Turbomachinery Degradation Monitoring Using Adaptive Trend Analysis ⁎

12th IFAC Symposium on Dynamics and Control of 12th Symposium on and 12th IFAC IFACSystems, Symposium on Dynamics Dynamics and Control Control of of Process including Biosystems 12th IFAC Symposium on Dynamics and Control of 12th IFACSystems, Symposium on Dynamics and Control of Process including Biosystems Process Systems, including Biosystems Florianópolis - SC,including Brazil, April 23-26,Available 2019 online at www.sciencedirect.com Process Systems, Biosystems Process Biosystems 12th IFACSystems, Symposium on Dynamics Control of Florianópolis - SC,including Brazil, April 23-26,and 2019 Florianópolis Florianópolis -- SC, SC, Brazil, Brazil, April April 23-26, 23-26, 2019 2019 Florianópolis - SC,including Brazil, April 23-26, 2019 Process Systems, Biosystems Florianópolis - SC, Brazil, April 23-26, 2019

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IFAC PapersOnLine 52-1 (2019) 679–684

Turbomachinery Degradation Monitoring Turbomachinery Degradation Monitoring Turbomachinery Degradation Monitoring ⋆⋆ Turbomachinery Degradation Monitoring Using Adaptive Trend Analysis Using Adaptive Trend Analysis ⋆ Turbomachinery Degradation Monitoring Using Using Adaptive Adaptive Trend Trend Analysis Analysis ⋆⋆ ∗ Using Adaptive Trend Analysis Marta Zagorowska Arne-Marius Ditlefsen ∗∗

