Pedro Stefanin Volpiani

Raphaella Fusita Bernardini

Lucas Franceschini

Pedro Stefanin Volpiani

Raphaella Fusita Bernardini

Neural network-based eddy-viscosity correction for RANS simulations of flows over bi-dimensional bumps

come

Neural network-based eddy-viscosity correction for RANS simulations of flows over bi-dimensional bumps

I. INTRODUCTION

Reynolds-Averaged Navier-Stokes (RANS) simulations are still a workhorse in the aerospace, mechanical, and chemical industries. Thanks to their low computational cost, they remain an indispensable tool in the design, analysis, and optimization of many aerodynamic components. In this approach, only the mean flow is considered and a mathematical model is required to capture the effect of turbulence via the unknow Reynolds stresses. The essence of RANS modeling is based on expressing these unknown terms as a function of known mean quantities. One of the most widespread assumption in RANS modeling is based on the Boussinesq hypothesis, which assumes a linear relationship between the turbulent Reynolds stresses and the mean-velocity gradient tensor. The proportionality constant is called the turbulence-eddy viscosity, ν t . Thus, the effect of turbulence is taken into account by increasing the fluid viscosity, based on local flow characteristics. Popular eddy-viscosity models are the Spalart-Allmaras (SA) model [START_REF] Spalart | A one-equation turbulence model for aerodynamic flows[END_REF], relying on a modeled transport equation for the modified eddy viscosity, the shear-stress transport (SST) model [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF], relying on two transport equations for the kinetic turbulence energy and the specific dissipation rate, and the kε model that transports the turbulent kinetic energy and the turbulent dissipation [START_REF] Jones | The prediction of laminarization with a two-equation model of turbulence[END_REF][START_REF] Edward | Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc[END_REF]. Eddy-viscosity RANS turbulence models gained attractiveness in industrial applications for three reasons: (i) they are relatively easy to implement, (ii) they present good convergence properties because of the additional dissipation and (iii) they have proven to give reliable predictions, especially when dealing with attached flows [START_REF] David | Turbulence modeling for CFD[END_REF][START_REF] Haase | FLOMANIA-a European initiative on flow physics modelling: results of the European-Union funded project[END_REF]. However, it is well known that complex effects such as flows with separations, high streamline-curvature, strong pressure gradients, etc. are poorly modeled by this approach.

With the undeniable success of machine learning (ML) in numerous fields, from image and speech recognition, up to self-driven cars and medical diagnosis, CFD aerodynamic calculations are also beginning to benefit from this technology [START_REF] Steven L Brunton | Machine learning for fluid mechanics[END_REF][START_REF] Duraisamy | Perspectives on machine learning-augmented Reynolds-averaged and large eddy simulation models of turbulence[END_REF][START_REF] Richard | Machine-learning for turbulence and heat-flux model development: A review of challenges associated with distinct physical phenomena and progress to date[END_REF]. It is therefore natural to use this technique to improve the performance of RANS turbulence models.

In early works, machine learning was applied with the goal of correcting directly the Reynolds stress tensor. Tracey et al. [START_REF] Tracey | Application of supervised learning to quantify uncertainties in turbulence and combustion modeling[END_REF] used kernel regression to correct the eigenvalues of the Reynolds stress anisotropy tensor by using input features from a low-fidelity model and training over a DNS dataset. Ling et al. [START_REF] Ling | Reynolds averaged turbulence modelling using deep neural networks with embedded invariance[END_REF][START_REF] Ling | Machine learning strategies for systems with invariance properties[END_REF] have proposed a novel neural network architecture to embed key physical modeling properties with Galilean invariance into the predicted Reynolds stress anisotropy tensor.

Wang et al. [START_REF] Wang | Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data[END_REF] proposed a data-driven, physics-informed machine learning technique based on random forests for reconstructing discrepancies in RANS modeled Reynolds stresses. They observed that small errors in the machine-learning-predicted Reynolds stresses could lead to large errors in the mean velocity field when solving the RANS equations. To fix this issue, Wu et al. [START_REF] Wu | Physics-informed machine learning approach for augmenting turbulence models: A comprehensive framework[END_REF] proposed to decompose the Reynolds stresses into linear and nonlinear parts and to treat them separately to enhance the learning procedure. Weatheritt and Sandberg [START_REF] Weatheritt | A novel evolutionary algorithm applied to algebraic modifications of the RANS stress-strain relationship[END_REF] created models for the Reynolds stress anisotropy in terms of invariants of the velocity gradient tensor using symbolic regression and gene-expression programming (GEP). Their method provides an algebraic expression, which can be straightforwardly implemented in RANS solvers and, depending on their form, easily interpretable from a physical point of view. These GEP-trained models have shown improved predictive accuracy in a posteriori tests such as rectangular ducts [START_REF] Weatheritt | The development of algebraic stress models using a novel evolutionary algorithm[END_REF] and turbomachinery flows [START_REF] Zhao | RANS turbulence model development using cfd-driven machine learning[END_REF].

In contrast, Duraisamy and co-workers [START_REF] Eric | A paradigm for data-driven predictive modeling using field inversion and machine learning[END_REF][START_REF] Singh | Machine-learning-augmented predictive modeling of turbulent separated flows over airfoils[END_REF] introduced a full-field multiplicative correction coefficient into the production term of the transport equations of turbulent quantities (e.g., ν in the SA model and ω in the kω models). The correction terms were inferred employing dataassimilation (DA) techniques, termed field-inversion, based on DNS and experimental data and transformed into model forms using machine learning. Good results were obtained for channel flows and flows over airfoils. Following Franceschini et al. [START_REF] Franceschini | Mean-flow data assimilation based on minimal correction of turbulence models: Application to turbulent high Reynolds number backward-facing step[END_REF], Volpiani et al. [START_REF] Volpiani | Machine learning-augmented turbulence modeling for RANS simulations of massively separated flows[END_REF] opted to introduce a correction to the Boussinesq-hypothesis by adding a forcing term in the momentum equations.

They employed variational data assimilation to infer the vectorial source correction from highfidelity numerical data and machine learning to reconstruct this quantity from the local mean-flow features. Their methodology showed excellent results when applied to periodic hill configurations.

