Thomas Lampert

Odyssée Merveille

Jessica Schmitz

Germain Forestier

Friedrich Feuerhake

Cédric Wemmert

STRATEGIES FOR TRAINING STAIN INVARIANT CNNS

Keywords: multi-stain analysis, digital pathology, whole slide images, deep convolutional networks

An important part of Digital Pathology is the analysis of multiple digitised whole slide images from differently stained tissue sections. It is common practice to mount consecutive sections containing corresponding microscopic structures on glass slides, and to stain them differently to highlight specific tissue components. These multiple staining modalities result in very different images but include a significant amount of consistent image information. Deep learning approaches have recently been proposed to analyse these images in order to automatically identify objects of interest for pathologists. These supervised approaches require a vast amount of annotations, which are difficult and expensive to acquire-a problem that is multiplied with multiple stainings. This article presents several training strategies that make progress towards stain invariant networks. By training the network on one commonly used staining modality and applying it to images that include corresponding but differently stained tissue structures, the presented unsupervised strategies demonstrate significant improvements over standard training strategies.

INTRODUCTION

The field of digital pathology emerged with the introduction of whole slide imaging (WSI) scanners and lead to the development of new tools for analysing histopathology slides [START_REF] Gurcan | Histopathological image analysis: A review[END_REF]. Meanwhile, deep learning architectures were developed that outperformed most state-of-the-art algorithms in image analysis. Deep learning has thus been quickly applied to solve digital pathology problems [START_REF] Bel | Automatic segmentation of histopathological slides of renal tissue using deep learning[END_REF][START_REF] Janowczyk | Deep learning for digital pathology image analysis: A comprehensive tutorial with selected use cases[END_REF][START_REF] Das | Multiple instance learning of deep convolutional neural networks for breast histopathology whole slide classification[END_REF]. Nonetheless supervised deep learning approaches require large amounts of annotated data. Unlike classic computer vision applications, annotating histopathology slides requires in-depth knowledge and therefore can only be done by trained experts such as pathologists. Consequently, the acquisition of large annotated histopathological datasets is a complicated task [START_REF] Schaadt | Role of task complexity and training in crowdsourced image annotation[END_REF].

Image datasets in digital pathology applications often consist of consecutive slides stained differently, each staining providing specific information on the same region of interest (see Figure 1, first row). Even though differently stained slides appear very different, there is often a significant amount of consistent information between them. For example, they may both share the same counterstain (e.g. haematoxylin), or they may highlight different parts of the same structure.

The analysis and integration of information from different stainings is usually performed with reference to a specific organ, structure, or pattern observed in the tissue. For example, to diagnose pathologies such as breast cancer or kidney allograft rejection it is necessary to study the inflammatory microenvironment of the organ. In these cases, the relevant information is the distribution of immune cells (e.g. macrophages or lymphocytes) in relation to important structures of the organ, such as glomeruli for the kidney, or lobules for the breast.

To automatically perform such an analysis, the structure of interest (glomeruli, lobules, etc.) should be detected in each section irrespective of the individual staining modality. This is a typical application of deep learning, assuming that annotations for each staining are available. Thus, there is a high demand for annotations in digital pathology, currently limited by the availability of experts and the complexity of the required expert knowledge [START_REF] Schaadt | Role of task complexity and training in crowdsourced image annotation[END_REF].

This article presents several training strategies that make progress towards stain invariant networks (i.e. a network that can generalise between stainings). By training the network using a commonly acquired staining in renal pathology Periodic acid-Schiff reaction (PAS, a histochemical reaction to detect certain types of suger-containing tissue components) and applying the same network to images with different stainings, the presented strategies demonstrate significant improvements over standard training strategies. These can be used in settings where there is not enough data or annotations for the desired application staining. Alternatives in the literature are concerned with transfer learning or domain adaptation strategies where some take advantage of networks pretrained on huge natural image datasets [START_REF] Bayramoglu | Transfer learning for cell nuclei classification in histopathology images[END_REF]. Most of these works consider the problem of tissue variability between images that have the same staining [START_REF] Gadermayr | Domain adaptive classification for compensating variability in histopathological whole slide images[END_REF][START_REF] Lafarge | Domainadversarial neural networks to address the appearance variability of histopathology images[END_REF] or between images from different tissues [START_REF] Kandemir | Asymmetric transfer learning with deep gaussian processes[END_REF]. These approaches are much more complex to implement as they require specific network architectures.

The adopted CNN architecture and proposed training strategies are presented in the next section, Section 3 presents inter-staining glomeruli segmentation results and analysis thereof, and Section 4 presents the study's conclusions.