∗∗ ∗∗ Marta Zagorowska ∗∗∗ ∗Arne-Marius Ditlefsen ∗∗ ∗∗ Marta Zagorowska Arne-Marius Ditlefsen ∗ ∗ ∗∗ Nina F. Thornhill Charlotte Skourup ∗∗ ∗ ∗∗ Marta Zagorowska Arne-Marius Ditlefsen Nina F. Thornhill Charlotte Skourup ∗ ∗∗ ∗ ∗ Charlotte Skourup ∗∗ ∗∗ Nina F. Thornhill Marta Ditlefsen NinaZagorowska F. Thornhill Arne-Marius Charlotte Skourup ∗ ∗ ∗∗ Chemical Engineering, Imperial College Ninaof F. Thornhill Charlotte Skourup ∗ Department ∗ Department of Chemical Engineering, Imperial College London, London, ∗ of Chemical Engineering, Imperial College London, ∗ Department South Kensington, SW7 2AZ London, UK (e-mail: Department of Chemical SW7 Engineering, Imperial College London, South Kensington, 2AZ London, UK (e-mail: ∗ [email protected], South Kensington, SW7 2AZ London, UK (e-mail: [email protected]) Department of Chemical Engineering, Imperial College London, South Kensington, SW7 2AZ London, UK (e-mail: [email protected], [email protected]) ∗∗ [email protected], [email protected]) Oil, Gas and Chemicals, Ole Deviks vei 10, 0666 Oslo, Norway ∗∗ South Kensington, SW7 2AZ London, UK (e-mail: ∗∗ ABB [email protected], [email protected]) ∗∗ ABB Oil, Gas and Chemicals, Ole Deviks vei 10, 0666 Oslo, Norway Oil, Gas and Chemicals, Ole Deviks vei 10, 0666 Oslo, ∗∗ ABB (e-mail: [email protected], [email protected], Oil, Gas and Chemicals, Ole [email protected]) Deviks vei 10, 0666 Oslo, Norway Norway (e-mail: [email protected], ∗∗ (e-mail: [email protected], [email protected]) ABB Oil, Gas and Chemicals, Ole Deviks vei 10, 0666 Oslo, Norway (e-mail: [email protected], [email protected]) [email protected]) (e-mail: [email protected], [email protected]) [email protected]) Abstract: Abstract: Performance Performance deterioration deterioration in in turbomachinery turbomachinery is is an an unwanted unwanted phenomenon phenomenon that that Abstract: Performance deterioration in turbomachinery is an unwanted phenomenon that changes the behaviour of the system. It can be described by a degradation indicator Abstract: Performance deterioration in turbomachinery is an unwanted phenomenon that changes the behaviour of the system. It can be described by a degradation indicator based based changes the behaviour of the system. It can be described by a degradation indicator based on deviations from expected values of process variables. Existing models assume that the Abstract: Performance deterioration in turbomachinery is an unwanted phenomenon changes the behaviour of the system. It can be described by a degradation indicator based on deviations from expected values of process variables. Existing models assume that that the on deviations from expected values of process variables. Existing models assume that the degradation is strictly increasing with fixed convexity and that there are no additional changes changes the isbehaviour of the values system. Itprocess can be variables. described by there a degradation indicator based on deviations from expected Existing models assume that the degradation strictly increasing with of fixed convexity and that are no additional changes degradation is strictly increasing with fixed convexity and that there are no additional changes during the considered operating period. This work proposes the use of an exponential trend on deviations from expected values of process variables. Existing models assume that the degradation is strictly increasing fixed convexity and thatthe there changes during the considered operating with period. This work proposes useare of no an additional exponential trend during the considered operating period. This work proposes the use of an exponential trend approximation with shape adaptation and apply it in a moving window framework. The degradation is strictly increasing with fixed convexity and that there are no additional changes during the considered operating period. This work proposes the use of an exponential trend approximation with shape adaptation and apply it in a moving window framework. The approximation with shape adaptation and apply in aa moving window framework. The suggested adjustment makes it for model to the of during themethod considered operating period. This work it proposes the use offollow an exponential trend approximation withof adaptation apply it framework. The suggested method of shape adjustment makesand it possible possible forinthe the moving model towindow follow the evolution evolution of suggested method of adjustment makes it possible for the model to follow the evolution of the indicator over time. The approximation method is then applied for monitoring purposes, approximation with apply itfor a moving framework. The suggested method of shape adjustment makesand it possible the model the evolution of the indicator over time. Theadaptation approximation method isinthen appliedtowindow forfollow monitoring purposes, the indicator over time. The approximation method is then applied for monitoring purposes, to predict future degradation. The influence of the tuning parameters on the accuracy of the suggested method of adjustment makes it possible for the model to follow the evolution of thepredict indicator overdegradation. time. The approximation is thenparameters applied foronmonitoring purposes, to future The influence method of the tuning the accuracy of the to predict future degradation. The influence of the tuning parameters on the accuracy of the algorithm is investigated and recommendations for the values are derived. Finally, directions for the indicator over time. The approximation method is then applied for monitoring purposes, to predict future degradation. The influence of the tuning parameters on the accuracy of the algorithm is investigated and recommendations for the values are derived. Finally, directions for algorithm is investigated and values are directions for further work are to predict degradation. The influence offor thethe tuning the accuracy of the algorithm isfuture investigated and recommendations recommendations for the valuesparameters are derived. derived.onFinally, Finally, directions for further work are proposed. proposed. further work are proposed. algorithm is investigated and recommendations for the values are derived. Finally, directions for further work are proposed. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. further work are proposed. Keywords: Trends, Function approximation, Device degradation, Fault detection, Prediction Keywords: Trends, Function approximation, Device degradation, Fault detection, Prediction Keywords: Trends, methods, Keywords:Compressors Trends, Function Function approximation, approximation, Device Device degradation, degradation, Fault Fault detection, detection, Prediction Prediction methods, Compressors methods, Compressors Keywords: Trends, Function approximation, Device degradation, Fault detection, Prediction methods, Compressors methods, Compressors 1. INTRODUCTION The changes of the shape of the degradation indicator indicator 1. INTRODUCTION The changes of theorigins. shape of the degradation 1. INTRODUCTION The of shape of indicator might have activities are 1. INTRODUCTION The changes changes of the theorigins. shape Maintenance of the the degradation degradation indicator might have multiple multiple Maintenance activities are one one might have multiple origins. Maintenance activities are of the main causes that influence the curvature (Madsen 1. INTRODUCTION changes of thethat shape of thethe degradation indicator Turbomachinery degradation is a complex