In this paper, we aim at performing machine-learning based RANS simulations of a family of two-dimensional bumps. The initial geometry was proposed by Webster et al. [START_REF] Webster | Turbulence characteristics of a boundary layer over a two-dimensional bump[END_REF], who investigated experimentally a turbulent boundary layer flowing over a 2D bump providing sufficient surface curvature and streamwise pressure gradient effects. A sketch of the baseline geometry is shown in The novelty of this work is that, instead of focusing on correcting the Reynolds-stress tensor or a specific term in the transport equation of turbulent quantities, we focus on correcting the turbulenteddy viscosity directly. A big advantage of this approach is that the new model is extremely simple and easy to implement. For example, full second-moment Reynolds stress models not only have a transport equation for each one of the 6 Reynolds stress components, but also a seventh transport equation for the length-scale-determining variable [START_REF] Cécora | Differential Reynolds-stress modeling for aeronautics[END_REF][START_REF] Eisfeld | Verification and validation of a second-momentclosure model[END_REF]. In the present formulation, there is no need to transport unclosed terms, which can reduce implementation, verification, and computational time. The inconvenience of using an eddy-viscosity-based turbulence model is that it is inherently restricted to the Boussinesq approximation. Even if this hypothesis is inaccurate in many cases [START_REF] François | About Boussinesq's turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity[END_REF], we still believe that such a correction can be useful for the fluid-mechanics community. Therefore, in the proposed methodology, that seeks a machine-learned correction of the SA model, we only need to solve mass and momentum conservation equations. However, input features of the neural-network architecture are derived from the baseline RANS-SA simulations, which are used as initial conditions in the computation of the corrected fields. Another interesting aspect of the current work is the absence of the field-inversion step. This is circumvented by computing directly the estimated eddy-viscosity field from statistics of the reference flow. This makes the model design easier, but has the inconvenience of relying on detailed information of the statistics. The high-fidelity reference dataset is the solution of large-eddy simulations of flows around a family of bidimensional bumps made available by Matai and Durbin [START_REF] Matai | Large-eddy simulation of turbulent flow over a parametric set of bumps[END_REF]. The baseline geometry,

primarily investigated in wind tunnel by Webster et al. [START_REF] Webster | Turbulence characteristics of a boundary layer over a two-dimensional bump[END_REF], is a circular arc of length 254.0 mm, with concave fillets in both sides, so that the total length of the arc is c = 305.0 mm. In the LES, the computational domain starts at x c = -1 3, ends at x c = 5 3 and the top wall is located at c 2. The bump starts at x/c = 0.0 and ends at x/c = 1.0, as it is represented in figure 1. The simulations were performed for five geometries, in which the bump heights h were set as different values (see figure 1), summarized in Table I.

The incompressible wall-resolved LES was computed using OpenFOAM with the dynamic Smagorinsky subgrid model. The inlet boundary condition is a fully developed turbulent flow, with a boundary layer momentum thickness of θ = 3.6 mm and free stream velocity of u ∞ = 16.8 m/s.

The Reynolds number based on these two quantities is Re θ = 2500, which leads to a kinetic viscosity of ν = 2.4192 ⋅ 10 -5 m 2 /s. More details about the simulation can be found in reference [START_REF] Matai | Large-eddy simulation of turbulent flow over a parametric set of bumps[END_REF].

The flow configuration studied here is characterized by different levels of curvature, pressure gradient and flow separation. For the lowest bump height (h20), the flow remains attached all along the bottom wall. Case h26 is characterized by a flow on the verge of separation. The other configurations (h31, h38, h42) develop a small separated region near the end of the bump and its length increases with the protuberance height.

Figure 2 displays the reference pressure and skin friction coefficients for cases h20, h31 and h42

and confirms these findings. As the height of the bump h increases, the minimum value of C p at the crest decreases, since the flow will be more accelerated. The opposite is observed for the maximum value of C f , which will increase with the bump height. In addition, for cases h31 and h42, in which there is separation, the C p curve between 0.8 ≤ x c ≤ 1.0 becomes flat. At the end of the bump, the adverse pressure gradient becomes stronger, and the C f curve progressively presents a region of negative values between for cases h31, h38 and h42.

III. GOVERNING EQUATIONS

The incompressible steady-state RANS system of equations is derived by decomposing the flow variables into mean and fluctuating terms, here formalized as

u i (x, t) = u i (x) + u ′ i (x, t)
for the velocity and p(x, t) = p(x) + p ′ (x, t) for the pressure. The resulting system of equations reads:

∂u i ∂x i = 0, (1)
u i ∂u j ∂x j = - ∂p ∂x i + ∂(2νS ij ) ∂x j + ∂τ ij ∂x j (2)
with the mean strain tensor S ij = (u i,j + u j,i ) 2 and the molecular viscosity ν. The term

τ ij = -u ′ i u ′ j
is the Reynolds stress tensor, and can be obtained through high-fidelity simulations.

In the RANS framework, the Reynolds stress tensor is frequently modeled following the Boussinesq approximation:

τ ij = 2ν t S ij - 2 3 kδ ij with k = 1 2 u ′ i u ′ i . (3)
The turbulent kinetic energy k is usually incorporated in the pressure. The kinetic-eddy viscosity ν t is determined by solving the one equation Spalart-Allmaras turbulence model for the quantity ν ( [START_REF] Spalart | A one-equation turbulence model for aerodynamic flows[END_REF]):

u j ∂ν ∂x j -∇ ⋅ σ -1 (ν + ν) ∇ν = P ν (ν, ∇u) -D ν (ν, ∇u) + C ν (∇ν) (4)
where the terms P ν , D ν and C ν are the production, destruction and cross-diffusion terms of the quantity ν, and are given by:

P ν = c b1 Sν, D ν = c w1 f w ν d 2 , C ν = c b2 σ ∂ν ∂x k ∂ν ∂x k . (5)
More details about the model variables and the physical definitions of each term are found in ref. [START_REF] Spalart | A one-equation turbulence model for aerodynamic flows[END_REF]. Further information about the numerical implementation in the finite-element software FreeFEM [START_REF] Hecht | New development in freefem++[END_REF] is available in [START_REF] Franceschini | Mean-flow data assimilation based on minimal correction of turbulence models: Application to turbulent high Reynolds number backward-facing step[END_REF][START_REF] Volpiani | Machine learning-augmented turbulence modeling for RANS simulations of massively separated flows[END_REF].

In our simulations, at the inlet, velocity profiles were set to be identical to the reference flow, and the modified turbulent viscosity was estimated from the LES solution (in the same way that will be discussed in the following section). This allows for the RANS solution to be developed in the same conditions as the LES. At the outlet, we impose zero normal stresses; and zero normal gradient of ν. No-slip boundary conditions were imposed at the lower wall, and initially, slip boundary conditions were used at the top. However, in the reference solution files available from the NASA website [START_REF] Matai | NASA turbulence modeling resource[END_REF], the boundary conditions on the top wall are questionably different from the article specification. It seems that there is a drastically reduction of the velocity at the top wall, yet the mesh size close to this boundary does not allow the correct representation of a boundary layer.

RANS simulations using slip boundary conditions give erroneous pressure distribution across the lower wall (see Fig. 2). To remedy any inconsistencies, we decided to use a shorter domain and impose the exact LES variables at the top boundary. Even though we know this fix is not ideal, at least we have compatible boundary conditions. We would like to highlight the importance of having high-fidelity data with well-defined numerical conditions. The results obtained with the new top boundary conditions are also shown in Fig. 2. The difference between RANS using the original and the shorter domain relies on the C p distributions. The new condition gives much closer results if compared to the original LES. As for the C f curve, this quantity was found to be much less sensitive with respect to the boundary condition specification. The major misprediction between RANS/LES resides above the bump: while the baseline RANS simulation presents an increase in C f , with its maximum value at the crest, followed by a decrease until the end of the bump, the reference C f curve presents a different behavior between 0.6 ≤ x c ≤ 0.8, right after the change in pressure gradient. The RANS-SA model differs from the reference results by overestimating the drop at x c = 0.9, and, after the bump, underestimating the recovery of C f for all three cases.