METHOD

CNN architecture

The U-Net architecture [START_REF] Ronneberger | U-Net: Convolutional networks for biomedical image segmentation[END_REF] is adopted as it has been proven successful in biomedical imaging [START_REF] Litjens | A survey on deep learning in medical image analysis[END_REF] and, in particular, glomeruli detection [START_REF] Bel | Automatic segmentation of histopathological slides of renal tissue using deep learning[END_REF]. Glomeruli segmentation is framed as a two classes problem: glomeruli (patches centred on a glomeruli), and tissue (that does not contain a glomeruli, randomly sampled). The slide background (non-tissue) is removed by thresholding each image by its mean value then removing small objects from the result and closing holes. Initial experiments revealed that the same training parameter values lead to convergence irrespective of the strategy used (batch size of 8, learning rate of 0.0001, 60 epochs, and the network that achieves the lowest validation loss is kept). The input patch size is 512 × 512 pixels.

To mitigate the lack of training samples, the use of elastic deformation [START_REF] Simard | Best practices for convolutional neural networks applied to visual document analysis[END_REF] is proposed by the original authors. In addition to this (using the parameters σ = 10, α = 100), the following augmentations are applied with an independent probability of 0.5 (batches are augmented 'on the fly'): affine: random rotation sampled from the interval [0 • , 180 • ], random shift sampled from [-205, 205] pixels, random magnification sampled from [0.8, 1.2], and horizontal/vertical flip; noise: additive Gaussian noise1 with σ ∈ [0, 2.55]; blur: Gaussian filter 1 with σ ∈ [0, 1]; brightness enhance2 with a factor sampled from [0.9, 1.1]; colour enhance 2 with a factor sampled from [0.9, 1.1]; contrast enhance 2 with a factor sampled from [0.9, 1.1]; stain variation by colour deconvolution [START_REF] Tellez | Whole-slide mitosis detection in H&E breast histology using PHH3 as a reference to train distilled staininvariant convolutional networks[END_REF], α sampled from [-0.25, 0.25] and β from [-0.05, 0.05]. These values were chosen as they result in realistic images. All samples are standardised to [0, 1] and normalised by the mean and standard deviation of the training set.

Training strategies

The standard approach for segmenting histopathological slides is to present the three colour channels to the network to exploit both colour and texture/structure information (referred to herein as RGB). Henceforth 'structure' refers to tissue appearance (i.e. is not used in a biological sense).

It is hypothesised that since histopathological slides of the same staining exhibit very low colour variance when com-pared to natural images, a deep learning approach will rapidly learn to rely upon structures being present in specific channels (or combinations thereof), which may limit the network's ability to generalise to unseen stainings. The effects of this are already known when dealing with inter-laboratory variance of the same staining [START_REF] Tellez | Whole-slide mitosis detection in H&E breast histology using PHH3 as a reference to train distilled staininvariant convolutional networks[END_REF][START_REF] Gallego | Glomerulus classification and detection based on convolutional neural networks[END_REF].

Objects to be segmented, e.g. glomeruli, are generally easily identified between stainings as globally they exhibit the same structure and texture, see Figure 1. It should therefore be possible to bias the network to learn stain invariant features. This work investigates this possibility by modifying the data presented to the network in an unsupervised manner. Borrowing domain adaptation terminology, herein the staining used for training and validation is referred to as as the source staining (irrespective of any transformations), and the stainings to which the network are applied as target stainings.

The simplest approach is to remove the colour component from the data (referred to herein as Greyscale). Each sample is converted to greyscale using the formula I = 0.2125R + 0.7154G + 0.0721B 1 . The result has all colour information compressed into one channel, forcing the network to learn structural features.

A more biologically motivated approach is to extract the haematoxylin channel from each staining by deconvolving the image using Ruifrok and Johnston's approach [START_REF] Ruifrok | Quantification of histochemical staining by color deconvolution[END_REF] (referred to herein as Haematoxylin). Haematoxylin is commonly used as a counterstain and should therefore highlight the same structures in each staining. This strategy should therefore remove stain specific information and force the network to learn features that generalise well.

The above-mentioned strategies either compress information contained in the colour channels (Greyscale) or focus on consistent inter-staining information (Haematoxylin) with the aim of simplifying the problem. They are therefore applied during training, validation, and testing. Another approach is to preserve the colour information, but modify it in such a way that the network does not learn to rely on colour specific structure. These are therefore only applied during training and validation. A brute-force approach to this is to randomly permute the colour channels. This strategy-Channel Swap-is simple to implement; however, the range of colours generated will be far greater than those encountered in application.

The last strategy, named Colour Transfer, restricts this to within meaningful bounds at the expense of being more complicated. The colour profile of each training patch is replaced with that of a randomly selected staining during training with probability N -1 N , where N is the number of stainings in the dataset. Therefore each 'staining' (including the source) is presented to the network with equal probability. The colour transfer is achieved by deconvolving the patch and applying its stain concentrations to stain vectors taken from another staining. Various approaches to this exist [START_REF] Macenko | A method for normalizing histology slides for quantitative analysis[END_REF][START_REF] Vahadane | Whole-slide mitosis detection in H&E breast histology using PHH3 as a reference to train distilled stain-invariant convolutional networks[END_REF][START_REF] Wemmert | Stain unmixing in brightfield multiplexed immunohistochemistry[END_REF] (including computer vision approaches [START_REF] Reinhard | Color transfer between images[END_REF]), in this work we use that proposed by Macenko et al. due to its simplicity (automatic determination of stain vectors) and realistic output (see Figure 1, bottom row). It is necessary, however, to have samples of the target stainings to determine their stain vectors (this requirement can be removed by using predetermined stain vectors and Ruifrok and Johnston's approach [START_REF] Ruifrok | Quantification of histochemical staining by color deconvolution[END_REF]). During training, a random patch from a random target staining is selected and the target stain vectors are determined.

