sensors Article

Sparse Reconstruction for Temperature Distribution Using DTS Fiber Optic Sensors with Applications in Electrical Generator Stator Monitoring João Paulo Bazzo 1 , Daniel Rodrigues Pipa 1 , Erlon Vagner da Silva 2 , Cicero Martelli 1 and Jean Carlos Cardozo da Silva 1, * 1

2

*

Graduate Program in Electrical and Computer Engineering (CPGEI)/Federal University of Technology-Parana, Curitiba 80230-901, Brazil; [email protected] (J.P.B.); [email protected] (D.R.P.); [email protected] (C.M.) Engie Brasil Energia, Saudades do Iguaçu 85568-000, Brazil; [email protected] Correspondence: [email protected]; Tel.: +55-41-3310-4690

Academic Editor: Vittorio M. N. Passaro Received: 4 July 2016; Accepted: 30 August 2016; Published: 7 September 2016

Abstract: This paper presents an image reconstruction method to monitor the temperature distribution of electric generator stators. The main objective is to identify insulation failures that may arise as hotspots in the structure. The method is based on temperature readings of fiber optic distributed sensors (DTS) and a sparse reconstruction algorithm. Thermal images of the structure are formed by appropriately combining atoms of a dictionary of hotspots, which was constructed by finite element simulation with a multi-physical model. Due to difficulties for reproducing insulation faults in real stator structure, experimental tests were performed using a prototype similar to the real structure. The results demonstrate the ability of the proposed method to reconstruct images of hotspots with dimensions down to 15 cm, representing a resolution gain of up to six times when compared to the DTS spatial resolution. In addition, satisfactory results were also obtained to detect hotspots with only 5 cm. The application of the proposed algorithm for thermal imaging of generator stators can contribute to the identification of insulation faults in early stages, thereby avoiding catastrophic damage to the structure. Keywords: generator stator temperature; distributed temperature sensing; sparse reconstruction algorithm

1. Introduction Stator temperature is one of the most influential parameters in the degradation of hydroelectric generators [1]. High temperatures, above 100 ◦ C, can accelerate the wear of the insulation layer of the windings, leading to premature failure and compromising the integrity of the generator [1,2]. Figure 1 shows some examples of faults that can occur due to insulation wear of the stator. Figure 1a presents a failure caused by defects in insulation between the core and the bars, which can cause partial discharges by corona effect due to potential difference between core and bars [3]. Figure 1b shows a failure caused by defects in insulation between bottom and top bars, which can also cause partial discharges due to the phase difference between the bars, increasing risk of short circuit in the stator [4,5]. In both examples, the faults cause areas of high temperature near the defect location. Thus, these faults can be identified as hotspots in certain parts of the structure. However, if the defect is not identified and repaired in the early stages, the hotspots can propagate over the structure, affecting the life expectancy of the insulation layer, leading in extreme cases to a catastrophic failure of the generator [3–5].

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(a)

(b)

Figure1.1.Examples Examplesofofinsulation insulation faults generator stator. (a) Isolation between corebars; and Figure faults in in thethe generator stator. (a) Isolation faultfault between core and bars; (b) Isolation fault between bars. (b) Isolation fault between bars.

Usually, stator temperature measurements are performed through conventional localized Usually, stator temperature measurements are performed through conventional localized sensors sensors such as PT100 (Platinum Thermo-resistance) or RTD (Resistance Temperature Detector). such as PT100 (Platinum Thermo-resistance) or RTD (Resistance Temperature Detector). These sensors These sensors are suitable for monitoring temperature changes during standard operation (no are suitable for monitoring temperature changes during standard operation (no failure), given that the failure), given that the temperature distribution is considered to be rather uniform [6,7]. On the other temperature distribution is considered to be rather uniform [6,7]. On the other hand, they are not able hand, they are not able to identify hotspots that may arise in the stator coils as a result of insulation to identify hotspots that may arise in the stator coils as a result of insulation breakdown. Generally, breakdown. Generally, the stator is instrumented with a few tens of localized sensors, which are not the stator is instrumented with a few tens of localized sensors, which are not sufficient to monitor sufficient to monitor the entire structure containing hundreds of bars. Furthermore, the use of the entire structure containing hundreds of bars. Furthermore, the use of conventional electronic conventional electronic transducers in generators presents additional drawbacks because of their transducers in generators presents additional drawbacks because of their sensitivity to electromagnetic sensitivity to electromagnetic interference [6,7]. These factors motivate the use of other sensing interference [6,7]. These factors motivate the use of other sensing technologies, aiming at monitoring technologies, aiming at monitoring the temperature distribution over the stator structure [8]. the temperature distribution over the stator [8].optical sensor technology (DTS) has great Recent research has shown that the structure distributed Recent research has shown that the distributed optical sensor technology (DTS)[6–10]. has great potential for applications related to monitoring of generator stator temperature DTSpotential systems for applications related to monitoring of generator stator temperature [6–10]. DTS systems measure measure temperatures by means of optical fibers. These optoelectronic devices provide a continuous temperatures means of optical fibers. These devices continuous profile of thebytemperature distribution alongoptoelectronic the fiber cable [11].provide Thus, athe completeprofile stator of the temperature distribution along the fiber cable [11]. Thus, the complete stator instrumentation instrumentation can be carried out using only one optical fiber as a sensor, which is also immune to can be carried outinterference using onlyfrom one optical fiberenvironment as a sensor, inside which the is also immune to electromagnetic electromagnetic the hostile generator [11,12]. The main DTS interference from the hostile environment inside the generator [11,12]. The main DTS technologies are technologies are those based on Raman and Brillouin scattering. Raman DTS systems have become those based Ramanapplication and Brillouin Raman DTSgreat systems havewhen become popularto for practical popular for on practical duescattering. to their low cost and stability compared equipment application due to their low cost and great stability when compared to equipment based on Brillouin based on Brillouin scattering [12]. Typically, commercial Raman DTS equipment is able to provide a scattering [12]. Typically, Raman DTS equipment is able to provide temperature profile alongcommercial an optical fiber with over 30 km length, with accuracya temperature of 0.1 °C andprofile spatial ◦ along an optical fiber with over 30 km length, with accuracy of 0.1 C and spatial resolution of 1 m resolution of 1 m [13]. The spatial resolution is defined as the spatial distance between the 10% and [13]. 90% The spatial resolution as thestep. spatial the 10% and 90% levels of response to levels of response to isa defined temperature In distance general, between for a temperature profile described by a step apulse temperature step. In general, for a temperature profile described by a step pulse with length smaller with length smaller than the spatial resolution, the measured temperature is lower than the real than the spatial the measured temperature lowerresolution than the real temperature by a ratio temperature by resolution, a ratio of temperature step length and is spatial [12,13]. This parameter can of temperature step length and spatial resolution [12,13]. This parameter can be a disadvantage of be a disadvantage of Raman DTS, and sometimes limits its use in certain applications where the Raman and sometimes limits its usedimensions in certain applications where thecase thermal variations occur thermalDTS, variations occur in regions with less than 1 m. In the of stator temperature in regions with dimensions less than 1 m. In the case of stator temperature monitoring, hotspots monitoring, hotspots with dimensions in the order of centimeters are either undetected by the Raman with the order of centimeters are undetected by theinsulation Raman DTS or at measured DTS dimensions or measuredinincorrectly, compromising theeither identification of some faults an early incorrectly, compromising the identification of some insulation faults at an early stage. stage. In studies forfor improvement in spatial resolution of Raman DTS systems have Inthe thelast lastyears, years,several several studies improvement in spatial resolution of Raman DTS systems been presented in the literature. From the hardware viewpoint, methods based on a more efficient use have been presented in the literature. From the hardware viewpoint, methods based on a more of optoelectronic devices have shown significant results regardingresults the DTS spatial resolution efficient use of optoelectronic devices have shown significant regarding the DTS (about spatial 10 cm) [14–17]. However, such techniques often present higher costs, besides other complications that resolution (about 10 cm) [14–17]. However, such techniques often present higher costs, besides other prevent their use in prevent commercial such as increments response time and in in measurement complications that theirequipment, use in commercial equipment,insuch as increments response time uncertainty. Therefore, another alternative that has been investigated is the use of signal processing and in measurement uncertainty. Therefore, another alternative that has been investigated is the use techniques [18–21]. These techniques have shown enhancement in DTS performance without increasing of signal processing techniques [18–21]. These techniques have shown enhancement in DTS equipment costs, as this does not require physical changes in the device. Recently, the method proposed performance without increasing equipment costs, as this does not require physical changes in the by BazzoRecently, et al. [21], based on Total Variation deconvolution, greatVariation potential deconvolution, with regard to device. the method proposed by Bazzo et al. [21], presented based on Total presented great potential with regard to spatial resolution. The results showed that it is possible to