IV. TURBULENT EDDY-VISCOSITY ESTIMATION

In the RANS framework, the most widespread models in the aerospace industry relies on the Boussinesq assumption and the computation of a turbulent-eddy viscosity. Classical models evaluate the turbulent-eddy viscosity by solving partial differential equations (PDE) that describe the transport of turbulent variables such as a modified viscosity ν, the turbulence kinetic energy k, its specific rate of dissipation ω, the turbulent dissipation ε, etc. The main goal of this work is to provide a PDE-free eddy-viscosity model that evaluates ν t by means of machine learning techniques. With this in mind, a good and realistic estimate of the turbulent-eddy viscosity is needed to perform the neural network training. The most immediate method to obtain the exact field of ν t is by calculating it directly from the reference mean and fluctuating fields. From the Boussinesq relation (3), we can derive the following turbulent-eddy viscosity expression [START_REF] Mettot | Quasi-laminar stability and sensitivity analyses for turbulent flows: prediction of low-frequency unsteadiness and passive control[END_REF]:

ν LES t = -(u i ′ u j ′ -2 3kδ ij ) ∂ j u i 2S ij S ij . (6)
To avoid divisions by zero or negative values of turbulent viscosites, instead of Eq. ( 6) we used

ν LES t = max 0, -(u i ′ u j ′ -2 3kδ ij ) ∂ j u i max (0, 2S ij S ij ) + . ( 7
)
where is a small parameter.

Yet, the insertion of a physical quantity obtained from DNS or LES into a RANS simulation does not guarantee an improvement in the solution. Thompson et al. [START_REF] Roney L Thompson | A methodology to evaluate statistical errors in DNS data of plane channel flows[END_REF], for instance, computed the true Reynolds stress tensor fields directly from DNS data and did not obtain sufficiently improved RANS solutions. Raiesi et al. [START_REF] Raiesi | Evaluation of turbulence models using direct numerical and large-eddy simulation data[END_REF] also did not reach better accuracy when calculating the turbulent kinetic energy and dissipation rate directly from DNS and LES data-sets. Therefore, it is possible that the direct extraction of the exact turbulent viscosity may not necessarily imply an augmented RANS solution. We believe that this evaluation step is important, hence, simulations were performed with the modified ν LES t to assess the effective improvement of the resulting flow.

The second way to estimate the turbulent-eddy viscosity is through a field inversion. In this approach, the estimation of the turbulent viscosity requires the correction of a RANS simulation, with the purpose of approximating the assimilated flow to the reference solution. However, Franceschini et al. [START_REF] Franceschini | Mean-flow data assimilation based on minimal correction of turbulence models: Application to turbulent high Reynolds number backward-facing step[END_REF] carried out a few of such inversions on a backward-facing step, which resulted in poor assimilated solutions, presenting strong unphysical distortions in separation regions. Even if other parameters could be assimilated to correct ν t , in this work we focus solely in the direct computation of the turbulent viscosity.

The turbulent viscosity fields obtained from Eq. ( 7), the baseline simulation and their differences turbulence model to find the right amount of ν t for each scenario. With this in mind, we expect to predict this quantity using artificial neural networks.

Once the turbulent-viscosity fields are estimated for each case, Eqs. ( 1) and ( 2) can be solved for the velocity and pressure with no further modeling. Most of our analysis are focused on the prediction of mean pressure and skin-friction distributions, because of their major importance in the aerospace industry. The comparison between LES, baseline RANS and RANS using ν LES t is shown in Fig. 4. Figure 4 puts in evidence that the RANS simulations done with ν LES t present significant improvement over the RANS-SA solutions. Both quantities of interest, C p and C f approximate very well to the reference curves, indicating that the extraction of the turbulent viscosity from the high-fidelity solution leads to a satisfactory correction of the flow, especially for flows without separation. As the bump height increases, it is observed that the C p and C f curves in figure 4, lose some accuracy, in particular between 0.8 ≤ x c ≤ 1.0. Yet, this change in precision is expected since the modified RANS simulation is still constrained to the Boussinesq hypothesis, which lacks accuracy in regions of separation. Figure 5 shows a comparison of streamwise velocity profiles between LES, baseline RANS-SA and RANS using ν LES t for cases h26 and h38. The baseline and modified RANS profiles are similar for the attached case h26. This indicates that the standard SA model manages to predict the resulting flow field in cases with small or no separation (at least faraway from the wall). However, very close to the wall the skin friction is not correctly predicted, as indicates Fig. 4. For case h38, we notice a better prediction of the mean velocity field for the simulation employing the modified eddy viscosity. It is worth mentioning that for flows presenting large separations, the approximation (7) may lose accuracy. Still, in general, the RANS results using ν LES t showed that it is a very good estimation of the true turbulent viscosity. We believe that trying to predict ∆ν t or ν LES t using machine learning can improve considerably the prediction of the mean RANS solutions. We highlight that Tan et al. [START_REF] Tan | Towards explainable machine-learningassisted turbulence modeling for transonic flows[END_REF] also tried to predict the eddy viscosity discrepancy but using the random forest algorithm. In their paper, however, the reference viscosity was given by the SST model and training and validation was done for a single case. They analysed the influence of several input features in the output prediction, but unfortunately did not test their algorithm in a RANS solver.

V. MACHINE LEARNING STRATEGY

The objective of an artificial neural network (also known as multi-layer perceptron) is to improve complex computational models. Artificial neural networks are nonlinear functions parameterized by weights w and biases b that can be learned from data. The concept of neural networks was inspired from the structure of biological neural circuits. Its basic unit is called perceptron. The ensemble of perceptrons organized in layers composes our neural network. The activation q (l) i of perceptron i in the l th layer can be represented as

q (l) i = φ ⎛ ⎝ j w (l) ij q (l-1) j + b (l) i ⎞ ⎠ . (8)
In other words, the input data from the (l -1) th layer are multiplied by a weight w, linearly combined and then passed through a nonlinear activation function φ. Therefore, ANN are built by consecutive composition of linear functions (matrix-vector multiplication) followed by nonlinear activation functions. Some of the widely used activation functions are the sigmoid function σ(x) =

1 (1 + exp -x ) or the rectifier linear unit (ReLU), σ(x) = max(0, x). For deep networks, ReLU is known to be a good candidate [START_REF] Nair | Rectified linear units improve restricted boltzmann machines[END_REF] and, for this reason, was kept in this study.

This feedforward algorithm is a mathematical model equivalent to a non-linear function with a high number of parameters (w, b), that relates the inputs in the first layer to the outputs in the last layer. This function is referred here as N (Q(x)), where Q(x) is the vector representing the input layer variables and N is the final neural network output.

f it = 1 - m ∑ k=1 (N (Q val k ) -N val k ) 2 m ∑ k=1 (N val -N val k ) 2 (10)
Here, m is the size of the validation data-set and k is one data point, to which is associated a set of reference inputs Q val k and output N val k . This fit parameter is evaluated during the training process to assess the quality of the network. It is taken into consideration for the choice of the hyper-parameters such as: number of hidden layers, number of neurons in each hidden layer and optimizer learning rate. The low memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) method was chosen as the optimizer with a learning rate of 0.1, and the back-propagation algorithm was applied for gradient calculation. The weights were initialized following the method proposed by

Glorot & Bengio [START_REF] Glorot | Understanding the difficulty of training deep feedforward neural networks[END_REF], and all the biases were initialized as 0.1. The open-source library PyTorch is employed in order to re-use well-implemented methods for these procedures. Appropriate hyperparameters for the network training are found by a grid search implementation. The training of the network was terminated using an early-stopping criterion. If validation errors did not improve after 5 epochs, the optimization was interrupted.