N.B. the intention is to present plausible colour profiles to the network. This does not necessarily correspond to biologically relevant image information but is explored for any potential for the tested staining modalities. Potential limitations are addressed, e.g. in Figure 1 the bright red colour (chromogenic reaction used to detect CD34, a marker of blood vessel inner lining) could be successfully transferred but it is not localised as in the original image.

EXPERIMENTS AND RESULTS

Data

Tissue samples were collected from a cohort of 10 patients who underwent allograft nephrectomy after complete loss of function of a transplanted kidney due to chronic graft failure. These kidneys displayed several signs of rejection. The paraffin-embedded samples were cut into 3 µm thick sections and stained with either Jones H&E, PAS or Sirius Red, in addition to two immunohistochemistry markers (CD34 and CD68), N = 5, using an automated staining instrument (Ventana Benchmark Ultra). Whole slide images were acquired using an Aperio AT2 scanner at 40× magnification (a resolution of 0.253 µm/pixel). All the glomeruli in each WSI were annotated and validated by pathology experts by outlining them using Cytomine [START_REF] Marée | Collaborative analysis of multi-gigapixel imaging data using cytomine[END_REF]. The dataset was divided into 4 training, 2 validation, and 4 test patients. The number of glomeruli in each staining dataset was: PAS -666 (train.), 580 (valid.), 1074 (test); Jones H&E -1005 (test); Sirius Red -922 (test); CD34 -1013 (test); CD68 -990 (test). The training set comprised all glomeruli from the source staining training patients (666) and 4662 tissue patches (to account for the variance observed in non-glomeruli tissue).

Experiments

For each strategy described in Section 2.2, a CNN was trained on the source staining (PAS) and applied to the target stainings (PAS, Jones H&E, Sirius Red, CD34, and CD68). Test patient whole slides were segmented and precision, recall and F 1 score of the glomeruli were taken. To evaluate robustness, five repetitions of each experiment were performed. Table 1 shows the means and standard deviations of the repetitions for each measure.

The segmentation results trained on and applied to the source staining (PAS) form the controls. State-of-the-art performance [START_REF] Bel | Automatic segmentation of histopathological slides of renal tissue using deep learning[END_REF][START_REF] Bukowy | Region-based convolutional neural nets for localization of glomeruli in trichrome-stained whole kidney sections[END_REF][START_REF] Govind | Glomerular detection and segmentation from multimodal microscopy images using a butterworth band-pass filter[END_REF] is achieved or surpassed and this establishes a baseline for the best achievable inter-staining perfor-mance. The segmentation results trained on the source staining and applied to the target stainings (RGB strategy) form the negative controls and, as the network essentially fails to detect anything, demonstrate the need for a transfer strategy.

The first strategy relies on compressing structural information into one band: Greyscale. This vastly increases target performance compared to RGB (with no affect on source performance) and leads to decent segmentation on Jones H&E and CD34, which are structurally the most similar to PAS.

Next is a strategy to extract biological information that is consistent across different stainings-the haematoxylin counterstain. Despite this biological plausibility, it does not result in good results. Indeed, the haematoxylin transformed images vary greatly between the different stainings (see Figure 1, second row). Several factors may explain this: 1) haematoxylin's concentration relative to the primary stain may vary from one staining to another, resulting in different shades of blue and fixation amount; 2) as a counterstain, haematoxylin may become mixed with another stain in structures that are targeted by both. These result in a color mixing that is in practice not perfectly unmixed by a color deconvolution algorithm.

Although training a network using compressed structural information vastly improves inter-staining generalisation, acceptable segmentation performance is not achieved. The third strategy attempts to preserve color information but force the network not to specify the structure to particular colours. The Channel Swap strategy provides results equal to or better than the previous strategies in all but one staining (CD34), however this is achieved at the expense of stability due to the randomised colour channels.

Finally, colour variance is restricted to a more realistic range using colour transfer, giving the best results in all cases. It should be noted that the quality of the result is highly dependent on the degree of structural similarity between the target and source staining. Visually, there is a large difference between histochemical staining methods (e.g. H&E, PAS, Jones) and immunohistochemistry (antibody-mediated detection of certain structures like immune cells of blood vessel inner lining) with weak blue counterstain. Therefore, some limitations of the approach are expected, and the degree of variation may explain why detection of glomeruli in the CD68 staining did not work well with PAS as source staining.