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spatial resolution. The results showed that it is possible to measure accurately temperature variations in lengths as short as 15 cm, and to obtain significant improvements for lengths down to 5 cm. This work proposes an image reconstruction scheme for improving the response of the thermal imaging system for generator stator using Raman DTS. The thermal images are generated by a reconstruction algorithm based on a DTS acquisition model and sparse representation theory. In this representation method, using a dictionary that contains prototype signal atoms, images are described by sparse linear combinations of these atoms. Lately, sparse representations have been successfully applied in many areas of image processing, such as denoising, inpainting and super-resolution [22,23]. To monitor the temperature distribution of the stator, we employ a sparse representation because the system can be considered a large structure at a uniform temperature, with occasional hotspots spread out in case of insulation failure (see Figure 1). The DTS readings can be viewed as degraded observation (blurred and downsampled) of the temperature distribution on stator surface. This distribution is assumed to have a sparse representation with respect to a dictionary of hotspots, which is built based on physical properties of the structure [23]. The principle of the image reconstruction ensures that under mild conditions, the sparse representation can be correctly recovered from the degraded observation (sensor readings) [23]. Due to the difficulty for reproducing insulation faults in real stator structure, the experimental tests were performed using a prototype to generate the hotspots in a similar structure. The proposed technique permits a more precise monitoring of the stator temperature distribution, facilitating the identification of insulation faults at an early stage, and preventing further damage to the generator. This paper is organized into seven sections: Section 2 presents an overview of a thermal imaging system for generator stator, with details of real stator structure and stator prototype used in experimental tests. The details of DTS acquisition model are presented in Section 3. The dictionary of hotspots used to generate thermal images is presented in Section 4. The details of the image reconstruction algorithm are presented in Section 5, and the results are shown in Section 6. Finally, Section 7 presents the main conclusions on the results obtained. 2. Overview of Thermal Imaging System In a previous work [6], Bazzo et al. presented a thermal imaging system for stator using DTS that was tested in a 200 MW hydroelectric generator. The main details of the structure and DTS installation on the stator surface are shown in Figure 2. As can be seen, the structure is basically composed of stacked 5 cm high bars with air gaps of 1 cm through which cooling air flow generated by the rotor circulates. The winding bars are installed in vertical slots spaced by approximately 10 cm. A distributed sensor based on fiber optics (DTS) was positioned on each slot that accommodates the winding bars, as the bars are the main source of heat structure [24,25]. Although this system has shown satisfactory results, the limitations of DTS spatial resolution impede the identification of hotspots with dimensions of less than 1 m. A similar work presented by Hudson et al. [7] also reports the need for a DTS equipment with spatial resolution about 10 cm for a more accurate thermal mapping of a generator stator. This motivates the development of an image reconstruction algorithm to improve system response, and to enable the identification of hotspots with dimensions at the order of centimeters. To evaluate the performance of the proposed method, we developed a prototype with similar structure of the stator surface, as shown in Figure 3. The tests on the prototype were necessary since it is not possible to generate insulation faults in the generator stator. Moreover, it is noteworthy that the generator was in perfect condition and in full operation at the power plant. The stator prototype was assembled with 35 aluminum plates with dimensions 200 × 5 × 1.5 cm, stacked with air gap of 1 cm, similar to the stator core plates (Figure 2). Each plate has holes spaced at 10 cm similarly to the stator slots, and the stator bars were represented by resistances that can be embedded into the holes. Resistances with different lengths were used, and these were driven by a Proportional Integral Derivative (PID) controller. Thus, it was possible to simulate hotspots with dimensions 5 to 209 cm, in a similar structure of stator surface. The optical fiber used as distributed sensor was installed

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the same way as the real stator structure. As the fiber must be positioned on the main heat sources of the structure, which must be previously known, the maximum lateral displacement between the fiber loops should be 10 cm, which is a critical system parameter to ensure the resolution of thermal Sensors 2016, 16, 1425 this structure is simple compared with the real stator structure, it reproduces 4 of 16the images. Although contact surface and the position where the sensor was installed in the real generator stator. Tests in Sensors 2016, 16, 1425 4 of fiber 16 structure, which must be previously known, the maximum lateral displacement between the ® the laboratory also allowed the use of a thermal camera Fluke Ti25 (Fluke Corporation, Everett, loops should be 10 cm, which is a critical system parameter to ensure the resolution of thermal MA, structure, USA) as awhich comparison the thermal imageslateral generated by the image reconstruction must be reference previouslyto known, the maximum displacement between the fiber images. Although this structure is simple compared with the real stator structure, it reproduces the loops should be 10 cm, which is a critical system parameter to ensure the resolution of thermal algorithm. In the real stator structure, this would not be possible, as there is not enough space to install contact surface and the position where the sensor was installed in the real generator stator. Tests in images. Although this structure is simple compared with the real stator structure, it reproduces the a camera after the rotor engagement. thecontact laboratory alsoand allowed the usewhere of a thermal camera Fluke® Ti25 (Fluke Corporation, Everett, MA, surface the position the sensor was installed in the real generator stator. Tests in Theasproposed imagereference reconstruction algorithm was based on abyDTS acquisition model and USA) a comparison to the thermal images generated the image reconstruction the laboratory also allowed the use of a thermal camera Fluke® Ti25 (Fluke Corporation, Everett, MA, a dictionary of hotspots to generate thermal images. 3 presents more details about sensor algorithm. the real stator structure, thisthermal would not Section be possible, not enough space to USA) as In a comparison reference to the images generatedasbythere the is image reconstruction model development. install a camera after the rotor engagement. algorithm. In the real stator structure, this would not be possible, as there is not enough space to install a camera after the rotor engagement.

Figure 2. Real structure of the 200 MW hydroelectric generator stator, and fiber optic installation for Figure 2. Real structure of of thethe 200 generatorstator, stator, and fiber optic installation Figure 2. Real structure 200MW MWhydroelectric hydroelectric generator and fiber optic installation for for the thermal imaging system using DTS. the thermal imaging system using DTS. the thermal imaging system using DTS.