B. Input features

For the prediction of our quantity of interest, a set of physical quantities is formulated to serve as inputs for the neural network. These input features are created from physical reasoning and are based only on information about the mean flow, in accordance with the nature of RANS solutions.

Knowing that the formulation of input features is equivalent to defining model variables for ν t ν, one important remark is that the inputs should be non-dimensional and rotationally invariant, in order to respect the turbulence modeling principles of dimensional homogeneity and objectivity.

Ling and Templeton [START_REF] Ling | Evaluation of machine learning algorithms for prediction of regions of high Reynolds averaged Navier Stokes uncertainty[END_REF] have composed a collection of 12 input features, among which we highlight: Q-criterion; ratio of pressure stresses to shear stresses; Gorlé et al. [START_REF] Gorlé | The deviation from parallel shear flow as an indicator of linear eddy-viscosity model inaccuracy[END_REF]'s marker for deviation from parallel shear flow; streamline pressure gradient; viscosity ratio; and turbulence intensity. In this paper, we also added turbulence quantities related to the model that we aim to fix. We employ the SA ratio of production to destruction and the ratio of production to diffusion. All input features are summarised in Table II. Since we do not transport the turbulent kinetic energy, we reconstruct this variable using the 2013 version of the Quadratic Constitutive Relation (QCR) [START_REF] Mani | Predictions of a supersonic turbulent flow in a square duct[END_REF], with

k qcr = 3 2 c cr2 ν t 2S ij S ij .
To avoid feature domination, input features are

q 1 Q-criterion Ω 2 -S 2 Ω 2 + S 2 q 2
Ratio of pressure normal stresses to shear stresses

∂ p ∂x k ∂ p ∂x k ∂ p ∂x k ∂ p ∂x k + 1 2 ∂ ū2 k ∂x k q 3 Gorlé et al. [37] marker ūi ūj ∂ ūi ∂xj ūi ūj ∂ ūi ∂xj + ūl ūl ūi ∂ ūi ∂xj ūk ∂ ūk ∂xj q 4 Streamline pressure gradient ūk ∂ p ∂x k ūk ∂ p ∂x k + ∂ p ∂x k ∂ p ∂x k ūi ūi q 5
Viscosity ratio

ν t ν t + 100ν q 6
SA ratio of production to destruction c b1 Sν

c b1 Sν + c w1 f w ν d 2 q 7 SA ratio of production to diffusion c b1 Sν c b1 Sν + c b2 σ ∂ν ∂x k ∂ν ∂x k q 8
Turbulence intensity k qcr k qcr + 1 2 ū2 i normalized in a way that all features exhibit a maximum value of approximately 1 and a minimum value of around 0 if the feature is positive (or -1 if the feature is both positive and negative).

C. Output quantity and assessment of the quality of the neural network

In this study, we explored two ways of correcting the turbulence-eddy viscosity field by means of ML techniques. In a first moment, we trained artificial neural networks to compute the normalized discrepancy between LES-and SA-based turbulence viscosities: ∆ν t ν; and in a second moment, we focus on predicting the normalized turbulence viscosity ν LES t ν directly. We then investigated the performance and robustness of each formulation.

Once the training was carried out using the input features from Table II, we compared the performance of each neural network taking into consideration the fit parameter given by Eq. [START_REF] Tracey | Application of supervised learning to quantify uncertainties in turbulence and combustion modeling[END_REF].

The resulting neural networks architecture and performance are shown in Table III for both approaches. Even though table III presents the fit parameter for the entire dataset, the choice of the best model performance cannot take into account the validation cases. This is because, in a realistic scenario, the high-fidelity data would not be available for bumps h26 and h38, therefore the evaluation of the resulting models must be made by considering only the training data. We observe that the neural network that predicts ν t ν has a slight advantage over the one that predicts ∆ν t ν. Nonetheless, both performances are still outstanding. The fit parameter is close to 99% in both cases.

Moreover, we note in Fig. 6(a) that it is easier to approximate ν t ν rather than ∆ν t ν since the ANN needs less iterations to maximize the fit function. To have a more visual representation, the predicted outputs from NN1 and NN2 are plotted respectively in Figs. 7 and8 together with their true fields for cases h26 and h38 that have not been used during training. We note that the neural network captures, in general, the physical behaviour of the expected quantities. Concerning the viscosity discrepancy, despite the noisy prediction, the ANN successfully predicts the right levels of ∆ν t in most of the domain. Some miss predictions still exist very locally. Regarding the total turbulence viscosity, the overall prediction is also excellent.

Close to the wall, the ANN was able to reproduce the regions of high ν t upstream and downstream of the bump, and at the crest. And towards the free stream, the predicted ν t becomes zero, as in the reference field.

Finally, Fig. 9 shows a visual representation of how the training process makes an effort to approximate the network's outputs to the training data. Most of the points reside in the vicinity of the reference line, yet they are not exactly identical, and there is still a part of the dataset that does not approach the line as desired. We also remark that the comparison between subfigures 9(a) and 9(b), for instance, confirms that the network that predicts ν t ν approximates slightly better to the reference datasets than the one that predicts ∆ν t ν.

VI. A POSTERIORI RESULTS

In this section, predictive tests are performed using the new NN-based model. The first step of the algorithm is to compute the input features from the initial baseline RANS-SA solution. Then, the neural-network routine generates the output parameter, which is subsequently introduced into the RANS solver. The converged state can be then compared with the reference and baseline solutions. We point out that the predictive test-cases should not include any data employed in the training phase. In a first stage, we want to predict the following non-dimensional correction term ∆ν t ν given by NN1. Figure 10 displays the distribution of the mean pressure and skin-friction coefficients for the reference solution, baseline RANS and NN1-based RANS. Despite the good prediction of the difference between the correct and modeled eddy viscosities (Fig. 7), when this quantity is given as an input for a RANS solver, the resulting flow field comes out noisy. However, we note that the shape of the C p and C f distributions are correctly captured by the augmented model. The new NN1-model also manages to predict the separation and reattachment locations in both scenarios, which is quite impressive taking into consideration that they present different flow physics. It is important mentioning that in order to have an exploitable model, two corrections needed to be implemented in the solver: (i) we needed to ensure that the total eddy viscosity was always positive and (ii) to enforce a zero viscosity at the wall as in the original SA model. These two considerations were necessary in order to bypass numerical instabilities or nonphysical results in the wall region (see Fig. 11). Reference LES simulation (symbols), baseline RANS-SA (solid lines), RANS-NN1 (dashed lines)