CONCLUSIONS

It has been shown that the efficacy of simple transfer strategies depend on the degree of structural similarity between the source and target staining. The presented work dramatically improves the inter-staining segmentation performance when compared to standard training approaches, and this gain is correlated with the complexity of the strategy used. Nevertheless, it appears that a limit has been reached and strategies other than modifying training data should be the focus of future research into developing stain invariant networks.

Fig. 1 :

 1 Fig. 1: Glomeruli patch examples with different transformations. Each column corresponds to a different staining, and each row a transformation. The colour transfer (4 th row) results are applied to a PAS patch, using each staining as a target.

Table 1 :

 1 Quantitative results for each strategy trained on PAS (source staining) and tested on different stainings (target stainings). The numbers in parentheses are the standard deviations of the corresponding scores.

	Training Strategy	Score	PAS	Jones H&E	Test Staining CD68	Sirius Red	CD34
		F 1	0.901 (0.006)	0.031 (0.015)	0.000 (0.000)	0.015 (0.012)	0.011 (0.009)
	RGB	precision	0.873 (0.014)	0.023 (0.014)	0.372 (0.292)	0.009 (0.007)	0.052 (0.017)
		recall	0.932 (0.021)	0.075 (0.031)	0.000 (0.000)	0.063 (0.067)	0.006 (0.006)
		F 1	0.899 (0.004)	0.599 (0.080)	0.030 (0.033)	0.301 (0.059)	0.596 (0.050)
	Greyscale	precision	0.867 (0.013)	0.444 (0.088)	0.154 (0.098)	0.464 (0.072)	0.703 (0.058)
		recall	0.934 (0.007)	0.944 (0.007)	0.017 (0.020)	0.238 (0.078)	0.523 (0.071)
		F 1	0.872 (0.010)	0.676 (0.036)	0.002 (0.001)	0.001 (0.000)	0.042 (0.019)
	Haematoxylin	precision	0.807 (0.019)	0.609 (0.096)	0.029 (0.024)	0.003 (0.003)	0.079 (0.041)
		recall	0.948 (0.003)	0.785 (0.075)	0.001 (0.001)	0.000 (0.000)	0.034 (0.020)
	Channel Swap	F 1 precision recall	0.889 (0.005) 0.846 (0.012) 0.936 (0.013)	0.671 (0.050) 0.659 (0.063) 0.700 (0.111)	0.106 (0.111) 0.167 (0.124) 0.090 (0.108)	0.508 (0.133) 0.658 (0.116) 0.457 (0.156)	0.379 (0.139) 0.605 (0.139) 0.299 (0.146)
	Colour Transfer	F 1 precision recall	0.882 (0.015) 0.834 (0.039) 0.938 (0.015)	0.813 (0.042) 0.748 (0.078) 0.899 (0.017)	0.153 (0.083) 0.417 (0.133) 0.099 (0.059)	0.739 (0.070) 0.742 (0.106) 0.746 (0.073)	0.709 (0.028) 0.726 (0.062) 0.706 (0.089)

This work was supported by the SysMIFTA project (FKZ:031L-0085A). The authors are grateful to Nvidia Corporation for donating a Quadro P6000 GPU, which was used to conduct this research.

Using the appropriate scikit-image functions.

Using the appropriate PIL functions.


Leguyadec

Sensitivity Analysis using Sobol indices for the thermal modelling of an Electrical Machine for sizing by optimization

Keywords: electrical machine, thermal modelling, LPTN, Sobol indices

Purpose -The thermal modelling of an electrical machine is difficult because the thermal behavior depends on its geometry, the used materials and its manufacturing process. In the paper, such a thermal model is used during the sizing process by optimization of a Hybrid Electric Vehicle (HEV). The paper deals with the sensitivities of thermal parameters on temperatures inside the electrical machine to allow the assessment of the influence of thermal parameters that are hard to assess.

Design/methodology/approach -A sensitivity analysis by Sobol indices is used to assess the sensitivities of the thermal parameters on electrical machine temperatures. As the optimization process needs fast computations, a lumped parameter thermal network (LPTN) is proposed for the thermal modelling of the machine, because of its fastness. This is also useful for the Sobol method that needs too many calls to this thermal model. This model is also used in a global model of a hybrid vehicle.

Finding -The difficulty is the thermal modelling of the machine on the validity domain of the sizing problem. The Sobol indices allow to find where a modelling effort has to be carried out.

Research limitations/implications -The Sobol indices have a significant value according to the number of calls of the model and their type (first order, total, etc.). Therefore, the quality of the thermal sensitivity analysis is a compromise between computation times and modelling accuracy.

Practical implications-Thermal modelling of electrical machine in a sizing process by optimization

Originality/value -The use of Sobol indices for the sensitivity analysis of the thermal parameters of an electrical machine.

