T Toullier

J Dumoulin
email: jean.dumoulin@ifsttar.fr

L Mevel

Study of complementary multi-sensors data influence on infrared thermography measurements for in-situ long-term monitoring

Keywords: infrared thermography, temperature, emissivity, calibration, data fusion, monitoring

Low cost solutions to infrastructure's thermal monitoring has become possible thanks to the development of uncooled infrared cameras over the past few years. However, in order to get a temperature value, one need to convert the radiative fluxes received by the camera. Such a step depends on spatial parameters, environment and the observed target characteristics which are usually unknowns or roughly estimated. To overcome this lack of knownledge, one solution is to combine the infrared camera measurements with other sensors. The purpose of this study is to examine the influence of the different parameters on the temperature conversion process and finally see to what extent complementary data can improve the estimation of the targeted object's temperature.

INTRODUCTION

Latest improvements on uncooled infrared cameras have brought new opportunities for non-invasive diagnostics in the Civil Engineering's field. In the context of thermal monitoring, such sensor can provide an interesting option thanks to its full-field observation capability allowing to embrace large areas. However, retrieving with time over long period, current targeted object's apparent surface temperature from those measurements is a challenge that needs to be addressed with care. [START_REF] Gaussorgues | Infrared Thermography[END_REF] In fact, the captured image depends on spatial parameters, the target's environment [START_REF] Adelard | Sky temperature modelisation and applications in building simulation[END_REF][START_REF] Dumoulin | Infrared thermography system for transport infrastructures survey with inline local atmospheric parameter measurements and offline model for radiation attenuation evaluations[END_REF] and its thermo-physical properties [START_REF] Howell | Thermal Radiation Heat Transfer[END_REF] that are usually unknowns [START_REF] Krapez | Radiative Measurements of Temperature[END_REF] or roughly estimated . Part of a possible solution is therefore to use complementary environmental sensors in order to get a better estimatation of time present measurement environmental conditions that will have an influence on the apparent surface temperature reconstruction. The objective of this paper is to list and examine the influence of different parameters on the temperature conversion process through infrared thermography implemented in outdoor environmental conditions for long term thermal monitoring. Moreover, the possibility of improving the estimation of the targeted object's temperature through the fusion of multi-sensors data is considered. Thanks to the Cloud2IR acquisition system, [START_REF] Crinière | Management of local multi-sensors applied to SHM and longterm infrared monitoring: Cloud2IR implementation[END_REF] an interesting amount of synchronized data have been gathered through years on various instrumented test sites, for instance: a small wood chalet in the framework of the SenseCity's experiments (IFSTTAR project). The temperature conversion model is improved by using meteorological reports (such as METAR) or in-situ data measurements acquired with various sensors such as pyranometers, pyrgeometers, weather stations, etc. In order to have proper acquisitions settings, the calibration process is also discussed. A radiative heat balance correction model for improving the temperature estimation through various sensors or local meteorological reports is firstly presented. Then, the tests sites involved in this study with their related instrumentation are presented. Finally, a summary of the sensors' data influence on the apparent temperature's scene estimation improvement, based on the availability and type of information is presented and discussed. Conclusion and perspectives are finally proposed.

TEMPERATURE'S DERIVATION FROM INFRARED MEASUREMENTS

Temperature retrieval from infrared measurements in outdoor conditions on built or natural targets is facing, in many real situations, lack of knowledge about the observed surfaces and surrounding environment. The irradiance received at camera's sensor level is a combination of all the scene contribution as illustrated in Fig. 1. Therefore, one needs to gather information about the geometry of the scene, the meteorological conditions, and the radiative properties of the observed surfaces in order to retrieve the temperature from those measurements. Such approach can be achieved by implementing multiple sensors in-situ. total to temperatures, the following simplified radiometric model is used:

L i,j total = obj τ atm τ opt L i,j obj + (1 -obj )τ atm τ opt (L i,j env + L i,j sun ) + (1 -τ atm )τ opt L atm + (1 -τ opt )L opt (1)
In outdoor conditions, environmental quantities can be derived from local weather station's data. The reflected, optics and atmospherics luminance values are computed by injecting a priori knowledge about the monitored objects, components of the infrared system and atmospherics conditions. Furthermore, a sky temperature correlation 7 combined with inputs from in-situ solar and meteorological measurements can be used to estimate the reflected radiance contribution and atmospheric transmission, in particular when no reference targets can be implemented on real site. In other case, when data are missing, envornmental informations can be retrieved from meteorological aerodrome report (METAR) data, if available and by taking care of the scene geographical location.

In order to get consistent measurements, a thermal calibration is performed prior to the experiment in laboratory conditions. In the presented results, the calibration is based on the RBF-function (2), as described in [START_REF] Toullier | Étude comparative de deux approches, thermocouples intégrés et thermographie infrarouge, pour la surveillance thermique d'une infrastructure de transport[END_REF] and derived from the Planck's law.

T i,j obj = B log R L i,j obj -F (2)
Figure 2: Optimized parameters results: the conversion is now near the objective line

The result of such calibration is given Fig. 2, where the temperature values are compared with and without the optimized parameters. On this figure, the new parameters enable a more reliable conversion from the digital levels of the measured signal to the estimated temperature.

EXPERIMENTAL SETUPS

Test site #1: instrumented road section

A road section that can be internally heated has been instrumented with different sensors. The monitoring is done with a FLIR SC655 thermal infrared camera (640×480 LWIR FPA, pitch 17µm, and a 13.1mm optical focal length). A weather station (Vaisala WXT 520), a net radiometer (Kipp & Zonen CNR4) and a pyranometer (Kipp & Zonen SPLite2) provides complementary measurements with meteorological data (rain, wind, temperature, etc.) and radiative heat fluxes. All those data are synchronized in time thanks to our multi-sensor data acquisition system Cloud2IR. [START_REF] Crinière | Management of local multi-sensors applied to SHM and longterm infrared monitoring: Cloud2IR implementation[END_REF] Due to the particular angle of view, distance and planar properties of the scene, those data are interesting to process and analyse spatial sampling issues, mixed pixels and transmissions effects. Moreover, the presence of thermocouples on the road surface enables us some comparison with infrared measurements.

PARAMETER'S MODEL INFLUENCE ON TEMPERATURE ESTIMATION ERROR 4.1 Spatial considerations

The positioning of the camera and the geometry of the scene will have an impact on the conversion process from digital levels recorded by the camera to the desired temperature. Nowadays, most infrastructures provides a BIM (Building Information Model) and in particular a 3D numerical model. In order to have knowledge on the actual camera view of the scene and thanks to this model, it is possible to retrieve the real world coordinates on the image, called resectioning (see Fig. 5). Let M be a calibration matrix such as for every (x, X) ∈ R 2 × R 3 of coupled coordinates points between the image and the real world we have:

x = M X (3)
By using the pin-hole model, we can define M with optical and geometrical parameters:

M =   f s c x 0 af c y 0 0 1     1 0 0 0 0 1 0 0 0 0 1 0   R 3x3 0 3x1 0 1x3 1
I 3 T 3x1 0 1x3 1 ( 4
)
Where those parameters are defined in Fig. 6. Since the system has 11 degrees of freedom, the knowledge of at least 6 couples of points (x, X) is necessary. The Levenberg-Marquardt algorithm is finally used to solve the system.

Extrinsics

• Translation T from the origin of world coordinates • Rotation R from the world coordinates As exposed previously, the road section test site #1 is mainly planar which is interesting for demonstrating the impact of the view angle to the conversion process. Once the resectioning has been applied, a distance map is obtained, representing the distance between the camera and the ground. In some cases, the model consideration is to use a constant distance over the entire image. However, the Fig. 7 shows this distance map (the range involved is [5; 10] meters) and also the impact of the distance on the atmospheric transmission, for varying environmental conditions. On this figure, the impact of the distance can be quite important depending on the perspective involved. Other spatial considerations must also be taken into account. Depending on the chosen optic and camera, images with varying spatial resolution and important angle between the observed surfaces and the camera may occur. In such cases, accurate temperature retrieval may be compromised due to non-square pixels, mixed pixels effects [START_REF] Jones | Scaling of Thermal Images at Different Spatial Resolution: The Mixed Pixel Problem[END_REF] and varying angle values along the same plane which may change the considered emissivity value. Moreover, when monitoring a particular plane or object in the scene, one may need to recover a given region of interest (ROI). The following figure shows an example of sampling issues due to a projection of a ROI on test site #2 and an illustration of the mixed pixel effect where different materials' radiation are integrated over the same sensor.