Results concerning the NN2-based model is shown in Fig. 12 for test cases h26 and h38. Despite not being perfect, this second model successfully predicts a flow with incipient separation for case h26 and the correct size of the separation bubble for case h38 and the results are much smoother than in the former simulation. Note also that, arguably, simulation h38 performed slightly better than h26. This can be explained by the fact that in the training database only one configuration presented attached flow, while the other two presented some kind of separation. Therefore, the majority of the training set was closer to configuration h38 rather than h26. We believe that the overall prediction could be improved if more cases were considered. We plot in Fig. 13 the streamwise velocity profiles for the LES, RANS-SA and RANS-NN2 for validation purposes. We note that the overall flow field is improved and resembles the one from simulation using ν LES t (Fig. 5). We conclude that predicting ν N N t using a neural network and subsequently adding the Reference LES simulation (symbols), baseline RANS-SA (solid lines), RANS-NN2 (dashed lines)

present separated flows). The performance of NN3 model is displayed in Table IV. Results are outstanding for the training cases (fit parameter over 99%) and for the validation case (fit parameter around 98 %). Testing the NN3 model in a RANS solver also showed excellent performance if compared to the baseline RANS-SA simulation (Fig. 14). Surprisingly, the results are even more accurate than the ones displayed in Fig. 5. So, in some way, the neural network managed to find an improved function that better described the validation set. We conclude that machine-learning assisted turbulence models can have a great impact in the prediction of key flow features. LES solution as the true field, it was observed that for a bump configuration the baseline model mispredicts the pressure and velocity fields, reflecting on a significant difference in the C f evolution above the bump. To enhance the model prediction, the neural network is treated as a non-linear approximator that corresponds to an estimated functional form of the turbulent viscosity or its discrepancy. This form is defined through supervised learning, where the network parameters are optimized such that the function fits to the given reference data-set. In this way, the novelty of the proposed model is that it corrects directly the eddy-viscosity field and no additional turbulence quantity needs to be transported, leading to a PDE-free eddy-viscosity model.

Therefore, in a first step, the reference turbulent viscosity was extracted from the true fields of LES simulations. This method lead to an important augmentation of the resulting RANS solution, generating a more accurate description of C p and C f . In a second step, a set of local input features were formulated following physical reasoning and modeling constraints. Two output quantities: the turbulence viscosity discrepancy ∆ν t = ν LES t ν SA t and the expected turbulence viscosity itself ν LES t were inferred using artificial neural networks. Both NN architectures seemed to give good prediction for each quantity of interest. The predicted field still contains noise due to the approximate nature of the neural networks, which might be a problem when injected in a RANS solver. Subsequently, both neural networks were tested in two scenarios never seen in the training process. Results obtained with NN1 that predicted the eddy-viscosity discrepancy did show an improved solution, despite the noisy skin-friction profile. On the other hand, NN2 that focused on predicting the correct eddy viscosity was able to reproduce the general flow-field behaviour with a much smoother solution. The method was proved to be robust even in predicting extrapolated flows. The results achieved using the proposed method are encouraging, but new analysis are still necessary to improve a posteriori computations.

Fig. 1 .FIG. 1 .

 11 Fig. 1. The bump profile consists of a circular arc of length L = 254 mm and height h = 20.1 mm. It was defined by three tangential circular arcs arranged in a way that the leading and trailing edges were tangent to the flat surface. The same geometry was used in the LES study carried out by Wu and Squires [23] at relatively lower inlet momentum-thickness Reynolds number (Re θ = 1500 in the LES and Re θ = 4030 in the experiment). The simulations showed good predictions of the mean flow and turbulence intensities in comparison with the experimental data of Webster et al.[START_REF] Webster | Turbulence characteristics of a boundary layer over a two-dimensional bump[END_REF]. Since the flow remained attached in the original setup, more recently, Matai and Durbin[START_REF] Matai | Large-eddy simulation of turbulent flow over a parametric set of bumps[END_REF] generated a LES database of flow over bi-dimensional bumps that exhibit different levels of flow separation by changing the bump height. This high-fidelity database is used in this study in the training and validation steps of our ML algorithm.

FIG. 2 .

 2 FIG. 2. Pressure (left) and skin friction (right) coefficients for cases h20 (a), h31 (b) and h42 (c). Reference LES simulation (symbols) and baseline RANS-SA with original domain and slip BC on the top wall (dashed lines) and with shorter domain (dash-dotted lines).

FIG. 3 .

 3 FIG. 3. Normalized turbulent viscosity ν T ν computed from the LES (left), computed using the SA model (middle) and their difference (right) for cases h20 (top), h31(middle) and h42 (bottom).

FIG. 4 .

 4 FIG. 4. Pressure (left) and skin friction (right) coefficients for cases h20 (a), h26 (b), h31 (c), h38 (d) and h42 (e). C p and C f curves are shown for the reference LES simulation (symbols), baseline RANS-SA (solid) and RANS-ν LES t

FIG. 5 .

 5 FIG. 5. Profiles of streamwise velocity component for cases (a) h26 and (b) h38. Profiles are shown for the reference LES simulation (solid), baseline RANS-SA (dashed) and RANS-ν LES t

Figure 6 (

 6 b) shows the evolution of the mean training and validation cost functions throughout the iterations for NN2. The figure confirms that training was stopped before the validation loss and fit parameter became a plateau, respecting the defined criteria to avoid overfitting.

FIG. 6 .

 6 FIG. 6. Evolution of the fit parameter (a) and of the mean losses (b) throughout the iterations.

FIG. 7 .FIG. 8 .FIG. 9 .FIG. 10 .

 78910 FIG. 7. Comparison of turbulent viscosities for cases h26 (top) and h38 (bottom). Left: ∆ν t ν. Right: ∆ν N N t

FIG. 11 .FIG. 12 .

 1112 figurations with larger heights. Unfortunately the original database does not cover cases larger than h42. So, to answer this question, we opted to train a new model (named NN3) using configurations h20, h26, h31 and h38. Having four cases (instead of three) in the training set enriches the resulting model and makes it unbiased (two configurations present attached and two configurations

FIG. 13 .FIG. 14 .

 1314 FIG.[START_REF] Wang | Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data[END_REF]. Profiles of streamwise velocity component for case h38. Profiles are shown for the reference LES simulation (solid), baseline RANS-SA (dashed) and RANS-NN2 (dashed-dotted line).

TABLE I .

 I Summary of configurations studied in this work. The reference data was performed byMatai and

	Durbin [24].
	Simulation Height (mm) Characteristics	Usage
	h20	20 (0.0659C)	No separation	training
	h26	26 (0.0878C) Incipient separation testing
	h31	31 (0.1032C)	Separated flow	training
	h38	38 (0.1259C)	Separated flow	testing
	h42	42 (0.1377C)	Separated flow	training
	II. CONFIGURATION AND REFERENCE FLOWFIELD

TABLE II .

 II Set #1 of local input features.

	Feature	Description	Formula

TABLE III .

 III Evaluation of the NN performance through the f it parameter. (T) indicates the training cases and (V) the verification cases not used during training. NN structure is in the form N inputs × (N layers × N neurons ) × N outputs , where N layers is the number of hidden layers and N neurons the number of neurons per hidden layer.