Figure 3.Stator Stator prototypedeveloped developed to to simulate simulate insulation and evaluate thethe imaging Figure 3. 3. Stator prototype to simulate insulationfaults faults and evaluate the imaging Figure prototype developed insulation faults and evaluate imaging reconstruction algorithm performance. reconstruction algorithmperformance. performance. reconstruction algorithm

The proposed image reconstruction algorithm was based on a DTS acquisition model and a

The proposed image reconstruction algorithm was based on a DTS acquisition model and a 3. Distributed Sensing (DTS) Acquisition dictionary ofTemperature hotspots to generate thermal images. Section 3Model presents more details about sensor model dictionary of hotspots to generate thermal images. Section 3 presents more details about sensor model development. In a previous work [21], we showed that a linear model is suitable to represent the DTS response development. if one aims to reconstruct hot steps. In this work, we employ the same acquisition model, which 3. Distributed Temperature Sensing (DTS) Acquisition Model 3. obtained Distributed Sensing (DTS) Acquisition Model was byTemperature linear system identification techniques [26]. The input f (z) is the real temperature In a previous work [21], we showed that a linear model is suitable represent the DTS profile In and the output g z is the DTS temperature readings. As we to are considering theresponse steady state, ( ) previous work [21], we showed that a linear model is suitable to represent the DTS response if oneaaims to reconstruct hot steps. In this work, we employ the same acquisition model, which was i.e.,if no time variations, the only independent variable is z which represents the distance (cm) along one aims to hotidentification steps. In this techniques work, we employ theinput same acquisition model, which wasthe ( ) is the real obtained byreconstruct linear system [26]. The temperature optical fiberby sensor. The response g (z) is obtained by[26]. the convolution DTSreal impulse response (of)the obtained linear system techniques The input is the temperature ( )identification profile and the output is the DTS temperature readings. As we are considering the steady state, h (profile z) i.e., withno input f z , as shown in Equation (1) [26]: ( ) ) is and thevariations, output (the theindependent DTS temperature readings. As represents we are considering the(cm) steady state, time only variable is which the distance along i.e.,the nooptical time variations, the The onlyresponse independent is by which represents the along ( ) variable fiber sensor. is obtained the convolution of distance the DTS (cm) impulse g ((zin))Equation =ishobtained f ([26]: z)by, the convolution of the DTS impulse(1) (z) ∗(1) ), as shown theresponse optical fiber sensor. The( response ℎ( ) with input response ℎ( ) with input ( ), as shown (1) [26]: ( in ) =Equation ℎ( ) ∗ ( ), (1) by applying the Laplace transform we obtain Equation (2): ( ) = ℎ( ) ∗ ( ), (1) by applying the Laplace transform we obtain Equation (2):

by applying the Laplace transform we obtain Equation (2):

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G (s) = H (s) F (s) . (2) ( ) = ( ) ( ). (2) The system identification consists in estimating the poles and zeros of a transfer function H (s), The system identification consists in estimating the poles and zeros of a transfer function ( ), as shown in Equation (3) [26]: as shown in Equation (3) [26]: Sensors 2016, 16, 1425

5 of 16 m m−1++….+ ∏ ∏(m −(s − + ) βi ) . . + b= m i =1 ( ()s)== b0 s + b1 s , (3) H = , (3) n + a sn−+ 1 …+ ∏ ∏(n −(s − ) αi ) s+ 1 ( ) =+ (. .). + i =1 ( ).an (2) where βi are the zeros and αi are the poles. consists in estimating the poles and zeros of a transfer function ( ), where βi areThe thesystem zeros identification and αi are the poles. The DTS equipment used in this work was an AP Sensing®® N4385B (AP Sensing GmbH, as shown in Equation (3) [26]: The DTS equipment used in this work was an AP Sensing N4385B (AP Sensing GmbH, Böblingen, Germany) model. This model features a spatial∏resolution of 1 m, acquisition time of 30 s, (resolution − ) + + … +a spatial Böblingen, Germany) model.( )This model features of 1 m, acquisition time of = = , sample interval down to 15 cm, and temperature 0.04) °C for fibers of up to (3) 2 km. To ∏ of ( − + + … +resolution 30 s, sample interval down to 15 cm, and temperature resolution of 0.04 ◦ C for fibers of up to evaluatewhere the βequipment response, an experimental test was carried out in a thermal bath i are the zeros and αi are the poles. 2 km. To® evaluate the equipment response, an experimental test®was carried out in a thermal bath LAUDA ®ECO model, used with in stabilized at 50 °C, providing hotspots of different TheRE415G DTS equipment this worktemperature was an AP Sensing N4385B (AP Sensing GmbH, ◦ C, LAUDA ECO RE415G model, with stabilized temperature at providing of different ( )50and ( hotspots ) were Germany) model. This model a spatial resolution of output 1 m, acquisition time of 30 s, lengths.Böblingen, The ambient temperature was 21.7features °C. The input obtained for ◦ C. The input f ( z ) and output g ( z ) were obtained for lengths.sample The ambient temperature was 21.7 interval down to 15from cm, and temperature resolution of 0.04 °C foras fibers of upintoFigure 2 km. To hotspots at 50 °C with lengths 5 cm to 4 m with intervals of 5 cm, shown 4. As can hotspots at 50 ◦ Cthe with lengths response, from 5 cmantoexperimental 4 m with intervals ofcarried 5 cm, as in Figure 4. As can outshown in a 1thermal be seen,evaluate for hotspotsequipment of 5 cm the measured temperaturetest waswas only 24.7 °C. From m andbath above, the ◦ ® ECO RE415G model, stabilized temperature at 50 °C, providing hotspots be seen,LAUDA for is hotspots of 5correctly cm thewith measured temperature was only 24.7 C. Fromof1different m and above, temperature measured (50 °C), ◦confirming the spatial resolution specification of the DTS ) and ( ) werespecification lengths. The ambient temperature was 21.7C), °C.confirming The input (the output obtained for of the the temperature is measured correctly (50 spatial resolution equipment. hotspots at 50 °C with lengths from 5 cm to 4 m with intervals of 5 cm, as shown in Figure 4. As can DTS equipment.

be seen, for hotspots of 5 cm the measured temperature was only 24.7 °C. From 1 m and above, the temperature is measured correctly (50 °C), confirming the spatial resolution specification of the DTS equipment.

Figure 4. Experimental results used to identify the DTS system model, where the hotspots were at Figure 4. Experimental results used to identify the DTS system model, where the hotspots were at 50 °C 50 ◦ C with lengths from 5 cm to 4used m, with intervals of system 5 cm. model, where the hotspots were at 50 °C Figure 4. Experimental results to identify the DTS with lengths from 5 cm to 4 m, with intervals of 5 cm. with lengths from 5 cm to 4 m, with intervals of 5 cm.

We employed employed the the prediction prediction error error minimization minimization (PEM) (PEM) approach approach to to estimate estimate the the transfer transfer function function We We employed the prediction error minimization (PEM) approach to estimate the transfer function coefficients. In this case, from initial estimates, the parameters are updated using a nonlinear least-squares coefficients. In thisIncase, fromfrom initial estimates, updated using a nonlinear coefficients. this case, initial estimates,the the parameters parameters areare updated using a nonlinear least- leastsearch method, where the objective isobjective to minimize the weighted prediction error norm [26]. As a result, squares method, where minimize the prediction error norm [26]. As[26]. squares search search method, where the the objective isistotominimize theweighted weighted prediction error norm As a result, we obtained a transfer function with nine poles and four zeros (determined empirically) and we obtained a transfer function with nine poles and four zeros (determined empirically) and 98% a result, we obtained a transfer function with nine poles and four zeros (determined empirically) and 98% accuracy. The comparison between experimental modelsimulation simulation isisshown in Figure 5. 5. accuracy. The comparison between experimental datadata andand model shown in Figure 98% accuracy. The comparison between experimental data and model simulation is shown in Figure 5.

Figure 5. Experimentalresults results vs. model. Figure 5. Experimental vs.DTS DTSacquisition acquisition model.

Figure 5. Experimental results vs. DTS acquisition model.