Intrinsics • Focal length f • Principle points (c x , c y ) • Aspect ratio a • Skew s

Emissivity considerations

In order to get information on thermo-optical wood house material's components and study their influence on apparent surface temperature estimation, emissivity measurements have been made on site with a portable measurement system, 10 as shown in Table 1 and Fig. 9. Figure 9: Emissivity map for the wood house, colors represents one particular emissivity measurement Fig. 10 shows the effect of putting different emissivity values during the conversion of the digital levels to temperature. In the first case, a constant emissivity value is chosen over the entire image. In the middle image, the emissivity map represented in Fig. 9 is used in the conversion. The last image represents the difference between the two previous images.

Material

Figure 10: Emissivity impact on the conversion process

Sky temperature considerations

The sky temperature (T sky ) can be obtained by using in-situ measurements by using pyrgeometers data for example. In the following Fig. the temperature obtained from infrared data with sky correction and without sky correction is compared to the thermocouple's data at surface. Taking into account the sky temperature from local measurements gives a best estimate of the object temperature. The difference between the thermocouples measurements and the estimated temperature from infrared data with sky correction is then reduced. When pyrgeometers data are not available, a possibility is to use a correlation from the literature that links the sky temperature with the air temperature, for example with the Swinbank's formula. [START_REF] Adelard | Sky temperature modelisation and applications in building simulation[END_REF] The following figure shows the sky temperature derived from the Swinbank's when the air temperature data are obtained by three different ways:

• From the local weather station • From Copernicus' data store [START_REF]Copernicus Data Store[END_REF] • From METAR data Even if differences exists in term of sky temperature (see Fig. 12), it represents small radiation quantities regarding the involved temperature levels. Online data are therefore an interesting way to enrich the radiative model conversion when no local measurements are available. In the following section, the sky temperature is derived from the Swinbanks correlation for two periods: in december and august. One can see that the sky temperature derived from the correlation will have an impact of apparent surface temperature images that depends on the period of observation for a same scene (here the wood chalet).

Sun's irradiance considerations

The sun's irradiance will also have an impact on the final temperature estimation. In fact, depending on the emissivity of the observed object, the irradiance at camera's sensor will contain the sun's reflected contribution, both specular and diffuse. An estimation of the solar irradiance can be obtained thanks to a pyrgeometer for example. In the case of the LWIR (7.5µm -13µm), even though the sun contribution may be small compared to the NIR, SWIR and MWIR bands, it stills represents a source of error in the conversion process. In particular for low object's temperatures with lower emissivity as illustrated in Fig. 14.

Rain consideration

The modification of emissivity due to fog or rain will also have an impact on the flux received at cameras sensors. In particular long-term measurement campaign involves variation of environment and climatic conditions (see Fig. 16). Accurate temperature estimation is therefore difficult to obtain. However, the knowledge of meteorological conditions can lead to a post-process of the temperature estimation to warn about strong uncertainty assumption. To overcome this effect, a black-body source and an automatic thermal calibration triggered with the weather station rains' detection could be used. The radiative transmission model could be updated with a more proper one such as in.

Results

The following figure represents the difference between a direct conversion of the digital levels to temperature by using the considerations presented previously in the radiative model or not over one chosen day. 17 shows that the error made by considering the environment contribution will be more or less important due to the fact that the errors come from different parameters that can compensate themselves. The peaks that we can observe in this figure represents the update of the camera's NUC (non-uniformity compensation) coefficients.

CONCLUSION

A review of different parameters involved into the long-term monitoring of structures through infrared thermography has been done. In such context, the temperature estimation for IR measurements needs to take into account many parameters or at least be aware of the involved difficulties to consider the estimation uncertainties. Using coupled measurements with those infrared data enables the temperature estimation to be more accurate.

The spatial considerations in this study could be extended to a stereovision system which would bring more information on both geometric and optic properties, in particular to retrieve the depth of the image. Even if such algorithms have been used on the computer vision field, they need to be adapted for the infrared spectrum since the detector's responses are different. Selecting a region of interest in order to extract particular information within the image should be made cautiously. In fact, the sampling of the image due to the projection into the ROI may vary. Moreover, the mixed pixel effect as stated in this study should be more considered for accurate temperature measurement since IR thermography usually consider each pixel as a homogeneous material. Finally, considering the sky temperature, mapping the emissivity to the image materials and using a distance map reduces the difference between the temperature estimation and actual surfaces measurements. An angle map derived from the spatial resectioning and prior knowledge on the involved materials could improve the emissivity map. However, other considerations such as rain, sun's specular part, optics properties, lens flare, sensors' drifts, etc. could be incorporated more deeply into the radiative model. For example, thanks to the knowledge of the geometry of the scene and the time, the specular part of the sun could be derived. Again, if there is no sensor available, meteorological data available on the internet could be used. Even more, models such as MODTRAN or SMARTS2 for example could provide an estimation of the solar spectral irradiance at ground. Moreover, one of our perspective is to combine such radiative model with a Bayesian model to estimate conjointly emissivity and temperature.

Figure 1 :

 1 Figure 1: Radiation interactions during IR measurements

  (a) Image of the road in the infrared spectrum (b) Image of the road in the visible spectrum

Figure 3 :

 3 Figure 3: Road section instrumentation: IR image (a) and test site overview (b)

Figure 4 :

 4 Figure 4: Wood chalet survey instrumentation: IR image (a) and test site overview (b)

Figure 5 :

 5 Figure 5: Resectioning principle, the conversion of the real world coordinates to the image ones

Figure 6 :

 6 Figure 6: Resectioning parameters with the pin-hole model

Figure 7 :

 7 Figure 7: Impact of the distance on the atmospheric transmission and its evolution with time

Figure 8 :

 8 Figure 8: From left to right: Mixed pixel effect illustration. Spatial sampling issues due to reprojection

Figure 11 :

 11 Figure 11: Thermocouples measurements compared to IR measurements temperature estimation with and without sky correction

Figure 12 :

 12 Figure 12: Sky temperature obtained with different air temperature data sources

Figure 13 :

 13 Figure 13: Swinbanks correlation for deriving the sky temperature

Figure 14 :

 14 Figure 14: Sun irradiance reflection on 7.5µm -13µm band for a normal object with different emissivity values

Figure 15 :

 15 Figure 15: Comparison of sun's irradiance measured and simulated on may 5th in the 0.3µm -2.8µm band

13

 13

Figure 16 :

 16 Figure 16: Rain episod issues for in-situ measurements

Figure 17 :

 17 Figure 17: Comparison of temperature conversion with and without environment considerations

Table 1 :

 1

		Color	Emissivity
	Wood wall (west) Blue	0.95
	Wood wall (east) Yellow	0.90
	Grass	Green	0.95
	Sidewalk	Red	0.89
	Tar	Grey	0.95
	Roof	Purple	0.92

Emissivity measurements values in the 7.5µm -13µm band.

ACKNOWLEDGEMENTS

Authors wish to thank for their support: French Region Bretagne and Sense-City Facility of Excellence of ANRs Future Investment Program.


