performance problems. However, it has not been demonstrated how the technique will perform under constant flow but varying pressure conditions. This can be investigated in future works.

The proposed technique is highly dependent on the use of baseline data, which might be difficult to ascertain in the field. Even though that is the case, the use of baseline data is widely acceptable in condition monitoring systems. Most predictive maintenance programs uses baseline data captured at installation or immediately after the completion of the first scheduled maintenance post the establishment of the program. The baseline data will then be used as a reference for comparison purposes with the machine condition at any future instant. Any variations observed can be an indication of a developing fault condition. Thus, baseline data provide historical context needed to identify subtle changes or trends indicative of impending failures.

From the practical applicability point of view due to operation in resource-constrained environments including embedded systems and edge devices, depending on baseline data can ensure that machine condition can be determined without the need for storing large amounts of machine condition data. In the proposed technique, only a single machine dataset is stored and used to compare against future collected data to decide the machine condition. It is envisaged that the amount of storage space required will be minimum, hence lowering the cost of the predictive maintenance program. From this baseline data, a selection of parameters could then be computed, including the DND, that could be used to monitor the machine fault condition. The technique would also allow implementation on edge devices, as the baseline data will only occupy a small storage space and future data will be used just to compute the DND that will be trended and monitored over time.

Moreover, the use of baseline data is important in condition monitoring, as it helps to reduce cases of false positives due to variations in operating conditions. Since baseline data will also be used as a reference for differentiating between faulty and healthy machine conditions, eliminating cases of healthy condition identified as faulty becomes an easy task of comparing with it, hence hastening the whole process. Also, with the existence of historical baseline data, there would be a benchmark to use when deciding whether to update the baseline with the current machine condition data. Even though the machine would not be faulty, updating the baseline could be necessary more so because some machine applications involve the accumulation of material on machine components and normal wear, which cannot be categorized as faulty.

Updating baseline data periodically should safeguard against baseline drift that could result in an increase in either false negatives or false positives. To ensure the robustness of baseline data, incremental learning algorithms can be applied to continually update the baseline data as machine operating conditions vary over time. Baseline data can also be computed from a set of data that represent the anomaly-free condition of the machine. In this study, four or five data sets were classified as without cavitation and represent ranges of operating conditions. Taking an average value of these datasets to be representative of the baseline data would help to guard against baseline drift in the system in the short term.

In comparison with the statistical approaches of proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) that have garnered wide coverage in fluid dynamics problems, the DND technique is applicable to systems that can be characterized by a known statistical distribution that experience subtle changes in the system behaviour. The DND method produces data that are relatively simple to interpret, and is particularly useful when understanding the underlying physics is not as important. In contrast, the POD and the DMD are well-suited to situations where spatial correlations are critical [29,30,31,32]. POD and DMD are data-intensive techniques and the proposed DND technique could be used to yield input data to these techniques in a machine learning application.

7. Conclusions

In this article, experiments were conducted to induce cavitation within a pump-as-turbine (PAT) and vibration data were collected coinciding with ‘no cavitation’, ‘inception cavitation’, and ‘full cavitation’ operating conditions. The captured vibration signals were used to propose an improved approach for detecting cavitation in PATs in the time domain. The approach involves modifying the computation algorithm for the deviation from normal distribution (DND) method. Instead of comparing the captured data’s actual distribution with their Gaussian distribution estimate, we use the Gaussian distribution values derived from selected baseline data at the designated operating speed.

Subsequently, the computed DND values were trended to illustrate the changes in the signals corresponding to variations in cavitation detection status. This modification of the original DND method relies on comparing the current machine state with its known healthy state, represented by baseline data at the best efficiency point. Our results indicate that a significant increase in the magnitude of DND change is detected when the PAT transitions into a cavitating state using the proposed method. Conversely, minimal changes in DND were observed when employing the original method, highlighting potential challenges in setting alarm thresholds for effective condition monitoring purposes.

The proposed approach was further validated by analysing data collected during PAT operation at different rotational speeds, including speeds exceeding and lower than the design speed. Consistently, our findings demonstrate the capability of the proposed approach to accurately detect cavitation states within the PAT, thereby enhancing condition monitoring effectiveness compared with the original method. Conversely, the minor DND value changes observed using the original method could make cavitation detection difficult, and with weaker detection capabilities. This paper also outlines the results of the first application of the DND method to fluid-based machinery failures. With no prior evidence of the DND method applied in the detection of fluid-based failures, our study underscores the responsiveness and reliability of the proposed approach in cavitation detection within PAT hydropower systems.

Furthermore, the simplicity and ease of implementation of the DND method make it suitable for monitoring by either domain experts or operators in a total productive maintenance framework. However, potential limitations such as the requirement for baseline data availability and the possibility of baseline drifts under varying operation conditions should be noted. Nonetheless, these do not invalidate the benefits of implementing this technique in a real-world application, as it also allows for real-time application.

While this study focused on detecting cavitation under stationary conditions, future research should explore the application of our proposed method under non-stationary conditions involving simultaneous changes in rotational speed and flow conditions. Overall, our findings suggest that the proposed approach offers a responsive and reliable technique for cavitation detection in PAT systems, with promising real-time application for condition monitoring in fluid-based systems.