process. Ac- The might have multiple origins. Maintenance activities are one one of the main causes influence curvature (Madsen Turbomachinery degradation is a complex Acof the main causes that influence the curvature (Madsen and 2014). As the timings the maintenance Turbomachinery degradation is aathe complex process. Acmight have origins. Maintenance activities are one cording to Tarabrin et al. (1996), loss of process. performance of theBakken, mainmultiple causes that influence the of curvature (Madsen and Bakken, 2014). As the timings of the maintenance Turbomachinery degradation is complex process. According to et al.i.e. (1996), the the loss of performance and 2014). As the the maintenance works are known, it is possible to the degradation cording to Tarabrin Tarabrin et (1996), loss of performance of theBakken, main causes influence the of (Madsen is mostly to forming on the Bakken, 2014). the timings timings ofcurvature the works are known, it that isAs possible to analyse analyse themaintenance degradation Turbomachinery degradation is athe complex cording Tarabrin et al. al.i.e. (1996), the loss deposits of process. performance is mostlytodue due to fouling, fouling, forming the deposits on Acthe and works are known, it is possible to analyse the degradation indicator in a given operating period, as described by is mostly due to fouling, i.e. forming the deposits on the and Bakken, 2014). As the timings of the maintenance blades and influencing the flow path. In industrial appliworks are known, it is possible to analyse the degradation indicator in a given operating period, as described by cording to Tarabrin et al. (1996), the loss of performance is mostly to fouling, forming deposits on the Cicciotti blades anddue influencing thei.e. flow path. the In industrial appliindicator in a given operating period, as described by (2015) or by Hanachi et al. (2017), especially for blades and influencing the flow path. In industrial appliworks are known, it is possible to analyse the degradation cations, degradation is measured usingthe anindustrial indicatoron based indicator in a given operating period, as described by Cicciotti (2015) or by Hanachi et al. (2017), especially for is mostly due to fouling, i.e. forming deposits the blades and influencing the flow path. In applications, degradation is measured using an indicator based Cicciotti (2015) or by Hanachi et al. (2017), especially for steady state operation. A similar approach was adopted by cations, degradation is measured using an indicator based indicator in a given operating period, as described by on deviations from expected values of process variables Cicciotti (2015) or by Hanachi etapproach al. (2017), especially for steady state operation. A similar was adopted by blades and influencing the flow path. In industrial applications, degradation is measured using an indicator based on deviations from expected values of process variables steady state operation. A similar approach was adopted by Salamat (2012), but for a wider range of operating points. on deviations from expected values of process variables Cicciotti (2015) or by Hanachi et al. (2017), especially for (Tarabrin et al., 1996; Li and Nilkitsaranont, 2009). Monsteady state operation. A similar approach was adopted by (2012), but for a wider range of operating points. cations, degradation is Li measured usingofanprocess indicator based on deviations from expected values variables (Tarabrin et al., 1996; and Nilkitsaranont, 2009). Mon- Salamat Salamat (2012), but for a wider range of operating points. Their moving window approaches assume that degradation (Tarabrin et al., 1996; Li and Nilkitsaranont, 2009). Monsteady state operation. A similar approach was adopted by itoring of these phenomena provides an insight into the Salamat (2012), but for a wider range of operating points. Their moving window approaches assume that degradation on deviations from expected values of process variables (Tarabrin al., phenomena 1996; Li andprovides Nilkitsaranont, 2009). itoring of et these an insight intoMonthe starts Their moving window approaches assume that immediately maintenance period. However, itoring of these phenomena provides an insight the Salamat (2012), butafter for athe wider range of operating points. condition of the equipment, and is used, e.g., for into mainteTheir moving window approaches assume that degradation degradation starts immediately after the maintenance period. However, (Tarabrin et al., 1996; Li and Nilkitsaranont, 2009). Monitoring of these phenomena provides an insight into the condition of the equipment, is used, e.g., for maintestarts immediately after the period. However, as by Li Nilkitsaranont (2009), degradacondition the equipment, and is e.g., for maintemoving approaches assume thatthe degradation nace planning et al., and 2016). The models starts immediately after the maintenance maintenance period. However, itoring of of these phenomena provides an available insight the Their as indicated indicated bywindow Li and and Nilkitsaranont (2009), the degradacondition of the(Xenos equipment, and is used, used, e.g., for into maintenace planning (Xenos et al., 2016). The available models as indicated by Li and Nilkitsaranont (2009), the degradation curvature might change during the operating period, nace planning (Xenos et al., 2016). The available models starts immediately after the maintenance period. However, assume that the degradation is a monotonic function of as indicated by Li and Nilkitsaranont (2009), the curvature might change during the operatingdegradaperiod, condition of the equipment, used, e.g., for maintenace planning (Xenos et al., and 2016). The available models assume that the degradation is ais monotonic function of tion tion curvature might change during the operating period, not necessarily because of the maintenance. They proposed assume that the degradation is monotonic of as indicated by might Li andchange Nilkitsaranont (2009), the proposed degradatime, and that the convexity is aafixed, thefunction geometric tion curvature the operating period, necessarily because of the during maintenance. They nace planning (Xenos et al., 2016). Thei.e. available models assume that the degradation is monotonic function of not time, and that the convexity is fixed, i.e. the geometric not necessarily because of the maintenance. They proposed a moving window approach where the approximating trend time, and that the convexity is fixed, i.e. the geometric tion curvature might change during the operating period, properties related toconvexity the graphis ofafixed, the function (Davidson not necessarily because of the maintenance. They proposed assume that the degradation monotonic function of a moving window approach where the approximating trend time, and that the is i.e. the geometric properties related to the graph of the function (Davidson anot moving window approach where the approximating trend is switched between linear and quadratic functions of time. properties related to the graph of the function (Davidson necessarily because of the maintenance. They proposed and Donsig, 2009) are constant. For the purpose of this a moving window approach where the approximating trend is switched between linear and quadratic functions of time. time, and that the convexity is fixed, i.e. the geometric properties related to the graph of the function (Davidson and Donsig, 2009) are constant. For the purpose of this is switched between linear and quadratic functions of time. Nevertheless, they