Cécile Riou

Bruno Colicchio

Jean-Philippe Philippe Lauffenburger

Olivier Haeberle

Olivier Haeberlé

Christophe Cudel
email: christophe.cudel@uha.fr

Calibration and disparity maps for a depth camera based on a four-lens device

Keywords: Multi-view cameras, depth cameras, homography, disparity map, camera calibration

Introduction

Multi-view imaging is a large domain, where numerous approaches can take place for computing the depths in a scene. Stereovision 1 , multi-camera array [START_REF] Vaish | Using plane + parallax for calibrating dense camera arrays[END_REF] , light-field imaging [START_REF] Birklbauer | Panorama light-field imaging: Panorama light-field imaging[END_REF][START_REF] Adelson | Single lens stereo with a plenoptic camera[END_REF] or coded aperture imaging [START_REF] Levin | Image and depth from a conventional camera with a coded aperture[END_REF] are the most well-known technics for capturing several point of views of a scene. Using a metric calibration on these systems allows for addressing recently growing applications in a research or industrial context. In multi-view imaging, the depth estimation problem is then related to the disparity between same pixels projected on the different coordinates in the different views. This task usually relies on the identification of the similarity between the different views, allowing for computing displacements of identified points and their corresponding 3-D positions.

The similarities identification is usually a time consuming task and is referenced as matching methods based on local or global methods [START_REF] Scharstein | A taxonomy and evaluation of dense two-frame stereo correspondence algorithms[END_REF] . Local methods compare two regions of interest by measuring similarity with well-known criteria such as block matching or feature approaches [START_REF] Szeliski | Computer Vision[END_REF] .

These approaches are efficient on images containing a large percentage of textured area, but fail for uniform regions, because these areas contain too few information for the matching process.

Occluded regions, due to the difference of point of views, are another cause of occurring errors in the matching process. Global approaches can overcome some of these issues, by estimating a disparity map that minimizes energy criteria on the whole image. Graph cuts [START_REF] Kolmogorov | Computing visual correspondence with occlusions using graph cuts[END_REF] , belief propagation [START_REF] Tseng | Low Memory Cost Block-Based Belief Propagation for Stereo Correspondence[END_REF] or dynamic programming [START_REF] Veksler | Stereo Correspondence by Dynamic Programming on a Tree[END_REF] methods are the most well-known global approaches.

Computational complexity of these methods is often high, and they are not able to properly solve the cases where images contain large uniform surfaces. For robotics or industrial control, lightpatterns are projected on whole objects and background of the scene, for helping the matching process. In this case, patterns modify the aspect of homogeneous objects, which seem to be textured. These active techniques are efficient for computing disparity map, but images are then unusable for any other controls such as edge detections, intensity measurements, barcode reading or OCR.

In order to deduce a depth from disparity, a metric calibration of the acquisition system is needed. Usually, calibration consists in determining intrinsic parameters for the acquisition device used for metric measurements, and permits to compute distances between scene objects and the camera device. An interesting survey of the reference methods is proposed by Zhang 11 , where approaches are divided in three categories following the reference object used for calibration. Highest accuracy can theoretically be obtained by using a 3-D object for calibrating, but in practice, calibration with a 2-D apparatus seems to be the best choice in most situations, because of its ease of use and good accuracy. In this case, calibration is performed by taking images of the 2-D pattern plane under different orientations by moving either the plane or the camera. A dozen of acquisitions is then a recommended minimum. Calibration of multi-camera devices has been studied by Vaish [START_REF] Vaish | Using plane + parallax for calibrating dense camera arrays[END_REF] . This work compares a non-metric approach, using plane and parallax, with metric calibration for computing synthetic aperture images. If the operating mode for calibrating is similar to our approach developed in this paper, the proposed affine model leads to generate images focused at different relative or uncalibrated depths.

We propose in this work a multi-lens device (Fig. 1) and a post-processing algorithm using contours in order to avoid computing a disparity map of homogeneous objects by using projected light patterns. A fast and flexible calibration step is also proposed. The calibration of this device could be qualified as an indirect method, because intrinsic parameters are not determined individually as is usually done, but in an overall manner using a reference homography. One of the main interests is to simplify the calibration step, which is then easily achievable by an operator for industrial applications. This method is an improvement of previous works [START_REF] Xu | Original method to compute epipoles using variable homography: application to measure emergent fibers on textile fabrics[END_REF] , in which we have extended the principles of "variable homography", defined by Zhang and Greenspan [START_REF] Zhang | Variable Homography Compensation of Parallax Along Mosaic Seams[END_REF] for measuring the height of emergent fibers on glass and non-wovens fabrics. This method has been defined for working with fabric samples progressing on a conveyor belt.

Triggered acquisition of two successive images was needed to perform 3D measurements. In this work, we have retained advantages of variable homography for measurements along the optical axis, but we have reduced the acquisitions to a unique one, by developing a device made of four lenses placed in front of a single image sensor. We have also adapted the variable homography formulation for this device, and we give a new formulation to calculate the depth. This method is presented in the first part of the paper and can be applied to any disparity maps calculated from various methods. In the second part, we propose also a new framework for computing disparity map adapted to our device, which then becomes a depth camera. The disparity map computation is an alternate to matching methods, exploring a contour based solution, where no correspondence matching is needed, drastically reducing the computation time. In 3-D industrial control by vision, most of stereo-vision systems project structured or random patterns on objects for providing texture on uniform surfaces. This method is efficient for measuring disparities between the multiple views, but in this case, only the depth images can be used for image analysis because the scene photometry is strongly modified by the projected pattern. On the contrary, we propose an algorithm preserving gray image levels. The camera could then be used to both measure object height and perform any other classic vision control, such as optical characters or pattern recognition. However, this method is not fully universal, and we have only studied cases for measuring flat objects, in a context of industrial control. This paper presents firstly the geometry of our device and describes the extension of variable homography for 3-D measurements. This approach leads to propose a new calibration scheme, for which only a reference homography is needed, instead of the intrinsic parameters, as usually.

Secondly, we have proposed an alternate manner of computing a depth map. Preliminary and promising results on synthetic and real images are presented in the last part.

Variable homography and depth measurements from disparity

The concept of variable homography has been defined by Zhang and Greenspan [START_REF] Zhang | Variable Homography Compensation of Parallax Along Mosaic Seams[END_REF] for parallax compensation in image mosaicking without metric calibration. This concept has then been extended by Xu et al. [START_REF] Xu | Original method to compute epipoles using variable homography: application to measure emergent fibers on textile fabrics[END_REF] , for 3-D measurements by using two successive acquisitions, of fabrics scrolling on a conveyor belt. This part presents how the variable homography can be used for modeling our multi-view device, described on the figure 1. Figure 2 gives a schematic representation of our system made out of mini-lenses and a single image plane. This figure is a 2D section representing only two neighboring projections, given the sub-images noted I i and I i+1 (we will use i=1 for next expressions). With four projections on the image sensor, each sub-image has three possible neighbors. If index i represents the SE view (South-East), index i+1 can be any view of NE (North-East), SW (South-West) or NW (North-West). This scheme simplifies our system to a parallel stereovision system, where triangulation could be used to determine the depth of point P when intrinsic and distortion parameters are known. Ideal situation is encountered when these parameters are identical for each camera, which is however never the case in practice. Precise calibration is then needed to determine numerical values of optical centers and focal distances. These parameters are necessary for the geometric re-projection, for computing depth by intersecting the two rays of each associated left and right image pixel. For example, on fig. 2,f An alternative approach is to take benefit of variable homography for modeling this device.

Variable homography definition for a pair of views

Variable homography formulation introduces a matrix K giving some interesting properties between both homography H A and H B defined for the A and B parallel planes and verifying p 1A =H A .p 2A and p 1B =H B .p 2B :

1 1 . . ) ( - + = i A i B K H K k H where           - - = 1 0 0 ) 1 ( 0 ) 1 ( 0 0 0 n n n n n n n v k k u k k K , ( 1
)
with n=i or i+1 and where u 0n and v 0n stand for the coordinates of the optical centers O n projected on the image sensor. The main parameter of K is the distance ratio k n defined by

k n =Z An /Z Bn ,

H A and H B expressions

Plane A is considered as the reference plane, and is used for calibrating the device. As our device uses 4 mini-lenses, it's interesting to study the case where these lenses don't have exactly the same focal distances. H A is then determined by assuming that homography is a combination of an intrinsic parameter and translation matrices, and is given in Ref. [START_REF] Hartley | Multiple view geometry in computer vision[END_REF] as:

                  - + - + =                   - + - + = 1 0 0 . 0 . 0 1 0 0 0 0 02 1 1 01 02 1 1 01 02 1 2 2 1 1 1 01 1 2 2 1 02 1 2 2 1 1 1 01 1 2 2 1 v Z f b v u Z f b u v Z Z f f Z f b v Z Z f f u Z Z f f Z f b u Z Z f f H A v A u A A A v A A A A A u A A A α α α α , ( 2
)
where b u and b v are horizontal and vertical distances between optical centers, respectively. By

considering 1 1 2 A A A Z Z Z ≈ + = ε
at the working distances, H A has been simplified in eq. 2 by

introducing the focal distance ratio α=f 1 /f 2 . Expressions ( 1) and ( 2) lead to give a simplified expression for H B , representing the geometric transformation for any point P located in a virtual plane B, parallel to the reference plane A:

                  - + - + = 1 0 0 . 0 . 0 02 1 1 1 01 02 1 1 1 01 v Z f b k v u Z f b k u H A v A u B α α α α . ( 3
)

P point depth computation from a pair of views

From H A and H B homographies, we propose to compute the depth of any point P, by the definition of disparity d given by

B A B B p H p H d 2 2 . . - =
. By considering that

B B B p H p 2 1 . = and
with expressions (2) and ( 3), d can be written as:

( )  ( )  2 / 1 2 1 1 1 2 1 1 1 1 2 2 1 1 ) , ( . . 1                           - +             - = = - =             v v u u A B d A C v d A C u A B p B A p B B Z f b k Z f b k p p dist p H p H d . ( 4
)
Assuming that

1 1 1 / B A Z Z k = and 2 2 v u b b b + =
, relations between depth Z B1 and disparity d are finally given by:

1 1 1 2 2 1 2 2 1 1 . A v u A v u A B Z bf d bf C C Z d C C Z Z + = + + + = or 1 1 1 1 A B Z bf Z bf d - = . ( 5
)
This definition of disparity is slightly different of the one usually given. In the next part, we study the differences and advantages to use this definition for 3-D measurements, especially in a 3-D industrial vision context.