		Output	NN			fit (%)
	Label	variable	structure	h20 (T) h26 (V) h31 (T) h38 (V) h42 (T) all
	NN1 ∆ν t ν 8 × (4 × 80) × 1 99.1	98.6	98.8	96.2	98.6 98.0
	NN2 ν t ν 8 × (4 × 80) × 1 99.6	98.9	99.6	98.7	99.6 99.3

ACKNOWLEDGMENTS

This work was supported by the European project HiFi-TURB (High Fidelity LES/DNS Data for Innovative Turbulence Models) -EU H2020 RGY under Grant Agreement No. 814837 and the internal project MODDA (Modèles Data-Driven pour l'Aérodynamique. ) * pedro.stefanin volpiani@onera.fr

A. Training and validation

The five available LES data sets are divided into: training cases, from which the neural network will learn; and testing cases, which will be used to validate our augmented RANS model. This separation, summarized in table I, aims towards a realistic scenario of data-driven models' usage in the future, since it represents a situation where high-fidelity data is available for a certain number of flows, and prediction is needed for an entirely new flow, somewhat similar to the training cases, but for which a reference solution is not available. This division reaches for the neural network's predictive capability of doing an interpolation between training cases, for a flow configuration that presents curvature, pressure gradient and separation. In particular, bump h26 represents a test case of a flow in the verge of separation, between an attached flow, h20, and a separated flow, h31. Even further, bump h38 represents an intermediate case between two separated flows, h31 and h42. Therefore, the neural network model must adapt to different types of flow conditions.

Once the training data is defined, the neural network learning process consists of an optimization problem, where the objective is to tune the weights and biases such that the resulting network is fitted to the training data. The goal is therefore to search for the optimal parameters (w, b) that minimize the cost function:

Here, n is the size of the training data and k is one data point, to which is associated a set of

predicted output for the given reference inputs, which evidently depends on the parameters (w, b) as well.

Consequently, the optimal parameters will lead to a neural network that minimizes the difference between the predicted and reference outputs. In other words, it will provide an approximation of the functional form N (Q(x)) that better describes the training data.

Validation of the training procedure is done by previously separating the dataset into training and validation data. Here, 75% of the h20, h38 and h42 datasets were used for training, and 25%

were used for validation. A fit function, relative to the standard deviation of the validation data, was defined to measure the predictive capability of the neural network during training.


Van Sang Nguyen
email: van-sang.nguyen@cea.fr

Anthony Bier

Hajar Es-Seghier

Ulrich Soupremanien

Gérard Delette

Stephane Catellani

A 20 kW, 3-level flying capacitor 1500 V inverter with characterized GaN devices for grid-tie applications

Keywords: », «Flying Capacitor Boost Converter», «Gallium Nitride (GaN)», «Photovoltaic» «

This work presents the static and dynamic characterizations of high voltage GaN power devices (GaN FET 900 V and GaN HEMT 1200 V) in order to implement a 3-level flying capacitor 1500 VDC inverter for high power density grid-tie applications with renewable energy sources such as solar and hydrogen energy. In the first part, the static characterizations are shown for two selected GaN power devices. Then these GaN devices were placed in a double-pulse-test-bench dedicated to the dynamic characterizations intended to observe the switching behaviors of the devices under the nominal voltage and current. Finally, in order to demonstrate the compactness of the converter, these GaN devices were implemented in a 20 kW, 3-level flying capacitor 1500 VDC inverter with the full-custom suitable passive elements of the output filters connected to the 800 VAC 3-phase grid.

Introduction

Wide-band-gap power devices take an important place in the power electronics systems design. Today, SiC devices become mature and emerge in many high power electronics products [START_REF]Yole report on Power SiC 2020: Materials, Devices and Applications[END_REF]. On the other hand, lower voltages GaN devices (≤ 650 V) are widely used for low power consumer applications, thanks to their efficient and compact designs [START_REF]Yole report GaN power 2021: epitaxy, devices, applications and technology trends[END_REF]. Due to their thermal and blocking voltage constraints, GaN devices are still in optimizing stage for achieving high power applications, in particular, for 800 V 3phase grid-tie applications with renewable energy power systems. Using 650 V GaN devices for applications with DC bus of 1500 VDC requires the consideration of a high multi-level topology, where challenges are: the reliability, the control and the auxiliary circuitry of the systems [START_REF] Farhangi | A Compact Design Using GaN Semiconductor Devices for a Flying Capacitor Five-Level Inverter[END_REF]. Following up a recent work of the flying-capacitor topology on a SiC based design [START_REF] Rodrigues | A 200 kW Three-level Flying Capacitor Inverter using Si/SiC based Devices for Photovoltaic Applications[END_REF] and a GaN based design with a DC bus of 800 VDC [START_REF] Bunin | Performance and cost benefits of D3gaN in 3-level 800V EV inverters[END_REF]; this article presents a design of a 3-level flying capacitor inverter working at 20 kW tied to the 800 V 3-phase grid where the nominal input voltage is 1500 VDC. This GaN based design's approach gives a potential compact solution to several renewable energy systems like solar, hydrogen or even battery power systems. Today's trend in the large PV plant is the increasing in the power of the string by using smart panels [START_REF] Catellani | GaN based panel-integrated, high-efficiency DC/DC optimizer for maximizing the yield of the large photovoltaic power plant[END_REF]. Another work in parallel is ongoing where the GaN based power electronics parts are integrated inside the junction box of the PV panel, which allows installing much more number of photovoltaic (PV) panels in a single string. In a conventional 1500 V string, the output power is typically at 10 kW, the power of the 1500 V smart string could excess 20 kW thanks to the higher number of the PV panels [START_REF] Catellani | GaN based panel-integrated, high-efficiency DC/DC optimizer for maximizing the yield of the large photovoltaic power plant[END_REF].

Working with 1500 VDC bus voltage, GaN devices have to perform consequently with a nominal drainsource blocked voltage of 750 V in steady-state operations. This work analyses and challenges the GaN FET 900 V devices and the GaN HEMT 1200 V in such operating conditions. The comparison of these two GaN devices firstly takes place on a Keysight B1506A power device analyzer for the static characterizations, and then dynamic test-bench is used to qualify these selected GaN devices for estimating the switching losses under the nominal configurations. This article includes the design of the grid-filters inductors with thermal consideration and the simulations of this 1500 V inverter. Finally, this work shows the experimental results of this GaN based 3-level flying capacitor inverter until 20 kW characterizations.