assumed that the monotonicity and and Donsig, 2009) are constant. For the purpose of this a moving window approach where the approximating trend work, this graphic interpretation is described as the shape is switched between linear and quadratic functions of they assumed that the monotonicitytime. and properties related theconstant. graph ofisFor the function and Donsig, 2009)to are the purpose this Nevertheless, work, this graphic interpretation described as(Davidson theofshape Nevertheless, they assumed that the monotonicity and convexity stayed the same during the operation period, work, this interpretation described as the is switchedstayed between quadratic functions of time. of a function. As shown Madsenis and Bakken the convexity Nevertheless, theythelinear assumed that the and and Donsig, 2009) are by constant. the purpose this sameand during the monotonicity operation period, work, this graphic graphic interpretation isFor described as(2014), theofshape shape of a function. As shown by Madsen and Bakken (2014), the convexity stayed the same during the operation period, and that there was only one transition moment. The apof a function. As shown by Madsen and Bakken (2014), the Nevertheless, they assumed that the monotonicity and assumptions about the shape of the degradation indicator convexity stayed theonly same during the moment. operation The period, and that there was one transition apwork, this graphic interpretation is described as the shape of a function. As shown by Madsen and Bakken (2014), the assumptions about the shape of the degradation indicator and that there was only one transition moment. The approach proposed in the current work is able to capture the assumptions about the shape of the degradation indicator convexity stayed the same during the operation period, are not always fulfilled. This work presents a modelling and that there was only one transition moment. The approach proposed in the current work is able to capture the of a function. As shown by Madsen and Bakken (2014), the assumptions about the shape of the degradation indicator are not always fulfilled. This work presents a modelling proach proposed in the current work is able to capture the behaviour of the degradation by adaptation of the shape are not always fulfilled. This work presents a modelling and that there was only one transition moment. The apapproach that captures the observed behaviour of the proach proposed in the current work is able to capture the assumptions about the shape the degradation indicator of the degradation by adaptation of the shape are not always fulfilled. This work presents a modelling approach that captures the of observed behaviour of the behaviour behaviour of the degradation by adaptation of the shape of the trend, and to adapt to the unknown timings of the approach that captures the observed behaviour of the proach proposed in the current work is able to capture the degradation indicator. The proposed framework is then behaviour of the degradation by adaptation of the shape of the trend, and to adapt to the unknown timings of the are not always fulfilled. This work presents a modelling approach that capturesThe theproposed observedframework behaviour isofthen the transitions. degradation indicator. of the to unknown the degradation indicator. The proposed behaviour of and the degradation by adaptation of the of shape used for monitoring to predict changes framework ofbehaviour the trendis ofthen the of the trend, trend, and to adapt adapt to to the the unknown timings timings of the transitions. approach that captures the observed of the degradation indicator. The proposed framework is then used for monitoring to predict changes of the trend of the transitions. used to changes of the theremainder trend, and to adapt toisthe unknown timings of the indicator. transitions. degradation indicator. The proposed The used for for monitoring monitoring to predict predict changes framework of the the trend trendisof ofthen the of indicator. The remainder of of the the paper paper is structured structured as as follows. follows. The The indicator. transitions. used for monitoring to predict changes of the trend of the degradation The of is as follows. indicator. of centrifugal is introduced, inThe remainder remainder of the the paper papercompressors is structured structured as follows. The The degradation of centrifugal compressors is introduced, inindicator. degradation of centrifugal compressors is introduced, including a description of degradation models and their ap⋆ The remainder of the paper is structured as follows. The degradation of centrifugal compressors is introduced, inFinancial support is gratefully acknowledged from the Marie Curie cluding a description of degradation models and their ap⋆ ⋆ Financial support is gratefully acknowledged from the Marie Curie Financial support is gratefully acknowledged from the Marie Curie cluding a description of degradation models and their ap⋆ proximation. Section 3 describes the trend approximation degradation of centrifugal compressors is introduced, inHorizon 2020 EID-ITN project acknowledged ”PROcess NeTwork Financial support is gratefully from theOptimization Marie Curie cluding a description of degradation models and their ap⋆ proximation. Section 3 describes the trend approximation Financial support is gratefully acknowledged from theOptimization Marie Curie Horizon 2020 EID-ITN project ”PROcess NeTwork Horizon 2020 project ”PROcess NeTwork Optimization proximation. Section 33ofdescribes the trend approximation with moving window and shape adaptation. The algorithm for efficient andEID-ITN sustainable operation of Europe’s process industries Horizon 2020 EID-ITN project ”PROcess NeTwork Optimization cluding a description degradation models and their ap⋆ proximation. Section describes the trend approximation with moving window and shape adaptation. The algorithm Horizon 2020 EID-ITN project ”PROcess NeTwork Optimization Financial support is gratefully acknowledged from the Marie Curie for efficient and sustainable operation of Europe’s process industries for and of process with moving window and shape adaptation. The algorithm is then applied to and evaluated for taking machinery condition operation and process performance intoindustries account for efficient efficient and sustainable sustainable operation of Europe’s Europe’s process industries proximation. Section 3 describes the trend approximation with moving window and shape Themonitoring algorithm for efficient andEID-ITN sustainable operation of Europe’s process Horizon 2020 project NeTwork Optimization is then applied to prediction prediction andadaptation. evaluated for monitoring taking machinery condition and ”PROcess process performance intoindustries account taking PRONTO”, Grant condition agreementand No process 675215. performance is then applied to prediction and evaluated for monitoring taking machinery machinery condition and process performance into into account account with moving window and shape adaptation. The algorithm is then applied to prediction and evaluated for monitoring taking machinery condition and process performance into account for efficient and sustainable operation of Europe’s process industries PRONTO”, Grant agreement No 675215. PRONTO”, Grant agreement No 675215. PRONTO”, Grant agreement No 675215. PRONTO”, Grant condition agreementand No process 675215. performance into account is then applied to prediction and evaluated for monitoring taking machinery