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Sensors 2016,the 16, 1425 Taking inverse Laplace transform of the transfer function, we get the impulse response6 of of16the system h z , presented in Figure 6. The DTS impulse response h z is used to assemble a sensitivity ( ) ( ) Sensors 2016, 16, 1425 6 of 16 Taking the inverse Laplace transform of the transfer function, we get the impulse response of matrix H, which represents the DTS acquisition model. The matrix-vector notation of H is presented the system ℎ( ), presented in Figure 6. The DTS impulse response ℎ( )the is impulse used to response assembleofa Taking of the transfer function, we get in Equation (4):the inverse Laplace transform  matrix-vector notation of H is sensitivity matrix H, which represents the DTS acquisition model. The the system ℎ( ), presented in Figure impulseh response h (z06.) The (z0−k ) ℎ( ) is used to assemble a h (zDTS 0−1 ) · · · presented in Equation (4): represents  sensitivity matrix H, which the DTS acquisition model. . .. The  matrix-vector notation of H is .. . ℎ( ) ℎ( h (z)0 )⋯ ℎ(· · · )  presented in Equation (4): H =  (4)  . ⋯ . .⋮ ..  = .. ⋮ ℎ( )... ⋯  . . (4) .  ℎ(. ⋮ ) ℎ( ⋮ ) ⋱ ℎ( ⋮ )  ℎ( h (z)k−) 1⋯ · ·⋮ · ) . h (z0 ) ) ℎ( ⋯ (z⋮k )) ℎ( =h ℎ( (4)

⋮ ⋮ ℎ( ) ℎ(

⋱ ⋮ ) ⋯ ℎ( )

Figure6.6.Experimental Experimentalresults results vs. vs. DTS Figure DTS acquisition acquisitionmodel. model. Figure 6. Experimental results vs. DTS acquisition model.

Although the matrix H has proven suitable for representing the sensor acquisition, the DTS model Although the matrix H has proven suitable for representing the sensor acquisition, the DTS model contains errorsthe and is itself a source of noise, as representing can be seen in Figure 5. To develop an efficient Although matrix H has provenof suitable the DTS contains errors and is itself a source noise, for as can be seen the in sensor Figureacquisition, 5. To develop an model efficient reconstruction algorithm, a statistical analysis of the noise is fundamental to set the norm in efficient the data contains errors and is itself a source of noise, as can be seen in Figure 5. To develop an reconstruction algorithm, a statistical analysis of the noise is fundamental to set the norm in the data term of the costalgorithm, function [22,27]. Thus,analysis an analysis of noise the residuals was performed bynorm the histogram greconstruction a statistical of the is fundamental to set the in histogram the data term of the cost function [22,27]. Thus, an analysis of the residuals was performed by the Hf, where is afunction vector formed thean sensor dataof(DTS readings), and f is a vector representing term of the gcost [22,27]. by Thus, analysis residuals was performed the histogramthe gg-Hf, where g is a vector formed by the sensor datathe (DTS readings), andthe f ishistogram aby vector representing temperature profile. Figure 7 summarizes the analysis results. Although presents Hf, where g is a vector formed by the sensor data (DTS readings), and f is a vector representing thea theslight temperature profile. Figure 7 summarizes theindicating analysis results. Although the histogram presents skew toward residual values, thereby largeAlthough model errors, misfit ispresents relatively temperature profile.high Figure 7 summarizes the analysis results. the this histogram a a slight toward highThis residual values, thereby indicating large model errors, this misfit is relatively rare skew (see height). behavior expected when large adopting linear models (for istractability slight skewbar toward high residual values,isthereby indicating model errors, this misfit relatively rarepurposes) (see bar height). This behavior physics is expected when adopting linear models (for tractability purposes) where the underlying is potentially nonlinear. accommodate this inaccuracy, rare (see bar height). This behavior is expected when adoptingTolinear models (for tractability where the underlying physics is potentially nonlinear. To accommodate this inaccuracy, we performed we performed thethe following statistical analysis, similarly to [28]. Assuming a generalized Gaussian purposes) where underlying physics is potentially nonlinear. To accommodate this inaccuracy, thewe following statistical analysis, similarly to [28]. Assuming a generalized Gaussian distribution, distribution, we obtained a shape parameter ≈ 1 using the method described in [27], indicating that performed the following statistical analysis, similarly to [28]. Assuming a generalized Gaussian the residuals have a Laplacian distribution. This information will be further exploited in Section 5. wedistribution, obtained a shape parameter p ≈ 1 using the method described in [27], indicating that the residuals we obtained a shape parameter ≈ 1 using the method described in [27], indicating that have Laplacian distribution. information will be further Sectionin 5. Section 5. thearesiduals have a LaplacianThis distribution. This information willexploited be furtherin exploited

Figure 7. Histogram of the residuals for the DTS model and a Laplacian curve fit.

Figure the DTS DTS model modeland andaaLaplacian Laplaciancurve curve Figure7.7.Histogram Histogramof ofthe theresiduals residuals for the fit.fit.

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4. Dictionary of Hotspots 4. Dictionary of Hotspots We assume that the thermal system for generator stator can be modeled through a sparse We assume the thermal generatora stator can be modeled throughdistribution a sparse representation. Thethat reason is that itsystem can befor considered large structure with uniform representation. The reason is that it can be considered a large structure with uniform distribution temperature, where occasional hotspots may arise in case of insulation failure (Figure 1). Using temperature, where arise in image case ofatoms insulation (Figure Using a a dictionary matrix D ∈occasional Rm×n thathotspots containsmay n prototype in thefailure columns { di } n1). j=1 , a thermal × m dictionary ∈ℝ that n prototype imageofatoms in the columns , a in image f ∈ R matrix can be represented by contains sparse linear combinations the dictionary atoms, as shown can be represented by sparse linear combinations of the dictionary atoms, as thermal image ∈ ℝ Equation (5) [29]: shown in Equation (6) [29]: f = Dα, (5) = , Each atom is the thermal image of the whole (6) where the vector α ∈ Rn contains the coefficients. structure when only one is active at atom a time. In this representation α iswhole sparse, where the vector ∈ ℝ possible containsheat thesource coefficients. Each is the thermal image of the when only one possible heatzeros. sourceThe is active at a time. In of this is sparse, i.e., or i.e.,structure it is assumed to contain mostly representation f representation may either beαexact f = Dα it is assumed contain mostly zeros. (6) The[29]: representation of f may either be exact f = Dα or approximate f ≈ to Dα, satisfying Equation approximate f ≈ Dα, satisfying Equation (7) [29]: k f − Dα k ≤ , (6) ‖ − ‖ ≤pϵ, (7) where is the minimumresidual residualerror errordesired, desired, and is the norm thethe deviation. where  is the minimum and p is normused usedfor formeasuring measuring deviation. Figure 8 shows exampleofofthe thesparse sparserepresentation representation of a dictionary. In In thisthis Figure 8 shows ananexample of an animaging imagingsystem systemusing using a dictionary. example, a combinationof ofthree three atoms atoms of used to to form thethe image f [29,30]. example, a combination of the thedictionary dictionaryDDwas was used form image f [29,30].

Figure8.8.Example Exampleof ofthe thesparse sparse representation representation of a dictionary. Figure ofan animaging imagingsystem systemusing using a dictionary.