Comparison with triangulation method

When two cameras are separated by lateral translation with no rotation, triangulation method leads to obtain the depth relative to p 1B and p 2B coordinates. From figure 2, depth Z B1 can be expressed as:

( )

1 1 1 2 2 2 2 2 2 1 ) ( ) ( f O p f O p p O f b Z B B B B - - - - + = ε (6)
This relation is usually simplified assuming f 1 =f 2 , and O 1 =O 2 in the coordinate systems of each sub-image:

d bf Z B 1 1 = with B B p p d 1 2 - = . ( 7
)
By comparing this expression with the one obtained with variable homography, there is an evident similarity, and variable homography translates the disparity d by a term of ) (

1 1 A Z bf - .
This is observed on fig. 3, comparing expressions ( 7) and (5). Curves are plotted according to the real dimensions of our device: pixels size of 10 μm, optical centers spaced by 9 mm and

f 1 = f 2 =8
mm. Triangulation curve is plotted in red, and the ones obtained from equation ( 5) are plotted in blue for several values of the reference distance Z A1 . In function of Z A1 , blue curves are thus shifted to the left. In fact, this translation offers some interesting properties. When the range of measurement is predicable, it would be then possible to work in the same range of disparity whatever the working depth, by choosing correctly Z A1 . It is quite easy to demonstrate that the depth error 1 B Z ∂ stays identical for both approaches, meaning this translation does not influence the accuracy of measurement:

d bf Z Z Z bf Z d B B B B ∂ - = ∂ → - = ∂ ∂ 1 2 1 1 2 1 1 1 (8)
There are some advantages to use variable homography for working within the same range of disparity. The same device and same algorithm can be used for a wide working distance range, from macroscopic applications to higher scales. For example, if Z A1 equal 7 cm, depth Z B1 is ranged between 7 and 5.8 cm for an interval of d of [0,20] pixels. For 3-D industrial vision inspection, when Z A1 is fixed at 90 cm, the corresponding measured depths are ranged from 90 cm to 26 cm, while keeping the same d interval. We can note that, for these different scales of measurements, we have used the same device. Switching between the different scales only requires adjusting the lenses focal distances. This last point is important, because in this case, accurate measurement would need to re-determine all intrinsic parameters using expression (6) when triangulation method is used. Next part explains how variable homography leads to an easier calibration step.

Simplified calibration for multi-view cameras

In practice, lenses are never perfectly aligned in multi-lenses cameras (lenses are not sorted), and lenses focal distances are also never strictly equal [START_REF] Dansereau | Decoding, Calibration and Rectification for Lenselet-Based Plenoptic Cameras[END_REF] . This is especially true for our prototype and this creates some undesirable distortions affecting depth measurements. In practice, imperfections are rarely taken into account and literature usually gives expression (7) as reference method to compute depth instead of the complete expression given by eq. ( 6). By considering a multi-view device, the complete determination of these intrinsic parameters could be an uncomfortable task, requiring multiple acquisitions [START_REF] Zhang | A flexible new technique for camera calibration[END_REF] , but nevertheless mandatory for providing accurate measurements.

A great advantage of variable homography is to include in matrix H A all intrinsic parameters between the views of 2 sub-images. By projecting measurement points from sub-image 2 to subimage 1 with H A , the number of required parameters for 3-D measurement is then reduced, and they are easy to determine via a simple calibration process. Sole parameters to recover during the calibration step are: Z A1 , H A , and the constant bf 1 . The easy and robust proposed calibration method is realized by using a reference chessboard pattern for detecting corners and by performing two successive acquisitions. The first acquisition is done at reference depth Z A1 , and the second one at depth Z B1 =Z A1 +d z . Corners of first acquisition are used to calculate all combinations of existing H A homographies between sub-images. For each acquisition, distances in pixels between the first and last corners detected on the chessboard are computed. We note d A the distance for the dataset of first acquisition and d B the distance for dataset of the second one.

Z A1 is then obtained by triangulation and is formulated as:

A B B z A d d d d Z - = . 1 . (9)
Finally, Z A1 , H A , d z and distances d, computed for corresponding corners p 1B and p 2B during the second acquisition, lead to find best value for bf 1 . In conclusion, interest of this calibration lies in its simplicity when calibrating, because it is easier to calibrate via two acquisitions than to determine the intrinsic parameters with several snap images, as usually done with well-known standard methods [START_REF] Zhang | A flexible new technique for camera calibration[END_REF] .

This calibration process for determining the variable homography parameters has been established by considering two views. This principle is easily extensible for a n-view camera, i.e our device. Calibration is then performed for each pair of working views, without additional acquisition.

Improved calibration with lens distortion correction

Our multi-view camera prototype uses low-cost lenses, presenting a distortion coefficient close to 1.5% (value is issued from lens datasheet). To improve measurements accuracy, we propose to take into account the distortion in our calibration scheme. By using standard approaches, such as corrections by using radial distortion models, the precise positions of the optic centers of each sub-image can be determined. As these intrinsic parameters are combined in H A , they are not immediately available at this step with our calibration process. We propose therefore an alternate approach. We consider that distortion effects are comparable to a non-uniform image magnification on image plane. This approach leads to adjust the computation of Z B1 as a function of the spatial location of correspondences in sub-images. This can be done by using the proportional relationship existing between magnification, focal distance and parameter bf 1 . The correction is performed by using the Γ operator, which transforms a vector representing a point P of coordinates (u,v,1) in the lifted coordinates [START_REF] Barreto | Epipolar Geometry of Central Projection Systems Using Veronese Maps[END_REF] P ~:

T v u v u v u P P P v u P T ) 1 , , , . , , ( ) , ( ) 1 , , ( 2 2
= Γ = → . ( 10
)
By using p B1 as reference location, bf 1 is then computed by the quadratic functions defined by:

( ) 2 1 . B p f e d c b a bf = . ( 11
)
Corners collected previously form a dataset and permit to compute the polynomial coefficients by using a least mean square method.

Measurement accuracy

The measurements obtained in this part have been obtained with our device presenting the following characteristics: CCD size: 15x15 mm 2 ; CCD resolution: 2048x2048 pixels; minilenses diameters: 8 mm; mini-lenses focal distance: 7.5 mm; distance between mini-lenses: 9 mm; sub-images resolution: 550x550 pixels. We present experimental measurements for three ranges of depth, in order to estimate the accuracy of our device in term of depth measurement.

Here, only measurements performed on SW and SE sub-images are presented for this experimentation. As shown by expression (8), depth error is a function of two factors: geometric parameters estimation and disparity measurement. Top: results of simplified calibration method. Bottom: results of improved calibration method.

At this point, we are interested in measuring the quality of the parameters estimated during calibration. For minimizing the role of disparity measurement in our estimation, we have used a pattern chessboard as depth target and corners detection is performed with the corresponding function of the OpenCV library. For each depth level, numerous acquisitions are performed, leading to about one thousand measurements. We have set the camera-chessboards distances at 4 cm, 33 cm and 88.5 cm, and the measured depth corresponds to Z B1 (distance separating the chessboard and the optical center). Figure 4 gives distributions of depth measured for each corners. By working at a short distance, captured images present important optical distortions. In this case, the second method offers a significant improvement, providing a standard deviation lower than 1%. For evaluating depth error, percentage is not significant here, because the uncertainty of measurements is directly linked to the depth range (see eq. 8), meaning depth error increases systematically with depth. This is the reason why our evaluation is presented by comparing theoretically predicted error with the measured standard deviation. In this case, the main difficulty concerns evaluation of the disparity variation d ∂ , which is a difficult to assess parameter. Even if we use a subpixel corner detector, d ∂ heavily depends on the pattern chessboard scale in sub-images and thus of the working distance. To provide best estimations, we have bounded its values between two realistic limit values, fixed here at 0.1 and 0.5 pixel. These values are then used for the result comparisons presented in table 1. This table shows that standard deviations observed during our measurements are comprised within the range of estimated depth error. This confirms that the proposed calibration is well-adapted to our measurement approach by variable homography.

These values of depth error allow for keeping a good uncertainty of measurements for applications in the field of metrology by vision.

Sub-images rectifications

Previous measurements were performed on coordinates of detected corners. To transform our device into a depth camera, disparity has to be computed on all pixels of the sub-image. As we have defined disparity as d=p 1B -H A .p 2B , a simple pre-processing steps can be performed, for rectifying each sub-image with homographies H A . In this case a reference sub-image must be chosen. Next examples consider SE as reference sub-image, as illustrated on fig. 5, but any other sub-image could be selected instead. By rectifying sub-images, we ensure that correspondences between images are well aligned along the epipolar line, as well-known in stereovision. By organizing lens positions in a square, epipolar lines are then horizontal, vertical or diagonal following positions of sub-images used for computing disparity. Multi-views approach is then interesting because in the reference sub-image, epipolar lines calculated from other views could be combined to improve the matching for establishing correspondences and thus enhanced the final disparity map.