PRONTO”, agreement No 675215. Copyright © Grant 2019 IFAC 679 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Copyright © 2019 IFAC 679 Copyright © 2019 IFAC 679 Copyright ©under 2019 responsibility IFAC 679Control. Peer review of International Federation of Automatic Copyright © 2019 IFAC 679 10.1016/j.ifacol.2019.06.141 Copyright © 2019 IFAC 679

2019 IFAC DYCOPS 680 Marta Zagorowska et al. / IFAC PapersOnLine 52-1 (2019) 679–684 Florianópolis - SC, Brazil, April 23-26, 2019

frag replacements

η

strictly monotonic, bounded function of time (decreasing), as depicted in Fig. 1. In consequence, the degradation indicator d calculated according to Eq. (1) would also be a strictly monotonic (increasing), function of time stabilising at a certain level.

Healthy value

ηD (t)

Operating point h performance loss

Such behaviour suggests using an exponential function of time for approximation of d(t), e.g. in form described by Tarabrin et al. (1996): f (t, β) = β1 − β2 exp(−β3 (t − T0 )) (3) where t denotes time, and T0 is the beginning of the degradation. It is usually assumed that T0 is known, but not necessarily that T0 = 0 (Cicciotti, 2015). Then β = [βi ], i = 1, . . . , 3 is a vector of constant parameters.

Degraded value Time

Fig. 1. Compressor efficiency as a function of time. If there is no degradation, healthy value of η is constant (solid black line). If the compressor degrades, then after a certain time η attains a new, degraded value (black dotted line). The transient ηD �= const (dash-dotted black line)

2.2 Degradation approximation

purposes in Section 4. Finally, conclusions and directions for future research complete the paper.

Exponential trend analysis was used, e.g., for monitoring by Cicciotti (2015) or maintenance planning by Xenos et al. (2016). However, in some industrial cases, the degradation indicator can be not monotonic and its convexity might change. An example of such case is depicted in Fig. 2. Figure 2a shows the actual efficiency of a compressor (black curve) compared with the expected value of 84% (blue horizontal line). The corresponding efficiency degradation indicator was calculated according to Eq. (1) and is presented as a function of time (in weeks) in Fig. 2b (black curve). The red curve shows the approximating function determined with Eq. (3).

2. TURBOMACHINERY DEGRADATION The degradation of rotating equipment, such as centrifugal compressors or turbines, is usually associated with fouling (Tarabrin et al., 1996). During steady-state operation, degradation can be measured by the deviation of the process variables from the expected, healthy values as described, e.g., by Loboda et al. (2007): Y − YD d= (1) Y where Y denotes the healthy value, and YD is the value in degraded state.

The efficiency degradation described in Cicciotti (2015) follows the assumptions about the monotonicity, convexity, and boundedness of the indicator. However, Fig. 2b shows an indicator that saturates at 0.012 starting in week two, but then increases to 0.025 in week eight. Moreover, starting in week nine, the monotonicity changes, and the indicator starts to decrease.

Usually, the undegraded value Y is delivered by the manufacturer of the compressor, whereas YD is estimated from thermodynamic relationships (Cicciotti, 2015) and shows the actual performance of the system. The ability to predict the value of YD would provide an insight into the future behaviour of the system. It could be estimated from (1), if the degradation indicator d was known, as: YD = (1 − d) · Y (2) Therefore, the question of modelling and predicting the degradation indicator d is considered, as it would provide knowledge on the degraded state of the system. It is usually assumed that the degradation is a function of only time in operation, d = d(t) (Tarabrin et al. (1996), Jasmani et al. (2013)). This approach is justified in compressors working at the same operating point for the whole period.

The red curve shows the approximating function of form (3), with T0 = 0 and parameters βi found by minimising the square of a norm of the residuals r multiplied by a vector of weights w = [wk ] n  wk rk2 (4) min β

k=1

with (5) r = [rk ] = [f (tk , β) − yk ] where (tk , yk ) are the observed data points, and wk denote the weights of the measurements, here wk = 1. The value of tk denotes time and yk is the corresponding value of efficiency degradation indicator, n indicates the number of measurements.

Typically, the variables used for the degradation of turbomachinery are the head or the efficiency (Tarabrin et al., 1996). Figure 1 shows the behaviour of a compressor efficiency, Y = η, going from the healthy value η (solid line) to a certain degraded value (dotted line), usually after up to 2000 hours of operation (Tarabrin et al., 1996). The transient curve (dash-dotted line) shows the efficiency ηD as a function of time, ηD (t) = (1 − d(t)) · η.

The approximating curve in Fig. 2b deviates from the actual value of the degradation indicator, starting from week three. This suggests that an approximation using a function with fixed shape is not sufficient. 3. APPROXIMATION WITH MOVING WINDOW AND SHAPE ADAPTATION

2.1 Degradation modelling This section presents a method of updating the parameters of (3) using a moving window. The minimisation problem (4) is solved repeatedly with the optimisation constraints adapted to the varying shape in each period.

One of the possible approaches to predict the behaviour of the degradation indicator is based on trend analysis. The efficiency ηD of the degraded compressor is assumed to be a 680

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The parameter T0i denotes the start of the ith approximation window. The signs of β2i and β3i define the shape in the ith window, i.e. the monotonicity and convexity of the function, as indicated by the first and second time derivative, respectively (Davidson and Donsig, 2009). The parameter β1i shifts the approximating function up and down the vertical axis.