High voltage GaN devices: static and dynamic characterizations

Static characterizations

By using a standard industrial test-bench (Keysight B1506A), the I/V curves, the threshold voltage and the parasitic capacitances of the selected GaN devices are extracted under the similar configurations. Fig. 1 shows large differences on the capacitance's characterization of DUT1 and DUT2 up to a drainsource voltage of 900 V. At this high voltage level, DUT1 has 958 pF of Ciss meanwhile this value of Ciss for DUT2 is 251 pF. A high frequency design up to a switching frequency of several MHz need to consider these capacitances; it is event more critical for the thermal issue in a high power design. Under the similar configurations of the drain-source current, the threshold voltage of DUT1 is approximately 3.4 V and for DUT2 about 1.5 V. As well known, this value is extremely critical for GaN based high power applications where the transient voltage dv/dt is usually very high due to the high voltage applied on the devices. This dv/dt might be the main reason at the origin of short-circuit within an inverter-leg where there are complementary transistors located at high side and low side. Design of the grid-tie flying-capacitor inverter need to take into account carefully this issue due to the DC bus voltage of 1500 V. Fig. 3 and Fig. 4 show the static resistances and the current/voltage curves of these two GaN devices, the DUT1 has a higher capacitance; however, its resistance value is much smaller than the DUT2. These static results suggest the correct driving voltage of each transistors in order to obtain the best performance of the devices. Under a drain current (at 25°C) of 30 A with different driving voltages, the on-state resistance of DUT1 varies between 36 mΩ and 48 mΩ where the typical value on the datasheet is 50 mΩ. This value of second device (DUT2) under the similar conditions is comprised between 76 mΩ and 80 mΩ where the maximum static on-state resistance value in its preliminary datasheet is 75 mΩ.

Dynamic characterizations

After these static characterizations, the dynamic characterizations use the high bandwidth voltage and current probes [START_REF] Nguyen | A high precision dynamic characterization bench with a current collapse measurement circuit for GaN HEMT operating at 175[END_REF], which are suitable for high-speed GaN devices. As in Fig. 5, an IZM current sensor with low insertion inductance of 0.3 nH and a bandwidth of 500 MHz [START_REF] Klein | Comprehensive AC Performance Analysis of Ceramic Capacitors for DC link usage[END_REF] from Fraunhofer-IZM, [START_REF] Klein | Power module design for utilizing of WBG switching performance[END_REF] allows to record precisely the rising and falling edges of the drain-source current of the device. An IsoVU probe [START_REF] Klein | Comprehensive AC Performance Analysis of Ceramic Capacitors for DC link usage[END_REF] with a bandwidth of 800 MHz is used to measure the drain-source voltage, a second low voltage probe with a bandwidth of 1 GHz measures the gate-source voltage. From these measurements, the turn-on and turn-off losses can be calculated. During the experiments, difficulties were noticed to perform the dynamic characterization of DUT2 at the nominal voltage, we found out the problem on the gate driver. The set-up of the dynamic characterizations for DUT2 is ongoing. In this article, only experimental results of the dynamic characterizations on the DUT1 are presented. Fig. 6 shows the experimental results of DUT1, which are the principal waveforms of the DPT set-up: the control signal on the gate VGS of 0 V/9 V, the rising -falling edges from IZM sensor, the drainsource voltage VDS of 0 V/760 V and the current on the inductance. In Fig. 8, GaN FET 900 V devices turns off to block a voltage of 760 V, the peak slew rate dv/dt of 50 V/ns is observed in this rising edge.

Simulations of the flying-capacitor 3-ph inverter

The electrical behavior of the 3-phase flying capacitor inverter is simulated using PLECS software. Fig. 9 shows a one-phase schematic representation model of the converter power stage. A 1500 VDC voltage source is placed at the input and the inverter is connected to the 800 VAC grid via a LCL-type filter. The current flowing through inductor L1 is controlled within a PI-based control. The current control loop and the grid voltage synchronization is implemented considering the rotating reference frame (DQ domain). Simulation results are focused in Fig. 10 on the voltage and the current waveform of the inductor L1.For a 20 kW 3-phase inverter, the peak value of the 50 Hz fundamental frequency sinusoidal AC current is 20 A and the RMS value of the signal 14.4 A. A 200 kHz triangular current ripple can be observed, which is twice the working frequency of the GaN devices. For the inductor design, its voltage must also be taken into account. The challenge in the design of the output filter, for a high power application, is the compliance to the high current, high voltage and high frequency at the same time.

Output filters design

In this work, the GaN based 3-level flying capacitor 1500 V inverter operates with an output LCL passive filter configuration. As we can see on the simulation, in this flying-capacitor topology, the output filters operate at a double switching frequency compared to GaN power devices. That is why the magnetic materials of these filters should be able to operate at high frequency and at the same time, the associated windings should have a low AC loss in high frequency

The targeted inductance value for the larger inductive filter element is 200 µH and should be held constant during operation within a wide range of current values. This performance are obtained providing that the magnetic core works in a linear regime, i.e. the partial saturation of the core has to be avoided. The inductance design has been performed using the standard rules with large margin regarding the maximum average induction allowed in the core. A special attention has been paid to the thermal heating control of both the core and winding. The temperatures have been calculated at the nodal position over a 3D meshing by considering the core and winding losses as heat sources and the cooling by natural convection regime at boundary condition. The thermal resistance network and its numerical implementation were described in a previous work [START_REF] Delette | Thermal Management Design of Transformers for Dual Active Bridge Power Converters[END_REF]. We considered two kinds of ferromagnetic material:

(i) a MnZn ferrite from TDK (N87) (ii) a powder core from Magnetics (Koolµ60) Commercial core geometries from the supplier catalogs have been screened, and for each material, a tradeoff has been found between the compactness and the heating. The characteristics of the designed inductors are listed in Table I. The calculated temperature maps are plotted on Fig11. It can be seen that the configuration with the Koolµ60 core is more prone to the winding heating due to a larger turn number compared to the N87 inductor. This is a consequence of the lower permeability of the powder core (µr=60) vs the ferrite one (µr = 1700). In the same time, cooling of the ferrite solution was improved thanks to a larger external surface. Finally, with a maximal operating temperature of 80°C, the Koolµ60 core volume is 25 % lower than the volume of its ferrite counterpart. Fig. 12 and Fig. 13 shows that the inductance value measured on inductors prototypes remains steady up to 40 A for the two selected materials.

Experimental results on a flying capacitor inverter with GaN devices

In this part, the experimental results with GaN devices are given within the 3-level 1500 V inverter. Fig. 14 shows the schematic of the 3-phase flying-capacitor inverter on the left side where the input voltage is 1500 VDC and the output is 3-phase 800 VAC. In order to mitigate the influences of the EMI perturbations, the optical communications (optical controls and optical measurements) with an Imperix controller were implemented, and the batteries supply all the auxiliary circuits. The size of this part is 150 x 150 x 80 mm including the underneath heatsink. This is the first implementation of the GaN based Flying-capacitor inverter within this article; we call it version 1 (V1). In this implementation, we could not increase the DC bus voltage up to the nominal value; we are working on the modification of this version for a further experiment, in particular, driving GaN HEMTs by negative base voltage to avoid the spike on the gate when the high dv/dt occurs.