Computation of disparity with the four mini-lenses device

With the set of rectified images, we propose to compute the corresponding disparity image (or disparity map), in order to use our device as a depth camera. The main idea is to subtract distances map between sub-images in order to highlight disparity values. This approach is to be used in a normal lightning environment with uniform objects. The next section shows that the proposed method doesn't need complex calculations, which is an advantage for real-time applications. However, this method is not universal, and can, up to now, be used for flat objects only.

Principle

The four sub-images delivered by our device can be combined for computing up to three disparity images. We have validated the principle by using only two associations with three subimages: disparity (SE, SW R ) and disparity (SE, NE R ). The reference sub-image is then SE, as illustrated on fig. 5. Our goal is to fill the content of any objects by the disparity existing between comparable contours of two sub-images. To do so, we propose an algorithm composed of four main steps and operating between two sub-images:

• computation of Distance Images DI xx ;

• computation of pseudo-Subtraction Images pSI xx ;

• application of a fusion process;

• computation of a disparity map by decoding the image given by the previous step.

As up to three disparity maps can be computed from the reference sub-image, a second and final step merges these maps in order to eliminate some artifacts. The depth image is then directly obtained, thanks to the previous calibration, and using expression (5).

• Distance images: DI xx

The We have defined two distinct operators for exploiting the property that subtraction between DI images can give the disparity. Furthermore, this property is not true on the whole image, especially when the flag × is encountered. We have established that the best result, where object is filled by the disparity value, is obtained by using two kinds of subtraction called here pseudo-subtraction, where the result value can be the flag × , the first operand or the true difference.

These operators are defined with arguments a and b , where a and b are pixel values of the reference image and the second input image, respectively. Definitions of these operators are for DI LR images:

( )

   × Ν ∈ - → otherwise b a if b a b a pSI LR , ) , ( , (12)
and for DI RL images:

( )

     × × = Ν ∈ - → otherwise b if a b a if b a b a pSI RL , ) , ( . (13)
Fig. 7 Examples of pseudo-subtractions pSI LR , pSI RL and FI for an object measuring l=8 width and for a d=4 disparity.

• Fusion image

The fusion uses pSI LR and pSI RL images for replacing on pSI LR image, the values appearing outside object by flag × . This operation is simply defined by:

   × Ν ∈ → otherwise b a if a b a FI , ) , ( . (15)
•

Decoded fusion

The goal of this final step is to decode the content of fusion image in order to identify each object by its disparity, position and dimension. Thanks to the previous steps, especially the distance propagation and pseudo-subtraction ones, a complete description of objects contained in original sub-images, can be extracted from the pixel values of the fusion image. These values are constituted of combinations of objects positions, sizes and disparities. For N separated objects in each view, we have established by simulations, that each pixel value in fusion image, can be decoded. We note and define:

d i , l i and p i , respectively, the disparity, length and first position of object i; . The number of occurrences of terms A i , B i and C i inside each sequence depends on the disparity, length and position of objects in the sub-images. This information is not used for the decoding. For the example of figure 7,with d Finally, the role of the decoding stage is to use this property for producing the disparity map.

- ; + -l -p -d =p C ; - -d =l B ; =-d A i- i- i i i i i i i i 1 1 1 1 - ; ... ... ... 1 1 1 1 1 1 1 A A B B A A Seq = - . ... ... ... ... i i i i i i i i i A A B B A A C C Seq = Each line
Depth images are then established with expression (5), linking depth and disparity.

Merging disparity maps

The algorithm presented above computes disparity map from two sub-images. A complete overview, resuming step connections, is shown on fig. 8. This scheme is designed work with any couple of sub-images. As this approach is only based on the contours detected in each subimages, the quality of initial contours is a primary initial pre-preprocessing step. For the moment, no matching process is performed between sub-images and this is an advantage of this approach.

Even if matching could be useful for ensuring that each contour point in sub-image i would have a corresponding point in sub-image i+1, we have decided to skip it, for increasing the speed of our method. But, with no matching process, some contours in sub-image i may have no correspondence in sub-image i+1. DI images are then affected and some artifacts in disparity images can appear. Following scanning direction, they are characterized by horizontal, diagonal or vertical trails. A solution for attenuating these artifacts has been found by exploiting the multiview capability of our device. As from a reference image (see fig. 5) three disparity maps can be computed, we propose to merge them for smoothing artifacts and retaining the most plausible disparity value for each pixel.

Results

Figure 9 presents some results on synthetic scenes, showing objects located at several depths.

The first case shows a basic configuration, where objects are located far away from each other. A second case shows superimposed objects. For these two cases, no final merging is necessary and only one disparity map is sufficient to generate the final depth image. For the last example, we consider a situation where all contours pixels have not correspondences in sub-images, as can happen in experimental conditions. The map image is then computed with the merging process using here disparity images (SE, SW R ) and (SE, NE R ) as inputs. Final depth image still contains residual errors, but stays very close to the expect result and is suitable for most of 3-D inspection processes in industrial vision. Figure 10 shows some typical height images computed on real scenes. We explore situations with objects having large uniform areas in natural lightning, for which standard matching methods often fail when no structured illumination is used. To do so, we have adapted the calibration distance Z A1 to the objects scale, as explained in paragraph 2.4. Heights measured for the four reference objects are then well recovered. One can however see some artifacts, which are still visible on the final depth map. The main characteristic of the proposed method is its ability to compute disparity map without requiring any matching. With a single image sensor snapping simultaneously four views, the captured images present similar brightness and contrast.

It could be expected that edge detection provides same contours for the four views. In practice, due to the image sensor sampling, it is not always the case and some no-exact correspondences between detected contours are sometime encountered. Figure 11 presents an example of this situation, when no-exact correspondences between detected contours corrupt the depth map.

Other artifacts could also be caused by the difference of point of views, resulting in occultations or hidden edges between the sub-images. With our method, this kind of artifact is significantly smoothed by the merging process and is not the most frequently seen during our experiments.

Our approach for computing depth map is therefore validated by these synthetic and experimental results. This work should provide some new perspectives for establishing disparity maps without matching procedure, which is the main interest of the approach. The existing artifacts encountered in some cases should disappear or be attenuated by improving and adapting an edge detection method for multi lenses device.

Conclusions and perspectives

We have described a multi-view device considered as a multi-view camera and used as a depth camera. This device can work for detecting objects to a depth of up to two meters and could find interests for industrial vision. Main contributions consist firstly in an alternate approach for modeling and calibrating this device with variable homography, and secondly in an original and new approach for computing disparity map or depth images.

Variable homography is an alternate approach for modeling multi-view systems. Main interest is to propose a rapid calibration procedure requiring only two acquisitions, without having first to determine intrinsic parameters. A comparison with the well-known triangulation method is performed, and we have demonstrated that the measurement uncertainties are then comparable, but calibration is easier in our case, especially if intrinsic parameters are not considered as identical for each sub-device. When optical distortions are significant, as for measurements performed at macroscopic scale, we have also proposed an improved calibration method, significantly reducing the measurement uncertainty. Calibration stays identical for normal and improved method and always requires only two acquisitions. The principles developed with the variable homography method lead to use a reference homography called H A , and measurement are based on a disparity d defined as

B A B B p H p H d 2 2 . . - = .
In the second part of this paper, a new approach has been presented for computing disparity map, according to the definitions given previously. The major interest of this approach is to propose a new framework requiring no matching process defeated by uniform or poorly textured areas. This approach is well adapted for measuring height of objects placed under a camera for industrial vision control. By using contours, this method is efficient on flat and uniform objects in natural lightning. Our solution can be considered as an interpolating method of the disparity or depth information in uniform areas, giving a realistic rendering of simple objects. As the scene photometry is preserved, unlike measurements performed with active illumination, some other standard controls as for example calipers, shape recognition or barcode reading, can be done conjointly with 3-D measurement.

This work has validated the interest of our approach, and some interesting perspectives are envisaged. First, we project to extend the calibration for other models of multi-views cameras, such as the focused plenoptic 17 ones. A study of the best compromise between resolution, depth accuracy measurement, working distance, pixel size and number of mini or micro lenses should be performed. Secondly, we would like to improve the original method proposed for computing disparity map. In the future, we will focus on edge detection, in order to reduce artifacts and also extend our approach for natural and complex scenes. Table 1 Comparison between theoretical and experimental depth measurement error. (bf 1 =7.2 pixels is obtained during the calibration stage). SC: normal calibration; IC: Improved calibration

Fig. 1

 1 Fig. 1 (a) Exploded view of the camera with its 4 mini-lenses (b) Assembled camera (c) Corresponding sub-images (North-West, North-East, South-West and outh-East).

1 ,f 2 ,

 12 O 1 , and O 2 must be known for recovering the depth of a point P from the rays (O 1 p 1B ) and (O 2 p 2B ).

Fig. 2

 2 Fig. 2 Geometric cross-section between two adjacent sub-images. O 1 , O 2 : optical centers of sub-images 1 and 2; b: distance between O 1 , O 2 ; f 1 , f 2 : lenses focal distances; ε difference between f 1 , f 2 . Z Ai , Z Bi : distances between O i and planes A and B. p 1A and p 2A : P A projections on both sub-images. p 1B and p 2B : P projections on both sub-images.

  p A : projection on sub-image 1 of the virtual point P' A .