Hassna Marzouk
email: marzouk.hsn@gmail.com

Abdelmajid Badri
email: abdelmajid_badri@yahoo.fr

Khadija Safi

PSNR Analysis of video transmission in VANETS, using NS2 and Evalvid Framework

Keywords: VANET, video transmission, road traffic, EvalVid, framwork, tool-set, NS-2, SUMO, YUV video, PSNR

. These networks are expected to offer new entertainment applications and traffic security services with intelligent equipment in a smart environment. A smart city is designed as a controlled and monitored environment using advanced technologies and new types of communication to improve the quality of life through innovative services. One of the key challenges of smart city communication is ensuring effective service delivery in economic, social and environmental conditions. As a result, multimedia communications, including streaming video, should be very beneficial for traffic management as well as for the provision of entertainment and advertising services. With video streaming services, real-time information will be used and provided by vehicle networks for safety and efficiency purposes. In this article, we focus on studying the effect of streaming low brightness video feeds on PSNR.

INTRODUCTION

Video transmission and communication in VANETs represents a major challenge because the video streaming [START_REF] Felipecunha | Data communication in VANETs: Protocols, applications and challenges[END_REF] in real time will considerably enrich the quality of experience perceived by the user as there will be more visibility on its environment, the conditions of the road traffic, the possibility of using applications such as emergency video calls, etc. However, the transmission of video data in the VANETs is a difficult task since the vehicular networks are a hostile environment for any kind of data especially video transmission (numerous packet losses, mobility of the nodes, etc.), as a result, researchers have focused on improving and optimizing video communication. Unfortunately, evaluating video transmission performance has not been an easy task. Therefore, finding a powerful and an easy tool is fundamental to advancing researches.

In this presentation, we propose an advanced simulation tool-set which integrates EvalVid framework [START_REF] Garrido Abenza | Simulation Framework for Evaluating Video Delivery Services Over Vehicular Networks[END_REF] [4] into NS-2 [START_REF] Issariyakul | Introduction to Network Simulator 2 (NS2)[END_REF], plus the addition of a mobility file generated from a real life map using SUMO [START_REF] Behrisch | SUMO -Simulation of Urban Mobility: An Overview[END_REF] which will make this tool-set credible while measuring delivered videos quality. With the enhancements, the toolset allows network-related researchers to evaluate real video streams using their proposed network designs or protocols as well as evaluate video quality of their de signed video coding mechanisms using a more realistic network.

The tool can adopt both raw YUV video files and downloadable video trace files as a video stream source, hence, researchers get to focus more on their area of research without worrying about having an available video source for the simulation. To evaluate the performance, the toolset uses PSNR which compares each pixel between the original and delivered video trace file. Furthermore, the tool-set combines the received video frames into real video that we can play before our eyes.

SYSTEM MODEL

Figure 1 shows the overall system model: Initially, a raw YUV video file is converted to an MPEG4 file. The latter is introduced in Evalvid to generate a video trace file which is the actual video object sent via a simulated transmission in NS2. The received video file, received at the receiving node in the simulation environment, is returned to Evalvid, which generates the PSNR quality model from among various other measures of quality of service.

Fig. 1: System Model: Evalvid Framework integration with NS2 [7]

The structure and main components of the Evalvid Evaluation Framework are described as follows:

Source: There are two video source formats, YUV QCIF (176x144) and YUV CIF (352x288).

Video Encoder and Video Decoder: EvalVid currently supports two MPEG4 codecs, the NCTU codec [START_REF]NCTU codec[END_REF] and ffmpeg [START_REF] Ffmpeg | [END_REF].

VS (Video Sender):

This component plays the compressed video file produced by the video encoder, fragments the largest video image into smaller segments, and then transmits these segments over a real or simulated network. For each packet transmitted, the timestamp, packet ID, and payload size of the packet are stored in the sender's trace file. After that, this component also generates a video trace file containing information about each image in a real video file. The video trace file and the sender trace file can be used to evaluate the video quality.

ET (evaluation of the trace): the evaluation is carried out on the side of the sender once the video transmission is finished. That is, the timestamp, packet ID, and packet payload information available to the recipient must be returned to the sender. Based on the encoded video file, the video trace file, the sender trace file, and the original receiver trace file, this AND component can report frame / packet loss and frame / packet jitter. and generate a reconstructed video file corresponding to the eventual reproduced video found to the end user.

In addition, this component considers a lost frame if it arrives later than its predefined reading time.

FV (Fix Video):

The measurement of the video quality is done frame by frame. Therefore, the total number of receiver-side video frames and the lost frames must be the same as the original transmitter-side video. If the codec cannot handle missing images, the FV component is an error concealment technique used to deal with this problem by inserting the last correctly decoded image in place of each lost image [START_REF] Wand | Error control and concealment for video communication: A review[END_REF].

PSNR (Peak

WORK APPROACH

Our working approach is summarized in four main steps, illustrated in Figure 2. Each step is explained separately in the following subsections.

Fig. 2 : Work steps

The Coding Process

The video file types used in our simulations are the standard H.261 Standard Interface Format (CIF) with a resolution of 352 × 288 because this format is commonly used in video conferencing, one of VANET's important applications. Before a CIF file can be used for simulation purposes, it is first encoded in MPEG, one of the widely used industrial standards for streaming video over the Internet.

Video Trace File Generation

The second step is to generate a video trace file from the original MPEG4 video file using the mp4trace tool.

The video trace file contains the number, type and size of the image, as well as the number of segments in case of image segmentation. The Evalvid tool was originally designed to evaluate actual video transmissions, which is why the mp4trace tool specifies the destination URL and the port number.

Simulation in NS2

In our simulation, we chose to use AODV protocol which has been determined to be the best in terms of packet delivery, jitter and throughput.

The other parameters are mentioned below:

RESULTS AND ANALYSIS

The video we used our simulation is based on "Highway CIF". Figure 3 shows the PSNR performance of the AODV protocol when the network sparsity is set at D = 100 m and its density is increasing. We used 100 frames to smooth the results to obtain a clearer analysis. The PSNR variation model is maintained at different network densities and it's affected by the brightness content of the images constituting the transmitted video. Since we are transmitting the same video file across different topologies, the brightness content of each frame in this video remains the same. We verified this by examining the two major drops in PSNR performance in Figure 5. The first occurred at approximately F = 500, corresponding to the approximate video playback time of T = 19s. We observed that during this period, the brightness decreased due to the appearance of a black car in excess, as can be seen in FIG. 4. The second major drop in PSNR occurred between F = 1300 and F = 1400, video playback time of T = 43 and T = 46. During this period, a black bridge appears first in the video, and then the car passes under its shadow, as shown in Figure 5. Therefore, the fall of the PSNR in both cases can be justified by the fact that when the brightness content of a frame decreases, the noise energy dominates the maximum signal energy and therefore degrades the PSNR It can also be noted in Figure 5 that the performance of the AODV on the PSNR degrades as the density of the network increases. The PSNR decreases by approximately 5 dB between Frame = 500 and Frame = 650 frames when the network goes from N = 4 nodes to N = 9 nodes. However, this attenuation is less important when the network is changing to N = 25 nodes and N = 64 nodes. This is because the 4-node topology allows direct communication between the sending and receiving nodes; but, when the network density increases to N = 9, 25 or 64 nodes, data is routed through intermediate nodes. In this case, the multiple jumps considerably reduce the performance of the PSNR.

CONCLUSION

In this paper, we have investigated the combined effect of the network density and the image britheness on the PSNR performance. Various network sparsity models were designed with varying network densities. Simulation and evaluation results presented interesting findings. PSNR performance worsens as the density of the network increases. We have also determined that PSNR is heavily affected when the network density increases because of packet loss.

  Signal Noise Report): The PSNR is one of the most widely used objective metrics for evaluating the quality of service of video transmissions at the application level. The following equation shows the PSNR definition between the luminance component Y of the source image S and the destination image D: where Vpeak = 2k-1 and k = number of bits per pixel (luminance component). The PSNR measures the error between a reconstructed image and the original.