I. Introduction

Sizing a hybrid vehicle and its components has proved to be a complex process. To be efficient, this requires considering the system level, the management of the system, and the component level [START_REF] Silvas | Comparison of Bi-Level Optimization Frameworks for Sizing and Control of a Hybrid Electric Vehicle[END_REF][START_REF] Akli | Integrated optimal design of a hybrid locomotive with multiobjective genetic algorithms[END_REF]. Moreover, multiphysics aspects have to be taken into account, such as the mechanical model of the vehicle or the engine, the electromagnetic model of the electrical machine (EM) and the thermal aspects of the components. To guarantee a good design, all these aspects are often considered in the same optimization process [START_REF] Silvas | Comparison of Bi-Level Optimization Frameworks for Sizing and Control of a Hybrid Electric Vehicle[END_REF][START_REF] Song | Component sizing optimization of plugin hybrid electric vehicles with the hybrid energy storage system[END_REF]. The drawbacks are the complexity of this approach and the associated computer effort. Because of this, simplified models of the components have to be developed to be inserted in a global optimization process.

In this scope, in [START_REF] Reinbold | Optimal sizing of an electrical machine using a magnetic circuit model: application to a hybrid electrical vehicle[END_REF][START_REF] Vinot | Comparison of different power-split architectures using a global optimization design method[END_REF], a global optimization method has been developed. It includes a cybernetics approach to represent the vehicle. The problem of energy management (electrical mode choice and battery power choice) is embedded as a sublevel of an optimization to assess the optimal fuel consumption of a given vehicle architecture. A sizing parameterized electromagnetic model of the electrical machine is used to calculate the losses and the maximum performances. A lumped thermal model (Mellor, 1991, Staton et al;, 2005) is linked to this electromagnetic model to assess the maximum available torque of the EM in steady state operating. Whereas the validation of the electromagnetic model is relatively easy by means of measurements and finite element method simulation, it is more difficult to develop accurate thermal models due to the uncertainty of many parameters. The accuracy of these models and their sensitivities to different parameters have then to be assessed. However, simple analytical models are sometimes difficult to define, mainly in the science of heat where some parameters may be empiric. So, the sensitivity analysis is useful to decide of the modelling effort (so a computation effort) and the parameters to be considered in the optimization process (which impacts the optimization problem size [START_REF] Delinchant | Derivative based global sensitivity analysis for screening and robustness studies of electromagnetic devices[END_REF]). This paper focuses on the thermal model of the electrical machine and its validation, and the use of the Sobol method [START_REF] Sobol | Sensitivity estimates for nonlinear mathematical models[END_REF] to study the parameter sensitivities for this model [START_REF] Delinchant | Derivative based global sensitivity analysis for screening and robustness studies of electromagnetic devices[END_REF].

The second part presents the principle of the Sobol sensitivity analysis approach. The third part describes the modelling and optimization levels referring to previous works. Then, it focuses on the thermal modelling. Finally, the fourth part presents a sensitivity analysis of the thermal model.

II. Sensitivity analysis using Sobol indices

II.1. Sensitivity analysis general framework

A simulation model can be represented by a function:

() yf  x (1)
Where y is one output of the model to analyze, and x=(x 1 , x 2 , …, x n ) is a vector whose size equals n. Each x i is a random variable defined by a distribution law (e.g.: Gaussian, Uniform, etc.).

The uncertainty analysis tries to give knowledge on the result obtained by a model, whereas the sensitivity analysis tries to give what are the main sources of uncertainty of the selected parameters. So, a sensitivity analysis can be useful to extract the more influential parameters among the x components that modify the model outputs (Y).

II.2. Sobol indices definition

There are different ways to measure the importance of an input variable on the outputs. To do that, one needs an indicator that can be qualitative (eg: Morris plan) or quantitative (eg: Sobol indices). The paper focuses on the Sobol indice method [START_REF] Sobol | Sensitivity estimates for nonlinear mathematical models[END_REF][START_REF] Sobol | Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates[END_REF]. These indices are global because they are not based on any hypothesis on the distribution of parameters (apart from their independence). These indices are quantitative because they are interpreted as the part of the variance due to an input or a group of factors of inputs.

 

| i E y x is the conditional expectation. It is the function depending only on the input x i , which best approaches the output y. The error of prediction is then characterized by the expectation of its square:

    2 | i E y E y x    (2)
The simple rules of calculation on the conditional expectations give:

        2 | ( ) | ii E y E y x Var y Var E y x      (3) So, the bigger     | i
Var E y x is, the better the prediction is. Therefore,

    | i
Var E y x would be the variance of the output y if it was only a function of x i . The closer this quantity is to () Var y , the better the x i factor explain the variance of y. In other words, in such a case, the factor x i is influential.