Fig. 3

 3 Fig. 3 Depths Z B1 as a function of disparity d. Triangulation and variable homography comparison.

Fig. 4

 4 Fig. 4 Depth measurement distributions for three ranges of measurements: 4 cm, 33 cm and 88.5 cm.

Fig. 5

 5 Fig. 5 Sub-images rectification: pre-processing before computing disparity maps. By choosing SE as reference subimage, disparity maps between (SE, SW R ), (SE, NE R ) and (SE, SE R ) can be computed.

  first step generates distance images (DI) following predetermined directions. When the working sub-images are horizontals, then DI LR (Distance Image Left to Right) and DI RL (Distance Image Right to Left) are generated. Similarly, Top/Bottom distance images are generated when vertical sub-images. Diagonal extension is also possible. Distance Images are constituted of values indicating the distances between pixels to the previous contour encountered following the scanning direction. A flag × is used when no contour has been yet encountered by scanning.Figure 6 illustrates examples to compute DI LR and DI RL images from SE and SW R views. On this example, a rectangular object is represented by its contour pixels, highlighted in red. A disparity d (or parallax d u ) of 4 pixels is simulated and the green points P 1 and P 2 represent a point having same coordinates on both original sub-images. The main property used for our method is based on the fact that the difference ) any pixel inside the object is a function of the disparity value d.

Fig. 6

 6 Fig. 6 Example of computation of distance image DI LR and DI RL

Fig. 8

 8 Fig. 8 Steps used for computing disparity map from SE and SW R sub-images. 1: sub-image acquisition; 2: binary contour extraction; 3: Distance images computation according scanning direction; 4: pseudo-subtraction images computation: 5; fusion step; 6: image fusion decoding; 7: final disparity map

Fig. 9

 9 Fig. 9 Synthetic images (only sub-image SE is shown) containing objects at different depths and corresponding depth maps. (a): 4 objects without superposition are placed in the scene; (b): some superposed objects are placed; (c): contours in sub-image are randomly degraded. Instead of presenting experimental results in the form of depth images, we prefer defining the height images, which facilitates comparison with reference objects: height image = Z A1 -depth image. Differences are just a translation of measurement values by the reference distance Z A1 .

Fig. 10

 10 Fig. 10 Experimental results images (only sub-image SE is shown) with four objects. Measured heights are: A: 15.6 cm; B: 19.4 cm; C: 8.7 cm and D: 3.9 cm.

Fig. 11

 11 Fig. 11 Zooms on no-similar contours detection and corresponding artifacts on disparity maps. (a) 4 contours sub-images; (b) artifact areas where the contours are not identical.

Cecile

  Fig. 1 (a) Exploded view of the camera with its 4 mini-lenses (b) Assembled camera (c) Corresponding sub-images (North-West, North-East, South-West and outh-East).

Fig. 2

 2 Fig.2Geometric cross-section between two adjacent sub-images. O 1 , O 2 : optical centers of subimages 1 and 2; b: distance between O 1 , O 2 ; f 1 , f 2 : lenses focal distances; ε difference between f 1 , f 2 . Z Ai , Z Bi : distances between O i and planes A and B. p 1A and p 2A : P A projections on both subimages. p 1B and p 2B : P projections on both sub-images. pA: projection on sub-image 1 of the virtual point P' A .

Fig. 3

 3 Fig. 3 Depths Z B1 function of disparity d. Triangulation and variable homography comparison.

Fig. 4

 4 Fig.4 Depth measurement distributions for three ranges of measurements: 4 cm, 33 cm and 88.5 cm. Top: results of simplified calibration method. Bottom: results of improved calibration method.

Fig. 5 29 Fig. 6 Fig. 7

 52967 Fig. 5 Sub-images rectification: pre-processing before computing disparity maps. By choosing SE as reference sub-image, disparity maps between (SE, SW R ), (SE, NE R ) and (SE, SE R ) can be computed.

Fig. 8 Fig. 9

 89 Fig. 8 Steps used for computing disparity map from SE and SWR sub-images. 1: sub-image acquisition; 2: binary contour extraction; 3: Distance images computation according scanning direction; 4: pseudo Subtraction images computation: 5; fusion step; 6: image fusion decoding; 7: final disparity map Fig. 9 Synthetic images (only sub-image SE is shown) containing objects at different depths and corresponding depth maps. (a): 4 objects without superposition are placed in the scene; (b): some superposed objects are placed; (c): contours in sub-image are randomly degraded.

Fig. 10

 10 Fig. 10 Experimental results images (only sub-image SE is shown) with four objects. Heights measured are: A: 15.6 cm; B: 19.4 cm; C: 8.7 cm and D: 3.9 cm.

Fig. 11

 11 Fig. 11 Zooms on no-similar contours detection and corresponding artifacts on disparity maps. (a) 4 contours sub-images; (b) artifact areas where the contours are not identical.

  giving the possibilities to compute H B from H A . By sweeping k around realistic

	values and by using a matching comparison method, it would be possible to find reliable
	correspondences where k=argmax(Matching(p 1B (k), p 2B )). This is the idea presented by Zhang
	and Greenspan 13 in their paper to compensate parallax in image mosaicking. We propose to
	extend this method, by developing the expressions of H A and H B , in order to simplify the re-
	projection expression to compute depth and easily include in our model all intrinsic parameters
	described on fig. 2.

Table 1

 1 Comparison

	Depth Z B1 (cm)	Predicted depth error d bf z Z ∂ = ∂ ). / ( 1 2

between theoretical and experimental depth measurement error. (bf 1 =7.2 pixels is obtained during the calibration stage). SC: simplified calibration; IC: Improved calibration

Range of estimated theoretical depth error Observed experimental standard deviation

		∂d	min =	1 . 0	pixel	∂d	max =	5 . 0	pixel	SC / IC
	4	(0.22 10 -3 ). d ∂ =0.05 mm	0.022 mm		0.111 mm	0.12 mm / 0.04 mm
	33	(15.1 10 -3 ). d ∂ =3.78 mm	1.51 mm		7.56 mm	2.47 mm / 2.29 mm
	88.5	(107 10 -3 ). d ∂ =2.69 cm	10.7 mm		53.8 mm	49.6 mm / 50.1 mm

  of the fusion image is then constituted of the following typical sequences arrangement:

	[ ×	× ; ...	1 Seq	;	2 Seq	;... Seq i	;... Seq N	× ... ;	] ×

  1 =4 and l 1 =8, where image fusion is constituted of the arrangement [ ]

	×	-4; ... ; ×	-	4;	-	4;	3;	3;	3;	-4; 3;	×	× ...	, one can
	recognize Seq 1 .


Ali Bechouche
email: ali_bechouche@yahoo.fr

Hamid Seddiki
email: sedikihamid2004@yahoo.fr

Djaffar Ould Abdeslam
email: djaffar.ould-abdeslam@uha.fr

Koussaila Mesbah
email: mesbahkoussaila@yahoo.fr

Magister Ali Bechouche Received Engineer

Magister Hamid Seddiki Received Engineer

Adaptive AC Filter Parameters Identification

Keywords: Adaptive linear neuron, ac filter parameters, dead time analysis, decoupled terms, experimental investigation, identification, proportional-integral controller, voltage-oriented control, voltage-source rectifier (VSR), VSR nonlinearity

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Three-phase voltage-source rectifiers (VSRs) are being increasingly used as high performance grid interface in power systems like distributed and renewable energy generation [START_REF] Rashid | Power Electronics -Circuits, Devices and Applications[END_REF]. VSR offers several advantages in its operation such as bi-directional power flow, unity power factor (UPF), low harmonic distortion of input currents and high quality dc voltage [START_REF] Rashid | Power Electronics -Circuits, Devices and Applications[END_REF][START_REF] Kolar | The essence of three-phase PFC rectifier systems-Parte I[END_REF][START_REF] Friedli | The essence of three-phase PFC rectifier systems-Parte II[END_REF].

The use of VSR requires incorporation of control system to achieve high performance. Voltage-oriented control (VOC) is the most classical method for VSR control [START_REF] Rashid | Power Electronics -Circuits, Devices and Applications[END_REF][START_REF] Milasi | Adaptive vector control for voltage source converters[END_REF]Antoniewicz and Kazmirkowski, 2008). This decoupled scheme, originally proposed in [START_REF] Schauder | Vector analysis and control of advanced static VAR compensators[END_REF], uses two proportional-integral (PI) controllers in an inner loop to track current references and one PI controller in an outer loop to regulate dc voltage. Although the classical VOC has a simple structure, its robust performance to model uncertainties can be of concern [START_REF] Malinowski | A comparative study of control techniques for PWM rectifiers in ac adjustable speed drives[END_REF]. For instance, if the ac filter parameters are not known exactly, the d-and q-axes current dynamics are not fully decoupled. Hence, the PI-based control scheme can be driven into instability [START_REF] Shaker | Stability analysis and output feedback control for a class of switched nonlinear systems[END_REF].