84

Efficiency [%]

83.5

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83 82.5 82

To allow adaptation to the varying shape of the degradation trend and in contrast to previous work where β i > 0, the constraints in the minimisation problem (4) are adjusted for the last element ofthe vector β i , i.e. β3i . using i the vector of differences Di = Dik in the period T0i , Tend calculated as yk − yk−1   Dik = (7)   i tk − tk−1 tk ∈ T0i , Tend

81.5 81

0

1

2

3

4

5 6 7 8 Time [weeks]

9 10 11 12

(a) Actual compressor efficiency (black line) and assumed efficiency if there was no degradation (blue line)

Formula (8) applies a switching rule for β3i  ≥ −0.0001, then β3i ∈ [0, ∞) Di < −0.0001, then β3i ∈ (−∞, 0]

0.035

Degradation indicator

0.03 0.025

0.015 0.01

0 0

gradation indicator

(8)

where Di is the mean value of Di . The rest of the parameters are in the interval [−3, 3]. The values are found by consideration of typical rates of degradation and loss of performance of real machines.

0.02

0.005

replacements

681

1

2

3

4

5 6 7 8 Time [weeks]

The constraints for the parameter β i allow the trend to follow the non-monotonic shape of the curves. The second issue is to force the system to follow the change in the trend of the indicator as soon as it happens, as the initial time is not known. This is done by adjusting the weighting vector, w, to give  less weight to older measurements in i period T0i , Tend . Therefore, the weights are distributed in the interval [0.9, 1] with the most recent measurements having the weights close to unity.

9 10 11 12

(b) Degradation indicator (black line) for data from Fig. 2a and approximating exponential function (red line)

Fig. 2. Example of compressor efficiency (Fig. 2a) and corresponding degradation indicator (Fig. 2b) from Brekke et al. (2009) Tf

4. PREDICTION AND MONITORING

Tf Tapp

In this part, the algorithm using trend analysis is applied for monitoring of the degradation due to fouling and shown in Fig. 2. Two features of the algorithm are analysed: how accurate the prediction is, and how early the algorithm is able to predict a change in the trend of the indicator, depending on the parameters of the algorithm, Tapp and Tf . The data come from a GE LM2500 engine operating offshore in the Norwegian Sea and were obtained from Brekke et al. (2009) using software developed by Rohatgi (2018). Accordingto Brekke et al. (2009), the data were collected during a period of operation without maintenance activities that could mitigate the loss of performance.

Tapp i+1 T0i T0

tik−1 tik tik+1

i+1 i Tend Tend

Time

Fig. 3. Explanation of moving window notation 3.1 Moving window idea The idea of a moving window approximation is depicted in Fig. 3. The ith approximation window starts at T0i , ends at i Tend and has length Tapp . The subsequent approximation i+1 , and has window, i + 1, is described with T0i+1 and Tend the same length, Tapp . The time difference between two consecutive windows, Tf = ∆T0 = ∆Tend , is also constant, and characterises the update rate of the approximation. The dots on the time axis denote the measurement instants tik in ith window. In this work, ∆tk = tik+1 −tik = 1 minute   i , for all i. for all tik ∈ T0i , Tend

The idea of the prediction is depicted in Fig. 4. The function  obtained after approximating the data in the i interval T0i , Tend (green lines) is extended into the next time period of length Tpred (blue lines). The end of the i i = Tend + Tpred . prediction window is marked with Tpred The arrows (red) show the differences between the actual values and the predicted curve.

3.2 Approximation with shape adaptation

4.1 Prediction accuracy

Rewriting Eq. (3) using the notation from section 3.1 yields: (6) f i (t, β i ) = β1i − β2i exp(−β3i (t − T0i ))

The differences between the actual values and the predicted curve are used for evaluating the accuracy of the 681

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Tf Tapp

d 1

mean SEP i

0.95 prediction differences Tpred

T0i

i+j i Tend Tpred T0j

i+j Tend

i+j Time Tpred

0.9

PSfrag replacements 0.85 0.8

Fig. 4. Prediction idea with moving window approach

0.75

prediction for a given window, for instance of the ith window on the left side of Fig. 4. An indicator is proposed to evaluate the accuracy in ith window as: n  i SEP = (yk − f (tk ))2 (9)

0.7

7

9

11

13

15

17

19

21

23

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27

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Fig. 5. The mean value of SEP as a function of Tapp , for k=1   i Tpred = 2 weeks and Tf = 2 days (blue points), with i i where tk ∈ Tend , Tpred . The higher the value of SEP , approximating quadratic function (orange line) the further the prediction is from the actual data points, i whereas SEP = 0 would mean that the prediction goes PSfrag replacements 0.05 through each data point, capturing also the noise. 0.045 Degradation indicator

4.2 Prediction accuracy - results and discussion The prediction results were evaluated as a function of Tapp . Figure 5 shows how the mean value of SEP i from Eq. (9) changes with the size of the approximating window (Tapp ) for a fixed value of the prediction window (Tpred = 2 weeks). An approximating window of less than 15 days resulted in high values of mean SEP i , where 0.85 and above is considered high. If Tapp is between 15 and 26 days, the mean SEP i is below 0.8, and then increases again if Tapp > 26. The quadratic approximation (orange line) in Fig. 5 suggests a robust minimum is at 20 to 21 days (three weeks). The explanation for this can be found by analysis of the black curve in Fig. 2b. It is visible that the two major changes of the trend of the degradation indicator last two weeks. They occurred in weeks one and two, and weeks eight and nine. Therefore, the approximation window of 21 days captures the entire transient behaviour and a short part of a neighbouring period. This allows the use of the properties of the approximating exponential function which follows this curvature.