In the middle of Fig. 15, the second version of the GaN based Flying-capacitor inverter (V2) with GaN FET 900 V; the gate drivers Si8271 with the isolated power supplies generate VGS of 0/9 V. Due to high dv/dt, once again, this set-up could not perform at the nominal DC bus voltage. On the left side of Fig. 3, the third version of the GaN based Flying-capacitor inverter (V3) with GaN FET 900 V; the gate drivers Si8271 with the isolated power supplies generate VGS of -5/14 V. To simplify the implementation, the Imperix's voltage and current sensors replace the optical onboard measures. By using this set-up, the GaN devices could perform at nominal input voltage, and generate the suitable sinusoidal output voltage for 3-phase grid of 800 V. In the designs with GaN FET 900 V, the GaN power in packaging TO-247 are between the board of the auxiliary circuits and the heatsink. For one-phase implementation in Fig. 16, bulky input capacitors (900 µF/1500 V) have been used to mitigate the 100 Hz fluctuant voltage on the input DC bus voltage, these bulky capacitors do not exist in the final 3-phase inverter. Using these high voltage GaN devices where GaN HEMT are still prototype devices for these implementations, the authors preferred to implement one-phase experiment before going further for the full implementation of the 3-phase inverter. Fig. 19 shows the thermal result of this one-phase inverter leg under 11.8 kW active output power, for a short duration of the experiment, the temperature on the packaging of GaN devices reach 40°C and keep increasing. We are working to demonstrate the complete 3-phase inverter with both GaN HEMT 1200 V and GaN FET 900 V with modifications on the gate driver circuitry for GaN HEMT 1200 V, on the thickness of PCB and betterments on the thermal perspectives. The grid-tie protocols implement as our control loops for a grid-tie three-phase current source inverter [START_REF] Bier | Three-phase grid-tied current-source inverter sizing and control for photovoltaic application[END_REF][START_REF] Bier | Control of a single-phase grid-tied GaN based solar microinverter[END_REF]. Eventually, in this demonstration of the complete 3-phase inverter, we are going to measure the stabilized temperature on the GaN power devices and the efficiency of the inverter.

Conclusion

This work presents the design of a 3-level flying capacitor 1500 VDC inverter with characterized GaN devices for 3-phase grid-tie applications where GaN devices perform under 750 V in steady-state operation. The static and dynamic characterizations take place to ensure the operation of the selected high voltage GaN devices in a nominal configuration. The paper include the PLECS simulations for this flying capacitor topology to identify the different working frequency of the elements in the inverter. The current and the voltage across the elements are presented in order to design the passive devices of the inverter; in particular, the output inductor filter is 3D-designed, optimized, characterized in this work.

In addition, the design, the detailed characterizations of this compact GaN based flying-capacitor inverter with its dedicated output filters are given. We show several difficulties due to the GaN devices themselves and issues on the design of gate driver and their power supplies. Finally, an experiment was performed at the nominal voltage of the PV string, 1500 V. The nominal power announced is 20 kW as the power of a smart PV string [START_REF] Catellani | GaN based panel-integrated, high-efficiency DC/DC optimizer for maximizing the yield of the large photovoltaic power plant[END_REF], however the experimental characterizations show that the output power of this flying capacitor 1500 VDC inverter can reach almost 36 kW. The realization of a complete 3-phase inverter is ongoing with improvements on the mechanical and thermal point-of-views.

Fig. 1 :

 1 Fig. 1: Static characterizations of capacitances for GaN FET 900V (DUT1) and GaN HEMT 1200V (DUT2)

Fig. 2 :

 2 Fig. 2: Threshold voltages of GaN FET 900 V (DUT1) and GaN HEMT 1200 V (DUT2)

Fig. 3 :Fig. 4 :

 34 Fig. 3: Static resistances of GaN FET 900 V (DUT1) and GaN HEMT 1200 V (DUT2)

Fig. 5 :

 5 Fig. 5: Double-pulse-test bench with combined footprint for DUT1 and DUT2

Fig. 6 :

 6 Fig. 6: Experimental waveforms of the DPT testIn Fig.7, GaN devices (FET 900 V) turns on from a blocking voltage of 760 V, a peak slew rate dv/dt of 100 V/ns is observed.

Fig. 7 :

 7 Fig. 7: Fall time of 6.8 ns of the drain source voltage (0 V/760 V) at turn-on

Fig. 8 .

 8 Fig. 8. Rise time of 20 ns of VDS at turn-off

Fig. 9 :

 9 Fig. 9: PLECS simulation of the 3-phase flying-capacitor inverter

Fig. 10 :

 10 Fig. 10: Simulation results: waveforms on the output filters with twice higher working frequency

Fig. 11 :

 11 Fig. 11: Temperature maps for Ferrite (left) and Koolµ (right) solutions

Fig. 12 :

 12 Fig. 12: Kool-Mµ based output filter with 22 turns of Litz wires: 105 x 75 x 55 mm

Fig. 14 :

 14 Fig. 14: Schematic of the 3-phase flying-capacitor inverterOn the right side Fig.13, GaN based active parts for one phase are demonstrated with GaN HEMT 1200V devices, so-called DUT2 in the previous sections, integrating gate driver Hey-1011 dedicated to GaN devices, GaN transistors are driven by VGS of 0/6 V. In order to mitigate the influences of the EMI perturbations, the optical communications (optical controls and optical measurements) with an Imperix controller were implemented, and the batteries supply all the auxiliary circuits. The size of this part is 150 x 150 x 80 mm including the underneath heatsink. This is the first implementation of the GaN based Flying-capacitor inverter within this article; we call it version 1 (V1). In this implementation, we could not increase the DC bus voltage up to the nominal value; we are working on the modification of this

Fig. 15 :

 15 Fig. 15: Three different versions of GaN based active parts for the implementation of one-phase

Fig. 16 :

 16 Fig. 16: Experiment of one-phase GaN FET 900V based inverter leg Fig.17 shows the 100 kHz control signals of four GaN devices in a leg of the flying-capacitor topology, also the 3-level voltage at the middle point of the leg connected to the AC grid filter. With all the three versions of the implementations, these control signals and the 3-level voltage perform correctly at the low DC bus voltage. The differences appear at a higher DC bus voltage when the dv/dt becomes critical to the low threshold voltage of GaN devices.

Fig. 17 :

 17 Fig. 17: Control signals of four GaN devices in a leg and the 3-level voltage of the legFinally, the experimental results in the nominal DC-Bus voltage are given in the Fig.18where the input DC-bus voltage is 1420 V (on channel 4 -green color), the flying-capacitor voltage is approximately 710 V (channel 2 -red color). At the output on a resistive load for this experiment, the output sinusoidal RMS voltage (on channel 3 -cyan color) is 462 VAC, this voltage is equal to 800 V/√3 as line to neutral value. The output RMS current is 25.6 A (on channel 1 -yellow color), where the maximum current goes through the output filter is up to 40 A. The output power in this single-phase or so-called one-phase experiment is 11.8 kW with a fan underneath the heatsink. Based on these experimental results, in the full configuration, the power of the 3-phase GaN based flying-capacitor could reach almost 36 kW.

Fig. 18 :

 18 Fig. 18: Waveforms of a one-phase inverter under DC bus voltage of 1420V

Fig. 19 :

 19 Fig. 19: Thermal image of one-phase under the output power of 11.8 kW

Table I : Characteristics of the designed inductors

 I

		Ferrite core (N87)	Powder Core (Koolµ60)
	Core + winding volume	421 cm 3	315 cm 3
	Turn number with 729 x 0.071 mm Litz wire	6	22
	Gap	0.5 mm	1 mm
	Iron loss	4.5 W	3 W
	Winding loss	10 W	14 W

Acknowledgements

The authors acknowledge CEA/LITEN -CARNOT project "FlyGaN" for funding the activities presented in this paper.