We built a dictionary of hotspots through simulation using the COMSOL® (Comsol, We built a dictionary of hotspots through simulation using the COMSOL® (Comsol, Stockholm, Stockholm, Sweden) multiphysics tool. In the simulations, we considered the physical properties Sweden) multiphysics tool. In the simulations, we considered the physical properties of the materials, of the materials, geometry and environment boundary conditions. Each atom was generated geometry and environment boundary conditions. Each atom was generated considering the position considering the position of the resistances in each plate. As the prototype has 35 plates at 5 cm high, of the resistances in each plate. As the prototype has 35 plates at 5 cm high, and 19 positions for and 19 positions for resistance, considering 1 atom for each 1 cm, we generated 3325 atoms (35 × 5 resistance, considering 1 atom for each 1 cm, we generated 3325 atoms (35 × 5 × 19) to model the × 19) to model the system. Although this representation of 1 cm results in a large number of atoms, system. this representation of 1with cm results in a large number of atoms, it was alignment necessary to it was Although necessary to reconstruct hotspots more accurate dimensions, also preventing reconstruct hotspots with more accurate dimensions, also preventing alignment problems of the sensor problems of the sensor installation. Since the total size of the image 209 × 200 cm, the vector length installation. Since the total size ofa the image 209 × 200 the ×vector of each atom is 41,800, forming dictionary matrix D ofcm, 41,800 3325. length of each atom is 41,800, formingThe a dictionary D of 41,800 × 3325. columns ofmatrix the dictionary matrix, or atoms, were formed by the temperature distribution 3 applied The columns dictionary matrix, or atoms, were formed temperature values generatedofbythe power of 1 W/cm to a resistance of 5 by cmthe in each position distribution along the 3 values generated by steady powerstate. of 1 We W/cm to aofresistance of 5 cm in each position along plates, considering set theapplied resistance 5 cm as a minimum condition, because of thethe plate considering dimensions steady and difficulty in using resistances length. However, the plates, state. We set thetubular resistance of 5 cmwith as a lower minimum condition, because data is sampled every 1 cminforusing forming atoms, as explained in the previous paragraph. of temperature the plate dimensions and difficulty tubular resistances with lower length. However, the vector α isrepresents the values of the thermal power that generate the hotspots in the theThus, temperature data sampled every 1 cm for forming atoms, as explained in the previous paragraph. structure. Therefore, by estimating α, we obtain the location andgenerate the amount of thermal that Thus, the vector α represents the values of the thermal power that the hotspots inpower the structure. causes each hotspot. Figure 9a shows details of the mesh geometry used in the simulations, and Therefore, by estimating α, we obtain the location and the amount of thermal power that causes each 3 at 26 °C ambient Figure 9b shows the temperature distribution for total power of 1 W/cm hotspot. Figure 9a shows details of the mesh geometry used in the simulations, and Figure 9b shows indistribution steady state. for Thetotal results showof a variation decreasing temperature of ◦ C with thetemperature, temperature power 1 W/cm3ofat2.62 26 °C ambient temperature, in steady / , where is the distance from the heat source. This decreasing function is−ax/45 state. The results show a variation of 2.62 ◦ C with decreasing temperature of e parametric , where xfitisofthe the numerical simulations, which was used to ease the construction of the dictionary. The application distance from the heat source. This decreasing function is a parametric fit of the numerical simulations, of the dictionary D in the image reconstruction algorithm is discussed in more detail in Section 5. which was used to ease the construction of the dictionary. The application of the dictionary D in the image reconstruction algorithm is discussed in more detail in Section 5.

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(a)

(b)

Figure 9. Multiphysics simulation of stator prototype. (a) Mesh geometry used in the simulations. (b)

Figure 9. Multiphysics simulation of stator prototype. 3(a) Mesh geometry used in the simulations; Temperature distribution for total power of 1 W/cm applied to resistance at 26 °C ambient (b) Temperature distribution for total power of 1 W/cm3 applied to resistance at 26 ◦ C ambient temperature. temperature.

5. Imaging Reconstruction Algorithm 5. Imaging Reconstruction Algorithm considering basic modelofofimage image reconstruction reconstruction theory, thethe acquisition system can be First,First, considering thethe basic model theory, acquisition system can be represented by Equation [22]: represented by Equation (7) (8) [22]: g== Hf + +, n, (8) (7) is a vector formed DTSreadings, readings, H is the matrix, representing wherewhere g is a vector formed byby DTS thesensitivity sensitivity matrix, isf aisvector a vector representing the temperature distribution statorsurface, surface, and and n all all sources of additive the temperature distribution inin stator n is is aavector vectorrepresenting representing sources of additive noise. Considering that the thermal image has a sparse representation in the constructed dictionary, noise. Considering that the thermal image has a sparse representation in the constructed dictionary, the acquisition system can be rewritten substituting Equation (6) into Equation (8), as shown in the acquisition system can be rewritten substituting Equation (5) into Equation (7), as shown in Equation (9) [22]: Equation (8) [22]: + + . n. (9) (8) g== HDα As shown in Figure 6, the DTS “spreads” the impulse, which is a characteristic of low-pass

As shown in Figure 6, the DTS “spreads” the impulse, which is a characteristic of low-pass systems. This is translated into an ill-conditioning of matrix H. Thus, the recovering of the systems. This is translated into an ill-conditioning of matrix H. Thus, the recovering of the temperature temperature distribution by simple inversion of Equation (9) yields high noise amplification, distribution bypoor simple inversion Equation (8) yields high noise amplification, generating poor generating results [21]. Thisof problem requires regularization, which stabilizes the reconstruction results [21]. This problem requires regularization, which stabilizes the reconstruction improves and improves the results. Although the dictionary has been built with the hotspots of 1 cm toand improve the results. Although the dictionary has been built with the hotspots of 1 cm to improve representation, representation, we expected them to occur with 5 cm or more (see Figure 9), because of the minimal we expected to occur with 5 cm morestate, (see as Figure 9), because of the minimal the condition conditionthem of structure prototype andor steady explained in Section 4. Therefore, most of appropriate regularization is Total as it privileges piece-wise constantthe signals. the structure prototype and steady state,Variation, as explained in Section 4. Therefore, mostThus, appropriate cost function to solve the inverse problem is given by Equation (10) [21,22]: regularization is Total Variation, as it privileges piece-wise constant signals. Thus, the cost function to solve the inverse problem is given = byargmin Equation ‖g − (9) [21,22]: ‖ + ‖ ‖ , (10) p

ˆ values = argmin g − HDα k p that +λ generate k Qα k1the , hotspots in the structure, (9) where is a vector with theα of thekthermal power α is the norm used in the data-fidelity term, λ is the regularization parameter which controls the of the solution to the noise, Q is apower finite difference matrix. approximation ˆ is a vector wheresensitivity α with the values of theand thermal that generate theInhotspots in the methods, structure, p is typical norms used for measuring the deviation are the -norms for = 1, 2 and ∞. It is common to the norm used in the data-fidelity term, λ is the regularization parameter which controls the sensitivity use the L2 norm in the data term because the noise is usually well represented by a normal of the solution to the noise, and Q is a finite difference matrix. In approximation methods, typical distribution [27,29]. However, according to the statistical analysis presented in Section 3, the DTS norms used for model measuring theresidual deviation arewith the Laplacian L p -normsbehavior, for p = 1,which 2 andindicates ∞. It is that common to use the acquisition contains errors the L2 norm L2 norm in the termbybecause the noise well represented normal [27,29]. should be data replaced an L1 norm, i.e., pis=usually 1 [27]. To solve Equation by (10)a with p = distribution 1, we used the However, according to the statistical analysis presented in Section 3, the DTS acquisition model contains Interactive Reweighted Least Squares (IRLS) approach. This method consists of approximating the residual with indicates thatthe thesolution L2 norm by an L1 costerrors function byLaplacian weighted behavior, quadratic which L2 norms, updating byshould solvingbea replaced least squares reiterating two steps untilp some stop criterion is attained, usually defined a norm,problem i.e., p = and 1 [27]. To solvethose Equation (9) with = 1, we used the Interactive Reweighted LeastbySquares ® (R2014a, MathWorks, Natick, MA, minimum update [31]. The implementation details in Matlab (IRLS) approach. Thisrate method consists of approximating the cost function by weighted quadratic L2 USA) are shown Algorithm 1. norms, updating the in solution by solving a least squares problem and reiterating those two steps until some stop criterion is attained, usually defined by a minimum update rate [31]. The implementation details in Matlab® (R2014a, MathWorks, Natick, MA, USA) are shown in Algorithm 1.