The first order Sobol indice is then defined by the ratio of these two quantities:

    | () i i Var E y x S Var y  (4)

II.2. Sobol indices calculation

These indices are based on the interpretation of the conditional expectation in terms of forecast. To calculate the Sobol indices, the variance of the outputs of the model is used. This calculation needs the independence of the inputs, but has no constraints on their distribution. Here, the indices are estimated using sampling such as Quasi Monte Carlo [START_REF] Grandjacques | Pick-freeze estimation of sensitivity indices for models with dependent causal processes inputs[END_REF]. The estimators of expectation and variances require a high number of samples. So, the used model has to be fast. If the model is a numerical one or requires high computation time, it is recommended to use meta-modelling such as Polynomial Chaos [START_REF] Sudret | Global sensitivity analysis using polynomial chaos expansions[END_REF] or a frequency technic named RBD-FAST (Random Balance Designs -Fourier Amplitude Sensitivity Test) [START_REF] Mara | Extension of the rbd-fast method to the computation of global sensitivity indices[END_REF]. In our case, the Peek and Freeze technic has been used directly on our fast model to approximate the conditional expectation [START_REF] Gamboa | Statistical inference for Sobol pick freeze Monte Carlo method, Statistics[END_REF]. This technic is based on the following equality.



           2 2 2 2 1 2 1 2 1 2 ( | ) ( | ) ( | ) ( | ) ( ) ( ) ( , ) i i i i Var E y x E E y x E E y x E E y x E y E Y Y E Y E Y Cov Y Y      (9)
Where Y 1 respectively Y 2 are obtained with the model simulation from random samples X 1 , resp. X 2 of x in which the values of x i (the parameter of interest) are kept the same for both X 1 and X 2 . 12 1 ( , )

() i Cov Y Y S Var Y  (10)
The Pick and Freeze estimator consists in taking an empirical estimator of the numerator and the denominator. This method is robust ([68]) and its speed of convergence does not depend on the number of input variables. The basic estimator of the first order Sobol indice is defined by:

     1 1 1 1 2 1 2 1 1 1 2 2 11 11 11 ˆN N N N N N i i i i NN NN ii Y Y Y Y S YY            (11)
The quality of the estimator is based on the used sampler. Sobol advocates the use of samplers with low discrepancies also named quasi-Monte Carlo samplers. In addition, to guarantee the quality of the estimator, it is recommended to use the Bootstrap technique [START_REF] Efron | An introduction to bootstrap[END_REF] which provides a confidence interval for the estimated magnitudes.

III. Global optimization and modelling of hybrid vehicle and the electrical machine

III.1. Global optimization process

As in many complex systems, Hybrid Electric Vehicle (HEV) design needs to address different levels of modelling (system and components) and optimization (system and control). Moreover, it is usually a multiobjective problem to find the best compromise between fuel consumption, battery size, driving comfort, system cost, component volumes… Here, a parallel hybrid vehicle architecture is chosen (see Figure 1). System level parameters (power of engine P ICE , battery size N BAT , DC voltage U DC and gear ratio k CPL and k RED ) are jointly designed with EM geometrical parameters, using an optimization algorithm (X EM ) [START_REF] Reinbold | Optimal sizing of an electrical machine using a magnetic circuit model: application to a hybrid electrical vehicle[END_REF]. The energy management control level which defines the vehicle operating mode (hybrid or all electric) and the battery current (charge or discharge) for each driving condition of a driving cycle have also to be addressed to assess the fuel consumption. In such a case, a sequential design leads to a suboptimal solution, and a global approach has proved to be more efficient. A good and frequently used approach to solve the bi-level optimization problem consists of an optimization level for the system and component parameters managed by a genetic algorithm and an optimization sublevel for the control part managed by a dynamic programming method (Silvas et al., Vinot, 2016[START_REF] Akli | Integrated optimal design of a hybrid locomotive with multiobjective genetic algorithms[END_REF]. Figure 2 presents the process developed in previous works and its connection with a thermal model. The genetic algorithm, NSGA-II, is used to design system parameters (gear ratio, engine size…).

There, the model of the electrical machine is a loss map linked to a maximum torque vs. speed characteristic. Using a Reluctance network (RN) model previously developed and validated [START_REF] Reinbold | Optimal sizing of an electrical machine using a magnetic circuit model: application to a hybrid electrical vehicle[END_REF], the losses are calculated on a sampled torque vs. speed area.

A first maximum torque vs. speed characteristic is then calculated based on a maximum DC voltage limitation with no thermal limits. It is used in the global process (Figure 2) to assess the vehicle performance constraints (maximum acceleration, "perfo. model" in Figure 2 using the direct system model).

A thermal model is used to generate a steady state maximum torque characteristic with a thermal limit ("steady state model" in figure 2) used in the fuel consumption assessment. For that, dynamic programming [START_REF] Bellman | Dynamic Programming[END_REF][START_REF] Vinot | Time reduction of the Dynamic Programming computation in the case of hybrid vehicle[END_REF] is used in order to compute the minimum fuel consumption of the vehicle in a known in advance driving cycle. As this operating cycle is representative of steady state driving conditions, the thermal constraint in the EM has to be respected in a steady state fashion.

A key point of such an approach is the accuracy of the model and the required computer efforts. To tackle this issue, a parameterized thermal resistance network validated with finite element method and/or measurement, offers a good solution (sections III.2 to III.4).