Fig. 3 :

 3 Fig. 3: PSNR Performance of AODV for various networkdensities[START_REF] Ke | An Evaluation Framework for More Realistic Simulations of MPEG Video Transmission[END_REF][START_REF] Ffmpeg | [END_REF] 25, 64)

Fig. 4 :Fig. 5 :

 45 Fig. 4: screenshot of the video at T=21s


Aziz Oukennou
email: oukennou@gmail.com

Achraf Berrar

Imane Belbhar

Noureddine El Hamri

Low Cost IoT System for Solar Panel Power Monitoring

Keywords: NodeMCU, IoT, Low Cost, Monitoring, Node-red, Solar Panel

In this work, we will present a low-cost system to monitor energy production from a solar panel. Based on simple devices, this solution made it possible to measure the current, voltage, power, and visualize them through an available and free IoT application called Node-Red. The project can be qualified having plenty of essential purposes : to be used in education field, research and even production monitoring in a photovoltaic system.

INTRODUCTION

The Internet of Things (IoT) is related, to the exchange of data between devices on an available network infrastructure. By 2020, the expected outcome is 50 billion devices connected [START_REF] Swan | Sensor mania! the internet of things, wearable computing, objective metrics, and the quantified self 2.0[END_REF]- [START_REF] Ansari | An Internet of things approach for motion detection using Raspberry Pi[END_REF]. IoT projects cover many areas and more specifically the remote monitoring of production and energy consumption of equipment. This category is necessary because we tend to take measures to reduce energy consumption and preserve the environment. A great deal of work has been done to create monitoring applications. In [START_REF] Kelly | Towards the implementation of IoT for environmental condition monitoring in homes[END_REF], the remote measurement and controlling of domestic devices over the Internet was designed. Health Monitoring and Management, was discussed as well in [START_REF] Hassanalieragh | Health monitoring and management using Internet-of-Things (IoT) sensing with cloud-based processing: Opportunities and challenges[END_REF]. Other applications have been realized, in [START_REF] Patchava | A smart home automation technique with raspberry pi using iot[END_REF] a Smart Home Automation Technique was presented, It is based on the use of Raspberry Pi using IoT as a module that can overcome the problem of consumption and also the high price of a computer. By focusing on the monitoring of renewable energy, other works exist in the literature. An successful development of an innovative system for a solar photovoltaic energy conditioning unit was described, it was based on the use of network GPRS (Global Positioning Radio Service), embedded system gateway and other components such as Arduino Uno R3 [START_REF] Shrihariprasath | A smart IoT system for monitoring solar PV power conditioning unit[END_REF]. The chosen solution is interesting; however, it seems to be expensive. Another solution was also proposed for the monitoring and the control of photovoltaic panels [START_REF] Adhya | An IoT based smart solar photovoltaic remote monitoring and control unit[END_REF], GPRS was integrated as a means of communication, and a hard solution was based on a PIC18F46K22 microcontroller. A control system was also designed in [START_REF] Salhaoui | An IoT Control System for Wind Power Generators[END_REF] using a smart IoT gateway to create a connection between an industrial case and the cloud. The aim of the work was to provide a solution to visualize in real-time the parameters of an existing wind energy system. With regard to solar energy management, a consistent work has been presented in [START_REF] Spanias | Solar energy management as an Internet of Things (IoT) application[END_REF], it is based mainly on artificial intelligence as well as a hard solution containing a microcontroller, a network radio, relays for reconnecting or bypassing panels, and sensors. In the majority of research, we find different monitoring configurations; the difference exists in both hard and soft terms. In the first aspect, some authors have tried to develop adapted cards with the necessary communication devices. Others have deployed existing platforms such as Arduino Uno R3. As for the second aspect, there are solutions based on the development of customized applications, whether on a computer or a mobile phone, or the use of existing applications that can communicate with the equipment under control and that are free or not. In this present work, we will propose a new architecture. The components are qualified as the cheapest in the market. Indeed, we will use Wifi as a communication solution and the monitoring application we have chosen Nod Red as a powerful and free tool to build applications of the Internet of Things (IoT). The whole system will be combined with a current, voltage and power sensor to monitor production from a solar panel. In short, the proposed solution will have various advantages, namely its low cost, which will make it accessible to anyone wishing to monitor production remotely; the second advantage is the ease of implementation, the flexibility of the solution. And it is an effective way to monitor solar panels generally installed in inaccessible locations. This paper consists of three sections: In section 2, the proposed architecture is introduced. Component specifications are given in section 3. Sections 4 and 5 will be devoted to present the obtained results, and estimated price of the solution. Section 6 discusses the conclusion.

PROPOSED ARCHITECTURE

This paper proposes a real time and low cost monitoring system for solar panel using NodeMCU which is connected to Node-Red. This system can help to measure PV system's productions: voltage, current and power.

The following figure shows an overview of the proposed system architecture.

Components required: Software

1) Node-Red, 2) IDE, 3) MQTT.

COMPONENT SPECIFICATIONS

NodeMCU

NodeMCU is an open source IoT platform, it is a wifi SOC (System On a Chip) produced by Espressif Systems. It is based ESP8266 -12E WiFi module. It is a highly integrated chip designed to provide full internet connectivity in a small package.

Fig. 2 : ESP8266 NodeMCU development kit

INA219

The INA219 is a power and current shunt controller with an I²C or SMBUS compatible interface. The instrument monitors both the shunt voltage drop and the bus supply voltage, with programmable conversion times and filtering. A programmable calibration value, combined with an internal multiplier, allows a direct reading of the current in amperes.

ARDUINO IDE:

Arduino software is an open source and free development environment (IDE), which can be downloaded from the official Arduino website [12]. It allows program edition (sketches) and its compilation, so it can be uploaded into the Arduino's memory and finally be able to communicate with the Arduino card through the terminal.

MQTT :

MQTT is a simple messaging protocol, designed for constrained devices with low-bandwidth. So, it's the perfect solution for Internet of Things applications. MQTT allows sending commands to control outputs, read and publish data from sensor nodes. Therefore, it makes it simple to establish a communication between multiple devices.

Fig. 4: MQTT connectivity

OBTAINED RESULTS

The microcontroller of NodeMCU gets the voltage, and current, and power output from the sensor. Once the NodeMCU is connected with MQTT via the Wi-Fi, we launch Node-Red and use the MQTT to receive the output values from the NodeMCU. To visualise the result, Node-Red dashboard must be launched. As we can see, the data measured by the sensor is transmitted to Node-Red in real time, any device connected to the same Wi-Fi, can visualise the results by using the same dashboard's address but he must have access to it by logging in with the user name and the password. The characteristics of the PV panel Figure 6 (current, voltage, power) can be presented graphically or by using a gauge. Two multimeters have been used to compare both the result and the sensor's result, one to measure the voltage and the other to measure the current. The power is obtained, by multiplying the values stated and obtained above. The Table 1 shows the results and the comparison between the values of Node-Red. Both the results are close to each other, there is a tiny error, due to the sensor's characteristics.

Fig. 5: Node-Red Flows

The obtained results are given in figure 6.

.

PROPOSED ARCHITECTURE PRICE

Table 2 gives an idea about the cost of performing the process using the stated architecture.

CONCLUSION

The proposed architecture provides Low Cost IoT System for Solar Panel Power Monitoring. A suitable association between software and hardware has been implemented to finally design an efficient and real-time measurement system. This solution can also be extended or applied in other applications, especially in research or for education purposes.