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Algorithm 1: Image Reconstruction Algorithm Require: Require: Require: Require: Require: Require:

g% DTS readings, H% sensitivity matrix, D% Dictionary of hotspots, Q% finite difference matrix λ % Regulariztion parameter (set empirically) e = 10–9 % avoids zero division HD = H × D % impulse response of the dictionary ˆ = HD 0 × g % intial solution α e % minimum update to stop

1: while stop > e 2: 3: 4: 5: 6:

ˆ0 = α ˆ α ˆ + e)) % data term weights Wh = diag(1 ./ (abs(g − HD × α) ˆ + e)) % penalization term weights Wl = diag(1 ./ (abs(Q × α) ˆ (HD 0 × Wh × HD + λ × (Q0 × Wl × Q))\(HD 0 × Wh × g) % least-squares α= ˆ −α ˆ 0 )/norm(α ˆ 0 ) % stopping criterion stop = norm(α

7: end while 8:

ˆ f = reshape(D × α,209,200) % thermal image

The proposed image reconstruction algorithm was evaluated with simulated data to assess the robustness to different noise levels, and with experimental data through the stator prototype (Figure 3). The results are shown in Section 6. 6. Results This section is organized into two subsections: Subsection 6.1 presents the results obtained simulating the response g by sensitivity matrix H for a given hotspot Dα, in order to assess the algorithm robustness applying different noise levels; Subsection 6.2 shows the results obtained by the experimental tests with the stator prototype, using resistances to emulate heat sources of different lengths. 6.1. Simulated Results The evaluation of the algorithm performance with respect to noise level was conducted simulating a hotspot with the dictionary D. The simulated hotspot covered a region of three plates, with a maximum temperature of 80 ◦ C and ambient temperature of 26 ◦ C, which represents a length of approximately 15 cm, considering the fiber installation (Figure 3). This length was chosen based on spatial resolution achieved with the Total Variation deconvolution method proposed in [21]. The simulated hotspot image f is shown in Figure 10. Based on the image of Figure 10, the response g was obtained using the sensitivity matrix H of the DTS acquisition model presented in Section 3. Thus, the algorithm performance was evaluated by adding white Gaussian noise at different levels to the simulated DTS readings. The images reconstructed for each noise level were compared with the ground-truth hotspot images. We employed the mean square error (MSE) and the maximum temperature difference (MTD) as figures of merit. The reconstructed images are shown in Figure 11, and a brief analysis of results is presented in Table 1. In the first test, Figure 11a, no noise was added to the simulated DTS readings, and the reconstructed image presented reduced errors relative to simulated hotspot, with only 0.1 ◦ C uncertainty. In tests adding a certain noise level (Figure 11b–f), it can be seen that is possible to obtain, from data with SNR up to 40 dB, acceptable images for application in the stator, with less than 4 ◦ C uncertainty for hotspots of 15 cm. However, from data with SNR between 30 dB and 10 dB (Figure 11d–f), besides large errors (up to −23.7 ◦ C) in the temperature estimation, other parameters

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such as location and dimension become affected. As can be observed, the reconstructed hotspot 10 of 16 covered four plates instead of threee plates of the original hotspot.

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Figure 10. Simulated hotspot image on three plates, where the ambient temperature was 26 °C, the maximum temperature was 80 °C and the length approximately 15 cm.

The reconstructed images are shown in Figure 11, and a brief analysis of results is presented in Table 1. In the first test, Figure 11a, no noise was added to the simulated DTS readings, and the reconstructed image presented reduced errors relative to simulated hotspot, with only 0.1 °C uncertainty. In tests adding a certain noise level (Figure 11b–f), it can be seen that is possible to obtain, from data with SNR up to 40 dB, acceptable images for application in the stator, with less than 4 °C uncertainty for hotspots of 15 cm. However, from data with SNR between 30 dB and 10 dB (Figure 11d–f), besides large errors (up to −23.7 °C) in the temperature estimation, other parameters ◦ C, Figure 10. Simulated Simulated hotspotimage image three plates, where ambient temperature 26 hotspot suchFigure as location and dimension become affected. As canthe bethe observed, the reconstructed 10. hotspot onon three plates, where ambient temperature waswas 26 °C, the ◦ the maximum temperature 80and Cplates and the approximately 15 cm. covered four plates instead ofwas threee of length theapproximately original hotspot. maximum temperature was 80 °C the length 15 cm. The reconstructed images are shown in Figure 11, and a brief analysis of results is presented in Table 1. In the first test, Figure 11a, no noise was added to the simulated DTS readings, and the reconstructed image presented reduced errors relative to simulated hotspot, with only 0.1 °C uncertainty. In tests adding a certain noise level (Figure 11b–f), it can be seen that is possible to (b) in the stator, with less obtain, from data with (a) SNR up to 40 dB, acceptable images for application than 4 °C uncertainty for hotspots of 15 cm. However, from data with SNR between 30 dB and 10 dB (Figure 11d–f), besides large errors (up to −23.7 °C) in the temperature estimation, other parameters such as location and dimension become affected. As can be observed, the reconstructed hotspot covered four plates instead (c) of threee plates of the original hotspot. (d)

(a) (e)

(f) (b)

Figure 11. plates, where thethe ambient temperature waswas 26 °C, ◦ C, Figure 11. Simulated Simulatedhotspot hotspotimage imageononthree three plates, where ambient temperature 26 the maximum temperature was 80 °C and the length approximately 15 cm. (a) Reconstructed hotspot ◦ the maximum temperature was 80 C and the length approximately 15 cm. (a) Reconstructed hotspot without adding hotspot with SNR 50 dB; (c) Reconstructed hotspot with with SNR without addingnoise; noise;(b) (b)Reconstructed Reconstructed hotspot with SNR 50 dB; (c) Reconstructed hotspot 40 dB; (d) Reconstructed hotspot with SNR 30 dB; (e) Reconstructed hotspot with SNR 20 dB; (f) SNR 40 dB; (d) Reconstructed hotspot with SNR 30 dB; (e) Reconstructed hotspot with SNR 20 dB; (c) (d) Reconstructed hotspot with SNR 10 dB. (f) Reconstructed hotspot with SNR 10 dB.

Another numerical analysis was performed to assess the minimum length Table 1. Algorithm performance with respect to the noise level.of a detectable hotspot given a maximum acceptable temperature difference of ±1 °C and different SNR levels. Table 2 summarizes the (SNR) results, where it can SQUARE be seen ERROR that the length for the givenDifference) conditions was Noise Level MSE (MEAN ) minimum MTD (Maximum Temperature (f) (e) 15 cm with SNR 50 dB. For other SNR levels, 40 dB and 30 dB, the performance Noiseless 0.5657 0.1 ◦ C was affected and the minimum length increased to 22 cm and 27 cm, respectively. Subsection 6.2◦ Cpresents the°C,results 50 dB 2.0921 − 0.5 Figure 11. Simulated hotspot image on three plates, where the ambient temperature was 26 the of ◦C 40 dB 6.9929 − 3.6 the experimental tests’ performance using the stator prototype. maximum temperature was 80 °C and the length approximately 15 cm. (a) Reconstructed hotspot 30 dB 9.3758 −12.1 ◦ C without adding noise; (b) Reconstructed hotspot with SNR 50 dB; (c) Reconstructed hotspot with SNR 20 dB 16.595 −20.9 ◦ C 40 dB; 10 (d)dB Reconstructed hotspot 21.8149 with SNR 30 dB; (e) Reconstructed hotspot with −23.7 ◦ C SNR 20 dB; (f) Reconstructed hotspot with SNR 10 dB.