Several methods have been proposed as alternatives to classical VOC scheme. Researchers have developed direct power control which leads to direct control of active and reactive powers [START_REF] Yan | Decentralised control for complex systems -an invited survey[END_REF][START_REF] Zhang | Model predictive direct power control of a PWM rectifier with duty cycle optimization[END_REF][START_REF] Norniella | Analytic and iterative algorithms for online estimation of coupling inductance in direct power control of three-phase active rectifiers[END_REF][START_REF] Kwak | Predictive-controlbased direct power control with an adaptive parameter identification technique for improved AFE performance[END_REF]. Control algorithms based on predictive approaches (Antoniewicz and Kazmirkowski, 2008;[START_REF] Zhang | Model predictive direct power control of a PWM rectifier with duty cycle optimization[END_REF][START_REF] Kwak | Predictive-controlbased direct power control with an adaptive parameter identification technique for improved AFE performance[END_REF][START_REF] Xia | Robust model predictive current control of three-phase voltage source PWM rectifier with online disturbance observation[END_REF][START_REF] Ramirez | A novel hybrid finite control set model predictive control scheme with reduced switching[END_REF][START_REF] Korelic | Predictive variable-structure system control of unity power factor rectifiers[END_REF] and input-output feedback linearization approaches [START_REF] Gensior | On some nonlinear current controllers for three-phase boost rectifiers[END_REF][START_REF] Burgos | Complex state-space modeling and nonlinear control of active front-end converters[END_REF] have been also developed. Although the above-mentioned methods can provide improved performances, they suffer from their complexity or ease of tuning. This paper improves the classical VOC by means of an adaptive method for ac filter parameters identification.

The proper identification of ac filter parameters has a significant influence on correct operation of VSR control. However, this relevant subject has not let to a large number of specific studies. Some works propose a methods based on recursive least squares algorithm [START_REF] Arriagada | On-line filtering reactance identification in voltagesource three-phase active-front-end rectifiers[END_REF]) and Lyapunov's function [START_REF] Milasi | Adaptive vector control for voltage source converters[END_REF] for inductance identification. Analytic and iterative methods are also presented in [START_REF] Norniella | Analytic and iterative algorithms for online estimation of coupling inductance in direct power control of three-phase active rectifiers[END_REF] for inductance estimation. The two methods are based on the discontinuities that appear in estimated grid voltage when there is an error in the inductance estimation. In (Antoniewicz and Kazmirkowski, 2008), the inductance was estimated from average absolute values of VSR ac-side voltage and current. This method is simple, but second order low-pass filters are required to dismiss the high frequency variations associated with those magnitudes. Hence, the dynamic method is relatively slow, which leads to high inductance estimation error. Recently, a least square algorithm-based method is proposed in [START_REF] Kwak | Predictive-controlbased direct power control with an adaptive parameter identification technique for improved AFE performance[END_REF] for adaptive resistance and inductance identification. Although the mentioned methods can provide acceptable results, some of them lead only to inductance estimation [START_REF] Milasi | Adaptive vector control for voltage source converters[END_REF]Antoniewicz and Kazmirkowski, 2008;[START_REF] Norniella | Analytic and iterative algorithms for online estimation of coupling inductance in direct power control of three-phase active rectifiers[END_REF][START_REF] Arriagada | On-line filtering reactance identification in voltagesource three-phase active-front-end rectifiers[END_REF] and, in low-power VSRs, the ac filter resistance cannot be neglected [START_REF] Kwak | Predictive-controlbased direct power control with an adaptive parameter identification technique for improved AFE performance[END_REF]. On the other hand, the used VSR terminal voltages in the identification process cannot be directly measured. They are usually estimated from output voltages of PI controllers and measured dc voltage. Since VSR nonlinearity due to the dead time and voltage drop exist in the used insulated-gate-bipolar-transistors (IGBTs) (Liu et al., 2012;[START_REF] Choi | Inverter output voltage synthesis using novel dead time compensation[END_REF][START_REF] Kim | Nonlinearity estimation and compensation of PWM VSI for PMSM under resistance and flux linkage uncertainty[END_REF][START_REF] Liu | Online estimation of the rotor flux linkage and voltage-source inverter nonlinearity in permanent magnet synchronous machine drives[END_REF], the mentioned methods [START_REF] Milasi | Adaptive vector control for voltage source converters[END_REF]Antoniewicz and Kazmirkowski, 2008;[START_REF] Norniella | Analytic and iterative algorithms for online estimation of coupling inductance in direct power control of three-phase active rectifiers[END_REF][START_REF] Kwak | Predictive-controlbased direct power control with an adaptive parameter identification technique for improved AFE performance[END_REF][START_REF] Arriagada | On-line filtering reactance identification in voltagesource three-phase active-front-end rectifiers[END_REF] should operate without considering the VSR nonlinearity. Hence, the identified parameters may suffer from negligence of the VSR nonlinearity. Some authors have already studied the influence of voltage-source inverter nonlinearity on parameter estimation [START_REF] Liu | Influence of nonideal voltage measurement on parameter estimation in permanent-magnet synchronous machines[END_REF], and rotor flux linkage estimation [START_REF] Liu | Online estimation of the rotor flux linkage and voltage-source inverter nonlinearity in permanent magnet synchronous machine drives[END_REF] in permanent-magnet synchronous machines. However, there is still no literature proposing a method for ac filter parameters identification of a VSR with VSR nonlinearity consideration.

In this paper, the proposed adaptive VOC maintains the simplicity of the classical VOC while providing robust performance to uncertainty in system parameters. A new method based on adaptive linear neuron (ADALINE) for ac filter parameters identification is first developed. The main advantage of this method is its simple structure, speed and ability to be trained online [START_REF] Bechouche | A novel method for identifying parameters of induction motors at standstill using ADALINE[END_REF][START_REF] Bechouche | Adaptive neural PLL for grid-connected DFIG synchronization[END_REF][START_REF] Du | A new online hybrid learning algorithm of adaptive neural fuzzy inference system for fault prediction[END_REF]. For accurate identification, the VSR nonlinearity is included in the ADALINE structure. Hence, correct ac filter parameters can be easily estimated online from the ADALINE weights. To guarantee the ADALINE's convergence, a discrete-time Lyapunov stability analysis [START_REF] Romdhane | Stability analysis of discrete input output second order sliding mode control[END_REF] is presented. The proposed ADALINE identifier is experimented and the influence of VSR nonlinearity on the ac filter parameters identification is investigated. Thereafter, the identified parameters are used to perform an adaptive VOC where the decoupled terms and PI current controller gains are updated online. Finally, a comparative study showed the superiority of the proposed adaptive VOC scheme compared to the classical one under parameter uncertainties.

The paper is organized as follows. Section 2 presents the classical VOC of a VSR. The VSR nonlinearity is analyzed in Section 3. Section 4 presents the developed ADALINE identifier. Experimentation of the proposed adaptive VOC is given in Section 5. Finally, Section 6 concludes this paper.

2 Classical VOC of a three-phase VSR

Three-phase VSR modeling

The power circuit of the three-phase VSR is depicted in Figure 1. The VSR is connected to the grid through inductance L and its serial resistance R. e a , e b , and e c are the balanced source voltages with their maximum amplitude E and angular frequency ω; i a , i b , and i c are the input currents; v an , v bn , and v cn are the VSR terminal voltages; V dc and C are, respectively, the dc-link voltage and capacitance; R L is the load resistance. S a , S b , and S c stand for the switching state of each phase leg. S i =1 (i=a,b,c) means that the upper switch of phase i is turned on, and the lower switch is turned off, while S i =0 denotes the opposite meaning. In this paper, S i (i=a,b,c) are generated by a sinusoidal pulse-width modulation (SPWM) scheme. SPWM is based on comparing a triangular carrier wave with a three-phase modulation signals β i (i=a,b,c). The phases of β i (i=a,b,c) are shifted by 2π/3 and by φ relative to the ac source. For three-phase VSR average model, S i =(1+β i )/2 (i=a,b,c) [START_REF] Rashid | Power Electronics -Circuits, Devices and Applications[END_REF]. The modulation signals can be considered as a control inputs to the system and they are directly related to the amplitude and the phase of the VSR terminal voltages. The neutral of the ac sources is not connected. The resistance and the inductance of the feeder cables are neglected. In abc coordinates, the three-phase VSR average model [START_REF] Rashid | Power Electronics -Circuits, Devices and Applications[END_REF] can be described as

0 0 0 0 0 0 a a a an b b b bn c c c cn i i e v R d L i R i e v dt R i i e v - -                 = - + -                 - -         (1) ( ) 1 2 dc dc a a b b c c L dV V C i i i dt R β β β = + + - (2)
where

2 1 1 1 2 1 6 1 1 2 an a dc bn b cn c v V v v β β β --             = - -             --       (3)
The VSR model (1) can be described in stationary αβ frame, using Clark's transformation, as 0

0 n n e i i v R d L e i i v R dt α α α α β β β β           = + +                     (4)
In Park's dq frame, the model ( 4) can be reasonably represented by the following equations:

d d d dn q q q qn e i i v R L d L e i i v L R dt ω ω         -   = + +                     (5)
where e d , e q , i d and i q are, respectively, the d-and q-axis components of the ac source voltages and the VSR input currents; v dn and v qn are the d-and q-axis components of the VSR terminal voltages.

The dc voltage dynamics can be expressed by ( )

3 4 dc dc d d q q L dV V C i i dt R β β + = + (6)
where β d and β q are the d-and q-axis components of the modulation signals.

The active P and reactive Q powers of the VSR are calculated as ( ) ( )

3 3 ; 2 2 d d q q q d d q P e i e i Q e i e i = + = - (7.a)
By setting the grid voltage vector according to the d-axis (i.e., e d =E and e q =0), the equations (7.a) become

3 3 ; 2 2 d q P Ei Q Ei = = - (7.b)
Under UPF operation (Q=0), and according to (7.b), it can be deduced that i q is null (i q =0).