Fig. 1 :architecture 2 . 1

 121 Fig. 1: System architecture 2.1 Components required: Hardware 1) ESP8266 based Wifi module NodeMCU, 2) INA219.

Fig. 3

 3 Fig. 3: INA219

Fig. 6 :

 6 Fig. 6: Sensor data's Dashboard and graphs

Table 1 :

 1 Data measurement and comparison

Table 2 :

 2 Price of the proposed instrumentation system

	NodeMCU	$2,38 (ww.aliexpress.com)
	INA 219	$4.67 (www.amazon.com)
	Node-Red	Free
	Total	$7,05

Multimeters


domain: spi phys

text:
on the wall opposite the liner and using a minimization method where experimental and numerical results are compared. In order to describe more precisely the local effects on the liner, ONERA applied a Laser Doppler Anemometry (LDA) technique to measure the acoustic velocity fields in the area above the liner, and used them as inputs for the eduction process. [START_REF] Primus | Validation of a direct propagation model for liner impedance eduction[END_REF][START_REF] Primus | Evaluation of an adjoint-based liner impedance eduction technique[END_REF][START_REF] Primus | An adjoint-based method for liner impedance eduction: Validation and numerical investigation[END_REF] The numerical resolution of the Linearized Euler Equations (LEE) was obtained with a discontinuous Galerkin scheme, accounting naturally for the presence of a shear flow in the simulation. The main challenge is to ensure the best reliability of the eduction process and high accuracy of the results. [START_REF] Zhou | A systematic uncertainty analysis for liner impedance eduction technology[END_REF] Recently, Roncen et al. [START_REF] Roncen | Statistical Inference Method for Liner Impedance Eduction with a Shear Grazing Flow[END_REF] dealt with a new approach based on Bayesian inference in order to have an evaluation of the uncertainties on the impedance results. For a classical perforate-overhoneycomb liner, it is usually assumed that the liner impedance is constant along the streamwise direction. However, when the incident Sound Pressure Level (SPL) is high, a phenomenon of vortex shedding appears in the vicinity of the perforations. [START_REF] Hersh | The Acoustic Behavior of Helmholtz Resonators Exposed to High Speed Grazing Flows[END_REF] This leads to a modification of the impedance (mainly an increase of the resistance). [START_REF] Zhang | Numerical investigation of a honeycomb liner grazed by laminar and turbulent boundary layers[END_REF][START_REF] Rienstra | Nonlinear asymptotic impedance model for a Helmholtz resonator of finite depth[END_REF][START_REF] Cummings | The response of a resonator under a turbulent boundary layer to a high amplitude non-harmonic sound field[END_REF] These studies show that the impedance depends on the incident sound pressure level in the presence of high SPL. In a lined duct configuration, the sound pressure level may be much higher upstream of the liner than downstream of the liner. It seems thus relevant to take into account the variation of the impedance as a function of the variation of the incident sound pressure level which decreases along the liner. Jones [START_REF] Jones | Impedance Eduction for Multisegment Liners[END_REF] recently educed the impedance of multisegment liners, where the studied liners had a piecewise impedance evolution along the streamwise direction.

The objective of this paper is to apply the same kind of eduction strategy on a classical Single Degree of Freedom (SDOF) liner to take into account the non-linear effects due to the incident sound pressure level with a shear grazing flow. Section II is dedicated to the presentation of the IFAR liner sample used for this study and the experimental set up, while Section III proposes a general presentation of ONERA eduction methods and its application on the IFAR liner. Section IV emphasizes the modifications on the eduction process in order to take into account the SPL effect.

II. Description of the experimental setup

A. Liner definition Single degree of freedom liners are typically made using a honeycomb structure that is backed by a rigid plate and topped by a perforated facesheet, as seen on Fig. 1. In this study, two of these classical SDOF liners are used. The first sample was designed in the framework of the IFAR benchmark challenge #1 and manufactured by an additive process. It has square core chambers, an integrated conventional facesheet, and a core depth of 50.8 mm. The second liner was manufactured for a previous study and already tested by NASA. [START_REF] Primus | ONERA-NASA cooperative effort on liner impedance eduction[END_REF] Its chambers are hexagonal and it is less thick than the IFAR sample. Table 1 sums up the characteristics of both samples.

Different empirical models exist to describe the behavior of liners at high sound pressure level or high flow speeds, 14 depending on their characteristics. Based on these models, both liner samples should exhibit non-linear effects due to incident SPL at around the same acoustic level. The aeroacoustic test bench at ONERA was developed and built in 2004. [START_REF] Minotti | Contribution à la mesure aéroacoustique par Vélocimétrie Laser Doppler[END_REF][START_REF] Minotti | Characterization of an acoustic liner by means of Laser Doppler Velocimetry in a subsonic flow[END_REF] It is made of a stainless steel tube of section 50 mm x 50 mm and has a total length of about 4 m. A 0.2 m-long test section is equipped with two silica windows for optical access. The termination is equipped with a quasi-anechoic outlet, which leads to a reflection coefficient smaller than 0.2 for frequencies higher than 500 Hz. A mean flow of bulk Mach number M b up to 0.5 can be provided. The flow temperature can be accurately regulated, from the ambient temperature up to 570 K. All along the study, the flow temperature was imposed to be equal to the ambient temperature (290 K). In the test section, this flow shows fully developed turbulent boundary layers, with a turbulence rate of a few percent at the center of the section. In Fig. 2, flow propagates from left to right. Two acoustic drivers are mounted upstream of the test section, and are used to generate tones (usually a multi-sine signal) at up to 150 dB over a frequency range of 0.3 to 3.5 kHz (i.e., the no-flow cut-off frequency of the duct for plane waves). When a multi-sine source is used, the overall SPL (OASPL) within the duct is much higher (usually 10 dB more) than the SPL at each tone frequency. The surface of the test liner forms a 150 mm-long portion of the lower wall of the flow duct and spans all the duct width. Flush-mounted GRAS 40SA microphone probes are used (probe diameter of 1.2 mm). Usually, two upstream locations are considered to measure the amplitude of the incident acoustic plane waves. [START_REF] Chung | Transfer function method of measuring in-duct acoustic properties. I. Theory[END_REF] However, for the pressure-based impedance eduction method, one microphone probe is moved along the whole set of microphone locations. The cross-spectrum between the microphone (or the LDA system) and driver signals is acquired, which provides a phase reference for the pressure measurements.

Two-microphone method for measuring the incident and reflected pressure waves

A two-microphone method is used to decompose the measured sound field in the duct into an incident and a reflected wave. [START_REF] Chung | Transfer function method of measuring in-duct acoustic properties. I. Theory[END_REF] This analysis is only performed below the first cut-off frequency of the duct, i.e. under 3.5 kHz, so that only plane waves are considered. This two-microphone method is used to assess the incident SPL onto the sample, i.e the SPL of the downstream wave denoted by A in Fig. 3, while the upstream reflected wave is denoted by B. Two microphone probes GRAS 40SA are placed flush to the wall, at a distance x 1 and x 2 of the sample and therefore of the wall impedance discontinuity. We note p 1 and p 2 the sound pressure they measure. The spacing between the two probes is 28 mm, which according to Boden & Abom 18 minimizes errors in the [600, 3450] Hz frequency range. In order to also minimize measurement errors at lower frequencies, the measurements are repeated by switching the two microphones and an average of the obtained frequency response function is taken: this makes it possible to overcome the intrinsic phase shift between the microphones. Using the complex notation, and by introducing the angular frequency ω = 2πf , the pressure field in the test section at a given position x reads p(x, t) = Re P (x)e jωt ,

with P (x) = Ae -jk + x +Be jk -x where k + and k -are the wave numbers, A and B are the complex amplitudes of the forward and backward waves, and j is the complex unit. The x-axis is oriented from left to right and has its origin at the liner upstream location, i.e. at the position of the impedance discontinuity. For a uniform flow of Mach number M , we have:

k ± = ± ω/c 0 1 ± M = ±k 1 ± M , (2)
with k = ω/c 0 . The two-microphone method allows the assessment of the sound pressure level S A of the downstream wave (i.e. 20 log P rms A 2.10 -5 with P rms

A = |A| √ 2
) from the sound pressure levels S 1 and S 2 measured at the microphones, by solving the following set of equations:

Ae -j k 1+M x + Be j k 1-M x = P 1 , (3)
Ae -j k 1+M x + Be j k 1-M x = P 2 , (4)
which yields the following expressions for the SPL of the downstream and upstream waves (in decibels, S B being defined from |B| in the same way as S A has been defined from |A|):

S A = S 1 -20 log 2| sin k∆x 1 -M 2 | + 10 log 1 + 2|H 21 | cos(φ 21 - k∆x 1 -M ) + |H 21 | 2 (5)
S B = S 1 -20 log 2| sin k∆x 1 -M 2 | + 10 log 1 + 2|H 21 | cos(φ 21 - k∆x 1 + M ) + |H 21 | 2 (6)
with ∆x = x 2 -x 1 and H 21 = |H 21 |e iφ21 = P 2 /P 1 , the frequency response function between the microphones. More details on the resolution process can be found in a previous paper. [START_REF] Mery | Experimental assessment of the effect of temperature gradient across an aeroacoustic liner[END_REF] In the present study, M is chosen equal to M b , the bulk flow Mach number.