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To show the importance of Tapp , two cases were chosen: Tapp = 14 days (2 weeks), and Tapp = 21 days (3 weeks). In both cases, shorter approximation period Tapp = 14 days yielded larger value of mean SEP i for two prediction periods, Tpred = 14 days and Tpred = 4 weeks. For Tpred = 14 days it was 0.8742 and 4.3072, whereas for Tpred = 4 weeks, it was 0.7661 and 3.455, respectively. The indicator was over four times higher when Tpred was longer. This suggests that trend analysis is better at short term predictions. It is confirmed by visual analysis of Fig. 2b. To accurately predict the efficiency degradation four weeks in the future, the trend would have to be constant for at least the summed lengths of the approximation and prediction windows, i.e. six weeks in this case. As there is no such period in Fig. 2b, the one month prediction is considered too long.

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Fig. 6. Degradation indicator for Tpred = 2 weeks and Tapp two and three weeks (solid black lines). The trend approximation and prediction are depicted in red dation indicator (black) and the approximating curves (red) for Tapp ∈ {14, 21} days, and Tpred = 14 days. The case Tpred = 4 weeks can be obtained directly by extrapolating the red curves and was omitted due to a space requirements. The period of two weeks is short enough to be contained almost entirely in a period with constant trend which results in incorrect predictions (Fig. 6a). Longer Tapp captures the curvature of the indicator and the approximation follows the exponential trend (Fig. 6b) and is therefore chosen for further analysis.

These results are also confirmed by the inspection of the efficiency indicators in Fig. 6 which presents the degra682

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Fig. 8. The moment of detection as a function of Tf (blue points) and approximating quadratic curve (orange line)

In this section, the properties of the method from the point of view of monitoring are considered, i.e. if the algorithm were used on-line, how early the operator would get a prediction indicating future change in the trend of the degradation indicator. The idea of the detection algorithm has been depicted in Fig. 7. The black curve shows the values of the data, approximated with exponential trend (solid green curves) given by Eq. (6), then extrapolated in the prediction period (dashed green lines). The green dots denote the values of the trend approximation at the end of the prediction period. In addition, the pink lines denote the mean value of the real data in approximation window (solid lines). For the purpose of the evaluation, a change has been defined using the difference hi (red arrows) between the final value of the approximating trend (green dot) and the mean extended in the prediction period (dashed pink lines): n  1  i , βi − yk (10) hi = f i Tpred n k=1   i , β i are given by Eq. where the parameters of f i Tpred i

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was set Tpred = 2 weeks, i.e. the algorithm is set to detect a change in the next two weeks. The parameter Tapp was set to 21 days (3 weeks). Figure 8 presents the moment when the increase of the trend has been detected as a function of Tf . The horizontal axis shows the value of Tf , and the vertical axis shows the detection moment as day in week eight (blue points). The results were then approximated with a quadratic function i (orange line), to emphasize the increasing value of Tend . Smaller values of Tf , below 44 hours, allow prediction earlier in week eight, at the expense of more frequent calculations. Larger Tf results in detecting the values up to three or four days later (Tf above 84 hours). Therefore Tf is chosen as 40 hours, as this allowed detection during the first two days of the week. Figure 9 presents the evolution of the differences hi as a function of time, with Tapp = 21 days, Tpred = 14 days, Tf = 40 hours, and the threshold of 0.006. Two changes were detected – one in week eight, and one at the beginning of week twelve. The moments of detection have been marked with red dots. The approximation windows corresponding to the detected changes (also indicated with red dots) are depicted in Fig. 10, T0i , T0j – dotted lines, j i , Tend – dash-dotted lines, i < j. The green curves Tend show the approximating function, and pink horizontal lines show the mean values in the approximated periods. The corresponding differences h are marked with arrows, upwards if h > 0, downwards if h < 0.

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(6), with β found solving Eq. (4), and denoting the end of the ith prediction window. The corresponding approximation window contains n values of the degradation indicator, yk , k = 1, . . . , n. The absolute value of the difference hi is compared with a predefined threshold θ. The change is detected if |hi | > θ. The choice of θ depends on the operator. In this work, a change of 0.5 percentage points of efficiency is detected, i.e. η − ηD = 0.5%. Taking η = 84% and using Eq. (1), the threshold θ is set to 0.006 in the degradation indicator. 4.4 Monitoring results and discussion

The analysis of Fig. 9 provides an insight into the duration of the change detected in week eight. The values |hi | above the threshold show that the prediction is different than the mean, i.e. the trend in the corresponding approximating window is not constant. Increasing |hi | indicates the period where the prediction moves away from the actual value. However, decreasing value of |hi | indicates that the trend follows the curvature of the indicator and the prediction is getting closer to the mean, with the maximum of hi indicating the transition. This implies that the change from week eight lasted two weeks, which is consistent with a visual analysis of Fig. 2, and thus confirms that the trend analysis might be used for monitoring.