Another numerical the minimum minimum length length of of aa detectable detectable hotspot hotspot Another numerical analysis analysis was was performed performed to to assess assess the ◦ given aa maximum C and and different different SNR Table 22 given maximum acceptable acceptable temperature temperature difference difference of of ± ±11 °C SNR levels. levels. Table summarizes the results, where it can be seen that the minimum length for the given conditions was 15 cm with SNR 50 dB. For other SNR levels, 40 dB and 30 dB, the performance was affected and the minimum length increased to 22 cm and 27 cm, respectively. Subsection 6.2 presents the results of the experimental tests’ performance using the stator prototype.

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summarizes the results, where it can be seen that the minimum length for the given conditions was 15 cm with SNR 50 dB. For other SNR levels, 40 dB and 30 dB, the performance was affected and the minimum length increased to 22 cm and 27 cm, respectively. Subsection 6.2 presents the results of the experimental tests’ performance using the stator prototype. Table 2. Algorithm performance with respect to the hotspot length and SNR level. SNR

Hotspot Length

MTD (Maximum Temperature Difference)

50 dB 50 dB 50 dB 40 dB 40 dB 40 dB 30 dB 30 dB 30 dB

15 cm 14 cm 13 cm 22 cm 21 cm 20 cm 27 cm 26 cm 25 cm

−0.5 ◦ C −3.7 ◦ C −5.1 ◦ C −0.9 ◦ C −1.6 ◦ C −2.4 ◦ C −0.7 ◦ C −1.8 ◦ C −2.6 ◦ C

6.2. Experimental Results The experimental results were obtained by tests with the stator prototype (Figure 3) and a thermal camera Fluke® Ti25 (Fluke Corporation, Everett, MA, USA), which was used as reference for the reconstructed images. In addition to the comparison with the thermal camera, the images generated by the sparse reconstruction algorithm were also compared with the linear interpolation method used in the thermal imaging system presented in [6]. Three resistances with different lengths were used as heat sources, which produced hotspots with approximately 100 cm, 15 cm and 5 cm. The test results for a hotspot of 100 cm is shown in Figure 12. In this test, the ambient temperature was 27.8 ◦ C; the total power applied to the resistance was 680 W (∼ =12 W/cm3 ), generating a hotspot ◦ with maximum temperature of 63.7 C after entering in steady state. The image taken by the thermal camera is presented by Figure 12a, where the temperature distribution in 17 plates of the stator prototype is observed. Figure 12b shows the image generated by linear interpolation of raw DTS readings, where it can be seen that the maximum temperature was measured in an approximate way for only seven plates (MTD of −2.8 ◦ C), while for the rest of the plates, the MTD was up to −14 ◦ C. This is because the unprocessed DTS readings are strongly influenced by spatial resolution (Figure 4), generating a blurred image. The image reconstructed by the proposed algorithm is represented in Figure 12c. In this result, the maximum temperature was 63.2 ◦ C (MTD of −0.5 ◦ C), and both length and location of the hotspot were in accordance with the image of thermal camera. As can be seen, the sparse reconstruction showed significant improvements when compared to the linear interpolation method, even for a hotspot with dimensions in line with the DTS spatial resolution (1 m). Although there are some differences in the temperature distribution along the plates, the main parameters of interest for application on stator structure are quite accurate, namely the location, length and maximum temperature. Another test was conducted using a hotspot with dimensions smaller than the DTS spatial resolution. In this experiment, we applied a total power of 120 W (∼ =15 W/cm3 ) at a resistance of 15 cm, for which the minimal length estimated in the tests is presented in Section 6.1. The generated hotspot was of 67.5 ◦ C and an ambient temperature of 25.9 ◦ C, as shown in Figure 13a. Figure 13b shows the image generated by the linear interpolation method, where the maximum temperature measured was only 46.1 ◦ C (MTD −21.4 ◦ C), and both the dimensions and the location of the hotspot were not correctly identified. This poor result is expected because this method uses the raw sensor readings and the hotspot length was only 15 cm, i.e., considerably lower than the spatial resolution of 1 m. The image generated by the proposed reconstruction algorithm is represented in Figure 13c. In this result, the maximum temperature was 66.8 ◦ C (MTD of −0.7 ◦ C), and location of the hotspot was in accordance with the image of the thermal camera. The length measured was about 22 cm, which can be considered a small difference, since that the dimension of the hotspot was six times smaller than DTS

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spatial resolution. Regarding the application in the generator stator, the proposed algorithm provides a great improvement when compared with the linear interpolation method, because the insulation faults can occur in just a few core plates (Figure 1), so it is important to perform reliable measurements 2016, with 16, 1425 12 of 16 forSensors hotspots dimensions less than 1 m.

(c) Figure 12. Experimental results for hotspot of 100 cm, with maximum temperature of 63.7 °C and ambient temperature of 27.8 °C; (a) Reference image taken by thermal camera; (b) Image generated by the linear interpolation method using raw DTS readings [6]; (c) Image reconstructed by proposed algorithm.

(b) (a) Another test was conducted using a hotspot with dimensions smaller than the DTS spatial resolution. In this experiment, we applied a total power of 120 W (≅15 W/cm3) at a resistance of 15 cm, for which the minimal length estimated in the tests is presented in Section 6.1. The generated hotspot was of 67.5 °C and an ambient temperature of 25.9 °C, as shown in Figure 13a. Figure 13b shows the image generated by the linear interpolation method, where the maximum temperature measured was only 46.1 °C (MTD −21.4 °C), and both the dimensions and the location of the hotspot were not correctly identified. This poor result is expected because this method uses the raw sensor readings and the hotspot length was only 15 cm, i.e., considerably lower than the spatial resolution of 1 m. The image generated by the proposed reconstruction algorithm is represented in Figure 13c. In this result, the maximum temperature was 66.8 °C (MTD of −0.7 °C), and location of the hotspot was in accordance with the image of the thermal camera.(c)The length measured was about 22 cm, which can be considered a small difference, since that the dimension of the hotspot was six times smaller than Figure Experimental results hotspot ofof 100 cm,cm, with maximum temperature of 63.7of°C and◦ C Figure 12.12. Experimental results for for maximum DTS spatial resolution. Regarding thehotspot application100 in the with generator stator,temperature the proposed 63.7 algorithm ambient temperature of 27.8 °C; (a) Reference image taken by thermal camera; (b) Image generated ◦ and ambient of 27.8 (a) Reference takeninterpolation by thermal camera; (b) Imagethe provides a greattemperature improvement whenC.compared withimage the linear method, because by the linear interpolation method using rawusing DTS readings [6]; (c) Image reconstructed by proposedby generated by the linear interpolation method raw DTS readings [6]; (c) Image reconstructed insulation faults can occur in just a few core plates (Figure 1), so it is important to perform reliable algorithm. proposed algorithm. measurements for hotspots with dimensions less than 1 m. Another test was conducted using a hotspot with dimensions smaller than the DTS spatial resolution. In this experiment, we applied a total power of 120 W (≅15 W/cm3) at a resistance of 15 cm, for which the minimal length estimated in the tests is presented in Section 6.1. The generated hotspot was of 67.5 °C and an ambient temperature of 25.9 °C, as shown in Figure 13a. Figure 13b shows the image generated by the linear interpolation method, where the maximum temperature measured was only 46.1 °C (MTD −21.4 °C), and both the dimensions and the location of the hotspot were not correctly identified. This poor result is expected because this method uses the raw sensor readings and the hotspot The Sensors 2016, 16, 1425length was only 15 cm, i.e., considerably lower than the spatial resolution of 1 m. 13 of 16 (a) (b)Figure 13c. In this result, image generated by the proposed reconstruction algorithm is represented in the maximum temperature was 66.8 °C (MTD of −0.7 °C), and location of the hotspot was in accordance with the image of the thermal camera. The length measured was about 22 cm, which can be considered a small difference, since that the dimension of the hotspot was six times smaller than DTS spatial resolution. Regarding the application in the generator stator, the proposed algorithm provides a great improvement when compared with the linear interpolation method, because the insulation faults can occur in just a few core plates (Figure 1), so it is important to perform reliable measurements for hotspots with dimensions less than 1 m. (c) Figure Experimental results 15 15 cm,cm, withwith maximum temperature of 67.5of°C67.5 and◦ C Figure 13.13. Experimental resultsfor forhotspot hotspotof of maximum temperature ambient temperature of 25.9 °C. (a)◦ Reference image taken by thermal camera; (b) Image generated and ambient temperature of 25.9 C. (a) Reference image taken by thermal camera; (b) Image by the linear interpolation method using raw DTS readings [6]; (c) Image reconstructed by proposed generated by the linear interpolation method using raw DTS readings [6]; (c) Image reconstructed by algorithm. proposed algorithm.