Currents control

In dq frame, the currents controller controls the ac side currents to achieve UPF operation and constant dc voltage. As currents controller, the PI-type is usually used [START_REF] Rashid | Power Electronics -Circuits, Devices and Applications[END_REF][START_REF] Malinowski | A comparative study of control techniques for PWM rectifiers in ac adjustable speed drives[END_REF]. However, PI controller has no satisfactory tracking performance, especially, for the coupled system described in (5). So, for high performance application with accuracy current tracking, decoupled currents control scheme for VSR should be applied. The decoupled scheme is shown in Figure 2. From Figure 2, the expression (8) can be deduced where K pc and K ic are, respectively, proportional and integral gains of the PI controllers. (•) * denotes a desired reference value.

V dc + - i a Sb Sa Sc i b i c R L C i L i C i dc R R R L L L e a e b e c v an v bn v cn n a S b S c S ( ) ( ) di v K i i K i i dt L Ri dt ∆ = -+ - = + ∫ (8.a) ( ) ( ) * * q q pc q q ic q q q di v K i i K i i dt L Ri dt ∆ = -+ - = + ∫ (8.b)
According to (8), under ideal decoupling between d-and q-axes, the two axis currents can be independently controlled. For K pc and K ic synthesis, a pole compensation method can be applied. Therefore, PI gains are designed as K pc =3L/t r and K ic =3R/t r , where t r is the desired time response.

dc voltage regulation

In designing the controller of the dc voltage dynamics, it is assumed that dc voltage varies on a time scale significantly slower than the d-axis current. Hence, the current i d tracks perfectly its reference i d * . The dc voltage control scheme is reported in Figure 3 where K pv and K iv are, respectively, proportional and integral gains of the voltage PI controller. From Figure 3, the equation ( 9) can be deduced. The gains K pv and K iv are designed using a pole placement method as K pv =2z 0v Cω 0v and K iv =Cω 0v 2 where z 0v is the damping ratio and ω 0v is the undamped natural frequency of the controlled system in closed-loop.

(

) ( )

* * * dc C pv dc dc iv dc dc dV i K V V K V V dt C dt = - + - = ∫ (9)
Once the VSR model and classical VOC are presented, the following section will analyze the VSR nonlinearity due to voltage drop and dead time effects in the used IGBTs.

Nonlinearity analysis of three-phase VSR

Figure 4 shows the nonlinearity for one leg of three-phase VSR, including two IGBTs. As can be seen in this figure, if i a flows to load (i a >0), the actual phase to center voltage v ao is 2

dc ao d V v V = + (when S a =1) (10.a) 2 dc ao ce V v V = - + (when S a =0) (10.b)
where V ce and V d are, respectively, the voltage drops of the active switch and freewheeling diode. If i a flows from load (i a <0), the actual phase to center voltage v ao is calculated as

2 dc ao ce V v V = - (when S a =1) (11.a) 2 dc ao d V v V = - - (when S a =0) (11.b)
Therefore, v ao can be represented in a general form as

( ) ( )( ) a 1 1 S sign 2 2 ao dc ce d a ce d v V V V i V V   = -+ -+ +     (12)
where,

( ) { } sign 1if 0; 1if 0 . a a a i i i = + > - <
In many cases, the voltage drops in the IGBT increase with current. Thus, they can be represented as follows [START_REF] Choi | Inverter output voltage synthesis using novel dead time compensation[END_REF]:

0 0 , ce ce ce a d d d a V V r i V V r i = + = + (13)
where V ce0 and V d0 are, respectively, the threshold voltages of active switch and freewheeling diode. r ce and r d are, respectively, the equivalent resistances of active switch and freewheeling diode. By substituting (13) in ( 12), v ao is rewritten as follows:

( ) ( )( ) ( ) a 0 0 1 1 S sign 2 2 1 2 ao dc ce d a ce d ce d a v V V V i V V r r i   = -+ -+ +     + + (14)
From ( 14), the average phase to center voltage can be derived as

( ) ( )( ) ( ) 0 0 sw 1 1 sign 2 2 1 2 a ao dc ce d a ce d ce d a T v V V V i V V T r r i   = -+ -+ +     + + (15.a)
where T a and T sw are, respectively, the effective time duration and the SPWM carrier period. Similarly, the average phase to center voltage of other phases can be obtained as follows:

( ) ( )( ) ( ) 0 0 sw 1 1 sign 2 2 1 2 b bo dc ce d b ce d ce d b T v V V V i V V T r r i   = -+ -+ +     + + (15.b) ( ) ( )( ) ( ) 0 0 sw 1 1 sign 2 2 1 2 c co dc ce d c ce d ce d c T v V V V i V V T r r i   = -+ -+ +     + + (15.c) i L P * V dc * i dc * - + + + i C * × ÷ V dc i d * 2 3 e d 1 pv iv K K s + L ω e d e q i d i q i d * i q * v dn * - + - + + + - - - + d v ∆ q v ∆ v qn * 1 pc ic K K s + 1 pc ic K K s + L ω
where T b and T c are the effective time duration of phases b and c, respectively. 5(c) for i a >0 and in Figure 5(d) for i a <0. The effective time duration T a is the area between t 5 and t 6 for i a >0 or between t 3 and t 4 for i a <0. From Figures 5(a)-(d), the time error T aerr [START_REF] Kim | Nonlinearity estimation and compensation of PWM VSI for PMSM under resistance and flux linkage uncertainty[END_REF] can be deduced as

( ) * err 2 1 a a a a T T T T t t = -= -- (16)
When i a is positive, T aerr = T d + T on -T off and when i a is negative, T aerr = -( T d + T on -T off ). The time error can be represented as

( )( ) err on off sign a a d T i T T T = + - (17)
From ( 16) and ( 17), the effective time duration can be deduced as

T T i M = + (18.b) ( ) * sign c c c T T i M = + (18.c)
To calculate the VSR average terminal voltages including the nonlinearity, the knowledge of the relationship between the terminal voltage and the phase to center voltage is required. This relationship can be written as follows:

ao an no bo bn no co cn no v v v v v v v v v             = +                   (19)
The absence of a neutral connection in the ac source forces the constraint that 0

a b c i i i + + = (20)
For any balanced ac filter parameters and grid voltages, the terminal voltages are constrained such that 0

an bn cn v v v + + = (21)
From ( 19) and ( 21), the neutral to center voltage becomes ( )

1 3 no ao bo co v v v v = + + (22)
Therefore, the relationship between the terminal voltage and the phase to centre voltage can be deduced as follows:

2 1 1 1 1 2 1 3 1 1 2 an ao bn bo co bn v v v v v v   --          = - -             --       (23)
For phase a, the average terminal voltage including VSR nonlinearity can be calculated from ( 15), ( 18) and ( 23) as

( ) ( ) ( ) ( ) ( ) ( ) ( ) * * * sw 0 0 sw 2 1 3 1 2 6 1 2sign sign sign 2 a b c an dc ce d dc ce d ce d a b c ce d a T T T v V V V T M V V V V V T i i i r r i   -- = -+       + -+ + +     × - - + + (24)
In ( 24), the term V ce -V d is neglected because V dc is much greater than V ce -V d [START_REF] Choi | Inverter output voltage synthesis using novel dead time compensation[END_REF]. Hence, ( 24) is simplified as

( ) ( ) ( ) ( ) ( ) ( ) * * * 0 0 sw sw 2 1 2 3 6 1 2sign sign sign 2 dc dc an a b c ce d a b c ce d a V MV v T T T V V T T i i i r r i   = -- + + +     × - - + + (25)
Similarly, the average terminal voltage including VSR nonlinearity of phase b and c can be easily deduced. Hence, the average terminal voltage vector is represented in the following matrix form:

( ) * * dist * 1 2 an an a a bn bn b ce d b cn c c cn v v D i v v V D r r i v D i v                 = + + +                         (26) When S a =1 When S a =0 If i a >0 If i a <0 a ia S a a S o vao Vce 2 dc V 2 dc V a ia S a a S o vao Vce 2 dc V 2 dc V a ia S a a S o vao Vd 2 dc V 2 dc V a ia S a a S o vao Vd 2 dc V 2 dc V where * * * * sw * * 2 1 1 1 2 1 3 1 1 2 an a dc bn b cn c v T V v T T v T     --         = - -             --       (27.a) dist 0 0 sw 2 1 6 dc ce d MV V V V T   = + +     (27.b) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2sign sign sign sign 2sign sign sign sign 2sign a b c a b a b c c a b c i i i D D i i i D i i i - -         = - + -         - - +     (27.c)
Therefore, the VSR model including nonlinearity is derived by substituting ( 26) in (1). In the next section, the VSR nonlinearity will be included in the ADALINE structure for accurately identifying R and L values.