III.2. Principle of LPTN modelling

To represent the thermal exchanges and to assess the maximum temperature in the steady state condition, a fast analytical thermal model, based on a lumped parameter thermal network (LPTN) is used [START_REF] Mellor | Lumped parameter thermal model for electrical machines of TEFC design[END_REF][START_REF] Staton | Solving the more difficult aspects of electric motor thermal analysis in small and medium size industrial induction motors[END_REF][START_REF] Boglietti | Evolution and modern approaches for thermal analysis of electrical machines[END_REF][START_REF] Nategh | Thermal analysis of a PMSRM using partial FEA and lumped parameter modelling[END_REF]. The LPTN is an analogy between thermal and electrical phenomena. For instance, temperature is equivalent to voltage and heat flow is equivalent to current. Thermal resistances depend on the considered thermal mode and the sizes of the exchange areas. The following equations are used to calculate the thermal resistances of the model, where  is the thermal conductivity coefficient and h is the thermal convection  conduction mode through a parallelepiped of length E and with a section S:

𝑅 𝑐𝑑,𝑝𝑙𝑝 = 𝐸 𝜆 × 𝑆
 conduction mode through a cylinder (radial), with an external radius R ext , an internal radius R int and an angle :

𝑅 𝑐𝑑,𝑐𝑦𝑙 = ln ( 𝑅 𝑒𝑥𝑡 𝑅 𝑖𝑛𝑡 ) 𝛼 × 𝜆 × 𝑆
 convection mode on a surface S:

𝑅 𝑐𝑣 = 1 ℎ × 𝑆 An analytical calculation is performed to find the temperature at every node of the equivalent circuit. In fact, it consists of solving the system of linear equations deduced from the Kirchoff's law applied at every circuit node.

III.3. Application of the LPTN modelling

The LPTN model has been applied to a permanent magnet synchronous machine commonly used in HEV. The machine used in this example is a 50kW machine (85 mm depth, diameter of 272 mm, see more details in [START_REF] Reinbold | Optimal sizing of an electrical machine using a magnetic circuit model: application to a hybrid electrical vehicle[END_REF]. Figure 3 shows a representation of the axial section of the EM and of the radial section of one slot. The sizes are not strictly respected in this figure which aims to show all the EM parts and the temperature points. The casing is cooled thanks to water channels whose temperature is externally controlled. The copper losses (CL) and the iron losses (IL) generate heat into the machine. The heat paths are represented by equivalent thermal resistances:

 conduction in dark blue  contact in light blue  convection in green.

Oblique lines represent orthoradial heat exchanges in the stator (between tooth and slot as an example). The internal cooling is ensured by an oil/air mixture (oil fog) which is splattered on the end-windings (yellow).

III.4. Validation of LPTN modelling

A finite elements model has previously been developed [START_REF] Le-Guyadec | Building of an electrical machine thermal model in the context of a hybrid electric vehicle global optimization[END_REF]. It has been useful to understand the main path of the thermal exchanges and then to adjust the topology of the LPTN model (coil-head and slot modeling). A comparison of the temperatures and the thermal exchanges in different areas of the machine has been performed for different operating points and different geometries of the EM (see Figure 4).

For the LPTN, the temperatures are those of the nodes, whereas for the finite element model (FEM) they are the average temperatures of the areas. Here, the sizes of the machine are from a reference machine described in [START_REF] Hsu | Report on toyota prius motor design and thermal manufacturing assessment[END_REF] with 935 W of Joule losses (assumed uniform in the winding) and 84 W of iron losses. The rotor iron losses are negligible in a permanent magnet machine thus the iron losses are uniformly assigned in the stator.

The differences between the temperatures computed with the two models are less than 5%. The thermal paths are also globally in good agreement with the LPTN model compared to the FEM. Figure 6 shows the power exchanged between different areas of the machine described in Figure 5 (the number in the x-axis Figure 6 corresponds to the thermal path numbered in Figure 5). The main difference is observed for the exchange between the slots and the stator (path 1): a difference of 15 % which remains acceptable. The same comparison has been performed for different operating points (i.e. involving mainly iron losses) and different geometries of the machine, mainly concerning the ratio length/diameter to simulate different machine shapes as in an optimization process. The differences between the model remain lower than 15% and the main disparities appear for "extreme" geometries not reachable in the design space (EM with high length compared to diameter). Even with the previous validations, some input parameters of the model (thermal exchange coefficients or thickness of certain areas) remain difficult to assess with accuracy. In order to know which parameters are critical to assess or which parameters have little effect on the temperatures, a sensitivity analysis was performed (see part IV).

IV. SENTITIVITY ANALYSIS

IV.1. Uncertainty analysis

An a priori study allowed the extraction of four influential parameters: -h oil , thermal convection coefficient between the end windings and the inner oil, -e ctc , thickness of the contact between the stator and the casing, -e ins , thickness of the insulation between slots and teeth, -λ radcd , equivalent thermal conduction of the windings in the plane normal to the winding axis.