A test to evaluate an extreme case for the image reconstruction algorithm was performed producing a hotspot of (a) only 5 cm. In this test we applied a total power (b) of 25 W (≅9 W/cm3) on a resistance positioned in one of the plates, which generated a hotspot with maximum temperature of 50.6 °C and an ambient temperature of 22.8 °C. The thermal image taken by the thermal camera is presented in Figure 14a. The result of the linear interpolation method is shown in Figure 14b. As can

Figure 13. Experimental results for hotspot of 15 cm, with maximum temperature of 67.5 °C and ambient temperature of 25.9 °C. (a) Reference image taken by thermal camera; (b) Image generated by the linear interpolation method using raw DTS readings [6]; (c) Image reconstructed by proposed algorithm. Sensors 2016, 16, 1425

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A test to evaluate an extreme case for the image reconstruction algorithm was performed producing a hotspot of only 5 cm. In this test we applied a total power of 25 W (≅9 W/cm3) on a A test to evaluate an extreme case for the generated image reconstruction algorithm wastemperature performed of resistance positioned in one of the plates, which a hotspot with maximum 3 ) on ∼ producing a hotspot of only 5 cm. In this test we applied a total power of 25 W ( 9 W/cm = 50.6 °C and an ambient temperature of 22.8 °C. The thermal image taken by the thermal camera is a resistance positioned in one of the plates, which generated a hotspot with maximum temperature of presented in Figure 14a. The result of the linear interpolation method is shown in Figure 14b. As can 50.6 ◦ C and an ambient temperature of 22.8 ◦ C. The thermal image taken by the thermal camera is be seen, the generated image is extremely blurred and it was not possible to identify the hotspot. This presented in Figure 14a. The result of the linear interpolation method is shown in Figure 14b. As can result shows that the use of Raman DTS becomes impractical for measurements in the order of 5 cm be seen, the generated image is extremely blurred and it was not possible to identify the hotspot. without using signal reconstruction techniques. The image generated by the reconstruction algorithm This result shows that the use of Raman DTS becomes impractical for measurements in the order of 5 cm is represented in Figure 14c. In this result, the maximum temperature was 41.5 °C (MTD of −9.1 °C). without using signal reconstruction techniques. The image generated by the reconstruction algorithm Besides the large hotspot temperature length was also incorrectly, is represented in difference Figure 14c.in Intemperature, this result, thethe maximum was measured 41.5 ◦ C (MTD of −9.1 ◦which C). was 25 cm instead of 5 cm. As can be seen, the hotspot is spread on five plates when it should be only Besides the large difference in temperature, the hotspot length was also measured incorrectly, which on one This of is 5mainly to seen, the DTS spatial is resolution the signal-to-noise ratio was 25 plate. cm instead cm. Asdue can be the hotspot spread onthat five affects plates when it should be only (SNR), spreading and attenuating the sensor response, as already shown in Subsection 6.1, more on one plate. This is mainly due to the DTS spatial resolution that affects the signal-to-noise ratio (SNR), specifically in attenuating Figure 11e,f. Although in as this case shown the algorithm does not provide precise spreading and the sensor response, already in Subsection 6.1, more specifically in measurements of temperature and dimension, it is possible to identify the existence of and approximate Figure 11e,f. Although in this case the algorithm does not provide precise measurements temperature location of a fault, for hotspots with upapproximate to 20 times location smaller of than the DTS and dimension, it iseven possible to identify thedimensions existence and a fault, even spatial for resolution. Section 7 presents mainsmaller conclusions onDTS the spatial proposed reconstruction hotspots with dimensions up tothe 20 times than the resolution. Section 7 method presents for the mainimaging conclusions on the proposed thermal of generator stators.reconstruction method for thermal imaging of generator stators.

(a)

(b)

(c) ◦ C. (a) Figure Figure 14. 14. Experimental Experimentalresults resultsfor foraahotspot hotspot of of 55 cm, cm, with with maximum maximum temperature temperature of of 50.6 50.6 °C. Reference image taken byby thermal generatedby bythe thelinear linear interpolation method (a) Reference image taken thermalcamera; camera;(b) (b) Image Image generated interpolation method using [6]; (c) (c)Image Imagereconstructed reconstructed proposed algorithm. usingraw rawDTS DTS readings readings [6]; byby proposed algorithm.

7. Conclusions This paper presented an image reconstruction method that can be a promising solution for thermal imaging systems based in Raman distributed temperature sensing (DTS). The reconstruction was based on sparse representations, which has proved suitable for the application. The main advantage is the possibility to correctly identify heat sources and hotspots smaller than the DTS spatial resolution (1 m). To reconstruct the thermal images, we employed a dictionary of hotspots built from a multiphysical model of the monitored structure. Tests were performed using a prototype with structure similar to surface of a 200 MW hydroelectric generator stator. This facilitated the laboratory tests and allowed the comparison between reconstruction images and images from a thermal camera used as reference.

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The evaluation of the algorithm performance with respect to noise level was conducted through simulations with the DTS model and dictionary of hotspots. These simulations shown that when signal-to-noise ratio (SNR) is up to 40 dB it is possible to obtain acceptable images for application in the stator, with less than 4 ◦ C uncertainty. However, with SNR between 30 dB and 10 dB besides the uncertainty in the temperature measurement, other parameters such as location and dimension become affected. This provides a lower bound to which the proposed method is applicable. The experimental results were achieved by generating hotspots of different dimensions in the stator prototype. These results show that it is possible to identify hotspots with dimensions as short as 15 cm, with temperature uncertainty of less than ±1 ◦ C, which represents a great advance considering the DTS spatial resolution. It was also observed significant improvements for hotspot down to 5 cm. In this critical case, despite a maximum temperature difference of almost −10 ◦ C, it was possible to identify the existence and approximate location of hotspots with dimensions up to 20 times smaller than the DTS spatial resolution. Regarding the application for imaging system for generator stators, improvements in image resolution can help identify wear on the insulation layer in early stages, facilitating maintenance and avoiding further damage to the structure, such as short circuit in the stator windings. The accurate temperature monitoring of the stator structure can be a fundamental tool of predictive maintenance, to ensure the performance and operational availability of the generator. Acknowledgments: The authors acknowledge the financial support of CNPq, CAPES, FINEP, ANEEL and Engie Brasil Energia. This project is developed under the Engie Brasil Energia’s R&D program (PD-0403-0028/2012) and regulated by ANEEL. Author Contributions: J.P.B. and E.V.S. developed the stator prototype and performed experimental tests; J.P.B. and D.R.P. worked in the development of image reconstruction algorithm and numerical analysis; C.M. and J.C.C.S. made the supervision and direction of the work. All authors participated in the research and review of the article. Conflicts of Interest: The authors declare no conflict of interest.

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