ADALINE for R and L identification

In stationary αβ frame, the equation ( 26) can be expressed as follows:

( )

* dist * 1 2 n n ce d n n v v D i V r r v D i v α α α α β β β β         = + + +                   (28.a) where ( ) ( ) ( ) ( ) ( ) 2sign sign sign 3sign 3sign a b c b c i i i D D i i α β - -     =     -       (28.b)
By substituting (28.a) in ( 4), the α-axis dynamic model of the VSR can be selected as follows:

( )

* dist 1 2 n ce d di e L Ri v V D r r i dt α α α α α α = + + + + + (29)
and ( 29) can be rewritten under the following form:

dist di Z L Ri V D dt α α α α = + + (30.a)
where

( ) * 1 2 n ce d Z e v r r i α α α α = - - + (30.b)
Using the Euler approximation with a sampling period T s , the derivative of i α is approximated as ( ) ( )

1 s i k i k di dt T α α α + - ≈ (31)
Then, i α is represented in the discrete-time domain as

( ) ( ) ( ) ( ) A 1 B 1 C 1 i k Z k i k D k α α α α = ⋅ -+ ⋅ -+ ⋅ - (32.a)
where

( ) ( ) ( ) ( ) ( ) * 1 1 1 1 1 2 n ce d Z k e k v k r r i k α α α α -= -- -- + -, A s T L = , B 1 s RT L = - , dist C . s T V L = - (32.b)
Therefore, the quantities R, L and V dist can be deduced from (32) in the following way:

dist 1 B C ; ; . A A A s T L R V - = = = - (33)
According to (32.a), the coefficients A, B, C can be easily identified online by three weights W 1 , W 2 and W 3 of the ADALINE. Indeed, by choosing the ADALINE with three inputs Z α (k -1), i α (k -1) and D α (k -1), the estimated α-axis current dynamics ( )

î k α can be computed as ( ) ( ) ( ) ( ) 1 2 3 ˆW 1 W 1 W 1 i k Z k i k D k α α α α = ⋅ -+ ⋅ -+ ⋅ - (34.a)
or in vectorial form

( ) ( ) ( ) ˆW 1 i k k X k α α = ⋅ - (34.b)
where

X α (k -1) = [Z α (k -1) i α (k -1) D α (k -1)] T and W(k) = [W 1 (k) W 2 (k) W 3 (k)
] are, respectively, the input vector and the weights vector of the ADALINE. (^) indicates the estimated quantities. When i α (k) is well estimated, the equality given by ( 35) is satisfied, with Hence, from ( 33) and ( 35), the parameters R, L and V dist are finally identified in the following way:

Y(k) = [A(k) B(k) C(k)]. ( ) ( ) W . k Y k = (35)
3 2 dist 1 1 1 W 1 W ˆˆ; ; W W W s T L R V - = = = - (36)
Figure 6 ADALINE structure for ac filter parameters identification

The topology shown in Figure 6 represents the identification process of R, L and V dist using ADALINE technique. This topology has the natural dynamic characteristics of the current i α dynamics. Therefore, the stability of the proposed method can be verified with an appropriate weight vector adaptation using the estimation error [START_REF] Romdhane | Stability analysis of discrete input output second order sliding mode control[END_REF]. This method has high immunity to measurement noise, because the feedback is used only in the weight adaptation algorithm [START_REF] Bechouche | A novel method for identifying parameters of induction motors at standstill using ADALINE[END_REF][START_REF] Du | A new online hybrid learning algorithm of adaptive neural fuzzy inference system for fault prediction[END_REF]. In this paper, V dist is identified just for the validation of identification algorithm and does not used in VOC. Indeed, V dist can indicate the convergence quality of ADALINE weights.

Using (34), the current estimation error ( ) ( ) ( )

k i k i k α α α δ = -
will be produced due to any mismatch between the estimated current and the actual one. This error is produced namely by parameter variation. Accordingly, it can be used to adaptively adjust the weight vector W

(k) = [W 1 (k) W 2 (k) W 3 (k)
] in a manner that minimizes the error.

To achieve this objective, a least mean square algorithm with learning rate η is used for the weights updating [START_REF] Bechouche | A novel method for identifying parameters of induction motors at standstill using ADALINE[END_REF][START_REF] Du | A new online hybrid learning algorithm of adaptive neural fuzzy inference system for fault prediction[END_REF]. Then, the ADALINE weight vector W(k) is recursively updated as follows:

( ) ( ) ( ) ( ) ( ) ( ) 1 W 1 W 1 1 T k X k k k X k X k α α α α ηδ ε - + = + + - - (37)
where η is the learning rate and ε is a small value to avoid division by zero if X α T (k-1)X α (k-1)=0.

K K s + 1 ˆpc ic K K s + L ω i q i d e d e q abc/αβ R R L L abc/dq i d i q i α i β e α e β θ e d L LSPWM block v an * v bn * v cn * S a S b S c - - + + - - - + + - + i abc PLL + i α (k) ( ) î k α - δ α (k) W 1 W 3 + i α (k-1) z -1 W 2
Learning Algorithm Eq. ( 37)

Eq. ( 36)

RL dist V D α (k-1) Z α (k -1) Eq. (32.b) V dc e α (k -1) v αn * (k -1) Eq. (28.b) i a (k-1) i b (k-1) i c (k-1)

Stability analysis of the ADALINE identifier

To prove the stability of the ADALINE identifier and to give the guideline in tuning the learning rate parameter η, Lyapunov function candidate for the parameter identifier (37) is used [START_REF] Romdhane | Stability analysis of discrete input output second order sliding mode control[END_REF]. This function is selected as

( ) ( ) ( ) W W T V k k k = % % (38)
where ( )

W k %
is the ADALINE estimation error, defined as

( ) ( ) ( ) W W k Y k k ≡ - % (39)
The Lyapunov's convergence criterion must be satisfied such that

( ) ( ) 0 V k V k ∆ < ( 40
)
where ∆V(k) is the change in the Lyapunov function. As ∆V(k) is defined positive as shown in ( 38), the stability condition ( 40) is satisfied when ∆V(k) < 0. The change in Lyapunov function is given by

( ) ( ) ( ) ( ) ( ) W 1 W 0 V k V k V k ∆ = + - < % % (41)
By using the parameters identifier error dynamics, which can be derived from ( 39), with the update law (37), ∆V(k) can be evaluated as

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 1 1 W W W W 1 1 W 1 1 1 2 1 1 1 1 T T T T T T T X k X k k V k k k k X k X k k X k X k X k X k X k X k X k α α α α α α α α α α α η ε η η ε ε - - ∆ = - - + - -   -   - -   = -+   + - - + - -     % % % % % ( 42
)
where ||•|| is the Euclidian norm in n ℜ . Assuming that ε >0 and 0< η <2, the bracketed term in ( 42) is negative and consequently the stability condition in ( 41) is satisfied. The following convergence properties are also satisfied:

( ) ( ) ( ) 0 0 0 W W W 1 W W 0 W , 1 k k k - ≤ -- ≤ - ≥ (43.a) ( ) ( ) ( ) ( ) W 1 lim 0 1 1 T k k X k X k X k α α α ε →∞ - = + - - % (43.b)
where W 0 is the vector obtained at perfect convergence of the estimator. According to (43), the augmented error is monotonically non-increasing, hence, the convergence is guaranteed.

Adaptive VOC of a VSR

In classical VOC, the current dynamics in dq frame is decoupled. The decoupling scheme requires an accurate knowledge of L. Hence, if L is not known, the current dynamics will not be decoupled perfectly. Although the integral term in the PI controller compensates for steady-state coupling disturbances, the transient system performance will be degraded [START_REF] Milasi | Adaptive vector control for voltage source converters[END_REF]. In addition, stability of the inner currents control is mainly affected by parameter uncertainties [START_REF] Shaker | Stability analysis and output feedback control for a class of switched nonlinear systems[END_REF]. As shown in Section 2.2, K pc and K ic are directly designed from the identified R and L values. Therefore, with uncertainties in R and L, the currents control performances will be degraded and the system can be driven into instability [START_REF] Milasi | Adaptive vector control for voltage source converters[END_REF]. To overcome this, an adaptive VOC scheme is proposed. Figure 7 shows the overall structure of the developed adaptive VOC of a VSR with online R and L identification. As R and L are updated online while the VSR runs, the proposed scheme guarantees asymptotic decoupling of the current error dynamics and stability of the PI currents controllers. As VOC strategy requires a phase-locked loop (PLL) for grid synchronization, an adaptive neural PLL is used [START_REF] Bechouche | Adaptive neural PLL for grid-connected DFIG synchronization[END_REF]. Note that the proposed adaptive VOC scheme can be implemented without any additional sensors compared to those already used in the classical scheme.

To validate in experiment the suggested ADALINE identifier and adaptive VOC strategy, the following section will present the experimental investigation.

Experimental investigation

The hardware platform of the three-phase VSR is first introduced. Then, the developed ADALINE-based algorithm for R, L and V dist identification is tested. At last, the identified R and L are used to perform an adaptive VOC of the VSR.

Experimental platform description

The developed adaptive VOC strategy is tested in a threephase VSR. The specifications of the experimental system 4 1 2 3 5 7 6 8 are given in Table 1 while the test bench is shown in Figure 8. It consists of a three-phase IGBT based VSR of 120 kVA (Mitsubishi) with anti-paralleling diodes (module CM100DY-24H). Three Hall-effect voltage sensors of LV25-P type and three Hall-effect current sensors of LA55-P type are used to measure dc-bus voltage V dc , two grid voltages (e a , and e b ), two input currents (i a and i b ) and load current i L , respectively. To test experimentally the developed identification and control method, a DS1104 dSPACE board based on TMS320F240 is used. The algorithms are programmed in MATLAB/Simulink environment and implemented using Euler resolution method and sampling period of 250 µs. Simulink's Real-Time Workshop is used to build real time code and execute it on dSPACE hardware inserted in a PC-Pentium. Slave DSP three-phase PWM generation block is used to convert modulation signals (β a , β b and β c ) provided by SPWM to PWM signals S a , S b and S c . The ADALINE identifier is implemented with a learning rate η=0.01. This value is experimentally adjusted to ensure stability and optimal speed of weights convergence. v alpha (V) V dc (2 V) i alpha (0.2 A) D alpha (x 0.1)

Online identification results of R, L and V dist

The proposed ADALINE technique for online identification of R and L is experimented. The VSR is controlled to achieve UPF operation using the classical VOC strategy. The PI gains K pc and K ic are calculated using the rated values shown in Table 1. V dc is regulated to 150 V and R L is selected to 56 Ω.