Non-intrusive measurement of the acoustic velocity field

The B2A duct is equipped with a LDA system, in order to measure the streamwise and vertical velocity components u and w of the flow. As this measurement technique is time-resolved, both the mean and fluctuating parts of the velocity can be measured. Especially, the acoustic velocity field can be obtained with a specific post-processing technique. [START_REF] Leon | Measurement of acoustic velocity components in a turbulent flow using LDV and high-repetition-rate PIV[END_REF] The acoustic velocity is defined here as the component of the signal that is correlated with the loudspeaker signal. It can be educed from the extraneous noise (mainly due to the turbulent flow) by a technique similar to the three-microphone signal enhancement technique. [START_REF] Chung | Transfer function method of measuring in-duct acoustic properties. I. Theory[END_REF][START_REF] Nance | Limitations of the three-microphone signal enhancement technique[END_REF] It consists in calculating the cross-spectral density function G Ui ,ls between the velocity signal U i and the loudspeaker signal ls (see [START_REF] Minotti | Characterization of an acoustic liner by means of Laser Doppler Velocimetry in a subsonic flow[END_REF][START_REF] Lavieille | Measurement of acoustic quantity fields in duct flow by Laser Doppler Velocimetry[END_REF] ). The auto-spectral density function of the acoustic velocity reads

G ui = |G Ui,ls | 2 G ls , (7)
where G ls is the auto-spectral density function of the loudspeaker signal. The acoustic velocity in the frequency domain is then given by

u i = G ui exp[jΦ(U i /ls)], (8)
where the phase of the acoustic velocity, referenced by the loudspeaker, is defined as

Φ(U i /ls) = arctan Im(G Ui,ls ) Re(G Ui,ls ) . ( 9
)
The uncertainty in the measurement of the velocity field is estimated with a Bootstrap approach which allows to evaluate the variance associated to the amplitude and phase of the measured field. [START_REF] Roncen | Statistical Inference Method for Liner Impedance Eduction with a Shear Grazing Flow[END_REF]

Test Matrix

The different tested configurations for the IFAR sample are presented in Table 2. The acoustic excitation consists of 12 pure tones distributed between 504 Hz and 2824 Hz, each having the same acoustic level. The multi-sine excitation is set in order to have the same incident sound pressure level for each tone. The 120 dB multi-sine excitation has thus an incident OASPL of 130 dB, and the 130 dB multi-sine excitation has an OASPL of 140 dB. The selected frequency for mono sine excitations is close to the resonance of the sample (i.e. close to the maximum of absorption): f = 1528 Hz.

The test matrix for the ONERA sample is similar to the test matrix for the IFAR sample. This time, the excitation is composed of 10 pure tones, distributed between 992 Hz and 3136 Hz. The resonance of the liner occurs around f = 1992 Hz, this frequency is thus chosen for the single tone excitation (case #8).

III. Impedance eduction approach

A. Formulation

In the B2A configuration, an impedance eduction method based on the two-dimensional time-harmonic linearized Euler equations is used. The underlying model is briefly summarized here for convenience, but all details can be found elsewhere. [START_REF] Primus | An adjoint-based method for liner impedance eduction: Validation and numerical investigation[END_REF] The LEE, written in non-conservative form, with an e jωt time dependence, assuming homentropic flow is

Lϕ = 0, (10)
where

Lϕ = jωϕ + A x ∂ϕ ∂x + A z ∂ϕ ∂z + Bϕ, (11)
and

A x =    U 0 c 0 0 U 0 c 0 0 U    , A z =    W 0 0 0 W c 0 0 c 0 W    , B =    ∂ x U ∂ z U 0 ∂ x W -∂ x U 0 0 0 0    .
Components of vector ϕ = u, w, p ρ0c0 , where ρ 0 is the density of the mean flow, represent the acoustic perturbations around the sheared mean flow of axial and transverse velocities U and W , respectively. The flow being assumed homentropic, the energy equation is replaced by the state equation p = c 2 0 ρ 0 . As we consider a shear mean flow profile, the impedance boundary condition on the lined wall is the standard one

p = ±ρ 0 c 0 Zw, (12)
where the sign is chosen depending on whether the upper or lower wall is considered. Z is first considered as constant all along the lined zone. To avoid any singularities in the rigid wall case, the boundary condition ( 12) is expressed as a function of the normal incidence reflection coefficient β = (Z -1)/(Z + 1). At the source plane, the state vector ϕ is taken to be equal to Cϕ 0 , where ϕ 0 is the plane-wave solution in a rigid duct with uniform flow Mach number M 0 , corresponding to the bulk Mach number of the actual shear flow U (z). As this plane-wave solution is chosen with an arbitrary amplitude taken to be equal to 1, the complex coefficient C stands for the source amplitude. C can be either set to the measured value of the acoustic pressure at the source plane, when available, or included into the eduction process to reproduce at best the measured acoustic field. At the exit plane, either a non-reflecting or an exit impedance boundary condition can be enforced. The first option is only used when the data suggests that this is proper, i.e. when no standing-wave can be observed in the downstream hard wall section. As the LDA measurement system does not extend to the exit plane, this information can only be provided by microphone measurements downstream of the lined section. As these additional measurements are not systematically performed, the exit impedance boundary condition is commonly used. This will be the case for all the computations shown in this paper. As a consequence, during the search procedure, both C and Z t are used as design variables, in addition to Z.

A discontinuous Galerkin (DG) scheme is chosen to solve Eq. ( 10) and the associated boundary conditions. In this method, discontinuities are allowed at the interface between two elements. An example of how such a method can handle discontinuities at hard-soft wall interfaces is found elsewhere. [START_REF] Pascal | Discontinuous Galerkin Method for acoustic modes computation in lined ducts[END_REF] A triangular mesh of Lagrange P 1 elements is used. To ensure the connection between elements and to apply the boundary conditions, a numerical flux is defined. An upwind numerical flux is used to ensure the connection between interior cells, while a centered flux is chosen at the boundaries (see previous papers 5, [START_REF] Piot | Liner impedance eduction technique based on velocity fields[END_REF] for more details).

The inverse problem is defined by the minimization of the following objective function:

J red = Ω obs {ϕ | DG -ϕ | Meas } {ϕ | DG -ϕ | Meas } dx dz Ω obs ||ϕ | Meas || 2 dx dz , ( 13
)
where Ω obs is the observation region. For this study, the impedance eduction method is based on LDA measurements above the test liner (see Fig. 4), Ω obs is a rectangular (x, z) region and the (u, w) velocity components of the state vector are considered in function (13).

In this study, the optimization problem is solved using a CMAES method, with the help of a surrogate model. A statistical inference method developed by Roncen et al. in 2018 can be also applied to compute uncertainties on the educed values. [START_REF] Roncen | Statistical Inference Method for Liner Impedance Eduction with a Shear Grazing Flow[END_REF] This method is based on a Bayesian approach, and uses a MCMC (Markov Chain Monte Carlo) algorithm instead of the CMAES technique. Coupled with the bootstrap method for estimating the errors on LDA measurements, it gives a good appreciation of the confidence intervals on the educed impedance.

B. Classical impedance eduction results

This section presents the results of the CMAES algorithm on the five cases presented in Table 2. The goal is to study the behavior of the educed impedance with respect to the bulk Mach number and the incident SPL.

Figure 5 highlights the effect of the Mach number on the impedance. The main effects are on the resistance which increases with the Mach number. The reactance is almost the same and on this liner configuration, no effect is noticeable on the reactance term, which is a known classical result.