The detection algorithm was applied to data from Brekke et al. (2009). Figure 10 shows the data used for the analysis (black curve) obtained by removing the first two weeks from Fig. 2b to start from a steady state of the degradation process. The influence of the update rate, Tf , on the detection of an increasing trend in weeks eight and nine is analysed. The value of Tf is sought that would allow the earliest prediction. At the same time, Tf represents how often the approximation is performed, so to minimise the number of calculations, the largest possible Tf should be chosen. Then an application for online monitoring with selected parameters is presented. The prediction window 683

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algorithm are evaluated: the accuracy of the prediction depending on the length of the approximation window, and the capability of early detection of the change of the trend as a function of the update rate. The influence of the tuning parameters is investigated and recommendations for the values evaluated in a monitoring framework. Further work includes increasing the robustness over a variety of datasets and comparison with other monitoring approaches.

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The authors would like to thank Dr James Ottewill from ABB Corporate Research Center in Poland for stimulating discussions and guidance.

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REFERENCES

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Brekke, O., Bakken, L.E., and Syverud, E. (2009). Compressor fouling in gas turbines offshore: Composition and sources from site data. In ASME Turbo Expo 2009: Power for Land, Sea, and Air, 381–391. Cicciotti, M. (2015). Adaptive Monitoring of Health-state and Performance of Industrial Centrifugal Compressors. Ph.D. thesis. Imperial College London. Davidson, K.R. and Donsig, A.P. (2009). Real Analysis and Applications: Theory in Practice. Springer Science & Business Media. Hanachi, H., Mechefske, C., Liu, J., Banerjee, A., and Chen, Y. (2017). Enhancement of prognostic models for short-term degradation of gas turbines. In 2017 IEEE International Conference on Prognostics and Health Management, 66–69. Jasmani, M.S., Van Hardeveld, T., and Bin Mohamed, M.F. (2013). Performance degradation monitoring of centrifugal compressors using deviation analysis. In Proceedings Of The 9th International Pipeline Conference, 543–551. Li, Y.G. and Nilkitsaranont, P. (2009). Gas turbine performance prognostic for condition-based maintenance. Applied Energy, 86(10), 2152–2161. Loboda, I., Yepifanov, S., and Feldshteyn, Y. (2007). A generalized fault classification for gas turbine diagnostics at steady states and transients. Journal of Engineering for Gas Turbines and Power – Transactions of the ASME, 129(4), 977–985. Madsen, S. and Bakken, L.E. (2014). Gas turbine operation offshore: On-line compressor wash operational experience. In ASME Turbo Expo 2014: Turbine Technical Conference and Exposition, V03BT25A008– V03BT25A008. Rohatgi, A. (2018). Webplotdigitizer. URL https://automeris.io/WebPlotDigitizer. Salamat, R. (2012). Gas Path Diagnostics for Compressors. Ph.D. thesis. Cranfield University. Tarabrin, A.P., Schurovsky, V.A., Bodrov, A.I., and Stalder, J.P. (1996). An analysis of axial compressors fouling and a cleaning method of their blading. In ASME 1996 International Gas Turbine and Aeroengine Congress and Exhibition. Xenos, D.P., Kopanos, G.M., Cicciotti, M., and Thornhill, N.F. (2016). Operational optimization of networks of compressors considering condition-based maintenance. Computers & Chemical Engineering, 84, 117–131.

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Fig. 10. The results of the monitoring with Tapp = 21 days, Tpred = 14 days, Tf = 40 hours The behaviour of hi could also be used for decision support regarding the degradation. Positive and increasing hi (detected in week eight in Fig. 9) means that the predicted value is larger than the mean in the ith window, and this suggests that the degradation would increase over the next prediction period. Negative value of hi , on the other hand, shows a decreasing indicator (detected in week 12 in Fig. 9 and confirmed by a decreasing trend in Fig. 10). This suggests that the degradation would decrease and the performance might improve in the next prediction period. 5. CONCLUSION Degradation monitoring is used with turbomachinery for purposes such as maintenance planning. Existing models of degradation assume that the degradation is strictly increasing with fixed convexity and that there are no additional changes during the considered operating period. Such behaviour is observed in certain industrial cases (Cicciotti, 2015), however, the shape of the degradation might change because of maintenance activities (Madsen and Bakken, 2014) or unexpected disturbances (Li and Nilkitsaranont, 2009). This work uses an exponential trend analysis with shape adaptation and applies it in a moving window framework to find the approximating model. The suggested adaptation method makes it possible for the model to follow the curvature of the degradation indicator. The method is applied to the analysis of the evolution of the trend of the degradation. Two features of the 684