The ranges of variation of these parameters are presented in Table 1 and correspond to typical values found in literature. The samples are generated by the semi Monte-Carlo method as explained in Section II.2. In our application, the sensitivity analysis is carried out for four parameters, i.e. p=4. N=10 6 samples are generated. For that, two p by N matrixes (X 1 and X 2 ) are generated in a completely random manner corresponding to N values of quadruplets [h oil , e ctc , e ins ,  adcd ]. Then, to assess the Sobol indices corresponding for example to h oil , the first column of X 2 is replaced by those of X 1 . The values of the temperature are then assessed using the LPTN model for its two sets of values providing Y1 and Y2 matrix used to calculate the Sobol indices with the method presented section II.2.

As the computation times of a LPTN model are really fast, such a process takes 285s to assess the indices corresponding to one parameter (2*N evaluations of the model).

For this distribution of samples, Figure 7 and Table 2 present the average value and the deviation of the temperatures for the four temperature points of the machine (orange points Figure 3) which are the most critical (see section III.3). The two temperatures which have the highest range of variation are the Coil-head and the Slot temperatures corresponding to the windings. They also present the higher deviation.

IV.2. Sensitivity analysis

A sensitivity analysis allows the identification of the thermal parameters that have the highest influence on the model outputs. So, these parameters have to be accurately estimated to obtain the temperatures of the critical areas of the machine with an acceptable accuracy (in the context of pre-sizing). The obtained results are presented in Table 3 and Figure 8 for three temperatures. h oil e ctc e ins λ radcd T coil_head 0,64 0,24 0,05 0,13 T oil 0,12 0,8 0,05 0,01 T mag 0,18 0,89 0,1 0,11 T slot 0.13 0.44 0.12 0.27 e ctc has a great influence on the temperatures of the magnets and the oil. Nevertheless, h oil has the strongest influence on the coil-head temperature, which is also the maximal temperature in the machine. On the other hand, λ radcd and e ins have less influence. So, these two parameters (e ctc and h oil ) have to be carefully chosen to correctly estimate the temperatures. The coil-head area and the contact between stator and casing have to be represented with good accuracy. The contact area between the stator lamination and the casing has also to be manufactured with good care.

IV. Conclusion

A LPTN model is developed to be implemented in a global optimization process for the sizing of the electrical machine of a hybrid vehicle. First it is validate using Finite Element Method. Then, it is used in the building of the efficiency map of the machine, by suppressing the operating points that cannot be thermally accepted.

As some parameters of this LPTN model can be difficult to assess, a sensitivity analysis using Sobol indices has been performed. After a short presentation of the Sobol method, it is used to assess the influence of four thermal parameters of an electrical machine on critical temperature (windings, magnet, copper). In the paper cases, it shows that the oil thermal convection coefficient has the highest influence on the coil-head temperature which is the most critical in our machines. The temperatures are also sensitive to the contact between stator and casing.

This model could now be used in future work to simulate and design different types of powertrain architectures and their components.

Figure 1 :

 1 Figure 1: Parallel Hybrid Vehicle power train and chosen sizing parameters

Figure 2 :

 2 Figure 2: Global optimization process

Figure 3 :

 3 Figure 3: Thermal model of the electrical machine (LPTN) The LPTN model gives the temperature of several areas in the machine. The critical areas (i.e. where the temperature has to be under a threshold) are:  the windings (T coil_head < 170 °C and T slot < 170°C),  the inner oil (T oil < 158 °C)  the permanent magnets (T mag < 220°C). The model inputs are geometrical variables and operating losses given by the electromagnetic model (see part II). Previously to its use with the global process, this model was validated by means of finite element model (FEM) (simulation).

Figure 4 :Figure 5 :

 45 Figure 4: Temperatures in different area of the machine obtained by FE model and LPTN model

Figure 6 :

 6 Figure 6: Thermal paths comparison between LPTN and FEM

Figure 7 :

 7 Figure 7: Distribution of the temperatures in coil-head, oil, magnet and slot considering. Table 2: Results of the uncertainty analysis on the temperatures (°C) inside the electrical machine

Figure 8 :

 8 Figure 8: Sobol indices of the LPTN model

Table 1 :

 1 Range parameters variations

	Parameters	Units	range
	h oil	W.m -2 .°C -1	[25; 100]
	e ctc	W.m -1 .°C -1	[1.6; 32]
	e ins	m	[6.10 -5 ; 12.10 -5 ]
	λ radcd	m	[19.10 -5 ; 50.10 -5 ]

Table 2 :

 2 Results of the uncertainty analysis on the temperatures (°C) inside the electrical machine

		T end	T mag	T oil	T slot
	Average value	146.3	122.2	122.5	128.6
	Deviation	5.1	2.6	2.2	3.6
	Maximal variation	13	6.4	5.4	8.4

Table 3 :

 3 First order Sobol indices.