Figure 6 is dedicated to show the SPL effect on the impedance, at a fixed grazing flow speed. One can notice an increase of the resistance when the OASPL is increased at the frequency where the absorption is maximum, i.e. near the liner resonance (X = 0).

In the case of mono-sine excitation, the incident SPL and the OASPL are the same as there is only one tone. Thus, the OASPL for an incident SPL of 140 dB and a mono-sine excitation is very close to the OASPL for a multi-sine excitation with 130 dB on each tone. For these two cases, the educed resistance values shown on Fig. 6 are very comparable: 0.307 and 0.345 for case #3 and #4 respectively. For an incident SPL of 150 dB, the effect is greater: the resistance increases to 0.52 around the liner resonance. However, the reactance is unresponsive to the sound pressure effect and remains the same for the considered frequency.

The incident OASPL is the driving parameter to exhibit the effect of the sound pressure level on the resistance. In the following sections, we will focus on the monosine excitation results since this excitation enables to reach higher levels (up to 150 dB), so as to ensure a maximum effect on the impedance. IV. Taking into account non-linear behavior in the eduction process

A. Streamwise varying liner impedance boundary condition

In the standard eduction process, the impedance boundary condition on the lined wall is constant all along the liner. High sound pressure level implies the creation of vortices and their detachment is done alternately on both sides of the perforation over two half-periods of the acoustic excitation. The discharge coefficient is defined by the ratio between the core inflow-to-outflow area on each orifice of the perforated plate. The inflow area is reduced due to the apparition of a "vena contracta" due to the vortex shedding. This effect was first exhibited by Hersh. [START_REF] Hersh | The Acoustic Behavior of Helmholtz Resonators Exposed to High Speed Grazing Flows[END_REF] Cummings 11 proposed a calculation of this discharge coefficient C D . The effect of the sound pressure level on the liner resistance is presented as an additive term related to a constant coefficient and the normal acoustic velocity |v| in each hole. The model on the resistance term can be written as:

R = R L + C|v|, (14)
where R L is the value of the resistance in the linear regime, C is a correction coefficient related to the discharge coefficient C D and |v| is the normal acoustic velocity. This last component is directly linked to the sound pressure level. Cummings proposed a way of calculating this discharge coefficient. Tonon et al. proposed other formulations for the calculation of the discharge coefficient. [START_REF] Tonon | Quasi-steady acoustic response of wall perforations subject to a grazing-bias flow combination[END_REF] Zhang & Bodony 9 performed DNS computation and confirmed that the discharge coefficient is directly linked to the sound pressure level. Very recently, Rienstra & Singh 10 managed to analytically calculate this non-linear effect and showed that this effect is even more important at the resonance of the liner. In the rest of the article, we will focus on excitation frequencies close to the resonance. The challenge here is to catch the impedance variation due to high sound pressure level with our eduction procedure with shear grazing flow. The idea is to educe the non-linear part of the resistance while taking into account the decrease of the sound pressure level along the lined region. In the stream-wise direction, the impedance of the liner is no more constant and reads

Z(x) = R L + jX + R N L (x), ( 15
)
where R N L (x) is a decreasing function.

Different models are proposed for R N L (x). All of them include a length parameter L τ which represents the length of the part of the liner over which the resistance is expected to change due to the SPL effect: when x > L τ , R N L (x) vanishes and only the constant term R L remains. Thus, the only difference between the models is the function R N L (x): one simple approach is to use decreasing polynomial functions of degree 1 or higher, another possibility is the use of exponential functions. The three models retained for the study are shown in Table 3. For simplicity, the liner is considered to start at x = 0 even if it not always exactly the case in the experimental setup. These three variable impedance boundary conditions are implemented to the LEE simulation and the educed parameters are modified accordingly.

# Formulation Z(x) Educed parameters (0) Constant R + jX (R, X, Z t , C) (I) Affine R L + jX + D(x -L τ ) × 1 0<x<Lτ (x) (R L , X, D, L τ , Z t , C) (II) Polynomial R L + jX + (D 1 (x -L τ ) + D 2 (x -L τ ) 2 ) × 1 0<x<Lτ (x) (R L , X, D 1 , D 2 , L τ , Z t , C) (III) Exponential R L + jX + A exp -x Lτ (R L , X, A, L τ , Z t , C).

B. Eduction results with a variable impedance value

In this section, the new eduction process is applied to the measurement on the IFAR liner sample.

Comparison of the three variable conditions

Figure 7 shows the results for the 3 new boundary formulations and the classical constant impedance for an incident SPL of 150 dB, with M b = 0.1 and f = 1528 Hz. The three impedance profiles are close in terms of evolution along the liner and starting value at x = 0.

To emphasize the efficiency of these new boundary conditions, we rely on the cost function. In Table 4, (I), (II) and (III) liner impedance boundary conditions are compared for the case with highest OASPL, where the effects are expected to be the most important. Changing from a constant to a variable impedance is clearly efficient to minimize the cost functions as the new cost is 6 times smaller than the original cost, however the improvement between conditions (I), (II) and (III) is small. Due to the way the calculation has to be computed, using either (II) or (III) liner impedance boundary conditions implies more time-consuming calculations with a low benefit on the cost function minimization.

In Table 5, the affine model (I) is applied for several sound pressure excitations. It can be noticed that as the incident SPL becomes greater the use of the variable liner impedance boundary condition is efficient to minimize the cost function. On the contrary, when the incident SPL is low (as in case #1), the gain is not significant. The evolution along the liner is thus indeed linked to SPL effects. This analysis of the cost functions shows that the affine liner impedance boundary condition (I) is sufficient. Moreover, it gives a very simple numerical formulation. In the following sections, we will therefore only focus on this boundary condition and compare it with the classical constant impedance. For this part, only the four cases with the same Mach number (#1 to #4) are studied. The numerical values of the relevant educed parameters for the affine condition (R L , D and L τ /L total ) are compared with the classical resistance R in Table 6. For the three cases with a high incident SPL (over 130 dB per tone), the values of D are very close. For case #1, D has a lower value, which suggests that the effect is slightly different in this case. The values of R L are very close for all four cases, and the ratio L τ /L T otal increases with the incident SPL. These results indicate that the affine boundary condition is able to take into account the effects of an increase of the incident SPL. The other educed values (the reactance and the incident source characteristics) remain close to each other even when the variable impedance formulations are used, as seen in Tables 7 and8. The reactance values with formulation (I) are always lower than the values with formulation (0), however the difference is constant, small (< 0.05), and independent from the SPL. This highlights the consistency of the new boundary condition model, as only the educed parameters relative to the impedance condition are modified.

Case #1 seems to be a limit case where the affine condition is not optimal (as the gain in cost function is small), even if the educed values of R L and L τ /L total are coherent with the other cases. More details on the limitations are presented in Part C.

Figure 8 illustrates the eduction results. For each incident SPL level, the new liner impedance condition (solid lines) and the constant impedance condition (dashed lines) are plotted. The three profiles are very similar, and the coherence of the new condition is clear: the resistance value at x = 0 increases with the incident SPL, but the values at the end of the affine evolution are very close for all three cases.

Figure 9 shows the evolution of the liner resistance value along the streamwise direction for the two monosine cases with OASPL 140 dB and 150 dB (#3 and #4). The curve is superimposed with the sound pressure field obtained by a LEE simulation with the educed impedance values imposed at the lined wall boundary. For high SPL on the liner, the resistance is high and decreases with the SPL. As soon as a threshold value of the SPL is achieved, the educed resistance is constant along the liner. The threshold value is around 130 dB for both cases. This behavior shows that with the affine impedance condition, the non-linear effects directly linked to the high SPL are visible. It is also interesting to note that on Fig. 7, the points when the decrease of the resistance stops are close for the two polynomial models. This indicates that the threshold is indeed linked to the physical effect regardless of the chosen formulation of the impedance model.

C. Limitations of the approach

In this section, in order to ensure the relevance of the impedance model in other cases, the affine boundary condition was tested with the ONERA liner sample. Previous studies [START_REF] Primus | ONERA-NASA cooperative effort on liner impedance eduction[END_REF] showed that this liner has a linear behavior regarding the incident SPL up to 130 dB: the affine impedance boundary condition is therefore expected to be unnecessary in this range. However, it should give a result coherent with the one given by the constant boundary condition.