Several experiments have been realized. First test consists to verify stability and accuracy of the identification method in steady state conditions. Second test is performed to show the effect of VSR nonlinearity in R and L identification. Two last tests are realized to show tracking ability of the identification algorithm with respect to step changes in L and R. In this paper, V dist is identified to verify the good convergence of the ADALINE weights. As analyzed in Section 4.1, identification of R, L and V dist consists of implementing (37). This method uses Z α (k -1), i α (k -1) and D α (k -1) as inputs in order to estimate i α (k). The estimation error δ α (k) is used in (37) for online updating W 1 , W 2 and W 3 . Therefore, these weights are exploited using (36) to reconstitute in real time R, L and V dist values. Figure 9 shows the used signals in identification process. From this, it can be observed that the VSR operates at UPF since e α and i α are in phase, D α has the six-step waveform and peaks of ±4, and V dc is well regulated to 150 V. Figure 10 illustrates the steady state performances of the proposed identification method. As shown in Figure 10(a), the ADALINE operates properly with good accuracy since δ a is very small. This proves the stability of the ADALINE identifier. Figures 10(b)-(d) show, respectively, the identified R, L and V dist values. According to these results, the ADALINE remain stable, since the identified parameters converge to stable and constant values. The identified R value is 1.01Ω and the identified L value is 10.45mH. Therefore, the identified values are very close to the nominal values provided Table 1. The identified value of V dist is also stable and close to 1.45V. On the other hand, the calculated typical value of V dist from Table 2 (from the Mitsubishi official datasheet) is 1.4V. Hence, the identified V dist value stays realistic since it is close the typical one. To study the influence of the VSR nonlinearity in R and L identification, tests with and without taking into account the VSR nonlinearity in the ADALINE structure are performed. The test without taking into account the VSR nonlinearity consists of implementing (37) with only two weights W 1 and W 2 , i.e., the input of W 3 is D α (k -1) = 0 (see Figure 6). Figure 11 illustrates the obtained experimental results. In Figure 11(a), it can be seen that the effect of the VSR nonlinearity in R identification is important. With taking into consideration the VSR nonlinearity the identified R is 1.01Ω, and without taking into consideration the VSR nonlinearity, the identified R value passes to 1.76Ω. Figure 11(b) shows the influence of the VSR nonlinearity in L identification. Without taking into account the VSR nonlinearity, the identified L value passes to 11.07mH. Compared to the results shown in Figure 11(a), the influence of the VSR nonlinearity in L identification is negligible. Consequently, the ADALINE interprets the VSR nonlinearity as an increased value in R.

Thereafter, two tests have been performed for illustrating the performances of the ADALINE identifier with respect to parameter variations.

Figure 12 presents the identified R, L and V dist in case of step change in L. At time t=3.47s, a three-phase inductance of 4.9mH has been connected in series with the ac filter. In Figure 12(b), L is identified within 0.09s after step change in L. The identified L value is close to 15.35mH. A slight deviation in R value is appeared and stabilized at 1Ω within 0.09s (Figure 12(a)). Figure 12(c) shows a slight deviation of V dist from its initial value and is stabilized within 0.09s. Figure 13 presents the identified R, L and V dist in case of step change in R. At time t=3.34s, a three-phase resistance of 0.4Ω has been short-circuited in the interconnected resistances (see Figure 8). In Figure 13(a), R is identified within 0.09s after step change in R. The identified R value is close to 0.61 Ω. In Figure 13(b), the identified L value remains stable (about 10.45mH) and it seems not affected by R variation. Figure 13(c) shows a slight deviation of V dist but it tends to converge to its initial value. Figures 10-13 reveal that the ADALINE identifier guarantees precise identification. Therefore, the identified parameters can be reliably used in different control blocks of VOC strategy.

Classical VOC of VSR with parameter uncertainties

As has been seen in Section 2.2, VOC requires knowledge of R and L for K pc and K ic computation. The terms Lωi q and Lωi d are needed for decoupling i d and i q dynamics, respectively. Hence, any deviation of R and L from their nominal values affects the control performances. This subsection presents the performances of the classical VOC of VSR with 60% mismatch in both R and L. The dc voltage is regulated to 150V. The load resistance R L is varied from 110Ω to 55Ω at time t 1 =2.28s and from 55Ω to 110Ω at time t 2 =6.45s. i q * is set to zero to ensure a UPF operation. The obtained results are illustrated in Figure 14. In Figure 14(a), oscillations in V dc are observed at times t 1 and t 2 . As L is mismatched, the terms Lωi q and Lωi d are not exactly calculated for i d and i q dynamics decoupling. In addition, the computed gains K pc and K ic cannot achieve the desired performances since the identified R and L are not exact. Figures 14(b) and (c) show that i d and i q are degraded and present oscillations at transient states. This degrades the power factor of the system since i q is not well maintained to zero. From Figures 14(d) and (e), the modulation signals β d and β q present also oscillations in transient states and tends to be saturated. Although the integral term in PI controller compensates the steady state error, oscillations in transient states are appeared. These degrade the power factor and can driven the system into instability. To improve the classical VOC performances, the PI current controller gains and decoupling terms must be updated. The following subsection presents the validation results of the proposed scheme, reported in Figure 7, for adaptive VOC.

Adaptive VOC of three-phase VSR

As shown in Figure 15, the adaptive VOC achieves good performances in transient and steady states with respect to R L variation. The dc voltage is regulated to 150V. The load resistance R L is varied from 110Ω to 55Ω at time t 1 =1.87s and from 55Ω to 110Ω at time t 2 =6.31s. i q * is set to zero to ensure a UPF operation. In Figure 15(a), transient performance of the regulated V dc is fast and zero steady state error is achieved. From Figures 15(b)-(e), the proposed adaptive VOC is more stable since the oscillations in transient states are removed. Indeed, the proposed adaptive control method can stay stable at any values of R and L. This is due to the self tuning feature, which enables the redesign of PI gains and decoupling terms in real time at different parameters. Therefore, in the proposed adaptive PI current controllers, the stability of VOC strategy becomes independent of system parameters.

The reported results indicate that the proposed adaptive scheme results in stable and robust grid-connected VSR operation and favorable dc voltage regulation response.

Conclusion

In this paper, an adaptive VOC of a VSR with ac filter parameters identification method was introduced. The proposed identification method, based on ADALINE, is robust and requires low computational cost. The VSR nonlinearity has been analyzed and included in the ADALINE structure. Hence, accurate ac filter parameters have been easily obtained from the ADALINE weights. Thereafter, the identified parameters have been used to realize an adaptive VOC where the decoupled terms and PI current controller gains have been updated online. Finally, the obtained experimental results have proved the ADALINE ability to track with accuracy the R and L values. It has been observed that the VSR nonlinearity consideration has significant influence on the identification of R value and has negligible influence on the identification of L value. A comparative study has showed the superiority of the proposed adaptive VOC scheme compared to the classical one under parameter uncertainties. The developed ADALINE identifier is independent on the type of the VSR control; therefore; it can be used to enhance the robustness of further existing control methods.

Figure 1

 1 Figure 1 Topological structure of a three-phase VSR

Figure 2

 2 Figure 2 Decoupled currents control of three-phase VSR

Figure 3

 3 Figure 3 dc voltage regulation of three-phase VSR

Figure 4

 4 Figure 4 Nonlinearity for one leg of three-phase VSR

  where M = T d + T on -T off .Similarly, the effective time duration of phases b and c can be deduced as follows:

Figure 5

 5 Figure 5 VSR input voltage and IGBT drive signals: (a) ideal IGBT drive signal, (b) actual drive signal with dead time, (c) actual output voltage with dead time and switch-on/switch-off times for i a >0, (d) actual output voltage with dead time and switch-on/switch-off times for i a <0

Figure 7

 7 Figure 7 Proposed scheme for adaptive VOC of a VSR with online ac filter parameters identification

Figure 8

 8 Figure 8 Different parts of the experimental platform: 1) threephase VSR, 2) PC-Pentium + dSPACE board, 3) dSPACE I/O connectors, 4) isolating transformer, 5) interconnecting resistances and inductances, 6) voltage sensors, 7) current sensors, 8) load resistance

Figure 9

 9 Figure 9 Used signals in identification of R, L and V dist

Figure 10

 10 Figure 10Steady state performances of the ADALINE identifier: (a) estimation error, (b) identified R, (c) identified L and (d) identified V dist

Figure 11

 11 Figure 11Influence of VSR nonlinearity on R and L identification values: (a) identified R and (b) identified L

  Figure 12Performances of the proposed method for step change in L

Figure 14

 14 Figure 14Performances of the classical VOC of a VSR with 60% mismatch in both R and L: (a) dc voltage V dc , (b) d-axis current i d , (c) q-axis current i q , (d) d-axis modulation signal β d and (e) q-axis modulation signal β q

Figure 15

 15 Figure 15Performances of the adaptive VOC of a VSR: (a) dc voltage V dc , (b) d-axis current i d , (c) q-axis current i q , (d) d-axis modulation signal β d and (e) q-axis modulation signal β q

Table 2

 2 Typical electrical characteristics of the used IGBT modules (from Mitsubishi CM100DY-24H datasheet)

	Turn-on delay time T on (µs)	0.1
	Turn-off delay time T off (µs)	0.3
	Switching control dead time T d (µs)	3.5
	Threshold voltage of the active switch V ce0 (V)	0.8
	Threshold voltage of the freewheeling diode V d0 (V)	0.7
	Equivalent resistance of the active switch r ce (mΩ)	9.4
	Equivalent resistance of the freewheeling diode r d (mΩ)	15