Only the two cases with incident SPL higher than 130 dB were included in the eduction process. Table 9 sums up the educed results for the two considered cases (#7 and #8) and for both boundary conditions. For both cases, the values of R and R L are very close. Comparing the cost functions shows that the gain obtained by changing the boundary condition is very small: for the multi-sine case, the cost function goes from 44 to 36, and it stays exactly the same for the mono-sine case.

Figure 10 shows the educed resistances with the constant boundary condition (0) and the affine boundary condition (I) for case #7. The constant educed resistance for case #8 is also plotted (dashed line) for comparison. The error margins computed using the statistical method developed by Roncen et al. [START_REF] Roncen | Statistical Inference Method for Liner Impedance Eduction with a Shear Grazing Flow[END_REF] are also added for the two constant cases (the inference method is not implemented with the affine boundary condition). One can notice that the educed values are close to each other, and that the variations all stay within the uncertainty margin. This means that there are no effects of the incident SPL on the resistance of the ONERA sample for this bulk Mach number condition. For the mono-sine case #8, D is positive and the ratio L τ /L total is inferior to 1 %. These values have little meaning compared to all the other results. However, as the resistance values and cost functions are exactly the same, this result confirms the consistency of our affine boundary condition: it is capable of giving back the constant impedance as a result when a variable impedance is not needed (in this case, due to the linear behavior of the sample). It also indicates that the variable impedance profile that was found in case #7 is questionable.

It should also be noted that the effect of the grazing flow is more significant than the effect of the SPL. Thus, at some point, the effect of the incident SPL may be present but too low compared to the flow to be correctly taken into account, which could explain the slight difference between case #1 and the other cases with the IFAR sample. This conflict between the two effects also makes it difficult to conclude on the results given by the variable boundary conditions without conducting a preliminary study on the behavior of the sample (using the classical constant conditions).

V. Conclusion

In this paper, a new impedance eduction technique has been presented. The direct problem of wave propagation in a flow duct subject to a grazing flow was represented by the two-dimensional Linearized Euler Equations and it was solved using a Discontinuous Galerkin scheme. The classical impedance boundary condition in the eduction process was modified to include a space-dependent variable term, based on the research by Cummings. 11 This modification has allowed to explain some aspects of the influence of incident SPL on the impedance of a liner. For liners with a non-linear behavior with respect to SPL, the new method has shown the existence of a threshold value above which the effects of the incident SPL on the resistance of the liner can be described with a space-dependent resistance. Under this threshold, the resistance becomes constant. The existence of this threshold confirms that SPL effects are well taken into account by using the affine condition. On the other hand, when the effect of the SPL is weaker than the grazing flow effect, a close match between the classical method and the new approach was observed, with no gain in efficiency. The new condition should thus not be used in these cases, even if its agreement with previous results is quite good. The use of this affine formulation should be useful in comparing similar samples tested in different setups. Indeed, as the educed resistance is fully independent from the sample length, different setups using different sample length are expected to find the same results for a given liner.

Figure 1 .

 1 Figure 1. Sketch of a conventional SDOF liner

Figure 2 .

 2 Figure 2. Sketch of B2A Aero-Thermo-Acoustic test bench.

Figure 3 .

 3 Figure 3. Sketch of the wave sorting procedure.

Figure 4 .

 4 Figure 4. Impedance eduction from LDA measurements

Figure 5 .Figure 6 .

 56 Figure 5. Effect of the bulk Mach number on the eduction result for multi-sine source with 130 dB incident SPL on each tone, (a) Normalized resistance and (b) Normalized reactance

Figure 7 .Figure 8 .

 78 Figure 7. Resistance eduction for a SPL of 150 dB for a single tone excitation f = 1528 Hz, comparison between the different boundary formulations Figure 8. Resistance eduction for a single tone excitation f = 1528 Hz, comparison between (0) and (I) boundary conditions for different incident SPLs

Figure 9 .

 9 Figure 9. Resistance evolution along the liner (in red) and sound pressure field simulated with the LEE (background color and isocontours) for M b = 0.1, single tone excitation, f = 1528 Hz, SP L = 140 dB (top) and SP L = 150 dB (bottom)

Figure 10 .

 10 Figure 10. Resistance eduction for the two multi-sine cases on the ONERA liner sample, for both boundary conditions

Table 1 .

 1 Characteristics of the liner samples

		IFAR sample	ONERA sample
	Facesheet thickness (δ)	0.86 mm	0.8 mm
	Hole diameter (φ)	1.04 mm	0.3 mm
	Percent Open Area (POA)	11 %	5 %
	Core cell size	10.16 mm x 10.16 mm 9.5 mm (diameter)
	Partition thickness	2.54 mm	0.1 mm
	Core depth	50 mm	20 mm
	B. Measurement
	1. ONERA B2A duct

Table 2 .

 2 Test matrix # Case Sample SPL [dB] on each tone OASPL [dB] M b

	Excitation type

Table 3 .

 3 Impedance boundary condition formulations

Table 4 .

 4 Comparison between (I), (II) and (III) formulations, OASPL = 150 dB, f = 1528 Hz, M b = 0.1 (case #4)

	Boundary formulation Cost function
	Constant impedance	1288.42
	(I)	217.60
	(II)	213.50
	(III)	200.89

Table 5 .

 5 Cost function comparison for M b = 0.1 for several incident SPL, f = 1528 Hz

	Case	Cost function
		Formulation (0) Formulation (I)
	#1 Multi sine 120dB incident SPL per tone	8.43	7.45
	#2 Multi sine 130dB incident SPL per tone	49.25	27.71
	#3 Mono sine 140dB incident SPL	362.21	154.17
	#4 Mono sine 150dB incident SPL	1288.42	217.60

Table 6 .

 6 Resistance value for M b = 0.1 for the constant (0) and the affine (I) boundary conditions, f = 1528 Hz

	Case	R	R L	D	L τ /L total
	#1 Multi sine 120dB incident SPL per tone 0.211 0.093 -5.95 0.41
	#2 Multi sine 130dB incident SPL per tone 0.307 0.114 -7.41 0.47
	#3 Mono sine 140dB incident SPL	0.345 0.112 -7.16 0.53
	#4 Mono sine 150dB incident SPL	0.52	0.14	-7.64 0.69

Table 7 .

 7 Incident source characteristics for both impedance boundary conditions

	Case	Amplitude |C|
		Formulation (0) Formulation (I)
	#1 Multi sine 120dB incident SPL per tone	0.089	0.090
	#2 Multi sine 130dB incident SPL per tone	0.333	0.331
	#3 Mono sine 140dB incident SPL	1.027	1.016
	#4 Mono sine 150dB incident SPL	2.241	2.193
		Phase θ c (in degrees)
		Formulation (0) Formulation (I)
	#1 Multi sine 120dB incident SPL per tone	-118	-120
	#2 Multi sine 130dB incident SPL per tone	-126	-127
	#3 Mono sine 140dB incident SPL	-119	-120
	#4 Mono sine 150dB incident SPL	-131	-132

Table 8 .

 8 Reactance values on liner for both impedance boundary conditions

	Case	Reactance
		Formulation (0) Formulation (I)
	#1 Multi sine 120dB incident SPL per tone	0.18	0.13
	#2 Multi sine 130dB incident SPL per tone	0.17	0.11
	#3 Mono sine 140dB incident SPL	0.17	0.12
	#4 Mono sine 150dB incident SPL	0.13	0.08

Table 9 .

 9 Resistance value for the ONERA sample, for M b = 0.2 for both liner impedance boundary conditions, f = 1992 Hz

	Case	R	R L	D	L τ /L total
	#7 Multi sine 130dB incident SPL per tone 1.432 1.259 -83.73	0.15
	#8 Mono sine 130dB incident SPL	1.345 1.345	37.2	0.007